So if you ignore atmosphere and just look at the albedo of a planet the incident radiation you can calculate the average temperature of a planet, though you do have to treat the Moon a bit differently. The moon doesn’t really store much energy so you really just have to treat it not as a planet but just calculate the temperature at any particular point and if the sun isn’t shining treat it like the moon goes close to absolute zero.

So you have the typical black body equation

SA = sigma T^4, where A = (1 – albedo)

a = albedo. albedo of the moon = .12, mars = .15 earth = .367, venus = .76

S = 1366 watts m-2 at 1 AU (for the moon and earth) and 590 at mars, 2620 at venus

For the moon you want to calculate the temperature of a particular latitude so

SA = sigma cosine(latitude) T^4

(sigma is the stefen boltzman constant = 5.6704×10^{−8} W m^{−2} )

For a planet like mars or the earth you can just treat the planet as a disk for incident radiation so that’s pi r^2 and the sphere emits proportional to the surface of a sphere which is 4 pi r^2, the pi r^2’s cancel out and you are left with this factor of 4

SA = 4 sigma T^4

T = [SA / (4 sigma)] ^ (1/4)

So anyway those are the numbers and formulas let’s crunch.

Formulas say earth average temperature is -25C = -13F. Real temperature is 59F

Formulas say mars average temperature is -56C, real average temperature is -55C

Formulas say moon noon temperature is 108 C, real noon temperature is 107C

And last we kind of assumed that the moon would be 0K without the sun but it’s really not that cold, something like 40K (-313C or -532F) which we kind of expected since a little bit of sun’s energy gets stored inside the moon.

so anyway, these black body stephen boltzman numbers are really close to reality except of course for the earth which has this greenhouse stuff going on. It’s good of course that we have greenhouse gases without it the earth would be 72F (22C) colder. It would be a pretty unpleasant place.

Good luck by the way ever trying to google the calculations I did above, your first twenty hits will be a bunch of global warming denier poets trying to explain physics with metaphors and analogies.

For comparison sake I’ll add venus too, its black body temp is -44C why is it supposed to be so cold? Because it’s the shiniest planet in the solar system and it actually has less radiation hitting the planet than we have in earth because it is so shiny.

Real temperature of Venus: 460C, that’s 860 degree F, more than hot enough to melt lead. Runaway greenhouse effect that’s what you have on Venus.

How about a comparison number for Venus since it’s solar system greenhouse champ?

sure i can add venus

what this tells me: earth going into a global cooling ice age, with, e.g., an approching magnetic reversal, was a real threat, and a reasonable part of our history. global warming is occuring, but it is really pushing the model parameters to increase the temp very much, which were things more blackbody like would be making us cooler.

One reason the numbers seems to match so well is that doing your calculation in reverse is how they calculated albedo, most likely. So you are getting what you put in when you pat yourself on the back you the temperature correct. A more independent method to get the albedo would be to integrate the solar spectral intensity, which is an extremely complex function with nearly infiniate entropy. The temperature is computed by numerically fitting the temperature parameter such that net integrated ir cooling = net non-reflected heating. The ir cooling is probably blackbody-like so there you are a-ok using the sigma. The heating function however is going to be very messing and frequency dependent. For instance as we kill off rainforest trees or weeds using herbicides like roundup that uv radiation is less efficiently chemically converted into entgy and is instead better modeled via the blackbody model you implicitly assume (bare soil).