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