Celestial Mechanics
Celestial mechanics, the study of motions of celestial bodies, together with spherical astronomy, was the main branch of astronomy until the end of the 19th century, when astrophysics began to evolve rapidly. The primary task of classical celestial mechan
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Celestial mechanics, the study of motions of celestial bodies, together with spherical astronomy, was the main branch of astronomy until the end of the 19th century, when astrophysics began to evolve rapidly. The primary task of classical celestial mechanics was to explain and predict the motions of planets and their satellites. Several empirical models, like epicycles and Kepler's law, had been employed to describe these motions. But none of these models explained why the planets moved the way they did. It was only in the 1680's that a simple explanation was found for all these motions - Newton's law of universal gravitation. In this chapter, we will derive some properties of orbital motion. The physics we need for this is simple indeed, just Newton's laws. (For a review, see *Newton's Laws, p. 149.) This chapter is mathematically slightly more involved than the rest of the book. We shall use some vector calculus to derive our results, which, however, can be easily understood with very elementary mathematics. A summary of the basic facts of vector calculus is given in Appendix A.3.
7.1 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 ml and m2 and the radius vectors in some fixed inertial coordinate frame, rl and r2 (Fig. 7.1). The position of the planet relative to the Sun, is denoted by r = r2 - rl. According to Newton's law of gravitation, the planet feels a gravitational pull proportional to the masses ml 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
o
Fig. 7.1. The radius vectors of the Sun and a planet in an arbitrary inertial frame are 'I and ' 2' and, ='2 is th e position of the planet relative to the Sun
H. Karttunen et al. (eds.), Fundamental Astronomy © Springer-Verlag Berlin Heidelberg 1994
r.
7. Celestial Mechanics
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(7.1) where G is the gravitational constant. (More about this in Sect. 7.5.) Newton's second law tells us that the acceleration T2 of the planet is proportional to the applied force: (7.2) Combining (7.1) and (7.2), we get the equation oj motion of the planet
m2 T2 =
-
r
Gmlm2-3 .
r
(7.3)
Since the Sun feels the same gravitational pull, but in the opposite direction, we can immediately write the equation of motion of the Sun: (7.4) 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 (7.3) and (7.4), and subtract (7.4) from (7.3) to get
..
r
r=-fl3 , r
(7.5)
where we have denoted (7.6)
The solution of (7.5) now gives the relative orbit of the planet. The equation involves the radius vector and its second time derivative. In principle, the solution shoul
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