Astrodynamics/N-Body Problem

N-Body Problem

Newton's law of universal gravitation only accounts for two bodies, m₁ and m₂. However, in space there are frequently more bodies to be concerned with. We can generalize the problem to say that there is a number N bodies to deal with, and create an equation that can deal with an arbitrary number of bodies.

All of the bodies together are known as a system of bodies, and it is assumed that no external forces are acting on the system.

When studying the N-body problem, it is important to focus on a single body, whose motion is of primary interest. We assume that this body, known as the ith body is able to move freely, but the other N - 1 bodies are stationary. The force of gravity between a random body n and the body i is given by the law of universal gravitation:

$F_{g}n=-\frac{G{m}_i{m}_n}{r_{ni}^3}{𝐫}_{𝐧𝐢}$

Where r_ni is the distance vector between object n and object i. We can express this vector in terms of the location vectors of the two bodies:

${𝐫}_{ni}={𝐫}_i-{𝐫}_n$

The total force of gravity on the ith object is given as:

${𝐅}_g=-\sum _{k=1}^{N-1}\frac{G{m}_i{m}_k}{r_{ki}}{𝐫}_{ki}$

We also know that the force of gravity on the ith body from the ith body (the force of gravity on itself) must be zero. We can also define a new force vector, F_O that consists of all the force elements that can act on a body, besides gravity. Some of these elements are drag, solar radiation pressure, magnetic field pressure, and thrust (in the case of a man-made satellite or rocket). We can define the total force on the object, F_T as a sum of the gravitational force and the other forces:

${𝐅}_T={𝐅}_{gn}+{𝐅}_O$

Motion of the ith Body

We can combine Newton's second law, and the law of universal gravitation:

$m_i{𝐚}_i={𝐅}_{Ti}$

We know that the acceleration is the derivative of the object's velocity. We can expand this as follows:

$\frac{d}{dt}{m}_i{𝐯}_i={𝐅}_T$

We can separate this equation into two terms, using the product rule of differentiation:

$m_i\frac{d}{dt}{𝐯}_i+{𝐯}_i\frac{d}{dt}{m}_i={𝐅}_{Ti}$

Now, we put everything in terms of r, remembering that v is the first derivative of r, and that a is the second derivative.

$m_i{{𝐫}_i}^{″}+{m_i}^{\prime }{{𝐫}_i}^{\prime }={𝐅}_T$

It is tempting to say that the second term, the time derivative of the ith bodies mass is zero, because the mass is a constant. However, consider the case of a rocket that is expelling fuel to travel, or a star that is radiating matter and energy. In these cases, it is obvious that the mass of an object is hardly constant. We will, therefore, not ignore this second term, although with the knowledge that in simple cases it does become zero.

The fact that many of the terms in this equation are vectors helps to hide the true complexity of this equation. It is worth mention here that this equation is highly complex, and nearly impossible to solve as it has been presented here. If we do assume that the time-derivative of the mass of the ith body is indeed zero, and if we assume that F_T = F_g (we are ignoring non-gravitational forces), we can simplify the equation as such:

${{𝐫}_i}^{″}=-\sum _{k=1}^{N-1}\frac{G{m}_k}{r_{ki}^3}{𝐫}_{ki}$

Gravitational Parameter

In a practical sense, whenever we are talking about man-made orbital objects, we are considering a large focus (the earth or the sun, typically) and a relatively small secondary body (a rocket, a satellite, etc). In the law of universal gravity, we can say that m₁ >> m₂, where the mathematical relation symbol ">>" means "much greater then". Because of this relationship, we can say that:

$G(m_1+m_2)=G{m}_1$

This is because the mass of the satellite is insignificant to the mass of the primary body. Because the two bodies are different, and they act differently, we can denote the relationship with a quick change of variables:

M = m₁

m = m₂

Where the bigger M is the bigger body, and the smaller m is the smaller body. Because the term GM is so common when studying orbits, it's common to simplify this term into a single variable μ. This variable, the 'gravitational parameter is defined as:

$\mu =GM$

μ is a constant for every M, although every different primary body will have a different value for μ.

Equation of Motion

In the case of man-made satellites, the effect of the satellites on each other is minimal, and the effects on the satellites from other large masses in space are also small because of the great distances in space. We can simplify the N-body equation into a 2-body problem. Also, we can substitute in our gravitational parameter to further simplify the equation:

${𝐫}^{″}+\frac{\mu }{r^3}𝐫=0$

This equation is the basic equation of motion, and will be our starting point for much of our analysis in the rest of the book. Notice that if we do not ignore the effects of other bodies, as we would when we are looking for a very precise calculation, we have to expand this equation to be:

${{𝐫}_{𝐢}}^{″}+\sum _{k=1}^{N-1}\frac{G{m}_k}{r_{ki}^3}{𝐫}_{ki}=0$

This equation is significantly more difficult to solve, and over long distances the majority of the terms are either minor or they tend to cancel.