SynchrotronSynchrotron RadiationRadiation ''Particles accelerated by a magnetic feld will radiate. For non – relativistic velocities the complete nature of the radiation is rather simple and is called cyclotron radiation. The frequency of emission is simply the frequency of gyration in the magnetic feld. However, 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'' George B. Rybicky & Alan P. Lightman in ''Radiative processes in astrophysics'' p. 167 Daniele Dallacasa Non-thermal Synchrotron Radiation on textbooks G. Ghisellini: “Radiative Processes in High Energy Astrophysics” § 4 M. Longair: “High Energy Astrophysics” § 8 G. B. Rybicky & A.P. Lightman: “Radiative Processes in Astrophyiscs” § 6 (+4.7 & 4.8) C. Fanti & R. Fanti: “Lezioni di radioastronomia” § 4 T. Padmanabhan: “Theoretical Astrophysics” § 6.10 & 6.11 Radiation from a charged particle the Lorentz force A charge q is defected by a uniform magnetic feld B according to: d⃗p q = ⃗v×H⃗ → v = v cosθ v = v sinθ dt c ∥ ⊥ θ is the ''pitch angle'' between the magnetic feld and the velocity of the charge θ Radiation from a charged particle the Lorentz force (2) d(mv ∥ ) = 0 → v is constant → The pitch angle remains constat dt ∥ d(mv ⊥ ) q = v × H dt c ⊥ In the direction perpendicular to the B feld we have a circular motion and in the non-relativistic case we have the well known cyclotron relations: mc r = v L qH ⊥ 2 πrL 2πmc TL = = v ⊥ qH x 2π qH ωL = = [c.g.s.units] TL mc Side view Top view Cyclotron radiation ''A charge moving in a magnetic feld is accelerated and then radiates [CYCLOTRON]'' - dipole in the plane of the circular orbit (dipole angular distribution) - linear polarization or circular or elliptical polarization Max amplitute 0.5 * Max amplitute 45o No radiation! 0.5 * Max amplitute Max amplitute Cyclotron radiation frequency of emission d⃗p q Lorentz Force: = m⃗a = ⃗v×H⃗ dt c q in modulus ma = v H sin θ = qβHsin θ c 4 dW 2 q 2 therefore in in the Larmor's formula ( − ) = β2 H2 sin θ dt 3 m2 c3 ωL qH at the frequency ν = = L 2π 2πmc H If we consider an electron, its gyration frequency is: ν ≃ 2.5 [MHz] L Gauss N.B. In general, the magnetic felds are weak, except in very particular stars, and collapsed bodies. Example: 34keV feature in Her-X1 (see Longair), → radiation emitted at low energies Towards synchrotron radiation: relativistic Larmor parameters The relativistic case is dealt with by considering the appropriate expression of the particle mass: mo m = = mo⋅γ √1−β2 γ mo c r = v = γ r L ,rel qH ⊥ L 2πrL ,rel 2πmo c γ TL ,rel = = = γ TL v ⊥ qH 2π qH ωL ωL ,rel = = = TL,rel γmo c γ if we consider v ∼ c ,we have: 2 γmo c ε r ≃ = L ,rel qH qH N.B. ''classical'' physics is ok, since for typical B felds strengths, h/p≪rL, rel Cyclotron radiation towards relativistic particles (1) [(ULTRA) - Relativistic] Charge ( γ≫1) moving in a uniform magnetic feld Recall Larmor's Formula in the invariand form (scalars are not affected) dt d τ = γ = [⃗p ,(i/c)W] FYI, i Computations never asked W2 = p2 c2+m2 c4 2 dW 2q dpi dpi P = = dt 3m2 c3 [d τ d τ ] 2 2 2 2 dpi dpi d⃗v 1 dW d⃗v dp = − = −β2 [d τ d τ ] (d τ ) c2 (d τ ) (d τ ) (d τ ) Already seen earlier 2 2 2 dW 2q 2 d⃗v 2 dp Two options: Prel = = γ −β dt 3m2 c3 [(d τ ) (d τ ) ] Cyclotron radiation towards relativistic particles (2) Classical, non relativistic, Larmor case → linear acceleration (no ac ) d⃗p dp ≃ ∣d τ ∣ d τ dW 2q2 1 dp 2 P = = dt 3m2 c3 γ2 (d τ ) d⃗p dp 1 dW → Centripetal acceleration ≫ β = ∣d τ ∣ ( d τ ) c ( d τ ) dW 2q2 d⃗p 2 2q2 d⃗p 2 P= = = γ2 dt 3m2 c3 (d τ ) 3m2 c3 (dt ) Relevant in relativistic cyclotron and synchrotron emission! (relativistic) Cyclotron radiation frequency of emission The energy is radiated in various harmonics of the gyration frequency: v ∥ cosθ ν = k ν 1 − k=1,2,3,4,....n Doppler shift along the LoS k rel ( c ) dW dW The energy in the various harmonics follows: ≈ β2 [ dt ] l+1 [ dt ] l (relativistic) Cyclotron radiation frequency of emitted radiation dW dW The energy in the various harmonics follows: ≈ β2 [ dt ]n+1 [ dt ]n As a result of a relativistic effect on electron motion and its emitted radiation... (Radiation) felds on the approaching side are amplifed, while they are attenuated on the receding side. The effective feld can therefore be represented as the superposition of a number of harmonics. Synchrotron radiation relativistic aberration sin(α')(√1−β2) tan(α) = cos α'+β 1 α o → (α) β 2 ' = 90 tan ≈√ 1− ≈ 2 γ ⇒ Amplifcation along the velocity direction ⇒(relativistic beaming) 2 Half of the radiation is within a cone wide γ Synchrotron radiation It is generated by ultra-relativistic electrons, for which the curvature of the trajectory plays the major role in determining the emitted power (single electron with its γ in a H feld) 4 dW 2q 2 − = β2 γ2 H2 sin θ H energy dt 3m2 c3 density UH 2 ifβ≃1 and ε=mo c γ then it can be rewritten as 4 2 dW 2q 2 2 2 2 H 2 −15 2 2 2 −1 − = ε H sin θ = 2c σ γ sin θ ≃ 1.62⋅10 γ H sin θ ergs 4 7 T π dt 3mo c 8 and σT is known as electron Thomson cross section defned as 2 2 8 π e 8 π 2 −25 −2 σT = = ro = 6.6524586...⋅10 cm 3 m c2 3 ( e ) Synchrotron radiation pulse duration (1) Pulse duration in the electron reference frame is me c γ 2 2 Δ t = Δ θ = = ωrel γ eH ωL In the observer frame, the duration of the pulse is shortened by propagation (Doppler) effects: the electron is closer to the observer in 2 travelling nearly at same speed of radiation emitted in 1 −8 v 1 1 1 5⋅10 Δ τ = (1− )Δ t = (1−β)Δ t ≈ 2 Δ t = 2 = 3 ≃ 2 s c γ γ ωL γ ωrel γ H[G] 2 = Synchrotron radiation pulse duration (2) −8 v 1 1 1 5⋅10 Δ τ ( )Δ ( β)Δ Δ = 1− t = 1− t ≈ 2 t = 2 = 3 ≃ 2 s c 2 γ 2 γ ωL 2 γ ωrel γ H[G] Foreword/consequences: 2 1. The duration of the pulse is ≈1/ γ shorter thanΔ t=TL 2. The factor (1 − v/c) is the same as appears in the L-W potentials and accounts for the beaming (more generally Doppler effect) in case of relative motion wrt the observer (pp. 200-202 Longair) Synchrotron radiation pulse angular distribution The radiation is emitted in a narrow cone aligned with v A given LoS receive pulsed fashes Istantaneous illumination Synchrotron radiation single electron spectral distribution (1) algebra is very complex; a detailed discussion is in Rybicki-Lightman p.175 Fourier analysis of the pulse provides the spectrum of the radiated energy. The spectrum is the same as the relativistic cyclotron but with an infnite number of harmonics. Beaming effects move the peak to frequencies higher than νL 3 1 3 2 3 2 eH The characteristic frequency is: νs ≃ = γ νL = γ = 4 π τ 4π 4 π me c 3 eH 2 18 2 −9 2 3 5 ε ≃ 6.24⋅10 ε H ≃ 4.2⋅10 γ H[μG] GHz 4 π me c 4 ⇒example: an electron with γ ∼ 10 in a H ∼ 1 μG feld has νs ∼ 0.4GHz Synchrotron radiation is generally emitted in the radio window H felds are generally 0.1 - 10 μG in space. Higher felds in ''condensed'' objects (e.g. stars) Synchrotron radiation single electron spectral distribution (2) The full expression of the emitted energy as a function of frequency is 3 dW(ν) dW∥ dW ⊥ √3e H sinθ ν = + ≈ F dt dt dt 8 π2 cm ν e ( s ) ν ν ∞ where F ν (y)dy = ∫ K5/ 3 ν ν νs ( s ) s K = modifed Bessel function 1/3 ν≪νs → F(ν) ∼ ν −ν/ νs ν≫νs → F(ν) ∼ e ν peak at ∼ 0.3 νs Nearly monochromatic emission, at 2 FYI, νm 0.3 νs depending on γ and H Computations never asked Synchrotron radiation typical radio spectrum of a radio source Which is the nature of the emitted energy? The cases of 3C219 and 3C273 HST + radio radio Synchrotron radiation the full sky at 408 MHz Synchrotron emission from relativistic electrons (positrons) within the Milky Way. In a feld of 1 μG, their energies correspond to γ γ 104. Sky @ 408 MHz (from skatelescope.org) Fast charges within a spiral galaxy: do they escape? Compute the Larmor radius of a relativistic proton in our galaxy B ∼ 10μ G mv c rL = γrL = γ rel qH alternative formulas γmc2 v rL = γrL = rel qH c γmc2 6 [GeV] v rL = 3.3×10 cm rel q H ( c ) [e] [Gauss] Synchrotron radiation a SNR Synchrotron emission from relativistic electrons (positrons) in the crab nebula Synchrotron radiation power – law energy distribution Let's consider an ensemble of relativistic electrons with energies distributed according to a power – law: −δ N(ε)dε = No ε dε The specifc emissivity of the whole population is dW (ν,ε) J (ν)d(ν) = s N(ε)dε (****) s dt ν ≈ F N ε−δ dε ν o ( s ) There are various (approximate) ways to derive the total emissivity: 1.