# 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.
```