I'm trying to figure out the time required for a blackbody to cool assuming it only looses heat via radiation.
I can estimate the mass, specific heat, surface area, emissivity, initial temperature, final temperature, etc.
I know the amount of heat which has to be lost is:
$ Q = mc\Delta T$,
where $m$ is mass, $c$ is specific heat, $\Delta T$ is a temperature_change
I know the rate of heat loss is:
$ P = \epsilon kST^4$,
where $k$ is Stefan Boltzmann, $T$ is the body temperature, $S$ is the surface area, $\epsilon$ is the emissivity.
The body temperature changes as the blackbody cools so I cannot simply use the initial or final temperature or I get very different results. I think I have to somehow integrate $P$ from $t = 0$ (when body is at initial temperature) to when the body is at the final temperature.
Any help would be greatly appreciated.