[Illustration: _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.
[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.
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 _t_₁ and _t_₂ 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 _r_₁ and _r_₂, and the angles _i_ and Ω. (The detailed
formulas for this are derived from analytic geometry.)
[Illustration: _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 _r_₂, _r_₁, 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 _t_₂ - _t_₁. 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.
[Illustration: _Figure 8_]
[Illustration: _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.
[Illustration: _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_