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:

d = r*( 2 * M/m )^(1/3)  

Where r is the radius of the satellite, m is 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 (p_m), the density of the parent body (p_M), and the radius of the parent body (R). (Hint: you may assume that each body is well approximated by a sphere.)

(b) Let's consider the Saturn system, and apply this equation. Saturn's moon Pan orbits the planet at a distance 1.34 x 10⁵ km; inside of a gap in the rings! Calculate the ratio of Pan's orbital radius to its calculated Roche Radius (p_Pan = 0.42 g/cm² , p_Saturn = 0.687 g/cm² , R_Saturn = 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 Saturn's rings is approximately 3 x 10¹⁹ kg. (Hint for calculating Saturn's mass: 0.687 g/cm² = 687 kg/m² )

Answer 1

(a) As given

Insering above two values in first expression

(b) Roche radius for the given data

Ratio is given as

(c) We have given the mass of saturn and its rings(=moon). To calculate the radius we need to find out the density of moon first.

Radius of moon