In its orbit each day, the International Space Station makes 15.65 revolutions around the earth. Assuming a circular orbit, how high is this satellite above the surface of the earth?

Verified step by step guidance

1. First, understand that the problem involves circular motion. The International Space Station (ISS) completes 15.65 revolutions around the Earth each day. We need to find the altitude of the ISS above Earth's surface.
2. Calculate the orbital period of the ISS. Since it makes 15.65 revolutions per day, the period \( T \) in hours is given by \( T = \frac{24}{15.65} \).
3. Use Kepler's Third Law, which relates the orbital period \( T \) to the semi-major axis \( a \) of the orbit: \( T^2 = \frac{4\pi^2}{GM}a^3 \), where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.
4. Rearrange the formula to solve for \( a \): \( a = \left(\frac{GMT^2}{4\pi^2}\right)^{1/3} \). Substitute the known values for \( G \), \( M \), and \( T \) to find \( a \).
5. Finally, calculate the altitude of the ISS above Earth's surface by subtracting Earth's radius \( R \) from the semi-major axis \( a \): \( ext{altitude} = a - R \). Use Earth's average radius \( R \approx 6371 \) km.

Key Concepts

Circular Orbit

A circular orbit is a path taken by an object around a celestial body where the distance from the center of the body to the object remains constant. In this context, the International Space Station (ISS) maintains a constant altitude above Earth, allowing for consistent orbital speed and period calculations.

Orbital Period

The orbital period is the time taken for a satellite to complete one full revolution around a celestial body. For the ISS, making 15.65 revolutions per day implies a period of approximately 92 minutes per revolution, which is crucial for determining the satellite's altitude using gravitational and centripetal force equations.

Gravitational Force

Gravitational force is the attractive force between two masses, such as Earth and the ISS. It provides the centripetal force necessary for the ISS to maintain its orbit. By equating gravitational force to centripetal force, one can derive the altitude of the ISS above Earth's surface, using the known values of Earth's mass and radius.