Unit 3 - Cartesian Equations for Ellipses

Cartesian Equations for Ellipses

It is hoped that you are already familiar with certain equations (if not please discuss these with your teacher):

• the equation for a circle centered at (0;0) and with radius r.

• the equation for a circle centered at (a;b) and with radius r

• the equation of an ellipse centered at (0;0) and with semi major axis of length a and semi minor axis of length b

• the equation of an ellipse centered at (h;k) and with semi major axis of length a and semi minor axis of length b

What we will be doing in this unit is using either a graphing calculator or another graphing utility to graph a number of orbits.

NOTE: For the purposes of this activity we will ignore inclination - that is we will assume that all orbits are in the Earth's Equatorial plane. It is perfectly acceptable to make these simplifications as we begin our study - mathematicians often make simplifications when they begin their study and in time they reintroduce the factors they ignored at first.

Task 1

In Spacewarn Bulletin 467 (September 25, 1992) we read that:

SATCOM C3 a U.S.A. communications satellite, was launched by an Ariane-4 rocket from Kourou in French Guiana. Initial transfer orbit parameters are period 627 min, apogee 35,705 km, perigee 197 km and inclination 6.9 deg.

Assuming the Earth to be circular and given the radius of the Earth to be 6000 km. Plot an outline of the Earth and of the SATCOM C3 orbit on the same screen.

Some remarks:

• Since we are dealing with an Earth centered orbit let us place the center of the Earth at the origin and let us assume that apogee is on the negative x-axis.
• NOTE - you will find that the use of the words apogee and perigee are not uniform in the literature you will come across. In common usage the words are used to refer to either the position of the point or the distance to the point. In the data provided above apogee and perigee are more accurately the apogee and perigee altitudes, i.e. the distances from the surface of the Earth to the points.
• The first task is to translate the given information into the information needed to plot the curves - that is we need a, b and c to plot the orbit of SATCOM C3. For a hint on finding a, b and c click here.

Did your sketch look like this?

Are there any aspects of the orbit that surprise you? For example it would seem as if the orbit almost intersects the Earth, explain this.

Task 2

• Visit the Spacewarn Bulletin homepage, and select an issue from each of the years 1997, 1996, 1995, 1994 and 1993.
• For each issue take down the apogee and perigee data for one launch (found in section B Text of Launch Announcements) as well as a short description of the vehicle launched taking into account the following:
  • You will notice that more vehicles than not have almost circular orbits - i.e. their apogee and perigee are very similar - for some of your data try to find vehicles whose apogee and perigee are significantly different!
  • Try to find at least one case where the apogee and perigee are given for BOTH a shuttle that launched a vehicle and for the vehicle that was launched and then plot both orbits on your graph (I have not yet found such data). Failing this try to find a shuttle that docked with the Mir space station and find the corresponding Mir data by searching the Mir page.
  • Select data for at least one shuttle.
• For each of your five sets of data draw a sketch of the Earth and the orbit of the vehicle as above.
• For one of your data sets write a short description of the mission purpose - that is write more than what is given in the Spacewarn Bulletin. Such information is available for the shuttles by visiting the NASA Space Shuttle Mission Chronology page. Your description should be written as if you are the NASA representative about to inform the press about the mission at an upcoming press conference (and should include your sketch).