Student Activity
Distance to the Horizon
(in Low Earth Orbit)


See the accompanying graph (below) showing
the relationship between h and d
Transparency Master 		It is possible to determine the maximum visual contact distance from a ground based observer to a satellite in orbit around the Earth.

This maximum distance occurs at the instant that the satellite appears above the horizon.

For low Earth orbit this distance approximates the radius of the visual footprint of the satellite.

The figure to the left shows a satellite in a circular polar orbit travelling from horizon to horizon and exactly through the zenith (the point in the sky directly overhead).

For convenience we have placed the observer at the North Pole, however the choice of location is arbitrary and the derived rules apply to all satellites, regardless of their orbit.

If the satellite is orbiting at height h above the Earth, whose radius is re then simple geometric arguments show that the distance d from the observer to the satellite as it "just" rises above the local horizon is given by

d = retan (eq.1)

The value of is defined as

= arccos(re/(re+h))

NOTE: arccos = cos-1 or inv cos on some calculators.

For any reasonable values of h, the satellite's altitude above the Earth's surface, the angle can be computed and then d determined. The graph below is a plot of equation (eq.1) above.


Transparency Master 		For low Earth orbits the distance to the horizon is the visual footprint of the satellite.

At the precise moment that the satellite comes into the view of the ground based observer, the ground based observer also comes into view from the satellite.

The only difference is that the horizon of the ground based observer is fixed, whereas the horizon of the satellite moves along with the satellite.

Assignment

  1. Using the accompanying graph (or from solving equation (1), determine the distance to the Earth's horizon as seen by astronauts aboard the International Space Station at an altitude of 380 kilometres.

  2. How much further can astronauts aboard the Space Shuttle see (assuming that they can see the horizon), if they are in a circular orbit at an altitude of 500 km?

  3. Ground based radar detects a satellite at the instant it appears above the horizon. The radar echo indicates the satellite is 2000 km away. What is the orbital altitude of the satellite?

  4. At what altitude must a satellite orbit so that its horizon is 3000 kilometres away?

  5. How high must you fly a satellite so that its ground track is 5000 km wide?

  6. The best photographs of the Earth's surface are taken through the thinnest part of the atmosphere; in other words...straight down. How high must you fly a satellite if you require photographs in a band 200 km wide, and assuming the best images use only the nearest 15% of the track width?

  7. Using the information from the question above, calculate the fraction of the Earth's surface that could be continuously photographed in a single orbit?

  8. An observer on the ground sees a satellite pass directly overhead at the same moment that it is seen setting by another ground based observer 1500 km away. What is the altitude of the satellite?

  9. Since a large fraction of the Earth can be seen from a high altitude, why are most earth surveillance satellites flown at the relativity low altitude of 300km to 500km?

  10. Explain which is the better orbital configuration for Earth surveillance satellites and which is better for telecommunications satellites - high altitude low inclination orbits (equatorial orbits), or low altitude high inclination orbits (polar orbits).
    What are the advantages and disadvantages of each?

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