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# Table of Contents

Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data.
• ## 1: Numerical Methods

This chapter is not intended as a comprehensive course in numerical methods. Rather it deals, and only in a rather basic way, with the very common problems of numerical integration and the solution of simple (and not so simple!) equations. Specialist astronomers today can generate most of the planetary tables for themselves; but those who are not so specialized still have a need to look up data in tables.
• ## 2: Conic Sections

A particle moving under the influence of an inverse square force moves in an orbit that is a conic section; that is to say an ellipse, a parabola or a hyperbola. We shall prove this from dynamical principles in a later chapter. In this chapter we review the geometry of the conic sections. We start off, however, with a brief review (eight equation-packed pages) of the geometry of the straight line.

• ## 5: Gravitational Field and Potential

This chapter deals with the calculation of gravitational fields and potentials in the vicinity of various shapes and sizes of massive bodies. The reader who has studied electrostatics will recognize that this is all just a repeat of what he or she already knows.
• ## 6: The Celestial Sphere

If you look up in the sky, it appears as if you are at the centre of a vast crystal sphere with the stars fixed on its surface. This sphere is the celestial sphere. It has no particular radius; we record positions of the stars merely by specifying angles. We see only half of the sphere; the remaining half is hidden below the horizon. In this section we describe the several coordinate systems that are used to describe the positions of stars and other bodies on the celestial sphere.
• ## 7: Time

In this chapter we briefly discuss the several time scales that are in use in astronomy, such as Universal Time, Mean Solar Time, Ephemeris Time, Terrestrial Dynamical Time, and the several types of second, hour, day and year that are or have been in use.  Some of the items in this chapter will be given only in short note form or single sentence definitions, particularly where they have already been discussed. Others will require a bit more discussion.
• ## 8: Planetary Motions

In this chapter, I do not attempt to calculate planetary ephemerides, which will come in a later chapter. Rather, I discuss in an idealistic and qualitative manner how it is that a planet sometimes moves in one direction and sometimes in another. That the treatment in this chapter is both idealistic and qualitative by no means implies that it will be devoid of Equations or of quantitative results, or that the matter discussed in this chapter will have no real practical or observational value.
• ## 9: The Two Body Problem in Two Dimensions

In this chapter we show how Kepler’s laws can be derived from Newton’s laws of motion and gravitation, and conservation of angular momentum, and we derive formulas for the energy and angular momentum in an orbit. We show also how to calculate the position of a planet in its orbit as a function of time. The discussion here is limited to two dimensions. The corresponding problem in three dimensions, and how to calculate an ephemeris of a planet or comet in the sky, is discussed elsewhere.

• ## 11: Photographic Astrometry

Astrometry is the art and science of measuring positions of celestial objects, and indeed the first step in determining the orbit of a new asteroid or comet is to obtain a set of good astrometric positions. For much of the twentieth century, most astrometric positions were determined photographically.

• ## 17: Visual Binary Stars

A visual binary is a gravitationally bound system that can be resolved into two stars. These stars are estimated, via Kepler's 3rd law, to have periods ranging from a number of years to thousands of years. A visual binary consists of two stars, usually of a different brightness.
• ## 18: Spectroscopic Binary Stars

There are many binary stars whose angular separation is so small that we cannot distinguish the two components even with a large telescope – but we can detect the fact that there are two stars from their spectra. In favorable circumstances, two distinct spectra can be seen. It might be that the spectral types of the two components are very different – perhaps a hot A-type star and a cool K-type star, and it is easy to recognize that there must be two stars there.
• ## Back Matter

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