Greg Egan on Nostr: But we also need to include the Coriolis force due to the rotation of the asteroid. ...
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!
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!