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Let me begin by saying that I'm not talking about the Goldilocks zone of life habitability on the planet earth, but rather human habitability here. The question is how far, or how close would we have to be to the sun to make our planet uninhabitable for us humans. Would 5,000 miles do it? 10,000.? Would it help to know the temperature of spacecrafts circling the earth and their various positions in relationship to the sun at different times as they circle?
Answer 1:

There're actually still a lot of questions that can come into play here, but I'll walk you through my thoughts...

As the Earth is warming due to global warming, we're seeing coastal areas of land become uninhabitable. So we know a 1 degree C increase in temperature could make portions of the Earth uninhabitable due to sea level rise. I imagine, however, that you're wondering at what point would the Earth become entirely uninhabitable by any human anywhere. We'd first have to ask a biologist how high of a temperature humans could stand. I'm not a biologist, but I'll throw out a guess that we could probably survive (quite uncomfortably) in temperatures near the highest that have been recorded on Earth, so something like 135 degrees F or 57 degrees C.

The next question is, how much closer would the Earth have to be to the sun for the entire planet to have temperatures over 57 degrees C (or whatever the biologist tells us would be an entirely unlivable temperature for humans)? Also, we'd have to consider other factors that might contribute to the temperature of the Earth as we move the planet closer to the sun -- for instance, the rising temperature allows more water to be stored as vapor, which is a greenhouse gas, which would allow temperatures to rise further. Also, we need that liquid water to live and we need oxygen in our atmosphere. It would be important to consider how those might be affected.

To start this or any other theoretical question about the Earth, I'm going to use information we already know. The planet Venus is the planet one closer to the sun than us. It's about 108,200,000 km from the sun and we're about 149,600,000 km from the sun. The temperature on the surface of Venus is about 460 degrees C. Clearly, that's way too hot for us to live and it's too hot for liquid water. So to start, I know that the Earth would probably be uninhabitable if it were 40,000,000 km closer to the sun.

That doesn't get us much closer to an answer, but next I'll consider what we know about the Earth's eccentricity, that is, how elliptical the Earth's orbit is around the sun. The Earth's orbit is nearly a perfect circle, but not quite. How circular it is changes over time and this plays a small part in climate change on Earth on 100,000-year and 400,000-year time-scales. If you find the difference in distance between the sun and Earth when the Earth is at its closest point and when it's at its furthest point, then find the increase in solar forcing this causes, we can get a better idea for temperature change over relatively short distances (at least compared to the last Venus example). Estimates for temperature changes due to the changes in solar forcing can be done, but you'll have to keep in mind that our atmosphere will be changing as well.

I hope you have enough information to start your road to discovery! This was an interesting question. I also like science for the sake of science and impractical theoretical questions, but you might also be interested in learning more about real climate change on our planet. Here's a link to some papers and presentations by James Hansen that might be of interest to you:

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