# Oort's formulae

## Oort's formulae

Two mathematical expressions derived by Jan Oort that describe the effects of differential galactic rotation on the radial velocities (v r) and tangential velocities (v t) of stars at an average distance r from the Sun. If each star is moving in a circular orbit about the galactic center, and if r is small in comparison with the Sun's distance from the center, then
v r = Ar sin2l
vt = Br + Ar cos2l
where l is the galactic longitude (see galactic coordinate system) and A and B are Oort's constants. Since v t is not directly measurable the second equation is usually replaced by the expression for proper motion, μ:
μ = 0.211(B + A cos2l )
The equations are valid for μ measured in arc seconds per year, velocities in km s–1, and distances in parsecs. The most widely accepted values for Oort's constants are
A : 15 (km s–1)/kiloparsec
B : –10 (km s–1)/kpc
Most stars, including the Sun, do not move in precisely circular orbits but once a correction has been made for the Sun's velocity with respect to the average velocity of stars in the solar neighborhood (assumed to be circular), the measured variation of v r and μ with l is sufficiently close to that predicted by the equations to support the theory that the general rotation pattern is circular and galactocentric.
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