I have downloaded a star database from this page (The HYG Database):
http://astronexus.com/node/34

And all the stars have (among other parameters) their XYZ coordinates in parsecs according to this definition:

X,Y,Z: The Cartesian coordinates of the star, in a system based on the equatorial coordinates as seen from Earth. +X is in the direction of the vernal equinox (at epoch 2000), +Z towards the north celestial pole, and +Y in the direction of R.A. 6 hours, declination 0 degrees

The problem is: I cannot figure out how to calculate the XYZ coordinates of the center of the galaxy (the Milky Way). So the question is: What are the coordinates of the Milky Way center? (Or is it not possible to calculate it just with that information?). This is an example of the data:
enter image description here

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    Note that your catalog only contains bright and/or nearby stars and the distances are from Hipparcos and are not likely to be trustworthy beyond a few hundred parsecs. So your stars are going to be in a pretty local bubble around the Sun and the furthest star will be less than ~10 % of the way to the Galactic center. A catalog based on Gaia-DR2 parallaxes will give a better idea of the structure of the Milky Way and the location of the galactic center. – astrosnapper Oct 11 at 21:27
  • +1 for a well explained and well-sourced first question by a new user! – uhoh Oct 12 at 5:53
up vote 6 down vote accepted

Distance estimates to the center of our galaxy vary around $r \approx$ 8000 $\pm$ 200 parsecs. Its spherical equatorial coordinates are $\alpha$ = 17h45m40s, $\delta$ = -29$^\circ$00'28" (J2000). Converting those to rectangular form, we have: $$\begin{align} X &= r \cos \delta \cos \alpha \approx -437 \pm 11 \ \mathrm{pc} \\ Y &= r \cos \delta \sin \alpha \approx -6983 \pm 175 \ \mathrm{pc} \\ Z &= r \sin \delta \approx -3879 \pm 97 \ \mathrm{pc} \end{align}$$ As with most celestial objects, its direction is known much more precisely than its distance. We can't pinpoint its 3D position, but we can narrow it down to a long, fuzzy ellipsoid pointing away from the Sun.

@MikeG's answer is excellent. This is a supplementary answer only.

Part of the problem finding the coordinates of the "center" is defining the center. One example might be the center of mass, but finding all the mass is quite a challenge as we can't even see all of the galaxy's mass due to dust clouds, as discussed in How was the galactic plane established?, links therein and @RobJeffries excellent answer, and of course dark matter being invisible and all.

But even though you can't see and identify all of the mass, you can try to have a look at the way that all of the stars that you can see seem to be moving via their radial Doppler shift, and at least try to estimate it.

However, for the purposes of your question, I'm not sure if that's the only definition of "center" that could work for you.

If instead you'd like to consider something more catchy and visual, how about the supermassive black hole that's almost for sure at the "center" of our galaxy, affectionately known as Sagittarius A*? That's just Sgr A* to her friends (see also Why is Sagittarius A* called so?). According to that article, the distance from our Sun to Sgr A* is 7,860 ±140 ±40 pc, which is a remarkable thing in itself because it has error bars on the error bars!

That number is linked to the ArXiv preprint An Improved Distance and Mass Estimate for Sgr A* from a Multistar Orbit Analysis, which is also an analysis of how some very special stars move.

Here's a GIF, found in this excellent answer to What is the evidence for a supermassive black hole at the center of Milky Way?

stars orbiting Sgr A*

The reason that I call this a supplementary answer is that you end up with very nearly the same answer as @MikeG. Here I've used the same procedure:

               X                     Y                    Z
parsecs:     -430±8,              -6861±122,           -3812±68
lightyears:  -1401±30,            -22,379±399,         -12433±221
AU:          -8.86(0.16)E+07,     -1.415(0.025)E+09,   -7.86(0.14)E+08
km:          -1.326(0.024)E+16,   -2.117(0.038)E+17,   -1.176(0.021)E+17
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    The second error bar is an additional systematic error. i.e. they have just separated out an additional source of error. – Rob Jeffries Oct 12 at 9:11
  • @RobJeffries thanks! Decades ago I did hear of a determination of uncertainty in the error in a different context. If I wanted to recombine them and recalculate, can I just do it in quadrature (e.g. ±100±10 → 100.5)? – uhoh Oct 12 at 9:17

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