# By how much does starlight heat the Earth?

According to this, the stars in the night sky have a cumulative magnitude of -6.5. This is very dim, so I expect the heat generated to be tiny, but I'm wondering how tiny.

Moonlight does measurably increase the Earth's temperature. According to this, the full moon heats the lower atmosphere by about 0.02°C, though a part of that difference may be from orbital effects. Is the heat from starlight significant enough to be detected above noise?

• SE.EarthScience might be another place to ask at.
– Nat
Commented May 18, 2018 at 21:49
• I wouldn't consider -6.5 very dim. That's brighter than Venus at the brightest (about -4 mag). Commented May 18, 2018 at 21:51
• Fair point. I was more comparing it to moonlight. I don't even know if -6.5 gets starlight over the combined magnitude of planetshine. EDIT: actually it does. Commented May 18, 2018 at 21:53
• en.wikipedia.org/wiki/Apparent_magnitude#Example:_Sun_and_Moon Commented May 18, 2018 at 22:36
• Though I don't know for sure, I'd expect it to be below the heat lost through the radiant thermal energy of the earth. Commented May 18, 2018 at 23:14

The average temperature of the Earth is the result of a radiative balance--the Sun adds energy, and the Earth radiates energy at the same average rate (call that rate the power $$P$$).

Approximating the Earth as a blackbody, the Stefan-Boltzmann law says: $$P \propto T^4$$

If you increase $$P$$ by a small amount $$dP$$ and call the resulting increase in equilibrium temperature $$dT$$, we see from the above relation that: $$\frac{dT}{T}=\frac{1}{4}\frac{dP}{P}$$

If we let $$P$$ represent the power from the Sun and $$dP$$ the additional power from the stars, and take the average apparent magnitude of the Sun as $$-26.74$$, and that of the stars as $$-6.5$$, we have: $$\frac{dP}{P}\approx 8 \times10^{-9} \implies \frac{dT}{T} \approx 2 \times10^{-9}$$

Thus, as the average surface temperature of the Earth, $$T$$, is around $$300K$$, you can expect to bump the temperature up and down by about $$0.0000006K$$ when you switch the stars on and off.

You could probably do worse than just assuming that the effect scales linearly with the incident flux (for low flux levels) and then just work out the ratio of fluxes.

If the cumulative magnitude is -6.5, then this compares with -12.5 for the full Moon (it varies of course with Earth-Moon distance). We can probably assume that the spectrum of moonlight is not too different from the average starlight spectrum.

Six magnitudes is a factor of $$10^{-6/2.5}= 0.004$$ fainter. So I would expect the effect of starlight to be 250 times less than the already small effect of reflected light from the full Moon.

The paper you reference cannot readily distinguish the heating effect of reflected sunlight from the Moon from the fact that the Earth tends to be slightly closer to the Sun at full Moon, and actually concludes that the latter is probably more important. I would conclude therefore that there is no chance of detecting any heating effects from starlight.