Into what shape does an orbiting sphere get distorted by tidal forces?

[Paulibus]

I'm having trouble imagining how an orbiting object would get distorted by tides --- in the first instance, say, the distortion of an initial uniform dense ball of coffee grounds (of the sort John Baez likes to imagine in his web pages). I know and understand why, if it were infalling radially towards a point mass, it would be distorted into an axially symmetric ellipsoid of rotation, with the axis along the radius of infall. The eccentricity of the ellipsoid would increase with time.

But how would this picture change with time if the ball were initially in a circular orbit, rather then falling radially? I can only imagine the distortion to be initially like that of our oceans as we orbit the center of mass of, say, the Earth-Moon system.

 

[Paulibus]

Belay that question. Just after I'd written it down I figured out that the ball would in time get spread out into a uniform (but less) dense ring of coffee grounds arond the attracting mass with the same circular cross section as the ball. As with Saturn and the other gas giants. And that the reason why such rings are circular is that elliptical rings are circularised over time by internal coffee ground collisions locally heating and dissipating energy from the ring preferentially in peri-Saturn regions? Maybe?

 

[DrClapeyron]

Prolate spheroid

In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via tidal forces when they orbit a massive body in a close orbit. The most extreme example is Jupiter's moon Io, which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing spectacular volcanism. It should be noted that the axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary.

http://en.wikipedia.org/wiki/Prolate_spheroid

Io immitates a prolate spheroid with its major axis lying parallel or within its orbital plane around Saturn. So eggshaped, maybe?

 

[Paulibus]

Thanks, Dr. Clapeyron. I'm still a bit puzzled, perhaps because I'm confusing a prolate spheroid with an ellipsoid of revolution. If they are the same thing then Wikipedia has Io's shape as always trying to be the same as if it were infalling radially. Simplistically Io is then volcanic because the radial direction rotates once every orbit and the resultant tides therefore kneads plastically and heats the material of which Io is made.

But I suspect there is more to Io's shape and vulcanism than Wikipedia suggests. Think for a moment of non-cohering coffee grounds starting off in orbit as a uniform ball, instead of a solid cohering ball. The grounds furthest from the primary object are free to orbit slower than those nearest it --- tidal forces will shear the ball. This is the distinction between radial infall and orbiting, I think. An imagined ball of coffee rounds would ultimately shear into a ring, not a prolate spheroid. Io must also be trying (but of course failing, because it coheres) to become a ring.

I imagine that even strongly cohering objects disintegrate into rings close to big planets. Certainly looks that way.

 

[Paulibus]

Here I go again, reinventing the wheel. Unless someone stops me it'll probably turn out to be square. That's what happens when one isn't prompty put right.

While on the subject of rings around massive objects being a product of gravity-induced shear, I want to point out here that shear and gravity are like a horse and carriage; we should for astronomical purposes always think of them as going together.

The simplest orbits that are ruled by Kepler's laws are circular. In this case the constant orbital speed is inversely proportional to the square root of the orbit radius. Which means that any arrangement of objects in circular orbits is continually being dynamically sheared in a tangential fashion (if the arrangement were to rotate rigidly without shearing, the orbital speed would be proportional to the radius). For example think of objects with circular orbits in an 'accretion' disc around a star. Imagine that such objects constitute a fluid, like a gas.

Fluids do not support static shear. Viscous drags are only pale shadows of the static shear stresses that steel can sustain. When fluids are appropriately sheared at low Reynold's numbers they respond by forming circular sub-structures. Our atmosphere provides examples like the well-defined vortex streets delineated by trade-wind clouds that trail from islands like Guadaloupe. The rolling doughnut-shaped clouds that enclose debris columns of nuclear explosions are another example. These structures facilitate shear, just as cylindrical rollers placed under heavy objects facilitate sliding them over the floor. (I''m now getting dangerously close to reinventing the wheel, as threatened).

It is then not surprising that stellar accretion discs that are being dynamically sheared by gravity develop circulating structures, and that gravity in time pulls these together into a hierarchy of planets, moons and minor planets (one of which, Aletta, is named for my mother) and suchlike debris, including the Earth. It seems to me that the sequence: gravitational collapse, conservation of angular momentum, dynamic fluid shear, fluid instability relieved by the formation of relatively stable circulating substructures, slowing n of further gravitational collapse... is important in the solar system for breeding its structural hierarchy.

And I can see no reaon why it shouldn''t be important on a larger scale, say with galaxy formation. It's a self-perpetuating structure-making system of the sort nature loves to invent, in other circumstances to make rivers and biological stuff.

It's just frustrating that fluid shear is such a difficult non-linear thing to model...bother those Navier-Stokes equations.