astrometricsastrophysicsearthnewtonian-gravityplanets
I know that there are several formulae that one can plug numbers into to arrive an estimate of Earth's speed around the sun (Kepler's third law for instance), but I'm wondering how these things are measured.
Since Earth is moving in an ellipse around the Sun, I thought doppler measurements of spectral lines would help tell where we were on the ellipse, since the velocity won't be constant everywhere.
But other than that, I'm not sure how one measures Earth's velocity.
Well, it depends... If you just made the sun much heavier, so the earth would have to move faster in it's orbit, you wouldn't feel any different. It's just that the year would be shorter and the tides higher.
If you just put a rocket behind the earth and somehow put it on rails so it couldn't go to a different orbit, then you'd feel it. You'd be heavier in the daytime and lighter at night. If it was faster enough, you'd fly off into space at night, along with the air, the ocean, your room, a few new volcanoes, and everything else :)
Let me try this way: the Sun isn't only pulling on the Earth, it's pulling on the Moon as well. The pull on the Earth is almost the same as the pull on the Moon, so the net effect of the Sun on the relative motion between the Earth and Moon is very small.
Recall Galileo's law of motion: if you drop two objects close together from the same height, they experience the same gravitational acceleration, so they stay together during their fall:
Now think of the ground as the Sun, the large object as the Earth, the small object as the Moon, and the fall as their orbit around the Sun: the Earth and the Moon stay together, because they experience (almost) the same gravitational acceleration from the Sun.
Notice that I said almost: the gravitational pull of the Sun decreases with distance, so the object closest to the Sun does feel a slightly larger acceleration. This means that if there wasn't any gravitational force between the Earth and the Moon, they would indeed slowly drift away from each other over time. But the gravitational force between them prevents this, and because this force is much bigger than the net effect of the Sun, the latter only causes small perturbations in the motion of the Earth and Moon around each other.
Best Answer
There is a lovely presentation by Terence Tao of UCLA, The Cosmic Distance Ladder, that explains this, and much, much more, in detail. Here's a youtube of it, and here are the slides he used.
As a brief summary, to measure Earth's velocity, you need to know the distance to the Sun, plus its orbital period of 1 year. The earliest method to do so is due to Aristarchus, and was based on the half moon not happening exactly midway between the full and new moons. It required knowing the Earth-Moon distance, which required knowing the Moon size, which required knowing the Earth size, hence the distance ladder.
I won't spoil the fun of reading/watching it by trying to reproduce any more of it here.