Oort constants

The Oort constants (discovered by Jan Oort) A {\displaystyle A} and B {\displaystyle B} are empirically derived parameters that characterize the local rotational properties of our galaxy, the Milky Way, in the following manner:

A = 1 2 ( V 0 R 0 − d v d r | R 0 ) B = − 1 2 ( V 0 R 0 + d v d r | R 0 ) {\displaystyle {\begin{aligned}&A={\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}-{\frac {dv}{dr}}{\Bigg \vert }_{R_{0}}\right)\\&B=-{\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}+{\frac {dv}{dr}}{\Bigg \vert }_{R_{0}}\right)\\\end{aligned}}}

where V 0 {\displaystyle V_{0}} and R 0 {\displaystyle R_{0}} are the rotational velocity and distance to the Galactic Center, respectively, measured at the position of the Sun, and v and r are the velocities and distances at other positions in our part of the galaxy. The rotational velocity means the average speed of stars in the circumferential direction, which differs from that of a particular star or the sun. As derived below, A and B depend only on the motions and positions of stars in the solar neighborhood. As of 2018, the most accurate values of these constants are A {\displaystyle A} = 15.3 ± 0.4 km s−1 kpc−1, B {\displaystyle B} = −11.9 ± 0.4 km s−1 kpc−1.

Historical significance and background

By the 1920s, a large fraction of the astronomical community had recognized that some of the diffuse, cloud-like objects, or nebulae, seen in the night sky were collections of stars located beyond our own local collection of star clusters. These galaxies had diverse morphologies, ranging from ellipsoids to disks. The concentrated band of starlight that is the visible signature of the Milky Way was indicative of a disk structure for our galaxy; however, our location within our galaxy made structural determinations from observations difficult. Classical mechanics predicted that a collection of stars could be supported against gravitational collapse by either random velocities of the stars or their rotation about its center of mass. For a disk-shaped collection, the support should be mainly rotational. Depending on the mass density, or distribution of the mass in the disk, the rotation velocity may be different at each radius from the center of the disk to the outer edge. A plot of these rotational velocities against the radii at which they are measured is called a rotation curve. For disk galaxies other than the Milky Way, one can measure the rotation curve by observing the Doppler shifts of spectral features measured along different galactic radii, since one side of the galaxy will be moving towards our line of sight and one side away. However, our position in the galactic midplane of the Milky Way, where dust in molecular clouds obscures most optical light in many directions, made obtaining our own rotation curve technically difficult until the discovery of the 21 cm hydrogen line in the 1930s. To confirm the rotation of our galaxy prior to this, in 1927 Jan Oort derived a way to confirm Galactic rotation from just a small fraction of stars in the local neighborhood. As described below, the values he found for A {\displaystyle A} and B {\displaystyle B} proved not only that the Galaxy was rotating but also that it rotates differentially, or as a fluid rather than a solid body.

Derivation

Consider a star in the midplane of the galactic disk with Galactic longitude l {\displaystyle l} at a distance d {\displaystyle d} from the Sun. Assume that both the star and the Sun have circular orbits around the center of the Galaxy at radii of R {\displaystyle R} and R 0 {\displaystyle R_{0}} from the Galactic Center and rotational velocities of V {\displaystyle V} and V 0 {\displaystyle V_{0}} , respectively. The motion of the star along our line of sight, or radial velocity, and motion of the star across the plane of the sky, or transverse velocity, as observed from the position of the Sun are then:

V obs, r = V star, r − V sun, r = V cos ⁡ ( α ) − V 0 sin ⁡ ( l ) V obs, t = V star, t − V sun, t = V sin ⁡ ( α ) − V 0 cos ⁡ ( l ) {\displaystyle {\begin{aligned}&V_{\text{obs, r}}=V_{\text{star, r}}-V_{\text{sun, r}}=V\cos \left(\alpha \right)-V_{0}\sin \left(l\right)\\&V_{\text{obs, t}}=V_{\text{star, t}}-V_{\text{sun, t}}=V\sin \left(\alpha \right)-V_{0}\cos \left(l\right)\\\end{aligned}}}

With the assumption of circular motion, the rotational velocity is related to the angular velocity by v = Ω r {\displaystyle v=\Omega r} and we can substitute this into the velocity expressions:

V obs, r = Ω R cos ⁡ ( α ) − Ω 0 R 0 sin ⁡ ( l ) V obs, t = Ω R sin ⁡ ( α ) − Ω 0 R 0 cos ⁡ ( l ) {\displaystyle {\begin{aligned}&V_{\text{obs, r}}=\Omega R\cos \left(\alpha \right)-\Omega _{0}R_{0}\sin \left(l\right)\\&V_{\text{obs, t}}=\Omega R\sin \left(\alpha \right)-\Omega _{0}R_{0}\cos \left(l\right)\\\end{aligned}}}

From the geometry in Figure 1, one 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:

R cos ⁡ ( α ) = R 0 sin ⁡ ( l ) R sin ⁡ ( α ) = R 0 cos ⁡ ( l ) − d {\displaystyle {\begin{aligned}&R\cos \left(\alpha \right)=R_{0}\sin \left(l\right)\\&R\sin \left(\alpha \right)=R_{0}\cos \left(l\right)-d\\\end{aligned}}}

and with these we get

V obs, r = ( Ω − Ω 0 ) R 0 sin ⁡ ( l ) V obs, t = ( Ω − Ω 0 ) R 0 cos ⁡ ( l ) − Ω d {\displaystyle {\begin{aligned}&V_{\text{obs, r}}=\left(\Omega -\Omega _{0}\right)R_{0}\sin \left(l\right)\\&V_{\text{obs, t}}=\left(\Omega -\Omega _{0}\right)R_{0}\cos \left(l\right)-\Omega d\\\end{aligned}}}

To put these expressions only in terms of the known quantities l {\displaystyle l} and d {\displaystyle d} , we take a Taylor expansion of Ω − Ω 0 {\displaystyle \Omega -\Omega _{0}} about R 0 {\displaystyle R_{0}} .

( Ω − Ω 0 ) = ( R − R 0 ) d Ω d r | R 0 + . . . {\displaystyle \left(\Omega -\Omega _{0}\right)=\left(R-R_{0}\right){\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}+...}

Additionally, we take advantage of the assumption that the stars used for this analysis are local, i.e. R − R 0 {\displaystyle R-R_{0}} is small, and the distance d {\displaystyle d} to the star is smaller than R {\displaystyle R} or R 0 {\displaystyle R_{0}} , and we take:

R − R 0 = − d ⋅ cos ⁡ ( l ) {\displaystyle R-R_{0}=-d\cdot \cos \left(l\right)} . So:

V obs, r = − R 0 d Ω d r | R 0 d ⋅ cos ⁡ ( l ) sin ⁡ ( l ) V obs, t = − R 0 d Ω d r | R 0 d ⋅ cos 2 ⁡ ( l ) − Ω d {\displaystyle {\begin{aligned}&V_{\text{obs, r}}=-R_{0}{\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}d\cdot \cos \left(l\right)\sin \left(l\right)\\&V_{\text{obs, t}}=-R_{0}{\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}d\cdot \cos ^{2}\left(l\right)-\Omega d\\\end{aligned}}}

Using the sine and cosine half angle formulae, these velocities may be rewritten as:

V obs, r = − R 0 d Ω d r | R 0 d sin ⁡ ( 2 l ) 2 V obs, t = − R 0 d Ω d r | R 0 d ( cos ⁡ ( 2 l ) + 1 ) 2 − Ω d = − R 0 d Ω d r | R 0 d cos ⁡ ( 2 l ) 2 + ( − 1 2 R 0 d Ω d r | R 0 − Ω ) d {\displaystyle {\begin{aligned}&V_{\text{obs, r}}=-R_{0}{\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}d{\frac {\sin \left(2l\right)}{2}}\\&V_{\text{obs, t}}=-R_{0}{\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}d{\frac {\left(\cos \left(2l\right)+1\right)}{2}}-\Omega d=-R_{0}{\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}d{\frac {\cos \left(2l\right)}{2}}+\left(-{\frac {1}{2}}R_{0}{\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}-\Omega \right)d\\\end{aligned}}}

Writing the velocities in terms of our known quantities and two coefficients A {\displaystyle A} and B {\displaystyle B} yields:

V obs, r = A d sin ⁡ ( 2 l ) V obs, t = A d cos ⁡ ( 2 l ) + B d {\displaystyle {\begin{aligned}&V_{\text{obs, r}}=Ad\sin \left(2l\right)\\&V_{\text{obs, t}}=Ad\cos \left(2l\right)+Bd\\\end{aligned}}}

where

A = − 1 2 R 0 d Ω d r | R 0 B = − 1 2 R 0 d Ω d r | R 0 − Ω {\displaystyle {\begin{aligned}&A=-{\frac {1}{2}}R_{0}{\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}\\&B=-{\frac {1}{2}}R_{0}{\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}-\Omega \\\end{aligned}}}

At this stage, the observable velocities are related to these coefficients and the position of the star. It is now possible to relate these coefficients 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:

Ω = v r d Ω d r | R 0 = d ( v / r ) d r | R 0 = − V 0 R 0 2 + 1 R 0 d v d r | R 0 {\displaystyle {\begin{aligned}&\Omega ={\frac {v}{r}}\\&{\frac {d\Omega }{dr}}{\Bigg \vert }_{R_{0}}={\frac {d(v/r)}{dr}}{\Bigg \vert }_{R_{0}}=-{\frac {V_{0}}{R_{0}^{2}}}+{\frac {1}{R_{0}}}{\frac {dv}{dr}}{\Bigg \vert }_{R_{0}}\\\end{aligned}}}

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

Measurements

As mentioned in an intermediate step in the derivation above:

V obs, r = A d sin ⁡ ( 2 l ) V obs, t = A d cos ⁡ ( 2 l ) + B d {\displaystyle {\begin{aligned}&V_{\text{obs, r}}=A\,d\,\sin \left(2l\right)\\&V_{\text{obs, t}}=A\,d\,\cos \left(2l\right)+B\,d\\\end{aligned}}}

Therefore, we can write the Oort constants A {\displaystyle A} and B {\displaystyle B} as:

A = V obs, r d sin ⁡ ( 2 l ) B = V obs, t d − A cos ⁡ ( 2 l ) {\displaystyle {\begin{aligned}&A={\frac {V_{\text{obs, r}}}{d\,\sin \left(2l\right)}}\\&B={\frac {V_{\text{obs, t}}}{d}}-A\,\cos \left(2l\right)\\\end{aligned}}}

Thus, the Oort constants can be expressed in terms of the radial and transverse velocities, distances, and galactic longitudes of objects in our galaxy - all of which are, in principle, obs

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