Relevant to this is the J2 Perturbation [1], commonly used when accurately modeling the Earth as an oblate spheroid (like in the article image, but less drastic) rather than a perfect sphere. This has resultant effects on orbits, such as the "gravity wells" in the GEO belt at 105degW and 75degE. There are higher-order perturbations [2] as you closer approximate the Earth's actual shape, such as J3, J4, etc.
That page didn't have a formula, in either cartesian or polar coordinates, for the shape of the object. Lots of formulas, but I didn't see anything I could use to create a 3d mesh and print one of these things out on my printer.
It's the general ellipsoid formula: x*2/a*2 + y*2/b*2 + z*2/c*2 = 1, where a, b and c are all unequal. The interesting part is actually that this shape could be made of a liquid, held by gravitation and maintain this asymmetrical shape. Normally, one would imagine it would be ellipsoid of revolution, where two of the axes are equal.
Hmm, that can't be true; it's of uniform density, and the thing it's rotating about has to be its center of mass.
What's odd about it (to me) is the optimal solution isn't symmetric (cylindrically symmetric). It's an intuition trap that you'd expect symmetric solutions. If the Wikipedia history is to be believed, Lagrange fell for this wrong assumption, and there was a 45-year gap before anyone noticed the subtle wrongness of it.
Relevant to this is the J2 Perturbation [1], commonly used when accurately modeling the Earth as an oblate spheroid (like in the article image, but less drastic) rather than a perfect sphere. This has resultant effects on orbits, such as the "gravity wells" in the GEO belt at 105degW and 75degE. There are higher-order perturbations [2] as you closer approximate the Earth's actual shape, such as J3, J4, etc.
[1] https://ai-solutions.com/_freeflyeruniversityguide/j2_pertur...
[2] https://oer.pressbooks.pub/lynnanegeorge/chapter/chapter-10-...
That page didn't have a formula, in either cartesian or polar coordinates, for the shape of the object. Lots of formulas, but I didn't see anything I could use to create a 3d mesh and print one of these things out on my printer.
It's the general ellipsoid formula: x*2/a*2 + y*2/b*2 + z*2/c*2 = 1, where a, b and c are all unequal. The interesting part is actually that this shape could be made of a liquid, held by gravitation and maintain this asymmetrical shape. Normally, one would imagine it would be ellipsoid of revolution, where two of the axes are equal.
https://en.wikipedia.org/wiki/Ellipsoid
To make a plot in software such as JMol, you really need parametric equations, which are also given in the above page.
This is surprising, but less surprising when you realise it is rotating about one of the foci, not the centre.
Hmm, that can't be true; it's of uniform density, and the thing it's rotating about has to be its center of mass.
What's odd about it (to me) is the optimal solution isn't symmetric (cylindrically symmetric). It's an intuition trap that you'd expect symmetric solutions. If the Wikipedia history is to be believed, Lagrange fell for this wrong assumption, and there was a 45-year gap before anyone noticed the subtle wrongness of it.
Remember the spontaneously flipping spinning hammer on the ISS?
...rotation for sufficiently non-spherical(-cow) objects in free fall is kinda weird.