Governing Equations

The actual equations governing the motion of satellite around the Earth is a more complex problem than the simple two-body problem. The gravitational forces of the Sun, Moon, and other planets, the nonspherical shape of the Earth, and the atmospheric drag are some of the factors that affect the motion of the satellite. However, in the beginning of this course, we will consider the motion of the satellite as a two-body problem. The two-body problem is a simplification of the actual problem where we consider the Earth and the satellite as the only two bodies in the system.

3.1 Two body problem

Assumptions:
1. The bodies are point masses.
2. There are no other forces acting on the system other than the gravitational forces which act along the line joining the two bodies.

Relative motion of two bodies of mass M and m

Figure 3.1: Relative motion of two bodies of mass M and m

Consider the system of two bodies of mass M and m illustratd in Figure 3.1. Let (X’, Y’, Z’) be the inertial set of cartesian coordinates and (X,Y,Z) be a set of nonrotating coordinates parallel to (X’,Y’,Z’) and having an origin coincident with the body of mass M. The position vectors of the bodies M and m with respect to the set (X’,Y’,Z’) are \(\mathbf{r}_M\) and \(\mathbf{r}_m\) respectively.
We define \(\mathbf{r}\) = \(\mathbf{r}_m\) - \(\mathbf{r}_M\)
Now, appling Newton’s law of gravitation in the inertial frame (X’,Y’,Z’) we get,

\[ m\ddot{\mathbf{r}}_m = -\frac{GMm}{r^2}\frac{\mathbf{r}}{r} \]

and

\[ M\ddot{\mathbf{r}}_M = \frac{GMm}{r^2}\frac{\mathbf{r}}{r} \]

where G is the universal gravitational constant which is equal to \(6.67 \times 10^{-11}\) \(Nm^2/kg^2\).

Dividing the above equations by m and M respectively, we get,

\[ \ddot{\mathbf{r}}_m = -\frac{GM}{r^3}\mathbf{r} \tag{1} \]

and

\[ \ddot{\mathbf{r}}_M = \frac{Gm}{r^3}\mathbf{r} \tag{2} \]

Subtracting equation (2) from equation (1) we get,
\[ \ddot{\mathbf{r}} = -\frac{G(M+m)}{r^3}\mathbf{r} \tag{3} \]

Note that since the coordinate set (X,Y,Z) is nonrotating with respect to the coordinate set (X’,Y’,Z’), the magnitudes and directions of r and \(\ddot{\mathbf{r}}\) as measured in the set (X,Y,Z) will be equal to that of measured in the inertial set (X’,Y’,Z’).Thus we can now discard the set (X’,Y’,Z’) and work with the set (X,Y,Z) only which has origin at the body of mass M.

Now, for our application to the problem of Earth-satellite system, we can assume M to be the mass of Earth and m to be the mass of satellite. The mass of satellite is very small compared to the mass of Earth. Hence, G(M+m) \(\approx\) GM. It is convienient to define a parameter, \(\mu\) = GM, which is called the standard gravitational parameter of Earth.

Then equation (3) becomes,

\[ \ddot{\mathbf{r}} = - \frac{\mu}{r^3}\mathbf{r} \tag{4} \]

Equation (4) is the vector differential equation of the relative motion for the two-body problem.

This equation is a second order ordinary differential equation where \(\mathbf{r}\) is a 3-component vector. Thus while solving, integrating this equation twice results in 6 constants of integration. Thus Six parameters are required to specify complete orbital state at any given time. These parameters are called the orbital elements.

Reference:
Bate, Mueller, White. (1971). Section 1.3, Chapter 1, “Fundamentals of Astrodynamics”.