and Cosmology

2. The Milky Way as a Galaxy
62
Consider a line-of-sight at fixed Galactic longitude l,
with cos l>0 (thus “inwards”). The radial velocity v r
along this line-of-sight for objects moving on circular
orbits is a function of the distance D, according to
(2.57). If Ω(R) is a monotonically decreasing function,
v r attains a maximum where the line-of-sight is tangent
to the local orbit, and thus its distance R from the Galactic
center attains the minimum value R min . This is the
case at
D = R 0 cos l, R min = R 0 sin l (2.66)
(see Fig. 2.18). The maximum radial velocity there,
according to (2.57), is
v r,max = [Ω(R min ) − Ω 0 ] R 0 sin l
= V(R min ) − V 0 sin l, (2.67)
so that from the measured value of v r,max as a function
of direction l, the rotation curve inside R 0 can be
determined,
( ) R
V(R) = V 0 + v r,max (sin l = R/R 0 ) . (2.68)
R 0
In the optical regime of the spectrum this method
can only be applied locally, i.e., for small D, due to
extinction. This is the case if one observes in a direction
nearly tangential to the orbit of the Sun, i.e., if
0