Neptune: The First Planet Discovered by Mathematics

When you lookup in the sky the most prominent celestial objects you see are the Sun and the Moon along with distant stars. What if I tell you that you can also see five planets of our solar system in the night sky starting from Mercury to Saturn? Yes, they just look like stars but don’t twinkle. This has been known since ancient times and incorporated into our calendar week of 7 days and each day is named after the objects of solar system which can easily be seen with the naked eye.

Moving visible solar system objects in the night & daytime sky, the Sun, Moon and the 5 naked eye visible planets (Mercury,Venus,Mars,Jupiter & Saturn)

So, are we limited to see only seven objects in our solar system?

Ofcourse not! Since the invention of telescope, the four largest moons of Jupiter were spotted by Galileo in 1610. Five decades later, Christiaan Huygens made the discovery of Saturn’s rings. More than a century later in 1781, the seventh planet Uranus was discovered by William Herschel. But, the discovery of 8th planet is remarkable in the sense that it was predicted with mathematical calculations rather than observational means, a pivotal event in the history of astronomy. So, let us take a journey through this mind bending discovery.

The problem with the motion of Uranus:

The discovery of Neptune resulted from the need to develop a theory explaining the erratic motion of the solar system’s 7th planet Uranus, which could not be completely accounted for by means of the gravitational effects of Jupiter and Saturn. Several astronomers since the planet’s discovery by William Herschel in 1781 had suggested that the perturbations in Uranus’s orbit could be caused by an unknown trans-Uranian planet, which could have been mistaken for a fixed star in earlier observations. However, the complex mathematics required for proving this hypothesis was so daunting that no one attempted the task. But there was a way, since Laplace already developed mathematical expressions for the mutual perturbations exerted by the planets as a result of their gravitational attraction. Using these expressions, one could carry out numerical calculations to produce tables of the positions of the planets over time. Bouvard, Laplace’s student, began the calculation of tables predicting the movements of Jupiter, Saturn and Uranus, where the calculation of Jupiter’s and Saturn’s tables proved to be relatively straightforward, but Uranus’ table however was highly intractable, even after taking into account the perturbations exerted by the other planets.

Irregularities in Uranus’s orbit: At position a, the outer planet gravitationally perturbs the orbit of Uranus, pulling it ahead of the predicted location. The reverse is true at b, where the perturbation retards(slows) the orbital motion of Uranus.

Bouvard was the first to speculate that the anomalous motion of Uranus could be occasioned by the gravitational action of a new planèt. Soon the idea got spreaded and in 1835, the astronomer Benjamin Valz, proposed to Arago carrying out a search for the planet. Arago evidently hoped that the problem of Uranus would be taken up at the Paris Observatory. Since there was no one else at the observatory he deemed capable of tackling such a difficult problem, he turned to Le Verrier. He had great faith in Le Verrier’s mathematical abilities, and so, at Arago’s request, Le Verrier abandoned the investigation of comets in which he was then involved and devoted himself to Uranus. At the same time, John Adams, a young British mathematician recently graduated from Cambridge, also started working on the same problem independently.


The work of Le Verrier and Adams:

In 1843, Adams did some calculations and predicted that the gravitational pull from an eighth planet was causing all the weird movement of Uranus. Unfortunately, Adams did not pursue the calculations with much eagerness until a much later date. He sent in his predictions to Royal Greenwich Observatory in September 1846, meanwhile Le Verrier completed an independent computation and predicted the exact position for Neptune. He sent the prediction to the French Academy on August 31, 1846. The two astronomers later became good friends.

Le Verrier wrote: It would be natural to suppose that the new body is situated at twice the distance of Uranus from the Sun using Bode’s law. Formulated as,

  where ‘a’ is semi major axis in AU and ‘m’ is consecutive integers from 0.

The law suggests that, extending outward, each planet would be approximately twice as far from the Sun as the one before.

Graphical plot of the traditional nine planets (including Pluto) and Ceres versus the first ten predicted distances using Bode’s law.

First, it is obvious that the new planet cannot come too close to Uranus [its perturbations would have been very evident]. However, it is also difficult to place it as far off, say, as 3 times the distance of Uranus, for then we should have to give it an excessively large mass. But then it’s great distance both from Saturn and Uranus would mean that it would disturb each of these two planets in comparable degree, and it would not be possible to explain the irregularities of Uranus without introducing very sensible perturbations of Saturn as well. We might add that since the orbits of Jupiter, Saturn, and Uranus all have a very small inclination to the ecliptic, it is reasonable to suppose, as a first approximation, that the same must apply to the new planet. Le Verrier had reduced the number of unknowns by two: he assumed the semi-major axis of the orbit and the inclination of the orbit. Nevertheless, there remained more than enough other unknowns, in part because the orbital elements of Uranus were themselves poorly determined. One can work out all the perturbations of the other planets except the new one, and finally establish the discrepancies between the calculated and observed positions so as to show the effects of the perturber. It is not possible in this way to obtain a unique solution to the problem since any number of other orbits remain possible for Uranus. As Le Verrier has already settled with 2 unknowns of new planet, he settled the same ones for Uranus; the semi major axis and orbital inclination. With this simplification, there remained eight unknowns in the orbital elements, to which he added a ninth, the mass of the perturbing planet.

Le Verrier needed only 3 months to specify the orbital elements of the perturbing planet, guess at its mass, and even provide an order of magnitude estimate of the apparent diameter it would present in the telescope. In his note of 31 August 1846, he summarized his methods and gave the predicted orbital elements for the new planet, to a degree of precision that would prove, however, to be entirely illusory. Le Verrier modified the semi-major axis of the orbit slightly from the initial hypothesis in which he had simply followed the Bode’s law, and taken it to be twice that of the orbit of Uranus, i.e., 38 AU. Le Verrier predicted that the planet lay in the sky about 5° east of the star delta in the constellation Capricorn, and also, as noted, indicated the approximate values of the apparent diameter and brightness of the planet, probably in an attempt to stimulate the imagination and ambition of an observer to look for it.


The Search & Discovery:

The two great mathematicians(Le Verrier and Adams) did their calculations manually. Up until independent confirmation from both, no one recognized the similarity of their solutions and their work had been just little more than a curiosity, but now it spurred to organize a secret attempt to find the planet. Adams continued to work on the problem, providing the British team with six solutions in 1845 and 1846, which resulted in searching the wrong part of the sky. Le Verrier was unaware that his public confirmation of Adams’ private computations had set in motion a British search for the purported planet. Le Verrier finally sent his results by post to Johann Galle at the Berlin Observatory. Galle received Le Verrier’s letter on 23rd September 1846 and immediately set to work observing in the region suggested by Le Verrier. Galle’s student, Heinrich Arrest, suggested that the recently drawn chart of the sky, in the region of Le Verrier’s predicted location, could be compared with the current sky to seek the displacement characteristic of a planet, as opposed to a stationary star.

Neptune was discovered just after midnight, after less than an hour of searching and the exact location varied a less than a degree from the computed position of Le Verrier, a remarkable match. After two further nights of observations in which its position and movement were verified, Galle replied to Le Verrier with astonishment: “the planet whose place you have computed really exists”. Shortly after the announcement of the discovery, the planet was viewed in Paris by Le Verrier himself, as well as by several other astronomers. Many wrote to congratulate Le Verrier.

New Berlin Observatory, where the planet Neptune was discovered in 1846

Thus the astronomers Le Verrier and John Adams are credited with the discovery of the 8th planet Neptune by using only mathematics in 1846. The discovery of Neptune not only represents the greatest triumph for Newton’s gravitational theory since the return of Halley’s Comet in 1758, but it also marks the point at which mathematics and theory, rather than observation, began to take the lead in astronomical research.

Clockwise from top left, John Adams, Le Verrier and Johann Galle. Rightside is Neptune in 1989 by the Voyager 2 probe 


Flushed with enthusiasm, Le Verrier wrote: “This success leads to the hope that, after observing the new planet for another 30 or 40 years, it will become possible to use it in turn to discover the orbit of the next one in order of distance from the Sun, and so on. Unfortunately, the more distant objects will be invisible because of their immense distance from the Sun. Nevertheless, over the course of centuries, their orbits will be traced out with great exactitude by means of their secular inequalities.”

Needless to say, his hope was not fulfilled in the way he expected. Other bodies in the solar system more remote than Neptune, such as Pluto, Eris, Haumea, Makemake, Sedna and more such worlds have been found; however, they are so remote and their masses are so small that the influence they exert on the orbit of Neptune is negligible. The discoveries of these “dwarf planets,” as they are now known, resulted not from mathematical investigations of the kind that led to Neptune’s discovery but from systematic photographic or CCD surveys.

So readers, encourage your children to study mathematics diligently. You do not know how they will make use of mathematics. Whenever your children ask you what’s the purpose of solving all the silly mathematics word problems, do tell them the story of discovering the planet Neptune. Mathematics is useful in all aspects of our lives. Even though most of us do not use mathematics to compute the celestial movements, we do use it to check our salary and various deductions.

Author: Vishweshwaran Murali, Mathematics Enthusiast.


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