Question

The Roche Radius, defined to the orbital distance at which a satellite tidally torn apart by the parent body, is named after Edward Roche, who first derived it in 1848. Recall that his radius is given by: )1/3 d=r (2 M m Where r is the radius of the satellite, m the mass of the satellite, and M is the mass of the parent body.

(a) Recast this equation in terms of the density of the satellite (m), the density of the parent body (M), and radius of the parent body (R).

(b) Let s consider the Saturn system, and apply this equation. Saturns moon Pan orbits the planet at a distance 1.34  105 km; inside of a gap in the rings! Calculate the ratio of Pans orbital radius to its calculated Roche Radius (Pan = 0.42 g/cm2, Saturn = 0.687 g/cm2, RSaturn = 58,000 km). Comment on whether this moon is safe from tidal disruption, or not.

(c) Using what you know about the Roche Radius, and the above example, calculate the radius of the moon required to create the rings of Saturn as seen today (assuming that it was just one moon, with the density of pan and near the present-day orbit of Pan). The total mass in Saturns rings is approximately 3  1019 kg. (Hint for calculating Saturns mass: 0.687 g/cm2 = 687 kg/m3)

Answer 1

The roche radius is given by ,

(a)

Dropping these in the roche radius eqaution,

is the desired equation.

(b)

The moon is well within the roche radius, and is quite safe from tidal disruption.

(c)

We could use the initial form of roche radius equation,

Mass of saturn is ,

Using roche radius calcualted in (b) ,

We calculate the radius for the mass of all saturn rings as a single moon,

is the radius where the moon of mass of saturn rings is going to orbit saturn.