Does it reach the Roche limit and then just break up and form a ring?
Does it reach the Roche limit and then just break up and form a ring?
Surely that depends on the size and tensional strength of the satellite?
Also: how small pieces does the satellite need to break into in order to stop further inspiralling?
If a satellite breaks into infinite number of pieces that form a ring of perfect central symmetry, the ring raises no tides and therefore loses no energy.
How about a finite number of pieces held together by cohesion?
If two satellites are orbiting at equal period 180 degrees then the tidal bulges they create on planet add up, not subtract. What would be the angle at which tides cancel?
Does the Lagrange stability analysis (unstable at 180 degrees, stable at 60 degrees) still hold if secondary and tertiary are inside Roche limit?
Yes, which is why there is a fluid Roche limit and a rigid Roche limit. Phobos is inside the fluid Roche limit but will stay intact for quite a while.
You might want to look at what happen with Shoemaker-Levy as it broke up into a number of large pieces, small as well, no doubt.Also: how small pieces does the satellite need to break into in order to stop further inspiralling?
If a satellite breaks into infinite number of pieces that form a ring of perfect central symmetry, the ring raises no tides and therefore loses no energy.
How about a finite number of pieces held together by cohesion?
We know time flies, we just can't see its wings.
Earth's rotational period is 24 hours; the Moon's orbital period (sidereal) is about 650 hours. One example in the Solar System is the case of Phobos, which has a period of less than a Martian day; it's likely to crash into the surface of Mars (or break up into pieces that will do so) in the next 10 million years or so* (https://arxiv.org/abs/0805.1454, https://www.universetoday.com/14258/...years-to-live/)
* The number depends on the internal properties of Mars, including the radial mass distribution and the internal damping of the materials that make up the Martian interior.