But we also need to include the Coriolis force due to the rotation of the asteroid. That contributes an acceleration of 2 ω y'(t) in the x direction and –2 ω x'(t) in the y-direction.
The final equations this gives us for the xy plane are:
x''(t) = 3ω^2 x + 2ω y'(t)
y''(t) = –2ω x'(t)
There are runaway solutions for these equations, but there are also bounded ones, where a free particle traces out an ellipse that is exactly twice as long in the y-direction as in the x-direction. (In my novel “Incandescence”, aliens in orbit around a black hole are able to tell that Newtonian gravity isn't a good fit for their situation, because they can measure this ellipse as departing from the 2-to-1 ratio.)
That still doesn't give us an equilateral triangle. But if we combine this 2-to-1 elliptical motion with just the right amount of bouncing up and down along the z-axis, we can make a free particle trace out a perfect circle, inclined at 60° to the xy-plane.
And then we can place three particles on that circle, spaced out at 120° intervals, so that they always form an equilateral triangle.
Of course, all of this is just a linear approximation to the exact orbits, only valid for very small distances. In reality, the LISA spacecraft will fly in a triangle that “breathes” and “flexes”, changing its shape and size. So unlike its ground-based counterparts, it will need to perform its extraordinary feats of measurement while the arms of its laser interferometer are not even fixed in relation to each other!
Greg Egan /
npub1qq…j5nhv
2024-10-05 06:56:41
in reply to nevent1q…5cam
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wss://relay.mostr.pubPublished at
2024-10-05 06:56:41Event JSON
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"content": "But we also need to include the Coriolis force due to the rotation of the asteroid. That contributes an acceleration of 2 ω y'(t) in the x direction and –2 ω x'(t) in the y-direction.\n\nThe final equations this gives us for the xy plane are:\n\nx''(t) = 3ω^2 x + 2ω y'(t)\ny''(t) = –2ω x'(t)\n\nThere are runaway solutions for these equations, but there are also bounded ones, where a free particle traces out an ellipse that is exactly twice as long in the y-direction as in the x-direction. (In my novel “Incandescence”, aliens in orbit around a black hole are able to tell that Newtonian gravity isn't a good fit for their situation, because they can measure this ellipse as departing from the 2-to-1 ratio.)\n\nThat still doesn't give us an equilateral triangle. But if we combine this 2-to-1 elliptical motion with just the right amount of bouncing up and down along the z-axis, we can make a free particle trace out a perfect circle, inclined at 60° to the xy-plane.\n\nAnd then we can place three particles on that circle, spaced out at 120° intervals, so that they always form an equilateral triangle.\n\nOf course, all of this is just a linear approximation to the exact orbits, only valid for very small distances. In reality, the LISA spacecraft will fly in a triangle that “breathes” and “flexes”, changing its shape and size. So unlike its ground-based counterparts, it will need to perform its extraordinary feats of measurement while the arms of its laser interferometer are not even fixed in relation to each other!",
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