page_title: Oort constants
section_title: Derivation
hierarchical_section_title: Oort constants / Derivation
caption_attribution_description: Nederlands: De formules van de Oortconstanten A en B voor de draaiing van melkwegsterrenstelsels op een huismuur van de Witte Singel 58-C, Leiden. A = 1 2 ( V 0 R 0 &#x2212; d v d r | R 0 ) {\displaystyle A={\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}-{\frac {dv}{dr}}|_{R_{0}}\right)} , B = &#x2212; 1 2 ( V 0 R 0 + d v d r | R 0 ) {\displaystyle B=-{\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}+{\frac {dv}{dr}}|_{R_{0}}\right)} English: Oort's constants formulas on a wall of Witte Singel 58-C, Leiden (the Netherlands). A = 1 2 ( V 0 R 0 &#x2212; d v d r | R 0 ) {\displaystyle A={\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}-{\frac {dv}{dr}}|_{R_{0}}\right)} , B = &#x2212; 1 2 ( V 0 R 0 + d v d r | R 0 ) {\displaystyle B=-{\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}+{\frac {dv}{dr}}|_{R_{0}}\right)}
context_page_description: The Oort constants and are empirically derived parameters that characterize the local rotational properties of our galaxy, the Milky Way, in the following manner:
  where and 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. 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 = 15.3 ± 0.4 km s⁻¹ kpc⁻¹, = -11.9 ± 0.4 km s⁻¹ kpc⁻¹. From the Oort constants, it is possible to determine the orbital properties of the Sun, such as the orbital velocity and period, and infer local properties of the Galactic disk, such as the mass density and how the rotational velocity changes as a function of radius from the Galactic center.
context_section_description: Consider a star in the midplane of the Galactic disk with Galactic longitude  at a distance  from the Sun.  Assume that both the star and the Sun have circular orbits around the center of the Galaxy at radii of  and  from the galactic center and rotational velocities of  and , 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:
  With the assumption of circular motion, the rotational velocity is related to the angular velocity by  and we can substitute this into the velocity expressions:
  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:
  and with these we get
  To put these expressions only in terms of the known quantities  and , we take a Taylor expansion of  about .
  Additionally, we take advantage of the assumption that the stars used for this analysis are local, i.e.  is small, and the distance d to the star is smaller than  or , and we take:
  .
  So:
  Using the sine and cosine half angle formulae, these velocities may be rewritten as:
  Writing the velocities in terms of our known quantities and two coefficients  and  yields:
  where
  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:
  so
   is the Oort constant describing the shearing motion and  is the Oort constant describing the rotation of the Galaxy.  As described below, one can measure  and  from plotting these velocities, measured for many stars, against the galactic longitudes of these stars.
caption: Oort constants on a wall in Leiden
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