The difference between Vhelio and Vlsr is maximally 20 km/s so that only when using the LWS03 or LWS04 does the velocity difference become significant. To assure that your wavelength of interest is centered as accurately as possible in the central resolution element, conversion of Vlsr to Vhelio is recommended. Note, however, that the Vearth and Viso will also contribute to offsetting the central wavelength within in the resolution elements, and therefore a minimum of 3 resolution elements on either side of the central element are required for the LWS04 AOT parameters.
Vhelio = Vlsr - V_sun o (xt,yt,zt)
where V_sun is a vector and "o" indicates a dot product. (xt, yt, zt) indicate the (x, y, z) directional cosines to the target.
The (xt, yt, zt) directional cosines of the target are:
xt = -cos(DEC) cos(RA), where x is the direction RA = 180d, DEC = 0 yt = cos(DEC) sin(RA), where y is the direction RA = 90d, DEC = 0 zt = sin(DEC), where z is the direction DEC = 90d
Vhelio = Vlsr - V_sun o (xt,yt,zt) Vhelio = Vlsr - [ ( -0.14 x xt ) + ( -17.32 x yt ) + ( 10.06 x zt ) ] Example: If one were looking at RA=0, DEC=0, one would subtract 0.14 km/s from Vlsr to get Vhelio, since xt = -1 and yt = zt = 0.
ut = -cos(b) cos(l), where u is the direction l = 180, b=0 vt = cos(b) sin(l), where v is the direction l = 90, b=0 wt = sin(b), where w is the direction b = 90
Vhelio = Vlsr - V_sun o (ut,vt,wt) Vhelio = Vlsr - [ ( -10.27 x ut ) + ( 15.32 x vt ) + ( 7.74 x wt ) ] Example: If one were looking at the Galactic Center, (l=0, b=0), one would subtract 10.27 km/s from Vlsr to get Vhelio, since ut = -1 and vt = wt = 0.
However the Mihalis and Binney also give a revised value for V_sun of 16.5 km/s in the (l=53d, b=25d) direction (u, v, w) = (-9, 12, 7) km/s (Equation 6-31). See also Binney and Tremaine 1987, "Galactic Dynamics," Princeton University Press, Equation 1-10. We do not use this revised value above, although it may in fact be more accurate, for the following reason. Since most observatories derive Vlsr from the Vhelio observed for a target, to convert back to Vhelio, it is best to use the same definition of V_sun as was used originally at the time of observation. Thus, the old "standard" value is thus most likely the best value.