# What is mathematical formula for finding the mean surface temperature on a planet in our solar system?

This question is to assume the common solar luminosity of approx $$4 \times 10^{26}$$ Watts.

The mean flux of Earth at, for example, 1.1 AU would be about 1176 watts--but how can I calculate mean surface temperature? Additionally, how might I reasonably calculate the temperature at more specific locations, such as at the poles and at the equator? This is to assume that all other conditions on Earth are the same EXCEPT for the distance from the Sun.

• The "mean temperature of the atmosphere" is an ill-defined and meaningless value. I think, but cannot be sure, that you want the average temperature on the surface of the planet. Aug 18, 2022 at 16:05
• That's exactly what I had in mind, thank you for your help! Aug 18, 2022 at 16:31
• I think this is what you are looking for. Since the average temp of a planet is wildly variable due to chemical composition of its atmosphere and the surface albedo. pveducation.org/pvcdrom/properties-of-sunlight/… Aug 18, 2022 at 17:33
• Thank you so much for your help, Gillgamesh! Much appreciated! <3 Aug 19, 2022 at 15:56

# The Proper Way

This gets really complicated, really quickly! Some factors:

• Albedo, or how much light the earth gives off. This is about 0.3 for back-of napkin efforts, but actually depends on what material is pointing to the sky (and atmosphere).
• Incoming solar energy, which starts as a baseline energy for the whole planet but also changes based on location (because the earth is round).

You then get to develop or find a climate model that takes the above factors in, calculates their effects on currents and wind, and allow for a variable incoming wattage, get a degree in astro-meteorology, and get the final answer.

# The Improper Way

It's time for radical approximations. We are talking spherical cows, frictionless planes, negligible air resistance sort of approximations! Nevermind that incident solar energy runs the weather and determines the climates of the world, which then in turn determines average temperature for a given time and season! We don't have time for that!

The general approach here would be...

1. Select a real city given city to use as your approximate stand in for your hypothetical earth
2. Get an average temperature and solar radiance for that city
3. Use the Stephan-Bolzmann law to figure out the heat flux out for that spot.
4. Use this to get a ratio of energy lost from the sun to energy gained from the sun
5. Figure out the new solar radiance for your hypothetical earth and location.
6. Use the ratio from 4 to determine how much energy the new hypothetical spot is putting back out into space
7. Use Stephan-Bolzmann to figure out the new temperature, using the energy flux from 6.

# Planet-Wide Calculation?

Using that Stephan-Bolzmann law for a whole planet is more accurate, because you do not have winds and waves doing things like transferring heat to or away from the area of consideration. We know Earth's albedo is about 0.3, so we can jump directly to taking solar flux and bringing it in to get temperature.

Make sure, once again, to use that grey body approximation!

• PipperChip, this is such a brilliant answer. I am grateful for your multi-layered post and useful links to the math involved here, which will keep me busy working everything out. Your time and efforts are very much appreciated, thank you so much! <3 Aug 19, 2022 at 16:00
• @JMYaden aww shucks. I am glad it's useful for you and hope you do not mind the occasional joke. I try to make these informative and entertaining. Aug 20, 2022 at 2:55
• I most certainly do not mind your humor, I got it! XD thank you again for taking the time to offer such a detailed response. Your expertise is appreciated and will not go to waste. Aug 20, 2022 at 6:21