KAIST Astrophysics (PH481) - Part 1 · KAIST Astrophysics (PH481) - Part 1 Week 2a Sep. 9 (Mon),...

KAIST Astrophysics (PH481) - Part 1

Week 2a Sep. 9 (Mon), 2019

Kwang-il Seon (선광일)Korea Astronomy & Space Science Institute (KASI)

University of Science and Technology (UST)

1

Thermal Bremstrahlung

2

Radiated electric vector• If a charged particle undergoes acceleration, electromagnetic wave is created.

- Electromagnetic waves are transverse. (The electric and magnetic vectors both are perpendicular to the direction of propagation.)

- The propagating electric vector � (due to the instantaneous acceleration at the retarded time or earlier time � ) lies in the plane defined by the acceleration vector � and the propagation direction � .

• Poynting vector

Et′� = t − r/c

a k

3

a

r

E

B

= c

Position of charge q

Acceleration of charge q at earlier time t = t – r/c

Electromagnetic wave at position r, at time t an = a sin

= c

= ck

Poynting vector Erad(r, t) ≈

qrc2

k × (k × a)

Brad(r, t) = k × Erad(r, t)

S =c

4πE × B

- The energy flux density carried by the electromagnetic wave is described by the Poynting vector:

The retarded time refers to conditions at the point � that existed at a time earlier than t by just the time required for light to travel from � to � .The Electric field responds to the changes after the “retarded time” delay.(For more details, see https://seoncafe.github.io/Teaching_files/2019_astrophysics/lecture2.pdf)

r′� r = 0 r

https://seoncafe.github.io/Teaching_files/2019_astrophysics/lecture2.pdf

• Larmor Formula (dipole approximation)- Energy emitted per unit time into unit solid angle

about the propagation vector � :

- Total power emitted into all angles:

- The power emitted is proportional to the square of the charge and the square of the acceleration.

- Pattern of the radiation: No radiation is emitted along the direction of acceleration, and the maximum is emitted perpendicular to acceleration.

- The instantaneous direction � is determined by � and � .

- The radiation will be linearly polarized in the plane of � and � .

k

Erad ak

a k

Larmor’s Formula 4

dPdΩ

=q2a2

4πc3sin2 Θ

P =2q2a2

3c3

1.13 The Electric Field of a Moving Charge 21

Fig. 1.11 The pattern of theradiation emitted by a chargewith its acceleration parallelto the velocity (top) andacceleration perpendicular tothe velocity (bottom)

1.13.2 Pattern

Equation 1.71 shows that the patternof the emitted radiation has a maximum per-pendicular to the acceleration, and vanishes in the directions parallel to it. Fig-ure 1.11 shows two examples: in the top part the particle has β̇ ∥ β while in thebottom part β̇ ⊥ β . The particle is non-relativistic.

Just for exercise, consider an antenna consisting of a linear piece of metal. Itis called a dipole antenna. What will be the pattern of the emitted radiation? See:http://en.wikipedia.org/wiki/File:Felder_um_Dipol.jpg.

What will be the pattern of an oscillating electron?We will soon see the modification occurring when the particle becomes relativis-

tic, both for the total emitted power and for the pattern of the emitted radiation.

v

v

a

a

Larmor’s formula

� = acceleration vectora

Brad

Erad

⇥

O

plane of � and �k a

k

a

Erad(r, t) ≈q

rc2k × (k × a)

Brad(r, t) = k × Erad(r, t)

r (for � )q > 0

• Bremsstrahlung (in German = “breaking radiation”) (or free-free emission)- Radiation due to the acceleration of a charge in the Coulomb field of another charge.- Bremsstrahlung is often called free-free emission because it is produced by free

electrons scattering off ions without being captured - the electrons are free before the interaction and remain free afterward.

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

- Then, the ratio between potential to kinetic energies is

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

Tne (cm−3) −e ni (cm−3) Ze

Bremsstrahlung 5

for typical � and � .ne ≪ 1 T ∼ 104 − 108K

Coulomb potential energythermal kinetic energy

≈Ze2/⟨r⟩

kT≈

Ze2n1/3ekT

= 1.67 × 10−7 × Z ( ne1 cm−3 )1/3 104 K

T≪ 1

← ne ≈1 electron

V≈

1 electron⟨r⟩3

• Thermal Bremsstrahlung is the radiation produced by thermal electrons distributed according to the Maxwell velocity distribution.- Thus, it is thermal emission, because it is produced by a source whose emitting

particles are in local thermodynamic equilibrium (LTE).

• 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 identical particles (electron-electron, proton-proton) is zero.- Because the accelerations of the two particles are equal in magnitude but opposite in

direction. Their radiated electric fields are equal in magnitude but opposite in sign, so that the net radiated electric field approaches zero at distances much larger than the collision impact parameter.

• The bottom line is that only the electron-ion collisions are important, and only the electrons radiate significantly.

6

• Assumptions- The plasma is assumed to be completely ionized and to consist of elections of charge � and ions of charge � , where � is the atomic number.

- In electron-ion bremsstrahlung, we treat the electron as moving in a fixed Coulomb field of the ion, since the relative accelerations are inversely proportional to the masses. Therefore, radiation from ions can be ignored.

- The electrons emit photons with energies � substantially less than their own kinetic energies.

- The energy loss of an electron, integrated over an entire collision, is only a small part of the electron kinetic energy.

−e +Ze Z

hν

7

aiae

=memi

⇠ (1800)�1 < 10�3AAACIHicbVDLSgMxFM34rPU16tJNsAh1YZnYQrtQKLhxWcE+oFNLJs20ocnMkGSEMsynuPFX3LhQRHf6NabtLLT1Qu49Oedeknu8iDOlHefLWlldW9/YzG3lt3d29/btg8OWCmNJaJOEPJQdDyvKWUCbmmlOO5GkWHictr3x9VRvP1CpWBjc6UlEewIPA+YzgrWh+nbV9SUmCe6z1CSaXs3vwkCTWOoqJmAR1Rzn7D45R+klckwtp3274JScWcBlgDJQAFk0+vanOwhJLGigCcdKdZET6V6CpWaE0zTvxopGmIzxkHYNDLCgqpfMFkzhqWEG0A+lOYGGM/b3RIKFUhPhmU6B9UgtalPyP60ba7/WS1gQxZoGZP6QH3OoQzh1Cw6YpETziQGYSGb+CskIG4e08TRvTECLKy+D1kUJlUvotlKoVzI7cuAYnIAiQKAK6uAGNEATEPAInsEreLOerBfr3fqYt65Y2cwR+BPW9w+GKqJt

Emission radiated per collision• Small-angle scattering approximation:

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

- Collision time: We assume that the interaction between the electron and the ion happens only when the electron passes close to the ion. The characteristic time � is

- Then, the radiation consists of a single pulse at an angular frequency of � . The frequency of the radiation due to a single collision is thus

τ

ω ≈1τ

=υb

8

b

Ze

−e

R

v

'AAAB7nicbVDLSgNBEOz1GeMr6tHLYBA8hV0N6DHgxWME84BkCbOTTjJkdnaYmQ2EJR/hxYMiXv0eb/6Nk2QPmljQUFR1090VKcGN9f1vb2Nza3tnt7BX3D84PDounZw2TZJqhg2WiES3I2pQcIkNy63AttJI40hgKxrfz/3WBLXhiXyyU4VhTIeSDzij1kmt7oRqNeK9Utmv+AuQdRLkpAw56r3SV7efsDRGaZmgxnQCX9kwo9pyJnBW7KYGFWVjOsSOo5LGaMJsce6MXDqlTwaJdiUtWai/JzIaGzONI9cZUzsyq95c/M/rpHZwF2ZcqtSiZMtFg1QQm5D576TPNTIrpo5Qprm7lbAR1ZRZl1DRhRCsvrxOmteV4KYSPFbLtWoeRwHO4QKuIIBbqMED1KEBDMbwDK/w5invxXv3PpatG14+cwZ/4H3+AHgQj5s=

ν =ω2π

≈υ

2πb

τ ≈bυ

The smaller the impact parameter, the higher the emitted frequency.

• Total emitted radiation by a single electron:- During the interaction we assume that the acceleration is constant and equal to

- From the Larmor formula, we get the instantaneous power:

- We assume that the acceleration is constant while it is in the vicinity of the ion for time duration � and to be zero before and after this period.

- The total energy emitted by the electron is then the sum (integral) for the entire duration of the collision.

- This is the total energy radiated by a single electron of speed � as it passes an ion of charge � with impact parameter � .

τ

υZe b

9

a ≈ −Ze2

meb2← F = mea ≈ −

Ze2

b2

P =2e2a2

3c3≈

2e2

3c3Z2e4

m2e b4=

2Z2e6

3m2e c3b4

W(b, υ) ≈ Pτ ≈2Z2e6

3m2e c3b3υ

Radiation from Single-speed electron beam• Consider only the electrons that intersect a narrow annulus of radius � and width � surrounding the ion. One can then calculate the power emitted by those electrons.

• If � the electron density, then the electron flux is � . The energy emitted by the annulus is

• Recall the relation between the impact parameter and frequency:

• The power per unit-frequency interval is

• Surprisingly, this is independent of frequency.

bdb

ne = neυ

10

𝒫b(b, υ)db = W(b, υ)neυ(2πb)db

b =υ

2πν|db | =

υ2πν2

|dν |

𝒫ν(ν, υ)dν = 𝒫b(b, υ)db

=8π2

3Z2e6

m2e c3neυ

dν

P1: RPU/... P2: RPU

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

5.4 Thermal electrons and a single ion 191

b

db

Area of annulus 2!b db

Ion

Electron flux

Fig. 5.4: Flux of electrons approaching an ion with the annular region representing the target areaat impact radius bin db. The number of electrons that pass through the annuals can be calculatedfrom the flux of electrons and the target area.

Single-speed electron beam

Consider that the ion is immersed in a parallel beam of electrons of speed v. Let us first con-sider only the electrons that intersect a narrow annulus of radius band width dbsurroundingthe ion (Fig. 5.4). One can then calculate the power emitted by those electrons as a functionof the emitted frequency n and the speed v.

Power from the annulus

If the density of electrons in the beam is ne, the electron flux is nev (electrons m−2 s−1). Thenumber per second that would strike an annulus of radius band width db is just this fluxtimes the area of the annulus, namely, nev 2πbdb. The energy emitted by these per unit time(emitted power per ion) is just this number times Q(b, v), the energy emitted by each electronwith impact parameter band speed v, namely (17),

Pb(b, v) db= Q(b, v)nev2πbdb. (W/ion in dbat b) (5.22)

This is the power coming from each ion due to electrons of density ne and speed v impingingon the ion at radius bin db.

The electrons actually arrive at the ion from all directions, and so the proper density to usein (22) would have been ne (d!/4π), but integration over all directions would directly yieldthe same result because

∫?d! = 4π sr.

Power per unit frequency interval

The quantity Pb(b, v) is the power emitted per unit impact-parameter interval. We now convertit to power per unit frequency interval, Pn (n , v), the unit used for photon spectra. The variablesband n have a one-to-one correspondence (20) as do their differentials (21). Thus, Pb(b, v)can be integrated over some range of bto obtain the emitted power from that range; similarly,Pn (n , v) can be integrated over the equivalent range in frequency. Because the ranges areequivalent, the integrated powers are the same:

∫ b2

b1Pb(b, v) db= −

∫ n2

n1

Pn (n, v) dn. (5.23)

neυ

Radiation from thermal electrons• Consider the electrons having the Maxwell distribution

• The volume emissivity is obtained by integrating the power per unit-frequency over the particle velocity and multiplying the number density of ions:

- The lower limit of the integral is the smallest velocity that an electron can still emit a photon of energy � . An electron cannot emit a photon of energy larger than it initially had.

- Because � is independent of frequency, the frequency dependence comes in only through � .

hν

𝒫νυmin

11

jν = ni ∫∞

υmin

f(υ)𝒫ν(ν, υ)4πυ2dυ

f(υ) = ( me2πkT )3/2

exp (− meυ2

2kT )

jν =323 ( 23 π

3

m3e k )1/2

Z2e6

c3neni

e−hν/kT

T1/2

hν =12

meυ2min → υmin =2hνme

Spectrum of Thermal Bremsstrahlung• We have approximately derived the result.

- There would be errors due to the approximations we have made.- However, it turns out to be correct except a minor factor, which is called the Gaunt

factor.

- where � is the velocity-averaged free-free Gaunt factor.- Summing over all ion species gives the emissivity:

gff

12

jffν =323 ( 23 π

3

m3e k )1/2

Z2e6

c3neni

e−hν/kT

T1/2gff

= 6.8 × 10−38gffZ2nineT−1/2e−hν/kT (erg cm−3 s−1 Hz−1)

jffν = 6.8 × 10−38 ∑i

gffZ2nineT−1/2e−hν/kT (erg cm−3 s−1 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.

jffν ∝ e−hν/kTν ∼ hT/k

13

P1: RPU/... P2: RPU


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 ⟨ ne2 ⟩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]

For a hydrogen plasma (Z = 1) with at low frequenciesGaunt factor is given byT > 3⇥ 105 K

AAACAnicbVDJSgNBEO2JW4zbqCfx0hgET2HGuJ0k4EXwEiEbZMbQ0+lJmvQsdNeIYQhe/BUvHhTx6ld482/sJHPQxAcFj/eqqKrnxYIrsKxvI7ewuLS8kl8trK1vbG6Z2zsNFSWSsjqNRCRbHlFM8JDVgYNgrVgyEniCNb3B1dhv3jOpeBTWYBgzNyC9kPucEtBSx9yr4UtcdoAHTGHbujt1gD1Aim9GHbNolawJ8DyxM1JEGaod88vpRjQJWAhUEKXathWDmxIJnAo2KjiJYjGhA9JjbU1Dole66eSFET7UShf7kdQVAp6ovydSEig1DDzdGRDoq1lvLP7ntRPwL9yUh3ECLKTTRX4iMER4nAfucskoiKEmhEqub8W0TyShoFMr6BDs2ZfnSeO4ZJdL9u1JsXKWxZFH++gAHSEbnaMKukZVVEcUPaJn9IrejCfjxXg3PqatOSOb2UV/YHz+AMHFlbc=

(h⌫ ⌧ kT )AAAB83icbVDLSgNBEOyNrxhfUY9eBoMQL2HXiHoMePEYIS/ILmF2MpsMmZ1d5iGEJb/hxYMiXv0Zb/6Nk2QPmljQUFR1090Vppwp7brfTmFjc2t7p7hb2ts/ODwqH590VGIkoW2S8ET2QqwoZ4K2NdOc9lJJcRxy2g0n93O/+0SlYolo6WlKgxiPBIsYwdpKfnXsC+Nzjiaty0G54tbcBdA68XJSgRzNQfnLHybExFRowrFSfc9NdZBhqRnhdFbyjaIpJhM8on1LBY6pCrLFzTN0YZUhihJpS2i0UH9PZDhWahqHtjPGeqxWvbn4n9c3OroLMiZSo6kgy0WR4UgnaB4AGjJJieZTSzCRzN6KyBhLTLSNqWRD8FZfXiedq5pXr3mP15XGTR5HEc7gHKrgwS004AGa0AYCKTzDK7w5xnlx3p2PZWvByWdO4Q+czx/8LZD0

g↵ =

p3

⇡ln

✓2.25kT

h⌫

◆

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

- Gaunt Factor• Note that the values of Gaunt factor for � are not important, since

the spectrum cuts off for these values.u = hν/kT ≫ 1

14

u = h⌫/kT = 4.8⇥ 1011⌫/T�2 = Z2Ry/kT = 1.58⇥ 105Z2/T

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

g↵ ⇠(1 for u ⇠ 11� 5 for 10�4 < u < 1

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

van Hoof et al. (2014, MNRAS, 444, 420)

log10 uAAAB8XicbVC7SgNBFL3jM8ZXYkqbwSBYhd1YaBmwsYxgHpiEMDs7mwyZnVlmZoWwpPMTbCwUsfUH/A47P8TeyaPQxAMXDufcy733BIngxnreF1pb39jc2s7t5Hf39g8OC8WjplGppqxBlVC6HRDDBJesYbkVrJ1oRuJAsFYwupr6rXumDVfy1o4T1ovJQPKIU2KddNcVatDPfG+S9gtlr+LNgFeJvyDlWvH7o4TCh3q/8NkNFU1jJi0VxJiO7yW2lxFtORVsku+mhiWEjsiAdRyVJGaml80unuBTp4Q4UtqVtHim/p7ISGzMOA5cZ0zs0Cx7U/E/r5Pa6LKXcZmklkk6XxSlAluFp+/jkGtGrRg7Qqjm7lZMh0QTal1IeReCv/zyKmlWK/55xb9xaVRhjhwcwwmcgQ8XUINrqEMDKEh4hGd4QQY9oVf0Nm9dQ4uZEvwBev8BSpiTkg==

log 1

0g ↵

AAACB3icbVC7TsMwFHV4lvJK6YiELCokpiopA4yVWBiLRB9SE0WO47RW7TiyHaQqytaFX2FhAKGu/AIbH8KO+xig5UiWjs+599r3hCmjSjvOl7WxubW9s1vaK+8fHB4d25WTjhKZxKSNBROyFyJFGE1IW1PNSC+VBPGQkW44up353UciFRXJgx6nxOdokNCYYqSNFNhnHhODIHedwhOmbjYmN3dPchjHRRHYNafuzAHXibsktWble1q1okkrsD+9SOCMk0RjhpTqu06q/RxJTTEjRdnLFEkRHqEB6RuaIE6Un8/3KOCFUSIYC2lOouFc/d2RI67UmIemkiM9VKveTPzP62c6vvFzmqSZJglePBRnDGoBZ6HAiEqCNRsbgrCk5q8QD5FEWJvoyiYEd3XlddJp1N2runtv0miABUrgFJyDS+CCa9AEd6AF2gCDCXgGr+DNerJerHdruijdsJY9VfAH1scPn9acsw==

Determination of the free–free Gaunt factor 425

temperature Te = 10 000 K, the longest wavelength that can bemodelled with the S98 data is ∼1.44 cm. Radio observations atlonger wavelengths are routinely made and CLOUDY should be ableto model those. In the view of the stated facts, we have used a muchlarger parameter space in our calculations: 10log γ 2 = −6(0.2)10and 10log u = −16(0.2)13. This is larger even than the current needsof CLOUDY and anticipates possible future modifications to the code,such as the addition of higher-Z elements and/or lowering the low-frequency cut-off.

The integration shown in equation (22) is carried out using anadaptive step-size algorithm based on equation 4.1.20 of Press et al.(1992) for carrying out a single step. This algorithm is open atthe left-hand side, thus avoiding the awkward evaluation of theintegrand at x = 0. During the evaluation of the integral, at everystep an estimate is made of the remainder of the integral to infinityby assuming that gff is constant. This estimate is reasonable as gffis only slowly increasing. The integration is terminated when thisestimate is less than 1 per cent of the requested tolerance. Therequested tolerance of the thermally averaged Gaunt factor is a freeparameter and the routine calculates an estimate of the actual errorin the final result taking into account both the imprecisions due tothe finite step-size and the error in the non-averaged Gaunt factor.For the electronic table, we used a requested relative tolerance of10−5. The data are presented in Fig. 2 and Table 3. The data canalso be downloaded in electronic form (see Section 5). Note thatthe data shown in Table 3 were calculated to a higher precision toassure that all numbers shown are correctly rounded. In additionto the electronic table, we also provide simple programs whichallow the user to interpolate the table. Testing of the interpolationalgorithm showed that the relative error was less than 1.5 × 10−4everywhere.

Comparing our results with those of S98, we noted the seriousproblem that the parameters 10log γ 2 and 10log u were transposedin table 2 of S98, as well as in the electronic version of that table.After correcting for this error, there were some minor discrepancieswhen we compared the numerical values in the electronic table ofS98 to our results. The largest relative error is for 10log γ 2 = −1.8and 10log u = 0.5 and amounts to almost 0.13 per cent. The medianrelative discrepancy is approximately 5 × 10−5. So it appears thatthe discrepancies we reported in Section 2.4 did not have a sig-nificant impact on the calculation of the thermally averaged Gauntfactor by S98.

4 TH E TOTA L FR E E – F R E E G AU N T FAC TO R

For completeness we will also include a calculation of the totalfree–free Gaunt factor which is integrated over frequency. Thisquantity is useful if one wants to calculate the total cooling dueto Bremsstrahlung without spectrally resolving the process. Theformula for this quantity is given by KL61 and S98

⟨gff (γ 2)⟩ =∫ ∞

0e−u⟨gff (γ 2, u)⟩du. (23)

Due to the similarity of the integrals in equations (22) and (23), wecan use the same adaptive step-size algorithm discussed in Section 3to calculate the data. For the evaluations of ⟨gff(γ 2, u)⟩, we used arelative tolerance of 10−6 to prevent them dominating the error in⟨gff(γ 2)⟩. The results are shown in Table 4 and Fig. 3. The computedvalues are also available in electronic form (see Section 5). The datain Table 4 show a small systematic offset w.r.t. the data in table 3 ofS98, ranging between +0.000 69 for 10log γ 2 = −4 and +0.000 21for 10log γ 2 = 4. This offset is likely due to the missing part of the

Figure 2. The base-10 logarithm of the thermally averaged free–free Gauntfactor as a function of u (top panel) and γ 2 (bottom panel). Thick curves arelabelled with the values of 10log γ 2 (top panel) and 10log u (bottom panel)in increments of 5 dex. The thin curves have a spacing of 1 dex. In the toppanel the Gaunt factors approach a limiting curve for 10log γ 2 < −2 and areindistinguishable in the plot.

integral below u = 10−4 in S98. The extended range in γ 2 of thedata presented here makes the limiting behaviour of the functionclear. Both for γ 2 → 0 and γ 2 → ∞ the function approaches anasymptotic value. Using our data, we determined the following fitsto the limiting behaviour of the function:

⟨gff (γ 2)⟩ ≈1.102 635 + 1.186γ + 0.86γ 2 for γ 2 < 10−6, (24)

and

⟨gff (γ 2)⟩ ≈1 + γ −2/3 for γ 2 > 1010. (25)

These extrapolations are expected to reach a relative precision of10−5 or better everywhere they are defined. The data in Table 4 canbe interpolated using rational functions

⟨gff (g)⟩ ≈a0 + a1g + a2g2 + a3g3 + a4g4

b0 + b1g + b2g2 + b3g3 + b4g4, (26)

where g = 10log γ 2. To limit the degree of the rational function, wemade two separate fits for the range −6 ≤ g≤ 0.8 and 0.8 ≤ g≤ 10.These fits achieve a relative error less than 3.5 × 10−5 every-where in its range for the first fit and 8.8 × 10−5 for the sec-ond. The coefficients are given in Table 5. We have implemented

MNRAS 444, 420–428 (2014)

Dow

nloaded from https://academ

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nras/article-abstract/444/1/420/1016620 by guest on 19 March 2019

log10(Z2Ry/kT ) =

AAACBHicbVC7TgJBFJ3FF+Br1ZJmIjHBBnex0MaExMYSlVcEJLPDABNmH5m5a1w3FDb+io2Fxtj6A3YWJv6Nw6NQ8CQ3OTnn3tx7jxMIrsCyvo3EwuLS8koylV5dW9/YNLe2q8oPJWUV6gtf1h2imOAeqwAHweqBZMR1BKs5g9ORX7thUnHfK0MUsJZLeh7vckpAS20z0xR+rx3b1jB3dV1oAruF+CIaHgzK+ydtM2vlrTHwPLGnJFtM3n19XJZSpbb52ez4NHSZB1QQpRq2FUArJhI4FWyYboaKBYQOSI81NPWIy1QrHj8xxHta6eCuL3V5gMfq74mYuEpFrqM7XQJ9NeuNxP+8Rgjd41bMvSAE5tHJom4oMPh4lAjucMkoiEgTQiXXt2LaJ5JQ0LmldQj27MvzpFrI24d5+1ynkUMTJFEG7aIcstERKqIzVEIVRNE9ekTP6MV4MJ6MV+Nt0powpjM76A+M9x+Y15pN

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

- For LTE, Kirchhoff’s law says:

- We can derive

- For � ,

- For � ,

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

hν ≫ kT

hν ≪ kT

15

↵↵⌫ =4e6

3mehc

✓2⇡

3kme

◆1/2nineZ

2T�1/2⌫�3⇣1� e�h⌫/kT

⌘g↵

= 3.7⇥ 108 nineZ2T�1/2⌫�3⇣1� e�h⌫/kT

⌘g↵ (cm

�1)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

↵↵⌫ = 3.7⇥ 108nineZ2T�1/2⌫�3g↵ (cm�1)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

↵↵⌫ =4e6

3mekc

✓2⇡

3kme

◆1/2nineZ

2T�3/2⌫�2g↵

= 0.018nineZ2T�3/2⌫�2g↵

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

⌧⌫ / ↵↵⌫ / ⌫�3AAACGnicbZBNSwMxEIazflu/qh69BIvgxbJrRT0KXjwqWBW6tcymWRvMJiGZFcrS3+HFv+LFgyLexIv/xmyt4NdA4OV5Z5jMmxgpHIbhezA2PjE5NT0zW5mbX1hcqi6vnDmdW8abTEttLxJwXArFmyhQ8gtjOWSJ5OfJ9WHpn99w64RWp9g3vJ3BlRKpYIAedapRjJB3YpXHxmqDOgZpelCCyyK2GU3TwZdToq3GoFOthfVwWPSviEaiRkZ13Km+xl3N8owrZBKca0WhwXYBFgWTfFCJc8cNsGu44i0vFWTctYvhaQO64UmXptr6p5AO6feJAjLn+lniOzPAnvvtlfA/r5Vjut8uhDI5csU+F6W5pKhpmRPtCssZyr4XwKzwf6WsBxYY+jQrPoTo98l/xdl2PWrUo5Od2sHuKI4ZskbWySaJyB45IEfkmDQJI7fknjySp+AueAieg5fP1rFgNLNKflTw9gFeoaJC

⌧⌫ / ↵↵⌫ / ⌫�2AAACGnicbZBNSwMxEIazflu/qh69BIvgxbKroh4LXjwqWBW6tcymWRvMJiGZFcrS3+HFv+LFgyLexIv/xmyt4NdA4OV5Z5jMmxgpHIbhezA2PjE5NT0zW5mbX1hcqi6vnDmdW8abTEttLxJwXArFmyhQ8gtjOWSJ5OfJ9WHpn99w64RWp9g3vJ3BlRKpYIAedapRjJB3YpXHxmqDOgZpelCCyyK2GU3TwZdToq3tQadaC+vhsOhfEY1EjYzquFN9jbua5RlXyCQ414pCg+0CLAom+aAS544bYNdwxVteKsi4axfD0wZ0w5MuTbX1TyEd0u8TBWTO9bPEd2aAPffbK+F/XivH9KBdCGVy5Ip9LkpzSVHTMifaFZYzlH0vgFnh/0pZDyww9GlWfAjR75P/irPterRTj052a429URwzZI2sk00SkX3SIEfkmDQJI7fknjySp+AueAieg5fP1rFgNLNKflTw9gFdHKJB

jffν = αffν Bν(T ) where Bν(T ) =2hν3

c21

ehν/kT − 1

Overall Spectral Shape• Spectral shape

- At low frequencies (optically thick emission),

- At high frequencies (optically thin emission),

• This spectrum shows the bremsstrahlung intensity from a source of radius � cm and temperature � .- The Gaunt factor is set to unity for simplicity.

- The density � varies from � to � increasing by a factor 10 for each curve.

- As the density increases, the optical depth also increases and the spectrum approaches the blackbody one.

R = 1015T = 107 K

ne = np 1010 cm−31018 cm−3

16

26 2 Bremsstrahlung and Black Body

Fig. 2.1 The bremsstrahlungintensity from a source ofradius R = 1015 cm, densityne = np = 1010 cm−3 andvarying temperature. TheGaunt factor is set to unity forsimplicity. At smallertemperatures the thin part ofIν is larger (∝T −1/2), even ifthe frequency integrated I issmaller (∝T 1/2)

Fig. 2.2 The bremsstrahlungintensity from a source ofradius R = 1015 cm,temperature T = 107 K. TheGaunt factor is set to unity forsimplicity. The densityne = np varies from1010 cm−3 (bottom curve) to1018 cm−3 (top curve),increasing by a factor 10 foreach curve. Note theself-absorbed part (∝ν2), theflat and the exponential parts.As the density increases, theoptical depth also increases,and the spectrum approachesthe black-body one

2.1.2 From Bremsstrahlung to Black Body

As any other radiation process, the bremsstrahlung emission has a self-absorbedpart, clearly visible in Fig. 2.1. This corresponds to optical depths τν ≫ 1. Theterm ν−3 in the absorption coefficient αν ensures that the absorption takes placepreferentially at low frequencies. By increasing the density of the emitting (and ab-sorbing) particles, the spectrum is self-absorbed up to larger and larger frequencies.When all the spectrum is self-absorbed (i.e. τν > 1 for all ν), and the particles be-long to a Maxwellian, then we have a black-body. This is illustrated in Fig. 2.2: allspectra are calculated for the same source size (R = 1015 cm), same temperature(T = 107 K), and what varies is the density of electrons and protons (by a factor 10)from ne = np = 1010 cm−3 to 1018 cm−3. As can be seen, the bremsstrahlung inten-sity becomes more and more self-absorbed as the density increases, until it becomesa black-body. At this point increasing the density does not increase the intensity

1010 cm−3

1011 cm−3

1012 cm−3

self-absorbedpart flat part

exponentialpart

Iν = Sν = Bν ∝ ν2

Iν = ∫ jνds ∝ e−hν/kT= constant if hν ≪ kT

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, � Hz, 300-30 � m) due to heated, but still “cold” dust grains in the nebula.

1012 − 1013 μ

17

P1: RPU/... P2: RPU


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 ḡ(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 ḡis 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)

ν ~1011 Hz→ λ ≈ 3 mm

The term H II is pronounced “H two” by astronomers. “H” indicates hydrogen, and “II” is the Roman numeral for 2.

Astronomers use “I” for neutral atoms, “II” for singly-ionized, “III” for doubly-ionized, etc.

An H II region in the Large Magellanic Cloud (observed with MUSE, VLT)

Astronomical Examples - X-ray emission from galaxy clusters• 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].

107

18

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

P1: RPU/... P2: RPU


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).

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.

19

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

Synchrotron Radiation

20

Synchrotron Radiation• Particles accelerated by a magnetic field will also radiate. Acceleration by a magnetic field

produces magnetobremsstrahlung, the German word for “magnetic braking radiation.”

• Cyclotron radiation: For non-relativistic 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.

• Synchrotron radiation is ubiquitous in astronomy.- It accounts for most of the radio emission from active galactic nuclei (AGN), which are thought to be

powered by supermassive black holes in galaxies and quasars.- It dominates the radio continuum emission from star-forming galaxies.

21

Galactic Synchrotron

• WMAP 23GHz emission and polarization map

WMAP 23GHz emission and polarization map

Puzzling radiation from the Crab nebula• A large part of the supernova remnant called the Crab

nebula appears as an bluish haze on the sky, and the origin of this light was, for a long time, a big mystery.- The bluish luminosity in the optical band could be considered

to indicate the presence of a hot, optically thin thermal plasma with a temperature of ~ 50,000 K.

- Such a hot plasma would radiate strong optical emission lines from excited atoms in the plasma. However, no strong spectral lines are observed. The spectrum is a smooth continuum.

- The puzzle of the source of radiation from the Crab nebula was solved dramatically when a Russian scientist (I. S. Shklovsk) postulated in 1953 that the bluish radiation might be the same as that previously discovered in man-made electron accelerators.

- He therefore suggested that optical astronomers look to see if the radiation from the Crab is polarized. They did (in 1954) and found it to be so.

- Nevertheless, this explanation was quiet a surprise; it required that the radiating electrons have energies in excess of � eV. This raised the question of how the electron attained such high energies. Later (1969), it was found that the energy source is a spinning neutron star, the Crab pulsar.

1011

22

Crab Nebula (HST mosaic image)The rapidly spinning (30 times a second) neutron star embedded in the center of the nebula is powering the nebula’s interior bluish glow. The blue light comes from electrons whirling at nearly the speed of light around magnetic field lines from the neutron star.

Pulsar Wind Nebula (HST in red + Chandra X-ray image in blue)

Larmor’s formula• Relativistic version of the Larmor’s formula

- Recall that

in an instantaneous rest frame of the particle.

- The power emitted by an accelerated electron is obtained to be

23

P =2q2

3c3γ4 (a2⊥ + γ2a2∥)

P′� =2q2

3c3a′�

2

β =υc

γ =1

1 − β2: Lorentz factor

a⊥: acceleration perpendicular to the electron velocity va∥: acceleration parallel to the electron velocity v

1.13 The Electric Field of a Moving Charge 21

Fig. 1.11 The pattern of theradiation emitted by a chargewith its acceleration parallelto the velocity (top) andacceleration perpendicular tothe velocity (bottom)

1.13.2 Pattern

Equation 1.71 shows that the patternof the emitted radiation has a maximum per-pendicular to the acceleration, and vanishes in the directions parallel to it. Fig-ure 1.11 shows two examples: in the top part the particle has β̇ ∥ β while in thebottom part β̇ ⊥ β . The particle is non-relativistic.

Just for exercise, consider an antenna consisting of a linear piece of metal. Itis called a dipole antenna. What will be the pattern of the emitted radiation? See:http://en.wikipedia.org/wiki/File:Felder_um_Dipol.jpg.

What will be the pattern of an oscillating electron?We will soon see the modification occurring when the particle becomes relativis-

tic, both for the total emitted power and for the pattern of the emitted radiation.

v

v

a

a

Beaming Effect• Imaging two frames K and K’. K’ moves with a velocity � relative to K.

• If a point has a velocity � in frame K’, what is its velocity � in frame K?

• Aberration formula: the directions of the velocities in the two frames are related by

• Aberration of light ( � , � )

υu′� u

u′� = c β = υ/c

24

uk =u0k + v

1 + vu0k/c2

u? =u0?

�⇣1 + vu0k/c

2⌘

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

tan ✓ =sin ✓0

� (cos ✓0 + �)

cos ✓ =cos ✓0 + �

1 + � cos ✓0

sin ✓ =sin ✓0

� (1 + � cos ✓0)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

tan

✓✓

2

◆=

✓1� �1 + �

◆2tan

✓✓0

2

◆! ✓ < ✓0

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

tan ✓ =u?uk

=u0?

�⇣u0k + v

⌘ = u0 sin ✓0

� (u0 cos ✓0 + v)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

� = angle of the velocity measured in K� = angle measured in K’θθ′�

xAAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokW9Fjw4rEF+wFtKJvtpF272YTdjVhCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8O/Pbj6g0j+W9mSToR3QoecgZNVZqPPVLZbfizkFWiZeTMuSo90tfvUHM0gilYYJq3fXcxPgZVYYzgdNiL9WYUDamQ+xaKmmE2s/mh07JuVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NeGNn3GZpAYlWywKU0FMTGZfkwFXyIyYWEKZ4vZWwkZUUWZsNkUbgrf88ippXVa8q4rXqJZr1TyOApzCGVyAB9dQgzuoQxMYIDzDK7w5D86L8+58LFrXnHzmBP7A+fwB4v+M8Q==

x0AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ6KokW9Fjw4rGK/YA2lM120i7dbMLuRiyh/8CLB0W8+o+8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6mfqtR1Sax/LBjBP0IzqQPOSMGivdP531SmW34s5AlomXkzLkqPdKX91+zNIIpWGCat3x3MT4GVWGM4GTYjfVmFA2ogPsWCpphNrPZpdOyKlV+iSMlS1pyEz9PZHRSOtxFNjOiJqhXvSm4n9eJzXhtZ9xmaQGJZsvClNBTEymb5M+V8iMGFtCmeL2VsKGVFFmbDhFG4K3+PIyaV5UvMuKd1ct16p5HAU4hhM4Bw+uoAa3UIcGMAjhGV7hzRk5L8678zFvXXHymSP4A+fzB0OGjSI=

yAAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0oMeCF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipOR2UK27VXYCsEy8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAmelfqoxoWxCR9izVNIItZ8tDp2RC6sMSRgrW9KQhfp7IqOR1tMosJ0RNWO96s3F/7xeasJbP+MySQ1KtlwUpoKYmMy/JkOukBkxtYQyxe2thI2poszYbEo2BG/15XXSvqp611WvWavUa3kcRTiDc7gED26gDvfQgBYwQHiGV3hzHp0X5935WLYWnHzmFP7A+fwB5IOM8g==

y0AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0m0oMeCF49V7Ae0oWy2k3bpZhN2N0Io/QdePCji1X/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHkyXoR3QoecgZNVZ6yM775Ypbdecgq8TLSQVyNPrlr94gZmmE0jBBte56bmL8CVWGM4HTUi/VmFA2pkPsWipphNqfzC+dkjOrDEgYK1vSkLn6e2JCI62zKLCdETUjvezNxP+8bmrCG3/CZZIalGyxKEwFMTGZvU0GXCEzIrOEMsXtrYSNqKLM2HBKNgRv+eVV0rqseldV775WqdfyOIpwAqdwAR5cQx3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzB0ULjSM=

u0AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0m0oMeCF49V7Ae0oWy2k3bpZhN2N0IJ/QdePCji1X/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0kJ73yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzXhjZ9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqaV1Wvauqd1+r1Gt5HEU4gVO4AA+uoQ530IAmMAjhGV7hzRk7L86787FoLTj5zDH8gfP5Az73jR8=

vAAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokW9Fjw4rEF+wFtKJvtpF272YTdTaGE/gIvHhTx6k/y5r9x2+agrQ8GHu/NMDMvSATXxnW/nY3Nre2d3cJecf/g8Oi4dHLa0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4fu63J6g0j+WjmSboR3QoecgZNVZqTPqlsltxFyDrxMtJGXLU+6Wv3iBmaYTSMEG17npuYvyMKsOZwFmxl2pMKBvTIXYtlTRC7WeLQ2fk0ioDEsbKljRkof6eyGik9TQKbGdEzUivenPxP6+bmvDOz7hMUoOSLReFqSAmJvOvyYArZEZMLaFMcXsrYSOqKDM2m6INwVt9eZ20riveTcVrVMu1ah5HAc7hAq7Ag1uowQPUoQkMEJ7hFd6cJ+fFeXc+lq0bTj5zBn/gfP4A3/eM7w==

KAAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KokW9FjwInhpwX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut7O2vrG5tV3YKe7u7R8clo6OWzpOFcMmi0WsOgHVKLjEpuFGYCdRSKNAYDsY38789hMqzWP5YCYJ+hEdSh5yRo2VGvf9UtmtuHOQVeLlpAw56v3SV28QszRCaZigWnc9NzF+RpXhTOC02Es1JpSN6RC7lkoaofaz+aFTcm6VAQljZUsaMld/T2Q00noSBbYzomakl72Z+J/XTU1442dcJqlByRaLwlQQE5PZ12TAFTIjJpZQpri9lbARVZQZm03RhuAtv7xKWpcV76riNarlWjWPowCncAYX4ME11OAO6tAEBgjP8ApvzqPz4rw7H4vWNSefOYE/cD5/AJ7LjMQ=

K 0AAAB6XicbVBNS8NAEJ34WetX1aOXxSJ6KokW9FjwInipYj+gDWWznbRLN5uwuxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWMbqZ+6wmV5rF8NOME/YgOJA85o8ZKD3dnvVLZrbgzkGXi5aQMOeq90le3H7M0QmmYoFp3PDcxfkaV4UzgpNhNNSaUjegAO5ZKGqH2s9mlE3JqlT4JY2VLGjJTf09kNNJ6HAW2M6JmqBe9qfif10lNeO1nXCapQcnmi8JUEBOT6dukzxUyI8aWUKa4vZWwIVWUGRtO0YbgLb68TJoXFe+y4t1Xy7VqHkcBjuEEzsGDK6jBLdShAQxCeIZXeHNGzovz7nzMW1ecfOYI/sD5/AH/Foz1

✓0AAAB7nicbZDLSgMxFIYz9VbrrerSTbCIrsqMCrqz4MZlBXuBdiiZ9EwbmskMyRmhlD6EG8GKuPUVfA13vo2Ztgtt/SHw8f/nkHNOkEhh0HW/ndzK6tr6Rn6zsLW9s7tX3D+omzjVHGo8lrFuBsyAFApqKFBCM9HAokBCIxjcZnnjEbQRsXrAYQJ+xHpKhIIztFajjX1AdtopltyyOxVdBm8OpZvPl0yTaqf41e7GPI1AIZfMmJbnJuiPmEbBJYwL7dRAwviA9aBlUbEIjD+ajjumJ9bp0jDW9imkU/d3x4hFxgyjwFZGDPtmMcvM/7JWiuG1PxIqSREUn30UppJiTLPdaVdo4CiHFhjXws5KeZ9pxtFeqGCP4C2uvAz187J3Ufbu3VLlksyUJ0fkmJwRj1yRCrkjVVIjnAzIE5mQVydxnp03531WmnPmPYfkj5yPHyPjk9k=

observer particle

• Beaming (“headlight”) effect:- Beam half-angle = the half-angle of the cone that includes one-half of the rays.- If photons are emitted isotropically in � , then half will have � and half � .

- Consider a photon emitted at right angles (� ) to � in � . Then the half-angle corresponds to the following angle in � .

- For highly relativistic speeds, � , � becomes small:

- Therefore, in frame � , photons are concentrated in the forward direction, with half of them lying within a cone of half-angle � .

K′� θ′� < π/2 θ′� > π/2θ′� = π/2 υ K′ �

K

γ ≫ 1 θb

K1/γ

25

beam half-angle:

✓b ⇡1

�AAACCHicbVA9SwNBEN3zM8avqKWFh0GwCncqqF3AxjKC+YBcCHObvWTJ7t2xOyfGI6WF/hELGwuDWOpPsPPfuPkoNPHBwOO9GWbm+bHgGh3n25qbX1hcWs6sZFfX1jc2c1vbFR0lirIyjUSkaj5oJnjIyshRsFqsGEhfsKrfvRj61RumNI/Ca+zFrCGhHfKAU0AjNXN7HnYYQtP3II5VdOsFCmjq9lOvDVJCv5nLOwVnBHuWuBOSL7qD86ePu4dSM/fltSKaSBYiFaB13XVibKSgkFPB+lkv0SwG2oU2qxsagmS6kY4e6dsHRmnZQaRMhWiP1N8TKUite9I3nRKwo6e9ofifV08wOGukPIwTZCEdLwoSYWNkD1OxW1wxiqJnCFDFza027YCJAk12WROCO/3yLKkcFdzjgntl0jghY2TILtknh8Qlp6RILkmJlAkl9+SZvJKB9Wi9WG/W+7h1zprM7JA/sD5/APT1nj0=

tan ( θb2 ) = ( 1 − β1 + β )2

sin θb =1γ

or cos θb = β

“ I )

Figutv 4. I la Dipole mdiation pattern for patii& at mst.

(11)

Figwp 4.llb Angular dktribution of mdiatim emitted by a partic& with parollel accelerariosl and wlocity.

( r ) Figutv 4 .11~ Same as a

(11)

Figwp 4.11d Angular distribution of mdhtion emitted by a particle with perpendicular acceleration and wlocity.

3-Extreme Relativistic Limit. When y>> I , the quantity (1 - Pp) in the denominators becomes small in the forward direction, and the radiation becomes strongly peaked in this direction. Using the same arguments as before, we obtain

“ I )


(11)



(11)



“ I )


(11)



(11)



“ I )


(11)



(11)



� particle’s rest frame:K′� : � observer’s frame:K :

parallel acceleration:

perpendicular acceleration:

Doppler Effect• In the rest frame of the observer � , imagine that the moving source emits one

period of radiation as it moves from point 1 to point 2 at velocity � .- Let frequency of the radiation in the rest frame � of the source to be � . Then the

time taken to move from point 1 to point 2 in the observer’s frame is given by the time-dilation effect:

- Difference in arrival times � of the radiation emitted at 1 and 2:

- Therefore, the observed frequency � will be

- Note that � appears even classically.

- The factor � is purely a relativistic effect.

Kυ

K′� ω′�

ΔtA

ω

1 − β cos θγ−1

26

�t = �t0� =2⇡

!0�

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

! =2⇡

�tA=

!0

� (1� � cos ✓)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

!

!0=

1

� (1� � cos ✓)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

Observer

1 2Photons emitted from point 2 will travel a smaller distance than photons from point 1 by the distance d.

ΔtA = tA(2) − tA(1) = Δt −dc

= Δt (1 − υc cos θ)

Angle for null Doppler shift• Angle for null Doppler shift is defined by:

- Relativistic Doppler effect can yield redshift even as a source approaches.

• Note �θb < θn

27

0.0 0.2 0.4 0.6 0.8 1.0β

0102030405060708090

angl

e (d

eg)

θb

θn

100 101 102 103γ

10-1

100

101

102

angl

e (d

eg) θb

θn

!

!0=

1

� (1� � cos ✓n)= 1

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

! cos ✓n =1� ��1

�=

✓1� ��1

1 + ��1

◆�1/2

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

cos ✓n =

✓1� ��1

1 + ��1

◆�1/2⇡ 1� 1

�for � � 1

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1� ✓2n

2⇡ 1� 1

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✓n ⇡r

2

�⇡

p2✓b

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source

b

n

vredshiftdeboosting blueshiftboosting

!

!0< 1 (✓ > ✓n)

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!

!0> 1 (✓ < ✓n)

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Equation of Motion in a uniform magnetic field• 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 � (or equivalently � ). Therefore, the second equation becomes

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

-

γ = constant |v | = constant

v = v⊥ + v∥

28

dE

dt=

d(�mc2)

dt= qv ·E = 0

dp

dt=

d(�mv)

dt=

q

cv ⇥B

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�mdv

dt=

q

cv ⇥B

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B ·✓�m

dv

dt=

q

cv ⇥B

◆!

8>>><

>>>:

dvkdt

= 0

dv?dt

=q

�mcv? ⇥B

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4.2 Total Losses 49

Fig. 4.1 A particle gyratesalong the magnetic field lines.Its trajectory has a helicoidalshape, with Larmor radius rLand pitch angle θ

The parallel and perpendicular components are

FL∥ = γmdv∥dt

= 0 → a∥ = 0

FL⊥ = γmdv⊥dt

= ev⊥c

B → a⊥ =evB sin θ

γmc

(4.5)

We can also derive the Larmor radius rL by setting a⊥ = v2⊥/rL, and so

rL =v2⊥a⊥

= γmc2β sin θeB

(4.6)

The fundamental frequency is the inverse of the time occurring to complete oneorbit (gyration frequency), so νB = cβ sin θ/(2πrL), giving

νB =eB

2πγmc= νL

γ(4.7)

where νL is the Larmor frequency, namely the gyration frequency for sub-relativisticparticles. Larger B means smaller rL, hence greater gyration frequencies. Vice-versa, larger γ means larger inertia, thus larger rL, and smaller gyration frequencies.Substituting a⊥ given in Eq. 4.5 in the generalized Larmor formula (Eq. 4.3) we get:

PS =2e4

3m2c3B2γ 2β2 sin2 θ (4.8)

We can make it nicer (for future use) by recalling that:

• The magnetic energy density is UB ≡ B2/(8π).• The quantity e2/(mec2), in the case of electrons, is the classical electron radius r0.• The square of the electron radius is proportional to the Thomson scattering cross

section σT, i.e. σT = 8πr20/3 = 6.65 × 10−25 cm2.Making these substitutions, we have that the synchrotron power emitted by a singleelectron of given pitch angle is:

PS(θ) = 2σTcUBγ 2β2 sin2 θ (4.9)

- Rearrange the above equations:

- Solution:

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

29

v = vk + v?AAACG3icbZDLSgMxFIYz9VbrbdSlLoJFEIQy0wp1IxTcuKxgL9CWkknPtKGZC0mmUIa+hwvd+Ayu3bhQxJUg6NuYaQva1h8CH/85h5zzOyFnUlnWt5FaWl5ZXUuvZzY2t7Z3zN29qgwiQaFCAx6IukMkcOZDRTHFoR4KIJ7Doeb0L5N6bQBCssC/UcMQWh7p+sxllChttc180yOq57jxYHTxi+1mSAThHDg+xTM2iLBtZq2cNRZeBHsK2dLhV/HxoXBXbpsfzU5AIw98RTmRsmFboWrFRChGOYwyzUhCSGifdKGh0SceyFY8vm2Ej7XTwW4g9PMVHrt/J2LiSTn0HN2Z7Cnna4n5X60RKfe8FTM/jBT4dPKRG3GsApwEhTtMAFV8qIFQwfSumPZ0KlTpODM6BHv+5EWo5nN2IWdf6zTO0ERpdICO0AmyURGV0BUqowqi6BY9oRf0atwbz8ab8T5pTRnTmX00I+PzB4Y4pfY=

dv⊥dt

=−e

γmecv⊥ × B = −

ωBB

v⊥ × B

dv∥dt

= 0

d2v⊥dt2

= − ω2Bv⊥ : harmonic oscillator

(Here, ωB ≡ eBγmec )

v(t) = v? (�x̂ sin!Bt+ ŷ cos!Bt) + ẑvkr(t) =

v?!B

(x̂ cos!Bt+ ŷ sin!Bt) + ẑ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

(initial conditions : v⊥ ∥ y and x⊥ ∥ x at t = 0)

vk = constant

|v?| = constant (since |v| = constant)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

~~~

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

a∥ = 0a⊥ = ωBυ

- The particle gyrates along the magnetic field lines with the angular velocity � .- Its trajectory has a helicoidal shape, with gyroradius radius � and pitch angle � .

ωBrB α

30

Larmor frequency: � Larmor radius: �

(Cychrotron frequency,non-relativistic gyrofrequency)

ωL =eBmec

rL =υ⊥ωL

ωB =17.6

γB

μG(Hz)

rB = 1.7 × 109 γβ⊥ ( BμG ) (cm)

gyrofrequency: �

gyroradius: �

pitch angle � : � , �angle between the magnetic field and velocityangle between � and �

ωB =eB

γmecrB =

υ⊥ωB

α υ∥ = υ cos α υ⊥ = υ sin α

v∥ v

~~~

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

pitch angle α

Synchrotron Power Radiated by a Single Electron• Relativistic version of the Larmor’s formula gives the radiated power:

• Total emitted power:

- Since � and � , we obtain � .

- Using the definitions:

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

a⊥ = ωBυ⊥ a∥ = 0 P =23

γ2q4B2

m2c5υ2⊥

31

⌦sin2 ↵

↵=

1

4⇡

Zsin2 ↵d⌦ =

2

3

�2 = 1� 1�2

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

P =2q2

3c3γ4 (a2⊥ + γ2a2∥)

P =43

σTc(γ2 − 1)UB

pitch angle : cos α =v ⋅ BυB

, classical electron radius : re =e2

mec

Thomson scattering cross − section : σT =8π3

r2e , magnetic field energy : UB =B2

8π

P = 2σTc(γβ)2UB sin2 α

Cooling Time• The energy balance equation becomes:

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

- for �� = 103AAAB8nicbVDJSgNBEO2JWxyXRD16aQyCXsKMEfQiBkTwGMEsMBlDT6cnadLL0N0jhCGf4cWDIl79A//Cm5/gX9hZDpr4oODxXhVV9aKEUW0878vJLS2vrK7l192Nza3tQnFnt6FlqjCpY8mkakVIE0YFqRtqGGkliiAeMdKMBldjv/lAlKZS3JlhQkKOeoLGFCNjpaDdQ5yjC9+7r3SKJa/sTQAXiT8jpWrh+/LY/biudYqf7a7EKSfCYIa0DnwvMWGGlKGYkZHbTjVJEB6gHgksFYgTHWaTk0fw0CpdGEtlSxg4UX9PZIhrPeSR7eTI9PW8Nxb/84LUxOdhRkWSGiLwdFGcMmgkHP8Pu1QRbNjQEoQVtbdC3EcKYWNTcm0I/vzLi6RxUvYrZf/WpnEKpsiDfXAAjoAPzkAV3IAaqAMMJHgEz+DFMc6T8+q8TVtzzmxmD/yB8/4DqkCTEQ==

32

7.3. COOLING TIME OR RADIATIVE LIFETIME 93

where re = e2/mc2 - classical electron radius, σT = 8π3 r2e =

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.

mc2dγdt

= − P ← E = γmc2

tcool =energy

cooling rate=

γmc2

P≈

23 yearsγ(B/Gauss)2 (if γ ≫ 1)

KAIST Astrophysics (PH481) - Part 1 · KAIST Astrophysics (PH481) - Part 1 Week 2a Sep. 9 (Mon),...

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