1. Rybicki & Lightman Problem 6.4 The following spectrum is observed from a point source at unknown distance d Assume the source is spherical, emitting synchrotron radiation R = radius of sphere B = magnetic field And the space between us and the source is filled with hydrogen which emits and absorbs by bound-free transitions, but is unimportant at Frequencies where the synchroton is optically thin.

2. Let the synchrotron source function be The absorption coefficient for synchrotron The bound-free absorption coefficient: Where A, B o, o, C and D are constants. p = power-law index of electron energies

3. a. Find R and B in terms of A, B o, o, C, D and the solid angle subtended by the source, When the source is optically thin Here, the source is optically thin in the part of the spectrum has so p = 2

4. At 2 the source becomes optically thick. So And implies(Equation 1) implies Substitute into (1), solve for R:

5. (b)Now find the solid angle and distance to the source The 2 part is the optically thick thermal emitter: the Rayleigh-Jeans tail So 1 = the frequency at which the hydrogen becomes optically thick so and So we have