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velocity along the line of sight is v cos 6. However, 6 = at, so vr = v cos(at) (5.6)

The radial velocity changes sign every half-cycle, and repeats periodically. This is shown in Fig. 5.6. The period of the motion is the circumference, 2-nr, divided by the speed v, so P = 2^/a. If we substitute equation (5.6) into equation (5.4), we find that the spectral lines shift back and forth, with a shift given by

So far we have been considering the situation in which the observer is in the plane of the orbit. If the observer is not in the plane of the orbit, the Doppler shift will be reduced (Fig. 5.7). If i is the angle between the plane of the orbit and the plane of the sky, then the projection of any velocity in the plane of the orbit into the line of sight is v sin i. The angle i is known as the inclination of the orbit. This gives us a radial velocity for an orbiting star vr = v sin i cos(¬ęt)

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