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1 A x 4 D r xx ' J x ' d 4 x ' no incoming fields c D r xx ' : the retarded Green function e U x 0 r 0 xr d J e c U 4 x ' r d xr 0 0 x r x x xr x r xr U f x x x i d f d x x xi A x e U Ux r 0 Lienard - Wiechert Potentials

2 UxrU 0 x 0 r 0 0 Uxr 0 c Rvn R c R 1n x 0 r 0 0 xr 0 R e e x,, A x, t1n t Rret 1n Rret x, t e R, A x, t ev xr x r Uxr c R d for 0 nonrelativistic motion d x r ret : evaluated at the retarded time 0, with r 0 0 x 0 R f f d d f d d f A e U x 0 r 0 x r d has no contribution e x 0 r 0 x r F e Uxr U c, c xr R, Rn, d d d d x r U x r U d d Ux r x r U Ux r 0 d Uc 4, c c 4 d d Ux rc x r d d U d d t

3 E x, te n 1n 3 R ret e c nn 1n 3 R ret, BnE ret 0 velocity field acceleration field R R U const F e c x r U x r U Uxr 3 0 Sec P ' Q R cos Rn, OQR 1n, b R sin R 1n r PQ r R sin b v t b b v t e b e E b v t 3 b the transverse component 1n 3 R 3 ret of the velocity field x x

4 1 E a e c nn R t ret S c 4 EB c 4 E n a d P d c 4 R E a e 4 c nn e v 4 c sin 3 P 3 e c 3 v Larmor's formula for a nonrelativistic, accelerated charge energy flux by Poynting vector t 3 e c 3 v 3 e dp m c 3 d t dp d t P 3 e d p p d m c 3 d d 1 generalization

5 d p p dp d d d d 1 c d E d dp d d p d e P 3 c 6 the Lienard result d t d e 3 m c d p 3 d d p d P e 3 m c d p 3 d t e 3 m c d E 3 d x d E d p d x d t Ex the radiated power power by external sources P d E d t 3 e 1 m c 3 v mc e mc d E d x 3 E m c, p e m c m c mv d Em 3 v d v, d pm 3 d v d E d x for 1

6 dp d p 1 c radiative-energy loss revolution d E d E P 3 c P watts10 6 E Mev J amp e m c 3 p 3 P 4 3 e e c 4 4 c E GeV meter 1 meter, E max 0.3 GeV E max 1 kev revolution 4 for 1

7 T T tt R T c E tt Sn 1 R T 1 c ret d t t 'T 1 power radiated solid angle d P t ' d Sn ret e nn S c 4 c R 1n 3 ret 4 E B c a a 4 E a n tttrtc t 'T Sn d t d t ' d t ' Sn d t d t ' : (power radiated)/area in the charge's time R Sn d t d t ' R Sn 1n e nn 4 c 1n 5

8 R d P t ' for a linear motion d e v sin cos n 4 c 3 1 cos 5 Larmor's result for 1 1 d P d max 8 max cos rest energy max total energy 1 0 d P t ' d 8 e v 8 c 3 1 5

9 rms 1 m c E P linear t ' d P t ' d d v 3 c 6 the Lienard result 3 d P t ' d e v 3 sin cos 1 4 c 3 1 cos 1 cos e d P t ' d e v 3 4 cos & c rms for 1 e P circular t ' v 3 c 4 3 p m vf P circular t ' e 3 m c 3 dp d t P linear t ' y

10 v c d P t ' v v d

11 1 d D c t t v pulse front's travelling distance L D d the length of the pulse T L 3 c c c L c the fundamental frequency 00 MeV synchrotron max 400, s 1 c s 1, c 10 3 A 10 GeV machine max 0000, s 1 c s 1 16 kev x-ray

12 d P t d A t A t c 4 E ret in the observer's time A 1 A t e i t d t A t 1 A e i t d d W d d P t d d t A t d t ' 1 e i ' t d t 1 0 d I,n d d d I,n A * 'A ' e i ' t d ' d d t A d d I,n d d d A A d d A if A t A A *

13 A e 8 c e 8 c e 8 c d I d d e nn 1n 3 ret e i t d t for an accelerated charge 0 nn 1n e i t ' R t ' c d t ' t t ' e i xc 4 c R t ' nn 1n e i tnr tc d t R t ' xnr t ' nn 1n given r t t & t d I d d A j for many particle d I d d e i tnr tc d t c nn 1n d I d d e d d t 4 c nn 1n nn e i tnr tc d t 4

14 T tt v e e i nr tc j1 d I d d N e j j e i nr j t c 1 c J x, t ei nx c d 3 x 4 c 3 nnj x, t ei t nxc d 3 x d t

15 nn cos v t sin sin v t 1 where nr t t c c sin v t cos 1 t c 6 t3 1 rms 1 d I d d e 4 c A A 3 A c A i 1 t e t c t 3 i 1 t c t 3 e 6 d t c 6 d t c x 3 x x3 i x e 3 x x3 i e d x d x y where x c t, 3 c 3

16 0 x sin 3 x x3 d x 1 3 K 3, d I d d e 1 3 c 3 4 d I d 0 d I d d d 7 16 e I I I I 7 I I 0 as 1 large c 0 K 3 cos 3 x x3 radiation polarized the plane of the orbit d x 1 3 K 13 1 K 5 13 radiation polarized the plane of the orbit 1 The radiation from a relativistically moving charge is very strongly polarized in the plane of motion. all c 3 c 3 3 E m c c n c 0 n c c critical harmonic frequency harmonic number fundamental harmonic frequency 3 c 4, 0 1 c

17 d I d d 0 e c e c c c c 1 c 3 c c c c 1 for c c c e c for c d I d d d I e 3 c d d 0 c 1 c 3 c 1 3 c 1 3 c c c rms rms

18 с d I d d I d d d sin с с d I d 3 e d I d d d c c c c d I d e c 3 K 53 x d x d I d d 0 c e c 6 e c 1 c 3

19 synchrotron radiation c c n d P n d 1 c d I, P n 1 d d n 0 c d I d n 0 B1 gauss, E e 5 MeV meters, s 10 3 significant harmonics radiated

20 d N d c I 9 3 c 8 c K 53 x d x I 4 e 4 3 total energy radiated per revolution mean number of photons N 5 mean energy I revolutionparticle 3 photon N c, GeVO 10 4 fundamental hundred of meters photon meter kev x-ray c

21 xa B wiggler, E particle sin z 0 d x d z z 0 k 0 a 0 k 0 0 fundamental wave number period T 0 c k 0, real k 0 for 1 k 0 k 0, real

22 O 0 0 c k 0 O 10 GHz for O centimeters 0 basic freq. c R : effective radius R min 1 R of curvature k 0 a 0 0 c 3 c 0 0

23 coherent superposition ON rest c 1 rest lab 1 cos lab lab c For 1 lab O with fixed K

24 length 0 s 1 d x 01 0 per cycle d z 1 d z 0 d x d z d z for for K K Lorentz d p x force eqn d e E B x x x e B y z m zc t z 1 m c B y z e d x d z B 0 sin k 0 z B 0 K k 0 a e B eb 0 k 0 m c 0 0 m c B y k, const y B z 0 m c k 0 a e 1 K

25 z t x x z x, x 1 k 0 a cos k 0 c t x k 0 a cosk 0 c t xa sin k 0 z a sin k 0 c t 1 4 k a 1 cos k 0 0 c t K 4 cosk c t 0 z t c z t d tc t 0 K x t c x t d t 0 K sin k 0 16 sin k 0 c t longitudinal c t transverse x 'x Lorentz z ' zc t c t transformation ' c t 1 K 8 k 0 sin 7 k c t ' c t z 0 c t t t ' 1 4k 0 c k 0 c t ' 1 4 K K sin k 0 K K sin k 0 c t ' 7 c t ' t t ' to the 1st approximation usually using the 1st term is good enough the nd term is used in differentiation

26 x ' t ' K k 0 sin t 'a sin t ' z ' t ' K 8 k 0 sin t ' K a sin t ' 8 1 k x ' z ' z ' max a 1 x ' K a z ' a max 8 1 K particle's speed in ' the moving frame 1 c d x ' d t ' d z ' d t ' ' K K cos K 4 4 K cos 1 K ' K cos for K 1 nonrelativistic SHM undulator '1 cos 1 for K 3 '1 relativistic wiggler 4 4 moving K K 1 -pattern K 1 1d SHM in x cos k 0 c t ' now

27 K x d P ' d ' e c 8 k ' 4 a sin k ' k wave number in 0 the moving frame e c 8 K k ' y k ' z k ' sin k ' y k ' z K k 0 a k 0 a 1 k P k d 3 P ' d P ' d k ' d P ' c d 3 k ' d ' k ' ' e c 8 K k ' y k ' z k ' k 0 d 3 k ' k 0 ' t ' 0 c 0 c No. of photon emitted d 3 k 'k ' d k ' d ' Inserting k ' k 0 to assure the monochromatic nature time for passing one period of the magnet structure in the moving frame t ' ' d 3 P ' d 3 k ' ' N ' N t d 3 P d 3 k invariant

28 d3 P d 3 k t ' ' t ' k ' y k y k sin sin k ' z k cos k ' k 1 cos d 3 P d d k d c e K k 0 K d 3 P ' d 3 k ' ' c e k 8 3 k k ' k ' y z k ' k 0 0 in the lab variables t t ' k 0 k 1 cos d 3 k k ' k 0 1 1, 1 1 k d k d c 1 4 sin 1 4 k 1 k 0 d P d d d 3 P d d k d d k c e K k 0 total radiated power P c e K k sin 1 5 P d P d d d d d 3 P 1 1 5

29 8 d P d d 3 P 1 d d d d d 3 P 1 1 k 1 1 k K 1

30 max k 0 c at 0 No. of photon per magnet period К K e B 0 k 0 m c e B 0 0 m c 93.4 B 0 T 0 m, max Typical undulator: B 0.5 T, 0 04 cm, E17 GeV Typical wiggler: B 0 1 T, 0 0 cm energy radiated per magnet period K0 EP t t 0 N P t max O K N 3 K E GeV ev 1 K 0 m K max 80 ev- 4 kev c

31 К B

32 E x, t 0 E 0 e i k x t 0 e v t 0 m E 0 ei k0xt 0 : polarization of EM wave d P d e 4 c 3 * v : polarization of radiation d P d c 8 d d e 4 m c 4 E 0 * 0 v 1 vv * Energy radiated/time/solid angle Incident energy flux in energy/area/time 1 cos e x cos e y sin e z sin, 4 d Pd ce 0 8 e 4 m c * 4 0 e x sin e y cos d d e m c 4 cos cos sin linear polarizationx -axis cos sin cos linear polarizationy -axis 4 d d e 1cos m c 4 unpolarized Thomson formula

33 e 4 T 8 Thomson cross section 3 m c 4 e mc classical electron radius cmc

34 k ' k kk m c m c 1 cos 4 d d e k ' m c 4 k * 0 spinless particle T m c m c for m c m c Compton formula : scattering angle in the lab spinless ln electron m c for m c Mc

Radiation processes and mechanisms in astrophysics I. R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 2009

Radiation processes and mechanisms in astrophysics I R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 009 Light of the night sky We learn of the universe around us from EM radiation, neutrinos,