Chapter 1

Systems and Constraints

1.1. Satellite systems

A remote sensing satellite placed in orbit around the Earth is subject to several gravitational forces that will define its trajectory and motion. We will see that orbit formalism dates as far as Kepler (1609), and the motion of satellites is modeled using Newton’s laws. The Earth has specific properties, such as being flat at the poles; these specificities will introduce several changes to the Kepler model: quite strangely, as we will see, the consequences will turn out be extremely beneficial for remote sensing satellites, since they will allow us to have heliosynchronous sensors; this will enable them to acquire data at the same time as the solar hour, which in turn simplifies the comparisons of the respective data acquisitions.

The objective of this chapter is to briefly analyze orbital characteristics in order to draw some conclusions regarding the characteristics of imaging systems that orbit the Earth. For more details, readers can refer to the work of Capderou [CAP 03].

1.2. Kepler’s and Newton’s laws

By studying the appearance of the planets around the Sun (and, in particular, that of Mars), in 1609, Kepler proposed (in a purely phenomenological manner) the following three laws describing the motion of planets around the Sun:

In 1687, Newton demonstrated these laws by giving us a model of the universal attraction. This model stipulates that two punctual masses m and M exercise a force F against each other, colinear to the line joining these two masses:

with G = 6.672 × 10–11 being the gravitational constant. Therefore, an interaction takes place between the two masses.

In the case of artificial satellites orbiting the Earth, it is obvious that the Earth’s mass (MT = 5.5974 × 1024 kg) is extremely large with respect to the mass of the satellite and we can easily assume that the center of the Earth may be mingled with the gravitational center of the Earth + satellite system. If we also assume that the Earth is a homogeneous sphere, we can consider it as a punctual mass by applying the Gauss theorem. The satellite with a mass m, located at a distance r from the Earth (with a mass MT), is then subjected to a so-called attractive “central force”:

with μ = G MT = 3.986 × 1014 m3s –2. Therefore, we can say that we have a central potential U (r):

[1.1]

This force being the only one that can modify the motion of the satellite, we can therefore show that this motion verifies the following essential properties:

– The trajectory of the satellite lies in a plane, the “orbital plane”. The distance r verifies, using polar coordinates, the equation of an ellipse:

[1.2]

described by two parameters: the eccentricity e and the parameter p of the ellipse. The Earth is at one of the foci of this ellipse: this is the first Kepler law. For an ellipse, a trajectory point is solely determined by the angle θ. Just like for θ′ = θ + 2π we find the same position values, we can say that the trajectory is closed in the orbital plane. The period of time that a satellite needs to pass from an angle θ to an angle θ + 2π is called a period: this is the period of time required for circling the Earth. An elliptic orbit has two main points:

– for θ – θ0 = 0, we see that value r is at its minimum. We say that we are at the “perigee”: the distance to Earth is denoted as rP;

– for θ – θ0 = π, we see that the value r is at its maximum. We say that we are at the “apogee”: the distance to Earth is denoted as rA.

We can easily deduce the relations:

[1.3]

– Since the attractive force is colinear to the distance vector , and there is no other force, the angular momentum

[1.4]

is conserved, so that:

where C is a constant that represents the law of equal areas, i.e. the second Kepler law.

– An ellipse can be characterized by its semimajor axis a defined by:

By applying the law of equal areas, we obtain the period T of the satellite:

[1.5]

which is the expression of the third Kepler law. The parameters of this period T are, therefore, only a – the semimajor axis – and μ (related to the Earth’s mass).

On an ellipse, the speed is not constant. We show that

[1.6]

except when we have a perfectly circular trajectory, for which we have:

[1.7]

The speed of a satellite varies along its trajectory around the Earth. The speed is, therefore, higher as the distance r becomes smaller. More specifically, if vP is the speed at the perigee (with rP = a(1 – e)) and vA is the speed at the apogee (with rA = a(1 + e)), we get:

From this, we may then deduce the following useful relation:

[1.8]

which shows that the ratio of the speeds to the perigee and apogee depends only on the eccentricity and therefore on the shape of the ellipse.

To conclude on the general aspects of orbits, we must emphasize the fact that these ellipses only need two parameters to be described accurately. We often choose a, the semimajor axis, and e, the eccentricity.

1.3. The quasi-circular orbits of remote sensing satellites

The satellite era started with the launch of the first satellite Sputnik in 1957. Some numbers of civilian remote sensing satellites have since been placed in orbit around the Earth. These orbits, whose eccentricity is very low (e < 0.001), are quasi-circular and, therefore, described either by the semimajor axis a or by their altitude h, defined by the relation:

with RT = 6, 378.137 km being the radius of the Earth at the equator. We often speak of a circular orbit for this type of orbit.

Choosing an orbit for a remote sensing satellite needs to consider several things. First and foremost, since we can show that the energy of an orbit only depends on the semimajor axis a, we must note that the choice in altitude is restrictive in terms of launch: a high altitude requires a launcher that is both heavy and expensive. Therefore, it does not seem to be appropriate to choose a high altitude for an optical imaging system: the resolution being proportional to the distance, a low altitude will allow us – with an identical sensor – to distinguish more details than a high altitude. A last crucial point comes from the atmospheric drags (associated with the effects of solar winds). Friction, more significant as the altitude is lower, is difficult to model and can slowly decrease the altitude and lead the satellite to burn up in the Earth’s low atmosphere. The gravity recovery and climate experiment (GRACE) satellite (a satellite devoted to geodesy), which was launched at an altitude of 485 km, was thus at only 235 km altitude in 5 years, or in other words, it had a loss in altitude of 250 km in 5 years. For the Satellite pour 1’Observation de la Terre (SPOT) satellites (h = 820 km), the daily loss in altitude is of the order of 1 m, which leads to monthly orbit corrections. Thus, remote sensing satellites are regularly subject to orbital maneuvers seeking to return them to their nominal altitude, which, in turn, needs more fuel in order to be performed (RADARSAT-1 was thus carrying 57 kg of propellant). Therefore, these are the multiple reasons that cause the majority of civilian remote sensing satellites to be placed in orbit at altitudes varying between 450 km (QuickBird 2) and 910 km (Landsat 1).

In the case of a perfectly circular orbit, the orbital period T is thus written as (equation [1.5]):

[1.9]

and the speed V is written as (constant on a circular orbit, relation [1.7]):

which gives the following table for different circular orbits:

It is important to remember the orders of magnitude: for a remote sensing satellite, the orbital period is of the order of 100 min, the speed is of 7.5 km/s and the number of orbits per day is around 15. More precisely, SPOT is at an altitude of 822 km: its period is 101.46 min, its speed is 7.424 km/s and it orbits 14.19 times a day.

We have seen that for an elliptic orbit, the distance to the focus and the speed are not constant. If we analyze the values of the eccentricities of the remote sensing satellite orbits, we notice that they are extremely small: for example, for SPOT, we have e = 1.14 × 10–3. In this case, the elliptical trajectories present some particularities:

– First, the shape of...