The question I was trying to solve was: What is the ecliptic north pole in galactic coordinates?
So this entry is about transformations between different systems of coordinates. So, what's the cast?
- There are good old spherical coordinates (theta, phi), where theta is the angle in radians between the point and the z-axis and phi is the (signed!) angle in radians of the projection of the point to the xy-plane an the x-axis. These spherical coordinates are such that the z axis is normal to the galactic plane and the x-axis points towards the galactic centre.
- Closely related are cartesian coordinates (x,y,z). There is some redundancy as x^2+y^2+z^2=1 but the advantage is that there is an easy expression for the scalar and cross product between to arbitrary points and this for example makes it easy to compute angles between points.
- What astronomers call galactic coordinates are pairs (l,b) which are closely related to theta and phi. At first you might think they are the same, but once you start computing things you realise that l corrsponds to phi and b to theta, so the order is reversed, then they are usually quoted in degrees (and decimal parts) and not in radians and last b is the angle between the point and the xy plane and not the z axis. So numerically (l,b)=(phi*180/Pi, 90-theta*180/Pi)
- Then there are equatorial coordinates (alpha, delta). Those are again similar to spherical coordinates but now the z axis points along the earth axis and the x-axis points to the vernal equinox. Again, people use the convention that delta, the declination, is the angle between the point and the equatorial plane (and not the earth axis) and just to make things a little more colorful, alpha, called the right ascession, is not quoted in degrees but in hours, minutes, and seconds (with decimal parts). The convention for delta that I found is degrees, minutes, seconds and I hope that 60 seconds is a minute and 60 minutes is a degree and not like some GPS receivers that use 100s instead of 60s.
- Actually, the earth axis is not exactly fixed in space but precesses (amongst other things), so there are actually at least two common versions of equatorial coordinates, those w.r.t. the earth axis at some moment in 1950 and those w.r.t. some moment in 2000 (Julian calender of course).
- Finally, there are ecliptic coordinates (lambda, beta), again of the spherical, degree type. So the ecliptic north pole that I am after has beta=90 degrees, lambda undefined. Wikipedia
gives formulas to convert between these and equatorial coordinates (which? 4. or 5.?)
Then cos(theta) is given in terms of the scalar product between ez and enp, the unit vector towards the ecliptic north pole. phi is slightly more complicated as giving it in terms of cos(phi) from a scalar product is not sufficient since this cannot tell the difference between phi and 2pi-phi. But the quadrant aware two argument version of arctan does the job: phi = arctan( (ez x ex).enp / ex.enp ) where x is the cross and . is the scalar product.
After all this pain, I arrive at (theta,phi) = (0.682213, -0.0287665) or if you prefer (l,b) = (-1.6482degrees, 50.9121degrees). Could anybody please confirm this?