Differentiating both sides tells us that dM(r)

If v(R) is a constant, say v0, then equation (16.3) tells us that v0r

Equating the two expressions for dM(r)/dr gives p(r) =

This means that the density falls as 1/r2. While this sounds like a rapid falloff, remember that the volume of a shell of thickness dr and radius r is 4-nr2 dr. So, in calculating the mass of each shell, the r2 factors cancel. This means that, as we go farther out, the mass of each shell stays constant. So, as far as we continue to add shells with this 1/r2 density falloff, the mass of the galaxy grows by the same amount with every shell we add.

Once we have a rotation curve for our galaxy, it is possible to use measured Doppler shifts to determine distances to objects. Since these distances are determined from the motions of the objects, they are called kinematic distances. For any particular object, we measure vr and t. We then determine the angular speed Q from n = (vr/R0 sin / ) + n0

ambiguity. Remember, there is no distance ambiguity for material outside the Sun's orbit, making this an interesting part of the galaxy to study.

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