In March 2006, two small satellites were discovered orbiting Pluto, one at a distance of 48,000 km and the other at 64,000 km. Pluto already was known to have a large satellite Charon, orbiting at 19,600 km with an orbital period of 6.39 days. Assuming that the satellites do not affect each other, find the orbital periods of the two small satellites without using the mass of Pluto

Verified step by step guidance

1. Step 1: Use Kepler's Third Law, which states that the square of the orbital period (T) of a satellite is directly proportional to the cube of the semi-major axis of its orbit (r). The formula is T^2 = \frac{4\pi^2}{G M} r^3, where G is the gravitational constant and M is the mass of the central body.
2. Step 2: Since we are not using the mass of Pluto, we can use the ratio of the orbital periods and distances of the satellites. Let T_C be the orbital period of Charon, and r_C, r_1, and r_2 be the distances of Charon, the first small satellite, and the second small satellite from Pluto, respectively.
3. Step 3: Apply the ratio derived from Kepler's Third Law: \frac{T_1^2}{T_C^2} = \frac{r_1^3}{r_C^3} and \frac{T_2^2}{T_C^2} = \frac{r_2^3}{r_C^3}, where T_1 and T_2 are the orbital periods of the first and second small satellites, respectively.
4. Step 4: Solve for T_1 and T_2 by taking the square root of both sides of each equation: T_1 = T_C \sqrt{\frac{r_1^3}{r_C^3}} and T_2 = T_C \sqrt{\frac{r_2^3}{r_C^3}}.
5. Step 5: Substitute the known values (T_C = 6.39 days, r_C = 19,600 km, r_1 = 48,000 km, r_2 = 64,000 km) into the equations to find T_1 and T_2.

Key Concepts

Kepler's Third Law of Planetary Motion

Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This law can be expressed mathematically as T² ∝ r³, where T is the orbital period and r is the average distance from the central body. This principle allows us to relate the distances of the satellites from Pluto to their orbital periods without needing to know the mass of Pluto.

Gravitational Force and Orbital Motion

The gravitational force between two bodies governs their motion in space. For satellites orbiting a planet, this force provides the necessary centripetal acceleration to keep them in orbit. The balance between gravitational force and the inertia of the satellite leads to stable orbits, which can be analyzed using the principles of circular motion and gravitational attraction.

Proportional Relationships in Orbital Mechanics

In orbital mechanics, the relationships between distance and period can be simplified using ratios. For satellites orbiting the same central body, the ratio of their periods can be derived from their distances. This allows for the calculation of unknown orbital periods by comparing them to known values, such as Charon's period, thus facilitating the determination of the smaller satellites' periods based on their respective distances from Pluto.