4. Keplerian Motion

The term “Coulombic” is usually reserved for electrostatic phenomena, where forces may be attractive or repulsive, while the word “Newtonian” refers to gravitational phenomena, where the forces are always attractive.

At the beginning of the space age, one finds the terms “first cosmic velocity”, “second cosmic velocity”, and so on, particular in the Soviet scientific literature. The third cosmic velocity is the one required to escape from the sphere of influence of the Solar System.

The subscripts p and a stand for the perigee and the apogee, respectively, for motion around the Earth (ἡ γῆ, ῆς), or perihelion and aphelion, for motion around the Sun (ὁ ἥλιος, ου). More generally, when the gravitational source is not specified, we speak of the periastron and apoastron, or pericenter and apocenter. The prefixes “peri” and “apo” come from the adverbs περί and ἀπό meaning “above” and “far away”, respectively. The names perihelion and aphelion were invented by Kepler (1596) as an extension of the terms perigee and apogee used by Ptolemy.

The apsidal line is the line joining the perigee and the apogee. This line segment is the major axis of the ellipse. The perigee and apogee are the two apsides, also called the inferior apsis and superior apsis, respectively. This word comes from the Greek ἡ ἁψίς, ῖδος, meaning “vault” or “celestial vault”, having lost the initial aspiration. The architectural term “apse”, which refers to the semicircular recess covered with a hemispherical vault, usually at the east end of a church, has the same origin.

The word “revolution” comes from low Latin revolutio, onis, meaning “unfolding” or “return”, from the Latin verb volvere, meaning “roll”, with the prefix re-. Since the Middle Ages, it has come to mean the periodic return of a heavenly body to its point of departure. Copernicus used it in the title of his celebrated treatise. The transition to its present use, to refer to a radical or significant change of political, economic, or cultural regime, as exemplified by the French Revolution, came only much later. The scientific meaning thus predates the everyday use of the term. So when we see references to the “Copernican revolution”, e.g., Kant in 1787, we may say that we have come full circle.

As we shall see later, when Keplerian motion is perturbed, we can define several periods relating to the actual motion, such as the nodal (or draconitic) period and the anomalistic period.

Kepler invented the term for these angles, from anomalia, æ in Latin. It originally comes from the Greek word ἡ ἀνωμαλία, ας, which means “irregularity” (privative prefix ἀν and adjective ὁμαλός, meaning “self-similar” or “regular”). The idea behind this was to express the irregular behaviour of this angle in time (since the motion does not appear to be circular or regular). Kepler first used the term to indicate the position of Mars with respect to the Sun and he defined several anomalies. Among these were the three described in this chapter: the true anomaly (anomalia coæquata vera), the eccentric anomaly (anomalia eccentri), and the mean anomaly (anomalia media). In his work Astronomia Nova, apart from the true anomaly, Kepler used the “artificial” anomaly (anomalia coæquata fictitia) and four other anomalies (anomalia circularis & elliptica, anomalia distantaria, anomalia scrupularia).

Eccentric means off-center. The center in question is not the center of the circle or the ellipse, but the focus of the ellipse, which is the center of attraction. In Chap. 1, this angle appears in the relation (1.​29).

In January 1900, the Bulletin Astronomique of the Paris Observatory provided a bibliography of 123 papers dealing with the solution of Kepler’s problem, either analytically or graphically. Some of the great names of astronomy and mathematics appear on the list, including Kepler (1609), Newton (1687), Cassini (1719), Simpson (1740), Euler (1747), Lalande (Astronomie, 1764), Lagrange (Sur le problème de Kepler, 1769), Gauss (Theoria motus, 1809), Littrow (Anomaliæ veræ per mediam determinatio, 1814), Delambre (1817), Bessel (Über das Keplersche Problem, 1818), Laplace (Mémoire sur le développement de l’anomalie vraie, 1823), Wallace (Two elementary solutions of Kepler’s problem, 1835), Encke (Auflösung des Keplerschen Gleichung, 1850), Cauchy (1854), Lehmann (Ueber eine definitive Lösung des Keplerschen Problems, 1855, followed by many other purportedly definitive publications over several years), Le Verrier (1855), and Radau (1882).

For v(M), the graph is discontinuous: v(π) = π ×σ(M) and v(0) = 0, denoting the sign of M by σ(M). For E(M), the representative curve is the reflection of \(M(E) = E -\sin E\) in the first bisector.

Emmy Noether (1882–1935) was a German mathematician, considered as the founder of modern algebra (inventor of rings and ideals). Noether’s theorem (1918) says that a conservation law is a consequence of the invariance of a physical law under a continuous transformation with one parameter. (This is proven using the Lagrangian formalism for the equations of classical mechanics.) As far as we are concerned here:

In the study of perturbed motion, the mass of the body is relevant in certain specific instances, such as the study of air resistance in the upper atmosphere or radiation pressure.

Claudius Ptolemaios (Κλαυδίος Πτολεμαῖος), who lived roughly from 90 to 168 AD, was a Greek mathematician, astronomer, and geographer from Alexandria (Claudius is a Roman first name, while Ptolemaios is a Greek name ὁ πτολεμαῖος, ου, meaning warlike). He wrote a number of things, the best known being the Geography and the Almagest. In the Geography (which means literally “drawing of the Earth”), he situates hundreds of places, including towns, mountains, and others, with the correct latitude and somewhat overestimated longitudes. In the Almagest (a title attributed to it later by Arab astronomers, from Al, “the”, the definite article in Arabic, and megistos, the superlative of mega, in Greek, meaning “very big”), he presents the geocentric planetary system, called the Ptolemaic system. To obtain good agreement between observation and model, he has each planet (and in particular Mars) describe a circle called an epicycle, whose center moves around another circle, called the deferent, centered on the motionless Earth. He refined this model with the eccentric (shifted deferent), the equant point, and other modifications. The work of Ptolemy, the last astronomer of the ancient world, was transmitted to Europe by the Arabs, where it formed the foundation of astronomy in the Middle Ages and the Renaissance.

Nicolaj Kopernik (1473–1543), in Latin Nicolaus Copernicus Torinensis, was a Polish astronomer. Following his studies in Italy, Copernicus returned to Poland and devised a cosmological system which replaced the Earth by the Sun as the center of the Universe. This was the heliocentric, or Copernican system, in which the orbit of each planet is a sphere centered on the Sun. He completed his treatise De Revolutionibus orbium cœlestium around 1530. However, fearing the reaction of both the Catholic hierarchy and the newly born Lutheran movement, he held back the publication. The book was finally printed in Nuremberg in 1543, the year of his death, thanks to the determination of his pupil Rheticus. The editor Petreius demanded a preface, but we do not know whether the author gave his agreement. Written by the theologian Osiander, it warned the reader that the system there described was merely a way of looking at the problem that could help to carry out calculations, and that it in no way put in doubt the Bible.

In those days, the authorities had radical ways of dealing with anyone foolish enough to contest religious dogma or biblical truths. The doctor Michel Servet, of Spanish origin, was the first to carry out a scientific investigation of the blood circulation. As a result, he was arrested in 1553 in the Protestant republic of Geneva, condemned to death, and burnt at the stake under the eye of Calvin himself.

Tyge Brahe, commonly known as Tycho Brahe, (1546–1601), was a Danish astronomer. He spent 20 years in his observatory in Uraniborg (“City of the Sky”), in Denmark, making very accurate astronomical measurements. He was the first to take into account the refraction of light. The accuracy of his observations was 1′ (1 arcmin, or 1/60 of a degree), while his contemporaries were not doing better than 10′. He measured the motion of the planet Mars, observing ten oppositions. His model of the Universe was a compromise between Ptolemy’s geocentric model and the heliocentric model of Copernicus. In 1597, he left for Bohemia where he worked with Kepler to set up the astronomical tables known as Tabulae Rudolphinae.

The German astronomer Johannes Kepler (1571–1630), in Latin Ioannes Keplerus, published the first two laws in 1609, in Astronomia Nova Αιτιολογητος seu Physica Cœlestis, and the third in 1619, in Harmonices Mundi. However, it would be wrong to think that the laws appear in a totally clear manner in these writings, as they would in today’s scientific papers. The mathematical terminology was heavy and the explanations hard to follow. There are even cases of one error of reasoning balancing out another.

The first of these two books is almost exclusively devoted to describing the orbit of Mars (whence the subtitle Tradita comentariis de motibus stellæ Martis ex observationibus G.V. Tychonis Brahe). The second law appears at the beginning of this work and the first at the end. Naturally, they concern only the elliptical trajectory of the planet Mars. The Greek word attached to the title Astronomia Nova is the substantive arising from the verb αἰτιολογέω, meaning “to seek causes”. But although these works seem difficult to follow nowadays, they nevertheless attest to the extraordinary discoveries made by their author. To demonstrate the eccentricity of the orbit of Mars or the Earth required a great level of trust in Tycho Brahe’s observations, made with the naked eye, and a considerable degree of mathematical ability. Many other moral qualities were also involved. Courage and self-confidence were essential to take such a revolutionary theory to its logical conclusions in the face of universal opposition, in a climate of family problems and widespread religious hostility, as the wars of religion tore Europe apart. Perseverance was another quality we may safely attribute: in Astronomia Nova, following fifteen pages of close calculations, Kepler tells us that he had to repeat them seventy times in order to arrive at the result. In his own words: “If this method seems possible and tedious, take pity on me, for I undertook these calculations 70 times, and you should not be surprised to find that I have spent 5 years on the theory of Mars. There will no doubt be several subtle geometers, like Viète, who will say that the method is not geometric. But let them solve the problem themselves if they are not satisfied.” This can help us to understand the astonishing clause in the title of Astronomia Nova (see Fig. 4.14): Plurium annorum pertinaci studio elaborata Pragæ (Written in Prague after several years of obstinate effort).

Kepler was deeply convinced that the cosmic and hence divine order had to be perfect, and had great difficulty renouncing the perfection of the circular orbit in favour of the ellipse, blemished as it was by the failings of the real world. Throughout his approach to science, Kepler was guided by this search for divine harmony. In his last work, this led to the musical harmony of the planets and the geometrical harmony of the regular polyhedra (Platonic solids) he fitted around the planetary orbits. In this context, Kepler wrote: “There are six planets because there are five regular polyhedra. I cannot begin to express my wonder before this discovery.”

Galileo Galilei (1564–1642) was an Italian physicist and astronomer. Founder of dynamics and the first genuine experimenter, he studied the free fall of bodies and parabolic motion. He propounded the principle of inertia, which corresponds to Newton’s first law. Shortly after the invention of the refracting telescope, he began to use this instrument to observe the sky. In 1610, he discovered four moons in orbit around Jupiter, and it was this observation that persuaded him that the Earth and the other planets were in orbit around the Sun. The discovery of the crescent of Venus (impossible in a geocentric system) confirmed this idea and he communicated it to Kepler. Galileo recorded all his astronomical discoveries in Sidereus Nuncius (the Celestial Messenger).

An unfailing advocate of the Copernican system, Galileo was condemned for the first time by the Inquisition on 16 February 1616, under the papacy of Paul V. In 1632, he published Dialogo sopra i due massimi sistemi del mondo, Tolemaico et Copernicano. This was not written in Latin, but in the local vernacular or volgare of Italy or Tuscany. That, too, was revolutionary. In this work, he made his preferences perfectly clear and was condemned a second time for heresy on 22 June 1633, under the papacy of Urban VIII, who proclaimed: “The opinion that the Sun sits motionless at the center of the world is absurd, a false philosophy, and formally heretical, because it explicitly contradicts the Holy Scriptures.” Only by getting down on his knees and abjuring did he avoid being burnt at the stake. But he was still imprisoned at the age of 70 and ended his days under house arrest.

At the end of the twentieth century, a Vatican commission was convened in 1981 under John Paul II to reconsider what had been done to Galileo. Its conclusions were announced on 31 October 1992: the Church recognised the errors of Galileo’s judges but was unable to proclaim his vindication.