Outflows from compact objects in supernovaeand novae

by

Andrey Dmitrievich Vlasov

Submitted in partial fulfillment of therequirements for the degree of

Doctor of Philosophy

in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2017

c© 2017Andrey Dmitrievich Vlasov

All Rights Reserved

Abstract

Outflows from compact objects in supernovae and

novae

by

Andrey Dmitrievich Vlasov

Originally thought of as a constant and unchanging place, the Universe is

full of dramas of stars emerging, dying, eating each other, colliding, etc.

One of the first transient phenomena noticed were called novae (the name

means ”new” in Latin). Years later, supernovae were discovered. Despite

their names, both novae and supernovae are events in relatively old stars,

with supernovae marking the point of stellar death. Known for thousands of

years, supernovae and novae remain among the most studied events in our

Universe. Supernovae strongly influence the circumstellar medium, enriching

it with heavy elements and shocking it, facilitating star formation. Cosmic

rays are believed to be accelerated in shocks from supernovae, with small

contribution possibly coming from novae. Even though the basic physics of

novae is understood, many questions remain unanswered. These include the

geometry of the ejecta, why some novae are luminous radio or gamma-ray

sources and others are not, what is the ultimate fate of recurrent novae, etc.

Supernova explosions are the primary sources of elements heavier than hy-

drogen and helium. The elements up to nuclear masses A ≈ 100 can form

through successive nuclear fusion in the cores of stars starting with hydro-

gen. Beyond iron, the fusion becomes endothermic instead of exothermic. In

addition, for A ∼> 100 the temperatures required to overcome the Coulomb

barriers are so high that the nuclei are dissociated into alpha particles and

free nucleons. Hence all elements heavier than A ≈ 100 should have formed

by some other means. All nuclear species heavier than A ≈ 100 are formed

by neutron capture on seed nuclei close to or heavier than iron-group nu-

clei. Depending on the ratio between neutron-capture timescale and beta-

decay timescale, neutron-capture processes are called rapid or slow (r- and

s-processes, respectively). The s-process, which occurs near the valley of

stable isotopes, terminates at Bi (Z = 83), because after Bi there is a gap

of four elements with no stable isotopes (Po, At, Rn, Ac) until we come to

stable Th. The significant abundance of Th and U in our Universe therefore

implies the presence of a robust source of r-process. The astrophysical site

of r-process is still under debate. Here we present a study of a candidate

site for r-process, neutrino-heated winds from newly-formed strongly mag-

netized, rapidly rotating neutron stars (”proto-magnetars”). Even though

we find such winds are incapable of synthesizing the heaviest r-process ele-

ments like U and Th, they produce substantial amounts of weak r-process

(38 < Z < 47) elements. This may lead to a unique imprint of rotation

and magnetic fields compared to such yields from otherwise analogous slowly

rotating non-magnetized proto-neutron stars.

Novae explosions are not as powerful as those of supernovae, but they occur

much more frequently. The standard model of novae assumes a one-stage

ejection of mass from the white dwarf following thermonuclear runaway. The

discovery by the Fermi space telescope of gamma-rays from classical no-

vae made the existence of shocks in novae outflows evident. The presence

of shocks in novae was considered well before the discovery of gamma-ray

emission; however, little previous theoretical work acknowledged the over-

whelming effect of shocks on observed emission and ejecta geometry. Here

we present the calculations of synchrotron radio emission from the shocks as

they propagate down the density gradient and peak at the timescale of a few

months. The model satisfactory fits observations and has several implications

for the physics of novae.

Contents

List of Figures iv

List of Tables vii

1 Introduction 1

1.1 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Neutrino-heated winds as the source of r-process . . . . 5

1.1.2 Neutrino-heated winds from rotating proto-magnetars . 9

1.2 Novae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Shock acceleration of protons and electrons . . . . . . . 15

2 Supernovae 24

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Hydrodynamic model description . . . . . . . . . . . . 28

2.2.2 Wind Equations . . . . . . . . . . . . . . . . . . . . . 31

2.2.3 Neutrino microphysics . . . . . . . . . . . . . . . . . . 32

2.2.4 Magnetosphere model . . . . . . . . . . . . . . . . . . 35

2.2.5 Conservation checks . . . . . . . . . . . . . . . . . . . 37

2.2.6 Shooting method . . . . . . . . . . . . . . . . . . . . . 37

2.2.7 Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.1 Validity of the Force-Free Approximation . . . . . . . . 48

i

2.3.2 Spherical versus Magnetized Rotating Trajectories . . 53

2.3.3 Spherical vs Magnetized Rotating Nucleosynthesis yields 59

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.4.1 r-Process Nucleosynthesis . . . . . . . . . . . . . . . . 71

2.4.2 Gamma-Ray Burst Outflows . . . . . . . . . . . . . . . 81

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Novae 94

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.2 Shocks in Novae . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.2.1 Evidence for Non-Thermal Radio Emission . . . . . . . 103

3.2.2 Condition for Radio Maximum . . . . . . . . . . . . . 107

3.2.3 Is the Forward Shock Radiative or Adiabatic? . . . . . 109

3.2.4 Thermal X-ray Emission . . . . . . . . . . . . . . . . . 111

3.3 Synchrotron Radio Emission . . . . . . . . . . . . . . . . . . . 113

3.3.1 Leptonic Emission . . . . . . . . . . . . . . . . . . . . 115

3.3.2 Hadronic Emission . . . . . . . . . . . . . . . . . . . . 117

3.4 Radio Emission Model . . . . . . . . . . . . . . . . . . . . . . 118

3.4.1 Pressure Evolution of the Post-Shock Gas . . . . . . . 118

3.4.2 Electron and Positron Spectra . . . . . . . . . . . . . . 119

3.4.3 Radiative Transfer Equation . . . . . . . . . . . . . . . 119

3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.5.1 Analytical estimates . . . . . . . . . . . . . . . . . . . 124

3.5.2 Radio lightcurves and spectra . . . . . . . . . . . . . . 130

3.5.3 Fits to Individual Novae . . . . . . . . . . . . . . . . . 134

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.6.1 Efficiency of Relativistic Particle Acceleration . . . . . 136

3.6.2 The slow matter in not main-sequence progenitor wind 138

3.6.3 Light Curve of V1723 Aquila . . . . . . . . . . . . . . 140

3.6.4 Radio Emission from Polar Adiabatic Shocks . . . . . . 142

ii

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Bibliography 150

Appendix: Non-Thermal Electron Cooling 181

iii

List of Figures

1-1 Schematic view of a shock . . . . . . . . . . . . . . . . . . . . 17

2-1 Geometry of neutrino-heated winds from proto-magnetar . . . 29

2-2 Magnetic field profiles for polar and last open field lines . . . . 34

2-3 Density trajectory with ρ ∝ t−3 extrapolation . . . . . . . . . 43

2-4 Ye,eq(t) and Lν(t) evolution from [22] used for nucleosynthesis

calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2-5 Energy density profiles of hydrodynamical trajectories . . . . . 51

2-6 Minimum surface B required for validity of the force-free ap-

proximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2-7 Pressure profiles in closed and open zones . . . . . . . . . . . 54

2-8 Radial profiles of v, ρ, T for the fiducial solution . . . . . . . . 60

2-9 Radial profiles of Ye, q and s for the fiducial solution . . . . . 61

2-10 texp versus r for polar, last open field line and spherical outflows 62

2-11 texp versus S for different models with threshold . . . . . . . . 63

2-12 M versus polar angle θ for P = 2 ms . . . . . . . . . . . . . . 64

2-13 r-process figure of merit η versus polar angle θ for P = 2 ms . 65

2-14 X(Z) for different polar angles θ for Lν = 1052 ergs s−1, P =1

and 3 ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2-15 Mej in individual elements and its ratio to spherical case . . . 69

2-16 A versus P and Ye for fixed Ye models . . . . . . . . . . . . . . 70

2-17 r-process figure of merit η versus Lν for different periods . . . 72

2-18 Mtot versus Lν for different periods . . . . . . . . . . . . . . . 75

iv

2-19 Rates of magnetar birth versus P . . . . . . . . . . . . . . . . 79

2-20 Nucleosynthesis yields compared to abundances of metal-poor

stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2-21 Magnetosphere-integrated Lorentz-factor versus Lν for differ-

ent P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2-22 Angular distribution of maximal Lorentz factor for P = 2 ms . 85

2-23 Time-integrated Xα, and Xn, Xp at 100 s versus period . . . . 87

3-1 Scheme of X-ray, UV and radio production regions in novae

outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3-2 Scheme of forward shock, reverse shock, cold shell and un-

shocked regions . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3-3 Radio light curves of V1324 Sco, V1723 Aql and V5589 Sgr . . 104

3-4 Brightness temperature of the early component of the three

radio novae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3-5 X-ray luminosities and temperatures kT of V5589 Sgr . . . . . 113

3-6 Peak thermal X-ray luminosity and peak 10 GHz radio lumi-

nosity νLν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3-7 Free-free optical depth τν versus gas temperature T behind

the shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3-8 Cooling function Λ(T ) from [247] for our fiducial ejecta com-

position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3-9 Energy spectrum and synchrotron emissivity of relativistic lep-

tons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3-10 10 GHz brightness temperature in the post-shock region versus

gas temperature for different models . . . . . . . . . . . . . . 127

3-11 Peak brightness temperature of non-thermal emission as func-

tion of the fraction of shock energy used for ionization fEUV,

density scaleheight H, shock velocity v8, and fraction of the

shock energy placed into relativistic electrons, εe . . . . . . . . 131

v

3-12 Peak brightness temperature of thermal emission as function

of density scaleheight H and shock velocity v8 . . . . . . . . . 132

3-13 Example model radio lightcurves and corresponding spectral

indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3-14 Synchrotron shock best-fit models for V1324 Sco, V1723 Aql

and V5589 Sgr . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3-15 Example light curves for a steep density gradient H = 5×1012

cm with smearings of different widths . . . . . . . . . . . . . . 141

3-16 Ratio of nonthermal peak radio luminosity to peak thermal

X-ray luminosity versus shock velocity vsh . . . . . . . . . . . 144

vi

List of Tables

2.1 Summary of Wind Models for RNS = 12 km, M = 1.4M . . . 49

2.2 Summary of magnetosphere-integrated quantities for RNS =

12 km, M = 1.4M . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3 Summary of Time-integrated Models . . . . . . . . . . . . . . 67

3.1 Commonly used variables and their definitions. . . . . . . . . . 100

3.2 Best-fit parameters for the thermal radio emission models . . . 106

3.3 Multiwavelength properties of novae . . . . . . . . . . . . . . . 106

3.4 Best-fit parameters of synchrotron model. . . . . . . . . . . . 135

Acknowledgement

I would like to thank our Lord and God and Saviour Jesus Christ for His

continued support throughout my life and throughout the PhD program. I

am greatly indebted to my advisor, Brian David Metzger, for his continuous

help and guidance ever since we first met. I learned from him not only great

deal of science, but also patience. I am grateful to my parents for everything

they have done for me. I am grateful to my wife Boguslava Bojidarova

Vlasova for her support. I am grateful to my kids for providing me the

opportunity to get distracted from science and then come back with fresh

mind. I am grateful to Andrei Beloborodov, Indrek Vurm, Yuri Levin for

fruitful discussions.

viii

1

Chapter 1

Introduction

Originally thought of as a constant unchanging place, space is full of evolution

which can be seen in human life, sometimes in hours, minutes and seconds.

One of the first variable phenomena noticed in European civilization was

supernova (SN) 1572, which was coined ’nova’ - new in Latin. Supernovae

and novae were not distinguished from each other for a long time. Only

much later it was realized that the events are clearly distinct in all param-

eters - total energy, luminosity, rate, ejected mass, etc. Novae constitute

frequent transients (around 50 per year in our Galaxy) with moderate total

energies of ≈ 1045 ergs, while supernovae are much rarer (1-5 per 100 years

in our Galaxy) and much more energetic (around 1049 ergs in radiation).

Supernovae and, to less extent, novae, are major factors contributing to evo-

lution of interstellar medium: they inject radiation, kinetic energy, products

of their nucleosynthesis, seeding the next generation of stars from the ashes

of the previous. The shock waves of supernovae are thought to trigger star

formation in giant molecular clouds [1].

1.1 Supernovae

The term ”supernovae” unites phenomena which are observationally similar,

but not necessarily similar in their mechanism of explosion. Supernovae reach

2

peak optical luminosities of ≈ 1043 ergs s−1 and typically last for 50 − 100

days. Historically, supernovae were divided into subclasses by their spectra

[2]. Type I supernovae do not exhibit hydrogen in their spectrum, but Type

II do. Type Ia displays intensive silicon lines, while Types Ib and Ic do not.

The supernovae of Type Ic, unlike Ib, lack also helium lines in their spectra.

Type II supernovae are divided on II-L (linear lightcurve decay) and II-P

(plateau in the lightcurve), IIn (narrow emission lines), IIb (after peak, their

spectrum transitions to Ib-like spectrum). So far, this classification has been

based mainly on observational properties and the association between the

above-mentioned classes and physics of the event has not been unambiguously

established for all classes [2].

Supernovae Ia are associated with thermonuclear runaway of a white

dwarf [2] which, by one mechanism or another, reached central density and

temperature enough for runaway carbon burning. Type II supernovae orig-

inate from collapse of the core of massive (M ∼> 10M) star [2]. An ambi-

guity still exists with physics of some further subclasses of SN Ia,Ib and Ic

[2]. Sometimes, the supernova ejecta runs into the circumstellar matter [3]

or into the material ejected by the same progenitor star prior to its explosion

[4, 5]. These interactions result in additional emission lines in the spectra

and increase bolometric luminosity by dissipating kinetic energy. The result

of the explosion is also notably different for core-collapse and white dwarf

runaway cases. Core-collapse supernova leaves behind either a neutron star

or a black hole, while in the case of thermonuclear runaway of a white dwarf

the star is fully destroyed. Hereafter, we focus on core-collapse supernovae.

Core-collapse supernovae occur when the core of massive (M ∼> 10M)

star runs out of nuclear fuel hence loses its energy source and pressure sup-

port. Typical progenitor star has a mass ≈ (10 − 15)M. The abundance

structure of the progenitor is onion-like: the layers outwards are iron, sul-

3

fur/silicon, neon/magnesium/oxygen, carbon/oxygen, helium. Type II su-

pernovae progenitors also have a hydrogen envelope on the outside, but Type

I supernova progenitors have lost the hydrogen envelope. For a long time,

the main paradigm was that the hydrogen envelope is stripped by line-driven

winds [6]. However, now the theories have shifted to the model where bi-

nary interactions strip the envelope (see [7] for review). Regardless of the

presence of a hydrogen envelope, fusion reactions terminate at iron-group nu-

cleus because iron-group nuclei are the most stable nuclei of all (they have the

most binding energy per nucleon). Hence, after iron group, nucleosynthesis

cannot produce energy by either fusion or fission anymore and gravitational

contraction of the core is inevitable.

The subsequent fate of the core depends on its mass. If its gravitation can

be balanced by degenerate electron pressure alone, then the star sheds outer

envelope and the core peacefully transitions to a cooling white dwarf. If the

mass of the core is too large to be supported by electron degeneracy pressure

(or, in other words, its mass is close to the Chandrasekhar limit ≈ 1.4M),

then the collapse will follow. Although by the time of collapse the star has

an age of few times 107 years, the collapse itself happens in fractions of

a second. As a result of collapse, iron-group nuclei are dissociated into free

nucleons. Electrons are ”pushed into” the protons (e+p→ n+νe). The core

becomes optically thick to neutrinos at this stage and they cannot carry out

energy efficiently. Dissociation of nuclei into nucleons greatly (by factor of

≈ 60) multiplies the number of particles and hence the number of degrees of

freedom. Consequently, the equation of state stiffens and collapse is stopped

at a density 2-3 times the nuclear density. This ”bounce” creates a sound

wave which travels down the density gradient and steepens into a shock at

radius of ≈ 10− 20 km. This shock produces a lot of neutrinos, but they do

not escape to infinity because the shock is optically thick initially. Neutrino

4

luminosity Lν experiences a spark of (2−3) ·1053 ergs s−1 for 10−20 ms when

the shock becomes optically thin to neutrinos as it propagates outward and

then Lν declines monotonically. The naive and beautiful idea that it is this

shock which produces a supernova has proven wrong. According to [8, 9],

the outward moving shock stalls into an accretion shock after having traveled

out to a distance of around 100 km. The shock stalls because of (1) neutrino

losses - even though the neutrino do not escape to infinity yet, they still carry

the energy away from immediate post-shock region and (2) the energy loss to

photodissociation of nuclei. How to revive this shock and make a supernova

explode is still an open question despite decades of intensive investigation

[10]. This problem is especially complicated because it involves almost exact

cancellation: only 1051 ergs come out in the form of kinetic energy of the

ejecta of the total 3 · 1053 ergs liberated by gravitational collapse and leaving

in form of neutrinos. Several scenarios have been proposed to facilitate the

supernova explosion. The scenarios which rely on neutrino-heating of pre-

shock region are called neutrino-driven mechanisms [11, 12, 13, 14]. Another

wide group of explosion scenarios are magneto-rotational scenarios [15, 16,

17, 18]. In those, the explosion is facilitated by centrifugal force and magnetic

pressure.

By one mechanism or another (or by some combination of them, or by

some completely different mechanism), in nature the core-collapse super-

novae do explode. The shock wave heats up the outer layers of the star and

sets them expanding. It is these outer layers that are observed as optical

supernova. Both observations of optical spectra line widths [19, 20] and sim-

ulations [21], show that velocity of the shock at the breakout from the star

is v ≈ 20, 000 km s−1 and ejection velocities ≈ 10, 000 km s−1.

After the core bounce, the hot proto-neutron star deleptonizes and cools

down on the timescale of ≈ 100 s [22]. The protoneutron star in the center

5

is left to cool and contract from the radius ≈ 50 km to normal neutron star

radius ≈ 12 km. Most of gravitational energy will be radiated as neutrinos

at this stage. Outer layers of proto-neutron star are heated up by neutrinos

and evaporate giving rise to a neutrino-driven wind. Neutrino-driven (or,

alternatively, neutrino-heated) winds are one of promising astrophysical sites

for rapid neutron capture, or r-process [23, 24]. Chapter 2 considers the

feasibility of neutrino-driven winds from rotating magnetized proto-neutron

stars as the sites for r-process.

1.1.1 Neutrino-heated winds as the source of r-process

The primary form of nuclear reactions in the Universe is thermonuclear fusion

occurring in the central regions of ordinary stars. It powers their radiated lu-

minosity. However, nuclear fusion, even in explosive environments, is unable

to produce nuclei heavier than A ≈ 100. The reason is that with increasing

nucleus charge, the Coulomb barrier also increases. For nuclei with A ∼> 100,

the temperatures required to overcome Coulomb barrier are so high that the

nuclei are dissociated into alpha-particles or free nucleons instead of fusing.

However, we measure non-zero abundance of A > 100 nuclei, up to 238U,

on Earth. That means that these heavy nuclei are produced by some other

process.

Seminal works [25, 26] establish the origin of the elements heavier than

A ≈ 100 (and some lighter isotopes as well) as neutron-capture process.

Neutrons do not have to overcome Coulomb barrier and can easily be cap-

tured by nuclei. After neutron capture, nuclei become neutron-rich and they

β-decay back to stability. So, in any environment where neutron capture

happens, β-decay also has to happen. Depending on speed ratio between

these two processes, rapid (r) and slow (s) neutron capture processes are dis-

tinguished. For s-process, the neutron capture rate is much slower than beta

6

decay rate. S-process is known to happen in convective shells of Asymp-

totic Giant Branch (AGB) stars [27]. Comprehensive models of s-process

predicting isotopic yields are available [28]. S-process track lies very close to

stability and s-process cannot step over the gap of A values with no stable

isotopes. For example, between Bi (Z=83) and Th (Z=90) there is gap with

no stable elements. So, U and Th can only be produced by r-process. In

addition, because r-process and s-process tracks are different, they produce

different isotopic patterns in lighter elements. Analysis of chemical compo-

sition of Earth [29], meteorites [30], other stars [31, 32] confirms necessity of

r-process for explaining measured abundances.

However, the astrophysical site where r-process occurs remains a matter

of debate [33, 34, 35, 36]. The two most promising sites are neutrino-driven

winds of core-collapse supernovae [37] or neutron star mergers (NSM) [38].

NSMs naturally eject neutron-rich matter which then undergoes r-process

nucleosynthesis up to A ∼> 200 and beyond [39, 40, 41, 42, 43, 44]. Here

we concentrate on another candidate site, the neutrino-heated winds from

rotating magnetized proto-neutron stars [37].

Neutrino-heated winds were proposed as most probable r-process site

within core-collapse paradigm [45, 37, 46, 47]. However, the current con-

sensus is that the neutrino-heated winds are unable to provide conditions

necessary for formation of third r-process peak at A ≈ 200 in non-rotating

non-magnetized spherically-symmetric models [37, 48, 49, 50, 51, 52, 53, 54,

22, 55]. Also, the studies of 244Pu in the deep sea sediments exclude the pos-

sibility of all core-collapse supernovae producing heavy r-process elements

[29], (however, see [30]). These two factors suggest that r-process happens

not in neutrino-heated winds of all core-collapse supernovae, but in some

subset of them. Several variations of standard scenario were proposed to

obtain successful r-process production by neutrino-heated winds like shock

7

waves leading to additional heating [56, 57], extreme mass neutron stars

(M ∼> 2M - see [58]) or nonconventional physics, like sterile neutrinos with

≈ eV mass [59, 60].

In chapter 2, we study another modification of standard neutrino-heated

wind, the neutrino-heated wind from rapidly rotating (P ∼< 10 ms) strongly

magnetized (Bsurface ∼> 1015−1016 G) protoneutron star, or ”proto-magnetar”

[23, 61]. Magnetar births account for at least 10 % of all neutron star births

in the Galaxy [62]. So, if the successful r-process happens in proto-magnetar

winds, this would not contradict [29] which find that production of heavy

r-process elements by all supernovae is ruled out.

We calculate hydrodynamic trajectories of outflows with different param-

eters (polar angle θ, luminosity Lν , period P ). In order to quickly esti-

mate feasibility of a particular hydrodynamic trajectory for r-process and

to compare our trajectories with trajectories from other works, we use a

simple analytic criterion developed by [63]: r-process will reach A ≈ 200

if r-process ”figure of merit” η ≡ s3/(τ0.5MeVY3e ) exceeds a threshold value

of ηthr = 2 · 103k3B s−1 nucleon−3, where s is entropy in kB per nucleon,

Ye = nprotons/(nprotons + nneutrons) is the electron fraction and τ0.5MeV is the

expansion time at the temperature of α-particle formation, Tα ≈ 0.5 MeV.

This criterion can be understood as following. At the conditions of interest,

the outflow is radiation-dominated and hence the entropy per nucleon is pro-

portional to T 3/ρ. To high precision, α-formation occurs at approximately

fixed temperature Tα ≈ 0.5 MeV. Formation of 12C from three α-particles

proceeds by the reaction sequence 4He(αn, γ)9Be(α, n)12C. Because 9Be

is much less bound that 12C, the second α-particle capture happens much

faster. Therefore this process is effectively four-body reaction (three α’s and

one neutron). Hence the reaction rate falls very fast with density. That’s

why we can assume that the reaction happens only during first τ0.5MeV after

8

α-formation. The amount of 12C produced can be thus estimated by:

X12C ∝ Xn(ρα)3τ0.5MeV ∝ Xn(Xαρ)3τ0.5MeV =

= Xn(2YeT3α/S)3τ0.5MeV ∝ XnY

3e τ0.5MeV/S

3 (1.1)

taking into account that Xα = 2Ye for neutron-rich winds (Ye < 0.5). For

r-process to proceed up to the nuclei with mass A and charge Z, the neutron-

to-seed ratio has to exceed mean number of captured neutrons in the resulting

nucleus,

Xn

X12C

> A− 2Z. (1.2)

r-process terminates at A ≈ 200 due to the lack of more massive stable

nuclei. For A ∝ 200, combination of (1.2) and (1.1) yields:

Xn

X12C

∝ Xn

XnY 3e τ0.5MeV/S3

= S3/(Y 3e τ0.5MeV) > ηthr = 2 · 103k3

Bs−1nucleon−3(1.3)

For Ye < 0.38, the final composition has non-zero neutron abundance and

the criterion (1.2) is modified. The modified version of the criterion is given

in the formula (2.24).

Neutrino-heated winds from rapidly-rotating magnetized proto-neutron

stars have been studied before [57, 64], but the geometry of the magnetic

field was approximated to be that of a simple monopole with B ∝ 1/r2. [57]

found that rotation and magnetic fields enhance η, but not to extent to imply

successful third-peak nucleosynthesis. In [57], part of the energy to escape

the star was supplied to the wind material by the rotational energy instead

of neutrino heating, hence reducing the amount of heating and asymptotic

entropy.

9

1.1.2 Neutrino-heated winds from rotating proto-magnetars

In Chapter 2 we use more realistic geometry motivated by force-free mag-

netospheric solution from [65]. Simulations of [65] study magnetosphere of

aligned rotator, that is, the pulsar with rotation and magnetic axes aligned.

Under this assumption, the magnetosphere is static. [65] also assume that

the matter inertia is negligible as compared to magnetic field density (this

assumption is called force-free condition). The characteristic size of the mag-

netosphere is the light cylinder radius, RLC = c/(2πΩ), where Ω is the an-

gular frequency of rotation. The light cylinder radius is the radius where

corotation speed equals to the speed of light. Hence, beyond light cylinder

no corotation of matter is possible and the field lines have to be open, leav-

ing to infinity. The region filled with field lines reaching infinity is called

open zone. Inside the light cylinder, there will be a region of corotating field

lines which do not leave to infinity. This region is called a closed zone. Very

close to the proto-neutron star (PNS) surface, the magnetic field is similar

to that of a standard magnetic dipole. The geometry of an aligned rotator

magnetosphere is shown at the Figure 2-1.

In aligned rotator geometry, the matter first moves along open lines which

all originate in the polar cap. Near the surface, all field lines point almost

radially outwards. Hence, the impact of centrifugal force in the heating re-

gion close to the surface is minimal. The resulting entropies range from

spherically-symmetric values for polar outflows almost to equatorial values

from [57] for last open field lines. τ0.5MeV is determined close to the light

cylinder. In that region, the last open field line direction has a substan-

tial horizontal component and the outflow is highly affected by centrifugal

force (see Fig. 2-1). The force-free geometry of an aligned dipole yields the

centrifugal reduction in τ0.5MeV and Ye which helps r-process nucleosynthe-

sis. The result is that the force-free outflows are more favorable for r-process

10

than equatorial ones, but still not enough for a successful third peak A ≈ 200

nucleosynthesis. Nucleosynthesis yields are also calculated using nucleosyn-

thesis network SkyNet [66], allowing us to quantitatively access the influence

of magnetic fields and rotations on yields of individual elements. Chapter 2

also addresses implications of proto-magnetars as central engines of GRB on

baryon loading of GRB jets and UHECR composition.

1.2 Novae

Novae explosions are the result of thermonuclear runaways (TNR) from the

surface of white dwarfs [67, 68, 69, 70, 71]. Novae occur in binary sys-

tems where one of the stars is a white dwarf, and the other one is main-

sequence star or a giant. The hydrogen-rich material from outer layers of

non-degenerate companion star accretes on the white dwarf, either through

the inner Lagrange point after Roche lobe overflow or through stellar wind of

giant star captured by white dwarf. After enough material has accumulated,

the temperature and pressure at the base of the accreted layer are sufficient to

ignite p-p hydrogen burning. At this time, the electrons in most of hydrogen

envelope are degenerate. Hence, the release of energy and increase of tem-

perature do not imply increased pressure and no rapid expansion occurs. In

addition, accreted matter is not governed by its self-gravitation, but rather

the gravitation of the central white dwarf. Hence, to reduce gravitational

force the shell has to expand substantial fraction of the WD radius, not the

substantial part of its own size (so-called thin-shell instability). Eventually,

the degeneracy is lifted and pressure raises, but at this point the temperature

is growing so fast that thermonuclear runaway occurs.

At T > 2 · 107 K, CNO cycle overtakes p-p reaction and at T > 108

K hot CNO cycle dominates. Hot CNO cycle dominates the total energy

11

production of TNR. During hot CNO cycle, large fraction of nuclei capable

of capturing a proton (CNONeMg...) do capture a proton and become β+

unstable nuclei (13N, 14O, 15O, 17F). The rate of energy generation is limited

then by temperature and pressure independent β+ decay rates [72]. During

hot CNO operation, convectional zone spans from the bottom of burning

zone almost to the surface, bringing β radioactive nuclei to cold outer layers.

Later β decay produces isotopic ratios which are very different from equilib-

rium operation of CNO cycle and from the solar abundances. Observational

evidence exists that the WD material gets mixed in the burning zone and

part of it is ejected together with nova ejecta [73, 74, 75]. Because of this

dredging, WD in classical novae systems decrease their mass in subsequent

novae explosions and accretion in between the explosions. Hence, they cannot

serve as a progenitor of Ia supernovae [76, 73].

As a result of TNR, ≈ 10−4M is ejected with v ≈ 103 km s−1. For a

long time, thermonuclear runaway was thought to produce single ejection of

matter which then expanded freely to large radii, interacting only with ISM

[72]. However, growing evidence from across the electromagnetic spectrum

suggests that the shocks are common, if not ubiquitous, in novae. Shocks

may originate from collision of two outflows: slower outflow launched ear-

lier in the outburst and faster outflow which then catches up with the slow

outflow [77, 24]. Alternatively, shocks may originate from the collision of

main nova outflow with pre-existing material around binary system, possibly

accelerated away by super-Eddington luminosity of white dwarf during the

outburst [78]. In either case, we call the outer ejecta Dense External Shell

(DES) while the faster outflow is called fast wind, following [79] and [24].

Optical spectroscopy reveals few system of lines corresponding to different

negative velocities [80],[81],[78]. The most prominent is wide (v ≈ (1−2)·103

km s−1) P Cygni profile which is visible on the timescale of few weeks after

12

the outburst. The P Cygni profile implies spherical outflow for fast wind.

Another system (sometimes multiple systems) shows narrow absorption lines

with negative radial velocities of order 200− 600 km s−1. These correspond

to DES. DES absorption systems are observed in the majority of novae [78]

implicating that DES possesses high covering fraction. In some novae, ab-

sorption line systems are seen to accelerate outwards with time [81]. Fast

spherical P Cygni wind collides with DES and that is indeed what is observed.

After 3-4 weeks, both P Cygni and absorption lines disappear and instead,

one wide system of emission lines appears [81, 78]. This phase is called neb-

ular phase of nova and it is accompanied by forbidden emission lines, which

require ionization source harder than the central white dwarf [82]. v ≈ 1000

km s−1 shocks naturally provide such ionization source. Velocity of nebular

system between P Cygni velocity and absorption lines velocities also nat-

urally arises from the shock model. IR observations of novae reveal dust

formation in some cases (see [83] for a review). The cold shell behind the ra-

diative shocks provides natural site for cold neutral gas shielded from ionizing

radiation where dust can form [84].

Hard thermal X-ray emission with kT ∼> few keV observed for some novae

[85] provide other evidence for shocks. This emission is too hard to originate

from a WD, which emits as a supersoft X-ray source with energies < 0.1

keV. The fact that hard X-ray luminosity LX ≈ 1033 − 1035 ergs s−1 is

much less than bolometric luminosity of the nova Lbol ≈ 1037 − 1038 ergs

s−1 seemed to imply that shocks play subdominant, almost negligible role in

novae energetics. For that reason, shocks in novae outflows have been studied

as an aside phenomenon for a long time with many comprehensive models of

novae explosions not mentioning shocks at all [86, 87, 72]. Shocks were used

to explain some features in the lightcurve [88] or spectrum [82],[89], but their

paramount importance for producing lightcurve was not acknowledged.

13

Dominant role of shocks in novae was fully realized following the discovery

of high-energy (∼> 100 MeV) γ-rays emission from novae by Fermi gamma-ray

space telescope [90]. While lower energy, ≈ 1 MeV, γ radiation resulting from

γ-decays of TNR products and Compton down-scattering of positron-electron

annihilation had been predicted beforehand [91], the only natural explanation

of ∼> 100 MeV γ-rays are high-energy particles from shocks [92, 93, 90, 79, 94].

First γ-ray detection was from nova V407 Cyg 2010 [95, 96, 93]. That

nova belongs to a rare class of symbiotic novae which are embedded in dense

wind of an M giant. Hence, the γ-ray emission from novae was thought to

be rare [95] and the material in which the nova outflow runs into (DES)

was thought to be the wind of M giant. However, since that time, Fermi-

LAT detected in ∼> 100 MeV γ-rays five classical novae (V959 Mon 2012,

V1324 Sco 2012, V339 Del 2013 - see [90], V1369 Cen 2013, V5668 Sgr - see

[97]). These discoveries show that classical novae also possess DES with high

covering fraction. Estimates based on both leptonic and hadronic scenarios

for γ-ray emission [79, 98] show that γ-luminosity cannot exceed ≈ 2 % of

total shock power. So, the γ-rays observations tell us lower limit on total

shock power Lshocks ∼> Lγ/0.02 ≈ 1038 − 1036 ergs s−1. Because of high

density environment, the shocks are radiative, i. e. they radiate all of the

shock power right away: Lopt,shock = Lshock. Then one finds that the shock

power and hence optical luminosity of shocks are a big fraction, up to 100 %

of optical luminosity of novae, Lopt,tot ≈ 1038 ergs s−1. Now, that we know

true power of shocks, [77] propose that hard X-ray luminosity of shocks is

small not because their overall power is small, but because of column of

neutral gas ahead of them which effectively absorbs X-ray emission. Indeed,

hard X-ray spectra of novae reveal absorbing columns 1021 − 1023 cm−2 (for

V5589 Sgr see [99], for V1723 Aql see [100], for review see [85]).

Evidence of shocks is also observed in radio waves. The most prominent

14

feature in radio lightcurve is late (t ≈ 200 − 500 days) peak produced by

thermal bremsstrahlung of electrons in expanding ionized spherical gas shell

with T ≈ 104 K [101]. Some novae show early-time ∼< 100 days non-thermal

radio peak [88, 100, 102, 103, 104] with brightness temperature Tb ≈ 105−106

K. This peak has been interpreted as shock-powered synchrotron emission

[88, 100, 102, 103, 104]. Its emergence at early times implies small size.

Together with flux comparable or greater than the late thermal peak, this

implies much higher brightness temperatures Tb ≈ 105 − 106 K. Note that

photoionization heat balance keeps the temperature of nova ejecta almost

constant at the level ≈ 104 K [105]. Hence, high brightness temperatures of

the early peak cannot be explained by gradually expanding spherical shell

of 104 K. Instead, they imply either nonthermal emission mechanisms or

gas temperatures 105 − 106 K, in either case being signature of shocks. In

addition, at early times the spectral index of optically thick expanding hot

sphere should be α = 2 and the flux should be raising as the observed surface

area ∝ t2. Therefore, the presence of early peak in numerous novae, for

example, Nova Vul 1984, V1324 Sco, V1723 Aql, V5589 Sgr and evolution of

spectral index: α > 2.4 in Nova Vul 1984 [88], α from 2.5 to 0.7 for V1324

Sco [106], from 1.5 to -0.2 for V1723 Aql [103] or from 0.77 to -0.08 for V5589

Sgr [99] cannot be reconciled with expanding hot sphere model.

Radio imaging using EVLA and other interferometers reveal the geometry

of nova outflow and the location of non-thermal emission. The geometry is

consistent with DES being the dense torus concentrated in equatorial plane

and fast wind being more spherically-symmetric outflow (see [107, 108] for

V959 Mon, [103] for V1723 Aql). Optical line profile modeling supports this

picture for V959 Mon [109]. For V959 Mon, the nonthermal emitting knots

are observed just where the DES and fast wind collide [107]. In the pic-

ture where DES is ejected during the outburst, DES of dense torus shape

15

can originate from gravitational interaction of companion star with origi-

nally spherical outflow [24]. Alternatively, the TNR itself can be asymmetric

ejecting more on the equator (Levin 2016, private communication). In the

picture [78] where DES is present even before the outburst, it naturally has

toroidal shape because DES is the gas which flew out of outer Lagrange point

of the companion. In all these scenarios, the orientation of dense torus is co-

incident with the orbital plane. In some novae this feature was confirmed

observationally (for example, V959 Mon - see [109]).

Early radio peak and nonthermal emission seen by EVLA can be ex-

plained as emission of shock-accelerated high-energy electrons ([24], see also

Chapter 3). Shock accelerated ultrarelativistic electrons produce synchrotron

radiation which is observed as early radio peak. Ultrarelativistic electrons

emit synchrotron radiation in the magnetic field amplified by the shock - this

is leptonic scenario. Alternatively, ultrarelativistic protons collide with ther-

mal protons and produce ultrarelativistic pions. Charged pions then decay

to electrons/positrons which, in turn, produce synchrotron radiation - this

is a hadronic scenario. In Chapter 3 we present the model of synchrotron

radiation of electrons in novae shocks. The model can explain observed in-

tensities of synchrotron radiation and, in some cases, the correct shape of

the lightcurve. For radio synchrotron radiation, quite high electron acceler-

ation efficiencies εe ≈ 0.01 and magnetic field energy fractions εB ≈ 0.01 are

implied, ruling out the hadronic scenario. The model also rules out the main

sequence (MS) stellar wind as DES, because MS stellar wind is too dilute to

produce observed radio emission.

1.2.1 Shock acceleration of protons and electrons

Most of the astrophysical studies of particle acceleration in shocks use su-

pernovae remnants (SNR). Numerous models of increased complexity were

16

proposed to explain cosmic rays (CR) by shock acceleration in SNR [110, 111,

112, 113, 114, 115, 116]. We will comment on SNR shock model explaining

CR origin because it is the main source of information about particle accel-

eration in astrophysical shocks, and then we will make connections of these

developments to the shocks in novae.

Energy budget of SNR

First, we consider if SNR can indeed supply required CR energy. CR energy

is injected in the Galaxy at a rate ≈ 4.3×1034 ergs s−1 [117]. The CR energy

budget is explained if the following fraction of SNR kinetic energy goes into

CR energy:

f = 0.14R−1100E

−151 , (1.4)

where R100 is supernovae rate per 100 years and E51 is ejected kinetic energy

in foe (1051 ergs). Taking into account that R100 is usually estimated to be a

few (1− 3) and given the acceleration efficiencies up to 10 % of shock power

seen in modern simulations (e. g. [113]), this number is not unreasonable.

Acceleration mechanism

The general outline of particle acceleration by successive bounces was first

proposed by Fermi [118, 119]. He explained the emergence of cosmic rays

by random wandering of energetic particles between moving magnetized in-

terstellar clouds [118, 119]. The particle energy gain for every bounce is

proportional to cloud velocity β = v/c, and the prevalence of head-on over

head-tail collisions is also proportional to β. Hence the acceleration efficiency

is proportional to β2 and the process is called second-order Fermi accelera-

tion. In shock environment however the particle gains energy every time it

17

Figure 1-1: Schematic view of a shock. v1 is the upstream and v2 is thedownstream region. The downstream region is more dense and the velocityin it is slower. For strong shock, downstream region is 4 times denser andvelocity is 4 times less than in the upstream.

bounces back and forth between upstream and downstream of the shock. This

is because the upstream region of the shock moves towards the downstream

region in downstream rest frame and vice versa, downstream moves towards

upstream in the upstream rest frame (see Figure 1-1). Because energy gain

of the particle every time is proportional to β, the Fermi acceleration at

the shock is called first-order Fermi acceleration. The particles are believed

to experience small deflections rather than reflections as a mechanism of

bouncing back and forth. Small deflections can be represented as diffusion in

the phase space. Hence another name for the mechanism is Diffusive Shock

Acceleration (DSA).

18

Upon relativistic particle collision with background particle of the same

type, substantial fraction of particle’s energy is lost. Hence, if the collisions

of particles are effective, then they will damp energy of CR-to-be much faster

than it is acquired. The necessary condition for CR acceleration is then that

the shock is collisionless, i. e. mediated not by collisions, but by some other

means. In astrophysics, collisionless shocks are usually mediated by magnetic

fields. Natural question then arises what are the values of magnetic fields in

SNR shocks. Tightly related question is what is the maximal energy of CR

accelerated by SNR shock.

Maximum energy of a CR in a shock

The maximum energy of CRs is limited by the ability of the accelerating

region to effectively scatter energetic particle, and the scattering is mediated

by magnetic field turbulence. [120] have argued that the maximum energy

attainable in shock acceleration SNR is:

Emax = sBRSNR(v1 − v2)Ze, (1.5)

where RSNR and B are the characteristic size and magnetic field in SNR, s is

the factor showing the efficiency of scattering s = rL/(mean free path) and

rL is the Larmor radius. In what follows, we adopt s = 1. After the time T0

when the mass swept by the ejecta equals the ejecta mass (”sweep time”),

the ejecta begins slowing down and shock power falls accordingly. Hence, it

is reasonable to calculate (1.5) at the sweep time T0:

Emax = 1.8 · 1014

(B

3µG

)E

1/251 M

−1/6ej, n

−1/3H Z eV (1.6)

Here Mej, is the ejected mass in solar masses and nH is interstellar density

in cm−3. The value of B is normalized to typical interstellar magnetic field:

19

the particles have to be confined not only in downstream region, but also in

the upstream region. Modest variations from the expectation value of the

equation (1.6) were obtained in [121, 122].

Observed CR spectrum has almost uniform power law dE ∝ E−2.7 up to

the ”knee” region, Eknee ≈ 4 × 1015 eV [117]. We see that simple estimate

(1.6) fails to reach this observed energy. Moreover, for E > Eknee, CR

spectrum falls steeper than E−2.7, but it is still measurable up to ≈ 5 · 1019

eV. A satisfactory theory of Galactic CRs acceleration has to provide CRs

at least up to the knee and possibly to ≈ 5 · 1019 eV. Based on (1.6), [120]

argued that shocks in SNR are unfavorable for producing CRs. The way out

of this difficulty is that the magnetic field to be substituted in the formula

(1.5) is much bigger than typical interstellar magnetic field.

CR-driven instabilities

The reason for the confining magnetic field to be bigger than interstellar

magnetic field is that cosmic rays themselves create instabilities leading to

magnetic turbulence and magnetic field amplification in the precursor layer

of the shock [123, 124, 125, 126, 127]. Precursor layer is the layer of upstream

media which is not disturbed by the shock itself yet, but it is disturbed by

propagating CRs and instabilities driven by CRs. At first, the resonant CRs

driven instability was thought to be the main mode for creating magnetic tur-

bulence [123, 124, 125, 126]. Later however the non-resonant Bell instability

[127] was shown to grow faster in linear regime than resonant CRs instabil-

ity. The magnetic fields in the shock precursor are observed to be amplified

by factors 30 − 60 as compared to far upstream both in SNR [128] and in

simulations [114]. This higher magnetic field is sufficient to push maximal

energy (1.6) little bit above the knee region. Above the knee, the CRs are

dominated by ions heavier than protons [117] with significant contribution

20

up to Fe. For heavy nuclei, additional factor Z increases the maximal energy

(1.6).

Predicted particle spectrum

DSA also makes prediction about the shape of accelerated spectrum. The

spectral shape can be understood from simple argument as follows [117]. If

we denote the compression ratio σ = ρ2/ρ1, then mass conservation implies

v1 = v2σ (see Figure 1-1). At every bounce, new energy E ′ of ultrarelativistic

particle is

E ′ = γ(E + p cos(θ)(v1 − v2)), (1.7)

where E, p are the energy and momentum of the particle before scattering,

θ is the angle between its momentum direction and the shock normal, and γ

is Lorentz factor for v1 − v2. Since it’s first order Fermi acceleration, we can

set γ = 1. After proper averaging over angles, we obtain

δp

p=

4

3

v1 − v2

c=

4

3(σ − 1)

v2

c(1.8)

for two reflections corresponding to one bounce back and forth. It is impor-

tant to note that the scattering has to be isotropic in the frame of the fluid in

order for this analysis to be true. On the other side, particles are lost at every

bounce by being advected downstream. The velocities of individual particles

are c: that is much bigger than the velocity of shock front in the upstream

frame, v1. However, the net flow of particles downstream through fixed fluid

element is zero because downstream is also filled with CRs and they come

from far downstream to shock surface, too. Hence, the CR particles are ad-

vected downstream with rate ∆n = nCRv1 per unit area of the shock with nCR

being the number density of CRs. Number of particles crossing the shock

21

surface per unit time per unit area is just n = nCRc〈cos(θ)〉/2 = nCRc/4.

Hence,

∆n

n= −4

v2

c(1.9)

Suppose the population of n0 particles starts with isotropically distributed

momenta of absolute value p0. Then, after N shock crossings, it will on

average evolve to:

n = n0

(1− 4

v2

c

)N(1.10)

p = p0

(1 +

4

3(σ − 1)

v2

c

)N(1.11)

Excluding N from the above equations, we can incorporate populations of

particles with different N ’s into single formula:

ln

(n

n0

)= − 3

σ − 1ln

(p

p0

)(1.12)

n ∝ p−3

σ−1 (1.13)

dn ∝ p−σ+2σ−1dp (1.14)

For strong shocks, σ = 4 and the last line becomes simply dn = p−2dp.

However, overall power going into CR acceleration can be as big as 10 % of

total shock power, altering the parameters of the shock itself and compression

ratio [113]. CRs make compression ratio smaller than 4, hence the resulting

spectrum of CRs is softer with slope ≈ −2.1. CRs observed on Earth have

spectrum dn ∝ p−2.7dp in the region 1011−4·1015 eV [117]. In fact, spallation

analysis of CRs yields travel times from the source ∝ E−γ with γ between

0.4 and 0.6 [117]. Injection spectrum E−2.1 times traveling (”trapping”) time

naturally yields the observed E−2.7 - this is one of attractive features of SNR

22

shock acceleration models.

Novae shocks as particle accelerators

In this work, we study the consequences of particle acceleration in novae

shocks. They are different from SNR shocks in a few aspects:

1. The density is 6-8 order of magnitude higher than in supernovae shocks

(but novae shocks are still collisionless, see [79])

2. The undisturbed magnetic field is not an ISM magnetic field, but the

frozen-in WD magnetic field drawn by DES from the surface of WD

High density of novae shocks precludes direct determination of magnetic field

by detecting self-absorption synchrotron frequency of relativistic electrons.

Free-free absorption of the medium dominates synchrotron self-absorption.

Hence, in radio synchrotron observations magnetic field amplification is de-

generate with particle acceleration efficiency (we can measure only εeε3/4B , see

Sect.3.6.1). Another consequence of high density is that far upstream of the

shock is neutral. Naively, this would preclude the possibility of particle accel-

eration by novae because non-resonant Bell instability is damped efficiently

by ion-neutral damping [129]. However, exactly because of high density, the

power of the shock ionizing radiation is enough to ionize quite wide layer

ahead of the shock [79, 94, 24]. Bell instabilities, magnetic turbulence, en-

ergetic particle acceleration upstream are all confined to this ionized layer

and hence its thickness should be used instead of ejecta radius in the formula

(1.5) for maximal achievable energy (see also [94]).

The magnetic field in DES is another factor different from SNR. In SNR

context, the upstream magnetic field is interstellar field which is often ho-

mogeneous on the scale of the remnants. So, the shocks probe all possible

inclination angles. Novae ejecta has traveled 103− 104 WD radii by the time

23

we observe shocks in radio, therefore the ”frozen in” field is stretched to be in

tangential direction irregardless of its original direction at the surface of WD.

The shocks propagate radially outwards, so the naive expectation is that the

shocks propagate perpendicularly to the magnetic field (quasi-perpendicular

shocks). Quasiperpendicular shocks do not accelerate protons or electrons,

according to simulation [130]. The matter of inclination angle is discussed in

more detail in Sect.3.3.1. The initial value of magnetic field in novae also can

be different than for supernovae shocks. However, the fact that we observe

high energy gamma-ray and radio synchrotron emission from novae tells us

unambiguously that particle acceleration does happen in at least some no-

vae.

24

Chapter 2

Supernovae

2.1 Introduction

All neutron stars are born as hot ‘proto-neutron stars’ (PNSs), which ra-

diate their gravitational binding energy and lepton number in neutrinos

over the first ∼ 10 − 100 seconds of their life [131]. A small fraction of

this luminosity is absorbed by matter in the atmosphere just above the

PNS surface, powering an outflow of mass known as the ‘neutrino-driven

wind’ [45, 37]. Initial interest in neutrino winds arose due to their poten-

tial as a primary source of rapid neutron capture (r-process) nucleosynthesis

[132, 46, 133, 47]. Neutrino-heated outflows are, however, also of great im-

portance in other contexts, such as their role in determining the maximum

Lorentz factor (baryon loading) and nuclear composition of gamma-ray burst

outflows [134, 135, 57, 136, 137, 138, 139].

Past studies of neutrino-driven winds have been primarily focused on

spherically symmetric, non-rotating winds accelerated purely by thermal

pressure [48, 49, 50, 51, 140, 53, 141, 54, 55]. This body of work has led

to the conclusion that normal PNS winds fail to achieve the conditions nec-

essary for nucleosynthesis to reach the third r-process peak at mass number

25

A ∼ 195. The latter requires an outflow with a combination of high en-

tropy S, short expansion timescale texp, and low electron fraction Ye [37, 63]

as it passes through the radii where seed nuclei form. Proposed mecha-

nisms to move beyond the standard scenario in order to achieve r-process

success include postulating additional sources of heating (e.g. damping of

convectively-excited waves; [56, 57]) or by resorting to extreme parameters,

such as massive ∼> 2.2M neutron stars [58].

The class of neutron stars known as ‘magnetars’ possess very strong mag-

netic fields ∼> 1014−1015 G and account for at least ∼ 10 per cent of neutron

star births [62]. Basic estimates show that magnetar-strength fields are dy-

namically important in PNS winds [142], even in the first few seconds after

core bounce when the neutrino heating and wind kinetic energy are highest.

Strong magnetic fields confine the outflow to only a fraction of the PNS sur-

face threaded by open field lines [142]. Rotation, when coupled with a strong

magnetic field, also provides an additional source of acceleration in the wind

via magneto-centrifugal forces [134]. Strong magnetic fields and rapid ro-

tation do not represent separate assumptions if rapid rotation at birth is

necessary to produce the strong field via dynamo action [143, 144].

[57] calculated the steady-state structure of rotating magnetized PNS

winds in ideal MHD under the assumption of an equatorial (one-dimensional)

outflow with an assumed monopolar radial field Br ∝ 1/r2. [57] found that

strong magnetic fields ∼> 1014 − 1015 G and rapid rotation (spin period P ∼<few ms) act to increase both the power and the mass loss rate of the wind as

compared to the spherical case (see also [134]). The conditions for a successful

r-process, as quantified by the critical ratio S3/texp we found to be moderately

improved from the spherical case. However, in some cases of extreme rota-

tion the conditions were in fact worse because although magneto-centrifugal

acceleration increases the outflow velocity (reducing texp), the residence time

26

of matter in the heating region−and hence its final entropy−was likewise

reduced in a compensatory way.

A successful r-process can take place even without high entropy if the

asymptotic electron fraction of the outflow Ye is substantially lower than the

equilibrium value Ye,eq ∼> 0.4 set by neutrino irradiation (eq. [2.10]). This is

conceivable if the outflow expands from the surface, where Ye starts much

lower, sufficiently rapidly as the result of centrifugal acceleration to avoid

entering such an equilibrium. In practice, however, both simple analytic

estimates (§2.3.2) and numerical simulations show that Ye Ye,eq is achieved

only for extremely rapid rotation (P ∼< 1 ms) near the centrifugal break-up

limit [136, 145]. Extreme rotation necessarily increases the total mass ejected

per event by a factor ∼> 10− 100 [136], placing such sources in tension with

studies of Galactic chemical evolution that suggest more frequent, lower-

mass pollution events are more consistent with the abundance distributions

of metal poor stars ([146, 147, 148]; however, see [149]). Most Galactic

magnetars cannot be born with extremely short spin periods P ∼ 1 ms,

as their high birth rate would overproduce the Galactic r-process [150] and

the energy of the explosion accompanying their birth would exceed those

measured for supernova remnants hosting magnetars [151].

Here we present calculations of neutrino-heated outflows from magne-

tized, rotating PNSs in the more realistic geometry of an aligned (axisym-

metric) dipole. Our solutions are calculated under the force-free assump-

tion, namely that mass flows along the magnetic field lines, with the lat-

ter prescribed by a self-consistent solution of the cross-field force balance

[152, 153, 65]. Once the hydrodynamical trajectories are obtained, they are

processed through nuclear reaction network SkyNet [66]. A series of one-

dimensional solutions calculated along flux tubes corresponding to different

polar field lines are then pieced together to determine the global properties

27

of the flow and total nucleosynthesis yields at a given neutrino luminosity

and rotation period. Although the rotation axes of pulsars are not in gen-

eral aligned with their magnetic fields, alignment is expected if the field is

generated as the result of a convective dynamo [143] and the results of our

calculations are expected to be insensitive to the inclination angle as long

as it is small. The case of aligned rotator is also the first step in exploring

generic inclination angles. The wind calculations presented here are more

accurate than previous works, which were one dimensional and assumed a

poloidal field the scaled with radius as ∝ 1/r2 or ∝ 1/r3 to mimick those

characterizing a monopolar or dipolar field structure [134, 57]. Our formal-

ism allow us to assess, for the first time, possible latitudinal dependencies of

the wind properties and nucleosynthesis yields.

A primary conclusion of this work is that proto-magnetar outflows are

more favorable for the r-process than slowly rotating, unmagnetized PNSs

by a factor up to ∼ 4 in S3/texp. This improvement results largely due to

the faster expansion rate of the outflow from centrifugal slinging, which is

accompanied by a more modest decrease in entropy than in the equatorial

monopole case. These improved r-process conditions are found to extend to

spin periods P ∼> 5 ms, indicating that core collapse events with extremely

high angular momentum are not necessary for a marked improvement. Nu-

cleosynthesis yields generally confirm this conclusion.

Beyond the r-process, our results have implications for the viability of

millisecond proto-magnetars as potential central engines of GRBs [154, 155,

134, 137] and the composition of UHECR, if UHECR are indeed produced by

GRB jest from proto-magnetars [156]. A long-standing theoretical mystery

is why GRB jets accelerate to bulk Lorentz factors Γ in the relatively narrow

range of ∼ 100 − 1000 [157], as this requires a finely tuned baryon loading

[158]. Here we confirm previous results that proto-magnetar outflows with

28

spin periods P ∼ 1 − 2 ms in the range necessary to explain the energetics

of GRBs come naturally ‘loaded’ with the correct baryon flux to achieve

the observed values of Γ. Our results also confirm the prediction that the

nuclear composition of GRB jets may be dominated by heavy nuclei instead

of protons [156]. There are some indications that ultra-high energy cosmic

rays (UHECRs; [159]) may be dominated by heavy nuclei ([160]; however,

see e.g. [161]). This would be naturally explained if millisecond magnetar

birth is the origin of UHECRs: in all our models nuclei with A ≈ 100 have

mass fraction of order of unity (see Fig. 2-23 and Fig.2-16).

2.2 Methods

2.2.1 Hydrodynamic model description

We calculate steady-state outflows along open field lines corresponding to

the solutions to the Grad Shafranov equations for an aligned dipole field [65]

and then we calculate nucleosynthesis yields on these trajectories using nu-

clear reaction network SkyNet [66]. The steady-state approximation is valid

for hydrodynamical trajectories because the properties of the PNS (rotation

period P , neutrino luminosity Lν , radius Rns) change slowly compared to

the time for matter to flow to the radii of interest. In force-free outflows

the matter flows along the prescribed field geometry. This approximation

is justified if the pressure P and kinetic energy density ρv2/2 are much less

than magnetic energy density B2/8π (where ρ and v are the density and

velocity of the outflow, respectively), a condition we check a posteriori from

our derived solutions.

The simplification that matter moves along the field lines reduces the

2D magneto-hydrodynamics problem to a series of 1D calculations along flux

tubes centered around each field line. Calculations are performed for different

29

Figure 2-1: Geometry of neutrino-heated winds from magnetized, rotatingproto-neutron stars. Outflows occur from the ‘open zone’, along field lineswith polar angles θ < θmax, where θmax = sin−1 [Rns/RY] is the angle ofthe last closed field line, Rns is the PNS radius, RY ∼< RL is the radius atwhich the last closed field line crosses the equator (the ‘Y point’), RL = c/Ωis the light cylinder radius, and Ω is the angular rotation rate of the PNS.In the illustration we have assumed RY = RL. The variable l defines thedistance measured along the field line from the surface, with the local fieldline direction denoted as l (the toroidal component of which is not shown).Note that l differs from the spherical radius r measured to the PNS center.The wind is composed of free nucleons at small radii, which recombine intoα-particles and heavier seed nuclei once the temperature decreases to ∼< 7GK several neutron star radii above the surface.

30

initial polar angles θ, with results then integrated over the open magneto-

sphere. The latter includes angles θ < θmax, where θmax = sin−1√Rns/RY

is the angle of the last closed field line, RY is the radius at which the last

closed field line crosses the equator (the ‘Y point’). Even if the force-free

approximation is valid in the wind itself, the location of the Y point, and

hence the size of the open zone, can be affected by the pressure of the closed

zone ([162]; §2.2.4, §2.3.1). The standard spherically-symmetric case is ac-

commodated within our framework by assuming a monopole magnetic field

Br ∝ 1/r2 and zero rotation. A schematic diagram of the wind configuration

is shown in Figure 2-1 for the case RY = RL, where RL = c/Ω = cP/2π is

the light cylinder radius.

The wind dynamics is calculated under the (Newtonian) approximation

of non-relativistic velocities, even though this assumption fails near the light

cylinder. This inaccuracy is tolerable because we aim to calculate hydrody-

namic trajectories interior to where T = 0.5 MeV, as typically occurs within

∼< 10 PNS radii above the surface where special relativistic effects play no

appreciable role. For the very same reason, we use nuclear statistical equi-

librium (NSE) equation of state for hydrodynamic trajectories calculations,

because NSE is good approximation before alpha particles formation. Nucle-

osynthesis network, however, uses density trajectories beyond our calculated

hydrodynamical trajectories and equation of state which takes into account

all nuclear species present. ρ(t) prescription and nuclear equation of state

used in nucleosynthesis network are described in detail in sect.2.2.7.

For simplicity we also neglect general relativistic (GR) effects, except as

included in an approximate way in a few test cases. Past studies have shown

that GR tends to increase the entropy of the flow [163, 50], but this effect

is expected to be similar for rotating and non-rotating stars, and our main

concern here is a comparison between these cases.

31

2.2.2 Wind Equations

The steady-state axisymmetric wind equations in cylindrical coordinates (x,

z) are given by

∂

∂l

(ρvB

)= 0; (2.1)

−v∂v∂l

+ xΩ2(l · x)− 1

ρ

∂p

∂l− GM

x2 + z2(l · r) = 0; (2.2)

vdYedl

= λ(ρ, T, Ye); (2.3)

vTdS

dl= q(ρ, T, Ye), (2.4)

where l is the distance measured along the field line. Equation (2.1) ex-

presses mass continuity, where ρ is the gas density, v is the velocity along

the magnetic field, B =√B2p +B2

φ is the magnetic field strength (where Bφ

and Bp are the toroidal and poloidal components, respectively). In §2.2.4 we

describe how the field strength and outflow geometry are calculated. Equa-

tion (2.2) expresses momentum conservation, where the second term is the

centrifugal force1, Ω = 2π/P is the rotation rate of the PNS, l is the unit

vector along the field line, and r is the unit vector from the center of the

PNS (see Fig. 2-1). The pressure p, entropy S and other thermodynamic

quantities are calculated from the Helmholtz equation of state [164].

Equation (2.3) evolves the electron fraction Ye due to weak interactions,

1The Coriolis force does not affect the motion of gas because it is perpendicular to thevelocity and hence to the magnetic field, the latter of which is parallel to the flow velocityin the force-free approximation.

32

which occur at the rate λ(ρ, T, Ye). Equation (2.4) evolves the entropy of

the outflowing gas due to neutrino heating and cooling, where q(ρ, T, Ye)

is the net heating rate. Tabulated values of λ and q are calculated as a

function of density, temperature, neutrino luminosity Lν , and neutrinosphere

temperature Tν (§2.2.3).

Equation (2.1) can be manipulated to read

∂ρ

∂l= −ρ

v

∂v

∂r+ ρ

d ln(B)

dl, (2.5)

which upon substitution into equation (2.2) gives

c2s − v2

v

dv

dl= c2

s

d ln(B)

dl+

1

ρ

(∂p

∂l

)ρ

+GM

r2 + x2(l · r)− rΩ2(l · x) (2.6)

where c2s ≡

(∂p∂ρ

)S,Ye

and the second term on the right hand side can be

written in terms of the other thermodynamic variables as follows

(∂p

∂l

)ρ

=(∂p

∂S

)ρ,Ye

dS

dl+

(∂p

∂Ye

)ρ,S

dYedl

= v

q

T

(∂p

∂S

)ρ,Ye

+ λ

(∂p

∂Ye

)ρ,S

,

(2.7)

where we have used equations (2.3) and (2.4).

2.2.3 Neutrino microphysics

Neutrino microphysics enters the wind evolution through the functions λ and

q (eqs. [2.3],[2.4]). Included are the dominant charged current reactions:

νe + p↔ n+ e+; νe + n↔ p+ e (2.8)

33

Cooling also includes neutral-current interactions [165]. We neglect the ef-

fects of strong magnetic fields on the rate of neutrino heating and cooling in

the PNS atmosphere [166, 167], as the corrections (primarily due to quanti-

zation of the lepton phase space into Landau levels) are found to be relatively

minor for surface field strengths ∼< 1016 G. The temperature of the neutri-

nosphere is calculated from the total electron neutrino luminosity according

to

Lνe + Lνe = 4πR2ns

7σ

8T 4ν . (2.9)

Competition between the absorption of neutrinos and antineutrinos by

nucleons in the wind (as dominate reactions well above the PNS surface)

drive the electron fraction towards the equilibrium value

Ye,eq =

(1 +

LνeLνe

〈ενe〉 − 2∆ + 1.2∆2/〈ενe〉〈ενe〉+ 2∆ + 1.2∆2/〈ενe〉

)−1

, (2.10)

set by the luminosities Lνe/Lνe and mean energies 〈ενe〉/〈ενe〉 of the elec-

tron neutrinos and antineutrinos, respectively, where ∆ = 1.293 MeV is the

proton-neutron mass difference [37].

Nucleosynthesis is extremely sensitive to the asymptotic value of Ye, which

in most of our wind solutions is found to approach Ye,eq to high accuracy.

The value and temporal evolution of Ye,eq, however depends on theoretical

uncertainties in the neutrino spectrum of the cooling PNS [54, 141]. Our

calculations of hydrodynamic trajectories take the mean neutrino and an-

tineutrino energies to be equal, 〈ενe〉 = 〈ενe〉, at the common value deter-

mined from the neutrinosphere temperature (eq. [2.9]). The ratio of Lνe and

Lνe is then chosen such that Y eqe ' 0.45 (eq. [2.10]), as motivated by recent

calculations which find a value of this order at early times in the PNS cooling

evolution (t ∼< 5 − 10 s after core bounce; e.g. [141], their Fig. 5). Please

34

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

106

107

Br3

, norm

aliz

ed to B

0R

ns3

Radius, cm

Maxim

um

heating

Sonic

poin

t

T=

0.5

Mev

Lig

ht cylin

der

r-1

Polar field line

Last open field line

Figure 2-2: Magnetic field strength B normalized to its surface value B0,along two field lines, corresponding to a polar outflow (θ = 0) and the lastopen field line (θ = θmax ≈ 21), calculated for a PNS with spin period P = 2ms. Near the surface both lines obey B ∝ 1/r3 (dipole), while at larger radiithe polar and last open field line follow B ∝ 1/r2 and ∝ 1/r, respectively.Shown for comparison are the locations of the sonic point (cross), maximumnet heating (plus), alpha particle formation radius R0.5MeV (asterisk), andlight cylinder radius RL for our solution with Lν = 1052 erg s−1.

note that neutrino microphysics described above is used for calculation of hy-

drodynamic trajectories only. Nucleosynthesis network runs after formation

of α-particles and hence above-mentioned processes are not relevant. Nu-

cleosynthesis network takes into account only neutrino emission reactions.

Neutrino microphysics and initial Ye values used in nucleosynthesis calcula-

tions are described in detail in sect.2.2.7.

35

2.2.4 Magnetosphere model

The magnetic field strength B and the field line trajectories (x,z) are calcu-

lated as a function of the distance along the field line l according to (e.g. [65])

B(l) = ξ/x (2.11)(dx

dl,dz

dl

)=

(l · x, l · z

)= ξ−1 (−∂zΨ, ∂xΨ) (2.12)

l · r =z∂xΨ− x∂zΨξ√x2 + z2

, (2.13)

where

ξ ≡√

(∇Ψ)2 +

(4π

c

)2

I2, (2.14)

and Ψ(x, z), I(x, z) are functions proportional to the magnetic flux and cur-

rent, respectively, flowing through a ring of constant cylindrical coordinates

(x, z). Coordinates (x, z), quantities l, l, r, r are illustrated at Figure 2-1. We

employ the solutions2 Ψ(x, z) and I(Ψ) of [65], focusing on the standard case

in which the last closed field line (θ = θmax) crosses the equatorial plane at

the light cylinder (RY = RL). As will be discussed further in §2.3.1, the

open zone may be larger than this (RY < RL) due to the high pressure of

the closed zone.

The geometry of the force-free magnetosphere is invariant to an overall

rescaling of Ψ and I. This implies that our solutions do not depend on the

strength of the magnetic field, as long as it is sufficiently high to justify

the force-free condition (§2.3.1). In contrast, our results do depend on the

PNS rotation period, as the latter determines the physical radius of the light

cylinder.

2Under force-free conditions the current can be written as a function of the flux, suchthat I(Ψ).

36

Figure 2-2 shows the magnetic field strength B as a function of spherical

radius r, calculated for a PNS spin period P = 2 ms along two different field

lines, corresponding to a polar outflow (θ = 0) and the last open field line

(θ = θmax ≈ 21). Close to the PNS surface, the poloidal component of the

field is dipolar (Bp ∝ 1/r3), with the toroidal component Bφ being small in

comparison. At larger radii near the light cylinder and beyond, the poloidal

field becomes monopolar (Bp ∝ 1/r2), while the toroidal field approaches the

dependence Bφ ∝ 1/r expected given the magneto-statics analogy between

the polar current and an infinite wire. These expectations are borne out

in Figure 2-2 by the magnetic field strength along the last open field line,

which varies as ∝ 1/r3 close to the PNS surface (poloidal dominated) before

transitioning to∝ 1/r (toroidal dominated) near the light cylinder. The polar

field line θ = 0, by contrast, is dominated by the poloidal component at all

radii, which slowly transitions from dipolar ∝ 1/r3 to monopolar ∝ 1/r2.

The faster the magnetic field strength decreases, the faster the matter

expands. The magnetic field near the surface decreases faster in the proto-

magnetar case B ∝ 1/r3 than for a spherical outflow B ∝ 1/r2. Non-polar

outflows also experience additional centrifugal acceleration, which causes the

matter to expand even more rapidly along field lines with larger θ (§2.3). In

§2.3 we show that centrifugal effects dominate the purely geometric effect of

the diverging field geometry in terms of its influence on the conditions for a

successful r-process.

The necessity of employing a realistic description for the field geometry

is made clear by the locations of various radii in the flow which are critical

to the nucleosynthesis. As shown by symbols in Figure 2-2, the sonic point,

location of maximum heating, and alpha particle formation radius often occur

at radii where the field geometry is not well-described by any limiting case.

For nucleosynthesis network calculations, we need to know ρ(t) beyond

37

calculated hydrodynamic trajectories, to several light cylinder radii. After

ρ(t) from the hydrodynamical trajectory ends, we extrapolate the density

dependence as ρ ∝ t−3 (see Fig. 2-3). The motivation for this extrapolation

is described in sect. 2.2.7.

2.2.5 Conservation checks

Equation (2.1) shows that the ratio of mass to magnetic flux,

M ≡ ρv

B, (2.15)

is a conserved quantity. It is automatically conserved at the level of our

numerical scheme because in our solution we express density from velocity

using (2.15). Conservation of momentum, energy, and lepton fraction, on

the other hand, serve as independent checks of our solutions. Momentum

and energy conservation (eqs. [2.2],[2.4]) can be combined into the integral

constraint (‘Bernoulli integral’) that

ρv

B

(v2

2+ ε+

p

ρ− GM√

r2 + z2− 1

2r2Ω2

)−∫ l

l0

ρ(l′)q(l′)

B(l′)dl′, (2.16)

is constant at each distance l across the radial grid, where ε is the internal

energy and l0 is an arbitrary starting point.

2.2.6 Shooting method

Equations (2.3), (2.4), and (2.6) are solved using a ‘shooting’ method by

which the eigenvalue is iteratively guessed and corrected in order to converge

upon an accurate solution. Equation (2.6) has a singularity at the sonic

point (v = cs), which can only be avoided for a transonic solution if the right

hand side of the equation simultaneously goes to zero. The value of M that

38

zeroes the right hand side at v = cs is not known a priori but must instead

be determined simultaneous with the full solution.

Several considerations are used to determine whether a given guess for

M is bigger or smaller than the critical value. First, if dv/dl is negative at

any point along the integration, then this implies that M is smaller than the

transonic value. On the other hand, if dv/dl exceeds an order-of-magnitude

estimate of its value by a large factor (taken to be∼ 50) then dv/dl is assumed

to be approaching infinity, indicating that M is bigger than the transonic

value. Such a ‘guess and check’ technique is used iteratively to determine

M and the structure of the subsonic solution up to the point where v is a

few percent less than cs. Above this velocity dv/dl is taken to be constant in

integrating the other equations (2.3,2.4) across the sonic point. Finally, once

v is a few percent above cs, then the full set of equations (2.3,2.4,2.6) are

again used to determine the supersonic structure. Our solutions are found

to conserve energy/momentum (eq. [2.16]) to an accuracy of ∼< 3%.

In addition to the transonic condition, boundary conditions on the tem-

perature, electron fraction, and density at the PNS surface are calculated

according to the following criteria [57]:

1. Zero net heating (q = 0.)

2. Weak equilibrium (dYe/dt ' λ = 0).

3. Density ρ = 1012 g cm−3, as found to produce an integrated neutrino

optical depth from infinity τν ∼ 1 (thus defining the surface as the

‘neutrinosphere’).

2.2.7 Nucleosynthesis

We start our reaction network calculations just after free nucleons recombine

into α-particles at a temperature of T ' 7 × 109 K. At small radii, where

39

the magnetic field is dynamically strong compared to the thermal or kinetic

energy densities, the force-free approximation is valid and we employ calcu-

lated density trajectories. At sufficiently large radii, matter inertia comes to

dominate the energy density of the magnetic field, and the outflows should

approach a spherical outflow. The density profile of a steady-state wind which

has reached a constant asymptotic velocity, should approach ∝ 1/r2 ∝ 1/t2,

while at larger radii where internal velocity gradients become important the

profile will approach ρ ∝ 1/r3 ∝ 1/t3 appropriate for a freely expanding

homologous outflow. As shown in Figure 2-3, we interpolate directly be-

tween our trajectories at small radii and the asymptotic 1/t3 dependence at

large radii. This simplification is motivated by the fact that the qualitative

features of the abundance patterns are robust to the detailed density tra-

jectory outside of the radii where charged particle processes cease and the

neutron-to-seed ratio has been determined. In order to test the importance

of an alternate density extrapolation, we have run a handful of trajectories

with ρ ∝ 1/r2 extrapolation. The difference in the asymptotic abundances

between the 1/r3 and 1/r2 density proles is less than 1% for most cases, and

none of the conclusions of this chapter are changed.

Another simplification is that we neglect heating or deceleration of the

wind by the reverse shock, which is produced as the wind interacts with the

surrounding supernova ejecta. This is justified in part3 because the radius

of the pulsar wind termination shock Rrs ' (vej/c)1/2Rej (e.g. [168]), where

vej is the mean velocity of the SN ejecta, is generally much larger than the

radius where the formation of seed nuclei occurs. The latter generally occurs

close to the light cylinder radius, which determines the radial scale of outflow

3In addition, highly magnetized winds experience much weaker compressional heating inthe shock than unmagnetized winds due to the additional support from magnetic pressure.

40

divergence. At time t since explosion we have Rej ' vejt and thus

Rrs

RL

≈ 37( vej

104 km s−1

)3/2(t

1s

)(P

1 ms

)−1

. (2.17)

We thus have Rrs RL for times t ∼> 1 s and rotation periods P ∼< 10 ms

of interest, implying that seed formation occurs well inside the termination

shock.

We start the nucleosynthesis network after the formation of α-particles.

By this time, the key parameters of the outflow (entropy s, expansion timescale

τdyn, electron fraction Ye and mass loss rate M) have already been determined

in the free nucleon zone at small radii, where nuclear statistical equilibrium

(NSE) provides a good approximation for equation of state. Furthermore,

after α-particle formation, neutrino heating and cooling have become neg-

ligible and hence are not included in the network reactions. Only neutrino

emissions are included in network calculations. Figure 2-8 shows that our

calculations begin at radii 5 − 7 times larger than the location where the

neutrino heating rate reaches its maximum just above the NS surface. The

locations of maximum neutrino heating are marked by stars in this figure,

while the locations where T = 0.5 MeV, which approximately corresponds

to the locations of α-formation, are marked by circles.

Hydrodynamic trajectories do not account for entropy gain from α-particle

formation, which we therefore increase by hand from its initial value s0 ac-

cording to

s = s0 +Xα(s, ρ, Ye) ·∆sα, (2.18)

where Xα(s, ρ) is mass fraction of α under NSE with given entropy, ∆sα =

Qα/Tα is entropy gain from α formation, Qα is the heat of α-formation, and

Tα is the temperature of α formation. Since the latter depends weakly on

41

other wind parameters, we adopt a fixed value of ∆sα = 10kB nucleon−1,

corresponding to Tα = 8 GK. The starting temperature for the reaction

network is determined from the density and the entropy according to equation

(2.18), with subsequent evolution tracked self-consistently from the density

trajectory and radioactive heating.

Finally, we must specify the initial outflow electron fraction used in our

network calculations, which equals the “final” electron fraction Ye,final set by

processes near the PNS surface. In standard thermally-driven PNS winds,

this value is determined by the competition between electron neutrino and

electron anti-neutrino capture reactions on free nucleons (2.8). These re-

actions have frozen-out (become slow compared to the expansion rate) at

the large radii where our calculations would begin, resulting in an electron

fraction close to the equilibrium value set by neutrino absorption reactions

(2.10).

For relatively slowly rotating proto-magnetars, P > 2 ms, the final

electron fraction is very similar to the normal thermally-driven case, i.e.

Ye,final ' Ye,eq. However, in the most rapidly spinning cases, P ∼< 2 ms, we see

from our hydrodynamical trajectories that Ye,final < Ye,eq due to rapid magne-

tocentrifugal acceleration (see also [134, 57, 169]), which causes a premature

freeze-out of the neutrino absorption processes before Ye is raised completely

from its low value near the PNS surface. More quantitatively, introducing

the definition (∆Ye)cent ≡ Ye,final − Ye,eq, we find that (∆Ye)cent ' −0.14 is

maximum in our trajectories. The values of (∆Ye)cent are reported in Table

2.1. We keep the dependence of (∆Ye)cent on Lν = 0.5(Lνe + Lνe) and θ and

we neglect the dependence on the other parameters (Ye,eq, Lνe , ενe , ενe).

In sect.2.2.3, we mentioned that for hydrodynamic trajectories we al-

ways set Lνe/Lνe so that Ye,eq = 0.45. However, in nucleosynthesis calcu-

lations we use Ye,eq(t) using the time evolution of Lνe , Lνe , 〈ενe〉, 〈ενe〉 from

42

PNS cooling calculations of [22]. This is justified because hydrodynamical

trajectories depend mainly on the amount of heating which is weakly de-

pendent on Ye,eq. Indeed, heating results from charged-particle reactions

(2.8). Lνe/Lνe in our calculations is never different from 1 by factor more

than 5, so if we replace some protons by neutrons, they will just capture

neutrino instead of antineutrino and heating effect would be roughly the

same. As the initial conditions for our network calculations we therefore

take Ye,final = Ye,eq(t) + (∆Ye)cent[Lν , θ] for P = 1, 2 ms and Ye = Ye,eq(t)

for all other periods. Given the uncertainties in neutrino radiation transport

[55, 22, 54], especially in the essentially unexplored case of very rapid rotation

(however, see [170, 171]) and given the effects of ultra-strong magnetic fields

on the microphysics [172, 166], we concentrate on comparing the properties

rotating proto-magnetar winds to the conventional unmagnetized spherical

wind case. In this way, we isolate the effects arising exclusively due to the

novel outflow geometry and magneto-centrifugal acceleration.

Reaction Network Calculations

We start our nucleosynthesis calculations at the point where hydrodynamical

trajectory reaches T = 7×109 K, corresponding to the approximate temper-

ature of α−particle formation. Actual starting temperature is slightly larger

because we take into account the entropy enhancement from α-formation

(eq. 2.18). We use the nuclear reaction network SkyNet [66] for the nucleosyn-

thesis calculation. The composition starts out in nuclear statistical equilib-

rium (NSE), a good approximation for the initial temperature T ∼> 7×109 K.

Given the extrapolated density as a function of time (see §2.2.1), SkyNet then

evolves the abundances of 7843 nuclear species under the influence of over

140,000 nuclear reactions. The evolved species range from free neutrons and

protons to 337Cn (Z = 112). SkyNet also evolves the entropy and tempera-

43

102

103

104

105

106

107

0.0001 0.001 0.01 0.1

ρ ~ t-3

ρ ~ t-3

Density, g c

m-3

Time from α-particle formation, s

Spherical

P=3 ms, θ=θmax

P=3 ms, θ=10

Figure 2-3: Density profiles of wind calculations for Lν = 1052 ergs s−1,for both rotating proto-magnetars with P = 3 ms (blue and green lines)and spherical wind (blue line). At large radii we extrapolate the densityprofile ρ ∝ t−3 (black lines), as expected at very late times for homologousexpansion.

44

ture, which change due to the expansion of the material and energy released

by the nuclear reactions. SkyNet uses a modified version of the Helmholtz

equation of state [173], which treats every nuclear species as a separate Boltz-

mann gas, including also electron-positron gas with arbitrary degree of rel-

ativism and photon gas. In some trajectories the initial temperature after

taking into account the entropy gain from α-formation is above 1010 K. In

these cases, SkyNet uses nuclear statistical equilibrium for evolution of the

network down to 1010 K, below which it switches to full network evolution

with all reactions.

The rates of the strong reactions come from the JINA REACLIB database

[174], but only the forward rates are used and the inverse rates are computed

from detailed balance. This is to ensure consistency with NSE, which de-

pends on the nuclear masses. Spontaneous and neutron-induced fission rates

are taken from [175], [176], [177], and [178]. Most of the weak rates come

from [179], [180], and [181] whenever they are available, and otherwise the

REACLIB weak rates are used. We used the nuclear masses and partition

functions from the WebNucleo XML file distributed with REACLIB, which

contains experimental data where available and finite-range droplet macro-

scopic model [182] data otherwise.

For Ye > 0.5, charged particle process nuclei (Z ' 38 − 40) can also be

formed through νp process (e.g. [183, 140]); however, even though SkyNet

supports including neutrino interactions, we do not include these in the

present calculations. We leave an exploration of charged particle process

nuclei formation in proton-rich proto-magnetar winds to future work.

Classes of Abundance Models

The nucleosynthesis products of proto-magnetar winds vary as a function

of the latitude of the open flux tube at a fixed time. They also vary in a

45

global sense, integrated over the entire magnetosphere at a fixed time, or

averaged over all times in the PNS cooling evolution. We cover the range of

possible diagnostics of the wind nucleosynthesis by calculating the abundance

patterns in three general cases:

1. Individual flux tubes along different latitudes θ at a fixed time (or

equivalently, neutrino luminosity), producing abundance yields as a

function of (θ, Lν , Ye, P ).

2. The entire wind at a fixed time, by integrating individual flux tubes

over the solid angle of the open magnetosphere, producing abundance

yields as a function of (Lν , Ye, P ).

3. The entire wind (integrated over the open magnetosphere) averaged

over the Kelvin-Helmholtz cooling epoch Lν(t) (Fig. 2-4), producing

abundance yields as a function of just the rotation period P .

In the angle-integrated cases, we weight the abundances by the mass loss

rate along each given open flux tube. In the time-integrated case, they are

weighted also by the total ejected mass from the entire magnetosphere at

each epoch t. In the latter case, by fixing the rotation period, we have as-

sumed that the magnetic spin-down time of the pulsar is longer than Kelvin-

Helmholtz cooling timescale of seconds; this approximately is justified for

surface magnetic dipole field strengths of Bd ∼< 1016 G.

In the time-integrated case, we also consider two methods for treating the

time evolution of the electron fraction Ye. In one case, we fix Ye at a constant

value throughout the cooling epoch. In the second case we derive its value as

described in §2.2.1 using the equilibrium electron fraction (eq. 2.10) from the

46

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

0.1 1 1010

50

1051

1052

1053

Ye

(Lν

e+

Lν_

e)/

2

Time, s

Ye, eq

(Lνe

+Lν

_e)/2

Figure 2-4: The blue line (right axis) shows the time evolution of the elec-tron neutrino luminosity Lν(t) = (Lνe + Lνe)/2 used in our calculations,based on the electron neutrino luminosity Lνe(t) and antineutrino luminos-ity Lνe(t) from the Kelvin-Helmholtz cooling calculations of [22]. A red line(left axis) shows the corresponding equilibrium value of the electron fractionYe,eq (eq. 2.10) used as input to our nucleosynthesis calculations (we alsodown-correct Ye for centrifugal effects for P ≤ 2 ms, but this is not shownhere).

time evolution of Lν , Lν , 〈ενe〉, 〈ενe〉 from the PNS cooling calculations of [22],

as shown on Figure 2-4. Since we are focused on the r-process, we integrate

only over epochs when Ye,eq < 0.5, i.e. at times 0.1 ∼< t ∼< 5 s, and all reported

wind properties (e.g., ejecta mass, mass fractions, etc. - all the data in Table

2.1) refer to just this time period. This is a reasonable approximation to the

total yield of the wind, since the bulk of the total mass loss is occurs at early

times t ∼ 1− 2 s.

47

2.3 Results

Solutions are calculated for a range of total neutrino luminosities Lν ≡ (Lνe+

Lνe)/2 = 1052, 6 × 1051, 3 × 1051, 1051 erg s−1, as are physically achieved at

times t ∼ 1, 3, 6 and 15 seconds after core bounce according to the PNS

cooling calculations of [184]. For each luminosity, solutions are calculated

for different values of the PNS spin period P = 1, 2, 3, 5, 10 ms and for

different angles θ of the open field lines as a fraction of the angle θmax of the

last open line (Fig. 2-1). For every hydrodynamic trajectory, corresponding

nucleosynthetic yield is also calculated using SkyNet network.

All results discussed in this section, as summarized in Figures 2-6−2-18

and Table 2.1, are calculated assuming a PNS of ‘standard’ mass M = 1.4M

and radius Rns = 12 km (except where otherwise noted).

Because only a fraction of the PNS surface is open to outflows in the mag-

netized rotating case, a spherically equivalent mass loss rate can be defined

according to

M ≡ 4πR2nsρ0v0, (2.19)

where ρ0 and v0 are the density and velocity, respectively, at the PNS surface.

The value of M for each field line solution is reported in Table 2.1 and Figure

2-12. The spherically equivalent mass loss rate M is not to be confused with

the true total mass loss rate

Mtot =

∫ρ0v0dS, (2.20)

where ρ0, v0 are surface density and velocity, and the integration extends

over the PNS surface threaded by open magnetic flux, θ ∈ [0, θmax]. The

magnetosphere-integrated values of Mtot are reported reported in Table 2.2

48

and Figure 2-18.

2.3.1 Validity of the Force-Free Approximation

Figure 2-5 compares the pressures associated with matter and radiation Pgas

to that of the magnetic field PB = B2/8π as a function of radius, calculated

for our fiducial solution (Lν = 1052 erg s−1; P = 2 ms) and an assumed surface

magnetic field strength B0 = 1015 G. Thermal pressure exceeds magnetic

pressure at the PNS surface, as expected since otherwise the field could

not be anchored to the star. Gas pressure decreases rapidly with radius,

such that magnetic pressure comes to dominate just ∼ 2 − 3 km above the

surface. Magnetic pressure continues to dominate until the kinetic energy

grows to a comparable size near the light cylinder, at which point the force-

free approximation breaks down (our assumption of non-relativistic velocity

is violated at a similar location). As already discussed, such inaccuracies

are tolerated at large radii because we are interested is the outflow structure

at temperatures T ∼> 0.5 MeV (marked as a circle in Fig. 2-5), at which

point the value of Pgas/Pmag is still ∼< 10 per cent. The ratio Pgas/Pmag at

T = 0.5 MeV for each of our solutions is provided in Table 2.1, for the

same assumed surface field strength B0 = 1015 G. For the nucleosynthesis

network calculations we use ρ(t) the hydrodynamic trajectory up to the light

cylinder. The part of ρ(t) near the light cylinder can be viewed as the

interpolation between the inner region where force-free approximation is valid

and ρ(t) ∝ t−3 approximation which we use after the end of hydrodynamic

trajectory.

Figure 2-6 shows the minimum surface field strength Bmin for which the

force-free criterion (defined as Pgas/Pmag < 0.1 at T = 0.5 MeV) is satisfied

for all outflow angles θ < θmax, as a function of the neutrino luminosity

and PNS spin period. At the highest neutrino luminosities ∼ 1052 ergs s−1,

49

Table 2.1: Summary of Wind Models for RNS = 12 km, M = 1.4M

P L(a)ν

θθmax

M(b) S(c) t(d)exp (∆Ye)

(e)cent η = S3/texpY

3e R

(f)0.5MeV

PgasPmag

|(g)R0.5MeV

(ms) (1051 erg s−1) - (M s−1) (kb n−1) (ms) - % of (η(h)thr

) (km) -

∗N/A 1 ∗N/A 3.8 · 10−6 100 0.1837 0.0039 0.7 47 ∗N/A∗N/A 3 ∗N/A 7.1 · 10−5 77 0.0424 0.0028 1.5 68 ∗N/A∗N/A 6 ∗N/A 4.5 · 10−4 67 0.0199 0.002 2.1 89 ∗N/A∗N/A 10 ∗N/A 1.7 · 10−3 61 0.013 0.0013 2.4 114 ∗N/A

2 1 0.0 5.8 · 10−6 94 0.1301 0.0038 0.9 36 2.0 · 10−2

2 3 0.0 1.1 · 10−4 74 0.0262 0.0025 2.2 48 1.0 · 10−1

2 6 0.0 7.4 · 10−4 65 0.0109 0.0017 3.5 58 3.3 · 10−1

2 10 0.0 2.9 · 10−3 60 0.0066 0.0008 4.6 70 9.5 · 10−1

2 1 0.3 7.4 · 10−6 68 0.0347 0.0034 1.2 28 5.4 · 10−3

2 6 0.3 9.3 · 10−4 48 0.0023 -0.0003 6.8 41 6.6 · 10−2

2 10 0.3 3.6 · 10−3 45 0.0017 -0.0017 7.9 49 2.4 · 10−1

2† 10 0.7 7.7 · 10−4 67 1.0 0.0019 40.6 35 3.3 · 10−2

2 3 1.0 2.5 · 10−4 37 0.0016 -0.0041 4.8 27 9.7 · 10−3

2 10 1.0 6.2 · 10−3 33 0.0006 -0.0102 9.0 38 2.7 · 10−1

2 1 1.0 1.3 · 10−5 45 0.0092 0.0006 1.4 23 2.4 · 10−3

2 3 1.0 2.6 · 10−4 37 0.0015 -0.0046 4.8 27 9.7 · 10−3

2 6 1.0 1.7 · 10−3 33 0.0008 -0.0085 7.4 32 4.7 · 10−2

2 10 1.0 6.5 · 10−3 32 0.0006 -0.011 9.1 38 2.8 · 10−1

3 1 0.1 5.8 · 10−6 95 0.132 0.0037 0.9 36 9.3 · 10−3

3 3 0.1 1.1 · 10−4 75 0.0267 0.0025 2.2 48 4.7 · 10−2

3 6 0.1 7.4 · 10−4 66 0.011 0.0017 3.6 58 1.5 · 10−1

3 10 0.1 2.9 · 10−3 61 0.0065 0.0009 4.8 69 4.4 · 10−1

3 1 0.3 6.7 · 10−6 77 0.0562 0.0036 1.1 31 3.9 · 10−3

3 3 0.3 1.3 · 10−4 61 0.0093 0.0019 3.3 39 1.5 · 10−2

3 6 0.3 8.4 · 10−4 54 0.0038 0.0007 5.8 46 4.7 · 10−2

3 10 0.3 3.2 · 10−3 51 0.0027 -0.0001 7.2 56 1.7 · 10−1

3 1 1.0 9.2 · 10−6 58 0.0198 0.0028 1.4 26 1.7 · 10−3

3 6 1.0 1.2 · 10−3 42 0.0014 -0.0025 7.6 36 2.4 · 10−2

3 10 1.0 4.5 · 10−3 39 0.0009 -0.0043 9.5 43 1.1 · 10−1

4 1 0.1 5.8 · 10−6 95 0.1332 0.0037 0.9 36 5.3 · 10−3

4 3 0.1 1.1 · 10−4 75 0.0269 0.0024 2.2 48 2.7 · 10−2

4 6 0.1 7.3 · 10−4 66 0.0111 0.0017 3.6 58 8.8 · 10−2

4 10 0.1 2.9 · 10−3 61 0.0065 0.0009 4.8 69 2.5 · 10−1

4 1 1.0 7.9 · 10−6 66 0.0311 0.0033 1.3 28 1.4 · 10−3

4 3 1.0 1.5 · 10−4 52 0.0049 0.001 4.1 34 5.0 · 10−3

4 6 1.0 10.0 · 10−4 47 0.0021 -0.0007 7.1 40 1.7 · 10−2

4 10 1.0 3.9 · 10−3 44 0.0014 -0.0022 9.0 47 6.7 · 10−2

5 1 0.1 5.8 · 10−6 95 0.1339 0.0037 0.9 36 3.4 · 10−3

5 6 0.1 7.3 · 10−4 66 0.0112 0.0017 3.6 58 5.7 · 10−2

5 10 0.1 2.9 · 10−3 61 0.0065 0.0009 4.9 69 1.6 · 10−1

5 1 1.0 7.4 · 10−6 71 0.0411 0.0035 1.2 29 1.1 · 10−3

5 3 1.0 1.4 · 10−4 56 0.0066 0.0015 3.8 36 4.3 · 10−3

5 6 1.0 9.3 · 10−4 50 0.0027 0.0001 6.5 42 1.4 · 10−2

5 10 1.0 3.6 · 10−3 47 0.0018 -0.0013 8.3 50 5.0 · 10−2

10 1 0.1 5.7 · 10−6 95 0.1311 0.0037 0.9 36 8.8 · 10−4

10 3 0.1 1.1 · 10−4 75 0.0264 0.0024 2.2 48 4.5 · 10−3

10 10 0.1 2.8 · 10−3 61 0.0065 0.0008 4.8 69 4.1 · 10−2

10 1 1.0 6.1 · 10−6 86 0.0876 0.0037 1.0 34 6.0 · 10−4

10 3 1.0 1.2 · 10−4 68 0.0159 0.0023 2.7 43 2.6 · 10−3

10 6 1.0 7.7 · 10−4 60 0.0064 0.0014 4.7 51 8.3 · 10−3

10 10 1.0 3.0 · 10−3 56 0.0039 0.0005 6.3 60 2.5 · 10−2

1 3 0.0 1.1 · 10−4 72 0.0233 0.0024 2.2 48 3.2 · 10−1

1 6 0.0 7.4 · 10−4 63 0.0099 0.0015 3.5 59 9.9 · 10−1

1 10 0.0 2.9 · 10−3 57 0.0062 0.0008 4.3 72 2.8 · 100

1 6 0.4 1.7 · 10−3 32 0.0007 -0.0094 6.9 32 1.6 · 10−1

1 3 1.0 5.4 · 10−3 14 0.0003 -0.1309 7.2 20 2.6 · 10−1

1 6 1.0 3.2 · 10−2 14 0.0002 -0.1367 9.8 25 2.8 · 100

1 10 1.0 1.1 · 10−1 14 0.0002 -0.1281 10.4 32 2.7 · 101

∗Spherical solutions; † Calculated using Pacynzsky-Wiita gravitational potential to approximate effects of GR. (a) Lν ≡

(Lνe+Lνe )/2; (b) Mass outflow rate (equivalent spherical; eq. [2.19]); (c)Final entropy (d)Expansion timescale at T = 0.5

MeV (eq. [2.23]); (e) Centripetal Ye lowering from equilibrium Ye = 0.45; (f)Radius at which T = 0.5 MeV, approximate

location of alpha particle formation; (g)Ratio of ‘matter pressure’ Pgas to magnetic pressure Pmag = B2/8π (for an

assumed surface field B0 = 1015 G) at T = 0.5 MeV, where Pgas includes gas pressure P , thermal energy density e,

kinetic energy density (rest energy subtracted); (h)Threshold ratio for outflow to attain third-peak r-process (eq. [2.24])

50

Table 2.2: Summary of magnetosphere-integrated quantities for RNS = 12km, M = 1.4M

P L(a)ν M

(b)tot M

(c)ej Γ

(d)max 〈η〉 = 〈S3/texpY 3

e 〉(e)

(ms) (1051 erg s−1) (M s−1) (M) - (% of ηthr)

∗N/A 1 3.8 · 10−6 3.5 · 10−5 ∗N/A 0.7∗N/A 3 7.1 · 10−5 2.6 · 10−4 ∗N/A 1.5∗N/A 6 4.5 · 10−4 9.4 · 10−4 ∗N/A 2.1∗N/A 10 1.7 · 10−3 3.3 · 10−3 ∗N/A 2.4

1 1 5.5 · 10−6 5.1 · 10−5 438.5 1.21 10 5.5 · 10−3 1.0 · 10−2 1.4 8.91 3 2.7 · 10−4 9.8 · 10−4 10.1 6.21 6 2.1 · 10−3 4.4 · 10−3 2.2 9.62 1 6.3 · 10−7 5.8 · 10−6 239.7 1.42 10 3.1 · 10−4 5.9 · 10−4 1.5 8.92 3 1.3 · 10−5 4.6 · 10−5 13.1 4.42 6 8.1 · 10−5 1.7 · 10−4 2.9 7.03 1 3.8 · 10−7 3.5 · 10−6 79.1 1.33 10 1.9 · 10−4 3.5 · 10−4 1.2 8.53 3 7.0 · 10−6 2.6 · 10−5 5.2 3.93 6 4.8 · 10−5 10.0 · 10−5 1.6 6.64 1 2.8 · 10−7 2.6 · 10−6 34.9 1.14 10 1.4 · 10−4 2.6 · 10−4 1.1 7.94 3 5.5 · 10−6 2.0 · 10−5 2.7 3.64 6 3.5 · 10−5 7.3 · 10−5 1.3 6.05 1 2.4 · 10−7 2.2 · 10−6 17.3 1.15 10 1.2 · 10−4 2.2 · 10−4 1.0 7.25 3 4.7 · 10−6 1.8 · 10−5 1.8 3.45 6 3.0 · 10−5 6.3 · 10−5 1.1 5.710 1 9.0 · 10−8 8.2 · 10−7 3.7 0.910 10 4.4 · 10−5 8.3 · 10−5 1.0 5.710 3 1.7 · 10−6 6.5 · 10−6 1.1 2.510 6 1.1 · 10−5 2.4 · 10−5 1.0 4.2

∗Spherical Solutions. (a) Lν ≡ (Lνe+Lνe )/2; (b)Total mass loss rate, integrated across the open zone from

θ = 0 − θmax (eq. [2.20]). (c)Total ejected mass at luminosities ∼ Lν , estimated using the PNS cooling

evolution from [184]. (d)Maximum Lorentz factor achieved by the proto-magnetar outflow, calculated

assuming a surface field B0 = 1016 G. (e)Threshold for r-process (eq. [2.24]), in a mass-weighted average

over the open zone from θ = 0− θmax.

51

1018

1020

1022

1024

1026

1028

1030

1032

106

107

Energ

y d

ensity, erg

s c

m-3

Radius, cm

Polar outflow of Lν=3 10

51 ergs s

-1, P=2 ms

B2/8 π

P

Rotational KE

Radial KE

Figure 2-5: Energy densities as a function of radius along the polar directionθ = 0 for our fiducial solution with Lν = 3 · 1051 erg s−1 and P = 2 ms,assuming a surface field B0 = 1015 G.

and hence earliest times, our solutions provide an accurate description of the

outflow at radii of interest only for extremely strong fields B0 ∼> 3 × 1015 G

(for P ∼> 2 ms). However, at later times when Lν ∼ 1051 erg s−1, even weaker

fields B0 ∼ 3× 1014 G are sufficient to satisfy the force-free condition.

Even if the force-free approximation is valid in the outflow for magnetar-

strength fields, the same may not be true in the closed zone due to the

large hydrostatic pressure of the co-rotating atmosphere. If the closed zone

pressure Pclosed exceeds that which can be confined by the magnetosphere,

the field will ‘tear’ open, enlarging the extent of the open zone (RY ∼< RL)

compared to the standard case RY = RL [134, 185].

Figure 2-7 compares the radial profile of Pclosed and Popen for Lν = 3×1051

erg s−1 and P = 2 ms along the last closed field line θ = θmax, to the pressure

52

1014

1015

1016

1017

1 2 3 4 5 6 7 8 9 10

Bm

in, G

Lν, 10

51 ergs s

-1

P=1 msP=2 msP=3 msP=4 msP=5 ms

Figure 2-6: Minimum surface magnetic field strength Bmin required for va-lidity of the force-free approximation at all angles and radii interior to thelocation of α-particle formation (T = 0.5 MeV), calculated for different val-ues of the neutrino luminosity Lν and spin period P . The force-free conditionis defined as Pgas/Pmag ≤ 0.1, where Pmag = B2/8π is the magnetic pressureand Pgas is the ‘matter pressure’, the latter of which includes gas pressure P ,thermal energy density e, and kinetic energy density (rest energy subtracted).

53

of the poloidal magnetic field PB = B2p/8π, calculated for B0 = 1015 G. The

closed zone pressure is determined by solving the equations of hydrostatic,

thermal and weak interactions equilibrium (v = 0; q = 0; dYe/dt = λ = 0).

The analytic model of [162] predicts that the closed zone RY extends to the

location where PB = Pclosed, which according to Fig. 2-7 occurs at a radius

r ∼ 30 km for B0 = 1015 G which is a factor of ∼ 3 times smaller than in

standard case (RY = RL = 100 km). Although most models in this chapter

assume RY = RL, the true extent of the open zone θmax could be larger than

our assumed value by a factor (RY/RL)1/2 ∼< 2.

2.3.2 Spherical versus Magnetized Rotating Trajecto-

ries

Based on S3/(τY 3e ) criterion of [63], the most promising period for nucleosyn-

thesis is P = 1 ms. However, the rotational energy of a P = 1 ms magnetar

exceeds the kinetic energy of a normal SN of order 1051 ergs - hence their

birth should be observable as a hyper-energetic supernova or a hypernova.

However, these events are rare [186, 187] and hence P = 1 ms magnetars

are rare. We then concentrate at P = 2 ms magnetars in this section. This

section provides a comparison between the properties of our rotating PNS

wind solutions (for fiducial parameters Lν = 1052 erg s−1, P = 2 ms) and

the spherical non-rotating solutions corresponding to the same neutrino lu-

minosity. In particular, Figures 2-8 and 2-9 show the radial profiles of wind

properties along two field lines, corresponding to a polar outflow (θ = 0) and

one along the last open field line (θ = θmax ≈ 21). This section also com-

pares the properties of our rotating solutions, such as the entropy/expansion

timescale (Fig. 2-11) and mass loss rate (Fig. 2-12), across a range of rotation

periods and neutrino luminosities.

54

1020

1022

1024

1026

1028

1030

1032

106

107

Energy density, ergs cm-3

Radius r, cm

B2/8π

Pclosed

Popen

RL

Figure 2-7: Comparison between the hydrostatic thermal pressure of theclosed zone Pclosed (red solid) and the poloidal magnetic field pressure B2

p/8π(green dashed) along the last open field line (θ = θmax), calculated for a PNSwith neutrino luminosity Lν = 3 × 1051 erg s−1, spin period P = 2 ms, andsurface magnetic field strength B0 = 1015 G. Also shown for comparison isthe thermal pressure in the outflow itself Popen (blue dotted). The closedzone pressure exceeds that of the confining magnetosphere at radii ∼> 30 kmsmaller than the light cylinder radius (RL ≈ 100 km), which according to thecriterion of [162] indicates that the true open portion of the magnetospheremay be larger than the force free assumption, i.e. the Y-point radius shouldobey RY < RL (see Fig. 2-1).

55

Centripetal lowering of Ye below Ye,eq

The electron fraction Ye quickly rises and saturates just a few kilometers

above the PNS surface (Fig. 2-9, top panel). The values of (∆Ye)cent are

reported in Table 2.1. For most of trajectories, |(∆Ye)cent| < 0.01. This is

because in each case the outflow composition enters equilibrium with neu-

trino absorptions, which drives Ye to Ye,eq (eq. [2.10]). Electron and positron

captures play no significant role in the freeze-out process because their rate

∝ T 6 decreases rapidly with temperature above the PNS surface, making

them quickly negligible compared to neutrino absorptions. Low value of

(∆Ye)cent justifies neglecting it for P > 3 ms (see Sect.2.2.7).

It is inevitable that the outflow composition enter equilibrium with the

neutrinos if the latter dominate at freeze-out. Each nucleon must absorb

an energy ∼> GMmp/Rns ≈ 160 MeV (for M = 1.4M and Rns = 12 km)

to escape the gravitational potential well of the PNS. Because this energy

greatly exceeds that of an average neutrino 〈εν〉 ' 4kTν ≈ 15 MeV, each

outflowing nucleon must absorb ∼ 10 neutrinos. This process renders its

initial identity as a proton or neutron irrelevant (e.g. [136]).

Centrifugal acceleration alters the above argument by providing an ad-

ditional source of wind energy independent of neutrino heating. In practice,

however, the asymptotic value of Ye is appreciably reduced below Y eqe only

for extremely rapid rotation near the centrifugal break-up limit [136]. In our

most rapidly rotating model with P = 1 ms, (∆Ye)cent reaches values as low

as −0.14 for the largest outflow angles θ = θmax (Table 2.1). However, for

slower rotation P ∼> 3 ms, Ye varies by less than a percent from its spherical,

non-rotating value across the entire open zone.

56

Mass loss rate

The spherically-equivalent mass loss rates of our fiducial polar (θ = 0) and

inclined (θ = θmax) solutions are M = 1 × 10−3M s−1 and 2 × 10−3M

s−1, respectively, as compared to M = 7 × 10−4M s−1 for the spherical,

non-rotating case. These nearly equal values again result because the mass

loss per unit surface area is set largely by the neutrino energy absorbed

near the surface [37]. The structure of the outflow, being subsonic in the

heating region, is determined mainly by hydrostatic equilibrium and hence

is relatively insensitive to the precise outflow geometry.

This expectation is borne out in the radial profiles of density (Fig.2-

8; middle panel) and heating (Fig.2-9; bottom panel) just above the PNS

surface, which are similar between the spherical and rotating magnetized

outflows. The slightly larger value of M in the θ = θmax case is the result

of centrifugal acceleration, which increases the density scale-height in the

net heating region, to which M is exponentially sensitive. This increase

is not as great as for a equatorial monopolar outflow with the same PNS

rotational period [57], because of the lower surface rotational velocity vφ,0 ∼<RnsΩ sin θmax as compared to the equatorial value vφ,0 = RnsΩ.

Figure 2-12 compares the values of M for different rotation rates and

neutrino luminosities. As expected, the mass loss rate depends sensitively

on the neutrino luminosity, approximately as M ∝ L5/3ν 〈εν〉10/3 ∝ L

5/2ν , with

a weaker dependence on other parameters [37]. The largest enhancement in

M due to rotation is for the shortest rotation period (P = 1 ms) and largest

angle (θ = θmax), for which M is a factor ≈ 10 times larger than the polar

or spherical cases. Although the mass loss rate per surface area is enhanced

by magnetic fields and rotation, this is offset by the smaller fraction of the

57

PNS surface fopen which is open to outflows in the magnetized case:

fopen ≈2πθ2

max

4π≈ 1

2

Rns

RY

≈ 0.13RL

RY

(P

ms

)−1

, (2.21)

where in the final two equalities we have approximated θmax = sin−1√Rns/RY ≈√

Rns/RY and have assumed Rns = 12 km.

A numerical estimate of the total mass loss rate Mtot integrated across the

open zone (eq. [2.20]) for each neutrino luminosity and spin period are given

in Table 2.2. The value of Mtot varies from a factor of ∼ 10 times larger to

∼ 30 times smaller than the spherical case as the spin period increases from

P = 1 ms to 10 ms. This transition occurs as the centrifugal enhancement

at small P is offset by the shrinking open fraction fopen ∝ P−1 for large P .

Final entropy

The asymptotic values of the entropy for our fiducial rotating solutions are

S = 87 kb nucleon−1 and 51 kb nucleon−1 for the θ = 0 and θ = θmax,

respectively, as compared to S = 87 kb nucleon−1 for the equivalent spherical

case. These similarities and differences can be understood from the fact that

the entropy increase of the outflowing matter is approximately given by

∆S ∝∫dQ

T≈ ∆Q

T, (2.22)

where dQ is the net heating per nucleon and T is the average temperature in

the heating region. To first order, the entropy of the polar rotating outflow

is similar to that in the spherical case for the same reason that M is similar:

heating is concentrated in the hydrostatic region and the value of ∆Q is

essentially fixed by that required to escape the gravitational well of the PNS.

In the case of inclined field lines θ = θmax the entropy is reduced because

less neutrino heating ∆Q is necessary to escape the potential well given the

58

additional source of centrifugal acceleration. The value of S in this case

approaches its value for an equatorial outflow in the monopolar case with

the same neutrino luminosity and spin period [57].

Figure 2-11 shows the asymptotic entropy across a range of PNS proper-

ties. The strongest trend is that entropy decreases with increasing neutrino

luminosity, approximately as S ∝ L−1/6ν 〈εν〉−1/3 ∝ L

−1/4ν [37]. This depen-

dence can also be understood from equation (2.22): the mean temperature

in the region of net heating is smaller for lower neutrino luminosities because

T is to first order set by the balance between neutrino heating ∝ Lν and

neutrino cooling ∝ T 6.

Expansion timescale

Matter flows across the range of radii at which alpha particles form on the

expansion timescale [63]

texp = 1.281

v

d lnT

dl

∣∣∣∣T=0.5MeV

(2.23)

The expansion timescale for our fiducial rotating solutions are texp = 11 ms

and 0.6 ms for the θ = 0 and θ = θmax, respectively, as compared to texp = 13

ms for the equivalent spherical case (Fig. 2-9; bottom panel). Faster expansion

in the magnetized case results from more rapid acceleration (Fig. 2-8; top

panel) due to the faster divergence of the dipolar areal function (Fig. 2-2)

and the related fact that for similar entropy a faster decrease in density

results in T = 0.5 MeV being achieved closer to the PNS surface, where

the radius (and hence expansion time) is likewise smaller (Fig. 2-8; bottom

panel).

The expansion time is further reduced by a large factor for θ > 0 due to

centrifugal acceleration, which increases the outflow velocity as compared to

59

the purely thermally-driven case. The value of texp in Table 2.1 shows that

at large angles θ ≈ θmax centripetal acceleration is more important than the

purely geometric effect of areal divergence for spin periods P ≤ 4 ms. For our

fiducial solution, for instance, areal divergence reduces texp from 13 ms for

spherical solution to 11 ms for polar outflows (θ = 0). In contrast, centrifugal

acceleration decreases the dynamical timescale from 11 ms for polar outflow

to 0.6 ms for outflows along the last open field line.

Figure 2-11 compares texp for solutions with different rotation rates and

neutrino luminosities. The strongest trends are that (1) texp decreases with

increasing neutrino luminosity (as also occurs in spherical, non-rotating winds;

[51]); (2) at fixed neutrino luminosity, texp is smaller for shorter spin periods

P and larger outflow angles θ, again due to centrifugal acceleration.

2.3.3 Spherical vs Magnetized Rotating Nucleosynthe-

sis yields

Variation in wind properties across the magnetosphere

P = 1 ms is found (see Sect.2.3.2) to be the most promising period for

heavy r-process nucleosynthesis based on s3/τexp criterion of [63], however,

even in P = 1 ms the third abundance peak (A ≈ 200) is not reached.

Nucleosynthesis calculations confirm this conclusion. We focus our analysis

in this section on models with P = 1 and 3 ms. The latter case plays

important role because the rotational energy of the magnetar with P = 3 ms,

Erot = IΩ2/2 ≈ 1051 ergs, is comparable to the kinetic energy of a normal

SN. Hence magnetars born with P ∼> 3 ms could in principle be ‘hidden’

among the normal population of normal SNe without violating constraints

on the observed kinetic energies of their ejecta from SN spectra and the total

energies of their remnants (as already discussed, hypernovae are known to

60

105

106

107

108

109

1010

1011

106

107

Velocity v, cm s-1

Radius, cm

L=1052

ergs s-1P=2 ms

θ= θmax

RL

T=0.5 MeVSonic point

Maximum heating

θ=0

Spherical case

103

104

105

106

107

108

109

1010

1011

1012

106

107

De

ns

ity

ρ,

g c

m-3

Radius, cm

Spherical case

θ=0

θ=θmax

10-2

10-1

1

10

106

107

Tem

pera

ture

T, M

eV

Radius, cm

-

Spherical case

θ=0

θ=θmax

Figure 2-8: Radial profiles of velocity v (top left), density ρ (top right), andtemperature T (bottom) for our fiducial solution with Lν = 1052 erg s−1 andP = 2 ms, shown for different field lines corresponding to a polar outflow(θ = 0), maximally inclined outflow (θ = θmax). Shown for comparison isa spherical, non-rotating outflow for the same neutrino luminosity. Criticalradii are marked with symbols, including the light cylinder RL (square),radius of α−particle formation R0.5Mev (circle), sonic point (diamond), andpoint of maximum net heating (asterisk).

61

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

106

107

Ye

Radius, cm

L=1052

ergs s-1

P=2 ms

Magnetar, θ=1.0 θmaxRL

T=0.5 MeVSonic point

Maximum heatingMagnetar, θ=0.0 θmax

Spherical case10

18

1019

1020

1021

1022

1023

106

107

Net heating q

, erg

s g

-1s

-1

Radius, cm

108

109

1010

106

107

En

tro

py s

, k

Bn

-1

Radius, cm

Figure 2-9: Radial profiles of the electron fraction Ye (top left), net heatingrate q (top right), and entropy s (bottom) for our fiducial solution with Lν =1052 erg s−1 and P = 2 ms (same as Fig. 2-8). Radii marked with symbolsare the same as in Figure 2-8.

62

10-4

10-3

10-2

10-1

100

101

106

107

t exp, s

Radius, cm

L=1052ergs s

-1P=2 ms

RLT=0.5 MeVSonic point

Maximum heating

Spherical case

θ=θmax

θ=0

Figure 2-10: Radial profiles of the local expansion timescale texp (eq. [2.23])for our fiducial solution with Lν = 1052 erg s−1 and P = 2 ms, shown fordifferent field lines corresponding to a polar outflow (θ = 0), maximally in-clined outflow (θ = θmax). Shown for comparison is a spherical, non-rotatingoutflow for the same neutrino luminosity.

63

10-5

10-4

10-3

10-2

10-1

100

30 40 50 60 70 80 90 100 110

t exp, s

S, kB/nucleon

P=2 ms

3rd

peak threshold for Ye=0.45

θ increase

GR, θ=140

1e+51

3e+516e+51

1e+52

Figure 2-11: Expansion timescale texp (eq. [2.23]) versus asymptotic entropyS, calculated for a PNS with spin period P = 2 ms and shown for differ-ent neutrino luminosities: Lν = 1052 erg s−1 (red square), 6 × 1051 erg s−1

(blue asterisk), 3× 1051 erg s−1 (green cross), and 1051 erg s−1 (purple plus).Different field line angles θ are shown for each luminosity, with θ increasingto the lower left hand corner of the plot. Circles of the same color showthe corresponding equivalent spherical, non-rotating solutions with the sameneutrino luminosity. The solid black line shows the threshold for a successfulthird peak r-process η = ηthr (for an assumed electron fraction Ye = 0.45).

64

10-6

10-5

10-4

10-3

10-2

0 0.2 0.4 0.6 0.8 1

M, M

⊙ s

-1

θ/θmax

P=2 ms

1051

3 1051

6 1051

1052

Figure 2-12: Spherically equivalent mass loss M = 4πr20ρ0v0 (eq. [2.19]) as

a function of the field polar angle, calculated for a PNS with spin periodP = 2 ms for different neutrino luminosities. Shown for comparison withhorizontal lines are the mass loss rates of the spherical solutions at the sameluminosities. Luminosities in ergs s−1 are written near each line.

65

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.2 0.4 0.6 0.8 1

η/η

thr

θ/θmax

P=2 ms

Figure 2-13: Critical ratio η ≡ S3/Y 3e texp of wind properties for a successful

r-process in units of the required threshold value ηthr (eq. [2.24]) as a functionof the field polar angle, calculated for a PNS with spin period P = 2 ms fordifferent neutrino luminosities: (from top to bottom) Lν = 1052 erg s−1 (redsquare), 6 × 1051 erg s−1 (blue asterisk), 3 × 1051 erg s−1 (green cross), and1051 erg s−1 (purple plus). Note that GR will act to enhance η/ηthr by afactor ∼ 3− 4 over the values shown. Shown for comparison with horizontallines are the values of η of the spherical solutions of the same luminosities.

66

be rare; e.g. [186, 187]).

Figure 2-14 shows the mass fraction as a function of nuclear charge Z for

different outflow latitudes θ, calculated for P = 1 ms (left panel) and P = 3

ms (right panel). For all values of θ, the abundance distribution extends

to higher masses Z than the otherwise equivalent spherical wind with the

same Lν and Ye, which is shown for comparison with a purple line. Part of

this effect is due to the shorter expansion time through the seed formation

region caused by the faster diverging outflow areal function ∝ 1/r3 in the

dipole magnetic field, as compared to the spherical wind case (areal function

∝ 1/r2), as well as centrifugal force from rotation.

Figure 2-14 also shows that the abundance distribution proceeds to heav-

ier elements with increasing θ, due to the additional acceleration caused by

magneto-centrifugal acceleration along field lines inclined with respect to the

rotation axis. The heaviest nuclei are synthesized in those outflows which

graze the closed zone and pass near the equatorial plane outside the light

cylinder.

Shown for comparison with a dashed blue line are the abundances in-

tegrated over the open zone of the entire magnetosphere, from which it is

apparent that the flow properties near the last open field line (θmax) also

dominate the total abundance of the wind. This is expected because the

mass loss rate per unit surface area is enhanced by magneto-centrifugal ac-

celeration for larger θ [134, 57] and, to a lesser extent, because outflows with

larger θ contribute a greater fraction of the total open solid angle of the

magnetosphere.

Time-integrated models

Figure 2-15 shows the total yield of proto-magnetar winds, integrated across

the entire Kelvin-Helmholtz cooling evolution using the Ye(t) evolution from

67

10-6

10-5

10-4

10-3

10-2

10-1

100

15 20 25 30 35 40 45 50 55

X

Z

P=1 ms, Lν=10

52 ergs s

-1, Ye, eq=0.45

Sphericalθ=0.9

0, Ye=0.45

θ=20.70, Ye=0.40

θ=30.60, Ye=0.33

Integrated

10-6

10-5

10-4

10-3

10-2

10-1

15 20 25 30 35 40 45 50 55

X

Z

P=3 ms, Lν=10

52 ergs s

-1, Ye=0.45

Sphericalθ=0.9

0

θ=5.40

θ=11.70

θ=18.00

Integrated

Figure 2-14: Mass fraction of nuclei X(Z) synthesized in the proto-magnetarwind of neutrino luminosity Lν = 1052 ergs s−1, and neutrino equilibriumelectron fraction Ye = 0.45. Results are shown for two rotation periods:P = 1 ms (left panel) and P = 3 ms (right panel); in the former case, thevalue of Ye is lower than its equilibrium value set by neutrino absorptionsdue to centrifugal acceleration effects, as marked in the key. Different linescorrespond to the abundances synthesized in the wind along field lines withdifferent polar angles θ at the NS surface, ranging from near the pole (θ ' 0)to the last open field line θmax (depending on the rotation period). Shownfor comparison with a blue dashed line are the mass-weighted abundancesintegrated over the solid angle of the open magnetosphere. A purple lineshows the abundances from an otherwise identical calculation of a sphericalnon-rotating non-magnetized wind.

Table 2.3: Summary of Time-integrated ModelsP, ms A(a) Z(a) Xα X†n X†p Mej, M

-∗ 87.34 37.71 0.561 9.24 · 10−7 3.65 · 10−7 9.98 · 10−4

1 98.94 42.61 0.429 4.83 · 10−4 5.58 · 10−5 3.45 · 10−3

2 98.73 42.53 0.685 3.42 · 10−4 4.07 · 10−5 1.79 · 10−4

3 97.81 42.13 0.719 2.18 · 10−4 2.55 · 10−5 1.06 · 10−4

4 97.36 41.94 0.714 1.90 · 10−4 2.22 · 10−5 7.80 · 10−5

5 95.07 40.96 0.688 1.62 · 10−4 1.90 · 10−5 6.64 · 10−5

10 94.07 40.53 0.675 9.34 · 10−5 1.08 · 10−5 2.52 · 10−5

(a) Mean mass number for all elements except H and He; ∗Spherical solutions; † Mass fraction of free

neutrons and free protons at t = 100 s, i.e. prior to the decay of free neutrons.

68

[22] (Fig. 2-4) and our calculated (∆Ye)cent for P = 1, 2 ms. We show both

the total mass (left panel) as well as in ratio of abundances to those calcu-

lated in the otherwise equivalent case of a spherical wind (right panel). One

clear trend is that the abundance distribution extends to heavier elements in

the rotating case, with a larger number of heavy elements synthesized with

decreasing rotation period as compared to the otherwise equivalent spherical

case.

However, strong magnetic fields also reduce the mass of the ejecta in light

elements. In part, this reduction is due to the fact that only a small fraction

of the PNS surface is open to outflows, such that the total ejecta mass from

each event is typically smaller than the spherical case by purely geometric

factor (2.21), which we show as a dashed line in Fig. 2-15. Another factor

decreasing the mass ejected in light elements is the nucleosynthesis pattern:

when the peak shifts to heavier elements, the abundances of light elements are

reduced. The only period that exhibits a significant enhancement compared

to geometric factor across most elements is P = 1 ms because of the large

centrifugal enhancement in the mass loss rate per unit surface area, which

overcomes the (comparatively modest) reduction due to fopen ≈ 0.1 in this

fastest spinning case.

Figure 2-16 shows the abundance-weighted mass A of synthesized nuclei

as a function of the magnetar rotation period (red crosses, bottom axis),

from which it is clear that A decreases monotonically with increasing spin

period P . For comparison a red line shows the result for a spherical wind

using the same neutrino-cooling evolution from [22] as in the magnetized

case. The mean value of A for P = 1 − 10 ms is 97, but the A in spherical

case is Asph = 87: hence, the rotation with P = 1 − 10 ms lowers A by 10

units. On the top axis we also show the value of A calculated in the spherical

non-rotating case assuming a temporally fixed value of Ye.

69

10-9

10-8

10-7

10-6

10-5

10-4

15 20 25 30 35 40 45 50 55

Eje

cte

d m

ass in

in

div

idu

al e

lem

en

t, M

⊙

Z

SphericalP=1 msP=2 msP=3 msP=4 msP=5 ms

P=10 ms

10-4

10-3

10-2

10-1

100

101

102

103

104

15 20 25 30 35 40 45 50 55

Eje

cte

d m

ass/E

jecte

d m

ass in

sp

he

rica

l ca

se

Z

P=1 msP=2 msP=3 msP=4 msP=5 ms

P=10 ms

Figure 2-15: Left: Total wind ejecta mass in individual elements, time-integrated across the Kelvin-Helmholtz cooling epoch using the Ye(t) evo-lution from [22] (Fig. 2-4) and our calculated (∆Ye)cent for P = 1, 2 ms.Right: Ratio of the wind ejecta mass of each element in proto-magnetarmodels relative to those in the otherwise equivalent spherical non-rotatingwind models. The dashed lines of same color shows the purely geometricfactor fopen (eq. 2.21) arising from fraction of the PNS magnetosphere opento outflows.

The fact that the ejecta mass in individual elements exceeds those pro-

duced in the spherical case by a factor up to 102 (or 103 in case of P = 1

ms) has the striking implication that millisecond magnetars possess unique

nucleosynthetic signatures, which would be measurable even if only one in

100(1000) neutron stars were born strongly magnetized with P ∼< 4 ms

(P ≈ 1 ms).

Table 2.3 summarizes important quantities from our time-integrated mod-

els. We see that centrifugal effects enhance the values of A and Z moving

to shorter periods. Enhanced centrifugal mass loss overpowers the geometric

factor for P = 1 ms, such that the total ejecta mass of Mej ≈ 3 × 10−3M

for P = 1 ms is about three times higher than in the spherical case.

70

70

75

80

85

90

95

100

2 4 6 8 10

0.445 0.45 0.455 0.46 0.465 0.47 0.475 0.48 0.485 0.49 0.495

Spherical case

A_

P, ms

Ye

Figure 2-16: Abundance-weighted mass A of synthesized nuclei, excludingH and He. Bottom Axis: Red crosses show the results for time-integratedmodels with Ye(t) evolution from [22] (Fig.2-4) for different rotation periods,while the solid line shows the result for the non-rotating spherical wind casefor the same Ye evolution. Top Axis: Blue asterisks show time-integratedmodels of a spherical wind which instead assume a temporally constant valueof Ye as marked on the top axis.

71

2.4 Discussion

2.4.1 r-Process Nucleosynthesis

Free nucleons recombine into α-particles once the temperature decreases to

T ∼ 5 × 109 K, as our solutions show occurs several tens of kilometers

above the PNS surface. Heavier elements start to form once the temperature

decreases further, via the reaction 4He(αn,γ)9Be(α,n)12C. After 12C forms,

additional α captures produce heavy ‘seed’ nuclei with characteristic mass

A ' 90 − 120 and charge Z [132]. The r-process occurs as remaining free

neutrons are captured onto these seed nuclei. The maximum mass Amax

to which the r-process proceeds depends on the ratio of free neutrons to

seed nuclei following completion of the α-process. Because 12C production

is the rate-limiting step to forming seeds, the neutron to seed ratio in turn

depends on the electron fraction Ye, entropy S, and expansion timescale texp

(eq. [2.23]) of the outflow [188].

The condition for r-process to reach the third mass peak (Amax ∼> 190)

can be expressed as [63]:

η ≡ S3

Y 3e texp

∼> ηthr ≈

8 · 109 (kBn−1)

3s−1, Ye ≥ 0.38

109 (1−2Ye)

Y 3e ( 0.33

Ye−0.17)2− 1

(2Ye)2

, (kBn−1)3s−1, Ye < 0.38

(2.24)

where the ratio η/ηthr thus serves as a ‘figure of merit’ for the potential success

of a given r-process site. Previous studies of the r-process in spherically

symmetric, non-rotating PNS winds typically find that η ηthr (e.g. [51]),

thus disfavoring such events as sources of heavy r-process nuclei.

Figure 2-13 shows η/ηthr calculated for our P = 2 ms solutions corre-

sponding to different field lines and neutrino luminosities. Figure 2-17 shows

the (mass flux weighted) average value of η/ηthr over the entire outflow for

72

10-3

10-2

10-1

1 2 3 4 5 6 7 8 9 10

S3 t

exp-1

Ye-3

, th

reshold

valu

e

Lν, 10

51 ergs s

-1

Increasing time

P=1 msP=2 msP=3 msP=4 msP=5 ms

Spherical case

Figure 2-17: Critical ratio η ≡ S3/Y 3e texp of wind properties required to

achieve third-peak r-process in units of the threshold value ηthr (eq. [2.24]),mass-averaged over the open field lines as a function of the PNS neutrinoluminosity for different rotation periods P = 1 ms (plus), 2 ms (cross), 3 ms(asterisk), 4 ms (open square), and 5 ms (filled square). Shown for comparisonwith dot-centered circles are the equivalent spherical wind solution for thesame neutrino luminosity. Note that GR will act to enhance η by a factor∼ 3− 4 over the values shown.

different periods and neutrino luminosities. Shown for comparison in each

case are the values of η/ηthr for our spherical solutions of the same luminosi-

ties.

Our most promising solutions for a PNS of mass 1.4 M are for Lν =

1052 ergs s−1, θ ≈ θmax and near break-up spin period P = 1 ms, for which

η is ≈ 10 per cent of the threshold value ηthr. Although these solutions are

still well below the threshold, they are a factor ∼> 4 times higher than the

spherical, non-rotating wind solution of the same neutrino luminosity. This

73

enhancement results from the combination of a much lower values of Ye and

texp due to centrifugal flinging and the dipolar geometry (§2.3.2). In most

promising cases, entropy was considerably smaller than the spherical case,

similar to that in the case of equatorial outflows of [57].

In light of our conclusion that magnetic fields and rotation enhance the

prospects for the r-process in PNS winds, we note several caveats. First, our

calculations do not include general relativity (GR). Past studies of neutrino-

heated winds have found that GR effects increase the final entropy4 and

decrease the dynamical timescale. If we adopt the ∼ 50% increase in entropy

found by [51] (see also [163, 50]), then the resulting increase in η by a factor

of 3 is nearly sufficient to push our most promising models for M = 1.4M

to within a factor of ∼ 2− 3 of the threshold. In order to explore the effects

of GR further, one of our most promising solutions Lν = 1052 ergs s−1, P =

2 ms, θ = 0.7θmax was recalculated with a [189] gravitational potential. The

value of η of the corresponding solution was found to be ∼ 4 times higher

than in the Newtonian case (see Table 2.1).

Another caveat is that we assume a small PNS radius of 12 km, even at

the highest neutrino luminosities corresponding to the earliest times (t ∼ 1

s) in the PNS cooling evolution. At such early stages after the explosion, the

PNS is inflated by thermal pressure support relative to its final size [184].

A larger PNS radius results in a lower value of η than would be calculated

assuming a fully contracted PNS (the rotation rate prior to full contraction

is also likely to be lower due to angular momentum conservation) due to

the shallower gravitational potential well (eq. [2.22]). We note, however,

that [184] do not include the effects of convective cooling, which may act to

enhance the rate of PNS cooling and contraction [190, 191].

4Higher entropy results primarily from the deeper effective gravitational potential wellof the PNS in GR, which requires a larger amount of neutrino heating ∆Q to escape, thusincreasing S according to equation (2.22).

74

Proto-magnetars as Galactic r-process site

In this subsubsection, we shall proceed under the assumption that the r-

process is indeed successful in proto-magnetar winds with moderately rapid

rotation P ∼ few ms, due to the inclusion of GR and other potential effects

not taken into account, and consider the resulting implications. Our most

promising models have luminosities L = 1052 ergs s−1 and P ∼< 5 ms.

Producing all of the heavy r-process elements in the Galaxy over its

lifetime requires a mean production rate Mr ∼ 5× 10−7M yr−1 [192], thus

requiring an ejecta mass per event of

Mej =Mr

frR≈ 1.5× 10−3M

(fr

0.1

)−1( R0.1RCC

)−1

(2.25)

for an event rate R, which we have normalized to the estimated rate of core

collapse supernovae of ∼ 3 per century [193], where fr ≈ 1− 2Ye ∼ 0.1− 0.2

is the fraction of the ejecta mass placed in r-process nuclei (as opposed to

4He).

Our models with P = 2 − 5 ms have total ejecta masses Mej ∼ 10−4M

(Table 2.2), which could be enhanced by a factor ∼> fopen ∝ RL/RY ∼ 3

(eq. [2.21]) due to the finite pressure of the closed zone (eq. [2-7]) or due to

coupling between the proto-magnetar outflow and the surrounding envelope

of the star [135].

Combining these estimates for the ejecta with equation (2.25), we con-

clude that ∼> 30% of the Galactic r-process could in principle originate from

proto-magnetars with birth periods P < 5 ms at a rate that is ∼> 10% of the

core collapse rate [62]. Their contribution could be greater if the r-process

receives contributions from extremely rapidly spinning neutron stars with

P = 1 ms (which contribute higher Mej but are probably much rarer; see

also [145]).

75

10-7

10-6

10-5

10-4

10-3

10-2

1 2 3 4 5 6 7 8 9 10

Mto

t, M

⊙ s

-1

Lν, 10

51 ergs s

-1

Increasing time

P=1 msP=2 msP=3 msP=4 msP=5 ms

Spherical case

Figure 2-18: Total mass loss rate Mtot integrated over open field lines(eq. [2.20]) as a function of the PNS neutrino luminosity for different rotationperiods P = 1 ms (plus), 2 ms (cross), 3 ms (asterisk), 4 ms (open square),and 5 ms (filled square). Shown for comparison with dot-centered circles arethe equivalent spherical wind solution for the same neutrino luminosity.

76

A population of moderately rapidly-rotating magnetar r-process sources

is compatible with a variety of other observations. A rotation period P ∼> 3

ms corresponds to a total rotational energy Erot ∼< 2×1051 ergs, which is too

low to be excluded as accompanying the birth of Galactic magnetars based on

the energetics of their SN remnants [151]. Magnetar birth, though likely not

as common as normal core collapse supernovae (however, see [194]), is more

common than binary neutron star mergers, the other commonly discussed

r-process site. An r-process source from magnetar birth would thus be in

better agreement with studies of Galactic chemical evolution that claim to

disfavor infrequent, high-Mej events [146, 195].

Magnetar birth in core collapse supernovae could also provide prompt

enrichment, as would accompany even the earliest stages of Galactic chemical

evolution, another observation in tension with the NS merger scenario. The

discovery of magnetars in massive clusters suggest that they result from

the core collapse of very massive stars ∼> 40M [196, 197, 198], which are

expected to produce massive neutron stars more favorable to the r-process

[58]. On the other hand, massive progenitors may be disfavored by the

lack of correlation between Fe production and third peak r-process elements

based on the observed abundances in metal-poor stars (e.g. [199]). Finally,

the birth of a millisecond magnetar is one of the leading theoretical models

for explaining recently discovered class of hydrogen-poor ‘super-luminous’

supernovae [200, 201, 202], events which are indeed observed to occur in very

metal poor galaxies (e.g. [203]) with conditions similar to those encountered

in the early history of the Galaxy.

Rate constraints on the birth of magnetars

If millisecond proto-magnetars are found to produce large quantities of rare

isotopes, one could in principle place constraints on their birth periods based

77

on their nucleosynthetic yields compared to those in our Galaxy as inferred

from the solar abundances. An upper limit on birth rate of magnetars of a

given rotation period is given by

R ≤ minXi,Mgas

Mej,it=

Xi,Mgas

Xi,windMejtgal

, (2.26)

where Xi,wind is mass fraction of an element (or an isotope) in the PNS wind,

Mej is the total wind ejecta mass, Xi, is the mass fraction of the same

element (or isotope) on the Sun, Mgas ≈ 3.3 × 1010M is the total mass of

gas in the Galaxy when the Sun formed, and tgal ≈ 1010 years is the age of

the Galaxy when the Sun formed. In other words, the total ejecta mass of

a given element or isotope under consideration cannot exceed the total mass

of that element contained in the gas from which the Sun formed (assuming

the Sun formed from “ordinary” gas with abundances representative of the

Galactic mean).

Figure 2-19 shows the result of such a calculation of the maximum allowed

birth rate in events per century, which is approximately also the fraction of

neutron star births. A purple line shows the rate derived from our time-

integrated calculation of proto-magnetar winds, as a function of the magnetar

rotation period. Shown for comparison with a blue line is the rate limit for

the standard non-rotating spherical wind case.

It may be surprising that, for standard spherical winds, the allowed event

rate is very low, ∼< 0.05 per century. This is due to the well-known fact that

standard spherical winds overproduce the charged particle process nuclei with

Z = 38− 40 (e.g. [140]) for the neutron-rich wind conditions Ye ∼< 0.5 found

using contemporary PNS cooling calculations [22, 54, 55].

It may also be surprising that the rate constraints are weaker on proto-

magnetar winds than on spherical winds, despite the fact that the nucleosyn-

78

thesis of proto-magnetars extends to higher Z elements, which are rarer in

the solar system. This is again explained by the fact that only a small frac-

tion of the PNS surface is open to outflows, such that the total ejecta mass

from each event is typically smaller than the spherical case by the geometric

factor fopen (eq. 2.21). The green line in Fig. 2-19 shows the rate constraints

one would derive if the composition of a spherical wind was attenuated by the

purely geometric factor fopen. The resulting rate constraint is weaker than

the proto-magnetar case including the full effects of the magnetized wind dy-

namics on the composition itself (purple line); this shows that the rare higher

Z elements which are synthesized proto-magnetar winds do tighten the rate

constraint significantly, but not enough to overcome the purely geometric

suppression factor fopen.

The net result of all of this is that the effects of magnetic fields and

rotation are sufficiently modest that - given also current uncertainties in

the electron fraction of the wind - one cannot at present place meaningful

constraints even on the birth rate of non-rotating PNS, much less on their

birth periods and magnetic field strengths. Still, our results show that at least

under our assumption for the electron fraction of the wind, the birth rate of

magnetars with millisecond periods cannot exceed ∼ 1 − 10% of the core

collapse SNe rate, depending on rotation period. Reassuringly, this number

exceeds the total birth rate of Galactic magnetars [62] and is consistent with

the lower millisecond magnetar birth rate inferred if they power hydrogen-

poor superluminous supernovae [204].

It is possible that neutron stars are generically born with magnetar-

strength fields. This could occur, for instance, if strong fields are generated

by a convective dynamo in the PNS, which later decay away in the majority

of cases by the time they are observed as radio pulsars. We note that, in

such a case the reduction in the open fraction of the PNS surface due to con-

79

0

0.2

0.4

0.6

0.8

1

1.2

1.4

2 4 6 8 10

Event ra

te, event per

centu

ry

P, ms

Magnetosphere rates

Spherical composition

Spherical

Figure 2-19: Upper limit on rates of magnetar birth as a function of thebirth rotation period P so as not to overproduce solar system abundancesof r-process nuclei (eq. 2.26). Purple crosses show the results based on ourabundance calculations of proto-magnetar winds (Fig. 2-15). Green crossesshow the rate constraint that would result if we used the composition fromthe spherical model, but down-correcting the ejecta mass Mej to account forthe purely geometric correction fopen (eq. 2.21) resulting from the small solidangle of the open magnetosphere.

finement by a dynamically-important magnetic fields [64] would be generic

to all winds and thus could help alleviate the current overproduction of the

charged particle process nuclei Ye < 0.5 wind models (even if rapid rotation

itself is comparatively rare). Also note that including GR will increase the

depth of potential well for matter to escape and lower neutrino energy be-

cause of gravitational redshift. Both these factors will lower the ejecta mass,

also alleviating constraints on the event rate.

80

As sources of light r-process nuclei in metal-poor stars

Figure 2-20 shows the abundances of our time-integrated models (Fig. 2-15)

compared to the abundances of two Galactic metal-poor stars, HD 122563

[31] and HD 88609 [32] which show a relative dearth of “heavy” r-process

nuclei (Z ∼> 56) compared to the solar system; all abundances have been

normalized to the Ru (Z = 44) abundance in our spherical model. Although

the spherical wind models produce large quantities of charged particle process

nuclei (Z = 38 − 40), they underproduce the solar abundances of the weak

r-process nuclei (Z = 41 − 55) in HD 122563 and HD 88609. By contrast,

our rotating magnetar models produce larger abundances of weak r-process

nuclei with Z ∼> 40, while comparatively underproducing the lighter charged

particle process nuclei.

Our results provide a potential explanation, within the PNS wind paradigm,

for the fact that light r-process nuclei with Z ∼< 56 in metal-poor stars show

greater star-to-star variation than the heavier r-process nuclei [205]. We

propose that such star-to-star variation could be understood if the charged

particle process and weak r-process nuclei are produced by a combination of

normal (slowly-rotating and/or weakly magnetized) NSs and a subclass of

magnetars with a range of birth rotation periods.

Magnetars with very short birth rotation periods of P ∼< 1 ms have also

been discussed as a source of heavy r-process nuclei, mainly by the ejection

of low-Ye matter during the early dynamical phases of the bipolar explosion

[169, 145, 206]. Although such extreme magnetars are probably rare, po-

tentially disfavoring this channel as the dominant Galactic source of heavy

r-process nuclei, magnetars with less extreme birth periods of ∼ 2 − 5 mil-

liseconds could produce light r-process nuclei in a larger fraction of events.

The rapid core rotation rates of the massive progenitor stars giving rise to

millisecond proto-magnetars are also likely to be more common at the low

81

metallicities which characterized early epochs in the chemical evolution of

our Galaxy [207].

This work considers the winds from aligned proto-magnetars, for which

the rotation axis coincides with magnetic axis. Studies of the more general

case of inclined rotators are less developed because the problem is inherently

non-stationary (for review, see e.g. [208]). Numerical studies have shown that

the equatorial region of the magnetar outflow in this case is characterized by

‘stripes’ of alternating magnetic fields, which are separated by current sheets

and are prone to magnetic reconnection [153]. Dissipation of magnetic energy,

for instance as these stripes reconnect outside the light cylinder radius [209],

provides an additional possible source of heating in the wind, which would

act to both increase the entropy of the flow and contribute to its acceleration.

If this heating occurs near the seed formation radius, typically close to the

light cylinder in our models, this would substantially enhance the prospects

for synthesizing third-peak r-process nuclei in misaligned rotators compared

to the aligned case studied here. The magnetic inclination angle therefore

provides an another parameter, in addition to the rotation period, which

could impart diversity to the r-process yields of proto-magnetar winds.

More work is clearly needed to distinguish the magnetar hypothesis from

other proposed sites of the light r-process elements, such as binary neutron

star mergers [210, 43, 211, 212, 44, 213, 214].

2.4.2 Gamma-Ray Burst Outflows

Lorentz Factors (Baryon Loading)

Millisecond proto-magnetars are considered promising central engines for

powering gamma-ray bursts (GRBs) [154, 155, 134, 137]. One striking fea-

ture of GRB outflows are their large Lorentz factors ∼ 100−1000 [157], which

82

10-5

10-4

10-3

10-2

10-1

100

25 30 35 40 45 50

X

Z

SphericalP=1 msP=2 msP=3 msP=4 msP=5 ms

P=10 msHD 88609

HD 122563

Figure 2-20: Abundances in time-integrated models compared to abundancesof Galactic metal-poor stars HD 122563 [31] and HD 88609 [32]. Abun-dances are normalized to the Ru (Z = 44) abundance of our spherical time-integrated model. After Z = 55, our models produce very small abundancesand they are not able to match the abundances observed in HD 122563 andHD 88609.

83

100

101

102

103

1 2 3 4 5 6 7 8 9 10

Γm

ax

Lν, 10

51 ergs s

-1

Increasing time

P=1 msP=2 msP=3 ms

Figure 2-21: Maximum angle-averaged Lorentz Γmax achieved by proto-magnetar outflows as a function of neutrino luminosity calculated accordingto equation (2.28) for an assumed surface field strength B = 1016 G and basedon the baryon loading determined from our wind solutions, shown for spinperiods P = 1 ms (plus), 2 ms (cross), and 3 ms (asterisk). Lower neutrinoluminosities correspond to later times following core bounce, indicating thatproto-magnetars with P ∼< 2 ms achieve Γmax ∼ 100− 1000 on timescales ofseveral seconds.

84

require a specific amount of entrained baryonic mass (‘baryon loading’).

In the proto-magnetar model the GRB jet is powered by the electromag-

netic extraction of the rotational energy of the PNS, as occurs at the rate

E =µ2Ω4

c3≈ 4× 1049

(B0

1015G

)2(P

ms

)−4

erg s−1 (2.27)

where µ = B0R3ns/2 is the magnetic dipole moment. The maximum Lorentz

factor (angle-averaged) that the outflow can achieve at a given time is set by

the ratio of the outflow power to its rest mass flux

σ = Γmax =E + Mtotc

2

Mtotc2. (2.28)

We have assumed RY = RL in equation (2.27), but Γmax will be moderately

higher if the open zone is larger (RY ∼< RL) because to first order M is pro-

portional to the open magnetic flux (neglecting centrifugal enhancements),

while E is proportional to the open flux squared.

Figure 2-21 shows the angle-averaged Γmax calculated using Mtot from our

solutions with P = 1− 3 ms (Fig. 2-18; Table 2.2) as a function of neutrino

luminosity for an assumed magnetic field B0 = 1016 G (this is the field

strength required to produce a jet with power according to eq. [2.27] similar

to those of the most luminous observed GRBs). Our results show that at high

luminosities, corresponding to early times after core bounce, Γmax is relatively

small ∼< 10. However, at lower luminosities corresponding to later times of

several seconds after core bounce, Γmax increases to ∼ 100. Our results thus

confirm previous findings [134, 57, 137] that proto-magnetars with short spin

periods P ∼ 1 − 2 ms are in principle capable of producing outflows with

the correct range of Lorentz factors over the appropriate timescale to power

long GRBs. By contrast, slower rotating magnetars with P ∼ 3 ms produce

outflows with lower luminosities and Γmax ∼ 10, which could be responsible

85

100

101

102

103

0 10 20 30 40 50 60 70 80 90

Γm

ax

Asymptotic θ

P=2 ms

Figure 2-22: Maximum Lorentz factor Γmax of the proto-magnetar outflowalong individual field lines, shown as a function of the asymptotic polar angleθ of the outflow, calculated for a PNS with spin period P = 2 ms and sur-face field strength B = 1016 G and shown for different neutrino luminosities:(from top to bottom) Lν = 1052 erg s−1 (red square), 6 × 1051 erg s−1 (blueasterisk), 3× 1051 erg s−1 (green cross), and 1051 erg s−1 (purple plus). TheΓmax(θ) distribution depicted only represents the true final angular distribu-tion of the outflow only in the case of ‘naked’ magnetar birth, with little orno surrounding matter at larger radii (as in the case of accretion-inducedcollapse, but unlike the core collapse case).

86

for less energetic or less relativistic phenomena such as low luminosity GRBs

or X-ray flashes [137].

Figure 2-22 shows the maximum Lorentz factor along each field line as a

function of asymptotic polar angle θ, calculated for a proto-magnetar with

B0 = 1016 G and P = 2 ms and shown for different neutrino luminosities.

In general Γmax peaks in the rotational equator (θ = π/2), with its value

falling to half of the peak value by θ ∼< 50. Keep in mind that for an aligned

rotator, the power per solid angle obeys dP/dΩ ∝ sin2 θ. Magnetar winds

are thus more powerful, and more highly relativistic, along the rotational

equator than in the polar direction.

However, translating these predictions for the bare magnetar wind into

the angular structure of the resulting relativistic jet in a GRB is non-trivial,

because the wide-angle magnetar outflow must interact with and become

collimated into a much narrower jet by the surrounding ejecta from the

exploding star. This is accomplished by magnetic stresses resulting from

the toroidal magnetic field that accumulates behind the outgoing supernova

ejecta, which redirect the equatorial wind into a collimated bipolar flow that

burrows its way out of the star along the polar rotation axis [135]. The

angular distribution Γmax(θ) shown in Fig. 2-22 should however, apply di-

rectly to the case of ‘naked’ magnetar birth, such as might occur following

the accretion-induced collapse of a white dwarf.

Ultra-High Energy Cosmic Rays

A surprising discovery by the Pierre Auger Observatory (PAO) is that the

composition of ultra high energy cosmic rays (UHECRs) appears to be dom-

inated by heavy nuclei (with mass similar to iron) at the highest cosmic ray

energies ∼> 5 × 1019 eV (e.g. [160]; [215]; however, see e.g. [161]). Such a

heavy-dominated composition is unexpected for most astrophysical sources

87

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

2 4 6 8 10

X

P, ms

Xα

Xn at 100 s

Xp at 100 s

Figure 2-23: Time-integrated mass fractions of α−particles, free neutronsand free protons in proto-magnetar winds. We report the mass fractions ofneutrons and protons at t = 100 s, since at that time free neutrons have notyet decayed to protons. The horizontal lines of corresponding colors showthese quantities in the spherical model. In spherical wind, the expansion isslower and hence neutrons have more time to capture onto seed nuclei duringthe r-process. This results in much lower value of Xn for spherical winds.

such as AGN, but it could arise naturally if UHECRs are accelerated in GRB

jets as a result of nucleosynthesis in proto-magnetar outflows [156, 216].

Figure 2-23 shows the composition of proto-magnetar jets is predicted to

be roughly XHe ≈ 0.6 by mass in helium, with the remainder Xh ' 1−XHe ≈0.4 in heavy nuclei of average mass A ≈ 95 (Fig. 2-16, Table 2.1). The

latter result is non-trivial, because if the expansion time of the outflow were

sufficiently short, then neutron capture reactions could in principle freeze-out

with an order unity mass fraction of free neutrons. However, we find that the

free neutron mass fraction at 100 s is always small Xn ∼< 10−3 (Fig. 2-23).

88

Such a low free neutron fraction would disfavor models in which the GRB

prompt emission is powered by the relative kinetic energy in the jet between

its neutral (neutron) and charged (protons, nuclei) constituents [217], unless

the nuclei are destroyed in the jet by photo-disintegration before the radius

at which neutrons collisionally decouple (see below).

The prediction of UHECRs dominated by nuclei with A ∼ 100 has im-

portant implications for the energy spectrum and pathlength of UHECRs

through the intergalactic medium. The normal Grezin-Zatsepin-Kuzmin

(GZK) cut-off in the cosmic ray spectrum occurs due to inelastic pion pro-

duction by high energy protons which interact with the cosmic microwave

background or extragalactic background light [218, 219]. At a fixed (mea-

sured) cosmic ray energy E, a nucleus of mass A has a bulk Lorentz-factor

which is a factor of A times smaller than a proton of the same energy. Naively,

one would therefore expect the effective GZK cut-off energy of a nucleus (in-

teracting with background radiation of a fixed temperature like the CMB) to

be A time larger than that for a proton. However, nuclei suffer from other loss

processes, the most important one being Giant Dipole Resonances (GDR),

which once excited cause the nucleus to shed free nucleons or α−particles,

reducing its energy.

[156] show that a nucleus of initial mass A = 56A56, energy E = 1020E20

eV travels a distance of

χ75 = 170E−1.520 A1.3

56 n−1ej Mpc (2.29)

before losing 25 % of its initial energy, where n−1ej ≈ 1 is the mean number

of nucleons ejected per GDR excitation. For an iron nucleus, the mean free

path with respect to EBL interaction is coincidentally the same as mean

free path for protons with respect to CMB pair production, resulting in a

89

similar GZK cut-off to the proton case. However, χ75 ∝ A1.3 implies that for

A = 95 nuclei from our models (Fig. 2-16), the mean free path is 2 times

larger than for protons. Thus we expect the GZK-like cut-off in the cosmic

ray energy spectrum to extend farther for proto-magnetar wind composition

than the usual cut-off predicted for protons or iron nuclei. By contrast, the

mean free path for helium is much shorter than iron, such the composition

arriving at Earth is expected to be dominated by heavy nuclei at the highest

energies. As already mentioned, the highest energy UHECRs do appear to

be heavier than protons [160], although the precise composition is model-

and calibration-dependent and hence remains uncertain.

2.5 Conclusions

We have calculated nucleosynthesis yields of steady-state neutrino-heated

outflows from magnetized, rotating proto-neutron stars (‘proto-magnetars’)

under the assumption of an axisymmetric force-free dipolar geometry (Figs. 2-

1, 2-2). The wind and abundances calculations presented here are more ac-

curate than previous one-dimensional models, that adopted a poloidal field

that scaled as either ∝ 1/r2 or ∝ 1/r3 at all radii [134, 57]. This allows us to

assess, for the first time, the latitudinal dependencies of the wind properties.

Our conclusions are summarized as follows:

• The force-free approximation is justified in magnetized proto-neutron

stars outflows at temperatures T ∼> 0.5 MeV relevant to heavy element

nucleosynthesis for surface magnetic fields B0 ∼> 1014−1015 G similar to

those of Galactic magnetars (Fig. 2-6) at neutrino luminosities achieved

over the first ∼ 10 seconds following core bounce. At later times (lower

neutrino luminosities), even lower magnetic field strengths ∼< 1013 G

similar to those of normal radio pulsars are sufficient for force-free con-

90

ditions.

• The force-free assumption breaks down in the closed zone due to the

high pressure of the co-rotating, hydrostatic atmosphere, as first pointed

out by [134]. This is estimated to increase the fraction of the surface

open to outflows (Y point radius RY ∼ RL/3 for B0 = 1015 G; Fig. 2-7)

as compared to the standard case RY = RL.

• Nucleosynthesis yields of magnetized rotating PNS outflows differ from

the standard spherical case as a result of the more rapid divergence of

the dipolar field geometry and the effects of magneto-centrifugal accel-

eration. The latter dominates the influence on the expansion timescale

of the outflow texp for spin periods P ∼< 4 ms.

• The asymptotic value of the electron fraction is substantially reduced

below its neutrino equilibrium value (eq. [2.10]) only for very short ro-

tation periods, P ∼ 1 ms, near centrifugal break-up. This confirms the

results of previous works employing a pure monopolar field geometry

[169].

• The mass loss rate per surface area is enhanced by centrifugal effects

along low latitude field lines (Fig. 2-12), although this enhancement is

not as great as in the equatorial monopole case [57]. The smaller frac-

tion of the PNS surface open to outflows as compared to the spherical

case (eq. [2.21]) more than compensates for this increase, resulting in

the total mass loss rate being smaller by a factor ∼> 10 than the equiv-

alent luminosity spherical case for P ∼> 2 ms (Fig. 2-18, Table 2.2).

Such suppression in the outflow fraction could be a generic feature of

PNS winds if strong, dynamically-important magnetic fields are tran-

sient but ubiquitous, e.g. as would be the case if they are present only

91

during the convective phases of the PNS cooling evolution. Such uncer-

tainties, in addition to the uncertain Ye evolution make it challenging

to place constraints on the birth rate of magnetars based on their nu-

cleosynthesis abundances (Fig. 2-19). For the most rapidly rotating

case P = 1 ms, the total mass loss is instead enhanced by a factor of 3

over the spherical case due to magneto-centrifugal acceleration (Table

2.3).

• Neutrino-heated winds from millisecond magnetars with rotation pe-

riods P ∼ 1 − 10 ms produce heavy element abundance distributions

that extend to higher atomic number than that from otherwise equiv-

alent spherical winds with the same Ye(t) (Fig. 2-15). This increase

in the neutron-to-seed ratio is driven mainly by the faster expansion

rate in proto-magnetar winds caused by the faster divergence of the

area function and due to additional magneto-centrifugal acceleration.

For the fastest rotation periods, P = 1 − 2 ms, it is also driven by

the lower value of Ye resulting from centrifugal acceleration. Unfor-

tunately, a direct detailed comparison of our predicted abundances to

data (e.g. in the solar system or on metal-poor stars) is hindered by

the larger uncertainties in the time evolution of Ye in the wind during

the PNS cooling phase.

• The heaviest elements are synthesized by outflows emerging along flux

tubes with latitude θ ≈ θmax, i.e. those in outflows which graze the

closed zone and pass near the equatorial plane outside the light cylinder

(Fig. 2-14). These fields lines also dominate the total mass budget of

the wind due to their larger fraction of the total solid angle and, to a

lesser degree, enhancements in the mass loss rate per unit area due to

centrifugal acceleration.

92

• Due to dependence of the charged particle process and weak r-process

pattern on the magnetic field strength and rotation rate of PNSs, natu-

ral variations in these quantities between different core collapse events

could contribute to the diversity of abundances observed on metal-poor

stars (Fig. 2-20). This r-process site in the PNS phase (post-explosion)

is notably distinct from that discussed in the context of MHD super-

novae [145]. The latter, which invoke low-Ye ejecta, have the poten-

tial to produce a greater total r-process yield; however, such extreme

magnetars are likely rarer than those discussed here due to the larger

angular momentum of the stellar progenitor core required to produce a

maximally-spinning PNS versus the slower ∼ 2−5 ms periods described

here.

Additional diversity in the r-process abundances of proto-magnetar

winds, not considered in detail here, could result from variations in the

magnetic inclination angle. Magnetic dissipation within the resulting

striped wind could result in additional wind heating and concomitant

entropy gain prior to seed formation, facilitating a heavier r-process

than the aligned case focused on here.

• Magnetars with birth periods P ∼> 4 ms represent an appealing site

for Galactic r-process source, consistent with a variety of observational

constraints, including (1) the requirement to not overproduce the ener-

gies of magnetar hosting supernova remnants; (2) within uncertainties,

the mass of r-process nuclei per event needed to explain observed galac-

tic abundances (eq. [2.25]); (3) low mass per event, consistent with pre-

vious studies of Galactic chemical evolution [195]; (4) recent evidence

for the birth of millisecond magnetars in low metallicity environments

based on the discovery of superluminous supernovae.

93

• If proto-magnetars are the central engines of GRBs, their relativis-

tic jets should contain an order unity mass fraction of heavy nuclei

with A ≈ 100. Subsequent particle acceleration in such a jet could

produce UHECRs with a heavy composition [160] and an energy spec-

trum that extends roughly a factor of 2 above the nominal GZK cut-off

for protons or iron nuclei [220]. Better statistics and modeling of the

UHECR energy spectrum, as well as a more firm measurement of the

composition-dependent UHECR spectrum, is needed to test this pre-

diction and its subtle differences with the normal proton-dominated

model predictions.

94

Chapter 3

Novae

3.1 Introduction

Classical and symbiotic novae are luminous transients, powered by runaway

thermonuclear burning of a hydrogen-rich layer accreted from a binary com-

panion (e.g., [67, 68, 69, 70, 71]). The resulting energy release causes the

white dwarf atmosphere to inflate, ejecting ∼ 10−5 − 10−4 M of CNO-

enriched matter at hundreds to thousands of kilometers per second (e.g.,

[221, 222]).

Radio and optical imaging [102, 223], and optical spectroscopy [109];[224],

suggests that a nova outburst proceeds in at least two stages (Figure 3-1).

The runaway is first accompanied by a low velocity outflow concentrated in

the equatorial plane of the binary, perhaps influenced by the gravity of the

companion star, as occurs in common envelope phases of stellar evolution

(e.g., [225, 226]). Given the uncertain nature of the slow ejecta, we use the

agnostic term ‘dense external shell’ (DES; [77]).

The outflowing DES is then followed by a more continuous wind (e.g.,

[227]) with a higher velocity and a more spherical geometry. A collision

between this fast outflow and the slower DES results in strong internal shocks,

95

which are most powerful near the equatorial plane where the density contrast

is largest. If the fast component expands relatively unimpeded along the

polar direction, this creates a bipolar morphology [102, 79]. Such a scenario

does not exclude fast shocks within the polar region, characterized by lower

densities and hard X-ray emission (Fig. 3-1).

Several lines of evidence support shocks being common features of nova

outbursts. Nova optical spectra exhibit complex absorption lines with mul-

tiple velocity components (e.g., [80, 81, 78]). In addition to broad P Cygni

lines indicating high velocity ∼> 1000 km s−1 matter, narrower absorption

features (≈ 500− 900 km s−1) are observed near optical maximum and may

be created within the DES. A large fraction of novae show such narrow lines.

This indicates that the absorbing material has a high covering fraction [78]

and hence is likely to impact the faster P Cygni outflow.

Many novae are accompanied by thermal X-ray emission of luminosity

LX ∼ 1033 − 1035 erg s−1 and temperatures ∼> keV (e.g., [228, 229, 230, 231,

232, 233, 234, 100, 102]; see [85] for a summary of Swift observations). This

emission is too hard to be thermal radiation from the white dwarf surface

[235], but is readily explained as free-free emission from∼> 1000 km s−1 shocks

(e.g. [236]). The X-ray emission is often delayed by weeks or longer after the

optical maximum, perhaps due to absorption by the DES [77].

Although the presence of shocks in novae have been realized for some

time, their energetic importance was only recently revealed by the Fermi

LAT discovery of ∼> 100 MeV gamma-rays, coincident within days of the

optical peak and last for weeks [90]. Gamma-rays were first detected in the

symbiotic nova V407 Cyg 2010, in which the target material for the shocks

could be understood as the dense wind of the companion red giant [95, 96, 93].

However, six additional novae have now been detected by Fermi-LAT, at least

four of which show no evidence for a giant companion and hence were likely

96

Figure 3-1: Proposed scenario for the locations of X-ray and radio emit-ting shocks. A slow outflow is ejected first, its geometry shaped into anequatorially-concentrated torus (blue). This is followed by a faster outflowor continuous wind with a higher velocity and more spherical geometry (red).The fast and slow components collide in the equatorial plane, producing pow-erful radiative shocks (Fig. 3-2) which are responsible for the gamma-rayemission on timescales of weeks and the non-thermal radio emission on atimescale of months. Adiabatic internal shocks within the fast, low densitypolar outflow power hard keV thermal X-rays and possibly radio emis-sion at very early times. Slower equatorial shocks produce the non-thermalradio peak on a timescale of months and softer X-rays (kT ∼< keV), whichare challenging to detect due to their lower luminosity and confusion withsupersoft X-rays from the white dwarf. The direction of the binary orbitalangular velocity Ω is indicated by an arrow.

97

ordinary classical novae with main sequence companions. The DES is clearly

present even in binary systems that are not embedded in the wind of an

M giant or associated with recurrent novae, supporting the internal shock

scenario.

The high gamma-ray luminosities Lγ ∼ 1035−1036 erg s−1 require shocks

with kinetic powers which are at least two orders of magnitude larger, i.e.

Lsh ∼ 1037−1038 erg s−1, approaching the bolometric output of the nova [79].

Gamma-rays are produced by the decay of neutral pions created by collisions

between relativistic protons and ambient protons in the ejecta (hadronic sce-

nario), or by inverse Compton or bremsstrahlung emission from relativistic

electrons (leptonic scenario). Hadronic versus lepton emission scenarios can-

not be distinguished based on the gamma-ray spectra alone [90], although

[79] cite evidence in favor of a hadronic scenario.

Additional evidence for shocks comes at radio wavelengths. Novae pro-

duce thermal radio emission from the freely expanding photoionized ejecta

of temperature ∼ 104 K, which peaks as the ejecta becomes optically thin to

free-free absorption roughly a year after the optical outburst [101]. However,

a growing sample of novae show an additional peak in the radio emission

at earlier times (∼< 100 days; [88, 100, 102, 103]; Fig. 3-4) with brightness

temperatures 105 − 106 K higher than that of photo-ionized gas. This addi-

tional early radio peak requires sudden heating of the ejecta (e.g., [237, 77])

or non-thermal emission [88, 103], in either case implicating shocks.

[77] (Paper I) developed a one-dimensional model for the forward-reverse

shock structure in novae and its resulting thermal X-ray, optical, radio emis-

sion. This initial work provided an acceptable fit to the radio light curves

of the gamma-ray nova V1324 Sco, under the assumption that the dominant

cooling behind the shock was provided by free-free emission. However, for

low velocity shocks ∼< 103 km s−1 line cooling of the CNO-enriched gas can

98

greatly exceed free-free cooling. For cooling rates determined by collisional

ionization equilibrium (CIE), this additional cooling reduces the peak bright-

ness temperature of thermal models to values ∼< 104 K, which are too low to

explain the early radio peak, unless line cooling is suppressed by non-LTE

effects.

In this work, we extend the [77] model to include non-thermal synchrotron

radio emission, which we demonstrate can explain the observed emission,

even for low shock velocities. In addition to providing information on the

structure of the ejecta and the nova outburst mechanism, synchrotron emis-

sion provides an alternative probe of relativistic particle acceleration in these

events, complementary to that obtained from the gamma-ray band. In lep-

tonic scenario, relativistic electrons accelerated directly at the shock power

the radio emission. Radio-emitting e± pairs are also produced in hadronic

scenarios by the decay of the charged pions. Radio observations can in prin-

ciple help disentangle leptonic from hadronic models.

This chapter is organized as follows. We begin with an overview of shocks

in novae (§3.2), including the collision dynamics, observational evidence for

shocks, the analytic condition for radio maximum, the radiative versus adi-

abatic nature of the shock, and thermal X-ray emission. In §3.3 we describe

key features of synchrotron emission, including leptonic and hadronic sources

of non-thermal particles and their cooling. In §3.4 we provide a detailed de-

scription of our model for radio emission from the forward shock. In §3.5 we

describe our results, including analytic estimates for brightness temperature,

and fits to the radio lightcurves of three novae: V1324 Sco, V1723 Aql, and

V5589 Sgr. In the discussion (§3.6) we use the radio observations to con-

strain the acceleration efficiency of relativistic particles and magnetic field

amplification in the shocks and their connection to gamma-ray emission. We

also discuss outstanding issues, including the unexpectedly monochromatic

99

light curve of V1723 Aql and the role of adiabatic X-ray producing shocks.

In §3.7 we summarize our conclusions.

3.2 Shocks in Novae

Following [77], we consider the collision between a fast outflow from the white

dwarf of velocity v1 and the DES of velocity v4 < v1 and unshocked density

n4, as shown in Figure 3-2. We assume that the DES of velocity v4 is ejected

at t = 0, corresponding to the time of the first optical detection. The fast

outflow of velocity v1 is ejected after a delay of time ∆t. The fast outflow

and DES collide at a radius and time given, respectively, by

t0 =v1

v1 − v4

∆t; R0 =v1v4

v1 − v4

∆t (3.1)

The mean density of the DES at the time of the collision is

n0 ≈MDES

4πR30fΩmp

∼ 108

(MDES

10−4M

)(t0

60 d

)−3 ( v4

103 km s−1

)−3

cm−3, (3.2)

where fΩ ∼ 0.5 is the fraction of the total solid-angle subtended by the

outflow and for purposes of an estimate we have assumed the thickness of

the DES is ∼ R0 ∼ t0v4. We define the ejecta number density as n ≡ ρ/mp,

where ρ is the mass density.

This interaction drives a forward shock through the DES and a reverse

shock back through the fast ejecta (see Fig. 3-2). We assume spherical sym-

metry and, for the time being, that the shocks are radiative (§3.2.3). For

radiative shocks the post shock material is compressed and piles up in a cen-

tral cold shell sandwiched by the ram pressure of the two shocks. Neglecting

non-thermal pressure, the shocked gas cools by a factor of ∼ 103, its vol-

ume becoming negligible. Hence the shocks propagate outwards at the same

100

Table 3.1: Commonly used variables and their definitions.

Variable Definitionv1 Velocity of fast outflow (Fig. 3-2)v4 Velocity of DESn1 Density of (unshocked) fast outflown4 Density of (unshocked) DEST4 Temperature of DES in the photo-ionized layer just ahead of the shockT3 Temperature immediately behind the forward shockn3 Density immediately behind forward shockvshock Velocity of forward shock in the white dwarf framevshell Velocity of central shell in the white dwarf frame

vf ≡ 108v8 cm s−1 Velocity of the shock in the frame of the DESH = 1014H14 cm Density scale height of DES

Tb Observed brightness temperature, corrected for pre-shock screeningby the photo-ionized layer

τff,4 Free-free optical depth of unshocked DESαff,4 Free-free absorption coefficient of unshocked DEStcool Cooling time of gas in the post shock region∆ion Thickness of ionized layer ahead of the shock (eq. [3.5])

η3 ≡ tcool/tfall Ratio of post-shock cooling timescale to shock expansion time (eq. [3.18])npk,∆ Density of unshocked DES at time of radio peak (τff,4 = 1)

for case when ∆ion < H (eq. [3.12], upper line)npk,H Density of unshocked DES at time of radio peak (τff,4 = 1)

for case when ∆ion > H (eq. [3.12], lower line)Mej Total ejecta massRsh Radius of cool central shell ∼ radius of shock

fEUV = 0.1fEUV,−1 Fraction of shock power placed into hydrogen-ionizing radiationεp = 0.1εp,−1 Fraction of shock power placed into relativistic protons

εe Fraction of shock power placed into relativistic electrons and positronsγpk Lorentz factor of the electrons which determine peak synchrotron emissivityTν Brightness temperature of emission at generic location behind the shock

Tν,sync Brightness temperature of synchrotron emission (eq. [3.38])τν Optical depth at arbitrary location behind the shockT thν,pk Peak observed radio brightness temperature due to thermal emissionT nthν,pk Peak observed radio brightness temperature of synchrotron emission

T thν,H,pk Thermal contribution to the peak brightness temperaturefor adiabatic shocks (eq [3.46])

T nthν,H,pk Synchrotron contribution at the peak brightness temperatureof adiabatic shocks (eq. [3.48])

101

Figure 3-2: Shock interaction between the fast nova outflow (Region 1) andthe slower dense external shell [DES] (Region 4). A forward shock is driveninto the DES, while the reverse shock is driven back into the fast ejecta. Theshocked ejecta (Region 2) and shocked DES (Region 3) are separated by acold central shell containing the swept up mass. Observed radio emissionoriginates from the forward shock, since emission from the reverse shock isabsorbed by the cold central shell. The ionized layer of thickness ∆ion (eq.3.5) in front of the forward shock is also shown.

velocity as the central shell, vshell = vshock ≡ vsh. The velocity of the cold

central shell is determined by equating the rate of momentum deposition

from ahead and from behind, as described in [77]. In what follows, we define

the shock velocity in the upstream frame,

vf = vsh − v4 = 108v8 cm s−1, (3.3)

normalized to a characteristic value of 1000 km s−1.

Radio emission is assumed to originate from the forward shock because

the reverse shock emission is highly attenuated by free-free absorption within

102

the cold central shell. The forward shock heats the gas to a temperature

T3 '3µmpv

2f

16k≈ 1.7× 107v2

8 K (3.4)

and compresses it to a density n3 = 4n4, where µ is the mean molecular

weight and v8 ≡ vf/108 cm s−1. We assume solar chemical composition

with enhanced abundances as follows: [He/H]=0.08, [N/H]=1.7, [O/H]=1.3,

[Ne/H]=1.9, [Mg/H]=0.7, [Fe/H]=0.7, typical of nova ejecta (e.g., [238]),

which corresponds to µ = 0.76. However, our qualitative results are not

sensitive to the precise abundances we have assumed.

Absent sources of external photo-ionization, gas well ahead of the forward

shock is neutral due to the short timescale for radiative recombination. The

upstream is, however, exposed to ionizing UV and X-ray radiation from the

shock, which penetrates gas ahead of the shock to a depth, ∆ion. The latter

is set by the balance between photo-ionization and recombination, similar to

an HII region ([77]; Fig. 3-2), and is approximately given by

∆ion ≈ 2× 1014fEUV,−1

( n4

107 cm−3

)−1

v38 cm, (3.5)

where fEUV,−1 = fEUV/0.1 is the fraction of the total shock power Lsh ∝ n4v3sh

placed into ionizing radiation and absorbed by the neutral layer. Although

the neutral gas upstream of the shock effectively absorbs soft UV and X-ray

photons, harder X-rays can escape from this region. A minimum ionizing

fraction of

fEUV ≈2 Ryd

kT3

= 0.02 v−28 (3.6)

is obtained in the limit that free-free emission is the sole source of ionization,

where Ryd = 13.6 eV. The value of fEUV can in principle greatly exceed this

103

minimum due to line emission and from the reprocessing of higher-energy

photons to lower frequencies by the neutral gas ahead of the shock or in the

central shell.1

3.2.1 Evidence for Non-Thermal Radio Emission

Figure 3-3 shows the radio light curves of three novae with early-time shock

signatures: V1324 Sco [239], V1723 Aql [100, 103], and V5589 Sgr [99]. Ther-

mal free-free emission from the photo-ionized nova ejecta peaks roughly a year

after the outburst [101]. In V1324 Sco and V1723 Aql, the light curve also

shows a second, earlier peak on a characteristic timescale of a few months.

This early peak is also present in V5589 Sgr, although the late thermal peak

in this event is only apparent at the highest radio frequencies [99].

Thermal emission from an expanding sphere is initially characterized by

an optically-thick, Rayleigh-Jeans spectrum Fν ∝ ν2, with the total flux in-

creasing with the surface area ∝ R2 ∝ t2. Then, starting at high frequencies,

the ejecta begins to become optically thin and the radio lightcurve decays as

the radio photosphere recedes back through the ejecta. The shape of the ra-

dio lightcurve near its maximum depends on the density profile of the ejecta,

which we take to be that of homologous expansion, n ∝ t−1r−2. Finally,

after the entire ejecta becomes optically thin, the spectrum approaches that

of optically thin free-free emission, Fν ∝ ν−0.1t−3.

In order to isolate the non-thermal, shock-powered contribution to the

radio emission, we first remove the thermal emission from the photo-ionized

ejecta, using a model for the latter as outlined in [240]. We assume that

shocks make no contribution to the emission at late times, t ∼> 100 − 120

1Ionizing X-rays from the white dwarf are likely blocked by the neutral central shellin the equatorial plane, although they may escape along the low density polar region(Fig. 3-1). Many novae are not detected as supersoft X-ray sources until after the earlyshock-powered radio emission has peaked [234].

104

0.1

1

10

10 100

F i, m

Jy

Time from optical outburst, days

36.5 GHz27.5 GHz17.5 GHz13.3 GHz

7.4 GHz4.6 GHz

0.01

0.1

1

10

10 100 1000

F i, m

Jy

Time from optical outburst, days

36.5 GHz28.2 GHz

6.8 GHz5.2 GHz

0.1

1

10

40 50 60 70 80 90 100 110 120 130 140

F i, m

Jy

Time from optical outburst, days

32.1 GHz14.0 GHz

7.8 GHz4.8 GHz1.9 GHz1.3 GHz

Figure 3-3: Radio light curves of V1324 Sco (top left panel; [239, 106]), V1723Aql (right panel; [100, 103]), and V5589 Sgr (bottom left panel; [103]). InV1723 Aql, only frequencies with full time coverage are shown. In V1324 Scoand V1723 Aql, we show for comparison our best-fit thermal models (Table3.2). In V1324 Sco and V1723 Aql, the maximum uncertainties in the dataare 0.3 and 0.1 mJy, respectively, and hence the error bars are not discerniblefor most of the data points. In V5589 Sgr, we do not attempt to fit a thermalmodel, even though some of the emission after day 100 could in principle bethermal.

105

103

104

105

106

20 40 60 80 100 120 140

T b, K

Time from optical outburst, days

36.5 GHz27.5 GHz17.5 GHz13.3 GHz

7.4 GHz4.6 GHz

104

105

106

20 40 60 80 100 120 140

T b, K

Time from optical outburst, days

36.5 GHz28.2 GHz

6.8 GHz5.2 GHz

Figure 3-4: Brightness temperature of the early component of the radio emis-sion for V1324 Sco (top panel) and V1723 Aquila (bottom panel), calculatedby subtracting the thermal emission component off the raw fluxes (Fig. 3-3)and assuming an emitting radius for the non-thermal emission correspondingto the fastest velocity of the ejecta inferred from the thermal fits. In V1723Aql, only frequencies with full time coverage are shown. In V1324 Sco andV1723 Aql, the maximum uncertainties in the data are 0.3 and 0.1 mJy, re-spectively, and hence the error bars are not discernible for most of the datapoints.

106

Table 3.2: Best-fit parameters for the thermal radio emission models. Thefits are based on observations at t ∼> 100 − 120 days in order to excludecontributions from the early radio peak.

Nova D v(a)min v

(b)max T(c) M

(d)ej ∆t(e)

kpc km s−1 km s−1 K M daysV1324 Sco 7.8 550 1300 1.2 · 104 2.7 · 10−4 15V1723 Aq 6.1 270 1600 1.0 · 104 2.2 · 10−4 -2.2

(a)Minimum velocity. (b)Maximum velocity. (c)Ejecta temperature. (d)Ejectamass. (e)Time delay between outflow ejection and optical outburst.

Table 3.3: Multiwavelength properties of novaeName D† Lnth10GHz, pk Lth10GHz, pk LX, pk

L10GHz,pkLX,pk

kT‡pk

Lγ, pk Orientationq

(kpc) (erg s−1) (erg s−1) (erg s−1) - (keV) (erg s−1) -

V1324 Sco 6.5(l) − 8(7.8) 3.4× 1030(d) 1.4× 1030(d) < 1034(h) > 3× 10−4 - 2.7× 1036(m) ?

V1723 Aq 5.3− 6.1(6.1)(e) 2.5× 1030(d) 6.3× 1029(d) 1.2× 1034(f) 2.1× 10−4 1.8− 3(f) - I

V5589 Sgr 3.2− 4.6(4.0)(b) 9× 1029(b) < 3.6× 1028(b) 1.2× 1034(b) 7.8× 10−5 0.14− 32.7(b) - ?

V959 Mon 1.0− 1.8(1.4)(a) < 1.1× 1029(c) 1.8× 1029(c) 2.4× 1033(g) < 5× 10−5 3.2(g) 5.3 · 1034(m) E

V339 Del 3.9− 5.1(4.5)(i) < 2.3× 1028(k) 9.2× 1029(k) 1.9× 1033(j) < 1.2× 10−4 > 0.8(j) 2.9× 1035(m) F†Estimated uncertainty range of distance, followed in paranthesis by the fiducial value adopted in calculating luminosities;‡Temperature of X-ray emission; qApproximate inclination of binary (E = edge on; F = face on; I = intermediate; ? =

unknown) (a) [108] (b) [99] (c) [107] (d) [241] (e) [103] (f) [100], correct to unabsorbed value (g) [242] (h) for kT ∼ 2− 25

keV, [243], (i) [244] (j) [245] (k) Justin Linford, private communication (l) [239] (m) [90]

days. Our best-fit parameters for the ejecta mass of Mej ≈ 2− 3× 10−4M

and maximum ejecta velocities of vmax ≈ 1300−1600 km s−1 are compiled in

Table 3.2. Our values are consistent with those found by [103] for V1723 Aql.

After subtracting the thermal component from the raw fluxes (Fig. 3-3), the

remainder should in principle contain only the shock-powered emission.

Our model for the late thermal emission is admittedly simplified, as it as-

sumes a homologous spherically symmetric and isothermal outflow. However,

we are primarily interested in modeling the shape of late thermal emission

near the time of early radio peak, at around 80-100 days. At this time the

ejecta is still optically-thick and thus our only essential assumption is that

the outer ejecta is isothermal and expanding ballistically.

Figure 3-4 shows the brightness temperature Tb of the early shock-powered

radio emission, which we have calculated assuming that the shock-powered

107

emission covers the entire solid angle of the outflow and originates from

a radius equal to that of the fastest ejecta inferred from the thermal fit,

R = vmaxt. The brightness temperature peaks at values ∼ 105−106 K which

greatly exceed the temperature ∼ 104 K of the photo-ionized ejecta inferred

at late times [103].

The outer DES through which the forward shock is propagating at time

t, can be modeled as an exponential density profile,

n4 = n0 exp[−vf (t− t0)/H], (3.7)

where n0 is a fiducial density at the time t0 of the collision (eq. [3.1]) and H

defines the density scaleheight. The light curves of V1324 Sco and V5589 Sgr

exhibit a rapid post-maximum decline on a timescale of tfall ∼ weeks much

shorter than the timescale of the collision ∼ t0 (Fig. 3-3). This suggests that

the forward shock is propagating down a steep radial gradient,

H ∼ vf tfall ≈ 6× 1013v8

(tfall

week

)cm, (3.8)

which is smaller than the collision radius R0 ∼ 1015 cm (eq. [3.1]).

3.2.2 Condition for Radio Maximum

The detection of gamma-rays within days of the optical maximum [90] demon-

strates that shocks are present even early in the nova eruption. However, at

such early times, the density of the shocked matter is highest, and radio

emission from the shocks is absorbed by the photo-ionized gas ahead of the

shock.

The observed brightness temperature, Tb, is related to the unscreened

108

value just ahead of the shock, Tν |shock, according to [77]

Tb = Tν |shocke−τff,4 + (1− e−τff,4)T4, (3.9)

where τff,4 = αff,4∆ion is the free-free optical depth of the ionized layer ahead

of the shock of thickness ∆ion (eq. [3.5]) and

αff,4 ≈ α0 ν−2T

−3/24 n2

4 (3.10)

is the free-free absorption coefficient [246], where T4 ≈ 2 × 104 K is the

temperature in the photo-ionized layer. For conditions in photo-ionized layer,

α0 = 0.1 cm5 K3/2 s−2.

Although the unscreened flux Tν is initially large, the emission reaching

the observer is suppressed by a factor of e−τff,4 1. Shock-powered radio

emission peaks at frequency ν once the shock propagates down the density

gradient until [77]

τff,4(ν) = αff,4(ν) ∆ion ≈ 1. (3.11)

Combining equations (3.5) and (3.11), radio emission peaks once the up-

stream density decreases to a value of

npk = max

npk,∆ ≡ 1× 106 ν210 f

−1EUV,−1 v

−38 cm−3

npk,H ≡ 5× 106 ν10H−1/214 cm−3

(3.12)

where ν = 10 ν10 GHz, H = H141014 cm, and the second line accounts for

cases when the entire scaleheight is ionized (∆ion > H).

The dependency npk ∝ νq, where q = 1 − 2, illustrates that the time

delay (∆tpk) between the maximum flux at different frequencies ν2 and ν1 is

109

typically comparable to the rise time,

∆tpk = qH

vflog

(ν2

ν1

), (3.13)

as was obtained by combining equations (3.7) and (3.12).

In summary, for characteristic parameters v8 ∼ 0.5 − 2, ν ∼ 3 − 30

GHz, the radio light curve peaks when the shock reaches external densities

of npk ∼ 106 − 108 cm−3. These are generally lower than the mean density

of the shell at the time of the collision, n0 (eq. [3.2]), and those required to

produce the observed γ−ray emission [79]. We now consider properties of

the forward shock at the radio maximum, defined by n4 = npk.

3.2.3 Is the Forward Shock Radiative or Adiabatic?

Whether the shock is radiative or adiabatic near the radio peak depends on

the cooling timescale of the post-shock gas,

tcool =3kT3

2µΛn3,pk

, (3.14)

where n3,pk = 4npk (eq. [3.12]) and

Λn2 = (Λlines + Λff)n2 (3.15)

is the cooling rate per unit volume (in ergs s−1 cm−3). For two-body cooling

processes, Λ depends on temperature only and it is called the cooling func-

tion. Λ receives contributions from emission lines, Λlines, and from free-free

emission

Λff ≈ 3× 10−27(T/K)1/2 erg cm3 s−1. (3.16)

110

Figure 3-8 shows tabulated cooling function from [247] for our assumed ejecta

abundances and power-law approximation to the line cooling rate,

Λlines ≈ 10−22Λc,−22

(T

1.7× 107K

)δerg cm3 s−1, (3.17)

with Λc,−22 = 1.5, δ = −0.7. The cooling rate is larger than for solar metal-

licity gas due to the enhancements of CNO elements.

The ratio of the cooling timescale (eq. [3.14]) at the time of radio maxi-

mum to the characteristic light curve decay timescale, tfall = H/vf (eq. [3.8])

is given by

η3 ≡tcool

tfall

∣∣∣∣n4=npk

=3kT3vf

8µΛnpkH≈ min

4.8 v7.48 H−1

14 fEUV,−1ν−210

1.3 v4.48 H

−1/214 ν−1

10

(3.18)

where the final equality uses equations (3.4), (3.12). Radiative shocks (tcool tfall) and adiabatic shocks (tcool tfall) are divided sharply at vf ≈ 1000 km

s−1, due to the sensitive dependence of tcool/tfall on the shock velocity.2

As we will discuss, in a radiative shock, only the immediate post-shock

material contributes to the thermal or non-thermal emission, allowing for

faster light curve evolution. The rapid post-maximum decline of V1324 Sco

and V5589 Sgr (tfall t) is consistent with radiative shocks in these systems

[77].

2Note that η also equals the ratio of the post-shock cooling length

Lcool = vf tcool = η3H (3.19)

to the scale-height, such that the condition for radiative shocks (η3 1) can also bewritten as Lcool H.

111

3.2.4 Thermal X-ray Emission

Thermal X-rays are diagnostic of the shock velocity and hence of their radia-

tive nature. Figure 3-5 shows that the X-ray light curve of V5589 Sgr peaks

at a luminosity of LX ∼ 1034 erg s−1 a few weeks after the optical outburst

[99]. The temperature is very high kT ∼> 30 keV initially, decreasing to ∼ 1

keV by the radio peak around day 50. Large velocities v8 1 are required

to produce the high X-ray temperatures at the earliest times, clearly impli-

cating adiabatic shocks. However, the X-ray and radio emission may not

originate from the same shocks, while even in a given band different shocks

may dominate the emission at different times.

The X-ray luminosity of the forward shock is given by ([77]; their eq. [38])

LX ≈ 4πfΩR2shvf

1 + 5η3/2

(3

2

n4

µkT3

)(Λff

Λ

)(3.20)

where fΩ is the fraction of the solid angle of the outflow covered by shocks.

The factor (1 + 5η3/2)−1 accounts for the radiative efficiency of the shock

and the factor Λff/Λ accounts for the fraction of the shock power emitted as

free-free emission in the X-ray band (line emission occurs primarily in the

UV).

X-ray emission can be attenuated by bound-free absorption. For our

adopted composition we find that the bound-free opacity is reasonably ap-

proximated as κ ≈ 2000(EX/keV)−2 cm2 g−1 across the range of X-ray en-

ergies EX of interest [248]. The X-ray optical depth at the energies corre-

sponding to the peak of the free-free emission (EX = kT3 ≈ 1.4v28 keV) is

thus given by

τX ≈ 3.3

(EXkeV

)−2 ( n4

107 cm−3

)H14 ≈ 1.8v−4

8

( n4

107 cm−3

)H14, (3.21)

112

Analogous to the radio light curve, the X-ray luminosity from the shock

propagating down a density gradient peaks at τX ∼ 1. This occurs once the

density of the pre-shock gas decreases to a value of

nX ≈ 6× 106H−114 v

48 cm−3. (3.22)

Figure 3-6 shows the maximum X-ray luminosity as a function of the

shock velocity, obtained by combining nX with equation (3.20) for fΩ = 1 and

assuming a shock radius of Rsh ∼ 1015 cm. Although this maximum value is

calculated assuming that the density of the shocked matter equals or exceeds

nX , this assumption is unlikely to be valid for the highest shock velocities

because the density required to achieve τX ∼> 1 becomes unphysically high.

For high velocity shocks (v8 ∼> 1), the predicted X-ray luminosity is several

orders of magnitude greater than the observed range in classical novae, LX ∼1033 − 1035 erg s−1 ([236, 233, 85]; see Table 3.3). This indicates that the

observed X-ray producing shocks either (a) cover a small fraction of the

outflow solid angle fΩ ∼ 0.01 − 0.1, or (b) are produced in regions of the

outflow with much lower densities, n nX , than the mean values n ∼107 − 109 cm−3 (eq. [3.2]).

The second condition is challenging to satisfy at early times when the

densities are probably highest, while both conditions are at odds with the

high covering fractions fΩ ∼ 1 and high shock densities needed to power the

gamma-ray luminosities [79]. We therefore postulate that the hard keV

X-ray emission may not originate from the same shocks responsible for the

gamma-ray and non-thermal radio peak, but instead from the low density

polar region (Fig. 3-1).

In such a scenario, the early radio peak could be powered either by the

same fast polar shocks, or by lower velocity (v8 ∼< 1) radiative shocks in

113

1x1033

1x1034

1x1035

15 20 25 30 35 40 45 50 55 60 65 70 0.1

1

10

100

Tim

e of

radi

o m

ax

L X, e

rgs

s-1

kT, k

eV

Time from outburst, days

Figure 3-5: X-ray luminosities (black circles; left axis) and temperatureskT (blue circles; right axis) of V5589 Sgr as measured by Swift XRT. Bluetriangles indicate temperature lower limits.

the higher density equatorial region (Fig. 3-1). In the latter case the radio-

producing shocks still produce thermal X-rays; however, being comparable

in brightness, yet much softer in energy, they may not be readily observ-

able. This would also be true if the X-ray luminosity of radiative shocks is

suppressed due to the role of thin-shell instabilities [249].

3.3 Synchrotron Radio Emission

Amplification of the magnetic field is required to produce the observed syn-

chrotron emission and is expected to result from instabilities driven by cosmic

ray currents penetrating the upstream region (e.g. [127, 114]). The strength

114

Figure 3-6: Thermal X-ray luminosity at the time of X-ray peak (black line -left axis; n = nx) and 10 GHz radio luminosity νLν at the time of non-thermalradio peak (red line - right axis; n = npk) as a function of shock velocity.Both luminosities are maximum allowed values, because they are calculatedassuming τX = 1 and τν=10 GHz = 1, respectively. We have assumed fiducialparameters for the density scale-height H = 1014 cm and shock microphysicalparameters εe = εB = 0.01, and a common radius Rsh = 1015 cm and coveringfraction fΩ = 1 of the shock. Also shown is the division between radiative andadiabatic shocks (vertical dashed blue line) and the range of observed radioluminosities (dashed red line) and X-ray luminosities and shock velocitiescorresponding to the observed X-ray temperature (grey region; Table 3.3,Fig. 3-5).

115

of the post shock magnetic field can be estimated as

Bsh = (24πP3εB)1/2 ≈ 0.22n1/24,7 v8ε

1/2B,−2 G, (3.23)

where n4,7 ≡ n4/107 cm−3, εB = 10−2εB,−2 is the fraction of the post-

shock energy density in the magnetic field. Substituting n4 = npk ≈ npk,∆

(eq. [3.12]) into equation (3.23) gives the field strength when the radio emis-

sion reaches its peak,

Bpk = 8.2× 10−2 ε1/2B,−2f

−1/2EUV,−1v

−1/28 ν10 G. (3.24)

An electron or positron of Lorentz factor γ produces synchrotron emission

mostly near a characteristic frequency

νsyn =1

2π

eBγ2

mec≈ 92 GHz ε

1/2B,−2v

−1/28 f

−1/2EUV,−1ν10

( γ

200

)2

. (3.25)

There thus exists a special electron Lorentz factor,

γpk ≈ 210 ε−1/4B,−2v

1/48 f

1/4EUV,−1, (3.26)

which determines the peak emission (νsyn = ν) and, remarkably, is indepen-

dent of the observing frequency.3

3.3.1 Leptonic Emission

The relativistic leptons responsible for synchrotron radiation originate either

from the direct acceleration of electrons via diffusive shock acceleration (e.g.,

3The ‘Razin’ effect suppresses synchrotron radiation below a critical frequency νR = νpγ

[246], where νp = (npke2/πme)

1/2. For γ = γpk we thus have ν/νR = 4.5 ε1/4B,−2v

5/48 f

1/4EUV,−1.

Since for parameters of interest we have ν ∼> νR, modifications from the Razin effect shouldbe weak and are hereafter neglected.

116

[112]), or from electron-positron pairs produced by the decay of charged

pions from inelastic proton-proton collisions. Both theory and simulations

predict the spectrum of accelerated electrons (dN/dE)dE ∝ E−pdE, with

p ∼> 2 (when E mec2) in the case of strong shocks (Figure 3-9, left panel).

Two key parameters are the fractions of the shock kinetic power placed into

non-thermal electrons and ions, εe and εp, respectively.

Particle-in-cell (PIC) and hybrid kinetic simulations (e.g., [250], [113],

[114], [115], [251], [252]) indicate that the values of εe and εp depend on

the Mach number of the shock, and the strength and the inclination angle

θ of the upstream magnetic field with respect to the shock normal. Shocks

which propagate nearly perpendicular to the direction of the magnetic field

(θ ≈ 90) do not efficiently accelerate protons or electrons [130], while those

propagating nearly parallel to the magnetic field (θ ≈ 0) accelerate both

[251, 252].

What is the geometry of the magnetic field in the ejecta? On timescales

of months, the ejecta has expanded several orders of magnitude from its

initial size at the base of the outflow [79]. This expansion both dilutes the

magnetic field strength via flux freezing and stretches the field geometry

to be perpendicular to the radial direction in which the shocks are likely

propagating. Based on the above discussion, such a geometry would appear

to disfavor hadronic scenarios. However, leptonic scenarios are also strained

because the values of εe ∼> 10−2 required to explain the γ-ray emission in

V1324 Sco [79] greatly exceed the electron acceleration efficiencies seen in

current numerical simulations [251, 252].

Due to global asymmetries, or inhomogeneities in the shocked gas caused

by radiative instabilities [79], the shocks may not propagate perpendicular

to the magnetic field everywhere, allowing hadronic acceleration to operate

across a fraction of the shock surface. This possibility motivates considering

117

alternative, hadronic sources for the radio-emitting leptons.

3.3.2 Hadronic Emission

Relativistic protons accelerated near the shock collide with thermal protons

in the upstream or downstream regions, producing neutral (π0) and charged

(π−, π+) pions. The former decay directly into γ−rays, while π± decay

into neutrinos and muons, which in turn decay into relativistic e± pairs and

neutrinos. In addition to electrons accelerated directly at the shock, these

secondary pairs may contribute to the observed radio emission.

Figure 3-9 shows the e± spectrum from pion decay, calculated for a flat

input proton energy spectrum (p = 2). Radio emission near the time of

peak flux is produced by electrons of energy γpkmec2 ≈ 100 MeV, where

γpk ≈ 200 (eq. [3.26]). By coincidence, this is close to the pion rest energy

mπc2 ≈ 140 MeV and hence to the peak of the e± distribution produce by

pion decay. Thus, secondary pairs can contribute significantly to the radio

emission if they are deposited in regions where their radiation is observable.

Although pion-producing collisions occur both ahead of the shock and in

the post-shock cooling layer, pairs produced in the cold central shell cannot

contribute to the radio emission due to the high free-free optical depth in this

region (see Fig. 3-7). Only e± pairs produced in the first cooling length be-

hind the shock contribute to the observable radio emission, but these contain

only a small fraction of the shock power,

εe ∼ εptcool

tπfπ = 4× 10−4 (εp/0.1)(fπ/0.1)v3.4

8 , (3.27)

where fπ is the fraction of proton energy per inelastic collision placed into

pions, tπ = (n3σπc)−1 is the timescale for pion production, and σπ ≈ 4×10−26

cm2 is the characteristic inelastic p-p cross section [253].

118

Pairs are also produced by p-p collisions upstream of the shock, which

then radiate after being advected into the magnetized downstream. However,

proton acceleration via Diffusive Shock acceleration (DSA, see Introduction)

is confined to the narrow photo-ionized layer ahead of the shock [94], the

narrow radial thickness of which, ∆ion (eq. [3.5]), limits the radial extent of

the pion production region ahead of the shock.4

3.4 Radio Emission Model

3.4.1 Pressure Evolution of the Post-Shock Gas

Gas compresses behind the shock due to radiative cooling at approximately

constant pressure (neglecting the effect of thermal instabilities; [254]). The

total pressure includes both the thermal gas and relativistic ions (‘cosmic

rays’),

Ptot = Pth + PCR = Pth,0

(n

n3

)(T

T3

)+ PCR,0

(n

n3

)4/3

, (3.28)

The energy fraction which goes into relativistic ions is twice the pressure

ratio between relativistic ions and total pressure εp = 2PCR,0/(Pth,0 +PCR,0).

Our numerical calculations use εp = 0.2, although our results are not sensi-

tive to this assumption because the observed emission is dominated by that

originating within the first cooling length behind the shock, i.e. before com-

pression becomes important.

4Very high energy protons of energy ∼ Emax ∼> 10 GeV leak out of the DSA cycle, es-caping into the neutral upstream with a nearly mono-energetic distribution with a compa-rable energy flux to the power-law spectrum ultimately advected downstream (e.g. [113]).However, the protons leaving upstream inject pairs at multi-GeV energies γpkmec

2

(eq. [3.26]) which radiate most of their synchrotron emission at higher frequencies thanthat responsible for the bulk of the radio emission.

119

3.4.2 Electron and Positron Spectra

The electron and e± energy spectra, N(E), in both leptonic and hadronic

scenarios are normalized such that a fraction εe of shock power is placed into

relativistic electrons. In the leptonic scenario, we assume that

N(E)= CEE−p, Emin < E < Emax (3.29)

where E = γmec2, Emin = 2mec

2 ≈ 106 eV and Emax = 1012 eV [94]. The

normalization constant is given by

CE =9mpn4v

2fεe

8

(n

n3

) p+23

×

(p− 2)Ep−2

min

1− (Emax/Emin)−p+2, p 6= 2

[ln(Emax/Emin)]−1, p = 2,

(3.30)

where the term ∝ (n/n3)(p+2)/3 accounts for adiabatic heating as gas com-

presses downstream. As shown in Appendix , most sources of cooling of

relativistic electrons in the post shock thermal cooling layer are negligible,

with the possible exception of Coulomb losses, which we include in a few

select cases as described below.

In the hadronic scenario, we model the spectrum of e± pairs from pion

decay following [253], assuming an input power-law spectrum of relativistic

protons Np(E) ∝ E−p with the same energy distribution in the leptonic case.

3.4.3 Radiative Transfer Equation

The radio emission downstream of the shock is governed by the radiative

transfer equation, which can be conveniently written in terms of the bright-

ness temperature Tν [246],

dTνdτν

= T − Tν + Tν,syn, (3.31)

120

where T is the gas temperature,

Tν,syn =jν,sync

2

2αffkν2. (3.32)

is the synchrotron brightness temperature, and τν =∫αffdz is the free-free

optical depth. Here z is the depth behind the shock and αff is the free-free

absorption coefficient (eq. [3.10]), where the relevant pre-factor in is now

α0 = 0.28 cm5 K3/2 s−2 for the higher temperatures appropriate to the post-

shock gas. We neglect synchrotron self-absorption, which is only relevant for

relativistic brightness temperatures of Tν ∼> 1011 K, much higher than those

observed in novae.

It is useful to change variables from free-free optical depth τν to temper-

ature, according to [77]

dτν = αffdT

dT/dz=

5kα0n4vf2ν2T 3/2Λ(T )

dT (3.33)

using the temperature gradient dT/dz = 2n2Λ(T )/(5kn4vf ) behind the shock

set by cooling. As both the free-free absorption coefficient and the cooling

rate scale as ∝ n2, this allows for a one-to-one (up to a coefficient) mapping

τν(T ) given Λ(T ). This mapping is notably independent of the details of how

the gas compresses behind the shock.

Figure 3-7 shows the evolution of the optical depth τν/τν,0 as a function

of temperature T behind the shock, where τν,0 = n4,7v8/ν210 and τν = 0 is

defined at the forward shock surface (T = T3; eq. [3.4]). If only free-free

cooling is included, then the radio photosphere (τν = 1) can occur at higher

temperatures of ∼> 105 K, depending on τ0,ν . However, if full line cooling is

included [247], then the photosphere temperature is much lower, ≈ 104 K.

The observed brightness temperatures ∼> 105 K of nova shocks (Fig. 3-4) can

therefore only be thermal in origin if line cooling is highly suppressed from

121

0.0001

0.001

0.01

0.1

1

10

4 4.5 5 5.5 6 6.5 7 7.5

Free-free cooling

Full line cooling

o i/o i

,0

Log10(T/K)

Figure 3-7: Free-free optical depth τν as a function of gas temperatureT behind the shock. The optical depth is normalized to a fiducial valueτ0,ν ≡ (n4/107cm−3)v8/ν

210. Different lines correspond to cases calculated for

a cooling function including full line cooling (purple) and one including justfree-free cooling (green).

its standard CIE value [77]. When line cooling is present, additional non-

thermal synchrotron emission above the photosphere is needed to reproduce

the observations.

The synchrotron emissivity is given by an integral over the electron dis-

tribution function,

jν,syn =1

4π

∫ Emax

Emin

P (ν, E)N(E)dE, (3.34)

where P (ν, E) is the emissivity for a single electron. For analytical estimates

122

10-23

10-22

10-21

10-20

10-19

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

~ T-0.7

R, e

rg c

m3 s

-1

Log10(T/K)

Figure 3-8: Cooling function Λ(T ) from [247], adopted for our fiducial ejectacomposition enriched in CNO elements. A green line shows the power-lawΛ ∝ T−0.7 (eq. [3.17]).

123

it is convenient to use the so-called delta-function approximation,

P (ν, E) ≈ 4

3cσT UBγ

2δ(ν − γ2νB

), (3.35)

where νB = eB/(2πmec). This yields reasonably accurate results for broad

and smooth electron distributions. Nevertheless, our numerical calculations

use the exact expression for synchrotron emissivity. For a power-law distri-

bution of electrons,

jν,syn =1

9CEhαfνB

(ν

νB

)− p−12

, (3.36)

where αf is the fine structure constant and CE = CE(mec2)−p+1 is a rescaled

normalization constant of the electron distribution.

We assume that the magnetic field does not decay as the plasma cools

and compresses downstream of the shock, its strength increasing as B =

Bsh(n/n3)Γ, where Γ = 2/3 for the flux freezing of a tangled (statistically

isotropic) field geometry. Our results are not sensitive to this choice because,

for any physical value of Γ, the integrated emission is dominated by the first

cooling length behind the shock (§3.5.1). Using CE ∝ n(p+2)/3 appropriate

for adiabatic compression, the emissivity scales as jν,syn ∝ n(2p+3)/3. Figure

3-9 compares the frequency dependence of the emissivity for electrons from

direct shock acceleration and pion decay.

124

0.01

0.1

1

101 102 103

E dN

/dE,

cm

-3

E, MeV

p=2.0 electron power-lawpion decay from p=2.0 protons

10-27

10-26

10-25

10-24

10-23

10-1 100 101 102 103

j i, e

rg c

m-3

s-1

str-1

Hz-1

i, GHz

p=2.0 power-lawpion decay from p=2.0 protons

Thermal

Figure 3-9: Left Panel: Energy spectrum of relativistic leptons, calculated fordirect shock acceleration of electrons (electron index p = 2) and for secondarypairs from pion decay (proton index p = 2). Right Panel: Synchrotronemissivity as a function of frequency for the particle spectra in the left panel.The thermal free-free emissivity is shown for comparison. Calculations wereperformed adopting the best-fit shock parameters for V1324 Sco (Table 3.4)of v8 = 0.63, εe = 0.08, fEUV = 0.05 at the 10 GHz maximum (t = 71.5 days,n4 = 4.5 · 106 g cm−3).

3.5 Results

3.5.1 Analytical estimates

The formal solution of equation (3.31) for the brightness temperature at the

forward shock surface (T = T3) is given by

Tν =

∫ 0

∞(Tν,syn + T )e−τνdτν , (3.37)

where, using equations (3.10), , (3.30), (3.32), and (3.36),

Tν,syn = 2× 109 n− 3−p

44,7 ν

1−p2

10 v11+p

28 ε

p+14

B,−2εe,−2

(T

T3

)3/2(n

n3

) 2p−33

K, (3.38)

and the prefactor varies between 1.4 − 2 × 109 for p = 2 − 2.5 due to the

complex dependence of CE (eq. [3.30]).

125

Radiative Shocks

For low velocity, radiative shocks (v8 ∼< 1) line cooling dominates free-free

cooling behind the shock, such that the radio photosphere τν = 1 is reached

at gas temperatures T ∼< 104 K (Fig. 3-7). Because the latter is significantly

lower than the observed brightness temperatures (Fig. 3-4), non-thermal ra-

diation with Tν,syn T must dominate the emission. This justifies neglecting

T compared to Tν,syn in equation (3.37). As justified below, a further simpli-

fication is allowed because∫Tν,syndτν is dominated by the first cooling length

behind the shock, where T ∼> 106 K and τν 1, allowing us to approximate

e−τν ≈ 1 in equation (3.37). Setting dT = T in equation (3.33), the optical

depth accumulated over a cooling length centered around temperature T is

given by

∆τν =5α0kn4vf

2ν2T 1/2Λ(T )= 2.3× 10−3 n4,7v

−2δ8 ν−2

10 Λ−1c,−22

(T

T3

)−δ− 12

, (3.39)

where we have used the power-law approximation Λ ∝ T δ for the line cool-

ing function in equation (3.17) assuming v8 ∼< 1. From equations (3.38)

and (3.39), the brightness temperature accumulated over a cooling length

centered around temperature T is given by

T nthν ≈ Tν,syn∆τν ≈

≈ 5× 106 n1+p

44,7 ν

− p+32

10 v11+p−4δ

28 ε

p+14

B,−2εe,−2Λ−1c,−22

(T

T3

)1−δ (n

n3

) 2p−33

K

∝n/n3=(T/T3)−1

(T

T3

) 6−2p−3δ3

. (3.40)

Because T nthν is an increasing function of temperature for p < (6−3δ)/2 ≈ 4,

this illustrates that most of the flux is accumulated over the first cooling

length behind the shock.

126

The thermal contribution to the brightness temperature at post-shock

temperature T is likewise approximately

T thν ≈ ∆τνT = 3.5× 104 n4,7v

2−2δ8 ν−2

10 Λ−1c,−22

(T

T3

) 12−δ

K. (3.41)

If free-free emission dominates the cooling (δ = 0.5), then the emission re-

ceives equal contributions from each decade in temperature behind the shock

[77]. For line cooling (δ = −0.7) the first cooling length again provides the

dominant contribution.

Figure 3-10 shows how the 10 GHz brightness temperature grows as a

function of temperature behind the shock for different assumptions about the

cooling and emission of the post-shock gas. At high optical depths, the radi-

ation and gas possess the same temperature (Tν = T ) because synchrotron

emission is not present in the cold central shell. However, differences in Tν

become pronounced as the shock front is approached at T = T3. When line

cooling is included, only the first cooling length behind the shock significantly

contributes to the final brightness temperature at the shock surface.

Coulomb losses of relativistic electrons only noticeably impact the non-

thermal emission well downstream of the shock, in regions where the gas

temperature is ∼< 105 K (eq. [A.2]). Coulomb losses therefore significantly im-

pact the observed radiation temperature only if free-free cooling dominates,

for which contributions to the emission from the post-shock gas Tν ∝ T 1/6

(eq. [3.40]) vary only weakly with gas temperature. Even in this case, how-

ever, the observed temperature reduced only by a factor of ∼ 2 compared to

an otherwise identical case neglecting Coulomb losses. Coulomb corrections

are completely negligible in our fiducial models when line cooling is included

because the bulk of the emission comes from immediately behind the shock,

where the temperature is high and the density is low.

127

104

105

106

107

108

104 105 106 107

Free-free cooling

Free-free cooling, Coulomb

Free-free cooling, thermal

Line cooling

Line cooling, thermal

T b=T

Col

d ce

ntra

l she

ll, T

shel

l

Shoc

k fro

nt, T

3

T i, K

T, K

Figure 3-10: 10 GHz brightness temperature in the post-shock region as afunction of gas temperature T < T3 behind the shock (T = T3). Differentlines correspond to different assumptions about (1) whether the emissionaccounts only for thermal free-free emission [dashed lines] or also includesnon-thermal synchrotron emission [dash-dotted lines], and (2) whether the as-sumed cooling function is just free-free emission [blue, purple] or also includesemission lines [red]. A purple line shows the non-thermal case with free-freecooling, including the effect of Coulomb losses on the emitting relativistic lep-tons. All calculations were performed for a pre-shock density of n4 = 4.5 ·106

g cm−3 corresponding to the 10 GHz peak time, adopting the best-fit shockparameters for V1324 Sco (Table 3.4) of v8 = 0.63, εe = 0.08, fEUV = 0.05.

128

Thus far, we have focused on the emission temperature at the forward

shock (the ‘unscreened’ temperature, Tν |shock in eq. [3.9]), and we have not

yet accounted for free-free attenuation by the ionized layer ahead of the

shock. The observed brightness temperature Tb only equals that of the shock,

Tν |shock, after the upstream density has become sufficiently low (n4 ∼< npk;

eq. [3.9]). The measured peak brightness temperature, Tν,pk, is thus obtained

by substituting n4 = npk (eq. [3.12]) into equations (3.40) and (3.41) for

T = T3, n = n3. Multiplying the resulting expression by a factor of 1/e

(since at the radio peak τν = 1) yields

T thν,pk ≈ max

1800 v−2δ−18 f−1

EUV,−1Λ−1c,−22 K

6900 v2−2δ8 ν−1

10 H−1/214 Λ−1

c,−22 K(3.42)

in the thermal case and

T nthν,pk ≈ max

3.9× 105 f− p+1

4EUV,−1ν

−110 v

19−p−8δ4

8 εp+1

4B,−2εe,−2Λ−1

c,−22 K

1.0× 106 ν− p+5

410 v

11+p−4δ2

8 εp+1

4B,−2εe,−2Λ−1

c,−22H− p+1

814 K

(3.43)

in the non-thermal case.

For p = 2, and for our fiducial power-law fit to the line cooling function

(Λc,−22 = 1.5, δ = −0.7), equation (3.43) becomes

T nthν,pk ≈ max

2.6× 105 f−3/4EUV,−1ν

−110 v

5.78 ε

3/4B,−2εe,−2 K,

6.9× 105 ν−7/410 v7.9

8 ε3/4B,−2εe,−2H

−3/814 K.

(3.44)

Figures 3-11 and 3-12 compare our analytic expressions for the peak observed

brightness temperature (eqs. [3.42, 3.43]) and the peak value resulting from

a direct integration of equation (3.37).

The top panel of Figure 3-11 shows how T nthν,pk depends on the density scale-

heightH and the ionized fraction of the preshock layer, fEUV. When the value

129

of H is small, or fEUV is sufficiently high, then the entire DES scaleheight is

ionized (H = ∆ion) and the peak brightness temperature is independent of

fEUV. On the other hand, when ∆ion < H, then the brightness temperature is

independent of H but decreases with increasing ionizing radiation as T nthν,pk ∝

f−(p+1)/4EUV =

p=2f−0.75

EUV (eq. [3.43]). The bottom panel of Figure 3-11 shows how

the peak temperature depends on the shock velocity, vf , and the electron

acceleration efficiency, εe.

Figure 3-12 compares the brightness temperature calculated in a purely

thermal model (εe = 0) to our analytic estimate of T thν,pk (eq. [3.42]). For

a broad range of parameters, the peak temperatures fails to exceed 105 K,

making thermal models of radiative shocks challenging to reconcile with ob-

servations (see Fig.3-4).

Adiabatic shocks

For higher velocity v8 ∼> 108 cm s−1, adiabatic shocks, the brightness temper-

ature is again estimated by assuming that the first scaleheight H behind the

shock dominates the emission, i.e. neglecting ongoing emission from mat-

ter shocked many dynamical times earlier. However, unlike with radiative

shocks, this assumption cannot be rigorously justified without a radiation

hydrodynamical simulation, an undertaking beyond the scope of this work.

The peak brightness temperature in the adiabatic case is estimated using

the expressions for radiative shocks from §3.5.1, but replacing the first cooling

length behind the shock tcoolvf with H. For purely thermal emission, the

brightness temperature is

T thν,H = 1.1× 105n24,7H14ν

−210 v

−18 K (3.45)

Substituting n4 = npk into equation (3.45) gives a peak thermal temperature

130

of

T thν,H,pk = 1.2× 104v−18 K, (3.46)

where we have assumed that the scaleheight is fully ionized (npk = nH,pk),

which is generally well satisfied for high velocity shocks.

For adiabatic shocks, the peak thermal brightness temperature is a de-

creasing function of the shock velocity due to the αff ∝ T−3/2 dependence

of free-free absorption. Comparing equations (3.44) and (3.46), a maximum

thermal brightness temperature of ≈ 104 K is thus obtained for the inter-

mediate shock velocity v8 ∼ 1 separating radiative from adiabatic shocks.

The fact that this falls well short of observed peak brightness temperatures

(Fig. 3-4) again disfavors thermal models for the early radio peak..

For non-thermal emission from adiabatic shocks, the brightness temper-

ature and its peak value are given, respectively, by

T nthν,H = 1.2× 107 n

5+p4

4,7 ν− p+3

210 v

p+52

8 εp+1

4B,−2εe,−2H14 K (3.47)

T nthν,H,pk = 1.4× 106 ν

− p+14

10 v5+p

28 ε

p+14

B,−2εe,−2H3−p

814 . (3.48)

3.5.2 Radio lightcurves and spectra

The shape of the radio light curve and spectrum are driven by the effects

of free-free opacity, which are independent of the emission mechanism and

hence apply both to thermal [77] and non-thermal models. Figure 3-13 shows

an example model light curve (left panel) and corresponding evolution of the

spectral index across two representative frequency ranges (right panel). At

high frequencies the light curve peaks earlier and reaches a larger maximum

flux, due to the frequency dependence of free-free absorption, αff ∝ ν−2,

131

105

106

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

H14=0.1

H14=0.5

H14=2.2

H14=10

T i, p

k, K

fEUV

106

107

108

109

1010

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

T i, p

k, K

Shock velocity, 108 cm s-1

¡E=10-4

¡E=10-3

¡E=10-2

¡E=0.1

Figure 3-11: Peak brightness temperature of non-thermal emission as func-tion of shock parameters. The top panel shows T nth

ν,pk as a function of thefraction of shock energy used for ionization fEUV and density scaleheight H.The bottom panel shows T nth

ν,pk as a function of the shock velocity, v8, andfraction of the shock energy placed into relativistic electrons, εe. Results ofthe full calculation are shown as symbols, while the analytic estimates fromequation (3.40) are shown as lines. The values of parameters not varied inthese figures are taken as best-fit values for V1324 Sco (Table 3.4).

132

1.5

2

2.5

3

3.5

4

4.5

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

T i, p

k, 10

4 K

Shock velocity, 108 cm s-1

H14=0.1H14=0.3H14=0.8H14=2.2

Figure 3-12: Same as Figure 3-11 but showing the observed thermal bright-ness temperature as a function of the shock velocity v8 and scale-height H14.Results of our full calculation are shown as symbols, while the analytic esti-mates from equation (3.41) are shown as lines.

133

0.01

0.1

1

10

30 40 50 60 70 80 90 100 110 120

F i, m

Jy

Time from optical outburst, days

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

30 40 50 60 70 80 90 100 110 120

Spec

tral i

ndex

`

Time from optical outburst, days

Max of 3 GHz

Max of 10 GHz

Max of 30 GHz

Figure 3-13: Left: Example model radio lightcurves at radio frequencies of3 GHz (black), 10 GHz (blue), and 30 GHz (red), calculated for our best-fit parameters for V1324 Sco (Table 3.4). The lightcurve does not includethermal emission from the cold central shell or other sources of photo-ionizedejecta. Right: Measured spectral index β between the 3 and 10 GHz bands(red) and the 10 and 30 GHz bands (blue). The time of light curve maximaare marked with symbols for 3 GHz (circle), 10 GHz (square), and 30 GHz(triangle). The plateau at β = 2 at early times corresponds to when thephoto-ionized layer ahead of the shock is still optically thick.

and the monotonically declining shock power. When such behaviour is not

observed, this indicates that our one zone model is inadequate or that free-

free opacity does not determine the light curve maximum (§3.6.3).

The spectral index β initially rises to exceed that of optically-thick isother-

mal gas (β > 2) because the higher frequency emission peaks first. Then,

once the lower frequency emission peaks, the shock becomes optically thin

and hence the spectral index approaches the spectral index of optically thin

synchrotron emission. Importantly, a flat spectral index does not itself pro-

vide conclusive evidence for non-thermal synchrotron emission, even though

the optically thin spectral indices are different for synchrotron and thermal

bremsstrahlung emission. The emission mechanism is instead more accu-

rately distinguished from thermal emission based on the higher peak bright-

ness temperatures which can be achieved by non-thermal models. Also, at

late times the radio emission should eventually come to be dominated by

134

0.1

1

10

20 40 60 80 100 120 140

F i, m

Jy

Time from optical outburst, days

36.5 GHz27.5 GHz17.5 GHz13.3 GHz

7.4 GHz4.6 GHz

0.001

0.01

0.1

1

10

20 40 60 80 100 120 140

F i, m

Jy

Time from optical outburst, days

36.5 GHz28.2 GHz25.6 GHz20.1 GHz

8.7 GHz6.8 GHz5.2 GHz

0.1

1

10

20 40 60 80 100 120 140 160

F i, m

Jy

Time from optical outburst, days

32.1 GHz14.0 GHz

7.8 GHz4.8 GHz1.9 GHz1.3 GHz

Figure 3-14: Synchrotron shock models fit to V1324 Sco (top left panel),V1723 Aql (right panel) and V5589 Sgr (bottom left panel). Parameters ofthe fits are provided in Table 3.4. In all cases the late-time thermal emissionhas been subtracted prior to the fit. Uncertainties are cited in the caption ofFig. 3-3.

thermal emission of the photo-ionized ejecta and hence the spectral index

will again rise to β = 2. This feature is not captured by Figure 3-13 because

we have not included thermal emission from the cool central shell or other

sources of photo-ionized ejecta.

3.5.3 Fits to Individual Novae

We fit our model to the radio light curves of three novae with early-time cov-

erage, V1324 Sco, V1723 Aql and V5589 Sgr (Figs. 3-3, 3-4). We employ a χ-

squared minimization technique across 9 free parameters: v1, v4, n4/n1,∆t,H, n0, p, εe, fEUV .

We assume a shock covering fraction of fΩ = 1 and take εB = 0.01, as the

135

Table 3.4: Best-fit parameters of synchrotron model.

Nova v(a)1 v

(b)4 n4/n

(c)1 v

(d)f

∆t(e) H(f) n0(g) εe

(h) fEUV(i)

(km s−1) (km s−1) - (km s−1) (days) (cm) (cm−3) - -

V1324 Sco 1700 670 0.43 630 12 3.7× 1013 7.8× 109 0.081 0.051

V1723 Aql 2300 770 0.66 830 3.2 9.5× 1013 6.7× 107 0.015 0.31

V5589 Sgr 2200 190 2.3 810 5 6.8× 1013 4.1× 108 0.014 0.07

(a)velocity of fast outflow, (b)velocity of slow outflow (DES), (c)ratio of densities of DES and fast outflow, (d)velocity of

the shock in the frame of the upstream gas, calculated from v1, v4, n4/n1 using equation (3.3), (e)time delay between

launching fast and slow outflows, (f)scale-height of slow outflow, (g)normalization of density profile of slow outflow

(eq. [3.7]), (h)fraction of shock power placed into power-law relativistic electrons/positrons, (i)fraction of shock powerplaced into hydrogen-ionizing radiation (eq. [3.5])

latter is degenerate with εe. Table 3.4 provides the best-fit parameters of

each novae. Although the electron power-law index is a free parameter, in

practice it always converges to a best-fit value of p ' 2.

Our best fit to V1324 Sco is shown in the upper panel of Figure 3-14.

The best-fit shock velocity of 640 km s−1 is within the range of radiative

shocks, consistent with the rapid post-maximum decline. For the parameters

of our best-fit model, npk,∆/npk,H = 0.96ν10, implying that for frequencies

above(below) 10 GHz the thickness of ionization layer is less than(greater

than) the scale height at the time of peak flux. The transition between these

two regimes can be seen as a small break in the 7.4 GHz light curve around

day 68. Substituting our best-fit parameters into equation (3.1), we find that

the collision occurred 20 days after the start of the outflow, corresponding to

7 days after the onset of the gamma-ray emission [90]. This discrepancy is

not necessarily worrisome, as the colliding ‘shells’ may be ejected over days

or longer.

V1723 Aql was more challenging to fit (Fig.3-14, middle panel), mainly

because different frequencies peak at nearly the same time. This is contrary

to the the expectation that high frequencies will peak first if free-free absorp-

tion indeed controls the light curve rise, as our model assumes (see §3.6.3 for

alternative interpretations). Although we cannot fit V1723 Aql in detail, we

can nevertheless reproduce the magnitudes of the peak fluxes for reasonable

136

parameters. Our best-fit model for V5589 Sgr is shown in the bottom panel of

Figure 3-14, although the data is more sparse than for the other two events.

3.6 Discussion

3.6.1 Efficiency of Relativistic Particle Acceleration

The non-thermal radio emission from novae probes relativistic particle ac-

celeration and magnetic field amplification at non-relativistic shocks, as pa-

rameterized through εe and εB. Considering all three novae, we infer electron

acceleration efficiencies in the range εe ∼ 0.01 − 0.08 for εB = 0.01 (Table

3.4). However, the peak radio luminosity is largely degenerate in εe, εB and

vf . By combining the peak 10 GHz luminosity of V1324 Sco, V1723 Aql,

and V5589 Sgr with analytic estimates of the peak brightness temperature

(eq. [3.43]) and assuming a shock radius equal to that of our best fit model

(Rsh = v4tpeak; Table 3.4), we obtain the following limits,

fΩv7.98 εeε

3/4B ≈ 7× 10−5 − 1.2× 10−4, (3.49)

where we have again assumed radiative shocks (v8 ∼< 1).

Now consider some implications of these constraints. Making the very

conservative assumption that εe + εB ∼< 1, i.e. εeε3/4B ∼< 0.5, we find

v8 ∼> 0.4f−0.13Ω ⇒ kT ∼> 0.2f−0.25

Ω keV. (3.50)

This lower limit on the shock temperature ensures, for example, that thermal

free-free emission from the shocks will fall within the spectral window of Swift

and Chandra. The radio emitting electrons should produce a measurable X-

ray signature, even if its too weak to detect or is overpowered by faster

137

adiabatic shocks.

Next, using our best fit parameters for the shock velocities v8 of each nova

(Table 3.4) and assuming an error on this quantity of 10 per cent, each event

separately provides a constraint

1.2× 10−3 ≤ fΩεeε3/4B ≤ 5.4× 10−3 V1324 Sco (3.51)

2.3× 10−4 ≤ fΩεeε3/4B ≤ 10−3 V1723 Aql (3.52)

2× 10−4 ≤ fΩεeε3/4B ≤ 9× 10−4 V5589 Sgr (3.53)

For physical values of εB < 0.1 [114], we thus require εe ∼> 10−3 − 10−2.

Such high acceleration efficiencies disfavor hadronic scenarios for the

radio-emitting leptons, which are estimated to produce εe ∼< 10−4 (eq. [3.27]).

They are also in tension with PIC simulations of particle acceleration at

non-relativistic shocks [251, 252] which find εe ∼ 10−4 when extrapolated to

shock velocities v8 ∼< 103 km s−1, modeling of supernova remnants ([116];

εe ∼ 10−4), and galactic cosmic ray emission ([255]; εe ∼ 10−3). On the

other hand, the inferred cosmic ray efficiencies are dependent on the shock

fraction of the accelerated electrons which escape the supernova remnant.

Modeling of unresolved younger radio supernovae typically find higher val-

ues of εe [256, 257], consistent with our results. Given these observational

and theoretical uncertainties, we tentatively favor a leptonic source for the

radio-emitting electrons.

In the above, we have assumed that the shocks cover a large fraction of

the outflow surface (fΩ ∼ 1). However, if instead we have fΩ 1, then the

required values of εe and εB would be even higher than their already strained

values. If the radio-emitting shocks are radiative, they must therefore possess

a large covering fraction fΩ ∼ 1.

Lower values of εe and εB are allowed if the radio emission is instead

138

dominated by high-velocity v8 ∼> 1, adiabatic shocks covering a large solid

fraction fΩ ∼ 1. The sensitive dependence of the brightness temperature on

the shock velocity implies that even a moderate increase in vf can increase

the radio flux by orders of magnitude for fixed εe, εB. On the other hand,

the adiabatic shocks responsible for the hard X-rays appear to require a

small covering fraction fΩ 1 so as not to overproduce the observed X-ray

luminosities (§3.2.4).

In V1324 Sco, [79] place a lower limit of εnth = εp+ εe > 10−2 on the total

fraction of the shock energy placed into non-thermal particles. Assuming that

the microphysical parameters of the radio and gamma-ray emitting shocks

are identical, and that the gamma-rays are leptonic in origin (εnth = εe ∼0.01− 0.1), then by combining this constraint with our constraints on V1324

Sco from equation (3.51), we find

3× 10−3 ≤ εB ≤ 2× 10−2 V1324 Sco, (3.54)

providing evidence for magnetic field amplification.

3.6.2 The DES is not a MS progenitor wind

Gamma-rays are observed not only in novae with red giant companions (sym-

biotic novae), but also in systems with main-sequence companions (classical

novae). Could the DES required for shock-powered radio and gamma-ray

emission be the companion stellar wind? In this section we estimate the radio

emission from the fast nova ejecta interacting with the (assumed stationary)

stellar wind of the binary companion. The density profile of a spherically

symmetric, steady-state wind is given by

nw 'M

4πr2vwmp

= 300M−10

r214vw,8

cm−3, (3.55)

139

where r = r141014 cm is radius, vw = vw,8108 cm s−1 is the wind speed, and

M = M−1010−10M yr−1 is the mass-loss rate normalized to one specific for

main-sequence stars.

The density and radius of the radio photosphere are given by equating

(3.55) with npk,H (eq. [3.12]) for a density scaleheight H ∼ r,

nw,pk = 1.4× 108ν4/310 v

1/3w,8M

−1/3−10 (3.56)

rw,pk = 1.5× 1011ν−102/3v−2/3w,8 M

2/3−10 (3.57)

Although nw,pk is several orders larger than in our fiducial DES models, the

radius of the photosphere is 3 orders of magnitude smaller. Under these

conditions the shock is adiabatic and hence the peak brightness temperature

is found by substituting nw,pk and rw,pk into equation (3.47), which gives (for

p = 2)

Tw,pk = 1.7× 106M1/12−10 v

3.58 ε

3/4B,−2εe,−2ν

−5/610 v

−1/12w,8 K (3.58)

Although the value of Tw,pk ∼ 106 K is comparable to those of observed radio

emission, the much smaller radius of the photosphere rw,pk compared to that

in our fiducial DES models of ∼ 1014 − 1015 cm would result in a peak radio

flux ∝ r2w,pkTw,pk which is approximately 8 orders of magnitude smaller than

the observed values.

Radio emission from the interaction of the nova ejecta with the main

sequence progenitor wind is thus undetectable for physical values of M , unless

the mass loss rate is comparable to that of the nova eruption itself.

140

3.6.3 Light Curve of V1723 Aquila

Our model does not provide a satisfactory fit to the light curve of V1723

Aql because all radio frequencies peak nearly simultaneously (Fig. 3-14),

contrary to the expectation if the light curve peaks due to the decreasing free-

free optical depth (§3.5.3). Here we describe modifications to the standard

picture that could potentially account for this behavior.

Steep density gradient

The V1723 Aql light curves are consistent with all frequencies peaking within

a time interval ∆tpk ∼< few days. From equation (3.13) this requires a scale

height of thickness

H ∼< ∆tpkvf = 2.6× 1013v8

(∆tpk

3 day

)cm, (3.59)

which is much smaller than the shock radius of Rsh ∼ 1014 − 1015 cm. Such

a steep gradient should also result in a short rise and fall time, which is

inconsistent with the relatively broad peak observed in V1723 Aql. This

contradiction could be alleviated if the shock reaches the critical density of

peak emission n = npk at different times across different parts of the ejecta

surface. In this case, the total light curve, comprised of the emission from all

locations, would be “smeared out” in time while still maintaining the nearly

frequency-independent peak time set by the steep gradient.

Figure 3-15 shows the light curves of our single-zone model (for an as-

sumed scale height of H = 5× 1012 cm) convolved with a Gaussian profile of

different widths σ, in order to crudely mimic the effects of non-simultaneous

shock emergence. The top panel shows the unaltered radio lightcurve, i.e.

assuming simultaneous shock emergence, which again makes clear that the

high frequencies peak first (§3.2.2). The middle and bottom panels show the

141

0.01

0.1

1

15 20 25 30 35 40

F i, m

Jy

Time from optical outburst, days

3 GHz10 GHz30 GHz

0.01

0.1

1

15 20 25 30 35 40

F i, m

Jy

Time from optical outburst, days

3.0 GHz10.0 GHz30.0 GHz

0.01

0.1

1

15 20 25 30 35 40

F i, m

Jy

Time from optical outburst, days

3 GHz10 GHz30 GHz

Figure 3-15: Example light curves (with late-time thermal emission sub-tracted) for a steep density gradient H = 5 × 1012 cm, which has beenconvolved with a Gaussian of different widths σ = 0 (top left panel), σ =2 days (right panel), and σ = 5 days (bottom left panel). This illustrateshow a shock propagating down a steep density profile, but reaching the DESsurface at different times at different locations across the ejecta surface, canproduce the appearance of an achromatic light curve.

same light curve, but smeared over time intervals of σ = 2 days and σ = 5

days, respectively. In this way, the difference between the times of maximum

at different frequencies can be substantially smaller than the rise or fall time,

as observed in V1723 Aql.

Collision With Optically Thin Shell

Another possibility is that the radio-producing shock occurs in a dilute thin

shell, which has a lower density than the mean density of the DES, i.e.

n0 npk ∼ 107 cm−1. Absorption then plays no part in the lightcurve rise,

142

which is instead controlled by the time required for the shock to cross the

radial width of the shell.

The free-free optical depth of a shell of radial thickness D = 1014D14 cm

and temperature Tsh = 2 · 104 K is given by

τsh = 3.5n24,7D14ν

−210 , (3.60)

The density and radiative efficiency of the shell can be written as

n4 = 1.7× 106τ1/2sh,−1D

−1/214 ν10 cm−3 (3.61)

η ≡ tcool

tsh= 3.1v4.4

8 τ−1/2sh,−1D

−1/214 ν−1

10 , (3.62)

where τsh = 0.1τsh,−1 is normalized to a sufficiently low value so as not to

influence the radio lightcurve significantly.

Although the parameter space for an optically thin shell (τsh < 0.1) which

is radiative (η ∼< 1) is not large, this represents a viable explanation for the

behavior of V1723 Aquila.

3.6.4 Radio Emission from Polar Adiabatic Shocks

Figure 3-6 shows the maximum X-ray luminosity and the maximum 10 GHz

radio luminosity as a function of the shock velocity, spanning the range from

radiative (v8 ∼< 1) to adiabatic (v8 ∼> 1) shocks. Both radio and X-ray lumi-

nosities increase by many orders of magnitude across this range. However,

their ratio LR/LX coincidentally varies only weakly with v8, as shown in Fig-

ure 3-16. Absolute luminosities depend on the covering fraction and radius

of the shock, but this ratio is obviously independent of these uncertain quan-

tities. Reasonable variations in the density scaleheight H, ionization fraction

fEUV, and the CNO abundances result in a factor ∼< 3 variation in this ratio

143

(Fig. 3-16, top panel), which is instead most sensitive to the microphysical

parameters, ∝ εeε3/4B for p = 2 (Fig. 3-16, bottom panel). Any shock produc-

ing X-rays will therefore also produce radio emission of intensity comparable

to the observed values. Fig. 3-16 shows for comparison the measured value of

LR/LX, or limits on its value, using the observed peak X-ray and non-thermal

radio luminosities for five novae.

The LR/LX ratio can deviate from its value shown in Fig. 3-16 if the

density of the shocked shell is sufficiently low that the optical depth of the

DES to radio or X-rays is ∼< 1, i.e. if its central density obeys n nX, npk.

The ratio of the density of maximum X-ray emission nX (eq. [3.22]) to that

of maximum radio emission npk (eq. [3.12]) is given by

nXnpk

≈ 1.2ν−110 H

−1/214 v4

8 (3.63)

For high velocity v8 1 adiabatic shocks we can therefore have npk ∼< n ∼<nX , in which case the X-ray emission is optically thin at peak and hence the

ratio LR/LX will be higher than its value in Figure 3-16. Likewise, when

n ∼< npk, nX (both X-rays and radio are optically thin at peak), we have that

LR/LX ∝ n−1/4 also increases with decreasing n. Thus, the LR/LX ratio in

Fig. 3-16 is a conservative minimum, achieved in the limit of a high density

DES.

The early-time X-rays in some novae are too hard to originate from ra-

diative shocks (Table 3.3, Figure 3-5) and thus could instead originate from

fast adiabatic shocks in the low density polar regions (Fig. 3-1; §3.2.4).

If all the observed X-rays originate from fast polar shocks, and all radio

emission from slower equatorial radiative shocks, then the measured ratio

LR/LX = LR,rad/LX + LR,ad/LX, comprised of contributions from both ra-

diative and adiabatic shocks, should exceed the ratio shown in Figure 3-16.

144

Figure 3-16: Ratio of nonthermal peak 10 GHz radio luminosity (eq. [3.44])to peak thermal X-ray luminosity (eq. [3.20]) as a function of shock velocityvsh. The X-ray luminosity is calculated for the preshock density n4 = nX(eq. [3.22]), the latter calculated using the bound-free cross sections from[248] for our assumed ejecta composition. The top panel shows that the ra-tio of luminosities varies by only a factor of a few for realistic variations in thedensity scale-height H, the ionized fraction fEUV, or if the CNO mass fractionis doubled. The bottom panel shows the more sensitive dependence of theluminosity ratio on the shock microphysical parameters εe and εB, calculatedfor H14 = 1, fEUV = 0.1, and standard CNO abundances. Shown for com-parison are measurements (squares), or limits (triangles) on the luminosityratio of the novae compiled in Table 3.3.

145

For V1324 Sco, V1723 Aql, V5589 Sgr this requires that εeε3/4B ≈ 10−3,

consistent with our best-fit values found earlier. The non-detections of non-

thermal radio emission in V339 Del and V959 Mon also do not contradict

εeε3/4B ≈ 10−3. We conclude that, within the uncertainties, the thermal X-ray

and non-thermal radio emission in novae is consistent with originating from

distinct shocks.

3.7 Conclusions

We have explored non-thermal synchrotron radio emission from radiative

shocks as a model for the early radio peaks in novae by means of a one-

dimensional model. Broadly speaking, we find that the measured brightness

temperatures can be explained for physically reasonable parameters of the

shock velocity and microphysical parameters εe and εB (Table 3.4). The

presence of a detectable early radio peak requires the presence of DES of

density ∼> 106 − 107 cm−3 on a radial scale of ≈ 1014 − 1015 cm from the

white dwarf. The emission from the nova ejecta colliding with a progenitor

wind of mass loss rate M ∼< 10−9M yr−1 is not sufficient to explain the

observed radio peaks (§3.6.2).

The thin photo-ionized layer ahead of the forward shock plays a key role

in our model, as its free-free optical depth determines the time, intensity and

spectral indices near the peak of the emission (see § 3.2.2). One-dimensional

models robustly predict that higher frequency emission peaks at earlier times,

imprinting a distinct evolution on the spectral index (Figure 3-13) which is

independent of whether the emission mechanism is thermal or non-thermal.

V1324 Sco and V5589 Sgr do exhibit this behavior, but in V1723 Aql the light

curves at all frequencies peak at nearly the same time. Possible explanations

include the nova outflow colliding with a shell of much lower density than the

146

mean density of the nova ejecta (§3.6.3). Alternatively, the evolution may

appear achromatic if the shock breaks out of the DES at different times across

different parts of its surface, as would occur if the DES is not spherically

symmetric (Fig. 3-15).

Because the light curve evolution is controlled by free-free opacity effects,

one cannot readily distinguish thermal from non-thermal emission based on

the spectral index evolution alone. Thermal emission models exhibit spec-

tral index behaviour [77] which is very similar to the non-thermal model

described here (Figure 3-13). Non-thermal emission is best distinguished by

the much higher brightness temperatures which can be achieved than for

thermal emission.

The relativistic electrons (or positrons) responsible for the non-thermal

synchrotron emission originate either from direct diffusive acceleration at the

shock (leptonic scenario), or as secondary products from the decays of pions

produced in proton-proton collisions (hadronic scenario). In either case, the

radio-emitting leptons are identical, or tightly related to, the particles which

power the observed γ-ray emission. For instance, if the observed ∼> 100

MeV gamma-rays are produced by relativistic bremsstrahlung emission (as

favored in leptonic scenario by [79]), then the energy of the radiating electrons

(also ∼> 100 MeV) are very close to those which determine the peak of the

radio synchrotron emission at later times (eq. [3.26]). Likewise, gamma-rays

from π0 decay in hadronic scenarios are accompanied by electron/positron

pairs with energies ∼> 100 MeV in the same range, as determined by the

pion rest mass. Once produced, relativistic leptons evolve approximately

adiabatically downstream of the shock for conditions which characterize the

radio maximum. The only possible exception is Coulomb cooling in cases

when free-free emission dominates lines in cooling the post-shock thermal

gas (Fig. 3-10).

147

In hadronic scenarios, only a small fraction of relativistic protons pro-

duce pions in the downstream before being advected into the central cold

shell (from which radio emission is heavily attenuated by free-free absorp-

tion). Relativistic protons therefore provide considerable pressure support in

the post-shock cooling layer, preventing the gas from compressing by more

than an order of magnitude (§3.4.1). The possible role of clumping due to

thermal instability of the radiative shock on this non-thermal pressure sup-

port deserves further attention, as the central shell may provide a shielded

environment which is conducive to dust and molecule formation in novae [84].

The small number of pions produced in the rapidly-cooling post-shock

layer implies that of the ∼< 10% of the total shock power placed into rel-

ativistic protons, only a tiny fraction εe ∼ 10−4 (eq. [3.27]) goes into e±

pairs capable of producing detectable radio emission. For physical values of

εB ∼< 0.1, the critical product εeε3/4B ∼< 10−5 which controls the brightness

temperature is a few orders of magnitude below that required by data of

∼ 10−4 − 10−3 (eq. [3.51-3.53]). We therefore disfavor the hadronic scenario

for radio-producing leptons. This does not rule out a hadronic origin for the

γ-ray emission, but it does suggest that the direct acceleration of electrons

is occurring at the gamma-ray producing shocks.

Within leptonic scenarios, the values of εeε3/4B required by the radio data

are typically higher than those measured from PIC simulations [113] and by

modeling Galactic SN remnants [116]. However, values of εeε3/4B ∼ 10−4 −

10−3 do appear similar to those inferred by modeling young radio supernovae

(e.g. [257]).

We confirm the finding of [77] that when CIE line cooling is included,

thermal emission from radiative shocks cannot explain the early high bright-

ness temperature radio peak. Perhaps the best evidence for non-thermal

radio emission comes from cases like V1324 Sco, where no X-ray emission

148

is detected. An X-ray non-detection implies a low velocity, radiative shock,

which would be especially challenged to produce a significant radio flux in

the early peak without a non-thermal contribution. We can also rule out

thermal emission from an adiabatic shock, at least under the assumption

that only the first scaleheight behind the shock dominates that contributing

to the observed emission (§3.6.4). Future radiation hydrodynamical simu-

lations of adiabatic shocks and their radio emission in the case of a steep

density gradient are needed to determine whether thermal adiabatic shocks

can be completely ruled out.

We have derived analytic expressions for the peak brightness temperature

of non-thermal synchrotron emission (eq. [3.44]). Our model can reproduce

the observed fluxes of V5589 Sgr, V1723 Aql, V1324 Sco and, in the case

of V1324 Sco, details of the radio lightcurve. Non-detections of early radio

peak from V959 Mon and V339 Del place upper limits on εe, εB. The upper

limits for V339 Del and V959 Mon are broadly consistent with the inferred

values of εe, εB values for novae with detected early radio peak.

In summary, radio observations of novae provide a important tool for

studying particle acceleration and magnetic field amplification in shocks

which is complementary to γ-rays observations. They also inform our under-

standing of the structure of the nova ejecta and internal shocks, which may

vary considerably with time and as a function of polar angle relative to the

binary axis.

Acknowledgements

We thank Laura Chomiuk, Tom Finzell, Yuri Levin, Koji Mukai, Michael Ru-

pen, Jeno Sokoloski, and Jennifer Weston for helpful conversations. We thank

Jeno Sokoloski for providing us with the numerical tools to fit the thermal

149

radio emission. ADV and BDM gratefully acknowledge support from NASA

grants NNX15AU77G (Fermi), NNX15AR47G (Swift), and NNX16AB30G

(ATP), and NSF grant AST-1410950, and the Alfred P. Sloan Foundation.

150

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Appendix: Non-Thermal

Electron Cooling in the

Post-Shock Cooling Layer

In calculating the radio synchrotron emission from nova shocks, we assume

that relativistic electrons and e± pairs evolve adiabatically in the post-shock

cooling layer. This assumption is justified here by comparing various sources

of cooling (Coulomb, bremsstrahlung, inverse Compton) to that of the back-

ground thermal plasma, tcool (eq. [3.14]). We again focus on electrons or

positrons with Lorentz factors of γpk ∼ 200 (eq. [3.26]), as these determine

the synchrotron flux near the light curve peak, where n3 = 4n4 = 4npk.

First note that thermal electrons and protons are well coupled behind the

shock by Coulomb collisions and thus share a common temperature. At radio

maximum (n4 = npk,∆) the Coulomb equilibration timescale for the thermal

plasma, te−p = 3.5T3/23 /npk s ([258]), is short compared to the thermal cooling

timescale tcool (eq. [3.14]),

te−p

tcool

≈ 0.03

(T

T3

)−0.2

v−0.48 . (A.1)

This ratio evolves only weakly as gas cools behind the shock.

Relativistic electrons experience Coulomb losses on thermal background

182

electrons on a timescale which is given by

te−etcool

=2γpk

3cσTn3 ln Λ tcool

= 10 v−3.48

( γpk

200

) ( ln Λ

20

)−1

. (A.2)

Coulomb losses can thus be potentially important behind the shock where

n npk.

The cooling rate due to relativistic bremsstrahlung emission from inter-

action with thermal protons and electrons is well approximated by γrb ≈(5/3)cσTαfsn3γ

1.2, an expression which is accurate to ∼ 10 − 20% between

γ ∼ 10− 103, where αfs ' 1/137 is the fine structure constant. This yields

trbtcool

=γpk

γrb tcool

' 3

5cσTαfsn3γ0.2pk tcool

≈ 45 v−3.48

( γpk

200

)−0.2

. (A.3)

The ratio of the synchrotron cooling timescale tsyn = 6πmec/(σTB2shγ) for

electrons with γ = γpk to the thermal cooling time is given by

tsyn

tcool

≈ 100 ε−1B,−2v

−5.48

( γpk

200

)−1

. (A.4)

Finally, the characteristic timescale for inverse Compton cooling on the

shock-produced thermal X-rays is tIC = 3mec/(4σTUradγpk), such that

tICtcool

=3mec

4σTUradγpktcool

≈ 1.7× 103 v−6.48

( γpk

200

)−1

, (A.5)

where Urad = (3/8)(vf/c)n3kT3/µ is the X-ray energy density.5

In summary, relativistic radio-emitting leptons cool in the post-shock

gas on a timescale which is generally much longer than that of the thermal

background plasma for the range of velocities v8 ∼< 1 of radiative shocks.

This justifies evolving their energies adiabatically behind the shock, with the

5Soft X-rays are in the Thomson regime for interacting with γ ∼ 100 electrons.

183

possible exception of Coulomb losses, which are unimportant immediately

behind the shock but may become so as gas compresses to higher densities

n n3.