ELLIPSE

as Planet’s Trajectory.

by ‘Literka’

See applet about planet's motion and Keppler's law – it may be helpful for this page.

Let us write the equation of ellipse with the center at (0,0)

Graphically it looks like this

It is an ellipse with half of length of major axis a=BO=OD and half of minor axis b=AO (we assume a>b).

It has two focuses E and F on the X-axis at the distance

from the center of coordinates.

It follows that the distance between focus and the most upper point of ellipse A is equal to a,

EA=FA=a.

Let us remind that the smallest distance from a focus to the point of ellipse is called periapsis, the largest distance – apoapsis. In our case these distances are achieved on X-axis, p=apoapsis=BE q=periapsis=ED.

From the above picture it follows that the point H belongs to the circle of center at (0,0) and radius a.

Hence an angle BHD is the right angle (90 degrees), so HE is the height of a right triangle and because of this

**HE²=BE*ED**

But HE=b and we see that

Theorem. Let p and q be periapsis and apoapsis of an ellipse.

Then b is the geometrical mean value of p and q,

(b is half of minor axis).

Then a is the arithmetical mean value of p and q, a=(p+q)/2 (a is half of major axis).

This theorem may be used to check if a point is a focus of ellipse.

There are several reasons causing that focuses of ellipse are important. Let us state three of them

Take a point A of an ellipse and focuses E and F of this ellipse. Compute the sum s=AE+AF (the sum of length of intervals). Then s is always the same and is independent of the choice of A. In fact s=2*a.

Additionally let us notice that the distance from a point A of an ellipse to a focus is a linear function of the x-coordinate of this point A.

Push a ball from one focus of an ellipse. It will bounce from the curve of ellipse and then it will cross another focus.

The most important fact for us will be the fact that an ellipse is a trajectory of a planet moving around a bigger planet, which is in a focus of this ellipse and this bigger planet is assumed to be motionless.

Let a planet V move along an ellipse (its trajectory). Let us notice that this ellipse can be parameterized by parameter t, which is time. Value of parameterization at time t – it is a position of V at time t.

Notice. ‘Literka’ strongly advice you to see a new applet prepared for better understanding, even if you familiar with the following definitions and remarks. See applet about planet's motion and Keppler's law and whole text will be more understandable.

Let us start with other parameterizations. The most obvious parameterization of the upper half of an ellipse is

and x=t.

Here t is a parameter.

Definition.

Radius vector is a vector starting at the center of coordinates and ending at some point of ellipse.

Another parameterization is x=a*sin(t), y=b*cos(t).

Here t is a parameter which is an angle between the positive part of X-axis and a radius vector of point of ellipse.

A parameterization in polar coordinates is given by

Here t is an angle defined as before and r is the length of radius vector, 0<e<1.

Unlike before, the center of coordinates is a focus of this ellipse. Denote by F another focus of this ellipse.

This parameterization reminds a motion of a planet assuming that it rotates around a planet placed at F. In fact, people are making mistakes thinking that this parameterization describes a motion of a planet. These mistakes are coming from a simple consideration suggesting that the Keppler’s law is preserved. Assume that the center of coordinates is the center of motionless planet. Then the radius vector is a vector starting at the center of motionless planet and ending at the center of the second planet. This law states that

Keppler’s Law of Equal Areas

Radius vector of a planet sweeps out equal areas in equal intervals of time.

Unfortunately, precise calculation shows that the above motion is not a motion of a planet, except for some special cases. The simplest way to check it is to compute angular momentum. Angular momentum is not a constant function in the above case, hence the result of calculation contradicts the law of preservation of angular momentum. This contradiction shows that motion is not a motion of a planet.

More or less, Keppler’s law states that in small and equal intervals of time changes of angles of radius vector are in the inverse proportion to the square of the length of radius vector.

For us also the inverse theorem is important:

Suppose that trajectory of a motion (clockwise or counterclockwise) is an ellipse having a focus at the center of coordinates. Assume that Keppler’s law is preserved (which means that radius vector sweeps out the same areas in equal intervals of time). Then this motion describes a motion of some planet (rotating around another).

Equivalently we can say that if a trajectory of a motion is an ellipse and the law of angular momentum is preserved with respect to a focus, then this motion describes a motion of some planet.

Notice. In the above statements we claim only that construction of such two planets is possible. They may be not real ones. If we say about motion of a planet here, we always assume that this planet rotates around another motionless planet and nothing else is disturbing this motion. Moreover we assume that always this motion is clockwise or always counterclockwise.

Simple computations show that for the last parameterization we have:

Let us turn to a parameterization of an ellipse by a motion of a planet.

‘Literka’ could not find such a parameterization of an ellipse. However, ‘Literka’ will describe time of a motion as a function of the X-coordinates. This can be regarded as a parameterization of an ellipse, but not in a direct way.

First, let us make some introduction. Consider a function t=f(x), where t is time, when a planet is in a position of X-coordinate equal to x. We assume that trajectory is an ellipse with a center at the center of coordinates. ‘Literka’ constructed examples of such functions, the graph of one of them looks like this

Nothing can be deduced from this graph. However, let us construct the other functions:

g(x)=[f(x)+f(-x)]/2, h(x)=[f(x)-f(-x)]/2

The function g is an even function, the function h is an odd function. Moreover f(x)=g(x)+h(x).

In fact f, as well as all function with symmetric domains with respect to point 0, can be written in a unique way as the sum of an odd and an even function (and g(x), h(x) give such split). Look at the graph of g and h:

First graph looks like upper half of an ellipse, the second one as an inverse function to the function sin(x). Some computations show that it is so. In fact

where sin^-1 means the inverse function to sin(x), a and c were defined before, C is a positive constant.

This function gives a parameterization of an upper part of an ellipse and this parameterization is defined by a motion of a planet. Equation shows the dependence of t with respect to x, hence also dependence of x with respect to t. Since y is a function of x, equation also describes the dependence of y with respect to t. The constant C cannot be found in a unique way – it depends on the mass of a planet being in the focus of an ellipse.

Notice that for c=0 it describes a motion of a planet along circle. In this case the motion is uniform (constant speed but not velocity vector).

Notice. ‘Literka’ was interested in motions of a planet because previously it wanted to write a software, which was supposed to be a game about navigating of a rocket in a space around Earth. After a while ‘Literka’ decided to terminate writing. Guys didn’t want to play this game, because often rocket was “escaping” and it was hard to turn back (to turn back they had to rotate a rocket about 180 degrees and start engines). Probably this game will never be finished, but even now it shows some interesting features. Among them – you can turn off engines and see an orbit of a rocket (in this game it is always an ellipse). At any time you can see the velocity, speed, distance from Earth, acceleration due to engines work etc.

If you are interested, contact ‘Literka’ at the mailing address given on the main page of ‘Literka’. Please, keep in mind that ‘Literka’ has only a limited access to the Internet.

Return to the main geometrical page of ‘Literka’.