Neutron Stars 4: Magnetism

Andreas ReiseneggerDepartamento de Astronomía y Astrofísica

Pontificia Universidad Católica de Chile

Bibliography• General books:

– Russell M. Kulsrud, Plasma Physics for Astrophysics– Leon Mestel, Stellar Magnetism

• Reviews:– Alice Harding & Dong Lai, Physics of strongly magnetized neutron stars,

Rep. Prog. Phys., 69, 2631 (2006): includes interesting physics (QED, etc.) that occurs in magnetar-strength fields - not covered in this presentation

– A. Reisenegger, conference reviews: • Origin & evolution of neutron star magnetic fields, astro-ph/0307133: General• Magnetic fields in neutron stars: a theoretical perspective, astro-ph/0503047:

Theoretical• Magnetic field evolution in neutron stars, arXiv:0710.2839: Theoretical, short

• Papers:– Goldreich & Reisenegger 1992, ApJ– Hoyos, Reisenegger, & Valdivia 2008, A&A– Reisenegger 2009, A&A

Outline

• Classes of NSs, evidence for B

• Magnetohydrodynamics (MHD) & flux freezing

• Comparison to other, related stars, origin of B in NSs

• Magnetic equilibria

• Observational evidence for B evolution

• Physical mechanisms for B evolution

– External: Accretion

– Internal: Ambipolar diffusion, Hall drift, resistive decay

Caution: Little is known for sure – many speculations!

Spin-down(magnetic dipole model)

Spin-down time (age?):

Lyne 2000, http://online.kitp.ucsb.edu/online/neustars_c00/lyne/oh/03.html

42

2

2

2

33

2 B

dt

d

cI

Magnetic field:

3

||

B

||2

st

Kaspi et al. 1999

“Magnetars”

Classical pulsars

Millisecond pulsars

Objects Emission B determination log B [G] log age [yr]

Classical pulsars Radio to gamma

Spin-down 11-13 3-8

Millisecond pulsars

Radio to gamma

Spin-down 8-9 8-10

Magnetars (SGRs & AXPs)

Gamma, X, IR

Spin-down, LX 14-15 (-16?) 3-5

RRATs Radio, X Spin-down 12-14 5-7

Isolated thermal “Magnificent 7”

X, optical Spin-down, cyclotron lines

13-14 4-6

Thermal CCOs in SNRs

X Spin-down 12.5??? 2.5-4.5

HMXBs X Cyclotron lines 12 young

LMXBs X Absence of pulsations, others

8-9? old

Note large range of Bs, but few if any non-magnetic NSs

Neutron star magnetic fields• Strongest B in the Universe, up to at least ~1015G.

• Persistent

• Cause rotational energy loss: accounts for bolometric luminosity of pulsars

• Soft gamma-ray repeaters (SGRs) & Anomalous X-ray Pulsars (AXPs):

X/gamma-ray luminosity >> rotational energy loss or cooling

Magnetically powered neutron stars or “Magnetars” (Thompson & Duncan 1993, 1995, 1996)

Quasi-periodic oscillations (QPOs) may be probing magnetic structure inside the star (Levin 2007)

• (Slight) deformation of NS due to B might cause:

– Precession (observed?)

– Gravitational waves (hope!)

Magnetic field

strengths

From R. Duncan’s “magnetar” web page, http://solomon.as.utexas.edu/~duncan/magnetar.html

Flux freezing

• tdecay is long in astrophysical contexts (r large), >> Hubble time in NSs (Baym et al. 1969) “flux freezing”

• Alternative: deform the “circuit” in order to move the magnetic field MHD

tL

R

eIRIdtdI

L

0

2

2

decay2~

1~~

c

r

R

Lt

rR

c

rL

MagnetoHydroDynamics

Assume 1 fluid moving with

Electrons have small mass: neglect their inertia, gravity, etc.:

Induction equation

(advection of field lines)

Current density is secondary, calculated by

01

Bvc

E

)( BvEct

B

Pc

Bj

dt

vd

Bc

j

4

Magnetic field origin?

• Fossil: flux conservation during core collapse:

– Woltjer (1964) predicted NSs with B up to ~1015G.

• Dynamo in convective, rapidly (differentially) rotating proto-neutron star (~ minutes)

– Scaling from solar dynamo led to prediction of “magnetars” with B~1016G (Thompson & Duncan 1993)

• Both?: Some memory of initial conditions, but strongly modified by differential rotation, etc.?

Sun

Highly disordered field: (random component~kG) >> (dipole component~50G)

Inversion every 11 yrs

Probably due to convection + differential rotation (dynamo effect)

http://solarscience.msfc.nasa.gov/3dfields.shtml

http://science.nasa.gov/ssl/pad/solar/maghstry.htm

Upper main sequence

(Ap, Bp stars)

Only small fraction detectably magnetic (Ap, Bp or CP=“Chemically Peculiar”)

Ordered field: low-order multipoles ~ kG

Convective core + stable, radiative envelope

A&A, 358, 929 (2000)

A&A, 358, 929 (2000)

Magnetic white dwarfs

Small fraction of all WDs

(Statistically) more massive than non-magnetic WDs

Ordered field, low multipoles ~ MG

Stars with long-lived, ordered B-fields

Radius

[solar units]

Bmax [G] Flux R2Bmax

Upper main sequence

3 3104 (“Ap” stars) 106

White dwarfs 10-2 109 3105

Neutron stars 10-5 1015 (magnetars) 3105

In all cases, (magnetic pressure) < 10-6 (fluid pressure). Weak B!!

All are stably stratified.

NS energies

• EG ~ GM2/R ~ Nn ~ 1054 erg

• E = I2/2 ~ 1053 Pms-2 erg

• ET ~ N(kT)2/ n ~ 1046 T82 erg

• EB ~ (B2/8)(4R3/3) ~ 1048 B152 erg

Generally E , ET , EB << EG: small perturbations

Stable stratification

Barotropic fluid: density = (P) [P = pressure]

Non-barotropic fluid: density = (P,Y), where Y = another, independent variable:

• Specific entropy in radiative zones of stars (upper MS & WDs)

• Composition (e.g., proton fraction) in neutron stars

(Pethick 1992; Reisenegger & Goldreich 1992; Reisenegger 2009)

• Like water with non-uniform temperature or salinity:

– Colder or saltier water stays at the bottom

– Weak B can’t force substantial, non-radial motions

Equilibrium only in non-barotropic fluid

cross section

Magnetic equilibria

• Force balance:

• B as small perturbation:

– Background

– Perturbation

(fluid perturbation described by 2 independent scalars)

Pc

Bj

0000 P

0

Pc

Bj

Stable magnetic field configurationsBraithwaite & Spruit 2004: simulation of ideal MHD in fluid, stably stratified star.B quickly reaches an equilibrium configuration with poloidal & toroidal components.

Equilibria & stability

• Poloidal-toroidal decomposition:

– Pure poloidal & pure toroidal field are unstable (Flowers & Ruderman 1977; Tayler 1973)

• Our current (semi-)analytic work

– Calculation of Flowers-Ruderman instability (P. Marchant)

– Construction of non-barotropic, poloidal + toroidal equilibria (A. Mastrano, T. Akgün)

– Find unstable modes of toroidal fields, study stabilizing effect of poloidal component (T. Akgün)

Braithwaite 2007

Evidence for B-field evolution

• Magnetars: B decay as main energy source?requires internal field ~10x inferred dipole

• Young NSs have strong B (classical pulsars, HMXBs), old NSs have weak B (MSPs, LMXBs).

Result of accretion?• (Classical) Pulsar population statistics: no decay? -

contradictory claims (Narayan & Ostriker 1990; Bhattacharya 1992; Regimbau & de Freitas Pacheco 2001)

• “Braking index” in young pulsars progressive increase of inferred B

32 n

||, ILX

Diamagnetic screening

Material accreted in the LMXB stage is highly ionized conducting magnetic flux is advected

Accreted material could screen the original B, which remains inside the star, but is not detectable outside (Bisnovatyi-Kogan & Komberg 1975, Romani 1993, Payne & Melatos 2004, 2007)

Questions:

• Do instabilities prevent this?

• Why 108-9 G, but not 0?

Speculation: Magnetic accretion?

Can the field of MSPs have been transported onto them by the accreted flow?

Force balance:

Mass transport:

Combination:

R

B

c

Bj

R

GM

4~~

2

2

R

GMRfvRfM

24'~4~ 22

G'

10~'2

~2

1

Edd84

1

52

2

f

MM

Rf

MGMB

Preliminary conclusions on magnetic accretion

The strongest magnetic field that might be forced onto a neutron star by an LMXB accretion flow is close to that observed in MSPs.

More serious exploration is required (S. Flores, PhD thesis in progress):

– Hydrodynamic model: transport through “turbulent viscosity” or wind

– Is the magn. flux transported from the companion star?

– Is it generated in the disk (“magneto-rotational inst.”)?

– Is it coherent enough?

B evolution inside NS

Terms:• Ambipolar diffusion: Driven by magnetic stresses (Lorentz force), protons &

electrons move together, carrying the magnetic flux and dissipating magnetic energy.

• Hall drift: Magnetic flux carried by the electric current; non-dissipative, may cause “Hall turbulence” to smaller scales.

• Ohmic or resistive diffusion: very small on large scales; important for ending “Hall cascade”. May be important in the crust (uncertain conductivity!).

Time scales depend on B (nonlinear!), lengthscales, microscopic interactions.

Cooper pairing (n superfluidity, p superconductivity) is not included (not well understood, but see Ruderman, astro-ph/0410607).

jc

Ben

jBv

tB

eA

Protons & electrons move through a fixed neutron background, colliding with each other and with the background (Goldreich & Reisenegger 1992):

Model conclusions

• Spontaneous field decay is unlikely for parameters characteristic of pulsars, unless the field is confined to a thin surface layer (Goldreich & Reisenegger 1992)

• Spontaneous field decay could happen for magnetar parameters (Thompson & Duncan 1996)

• Simulations (include moving neutrons):

– 1-d: Hoyos, Reisenegger, & Valdivia 2008

– 2-d: in progress

Conclusions

Magnetic fields have:

– Very small effect on structure of stars

– Strong effect on NS appearance & evolution (pulsar braking, magnetars)

– Source currents due to moving p, e, or other charged particles

– Uncertain origin: fossil – dynamo – both ?

– (possibly) Stable equilibrium configurations with linked toroidal & poloidal components, thanks to stable stratification

– Non-trivial evolution, even in the most “prosaic” NS models (no need for ferromagnetism, quarks, Cooper pairs, etc. ...):

• Internal (ambipolar diffusion, weak interactions) in magnetars

• External (diamagnetic screening, flux accretion) in LMXBs MSPs