# 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’ll demonstrate how to perform this transformation using a sky position and barycentric radial-velocity.

Make print work the same in all versions of Python and import the required Astropy packages:

import astropy.units as u
import astropy.coordinates as coord


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

icrs = coord.ICRS(ra=258.58356362*u.deg, dec=14.55255619*u.deg,


We next need to decide on the velocity of the Sun in the assumed GSR frame. We’ll 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’ll 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)


Out:

114.8831448932284 km / s


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

def rv_to_gsr(c, v_sun=None):
"""Transform a barycentric radial velocity to the Galactic Standard of Rest
(GSR).

The input radial velocity must be passed in as a

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

v_proj = v_sun.dot(unit_vector)


114.8831448932284 km / s