# Calculation of Gauss leading to 18:7 resonance between orbits of Jupiter and Pallas

After Gauss helped relocate Ceres, he studied the orbit of the asteroid Pallas and discovered (1812) that Jupiter and Pallas have an orbital resonance that is nearly equal to 18:7. For instance, using the modern estimates of their orbital periods as 4332.59 days and 1684.87 days respectively, their ratio has continued fraction expansion $[2,1,1,2,1,511,2,\ldots]$, which is quite clearly approximated well by $[2,1,1,2,1] = 18/7$. Of course Gauss did not have the modern values for the orbital periods. If we use the cruder estimates of 4333 and 1685 we get a ratio with continued fraction expansion $[2, 1, 1, 2, 1, 240, 113, \ldots]$, which again is begging to be approximated by $[2,1,1,2,1]$.

My questions:

1. What was the ratio computed by Gauss that he then determined is nearly 18/7? Maybe it was not the orbital period directly but some other astronomical measure whose ratio would come out to the same thing.
2. Did he use continued fractions, or was his great familiarity with decimal expansions of "small" fractions enough? I think this example is a really super illustration of continued fractions, but I would like to know for sure if it's how Gauss attacked the problem.

I can find a number of sources discussing this resonance, but they mention the value 18/7 without giving the Gaussian calculation behind it (i.e., what ratio did he estimate as 18/7), and they don't indicate his method either.

• 7 and 18 are Lucas number, so I guess 18/7 was familiar to Gauss May 13, 2015 at 21:03
• And Gauss was (much more than) familiar with continued fractions, see for example en.wikipedia.org/wiki/Gauss%27s_continued_fraction May 13, 2015 at 21:05
• @VicAche, this work of Gauss was in the early 1800s and Lucas worked in the second half of the 1800s, so your comment about Lucas numbers is not plausible. I am well aware that Gauss was very familiar with continued fractions (I am a number theorist), but just because continued fractions are an ideal way to discover a good rational approximation I don't know whether that is how Gauss really approached this problem. Gauss knew decimal expansions very well too (his Disq. Arith. has a major treatment of them), so if data led him to 2.57151 he could have "seen" it's close to 18/7 = 2.57142...
– KCd
May 13, 2015 at 21:19
• Given Gauss work on Fibonacci, you will admit that it's very improbable he didn't knew about Lucas number, of course by another name. May 13, 2015 at 22:38
• Answer is here (or so its seems, it's in german) groups.google.com/forum/#!topic/de.sci.mathematik/s3khftjGVS8 May 13, 2015 at 22:45

Gauss's ratio was that of the mean motions $\mathit{[M]}$ (⚴) and $\mathit{[M']}$(♃) of Pallas and Jupiter. (Instead of these letters he used planet symbols, which are shown in their unicode version in brackets.)

As can be seen in his Nachlass (Werke, vol. 7, e.g. p. 553), he was expressing the perturbations of Pallas elements as sums of dozens of trigonometric terms $A\sin(k\mathit{[M']-\ell [M]}+\delta)$. On p. 604 the editors comment:

On one of the sheets which contain the integration of the perturbations of the epoch $\varepsilon$, one finds the following small computation, in which the first number given is Pallas's mean motion $\mathit{[M]}$: $$\begin{gather} \mathit{769'',202079}\\ \textit{das 7-fache} = \mathit{5384'',414553}\\ \underline{\mathit{18\ m. m. [M'] = 5384'',392272}}\\ \mathit{18 [M']- 7[M] = -0,022281} \end{gather}$$ This seems to be the only hint at the first step in Gauss's discovery of the commensurability of both periods; from this one can't however conclude anything more than that he had just found the quantity $\mathit{18 [M']- 7[M]}$ extremely small.

I read this as saying that this ("small divisor"?) stood out enough as it is, numerically among the $\mathit{k[M']- \ell[M]}$'s he was looking at, with no need for an appeal to, e.g., continued fractions.

• Thanks for this information! Starting from p. 553 at your link, I find on pp. 557-558 the numbers 18 and 7 floating around. What is being discussed in the first paragraph on p. 557? On p. 558 I see in the second paragraph he singles out the rational ratio 7/18 and I see in the middle of the page $n' = 299.12817$ and $n = 769.16512$ (not quite the $769.202079$ you cite from the editor comment on p. 604), where $18n'$ and $7n$ are both 5384 plus a small bit.
– KCd
May 14, 2015 at 0:17
• The impression I have gotten from looking at various references discussing resonances is that for the most part working scientists really don't discover them using continued fractions.
– KCd
May 14, 2015 at 0:29
• @KCd: The top of p.557 says approximately, "It was already observed in Article 19 of the Exposition that Pallas offers an example of rational relationship of the mean motions n and n'; indeed Pallas and Jupiter's mean motions are in precisely the ratio 18:7, so that this relationship keeps reproducing itself exactly. In this Article the terms depending on the argument 18[M'] - 7[M] will be given in secular form for practical computation, whereas the exact integration, through which the periodic form reproduces itself, is the object of the next Article." Alas, I can't find this "Art. 19"... May 14, 2015 at 1:30