I am trying to replicate the results found for Gliese 710's closest approach of ~0.05 parsecs in 1.3 million years approximately. I thought that by plotting the sun at (0,0) and using the stars ra,dec, and distance i could plot its position in cartesian. This was simple enough using the following:

x = dist_pc * np.cos(dec_rad) * np.cos(ra_rad)
y = dist_pc * np.cos(dec_rad) * np.sin(ra_rad)
z = dist_pc * np.sin(dec_rad)

I get more or less where i expect gliese 710 to be. I then thought i could determine the velocity vector of gliese 710 and plot its trajectory over 1.5 million years and find the point on this trajectory that is closest to the sun. The problem arises when I determine the cartesian velocity vector. I use the following:

vx = (-radial_velocity * np.cos(ra_rad)*np.cos(dec_rad)) + (v_tan_b * np.cos(ra_rad)*np.sin(dec_rad)) + (v_tan_l*np.sin(ra_rad))
vy = (-radial_velocity * np.sin(ra_rad)*np.cos(dec_rad)) + (v_tan_b*np.sin(ra_rad)*np.sin(dec_rad)) - (v_tan_l*np.cos(ra_rad))
vz = (-radial_velocity * np.sin(dec_rad)) + (v_tan_b*np.cos(dec_rad))

For reference here are my variables:

ra_deg = 274
dec_deg = -1.9386

ra_rad = np.radians(ra_deg)
dec_rad = np.radians(dec_deg)

dist_pc =  19.0

pm_mas_ra = -.414
pm_mas_dec = -0.108


radial_velocity = -14.53 #km/s
years = np.linspace(0, 130000, 1000)

v_tan_l = 4.74 * (pm_mas_ra/1000) * dist_pc #km/s
v_tan_b = 4.74 * (pm_mas_dec/1000) * dist_pc #km/s

Finding the position of the star every 100 years for 1.5 million years and plotting results in a trajectory that goes straight through the Sun. Of course this is unphysical but makes sense because if Gliese-710 has a radial velocity of -14.53 km/s and you're propagating it for 1.5 million years, at some point it will have a distance of zero. At this point doesn't matter where it is on the sky, it will end up at (0,0) in your x/y coordinate system.

So this has left me pondering the issue and i am at a standstill as to how to progress. Have i overlooked something in my methods? Would i benefit from converting to a galactic frame using astropy possibly? I'd appreciate insight from someone more educated in this than me.