Question 9: Nov 23, 2017
Hand-in: Dec 7, 2017

Accretion around stellar-mass black holes produces radiation that
peaks in x-rays.  From accretion around supermassive black holes, the
spectrum peaks in the ultraviolet.  The task is to derive an
order-of-magnitude formula relating the peak wavelength (or frequency)
to the mass of the accreting black hole.

Radiation comes from near the inner edge of the accretion disc.  For
this problem, let us drastically approximate that region by a sphere
with radius corresponding to the ISCO, and radiating as a blackbody.
If we can estimate the blackbody temperature, the result will follow.

If the temperature is too high, the radiation pressure will blow away
accreting gas.  So the expected temperature will be such that
radiation pressure times gas-particle cross-section equals the weight
of the gas particles.  The gas will certainly be ionised, hence the
cross-section will be that of Thomson scattering.

Feel free to use h = c = G = 1 for the calculation.  You can always
put the factors back in the last step, using dimensional arguments.