The simplest orbit to consider is circular. It can be characterised by stating the orbital altitude (height of the spacecraft above the Earth's surface) and the orbital inclination (the angle of the satellite's orbital plane to the Earth's equatorial plane). It is the values of these parameters (primarily) that dictate whether an orbiting satellite can be seen by a particular observer (an orbit actually requires several more values to describe it more accurately, such as in the NORAD SGP4 models used in a large number of tracking programs, but for initial calculations of visibility, altitude and inclination suffice). In fact most orbits are elliptical in nature; the height varies between the apogee (farthest point from the Earth), down to the perigee (closest point on the orbit to the Earth).
For a satellite of given size, the higher the orbital altitude, obviously the dimmer the satellite will appear - according to the spacecraft type and it's intended mission the altitude may vary between (roughly) 150 km and 40000km. This brightness is often expressed in magnitudes (a logarithmic scale which follows the response of the eye), used in astronomy to indicate the brightness of stars, planets, and other heavenly bodies. The brightest stars visible are around magnitude (mag.) 0 to -1.
The brightness is complicated by at least three further factors:
Local elevation above the horizon also affects brightness; a pass low near the horizon will see the satellite one or two magnitudes dimmer than for a similar pass close to the observer's zenith, due to atmospheric extinction of the light - it has to travel through more of the Earth's atmosphere to reach the observer.
It is common to see the satellite gradually fade from view during a pass (over several seconds), it's light extinguished as it enters the Earth's shadow (equally for a satellite to emerge from the Earth's shadow). Such eclipses are influenced by the next factor we must consider.
The orbital inclination dictates over which areas of the Earth the satellite will 'fly'. In an orbit of 25 degrees inclination the ground track (the point on the Earth's surface directly below the satellite which is traced out during its orbit) will never exceed 25 degrees North or 25 degrees South in latitude. This satellite would never be visible from Northern Europe for example, unless it's orbital altitude were some 1500km or so (and thus would then appear considerably dimmer than if it were in low Earth orbit or at a higher elevation in the local sky). Based on inclination we can place orbits in one of three categories:
Thus far we can see that for a satellite to be easily visible to an observer it should be in low Earth orbit at an inclination that is almost equal to or greater than the observer's latitude.
Earlier I alluded to the problem of the Earth's shadow - when eclipsed the satellite is naturally not visible. Such events are dependent upon the satellite's altitude, inclination, the time of year and the observers location. The Earth's shadow is for example 'longer' in the local sky for an observer at the equator than for, say, an observer in the Northern polar region during June. Thus the fraction of the night available for observing low Earth orbiting satellites is shorter in Ecuador then in Greenland at that time of year. In fact our Arctic observer may seldom see satellites disappear into eclipse.
Looking at the over simplified situation presented in this figure (click on it for details) we can see the situation during a northern summer. The satellite can be seen by northern observers (region 1) when passing along the arc NA of its orbit. It is visible to southern observers between points B and S for a shorter time. The situation is reversed in the northern winter. During their respective summers the region of visibility can be seen to extend over a greater range of latitudes, with the observing period extending well into the night. Observers at middle latitudes have a shorter visibility window around sunrise and sunset which varies little in length throughout the year.
There are however two exceptions to these visibility constraints, though both are not exactly common methods of observation. First, it is possible to view the brightest satellites, such as Mir and the shuttle during the daytime. Daytime sightings of the shuttle have been reported by at least one observer. It obviously helps to know exactly where to look (courtesy of one of the many prediction programs available) and to look under optimum lighting conditions. That is to say when the sun-observer-satellite is at a maximum; either the satellite is quite low in the west just after sunrise, or low in the east shortly before sunset. Binoculars would be a great help here, but be wary of the sun! One trick which may be of some use is the use of a polarising filter. Sunlight when scattered in the atmosphere becomes polarised, thus some contrast improvement may be gained by using an appropriately aligned filter. When using this for daytime observation of Jupiter during the recent comet Shoemaker-Levy9 impact, I found some improvement.
A second exception lies in the fiery death of an orbiting body; you may find yourself in the right place at the right time to witness a re-entry, as the satellite experiences frictional heating in the upper atmosphere, leaving a fiery trail across the night (or even day) skies.
Of course it is not simply a question of watching for a given satellite at the same time each night. Few satellites have an orbital period which is a simple fraction of one day (geostationary satellites being the obvious exception). The orbital period is dictated by the satellites altitude. The higher the altitude, the further it has to travel around the Earth and the longer it thus takes. Satellites in low Earth orbit (say 300 kilometres) perform one orbit in around 90 minutes. By the time we are at geostationary altitudes (about 36000 kilometres) one orbit takes 24 hours.
Thus the satellite arrives later (or earlier) on successive nights. With each delay/advance in arrival time the Earth will have rotated a little farther (or less) with respect to the satellite's orbit. The consequence of this is that each night the satellite will appear in a different portion of the sky during each pass and the number of visible passes will vary. In the longer term (days to weeks) the passes will drift from evening to daylight hours, then into the morning before returning to the evening once more. Imagine that you tried to live a 22 hour day. As the days passed you would gradually wake earlier and earlier till you were having breakfast when others were off to bed. With more time your waking hours would re-synchronise with everyone else before beginning this cycle once more. Thus windows of visibility are created.
Take for example the International Space Station. This will be visible for two weeks or so in the evening sky, the best passes (those of highest local altitude above the horizon) will occur earlier each day. Eventually they are lost in daylight for the next several days before emerging into the pre-dawn sky. After a series of good morning passes for a week or two, passes are lost due to them being eclipsed as they occur around midnight, before reappearing in the evening sky, to repeat the cycle once more. Many satellites in low Earth orbit go through such a cycle. In the case of the shuttle, due to the short term nature of the missions (7-10 days typically), an entire mission can occur entirely outside of one of these windows of visibility.
The simple idea of circular/elliptical orbits presented here belies the complications which arise from the fact that the satellite suffers greater air resistance the lower its orbit is. This bleeds off the orbital energy lowering the orbit yet further as the satellite begins to brush the upper atmosphere at perigee. The forces on the satellite due to the Earth (and Moon, Sun, etc) vary throughout it's orbit (the Earth is not a nice spherical shape!) giving rise to continual change in the orbit.
Fortunately, advanced orbital models such as SGP4 and SDP4 take into account these and other effects. They are the basis for many software packages for satellite tracking, and when used with suitable orbital data can yield quite accurate predictions which observers can easily use to aid observing. Naturally the software needs accurate and recent data; this comes in the form of keplerian or two-line element sets, which can be found at several resources.
Having located and observed your satellite, it is entertaining to know what it's up to - what is it's mission, it's current status? How long has it been up there?
If you want to search out a particular satellite or start with an 'easy' object, details of various satellites and rockets which are visible, and their current situations can be found here.
Of course the amateur can also make a contribution with their observations through either
Besides specific studies as mentioned above, these measurements can provide important clues as to the mission of a satellite in the absence of any official information, as is the case with a number of military satellites.
Links: to the VSO Home Page, Satellite Predictions.