 # How Kepler’s Laws Help Calculate the Radius of the Synchronous Orbit of an Earth Orbiting Satellite

When thinking of a planet orbiting another planet, people first assumed that the planet circulates a planet in a perfect circle. Turns out that was not the case. Planets such as Pluto and every other planet did not orbit in a circle. Johannes Kepler published a significant breakthrough in calculating orbiting objects.

Well before Kepler evolved his laws of planetary motion, the philosopher Ptolemy described the epicyclical motion of planets in orbit around the earth. This geocentric model roughly explained the retrograde motion of planets, which appear to move backwards at times, when viewed from the earth. Unfortunately, for Ptolemy, the Copernican heliocentric view proved more correct and invalidated the epicyclical motion model.  Still, Copernicus’s theory had a serious flaw. Planets don’t travel in circular orbits. It took Kepler to discover how they really do move. During the solar day under discussion, the line would have swept out a piewedge-shaped space as the planet moved. When the planet is far from the sun, the pie wedge is long and narrow. When the planet is close to the sun, the pie wedge is short and squat. According to Kepler’s second law, the area defined by the long narrow wedge and the short squat wedge are equal.

His study show three laws of planetary motion, which the first law of motion shows orbiting object would travel through using the two internal point to find out the eclipse. The second law of motion shows the objects speed changes as it orbits an object. The closer the object orbit around the foci the faster the object travels. Kepler’s third law shows “The square of the orbital period of a planet is proportional to the cube of the mean distance from the Sun (or in other words–of the”semi-major axis” of the ellipse, half the sum of smallest and greatest distance from the Sun)” (Stern, 2005). Kepler’s third law shows a comparison of two or more motion of an orbiting object. This third law shows that object orbit farther from the foci has larger equilibrium. Meaning the object would travel a longer path, slower speed, and takes longer to complete a full orbit. Making these rules in place, thinking about a satellite orbiting around the earth, we would be able to figure exactly how the satellite would orbit the earth. The Kepler law allows us to understand the shape of the equilibrium, the speed, velocity, and differences in orbiting the earth at a various distance from the foci from the mass of the satellite.

Many satellites operate at roughly the same distance from earth; a sweet spot called the GEO (Geostationary equatorial orbit).  This is a sweet spot because the time it takes for a full orbit around earth is equal to one earth day.  This can be important for numerous reasons, surveillance is among them.

Kepler’s 3rd law teaches us “The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit” (HyperPhysics).  This gives scientists the ability to determine an orbital pattern just by recording how long it takes to make a full orbit.  Interestingly this is not exactly true.  “Proportional” is a key word here.  Because the Sun is so massive, adding any planet’s mass to it is close to negligible, this is why the equation is simplified to T^2=A^3.

Detailed examination of Brahe’s measurements over an additional 10-year period led Kepler to uncover the constant relationship between the time of one planetary orbit (orbital period) and the average orbital radius. The constant relationship stands as the square of the orbital period (T) divided by the cube of the average orbital radius (R), or K = T2/R3.

These discoveries made by Kepler and Galileo has impact strongly in Isaac Newton’s study of the law of motion and principle of mathematics in astronomy. Kepler’s law of planetary motion set a numerical rule that sets the ground for many future discoveries in the world of wonder. As Newton has expanded in Galileo and Kepler’s studies, by showing discoveries in changes in velocity, momentum, inertia, and more in-depth relationship in motion in the law of gravity.

The impact of Kepler’s discoveries is penetrative to many sciences, most obviously space travel.  Studying the behavior of entities in space, including planets, provides insights into spacecraft travel. It has also impacted Lagrange, Laplace, Hamilton, Gibbs, and many great scientists came after to finish his work up till today.

References