"Gauss' planetary equations" are a set of equations that describe the rate of change of the six elements of a planetary orbit in the presence of a disturbing force. I just intend to ask in which of his works did Gauss first write his planetary equations. Did he write them down in the Theoria motus (1809) or perhaps in some of his shorter papers?


2 Answers 2


I agree with the conclusion that the planetary equations are not in Theoria motus (1809). I believe Gauss first wrote them down in §14 of the essay Exposition d'une nouvelle méthode de calculer les perturbations planétaires avec l'application au calcul numérique des perturbations du mouvement de Pallas, drafted (c.1812) for a Paris prize competition but only published posthumously (1906).


Gauss was not quite so explicit, but it does seem that his "planetary equations" first appear in Theoria Motus.

In Planetary Orbital Equations in Externally-Perturbed Systems: Position and Velocity-Dependent Forces by Veras and Evans we read:

"The work of Burns (1976), subsequently popularised by Murray & Dermott (1999, pp. 54-57), provides a mechanism to obtain analytic equations for $da/dt$ , $de/dt$ , $di/dt$ , and $d\Omega/dt$ arising from a small perturbative force with a given prescription for radial, tangential and normal components. This line of attack can be traced back ultimately to Gauss (see e.g., section 9.13 of Brouwer & Clemence 1961)."

This apparently refers to four of the Gauss planetary equations. The cited book Methods of Celestial Mechanics by Brouwer and Clemence does not have a section IX.13. It does, however, have section XI.13, titled Gauss's Form, which reads:

"Equations in this form were first derived by Gauss and applied to the calculation of first-order perturbations by Jupiter on Pallas. Gauss made use of these equations also for deriving secular perturbations in the elements. Finally, the equations have been used extensively for computing by numerical integration perturbations in the elements of comets and minor planets."

Gauss and Ceres informs us that Gauss published his methods in "Theoria motus corporum coelestium in sectionibus conicus solem ambientium" of 1809 (apparently, Gauss had to translate the book from German into Latin at publisher's request... for greater accessibility). But it has a reproduction of Gauss's sketch of orbits of Ceres, Pallas and Vesta referenced to

Astronomische Untersuchungen und Rechnungen vornehmlich über die Ceres Ferdinandea [Astronomical investigations and calculations mainly on Ceres Ferdinandea], 1802, SUB Göttingen: Cod. Ms. Gauß Handbuch 4, Bl.1

The accompanying text reads:"Gauss was able to calculate the orbit of the planetoid. On December 7, 1801, the planetoid Ceres reappeared exactly at the location predicted by Gauss". Apparently, Pallas and Vesta were less cooperative. The linked biography page adds that "at the time" of Theoria Motus "he was able to calculate the exact path of the first three dwarf planets Ceres, Pallas and Juno as well as the to discover the first resonance relation in the asteroid belt."

There is Davis's English translation of Theoria Motus, Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections. Perturbations of Pallas by Jupiter are mentioned at the very end, on p.277. On the resonance see Calculation of Gauss leading to 18:7 resonance between orbits of Jupiter and Pallas.

  • $\begingroup$ Thank you very much Conifold for your long answer. I learned a lot from your answer, but i'm still not sure the equations first appeared in the Theoria Motus (i don't believe until i see the equations written down). Your answer convinced me that Gauss is indeed the originator of the equations (so thet are not mistakenly named after him), but not about when did they first appear. So can you give a page in the Theoria motus where the equations are written down? $\endgroup$
    – user2554
    Dec 19, 2016 at 17:34
  • $\begingroup$ @user2554 I can not see all pages from Davis's translation in Google view, but I did not see them on the pages that are shown. Brouwer and Clemence have them on pp.305-306 with quoted credit to Gauss but the phrasing "this line of attack can be traced back ultimately to Gauss" might mean that he did not write them explicitly but used similar approach. The last sections of Theoria Motus seem like the best guess for what modern authors take to be the source of "Gauss's equations". $\endgroup$
    – Conifold
    Dec 20, 2016 at 1:02

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