If the planet and star in question are very similar to the Earth and the Sun, how can I calculate the planet's surface temperature by knowing the distance from the star?
This question was originally asked on the Worldbuilding stack exchange, and someone suggested I ask it here.
Neat and tidy calculations for this aren't possible.
The math behind effective temperature of a planet can be found here and that's a straight forward calculation where you can input distance, solar energy, planet's albedo and get an average temperature. Earth's effective temperature calculation comes to -21 °C, which obviously isn't accurate, but that's as far as the calculation can take you.
Feedback mechanisms:
With essentially no change to distance from the sun, and only relatively small changes in tilt and eccentricity, Earth's temperature can fluctuate from a current 15 °C global average to about 5 °C as it goes in and out of ice ages.
Feedback mechanisms like albedo (ice cover is the key driver, but also deserts, surface area of the oceans), CO2 capture, which increases with colder oceans, can amplify small changes, leading to significant changes in surface temperature. An orbital change, which should account for maybe 1 degree C, or an increase in CO2, which, by itself, captures less than 1 degree C, through feedback mechanisms, can lead to much larger temperature changes.
More on feedback mechamisms here.
Land placement and Ocean circulation
5 or 10 million years ago, Earth's average temperature was about 18-20 °C and that change was driven, primarily, by the closing of the Isthmus of Panama which affected ocean circulation. Just the closing of the Isthmus made ice ages possible. That's a pretty small change in the grand scheme of things, whether your planet has a near equatorial passage where oceans can flow or whether it doesn't, but that one thing can mean more than 10 degrees.
As Antarctica drifted over the south pole (the last 30-35 million years or so) glaciers formed and the earth cooled, because as ice forms, more sunlight is reflected into space and oceans levels drop. Oceans having the lowest albedo and ice the highest. Also, as Antarctica driffs away from the south pole over the next 20-30 million years, Earth is expected to warm, unless another change occurs, like the continued absorption into land and the oceans of CO2.
Earth can only enter an ice age when there's land near the poles. The glacial period of 440-460 million years ago most of the land was over the south pole. The sun was about 4% less luminous and the earth had much more CO2 (though CO2 levels did drop during that time, they didn't drop close to where they are now) and the orbit of the earth, 440 million years ago has some unknowns, but a large land mass over the south pole is thought to have played a key role in making that glaciation possible. The tipping point of ice formation at the poles is particularly sensitive and a significant variable.
So, even with very similar to earth like planets. Just where the land is and how well the oceans circulate and whether the planet can form glaciers and sea ice, or not, can fluctuate perhaps 20 degrees C in global average temperature. How much CO2, how much active volcanism, how much reflective material in the upper atmosphere - all variables.
Even with identical earths, the feedback mechanisms make it difficult to predict what Earth's temperature would be if you pushed Earth 1 million or 1/2 million miles further from the sun. The calculations for surface temperature wouldn't follow a simple mathematical formula. Now, you'd still get colder as you moved the planet away, but one push might give you 1-2 degree and the next push might give you 5-10 and the one after that 0-1. There would be no way to do a clean formula.
