# Moon's libration from Mathematica data

Wolfram Mathematica provides "TransitTime" option (for the Moon in particular).
Knowing this time moment, you can get the position of the moon at upper culmination
(for example, for each day of the year). So I got the following plot:

and the question arose:

• Is it real picture of Moon's librations?
• Or it may be inaccuracy in Mathematica data?
• Or it consequence of incorrect "experiment" (equation of time was not taken into account etc)
• Please label the axes. What to the horizontal and vertical axes represent? Commented Oct 24, 2023 at 8:12
• @PM2Ring, so, variation of azimuth is consequence of data errors? Ok, but is it possible to get something like this for real librations? Commented Oct 24, 2023 at 8:22
• @JamesK horizontal coordinates of the Moon (for London geoposition) in moment of transit Commented Oct 24, 2023 at 8:25
• You can get real libration data from Horizons (also see Horizons API), but it will involve a bit of coding. Commented Oct 24, 2023 at 8:47
• With the Moon, all sorts of complicated things are possible. ;) Here's one that's kind of an analemma. Use my 3D orbit plotter to plot the Moon (id 301) relative to the Earth (id 399) over a Saros cycle, eg between the solar eclipses on 1973-Jun-30 & 1991-Jul-11, with a step size of 39343 minutes (the mean sidereal month). Commented Oct 24, 2023 at 9:31

This isn't libration.

The moon's average rotation matches exactly to its orbital period, so it always shows the same face towards Earth, but there is a slight "wobble" and you can see up to about 5 degrees around the "back" of the moon.

What you are plotting cannot be the position at the moment of transit, since by definition this is at 180 degrees and all you points would be in a vertical line in the middle.

Your plot coulds be the position of the moon above the horizon at the point when it reaches its highest point. This will always be very close to culmination (which is when the moon is at exactly 180 degrees, due south) but could be very slightly different, due to the fact that the moon is also moving along the ecliptic, which may be tilted to the horizon. (Or it could be some other source of error in the Mathematica data)

So the highest point can be a few tenths of a degree different to due south. This effect would be greater when the moon is higher, as the meridians are closer together near the zenith than near the horizon. This explains the trapezium-shaped distribution of points.

• Thank you, so it's not just error anyway, it is consequence of complex orbital motion? Commented Oct 24, 2023 at 8:29
• Not particularly complex, just the fact that the orbit is inclined to the equator. (the orbital motion of the moon is complex, but this isn't a consequence of that) Commented Oct 24, 2023 at 8:30
• Hmm, so why points looks like "dynamical chaos", not as Lissajous curves? Commented Oct 24, 2023 at 8:33
• Looks like random sampling to me, Not chaos. Don't know why you'd expect anything else. Commented Oct 24, 2023 at 8:38
• I agree with you on transit. Upper culmination occurs slightly before or after transit because of the Moon's motion in declination. I saw a web site discussing this year's ago. Mathematica's plot shows a real effect. Whether their plot is accurate is unknown (but probably is). Commented Oct 24, 2023 at 12:14