Kepler's laws of planetary motion. First law. The planetary orbits are conic sections. The kind and the shape of the conic section depend on the initial parameters of the planet's trajectory.

One may vary the absolute value of the initial velocity, keeping its direction and the solar distance constant.

Here, the Sun is in the left focus of a small ellipse. A certain velocity results in a circular orbit.

These examples show that even if all motions follow from one differential equation, each single trajectory is dependent on the initial position and initial velocity of the planet. Scale variation 1 to 2.

Here, a somewhat elongated ellipse develops with the Sun in the focus on the right. It can clearly be seen that the planet moves faster near the Sun than far from it. Starting with still higher velocity, the planet eventually leaves the solar system on a hyperbole, here compared to an ellipse.

Again, the starting direction makes a right angle with the line connecting Sun and planet. However, this angle may also be acute, for instance, plus 60 degrees. In this case, a much more elongated ellipse results.

Here, both can be compared. Obviously, the same is true in the other direction, that is, minus 60 degrees. The absolute value of the velocity is still the same, only the directions vary.

Again, a flat ellipse results, here compared to the first one. The third parameter to be varied is the initial solar distance. In this and the following examples, the initial velocity vector is kept constant. In the first example, the Sun is in the right focus.

The next orbit is circular. If one starts the trajectory at a still greater distance, again an ellipse results, but the Sun is now in the left focus.

Increasing the initial distance further eventually results in a hyperbolic trajectory.

Kepler's second law, the angular momentum with respect to the Sun is constant. This means that the position vector of a planet relative to the Sun sweeps out equal areas of the ellipse in equal times.

These areas are shown here. A modern physicist generally formulates this law differently. The angular momentum of the planet with respect to the Sun is constant. This follows from the existence of a central force between Sun and planet.

Kepler's third law tells us something about the periods of planetary revolution. The squares of the periods of revolution are proportional to the cubes of the semi-major axes of planetary orbits.

Here, the ratio of the two periods is two. For the ratio of the semi-major axes, one gets two to the power of two-thirds, or cube root of two squared. The planets are moving with corresponding velocities. To make the situation clearer in the picture, they are stopped after each revolution.