A planet's orbit is not a circle but an ellipse. And the central star does not sit at the centre of that ellipse; it sits at one of the two foci. This is Kepler's first law.
An ellipse is the curve on which the sum of the distances from the two foci is everywhere the same. Add the two lengths in the figure and the total is wherever the planet happens to be. That is the semi-major axis, the quantity that expresses the size of the orbit.
There is nothing at the other focus. No partner sits there pulling; it is simply how the shape of the orbit is defined.
Follow the planet with your eye and you notice that its speed varies with position. Near the central star it is fast, far away it is slow. This is where it differs from uniform circular motion. From the next scene we look into how it varies.
Draw the line joining the central star to the planet. As the planet moves, that line sweeps out a fan-shaped area.
The area swept in a given time is the same wherever the planet is on its orbit. This is Kepler's second law, the law of equal areal velocity.
The two shaded regions in the figure are both swept in the same length of time. The fan near the central star is short and wide, the far one long and narrow. The shapes are utterly different, and the areas are equal.
If the area holds while the shape changes, then what is changing is the distance travelled. Near in, a long arc is covered in a given time; far out, a short one. That is what it means to say the planet is faster nearer in.
Let us put the swept area into an equation. In a short time the planet travels a distance , and the area swept can be taken as a thin triangle on that base.
The area of a triangle is × base × height. The base is , and writing for the angle between the radius and the velocity, the height is , so .
Divide by the time and the areal velocity is . That this value is the same everywhere on the orbit is the second law in the form of an equation.
If falls, must rise for the balance to hold. That speed and distance are tied together in inverse proportion is contained in this one equation.
In practice it is nearly always the nearest and farthest points that are used. At those two places the radius and the velocity meet at a right angle, so and the areal velocity takes the simple form .
Setting the two equal gives , that is, . The ratio of the speeds is the inverse of the ratio of the distances.
The lengths of the two arrows in the figure are the ratio of the speeds. If the near distance is half the far one, the speed there is twice as great.
This equation alone cannot be solved with two unknowns in it. Pairing it with the conservation of mechanical energy is the standard method in examinations. How the potential energy is defined is treated in the lecture on escaping for good.