Lunar Theory. The birth of the Three Body problem

Until Newton realised and was able to show that the way gravity works was solvable methematically , what passed for alamanacs were written with the aid of Greek styled models. The following is an account of Newton’s ideas compared to the ancient ways and how some of his historians have fumbled with his efforts some going so far as to abuse him.

I have stripped most of such arguments from the essay here as I imagine those reading this are like me not interested in how he was wrong so much as in trying to understand what he meant when he was right.

It is a subject too wonderful for me:

The anomalies or inequalities in the orbit of the moon around the earth have been a major challenge to astronomers since Antiquity. Hipparchus pointed out that the moon’s position at quadrature deviated in longitude by over two and a half degrees from the Greek model of epicyclic motion.

And yet, this model accounted for its position at conjunction and opposition from the sun.

Ptolemy proposed the “Evection” mechanism which predicted a near doubling of the apparent size of the Moon. Nevertheless, Ptolemy’s lunar model was not challenged until the 15th century astronomer Ibn-al-Shatir developed an alternative mechanism that accounted for the evection without introducing the large unobserved variations of the lunar size in Ptolemy’s model.

In the 17th century alternative models were developed by Tycho Brahe and Johannes Kepler who incorporated his empirical law of areas for planetary motion into his lunar model. In 1640 Jeremiah Horrocks refined Kepler’s model, predicting correctly the inequalities in the distance of the moon from the earth which was being determined at that time by a micrometer.

By setting the centre of the moon’s elliptic orbit on an epicycle, Horrock’s model gave rise to an oscillating eccentricity and line of apses. Moreover, for the angle of rotation of this centre, Horrocks adopted Kepler’s choice of twice the angle between the earth-sun axis and the mean line of apses of the orbit of the moon.

Since Greek times, the corresponding angle chosen to describe Ptolemy’s evection inequality was twice the elongation of the moon from the sun, but it is the Kepler-Horrocks angle which turns out to be correct as shown by Newton in his dynamical formulation of this inequality.

In the Principia Newton demonstrated that the regular motion of the Moon around the Earth was caused by the gravitational attraction between these two bodies, but that the above mentioned inequalities in this motion were due to perturbations by the gravitational force of the Sun.

Detailed aspects of the calculations which involve the notoriously difficult three-body problem, however, remained a major challenge to astronomers after Newton’s pioneering work.

Over the past centuries Newton’s lunar work has been received with immense admiration by those who have been able to understand the profound mathematical innovations in his theory. An early reviewer of the second edition of the Principia stated that “ the computation made of the lunar motions from their own causes, by using the theory of gravity, the phenomena being in accord, proves the divine force of intellect and the outstanding sagacity of the discoverer”.

The French astronomer Laplace asserted that the sections of the Principia dealing with the motion of the moon are “ one of the most profound parts of this admirable work”, and the British physicist and Astronomer Royal George Airy regarded it “as the most valuable chapter that has ever been written on physical science”.

A close relation that exists between the kinematical model of Horrocks, and the dynamical lunar theory of Newton, already understood in the 18th century. Leonhard Euler asserted that Horrocks’s model had been the inspiration for his well known method of variation of parameters -the basis for the modern formulation of Newton’s lunar perturbation theory.

According to Newton’s theory (in the absence of the solar perturbation) the orbit of the Moon would be an ellipse with the Earth at one of the foci satisfying the area law or what is now called conservation of angular momentum.

Such elliptical motion had been proposed earlier by Kepler, who followed the kinematical tradition of the ancient Greeks. By fitting Tycho Brahe’s observational data, Kepler found that an ellipse gave the best results for the orbit of the planet Mars, and naturally he assumed that such an orbital curve applied also to other planets.

Newton in the Principia demonstrated that this orbital motion was a consequence of a gravitational force which varies inversely as the square of the distance between the Sun and the planets. Likewise, by a veritable tour de force. Newton also demonstrated that Horrock’s model was a consequence of the perturbing effects due to the gravitational force of the sun.

In the Principia, Newton presented most of his arguments in a qualitative manner in some of the 22 corollaries following Proposition 66. Some of the mathematical methods underlying his arguments can be found, although in a rather succinct form, in Cor. 3 and 4 following Proposition 17, in Book 1.

Details of his perturbation theory appear in a manuscript in the Portsmouth collection of Newton’s papers show that Newton’s method anticipated the perturbation methods of Euler and Laplace.

Some of Newton’s explanations for the rules in the lunar motion had been presented in the first edition of the Principia (1687). Subsequently, in the second (1713) and third (1726) editions, he added a new Scholium to Proposition 35 of Book 3 where some the contents of his 1702 booklet are included, and the gravitational basis of his rules are described.

The lunar theory in the Principia and the rules for lunar tables presented in the lunar motion are intimately related, although Newton’s lunar method, which we now characterize as a theory to lowest order in the perturbing force, was inadequate to obtain the magnitude of some of the lunar inequalities, the periods of these inequalities were not simply “surveyed” but calculated from first principles.

Newton was evidently aware of shortcomings of his theory, and for the purpose of providing a useful source for tables of lunar positions, he judiciously adjusted amplitudes to obtain agreement with the excellent lunar data provided to him by Flamsteed. Unfortunately, he admitted only occasionally to this procedure ,thereby causing confusion to his readers.

For example, in discussing the amplitude of the semiannual inequality of the line of apsides in the Scholium to Prop. 35, Newton stated that “ it comes to about 12^0 18′ as nearly as I could determine from the phenomena”. It can be readily verified, however, that Newton’s approximate perturbation theory gives only about 8^0, which leads to a disagreement with the observed magnitude of the evection well known since Greek times.

The Porstmouth manuscript shows that Newton was able to do this calculation, but evidently he suppressed this result. In fact, the amplitude of other inequalities, which Newton claimed to have obtained “by the theory of gravity” were also adjusted to fit observation, e.g. the annual inequality which Newton claimed to have found “ by the theory of gravity” to be 11′ 49” (11 minutes and 49 seconds of arc) which is close to the modern value, is actually found to be 13′ (13 minutes) according to his lowest order theory.

Newton was aware of the difficulties in carrying out higher order calculations of his lunar equations, and it took another century of arduous labour by mathematical astronomers of the caliber of Clairut, Lagrange, Euler, Laplace and Hill before these calculations were carried out successfully.

In addition to deriving the lunar inequalities which were known from observations, Newton deduced four new inequalities from his gravitational theory which he included in the lunar motion although these had not been recorded previously. One of these inequalities, referred to by Kollerstrom as the “sixth equation”, also gives further evidence that Newton derived from his dynamical theory the model introduced by Horrocks.

In Newton’s approximate theory, the radius of Horrocks’ epicycle for the centre of the lunar orbit is proportional to the average of the difference between the solar gravitational force acting on the moon and on the earth. Hence this radius depends inversely on the cube of the distance between the earth and the sun, and because of the eccentricity of the earth’s orbit around the sun, this leads to an annual variation of this radius.

For example, in the summer when the earth is farther from the sun, the radius of the epicycle decreases relative to its value during the winter. Newton took this effect into account by adding an epicycle to the Horrocksian model, which he described in detail in the Scholium after Proposition 35, Book 3. It is this epicycle which gives rise to Newton’s sixth equation. Kollerstrom reports that the inclusion of this equation gives an improvement after a wrong sign for this effect in the lunar motion is corrected, as Newton did in the second edition Principia.

Subtle connections which often exist between theoretical ideas before and after the occurrence of a scientific revolution are very important to our understanding of the history and philosophy of science. As mentioned earlier, Ptolemy’s model of an eccentric motion for planetary motion which is governed in time by an “equant”(?) is mathematically equivalent to the epicycle models Ibn-al-Shatir and Copernicus later introduced to replace this equant.

Likewise, Kepler’s elliptical orbit for the planets satisfying the area law is also equivalent to these models to first order in the eccentricity. These various models could not be distinguished until the accuracy of astronomical observations exceeded about ten minutes of arc. This accuracy was necessary to observe effects due to quadratic terms in the eccentricity of Mars, which was first achieved by Tycho Brahe and culminated in Kepler’s discovery of his three empirical laws.

Applying his gravitational theory, Newton then derived Kepler’s laws. His metamorphosis of Horrock’s model, previously developed within the Greek kinematical tradition, into a dynamical model based on gravitational theory is another remarkable example of deep connections between these two seemingly unrelated traditions in science.

In conclusion, it is worthwhile to quote Newton’s 1694 reply to Flamsteed who evidently also misunderstood Newton’s approach:

I believe you have a wrong notion of my method in determining the Moons motions. For I have not been about making such corrections as you seem to suppose, but about getting a general notion of all the equations on which her motions depend and considering how afterwards I shall go to work with least labour and most exactness to determine them.

For the vulgar way of approaching by degrees is bungling and tedious. The method which I propose to my self is first to get a general notion of the equations to be determined and then by accurate observations to determine them. If I can compass the first part of my design I do not doubt but to compass the second. And to go about the second work till I am master of the first would be injudicious.

The original essay was written by Michael Nauenberg and published as Newton’s Lunar Theory by UC Santa Cruz – Physics Department Nicholas Kollerstrom, Newton’s Forgotten Lunar Theory: His Contribution to the Quest for Longitude, (Green Lion Press, Santa Fe, 2000)

http://physics.ucsc.edu/~michael/koll.html