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. [Illustration: _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, 2_a_, 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√((_a_³)/(_k_)), where _k_ again was (using Newton’s work) essentially the mass of the earth. [Illustration: _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 _t_₁ and _t_₂ (see _Figure 6_). Then another of Kepler’s laws says that the ratio between the time difference _t_₂ - _t_₁ and the period _T_ equals the ratio between the sector of the ellipse OP₁P₂ and the area of the entire ellipse.