Preliminaries to the Neptune Discovery: Newtonian Gravitational Theory

Abstract

The discovery of Neptune stunned the 19th century world. It involved the mathematical prediction of the existence of a major planet never yet seen through a telescope, because of the way it seemed to be pulling on a known planet, Uranus, away from its predicted orbit. When an actual planet was discovered near the predicted position, it was famously hailed as the greatest triumph of Isaac Newton’s theory of gravitation.

How did they do it? Largely by standing on the shoulders of giants—a dazzling succession of 18th century mathematicians, many of them French—who had to overcome many roadblocks in solving difficult problems, some of which had given Newton himself headaches. In so doing they developed celestial mechanics, the branch of mathematical astronomy that would later point the way to Neptune.

Appendix 1.1

1.1.1 Elements of a Planetary Orbit

By way of background, at the heart of the mathematical investigation that led to the discovery of Neptune are various concepts of celestial mechanics, beginning with the basic Keplerian ellipse (the path a planet travels round the Sun at one focus when unperturbed) which is described by six parameters known as elements. The subject matter proceeds to the case where this simple orbit is perturbed by the gravitational attraction of another planet, and so to several other planets, with the resulting orbit defined by a series of complicated formulae. This is known as the forward perturbation problem.

In the hands of Laplace, Lagrange, and others, 18th century astronomers had become very skillful in representing the perturbation of planet P’s orbital elements by another planet P′ as a periodic series representing the elements of an orbit as a function of time. Let the orbital elements, e, a, i, Ω, ω, and ν of planet P, which is that to be perturbed, be defined as follows, with the orbital elements of the perturber represented by the same symbols accented:

• e, the eccentricity, defining the shape (elongation) of the ellipse, where e = 0 is a circle and e = 1 is a line;

• a, the semimajor axis; the sum of the perihelion and aphelion distances divided by 2;

• i, the inclination, or vertical tilt of the ellipse with respect to the reference plane (e.g., plane of the Earth’s orbit, known as the ecliptic);

• Ω, the longitude of the ascending node of the ellipse, where the orbit tilts upward relative to the reference point, the First Point of Aries, symbolised by ♈;

• ω, the argument of the perihelion, defining the orientation of the ellipse in the orbital plane, as the angle measured from the ascending node to the perihelion;

• ν, the true anomaly at epoch M_o, which defines the position of the body at the epoch, a specific point in time.

In addition, to compute the perturbations, it is necessary to know the masses of the two planets, m and m′, and n and n′, the mean motion of the two planets. A significant role in these computations is played by the perturbation function, introduced by Lagrange, which can be developed into a (very complicated) power series in e₁, e₂, and sin²I/2, where I is the angle between the two planetary orbits.

Applications of perturbation theory are mentioned throughout the book, especially as related to Adams’s and Le Verrier’s calculations of the position of the unknown planet disturbing Uranus described in Chap. 4 and 5. The literature is vast, highly technical, and mostly in French, with the most significant contributions including those of Lagrange, Laplace, Pontécoulant, and Tisserand. Most of the more recent treatments have recast the classical methods in terms of vectorial notation. The best single-volume text for English-speaking readers with a good knowledge of differential equations that follows the classical methods is probably still (Moulton, 1914) (Fig. 1.16).

Fig. 1.16

Diagram showing the elements of a planetary orbit (Credit: Wikipedia Commons)