  CHAPTER 24, Inc., MADISON, WI
SBE 24 GeoSat Pt1

## GEOSTATIONARY ORBITS PART 1:PHYSICAL PRINCIPLES

by Neal McLain, CSBE

This is the first of a series of articles about geostationary orbits; i.e., the orbits occupied by communications satellites which remain at fixed points in the sky. In this series, we will cover some basic physical principles, orbital geometry, antenna mounts, and pointing angles.

This first article deals with basic physical principles: time, the geographic coordinate system, and Kepler's Laws.

### TIME

The earth moves through space in two ways:

• The earth rotates about its polar axis. The time interval required for the earth to rotate exactly once is called one sidereal (sigh-DEAR-e-al) day.

• The earth revolves round the sun once per year.

The rotation of the earth about its axis causes the sun to appear to rise in the east and set in the west. At one critical instant during the sun's daily course across the sky, it reaches its maximum height; this instant is called solar noon.

The time interval between two successive solar noons is called one solar day. During one solar day, the earth rotates slightly more than once about its polar axis. Thus, one solar day is slightly longer (by about four minutes) than one sidereal day.

The reason for this phenomenon is explained in the following figure: Note that as the earth moves from Day 1 to Day 2, it revolves about 1/365th (or about one degree) of its annual orbit around the sun. Thus, the earth must rotate slightly more than once in order to reach local noon at the Day 2 position.

A solar day is subdivided into solar hours, solar minutes, and solar seconds:

• One Solar day = 24 solar hours.
• One Solar hour = 60 solar minutes.
• One Solar minute = 60 solar seconds.

Measured in solar units, one sidereal day = 23 hours 56 minutes 4.091 seconds.

### GEOGRAPHIC COORDINATES

Any point on the earth's surface can be specified by two geographic coordinates, called latitude and longitude. Latitude and longitude are measured in arc degrees, or simply degrees. One arc degree equals 1/360th of the circumference of a circle, and is represented by the symbol °.

Latitude is measured in arc degrees north or south of the equator. The equator itself is defined to be 0° latitude; the North Pole is at 90° north latitude, and the South Pole is at 90° south latitude. Lines of equal latitude are called parallels.

Longitude is measured in arc degrees east or west of the Prime Meridian at Greenwich, England.

The Prime Meridian itself is defined to be 0° longitude; points to the west of the Prime Meridian are called west longitude, and points to the east are called east longitude. Lines of equal longitude are called meridians.

The intersection of the 38th parallel north and the 77th meridian west is represented as follows:

```              38° North Latitude
77° West Longitude```

These are the geographic coordinates for a point near Washington, DC.

Divisions within an arc degree can be specified in either of two ways:

• By decimal notation, such as:
```              38.8978° North Latitude
77.0367° West Longitude```
• By arc minutes and arc seconds:
```              One arc minute (') = 1/60 arc degree.
One arc second (") = 1/60 arc minute.```

Using this notation, a point near Washington might be represented:

```              38° 53' 52" North Latitude
77° 02' 12" West Longitude```

The actual physical distance of one second of arc, measured on the ground, is approximately:

```              COORDINATE    METERS                 FEET
Latitude      30.8                   101
Longitude     30.8*(cos(latitude))   101*(cos(latitude))```

### SATELLITES

Satellite is the name given to any body which revolves around the earth in the space above the earth's atmosphere.

Satellites may be natural or artificial:
The earth has one natural satellite: the moon .

The earth's first artificial satellite, Sputnik 1, was launched by the USSR in 1957. In the years since, hundreds of other artificial satellites have been launched.

Satellites move about the earth in paths called orbits..

From the study of astronomy, we know three important facts about orbits. These facts, known as Kepler's Law's, were first set down in the early 1600s by the German mathematician and astronomer Johannes Kepler. Kepler defined these laws to describe the motion of the planets about the sun; however, these laws apply equally to the motion of satellites about the earth.

### KEPLER'S FIRST LAW

Kepler stated the first law as follows: Each planet moves in an ellipse with the sun at one focus. For a graphic illustration of this law, click here.

More generally, this law can be stated as follows: Every orbit is an ellipse with the primary at one focus.

From the study of geometry we know that every ellipse has two foci. In the case of an artificial satellite moving about the earth, the earth is the primary, and the center of the earth is one focus.

The location of the other focus (the empty focus) depends on the shape, or eccentricity, of the orbit:

• If the orbit is long and narrow (high eccentricity), the empty focus is a point out in space.
• If the orbit is more nearly circular (low eccentricity), the empty focus is a point inside the earth.
• If the orbit is a circle (zero eccentricity), both foci merge to a single point at the center of the earth.

The Space Shuttle is an example of an artificial satellite with low eccentricity. When watching the NASA Television channel, one frequently hears Houston Mission Control say things like "an orbit 360 miles by 430 miles."

### KEPLER'S SECOND LAW

Kepler stated the second law as follows: The radius vector from the sun to a planet sweeps out equal areas in equal times.

More generally, this law can be stated as follows: The radius vector from the primary to a satellite sweeps out equal areas in equal times.

Stated non-mathematically, this law simply says: the farther a satellite is from its primary, the slower it moves. Again, the shape of the orbit is important:

• If the orbit is long and narrow (high eccentricity), satellite velocity varies over a wide range.
• If the orbit is more nearly circular (low eccentricity), satellite velocity is more nearly constant.
• If the orbit is a circle (zero eccentricity), the satellite moves at constant velocity.

The highly-eccentric orbit of a comet moving around the sun illustrates this idea. When the comet is moving toward the sun, its velocity increases: it's "falling" toward the sun. At the point nearest the sun, it whips around the sun (one focus), then moves off into space again, slowing as it moves. When it reaches the point farthest from the sun, its velocity is minimum. It passes around the empty focus (simply a point in space), then begins to "fall" toward the sun again, gaining speed as it moves.

### KEPLER'S THIRD LAW

Kepler stated the third law as follows: The relationship R3/T2 is the same for all planets, where:

 R = average orbit radius T = orbit period

More generally, this law can be stated as follows: For any given satellite system, the relationship R3/T2 is the same for all satellites.

The following table illustrates this law for the Solar System. Note the remarkable agreement among the figures in the R3/T2 column.

 PLANET AVERAGE RADIUS (kms) ORBIT PERIOD (days) R3 (kms)3 T2 (days)2 R3/T2 (kms)3/(days)2 Mercury 5.791E+07 8.797E+01 1.942E+23 7.739E+03 2.510E+19 Venus 1.082E+08 2.247E+02 1.267E+24 5.049E+04 2.510E+19 Earth 1.496E+08 3.653E+02 3.348E+24 1.334E+05 2.510E+19 Mars 2.279E+08 6.870E+02 1.184E+25 4.719E+05 2.509E+19 Jupiter 7.783E+08 4.333E+03 4.715E+26 1.877E+07 2.512E+19 Saturn 1.427E+09 1.076E+04 2.906E+27 1.158E+08 2.510E+19 Uranus 2.870E+09 3.069E+04 2.363E+28 9.416E+08 2.510E+19 Neptune 4.497E+09 6.019E+04 9.092E+28 3.623E+09 2.510E+19

Source: Computations by the author, based on data taken from Astronomy
Data Book
by J. Hedley Robinson (New York: John Wiley & Sons, 1972).
Data for Pluto were not available. "E+" represents the exponential
function; example: 3.653E+02 means 3.653 x 10
"Day" means one earth solar day.

The real significance of this law is this: there is a fixed relationship between average orbit radius R and orbit period T. If we know either, we can calculate the other.

We can apply this law to earth satellites. Let's consider three examples:

The Moon: The moon moves in a very high orbit (R = about 383,000 km.) and has a very long orbit period (T = about 27.3 days). The moon moves so slowly that the earth rotates under it faster than the moon moves about it. From the point of view of an observer on the earth's surface, the moon appears to rise in the east and set in the west, just like the sun and the stars.

If we plug the above values for R and T into the third-law equation, we can determine

R3/T2 = 7.532 x 1013

Knowing this, we can now work backwards to determine R or T in other situations.

The Space Shuttle: The shuttle's typical orbit height (above the earth) is about 600 Km. Adding the earth's radius (6370 Km) gives us the shuttle's orbit radius R = 6970 Km. Applying the third law equation yields T = 0.067 days, or about 1.6 hours. From the point of view of an observer on the earth's surface, the shuttle streaks across the sky every 1.6 hours.

Geosynchronous Satellite: Now suppose we wish to know the orbit radius of a satellite whose period T equals one sidereal day. This satellite would move at the same rate at which the earth rotates; this, its motion would be synchronous with the earth's rotation. We call this a geosynchronous satellite.

• Using the third-law equation, we discover that R = 42,155 Km. Subtracting the earth's radius (6370 Km) yields the orbit height above the earth: 35,785 Km., or about 22,236 miles.

We will discuss geosynchronous satellites in detail in Part 2.

CONTINUE TO PART 2 - ORBITAL GEOMETRY
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