14.14:
Energy of a Satellite in a Circular Orbit
Consider an artificial satellite orbiting the Earth in a perfectly circular orbit under the influence of the Earth's gravitational force.
Therefore, it orbits with critical velocity vc, which is equal to the square root of the product of the gravitational constant and the Earth's mass, divided by its total distance from the Earth's center.
Now, squaring and multiplying the critical velocity equation with half times the satellite's mass, the kinetic energy of the satellite in a circular orbit is obtained.
Recall that the potential energy of the satellite is expressed as the negative product of the gravitational constant and the two masses, divided by the distance between them.
Therefore, the potential energy of the satellite equals minus two times its kinetic energy.
Since the total energy is a sum of kinetic and potential energy, it equals the negative of its kinetic energy for the satellite in circular orbit. The negative sign in the total energy equation indicates that the satellite is bound to the Earth.
14.14:
Energy of a Satellite in a Circular Orbit
Thousands of artificial satellites orbit the Earth every day at various distances from the Earth. Satellites that orbit the Earth below an altitude of 1,600 km are considered to be orbiting in low-Earth orbit (LEO). Research satellites and Earth observation satellites are usually placed in LEO, and mostly orbit the Earth in elliptical orbits. Navigation satellites are placed in medium-Earth orbit (MEO), ranging from 2,000 km to 36,000 km from the surface of the Earth. Meanwhile, communication satellites orbit the Earth at approximately 36,000 km in geostationary or geosynchronous Earth orbits (GEO). These satellites mostly have circular orbits, and their orbital periods are equal to the Earth's rotation period.
The total energy of satellites in circular orbits is conserved and can be derived using Newton's law of gravitation. In such an orbit, the kinetic energy of the satellite is numerically half of its potential energy, and hence the total energy becomes equal to the negative of kinetic energy. The negative sign here indicates that the satellite is gravitationally bound to the Earth.
In elliptical orbits, since the distance of the satellite from the Earth changes every instant, its kinetic energy and potential energy vary. However, its total energy, the sum of its kinetic and potential energy, remains constant.
This text is adapted from Openstax, University Physics Volume 1, Section 13.4: Satellite Orbits and Energy.