I saw a program awhile back and they speculated how in the future Andromeda will be visible in the night sky. I forget their prediction in terms of years, but it is long before the earth is too hot for humans as we know it. It is interesting that our galaxy will change into a giant elliptical galaxy before the sun expands enough to take out the earth.

As far as the calculation, while I understand what you are saying I am not aware of the formula. It is so far off, I will just enjoy the simulations :-)