# What was the “libration of Pallas” that Gauss studied, when Pallas is not a moon of Jupiter?

This question is a request for general explanation of an astronomical phenomenon known as "Pallas libration", and isn't much about the details of Gauss's mathematical model of this libration. As is well known (see this post: Calculation of Gauss leading to 18:7 resonance between orbits of Jupiter and Pallas), Gauss discovered a $$18:7$$ resonance of Pallas with Jupiter, and according to the source given in https://aas.org/archives/BAAS/v31n5/aas195/431.htm ("On the Commensurate Motion of Pallas") he developed a theory of the periodic libration of Pallas with a period of $$737$$ Jupiter sidereal periods (Werke v. 7, p. 421, p. 559). Martin Brendel, in his essay on Gauss's astronomical works, also commented about Gauss's theory of libration (he actually devoted an entire section to it in his essay).

The basic point i don't understand is the use of the word "libration" within the context of the motion of asteroids - as far as i know, libration is a phenomenon in which an observer on a planet (Earth, for example) is able to see more then a half of an orbiting moon which is in a "tidal lock" with the planet (the effect in which tidal forces and moments synchronize the rotational angular motion of the moon with his orbital angular motion). This is caused by visual effects such as the parallax derived from the change in positions of the observer. But Pallas isn't even a moon of Jupiter, needless to say in "tidal-lock" with Jupiter. So i'd really like to get an explanation on this.

Although this question is more about the physics involved then about the historical circumstances, I chose to mention Gauss and ask it on HSM stackexchange, simply because he was one of the few astronomers that attempted to formulate such a theory (which is quite a "niche" in astronomical sciences), so one can't really seperate the science from it's history and discoverers in this case.

• The word "libration" simply means oscillation, no matter what the cause is. Comes from the Latin "librare", to sway. So libration of Pallas has nothing to do with libration of the Moon or satellites. – Alexandre Eremenko Aug 27 at 20:44
• Thanks! I guess i wrote a very long question which has a very simple answer... just another question: what is "Pallas libration" ? If libration is simply oscillation than what is the physical quantity that oscillates here? and why is this important? – user2554 Aug 27 at 20:55
• I have not read these works of Gauss, but I suppose it is longitude that oscillates. When we observe a star-like object in the Solar system, there are two things we want to know about it: how longitude and latitude change with time. (Here longitude and latitude are ecliptic coordinates, related to the Sun). In the absence of Jupiter, an asteroid would move on Kepler's orbit. Jupiter causes small oscillations (librations) of its coordinates WITH RESPECT TO the Kepler's orbit. – Alexandre Eremenko Aug 27 at 22:07

"Libration" loosely refers to longitudinal oscillations due to orbital resonances. The tidal lock and the trojans, the only situations that Wikipedia mentions, are just special cases. The existence of resonances does not require one body to revolve around the other, and Jupiter exerts a major pull on the asteroid belt just as a planet would on its moons.

However, according to Woltjer's On the supposed libration in the motion of Pallas, Gauss's "libration" of Pallas was likely a numerical artifact:

"One of the results of GAUSS' investigation on the motion of Pallas as disturbed by the attraction of Jupiter consists in the commensurability of the mean motions $$18n'-7n=0.\ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ This equation shows a remarkable difference with the corresponding equations for some of the best known cases of libration in the solar system. In all these cases the algebraic sum of the integral numbers from the coefficients of the mean motions in the equations analogous to $$(1)$$ is zero; here it is $$+11$$...

The origin of this strange result contained in $$(1)$$ lies in the fact that GAUSS developed the perturbative function for definite numerical values of the elements. If $$\lambda, \varpi, \Omega$$ denote the mean longitude, longitude of perihelium, longitude of ascending node of Pallas, and the corresponding quantities for Jupiter are denoted by accented symbols, we have the following arguments in the perturbative function associated with the commensurability $$18:7$$: $$18\lambda'-7\lambda-11\varpi\\ 18\lambda'-7\lambda-10\varpi-\Omega\ \ \ \ \ \ (2)$$ All the corresponding terms have been coalesced into one by GAUSS' numerical development; If however one of these terms, say the first; represents a case of libration, then the other's arguments have mean motions, such as for the second term $$\varpi-\Omega$$, which in general are different from zero.

Thus the coefficient $$\alpha$$ in the differential equation for the argument of the libration u: $$\frac{d^2u}{dt^2}=-7\alpha n^2\sin u$$ should correspond to one of the arguments $$(2)$$ only, and not to the coefficient resulting from the addition of all the terms $$(2)$$ for definite numerical values of $$\varpi$$, $$\varpi'$$ and $$\Omega$$. Consequently it seems to me that we have no reason to assume the existence of libration in the motion of Pallas, until a more detailed investigation of the problem has been made by considering each of the terms $$(2)$$ separately.

Taylor in The secular motion of Pallas summarizes this as "Woltjer (1922) showed that the $$18:7$$ commensurability was unlikely to lead to a libration in the motion of Pallas", and discusses the effect of resonances with Jupiter on the motion of asteroids more generally.

• Thanks @Conifold! i appreciate your effort. Your answer and Alexandre Eremenko's cooments helped me understand the context in astronomy for Gauss's work on the supposed libration of Pallas. It really saved me a lot of time! Just as a byside comment - i tend to like answers that open doors for understanding the work of great scientific explorers of the past, not closing doors (and don't be offended by this, i have huge appreciation for your work here on HSM stackexchange)... – user2554 Aug 28 at 13:00
• In other words, can you tell something about the significance of Gauss's mathematical theory of libration of Pallas? because even if it's a model for a hypothetical phenomena that doesn't exist at all, it's still a mathematical theory that might be relevant to other cases than Pallas. I say so because from looking at the relevant pages of Gauss's nachlass (p. 557-560 in volume 7) i gained the impression that he really developed a sophisticated model for Pallas involving his arithmetic-geometric mean and his work on perturbation theory. – user2554 Aug 28 at 13:00
• @user2554 I am the opposite, like to narrow things down and close the case :) You'll be disappointed, I am afraid. Gauss's work (along with Laplace's and Poisson's) helped turn perturbative orbital calculations into a routine, Taylor still uses some elements of his approach to averaging, in 1981. But on libration specifically the problem is mathematical, and not specific to Pallas, his way of averaging does not approximate the solutions correctly long term. Secular motions and singular perturbation theory require some more global and qualitative methods that Poincare developed later on. – Conifold Aug 29 at 3:04