Is there a practical way to determine the galactic coordinates directly from the equatorial system?

Specifically, I can't find the $(l,b)$ for the Pleiades star cluster (M45) anywhere. I am trying to determine if they reside in the thick disk or in the thin disk. (I am assuming these are the possible options, since the halo mainly contains globular clusters and the bulge is too messy so that the Pleiades wouldn't be so noticeable.)

– user15104
Apr 8, 2017 at 10:35

The galactic coordinate system is centred on the Earth (not the galactic centre) with 0 degrees longitude in the direction of the galactic centre. (as measured by observation of the distribution of neutral hydrogen). The physical centre, marked by Sagittarius A* is offset by less than 0.1 degrees.

The coordinate transform from equatorial (right ascension, declination) to galactic is just spherical trigonometry

The formulae, copied from wikipedia:

$$l = 303^\circ - \arctan\left({\sin(192.25^\circ - \alpha) \over \cos(192.25^\circ - \alpha) \sin 27.4^\circ - \tan\delta \cos 27.4^\circ}\right)$$

$$\sin b = \sin\delta \sin 27.4^\circ + \cos\delta \cos 27.4^\circ \cos (192.25^\circ - \alpha)$$

This uses the B1950 coordinates, which differs slightly from the J2000 coordinates, due to axial precession (of the Earth axis of rotation).

To determine the 3D position of a star cluster relative to the disk, you would also need distance information.

You can use a coordinate converter, such as the one found here to change from RA, Dec to $$l, b$$.

The coordinates of an object do not directly tell you whether something is a thick/thin disk object, though you can say that an object that lies far (more than a few hundred parsecs) from the Galactic plane is more likely to be part of the thick disk, but members of the thick disk can be in the Galactic plane. The definition is not straightforward (or agreed) and can include position, kinematics, age and chemical composition.

However, I can tell you straight away that the Pleiades is a thin disk object. That is because it is only 100 million years old and has a chemical composition that is quite similar to the Sun and other stars that belong to the thin disk.