Radiative Processes in Astrophysics

!

Lecture 8 Nov 4 (Mon.), 2013

(last updated Nov. 4)

Kwang-Il Seon UST / KASI

[Bremsstrahlung]• Bremsstrahlung (= “breaking radiation”) (or free-free emission): radiation due to the

acceleration of a charge in the Coulomb field of another charge.

Consider bremsstrahlung radiated from a plasma of temperature T and densities electrons with charge -e and ions with charge Ze.

We calculate an important ratio:

!!!!!!for typical and .

Therefore, Coulomb interaction is only a perturbation on the thermal motions of the electrons.

Coulomb potential energythermal kinetic energy

≈Ze2 / rkT

≈ Ze2 / ne

1/3

kT

= 1.670 ×10−7Z 1 cm-3

ne

⎛⎝⎜

⎞⎠⎟

1/3104 KT

≪1

ne (cm−3)

ni (cm−3)

T ~104 −108 Kne < 1 cm−3

A full understanding of this process requires a quantum treatment. However, a classical treatment is justified in some regimes, and the formulas so obtained have the correct functional dependence for most of the physical parameters.

• Bremsstrahlung due to the collision of like particles (electron-electron, proton-proton) is zero in the dipole approximation,

because the dipole moment is simply proportional to the center of mass (a constant of motion).

!!

• Approximations:

(1) In electron-ion bremsstrahlung, we can treat the electron as moving in a fixed Coulomb field of the ion, since the relative accelerations are inversely proportional to the masses.

!!(2) A series of small-angle scatterings

(3) Classical calculation => Quantum correction

(4) Non-relativistic => Relativistic

ei∑ ri = e ri∑ ∝m ri =∑ miri∑

aiae

= me

mi

<10−3

[Emission from single-speed Electrons]• Small-angle scattering approximation:

The electron moves rapidly enough so that the deviation of its path from a straight line is negligible.

!!!!Take Fourier transform of the second derivative of the dipole moment.

!!Collision time: the electron would be in close interaction with the ion over a time interval.

!For , the exponential in the integral oscillates rapidly.

For , the exponential is essentially unity, so we may write

b

Ze

−e

R

v

−ω 2d(ω ) = − e2π

!veiωt dt−∞

∞

∫

dipole moment:

d = −eR!!d = −e!v

τ = bvωτ ≫1

ωτ ≪1

d(ω ) ≈e

2πω 2 Δv if ωτ ≪1

0 if ωτ ≫1

⎧

⎨⎪

⎩⎪

where is the change of velocity during the collision.

Δv

• Spectrum of the emitted radiation by a single electron:

!!!Let us now estimate . Since the path is almost linear, the change in velocity is predominantly normal to the path.

!!!!!Thus for small angle scatterings, the emission from a single collision is

dWdω

= 8πω4

3c3d(ω )

2=

2e2

3πc3Δv 2 if ωτ ≪1

0 if ωτ ≫1

⎧

⎨⎪

⎩⎪

Δv

Δv ≈ Δv⊥ = 1me

F⊥ dt∫

= Ze2

me

bb2 +v2t 2( )3/2

dt−∞

∞

∫ = 2Ze2

mebv

dW (b)dω

=8Z 2e6

3πc3me2v2b2 if b≪ v /ω

0 if b≫ v /ω

⎧

⎨⎪

⎩⎪

dxb2 + x2( )3/2

= xb2 b2 + x2( )1/2∫

• Total spectrum for a medium with ion density , electron density and for a fixed electron speed.

!!!!!!!

• Upper limit

The integral is negligible for .

ni ne

nev- flux of electrons (per unit area per unit time) incident on one ion =

dWdωdVdt

= nenivdW (b)dω

2πbdbbmin

∞

∫

= 16e6

3c3me2v neniZ

2 dbbbmin

bmax∫ = 16e6

3c3me2v neniZ

2 ln bmaxbmin

⎛⎝⎜

⎞⎠⎟

- element of area = 2πbdb

bmax ~ v /ω

- a good approximation is obtained in low-frequency regimes:

b≫ bmax ~ v /ω

• Lower limits

by the small-angle approximation:

by the uncertainty principle:

!When

a classical description of the scattering process is valid. Then,

When

the uncertainty principle plays an important role. Then,

!• For any regime the exact results are conveniently stated in terms of correction factor or Gaunt

factor. Precise expression of the Gaunt factor comes from QED (Quantum Electrodynamics) computation.

!!!Typically . Tables and plots are available by Bressaard and van de Hulst (1962) and Karzas and Latter (1961).

Δv / v ~ 2Ze2 /bmev2 <1 → bmin >bmin(1) ≡ Ze2 /mev2

Δx / Δp ≥ ! → bmin > bmin(2) ≡ ! /mv

bmin(1) ≫ bmin

(2) , or 12mev2 ≪ Z 2Ry

bmin = bmin(1)

bmin(1) ≪ bmin

(2) , or 12mev2 ≫ Z 2Ry

bmin = bmin(2)

Ry ≡ mee4

2!2= 13.6 eV

⎛⎝⎜

⎞⎠⎟

4π jω (v,ω ) ≡dW (b)dωdVdt

= 16πe6

3 3c3me2vneniZ

2gff (v,ω ) gff (v,ω ) =3πln bmax

bmin

⎛⎝⎜

⎞⎠⎟

gff (v,ω ) ~ 1 to few

[Thermal Bremsstrahlung Emission]• We now average the above single-speed expression over a thermal distribution of electron speeds.

!!At frequency , the incident velocity must be at least such that , because otherwise a photon of energy could not be created.

This cutoff in the lower limit of the integration over electron velocities is called a photon discreteness effect.

!!!!!!The exponential factor has

dP = me

2πkT⎛⎝⎜

⎞⎠⎟3/2

e−mev2 /2kT d 3v = me

kT⎛⎝⎜

⎞⎠⎟3/2

21/2π −1/2e−mev2 /2kTv2dv

where vmin ≡2hνme

⎛

⎝⎜⎞

⎠⎟

ν hν ≤ 12mv2

dWdVdtdω

= me

kT⎛⎝⎜

⎞⎠⎟3/2

21/2π −1/2 dW (v,ω )dVdtdω

v2e−mev2 /2kT dvvmin

∞

∫

= me

kT⎛⎝⎜

⎞⎠⎟3/2

21/2π −1/2 16πe6

33/2c3me2 nineZ

2 gff (v,ω )v v2e−mev2 /2kT dv

vmin

∞

∫

= me

kT⎛⎝⎜

⎞⎠⎟3/2

21/2π −1/2 16πe6

33/2c3me2 nineZ

2 gff (v,ω )e−mev2 /2kT d(v2 / 2)vmin

∞

∫

exp − mv2

2kT⎛⎝⎜

⎞⎠⎟= exp − mvmin

2

2kT⎛⎝⎜

⎞⎠⎟exp − m(v

2 −vmin2 )2kT

⎛⎝⎜

⎞⎠⎟= exp − hν

kT⎛⎝⎜

⎞⎠⎟ exp − m(v

2 −vmin2 )2kT

⎛⎝⎜

⎞⎠⎟

hν

!!!In terms of , the volume emissivity is

!!!!!!!!where the velocity-averaged free-free Gaunt factor.

Summing over all ion species gives the emissivity:

dWdVdtdω

= me

kT⎛⎝⎜

⎞⎠⎟3/2

21/2π −1/2 16πe6

33/2c3me2 nineZ

2e−hν /kT me

kT⎛⎝⎜

⎞⎠⎟−1

gff (v,ω )e−u du0

∞

∫

ενff ≡ dW

dVdtdν= 2π me

kT⎛⎝⎜

⎞⎠⎟1/2

21/2π −1/2 16πe6

33/2c3me2 nineZ

2e−hν /kT g ff (v,ω )e−u du0

∞

∫

= 2kT

⎛⎝⎜

⎞⎠⎟1/2 32π 3/2e6

33/2c3me3/2 nineZ

2e−hν /kT g ff (v,ω )e−u du0

∞

∫

= 25πe6

3mec3

2π3kme

⎛⎝⎜

⎞⎠⎟

1/2

T −1/2nineZ2e−hν /kT g ff

ενff = 6.8 ×10−38nineZ

2T −1/2e−hν /kT g ff (erg s−1 cm-3 Hz-1)

where u ≡me(v2 −vmin

2 )2kT

⎛

⎝⎜⎞

⎠⎟

ν =ω / (2π )

gff ≡ gff (v,ω )e−u du0

∞

∫gff (v,ω ) = 3 /π( )ln bmax /bmin( )

gff

ενff = 6.8 ×10−38 nineZi

2

i∑ T −1/2e−hν /kT g ff (erg s

−1 cm-3 Hz-1)

Note that main frequency dependence is ,which shows a “flat spectrum” with a cut off at . The spectrum can be used to determine temperature of hot plasma.

ενff ∝ e−hν /kT

ν ~ kT / hP1: RPU/... P2: RPU

9780521846561c05.xml CUFX241-Bradt September 20, 2007 5:19

5.5 Spectrum of emitted photons 195

Log ! (Hz)10 12 14 16 18 20

–46

–44

–42

T = 5 × 10 K

Gaunt factor effect

7

PureExponential

n i = ne = 106 m–3 Log

j( !

)

(J m

–3 H

z–1)

Fig. 5.5: Theoretical continuum thermal bremsstrahlung spectrum. The volume emissivity (37) isplotted from radio to x-ray frequencies on a log-log plot with the Gaunt factor (38) included. Thespecific intensity I(n , T) would have the same form. Note the gradual rise toward low frequenciesdue to the Gaunt factor. We assume a hydrogen plasma (Z = 1) of temperature T = 5 ×107 K withnumber densities ni = ne = 106 m−3.

! (Hz)

Linear-linear plot(a)

j (W

m–3

Hz–

1 )

T2

T2 > T1

Log-log plot(c)

log ! (Hz)

CDlo

g j(

!)

T2

T1

T1

! (Hz)

Semi-log plot(b)

C D

log

j(!)

T2

T1

Fig. 5.6: Thermal bremsstrahlung spectra (as pure exponentials) on linear-linear, semilog, andlog-log plots for two sources with the same ion and electron densities but differing temperatures,T2 > T1. Measurement of the specific intensities at two frequencies (e.g., at C and D) permits oneto solve for the temperature T of the plasma as well as for the emission measure ⟨ ne

2 ⟩av !. [FromH. Bradt, Astronomy Methods, Cambridge, 2004, Fig. 11.3, with permission]

exp(−hn/kT) ≈ 1.0. The dashed curve in Fig. 5.5 is thus flat as it extends to low frequencies.The effect of the Gaunt factor is shown; it modifies the exponential response noticeably butmodestly over the many decades of frequency displayed.

The curves in Fig. 5.6 qualitatively show the function jn (n , T) on linear, semilog andlog-log axes for two temperatures T2 > T1. The exponential term causes a rapid reduction(“cutoff”) of flux at a higher frequency for T2 than for T1. At low frequencies, becausethe exponential is essentially fixed at unity, the intensity is governed by the T −1/2 term if

Bradt, Astrophysics Processes

gff =3πln 2.25kT

hν⎛⎝⎜

⎞⎠⎟

For a hydrogen plasma (Z=1) with at low frequencies Gaunt factor is given byT > 3×105 K (hν ≪ kT )

- Gaunt Factor• Karzas & Latter (1961, ApJS, 6, 167):

Note that the values of Gaunt factor for are not important, since the spectrum cuts off for these values.

1961ApJS....6..167K

u = hν / kT = 4.8 ×1011ν /Tγ 2 = Z 2Ry / kT = 1.58 ×105Z 2 /T

1×10−2 ?

1961ApJS....6..167K

u = hν / kT ≫1

gff ~1 for u ~1

~1− 5 for 10−4 < u <1

• Novikov & Thorne (1973, in Black Holes, Les Houches)

U.P. = Uncertainty principle

• To obtain the formulas for non thermal bremsstrahlung, one needs to know the actual distributions of velocities, and the formula for emission from a single-speed electron must be averaged over that distribution. One also must have the appropriate Gaunt factors.

• Integrated Bremsstrahlung emission per unit volume:

!!!!!!!!!frequency average of the velocity averaged Gaunt factor:

ε ff ≡ ενff∫ dν = 2

5πe6

3mec3

2π3kme

⎛⎝⎜

⎞⎠⎟

1/2

T −1/2nineZ2 e−hν /kT g ff dν∫

= 25πe6

3mec3

2π3kme

⎛⎝⎜

⎞⎠⎟

1/2kT 1/2

h⎛⎝⎜

⎞⎠⎟nineZ

2 e−u g ff du0

∞

∫

= 2πkT3me

⎛⎝⎜

⎞⎠⎟

1/225πe6

3hmec3 nineZ

2gB

ενff == 1.4 ×10−27nineZ

2T 1/2gB (erg s−1 cm-3)

where gB ≡ e−u g ff du0

∞

∫ , u = hν / kT

gB = 1.3± 0.2

[Thermal Bremsstrahlung (free-free) Absorption]• Absorption of radiation by free electrons moving in the field of ions:

For thermal system, Kirchoff’s law says:

!!!We have then

!!!For ,

!For ,

!!Bremsstrahlung self-absorption: The medium becomes always optically thick at sufficiently small frequency. Therefore, the free-free emission is absorbed inside plasma

14π

dWdtdVdν

= jνff =αν

ff Bν (T ) Bν (T ) = 2hν 3 / c2( ) exp hν / kT( )−1⎡⎣ ⎤⎦−1

ανff = 4e6

3mehc2π3kme

⎛⎝⎜

⎞⎠⎟

1/2

T −1/2Z 2nineν−3 1− e−hν /kT( )gff

= 3.7 ×108Z 2nineT−1/2ν −3 1− e−hν /kT( )gff (cm-1)

hν ≫ kT

hν ≪ kT ανff = 4e6

3mekc2π3kme

⎛⎝⎜

⎞⎠⎟

1/2

T −3/2Z 2nineν−2gff

= 0.018Z 2nineT−3/2ν −2gff (cm

-1)

ανff = 3.7 ×108Z 2nineT

−1/2ν −3gff (cm-1) τν ∝αν

ff ∝ν −3

τν ∝ανff ∝ν −2

Overall Spectral Shape

6.1. FREE-FREE EMISSION 89

We get

α f fν =23

!

23π

"1/2 !

mec2

kT

"1/2

αr2ene

#

i

Z2i n(Zi)gf f (ν)h

$ chν

%3(1 − e−hν/kT )

= 3.7 108T−1/2ne#

i

Z2i n(Zi)ν−3gf f (ν)(1 − e−hν/kT ) cm−1 (6.22)

At hν≫ kT , the exponential is negligible and α f fν ∝ ν−3. For hν≪ kT , we get

α f fν = 0.018 T−3/2neν−2#

i

Z2i n(Zi)gf f (ν). (6.23)

The optical depth of a cosmic gas cloud to free-free absorption τ f fν = α f fν R,where R is the size of the source. Since τ f fν ∝ ν−2 at small ν, the source is alwaysoptically thick at sufficiently small frequency. It is optically thin at large frequen-cies. Let us fill a cloud of a fixed temperature T with more and more material.The evolution of the resulting spectrum is presented below.

log dWdt d

=kT/h

τ

Planck for temp T

star for all (a lot of matter)

HII regions (little matter)

log ν

ν

ν

τν>>1

>> 1ν

ν

ν

<<1

frequencyself-absorption

τ

In optically thick objects, e.g. stars, the photons have a distribution close tothe Planck distribution. When computing stellar structure, one does not considerFigure from the Lecture Note of J. Poutanen

Astronomical Examples - H II regions• The radio spectra of H II regions clearly show the flat spectrum of an optically thin thermal

source. The bright stars in the H II regions emit copiously in the UV and thus ionize the hydrogen gas.

• Continuum spectra of two H II regions, W3(A) and W3(OH):

Note a flat thermal bremsstrahlung (radio), a low-frequency cutoff (radio, self absorption), and a large peak at high frequency (infrared, ) due to heated, but still “cold” dust grains in the nebula.

P1: RPU/... P2: RPU

9780521846561c05.xml CUFX241-Bradt September 20, 2007 5:19

5.5 Spectrum of emitted photons 197

–22

–24

–26

9 10 11 12 13 14log ! (Hz)

W3(A)

W3(OH)

log

S (W

m–2

Hz–

1 )

Fig. 5.7: Continuum spectra (energy flux density) of two H II (star-forming) regions, W3(A) andW3(OH), in the complex of radio, infrared, and optical emission known as “W3.” The data (filledand open circles) and early model fits (solid and dashed lines) are shown. In each case, there is a flatthermal bremsstrahlung (radio), a low-frequency cutoff (radio), and a large peak at high frequency(infrared, 1012−1013 Hz) due to heated, but still “cold,” dust grains in the nebula. The models fitwell except at the highest frequencies. [P. Mezger and J. E. Wink, in “H II Regions & RelatedTopics,” T. Wilson and D. Downes, Eds., Springer-Verlag, p. 415 (1975); data from E. Kruegeland P. Mezger, A & A 42, 441 (1975)].

shows the expected emission lines. Comparison with real spectra from clusters of galaxiesallows one to deduce the actual amounts of different elements and ionized species in theplasma as well as its temperature. It is only in the present millennium that x-ray spectra takenfrom satellites (e.g., Chandra and the XMM Newton satellite) have had sufficient resolutionto distinguish these narrow lines.

Integrated volume emissivity

Total power radiated

The total power radiated from unit volume is found from an integration of (37) over frequencyand may be expressed as (Prob. 53)

j(T ) =! ∞

0j(n) dn = C2 g(T, Z ) Z2 ne ni T 1/2,

C2 = 1.44 ×10−40 W m3 K−1/2 (W/m3) (5.39)

where T is in degrees K, and ne and ni, the number densities of electrons and ions, respectively,are in m−3. The integration is carried out with g = 1, and a frequency-averaged Gaunt factor gis then introduced. Its value can range from 1.1 to 1.5 with 1.2 being a value that will giveresults accurate to ∼20%. Note that the total power increases with temperature for fixeddensities, as might be expected.

Figure from Bradt, Astrophysics Processes Data from Kruegel & Mezger (1975, A&A, 42, 441)

1012 −1013 Hz

ν ~1011 Hz→ λ ≈ 3 mm

Astronomical Examples - X-ray emission• Theoretical spectrum for a plasma of temperature K that takes into account quantum effects.

Comparison with real spectra from clusters of galaxies allows one to deduce the actual amounts of different elements and ionized species in the plasma as well as its temperature. It is only in the present millennium that X-ray spectra taken from satellites (e.g., Chandra and the XMM Newton satellite) have had sufficient resolution to distinguish these narrow lines The dashed lines show the effect of X-ray absorption by interstellar gas (Bradt, Astrophysics Processes).

P1: RPU/... P2: RPU

9780521846561c05.xml CUFX241-Bradt September 20, 2007 5:19

198 Thermal bremsstrahlung radiation

Fig. 5.8: Semilog plot of theoretical calculation of the volume emissivity jn , divided by electrondensity squared, of a plasma at temperature 107 K with cosmic abundances of the elements as afunction of hn/kT. The abscissa is unity at the frequency where the exponential term equals e−1.The various atomic levels are properly incorporated; strong emission lines and pronounced “edges”are the result. The dashed lines show the effect of x-ray absorption by interstellar gas. The straight-line portion of the plot falls by about a factor of ∼3 for each change of u by unity, as expected forthe exponential e−u. [From W. Tucker and R. Gould, ApJ 144, 244 (1966)]

White dwarf accretion

One can use the expression (39) for j(T) to deduce the equilibrium temperature of an opticallythin plasma into which energy is being injected. An example is gas that accretes onto thepolar region of a compact white dwarf star from a companion star (Section 2.7). As the matterflows downward, it is accelerated by gravity to very high energies. Just above the surface, itmay encounter a shock, which abruptly slows the material and raises it to a high density; thekinetic infall energy is converted into random motions (i.e., thermal energy). The material isthen a hot, optically-thin plasma that slowly settles to the surface of the white dwarf.

This plasma radiates away its thermal energy according to the expressions (36) and (39)above. At the same time it is continuously receiving energy from the infalling matter. Inequilibrium, the energy radiated by the plasma equals that being deposited by the incomingmaterial. In effect, the temperature will come to the value required for the plasma to radiateaway exactly the amount of energy it receives.

One can thus use the deposited energy as an estimate of the radiated energy. That is, ifvalues are adopted for the accretion energy being deposited per cubic meter per second andfor the densities ne and ni, the temperature of the plasma may be determined from (39).

bremsstrahlung practical applications(2)

A cluster of galaxies: Coma (z=0.0232), size ~ 1 Mpc

Typical values of IC hot gas have radiative cooling time exceeding 10 Gyr

all galaxy clusters are X- ray emitters

Coma cluster (z = 0.0232), size ~ 1 Mpc

107

Astronomical Examples - Supernova Remnants• SNR G346.6-0.2

X-ray spectra of the SNR from three of the four telescopes on-board Suzaku (represented by green, red and black). The underlying continuum is thermal bremsstrahlung, while the spectral features are due to elements such as Mg, S, Si, Ca and Fe. The roll over in the spectrum at low and high energies is due to a fall in the detector response, which is forward-modeled together with the spectrum.

Sezer et al. (2011, MNRAS, 415, 301)

Cooling function

88 CHAPTER 6. BREMSSTRAHLUNG (FREE-FREE RADIATION)

log T

log dt dVdW cooling function denoted by (T)Λ

10 104 9non-relativistic

electrons

dipole brems. p-e e-e quadrupole brems.

relativistic eT

T

T

1/2

3/2

-

- -

line coolingat small T

6.1.1 Free-Free Absorption

This is a 3-body interaction

e

Z ei

-

A useful trick to compute the absorption coefficient when you know the emis-sion coefficient is to use the fact that in complete thermodynamic equilibrium wehave emission=absorption at each ν:

j f fν = α f fν Bν(T ), (6.20)

where the lhs is the emission coefficient [erg/g/sec/ster/Hz], α is the absorptioncoefficient [cm−1], and Bν is the Planck function [erg/cm2/sec/ster/Hz].

Planck function:

Bν(T ) = 2!ν

c

"2 hνehν/kT − 1

. (6.21)

Figure from the Lecture Note of J. Poutanen

[Relativistic Bremsstrahlung]• Normally, the ions move rather slowly in comparison to the electrons.

In a frame of reference in which electron is initially at rest, the ion appears to move rapidly toward the electron. The electrostatic field of the ion appears to the electron to be a pulse of electromagnetic radiation. This radiation then Compton (or Thompson) scatters off the electron to produce. Transforming back to the rest frame of the ion (or lab frame) we obtain the bremsstrahlung emission of the electron.

• In the (primed) electron rest frame, the spectrum of the pulse:

!!In the low-frequency limit, the scattered radiation is

!!Transverse lengths are unchanged, , and . The scattering is forward-backward symmetric, we therefore have the averaged relation .

dW ′dA′dω ′

= q2

π 2b′2v2b′ω ′γ v

⎛⎝⎜

⎞⎠⎟

2

K12 b′ω ′

γ v⎛⎝⎜

⎞⎠⎟= (Ze)2

π 2b′2c2b′ω ′γ c

⎛⎝⎜

⎞⎠⎟

2

K12 b′ω ′

γ c⎛⎝⎜

⎞⎠⎟

dWdω

=σ TdW ′dA′dω ′

σ T = 8π3

e4

me2c4

⎛⎝⎜

⎞⎠⎟

← v ≈ cin the ultrarelativistic limit

b = b′ ω = γω ′(1+ β cosθ ′)ω = γω ′

∴ dW (b)dω

= 8Ze6

3πb2c2me2

bωγ 2c

⎛⎝⎜

⎞⎠⎟

2

K12 bω

γ 2c⎛⎝⎜

⎞⎠⎟

• For a plasma with a single-speeds

!!!!!!

• For a Maxwell distribution of electrons, a useful approximate expression for the frequency integrated power is given by Novikov & Thorne (1973).

!!See also Itoh et al. (2000, ApJS, 128, 125), Zekovic (2013, arXiv:1310.5639v1)

!• At higher frequencies Klein-Nishina corrections must be used.

dWdωdVdt

= nenivdW (b)dω

2πbdbbmin

∞

∫

= 16Z2e6

3c3me2 neni

bωγ 2c

⎛⎝⎜

⎞⎠⎟K1

bωγ 2c

⎛⎝⎜

⎞⎠⎟db

bmin

bmax∫

= 16Z2e6

3c3me2 neni ln

0.68γ 2cωbmin

⎛⎝⎜

⎞⎠⎟

ενff == 1.4 ×10−27nineZ

2T 1/2gB 1+ 4.4 ×10−10T( ) (erg s−1 cm-3)

[Synchrotron Radiation]• Particles accelerated by a magnetic field will radiate.

• Cyclotron radiation: For nonrelativistic velocities the radiation is called cyclotron radiation. The frequency of emission is simply the frequency of gyration in the magnetic field.

• Synchrotron radiation: For extreme relativistic particles the frequency spectrum is much more complex and can extend to many times the gyration frequency. This radiation is known as synchrotron radiation.

~~~

Fipw 61 Helkal motion of a partick in a nni~om magnetic fiU

which may be written

4 3

P = - OTC/32 &J,.

Here u T = 8 r r i / 3 is the Thomson cross section, and U, is the magnetic energy density, U, = B 2 / 8 n .

6.2 SPECTRUM OF SYNCHROTRON RADIATION: A QUALITATIVE DISCUSSION

The spectrum of synchrotron radiation must be related to the detailed variation of the electric field as seen by an observer. Because of beaming effects the emitted radiation fields appear to be concentrated in a narrow set of directions about the particle’s velocity. Since the velocity and acceleration are perpendicular, the appropriate diagram is like the one in Fig. 4.1 Id.

The observer will see a pulse of radiation confined to a time interval much smaller than the gyration period. The spectrum will thus be spread over a much broader region than one of order we/2r . This is an essential feature of synchrotron radiation.

We can find orders of magnitude by reference to Fig. 6.2. The observer will see the pulse from points 1 and 2 along the particle’s path, where these points are such that the cone of emission of angular width -l/y includes

• Consider a particle of mass m and charge q moving in a uniform magnetic field, with no electric field.

• Equations of motion:

!!The first equation implies that , Therefore, it follows that

!!Decompose the velocity into , and take dot product with B.

!!!Therefore,

!Helical motion: The perpendicular velocity component processes around B. Therefore, the motion is a combination of the uniform circular motion and the uniform motion along the field.

[Total Emitted Power]

dEdt

= d(γ mc2 )

dt= qv ⋅E = 0

dpdt

= d(γ mv)dt

= qcv ×B

γ = constant or equivalently v = constant

γ m dvdt

= qcv ×B

B ⋅ γ m dvdt

= qcv ×B⎛

⎝⎜⎞⎠⎟ →

dv!dt

= 0

→ dv⊥

dt= qγ mc

v⊥ ×B

v = v! + v⊥

v! = constant, and also

v⊥ = constant since v = constant.

• Gyrofrequency and gyroradius:

!!!!!Note that the nonrelativisitic gyrofrequency is independent of velocity.

• Total emitted power:

Since ,

!!where is the pitch angle, the angle between magnetic field and velocity.

!!For an isotropic distribution of velocities, it is necessary to average the formula over all angles.

!

d 2v⊥

dt 2= qγ mc

dv⊥

dt×B = q

γ mc⎛⎝⎜

⎞⎠⎟

2

v⊥ ×B( )×B = qγ mc

⎛⎝⎜

⎞⎠⎟

2

−v⊥ (B ⋅B)+B(B ⋅v⊥ )[ ]

= − qBγ mc

⎛⎝⎜

⎞⎠⎟

2

v⊥ gyrofrequency :ω B =qBγ mc

, gyroradius :R = v⊥ω B

ω B,nr =qBmc

P = 2q2

3c3γ 4 a⊥

2 + γ 2a!2( ) = 23γ

2 q4B2

m2c5v⊥2 =

23γ 2 q4B2

m2c5v2 sin2α

= 23re2cβ 2γ 2B2 sin2α = 2σ T c(γβ )

2UB sin2α

a⊥ =ω Bv⊥ , and a! = 0

α

cosα ≡ v ⋅BvB , re =e2

mc2,σ T = 8π

3re2 ,UB =

B2

8π

P = 43σ T c(γβ )

2UB ← sin2α = 14π

sin2α dΩ = 23∫

- Cooling Time• The energy equation becomes:

!!

• cooling time: the typical timescale for the electron to loose about of its energy is approximately

!!

• for

mc2 dγdt

= −P

tcool =energy

cooling rate= γ mc2

P= 4πmc

σ T

1γ B2 sin2α

= 15 yearsγ B2 sin2α

γ = 103

7.3. COOLING TIME OR RADIATIVE LIFETIME 93

where re = e2/mc2 - classical electron radius, σT = 8π3 r

2e =

2310

−24 cm2 - Thomsoncross-section and UB = B2/(8π) is the magnetic energy density. One can considerUB as the energy density of virtual magnetic field photons of energy !ωB withnumber densityUB/(!ωB). The emitted power= cross-section × velocity× energydensity.

One can view the process quantum-mechanically as if the electron collides(scatters) with virtual B-field photons and ”knocks” them free, this produces radi-ation.

If the electron velocity distribution is isotropic then one can average over thepitch angle (

!

sin2 αdΩ4π =23):

Pemitted =43σTcβ2γ2UB. (7.11)

This formula is valid for any velocity β.

7.3 Cooling time or radiative lifetimeConsider how the electron loose energy. The energy equation becomes:

mc2dγdt= −Pemitted = −2σTc(γ2β2)UB sin2 α. (7.12)

One can solve this ODE. (At home: assume β = 1 and solve this equation!)The typical timescale for the electron to loose about half of its energy (i.e.

cooling time) is approximately

tcool =Energy

cooling rate=γmc2

−mc2 dγdt=γmc2

Pemitted=4πmc2

σTc1

γB2 sin2 α=15 yearsγB2 sin2 α

,

(7.13)thus for γ = 103 this results in the following cooling times:

Location Typical B tcool cooling length size of object≈ ctcool

Interstellar medium 10−6 G 1010 years 1028 cm 1022 cmStellar atmosphere 1 G 5 days 1015 cm 1011 cmSupermassive black hole 104 G 10−3 sec 3 107 cm 1014 cmWhite dwarf 108 G 10−11 sec 3 mm 1000 kmNeutron star 1012 G 10−19 sec 3 10−9 cm 10 km

In strong B-fields, the electron loses its energy before it can cross the source.