A BIG thank you to Starry Night's lead developer Dave Whipps for taking the time to create this KB article.
"Precession" refers to the precession of the equinoxes, and is most easily seen by comparing the North Celestial Pole with a reference star (which is not thought to move very much with respect to the background stars.)
For example, right now, the North Celestial Pole (the pole of the "Celestial" coordinate system) points at a particular star, which we now call "Polaris" (because of how close it is to the pole), but it wasn't always the North star.
Here's how to calculate one cycle of precession in Starry Night Pro:
1. Open Starry Night Pro
2. In the Options Panel, turn on: Celestial Guides > Poles and Pole Precession Circles
The Starry Night window will now look like this:
Note that the dates on the circle are approximate.
3. Set the date for today, around midnight
4. Right-click the bright star closest to the North Celestial Pole. It should be Polaris
5. Choose "Select Polaris"
6. Also choose "Center"
7. In the menu bar, choose Options > Orientation > Equatorial
This isn't completely necessary, but it keeps things from jumping around a lot while we're stepping dates in the next stage.
8. In the "Time Flow Rate" control, set the steps for 500 years
You can make it smaller if you want, but much larger and it gets annoying.
9. Click the Step Time Forward button a few times
This is where the North Celestial Pole appears relative to Polaris today.
Notice how the North Celestial Pole moves away from Polaris after 500 years (pretty quickly even!)
In 1000 years (a virtual blink of the eye in terms of the life of the universe) Polaris is moving far away from the North Celestial Pole.
10. Keep stepping the time until Polaris is AGAIN right near the North Celestial Pole
December 11th, 2012 AD
That is, step forward in time until the North Celestiai Pole has completed one full cycle around and is back where it began
December 11th, 27485 AD
Note, when closing in upon the complete precessional cycle you will need to reduce the number of years stepped from 500 to 10 (approximately) in order to achieve a very close proximity match.
Now, subtract todays year from the year it arrives back, that is, 27,485 - 2012 = 25,473!
See also Axial Precession in Wikipedia
In practical terms, calculating an exact value for this is extremely difficult given the multiple factors affecting it.