Oort's constants

Figure 5.1: (Taken from http://www.csupomona.edu/~jis/1999/kong.ps) The observer is moving on a circular orbit with velocity $\Theta _0$ and radius $R_0$, and observes a star at distance $r$, and galactic longitude $l$, on circular orbit with velocity $\Theta$ and radius $R$. In the text, $r$ is called $d$ and the circular velocities are $V_c$ and $V_{c,0}$ for the star and the Sun, respectively.

Suppose you're on a circular orbit with radius $R_0$ around the MW, and you measure the velocities of stars around you. For simplicity, assume all stars are also on circular orbits. Stars on orbits with radius $R<R_0$ will go faster than you do, and so will over take you, and the opposite for stars with $R>R_0$. Jan Oort computed how the velocities you measure for these stars depend on their distance and the direction you see them in.

For the geometry, look at Figure (5.1)^5.2. To keep the notation the same, let's call the circular velocity at the position of the observer $V_{c,0}$, and at the position of the star $V_c$. The radial and tangential velocity of the star with respect to the observer (at the position of the Sun) are

$$V_r = V_c\cos(\alpha) - V_{c,0}\sin(l)$$

$$V_t = V_c\sin(\alpha) - V_{c,0}\cos(l)\,.$$ (5.2)

Now, in the right-angled triangle on the figure, you can convince yourself that

$$d+R\sin(\alpha) = R_0\cos(l)$$

$$R\cos(\alpha) = R_0\sin(l)$$

$$R_0 = d\cos(l) + R\cos(\beta)\approx d\cos(l) + R\,{\hbox{\rm when $d\ll R_0$}}\,,$$ (5.3)

where $\beta$ is the angle Sun-MW Centre-Star. Combining these equations gives

$$V_r = (\Omega-\Omega_0) R_0\sin(l)$$

$$V_t = (\Omega-\Omega_0) R_0\cos(l)-\Omega\, d\,,$$ (5.4)

where $\Omega_0=V_{c,0}/R_0$ is the angular velocity of the observer, and $\Omega=V_c/R$ the angular velocity of the star. If $\vert R-R_0\vert$ is small, we can perform a Taylor expansion to find $\Omega\approx\Omega_0+(d\Omega/dR)(R-R_0)$ and substitute this in the previous expression. Finally we can introduce Oort's constants

$$A \equiv -{1\over 2} \left[ {dV_c\over dR}\vert _{R_0} - {V_{c,0}\over R_0... ...]\approx 14.4\pm 1.2{\hbox{\rm km}}\,{\hbox{\rm s}}^{-1}\,{\hbox{\rm kpc}}^{-1}$$

$$B \equiv -{1\over 2} \left[ {dV_c\over dR}\vert _{R_0} + {V_{c,0}\over R_0... ...prox -12.0\pm 2.8{\hbox{\rm km}}\,{\hbox{\rm s}}^{-1}\,{\hbox{\rm kpc}}^{-1}\,.$$ (5.5)

We'll discuss in a moment how the measured values are found. With these definitions we obtain after some algebra,

$$V_r = A\,d\,\sin(2l)$$

$$V_t = A\,d\,\cos(2l)+Bd\,.$$ (5.6)

Let's see what this means. For a star toward the galactic centre or anti-centre ($l=0$ and $l=180^o$ respectively), we find the radial velocity to be zero, and the tangential velocity to be $(A+B)\,d$. For example with $d=1{\hbox{\rm kpc}}$, $V_t\approx 2.4{\hbox{\rm km s$^{-1}$}}$.

To measure $A$ and $B$, we need to measure the radial and tangential velocities of stars of known distance.

Finally, let's see what we would expect to find for a Keplerian disk, for which $V_c\equiv V_{c,0} (R_0/R)^{1/2}$, and hence $dV_c/dR=-(1/2)V_c/R$. At the position of the Sun, we know $V_{c,0}\approx 220{\hbox{\rm km s$^{-1}$}}\,$ and $R_0\approx 8.5{\hbox{\rm kpc}}$ hence

$$A_{\rm Kepler} = {3\over 4} {V_{c,0}\over R_0}= 19.4{\hbox{\rm km s$^{-1}$}}\,{\hbox{\rm kpc}}^{-1}$$

$$B_{\rm Kepler} = -{1\over 4} {V_{c,0}\over R_0}=-6.5{\hbox{\rm km s$^{-1}$}}\,{\hbox{\rm kpc}}^{-1}\,.$$ (5.7)

This does not compare well with the measured values in Eqs.(5.5). Assuming we got $V_c/R$ correct, let's compare the measured and computed values for $dV_c/dR$,

$${dV_c\over dR}\vert _{\rm measured} = -(A+B) = -2.4\pm 3.0 {\hbox{\rm km s$^{-1}$}}\,{\hbox{\rm kpc}}^{-1}$$

$${dV_c\over dR}\vert _{\rm Keplerian} = -{1\over 2}{V_c\over R} = -13.0{\hbox{\rm km s$^{-1}$}}\,{\hbox{\rm kpc}}^{-1}\,.$$ (5.8)

So what appears to be wrong is that the circular velocity, as computed for Keplerian fall-off, decreases much more rapidly with increasing $R$, than the values measured from Oort's constants. We've assumed $R_0=8.5{\hbox{\rm kpc}}$, but for the two values to agree, we would require $R_0$ to be larger by a factor 6! Clearly, the discrepancy is much larger than the uncertainties in our values of either $V_c$ or $R_0$. We shall see later that there is other evidence that suggests a much more dramatic failure of our underlying assumptions.

We've so far assumed stars to be on exactly circular orbits. This is of course only an approximation, and real stars will have velocities which differ slightly from $V_c(R)$: they will have (small) peculiar velocities in all three Cartesian directions. The reference system that is on a circular orbit with velocity $V_c(R)$ is called the local standard of rest (LSR)^5.3, and, for example the Sun moves with a velocity of $\sim 16 {\hbox{\rm km s$^{-1}$}}$ with respect to its LSR. This is called the solar motion. To determine $A$ and $B$, we need to measure $V_r$ and $V_t$ as a function of $d$ and $l$ statistically, and take into account that we also need to determine the solar motion.

When Oort did his measurements of the constants $A$ and $B$, he discovered that some stars had very large deviations from the expected values of $V_r$ and $V_t$. He called them high velocity stars. He also correctly identified their origin, they are stars in the MW halo. Unlike the disk, the halo does not rotate, and so the high velocities of these stars are due to the rotation velocity of the Sun.