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# Convert a radial velocity to the Galactic Standard of Rest (GSR)

Radial or line-of-sight velocities of sources are often reported in a Heliocentric or Solar-system barycentric reference frame. A common transformation incorporates the projection of the Sun’s motion along the line-of-sight to the target, hence transforming it to a Galactic rest frame instead (sometimes referred to as the Galactic Standard of Rest, GSR). This transformation depends on the assumptions about the orientation of the Galactic frame relative to the bary- or Heliocentric frame. It also depends on the assumed solar velocity vector. Here we will demonstrate how to perform this transformation using a sky position and barycentric radial-velocity.

Use the latest convention for the Galactocentric coordinates:

    >>> import astropy.coordinates as coord
    >>> coord.galactocentric_frame_defaults.set("latest")

For this example, let’s work with the coordinates and barycentric radial velocity of the star HD 155967, as obtained from Simbad:

    >>> from astropy import units as u
    >>> icrs = coord.SkyCoord(
    ...     ra=258.58356362 * u.deg,
    ...     dec=14.55255619 * u.deg,
    ...     radial_velocity=-16.1 * u.km / u.s,
    ...     frame="icrs",
    ... )

Next, we need to decide on the velocity of the Sun in the assumed GSR frame. We will use the same velocity vector as used in the `Galactocentric` frame, and convert it to a `CartesianRepresentation` object using the `.to_cartesian()` method of the `CartesianDifferential` object `galcen_v_sun`:

    >>> v_sun = coord.Galactocentric().galcen_v_sun.to_cartesian()

We now need to get a unit vector in the assumed Galactic frame from the sky position in the ICRS frame above. We will use this unit vector to project the solar velocity onto the line-of-sight:

    >>> gal = icrs.transform_to(coord.Galactic)
    >>> cart_data = gal.data.to_cartesian()
    >>> unit_vector = cart_data / cart_data.norm()

Now we project the solar velocity using this unit vector:

    >>> v_proj = v_sun.dot(unit_vector)

Finally, we add the projection of the solar velocity to the radial velocity to get a GSR radial velocity:

    >>> rv_gsr = icrs.radial_velocity + v_proj
    >>> print(rv_gsr)
    123.30460087379765 km / s

We could wrap this in a function so we can control the solar velocity and reuse the above code:

    >>> def rv_to_gsr(c, v_sun=None):
    ...     """Transform a barycentric radial velocity to the Galactic Standard of Rest
    ...     (GSR).
    ...
    ...     Parameters
    ...     ----------
    ...     c : `~astropy.coordinates.BaseCoordinateFrame` subclass instance
    ...         The radial velocity, associated with a sky coordinates, to be
    ...         transformed.
    ...     v_sun : `~astropy.units.Quantity`, optional
    ...         The 3D velocity of the solar system barycenter in the GSR frame.
    ...         Defaults to the same solar motion as in the
    ...         `~astropy.coordinates.Galactocentric` frame.
    ...
    ...     Returns
    ...     -------
    ...     v_gsr : `~astropy.units.Quantity`
    ...         The input radial velocity transformed to a GSR frame.
    ...     """
    ...     if v_sun is None:
    ...         v_sun = coord.Galactocentric().galcen_v_sun.to_cartesian()
    ...
    ...     gal = c.transform_to(coord.Galactic)
    ...     cart_data = gal.data.to_cartesian()
    ...     unit_vector = cart_data / cart_data.norm()
    ...
    ...     v_proj = v_sun.dot(unit_vector)
    ...
    ...     return c.radial_velocity + v_proj

    >>> rv_gsr = rv_to_gsr(icrs)
    >>> print(rv_gsr)
    123.30460087379765 km / s