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this way, the motion of the LSR depends only on M(R0).

(2) Kinematic. The origin of the coordinate system moves with the average velocity of all the stars in the vicinity of the Sun. This averages out the effects of the random motions of these stars.

The two definitions should result in the same velocity system. However, there are small differences, which we will ignore. (The difference tells us about the dynamical properties of the galaxy.) With respect to the LSR, the Sun is moving at about 20 km/s towards a right ascension of 18 h, and a declination of 30°. (In galactic coordinates, this is € = 56°, b = 23°.) Once we know the Sun's motion, we can use it to correct Doppler shift measurements to give us the radial velocity of the object with respect to the LSR.

We now look at the Doppler shifts we will observe for some material a distance R from the galactic center, moving in a circular orbit with a speed v(R). The situation is shown in Fig. 16.6 for R < R0, but the result holds for R > R0 (see Problem 16.12). The relative radial velocity is given by

Using the relationship between sines and cosines, gives vr = v(R) sin 0 - v0 sin €

We can measure €, but not 0, so we must eliminate it using the law of sines:

sin(180° - 0)/R0 = sin €/R Simplifying the left-hand side gives

Substituting into equation (16.7) gives vr = R0^(R)sin € - R0n0 sin € Factoring out the R0 sin € gives vr = [fl(R) - fi0]R0sin €

In tracing the behavior of vr, it is convenient to divide the galaxy into quadrants, based on the value of €. This is illustrated in Fig. 16.7.

Let's look along a line of sight at some galactic longitude € and see how the radial velocity changes with increasing distance d from the Sun. This is illustrated in Fig. 16.8. We first look at the case for € < 90° (first quadrant). As we look at material closer to the galactic center, the quantity [^(R) - 00] becomes larger. This means that vr 0 0