HIGH ENERGY ASTROPHYSICS - Lecture 5

PD Frank Rieger

ITA & MPIK Heidelberg

Wed

1

Synchrotron Emission I

1 Overview

• Radiation of charged particles gyrating in magnetic fields.

• Total emitted power for single particle (Larmor formula), spectral distribu-tion.

• Characteristic Emission at ν ∝ γ2B, nearly linearly polarised.

• For a non-thermal, power law electron distribution, n(γ) ∝ γ−p, emittedspectrum a power law jν ∝ ν−(p−1)/2 too.

• Believed to be the fundamental radiation process at the lower end of the HEspectrum.

2

2 Synchrotron Radiation

”Magnetobremsstrahlung” = Radiation emitted by relativistic charged particles

(mostly electrons) due to acceleration (”gyro-motion”) in a static magnetic field.

Note: In the sub-relativistic regime this process is called ”Cyclotron emission”

Approach: Semi-quantitative analysis of spectral features as detailed calcula-

tions are lengthy (see, e.g., Rybicki & Lightman, Chap. 6).

Structure:

• Motion of electrons in magnetic fields

• Look at emission from a single electron

• Consider electron distribution and opacity effects to obtain final spectrum

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3 Motion of a Charged Particle in Static Magnetic Field

Remember: Particle gyrates around magnetic field with angular frequency (Lar-

mor frequency)

ΩL =qB

γmc

and associated radius of gyration orbit (Larmor radius), γmv2⊥/rL = (q/c) v×B,

rL =v⊥ΩL

=γmcv sin θ

qB' γmc

2

qBNumerically:

ΩL ' 1.8× 107(B

1 G

)(mem

)(1γ

)Hz

rL 'c

ΩL' 1.7× 103 γ

(1 G

B

)(m

me

)cm

' 1.7× 109 γ(

10−6 G

B

)(m

me

)cm

⇒ rL is small on cosmic scales (B ∼ 10−6 G in interstellar medium, B ∼ 1 G incenter of AGN [but away from BH]).

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4 Total Emitted Power for a Single Particle

• Remember: Larmor’s Formula for relativistically moving particle (lecture 4):(dE

dt

)K

=2

3

q2γ4

c3(a2⊥ + γ

2a2‖)

with a⊥, a‖ = acceleration components perpendicular and parallel to direction

of motion (not to magnetic field!).

• For gyro-motion, a‖ = 0 and a⊥ = v2⊥/rL = ΩLv⊥ (centrifugal), with v⊥ =v sin θ, so

Psyn =2

3

e2γ4

c3v2⊥e

2B2

γ2m2c2=

2

3

β2e4B2

m2c3γ2 sin2 θ

• For isotropic distribution of particles, average pitch angle

< sin2 θ >=1

4π

∫ 4π0

sin2 θdΩ =1

4π

∫ 2π0

dφ

∫ π0

sin2 θ sin θdθ

=1

2[cos(3θ)− 9 cos(θ)]π0/12 =

2

3

5

• Average single particle synchrotron power [erg/s] for β → 1

〈Psyn〉 =4

9

e4B2

m2c3γ2 =

4

3cσT

(mem

)2γ2 uB

with magnetic energy density

uB :=B2

8π,

and Thomson cross-section

σT :=8π

3

(e2

mec2

)2= 6.65× 10−25 cm2

• Characteristic electron cooling timescale:

tcool(γ) :=E

|dE/dt|=

γmec2

< Psyn >= 7.8× 108 1

B2γsec

Note:

1. Psyn ∝ 1/m2 ⇒ Synchrotron radiation from charged particles with largermass (e.g. protons) is usually negligible.

2. E = γmc2 ⇒ Psyn ∝ E2uB, so more energetic particles radiate more.

6

5 Energy Evolution for Electrons due to Synchrotron Losses

Start at t0 with initial energy of electron = E(t0):

• From Psyn = −dE/dt and energy E = γmec2

dE

dt= −c1E2 with c1 :=

4

3cσT

1

(mec2)2B2

8π= 1.6× 10−3B2 [1/erg/s]

• Integrate (dE/dt)/E2 = −c1 from t0 to t (t > t0) to give[− 1E

]tt0

= −c1(t− t0) viz.1

E(t)− 1E(t0)

= c1(t− t0)

E(t0)− E(t)E(t)E(t0)

= c1(t− t0)

so finally

E(t) =E(t0)

1 + c1(t− t0)E(t0)∝ 1t

for large times

• Note: Timescale tcool for cooling down to 1/2 of initial energy E0 at t0 = 0:E(tcool) =

E02 =

E01+c1tcoolE0

to give tcool(E0) =1

c1E0.

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6 Single Electron Spectrum

Remember: Radiated power as a function of time, P (t) ∝ |~a(t)|2, reflects time-dependence of acceleration a(t). Distribution of this radiation over frequency,

Pν is called spectrum and reflects power (Fourier) spectrum of a2(t).

For synchrotron emission, observed radiation varies with time due to relativistic

beaming (in addition to fundamental gyro-motion), so expect observed spectrum

to contain power at frequencies much higher than ΩL/2π.

�

��

Figure 1: A charged particle in slow gyro-motion around a magnetic field will emit an approximate dipolepattern with maximum in the direction of motion v. For higher speeds, aberration (beaming) changes patternto an asymmetric shape, with most emission contained in forward cone of half opening angle ∼ 1/γ.

8

• Aberration (Lecture 4): Radiation of relativistic particle is strongly beamedin forward direction, focused into cone of opening angle ∆α ' 2/γ [rad].

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���

21

Observer

r

• Observer only sees radiation once per obit when particle velocity is within2/γ of line of sight. ∆tobs:=duration for which beam stays within line of sight

⇒ Observer sees narrow pulse of duration ∆tobs once per gyro-period

⇒ Most of radiation power must appear at frequency ωc ∼ 1/∆tobs.

(Basic motion is periodic, t ∝ 1/ΩL (discrete spectrum), but frequency spacing ΩL between successive harmonics

becomes so narrow as to be negligible, so can treat as continuous spectrum)

9

Estimate Characteristic Frequency νc or ωc = 2πνc:

• Path length ∆s for which we see radiation: ∆s = r ∆α = r (2/γ).

• Radius of curvature r of path given by equation of motion:

γm∆~v

∆t=q

c~v × ~B

With |∆~v| ' v ∆α and ∆s ' v∆t, i.e. ∆t = ∆s/v, this gives:∆α

∆s' qB sin θ

γmcv

so that

r =∆s

∆α' γmcvqB sin θ

(Note: 3-dim r is different from projected/perpendicular Larmor radius)

• Path length ∆s ' r ∆α = r (2/γ) = 2vmcqB sin θ .

10

• Times t1 and t2 at which particles passes point 1 and 2 related by ∆s =v(t2 − t1), thus

t2 − t1 '2mc

qB sin θ

• Travel time effects: Arrival time difference is less than (t2 − t1) by ∆s/c:

∆tobs = (t2−t1)−∆s

c= (t2−t1)

[1− v

c

]=

2mc

qB sin θ

[1− v

c

] 1 + vc1 + vc

' mcγ2qB sin θ

⇒ Observed pulse has duration shortened by 1/γ2Note: Converting from rad to Hz, observed duration would be = 2π∆tobs

• Spectrum (in terms of frequencies) will be broad with characteristic angularfrequency (ωch = 2πνch) at which significant part of power is radiated:

ωch '1

tobs= γ2

qB

mcsin θ = γ2Ω0 sin θ = γ

3ΩL sin θ

with cyclotron frequency Ω0 := qB/(mc).

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7 Resulting Observed Electric Field

intensity

t ime

�����

width: ������

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��

The observed time-dependent E-Field, E(t), from an electron is a sequence of

pulses of width ∆tobs ' 1γ2Ω0 =1

γ3ΩL, separated in time by ∆t ∼ 2π/ΩL. Note:

Larmor frequency ΩL = eB/γmc =: Ω0/γ.

12

8 Single Particle Synchrotron Spectrum Pν(γ)

Approach:

Notation alert: Note E here refers to electric field, not particle energy!

• Want spectrum=power per unit frequency, Pν(γ) [erg/s/Hz],

satisfying∫∞

0 Pν(γ)dν = Psyn =23β2e4B2

m2c3γ2 sin2 θ.

⇒ Find dimensionless frequency distribution function F̃ (ν/νc) with

Pν(γ) = Psyn1

νcF̃ (ν/νc)

satisfying1

νc

∫ ∞0

F̃ (ν/νc)dν =νcνc

∫ ∞0

F̃ (x)dx!

= 1.

• Start by Fourier-transforming E(t) ∝ g(ωct) (get from full L-W potentials),

Ê(ω) =1

2π

∫ ∞−∞

E(t)eiωtdt ,

where inverse transform E(t) =∫∞−∞ Ê(ω)e

−iωtdω.

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• Remember Parseval’ theorem:∫ ∞−∞

E2(t)dt = 2π

∫ ∞−∞|Ê(ω)|2dω

• Total energy W per unit area in pulse (Poynting theorem):dW

dA=

c

4π

∫ ∞−∞

E2(t)dt = c

∫ ∞0

|Ê(ω)|2dω

so total energy per area per unit time per unit frequency:

dW

dAdtdω≡ 1T

dW

dAdω=c

T|Ê(ω)|2

• Spectrum Pν(γ) = 2πPω(γ) from

Pω(γ) ≡1

T

dW

dω=c

T

∫r2|Ê(ω)|2dΩ

where r distance, T = 2π/ΩL period and solid angle element dΩ = dA/r2,

using ∆θ ∼ 1/γ.

14

⇒ Result: Total synchrotron power per unit frequency [erg/s/Hz] for a singleelectron with Lorentz factor γ (relativistic case β ' 1)

Pν(γ) =√

3e3B sin θ

mc2F

(ν

νc

)with F (x) := x

∫∞x K5/3(x

′)dx′ ' 1.8 x0.3e−x, K5/3 modified Bessel function oforder 5/3, and γ entering via

νc =3

4πγ2eB

mcsin θ =

3

4πγ2 Ω0 sin θ

Note: By convention, F is normalized somewhat differently compared to F̃ , i.e.∫∞0 F (x)dx =

8π9√3, introducing the

slightly different numerical value in Pν .

0.01

0.1

1

0.001 0.01 0.1 1

F(!/! c

)

!/!c

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

F(!/! c

)

!/!c

Figure 2: Single Particle Spectrum F (ν/νc) as function of normalized frequency ν/νc. Left: Log-log repre-sentation. Right: Normal representation. The spectrum is continuous and has a maximum at νmax = 0.29νc.

15

9 Example: Synchrotron Emission from the Crab Nebula

Figure 3: Crab Nebula (d ∼ 2 kpc) caused by SN explosion in 1054 A.D. - composite image: ChandraX-ray [blue], HST optical [red and yellow], Spitzer infrared [purple]. X-ray image is smaller than othersas extremely energetic electrons emitting X-rays radiate away their energy more quickly than lower-energyelectrons emitting in optical and infrared [Credits: NASA].

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Emission as synchrotron radiation of relativistic electrons, characteristic frequency:

νc '1

2πγ2Ω0 =

1

2πγ2eB

mec' 280

(B

10−4 G

)γ2 Hz

for average B ∼ 10−4 G in Crab Nebula.

• Optical Emission:Optical emission (HST) at ν ∼ 5× 1014 Hz requires electrons with γ ∼ 106.⇒ Cooling time scale tcool ∼ 2500

(106

γ

)yr >∼ tage age of Nebula.

• X-ray Emission:Chandra (ACIS, 0.2-10 keV) X-ray emission ν ∼ 1017 Hz requires γ ∼ 107,electrons cool quicker by factor ∼ 10.⇒ X-ray emission spatially less extended.⇒ tcool < tage ∼ 950 yr of Nebula, need continuous supply of fresh electrons.

• Radio Emission:Crab Nebula also bright in radio (NRAO, ν ∼ 5 × 109 Hz), less energeticelectrons needed, γ ∼ 5× 103, size constrained by age of Nebula.

⇒ See synchrotron emission from broad distribution of electrons in energyspace.

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10 Synchrotron Spectrum for Power-law Electron Distribution

In many case, synchrotron radiation is emitted from relativistic electrons with

power-law density distribution in energy, i.e.

n(γ)dγ = n0γ−pdγ [cm]−3

for γmin < γ < γmax, typically p ∼ 2−3⇒ ”Nonthermal synchrotron radiation”.

Resultant emission spectrum = frequency distribution of emitted power [erg/s/Hz/cm−3]:

jν =

∫ ∞1

< Pν(γ) > n(γ) dγ .

Hard to do analytically using exact < Pν(γ) >⇒ Approximation: Photons areonly emitted at frequency νc ' γ2νL, νL = Ω0/2π = eB/(2πmec),

Pν(γ) '< Psyn > δ(ν − νc) =< Psyn > δ(ν − γ2νL)

So

jν '4

3cσTuBn0

∫ γmaxγmin

γ2

γpδ(ν − γ2νL) dγ

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Substituting ν ′ = γ2νL, i.e., dν′ = 2νLγdγ ↔ dγ = dν ′/(2γνL)

jν ∝1

2νL

∫ γ2maxνLγ2minνL

1

γp−1δ(ν − ν ′) dν ′ = 1

2νL

∫ γ2maxνLγ2minνL

(νLν ′

)(p−1)/2δ(ν − ν ′)dν ′

since γ = (ν ′/νL)1/2. Thus

jν =2

3cσTn0

uBνL

(ν

νL

)−p−12for γ2minνL < ν < γ

2maxνL.

Synchrotron spectrum of a power-law electron distribution of in-

dex p, n(γ) ∝ γ−p, is a power-law jν ∝ ν−α with spectral index

α =p− 1

2.

Note: Outside limits, emissivity will be dominated by that of particles at γminand γmax, respectively. Thus for ν < νmin := γ

2minνL (using asymptotic for F (x))

jν ∝ ν1/3 ,

and for ν > νmax := γ2maxνL

jν ∝ e−ν/νmax .

19

log

flu

x

log frequency

individualelectronspectra

power- law superposition

20

11 Polarization

• Electron acceleration vector changes only slightly during a pulse⇒ quasi-constant polarization state.

• Synchrotron radiation turns out to be highly elliptically polarized, i.e. verynearly linearly.

• Polarization allows to measure magnetic field direction (⊥ to observed electricfield vector; but note caveat: Faraday rotation and B-field inhomogeneities).

• For a power-law energy distribution of electrons with index p, maximumdegree of linear polarization (polarized flux/total flux):

π(p) =(3p + 3)

(3p + 7)' 0.7

for typical p = 2− 3.⇒ Observation of highly polarised emission as argument for synchrotronorigin.

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Cyclotron Radiat ion Synchrotron Radiat ion

LinearPolarizat ion

CircularPolarizat ion

CircularPolarizat ion

Figure 4: Left: Non-relativistic cyclotron motion: When viewed in orbital plane, radiation is 100% linearlypolarized with electric vector oscillating perpendicular to magnetic field B. Viewed from along B, emission is100% circularly polarized. Right: For relativistic motion, radiation is beamed into direction of motion. Thetwo components of circular polarization effectively cancel almost, whereas linear polarization largely survives.

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Figure 5: Decomposition of synchrotron polarisation vectors on the plane of the sky. The radiation is almostnearly linearly polarised, and dominated by the component perpendicular to the projected magnetic fielddirection. From Rybicki and Lightman, Fig. 6.7.

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12 Example: B-Field Direction Inferred from Observations

Figure 6: B-field vectors inferred from degree of radio (2 × 109 Hz) synchrotron polarization for the spiralgalaxy M51 [distance ∼ 10 Mpc, B ∼ O(10µG)] by rotation of the observed electric field vector by 90◦. Thereis a tendency for B to run parallel to the spiral arms.

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13 Exact Calculations

Evaluating spectrum for single electron, radiation can be split into two compo-

nents linearly polarised across and along magnetic field as projected on the sky

P⊥(ν, γ) =

√3

2

e3B sin θ

mc2

[F

(ν

νc

)+ G

(ν

νc

)]P‖(ν, γ) =

√3

2

e3B sin θ

mc2

[F

(ν

νc

)−G

(ν

νc

)]with P⊥ > P‖, where

G(x) := x K2/3(x) and F (x) = x

∫ ∞x

K5/3(x′)dx′ ,

with Ki modified Bessel function of ith order. Total power for single electron:

Pν(γ) = P⊥(ν, γ) + P‖(ν, γ) ∝ F (ν/νc) .Fractional degree of linear polarization:

π(ν) :=P⊥(ν, γ)− P‖(ν, γ)P⊥(ν, γ) + P‖(ν, γ)

For single particle π(ν) = G(ν/νc)/F (ν/νc). For a power law of index p, P⊥ and

P‖ must be integrated over particle energy. Result gives π(p) = (3p+3)/(3p+7).

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0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

F(!/! c

)

!/!c

FG

Figure 7: Comparison of the synchrotron distribution functions F and G, noting that we have P⊥(ν, γ) ∝F(ννc

)+G

(ννc

)and P‖(ν, γ) ∝ F

(ννc

)−G

(ννc

)

26