Celestial mechanics

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Sylvio Ferraz-Mello (2009), Scholarpedia, 4(1):4416. doi:10.4249/scholarpedia.4416 revision #91107 [link to/cite this article]
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Figure 1: 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.

Contents

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.

Figure 2: 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 \( m_1\) and \( m_2\) and are separated by a distance \(r\ ,\) they attract one another with the force \(|\vec{f}|=\frac{Gm_1m_2}{r^2}\) where \(G\) is a constant (\(G=6.678 \times 10^{-8}cm^3g^{-1}s^{-2}\)). 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\[ \vec{\mathcal{A}} = m\vec{r} \times \vec{v} \]

where \(\vec{r}\) is the heliocentric position vector of the planet, \(\vec{v}\) the velocity of the planet, and \( m \) its mass. 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\[ E = \frac{1}{2} m v^2 - \frac{G(M+m)m}{r} \]

where \( M \) is the mass of the Sun. 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

\( r=\frac{p}{1+e cos\theta} \)

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

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

\( p = \frac{{\mathcal A}^2}{{G(M+m)m^2} } \)

This conic section is an ellipse when \(0<e<1\ ,\) a hyperbola when \(e>1\ ,\) a parabola when \(e=1\) and one circle. Therefore, Newton’s result generalizes the first of Kepler’s laws 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 section 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 \(-\frac{G^2(M+m)^2 m^3}{2 \mathcal A^2}\) or zero, respectively.

In the case of an ellipse, the semi-major axis may be obtained from the parameter \(p\) through \(a=p\sqrt{1-e^2}\ .\)

In addition, in the case of elliptic motion, the combination of the various equations allows us to find the relationship between the semi-major axis of the ellipse and the orbital period \(P\ :\)

\(\frac{a^3}{P^2}=\frac{G(M+m)}{4\pi^2}\ ,\)

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 the masses \(M+m\ .\) However, planetary masses are small 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 greatest 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 the nature of these objects was not yet known. He analyzed the list of bright comets observed since the antiquity and verified that many of these observations were separated, one from another, by about 3/4 of century. He 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 brought great renown to Newton’s theory. However, the mathematics invented by Newton to compose his work (the theory of fluxions, or today, the differential calculus) was not understood by many other scientists and its adoption was only gradual. Even so, his gravitation theory was successfully used in next century for the construction of theories of the motion of planets, satellites, and comets. His theory was able to explain the polar flattening of the Earth, the variations of the gravity acceleration \(g\) on the surface of the Earth, the tides, etc. The success was so extensive that many people started to believe that it would be able to explain everything. The deterministic view of natural phenomena grew. After this view, it would be enough to know exactly 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 motions 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 noticed with a telescope and thus, the first planet discovered since remote antiquity. It was observed in the sky year after year for many decades and eventually 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 gave details on its orbit. However, his results did not get the attention it deserved 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 was not kept fixed, but was precessing. The major axis of 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. Just as for Neptune, Leverrier looked for the possibility of an extra, intra-mercurial, planet which could be responsible for the excess of approximately 40 arc seconds per century. This planet was several times “discovered” and even got a name: Vulcan. 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 him 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 a matrix giving the curvature of the space-time at every point, and GRT predicts that the motion of bodies will follow space-time geodesics.

This is a very intricate mathematical problem that 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 hypothesis necessary to reach these results is the spherical symmetry of the field. If it is deformed (inflated for instance) in such a way that the spherical symmetry is kept, the results may not change. The way in which the distance from one point in space to the center of the field is defined can not affect the solutions. To solve the problem, it is necessary to construct, in parallel to the theory of the motion, 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.

Until the second half of the 20th century, astronomers did not need 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 astronomers 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 talk of Celestial Mechanics without mentioning chaos. Many problems in Celestial Mechanics 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. 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 Fig. 3. 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 than 1 million years. The transition from the regime (a) to the regime (b), the first discovered in this problem, allowed the explanation of 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, in one of these occasions, the asteroid crosses the orbit of Mars when the planet is close to the crossing point, the attraction of Mars will disturb 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). Further progress was made with 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's fate. Mars has a very small mass and an asteroid may remain in the regime (b) for millions of years without having an approach to Mars close enough to disturb its motion. But in 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 have its motion disturbed 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 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 also its physical destruction.

Figure 3: Regimes of motion of an asteroid whose mean orbital period is 1/3 of Jupiter’s period (cf. Ferraz-Mello and Klafke). In this figure are represented, in polar coordinates, the eccentricity (radius vector) and the longitude of the perihelion (polar angle) of the asteroid’s orbit. The curves are evolutionary paths. The longitudes are reckoned from the direction of the perihelion of Jupiter’s orbit. Regimes: (a) The orbit of the asteroid is almost circular (small eccentricities) and the perihelion rotates; (b) The orbit has great variations in eccentricity (between 0.2 and 0.4) and the perihelion remains oscillating around 0 (i.e about the direction of Jupiter’s perihelion); (c) The eccentricity of the asteroid shows large variations and may become very large (in the outermost curves, the eccentricity goes beyond 0.9). The asteroid perihelion remains oscillating around 180 degrees (i.e the direction opposite to Jupiter’s perihelion). The three regions are not perfectly insulated and the asteroid may chaotically transit from one regime of motion to another. (Courtesy of EDUSP/USP)

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, will not last forever. Chaotic instabilities act very slowly as in the example above described. The rotation axis of Mars and Venus undergo large chaotic variations. Fortunately for us, the Moon forces the rotation of the Earth to be more regular thus keeping the delicate climatic equilibrium of our planet. Mercury, which has been parading with the other planets for 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 such as the collisions of bodies creating big craters, the formation of the Moon around the Earth, and other events that are difficult to imagine, may have occurred because minor bodies under the continuous disturbing influence of the larger 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 the 20th century when the theory of the motion of artificial satellites was established, as well as the theories of the maneuvers necessary to transfer a 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.

Bibliography

  • Brouwer, D. and Clemence, G.: Methods of Celestial Mechanics (Academic Press, New York, 1961)
  • Brumberg, V.: Essential relativistic Celestial Mechanics (Adam Hilger, Bristol, 1991)
  • Ferraz-Mello, S.: Canonical Perturbation Theories. Degenerate Systems and Resonance (Springer, New York, 2007)
  • Ferraz-Mello, S. and J.C. Klafke, J.C.: Mecanica Celeste, In: Friaça et al., (eds) Astronomia: Uma visão geral do Universo, (EDUSP, São Paulo, 200) pp. 51-80.
  • Ferraz-Mello, S. and Klafke, J.C.: A Model for the Study of Very-High-Eccentricity Asteroidal Motion.The 3:1 resonance In: A.E.Roy, (ed.), Predictability, Stability and Chaos in N-body Dynamical Systems (Plenum Press, New York, 1991) pp. 177-18
  • Ferraz-Mello, S. Klafke, J.C. Michtchenko, T.A. and Nesvorny, D.: Chaotic Transitions in Asteroidal Resonances, (Review Paper) Celestial Mechanics and Dynamical Astronomy, 64 (1996) pp. 93-105.
  • Ferraz-Mello, S.: Slow and Fast Diffusion in Asteroidal-Belt Resonances. A Review. Celestial Mechanics and Dynamical Astronomy, 73 (1999), pp. 25-37.
  • Kovalevsky, J.: Introduction à la Mécanique Céleste (Armand Colin, Paris, 1963); Introduction to Celestial Mechanics (D. Reidel, Dordrecht, 1967)
  • Kurth, R.: Introduction to the Mechanics of the Solar System, (Pergamon, London, 1959)
  • Morbidelli, A.: Modern Celestial Mechanics. Aspects of Solar System Dynamics (Taylor and Francis, London, 2002)
  • Murray, C. and Dermott, S.F.: Solar System Dynamics, (Cambridge Univ. Press, Cambridge, 1999)
  • Wisdom, J.: Chaotic behavior and the origin of the 3/1 Kirkwood gap, Icarus, 56 (1983), pp. 51-74.

Internal references

  • Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
  • Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.


See also

KAM theory, Three body problem

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