Slide 43 of 48
Notes:
The number density ratios (ie numbers per unit volume) are multiplied by the ratio of gamma^2 because the power produced is proportional to gamma^2 in both cases, and we want to find the ratio of power produced by the IC and synchrotron processes.
Since we are assuming that the energy distribution of cosmic ray particles within a unit volume has the form N(E) which is proportional to E^-(3/2), the ratio of IC to synchrotron produced X-rays reduces to the equation shown.
[Recall that cosmic rays are required to produce X-rays by the IC and synchrotron processes].
