# How to calculate true equinox position?

I'm trying to calculate the state of the Earth at a given time using Keplerian orbital elements from this page. It has the following instruction:

1. Compute the coordinates, $\mathbf r_{ecl}$, in the J2000 ecliptic plane, with the x-axis aligned toward the equinox:

So after this calculation we are supposed to know direction of some equinox. But there exists the phenomenon of precession of equinoxes, so this direction shouldn't in general be fixed. I assume the equinox mentioned in that document is some "mean" equinox for the range of dates the data are valid for.

So, how do I calculate the true equinox position/direction, given these data?

• It does say "J2000 ecliptic plane", so this would be the J2000 equinox. – barrycarter Sep 21 '17 at 15:55
• @barrycarter could you point me how to proceed from this? Do I have to find some additional data to calculate the offset from J2000 equinox? – Ruslan Sep 21 '17 at 17:04
• I sort of see what you're trying to do, but could you give me/us more details? Are you asking for the Earth's position in the ICRF reference frame? BTW, feel free to contact me directly (we can post the results back here), contact info in profile. – barrycarter Sep 21 '17 at 17:10
• @barrycarter I'm trying to get the basic model of Earth's motion around the sun working numerically. I.e. I do understand the basic facts like existence of obliquity of the Earth's axis of rotation, non-circularity of the orbit etc., but I've never managed to find out actual configuration of the Earth to be able to at least predict the time of sunrise accurately to a couple of minutes, and maybe even nearest solar eclipses. So I guess I don't actually need the position in ICRF, more like in the ecliptic reference frame or what it's called. – Ruslan Sep 21 '17 at 17:19
• You might look at CSPICE and other tools mentioned in astronomy.stackexchange.com/questions/13488 – barrycarter Sep 21 '17 at 18:11

Caption of Table 1 in the OP's reference contains:

Keplerian elements and their rates, with respect to the mean ecliptic and equinox of J2000

So, "the equinox" mentioned in step 5 cited in the OP refers to the mean equinox of J2000. Now the main problem is what the mean equinox actually is.

According to Oxford Reference, mean equinox is

The direction to the equinox at a particular epoch, with the effect of nutation subtracted. The mean equinox therefore moves smoothly across the sky due to precession alone, without short-term oscillations due to nutation.

This means that the term equinox, as used in astronomy, doesn't only mean the discrete events of vernal and autumnal equinoxes, but also the direction which the perpendicular$^\dagger$ to the Earth's axis points to at any moment of time. Thus we can conclude that the equinox of J2000 is the direction this perpendicular points to at the epoch J2000, not at 20 March 2000 or any other.

So to get the true orientation of the Earth at the event of equinox we should use a precession+nutation model to advance the orientation at the Epoch to the date required.

$^\dagger$ the one lying in the ecliptic plane, of course