1. March 28, 2011 HW 7 due Wed. Midterm #2 on Monday April 4 Today: Relativistic Mechanics Radiation in S.R.

2. Fields of a Uniformly Moving Charge If we consider a charge q at rest in the K’ frame, the E and B fields are where

3. Transform to frame K We’ll skip the derivation: see R&L p.130-132 The field of a moving charge is the expression we derived from the Lienard-Wiechart potential: We’ll consider some implications 

4. Consider the following special case: (1) charged particle at x=y=z=0 at t=0 v = (v,0,0) uniform velocity (2) Observer at x = z = 0 and y = b sees

5. What do E x (t), E y (t) look like?

6. Ex(t) and Ey(t) (1) max(Ey) >> max (Ex) particularly for gamma >>1 (2) Ey, Bz strong only for 2t 0 (3) As particle goes faster, γ increases, E-field points in y-direction more

7. Ex(t) and Ey(t) (4) The observer sees a pulse of radiation, of duration (5) When gamma >>1, β~1 so (6) To get the spectrum that the observer sees, take the fourier transform of E(t)  E(w) We can already guess the answer 

8. The spectrum will be whereis the fourier transform The integral can be written in terms of the modified Bessel function of order one, K 1 The spectrum cuts off for

9. Rybicki & Lightman give expressions for and some approximate analytic forms -- Eqns 4.74, 4.75

10. Bessel Functions see Numerical Recipes Bessel functions are useful for solving differential equations for systems with cylindrical symmetry Bessel Fn. of the First Kind J n (z), n integer Bessel Fn. of the Second Kind Y n (z), n integer Jn, Yn are linearly independent solutions of

12. Modified Bessel Functions:

14. Relativistic Mechanics 4-momentum where m 0 = rest mass, i.e. the mass in the inertial reference frame in which the particle is at rest. For particles with mass: where

15. For photons:

16. Can show that p and E transform in the following way: (v in x-direction) where

17. Relativistic Mass For what value of v/c will the relativistic mass exceed its rest mass by a given fraction f? 0.001 0.014 0.01 0.14 0.1 0.42 1.0 0.87 10 0.994 100 0.999 At high beta, m >> m 0

18. The invariant quantity is or since where

19. 4-acceleration Newton’s Second Law 4-Force F=ma

20. For the Lorentz force The corresponding 4-vector is So the equation of motion for a charge is

22. Does this make sense? repeat  W = particle energy so OK! t, not tau

23. so for μ=1 similarly for y,z

24. Emission from Accelerated Relativistic Particles Basic Idea: (i) We know how to calculate the emission from an accelerating particle if << c  Larmor result (ii) We can transform to an inertial frame, K’, in which the particle is instantaneously at rest  calculate the radiation field  transform back to lab frame, K

25. Consider a particle with change q in its instantaneous rest frame K’ Let momentum=0 since the particle is at rest In K frame: So power so emitted energy / time =P is a Lorentz invariant

26. In rest frame, the Larmor result: or In terms of components of the acceleration parallel and perpendicular to the direction of motion between frames, recall that we derived: provided the particle is instantaneously at rest in K’

28. Angular Distribution of Radiation: In the rest frame K’, consider emission of energy dW’ into solid angle dΩ’ How do dW’ and dΩ’ transform?

29. dW’ Since energy and momentrum form a 4-vector, dW transforms like dt For photons, so if we let

30. Let aberration formula  so Since

31. so... Now of course we are interested in the power, i.e. the energy/time: so how does dt’ transform?

32. There are two possibilities for relating dt and dt’ (i) This is the interval as seen in K frame  emitted power P e so can also write

33. (ii)c.f. Doppler formula This is better – it’s what you would actually measure as a stationary observer in K  received power P r or We’ll adopt this

34. Suppose the acceleration of the particle is caused by a force having components with respect to the particle’s velocity (see R&L problem 4.14) since dE’ =dx’ =0

35. and Sois more effective than in producing radiation

36. Beaming As β  1, photons which are isotropic in the rest frame are “beamed” forward In K: so Strongly peaked at θ=0

37. What happens to a dipole? Recall that is the rest frame of the emitting particle, Larmor’s result had a dipole angular distribution in K’ angle between the acceleration and the direction of emission writing Working out the result is messy; see R&L Eqn. 4.101 and 4.103

38. Qualitative Picture

39. Transformation of the Equation of Radiative Transfer How does the specific intensity I ν transform? Recall the radiation density We showed that so Now, the energy per volume is also where f = phase space density of photons f = # photons / dV’ where dV’ = dx’dy’dz’ dp x ’ dp y ’ dp z ’

40. How does the phase space volume transform? so phase space volume is an invariant But since f = Number of photons/dV f is a Lorentz invariant

41. OK: since

42. What about the source function S ν ? Recall sotransforms like

43. Optical depth gives fraction of photons absorbed so

44. Absorption coefficient K θ v l l =distance perp. to v Both l and ky are perpendicular to v  transform the same way is Lorentz invariant So

45. Emission coefficient j ν So

46. So, to solve a radiative transfer problem: set up problem in emitter rest frame solve the equation of radiative transfer transform the result