Kepler’s laws

retrograde_mars_tezel_2007-2008

Figure 1: Many ancient and medieval cultures believed the stars and the planets rotated around a fixed Earth. The complex motions of the planets—which sometimes move against the perceived rotation of the rest of the sky (retrograde motion, shown in the photo)—led Renaissance astronomers to question this geocentric theory. These astronomers discovered the laws of orbital mechanics, transforming natural philosophy into the practice of science. (Photograph ©2007–08 Tunç Tezel.)

Copernicus was met with rage from the church when he suggested in 1543 that, based on observations, that the planets were moving in circular orbits around a stationary sun. The idea that the sun was the center of the Universe, and not the Earth, was unheard of in a world that was based heavily on the Holy Bible. The findings of Copernicus ignited the heliocentric Copernican Revolution in Europe.

Johannes Kepler, based on more accurate observations, corrected what Copernicus had found by stating that the orbits of the planets are not circular, but elliptic. That means that the Sun is not the center of the Solar System, but is located in one of the focal points of each of the planets ellipses. It was later, when Isaac Newton and Gottfried Leibniz discovered calculus, that it was shown mathematically that Kepler’s laws of planetary motion were shown to be correct.

Kepler stated three laws, now known as Kepler’s laws of planetary motion.

The first law says that, as already mentioned, that the satellite (artificial or natural) moves in elliptical orbits around its central body. The central body is located in one of the focal points of the satellites ellipse, and the other focal point is empty.

keplers_second_law

Figure 2: A satellite is orbiting a star. The red and the green areas are equal in size, so the satellite must use the same time from t1 to t2, as from t3 to t4.

The second law states that the area speed is constant. The area speed is the area swept by the line between the central body and the satellite over a certain time span. See Figure 2. The satellite spends equal amounts of time to span the two areas, since their sizes are equal. The speed of the satellite is therefore high close to the central body and lower when the satellite is further away.

The third law states that the square of the period T for an orbiting satellite is proportional to the cube of semi-major axis (see next section) a of the ellipse: T^2 = \mathrm{const} \cdot a^3. The constant is equal for all satellites orbiting the same central body.

It is important to know that the Kepler orbits assume one central, stationary body with a large mass and an orbiting satellite with a much smaller mass. Other effects such as solar radiation pressure, atmospheric drag, and influence from the gravity of other planets are neglected. Studying the orbit of Mercury, the general theory of relativity has to be included to calculate satellite orbits exactly. By making the above simplifications though, as we will see, the mathematical description of the orbits becomes quite simple and easy to work with.

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This article is a part of a pre-course program, used by NAROM in different courses, for example Fly a Rocket!