March 28, 2011

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 Ex(t), Ey(t) look like?

6. Ex(t) and Ey(t) (1) max(Ey) >> max (Ex) particularly for gamma >>1 (2) Ey, Bz strong only for 2t0 (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 where is the fourier transform The integral can be written in terms of the modified Bessel function of order one, K1 The spectrum cuts off for

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

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

12. Modified Bessel Functions:

14. Relativistic Mechanics 4-momentum where m0 = 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 >> m0

18. The invariant quantity is or since where

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

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

22. repeat  Does this make sense? 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 <v> << 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 Pe 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 Pr 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 So is 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 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 Iν transform? Recall the radiation density so We showed that Now, the energy per volume is also where f = phase space density of photons f = # photons / dV’ where dV’ = dx’dy’dz’ dpx’ dpy’ dpz’

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 functionSν? Recall so transforms like

43. Optical depth gives fraction of photons absorbed so

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

45. Emission coefficientjν 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