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