# Temperature of an opaque body with known reflectance

Assume we have an opaque body, irradiated from all sides with a spectral irradiance $$E(\lambda)$$ [W/m2]. Furthermore, we know its reflectance $$R(\lambda)$$. (It's in vacuum, and in no thermal contact to anything).

The heat absorbed per time will be $$dQ_{irrad}/dt = (1-R(\lambda))*A*E(\lambda)$$. Finally, we assume the situation has reached a steady state.

I would have assumed that in the steady state:

• The body radiates a certain amount of heat per time $$dQ_{rad}/dt$$ and reaches a certian temperature $$T$$.

• The same amount of power enters and leaves the body $$dQ_{irrad}/dt = dQ_{rad}/dt$$

I struggle, starting from this scenario:

• to derive the temperature that the body will reach
• to relate to the notion of (spectral) emissivity (how can I look at this to see $$\epsilon(\lambda) = \alpha(\lambda)$$)

Let me know if the problem if underspecified and even better how to solve it. Thank you for your help. Regards