User:LemonBalmer/sandbox

Derivation of the Oort constants

The Oort constants were first derived by Jan Oort in 1927 as a way to determine the rotation properties of the Milky Way from only the observed radial and transverse velocities and positions of local stars.

Consider a star in the midplane of the Galactic disk with Galactic longitude $l$ at a distance $d$ from the Sun. Assume that both the Sun and the star have circular orbits around the center of the Galaxy at radii of $R$ and $R_{0}$ from the galactic center and rotational velocities of $V$ and $V_{0}$, respectively. The radial velocity and transverse velocity of the star as we observe from the position of the Sun are then:

$V_{obs,r}=V_{star,r}-V_{sun,r}=Vcos\left(\alpha \right)-V_{0}sin\left(l\right)$

$V_{obs,t}=V_{star,t}-V_{sun,t}=Vsin\left(\alpha \right)-V_{0}cos\left(l\right)$

Since we have assumed circular motion, the rotational velocity will be related to the angular velocity by $v=\omega r$ and we can substitue this into our velocity expressions:

$V_{obs,r}=\omega Rcos\left(\alpha \right)-\omega _{0}R_{0}sin\left(l\right)$

$V_{obs,t}=\omega Rsin\left(\alpha \right)-\omega _{0}R_{0}cos\left(l\right)$

where we have used the angle difference identities to simply the expressions involving $l$. From the geometry in Figure 1, we can see that the triangles formed between the galactic center, the Sun, and the star share a side or portions of sides, so the following relationships hold and substitutions can be made:

$Rcos\left(\alpha \right)=R_{0}sin\left(l\right)$

$Rsin\left(\alpha \right)=R_{0}cos\left(l\right)-d$

and

$V_{obs,r}=\left(\omega -\omega _{0}\right)R_{0}sin\left(l\right)$

$V_{obs,t}=\left(\omega -\omega _{0}\right)R_{0}cos\left(l\right)-\omega d$

To put these expressions only in terms of the known quantities $l$ and $d$, we take advantage of our assumption that the stars used for this analysis are local, i.e. $R-R_{0}$ is small, and take a Taylor expansion of $\omega -\omega _{0}$ about $R_{0}$.

$\left(\omega -\omega _{0}\right)=\left(R-R_{0}\right){\frac {d\omega }{dr}}|_{R_{0}}+...$

Additionally, if the distance to the star is much less than $R$ or $R_{0}$ then $R-R_{0}=-dcos\left(l\right)$. So:

$V_{obs,r}=-R_{0}{\frac {d\omega }{dr}}|_{R_{0}}dcos\left(l\right)sin\left(l\right)$

$V_{obs,t}=-R_{0}{\frac {d\omega }{dr}}|_{R_{0}}dcos^{2}\left(l\right)-\omega d$

Using the sine and cosine half angle formulae, we can rewrite these velocities as:

$V_{obs,r}=-R_{0}{\frac {d\omega }{dr}}|_{R_{0}}d{\frac {sin\left(2l\right)}{2}}$

$V_{obs,t}=-R_{0}{\frac {d\omega }{dr}}|_{R_{0}}d{\frac {\left(cos\left(2l\right)+1\right)}{2}}-\omega d=-R_{0}{\frac {d\omega }{dr}}|_{R_{0}}d{\frac {cos\left(2l\right)}{2}}+\left(-{\frac {1}{2}}R_{0}{\frac {d\omega }{dr}}|_{R_{0}}-\omega \right)d$

Now we can write the velocities in terms of our known quantities and two coefficients $A$ and $B$:

$V_{obs,r}=Adsin\left(2l\right)$

$V_{obs,t}=Adcos\left(2l\right)+Bd$

where

$A=-{\frac {1}{2}}R_{0}{\frac {d\omega }{dr}}|_{R_{0}}$

$B=-{\frac {1}{2}}R_{0}{\frac {d\omega }{dr}}|_{R_{0}}-\omega$

At this stage, our observable velocities are related to these coefficients and the position of the star. How do these coefficients relate to the rotation properties of the galaxy? For a star in a circular orbit, we can express the derivative of the angular velocity with respect to radius in terms of the rotation velocity and radius and evaluate this at the location of the Sun:

$\omega ={\frac {v}{r}}$

${\frac {d\omega }{dr}}|_{R_{0}}={\frac {d{\frac {v}{r}}}{dr}}|_{R_{0}}=-{\frac {V_{0}}{R_{0}^{2}}}+{\frac {1}{R_{0}}}{\frac {dv}{dr}}|_{R_{0}}$

so

$A={\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}-{\frac {dv}{dr}}|_{R_{0}}\right)$

$B=-{\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}+{\frac {dv}{dr}}|_{R_{0}}\right)$

$A$ is the Oort constant describing the shearing motion and $B$ is the Oort constant describing the rotation of the Galaxy. As described below, we can measure $A$ and $B$ from plotting these velocities, measured for many stars, against the galactic longitudes of these stars.