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Let's say you've calculated the tidal heating of a moon in Watts using the method described in How does one calculate the tidal heating of a satellite?, how do you then convert this to surface temperature?
In my case, the moon orbits a gas giant which orbits a star.
Equilibrium assumption. Once you have the heating power generated inside, $\dot E$, a reasonable way to calculate (or at least approximate) the surface temperature, is to simply assume that: all the power generated by the tidal heating, is radiated outwards and lost to space.
Why assumption is reasonable. Note such assumption is indeed reasonable: assume the opposite: assume that the power generated is greater than radiated power, $\dot E > P$. This means that more heat is generated than the system is able to dump, meaning, an increase in temperature, meaning, $P$ will get larger because $P$ increases with $T$, and this will proceed until $P = \dot E$. Assume the opposite, assume $\dot E < P$, in this case, more power is being radiated outwards than being generated, meaning, temperature will decrease, and thus $P$ will decrease, until $P = \dot E$. Indeed, $P = \dot E$ is the equilibrium case.
Calculating surface temperature. Using Stefan-Boltzmann Law, the power radiated by a surface of temperature $T$ is: $$ P = A\epsilon\sigma T^4 $$
where $A$ is the surface area of the object, $\epsilon$ is the emissivity of the object [for perfect blackbodies, $\epsilon=1$], $\sigma$ is a constant, known as Stefan–Boltzmann constant, and $T$ is the temperature of the surface (after all, power is being radiated outwards from the surface).
Since you claim you have $\dot E$, then just make said assumption above: $P = \dot E$. That said, temperature becomes trivial to find:
$$ T = \left(\frac{\dot E}{A\epsilon\sigma}\right)^{\frac{1}{4}} $$
You might also want to include $P_0$, the power radiated inwards towards the satellite [say, by a star, or whatever]. In such a case, the equation would be: $P - P_0 = A\epsilon\sigma T^4$. The calculation of $P_0$ is not complicated and can be done simply using geometrical reasoning.
This is just an approximation: A similar calculation than this is used to estimate temperature of planets, and to calculate Goldilocks zone (or habitable zones) around a star: the power received by the planet [Stefan-Boltzmann] + generated inside [tidal locking + etc] = power radiated outwards [Stefan-Boltzmann].
Above procedure is also used to calculate temperature of stars based on the radiated power [it is reasonable to assume stars are perfect blackbodies].
However, this calculation complete ignores absorption and re-emission of thermal radiation by atmospheric gasses [like, the greenhouse effect]. The more dense and atmosphere of a planet, the more this calculation risks being in error.
This article:
"Exomoon Habitability Constrained by Illumination and Tidal Heating", Rene Heller and Roy Barnes, Astrobiology 2013 - https://arxiv.org/vc/arxiv/papers/1209/1209.5323v2.pdf - has a lot info for anyone interested in the potential habitability of hypothetical giant exomoons orbiting giant exoplanets.
What you are looking for is called heat equation, and is a well established way to calculate the spatial and temporal variation of the temperature of an extended body knowing the energy flow.
$\partial u \over \partial t$$-a \nabla^2u=0$
In your case it can be assimilated to the case of internal heat generation.
$1 \over \alpha$$\partial u \over \partial t$$=($$\partial^2u \over {\partial x^2}$$+$$\partial^2u \over {\partial y^2}$$+$$\partial^2u \over {\partial z^2}$$)$$+ $$1\over k$$q$
where $\alpha$ is the thermal diffusivity $\alpha = $$k \over {c_p \rho}$
By solving that equation after having assigned the boundary conditions you can get the temperature distribution in the entire body and thus also on its surface.
