January 24, 1964 Electronics
[Table of Contents]
Wax nostalgic about and learn from the history of early electronics.
See articles from Electronics,
published 1930  1988. All copyrights hereby acknowledged.

One of the major advantages
of the age of powerful personal computers  be they in the form of desktop systems,
tablets, or smartphone apps  is that for most computationintensive tasks there
only needs to be one or maybe at most a few people smart enough to know how to do
them. Everyone else who has to perform the task just needs to be able to input the
proper parameters to ensure a useful output. That is a significant statement, because
in the days before ubiquitous computer availability and incredible computing power,
highly capable engineers, scientists, analysts, and mathematicians either had to
be on staff or an expert external resource was used for difficult and/or timeintensive
tasks. Over time, fewer and fewer people are needed to produce very precise and
reliable results. In many ways, other than the creative intuition involved in concept,
creation, and execution, a large part of the product design and planning phases
have been automated with the help of very capable software. This 1964 Electronics
magazine article about plotting of satellite ground coverage maps is a good example.
Plotting Coverage Circles for Satellite Communications
Fig. 1  Coverage of a ground station located at 55 degrees
north latitude.
Graphical method solves most circularorbit cases to determine possibility of
communications.
By Daniel Levine and William H. Welch
Lockheed Missile and Space Co., Sunnyvale, California
Circle of Influence
Continuing announcements of communication satellite plans give point to an article
that shows how, with simple computation and available charts the planner, engineer
and even the space hobbyist can determine the coverage circle of a ground station
for a satellite in a nearly circular orbit
Plotting the coverage circle of a ground station that is in communication with
a satellite is a tedious task because of the extreme distortion of many map projections,
as illustrated in Fig. 1. A graphical technique has been developed that greatly
facilitates this operation. It involves drawing the coverage circle as a semicircle
on a work sheet and then transferring coordinate values to the map projection in
use.
For this purpose, the surface station is taken to be located at 0 degrees longitude
and Φ_{s} degrees latitude. The radius of
its coverage circle subtends the angle a at the center of the earth. The latitude
and longitude, Φ and λ, of points on the circumference of this circle satisfy the
equation^{1}
cos λ = (cos α  sin Φ_{s} sin Φ)/cos Φ_{s}
cos Φ
Rather than solve this equation for specific pairs of coordinates, the engineer
can use a compass to draw the coverage circle on the work sheet of Fig. 2A
or 2B, depending upon the station coordinates. Figure 2A presents a polar stereographic
projection of the earth and Fig. 2B shows a meridional stereographic projection
of the earth with the pole of the projection centered on the equator. A circle drawn
on the earth about a point representing the station appears on the stereographic
projection as a circle with its center displaced in latitude from the ground site.
Fig 2.  POLAR stereographic projection (A) stereographic meridional
projection (B) and representative radius scale (C).
The central angle of the coverage circle a can be determined from Fig. 3
for a particular elevation angle above the horizon for a satellite in a nearcircular
orbit at an altitude of H nautical miles. For example, a central angle of 29 deg
occurs for 0 deg elevation angle when the orbital altitude is approximately 490
nautical miles.
Example  The circle drawn on Fig. 2A is for a station
latitude of 55 deg and circular coverage α = 29 deg.
The procedure for drawing this circle is as follows
1. Station latitude Φ_{s} = 55 deg N.
2. α = 29 deg determines the northern and southern
limits of the circle on the zeroth meridian as 55 deg + 29 deg
= 84 deg N and 55 deg  29 deg = 26 deg N.
3. A caliper is set at the diameter length between 84 deg N and 26 deg N on the
zeroth meridian of Fig.2A.
4. The caliper diameter AD is marked on scale AB of Fig. 2C and the corresponding
circle radius AE is measured on the associated radius scale AC, as illustrated.
5. Measure the radius length AE from either the 84 deg N or the 26 deg N limit
on the zeroth meridian of Fig. 2A to locate the center of the circle on the
map.
6. The semicircle is drawn on Fig. 2A using the radius AE and the center
location of step 5 above.
The coordinates of points on the circle read from Fig. 2A are shown in Table
I. If the actual longitude of the station is 130 deg W, the table leads to the distorted
coverage zone shown in Fig. 1 for a Mercator projection.
When the ground station is nearer the equator, the semicircle may extend beyond
the limits of Fig. 2A. For this case, it should be drawn on the meridional
stereographic projection of Fig. 2B, using the same sequence of steps outlined
above. Both Fig. 2A and 2B are usable for stations in southern latitudes by
reading all latitude values as south, rather than north. When the central angle
exceeds 30 deg, a different graphical procedure may be employed.^{2} Neither
the technique described here, nor that of the footnote reference, applies for the
coverage in a highly elliptic orbit, since the limits on the surface of the earth
do not lie on a circle about the ground station.
Fig. 3  Central angle upon which coverage circle depends
is determined from graph.
Scaling Radius  Figure 2C provides a convenient means of dividing
a length (the measured diameter of the circle) by two to obtain a direct measure
of the radius. The figure comprises two lines that intersect at an angle of 60 deg.
This can be redrawn enlarged with intermediate ticks to facilitate accuracy.
Alternatively, the latitude of the center of the circle may be computed by means
of the relation
Φ_{center} = 90 deg  2 tan^{1} (cos Φ_{s})/(sin Φ_{s}
+ cos α)
Thus, for the example presented, Φ_{s} = 55 deg, α
= 29 deg, and slide rule computation gives Φ_{center}
= 52.5 deg, in agreement with the location on Fig. 2A.
Another simple means of locating the center of the circle is to measure the distance
between the northern and southern limits on the zeroth meridian, established in
step 2 of the example. Division of this distance by 2 then gives the linear measure
of the separation of the center from either of the established limits.
References
(1) D. Levine, "Radargrammetry", p 300, McGrawHill Book Co., New York, N. Y.,
1961.
(2)"The Microwave Engineer's Handbook and Buyer's Guide," p 154, 1963.
Coordinates for Example  Table I
Maps as Work Sheets  Table II
Since the Fig. 2A is a polar stereographic projection, the coverage circles
may be drown on any standard map that employs this projection. The following maps
are listed in order of decreasing scale factor:
(1) Scale 1:10,500,000, sixsheet wall map of the world, size 108 in. by 126
in., USAF Strategic Outline Charts, No. SO17.
(2) Scale 1:17,750,000, size 51 in. by 30 in., USAF Strategic Planning Charts,
Northern hemisphere  No. SP2; Southern hemisphereNo. SP3.
(3) Scale 1:24,000,000, size 38 in. by 36 in., Northern hemisphere USAF Strategic
Outline Chart, No. SO16.
(4) Scale 1:48,000,000, size 19 in. by 18 in., Northern hemisphere, USAF Strategic
Outline Chart, No. SO16a.
(5) Scale 1:63,000,000 at 60 deg. N. lat., Weather Plotting Chart, Northern hemisphere,
No. AWSWPC6631, (Aeronautical Chart and Information Center).
The meridional stereographic projection, Fig. 2B, also corresponds to a
standard projection, so the coverage circle may be drawn on any appropriate map.
The choice now is more limited than for the first figure. For this purpose, there
is available the National Geographic map of the Atlantic Ocean, scale 1:20,000,000,
size 25 in. by 31.25 in. (Sept. 1941).
For this chart, the circle is centered on the 40th meridian, which corresponds
to the vertical line on the left side of Fig. 28. Continental outlines and
other geographic features are ignored when using this map as a work sheet.
Posted November 29, 2023 (updated from original
post on 11/20/2018)
