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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.

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Earth doesn't revolve around Sun.. It moves in straight line with uniform speed.. Seriously! –  Sachin Shekhar Dec 31 '12 at 14:34
    
@SachinShekhar: Then, I'd have to become a layman (in astronomy) now :-) –  Waffle's Crazy Peanut Dec 31 '12 at 15:07
    
@CrazyBuddy I am seriously serious... Its the local geometry of Spacetime which creates the confusion. Recognize the truth.. –  Sachin Shekhar Dec 31 '12 at 15:27
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@SachinShekhar - I suspect you're confused about the realm of applicability of general relativity. The earth does move in a circle around the sun. Space time isn't that warped around the sun. We can happily consider the solar system to be completely flat space and use newtonian dynamics and be completely correct to a very large degree of accuracy. –  Kitchi Dec 31 '12 at 20:06
    
@Kitchi What!!! Then, you don't know anything about General Relativity. Its not a matter of discussion at all. Force of gravity doesn't exist, so there's nothing to accelerate Earth (change direction). –  Sachin Shekhar Jan 1 '13 at 2:52
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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.

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You say that the measurement involves computation. The speed cannot be measured directly :) –  Val Dec 31 '12 at 13:27
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One might say that nothing can be measured directly. Every measurement contains instrument. The only difference is that sometimes, the model to go from instrument to quantity is simple, and sometimes it's very complicated... –  gerrit Jan 10 '13 at 22:59
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The orbital velocity of the Earth can be measured via the annual Doppler effect, or by aberration. The Doppler effect measurement requires a star near the ecliptic, and the aberration measurement requires a star normal to it. Both techniques involve comparing two observations taken six months apart.

For example the aberration constant is 20.496 arcseconds, or .000099365 radians. Multiplying the aberration constant in radians by the speed of light gives the mean orbital velocity of the Earth-Moon barycenter: $ .000099365 \, c = 29.79 \ \mathrm{km}/\mathrm s $.

The annual Doppler effect must be removed from all spectroscopic radial velocity measurements in order to find the radial velocity of the observed star with respect to the solar system barycenter.

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