is there a reason why an accurate measurement of $G$ cannot be performed by measuring the distance and rotation period between 2 orbiting 1 kg masses in free fall? Using a simple estimation for 2 Pt spheres of 1 kg each (radius 22 mm) separated by 4 radiae centre-to-centre I estimate a period of 20500 s or about 5.7 hours. Could it be performed in ISS? Could the sources of error be managed so that the accuracy of torsion balances is surpassed?
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$\begingroup$ I'm not sure I understand what you're trying to do, are you trying to find the rotation period of one mass around the other? $\endgroup$– PhilipCommented Aug 26, 2020 at 13:24
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$\begingroup$ @Philip: yes, rotation rate (angular velocity) and separation can be measured to high precision. From there measure G. $\endgroup$– Wouter M.Commented Aug 26, 2020 at 13:30
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2 Answers
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Two platinum spheres in free space might orbit each other in six hours, but near the Earth they would orbit Earth every 1.5 hours. In practice you would be measuring their orbits around the Earth, which interact with each other in surprising ways.
Many years ago I was adjacent to an effort to design such a satellite-based gravity experiment. (If it ever had a website, it's long gone.) The idea was that small test masses would be released in a cavity in a satellite in low-Earth orbit, so that their primary interaction was mutual gravitation. It turns out that mutual attraction would not make them orbit each other; instead they would undergo what's known as horseshoe orbit interaction, and be repelled from each other in the accelerated reference frame of the satellite.
I would guess that if you wanted your two bodies to primarily orbit each other, you would have to move them far enough from Earth that their mutual period is much briefer than their orbital period around Earth. You might be able to do your six-hour experiment in a geostationary orbit, with a twenty-four hour period, or you might have to go higher. You might compute the size of the Hill sphere for your test masses for different Earth orbits.
Furthermore, everything about doing a space-based gravitation experiment was at least ten times harder than you might have expected. In low-Earth orbit, my friends learned that their mutual-gravitation experiment was going to be sensitive to things like the location of sufficiently large herds of cattle relative to the orbital path. That's probably a reason why the GRACE mission happened before Gravity Probe B, and why my friends' GPB-beating experiment never happened at all.
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$\begingroup$ In low-Earth orbit, my friends learned that their mutual-gravitation experiment was going to be sensitive to things like the location of sufficiently large herds of cattle relative to the orbital path. So you're saying such an experiment could potentially be an udder failure? $\endgroup$ Commented Aug 26, 2020 at 17:37
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1$\begingroup$ Puns aside: the most important lesson I got from this experience in the long run was that we were all wearing our what-is-gravity blinders and thought of the business with the cattle as a silly annoyance, while the GRACE folks realized that we knew enough about gravity to use it to continuously measure flows of glaciers and groundwater on size scales like cattle herds. We were smart enough to see the effect, but not smart enough to see the opportunity. $\endgroup$– rob ♦Commented Aug 26, 2020 at 18:15
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$\begingroup$ @rob: I understand that the orbital dynamics would be complex. Maybe out at L2 it would be manageable; the main gain would be to know both masses to µg accuracy and their instantaneous distances to 0.5 µm. Would that allow extraction of a better value for G? $\endgroup$ Commented Aug 27, 2020 at 9:30
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1$\begingroup$ @WouterM. There are so many sneaky problems in gravity experiments. Another is that electrostatic attraction mimics gravity (and repulsion, antigravity), and that isolated objects in space tend to become charged as cosmic rays knock ions free. I'm not prepared to give you a proper uncertainty analysis in a comment. $\endgroup$– rob ♦Commented Aug 27, 2020 at 11:23
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You would not want to do this in the ISS. It would need to be in a vacuum, and the gravity from the station and the people moving around inside could have an effect. Placing it a couple of hundred yards away might work. Out there you might need to worry about the gradient of the earth's field. In terms of measurements, getting an accurate distance between the masses could be a limiting factor.
