Solar Motion Corrections to μ and v_r                                               

 

Suppose a star has known α, δ, μ_α , μ_{δ ,}π and v_r. 
Furthermore suppose that the known celestial
coordinates of the solar apex are A and D
and the solar motion is v_¤ .  Unknown
quantities to be determined are τ, the
proper motion component normal to the
great circle passing through the star,
apex and antapex; υ, the proper motion
component in the direction of the
antapex and τ', υ' and v_r.', the values
of τ, υ and v_r after corrections for solar
motion.  Also initially unknown are the
auxiliary quantities λ, the angular
separation of the star and the apex, and
-ψ, the position angle of the apex with
respect to the star.

One can use the following three relationships,
derived from spherical trigonometry, to obtain the
values of λ and ψ:

                cos λ = sin D sin δ + cos D cos δ cos (α - A)                                                                                                           (1)

                sin λ sin ψ = cos D sin (α - A)                                                                                                                                  (2)

                sin λ cos ψ = sin D cos δ - cos D sin δ cos (α - A).                                                                                                 (3)

Equation (1) yields a unique value for λ (since 0° £ λ £  180°).  Equation 2 yields a value for sin ψ and equation (3) yields a value for
cos ψ, from which a unique solution can be obtained for ψ (0° £ ψ < 360°).  Once λ and ψ are determined the following five relationships, obtained from the above figure, yield the values of τ, υ, τ', υ' and v_r':

                τ = 15 μ_α cos δ cos ψ + μ_δ sin ψ                                                                                                                              (4)

      υ = 15 μ_α cos δ sin ψ - μ_δ cos ψ                                                                                                                               (5)

      τ' = τ                                                                                                                                                                         (6)

      υ' = υ - v_¤ sin(λ)π / 4.74                                                                                                                                            (7)

and

      v_r.' = v_r. + v_¤ cos λ.                                                                                                                                                   (8)

Note the unit conventions assumed in these equations:    μ_α [s/yr], μ_δ ["/yr], π ["], v_r [km/s], v_¤ [km/s], τ ["/yr], υ ["/yr], τ' ["/yr], υ' ["/yr] and v_r' [km/s].