Synchrotron Radiation: II. Spectrum
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1 Synchrotron Radiation: II. Spectrum Massimo Ricotti University of Maryland Synchrotron Radiation: II. Spectrum p.1/18
2 ds=v dt_em dt=ds cos(theta)/c=v/c cos(theta)dt_em Synchrotron Radiation: II. Spectrum p.2/18
3 Spectrum. I. Mono-energetic CRs Effect of beaming: Opening angle δθ = 2/γ dt em = (δθ/ω B ) 1/γω B ω L Thus the frequency of the pulse is shorter by a factor of γ Synchrotron Radiation: II. Spectrum p.3/18
4 Form cyclotron to synchrotron time frequency Synchrotron Radiation: II. Spectrum p.4/18
5 Form cyclotron to synchrotron time frequency Synchrotron Radiation: II. Spectrum p.4/18
6 Form cyclotron to synchrotron time frequency Synchrotron Radiation: II. Spectrum p.4/18
7 Relativistic effects make the frequency higher by another factor γ 2 In the observer frame of reference: δt rec = δt em (1 β cos θ) where θ δθ is the viewing angle Thus, dt rec = dt em (1 β + βδθ 2 /2) = = dt em (1/2γ 2 + 1/2γ 2 ) = dt em /γ 2 Similarly can derive superluminal motions proposed by M.J. Rees: v obs dsδθ/dt rec v em γ, that can be > c. Including the pitch angle: ω c = 3/2γ 3 ω B sinα = 1.5γ 2 ω L sinα where ω L = qb 0 mc Larmor freq. Synchrotron Radiation: II. Spectrum p.5/18
8 Spectrum. I. Mono-energetic CRs Details of the spectral shape are not important as we will see later. x 1/3 for x 1 F(x) x 1/2 exp( x) for x 1 where x = ω/ω c. Maximum of F(x) at x = 0.29 What is the normalization of P(ω)? dp dω P (2/3)γ2 β 2 sin 2 α(q 4 /m 2 c 3 )B0 2 ω c (3/2)γ 2 (qb 0 /mc) sinα = 4 9 B 0 q 3 sinα mc 2 Indeed consedering the correct normalization of F(x) we have, dp(ω) dω = 3 2π B 0 q 3 sin α F(x) mc 2 Synchrotron Radiation: II. Spectrum p.6/18
9 Spectrum. II. Power-law distribution of CRs with p 2.5. n γ dγ = n 0 γ p dγ P 1 P(γ)n γ dγ where P(γ) γ 2. Assume for simplicity F(x) = δ(x 1). Set ν = ν c = γ 2 ν L, dν = 2γdγν L, P(ν) ( ) (1 p)/2 ν δ(ν ν dν ) ν (p 1)/2 2ν L ν L Thus, Synchrotron is characterized by a power law spectrum with slope (p 1)/ The flux now depends on the combination of n 0 and B 0. Need more info to measure the magnetic field! Synchrotron Radiation: II. Spectrum p.7/18
10 Synchrotron self-absorption We have seen the synchrotron emission mechanism: what about absorption? and stimulated emission? We can have both: absorption is important at low frequencies. Why? For synchrontron the source function is S ν B 1/2 0 ν 5/2. Here is the qualitative derivation. For BB: S ν = 2ν2 c 2 ( hν e hν/kt 1 ) 2ν2 c 2 KT kt is energy of thermally excited harmonic oscillator. Replace kt with appropriate energy. ǫ = γm e c 2 = me 2 c(ν/ν L ) 1/2. Thus, S ν B 1/2 0 ν 5/2. What is the flux in the optically thick regime? Synchrotron Radiation: II. Spectrum p.8/18
11 First of all we have I ν = S ν in optically thick regime. F max ν = πi ν (R source /dist) 2 B 1/2 0 ν 5/2 max(r source /dist) 2 Can break the degeneracy n 0 B 0 and measure magnetic field. Synchrotron Radiation: II. Spectrum p.9/18
12 In addition there seems to be a maximum flux of synchrotron radiation from compact radio sources. Why? Synchrotron Radiation: II. Spectrum p.10/18
13 Inverse Compton losses T b = c2 I ν 2k B ν 2 brightness temperature In 1969 Kellermann and Pauling-Tohth noted that in compact radio sources T b (max) < K (clearly this is non-thermal emission) How can this observation be explained? Recall that L c L s = U ph U B where U ph F ν ν For B 0 = 10 3 Gauss, γ = 10 3 γ 2 ω L 10 9 Hz Compton scattering with ultra-rel electrons of GHz photons γ 2 ν Hz (optical wavelengths) Synchrotron Radiation: II. Spectrum p.11/18
14 Astrophisical sources of synchrotron radiation Pulsars SN remnants (for example, the Crab nebula) Gamma ray bursts Radio Galaxies (jets from AGN) Synchrotron Radiation: II. Spectrum p.12/18
15 Synchrotron Radiation: II. Spectrum p.13/18
16 Synchrotron Radiation: II. Spectrum p.14/18
17 Synchrotron Radiation: II. Spectrum p.15/18
18 Synchrotron Radiation: II. Spectrum p.16/18
19 Application to Radio Galaxies In 1959 Burbidge noticed a problem with the energetic requirements of radio galaxies In radio galaxies synchrotron dominates the entire spectrum from 10 meter to mm wavelengths Near equipartition of U CR and U B minimizes the energy requirement to produce the observed luminosity R lobes 30 kpc, B Gauss (from peak of spectrum near optically thick synchrotron) E tot = E CR + E mag 2E mag = (4π/3)R 3 lobes B 2 0 8π 1058 ergs This is an enormous amount of energy: about energy generated by 10 7 SN explosions Synchrotron Radiation: II. Spectrum p.17/18
20 In addition synchrotron cooling of lobes is extremely short: m e c 2 γ = P Sync = 2β 2 γ 2 cσ T U B sin 2 α t cool = γ γ m e c 2σ T U B γ sin 2 α t cool 10 7 yrs forγ = 10 4 andb 0 = 10 5 Gauss Need engine that keeps pumping energetic electrons: SMBH at the center of galaxy. However, our assumption of γ = const in the derivation of the sychrotron radiation is valid because the period of gyration is typically of the order of seconds: t cool. Synchrotron Radiation: II. Spectrum p.18/18
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