Johannes Kepler's 3 Laws of Planetary Motion

Based on Kepler's third law, which states that the farther the planet is from the Sun, the weaker will be its forces; Newton formulated The Law of Gravitation.

Johannes Kepler, a German mathematician, has contributed a great deal to the field of astronomy and astrology. The Laws of Planetary Motion that are formulated by Kepler, proves that the orbit of the planets are ellipses and not circles, as believed by many.

The ellipse is a geometrical shape that has two foci, such that, the sum of the distance from the focus to any point on the surface of the ellipse is constant. The orbits of planets have small eccentricities (flattening of ellipse), and so, they appear as circles.

Based on the properties of ellipses, Johannes Kepler devised three laws that explain the motion of planets around the Sun.

As discussed earlier, an ellipse has two foci. While studying the motion of planets around the Sun, Kepler explained that the path followed was elliptical, with the Sun as one of the two foci. In simple terms, the law can be stated as:

All planets move in elliptical orbits, with the Sun at one focus.

This indicates that the Sun is one focus, while the other focus is known as the vacant or empty focus. As seen in the diagram, the Sun and the empty focus lie on the major axis of the ellipse, and the planet lies on the surface of the ellipse.

As the planet is continuously moving around the Sun, and as the Sun is not at the center of the ellipse, the Planet-Sun distance will always keep on changing.

As the Sun is one of the foci, it is clear that the Planet-Sun distance will be changing. But, the planet covers up for the increase in the distance by moving faster when it is closer to the Sun.

This indicates that planets do not move at a uniform speed. The second law states that:

The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.

The point at which the speed of the planet is fastest is known as Perihelion, while the distance where the speed is the slowest is known as Aphelion. The distance measured from the perihelion to the position of the Sun is known as perihelion distance, while the distance from the Sun's position to the aphelion is known as the aphelion distance.

The law says that, while moving in an elliptical path, the planet moves faster when it is closer to the Sun. This way, the radius sweeps equal areas in equal amount of time.

If the planet is observed at successive times (P1, P2, P3, P4), we draw the radius vector during the first second observations, we find the two radius vectors having the same area. So, the area swept during the time 't' by the planet to move from P1 to P2 is the same as the area swept while moving from P3 to P4.

This law relates the time required by a planet to make a complete trip around the Sun to its mean distance from the Sun. This law can be simply stated as:

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

The symbolic representation of the law is as follows,

P² ∝ a³

Here, 'p' is the orbital period while 'a' is the semi-major axis of the orbit.

To convert this equation into a ratio, we introduce a constant of proportionality

p² / a³  = 1 (yr² / AU³)

where 'yr' is the sidereal year * and 'AU' is the astronomical unit **.

* - Time taken by the Earth to complete one trip around the Sun
** - Average Earth-Sun distance, i.e., approximately 149.6×10⁶ km

The law is commonly represented as,
T² = (4Π² / GM) R³
where,
T - Earth years
G - Constant of Gravitation
M - Mass of the larger body
R - Center of mass of two bodies

The third law was applied to Earth satellites. The law stated that, when a satellite is farther from the Earth, it will take a longer time to complete the orbit, and its average speed will be slower.

In the early fifteenth century, there were debates whether planets actually go around the Sun or not. Tycho Brahe, a Danish nobleman, believed that, if the actual positions of planets would be known, this debate would be resolved. He then began a study on the positions of the planets and recorded the data.

This data was later studied by Kepler, and based on that, and Kepler's research, the three simple laws of planetary motion were formulated. These laws are the basis of many other physics laws formulated by Newton, Galileo, and the like.