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3. We will now show that the system of orbiting bodies moves with a constant velocity.

We will begin by setting an arbitrary fixed point in space, O, which our two bodies, M and m, will move in relation to. Note that, for ease of calculation, the central body M can be placed at this fixed reference point.

figure 3.1

*note that r = r ₂-r₁.

By Newton's second law, the equations for the gravitational force exerted by each body on the other can be expressed by the equations

where r / r is a simply a unit vector pointing in the direction of r, vectorizing the force.

It should be noted that, using Matlab software, a qualitative analysis of this system is possible without going any further. M-files utilizing the numerical ODE solvers in Matlab produced the following image. These m-files are available in the reference section of this paper.

figure 3.2

We return now to our two second order differential equations from [3.1]. Adding these two equations together and integrating twice gives

We now define the center of mass of the system to be R , determined by the equation

From equation [3.4] above, we can conclude that

Differentiating [3.6] gives

Since R has a constant velocity, the center of mass of the orbiting system will move in a straight line path at a constant speed for all time, unless acted upon by an outside force.

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