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A single flyby doesn't cut it: It gives you a calculation of the mass of the body at that distance from the sun. If you fly by the same body at two different distances from the sun, you will come up with two different masses. (There were at least two flyby's of Halley planned, but one didn't make it.)
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So we shouls be able to measure a difference right here on earth as we move away from the sun in our own orbit?
Why can't we?