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Definitely radii, with one notable exception... When observers talk about how large an object is on the sky, they usually discuss angular size, which is related to the diameter of an object, not the radius. So when discussing the angular size of, say, Alpha Centauri A (a few milliarcseconds), this is related to the diameter of the star, not the radius.

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The formula $$4 \pi R ^ 2 ơ T ^ 4 = \frac{\pi R ^ 2 L_{sun}(1 - a)}{4 \pi d ^ 2}$$ is correct, if you want to calculate the radiative equilibrium temperature. You only need to use the right units. We can further simplify the formula to $$T ^ 4 = \frac{ L_{sun}(1 - a)}{16 \pi d ^ 2 ơ}\;.$$ You should input the luminosity in watts, the ...

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As of March 6, 2015, Dawn has entered orbit around Ceres. But it's really so far only "captured by Ceres' gravitational pull". Then, it was still 61,000 km from Ceres. It's slowly spiraling down into an orbit that JPL's Dawn Journal calls "RC3", which will be 13,500 km above Ceres. This orbit will last 15 days per revolution. It will take about 15 ...

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Your argument (large temperature leads to greater mixing) is correct so long as there are no other large scale forces acting on the system. This isn't true in planet formation, because gravity plays a very important role. I'm not an expert on planet formation, but I think the argument goes something like this: As a planet forms from material from the ...

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