Calculating the Orbit of Telstar I

Figure 4

In Project Telstar we had to calculate the satellite’s orbit from observations made by our precision trackers. This introduced a few problems in addition to the ones we encountered with Project Echo. In the first place, the orbit of the Telstar satellite is a elongated ellipse, as indicated in [Figure 4], rather than being almost circular, as in the case of Echo I. We mentioned earlier that a precision tracker can furnish data on a satellite’s elevation angle, E, and azimuth, A (see [Figure 1]). It can also give us a reading for ρ, the distance from the tracker to the satellite ([Figure 4]). If we know the position of the tracker on the earth, we can reduce the quantities A, E, and ρ to the angle θ and the distance r (measured from the center of the earth to the satellite). These two quantities locate the satellite in the plane of its orbit, but in order to describe its position completely we must also specify this orbital plane. In [Figure 5] the orbital plane is shown as a shaded surface, with θ and r being the same as before. You will recall that the line OM represents the intersection between this plane and the equatorial plane; we call the angle i between the two planes the inclination of the orbit. Finally, we have the angle Ω between the line OM and some line OA to the point A, which we can choose as any convenient spot in the equatorial plane. Now we have specified the orbital plane completely. The point A can be found from day to day by fixing its position relative to a certain star in the sky.

Figure 5

Figures [4] and [5] tell us something about the geometry of the satellite’s position in space, but for the complete story we must also give the time at which it can be found there. For this purpose, there are some astronomical laws that relate position on an elliptic orbit to time. Two of these are illustrated in [Figure 6]; in looking at this figure, you should imagine that you are standing off to one side of the orbital plane to get a good view of the entire orbit. The longest dimension of the ellipse, 2a, is called the major axis; this dimension is related to the satellite’s period—the time it takes to go once around the ellipse. More than three hundred years ago the astronomer Johannes Kepler observed that the period T, of an ellipse is

T = 2π√

a³
k

where k again was (using Newton’s work) essentially the mass of the earth.

Figure 6

Instead of a complete revolution, we may only be concerned with part of one orbit. Let’s say that this part lies between the two positions P₁ and P₂ that the satellite occupies at the two times t₁ and t₂ (see [Figure 6]). Then another of Kepler’s laws says that the ratio between the time difference t₂ - t₁ and the period T equals the ratio between the sector of the ellipse OP₁P₂ and the area of the entire ellipse.

Now let us see how we can use the quantities r, θ, i, and Ω as well as Kepler’s two time laws to determine the motion of the satellite in space. Suppose that we have made observations of the Telstar at two times t₁ and t₂ and that we have measured its distance along lines ρ₁ and ρ₂ in [Figure 7]. In other words, we know that at these two times the satellite was at the points P₁ and P₂. Since three points determine a plane, we know in this case that P₁, P₂, and O define the satellite orbital plane. Knowing this, we can now calculate the angles θ₁ and θ₂, the distances r₁ and r₂, and the angles i and Ω. (The detailed formulas for this are derived from analytic geometry.)

Figure 7

However, we still do not know the length and the width of the particular ellipse the satellite is following and how this orbit is oriented within the orbital plane. Let us imagine again that we can stand off to one side of the orbit and take a good look at it; [Figure 8] shows us what we would see. There are the two points P₂ and P₁ at which we have observed the satellite. We know the positions of these points relative to each other and in relation to the center of the earth, because we have already calculated r₂, r₁, θ₂, and θ₁, But any number of ellipses could be made to pass through these two points. Some might be very large, others might be so narrow that they would intersect the earth and thus be impossible. However, only one of these ellipses will satisfy the time difference that we observed between P₁ and P₂. In other words, the shape and period of this particular ellipse must be such that it will cause the satellite to pass through P₁ and P₂ in exactly the time interval t₂ - t₁. If we work out our time formulas, we will convince ourselves that there is only one such ellipse. When we have found it, we have determined the orbit of the Telstar satellite from the two observed positions and times.

Figure 8

Figure 9

In principle then, we could keep track of the Telstar satellite by making a pair of observations P₁ and P₂, and then predicting ahead a short segment of an orbit that is the ellipse we have computed. After a while we must verify this ellipse with two more observations P₃ and P₄, predict ahead over another segment, and verify again with P₅ and P₆ (see [Figure 9]), The reason we have to keep taking new measurements is that the elliptic orbit does not remain the same. As we discussed in connection with Echo I, the orbital plane will “wobble” about the earth because of the equatorial bulge. We also know that the orbit’s major axis will revolve within the orbital plane. As we have seen before, these effects are small and can be represented by appropriate mathematical formulas. If we calculate them, we will see the connection between one pair of observations and a later one, and eventually we can increase the time interval between successive pairs of observations. There are also mathematical formulas that we can use to predict the position of the satellite for many revolutions in its elliptic orbit.

In order to predict orbits successfully, we must also realize that the measurements we obtain from a precision tracker, such as the angles A and E and the distance ρ, are always subject to small inaccuracies. Thus it is not really possible to take just two measurements like P₁ and P₂ and determine a satisfactory orbit from them. In reality, our tracker takes many readings, and these are averaged to give adequate information about the orbit. Therefore, the picture we have in mind is not quite like [Figure 7], but rather like [Figure 10]. Here the trackers have established a series of points that are somewhat scattered, and by taking averages we can calculate an orbit that passes through them in a smooth fashion.

The trackers we have mentioned so far have given us azimuth and elevation angles and also the distance to the satellite at every instant. Sometimes we must use simpler instruments that do not yield all this information. They might, for instance, only give us the two angles. The mathematics of calculating an orbit from such measurements is somewhat different, but the process is fundamentally the same as we have discussed here.

When you do these calculations for the Telstar satellite from one day to the next—and especially if you have more than one satellite to keep track of—the amount of work will become quite large. Nowadays our calculations are done for us on electronic computers, which both receive information from the tracking instruments automatically through Teletype or DataPhone channels and send back information concerning future positions of the satellite to the ground stations. There are still quite a few problems to be solved, and we are presently working on ways of making all this equipment perform the orbit predictions for the Telstar satellites automatically and efficiently.

Figure 10

Franz T. Geyling was born in Tientsin, China, and received a B.S. in 1950, an M.S. in 1951, and a Ph.D. in 1954 from Stanford University. He joined Bell Telephone Laboratories in 1954, and has been engaged in celestial mechanics studies of rockets and satellites, as well as stress analysis of submarine cables.

CASE HISTORY NO. 2
What Color Should a Satellite Be?

Peter Hrycak
Mechanical Engineer—Member of Staff, Electron Device Laboratory