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. Determine the radius of the satellite's orbit by adding the altitude of the orbit to the Earth's radius. The Earth's radius is approximately 6371 km, and the altitude of the orbit is 705 km.
2. Use the formula for the orbital period $T$ of a satellite in a circular orbit: $T = 2\pi \sqrt{\frac{r^3}{GM}}$, where $r$ is the radius of the orbit, $G$ is the gravitational constant (approximately $6.674 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}$), and $M$ is the mass of the Earth (approximately $5.972 \times 10^{24} \, \text{kg}$).
3. Convert the radius of the orbit from kilometers to meters by multiplying by 1000, as the gravitational constant is in cubic meters per kilogram per second squared.
4. Substitute the values of $r$, $G$, and $M$ into the formula to calculate the orbital period $T$ in seconds.
5. Convert the orbital period from seconds to hours by dividing by 3600 (the number of seconds in an hour).

Key Concepts

Circular Orbit

A circular orbit is a path in which an object moves around a central body at a constant distance. In this case, the Aura spacecraft orbits the Earth at a height of 705 km. The gravitational force between the Earth and the satellite provides the necessary centripetal force to maintain this circular motion.

Orbital Period

The orbital period is the time it takes for an object to complete one full orbit around a central body. It can be calculated using Kepler's Third Law, which relates the period of orbit to the radius of the orbit and the mass of the central body. For satellites, this is often expressed in hours or minutes.

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's Law of Universal Gravitation. For satellites, this force is crucial as it keeps them in orbit around the Earth. The balance between gravitational force and the satellite's inertia determines the characteristics of its orbit, including the orbital period.