Synchrotron Power Cosmic rays are astrophysical particles (electrons, protons, and heavier nuclei) with extremely high energies. Cosmic-ray electrons in the galactic magnetic field emit the synchrotron radiation that accounts for most of the continuum emission from our Galaxy at frequencies below about 3 GHz. We can use Larmor's formula to calculate the synchro tron power and synchrotron spectrum of a single electron in an inertial frame in which the electron is instantaneously at rest, but we need the Lorentz transform of special relativity to transform these results to the frame of an observer at rest in the Galaxy. (x; y; z; t) (x ; y ; z ; t ) v x The Lorentz transforms relating the event coordinates in the unprimed frame and the coordinates in the primed frame moving with velocity in the direction are: x = (x + vt ) y = y z = z t = (t + Ìx =c) (5B1) x = (x À vt) y = y z = z t = (t À Ì x=c) (5B) where Ì Ñ v =c (5B3) and Ñ ( 1 À Ì ) À1= (5B4) is called the Lorentz factor. (Áx; Áy; Áz; Á t) (Áx ; Áy ; Áz ; Át ) If and are the coordinate differences between two 1 of 6 1/16/8 11:49 AM

events, the differential form of the (linear) Lorentz transforms is: Áx = (Áx + vát ) Á y = Áy Á z = Áz Á t = (Át + ÌÁx =c) (5B Áx = (Áx À vát) Áy = Áy Áz = Áz Át = (Át À Ì Áx=c) (5B6 Here is a derivation of these results that you should review before proceeding. Using the famous equation m e of an electron: E = mc we can calculate the energy equivalent to the rest mass E = m c e = 9:1  1 À8 g  ( 3  1 1 À1 cm s ) = 8:  1 À7 erg 8:  1 À7 erg E = :1 ev :51 MeV 1:6  1 À1 À1 erg (ev) = 5  1 5 = 14 Cosmic-ray electrons with energies in the range to ev have and such cosmic-ray electrons are called ultrarelativistic. These electrons still move on spiral paths along magnetic field lines, but the angular frequencies of their orbits are lower because the inertial masses of the electrons are higher by a factor of :! 9 1 1 9 14 1 to 1 Ù Ù 1 3 to 1 8 µ 1 :51  1 6 B = eb! = G m e c = 1 5 B Ù 5  1 À6 Example: A cosmic-ray electron with in the Galactic magnetic field G will have an orbital frequency B Ñ! B Ù 14  1 À5 Hz Ù 1 cycle in two hours: Ù v Ù c µ 1 R Since whenever, the orbital radius of an ultrarelativistic electron is quite large: À1 R Ù c Ù 3  1 1 cm s Ù 3:4  1 13 cm Ù AU! B Ù Â 14  1 À5 Hz At first glance, these results are not very promising for the production of radio radiation: the high relativistic masses of cosmic-ray electrons reduce their orbital frequencies and accelerations to extremely low values. However, the Larmor radiation formula is only valid at of 6 1/16/8 11:49 AM

low velocities; that is, in inertial frames in which the electron is nearly at rest. In the observer's frame, two relativistic effects account for the strong radio radiation: (1) the total power is multiplied by and () beaming turns the slow sinusoidal radiation into a series of sharp pulses containing power at much higher frequencies. We proceed to calculate these relativistic corrections. Synchrotron Power From a Single Electron Nonrelativistic equations such as Larmor's equation describing the electromagnetic radiation from an accelerated charge are correct only in inertial frames where the electron velocity v Ü c, but the results can be transformed to any other inertial frame by the Lorentz transform. In this way, it is possible to calculate the total power radiated by an ultrarelativistic electron in a magnetic field parallel to the x -axis. We use primed coordinates to describe an inertial frame in which the electron is (temporarily) nearly at rest. Then Larmor's equation correctly gives What is e (a ) P? = : 3c 3?, the magnetic acceleration of the electron in the galaxy frame? The differential form of the Lorentz transform yields so a ; vy v y = : This factor of is a consequence of relativistic time dilation clocks in moving frames appear to run slow by a factor. Consequently, Similarly, a z = a z= so a? = : Ø 3 B = G dy dy dt dy dt v y Ñ = = = v y dt dt dt dt dt dt dt dt = 1 dt dv y dv y dt a y Ñ = = 1 dt = : dt dt dt dt dt a? dv y a y Thus 3 of 6 1/16/8 11:49 AM

P = How do we transform P to P, the power measured by an observer at rest in the Galaxy? The following argument is from Rindler's Essential Relativity, p. 98. Imagine two identical electrons of rest mass m e, one at rest in the unprimed frame and the other at rest in the primed frame. If one electron is slightly displaced from the other along the y -axis, they will interact as they pass each other and be accelerated in the Æy direction. Observers at rest in each frame see "their" electron move with some small, but the "other" electron will appear to move in y the opposite direction by a factor more slowly because of time dilation recall the result above. Invoking momentum conservation, observers in each frame conclude that v = y = v y the "other" electron has inertial mass. Thus? e (a ) 3c 3 v y Ü c = e a? 3c 3 that is, power is the same in all frames. Consequently, m e and hence its energy is greater by the same factor de de dt P Ñ = = de de dt = P 1 ; dt dt dt de dt dt = P P = e a? 4 ( a ) 3c 3 k = 4 Recall that and, by force balance,! B = eb mc so Ë a? Ñ dv? v dt =! B? a? = ebv? ebv sin Ë = ; mc mc ~ v B ~ Ë where the angle between and is called the pitch angle. For a given pitch angle, the time-averaged radiated power of a single electron is e P = eb v sin Ë 3c 3 mc 4 of 6 1/16/8 11:49 AM

Ë B ~ ~ v The pitch angle between the directions of the magnetic field and the electron velocity. We can express this power in terms of the Thomson cross section of an electron, T. The Thomson cross section is the classical scattering cross section for electromagnetic radiation. If a plane wave of electromagnetic radiation is incident on a charge at rest, the electric field of that radiation will accelerate the charge, which in turn will radiate power in other directions according to Larmor's equation. This process is called scattering, not absorption, because the total power in electromagnetic radiation is unchanged: all of the power lost from the incident plane wave is reradiated in other directions. In one of the problem sets, you show that the geometric area that would intercept this amount of incident power is Û Û T Ñ 8Ù Ò e Ó 3 m e c (5B7) Numerically, Û T = 8Ù Ô (4:8 Â 1 À1 statcoul) Õ Ù 6:65 Â 1 À5 cm 3 9:1 Â 1 À8 g (3 Â 1 1 À1 cm s ) The reason for using the Thomson cross section will become clear when we discuss inverse- Compton scattering of radiation by the same cosmic rays that are producing synchrotro n radiation. Also, we can replace B by the magnetic energy density B UB = 8Ù (5B8) to get 5 of 6 1/16/8 11:49 AM

Ô 8Ù Ò e Ó Õ Ò B Ó v P = c sin Ë 3 mc 8Ù c P = Û T Ì c U B sin Ë (5B9) Ì Ñ v=c where. The radiated power depends only on physical constants, the square of the electron energy (via ; Ì Ù 1 for all µ 1), the magnetic energy density, and the pitch angle. The relativistic electrons in radio sources can have lifetimes of thousands to millions of years before losing their ultrarelativistic energies via synchrotron radiation or other pro cesses, so they are scattered repeatedly by magnetic-field fluctuations and charged particles in their environment, and the distribution of their pitch angles Ë gradually becomes random. The average synchrotron power per electron in an ensemble of electrons with the same Lorentz factor hp i but random pitch angles is therefore hp i = Û Ì T c U hsin B Ëi : Z Z hsin Ëi Ñ sin ËdÊ d Ê = 1 Z sin ËdÊ 4Ù Z Ù Z Ù hsin Ë i = 1 sin Ë sin Ë dë d 4Ù = Ë= hsin Ë i = 1 4 Ù = 4Ù 3 3 4 hp i = Û Ì cu 3 T B (5B1) µ 1 This is the average synchrotron power emitted by a relativistic electron. When, Ì Ù 1 and the Ì factor may be ignored. Relativistic effects make the synchrotron power a factor larger than in the limit v Ü c, so for electrons with Ø 1 4, the power radiated by 8 each electron is multiplied by, a huge amount. 1 6 of 6 1/16/8 11:49 AM