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8. Synchrotron Radiation (contd)

Electrodynamics of Radiation Processes

http://www.astro.rug.nl/~etolstoy/radproc/

Chapter 6: Rybicki&LightmanSections 6.3, 6.4, 6.5, 6.6, 6.8

Synchrotron Radiation

Emission by ultra-relativistic electrons spiraling around magnetic f ield lines

(magneto-bremsstrahlung)

Synchrotron Radiation

First observed in early terrestrial particle accelerator experiments, whereelectrons were moving in a circular path.

It is the dominant emission mechanism in astrophysical radio sources, and also important at optical and X-ray wavelengths in AGN.

² radio continuum emission of the Milky Way. ² non-thermal continuum emission of SNRs such as the Crab nebula ² optical and X-ray continuum emission of quasars and AGN. ² transient solar events ² Jupiter

Synchrotron emission depends on and thus reveals the presence of a magnetic f ield, and the energy of the particles interacting with it.

Space is full of magnetic fields

location Field strength (gauss)interstellar medium 10-6

stellar atmosphere 1

Supermassive Black Hole 104

White Dwarf 108

Neutron star 1012

this room 0.3

Supernova remnants/Crab Nebula 10-3

1 gauss (G) = 10-4 tesla (T)

1 tesla (T) = 1 Wb m-2

typically very weak magnetic fields, but there is a plentiful supply of relativistic electrons even in low density environments

log P

log!!c

Synchrotron emission -e v

x

xx

x

x

x

v?

a

�tobs ⇠1

�2

1

!cyc · sin↵P / !1/3

P / !1/2 e�!

for a single electron

!cyc =eB

mc

!sync =eB

�mec

!c =3

2�2!cyc sin↵

!c =3

2�3!sync sin↵

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Basics of Synchrotron Emission: spectrum Synchrotron power radiated by electrons

cyclotron

P0 =2

3

e2a2

c3

rest frame of electron

synchrotron laboratory frame, sees a relativistic electron

power is a Lorentz invariant (unchanged under LT)

P = P0 just need a ʹ

Need Eʹ in electron’s frame - Lorentz shifted B-f ield, what the lab see’s as a B-f ield is Lorentz shifted to Eʹ -f ield

the transformation law of E&B f ields(∥and ⟂ to velocity)

We don’t care about B-f ield in electron rest frame, no motion, no B-f ield.

In lab frame there is no electric f ield and we ignore the interaction of electron with own E f ield.

E0? = � · v

c⇥B =

�Bv

csin↵

Synchrotron power radiated by electrons

E0? = � · v

c⇥B =

�Bv

csin↵

| a0 |= eE0

me=

eB

mec· �v · sin↵

P0 =2

3

e4B2�2v2

m2ec

5· sin2 ↵ highly relativistic, so v~c

P0 = P =2

3

e4B2

m2ec

3· �2 sin2 ↵

Synchrotron radiates power much faster, by a factor γ2, and (c/v)2. Simply moving faster

P =2

3

e4

m2ec

3·⇣vc

⌘2

Cyclotron:

Synchrotron power radiated by electrons: timescales

P0 = P =2

3

e4B2

m2ec

3· �2 sin2 ↵ �T =

8⇡

3·⇣ e2

mec2

⌘2

Thomson cross-section

P = 2�T cB2

8⇡�2 sin2 ↵

UB Energy density in the magnetic f ield

P = 2�T c UB �2 sin2 ↵

Lifetime of an electron emitting synchrotron emission, tlife =total energy of electron

rate loosing energy

tlife =�mec2

�2⇣

e4B2

m2ec

3

⌘ =m3

ec5

�e4B2

Faster electron moves the more quickly it decays, the stronger the magnetic f ield the shorter it lives

Synchrotron power radiated by electrons: timescales

P = 2�T c UB �2 sin2 ↵

Cooling time of an electron tcool =�mec2

2�T c UB �2 sin2 ↵2/3

tcool ⇠ 16 yr⇣1 Gauss

B

⌘2 1

�

Electrons in a plasma emitting synchrotron radiation cool down. The time scale for this to occur is given by the energy of the electrons divided by the rate at which they are radiating away their energy. The energy E = 𝛾mc2

Crab Nebula B~1mG E~4kev photons

Instead of assuming a γ we pick the energy of the photons we are observing

Using critical synchrotron frequency:

!sync ⇠ �2 · !cyc

E ⇠ ~! ⇠ ~�2 eB

mec

!cyc =eB

mc

tcool ⇠ 16 yr⇣1 Gauss

B

⌘2 1

�

Putting this γ into tcool

tcool ~ 2 years

~1000yrs old (1054 BCE)

Must be new source of relativistic electrons

> pulsar, sitting in middle

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Electron Energy Lifetime

Synchrotron Spectra steepen with age…

Break/steepening in Spectral Energy distributionno sharp cut off due to continuous injection of particles

from t=0 (supernova!) to the current time (T)

value of γ at the break frequency

where

interpret break frequency νb to be point where t1/2~T for the injected electrons

yrs

typical cooling times

location Typical B (gauss) tcool

interstellar medium 10-6 1010yrs

stellar atmosphere 1 5days

Super-massive black hole 104 10-3sec

white dwarf 108 10-11sec

neutron star 1012 10-19sec

Spectrum and Polarisation of Synchrotron radiation

to obtain a realistic synchrotron spectrum need to convolve the mono-energetic electron spectrum with an energy distribution function.

The shape of the frequency spectrum of a synchrotron source is determined by the emission properties and the energy distribution of the emitting particles.

In the simplest case - the frequency distribution is a declining power law over the range of interest, and so

due to a power-law electron distributionP(!) / !�s

N(E) / E�P

Power Law Energy DistributionIn a wide range of astrophysical applications, the energy spectrum of relativistic electrons is a power-law as might be produced by a stochastic acceleration mechanism.

P0 = P =2

3

e4B2

m2ec

3· �2 sin2 ↵ / B2�2 / B2E2

⌫ / �2B / E2B

Convert E (and dE) into ν and dν E = h⌫ = h! / �2 eB

mec2

E /⇣ ⌫

B

⌘1/2

dE / ⌫�1/2 B�1/2 d⌫

dE

d⌫/ (⌫B)�1/2

Fermi mechanism, in supernovae remnants: numbers of electrons (N) scatter off turbulent magnetic “bubbles” and escape with specif ic energies (E), resulting in distribution:

dN

dE/ E�P

dP

dE/ dN

dE· PE / E�P · E2 · B2dP

dE/ dN

dE· PE / E�P · E2 · B2/ B2 · E2�P

Power Law Energy DistributiondE

d⌫/ (⌫B)�1/2 E /

⇣ ⌫

B

⌘1/2

Note: sign of P is something that can defined as + or -

dP

dE/ dN

dE· PE / E�P · E2 · B2/ B2 · E2�P

dP

dE· dEd⌫

/ B2E2�P⌫�1/2B�1/2

dP

d⌫/ B3/2 ⌫

2�P2 B

�2�P2 ⌫�1/2 / B

1+P2 ⌫

1�P2

dP

d⌫/ B

⇣ ⌫

B

⌘P�12

in Milky Way

Spectral index is something that can be measured 1 . P� 1

2. 0.5

empirical result3 . P . 2

s =p� 1

2

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log P

log!!c

Synchrotron Emission Spectrum

There is a very simple relation between

p = spectral index of particle energies

s = spectral index of observed radiation

s =�(p� 1)

2

Synchrotron vs. thermal radiation

(ii) Select a suitable set of coordinates to work out the f ield components radiated by the electron spiraling in a magnetic f ield,

Spectrum of Synchrotron Radiation

There is no simple way to derive the spectral distribution of synchrotron radiation.

(i) Start with the energy emitted per unit bandwidth for an arbitrarily moving electron, (dW/dν dΩ)

(iii) battle away at the algebra to obtain the spectral distribution of the f ield components.

Rybicki & Lightman – section 6.4

R&L, Eqn 3.13

Spectrum and Polarisationx = !/!c

The total emitted power per frequency is the sum

P(!) =

p3

2⇡

e3B sin↵

mec2F(x)P?(!) + Pk(!) =

Synchrotron emission from an electron with pitch angle α. The radiation is confined to the shaded solid angle.

Synchrotron Emission Spectrum x = !/!c

P(!) =

p3

2⇡

e3B sin↵

mec2F(x)For a highly relativistic case

F is a dimensionless function

F(x) ⇠⇣⇡2

⌘1/2e�x x1/2x � 1

F(x) ⇠ 4⇡p3 �( 13 )

⇣x2

⌘1/3x ⌧ 1

p >1/3Longair, High Energy Astrophysics f igure 8.8

Synchrotron Emission Spectrum

linear

log-log

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Polarisation of Synchrotron Radiation

The radiation from a single relativistic charge will be elliptically polarised.

For any reasonable distribution of particles that varies smoothly with pitch angle, the elliptical component will cancel out. Thus synchrotron radiation will be partially linearly polarised.

Cyclotron radiation will be circularly polarised

Polarisationparallel and perpendicular components define POLARISATION

We can characterise the radiation from its powers parallel and perpendicular to the projection of the magnetic f ield.

the amount of linear polarisation for particles of single energy, γ:

this value is quite high, ~75%

Polarisation of Synchrotron

for particles with a power-law distribution of energies

this is a clear prediction for determining if synchrotron radiation is present in an astronomical source

⇧ =p + 1

p + 73

p > 1/3

Total Energy

U = Ue +UB

Equipartition ⇠ particle energy

field energy⇠ 1

How much relativistic particle energy there is in a synchrotron source: minimum energy requirements.

Minimum Energy RequirementsGiven a source, volume V, where the synchrotron radiation has luminosity Lν at frequency ν. The spectrum of the radiation is of power-law form, Lν ∝ ν−α. The radio luminosity can be related to the minimum energy of the ultra-relativistic electrons and the minimum magnetic flux density B.

minimum energy requirement

minimum magnetic f lux density

η ion/electron energy ratio

The “invisible” cosmic-ray protons and heavier ions emit negligible synchrotron power but they still contribute to the total cosmic-ray particle energy

Cosmic rays collected near the Earth have η ≈ 40, but η in radio galaxies and quasars has not been measured.

The Crab Nebula atypically small, most SNR have steeper slopes (-0.6 to -0.8).

minimum energy

substantial fraction of blast energy (~1051 erg)

There must exist a highly eff icient means of converting the gravitational energy of collapse into high energy particle and magnetic f ield energy. Supernova remnants are very powerful sources of high energy electrons.

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B0.6 0.8 1.0 1.2 1.4 1.6 1.8

0.8

0.7

0.6

0.5

0.4

non-thermal energy curve for a SNR

given the large amount of kinetic energy associated with SNRs is it possible that SNRs are the source of most of the relativistic particles in our Galaxy?

we detect on Earth as cosmic rays

from all sky radio surveys we know the Galaxy is f illed with cosmic rays because the radio emission is almost entirely synchrotron from highly relativistic electrons, which are also observed in SNR

Bonn 408MHz all sky survey

SN as particle accelerators

SN convert a good fraction of their blast energy into CRs around these radio frequencies

Turnover from self-absorption

S⌫ =2me

3 !1/2cyc

⌫5/2 / ⌫5/2

B1/2S⌫ =

2me

3 !1/2cyc

⌫5/2 / ⌫5/2

B1/2

S⌫ / ⌫�↵

Synchrotron Self-Absorption spectrum

for optically thin synchrotron emission the observed intensity is proportional to the emission function

for optically thick synchrotron emission the observed intensity is proportional to the source function

the synchrotron spectrum of a source with a power law electron distribution

log Sν

-

F⌫ / ⌫5/2

B1/2

Synchrotron Self-Absorption

h⌫ = h!c = �2 eB

mec2

This means, more electrons = more power synchrotron emission

while system is optically thin, when it becomes OPTICALLY THICK (we get self-absorption)

I⌫ = S⌫(1� e�⌧ ) I f you can emit at a certain frequency you can also absorb at that frequency

This means, more electrons, the more absorption, adding energy to other electrons instead of observable radiation

Need to consider in more detail the form of the source function, Sν= jν / ανto understand WHEN we need to consider self-absorption.

Consider the brightness temperature of Synchrotron emission

Synchrotron Self-Absorption

at low frequencies Tb may approach the kinetic temperature of the radiating electrons

temperature of a black-body which would produce the observed surface brightness of the source at the frequency ν in the Rayleigh-Jeans limit, hν << kT.

for a self-absorbed source, Tb = Te in the Rayleigh-Jeans limit

Te ⇠�mec2

3k⇠ !1/2

c mec2

3k !1/2cyc

defines an upper limit for what synchrotron emission could be in the optically thick case.

S⌫ =2me

3 !1/2cyc

⌫5/2 / ⌫5/2

B1/2

Tb =c2

2⌫2kS⌫

S⌫ =2me

3 !1/2cyc

⌫5/2 / ⌫5/2

B1/2

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Cygnus A radio galaxythe brightest extragalactic radio source in the northern sky

Luminosity ~8 × 1028 W Hz−1 at 178 MHz; Size ~100kpc; distance 232Mpc

the minimum total energy is 2 × 1052η4/7 J which corresponds to a rest-mass energy of 3 × 105η4/7 M⊙ of matter.

These are enormous energy demands in the form of relativistic particles and magnetic f ields. A considerable amount of mass has to be converted into relativistic particle energy and ejected from the nucleus of the galaxy into enormous lobes well outside the body of the galaxy.

The super-massive black hole at the core has a mass of ~2.5 x 109 M⊙

perhaps the most powerful objects in the Universe are radio galaxies. they contain large reservoirs of non-thermal energy

located 100s of kpc from the nuclei, the power centres of these galaxies

It was f irst detected in 1956 by Burbidge in a jet emitted by M87, as a confirmation of a prediction by Iosif S. Shklovsky in 1953

1.5 light-months diameter6000 light-years longpolarized emissionblobs move & change in intensity & polarisation

Pmin ⇠ 1060ergs > 109 SNe

Radio Galaxies

electrons must be ultra-relativistic to produce a similar power in the radio - where ωB can be Doppler boosted by the potentially large factor 𝛾2.

for typical Galactic HII regions, B~3 x 10-6 gauss

HII regions

ordinary electrons at 104 K (HII regions for example) are much stronger sources of thermal (radio) emission than synchrotron.

in situations where particle velocities are relativistic the densities are generally low and so synchrotron emission tends to dominate thermal bremsstrahlung (the emissivity goes with square of density)

1. particles radiate into a cone 1/𝛾 from direction of motion.

2. a particle spectrum extends up to the critical frequency, ωc

(spectrum is a function of ω/ωc).

3. for a power-law distribution of particle energies with index p over a suff iciently broad range of energy, the spectral index of the radiation is: s=(p - 1)/2.

4. radiation is highly polarised

SUMMARY – Synchrotron Radiation