Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

A geostationary satellite has an orbit that lies within the Earth's equatorial plane. The shape of the orbit is circular and the angular velocity of the satellite in the orbit matches the rate of rotation of the earth. Viewed from Earth, the satellite would appear stationary in the sky. Since orbital period is proportional to orbital altitude, there is only one specific altitude that can be used by geostationary satellites. The geostationary orbit is said to be geosynchronous. See illustrative drawing next:

Geostationary orbits are useful mainly for communications. The altitude for these satellites turns out to be quite high, approximately 36 * 106 m. Accordingly, the delay of radio signals to and from the satellites is large compared to that of paths using either land-based facilities or near-Earth orbiting satellites. Such delay is unsuited for transaction-oriented communications that rely on a rapid exchange of two-way communications.

The satellite does have a large view of the Earth. Consequently, geostationary satellites are used predominantly for one-way broadcast service where the signal transmitted to Earth would cover a large geographic area. International agreements assign space in the orbit so that the satellites do not interfere with one another and global coverage is satisfied by parties wanting to use the technology.

Geostationary orbits are not achieved precisely in practice and the satellites therefore show a small amount of wander. Also, perturbations from such effects as the gravitational pull of the Sun and Moon cause such a satellite to wander ever further from its assigned position. Before the wandering can go too far astray, small rocket jets are used to perform orbital maintenance by nudging the satellite back to within its proper space.

Celestial Mechanics provides the relationship:

P^{2} = [4*π^{2}/G*M]**a ^{3}** ,

where P is the period of an elliptic orbit,

**a** is the average of the apogee and perigee distances i.e.
the
radius r of a circular orbit,

and G*M is the standard gravitational parameter discussed in the second topic of Chapter 5.

The foregoing relationship has the importance that it can be used to calculate G*M,
given the P and **a **of an elliptical orbit.

Our current use of the relationship is find r for a circular orbit given P. Rearranging the relationship we find:

a = r = (P2 * G*M / (4 * π2)) 1/3

The determination of orbital parameters for a geostationary orbit is straightforward. The orbital period, P, is the time for one complete revolution of the Earth on its axis which is called a sidereal day and is 86,164.09054 s, more or less. The altitude of the orbit is the radius of the orbit minus the Earth's radius.

r = (86,164.090542 * 398,600,441,800,000 / (4 * π2)) 1/3

= 42,164,169.6371354 m

Alt = r - earthRadius = 42,164,169.6371354 - 6,371,010.00 = 35,793,159.6371354 m

Orbital velocity is the tangential velocity of the circular orbit and is such that one revolution is completed in one sidereal day. It is the same value everywhere along the orbit, thus we can equate the tangential velocity to the orbital insertion velocity, launch velocity as used by the spreadsheet:

Launch velocity = tangential velocity = distance / time = 2 * π * r / P

= 2 * π * 42,164,169.6371354 / 86,164.09054

= 3,074.6600990408 m/s

We will assume the orbit is such that at t = 0, the Latitude and Longitude of the satellite are both 0.0. Therefore:

Latitude, Longitude, Altitude = 0.0, 0.0, 35,793,159.6371354

The Elevation must be 0.0 degrees to specify a strictly tangential velocity (no radial component). The inclination of the orbit is the angle of the plane of the orbit measured from the equator. Since Azimuth on the worksheet is measured from North and the satellite travels due East, we have:

Elevation, Azimuth = 0.0, 90.0 degrees, respectively.

The altitude is sufficiently high that a drag coefficient, Cd = 0.0 can be assumed. The orbit is not dependent on the mass of the satellite since its mass is negligibly small compared to that of the Earth.

To get one complete orbit, set up the integrations with a Step Size corresponding to a fraction (1/100,000) of a sidereal day. With a Steps per Output = 500 and Number of Outputs = 200 we can get exactly one sidereal day of orbit, that is one complete orbit.

A typical plot is shown in the following figure and as expected the plot indicates a circular orbit.

The same laws of physics that govern the motion of satellites in orbit around the Earth apply to planets that orbit the Sun. For the purpose of calculating planetary orbits, a slightly different spreadsheet is used. This allows for certain parameters to be used differently than in the case of Earth orbiting satellite spreadsheets.

The planet Earth orbits the sun in a plane called the ecliptic. The ecliptic plane is analogous to the equatorial plane of the Earth except that we are dealing with the Earth orbiting the Sun in the ecliptic plane rather than a geostationary satellite orbiting the Earth in the equatorial plane. Other planets have orbits that are slightly inclined to the ecliptic. If we wanted to examine their orbit, we could set the azimuth of the orbit to a suitable value that reflected this inclination but we will limit our illustration here to that of Earth's orbit.

Latitude is used to specify a launch point not in the ecliptic plane but we leave this at 0.0 since we will explore orbits in the ecliptic plane.

In the case of Earth satellites we use Altitude (distance above the surface of the Earth) but for planets, it is more convenient to specify centre-to-centre distance between the Sun and the planet, so instead of Altitude, we specify Distance which means centre-to-centre distance for Sun to planet.

The Elevation and Azimuth are used to specify the angle of the planetary orbit at its starting point. We will assume that the planet starts at perigee and is in the ecliptic plane. For planets in our solar system, the orbit is oriented such that the planet travels counter clockwise when viewed from above (looking down on the North Pole of the Earth) therefore, Elevation = 0.0 and Azimuth = 90.0.

It is interesting to look at the orbit of Earth around the Sun. We can look at the details of the orbital characteristics and relate them to some observations that can be made from Earth. For the Earth's orbit, the parameters we use are summarized below:

Latitude = 0.0

Distance = 147,090,000,000 km (Earth to Sun perigee distance)

Launch Velocity = 30,290 km/s

Elevation, Azimuth = 0.0, 90.0 (to place orbit in equatorial plane)

G*M = 132,712,440,020,000,000,000 (this is the value for the Sun)

The following figure shows the orbit for Earth with the parameters set to values discussed in the foregoing paragraphs.

As can be seen, the Earth's orbit is very nearly circular. Eccentricity is a measure of how far from a circle the orbit is. With an eccentricity of 0.01688 the apogee distance is about 3.4% greater than the perigee distance.

The variation in the rate at which the Earth orbits the Sun can be seen in the above diagram. Note that at one quarter of a year away from perigee, the earth is more than halfway closer to the apogee. This is to be expected since the portion of the orbit around perigee has the Earth traveling at its fastest.

Although the variation from a circular orbit is small, the consequences of the elliptic nature of the orbit are easily observed on Earth. One of the principal observations is the variation of the transit of the Sun across the observer's meridian (the line that goes from North to South Pole passing through the observers position). If the Sun moved at a uniform rate through the sky, this passage would occur at the same time each day (at 12 noon in the centre of a time zone). But because the Earth travels faster at perigee (during winter in the Northern Hemisphere) than at apogee, it rotates around the Sun at a faster rate during this time.

The effect is easy to see on the output of the integration for the orbit. At perigee, the sun reaches the meridian before 12 noon while at apogee position it does not reach the meridian position until after 12 noon. For the former position the sun is said to be "fast of the clock" and for the latter it is said to be "slow of the clock".

Clocks on Earth always mark the same number of seconds between 12 noon on successive days (86400 s). As a consequence of the higher rate of rotation around the Sun near perigee, at 12 noon from day to day the sun appears to be further and further past the meridian. As the Earth approaches apogee, it slows in its orbital velocity, the effect reverses, and eventually the Sun will appear to cross the meridian before 12 noon. This cycle repeats from year to year. The terms used for this effect are "sun slow" when it transits after 12 noon and "sun fast" when it transits before 12 noon. Through the year, the magnitude of the effect amounts to about 15 minutes slow to 15 minutes fast, a total variation of about 30 minutes.

In the above discussion, we examined Earth's journey around the Sun and its rotation on its axis. These cycles have been used to define time and its duration for most of recorded history. In the last half of the 20th century, methods for the precise measurement of time showed that the rate at which the earth rotates on its axis and other astronomical phenomenon such at the solar year were irregular in duration. These irregular and unpredictable perturbations are in addition to the regular ones such as discussed above which are due to the elliptical shape of the Earth's orbit. Factors that cause such irregularities are the tidal friction on the Earth due to the Moon, Sun, and to a lesser extent other solar system objects, as well as core-mantle interaction and plate tectonics causing shift in the mass distribution of the Earth.

In 1967, the SI (Syste`me International) second was defined as 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the caesium 133 atom. This duration of a second was chosen to correspond to the duration of one second based on ephemeris time, the time based on planetary motion, to the precision of measurement available. Further developments have spawned a large number of time scales that are designed to run at a rate based on the SI second.

For example: Barycentric Coordinate Time (TCB) is the time base located at the centre of mass of the solar system, but without the solar system present; Geocentric Coordinate Time (TCG) is the time base for the centre of mass of the Earth without the Earth being present. The reason for removing mass bodies in these two definitions is to remove an effect predicted by the Theory of Relativity. This effect manifests itself as a slowing of the clock rate for a clock that is close to a massive body, and therefore in a stronger gravity field, as observed by someone further removed from the massive body. The gravitational effect is substantial. A caesium atomic clock running at the surface of the Earth will lose about 22 ms per year relative to TCG due to the presence of Earth's gravity. In fact, the Global Positioning System (GPS) is designed to account for the difference in gravitational field (and hence clock rate) at the satellite altitude relative to observers at sea level. This is necessary since these satellites beam down precise time of day information that would otherwise drift and make position calculations inaccurate should the difference in time be allowed to accumulate.

Coordinated Universal Time (UTC) has been defined for the purpose of maintaining civil time, the time we use on our watches and other timepieces. UTC is based on caesium clocks and can be accurately maintained to run at the SI rate, but our civil clocks need to keep pace with the daily motions of the solar system so that the Sun, for example, is always highest at around 12 noon every day. Thus another time scale is introduced for use with timepieces: Universal Time (UT also called UT1). Since UT keeps pace with the irregular motion of the Earth, a process is followed where leap seconds are added/deleted to/from UTC where the difference between UT and UTC is kept to less than 0.9 s. The last leap second adjustment (at this writing) occurred on Dec 31, 2005. The previous adjustment was seven years earlier.

Earth observation satellites such as Radarsat-1 and, more recently, Radarsat-2, occupy Polar Orbits as are explored in the next topic.

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