Saturn has a mass of 5.7 × 1026 kg and a radius of 60,300 km.
a. Find an image of Saturn and estimate the radius of the outer edge of the rings in units of Saturn’s radius.
b. Compute Saturn’s average mass density.
c. What is the minimum density that a moon of Saturn orbiting at the outer edge of the rings must have to resist tidal disruption?
d. It is thought that Saturn’s rings are composed of bodies made mostly of water ice. Is this consistent with your answer to part c?
The Correct Answer and Explanation is :
iturn0image0turn0image1turn0image3turn0image8a. Estimating the Radius of Saturn’s Outer Rings in Units of Saturn’s Radius
Saturn’s main rings extend from approximately 7,000 km to 80,000 km from the planet’s equator. citeturn0search1 The outermost major ring, known as the F ring, has an outer edge at about 140,180 km from Saturn’s center. citeturn0search4 Given that Saturn’s radius is 60,300 km, the outer edge of the F ring is approximately 2.33 times Saturn’s radius.
b. Computing Saturn’s Average Mass Density
The average mass density (ρ) of a spherical object is calculated using the formula:
[ \rho = \frac{M}{\frac{4}{3} \pi R^3} ]
Where:
- ( M = 5.7 \times 10^{26} \, \text{kg} ) (mass of Saturn)
- ( R = 60,300 \, \text{km} = 60,300,000 \, \text{m} ) (radius of Saturn)
Calculating the volume of Saturn:
[ V = \frac{4}{3} \pi (60,300,000 \, \text{m})^3 ]
[ V \approx 9.18 \times 10^{22} \, \text{m}^3 ]
Now, calculating the density:
[ \rho = \frac{5.7 \times 10^{26} \, \text{kg}}{9.18 \times 10^{22} \, \text{m}^3} ]
[ \rho \approx 6.21 \, \text{kg/m}^3 ]
Therefore, Saturn’s average mass density is approximately 6.21 kg/m³.
c. Minimum Density for a Moon to Resist Tidal Disruption
A moon can resist tidal disruption if its self-gravity is strong enough to counteract the tidal forces exerted by Saturn. This condition is met when the moon’s density is greater than a critical value, which can be estimated using the Roche limit formula:
[ \rho_{\text{min}} = \frac{3 M_{\text{Saturn}}}{4 \pi R_{\text{Saturn}}^3} ]
Where:
- ( M_{\text{Saturn}} = 5.7 \times 10^{26} \, \text{kg} )
- ( R_{\text{Saturn}} = 60,300 \, \text{km} = 60,300,000 \, \text{m} )
Calculating the minimum density:
[ \rho_{\text{min}} = \frac{3 \times 5.7 \times 10^{26} \, \text{kg}}{4 \pi (60,300,000 \, \text{m})^3} ]
[ \rho_{\text{min}} \approx 1.86 \times 10^3 \, \text{kg/m}^3 ]
Therefore, the minimum density required for a moon to resist tidal disruption near Saturn is approximately 1,860 kg/m³.
d. Consistency with the Composition of Saturn’s Rings
Saturn’s rings are primarily composed of water ice particles, with densities around 1,000 kg/m³. citeturn0search1 This is significantly lower than the minimum density of 1,860 kg/m³ required to resist tidal disruption near Saturn. Therefore, the low-density ice particles that make up Saturn’s rings are not capable of forming large, cohesive moons within the Roche limit, as their self-gravity is insufficient to withstand Saturn’s tidal forces. This is consistent with the observed absence of large moons within the main ring system.
In summary, the low density of Saturn’s ring particles explains why they cannot coalesce into larger moons within the Roche limit, aligning with the minimum density calculations for tidal disruption.