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Re: [Phys-L] Ex: Re: Roche limit

Regarding the punchline of Dr. Mallinckrodt's recap some simplified Roche limit calculations:

But this result is still at VERY significant odds with the original calculation of
Roche in 1849

S = 2.44 (rho_primary / rho_satellite )^(1/3) R

which assumes a fluid satellite in hydrostatic equilibrium … and makes the
calculation significantly less straightforward!

Here "significantly" is the key word.

It’s easy to understand why a fluid satellite will be torn apart at much larger
distances from the primary because of the fact that the tidal force
themselves will deform the satellite into a radially elongated ellipsoidal
shape which, in turn, further increases the tidal forces.

It's even worse than this. This is because the centrifugal + gravitational field seen in a frame revolving with the satellite (assumed to be in a circular orbit) has a *triaxial* anisotropy, and the equilibrium surface is *not* simply a prolate spheroid pointing radially toward/away from the parent body. The equilibrium shape is indeed still prolate, but it is a triaxial ellipsoid, not a prolate spheroid of revolution. And this extra anisotropy affects the matter distribution of the satellite, and this affects the strength its gravitational field at the satellite's the ends in the prolate direction, and this affects the orbital distance where the surface g-field goes to zero. The external field in which the satellite finds itself in its own rest frame has both a tidal stretching component along the orbital radial direction, and *also* a *compressive* component acting along the direction perpendicular to the satellite's orbital plane. The strength of the compressive component is 1/3 of the strength of the stretching component. This compressive component has a SHO potential form (restoring force proportional to the distance from the satellite's center of mass. The radial prolate "stretching" component of the external tidal combo (gravitational+ centrifugal) field is like a SHO with a negative spring constant with a force strength that pulls away from the satellite's center of mass with a strength that is proportional to the amount of the radial separation from it. But in the 3rd perpendicular direction (parallel to the satellite's orbital direction) there is no combo tidal field at all (to leading order in the small satellite approximation). Solving a Laplace/Poisson equation with triaxial ellipsoidal anisotropy is, I believe, well beyond the abilities of nearly all high school students, and lower level undergraduates. This means the simplified calculations referred to earlier are essentially the only game in town for trying to understand at the lower level the tidal disruption at the Roche limit--even though those calculations are mostly only hand-waving order of magnitude estimates, and do not have all the detailed nuances of Roche's calculation. And even Roche's calculation itself only applies to situations where the ratio of the satellite's mean radius is tiny compared to its orbital radius, *and* the satellite's spin is tidally locked to the parent body (prograde rotation at the same rate as revolution). If the satellite's spin is not tidally locked then there is another *internal* centrifugal field to consider and the external field as seen in a frame rotating with the fluid of the satellite now becomes time dependent to boot. If the "satellite" has a mass and size comparable to the "parent" body then this small satellite approximation breaks down, and the external field seen in either body's rest frame is not quadratic in the separation from the body's center of mass (owing to non-quadratic nature of the inverse square law for Newtonian gravity). This is why the Roche lobes for a pair of close nearly contacting binary stars are not prolate ellipsoids (even triaxial ones).

David Bowman