Where I can find obliquity direction?

Question

I can find axial tilt of planets easily, but that doesn't specify the direction of that tilt, i.e. planet's rotation axis may be anywhere in circle defined on a sphere by axial tilt value. And I can't google for obliquity direction as it only gives me obliquity value, not it's direction. Even NASA's HORIZONS only gives obliquity value. I expect there should be another angle, either from main body equinox, Earth's equinox, ascending node or maybe periapsis, that with axial tilt and orbital elements would define direction of planet's rotation axis. I know that rotation axis is often if not usually precesses, but that takes thousands of years, and compared to time from J2000 it's practically neglectable (or at least it's easier to find precession rate and make a correction for it, knowing some starting value).

Answer 1

Essentially, the idea is to calculate de right ascension and declination values of the north pole of each planet using special formulas. Each formula is planet-specific, due to its unique interactions with other bodies in the solar system. The values of the right ascension and declination are expressed using the planet's ecliptic equator (the intersection of the body's equatorial plane and the International Celestial Reference Frame (ICRF), which is fixed to the stars and centered on the solar system barycenter).

(Credit IAU Working Group)

Once, the north pole is determined, the equatorial plane can be deducted (the plane orthogonal to the vector defined between the north pole and the planet center and containing the planet center) as well as the line of nodes that define the intersections of the ICRF plane and the planet's equatorial plane.

The last step is to properly orient the north pole to a given direction. This is achieved by calculating the angle W that specifies the position of the line of nodes. Again, the formula to calculate the value for W is planet specific. Using the value of W, the north Pole axis can be rotated around the ICRF z-axis (0,0,1), to position the line of nodes properly.

In each formula, the right ascension, the declination, and the position of the prime meridian are calculated using:

T = interval in Julian centuries (of 36525 days) from the standard epoch J2000
d = interval in days from the standard epoch J2000
The standard epoch is JD 2451545.0, i.e. 2000 January 1 12 hours TDB

To learn more about this works for each planet, take a look at the following document:

IAU Working Group on Cartographic Coordinates and Rotational Elements: 2009