It's clear enough if you've looked at the page dealing with the two sorts of weather satellite that, although many of the essential features of satellite orbits are straightforward enough, if we want to 'observe' them, then it's necessary to look a lot more deeply into the matter.

For sure, with geostationary satellites, this isn't a serious problem as they tend to be parked to coincide with particular points of longitude. For instance, METEOSAT 7 which is operated by the European Space Agency, occupies a position nominally at 0° longitude (on the Greenwich meridian) and does a pretty good job of staying there - the satellites contain the means to hold their position for long periods or to be moved into alternative locations when required.

The elevation isn't hard to calculate, applying a little trigonometry. Using the earlier definition of 'r' for the radius of the Earth and 'h' for the satellite elevation, if 'q' if the angular latitude, and 'f' is the perceived elevation of the satellite above the observer's local horizon, then it can easily be shown that the elevation is given by:

This figure will perhaps help to make things clearer:

So for example, at my local latitude of 55° a geostationary satellite 35,000 km from the Earth would appear at about 16° 30' or so above the horizon - not very high in the sky at all in fact. As 'h' becomes very large compared to 'r' (about 6300 km) we can see that 'f' tends to be given by:

It's also possible to see that given a particular satellite altitude, there will exist some latitude above which the satellite will not be visible, but below the horizon - the particular latitude 'f_{c}' for which this occurs is given by:

For our particular case of geostationary satellites at 35,000km altitude, this works out at about 81° N or S. Beyond those latitudes, the satellite cannot be observed. Fortunately this does not exclude too many people in these relatively unpopulated polar regions !

The situation for polar orbiters is a good deal more complex and some reasonably clever mathematics is needed here to calculate when and where in the sky a particular satellite will be found. First of all, have a look at Figure II which contains a representation of a satellite in a circular orbit around the Earth:

There are quite a few elements to note here. First of all, the orbit plane is inclined at an angle 'a' to the equatorial plane, unlike the geostationary satellites. When this angle reaches 90°, the satellite passes over the north and south poles i.e. it's in a polar orbit. That's the first element to note. To specify it's position at any particular moment, it's necessary to define some reference point. This could be taken as the point at which it crosses the equator (the *ascending node* when it is moving northwards - 'A' in this case) and the angles would be measured, starting from here, much as degrees of latitude and longitude are. Together with the height 'h', the two angles f (the azimuth) and q (the elevation) are sufficient to define the satellite's position. This is the kind of approach required to characterise satellite orbits.

The general strategy for determining when and where a satellite may be observed depends on first characterising the size and shape of the orbit, establishing exactly where a satellite is at some precise time, and using the Newtonian Laws of Motion to extrapolate into future time. Provided this period isn't too long during which the motional characteristics might change, this method works very well. How is it done ?

A detailed description is beyond the scope of what I'd like to show here, but some points do call for explanation. First, although orbits have been so far described as circular, in practice this is never the case exactly - all orbits are elliptical to some degree. This is illustrated in Figure III (somewhat exaggerated as far as weather satellites go). The long and short semi-axes 'a' and 'b' are related by the expression b^{2} = a^{2}(1-e^{2}). Here e is known as the *eccentricity*. For a circle, e is zero, for an ellipse 0<e<1. Open-ended orbits result for e³1. The satellite moves in this orbit about the focus 'E' where its closest approach, known as *perigee* is at 'P'; the most distant point, the *apogee*, is at 'A'.

The angle f is known as the *true anomaly* and is measured anti-clockwise around the orbit from perigee. The laws of elliptical planetary motion were first set out by the early 17th. century astronomer Kepler. In recognition of his work, the set of data which characterises orbital motion is nowadays often known as the Keplerian elements. These parameters, besides those already mentioned above, also include others establishing the exact time when the elements were calculated (*epoch*), the period [more accurately the *mean motion* or the number of revolutions per day), the *decay rate* (resulting from atmospheric resistance). Angles are measured with reference to an imaginary axis originating at the centre of the Earth extending through the sun to infinity at the vernal (Spring) equinox. The *right ascension of the ascending node* RAAN (point 'A' in Figure II above) is measured relative to this, and the *argument of perigee* - the angle between RAAN and perigee follows from this definition. The other element, the *mean anomaly* is analogous to f above but instead refers to a circular orbit with the same period as the elliptically orbiting satellite.

All orbiting satellites have their own individual Keplerian elements which are calculated once the orbit has been characterised, and which are updated from time to time to correct for orbital drift. They are widely and readily available for most well known satellites (See the links page for more details). For calculation purposes, they are often given in what is known as a *two line set*. One of these might look something like this:

NOAA 15 1 25338U 98030A 02214.01740111 .00000324 7624 2 25338 98.5657 236.4501 0010294 176.2532 183.8718 14.24092376 21925

The exact formatting is important and all spaces are significant if computer programs are able to take the numbers in a consistent way. There are quite a few such programs (see the links page) many of which will run on quite modestly resourced computers. These can be used to predict when and where a given satellite can be observed with a high degree of accuracy over quite a long period - say a month or more. This data can be used in turn to program suitable receiving gear so that satellite images can be received quite automatically - as were all the images shown on these pages.

Before finishing this section, there is one particular feature of polar orbiters worth noting. If the inclination is made a little more than 90° (usually about 98°) and other orbital trajectory elements are chosen correctly, such a satellite can be made *sun synchronous*. That is to say, it passes overhead at around the same local time each day; this feature is particularly useful if it is desired to make observations at the same time daily, e.g. about 8:00 a.m. This will ensure that there is a satellite pass not too far from that time which will have a sufficiently useful elevation to give useable image data. The American NOAA (National Oceanographic and Aeronautical Administration) series are all of this type; most of the earlier Russian METEOR series aren't and the optimum orbital times change daily in a regular way. More powerful launch rockets are required to achieve polar orbits which were not available when the METEOR series (to date) were launched.

A typical NOAA satellite pass generally lasts of the order of only 10 to 15 minutes or so, hence it's clearly important to be ready for it ! I hope the outline I've given will show some of the details of what's called for in order to achieve this.

© Stuart Hill [Updated November 2002]