I'm interested in calculating the maximum surface temperature for a rotating planet from the solar constant. I can calculate the solar constant for a planet at any distance from the sun, no problem, deriving 1366 W/m$^2$ arriving at earth/ moon. I can then derive the average black body temperature for said planet, e.g. 255 K for earth. However, for a body such as the moon, with no atmosphere or water, it has a max daytime temp of approximately 370 K and night time temp of 100 K. Whilst the black body temperature theoretically must be the same as earth, this gives no real information on the possible temperature range.
I became interested in working out what temperatures on earth would be like if it was just rock - no water or atmosphere, with just a rock surface. I quickly realised that this is dependent on the rate of rotation: the moon rotates slowly enough that the day time surface at the equator will reach the maximum temperature possible with 1366 * (1 - albedo) W/m$^2$ heating the surface, which does give a temperature around 370 K, using the equation $\sigma T^4$.
For ease of calculations, I am making the assumption that the surface of both the moon and earth are made of basalt with an albedo of 0.85, and basalt has a specific heat of 840 K/kg* K and a thermal conductivity of 1.5 W/mK. From my very basic calculations, it seems the top few inches of the (basalt) lunar surface will reach thermal equilibrium within a few hours, and with a day length of around 15 days, the thermal radiation from the surface will quickly reach the scorching temperatures of 100 C as measured by the Apollo missions and measured from thermal imaging cameras.
However on earth, I think it is rotating too quickly to reach that temperature, but with my maths skills I don't know how to work out the rates of change involved to calculate the temperature at any given point during the day. For this, I am assuming the earth isn't tilted on its axis.
Can anyone help with the maths to calculate this? The black body temperature that is usually derived from the solar constant doesn't take into account the maximum temperature possible during the day, on the equator where the maximum solar energy is hitting the earth and how long it takes the rock surface to heat up and reach thermal equilibrium.