Oort rotational formulas

Oort rotational formulas in Leiden

The Oort rotation formulas for the differential rotation of the star system of the Milky Way were developed by the Dutch astronomer Jan Hendrik Oort (1900–1992).

In 1927 Oort succeeded in proving the rotation of our galaxy. With the help of stellar statistics , he observed the stars in the vicinity of the sun and described the differential rotation of the spiral arms . The main object of investigation was the spatial distribution of radial speeds and intrinsic movements.

Since the stars do not exactly follow the differential rotation of the Milky Way, but rather have additional peculiar velocities , the Ortian rotation formulas do not apply to each individual star, but only on average over many stars (Figure 2).

formulation

Figure 1: Geometry in the plane of rotation of the Milky Way

The Ort rotation formulas are:

${\displaystyle v_{\text{r}}\approx A\cdot R\cdot \sin(2l)}$ for the radial velocity of a star (towards or away from the sun) and

${\displaystyle EB\approx A\cdot R\cdot \cos(2l)+B\cdot R}$for the proper motion of a star (more precisely: its component in the plane of rotation of the Milky Way)

with the Oort constants (current numerical values, determined from the results of Hipparcos )

${\displaystyle A={\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}-\left.{\frac {\mathrm {d} v}{\mathrm {d} r}}\right|_{R_{0}}\right)\approx (+14{,}8\pm 0{,}8)\,\mathrm {km/s/kpc} }$( Shear ) and

${\displaystyle B=-{\frac {1}{2}}\left({\frac {V_{0}}{R_{0}}}+\left.{\frac {\mathrm {d} v}{\mathrm {d} r}}\right|_{R_{0}}\right)\approx (-12{,}4\pm 0{,}6)\ \mathrm {km/s/kpc} }$ ( Vortex strength )

as well as for the galactic longitude of the star and its distance from the sun. ${\displaystyle l}$ ${\displaystyle R}$

interpretation

Figure 2: Double wave of proper motion, determined from observation data;
due to the negative sign of , the curve must, strictly speaking, be shifted downwards, cf. ${\displaystyle B}$ ${\displaystyle 2\left|B\right|}$

Radial velocity and proper motion describe a double wave with two maxima and minima over 360 ° of the galactic length (Figure 2).

A + B

${\displaystyle A+B=-{\frac {\mathrm {d} v}{\mathrm {d} r}}{\Big |}_{R_{0}}\approx (+2{,}4\pm 1{,}4)\,\mathrm {km/s/kpc} ,}$

d. H. the rotation curve of the Milky Way is almost flat ( rising slightly ) near the sun . ${\displaystyle v(r)}$

A - B

${\displaystyle A-B={\frac {V_{0}}{R_{0}}}\approx (+27{,}2\pm 1{,}4)\,\mathrm {km/s/kpc} }$

is the angular velocity for the rotation of the sun around the center of the Milky Way . ${\displaystyle \Omega _{0}}$

This corresponds to an orbital period of the sun around the center of the Milky Way of years (i.e. 230 million years), also called the galactic year . ${\displaystyle T_{0}={\tfrac {2\pi }{\Omega _{0}}}\approx 230\cdot 10^{6}}$

With the distance of the sun from the center of the Milky Way, this results in a speed of revolution for the sun , which agrees relatively well with other observation data. ${\displaystyle R_{0}\approx 8\,\mathrm {kpc} }$ ${\displaystyle V_{0}\approx 220\,\mathrm {km/s} }$

The other way around, the distance of the sun from the center of the Milky Way can also be determined. To do this, the speed of the sun relative to objects that do not follow the rotation of the Milky Way (e.g. globular clusters ) must be known. ${\displaystyle A-B}$ ${\displaystyle R_{0}}$ ${\displaystyle V_{0}}$

swell

1. ↑ http://people.virginia.edu/~dmw8f/astr5630/Topic06/Lecture_6.html#sec2
2. ↑ http://people.virginia.edu/~dmw8f/astr5630/Topic06/t6_oort_hipparcos.html