Radiation from a Moving Charge

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1 Radiation from a Moving Charge 1 The Lienard-Weichert radiation field For details on this theory see the accompanying background notes on electromagnetic theory (EM_theory.pdf) Much of the theory in this chapter can be applied to many cases of radiating charges. Its main use will be in the application to synchrotron radiation. Other reading: Rybicki & Lightman, Radiative Processes in Astrophysics, on which the following development is mainly, but not entirely, based. In particular, the emissivity in the Stokes parameters is not dealt with in R&L. High Energy Astrophysics: Radiating charges Geoffrey V. Bicknell

2 1.1 Retarded time and position x X t Field point P Source point X t Xt x x Xt O Trajectory of charge High Energy Astrophysics: Radiating charges 2/42

3 In the above diagram: O Arbitrary coordinate origin P Field point t time t Retarded time (1) x Position vector of field point X t Position vector of moving charge (source point) Xt Position vector of moving charge at retarded time High Energy Astrophysics: Radiating charges 3/42

4 Retarded time and retarded position vector The retarded time is defined as the time of emission of an electromagnetic wave from the particle which arrives at the field point at time t, i.e. x Xt t t c x Xt t t c The retarded position vector is: r x Xt (2) (3) High Energy Astrophysics: Radiating charges 4/42

5 The corresponding unit vector pointing from the retarded source point to the field point is: Velocities n x Xt x Xt Velocity of charge X t X t c t X t c t (4) (5) High Energy Astrophysics: Radiating charges 5/42

6 1.2 The scalar and vector potential (Lienard- Weichert potentials) Scalar potential Vector potential Electromagnetic field E t x At x q r1 t n 0 q X t r1 t n A B curl A t (6) (7) High Energy Astrophysics: Radiating charges 6/42

7 The result of the differentiations is: Ext q r 2 n 3 n 1 2 r n r 1 n c c B c 1 n E (8) High Energy Astrophysics: Radiating charges 7/42

8 r 2 Many of the terms in these expressions decrease as. These correspond the Coulomb field of the moving charge. However, the terms proportional to the acceleration only decrease as r 1. These are the radiation terms: E rad q c 0 r n n 1 n n 3 (9) Note that E rad n 0 (10) High Energy Astrophysics: Radiating charges 8/42

9 Poynting flux The Poynting flux of the Lienard-Weichert electromagnetic field is given by: S E B E n E c 0 c 0 E 2 n E ne 0 c 0 E 2 n for the radiation component (11) This can be understood in terms of equal amounts of electric and magnetic energy density ( 0 2E 2 ) moving at the speed of light in the direction of n. This is a very important expression when it comes to calculating the spectrum of radiation emitted by an accelerating charge. High Energy Astrophysics: Radiating charges 9/42

10 Trajectory of particle when 2 n Illustration of the beaming of radiation from a relativistically moving particle. 2 Radiation from relativistically moving charges Relativistic beaming Note the factor 1 n 3 in the expression for the electric field. Expressing 1 n 1 cos (12) where is the angle between and n, we can see that this factor is small 1 the particle is relativistic 0 the field point is in the direction of the particle High Energy Astrophysics: Radiating charges 10/42

11 When 1 n 0 the contribution to the electric field is large because of the factor 1 n 3 in the expression for the electric vector. We quantify this further as follows: 1 n cos (13) High Energy Astrophysics: Radiating charges 11/42

12 The minimum value of 1 n is and the value of this quantity only remains near this for 1. This means that the radiation from a moving charge is beamed into a narrow cone of angular extent 1. This is particularly important in the case of synchrotron radiation for which 10 4 (and higher) is often the case. High Energy Astrophysics: Radiating charges 12/42

13 3 The spectrum of a moving charge 3.1 Fourier representation of the field Consider the transverse electric field, E t, resulting from a moving charge, at a point in space and represent it in the form: E t E 1 e t 1 + E 2 e t 2 E e t 12 (14) where e 1 and e 2 are appropriate axes in the plane of the wave. (Note that in general we are not dealing with a monochromatic wave, here.) We approach the problem of determining the spectrum by using Fourier analysis. High Energy Astrophysics: Radiating charges 13/42

14 The Fourier transform relations for the electric field: The condition that E e it E t dt E t E t E e 2 it be real is that E E * d (15) (16) Note: We do not use a different symbol for the Fourier transform. The transformed variable is indicated by its argument. High Energy Astrophysics: Radiating charges 14/42

15 3.2 Spectral power in a pulse Outline of the following calculation E t t Diagrammatic representation of a pulse of radiation with a duration t. t Consider a pulse of radiation Calculate total energy per unit area in the radiation. Use Fourier transform theory to calculate the spectral distribution of energy. Show that this can be used to calculate the spectral power of the radiation. High Energy Astrophysics: Radiating charges 15/42

16 Nitty-gritty The energy per unit time per unit area of a pulse of radiation is given by: dw Poynting Flux c dtda 0 c 0 E 2 1 t E 2 2 t E 2 t (17) where E 1 and E 2 are the components of the electric field wrt (so far arbitrary) unit vectors e and in the plane of the wave. We refer 1 e 2 to the two components of the transverse electromagnetic wave as different modes of polarisation. + High Energy Astrophysics: Radiating charges 16/42

17 Note that there is a difference here from the Poynting flux for a pure monochromatic plane wave in which we pick up a factor of 1 2. That factor results from the time integration of from, in effect, 0 E 2 d cos 2 t which comes. This factor, of course, is not evaluated here since the pulse has an arbitrary spectrum. The total energy per unit area in the component of the pulse is (The reason for the dw c da 0 E 2 t dt subscript is evident below.) (18) High Energy Astrophysics: Radiating charges 17/42

18 From Parseval s theorem, E 2 t dt E 2 2 d (19) The integral from to can be converted into an integral from 0 to using the reality condition: For the negative frequency components, we have E E* E * E E 2 (20) so that E 2 t dt E 2 d (21) High Energy Astrophysics: Radiating charges 18/42

19 The total energy per unit area in the pulse, associated with the component, is dw c da 0 E 2 t dt c E 0 2 d (22) We identify the spectral components of the contributors to the Poynting flux by: dw dda c E 2 (23) High Energy Astrophysics: Radiating charges 19/42

20 dw The quantity represents the total energy per unit area per unit dda circular frequency in the entire pulse, i.e. we have accomplished our aim and determined the spectrum of the pulse. We can use this expression to evaluate the power associated with the pulse. Suppose the pulse repeats with period T, then we define the power associated with component by: dw daddt dw TdAd c E T 2 (24) This is equivalent to integrating the pulse over, say several periods and then dividing by the length of time between pulses. High Energy Astrophysics: Radiating charges 20/42

21 3.3 Emissivity The emissivity is defined in the folowing way: Energy radiated in polarisation mode into solid angle d, circular frequency range d in time dt The total emission coefficient dw d d dt dddt j is defined by (25) dw 11 j dddt dw dddt (26) High Energy Astrophysics: Radiating charges 21/42

22 Variables used to define the emissivity in terms of emitted power. r da d r 2 d Region in which particle moves during pulse Consider the surface da to be located at a distance that is large compared to the distance over which the particle moves when emitting the pulse of radiation. Then da dw daddt dw dddt r 2 d and dw r dddt r 2 dw daddt (27) High Energy Astrophysics: Radiating charges 22/42

23 Emissivity corresponding to the e component of pulse. dw dddt c 0 r E T 2 c 0 r E T E * (Summation not implied) (28) 3.4 Independence of radius Note that dw r dddt 2 E 2 (29) High Energy Astrophysics: Radiating charges 23/42

24 However, we know that for the electric field of a radiating charge so that E t 1 -- r E 1 -- r (30) dw is independent of r dddt (31) High Energy Astrophysics: Radiating charges 24/42

25 3.5 Relationship to the Stokes parameters We generalise our earlier definition of the Stokes parameters for a plane electromagnetic wave to the following: c 0 I E T 1 E * 1 + E 2 c 0 Q E T 1 E * 1 E 2 c 0 U E T * 1 E 2 + E 1 1 V -- c E i T * 1 E 2 E 1 E * 2 E * 2 E * 2 E * 2 (32) High Energy Astrophysics: Radiating charges 25/42

26 The definition of I is equivalent to the definition of specific intensity in the Radiation Field chapter. Also note the appearance of circular frequency resulting from the use of the Fourier transform. To aid the following theoretical development, we define the polarisation tensor by: I I 1 + Q -- U iv 2 U + iv I Q c E T E * (33) High Energy Astrophysics: Radiating charges 26/42

27 In the above development, we calculated the emissivities, dw dddt c r T 2 E E * corresponding to each wave mode. More generally, we define: (34) dw dddt c r T 2 E (35) and these are the emissivities related to the components of the polarisation tensor. I E * High Energy Astrophysics: Radiating charges 27/42

28 In general, therefore, we have dw Emissivity for 1 dddt 2 -- I + Q dw Emissivity for 1 dddt 2 -- I Q dw Emissivity for 1 dddt 2 -- U iv dw dddt dw* Emissivity for 1 dddt 2 -- U + iv (36) High Energy Astrophysics: Radiating charges 28/42

29 Consistent with what we have derived above, the total emissivity is I and the emissivity into the Stokes dw dddt dw dddt Q is (37) Q dw dddt dw dddt (38) High Energy Astrophysics: Radiating charges 29/42

30 Also, for Stokes U and V: U V Note that the expression for is independent of radius, r, because of the r 1 dw dddt dw 12 i dddt dw* dddt dw* dddt dw dddt (39) dependence of the Electric field. High Energy Astrophysics: Radiating charges 30/42

31 4 Fourier transform of the Lienard-Weichert radiation field The emissivities for the Stokes parameters depend upon the Fourier transform of where re t t q r --- n n c 0 r 1 n 3 r t --- r x Xt c (40) (41) High Energy Astrophysics: Radiating charges 31/42

32 The aim of this section is to find the most convenient way of expressing the Fourier transform of re t in terms of the motion of the charge. To begin with we ignore the difference between r and r since the distance to the field point is large compared to the distance over which the charge moves, i.e. r r 1. Transformation to retarded time The Fourier transform involves an integration wrt t. We transform this to an integral over t as follows: To prove that dt t -----dt 1 ndt t t n, we proceed as follows: t (42) High Energy Astrophysics: Radiating charges 32/42

33 The relationship between field point time t and source point time (retarded time) is given by: x Xt t t c Differentiate wrt t: (43) t t x c t Xt (44) Now, x t Xt x t i X i tx i X i t 12 / High Energy Astrophysics: Radiating charges 33/42

34 x Xt 2 x X t i i ( X t i t) t 1 t i tn i 1 n (45) Hence, re q c 0 q c 0 n n n 3 e it 1 n n n n 2 e it dt dt (46) High Energy Astrophysics: Radiating charges 34/42

35 Integrand in terms of retarded time The next part is to express e it exp it r c (47) in terms of retarded time t. Since r x Xt x when x» Xt (48) High Energy Astrophysics: Radiating charges 35/42

36 then we expand r to first order in X around X i 0. Thus, r x j X j t r0 r X X i i r X i x i X i t x i at X r r i 0 (49) r Note that it is the unit vector the retarded unit vector r x i ---X r i r n i X i r n Xt n n r - r which enters here, rather than High Energy Astrophysics: Radiating charges 36/42

37 Hence, expit exp exp it + r - c n Xt c n Xt i t c exp ir c (50) ir The factor exp is common to all Fourier transforms re c and when one multiplies by the complex conjugate to obtain E E * this factor gives unity. This also shows why we expand the argument of the exponential to first order in leading term is eventually unimportant. Xt since the High Energy Astrophysics: Radiating charges 37/42

38 Convenient form of integrand The remaining term to receive attention in the Fourier Transform is n n n 2 (51) We only need to expand the unit vector n to zeroth order in because the zeroth order term does not disapear. Hence n n Therefore, X i (52) n n n 2 n n n 2 (53) High Energy Astrophysics: Radiating charges 38/42

39 It is straightforward (exercise) to show that d n n dt 1 n n n n 2 (54) Hence, re q e ir c 4c 0 d n n dt 1 n exp it n Xt c dt (55) High Energy Astrophysics: Radiating charges 39/42

40 One can integrate this by parts. First note that n n n 0 since we are dealing with a pulse. Second, note that, (56) d dt exp i t n Xt c exp it n Xt c (57) i1 n High Energy Astrophysics: Radiating charges 40/42

41 and that the factor of 1 n cancels the remaining one in the denominator. Hence, our final result: re iq e ir c 4c 0 n n exp it n Xt c (58) In order to calculate the Stokes parameters, one selects a convenient coordinate system ( e 1 and e 2 ) adapted to the physical situation. The motion of the charge enters through the terms involving t and Xt in the integrand. dt High Energy Astrophysics: Radiating charges 41/42

42 Remark on relativistic beaming The feature associated with radiation from a relativistic particle, namely that the radiation is very strongly peaked in the direction of motion, shows up in the previous form of this integral via the factor 1 n 3. This dependence is not evident here. However, when we proceed to evaluate the integral in specific cases, this dependence resurfaces. High Energy Astrophysics: Radiating charges 42/42

Electromagnetic Theory

Summary: Electromagnetic Theory Maxwell s equations EM Potentials Equations of motion of particles in electromagnetic fields Green s functions Lienard-Weichert potentials Spectral distribution of electromagnetic