## Elliptical orbits

Geometry of an ellipse.The length of the semimajor axis is a; the length of the semi-minor axis is b.The two foci are at F and F'.The eccentricity is e, and the distances from the two foci to points on the ellipse are r and r'.

Geometry of an ellipse.The length of the semimajor axis is a; the length of the semi-minor axis is b.The two foci are at F and F'.The eccentricity is e, and the distances from the two foci to points on the ellipse are r and r'.

r + r' is a constant. We can see that for a point on the semi-major axis (and the ellipse), this sum is 2a, so it must be 2a everywhere. That is r + r = 2a

The eccentricity of an ellipse is the distance between the foci, divided by 2a. A circle is an ellipse of eccentricity zero (both foci are at the same point, the center of the circle). The eccentricity of any ellipse must be less than unity. From the point where the curve crosses the minor axis, r = r' = a, so

### 5.4.1 Geometry of ellipses

In general, orbiting bodies follow elliptical paths. A circle is just a special case of an ellipse. In this section, we generalize the results from the previous section from circular orbits to elliptical orbits. The basic underlying physical ideas are the same.

We first review the geometry of an ellipse, as shown in Fig. 5.9. We describe the ellipse by its semi-major axis a and semi-minor axis b. Each point on the ellipse satisfies the condition that the sum of the distances from any point to two fixed points, called foci (singular focus), is constant. If r and r' are these two distances then

In a binary system, the center of mass of the two stars will be at one focus on the ellipse. The farthest point from that focus is called the apas-tron. From the figure, we see that the distance from the focus to this point is r(apastron ) = a(1 + e )

The closest point to the focus is called the peri-astron. Its distance from the focus is r(periastron ) = a(1 — e )

The average of these two values is a, the semimajor axis. This is the quantity that replaces the radius of a circular orbit in our study of binary stars.

It is useful to have an expression for the ellipse, relating the variables r and 0. From the law of cosines, we see that r'2 = r2 + (2ae )2 + 2r(2ae ) cose

5.4.2 Angular momentum in elliptical orbits

The gravitational force between two objects always acts along the line joining the two objects. The center of mass also lies along this line. This means that the force on either object points directly from that object towards the center of mass. Therefore, these forces can exert no torques about the center of mass. If there are no torques about the center of

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