Polar Curves and their Moving Frames

Now we will study satellite motion using polar coordinates. We will show in Harmony of the Spheres that for a single satellite, ignoring all others, if we could assume the Earth did not move, then the satellite must move in a plane that contains the Earth if it is governed by Newton's gravitational law. For now, we will simply assume that the satellite moves in a plane. Of course, Newton was interested in the motion of the planets around the Sun, but the principle will be the same.

The effect of the satellite on the Earth is, however, not entirely negligible -- especially if that satellite is the Moon! The Earth does wobble just a little because of the satellite's pull. Newton therefore imagined a fictitious "center of mass" of the satellite-Earth system (recall our hockey puck experiment in the introduction) and showed that his slightly modified equations with that center of mass at the origin (instead of the Earth) gave the Keplerian prediction of elliptical orbits. Now this wobble is in any case so vanishingly small for Earth that we may safely ignore it in our calculations, and we will. I only mention the correction for completeness.

Polar Functions and Curves

So we may use a 2-dimensional polar system of coordinates to describe the motion of the satellite around the Earth. We will, naturally, place the Earth at the origin of this system of coordinates. For the planets orbiting the Sun, this polar plane is essentially the observed ecliptic. We therefore assume that the "satellite," moving under the influence of gravity alone, traces a plane curve parametrized by an angle q,

We will write instead of the more correct (but pedantic) . We do not know what the polar function r(q) is. Our job will be to find it. We will prove later using Newton's law, that if we assume the satellite is not moving on a straight line away from or towards the Earth, at a single point in time, then it will never move that way at any instant of the past or future, and in fact, r(q) will never be 0.

The next series of pictures and equations will sum up the kinematics of our problem. That is, assuming only that the planet moves in a plane, and making no assumption at all yet about the nature of the accelerations, we will say all that we can say about the motion. After that, we will introduce the dynamics of the problem. It is only when we reach that point that the role of the gravitational force comes into play. So, until we say otherwise, all remarks below are quite general, and do not depend on Newton's gravity. But remember that the fact that motion is in a plane will depend on the property of the gravitational law that the force of attraction (essentially the acceleration) is always towards the Earth.

Now the trick is to define a new polar curve, u(q) in the following way. For each angle q, there is a point along the curve at that angle. Its polar coordinates are: , and its Cartesian coordinates are:

where

If we connect this point to the origin (the Earth) by an imaginary line, that line intersects the unit circle, the set of points at distance 1 from the origin, in a unique point (since r(q) is never 0). This point, for each q, is denoted . So the curve is the projection of the curve (x(q),y(q)) onto the unit circle. We then have an important and rather obvious identity for

the Cartesian curve that describes the planetary orbit as a function, not of time, but of q.

If we represent the polar curve also in Cartesian coordinates then

We represent the point on the unit circle with the symbol: , which is, apparently, nothing but a "wrapping function" discussed in most trigonometry texts,

so we may write, finally:

The important point here is that has two Cartesian components for each angle q. It is a point on the unit circle.

We are also assuming that we can actually use to parametrize the curve . It is important to know that as time increases, increases. The planet does not "back up" as Ptolemy thought. Actually Kepler's second law indicates that this should be so. This is why.

as functions of time. Thus (as you will show below)

Now,

so, since is a function of time, it is clear that if became 0, then at that time, would be 0 also.

But then we would have

and so velocity would be a multiple of position. But we have ruled this possibility out. So from now on, differentiation will be with respect to the variable q, rather than with respect to time. In particular, we may calculate the derivative of .

by differentiating each coordinate separately as we do for all curves. This gives a new curve that we shall call

Question 1: Show that

and also show that for each , is perpendicular to .

End of Question

Moving Frames of Polar Curves

Summing up Newton's observations so far, we have plane curves, parametrized now by the angle q,

all associated with our satellite. Our job is to learn what the function r(q) is.

Question 2: Show that if is a real-valued function of , and if is a plane curve, then we may define a new plane curve:

by multiplying each of the coordinates of by for each . Explain carefully why we have an analog of the "product formula"

End of Question

The pair of vectors are perpendicular to each other, and each has length 1. They define a "moving frame" along the curve . Imagine that you ride along the orbit, and at each point , you construct the perpendicular pair: (blue) and (dark red) at that point, as pictured below:

The bright red vector is the "velocity" as we traverse the curve by covering equal angles in equal times in the counterclockwise sense.

Now Newton's ingenious idea is to use this "moving frame" as our reference for the measurement of velocities. We measure the components of velocity with respect to this rotating frame at each "time" . Now satellites generally do not traverse equal angles in equal times. Kepler's second law says that they do something more interesting, as we shall see. So we are not pretending that this curve, parametrized as it is, by , represents an actual satellite orbit on this page, although our geosynchronous satellite will behave that way. On the next page, we will simulate more natural satellite orbits using Newton's equations! Here, we just want to present the basic idea.

Now, if we represent the velocity at the point above in this moving frame we will see a picture like this:

Imagine then, that as you ride the orbit, you record how the velocity (bright red vector) changes as changes. We will trace the curve that makes in the exercise on this page. You will be able to provide your own examples, but in general it will have no easily recognizable form:

This is not so with real satellites, as we shall see.

Now, with these pictures understood, and with the ones you will create yourself, we are prepared to derive some new relations for this velocity curve.

To make the analysis more realistic, and to prepare for the study of real satellites, we let q itself be an increasing function of time t. That is, for the satellite, the angle q increases with t. So we may write the curve that describes the planet's position also as

Differentiation with respect to time, t, will now be denoted by a "dot" rather than a "prime", but remember that

First, let us calculate the velocity,

This is why. We calculate

using the chain rule and our generalized product rule derived above. This is

but by an earlier definition:

Thus, in the coordinates of our moving frame:

Acceleration of a Curve in Moving Frame Coordinates

Now we differentiate again.

Question 3: Show that the acceleration of the planet is given by

(0.1)

or, in the coordinates of our moving frame:

Do the detailed calculation using the chain rule as above. The first coordinate:

is called the "radial" component of acceleration.

For example, if r is constant and is constant (we are not considering gravity yet) then is the speed

You may recognize that this radial component then becomes the entire acceleration.

which is called the "centripetal acceleration" for uniform circular motion. This will in fact represent the motion of our geosynchronous satellite that moves at constant speed and at constant distance from the Earth. The other terms have similar interpretations.

End of Exercise

This finishes the discussion of the kinematics of satellite motion. On the next page, we will begin to study the dynamics, following Newton, and assuming that the motion is governed by his gravitational law.

Exploration: Graphing polar curves

On this page, you are invited to experiment with a few of these ideas. First, use the control panel

to define a polar curve: . We use A for the angle, here measured in radians.

Initially, we supply R = 2/(1+0.5*cos(A)), which, as you may recall from the previous page, is an ellipse with the origin at one focus. This ellipse has eccentricity 0.5. If you would like to work with more exotic polar curves, just enter the curve in the top yellow field. You may also specify the interval on which A is defined.

The curve will be traversed in equal steps in A, and you can define the size of the step in the Step for A field.

Next, press Graph Curve to see the curve in the large Graph2D on the right. You may now watch the frame move around the curve by repeatedly pressing the Plot Next Point button. The bright red vector is the "velocity" vector: . On this page, .

In the left hand Graph2D you will see the curve referred to our moving frame: traced out as you step along. It might be interesting to examine this for a circle, given, say, by R = 2. Now change it to a slightly eccentric ellipse, say, 2/(1+0.08*cos(A)). Do you see what is going on? We will have much more to say about this picture on the next page, when we begin to study the dynamics of satellites.

Now, if you want to speed things up, just press the Animate ! Button. and watch the show!