Convert Ecliptic to Galactic Coordinates

Question

How to convert Ecliptic co-ordinates to Galactic? The wikipedia article on Celestial Coordinate systems contains various conversions, but this one is currently absent.

For a more specific example: I would like to convert Proper motion in ecliptic latitude and longitude from the ATNF catalog into proper motion in galactic coordinates. Using the following notation from the catalog documentation:

PMElong:     Proper motion in the ecliptic longitude direction (mas/yr) 
PMElat:      Proper motion in ecliptic latitude (mas/yr) 
...
ZZ:          Distance from the Galactic plane, based on Dist
XX:          X-Distance in X-Y-Z Galactic coordinate system (kpc)
YY:          Y-Distance in X-Y-Z Galactic coordinate system (kpc)

How would I calculate the following (my own addition) from PMElong & PMElat?

PMZZ:        Proper motion perpendicular to Galactic plane
PMXX:        X-direction proper motion in X-Y-Z Galactic coordinate system (kpc/yr)
PMYY:        Y-direction proper motion in X-Y-Z Galactic coordinate system (kpc/yr)

Answer 1

Probably it's easiest to use some package for this, for example, if you use Python, you can use astropy: http://docs.astropy.org/en/stable/generated/examples/coordinates/plot_galactocentric-frame.html. Or if you are interested, here is a description on how to do it by hand, including the propagation the uncertainties: http://adsabs.harvard.edu/abs/1987AJ.....93..864J

Of course, you also need radial (line-of-sight) velocities, but I don't know if that is in the ATNF catalogue...

Edit: About the radial velocities: we only see the pulsars move across the sky and know the distance, so we get two velocity components perpendicular to our line of sight, a $v_\alpha$ and a $v_\delta$, from the proper motion in Right Ascension ($v_\alpha=d\cdot \mu_\alpha$) and the propermotion in Declination ($v_\delta=d\cdot\mu_\delta$).

Probably, it is useful for you to convert these proper motions from $\mu_\alpha, \mu_\delta$ to galactic longitude ($l$) and latitude ($b$) $\mu_l,\mu_b$, so you can get the velocities in cartesian coordinates: $v_y=d\cdot\mu_l$ and $v_z=d\cdot\mu_b$. This way, you have at least the two velocity components perpendicular to our LOS. Oh, and you maybe also want to subtract the rotation of the sun around the MW center.

There are ways to estimate the radial velocity sometimes: if the pulsar lives in a binary system, a spectrum of the companion maybe. Or a very crude way would be maybe something like http://adsabs.harvard.edu/full/1977ApJ...216..842H... But I don't know much about it.