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.