# What causes the variation in the length of the synodic month, BESIDES the eccentricity of the Moon's orbit? And how much do they contribute?

When explaining the reason for the variation in the length of the synodic lunar month, usually the only explanation given is that the Moon's orbit is elliptical, and the portion of its elliptical orbit which the Moon has to cover after finishing one full sidereal month, is different in different months. For example see here: https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html#synodic.

Lately I found here: Is the difference in time between the sideral and synodyc month - constant?, that the length of the sidereal month also varies, which would obviously be an additional cause for the variation in the length of the synodic month. In the source cited there (https://encyclopedia2.thefreedictionary.com/sidereal+month), there's no further explanation for the variation of the length of the sidereal month other than:

Because of the irregularities of the moon’s motion—that is, deviations from the motion prescribed by Kepler’s laws—the length of a sidereal month is not constant and can vary by a few hours.

I didn't find anywhere else mentioning these variations, let alone explaining it. So is it true that the length of the sidereal month is not constant? If yes, to what degree? What causes it?

Another cause for the variation in the length of the synodic month, is probably the ecentricity of Earth's orbit. And although the ecentricity of Earth's orbit is less then that of the Moon's, it requires less percentage of a whole month (the time Earth traveled a portion of its orbit) to make a significant difference, than the percentage required of only two days (the time the Moon traveled the corresponding portion). The only place I found mentioning that effect was here: https://www.convertunits.com/from/synodic+month/to/sidereal+month (Edit: It's probably quoted from Wikipedia), stating the following:

Due to the eccentricity of the lunar orbit around Earth (and to a lesser degree, the Earth’s elliptical orbit around the Sun), the length of a synodic month can vary...

But they don't say how much it contributes.

So I'm wondering how much of the variations are really due to the eccentricity of the Moon's orbit, which is viewed like the only reason.

Also, are there any other reasons for the variation in the length of the synodic month that I didn't mention here? If yes, how big is their effect?

As a bonus, how much does each of these causes add up to make the time real moon phases happen deviate from the average time?

• Excellent question (+1). I think Nodal precession also might play a role here, as it causes slight variations in the orbit as a whole in large timescales, here you can refer to the Milankovitch eccentricity cycles for more details, perhaps Orbital inclination of Moon's orbit relative to Earth's axis Oct 19, 2023 at 6:24
• @mike g: After all I'm figuring that you're probably right! PM 2Ring says in his answer below "Interestingly, the variation from New Moon to New Moon in this system, is only ~±4 hours, and between First or Last Quarters it's ~±6 hours". I also found that longer synodic months coincide with shorter sidereal months, as in my comment to PM's answer. Truth wins after all!! Oct 31, 2023 at 22:11

Q: What causes the variation in the length of the synodic month (besides the moon's orbital eccentricity), and how much do the other causes contribute? (Also) Is it true that the length of the sidereal month is not constant? If yes, to what degree? What causes it?

The question quotes a number of partial or simplified answers, and finds them unsatisfactory. This is understandable, the usual verbal answers are very heavily simplified relative to the real complexities of the perturbed motions of the moon, plus, to a lesser extent in the case of the synodic month, those of the sun. The resulting motions are complicated enough that they seem to defy simple verbal explanation. Also, the moon's true motions are really not very close to a Keplerian ellipse. The largest of its perturbations have names (not counting the inequality of its elliptic motion), especially evection, variation, etc., see also the other wiki information in 'Results of lunar theories'.

The subject of lunar theory used to be taught to math students in some universities especially during the 19th century. But the tedious volume and complexity of the trigonometry was one of several reasons for its unpopularity, and both teachers and students eventually found other topics they considered better worth their time and efforts. Still, for those interested, the textbooks of that time contain much relevant information: e.g. H Godfray (1859), Elementary Treatise on the Lunar Theory , with useful explanations of the larger effects.

Not the least of the difficulties in answering practical or numerical questions about this topic is that the effects are not perfectly periodic, and the amplitudes of the variations in the aggregate are not uniform. (Sea) tides have several features of complexity in common with the moon's motions (which are also well known to influence the tides). The American Mathematical Society's webpages on sea tides states: "The tidal force is governed by ... astronomical motions which are themselves periodic, but since the various frequencies have no whole-number ratios between them, the whole configuration never repeats itself exactly." (http://www.ams.org/publicoutreach/feature-column/fcarc-tidesi1) Likewise, and for the same reason, although there are approximations, there is no exactly repeatable pattern of intervals between successive (true) sun-moon conjunctions in ecliptic longitude (i.e. new moons), or between oppositions in ecliptic longitude (i.e. full moons).

A considerable quantity of specific example information about the variability of lunations and their irregularity is given by Jean Meeus in 'More Mathematical Astronomy Morsels' (2002), pages 19-31. He identifies specific variations in month-lengths, with some periodic effects including the ~8.8-year period of the rotation of the moon's axis, and a 184-year period, and identifies some other periodicities, plus an effect of the earth's orbital eccentricity (as it reduces, the extent of the variabilities in lunations also becomes a little less).

He writes for example (no doubt after fearsomely lengthy calculations) that during the years 1760-2200 the shortest and longest intervals between full moons are to be

shortest: 29 days, 06 hours, 34 minutes (between the Full Moons of 1783 June 15 and July 14); and

longest: 29 days, 19 hours, 58 minutes (between the Full Moons of 1880 Dec. 16 and 1881 January 15).

As for intervals between new moons 1760-2200:

shortest: 29d 06h 34m, from 1885 June 12 to 1885 July 12

longest: 29d 19h 58m, from 1787 Dec. 9 to 1788 Jan. 8.

The Meeus article on month lengths is well worth reading, but it has too much information to incorporate here.

True month-lengths for particular specific times can be found in a number of ways, e.g. by algorithms constructed to show times of successive new and/or full moons starting from a given date, or for the successive passages of the moon past a point with given ecliptic longitude after a given date, etc.

Building-blocks for such algorithms can be found in

(a) listings of polynomials in the time-interval from a standard epoch for the moon's mean longitude and for the mean values of the moon's four principal arguments D, l, l' and F, these are 'Delaunay arguments' mentioned under results of lunar theories: and

(b) listings of the moon's principal inequalities in longitude, and

(c) listings of the sun's principal inequalities in longitude.

Mean elements (a) can be found for example in "Numerical expressions for precession formulae and mean elements for the Moon and the planets", J L Simon et al., 1994.

Lunar inequalities of longitude (b) can be found for example in J Meeus, 'Astronomical Algorithms (1998) (calculating trigonometrical longitude terms from those in the table on p.339), alternatively calculating the longitude terms given by "ELP 2000-85: a semi-analytical lunar ephemeris adequate for historical times", Chapront-Touze', M.; Chapront, J., 1988.

Solar inequalities of longitude (c) can be found for present purposes from the formula for 'C' on p.164 of Meeus (1998).

The calculations can be arranged e.g. by iterative computation:

To find accurate new and full-moon times, compute for successive trial times (i) the Delaunay mean elongation argument D from the data cited at (a) above, and apply (ii) an additive correction for the moon's inequalities of longitude as far as the precision required, calculated from either of the sets of trigonometrical terms cited at (b) above, and also (iii) subtract a correction for the sun's inequalities as in Meeus's formula cited at (c) above; then iterate to seek times when the result is either zero or an even multiple of 180 degrees for new moons, or an odd multiple of 180 degrees for full moons.

The moon's daily variation in elongation can be used to converge on each wanted result, it is approximately given by the following data given by Meeus (in degrees and decimals), a constant plus the largest periodic terms:

 12.190749
+ 1.434006 cos ***l***
+ 0.280 135 cos 2***D***
+ 0.251632 cos (2***D*** - ***l***)


(where D and l are two of the Delaunay variables, the moon's mean solar elongation and mean anomaly), plus many other smaller terms.

To find successive times when the moon passes a chosen degree Z of ecliptic longitude, to find the length of the (tropical) lunar month starting at that point, compute for successive trial times (i) the moon's mean longitude from the data cited at (a) above, and apply (ii) as before the additive correction for the moon's inequalities of longitude as far as the precision required, calculated from the data cited at (b) above; then iterate to seek times when the result is either zero (+ Z ) or an even multiple of 180 degrees (+ Z ).

I hope that this sketch, with sources for data, calculations, and further explanations, will be found anyhow somewhat useful.

• There's definitely a lot of useful information in this answer, but it has no final answer for any part of my question. For the "What causes..." parts, I guess that real explanations would be very long and complicated, and probably that's the reason why Wikipedia also doesn't offer them in the articles quoted in this answer. But for the "how much..." parts, it shouldn't be a big deal (for one who knows how to deal with the proper software) to post the end results, which is what I'm looking for. Oct 22, 2023 at 20:21
• As for the total of the variations in the length of the synodic month, there's an excellent database for that here: astropixels.com/ephemeris/moon/synodicmonth2001.html, and that wasn't part of my question. Oct 22, 2023 at 20:21

The Moon's orbit around the Earth is not a simple Kepler ellipse. It's dynamically distorted by tidal forces, primarily from the Sun.

Newton showed that Kepler's 3 laws arise from the inverse square law of gravity. When the mass of the orbiting body is negligible compared to the primary, its motion follows Kepler's laws. This is known as a one-body system. Newton also showed that a two-body system, where the mass of the orbiting body is not negligible, can be reduced to an equivalent one-body system.

However, when there are three or more bodies, things become complicated, especially when the mass ratios are not negligible. Much of the development of celestial mechanics in the centuries after Newton involved finding mathematical techniques for dealing with these complexities. The Moon is important for navigation and tide prediction, so considerable time and energy was devoted to developing lunar theory.

Fortunately, the Sun is a lot more massive than the Earth-Moon system, so we get a reasonable first approximation by treating the motion of the EMB (Earth-Moon barycentre) around the Sun as a Kepler ellipse. But the orbit of the Earth and Moon about their barycentre is not a nice static ellipse. The shape and size of the orbit is distorted by the Sun, with the strength of the distortion depending on the distance and direction to the Sun.

In traditional celestial mechanics, orbits are specified as ellipses given by orbital elements.

For the Earth and Moon orbits, the reference plane is the ecliptic. The Moon's orbit has a mean inclination of ~$$5.14°$$, but its actual inclination varies from that by ~$$0.16°$$. It undergoes both kinds of orbital precession. The major axis precesses (apsidal precession) with a period of ~$$3232$$ days ($$8.85$$ years), relative to the equinox direction. The line of nodes precesses backwards with a period of ~$$6798$$ days (18.6 years). Both of these precessions are roughly linear, but both have noticeable non-linear wobbles. robjohn has a nice diagram of smooth nodal precession in this answer:

The largest perturbation of the Moon's orbit is the evection:

It arises from an approximately six-monthly periodic variation of the eccentricity of the Moon's orbit and a libration of similar period in the position of the Moon's perigee, caused by the action of the Sun.

The next largest perturbation is the variation, which compresses the orbit in the direction towards the Sun and expands the orbit in the perpendicular direction.

The other major perturbations have smaller effects, but those effects are less symmetrical. By the mid-nineteenth century, it was realised that the major perturbations could be expressed in terms of a small number of angular terms (the Delaunay arguments). Equations involving 15-20 trigonometric terms are adequate for determining the lunar orbit to a precision of around 30 seconds (of time), and predicting the occurrence of eclipses. Predicting eclipse times to higher precision, and determining the tracks of solar eclipses requires considerably more terms. The lunar theory of E. W. Brown, developed around the turn of the twentieth century, used over 1400 terms. As you can imagine, computing accurate lunar positions for navigation involved considerable labour in the days before electronic computers.

These perturbations combine together in complex ways. However, at any point in time we can take a "snapshot" of the Moon's position and velocity vectors, and find an ideal Kepler orbit which matches it at that moment. This is known as an osculating orbit:

the osculating orbit of an object in space at a given moment in time is the gravitational Kepler orbit (i.e. an elliptic or other conic one) that it would have around its central body if perturbations were absent. That is, it is the orbit that coincides with the current orbital state vectors (position and velocity).

The JPL Development Ephemeris doesn't use orbital elements to do its calculations. It performs numerical integration of the equations of motion of the major masses in the Solar System, including individual masses of 343 asteroids. However, it can compute osculating elements for the bodies in its database. We can get a feel for the true orbit by seeing how the osculating elements vary over time.

The Moon's true trajectory quickly deviates from the trajectory calculated from its osculating elements. It's quite close for a day or so, but after 6 days the error in the ecliptic longitude is ~$$0.5°$$.

Before we examine these osculating elements, I'll digress slightly to explain how the various elements affect the orbital period. The synodic period of the Moon is defined in terms of ecliptic longitude. Specifically, the lunar phase is proportional to the Moon's elongation, the difference between the Moon's and the Sun's ecliptic longitudes. Thus the New Moon occurs when the elongation is $$0°$$, First Quarter is when the Moon is $$90°$$ ahead of the Sun, Full Moon is when the Moon is $$180°$$ ahead of the Sun, etc.

The Sun's apparent motion is in the ecliptic plane, so its ecliptic longitude equals its true anomaly plus the longitude of perihelion. But the Moon's orbit is inclined to the ecliptic, which modifies the speed of its ecliptic longitude slightly, making it slower at the nodes.

I should mention that the ecliptic frame precesses backwards with a period of ~26,000 years, so cycles measured relative to the equinox differ slightly from those measured relative to the stars.

The fundamental equation of an ellipse is: $$r = \frac{p}{1 + e\cos\nu}$$ where $$r$$ is the orbital radius, $$p=a(1-e^2)$$, $$\nu$$ is the true anomaly, $$a$$ is the semi-major axis, and $$e$$ is the eccentricity. The semi-minor axis is $$b=a\sqrt{(1-e^2)}$$.

The orbital period and speed are given by Kepler's 3rd law: $$n^2 = \left(\frac{2\pi}{P}\right)^2=\frac{\mu}{a^3}$$

where $$P$$ is the (anomalistic) period, the time between successive periapsis passages, $$n$$ is the mean motion, and $$\mu$$ is the gravitational parameter. For a two-body system, $$\mu=G(m_1+m_2)$$, where $$G$$ is the universal gravitational constant and $$m_i$$ are the masses. (As Wikipedia explains, it's very hard to measure $$G$$, but we know $$\mu$$ for major Solar System bodies to quite high precision).

Another important parameter is $$\mathbf{h}$$, the specific angular momentum. This vector is perpendicular to the orbit, and it's a constant of motion in a one- or two-body system. From Kepler's 2nd law, its magnitude $$h$$ is half the areal velocity. Thus $$h=nab$$. Also, $$h^2=\mu p$$, and $$h=r^2\dot\nu$$, where $$\dot\nu$$ is the rate of change of the true anomaly, i.e., it's the body's angular speed in the orbital plane. So if $$h$$ is constant, the angular speed of an orbiting body is inversely proportional to the square of the distance from the body to its primary. The Moon's specific angular momentum isn't constant, but it only varies by ~1%.

Here are some plots of the main osculating elements that relate to the angular speed. These plots cover the period between two New Moons, 2022-Jan-02 UTC and 2023-Jan-22 UTC, a span of ~13 synodic months, and ~14 of the other kinds of lunar month.

Eccentricity

Semi-major axis

Inclination

Mean motion

Sidereal orbit period

Specific angular momentum

Here's the Python / Sage script that generated those plots (except the angular momentum plot), using data from JPL Horizons. It can generate plots for a few other osculating elements, for any body that Horizons knows.

As you can see, all of these elements vary quite quickly, and the amount of variation is not constant. However, the total amount of variation is mostly fairly small. The semi-major axis varies by ~1%. The inclination varies by ~3%, but that's not so bad because the inclination is fairly small. The worst "offender", by far, is the eccentricity, which is why it's commonly blamed for the variation in the synodic month length.

The size of the Moon's mean eccentricity does make a significant contribution to the synodic (and sidereal) month length variation. However, because the Moon's orbit is precessing, changing eccentricity, and changing size, the pattern of synodic month length variation is even more complex.

The maximum variation in the synodic month length purely due to the eccentricities is around 8 hours. Here's a plot showing the month length (from Full Moon to Full Moon) covering 235 synodic months (a Metonic cycle, almost exactly 19 tropical years, and 254 tropical months), in a simplified system where the orbits have the same eccentricities as the real system, and the same tropical & synodic periods, but the orbits are coplanar, with no perturbations.

The vertical axis shows the difference in month length from the mean synodic length, in hours.

Here's the plotting script, which can also print that data in CSV format, with a few extra items: the month length (in days), and the true anomaly of the Moon at the moment of Full Moon (in degrees).

Interestingly, the variation from New Moon to New Moon in this system, is only ~±4 hours, and between First or Last Quarters it's ~±6 hours.

You may like to look at elements plots of the Earth, and of other Solar System bodies, to see how well-behaved they are, compared to the Moon.

• Really a beautiful answer! But I would also like to know the final numbers, like: Total variation of the synodic month ~13.5 hours. of which: total variations of the sidereal month =__ hours, cause A =__ hours, etc. Oct 27, 2023 at 19:56
• Thanks, @George. I'm not sure if it makes sense to try and do a breakdown like that. It's not like the Equation of Time, where we can cleanly separate the effects of eccentricity and declination. Eg, the tidal stretching captured by the Variation changes both a and e, and is sensitive to the direction of the Sun relative to the major axis orientation. These effects combine in a complex, non-linear, time dependent way, so the total effect isn't just the sum of the parts. Also, it would be rather tricky to calculate in a way that was realistic and accurate. But I'll think about it... Oct 27, 2023 at 21:26
• What does the Sideral orbit period plot show? To which sec does it refer? Oct 29, 2023 at 12:49
• "If the EMB and lunar orbits were fixed Kepler ellipses the synodic period would be constant" - Not exactly, because the most talked about cause, which is "which part of the ellipse has to be covered twice", would still be present. And that includes many sub-causes. Oct 29, 2023 at 12:54
• It's truly surprising, because, as I stated in the beginning of my question, that's what they teach us all around. See for example eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html#synodic, timeanddate.com/astronomy/moon/lunar-month.html. Oct 29, 2023 at 20:59