So what sorts of satellite are there ?


There are all sorts of satellites up aloft, serving all sorts of functions. Many are familiar, such as satellite television systems or long distance telephone communications. As far as weather satellites go, there are essentially two kinds, known as polar orbiting and geostationary. The former travel in an orbit the plane of which approximately passes through the north and south poles of the Earth. As they travel round the Earth (which rotates beneath them), it follows that all points on the planet's surface will lie beneath the satellite at some point, so that the weather systems that exist can be seen from space, wherever they are.

The geostationary satellites on the other hand, occupy a fixed point in the sky and always command the same view of the Earth. Anticipating a little, they are much farther away so are able to view the entire face of the Earth pointing in their direction. Thus a series of them suitably arranged can view the entire surface of the Earth continuously and seamlessly. So how is this done ?

Figure I on the left represents in a very simple way a satellite travelling round the Earth. Let's represent the Earth's radius by 'r' and the satellite's height by 'h'. The simple maths which follows will have to be taken on trust, but if we apply the laws of motion first set out by Isaac Newton over 300 years ago, if 'v' is the velocity of the satellite, then it can be shown that the satellite has a constant acceleration toward the Earth given by the expression v2/(r+h) and the force that's keeping it in its orbit is just mv2/(r+h). [Force = mass x acceleration] If there weren't any such force, it would just fly away into space ! (It's no different to the Moon in that respect.)

The force that keeps it there is gravity and again, courtesy of Isaac Newton, it can be shown that this is given by the expression GmM/(r+h)2. Here M is the mass of the Earth and G is a number known as the Universal Gravitational Constant. Since the satellite stays in constant relation to the Earth, these two forces must be equal. So we can say:

GmM/(r+h)2 = mv2/(r+h)

This expression will simplify quite a lot - we can take out 'm' from both sides as well as an (r+h) giving:
v2 = GM/(r+h)

Now let's suppose the time it takes go right round one orbit is 'T' - and since Velocity = Distance/Time, it follows that 'v' is given by:
v = 2p(r+h)/T 
  
After substituting this expression for 'v' we end up with another somewhat more complex one:

T2 = 4p2(r+h)3/GM

From this we could in principle get 'T' - but we need to take this further. How much does the Earth weigh ? We need to know this to get M. If s is the density of the Earth, then from mass = density x volume, we can get:
M = s x  4pr3/3

Putting this back into our expression for 'T' leads to:

T2 =  3p(r+h)3/Gsr3

This leads to some interesting results. Let's suppose the satellite is in a very low orbit. In that case 'h' is very small compared to 'r' - we can approximate (r+h) with 'r' and it disappears entirely from the equation. We are left with:

T2 = 3p/Gs

or
T = Ö(3p/Gs)

In other words, as T is only dependent on G and s, it's independent of the particulars of the satellite or how high it happens to be orbiting (as long as it's not too high, of course). Inserting the appropriate numbers shows that the period 'T' is round about 95 minutes or so. So all low Earth orbiting [LEO] satellites go round at roughly the same rate, determined only by the density of the Earth and the Universal Gravitational Constant. A remarkable result ! The polar orbiting weather satellites are in this category with an altitude of around 400 km or so and a period about 100 minutes. They go round the Earth about 14 times per day, and as the Earth is also rotating beneath them, it turns out that every point on the Earth's surface is passed roughly overhead at least twice a day. If the timing is such that one pass is in the morning and the other in the evening, one can build up a progressing picture of weather developments in some detail.

Suppose on the other hand, we make T to be 24 hours so that the satellite goes round at the same rate as the Earth rotates - it therefore stays in the same relation to the Earth's surface. Viewed from the Earth, it always stays in the same point in the sky. The equations can again be solved to give 'h' which now turns out not to be small relative to 'r' at all. In fact it works out at about 35,000 km which is quite a long way from the Earth. From that distance, we can see an entire face of the planet much as it would be seen from the Moon only much larger. Three such satellites spaced at 120° apart in a common plane would suffice to see all of the Earth at the same time. For this to work properly, it should be remembered that any geostationary satellites have to be in the equatorial plane. The progress of all the Earth's weather systems can be kept under constant surveillance from which accurate weather forecasting becomes a real possibility. This is exactly how it's done today, and it's possible for an interested observer to make use of both these systems for themselves - more on other pages...

To see how the Earth appears from space, have a look here (courtesy of NASA) whilst a polar orbiting satellite might look something like this.


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© Stuart Hill [Updated November 2002]