Lunar Temperature
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With no atmosphere and a surface made up almost entirely of rocky materials with low thermal conductivity and relatively low heat capacity, during the lunar day the surface temperature quickly reaches equilibrium with incoming solar radiation. The Stefan-Boltzmann equation sets the numbers:
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<math> I = \epsilon\sigma T^{4}</math>
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where I represents absorbed solar energy per unit area, T is the absolute surface temperature (kelvins), <math>\epsilon</math> is the emissivity, and <math>\sigma</math> is Stefan's constant, 5.67x10^-8 in metric units.
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For a surface with the sun directly overhead, for example a horizontal region near the equator at lunar noon, I is the solar constant in Earth's neighborhood, about 1366 W/m^2, minus the portion reflected. Since the emissivity is close to 1 minus the reflectance, those two terms cancel out, and inverting the equation gives the maximum day-time high on the Moon: 394 K or about 120 degrees C.
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When the sun's not directly overhead whether you are at the equator during lunar morning or evening, near the poles, or looking at a rock face sharply angled to the horizontal, the surface temperature will be lowered because the same solar energy is spread over a larger area.
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<math>I = 1366 \cos(\theta) W/m^2</math>
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where <math>\theta</math> is the angle of the Sun's position relative to a line perpendicular to the surface. Because the lunar rotational axis is tilted only 1.5 degrees from the ecliptic, solar angles at noon are always within 1.5 degrees of the lunar latitude value.
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For an angle of 30 degrees, (maximum temperature for a horizontal surface at latitude 30 degrees N or S, or equatorial temperature at roughly plus or minus two Earth days from lunar "noon"), T is then 380 K, or 107 degrees C. At 60 degrees, the temperature is still 331 K or 58 degrees C. At 75 degrees we reach about 281 K or 8 degrees C. At 85 degrees the equilibrated temperature drops to 214 K or -59 degrees C. At the lunar poles there are believed to be regions which never receive direct sunlight. If they don't receive significant warming from higher elevation surfaces that are in direct sunlight, they would be equilibrated only with the thermal background radiation of deep space at 2-3 K (-270 degrees C), and would likely form cold traps holding volatile materials.
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During the night the surface temperature drops further as the rocks radiate away the energy they've absorbed during the day time, with regions near the lunar equator dropping to about 120 K or -150 degrees C by the end of the night.
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The temperature drop is limited by conduction of heat from layers several meters below the surface, which maintain a roughly steady average temperature that can also be determined from the Stefan-Boltzmann law. In this case 'I' represents the incoming solar energy averaged over a full day-night cycle
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<math>I_{ave} = 1366 \cos(\theta)/\pi W/m^2 </math>
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so at the equator T is about 296 K, or a comfortable 23 degrees C if you bury yourself sufficiently. At 60 degrees that drops to 249 K or -24 degrees C. The average subsurface temperature near the poles (85 degrees and higher) would be below 160 K or -110 degrees C.
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