# What is this 877-year cycle in the orbits of Jupiter & Saturn, and this multimillion-year cycle in the lunar orbit?

The book The Theory That Would Not Die (by Sharon Bertsch McGrayne, 2011) states the following on page 28:

He [Pierre-Simon Laplace] used other methods between 1785 and 1788 to determine that Jupiter and Saturn oscillate gently in an 877-year cycle around the sun and that the moon orbits Earth in a cycle millions of years long.

What are the two cycles mentioned here? What exactly happens every 877 years in the orbits of Jupiter & Saturn, and every so many millions of years in the lunar orbit?

• For Jupiter & Saturn, my searching finds no cycles longer than the 20-year cycle of great conjunctions.

• For the moon, my searching finds no cycles in the lunar orbit longer than 19 years.

• FWIW, here's a current question related to computing good rational approximations to cycles: math.stackexchange.com/q/4412720/207316 Mar 30 at 5:10
• At some point, "millions of years" runs into a putative non-repeating pattern which almost repeats at some absurd level of precision. Mar 30 at 13:15
• Maybe the "millions of years" is just all the cycles of the lunar orbit taken together. It probably is just shoddy data/math. Mar 31 at 13:07

A biography of Laplace mentions his derivation of an 877 year cycle in the motions of Saturn and Jupiter.

The periods of the planets are about 30 and 12 years respectively so they will approximately return to the same relative position every sixty years. Laplace proposed that this periodicity should create a perturbation that should be found when the differential equations that govern their motion are integrated. His initial calculation of the variations in the mean motion of the planets found that the motion of Saturn would vary sinusoidally with a period of 877 years and an amplitude of 47 arcminutes (¾ of a degree), although he later changed this to 924 years.

I can't find a reference to a million-year secular motion of the moon, but the moon has multiple motions, there may be such a motion described in the works of Laplace, somewhere.

• Thanks for that reference! The lunar cycle also is discussed on p. 144: "In fact [...] the period of the moon's secular variation in mean motion of revolution was so long, and the earth's gravity by comparison so powerful a force, that the major axis of the lunar equator is always drawn toward the center of the earth, subject only to the libration that shows a tiny rim of the hidden hemisphere, now on one side and now on the other. The period of the moon's long inequality was the greatest that Laplace had yet studied, amounting to millions of years." Mar 30 at 4:00
• I only understand gravitation and elliptical orbits in a two body system so I fail to appreciate how the "return to the same relative position every sixty years" would produce an "oscillation", let alone one with a period of 877 or 924 years. Mar 31 at 13:55
• That is a three body effect. There are perturbing forces and the result of integrating the equations of motion is that there is a periodic effect that has a very long period, of about 900 years. You its not obvious, and there is no intuitive explanation of why that I am aware of. Mar 31 at 16:24
• Did he by any chance solve those differential equations using a Laplace transform? Mar 31 at 20:36

After a cursory search, I found a reference which explained how Laplace came to determine the oscillation cycle of 877 years.

What Laplace found was that peculiarities arose in the Jupiter-Saturn system because their orbits of near approach to commensurability of their mean motion. The term "commensurability" means that the ratio of the mean motions of the planets could be expressed in terms of ratio of small whole numbers. This is reminiscent of the Pythagorean view of Nature. In the case of Saturn and Jupiter, Saturn made two trips around the Sun while Jupiter made five trips. Laplace took advantage of this relationship and designated the two mean motions in terms of the difference $$\mathrm{5n^{'}-2n}$$. Laplace also examined the effect of the ellipcity of the orbits by setting $$\mathrm{5n^{'}-2n=0}$$ and including pertubation terms due to variation in the eccentricity of the orbits. According to the calculation, Jupiter and Saturn were engaged in a cosmic dance in which one orbit expanded while the other contracted in a cyclic manner. Laplace first discovered that the period of this cycle of acceleration/deceleration was 877 years, but later calculations gave 929 years (Gillispie, 1997, p. 127)

Ref.: Physical Chemistry: Multidisciplinary Applications in Society p. 443, by Kenneth S Schmitz, Elsevier

FMI, you can check various cycles that happens in planetary system: https://www.aipro.info/wp/wp-content/uploads/2017/08/PLANETARY_-RESONANCES.pdf

• @uhoh page 443 of the first entry in Google books when you search "jupiter oscillation cycle 877 years" Apr 3 at 3:14
• There it is on pages 443 and 262 for some reason! i.stack.imgur.com/7Lr9J.png
– uhoh
Apr 3 at 5:01

(This is just to supplement the existing answers.)

A couple of additional useful references:

1. George E. Smith, Closing the Loop: Testing Newtonian Gravity, Then and Now, appearing as Ch. 10 in "Newton and Empiricism" (2014) by Zvi Biener and Eric Schliesser.

On p. 282 there's a neat plot showing the approx. 900-yr Jupiter-Saturn "Great Inequality" over the past 2000 yrs. (Fluctuation in Saturn's motion peaks at about 60 minutes of arc and Jupiter's peaks at about 20 minutes of arc.) Here's the gist of Smith's explanation, in my own words:

A unique line L (a "diameter") can be drawn in the plane of the orbits of Jupiter and Saturn, passing through their points of least- and greatest- separation. Conjunctions are then of two types: (a) for conjunctions that occur on one side of line L, the planets have greater separation after the conjunction than before, and (b) for conjunctions on the other side of line L, the planets have less separation after the conjunction than before. Conjunctions occur about once every 20 years, with Saturn covering about 2/3 of its orbit in that time; consequently, about 2 of every 3 conjunctions occur on the same side of line L, producing a net perturbation of both planets in the course of every 3 conjunctions -- about every 60 years. It so happens that about 450 years of conjunctions are required before the "2 of every 3" switch types (i.e. change the side of line L on which they occur), thus reversing the effect and resulting in a cycle whose period is about 900 years.

And pp. 298ff. discuss the "secular" acceleration of the moon's mean motion. The gist here is that the moon's mean motion is observed to be accelerating about 12 arc-seconds/century, an amount explained by two theoretical components:

• About 6 arc-seconds/century due to planetary perturbations of Earth's orbit. (This is a sum of perturbation terms whose dominant periodic part was found by Laplace to be about 10 arc-seconds/century; Adams later included higher-order terms whose effect was to reduce the total to the currently accepted 6 arc-seconds/century.)
• About 6 arc-seconds/century due to tidal effects via the moon's gravity.

For the period as found by Laplace, I've been unable to find a source that provides anything more precise than just "millions of years". I had hoped to find the value in Laplace's actual derivation at the Smithsonian Library, but alas, my French is lacking.

1. Kushner, David. The Controversy Surrounding the Secular Acceleration of the Moon’s Mean Motion, Archive for History of Exact Sciences, vol. 39, no. 4, Springer, 1989, pp. 291–316.

I didn't know this when I asked, but apparently the second part of my question was at one time a very hot potato:

But one of these secular inequalities has particularly engaged the attention--and enraged the passion--of astronomers: the secular variation of the moon's mean motion. Indeed the international controversy which flared up circa 1860 was one of the largest and most active of the century.

Finally, a quote that I enjoyed (from Smith, p. 299):

"For I find this Theory [of the Moon] so very intricate & the Theory of Gravity so necessary to it, that I am satisfied it will never be perfected but by somebody who understands the Theory of gravity as well or better than I do." -- Isaac Newton (in a letter to Flamstead, February 16, 1695)