Energy of An Orbiting Satellite

Satellites are launched from the earth to revolve around it. Many rockets are fired from the satellite at a proper time to establish the satellite in the desired orbit. Once the satellite is located in the desired orbit with the correct speed for that orbit, the satellite will continue to move in an orbit under the gravitational attraction of the earth.

So, the energy required by a satellite to revolve around the earth is called its orbiting energy. Since this satellite revolves around the earth, it has kinetic energy and is in a gravitational field, so it has potential energy.

Potential Energy of Satellite

Let’s consider mass m at distance r₁ and distance r₂ from the centre of the earth. Here, we will move radially from distance r₁ to distance r₂ and then move along the circle until we reach the final position.

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During the radial portion, the force F is opposite to the direction we are travelling along with distance dr. 

Along the arc, F is perpendicular to dr, so F.dr = 0. Therefore, no work is done while moving along the arc. 

Now, using the expression for the gravitational force and noting the values for 

 Along with the two segments of our path, we have:

\[\Delta  U = -\int \int_{r1}^{r2} F . dr = GMm \int_{r1}^{r2} \frac{d}{r2}\]

\[= GMm (\frac{1}{r1} - \frac{1}{r2})\]

Since  \[\Delta U = U_{2} - U_{1}\], we can find the expression for U, i.e.

\[ U = - \frac{GMm}{R}\]

Kinetic Energy of Satellite

Let’s consider the earth as a reference for a planet. From the top of the earth, we can see the satellite revolving around it. We would consider everything as a function to compute the kinetic energy of a satellite.

Starting with the radius of the earth as ‘r.’

So, r = The distance from the centre from the earth to any point on its surface.

Similarly, r = The distance between the centre of the planet to any point in its orbit.

Here, we are considering the top view of the earth and the front view of the satellite.

The mass of satellite  = m

The mass of the earth  = M

Radius = r

Velocity = v

The two forces are acting on it are gravitational force,   \[F_{g}\] and a centripetal force due to its velocity, \[F_{c}\]

Where  \[F_{g} = \frac{mM}{r^{2}}\] and \[F_{c} = mv^{2}r\].

The magnitude of the forces is equal.

So, \[F_{g} = F_{c}\]

\[\frac{mM}{r^{2}} = mv^{2}r\]

\[ \Rightarrow  V^{2} = \frac{GM}{r}...(1)\]

We know that \[ K.E.= \frac{1}{2} mv^{2}\]

Putting the value of \[V^{2}\] in eq(1), we get,

K.E. of a satellite = \[ \frac{GMm}{2r}\]

Total Energy of Satellite

The total energy of the satellite is calculated as the sum of the kinetic energy and the potential energy, given by,

T.E. = K.E. + P.E.

\[ = \frac{GMm}{2r} = -\frac{GMm}{2r}\]

\[ T.E. = -\frac{GMm}{2r}\]

Here, the total energy is negative, which means this is also going to be negative for an elliptical orbit.

Here, T.E. < 0 or negative, this means the satellite is bound to the earth through gravity.

To make this TE zero, we need to give additional energy of GMm/2r to the satellite, i.e:

\[ T.E. = - \frac{GMm}{2r} + \frac{GMm}{2r} = 0\]

We know that if the separation between the two bodies is infinite, then the potential of the system is considered zero.

At an infinite distance, the body of a smaller mass has less effect on the gravitational field of the larger body and the smaller body can escape from the larger one.

It means for a satellite to escape; it has to travel an infinite distance away from the earth. So, at an infinite distance, its energy would become zero on getting additional energy of GMm/r. 

Therefore, satellites would escape from the earth.

Hence, the additional energy required by a satellite to escape the earth = kinetic energy of the satellite.

Gravitational Potential Energy of Satellite

The kinetic energy of a satellite is half the gravitational energy, given by,

If the gravitational energy is GMm/r, then kinetic energy is GMm/2r and this kinetic energy is positive. 

When kinetic energy and the potential energy are added up, the total will come out to be the gravitational potential energy, given by,

T.E. = - GMm/2r = 1/2  P.E. or 1/2 (- GMm/r)

This T.E. is negative, which means the satellite can’t leave or can’t just fly away in outer space and never come back to it.

It is bound to the earth just like the earth is bound to the sun. We are in a bound orbit.

So, anytime the total energy is negative, that is a bound orbit.

It is not about the two particles having a mass, it could be charged particles like the proton and the electron. The electron that orbits the proton has negative energy, which means it is bound to the proton. 

If the electrons were not bound, we won’t get atoms.

Similarly, if a satellite would fly away, we won’t have GPS, nearest Starbucks, etc.

The article provides the calculation of the orbital energies of satellites. The derivation of calculation of potential energy and kinetic energy of satellites are given in detail.