What is the mass of Saturn based on the orbital data of its moon Tethys?

What is the mass of Saturn based on the orbital data of its moon Tethys?

Final answer: 5.89 x 10^26 kg Explanation: To determine the mass of Saturn based on the orbital data of its moon Tethys, we can use Kepler's third law of planetary motion. Kepler's third law states that the square of the orbital period (P) of a planet is proportional to the cube of the semi-major axis (r) of its orbit, which in the case of a circular orbit is equivalent to the orbit's radius. The law can be formulated as P^2 = (4π^2)/(G*M) * r^3, where G is the gravitational constant and M is the mass of the central body, which in this case is Saturn. Given Tethys' orbital period (P) is 1.888 days (converted to seconds) and its average orbital radius (r) is 294,700 km, we can rearrange the formula to solve for M (Saturn's mass): M = (4π^2 * r^3)/(G*P^2). Plugging in the values, we find Saturn's mass to be approximately 5.68 x 10^26 kg, which corresponds to the closest option provided, 5.89 x 10^26 kg. It's important to note that G is 6.67430 x 10^-11 m^3kg^-1s^-2 and we need to be consistent with units throughout our calculation (converting days to seconds and kilometers to meters as necessary).