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. Start by understanding Kepler's Third Law, which states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit: \( T^2 \propto a^3 \). This can be expressed as \( \frac{T_1^2}{a_1^3} = \frac{T_2^2}{a_2^3} \) for two satellites orbiting the same body.
2. Identify the known values: For Charon, the semi-major axis \( a_1 = 19,600 \) km and the orbital period \( T_1 = 6.39 \) days. For the first small satellite, \( a_2 = 48,000 \) km, and for the second small satellite, \( a_3 = 64,000 \) km. We need to find \( T_2 \) and \( T_3 \).
3. Apply Kepler's Third Law to find the orbital period of the first small satellite: \( \frac{T_2^2}{48,000^3} = \frac{6.39^2}{19,600^3} \). Rearrange this equation to solve for \( T_2 \): \( T_2 = 6.39 \times \sqrt{\frac{48,000^3}{19,600^3}} \).
4. Similarly, apply Kepler's Third Law to find the orbital period of the second small satellite: \( \frac{T_3^2}{64,000^3} = \frac{6.39^2}{19,600^3} \). Rearrange this equation to solve for \( T_3 \): \( T_3 = 6.39 \times \sqrt{\frac{64,000^3}{19,600^3}} \).
5. Calculate the values of \( T_2 \) and \( T_3 \) using the equations derived in the previous steps. This will give you the orbital periods of the two small satellites in days.

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 applied to any object orbiting a central body, such as satellites around Pluto, allowing us to compare their orbital periods based on their distances from Pluto.

Proportional Relationships

A proportional relationship is a relationship between two quantities where their ratio remains constant. In the context of orbital mechanics, the ratio of the square of the orbital period to the cube of the orbital radius is constant for all satellites orbiting the same central body, allowing us to calculate unknown periods if one is known.

Orbital Radius and Period Relationship

The relationship between orbital radius and period is crucial for determining the period of satellites. Given that the orbital period is proportional to the 3/2 power of the orbital radius, knowing the period of one satellite allows us to find the periods of others by comparing their orbital radii, assuming the central body's mass remains constant.