3145 Topics in Theoretical Physics - Astrophysical Radiation Processes PHY3145 Topics in Theoretical Physics Astrophysical Radiation Processes 4: Relativistic effects II Dr. J. Hatchell, Physics 407, [email protected] 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Course structure 1. Radiation basics. Radiative transfer. 2. Accelerated charges produce radiation. Larmor formula. Acceleration in electric and magnetic fields – non-relativistic bremsstrahlung and gyrotron radiation. 3. Relativistic modifications I. Doppler shift and photon momentum. Thomson, Compton and inverse Compton scattering. 4. Relativisitic modifications II. Emission and arrival times. Superluminal motion and relativistic beaming. Gyrotron, cyclotron and synchrotron beaming. Acceleration in particle rest frame. 5. Bremsstrahlung and synchrotron spectra. 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 1 3145 Topics in Theoretical Physics - Astrophysical Radiation Processes Motion in the particle’s rest frame Aim: Calculate motion in particle’s rest frame (usually so that we can apply Larmor formula) Method: Lorentz Transform position, fields etc. from observer’s frame S to comoving frame S’ (standard relativity convention) S y S’ y’ v L.T. v x x’ Observer’s frame Particle’s rest frame Examples • Acceleration in E-field: Bremsstrahlung (relativistic) - HOMEWORK • Acceleration in B-field: Cyclotron/Synchrotron 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Example: Synchrotron particle motion and acceleration in rest frame Aims: i. Relativistic motion of charged particle in magnetic field; ii. Acceleration in particle rest frame (needed for Larmor formula) iii. Total power Method: i. Lorentz force in lab. frame ii. Lorentz transform B into instantaneous rest frame. Use Lorentz force to calculate acceleration. iii. Larmor’s formula in particle rest frame 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 2 3145 Topics in Theoretical Physics - Astrophysical Radiation Processes Synchrotron motion (continued…): i. Relativistic motion of charged particle S B in magnetic field θ Use Lorentz force in lab. frame r a γvma = qvB sin θ v Equate with centripetal acceleration v 2 v2 sin2 θ qvB sin θ θ is the pitch angle ⊥ = = (angle between B r r γvm and v) Angular velocity – relativistic gyrofrequency v qB ωg ωr = ⊥ = = r γvm γv 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Synchrotron acceleration (continued…): z B Instantaneously take v along x and B in x-z plane y B =(Bx, 0,Bz)=(B cos θ, 0,Bsin θ) v x E = (0, 0, 0) S ii. Acceleration in charge rest frame. Lorentz transform E and B and apply Lorentz force remembering that in rest frame v! =0 Ex! = Ex =0 Bx! = Bx = B cos θ 2 E! = γ(E vB )=γvB sin θ B! = γ(B + v/c E ) = 0 y y − z − y y z 2 E! = γ(Ez + vBy) = 0 B! = γ(B v/c E )=γB sin θ z z z − y γ! ma! = q(v! B! + E!) only has non-zero terms in E ! v × γvqvB sin θ a! = m 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 3 3145 Topics in Theoretical Physics - Astrophysical Radiation Processes Emitted power in rest and observer’s frame iii) Emitted power. Use Larmor’s formula to get the emitted power in the particle rest frame Total emitted power in observer’s frame is the same as in charge rest frame (dW/dt Lorentz invariant): 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Gyroradiation (non-relativistic) 2 Power varies as sin (ωgt) so electric field varies as sin(ωgt). All radiation emitted at gyrofrequency of the charged particle. (NB independent of velocity) 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 4 3145 Topics in Theoretical Physics - Astrophysical Radiation Processes Relativistic aberration Results: S’ y’ S y v u’ L.T. u θ’ θ x’ x Observer’s frame Aberration formula For photons u’=c also 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Relativistic beaming Aim: How does the viewed power pattern of emission from an accelerated particle change when the particle moves relativistically with respect to an observer? Quick estimate: put θ’ = π/2 in aberration formula. S’ v S a Particle rest frame Observer’s frame 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 5 3145 Topics in Theoretical Physics - Astrophysical Radiation Processes 2b (ii) Relativistic beaming v a a Rest frame Lab frame 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Gyroradiation (non-relativistic) 2 Power varies as sin (ωgt) so electric field varies as sin(ωgt). All radiation emitted at gyrofrequency of the charged particle. (NB independent of velocity) 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 6 3145 Topics in Theoretical Physics - Astrophysical Radiation Processes Cyclotron (mildly relativistic) Aim: Cyclotron frequency spectrum Method: Consider increasing narrowing of pulse due to beaming (qualitatively). Result (without derivation): Frequency spectrum consists of a series of harmonics lω ω = g l γ(1 (v /c) cos θ) − ! θ is angle of B to LOS Doppler factor for motion parallel to B lω simplifying to ω = g for motion perpendicular to B l γ 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics X-ray pulsar cyclotron harmonics X-ray pulsar V0332+53 Image: Tsygankov et al. 2006 MNRAS 371, 19 in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 7 3145 Topics in Theoretical Physics - Astrophysical Radiation Processes Cyclotron harmonics: demonstration amplitude time 7 cos(ωgt) cos(ωgt)+a1 cos(2ωgt) al cos(lωgt) !l=1 Gyrofrequency Gyrofrequency + Gyrofrequency + only first harmonic first six harmonics l=1,2 l=1,2,3,4,5,6 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Synchrotron beaming (γ >> 1) Aim: Understand (qualitatively) pulse length of received power and therefore frequency of received radiation Method: Combine relativistic beaming (Lecture I) with relativistic motion around circle. Result (see lectures): duration of pulse S A B 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 8 3145 Topics in Theoretical Physics - Astrophysical Radiation Processes Synchrotron spectrum for single charge Maximum frequency component log(P) log(ν) Exact result 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Diamond Synchrotron High power X-ray / UV synchrotron radiation used experimentally to probe properties of materials. Images: www.diamond.ac.uk 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 9 3145 Topics in Theoretical Physics - Astrophysical Radiation Processes i. Acceleration of electron and total power Bremsstrahlung = free-free radiation = radiation by an unbound charged particle due to acceleration by another charged particle S y S’ y’ e- v E b x’ x Relativistic accelerations Ze Total radiated power from a single electron-ion encounter 3145 Topics in Theoretical Physics - radiation processes Dr J Hatchell 3145 Topics Dr J. Hatchell 10 .