Kepler’s Laws for Satellites – Space Segment

If you are given a task to design a satellite right now, an important consideration would be its orbit. You need to know how your satellite would move in space. To do that, you need to understand Kepler’s Laws. Kepler’s laws for satellites are basic rules that help in understanding the movement of a satellite.

Kepler postulated these laws based on empirical evidence he gathered from his employer’s data on planets. His employer, Tycho Brahe, had extremely accurate observational and record keeping skills. A few years later, Newton came along and proved that these laws hold true mathematically. These laws describe the fundamental nature of any orbiting object.

Please note that these statements have been modified to present the underlying concept in terms of communication satellites.

Kepler’s First Law

Statement: Closed orbits have the shape of an ellipse, and the ellipse lies in a single plane. 

This statement implies that all orbits take the form of an ellipse. An ellipse is a shape that appears to resemble an elongated circle. In fact, a circle is a special type of an ellipse. An ellipse has two focus points or foci.

Another property of an ellipse is its eccentricity, which is a measure of its elongation/thinness. The eccentricity of an ellipse can assume any value between 0 and 1. In the case where eccentricity is 0, the ellipse has no elongation and is essentially a circle.

The two axes of an ellipse are known as major axis and minor axis. You can determine their location in the ellipse based on their names.

For a satellite in orbit, the body it revolves around will always be situated at one of the foci. In the case of a circular orbit, the two foci combine to form the center of the circle.

There are two important spots on an elliptical orbit which are useful to describe the movement of a satellite. The apogee is the farthest distance that a satellite can be from Earth. Whereas, the perigee is its closest point from Earth. These distances are measured from the surface of the planet.

The sum of the distance of an object in an elliptical orbit from both the foci is always constant.

Kepler’s Second Law

Statement: A satellite does not have a uniform velocity in an elliptical orbit. It will move faster when close to Earth and slower when far from Earth. The satellite will have a constant speed in the case of a circular orbit.

U shaped roller coaster analogy to understand Kepler's laws for satellites
The roller coaster will gather speed as it comes down towards the bend. This is similar to how a satellite gains momentum as it nears its perigee.

Imagine a rollercoaster at the top of a U-shaped track. As the roller coaster starts its descent, it picks up speed and is at its fastest when it passes the U-bend. As the roller coaster climbs back up towards the other end, it loses its speed.

The movement of the roller coaster is similar to how a satellite moves in an elliptical orbit. At its apogee, the satellite is at its slowest speed. However, it is its fastest speed at the perigee of its orbit.

An alternate method to state this phenomenon is that in an elliptical orbit, a satellite sweeps equal areas in equal times. Hence, this law is also known as the Law of Areas.

Satellite operating companies use this to their advantage. They position the orbit in a way that would help them maximize the time a satellite spends beaming signals to a particular region.

Kepler’s Third Law

Statement: The period of the orbit depends only on the average distance between the satellite and Earth. 

Mathematical representation:

{ a }^{ 3 }\quad =\quad \frac { \mu  }{ { n }^{ 2 } }

  • a = mean distance between the two bodies
  • \mu = Earth’s geocentric gravitational constant
  • n = periodic time in rad/s

Kepler’s third law basically means that the square of the periodic time of orbit is inversely proportional to the cube of the distance between them.

The third law is really cool. It shows that the orbital period of a satellite in orbit depends only on its distance from Earth. And some constant that depends on Earth’s parameters. No other parameter of the satellite (like mass/shape) matters in calculating its orbital period. In other words, if the distance is large, the orbital period reduces, and if it is small, the orbital period increases.

The following equation can calculate the orbital period (in seconds)

P\quad =\quad \frac { 2\pi  }{ n }

We will use the two equations above to calculate the distance for a geostationary orbit in the next post.

Summary

  • Kepler’s first law: Elliptical orbits with the primary body at one of the foci. The primary body is Earth in this case.
  • Kepler’s second law: Satellites will cover equal areas in equal intervals of time; they will move faster when closer to Earth and slower when far away from it.
  • Kepler’s third law: The orbital period of a launched satellite depends on only on of its parameters, i.e. its distance from Earth. The orbital period is the time taken by satellites to complete one orbit.

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