Selected Relationships for Synchrotron Radiation
In these relationships, e = |electron charge| = 4.80´10-10 statcoulombs, me = electron rest mass = 9.11´10-28 g, c = light speed = 2.998´1010cm/s, B is the magnetic induction in gauss, E = total electron energy in erg = γmc2 where 1/γ = Ö (1-v2/c2) and v = electron speed, θ = pitch angle of electron's helical path.
| Q = Quantity | Q = f (constant, variables) |
constant = g(fundamental constants |
Numerical value of constant |
| Critical frequency = frequency of maximum synchrotron intensity | νc = C1BE2 | C1 = 3e/(4πme3c5) |
C1=
6.266´1018,
[B] = gauss, [E] = erg, [νc] = Hz, or C1= 16.08, [B] =μgauss, [E] = GeV, [νc] = MHz |
| Observed power spectrum | P*(E,ν) = 4πC2(B/sin2θ)F(ν/νc) |
C2 =
(Ö 3)e3/(4πmec2) F(x) = xòx¥K5/3(η) dη |
C2
[cgs] = 1.865´10-23 K5/3 is a modified Bessel function of fractional order 5/3 |
| Electron energy depletion rate | dE/dt = -C3B2E2 |
C3 = 32π2C1C2/(9Ö
3) = 2e4/(3me4c7) |
C3[cgs]=
2.368´10-3 1/C3 = 8.4×109 yrs·μgauss2·Gev |
We note that the final equation above, dE/dt = -C3B2E2 , has the solution E(t) = E0/(1 + t / t½), where E0 is the electron's initial total energy and
t½ = 1/(C3B2E0) is the electron energy "half life." This "half life" is very different than an exponential decay half life. With exponential decay, after
n half lives, the decaying quantity is reduced to (½)n of its original value. With the decay scheme described here, after n "half lives," the decaying
energy is reduced to 1/(1 + n) of its original value. Only for n = 1 half life do these two very different decay functions yield the same result.