We can use equation (5.38) to eliminate the ratio vp/va. Solving for va gives

If we put these into equation (5.36), the total energy simplifies to

5.4.4 Observing elliptical orbits

In studying the Doppler shifts of elliptical orbits as compared with circular orbits, there are three important differences:

(1) The speed along an elliptical orbit is not constant.

(2) In an elliptical orbit the velocity is not perpendicular to the line from the center of mass to the orbiting object.

(3) Even if you are in the plane of the orbit, the radial velocity curve depends on where you are relative to the major axis of the ellipse.

Now that we have the ratio vp/va, we need another relation between them to be able to solve for va and vp individually. We can use conservation of energy to equate the energies at the apas-tron and periastron. Using equation (5.36) gives va___= v2___g

A To Observer

A To Observer

We can now use this in the left-hand side of equation (5.36). We can then solve for v at any point r:

(a) Radial velocity vs. time, t, for an elliptical orbit. (b) In contrast to the circular orbit both the magnitude and direction of v change throughout the orbit. (We assume for this figure that the observer is in the plane of the orbit, or that i = 90°.) Four points are shown in the orbit and in the radial velocity curve. At points 2 and 4 the motion is perpendicular to the line of sight, so vr = 0. For point 1 the motion is directly toward the observer, producing the maximum negative vr, and, for point 3, the motion is away from the observer, producing the maximum positive vr .The motion is also faster at 1 than at 3. In addition, going from 4 to 1 to 2 takes less time than going from 2 to 3 to 4.This accounts for the distorted shape of the radial velocity curve.

(a) Radial velocity vs. time, t, for an elliptical orbit. (b) In contrast to the circular orbit both the magnitude and direction of v change throughout the orbit. (We assume for this figure that the observer is in the plane of the orbit, or that i = 90°.) Four points are shown in the orbit and in the radial velocity curve. At points 2 and 4 the motion is perpendicular to the line of sight, so vr = 0. For point 1 the motion is directly toward the observer, producing the maximum negative vr, and, for point 3, the motion is away from the observer, producing the maximum positive vr .The motion is also faster at 1 than at 3. In addition, going from 4 to 1 to 2 takes less time than going from 2 to 3 to 4.This accounts for the distorted shape of the radial velocity curve.

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