# Kepler's Laws of Planetary Motion

## Introduction

The astronomer and philosopher Johannes Kepler published his findings on the planetary orbits in three laws, known as Kepler's laws of planetary motion.

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

An ellipse is a special closed oval curve.

An ellipse has two special points called focal points (the focus $$F_1$$ and the focus $$F_2$$). For any point on the ellipse, the sum of the two distances from these two focal points is constant.

The line through the two focal points is called major axis and is separated by the center $$M$$ into its two semi-major axes $$\overline{MS_1}$$ und $$\overline{MS_2}$$. The length of each of the two semi-major axes is denoted by $$A$$.

Similarly, we define the minor axis, consisting of the semi-minor axes $$\overline{MS_3}$$ und $$\overline{MS_4}$$. The length of the minor axes is denoted by $$b$$.

The four points where the axes cross the ellipse are called vertices.

The degree of flattening of an ellipse is called eccentricity. In the figure below, ellipses with declining eccentricity. A cirlce may be regarded as an ellipse with eccentricity zero. The eccentricities of ellipses are between 0 to 1.

## Kepler's first law

The orbit of every planet is an ellipse with the Sun at one of the two foci.

The sun is thus located at a focal point of a planet's orbit and not as one might expect in the center. The other focal point is empty. As the planet moves along the ellipse, its distance from the Sun changes continually. The point next to the Sun is called perihelion, the point farthes away from the Sun is called aphelion.

Semi-major axis
$$\rm AE$$
Num. eccentricity

Distance
from the Sun
$$\rm AE$$

Velocity
Current
$$\rm \frac{km}{s}$$
Min./Max.
$$\rm \frac{km}{s}$$
Average
$$\rm \frac{km}{s}$$

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

## Kepler's second law

$$\dfrac{\Delta A}{\Delta t} = \rm{konst.} \qquad \dfrac{\Delta A_1}{\Delta t_1} = \dfrac{\Delta A_2}{\Delta t_2}$$ A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

The surfaces swept by the connection line in perihelion (green) and aphelion (orange) are shown in the animation.

If the planet is closer to the Sun on its orbit, it moves faster, if it is located further away from the Sun, it moves slower. Therefore, the Earth is slower in the summer (on the northern hemisphere) because it is farther away from the Sun. For this reason and because of the longer way the summer is about 9 days longer than the winter.

Semia-major axis
$$\rm AE$$
Num. eccentricity

Distance
from the Sun
$$\rm AE$$

Velocity
Current
$$\rm \frac{km}{s}$$
Min./Max.
$$\rm \frac{km}{s}$$
Average
$$\rm \frac{km}{s}$$

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

## Kepler's third law

$$\dfrac{(T_1)^2}{(T_2)^2} = \dfrac{(a_1)^3}{(a_2)^3}$$ The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

This law shows that planets farther away from the Sun need longer to orbit the Sun than planets near to the sun. For example our Earth needs 365 days for one orbit. However the planet Neptune which is very far away from the Sun needs about 165 years.

Mars
Semi-major axis
$$\rm AE$$
Orbit duration
Distance
from the Sun

$$\rm AE$$

Erde
Semi-major axis
$$\rm AE$$
Orbit duration
Distance
from the Sun

$$\rm AE$$

### Sources

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