Use astropy to convert from LSRK to BARY reference frame.

convert-frames.py

from astropy.coordinates import SkyCoord, ICRS, LSR
import astropy.units as u
import numpy as np
from numpy.linalg import norm

# convert from degrees to radian
deg = np.pi / 180.0

# coordinates of AK Sco
# 16 54 44.8497760618 -36 53 18.560813962
sc_icrs = SkyCoord(ra=253.686874, dec=-36.88849, unit=(u.deg, u.deg))

# print(sc_icrs.ra.hms)
# print(sc_icrs.dec)

ua = sc_icrs.ra.degree
ud = sc_icrs.dec.degree

# define "cartesian" coordinates where Z = North, X = 0 deg RA.
x = np.cos(ud * deg) * np.cos(ua * deg)
y = np.cos(ud * deg) * np.sin(ua * deg)
z = np.sin(ud * deg)

# from Alencar+03
# barycentric velocity in cartesian coordinates
u_uvec = np.array([x, y, z])
ak_bary = -1.97
u_vec = ak_bary * u_uvec


print("AK Sco Vec: v = {:.3f}x + {:.3f}y + {:.3f}z km/s".format(*u_vec))
# print("AK Sco Norm: {:.3f} km/s".format(norm(u_vec)))
print("AK Sco BARY velocity: {:.3f} km/s".format(ak_bary))

# the direction of the LSRK frame
sc_lsrk = SkyCoord("18:03:50.29 +30:00:16.8", unit=(u.hourangle, u.deg))
# from: http://www.gb.nrao.edu/~fghigo/gbtdoc/doppler.html

za = sc_lsrk.ra.degree
zd = sc_lsrk.dec.degree

x = np.cos(zd * deg) * np.cos(za * deg)
y = np.cos(zd * deg) * np.sin(za * deg)
z = np.sin(zd * deg)

lsrk_vec = 20.0 * np.array([x, y, z])
print()
print("LSRK Vec: v = {:.3f}x + {:.3f}y + {:.3f}z".format(*lsrk_vec))
# print("LSRK Norm: {:.3f} km/s".format(norm(lsrk_vec)))

print()
print("LSRK Vec dotted on to AK Sco norm")
# ak_lsrk = np.dot(lsrk_vec, u_uvec)
lsrk_dot = np.dot(lsrk_vec, u_uvec)
print("v_LSRK = v_BARY + {:.4} km/s".format(lsrk_dot))

print("AK Sco LSRK velocity {:.4f} km/s".format(ak_bary + lsrk_dot))


# My confusion in UZ Tau resulted from seeming to add the two vectors (which is true in a 3D space sense).
# However what we're really concerned about is the projection of the LSRK vector *onto* the vector directed
# towards UZ Tau E, since this is how much the *radial velocity* of UZ Tau will change. Testing this with
# NED helped sort things out, including the following equation
# V_con = V + V_apex [ sin(b) sin(b_apex) + cos(b) cos(b_apex) cos(l - l_apex)] which made me think this was
# a dot-product operation and got me thinking in the right direction towards radial velocities rather than
# true 3D space motions.