Question

One of Saturn's moons has an orbital distance of 1.87 x 108 m. The mean orbital period of this moon is approximately 23 hours. Use this information to estimate a mass for the planet Saturn.

Answer 1

Answer on Question #68081, Physics / Astronomy | Astrophysics

Question:

One of Saturn's moons has an orbital distance of $1.87 \times 10^{8} \, \text{m}$. The mean orbital period of this moon is approximately 23 hours. Use this information to estimate a mass for the planet Saturn.

Solution:

According to Kepler's $3^{\mathrm{rd}}$ law

$$\frac{a^3}{T^2} = \frac{G \mathfrak{W}_{Sat}}{4\pi^2},$$

where

- $a$ — moon’s mean orbital distance;

- $T$ — mean orbital period of the moon;

- $G$ — gravitational constant;

- $\mathfrak{W}_{Sat}$ — Saturn’s mass.

Therefore $\mathfrak{W}_{Sat} = \frac{4\pi^2 a^3}{GT^2}$.

$$a = 1.87 \times 10^{8} \, \text{m}$$ $$T = 23 \, h = 82800 \, \text{s}$$ $$G = 6.67408 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}$$ $$\mathfrak{W}_{Sat} = \frac{4\pi^2 (1.87 \times 10^{8})^3}{6.67408 \times 10^{-11} \, 82800^2} = 5.64 \times 10^{26} \, \text{kg}$$

Answer:

$$5.64 \times 10^{26} \, \text{kg}$$

Answer provided by https://www.AssignmentExpert.com