What Is the Luminosity of Emitted Radiation from a Black Hole Accreting Mass?

[bowlbase]

Homework Statement

Suppose that a black hole of mass M accretes mass at a rate $\dot{M}$. Further suppose that accretion of mass Δm leads to the radiation of energy $\Delta E= \eta Δmc^2$, for some effeciency of energy conversion, $\eta$. What is the luminosity of emitted radiation in terms of $\dot{M}$ and $\eta$?

Homework Equations

Luminosity equation, sort of?

The Attempt at a Solution

The only equation I've ever seen for luminosity has been in terms of temperature and radius. So I'm at a loss here where to go with this. I understand that luminosity is the amount of energy an object gives off over time but I'm sure I can't just put that ΔE over a Δt and call it good. This is suppose to be a simple question but I'm just not making the connections I need to solve it.

 

[bowlbase]

Edited for clarification. Currently my answer is just the ΔE over 4pi. Anyone have a suggestion for this?

 

[Dick]

Edited for clarification. Currently my answer is just the ΔE over 4pi. Anyone have a suggestion for this?

Ok. I'm not an expert but I really don't see any other way to go with this. Luminosity is dE/dt as far as I know. So dE/dt=η*dM/dt*c^2=η*$\dot M$*c^2. You don't quite divide them both by Δt, but it's really almost the same thing. You said this is an easy question, right? Maybe it is that easy. At least I don't see what else to do. I don't see what sense ΔE over 4pi makes, it has the units of energy, not energy/second.

 

[bowlbase]

Maybe I'm just thinking too hard about this. What you have must be correct I would assume, but it makes me question why he'd even make that part of the problem set. Unless just to check if we recall what luminosity is. Anyway, thanks for the assistance.

 

[Dick]

Maybe I'm just thinking too hard about this. What you have must be correct I would assume, but it makes me question why he'd even make that part of the problem set. Unless just to check if we recall what luminosity is. Anyway, thanks for the assistance.

Yes, it does seem too easy. Let me know if I've missed something important.