Neptune’s Orbit: Reassessing Celestial Mechanics

Abstract

The almost miraculous discovery of 1846 and at first bitter priority dispute ended in March 1847, with the piece by Jean-Baptiste Biot that was reprinted in The Athenaeum, followed by the orchestrated, highly public display of men of men of science getting along amicably at the Oxford meeting BAAS meeting in June, where Le Verrier and Adams met. Meanwhile, Adams had swiftly refined Neptune’s orbit, and Lassell’s prompt discovery of Triton had enabled Neptune’s mass to be determined. However, in that decade, American astronomy was coming of age, and for lack of large telescopes was led by elite mathematicians who now fastened on Neptune to prove their credentials. Benjamin Peirce of Harvard, with claims that Neptune had been found by “happy accident”, re-ignited controversy by challenging whether a theoretical discovery had been made at all. This, and the potential for further discoveries, sustained a century’s international focus and effort to revise celestial mechanics.

In the case of Neptune, the predictions of Adams and Le Verrier would be partially affirmed, though it would also become clear that their success was owing to very special conditions. Meanwhile, Le Verrier attempted to use the same methods of perturbation theory to explain an unexplained advance of the perihelion of Mercury on the basis of an intra-mercurial planet (Vulcan). After briefly covering Le Verrier’s personal obsession with Vulcan, we explain how this and the analyses of other practitioners of celestial mechanics illuminate the struggle to transcend Bode’s law by pushing celestial mechanics in new directions in the pre-computer era. The advent of photography for survey work, epitomized by Percival Lowell’s (1855–1916) dedication to the search for “Planet X” early in the 20th century, fueled the belief that more distant planets awaited discovery.

Despite the discovery of Pluto being hailed as a late-ripening fruit of Lowell’s obsessive search, we now know it as only the first of a number of objects of dwarf-planet class populating the outer Solar System (the region of the Kuiper belt). As related in Chap. 10, the methods of celestial mechanics have been largely abandoned for empirical methods of searching for planets, leading to greatly extended knowledge of the region beyond Neptune—an unexpected coda to 1846.

Appendices

Appendix 9.1

9.1.1 The Perturbation of Uranus by Neptune: A Modern Perspective

In the 19th century, mathematical astronomers were able to see this, at best, in a glass darkly. In recent years, investigators have returned to the problem, with new methods of analysis.

The starting point must be, now as then, the deviation, first discerned by Bouvard, in heliocentric longitude from a Kepler orbit even after known perturbations of other planets had been subtracted out. The values of this discrepancy, Δϕ, have been summarised in the Table 9.3 below. These were, of course, ascribed by Adams and Le Verrier to the unknown planet, and served as the basis of their attempts to work out the orbital elements of that unknown.

Table 9.3 Discrepancy between observed and calculated heliocentric longitude of Uranus (Δϕ) after known causes of perturbation have been subtracted out, 1690–1840. Note that the observations for 1690–1780 correspond to the “ancient” observations, those from 1781 (Herschel’s discovery) to the “modern” ones (After Lodge, 1960)

In 1990, H.M. Lai, C.C. Lam, and K. Young of the Chinese University of Hong Kong published a new analysis using a simple model, in which the unperturbed orbits of Uranus and Neptune were regarded as circular and co-planar, in order to explain the main features of Δϕ as given above. (Lai, Lam, and Young, 1990). Those features included:

• 1) A typical magnitude of Δϕ is 50 to 100 secs of arc.

• 2) An approximate periodicity of about 100 years (e.g., measured between peaks in 1705 and 1815).

• 3) An apparent secular variation, which may be described as a linearly decreasing term.

• 4) Phase such that Δϕ rises through zero in the year 1777 and descends back through zero in 1830–31.

Lai et al. carried out two calculations using their simple model: (1) the forward problem of calculating the deviations in the position of Uranus and (2) the inverse problem of using the observed deviations to infer the elements of Neptune (Fig. 9.21).

Fig. 9.21

Upper: Comparison of the forward perturbation problem with the data. Lower: The solutions of the forward perturbation problem over a longer time period (Credit: Kenneth Young)

The authors noted that Adams and Le Verrier, by assuming a Bode’s law orbital radius and hence a period for the unknown planet, had needed only to find the phase of Neptune in its orbit and the mass in the inverse problem. This had been a reasonable approach at the time, though of course since Bode’s law was later completely discredited—indeed discredited by the discovery of Neptune itself—a more general approach to the problem would have involved allowing the orbital radius of Neptune to vary as well. Adams, indeed, had realised this, and as noted earlier, was proceeding along this line late in the course of his calculations. The most important results of the paper of Lai et al. are as follows:

• 1) The deviations in Uranus’s longitude caused by perturbation by Neptune are actually an order of magnitude larger than those found by Adams and Le Verrier. Since, historically, in the construction of a planetary theory, the biggest terms in the perturbation are those with long periods, or terms having nearly the period of the disturbed body (which happens because of the near-resonance in this case) these can in large part be hidden by throwing them upon the mean motion, the epoch, the eccentricity and the place of the perihelion of the disturbed body. In the case of Uranus, this was mostly accomplished by an incorrect choice of the semi-major axis and eccentricity, especially the latter, assumed for the unperturbed orbit of Uranus. The near-resonance was such that a component of the perturbing force is in very nearly 2:1 resonance with the unperturbed motion of Uranus.

• 2) There is another problem with the historical calculation. Because the motion is responding to the second harmonic of force, the inverse problem suffers from a possible ambiguity in phase (where phase refers to the particular point in time in the cycle). One solution was the one found by Adams and Le Verrier, the other is displaced 180 degrees opposite. Thus there are actually two solutions for the heliocentric longitude of Neptune, diametrically opposite each other, which are not too different in terms of the quality to the fit to the data. The fact that Adams and Le Verrier found the correct solution (the one actually occupied by the planet at the time) would seem to be rather fortuitous.

• 3) The interesting features described under (1) and (2) are consequences of the fact that Uranus and Neptune form a nearly resonating system, as a result of the fact that, as noted above, a component of the perturbing force is in very nearly 2:1 resonance with the unperturbed motion of Uranus. Since the resonance is not quite exact (the period of Neptune differs from twice that of Uranus by 2 percent), this generates the well-known phenomenon of “beats” over a long-period rhythm. The choice of (an incorrect) mean motion that absorbs most of the perturbation means that, during the historical period, the two sinusoidal terms are nearly out of phase and come close to canceling out; thus, the reported deviations, i.e., the difference from the (incorrect) mean motion actually reached a minimum during this period, not a maximum as assumed by Herschel and others after the discovery. Their mostly incorrect understanding of perturbation theory made the deviations symmetric around the 1822 Uranus-Neptune conjunction. Nevertheless, Le Verrier’s and Adams’s calculations were valid within the limitations of the theory they used, insofar as they did succeed in finding the local minimum in the inverse problem—a not inconsiderable achievement, since, according to the authors, “even a search with many fewer parameters [than those actually involved] would have been a major computation endeavor” in the days before electronic calculators. Also, whereas the true frequency as determined by the modern knowledge of the Neptune period leads to a driving force slightly under resonance, the fitted frequency [Δϕ] corresponds to a driving force slightly above resonance. The fact that the perturbation was close to resonance, either slightly above or slightly below, sufficed for Adams and Le Verrier to determine the heliocentric longitude of Neptune during their era, not far from the time when Uranus and Neptune shared the same heliocentric longitude (their conjunction in 1822).

Appendix 9.2

9.1.1 Note on Celestial Mechanics After Poincaré: Phase Space and Chaos

The discipline of celestial mechanics has undergone significant conceptual changes since the days of Adams and Le Verrier, especially following the publication of Henri Poincaré’s Les Méthodes Nouvelles de la Mécanique Céleste (Poincaré , 1892) . The classical searchers for the disturber of Uranus would hardly recognise their own field.

Among the most important innovations was recasting the formulae of celestial mechanics in terms of the Hamiltonian function (introduced in 1835 by the Anglo-Irish mathematician and astronomer Sir William Rowan Hamilton [1805–1865]), which shifted the focus from velocities to momenta of particles, and allowed the differential equations for position and momentum to be derived from the total energy of the system.

This innovation eventually led to the concept of the phase space—an abstract space of a large number of dimensions in which one dimension for each of the coordinates describes a physical system comprising a certain number of particles (e.g., planets, as in the three-body problem). In this way, numerical data was transformed into a contoured map of all allowed possibilities “It was in these investigations,” notes mathematics writer Ivars Peterson (1993:160), “that Poincaré first caught a glimpse of, and to some degree appreciated, what we now know as dynamical chaos.” Dynamical chaos entered in precisely because, as Poincaré realised (pp. 185–186), “small differences in initial conditions could lead to great disparities at future times,” making it “impossible to predict with any degree of assurance the changes an orbit might undergo at some point in the distant future.” Admittedly, the deviations remained quite small over time spans of a few hundred years, such as those that had been considered by Lagrange and Laplace in their celebrated investigations of the “stability” of the Solar System, but it was clear that an orbit might undergo drastic changes in the distant future. It would not be until the advent of digital computers that astronomers such as Jack Wisdom, in the 1970s, began to systematically explore the implications for the evolution of orbits of Poincaré’s phase space in detail, and when they did so they identified that planetary positions and velocities could be plotted out into regions of regular, predictable behavior and regions of chaotic, unpredictable behavior, where the zones of chaos were found around resonance positions. (Wisdom, 1987).