The energy of an orbiting satellite

Introduction

Satellites are small orbiting objects that circle massive celestial bodies like planets and other natural satellites. Satellites can be natural and artificial. An object that is created by humans and intentionally launched into space and placed into an orbit to revolve around a body is known as an artificial satellite. India launched its first-ever artificial satellite – Aryabhata, in 1975. On the other hand, natural satellites are astronomical objects or bodies that orbit around other astronomical bodies such as planets, dwarf planets, etc. The orbit is a trajectory of an object that curves with the gravitational force. For a planet to revolve within the trajectory and stay in orbit, a centripetal force acts on the body, and the transformation of kinetic energy and potential energy takes place. 

Kepler’s law:

Tycho Brahe gathered data about the behavior of the planets and their movement, which was later analyzed by his assistant Johannes Kepler. Kepler then formulated three laws that supported the formulation of Newton’s universal law of gravitation. The three laws formulated by Kepler are known as Kepler’s laws of planetary motion and are as follows:

Law of the orbit:

All planets are moving along an elliptical orbit with the sun at one of the ellipse’s two foci.

Law of area:

A line connecting the planet and the sun will sweep an equal area at equal time intervals.

Law of periods:

The time required for a revolution of a plane squared will be proportional to the cubed value of the semi-major axis of the elliptical orbit traced by the planet.

Satellites – Natural and Artificial:

Satellites revolve around a planet and follow a similar motion as that of the planets around the sun. Thus, Kepler’s law will apply to satellites as well. The satellites may follow an elliptical or circular orbit.

Natural satellites are astronomical bodies that orbit around another massive celestial body such as a planet, a dwarf planet, etc. For example, the only natural satellite of the earth is the moon.

Artificial satellites are manufactured objects that are launched into orbits using rockets. More than one thousand artificial satellites are orbiting the earth, all launched for specific purposes.

Uses of artificial satellites:

Navigation satellites: The global positioning system enables satellites to orbit around the earth and calculate the accurate location

Communication satellites:

These are for transmission through television, the internet, and mobile phones.

Weather satellites:

These orbiting satellites image temperature, cloud, and rainfall used for weather forecasting.

Let us consider a satellite in a circular orbit at a distance of h from the earth, and the total distance from the center of the earth will be Re+h, where Re is the radius of the earth. 

Considered the mass of the satellite to be m and its speed to be v then the centripetal force acting on the orbit will be calculated as:

Fc = mv2/(Re+h) —- (eq 1)

When the satellite orbits around the earth, the centripetal force will be provided by the gravitational force of the earth thus.

Fg= GmMe/(Re+h)2 —- (eq 2)

Me = Mass of Earth

Equating both the equation we get:

v2 = GMe/(Re+h) —-(eq 3)

The formula shows that v is inversely proportional to the satellite’s distance from the earth i.e., h.

 If this is considered to be 0, the distance of the satellite from the earth is Re.

v2 = GM/(Re) —-(eq 4) 

The area traversed by an orbiting satellite is 2π(Re+h)  with the speed of v. Thus to calculate the period of traversing the area is:

T = 2π(Re+h) /v

= 2π(Re+h)3/2/√GMe  —– (eq 5)

The energy of an orbiting satellite:

The kinetic energy of the satellite formula is:

KE = 1/2 mv2

= GmMe/2(Re+h) —- replacing the value of v2 from eq 3

Where the orbiting satellite has the speed of v and mass of m

The potential energy of the orbiting satellite at a distance of Re+h, considering the gravitational potential energy to be zero in space, is : 

PE= (-) GmMe/(Re+h)

The kinetic energy (KE) is positive, and potential energy (PE) is negative. The magnetic kinetic energy is half of that of the potential energy, and thus the total energy (E) is calculated as:

E=KE+PE

= (-)GmMe/2(Re+h)

When the orbiting satellite moves in a circular orbit, the total energy is always negative. This is because the total negative potential energy is twice the magnitude of the positive kinetic energy.

Even when the orbit is elliptical, the total energy remains negative, but the kinetic energy and potential energy vary from point to point. The total energy of an orbiting satellite will always be negative, as when the total energy is positive or zero, the satellite will escape to infinity.

Geostationary satellite:

Geostationary satellites seem to be fixed when viewed from the earth as the satellites are in sync with the rotation of the earth.

If we consider the value of Re+h in eq 5 in a way that time of traversing the orbit by a satellite as 24 hrs, we see that the satellite will have the same period as that of the earth to complete rotation and thus when viewed from earth at any point, it seems to be fixed.

Conclusion:

Whether natural or artificial, all orbiting satellites are kept into their orbit by centripetal force. Orbits are gravitationally curved paths. The energy for orbiting satellites combines kinetic energy and potential energy. The negative potential energy being twice the magnitude of the positive kinetic energy results in negative total energy. This negative total energy ensures that the satellite keeps moving along the elliptical or circular orbit and does not escape infinity. Satellites that orbit in the same direction of the earth’s rotation seem to be fixed from any point of the earth and are termed geostationary satellites. In these satellites, the period for traversing the orbit is 24 hours, the same as the earth’s rotation period.