Approximating the sun as a black body effectively means that it's surface temperature is 5800k. The reason we say the earth moves around the sun is because the effects are more visible on a macro scale, and easier to. The sun is currently brightening at about one percent per hundred million years, and that rate is gradually increasing.
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Relatively recent measurements indicate that the sun is nearly the roundest object ever measured.
How would you prove to them that earth orbits the sun?
1) the sun seems brighter (more dazzling) if there is more scattering in the atmosphere. And, indeed, if we look at the most dominant wavelength of the sun. The sun would actually look very small to us in the sky if there were no atmosphere (it's the same. You say that sun rises in the east (with a certain degree of oscillations due to the tilt of the axis) just because the earth.
The sun does not rise, it is the horizon that goes down. Well, from the sun's perspective, one square km of land near the poles looks a lot smaller than one square km of land near the equator for the same exact reason. If scaled to the size of a beach ball, it would be so round that the difference. Those are completely different things.
3 the sun, moon, earth (and so on) all move around each other.
Similarly, how would you prove to them. Assume you're talking to someone ignorant of the basic facts of astronomy.
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