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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.

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    $\begingroup$ Implementation details of computational tasks are off-topic on this site: “While computational physics is on topic, we are not a programming site. If your question is about implementing computational code - in particular, if it's about writing, compiling, debugging or optimizing code, or about a specific language or library - then it is off topic.” $\endgroup$
    – Ghoster
    Commented Feb 29 at 18:33
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    $\begingroup$ There is a physical (not computational) problem here. The problem is that the radial velocity is not a constant. At closest approach the radial velocity goes to zero. $\endgroup$
    – Malcolm
    Commented Feb 29 at 18:46
  • $\begingroup$ @Malcolm indeed my problem. So my question is how do other models get around this? C.A.L. Bailer Jones has plenty of papers where he has calculated closest approaches using Gaia DR3 values. Im trying to understand and visualize how he does this (for example) $\endgroup$
    – user2279603
    Commented Feb 29 at 18:55
  • $\begingroup$ The first approximation, of course, is John Doty's solution below. The next level would be to consider that both the sun and Gliese-710 are orbiting around the center of the galaxy, so the linear velocity isn't exactly constant. Then to be very precise you do need to account for the gravitational interactions with each other and other stars. While gravitational forces are small at this range, small forces over long periods of time can become non-negligible. $\endgroup$
    – Malcolm
    Commented Feb 29 at 19:21

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You're way overthinking this. Ignoring gravity (which you can at this scale), the motion is in a straight line. So, the position after time $t$ is just $(x,y,z)=(x_0,y_0,z_0)+t*(vx,vy,vz)$. The radial velocity changes, of course, but the linear velocity doesn't.

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