The LISA gravity wave detector will comprise three spacecraft, all in free-all in heliocentric orbits, that always lie on the vertices of an equilateral triangle. But how can they do that?
The answer is: they can't! Not precisely. Of course, 3 spacecraft at 120° intervals around a perfect circular orbit could do it, but the LISA craft need to be much closer together.
So, the question is: how do they fly in a formation that *approximates* this goal?
To see how, the best way is to imagine what forces you would feel if you were inside a hollowed-out asteroid in a circular orbit, tidally locked so that one axis always pointed towards the sun.
Call the sunwards axis the x-axis, the direction pointing around the orbit the y-xis, and the direction out of the plane of the orbit the z-axis.
Along the z-axis, you would feel a “tidal squeeze” of –GM/r^3 z pulling you back towards the plane of the orbit. A free particle displaced along the z-axis would bounce up and down as if it was on a spring, with a periodic motion (not coincidentally) the same as the period of the orbit.
In the xy plane, things would be a little more complicated. You would feel a “tidal squeeze” of –GM/r^3 y along the y-axis, but that would be exactly cancelled out by a centrifugal force of GM/r^3 y due to the tidally locked rotation of the asteroid, which has an angular rate of ω = √(GM/r^3).
Along the x-axis, you would feel a “tidal stretch” of 2GM/r^3 trying to tear you away from the centre of the asteroid, *plus* a centrifugal force of ω^2 x = GM/r^3 x, for a total of 3GM/r^3 x.
Greg Egan /
npub1qq…j5nhv
2024-10-05 06:56:15
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npub1qqm7nu2qf25xd3mw6y6cyp4v2wr7ktf43ymp5wqz4u8jvz76ymtsjj5nhvPublished at
2024-10-05 06:56:15Event JSON
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"content": "The LISA gravity wave detector will comprise three spacecraft, all in free-all in heliocentric orbits, that always lie on the vertices of an equilateral triangle. But how can they do that?\n\nThe answer is: they can't! Not precisely. Of course, 3 spacecraft at 120° intervals around a perfect circular orbit could do it, but the LISA craft need to be much closer together.\n\nSo, the question is: how do they fly in a formation that *approximates* this goal?\n\nTo see how, the best way is to imagine what forces you would feel if you were inside a hollowed-out asteroid in a circular orbit, tidally locked so that one axis always pointed towards the sun.\n\nCall the sunwards axis the x-axis, the direction pointing around the orbit the y-xis, and the direction out of the plane of the orbit the z-axis.\n\nAlong the z-axis, you would feel a “tidal squeeze” of –GM/r^3 z pulling you back towards the plane of the orbit. A free particle displaced along the z-axis would bounce up and down as if it was on a spring, with a periodic motion (not coincidentally) the same as the period of the orbit.\n\nIn the xy plane, things would be a little more complicated. You would feel a “tidal squeeze” of –GM/r^3 y along the y-axis, but that would be exactly cancelled out by a centrifugal force of GM/r^3 y due to the tidally locked rotation of the asteroid, which has an angular rate of ω = √(GM/r^3).\n\nAlong the x-axis, you would feel a “tidal stretch” of 2GM/r^3 trying to tear you away from the centre of the asteroid, *plus* a centrifugal force of ω^2 x = GM/r^3 x, for a total of 3GM/r^3 x.\n\nhttps://media.mathstodon.xyz/media_attachments/files/113/253/504/608/004/523/original/9ed4207940b0e5f4.mp4",
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