Kepler's laws of planetary motion

Johannes Kepler (1571-1630) was a German mathematician and astronomer. Might the best known Kepler's discoveries are three laws of planetary motion. In the 16^th century, Polish astronomer Nicolaus Copernicus (1473-1543) published the work De revolutionibus orbium coelestium. In this book Copernicus presented the heliocentric theory as an alternative theory to Ptolemy's geocentric model, which made a revolution in understanding the universe.

However, Copernicus theory was more qualitative than quantitative. Further progress was done thankfully to Danish astronomer Tycho Brahe (1546-1601) – who collected very accurate data on planetary motion and Johannes Kepler – who interpreted it. Kepler established that planetary orbits are not circles but ellipses! It was impressive discovery, especially having in mind very small eccentricities of orbits. First two laws Kepler published in 1609 in the book Astronomia nova, whereas the third low was published in 1619 in the book Harmonices Mundi. Kepler's laws appeared as observational facts. On the the other hand, they follow from the mechanical laws.

Tycho Brahe is known as the greatest astronomer in the pre-teleskopic era. He was both maker of instruments and observer. His measurement of celestial bodies motion and other astronomical phenomena was very detailed and precise. In 1572 he observed supernova star named Tycho's Supernova after him. The same type of phenomena Kepler discovered in 1604. This supernova is known as Kepler's Supernova or Kepler's star. It is believed that these discoveries, showing that there are changes within stars, contribute to the new ideas about motion and location of the Earth in the universe.

In this article we are going to show all three laws of planetary motion. The matter will be presented on a high school level. We begin with a consice review of the ellipse, that is essentioal geometric curve in the context of Kepler's laws.

The size and the shape of an ellipse is defined by the eccentricity e and one of the semi-axes, either major semi-axis a or minor semi-axis b.

The eccentricity is sometimes called numerical in contrast to the linear eccentricity c. As Figure 1 shows major and minor axis are the lengths connecting pairs of extreme points on the ellipse. Linear eccentricity is the length between the center and a focus.

Numerical eccentricity is defined as ratio of linear eccentricity to major semi-axis (equation 2). The major semi-axis, minor semi-axis and linear eccentricity are connected by the Pythagorean theorem (equation 1).

One of the basic characteristic of the ellipse is that sum of lengths between a point T to foci is constant and equal to the major axis (equation 3). It can be shown that the average length between a point T and a focus is equal to the major semi-axis.

As an exercise let calculate how far from the Sun will Halley's comet be in the moment when it stars its return journey? The length of major semi-axis is 17.8 au and the eccentricity is 0.96.

Firstly, using equation (2) we can calculate liner eccentricity c. The final result is the sum of c and major semi-axis a.

Haw to draw an ellipse knowing values of semi-axes?

We can easily draw an ellipse knowing relation (3). The following is the algorithm.

1. Draw two perpendicular lines and mark points A, B, C, D (as in the Figure 1).
2. The length OB take with compass, then put compass in point C and mark foci.
3. Randomly choose few points T₁,T₂,... on the length CD.
4. Put compass in F₁ and draw part of circle of radius A-T₁.
5. Put compass in F₂ and draw part of circle of radius B-T₁, and mark both interactions with previous part of circle.
6. Repeat step 4 and 5 for other chosen points. The more points are chosen the more precise ellipse.
7. Connect obtained intersections.

1. A planet is moving around the Sun along an ellipse, with the Sun at one of the focuses.

So, the first law tells us that the orbit of a planet is an ellipse. The point of ellipse at which two bodies (the Sun and a planet) are in the closest position is called the periapsis (or pericentre). Contrary, the point of farthest approach is called apoapsis (or apocentre). A planet change its velocity during a cycle. The highest speed is in the periapsis, while the minimal speed is in the apoapsis. This follows from the fact that gravitational interaction between a planet and the Sun is the strongest in the periapsis and gradually decline towards the apocentre.

Planets in our Solar system have very small eccentricities of its ellipses. For example, our ellipse has eccentricity of 0.017, neighboring Venus has 0.007, Jupiter 0.048. Apart of Pluto, Mercury's path has the highest eccentricity, 0.206.

Although this laws is usually formulated specifically for the Sun system, it worth generally. Thus, for any other planetary system. Also, it is possible that a celestial object move along an parabola or hyperbola instead of an ellipse. In every case, the central body is in a focus. These conic sections are called Keplerian orbits.

As an illustration of this consideration, Table 1 shows basic orbital elements for several celestial bodies in the Solar System. Apparently, eccentricities of an orbit could be very different from almost a circle like in case of Venus to the almost borderline case of Helley's comet path. Namely, when the value of an eccentricity is 1 than the path is not an ellipse but parabola. In case that eccentricity is grater than 1 the path is hyperbola. Astronomical unit (abbreviation: au) is the length between the Sun and Earth.

Figure 4 roughly compares orbit shapes. There is nothing physically neither in the center of ellipse nor another focus.

Video 1. An animation of motions in the Solar System. One can see orbits of inner and outer planets as well as of some comets, including Halley's comet. Distances between bodies and orbital periods are nicely compered.

2. Areal velocity of a planet is constant.

The "normal" velocity can be understand as a line distance which is swept out by a particle per unit time. Imagine now the area which is sweep out by a line connecting the Sun and a planet. Areal velocity of a planet is this area per unit time. In other words, a line connecting the Sun and a planet sweeps out equal areas during equal intervals of time. Figure 5 illustrates this facts: if times required for sweep out the marked areas are equal than these areas are equal as well.

3. The cubes of the major semi-axis of an orbit is proportional to the square of the orbital period.

This law put in a relation the distance of a planet from the Sun and its orbital period. More precisely, in case of Solar system it holds equation (4). So, the constant in the law is 1 (with the declared dimension) which follows from the general form (5). This general form holds for any planetary system, where M is the mass of the central body and G is the gravitational constant.

If we actually deal with two body problem, then in equation we have masses of both bodies (6). For example, in case of the system Pluto and its moon Charon we have to use (6) since the mass of Charon by Pluto is not negligible.

Exercise 1. Orbital period of Europe, Jupiter's satellite is 3.551 day. The value of major semi-axis of this Galilean moons is 670900 km. Calculate the mass of Jupiter.

Solution: The result 1.89*10²⁷ kg follows from the relation (5).

Exercise 2. The mass of planet Uranus is 8.66*10²⁵ kg. Its satellite Ariel has the major semi-axes of 191020 km. What is the orbital period of Ariel?

Solution: Again, we will use (5) which lead to the result P=2.52 day.

Video 2. A planet's elliptical orbit around the Sun.

The highest velocity of the planet is periapsis, the point on the ellipse that is closest to the Sun.

Contrary, the smallest velocity is in the apsis, a point oppositee of the periapsis.

Kepler's lows of planetary motion:

1. A planet is moving around the Sun along an ellipse, with the Sun at one of the focuses.
2. Areal velocity of a planet is constant.
3. The cubes of the major semi-axis of an orbit is proportional to the square of the orbital period.