<p>I discuss the variation in the length of the synodic month in <a href="https://astronomy.stackexchange.com/a/55112/16685">this previous answer</a>.</p>
<blockquote>
<p>When is the ideal reference point when the actual lunar phase exactly matches the mean, so the errors are roughly the same in both directions?</p>
</blockquote>
<p>There isn't a simple reference point like that because the major phases (New Moon, First Quarter, Full Moon, Last Quarter) can occur at any point on the Moon's orbit relative to its perigee. The Moon's motion is fastest at perigee and slowest at apogee. Similarly, the Sun's apparent motion is fastest when the Earth-Moon system is at perihelion. However, the eccentricity of the Moon's orbit is roughly 3× greater than the eccentricity of the Earth's orbit, so the Moon's position has a greater impact on the month length.</p>
<p>Although the eccentricities are the dominant factors in the month length variation, causing a variation of ~±8 hours, other factors add a few more hours to the range, as discussed in my previous answer.</p>
<p>Approximating the Moon's position at any point in its orbit is quite hard. However, if you just need the times of the major phases over a limited time span (say, a century or two) it's not too bad, if you can tolerate an error of a minute or so. Analytical formulas using only a dozen or so trigonometric terms perform quite well.</p>
<p>Here are a couple of graphs produced using JPL <a href="https://ssd.jpl.nasa.gov/horizons/" rel="nofollow noreferrer">Horizons</a>, showing the difference (in hours) from the mean month length of 29.53059 days for each of the major phases. Each data point shows the time difference for the upcoming month, that is, how long it will take until the Moon returns to that phase, minus 29.53059 days. I use the <a href="https://en.wikipedia.org/wiki/Terrestrial_Time" rel="nofollow noreferrer">Terrestrial Time</a> scale (TT) to avoid the complication of leap seconds.</p>
<p>Firstly, here's the graph for the current year. The graph also shows the times for each New Moon. The times are shown to the nearest millisecond, but I actually calculated the positions to better precision than one millionth of a degree. The JPL data for the Moon is <em>very</em> good.</p>
<p><a href="https://i.sstatic.net/yoV4yA0w.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yoV4yA0w.png" alt="Moon phase deltas, 2024" /></a></p>
<p>Here's a more extensive graph covering 19 years, the <a href="https://en.wikipedia.org/wiki/Metonic_cycle" rel="nofollow noreferrer">Metonic cycle</a>.</p>
<p><a href="https://i.sstatic.net/Wxdvg5tw.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Wxdvg5tw.png" alt="Moon phase deltas, 2000-2019" /></a></p>
<blockquote>
<p>So basically if a clock has a lunar phase indicator that uses the constant 29.53059 day cycle, then it can drift several days off from the real lunar phase within a few months?</p>
</blockquote>
<p>No, the drift is only a day or so. Here's another graph for 2000-2019, but starting from the March New Moon, since the previous graph shows that the errors in the New &amp; Full Moons are both small at that epoch.</p>
<p>In this graph, I use the initial times of each of the four major as the starting points of the calculations. Then I predict the succeeding phases by adding multiples of 29.53059 days to each of those four starting times, and then subtract the actual times of those events to calculate the errors.</p>
<p><a href="https://i.sstatic.net/AJrRzdd8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/AJrRzdd8.png" alt="Moon phase, cumulative error, 2000-2019" /></a></p>
<p>As you can see, the error is generally under a day, and never exceeds 30 hours.</p>

CC BY-SA 4.0
2024-10-26T22:22:14.240Z
So basically if a clock has a lunar phase indicator that uses the constant 29.53059 day cycle, then it can drift several days off from the real lunar phase within a few months.
Calmarius
CC BY-SA 4.0
2024-10-27T06:48:42.430Z
@Calmarius No, it's not that bad because the errors alternate between positive and negative every 6 months or so. The cumulative error is usually under a day, and never more than 30 hours. I've added a new graph to illustrate that. Also note that Full & New Moon have less drift than First & Last Quarter.
PM 2Ring
CC BY-SA 4.0
2024-10-28T09:02:48.900Z
And then if we find a starting point that centers one of error functions, eg. the red one, then we can bring down the error to about 15 hours, right?
Calmarius
CC BY-SA 4.0
2024-10-28T09:56:42.290Z
@Calmarius Sure, if you just want New Moon (or Full Moon), you can get the error to be within ±14 hours; First & Last Quarter have a larger range.
PM 2Ring
CC BY-SA 4.0
2024-10-28T10:06:44.727Z
@Calmarius You may enjoy https://en.wikipedia.org/wiki/Eclipse_cycle It has a lot of good info on the various lunar months and the cycles that arise from them.
PM 2Ring
CC BY-SA 4.0
2024-10-26T19:17:53.390Z
2024-10-27T06:44:33.767Z
2024-10-27T06:44:33.767Z
PM 2Ring
<p>The average length of the lunation is 29.53059 days, but the Moon doesn't go around with an uniform speed with respect to the Sun. Sometimes it's ahead of the mean sometimes it's behind. So there is kind of analemma-like motion, where the Moon appears to swing back and forth around the mean.</p>
<p>I know the actual formula for lunar phases is way more complex. But if I wish to stick with the simple 29.53059 day constant lunation length, how accurate is it for predicting lunar phases?
When is the ideal reference point when the actual lunar phase exactly matches the mean, so the errors are roughly the same in both directions?</p>
<p>I've searched in the topic but the only thing I've found is that the synodic months I sometimes some hours shorter or longer than the average, but no info about what's the maximum error these differences accumulate to.</p>

<the-moon><synodic-period>
When does the mean lunar phase aligns with the actual lunar phase? What's the maximum difference between the two?
Calmarius
# When does the mean lunar phase aligns with the actual lunar phase? What's the maximum difference between the two?

(Asked by: username_0 on 2024-10-26)

The average length of the lunation is 29.53059 days, but the Moon doesn't go around with an uniform speed with respect to the Sun. Sometimes it's ahead of the mean sometimes it's behind. So there is kind of analemma-like motion, where the Moon appears to swing back and forth around the mean.

I know the actual formula for lunar phases is way more complex. But if I wish to stick with the simple 29.53059 day constant lunation length, how accurate is it for predicting lunar phases?
When is the ideal reference point when the actual lunar phase exactly matches the mean, so the errors are roughly the same in both directions?

I've searched in the topic but the only thing I've found is that the synodic months I sometimes some hours shorter or longer than the average, but no info about what's the maximum error these differences accumulate to.

**Tags:** the-moon, synodic-period


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## 1 answers

### Answer by username_1 on 2024-10-26. Score: 1

I discuss the variation in the length of the synodic month in [this previous answer](https://astronomy.stackexchange.com/a/55112/16685).

> When is the ideal reference point when the actual lunar phase exactly matches the mean, so the errors are roughly the same in both directions?

There isn't a simple reference point like that because the major phases (New Moon, First Quarter, Full Moon, Last Quarter) can occur at any point on the Moon's orbit relative to its perigee. The Moon's motion is fastest at perigee and slowest at apogee. Similarly, the Sun's apparent motion is fastest when the Earth-Moon system is at perihelion. However, the eccentricity of the Moon's orbit is roughly 3× greater than the eccentricity of the Earth's orbit, so the Moon's position has a greater impact on the month length.

Although the eccentricities are the dominant factors in the month length variation, causing a variation of ~±8 hours, other factors add a few more hours to the range, as discussed in my previous answer.

Approximating the Moon's position at any point in its orbit is quite hard. However, if you just need the times of the major phases over a limited time span (say, a century or two) it's not too bad, if you can tolerate an error of a minute or so. Analytical formulas using only a dozen or so trigonometric terms perform quite well.

Here are a couple of graphs produced using JPL [Horizons](https://ssd.jpl.nasa.gov/horizons/), showing the difference (in hours) from the mean month length of 29.53059 days for each of the major phases. Each data point shows the time difference for the upcoming month, that is, how long it will take until the Moon returns to that phase, minus 29.53059 days. I use the [Terrestrial Time](https://en.wikipedia.org/wiki/Terrestrial_Time) scale (TT) to avoid the complication of leap seconds.

Firstly, here's the graph for the current year. The graph also shows the times for each New Moon. The times are shown to the nearest millisecond, but I actually calculated the positions to better precision than one millionth of a degree. The JPL data for the Moon is *very* good.

[![Moon phase deltas, 2024](https://i.sstatic.net/yoV4yA0w.png)](https://i.sstatic.net/yoV4yA0w.png)

Here's a more extensive graph covering 19 years, the [Metonic cycle](https://en.wikipedia.org/wiki/Metonic_cycle).

[![Moon phase deltas, 2000-2019](https://i.sstatic.net/Wxdvg5tw.png)](https://i.sstatic.net/Wxdvg5tw.png)

> So basically if a clock has a lunar phase indicator that uses the constant 29.53059 day cycle, then it can drift several days off from the real lunar phase within a few months?

No, the drift is only a day or so. Here's another graph for 2000-2019, but starting from the March New Moon, since the previous graph shows that the errors in the New & Full Moons are both small at that epoch.

In this graph, I use the initial times of each of the four major as the starting points of the calculations. Then I predict the succeeding phases by adding multiples of 29.53059 days to each of those four starting times, and then subtract the actual times of those events to calculate the errors.

[![Moon phase, cumulative error, 2000-2019](https://i.sstatic.net/AJrRzdd8.png)](https://i.sstatic.net/AJrRzdd8.png)

As you can see, the error is generally under a day, and never exceeds 30 hours.

> **Comments:**
> **username_0**: So basically if a clock has a lunar phase indicator that uses the constant 29.53059 day cycle, then it can drift several days off from the real lunar phase within a few months.  
> **username_1**: @username_0 No, it's not that bad because the errors alternate between positive and negative every 6 months or so. The cumulative error is usually under a day, and never more than 30 hours. I've added a new graph to illustrate that. Also note that Full & New Moon have less drift than First & Last Quarter.  
> **username_0**: And then if we find a starting point that centers one of error functions, eg. the red one, then we can bring down the error to about 15 hours, right?  
> **username_1**: @username_0 Sure, if you just want New Moon (or Full Moon), you can get the error to be within ±14 hours; First & Last Quarter have a larger range.  
> **username_1**: @username_0 You may enjoy https://en.wikipedia.org/wiki/Eclipse_cycle It has a lot of good info on the various lunar months and the cycles that arise from them.  

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