Question

Calculate the Roche limit of the satellite Ophelia assuming that its density is 1.0 g/cm³ (i.e., water ice) and compare that limit to the radius of its orbit. Derive a conclusion from this comparison. 

Answer 1

r_R = 2.5(ρ_M/ρ_M)^1/3 x R

ρ_M = 1.290g/cm³

ρ_m = 1.0g/cm³

R = 25,559km

r_R = 2.59(1.290/1.00)^1/3 x 25, 559 = 2.5 x 1.0885 x 25, 559

r_R = 69, 552km

Radius of the orbit of Ophelia = 53.8 × 10³km

The orbit of Ophelia is located inside the Roche limit as calculated. Therefore, Ophelia would be unstable if its density actually is 1.00g/cm (i.e., it is composed of water ice).

If it is a rocky object with a bulk density of about 3.2g/cm³, its Roche limit would be r = 47,200 km. In that case, Ophelia’s orbit lies outside the Roche limit and the satellite would be stable.

Given that Ophelia does appear to be stable, its Roche limit is less than 53,800 km, which indicates that it is not composed of ice.

Note: This calculation can be extended to determine the density of Ophelia that yields a Roche limit of 53,800 km

The resulting density may indicate that Ophelia has a rocky core covered by an ice mantle.

The radius of the hypothetical core and the thickness of the ice mantle can be calculated based on assumed densities.