Celestial Mechanics

Previously, we explored the basics of binary star systems and their orbital dynamics. Now, we will formally develop how these dynamics arise from Newtonian mechanics and gravitation. This is extremely important; understanding celestial mechanics is crucial for studying stellar evolution, galaxy formation, and cosmology. When we later study the Schwarszschild solution, we will see how celestial mechanics is modified by general relativity.

Table of Contents

• Table of Contents
• Introduction
  • Review of Center of Mass Frame
  • Energy and Angular Momentum Conservation

Introduction

Suppose we have two bodies of masses and orbiting each other under their mutual gravitational attraction. This is a Keplerian binary system, and we can analyze its motion using Newton's laws of motion and gravitation.

Review of Center of Mass Frame

We start by defining the center of mass (cm) of the system, so that we can simplify the two-body problem into an equivalent one-body problem. We have already done the basics in the section on the history of astronomy, but we will review it here for completeness. Let the position vectors of the two bodies be and . The center of mass is given by

We will operate on the vector , which points from mass to mass . Newton's second law for the two-body system can be written as

where is the acceleration of the center of mass. As there are no external forces acting on the system, the center of mass moves with constant velocity, i.e., . This means that the center of mass is at rest or moving with constant velocity, and we can choose a reference frame where the center of mass is . So

We can recover the individual position vectors and in terms of as follows:

where is the reduced mass of the system. Often we can treat one mass as much larger than the other (e.g., a star and a planet), in which case if . In the center of mass frame, Newton's second law becomes (see the section on the history of astronomy for details)

where is the force on mass due to mass . If we plug in the gravitational force

we have

Finally, dropping the subscripts, we have the equation of motion for the relative position vector :

This is equivalent to a one-body problem where a single body of mass moves under the influence of a central gravitational potential due to a fixed mass located at the origin.

Energy and Angular Momentum Conservation

The next step is to derive the conserved quantities of the system, namely the total energy and angular momentum. The total mechanical energy of the system is given by the sum of the kinetic and potential energies:

where is the relative speed of the two bodies. In polar coordinates , the velocity can be expressed as

meaning that