<p>This isn't libration.</p>
<p>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 &quot;wobble&quot; and you can see up to about 5 degrees around the &quot;back&quot; of the moon.</p>
<p>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.</p>
<p>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)</p>
<p>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.</p>

CC BY-SA 4.0
2023-10-24T08:29:58.787Z
Thank you, so it's not just error anyway, it is consequence of complex orbital motion?
lesobrod
CC BY-SA 4.0
2023-10-24T08:30:57.453Z
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)
James K
CC BY-SA 4.0
2023-10-24T08:33:06.363Z
Hmm, so why points looks like "dynamical chaos", not as Lissajous curves?
lesobrod
CC BY-SA 4.0
2023-10-24T08:38:46.730Z
Looks like random sampling to me, Not chaos.  Don't know why you'd expect anything else.
James K
CC BY-SA 4.0
2023-10-24T12:14:07.227Z
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).
JohnHoltz
CC BY-SA 4.0
2023-10-24T08:24:52.620Z
2023-10-24T08:30:18.363Z
2023-10-24T08:30:18.363Z
James K
<p>Wolfram Mathematica provides &quot;TransitTime&quot; option (for the Moon in particular).<br />
Knowing this time moment, you can get the position of the moon at upper culmination<br />
(for example, for each day of the year).
So I got the following plot:</p>
<p><a href="https://i.sstatic.net/St0vg.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/St0vg.png" alt="enter image description here" /></a></p>
<p>and the question arose:</p>
<ul>
<li>Is it real picture of Moon's librations?</li>
<li>Or it may be inaccuracy in Mathematica data?</li>
<li>Or it consequence of incorrect &quot;experiment&quot; (equation of time was not taken into account etc)</li>
</ul>

CC BY-SA 4.0
2023-10-24T08:12:00.623Z
Please label the axes.  What to the horizontal and vertical axes represent?
James K
CC BY-SA 4.0
2023-10-24T08:22:41.437Z
@PM2Ring, so, variation of azimuth is consequence of data errors?
Ok, but is it possible to get something like this for real librations?
lesobrod
CC BY-SA 4.0
2023-10-24T08:25:58.017Z
@JamesK horizontal coordinates of the Moon (for London geoposition) in moment of transit
lesobrod
CC BY-SA 4.0
2023-10-24T08:47:49.673Z
You can get real libration data from [Horizons](https://ssd.jpl.nasa.gov/horizons/) (also see [Horizons API](https://ssd-api.jpl.nasa.gov/doc/horizons.html)), but it will involve a bit of coding.
PM 2Ring
CC BY-SA 4.0
2023-10-24T09:09:29.973Z
@PM2Ring, Now I see that I'd like to get something like 
[here](https://astronomy.stackexchange.com/questions/50935/how-do-times-of-lunar-standstills-vary-over-time-if-we-folded-them-back-into-a?rq=1)
(especially "modulus" plot) but for some position in the sky. Like "analemma" of the Moon, that show it's complicated orbit
lesobrod
CC BY-SA 4.0
2023-10-24T09:31:57.687Z
With the Moon, all sorts of complicated things are possible. ;) Here's one that's kind of an analemma. Use my [3D orbit plotter](https://astronomy.stackexchange.com/a/49823/16685) 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).
PM 2Ring
<the-moon><data-analysis><astrometry>
Moon's libration from Mathematica data
lesobrod
# Moon's libration from Mathematica data

(Asked by: username_0 on 2023-10-24)

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:

[![enter image description here](https://i.sstatic.net/St0vg.png)](https://i.sstatic.net/St0vg.png)

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)

**Tags:** the-moon, data-analysis, astrometry

> **Comments:**
> **username_1**: Please label the axes.  What to the horizontal and vertical axes represent?  
> **username_0**: @PM2Ring, so, variation of azimuth is consequence of data errors? Ok, but is it possible to get something like this for real librations?  
> **username_0**: @JamesK horizontal coordinates of the Moon (for London geoposition) in moment of transit  
> **username_2**: You can get real libration data from [Horizons](https://ssd.jpl.nasa.gov/horizons/) (also see [Horizons API](https://ssd-api.jpl.nasa.gov/doc/horizons.html)), but it will involve a bit of coding.  
> **username_0**: @PM2Ring, Now I see that I'd like to get something like  [here](https://astronomy.stackexchange.com/questions/50935/how-do-times-of-lunar-standstills-vary-over-time-if-we-folded-them-back-into-a?rq=1) (especially "modulus" plot) but for some position in the sky. Like "analemma" of the Moon, that show it's complicated orbit  
> **username_2**: With the Moon, all sorts of complicated things are possible. ;) Here's one that's kind of an analemma. Use my [3D orbit plotter](https://astronomy.stackexchange.com/a/49823/16685) 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).  

---

## 1 answers

### Answer by username_1 on 2023-10-24. Score: 1 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you, so it's not just error anyway, it is consequence of complex orbital motion?  
> **username_1**: 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)  
> **username_0**: Hmm, so why points looks like "dynamical chaos", not as Lissajous curves?  
> **username_1**: Looks like random sampling to me, Not chaos.  Don't know why you'd expect anything else.  
> **username_3**: 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).  

---

