There is an observed relation, between the brightness & power of relativistic jets, common to Quasars & GRBs:
Could the following calculations help explain the same?
By partial fractions:
So the integration yields:
For relativistic jets , and most of the power is radiated early on, at high . So:
But the initial electron energy was also proportional to . So, the average power of emissions is (calculated to be) constant at all energy scales:
Perhaps some similar sort of scale invariance, whereby the power emitted by decelerating electrons is quasi-constant, could account, for the observed Jet brightness / power relation.