and Cosmology

2.4 Kinematics of the Galaxy
where the derivative has to be evaluated at R = R 0 .
Hence
( ) dΩ
v r = (R − R 0 ) R 0 sin l,
dR |R 0
and furthermore, with (2.56),
( ) [ (dV
dΩ
R 0 = R ) ]
0
− V dR |R 0
R dR |R 0
R
( ) dV
≈ − V 0
,
dR |R 0
R 0
in zeroth order in (R − R 0 )/R 0 . Combining the last two
equations yields
[ (dV )
v r = − V ]
0
(R − R 0 ) sin l ; (2.60)
dR |R 0
R 0
in analogy to this, we obtain for the tangential velocity
[ (dV )
v t = − V ]
0
(R − R 0 ) cos l − Ω 0 D .
dR |R 0
R 0
(2.61)
For |R − R 0 |≪R 0 it follows that R 0 − R ≈ D cos l; if
we insert this into (2.60) and (2.61) we get
v r ≈ ADsin 2l , v t ≈ ADcos 2l + BD ,
(2.62)
where A and B are the Oort constants
A := − 1 [ (dV )
− V ]
0
,
2 dR |R 0
R 0
B := − 1 [ (dV )
+ V ]
0
. (2.63)
2 dR |R 0
R 0
The radial and tangential velocity fields relative to the
Sun show a sine curve with period π, where v t and v r
are phase-shifted by π/4. This behavior of the velocity
field in the Solar neighborhood is indeed observed (see
Fig. 2.17). By fitting the data for v r (l) and v t (l) for stars
of equal distance D one can determine A and B, and
thus
Ω 0 = V 0
R 0
= A − B,
( ) dV
=−(A + B) .
dR |R 0
(2.64)
Fig. 2.17. The radial velocity v r of stars at a fixed distance
D is proportional to sin 2l; the tangential velocity v t is a linear
function of cos 2l. From the amplitude of the oscillating
curves and from the mean value of v t the Oort constants A
and B can be derived, respectively (see (2.62))
The Oort constants thus yield the angular velocity of
the Solar orbit and its derivative, and therefore the
local kinematical information. If our Galaxy was rotating
rigidly so that Ω was independent of the radius,
A = 0 would follow. But the Milky Way rotates differentially,
i.e., the angular velocity depends on the radius.
Measurements yield the following values for A and B,
A = (14.8 ± 0.8) km s −1 kpc −1 ,
B = (−12.4 ± 0.6) km s −1 kpc −1 . (2.65)
Galactic Rotation Curve for R