Michel Capderou

Satellites of Other Celestial Bodies

In the present chapter, we shall examine the motion of a satellite around another celestial body, but more concisely than we have done for the Earth and Mars. The chapter is divided into two parts, each following practically the same plan. There is a clear separation between Part A, which concerns the satellites of planets, and Part B, which deals with satellites of natural satellites. In the second part, to avoid confusion, a natural satellite or moon of a planet will be referred to by the term “natural satellite,” and the term “satellite” will be reserved for an artificial or technological satellite.

Global Positioning Systems (GPS)

We first explain the detailed calculation for determining the position and velocity of the GPS receiver. We then give the characteristics of different constellations of navigation satellites. We conclude with an appendix on GPS and relativity.

Satellite in Real (Perturbed) Orbit

We first examine all the forces (perturbing forces) that separate the real motion of the satellite from its theoretical ideal motion (Keplerian). We the apply the theories of Lagranges equations of the perturbing method to determine the changes to each of the orbital elements.

Orbit Relative to the Sun: Crossing Times and Eclipse

We begin by studying the position of the orbital plane of an arbitrary satellite relative to the direction of the Sun, focusing on the notion of crossing time. We then turn more specifically to Sun-synchronous satellites for which this relative position provides the very definition of their orbit. We end the chapter with a more theoretical question, calculating the angle between the direction of the Sun and the plane of the orbit, and this will lead us to the study of solar eclipses, when the satellite is in the Earth’s shadow.

Satellite in Keplerian Orbit

The position of a body orbiting in a Keplerian motion is defined by the 6 Keplerian elements. From now on, we shall be mainly concerned with the periodic motion of a body, the artificial satellite, in the gravitational field of the Earth.

Spatiotemporal and Angular Sampling

Once again, we shall change our point of view! From an arbitrary point P on the Earth, we now note the time and angular conditions of our view of the satellite S. This is the opposite problem to determining the ground track of the swath: we must now establish the satellite sampling for a given instrument. We shall also determine, for this point P, the direction of the Sun at the instant of time when P is viewed by the satellite.

Orbit Relative to the Earth: Recurrence and Altitude

In this chapter, we discuss the position of the satellite orbit relative to the Earth. There are two distinct parts. The first concerns the position of the satellite ground track relative to the Earth, and the second the altitude of the satellite measured from the terrestrial ellipsoid.

View from the Satellite

In the preceding chapters, we have discussed the satellite orbit, position, and ground track. All this can be deduced from the position S of the satellite as viewed from the center of attraction O, which is the center of the Earth. The time has come to look at things from a different standpoint: we shall now be concerned with the view from an instrument carried aboard the satellite. The main difference is that we are now looking at things from the point of view of the satellite S. As a consequence, this chapter is principally concerned with observation satellites.

Ground Track of a Satellite

If the calculations of the motion are made in a Galilean frame (Earth-centered space-fixed), in most practical cases, one needs to know the position of the satellite relative to the Earth (Earth-centered Earth fixed). The satellite track is well defined by its geographical coordinates, longitude and latitude. We take this opportunity to make a brief statement on map projections.

Orbit and Mission

Drawing on examples from actual cases, we show how each mission requires a specific orbit. We explore all types of diverse orbits.