Celestial mechanics

Category:Celestial mechanics Planetary motion according with Kepler laws. The speed of the planet is such that the area swept by the radius vector SP grows uniformly. The speed of the planet is larger when it is closer to the Sun.(Courtesy of EDUSP/USP) Celestial Mechanics is the science devoted to the study of the motion of the celestial bodies on the basis of the laws of gravitation. It was founded by Newton and it is the oldest of the chapters of Physical Astronomy.

Table of Contents The pre-Newtonian Celestial Kinematics Newton’s Celestial Mechanics Celestial Mechanics after Einstein Chaos Concluding Remarks Bibliography See also

The pre-Newtonian Celestial Kinematics

The story of the mathematical representation of celestial motions starts in the antiquity and, notwithstanding the prevalent wrong ideas placing the Earth at the center of the universe, the prediction of the planetary motions were very accurate allowing, for instance, to forecast eclipses and to keep calendars synchronized with the motion of the Earth around the Sun. The epicycles, introduced by Apollonius of Perga around 200 BC, allowed the observed motions to be represented by series of circular functions. They were used to predict celestial motions for almost two millennia. Their long life was certainly related to the stagnation that prevailed in the western world during the dark ages between the end of the Helenic civilization and the Renaissance. In the 16th century, the Copernican revolution put the Sun in center of the Universe. However, the breakthrough in our knowledge of celestial motions was rather related to Tycho Brahe and Johannes Kepler. Tycho, in his Uraniborg observatory, accurately measured the position of the planets in the sky for more than 20 years. Kepler inherited the data gathered by Tycho and used them to discover the three laws that bear his name (see Fig. 1).

• First Law or Elliptical Orbits Law (1609): The planets move on ellipses with one focus in the Sun;
• Second Law or Law of Equal Areas (1609): The planets move with constant areal velocity (equal areas are swept in equal times); in modern words: with constant angular momentum;
• Third Law or Harmonic Law (1619): The ratio of the cube of the semi-major axes of the ellipses to the square of the periods of the planetary motions is constant and the same for all planets.

The work of Kepler is a monument to the human genius. First of all, Tycho’s data on Mars could not be fitted to a heliocentric uniform motion. With respect to a uniform motion, sometimes Mars was in advance, sometimes in retard. Kepler decided to tackle the problem from scratch! Remember that mathematics had remained stagnant since antiquity and the tools inherited from the Greeks, geometry and arithmetic, were the only available. Kepler considered as working hypotheses that the Earth was uniformly moving on a circle and that the motion of Mars was periodic and coplanar with the motion of the Earth. Then he used Tycho’s observations to determine the orbit of Mars. Tycho’s observations were apparent positions of the planets on the celestial sphere. The resulting datum is a direction (only recently had celestial distances been measured, and only in a few cases). Kepler constructed triangles (Fig. 2). After having determined the period of the motion of Mars around the Sun, he looked for observations in dates separated by just one period. Then he constructed triangles, each having as vertices one position of Mars in space (assumed to be the same – after one period Mars returns to the same position) and the position of the Earth in the two dates. The angles of the triangle were obtained from the measurements done by Tycho, and these triangles allowed the position of Mars relative to the Earth to be determined. He thus found that the orbit of Mars was not a circle but rather an ellipse with one focus in the Sun. After that, he inverted the process. He assumed that the real motion of Mars followed an ellipse with constant areal velocity, and started looking for observations separated by one year (in one year the Earth is back to the same place). The triangles now have two vertices on the orbit of Mars (assumed as known) and one vertex on the position of the Earth at those dates. The triangles thus obtained allowed one to determine the position of the Earth with respect to the orbit of Mars. In this way Kepler discovered that the Earth, also, was moving on an ellipse with constant areal velocity. The two first laws were thus discovered. The third law remained elusive for about one more decade, but was finally unraveled.

Kepler’s technique to determine the orbits of the Earth and Mars. (a) The position of Mars is determined from two observations of Mars done at times separated by one integer number of Mars revolutions (i.e. in the two points Mars is in the same position in its orbit). (b) The position of the Earth is determined from two observations of Mars done at times separated by one year (i.e. in the two points, the Earth is in the same position in its orbit). The figure shows 2 triangles constructed in this way. The observed directions of Mars were taken from Tycho Brahe's observations. (Courtesy of EDUSP/USP)

Newton’s Celestial Mechanics

Newton’s theory of universal gravitation resulted from experimental and observational facts. The observational facts were those encompassed in the three Kepler laws. The experimental facts were those reported by Galileo in his book Discorsi intorno à due nuove scienze (“Discourses Relating to Two New Sciences”, which should not be confounded with his most celebrated “Dialogue Concerning the Two Chief World Systems”). The basis of Newton theory arose from the perception that the force keeping the Moon in orbit around the Earth is the same that, on Earth, commands the fall of the bodies.

• Law of Universal Gravitation (1687) – Bodies attract themselves mutually with a force proportional to their masses and inversely proportional to the square of the distance between them.

In other words, if two bodies have masses <math> m_1</math> and <math> m_2</math> and are separated by a distance <math>r\ ,</math> they attract one another with the force <math>|\vec{f}|=\frac{Gm_1m_2}{r^2}</math> where <math>G</math> is a constant (<math>G=6.678 \times 10^{-8}cm^3g^{-1}s^{-2}</math>). This constant is universal and does not depend of the nature of the bodies or on where in the Universe they are found, here or elsewhere.

This law inaugurated the Celestial Mechanics (even if the name came to be used only after Laplace’s work). Newton initially studied the problem of the motion followed by two bodies in mutual attraction (for instance, the Sun and one planet). He showed that under ideal conditions (when no other forces disturb the motions of the two bodies), the relative motion obeys laws which, in some sense, include the first two laws of Kepler.

The first result, easy to obtain, was that the angular momentum of the planet is conserved. The angular momentum is the vector:

<math> \vec{\mathcal{A}} = m\vec{r} \times \vec{v} </math>

where <math>\vec{r}</math> is the heliocentric position vector of the planet, <math>\vec{v}</math> the velocity of the planet, and <math> m </math> its mass. If the vector <math>\vec{\mathcal{A}}</math> remains constant, this means that the plane formed by <math>\vec{r}</math> and <math>\vec{v}</math> is always the same (the motion of the planet is planar) and the areal velocity <math>\frac{1}{2}\vec{r}\times \vec{v}</math> is constant as given by Kepler’s second law. The interesting point concerning this result is that it does not depend on the explicit form of the attraction forces. They arise for all attraction laws in which the two bodies attract themselves with forces aligned with the line passing by them (the so-called central forces). Another result found by Newton is that the mechanical energy is conserved. The mechanical energy of the planet is the sum of its heliocentric kinetic and potential energies:

<math> E = \frac{1}{2} m v^2 - \frac{G(M+m)m}{r} </math>

where <math> M </math> is the mass of the Sun. These two conservation laws may be combined into a first-order differential equation in the distance <math>r</math> having as independent variable the position angle of the planet in the plane of its heliocentric motion. This equation is easily solved and gives

<math> r=\frac{p}{1+e cos\theta} </math>

This equation is the equation of a conic section in the polar coordinates <math> r,\theta </math> and the constants <math>p</math> and <math>e</math> are its parameter and eccentricity, which are related to the planet energy and angular momentum through

<math>e=\sqrt{1+\frac{2E\mathcal A^2}{G^2(M+m)^2m^3}}\ ,</math> and