# Solution: Synchronous Orbit Height around Saturn Let's solve step-by-step using **Kepler's Third Law** (for a circular orbit): ## 1. **Given Data** - **Saturn Mass (\(M\))** = \(5.685 \times 10^{26}\) kg - **Saturn Radius (\(R_{Saturn}\))** = 58,232 km = \(5.8232 \times 10^7\) m - **Gravitational Constant (\(G\))** = \(6.67408 \times 10^{-11}\) SI - **Rotation period (\(T\))** = 10.7 hours = \(10.7 \times 360 = 38,520\) s ## 2. **Kepler’s Third Law (for Synchronous Orbit)** For a synchronous orbit, the orbital period of the satellite equals the rotation period of Saturn. \[ T^2 = \frac{4\pi^2 r^3}{GM} \] Where: - \(T\) = orbital period (s) - \(r\) = orbital radius (distance from Saturn's center to satellite, in m) - \(G\) = gravitational constant - \(M\) = mass of Saturn Rearrange to solve for \(r\): \[ r^3 = \frac{GMT^2}{4\pi^2} \] \[ r = \left(\frac{GMT^2}{4\pi^2}\right)^{1/3} \] ## 3. **Substitute Values** \[ r = \left[\frac{(6.67408 \times 10^{-11}) \times (5.685 \times 10^{26}) \times (38,520)^2}{4\pi^2}\right]^{1/3} \] Let's calculate step by step: #### a. Calculate \(GMT^2\): - \(G \times M = 6.67408 \times 10^{-11} \times 5.685 \times 10^{26} = 3.794 \times 10^{16}\) - \(T^2 = (38,520)^2 = 1.4842 \times 10^9\) So, \[ GMT^2 = 3.794 \times 10^{16} \times 1.4842 \times 10^9 = 5.634 \times 10^{25} \] #### b. Calculate denominator: \[ 4\pi^2 = 4 \times (3.1416)^2 \approx 39.4784 \] #### c. Full fraction: \[ \frac{GMT^2}{4\pi^2} = \frac{5.634 \times 10^{25}}{39.4784} = 1.4276 \times 10^{24} \] #### d. Take cube root: \[ r = (1.4276 \times 10^{24})^{1/3} \] The cube root of \(10^{24}\) is \(10^8\): \[ r \approx (1.4276)^{1/3} \times 10^{8} = 1.110 \times 10^8 \text{ m} \] ## 4. **Calculate Height Above Saturn’s Surface** \[ h = r - R_{Saturn} \] \[ h = 1.110 \times 10^8\ \text{m} - 5.8232 \times 10^7\ \text{m} = 5.277 \times 10^7\ \text{m} \] Or, in kilometers: \[ h = 52,770\ \text{km} \] --- ## **Final Answer** **Height of a satellite on a synchronous orbit around Saturn:** \[ \boxed{52,770\ \text{km}} \] --- ### **Approximation Used** - Saturn is treated as a perfect sphere, ignoring oblateness and atmospheric drag. - The mass of the satellite is negligible compared to Saturn. - The orbit is assumed to be circular. --- **Let me know if you need further clarification or details!**