{S:X;X;0;5; Satellite orbit and longitudinal increments usually are fairly constant which could easily give you the impression that predictions are straightforward. Over a couple of orbits this would hold true but over a period of time you would encounter errors. The increments are due to the rotation of the earth but these orbits or nodes drift due to the earth orbiting the sun.}
{S:X;X;0;6; The next problem is to determine where the satellite is situated at any time in that orbit which again appears to be a simple problem. Wrong as first the earth is oblate and next the orbit is probably elliptical which means the satellite altitude is constantly changing. When the altitude decreases the speed increases (yes this is correct) and when the altitude increases the satellite slows down.}
{S:X;X;0;7; Now if you want very accurate results you should also consider the effects of atmospheric drag, the Sun and moon gravitational pull plus any effects due to solar storms.}
{S:X;X;0;9;Background}
{S:X;X;0;11; In the early 17th century a German astronomer Johann Kepler observed over many years the effects of planetary motion from which he derived some basic laws of motion. These laws went unproven for many years till Newton proved them mathematically and also went on to derive the universal law of gravitation.}
{S:X;X;0;12; These laws and their derived equations are the basis of most prediction programs used today and although one program can look completley different to another this is often only due to the presentation of data.}
{S:X;X;0;14;Orbits}
{S:X;X;0;16; Their are three types of basic orbit used today the first being a geosynced or more commonly known as geostationary. This is the most expensive orbit to launch and requires greater station keeping control but it has the advantage of being able to provide 24 hr sevice. These satellites operate at an altitude of approx 65000Km and can provide a service to half the globe and often for communication three evenly spaced satellites are used to give 24hr total globe coverage with inter-satellite communication.}
{S:X;X;0;17; Due to the Van Allen belt which is at an altitude of 1500-20000Km medium height satellites are out of the question as the high energy particles would soon damage the electrical systems. To overcome this a highly elliptical orbit is used with a Perigee around 3000Km and the apogee between 30-40000Km. As the satellite travels towards perigee it accelerates to such a high speed that when it travels through the Vann Allen belt the effects are minimal. Then as it travels to apogee it slows to a standstill and at this part of the orbit it can appear to be geostationary for several hours.}
{S:X;X;0;18; The final type of orbit is the low earth orbit which is the one most prediction programs are aimed at. Most amatuer and remote imaging satellites (weather , earth resources etc) use this orbit as its cheap to launch. But it has the problem that the period is short and the orbit parameters are often changing making this the most difficult to predict.}
{S:X;X;0;20;Geometry of the orbit}
{S:X;X;0;22; The ellipse is made up of three dimensions its length (Major Axis) ,the width (Minor Axis) and the distance to the focal point from the origin. Although the ellipse has two focal points in the the context of an orbit it only has one which occurs at the earths geocenter and we use half the axis length which is known as the Semimajor and Semiminor axis. The point where the semimajor axis coincides with the orbit is known as the Perigee (minimum altitude) and at the opposite end is the Apogee (maximum altitude). The shape thus requires two paramters but if we use a ratio of the axis to describe the shape only one would be required. This is called the Eccentricity and with a value of 0 we obtain a circle and a value of 1 well watch it crash back down on the lanch pad. To describe the size we can specify the semimajor axis or derive it from the speed often given in orbits per day which is known as Mean Motion.}
{S:X;X;0;23; The next problem is to determine the position of the orbit , here we use the perigee to indicate the start of the orbit. To specify the position we measure the angle from the equator when the satellite is ascending till it reaches the perigee and this is known as the Argument of Perigee. We now know the size and shape and position of the orbit but as yet not the orientation, this is known as the Inclination. The easiest way to describe inclination is the angle between the equator and the satellite path as it enters the northern hemisphere. This parameter is probably the most important as it has the most dynamic effect on the orbit path.}
{S:X;X;0;25;Time}
{S:X;X;0;26; When you look at you watch you are working in solar time which is the time the sun passes the meridian to when it next passes the meridian which is 24 hours or 1440 minutes. But due to the earth orbiting the sun this is just over 360 degrees (Solar day=360.985 degs) but to predict the orbit we need to now how long it takes to rotate 360 degrees. The time for the earth to rotate 360 degrees is approximately 1436.07 minutes and is known as the Sidereal day. The time the satellite takes to complete one orbit of the earth is known as Mean Motion. The most common unit for Mean Motion is revolutions/day where the day is a solar day and thus for preditctions this has to be converted to sidereal time.}
{S:X;X;0;27; If this is confusing try this, the orbit period is the time the satellite orbits from perigee to perigee and is given in Mean Motion which is the Anomalistic period. This period is not the true period of one orbit of the earth as the position of perigee is moving due to the effects of the earths equatorial bulge. This change is known as the rate of change of argument of perigee and is a function of the inclination. To make long term predictions we need to know the time from one node to the next ie one orbit around the earth and this is known as the Nodal period. This can be mathematically calculated from Mean Motion, Inclination and Eccentricity.}
{S:X;X;0;29;Increment and Precession}
{S:X;X;0;30; Longitude increment is basically the difference in longitude between two successive orbits and is the angle the earth rotates during one orbit. This can be calculated from the period of the orbit where the period is the nodal period.}
{S:X;X;0;31; If the earth was spherical the orbit would remain fixed in space but due to the equatorial budge the satellites orbit plane precesses about the earths N-S axis which is a function of inclination and altitude. One useful orbit is the Sun-synchronous orbit where if the inclination is around 98degs the total precession for the year is 360degs. This has the advantage in that the satellite follows the terminator all year round and thus the solar cells see sunlight all day long which reduces the battery requirements. Also with this type of orbit the satellite passes the same part of the earth at about the same time each day. This is a common orbit used by remote imaging or weather satellites and are often referred to as a polar orbiter.}
{S:X;X;0;32;Position in Space}
{S:X;X;0;33; We have looked at how the orbit behaves but we need a method of referencing to a fixed point. To achieve this we use an inertial co-ordinate system (fixed directions with respect to stars) which is known as the Right Ascension-Declination Co-ordinate system.}
{S:X;X;0;34; Consider a sphere of infinite radius were the earth is the centre this is known as the Celestial sphere. Now if we extend the equator in all directions it becomes the celestial equator. Next we extend the earths N-S axis to become the celestial N-S axis. Finally the last axis is a line extended to a fixed star in space known as the First Point of Aries or the Vernal Equinox. The First Point of Aries is derived by extending a line from the earth through the Sun on the first day of spring.}
{S:X;X;0;35; We can now define the position of the orbit plane at any point in space simply by specifying the angle between the orbit plane and the First Point of Aries. This is called the Right Ascension of Ascending Node (RAAN). Now we can specify the orbit parameters and its position in space but when these are measured one thing is missing and that is where is the satellite in the orbit. To define the position of the satellite we use Mean Anomaly which is the angle between the satellite and the start of the orbit where it crossed the equator.}
{S:X;X;0;36;Sub Satellite Point}
{S:X;X;0;37; Now we can establish the position of the satellite we must now covert this to an earth position and perhaps determine the azimuth and elevation to the observer. We may also wish to show the Footprint (coverage area) as well as range and direction to point the aerial.}
{S:X;X;0;38; To convert the satellites space position to that of a terrestrial position requires the use of Spherical Trigonometry. It should be noted that in Spherical Trigonometry the internal angles do not add up to 180 degrees and the square of the hypotenuse does not usually equal the sum of the squares of the other two sides. I will not go in to the mathematics as these are beyond the scope of this article and as it uses Napier's rules and my word processor can not cope with the funny shaped symbols (I know this is a cope out but thats my excuse).}
{S:X;X;0;39;Element Sets}
{S:X;X;0;40; The data that defines the satellite orbit and position is often referred to as the Keplerian elements and comes in one of two formats. The first is just a list of the parameters and its values as shown below :-}
{S:X;X;0;41;Satellite: NOAA-10}
{S:X;X;0;42;Catalog number: 16969}
{S:X;X;0;43;Epoch time: 93259.71206390}
{S:X;X;0;44;Element set: 303}
{S:X;X;0;45;Inclination: 98.5179 deg}
{S:X;X;0;46;RA of node: 272.0121 deg}
{S:X;X;0;47;Eccentricity: 0.0014229}
{S:X;X;0;48;Arg of perigee: 49.3707 deg}
{S:X;X;0;49;Mean anomaly: 310.8720 deg}
{S:X;X;0;50;Mean motion: 14.24831329 rev/day}
{S:X;X;0;51;Decay rate: 5.1e-07 rev/day^2}
{S:X;X;0;52;Epoch rev: 36366}
{S:X;X;0;53;Checksum: 295}
{S:X;X;0;54; The next format is two line NASA or AMSAT format which is a compressed form and sometimes a key is given so you know which numbers are for which element. A number of programs will allow you to update automatically from these 2 line elements if they are saved as a Text file which can save you a lot of time typing. A typical 2 line set is shown below :-}
{S:X;X;0;56;DECODE 2-LINE ELSETS WITH THE FOLLOWING KEY:}
{S:X;X;0;66; You may have noticed a couple of parameters not yet mentioned the first being Epoch Time which is the time the obit was measured. It takes the form year then days from Jan 1st and the time during the day in a fraction of a day.}
{S:X;X;0;67; The next is Catalog number which is a unique NASA reference number which can be used by the program to identify which satellite the elements belong to. Often element sets will also include a checksum at the end for error checking. Finally you may encounter Epoch Revs which is simply the number of orbits since launch.}
{S:X;X;0;69;Pitfalls}
{S:X;X;0;70; Although I have not specified any equations some use a unitless number so the angle should be expressed in radians. The next problem is that most angles are between 0 and 360 degrees but most computers only calculate one quadrant (0-90 degrees) so you must remember to take this into account.}
{S:X;X;0;71; Many programs use a sidereal time constant to convert between solar and sidereal time but this constant is different each year.}
{S:X;X;0;73;Conclusions}
{S:X;X;0;74;If you have found this heavy going then be thankful I left the mathematics out as this can reach degree level. As you can see predicting satellite orbits is by no means a simple one and when you look at a program that only prints a table you only see the effect of about 5% of the code. If you have found this all rather confusing then you can forget writting your own program. Probably if you require a specific type of output the best bet is to buy one and then customize the output to your requirements.}
