Unit 4 - Polar Equations for Ellipses

Remarks

Some of you may have experienced some problems with sketching your graphs in the tasks of unit 3 since your graphing devices may have wanted you to give the equation in the form y = ..., that way you would have ultimately had to draw two graphs (equations below for unit 3 - task 1) to get the illusion of the orbit and the Earth in your sketch.

or (Unit 2, task 1)

The Cartesian representation of the ellipse has one further shortcoming for the application of space flight. In space flight and in particular the scenario we have sketched, it is critical to know exactly where the vehicle is at any point in time (both if you are trying to avoid it and if you are trying to meet it!). The Cartesian representation has the limitation that for any value of x there are two corresponding values for y and for every value of y there are two corresponding values of x (with the exception of the apogee, perigee and nodes), so in the application of space flight even knowing one of the coordinates and given the representation of the orbit in Cartesian form only narrows the position of the vehicle down to two possible place (which are more than likely very far apart!).

One way of addressing this matter is to use a different representation for the ellipse, there is one that we will consider, the polar representation.

Polar Coordinates

We are familiar with finding a point in the plane by specifying its rectangular coordinates (x, y). In space flight (and other applications) for reasons including those sited above it is often more useful to locate points by means of their polar coordinates. The polar coordinates of a point give its position relative to a fixed reference point O (the pole) and a given ray beginning at O (the polar axis). To start with let us consider the familiar x-y plane and let O be the origin and the positive x-axis the polar axis, then the point P with polar coordinates r and t (where t is often given as the angle theta) written (r, t) is located as follows:

• First find the angle t (measured counter-clockwise) with the positive x-axis as one side, then
• Determine P as follows:
  • if r is positive then P is simply the point a distance r from O measured along the side of the angle,
  • if r is negative then P is the point a distance r from O measured along a ray opposite to the side of the angle.

It may (or may not be interesting) to visit a discussion of trigonometric graphs drawn using polar coordinates that I prepared during a class that I took at the University of Georgia (visit the discussion).

Polar Representation of the Ellipse

We are now ready to describe the ellipse by means of polar coordinates. To do this we will use the second definition used in unit 2 namely:

A conic section with eccentricity e is the locus of a point P whose distance from the focus (F) is e times its distance from the directrix.

• Furthermore the conic section is an ellipse when e < 1, a parabola for e = 1 and a hyperbola for e > 1.

We begin by finding the general equation for a conic in polar coordinates:

Note when the directrix goes through (d; 0) the equation becomes:

If we now consider the figure alongside for the ellipse (with e<1) and use the equations derived above then:

for t = 0 we get and t = pi we get

furthermore which implies

and we get the following equations for an ellipse with eccentricity e, major axis a and focus at the origin:

or

Student Task

• What remains to do before we can use the equation just derived is to see if we can find the eccentricity e for an ellipse using no more information than we previously used to draw the ellipses, that is given the apogee and perigee alone.
  • Certainly we know that e = c/a (where c is the distance from the center to the focus and a the length of the semi major axis), and as we have already seen c can be established easily
  • Consider the diagram alongside with A the distance from the focus to the point of apogee and P the distance from the focus to the point of perigee.

Determine e in terms of A and P alone.

• Return to the data you gathered in unit three and redraw the five ellipses using your graphing utility but this time in their polar form. You will notice that your graphing utility will most probably ask you to give a range of values for t (the angle) - we will use this range to our advantage in the next problem.
• We will be looking at orbit transfers in more detail in the next unit, for the moment, however, consider only the following data and draw the orbits.
  • In the Space warn bulletin (499) we read that: SPEKTR, a Russian module to be docked with MIR station, was launched from Baykonur cosmodrome by a Proton-K rocket at 05:33 UT. The 23.5 tonne spacecraft will remain with MIR for at least three years. Initial orbital parameters were period 89.8 min, apogee 337 km, perigee 221 km, and inclination 51.7 deg.
  • We now add the following: after maintaining this initial orbit for four revolutions (t = 8 pi), the SPEKTR will use its thruster engines to change its orbit to a new orbit with parameters: 93.5 min, apogee 942 km, perigee 221 km, and inclination 51.7 deg.
    • When you have tried this problem click here for a look at my results.
• Be sure that you understand the derivation of the polar form of the ellipse, so that you could explain it to a group of students who will be visiting you in the navigation department of NASA in a few days time.