# What is Kepler's 3rd law

## Kepler's laws

### introduction

The astronomer and natural philosopher Johannes Kepler published his findings about the planetary orbits in three laws, the so-called Kepler's laws.

According to older theories of Ptolemy and Copernicus, the planets moved on nested circular orbits. Kepler discovered that the planets move on elliptical orbits.

### Brief information on ellipses

An ellipse is a special closed oval curve.

The two points \ (F_1 \) and \ (F_2 \) are the Foci the ellipse. For any point on the ellipse, the sum of the two distances to the two focal points is constant.

The axis through the two focal points is called Main axis and is divided by the center \ (M \) into its two major semi-axes \ (\ overline {MS_1} \) and \ (\ overline {MS_2} \). The points \ (S_1 \) and \ (S_2 \) are called the main vertex. The length of one of the two major semi-axes is denoted by \ (a \).

Analogously, one speaks of the side vertices \ (S_3 \) and \ (S_4 \) and the Minor axis, consisting of the small semi-axes \ (\ overline {MS_3} \) and \ (\ overline {MS_4} \). The length of the small semiaxes is denoted by \ (b \).

The degree of flattening of an ellipse is called eccentricity. In the figure below you can see ellipses with eccentricity decreasing to the right. A circle can be understood as an ellipse with zero eccentricity. The eccentricities of ellipses are between 0 and 1.

### 1. Kepler's law

The planets move on elliptical orbits with the sun in one focal point.

The sun is therefore in a focal point of the planet's orbit and not, as one might assume, in its center. The other focus is empty. As the planet moves along the ellipse, its distance from the sun is constantly changing. The point closest to the sun is called perihelion, the point furthest from the sun is called aphelion.

distance

speed

This animation is based on a Java applet by Walter Fendt (http://www.walter-fendt.de/ph14d/kepler1.htm).

### 2. Kepler's law

\$\$ \ dfrac {\ Delta A} {\ Delta t} = \ rm {const.} \ qquad \ dfrac {\ Delta A_1} {\ Delta t_1} = \ dfrac {\ Delta A_2} {\ Delta t_2} \$\$ The connecting line between sun and planet covers the same area at the same time

The areas near and far from the sun covered by the connecting line are shown in green and orange in the animation.

If the planet is closer to the sun in its orbit, it will move faster; if it is further away from the sun, it will move more slowly. Therefore, the earth is slower in summer (in the northern hemisphere) because it is further away from the sun. For this reason and because of the greater distance, summer is 9 days longer than winter.

distance

speed

This animation is based on a Java applet by Walter Fendt (http://www.walter-fendt.de/ph14d/kepler2.htm).

### 3. Kepler's law

\$\$ \ dfrac {(T_1) ^ 2} {(T_2) ^ 2} = \ dfrac {(a_1) ^ 3} {(a_2) ^ 3} \$\$ The ratio of the squares of the orbital times and the 3rd powers of the major semiaxes is constant for all planets

This law shows that planets far from the Sun need more time to orbit than planets near the Sun. For example, our earth only needs 365 days for one orbit, whereas the planet Neptune, which is much further away from the sun, needs about 165 years.

Mars

earth

### literature

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