Dr. Sylvio Ferraz-Mello
Universidade de São Paulo, IAG
Curator and author
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 is the oldest of the chapters of Physical Astronomy.
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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 predict eclipses and to keep calendars synchronized with the motion of the Earth around the Sun. The epicycles, introduced by Apolonius of Pergamon, around 200 BC, allowed the observed motions to be represented by series of circular functions. They were used to predict celestial motions for almost 2 millennia. Its 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, for the knowledge of celestial motions the great progress 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 figure <ref>F1</ref>).
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 intorni a due nuove scienze (which should not be confounded with his most celebrated “Dialogues” on the world systems). The basis of Newton theory arose from the perception that the force that keeps the Moon in orbit around the Earth is the same that, on Earth, commands the fall of the bodies.
- Universal Gravitation’s Law (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 \(m_1\) and \(m_2\) and are separated by a distance \(r\), they attract one another with the force
- <math Eq1>
|\vec f|=\frac{Gm_1m_2}{r^2} </math>
where \(G\) is a constant (\(G=6.678 \times 10^{-8}cm^3g^{-1}s^{-2}\)). This constant is universal and do not depend of the nature of the bodies or on where they are, here or elsewhere.
This law inaugurated Celestial Mechanics (even if the name come to be used only after Laplace’s work). Newton initially studied the problem of the motion followed by two bodies in mutual attraction (say, the Sun and one planet). He showed that under ideal conditions (if no other forces disturb the motions of the two bodies) the relative motion obey to laws which, in some sense, include the first two laws of Kepler.
The first result, easy to obtain, is that the angular momentum of the planet is conserved. The angular momentum is the vector:
- <math Eq2>
\vec{\mathcal{A}} = m\vec{r} \times \vec{v} </math>
where \(\vec{r}\) is the heliocentric position vector of the planet and \(\vec{v}\) is the velocity of the planet.
If the vector \(\vec{\mathcal{A}}\) remains constant, this means that the plane formed by \(\vec{r}\) and \(\vec{v}\) is always the same (the motion of the planet is planar) and the areal velocity \(\frac{1}{2}\vec{r}\times \vec{v}\) 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 Eq3>
E = \frac{1}{2} m v^2 - \frac{G(M+m)m}{r} </math>
These two conservation laws may be combined into a first-order differential equation in the distance \(r\) 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 Eq4>
r=\frac{p}{1+e cos\theta} </math>
This equation is the equation of a conic and the constants \(p\) and \(e\) are its parameter and eccentricity, which are related to the planet energy and angular momentum through
- <math Eq5>e=\sqrt{1+\frac{2E\mathcal A}{G^2(M+m)^2m^2}}</math> and \(p={\mathcal A}^2/{G(M+m)m^2}\)
This conic is an ellipse if \(0<e<1\), a hyperbola if \(e>1\), a parabola if \(e=1\) and one circumference if \(e=0\). Therefore, Newton’s result generalizes the first Kepler’s law showing that, indeed, the motion of one body attracted by the Sun may be an ellipse, as the orbit of the planets, but may also be a hyperbola as the motion of some comets. The type of the conic will be determined by its energy. If the energy is negative the above equations give \(e<1\) and the motion is an ellipse. If the energy is positive the above equations give \(e>1\) and the motion is a hyperbola. The circumference and the parabola are the limiting cases in which the energy is exactly equal to \(-\frac{G^2(M+m)^2 m^2}{2 \mathcal A}\) or zero, respectively.
In the case of an ellipse, the semi-major axis may be obtained from the parameter \(p\) through
- <math Eq6>
a=p\sqrt{1-e^2}. </math>
In addition, in the case of elliptic motion, the combination of the various equations allows to find the relationship between the semi-major axis of the ellipse and the orbital period \(P\):
<math Eq7> \frac{a^3}{P^2}=\frac{G(M+m)}{4\pi^2} </math>
which is not the result given by Kepler’s third law! In Kepler’s third law, it is said that the ratio of the cube of the semi-major axes to the square of the periods is the same for all planets. Newton’s theory shows that this quantity is in fact proportional to the sum of masses \(M+m\). However, planetary masses are small as compared with the mass of the Sun (the mass of the largest planet is 1/1047 of the mass of the Sun) and Kepler’s third law is a very good approximation of the actual result. It is enough to adopt the approximation \(M+m \sim M\), to transform the above equation in Kepler’s third law.
Newton did not limit himself to the problem of the motion of two attracting bodies. He considered in his work also the problem of the motion of the Moon around the Earth under the joint attractive forces of the Earth and the Sun. One of the great achievements of Newton’s theory was due to one of his disciples, Edmond Halley. Halley understood that Newton’s theory should be universal and applied it to study the motion of comets at a time in which it was not yet clear what these objects are. He analyzed the list of bright comets observed since the antiquity. He verified that many of these observations were separated by about 3/4 of century one from the next. He studied the records and concluded that the comets of 1531, 1607 and 1682 were, in reality, one and the same comet and predicted its return in 1758. The comet that is today known as Halley’s actually appeared at the predicted time. This event was brought big renown to Newton’s theory. However, the Mathematics invented by Newton’s to compose his work (the theory of fluxions, or today, the differential calculus) was intimidating to many and its adoption was only gradual. Even though, his gravitation theory was successfully used in next century for the constructions of theories of the motion of planets, satellites and comets, of theories able to explain the polar flattening of the Earth, the variations of the gravity acceleration \(g\) on the surface of the Earth, the theory of tides, etc. The success was so big that many people started believing that it would be able to explain every thing. The deterministic view of the natural phenomena grew. It would be enough to exactly know the present situation to determine the future evolution. With some humor, the imaginary being which would be determining in an unequivocal way the motion of all bodies, is sometime times called Laplace’s demon! But, as we know today, one of the tricks of gravitation is that the determinism of its equations is not enough to make their solutions predictable for ever.
Newton’s gravitation theory allows the construction of sets of differential equations whose solutions are the motion followed by the bodies. These equations are not easy to solve, and the great achievement of 18th century, to which we may associate the names of Leonhard Euler and Joseph-Louis Lagrange, among many others, was the construction of theories to obtain the solution of Newton’s equations, the Theory of Perturbations. The most notorious achievement of the Theory of Perturbations was recorded in 1846. Its story began in 1781, when a new planet, Uranus, was discovered by William Herschel. It was the first planet discovered with a telescope and thus, the first planet discovered since the remote antiquity. He was observed in the sky year after year for many decades and at some moment, the observations were enough to allow the construction of an accurate theory of its motion, which should be fully explained with Newton’s equations. But this was not so. The motion of Uranus did not follow the results given by the theory. Bessel was the first to consider the problem posed by these discrepancies, but the arc drawn in the sky by the motion of Uranus since its discovery was too short and he could not find the explanation. His ideas were reconsidered some decades later, around 1840, by Adams in England and Leverrier in France. Adams concluded, in 1845, that the observed discrepancies in the motion of Uranus were due to a yet unknown planet and give details on its orbit. However, his results did not deserve attention from English astronomers. In France, Leverrier obtained the same results and published them in 1845 and 1846. In addition when he believed that his results were accurate enough to allow a search, he sent a letter to Galle, in Berlin’s observatory, and asked him to look for the planet. His request was promptly accepted and in the next evening Galle discovered Neptune less than one degree afar from the position indicated by Leverrier!
Uranus was not the only problematic planet. The motion of Mercury also was showing discrepancies. The very accurate calculations done by Leverrier showed that the perturbations of the other planets on the motion of Mercury were such that its ellipse did not kept fixed but was precessing. The major axis ot the ellipse should have a slow rotation: 530 arc seconds per century (277 arc seconds due to the attraction of Venus, 153 due to Jupiter, 90 due to the Earth and 11 due to Mars and the other planets). But the observed motion of Mercury was showing not 530 but about 570 arc seconds per century. As in the case of Neptune, Leverrier looked for the possibility of an extra, intra-mercurial, planet which could be responsible for the excess of 40 arc seconds per century. This planet was several times “discovered” and even got a name: Vulcano. However, these “discoveries” were never confirmed and the discrepancy could not be explained by Newton’s Celestial Mechanics and had to wait for the next progresses on our knowledge of gravitation.
Celestial Mechanics after Einstein
In 1915 Einstein published his first results on a new theory of gravitation which became known as General Relativity Theory (GRT). The application to Celestial Mechanics done by himself, showed that the two-body motion laws introduced by Newton (and Kepler) should be corrected. The planets were not moving on fixed ellipses but on ellipses whose axes were slowly rotating. In the case of Mercury, the rotation indicated by Einstein’s theory was 43 arc seconds per century. This was almost exactly the value of the discrepancy between the theories founded on Newton’s gravitation and the observations.
Celestial Mechanics was one of the first branches of science to explore the consequences of the GRT. As soon as 1916, the first articles on the construction of the equations of planetary motion were published by Droste and by De Sitter. In Newton’s theory, the law expressing the attraction force between two bodies is fully independent of the equations of the motion of these bodies. In GRT, the field and the motion appear together in only one set of equations: Einstein’s field equations. These equations allow the calculation of one matrix giving the curvature of the space-time in every point and GRT says that the motion of the body will follow the space-time geodesics.
This is a mathematically very intricate problem which could be completely solved only in particular cases as the field created by a fixed spherical body (Schwarzschild, 1916), and later on a fixed spheroid (Levi-Civita, 1917) and a rotating sphere (Kerr, 1943). The GRT equations of the motion of point masses in gravitational interaction was published in 1938, almost simultaneously in 2 papers, one by Eddington and Clark and the other by Einstein, Infeld and Hoffmann.
In the simplest of these cases, the Schwarzschild solution, the field has spherical symmetry and the distance between two points in this field is simply written using spherical coordinates. But this brings embedded one of the most difficult problems of GRT. To give an example, we may compute the planetary orbits and find Einstein’s precessing ellipses. We may also look for the relationship between the size of the ellipse (its semi-major axis) and the period of the motion and find a given function, a relativistic version of Kepler’s harmonic law. But the only hypothesis done to reach these results is the spherical symmetry of the field. If it is deformed (inflated for instance) is such a way that the spherical symmetry is kept, the results do not change. The way in which the distance from one point in space to the center of the field is defined is totally irrelevant and do not affect the solutions. To solve the problem is necessary to construct in parallel the theory of the processes used to measure the distances – e.g. using light rays propagating between the points – and use the distances measured with the same “rule”. Only after that step it is possible to compare the complete results of the theory to those observed.
Untl the second half of the 20th century, there was no need for astronomers to calculate any relativistic effects beyond the precession of the planetary ellipses given by GRT (42.98 arc seconds per century for Mercury, 8.62 arcseconds per century for Venus, 3.84 arc seconds per century for the Earth, 1.35 arc seconds per century for Mars). However, the progresses of Astrometry, including laser, radar, and VLBI (very-long baseline interferometry) forced to tackle the problem of the full consideration of the equations of GRT for the motion of the celestial bodies and also the measurement of times and distance.
Chaos
Nowadays, it is no longer possible to written on Celestial Mechanics without mentioning chaos. Many problems in CM are characterized by an evolution due only to gravitational forces with conservation of total energy and angular momentum for times of the order of millions or billions of years. So long evolutions under these conditions propitiate the rise of chaotic phenomena. The solutions of the equations of CM show great sensitivity to initial conditions: very close initial conditions may lead to totally different evolutions. We illustrate this fact presenting one example.
Many cases of chaotic evolution are found among the asteroids when they move in orbits with periods commensurable with the orbital period of Jupiter (11.8 years). For instance, the asteroids of the 3/1 resonance (i.e. with periods equal to 1/3 of the period of Jupiter) show three main regimes of motion as shown in figure X. Chaos means, in this case, that these three regimes of motion are not strictly separated and that there are solutions transiting from one regime to the next in time scales shorter than1 million years. The transition from the regime (a) to the regime (b), the first discovered in this problem, allowed to explain the almost non existence of asteroids in this resonance. While the asteroid is in the low-eccentricity regime (a), the motion is nearly regular on a precessing ellipse whose eccentricity may show small periodic variations, but remains below 0.2 – 0.3. In the medium-eccentricity regime (b), on the contrary, the orbital eccentricity may reach values as high as 0.6 – 0.7. When the eccentricity becomes higher than 0.3 – 0.4, the minimal distance of the asteroid to the Sun becomes smaller than the average radius of the orbit of Mars. This means that once per period the asteroid will cross the orbit of Mars. If by chance, one of these occasions the asteroids crosses the orbit of Mars when the planet is close to the crossing point, the attraction of Mars will perturb the motion of the asteroid and may greatly change its period so that it will leave the resonance (after the close approach to Mars it will be on an orbit whose period is no longer 1/3 of the period of Jupiter). A further progress was the discovery of the existence of the high-eccentricity regime (c) in which the eccentricity of the asteroid may reach 0.9. Even if the transitions to this regime are not expected to have the same frequency as the transitions to (b), they are full of consequences for the asteroid fate. Mars has a very small mass and an asteroid may remain in the regime (b) for millions of years without having a close approach to Mars. But in the regime (c), the asteroid will not only cross the orbit of Mars but also the orbits of the Earth and Venus which are 10 times more massive than Mars and can be deviated by these planets even in a less close approach. These results were obtained with simplified models considering Jupiter moving in accordance with Kepler’s laws. However, more complex models, where the real motion of Jupiter was considered, showed that the reality is still more drastic: the regime (c) is not bounded, and the asteroid may enter into so stretched orbits, crossing the orbits of the inner planets and allowing the asteroid to collide with the Sun. In this case we have more than a change in the orbit of the asteroid but its physical destruction.
The possibility of chaos, i.e of transitions between different regimes of motion affects directly the evolution of the Solar System. The apparent harmony that we observe, resulting from 5 billion years of evolution, is not eternal. Chaotic instabilities act very slowly as in the example above described. Would not the Moon be orbiting the Earth, the rotation axis of our planet would undergo large chaotic variations able to destroy the delicate seasonal equilibrium of our planet as it is know to happen with Mars and Venus. Mercury, which is parading with the other planets since 5 billion years can have a close approach to Venus in the next 10 billion years with unpredictable consequences for its future motion. Violent events as collisions of bodies creating big craters in the planets, the formation of the Moon around the Earth, and other that are even difficult to imagine, may have occurred because minor bodies under the continuous disturbing influence of the largest ones evolved chaotically from very regular primordial motions
Concluding Remarks
Celestial Mechanics is a four centuries old science. During this long time it developed in many directions, impossible to consider in a short introduction. We may mention that several chapters of Mathematics appeared aiming at the study of the solutions of the motion of N bodies under mutual gravitational attraction (q.v.). The three body problem became a classic (q.v.). The KAM theory founded on the famous Kolmogorov’s theorem, aimed at solving problems raised by the Theory of Perturbations. In the domain of applications, we have to mention Astrodynamics, one chapter of Celestial Mechanics that experienced a great development in 20th century when the theory of the motion of artificial satellites was established as well as the theories of the maneuvers necessary to transfer one spaceship from one orbit to another. One last point to keep in mind is that present-day Celestial Mechanics can not be restricted to gravitational forces. There is a panoply of non-gravitational forces acting on natural and artificial celestial bodies that perturb their motion in a significant way: gas drag, thermal emissions, interactions between radiation and matter, comet jets, tidal friction, etc.