# Vhelio vs. Vlsr

## When is the Conversion Needed?

The LWS01 and LWS02 AOTs use resolution elements 900 km/s wide or greater,
and the LWS03 and LWS04 AOTs use resolution elements 30 km/s wide or greater.
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.

## The Conversion

To convert from Vlsr to Vhelio we
subtract the sun's motion with respect to the
local standard of rest, V_sun, as projected into the direction of the target,
from the target's radial motion with
respect to the local standard of rest, Vlsr:

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 Conversion in Equatorial Coordinates

The Sun's motion with respect to the
lsr is taken to be 20 km/s in the (B1950.0) equatorial direction (18h, 30d)
(see references below).
In the (x, y, z) directions this equals (-0.14, -17.32, 10.06) km/s.
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

Therefore:

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.

## The Conversion in Galactic Coordinates

The Sun's motion with respect to the
lsr is taken to be 20 km/s in the galactic direction (l=56d ,b=23d)
(see references below).
In the (u, v, w) directions this equals
(-10.27, 15.32, 7.74) km/s.
The (ut,vt,wt) directional cosines of the target are:
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

Therefore:

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.

## References

A very complete discussion of the determination of V_sun is found in
Chapter 6 of Mihalis and Binney 1981, "Galactic Astronomy," Freeman Press,
San Francisco. Here the authors cite the "standard" V_sun (used above) in
equation 6-27 and reference Blaauw and Schmidt 1965, "Galactic Structure,"
Chapter 4. We note that most radio observatories still use this old
standard value, often expressed as V_sun = 20 in the direction (18h,30d)
in B1900.0 coordinates. (The precession from B1900.0 to
B1950.0 equinox is insignificant
for this discussion. )
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.