Equations of Motion

We shall concentrate on the systems of only two bodies. In fact, this is the most complicated case that allows a neat analytical solution. For simplicity, let us call the bodies the Sun and a planet, although they could quite as well be a planet and its moon, or the two components of a binary star.

Let the masses of the two bodies be m 1 and m2 and the radius vectors in some fixed inertial coordinate frame r1 and r2 (Fig. 6.1). The position of the planet relative to the Sun is denoted by r = r2 — r1. According to Newton's law of gravitation the planet feels a gravitational pull proportional to the masses m 1 and m2 and inversely proportional to the square of the distance r. Since the force is directed towards the Sun, it can be expressed as

where G is the gravitational constant. (More about this in Sect. 6.5.)

Newton's second law tells us that the acceleration r2 of the planet is proportional to the applied force:

Fig. 6.1. The radius vectors of the Sun and a planet in an arbitrary inertial frame are ri and r2, and r = r2 — ri is the position of the planet relative to the Sun

Since the Sun feels the same gravitational pull, but in the opposite direction, we can immediately write the equation of motion of the Sun:

We are mainly interested in the relative motion of the planet with respect to the Sun. To find the equation of the relative orbit, we cancel the masses appearing on both sides of (6.3) and (6.4), and subtract (6.4) from (6.3) to get

Combining (6.1) and (6.2), we get the equation of motion of the planet r = —\\-

where we have denoted

Hannu Karttunen et al. (Eds.), Celestial Mechanics.

In: Hannu Karttunen et al. (Eds.), Fundamental Astronomy, 5th Edition. pp. 113-130 (2007) DOI: 11685739_6 © Springer-Verlag Berlin Heidelberg 2007

The solution of (6.5) now gives the relative orbit of the planet. The equation involves the radius vector and its second time derivative. In principle, the solution should yield the radius vector as a function of time, r = r(t). Unfortunately things are not this simple in practice; in fact, there is no way to express the radius vector as a function of time in a closed form (i. e. as a finite expression involving familiar elementary functions). Although there are several ways to solve the equation of motion, we must resort to mathematical manipulation in one form or another to figure out the essential properties of the orbit. Next we shall study one possible method.

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