On July 15, 2004, NASA launched the Aura spacecraft to study the earth's climate and atmosphere. This satellite was injected into an orbit 705 km above the earth's surface. Assume a circular orbit. (a) How many hours does it take this satellite to make one orbit?

Verified step by step guidance

1. First, understand that the satellite is in a circular orbit around the Earth. The radius of the orbit is the sum of the Earth's radius and the altitude of the satellite. The Earth's average radius is approximately 6371 km, so the orbit radius is 6371 km + 705 km.
2. Next, use the formula for the orbital period of a satellite in a circular orbit: \( T = 2\pi \sqrt{\frac{r^3}{GM}} \), where \( r \) is the orbit radius, \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2} \), and \( M \) is the mass of the Earth \( 5.972 \times 10^{24} \text{ kg} \).
3. Convert the orbit radius from kilometers to meters to match the units of the gravitational constant: \( r = (6371 + 705) \times 10^3 \text{ m} \).
4. Substitute the values of \( r \), \( G \), and \( M \) into the orbital period formula to calculate \( T \). This will give you the period in seconds.
5. Finally, convert the period from seconds to hours by dividing the result by 3600 (since there are 3600 seconds in an hour). This will give you the time it takes for the satellite to complete one orbit in hours.

Key Concepts

Circular Orbit

A circular orbit is a path followed by a satellite around a celestial body where the distance from the center of the body to the satellite remains constant. In this scenario, the satellite maintains a fixed altitude of 705 km above Earth's surface, implying a uniform orbital speed and gravitational force acting as the centripetal force.

Orbital Period

The orbital period is the time taken for a satellite to complete one full orbit around a celestial body. It depends on the radius of the orbit and the mass of the central body. For a satellite in a circular orbit, the period can be calculated using Kepler's third law, which relates the orbital period to the radius of the orbit and the mass of the Earth.

Gravitational Force

Gravitational force is the attractive force between two masses, such as Earth and a satellite. It provides the necessary centripetal force to keep the satellite in orbit. The strength of this force depends on the masses involved and the distance between them, and it is crucial for determining the satellite's orbital speed and period.