Sun-Jupiter-Saturn

Abstract These lectures are devoted to the main results of classical perturbation theory. We start by recalling the methods of Hamiltonian dynamics, the problem of small divisors, the series of Lindstedt and the method of normal form. Then we discuss the theorem of Kolmogorov with an application to the Sun-Jupiter-Saturn problem in Celestial Mechanics. Finally we discuss the problem of long-time stability, by discussing the concept of exponential stability as introduced by Moser and Littlewood and fully exploited by Nekhoroshev. The phenomenon of superexponential stability is also recalled. 

Using the classical methods of Celestial Mechanics, we can expand the distance A in the Hamiltonian (19) as a function of the Poincare variables, and we can calculate the so called secular system at order two in the masses (used, for instance, in Laskar 1988 in a model with 8 planets, to study the long term evolution of the solar system). In the secular system the dependency on the angles Ai, A2 (which evolve much faster than the other Poincare variables) is dropped out by simply averaging the Hamiltonian over the angles themselves. Thus, the actions Ai, A2 are first integrals for the secular system, which are replaced with their numerical values corresponding to the data for the real system Sun-Jupiter-Saturn at a fixed initial time. Therefore, we can actually expand the secular Hamiltonian as a power series in the form... 

The procedure above allows us to prove only that there is an invariant torus close to the initial conditions of the Sun-Jupiter-Saturn system, not that the orbit of the system actually lies on a torus. Since we can not exclude the possibility of Arnold s diffusion, this is not enough to prove the perpetual stability of the orbit of the secular system. Therefore, we make a more accurate analysis in order to prove that the orbit is actually confined in a gap between two invariant tori. The procedure is illustrated in Figure 3. 

A large majority of known triple stellar systems have the Hill-type stability, and so is the Sun-Jupiter-Saturn system (99.99% of the mass of the Solar System). The close binary is then the Sun and Jupiter, while Saturn is the third body. 

Our solar system consists of the Sun, the planets and their moon satellites, asteroids (small planets), comets, and meteorites. The planets are generally divided into two categories Earth-like (terrestrial) planets—Mercury, Venus, Earth, and Mars and Giant planets—Jupiter, Saturn, Uranus, and Neptune. Little is known about Pluto, the most remote planet from Earth. 

The density estimates in Table 7.1 show a distinction between the structures of the planets, with Mercury, Venus, Earth and Mars all having mean densities consistent with a rocky internal structure. The Earth-like nature of their composition, orbital periods and distance from the Sun enable these to be classified as the terrestrial planets. Jupiter, Saturn and Uranus have very low densities and are simple gas giants, perhaps with a very small rocky core. Neptune and Pluto clearly contain more dense materials, perhaps a mixture of gas, rock and ice. 

Hydrogen isotopic compositions, expressed as molar D/H ratios, of solar system bodies. The relatively low D/H values in the atmospheres of Jupiter and Saturn are similar to those in the early Sun, whereas D/H ratios for Uranus and Neptune are intermediate between the Jupiter-Saturn values and those of comets and chondrites. The Earth s oceans have D/H shown by the horizontal line. Mars values are from SNC meteorites. Modified from Righter et al. (2006) and Lunine (2004). 

When thinking about how our solar system may have evolved from proplyds (protoplanetary disks), we must remember that the violence of the early Solar System was tremendous as huge chunks of matter bombarded each other. In the inner Solar System, the Sun s heat drove away the lighter-weight elements and materials, leaving Mercury, Venus, Earth, and Mars behind. In the outer part of the system, the solar nebulas (gas and dust) survived for some time and were accumulated by Jupiter, Saturn, Uranus, and Neptune. 

Modern telescopic and spacecraft study of Jupiter, Saturn, Uranus, and Neptune, their properties, and their systems of rings, moons, and magnetospheres, has been the purview of the planetary scientist with little connection to the universe beyond until 1995, when the first extrasolar giant planet was discovered. Now the solar system s giants are the best-studied example of a class of some 100 objects which—while only one has been measured for size and hence density—may be present 10% of Sun-like stars. 

Neptune is the eighth planet from the Sun and about four times the size of Earth. Astronomers consider Neptune to form with Uranus a subgroup of the Jovian planets (Jupiter, Saturn, Uranus, and Neptune). Neptune and Uranus are similar in size, mass, periods of their rotation, the overall features of their magnetic fields, and ring systems. However they differ in the structure of their atmospheres (perhaps the more conspicuous features of Neptune s clouds are caused by its significant internal energy source, which Uranus lacks), the orientations of their rotation axes, and in their satellite systems. 

The Sun O and the Moon 3 are rendered with gold and silver, while Venus and Mars o provide the copper and iron pigments for the green robe of the angel below the Sun O. Saturn and Jupiter in turn, supply lead-tin yellow for the garment of the angel below the Moon 3. In the middle, Christ s blood and loincloth are painted with the quicksilver-sulphur compound vermiUon, the pigment colour held by the Chinese to represent eternal Hfe. 

Body Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto... 

At the time, more than a dozen planetary satellites had already been discovered for Jupiter, Saturn, Uranus, and Neptune. None had been found for Venus or Mercury, nor were they likely to be found, given the proximity of these planets to the Sun. Mars likewise had no satellites. .. or, at least, none that had yet been discovered. 

The sign that the Sun occupied at the moment of your birth is the most basic astrological fact about you. It defines your ego, motivations, needs, and approach to life. But the Sun isn t the only planet that affects you. (For astrological purposes, both luminaries — the Sun and the Moon — are called planets. Do yourself a favor and don t use this terminology when talking to astronomers.) Mercury, Venus, Mars, Jupiter, Saturn, Chiron, Uranus, Neptune, and Pluto, not to mention the Moon, represent distinct types of energy that express themselves in the style of the sign they re in. 

It takes 84 years — a human lifetime — for Uranus and its 15 moons to travel through the zodiac. Like Jupiter, Saturn, and Neptune, Uranus is a gas giant. But while every other planet rotates on its axis in a more-or-less upright fashion, unconventional Uranus seems to roll around on its side with its north pole pointing to the Sun. As a result, the Uranian day is 42 years long, and the night is the same length. 

I consider the planets in this order first the Sun and Moon, then the planets in order of their distance from the Sun Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto. When looking up an aspect, be sure to look for it under the planet that comes first in the list. An aspect between Mercury and Uranus, for example, appears under Mercury an opposition between Venus and Pluto is discussed under Venus, and so on. 

Mercury orbits closest to the sun, followed by Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. Neptune is farthest from the Sun for 20 of every 248 years. Planets will never collide because one is always higher than the other, even when their orbits do intersect. 

Nineteenth-century spectroscopes allowed astronomers to record emission spectra of stars, comets, and planets, and even to discover a new element (helium) on the Sun. In Germany, Hermann Vogel (1842-1907) discovered that the major components of Jupiter s atmosphere are hydrogen and helium, very similar to the Sun. In the earlier years of the 20th century, Vesto M. Slipher (1875-1969), at the Lowell Observatory, recorded spectra from Jupiter, Saturn, Uranus, and Neptune and in the early 1930s German astronomer Rupert Wildt (1905-76) analyzed them and discovered traces of methane (CH4) and ammonia (NH3) in the atmospheres of these frigid outer planets. 

Until the summer of 2006, there were nine recognized planets in our solar system Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto. These nine planets are divided into two groups based on distance from the sun. The inner planets include Mercury, Venus, Earth, and Mars. The outer planets include Jupiter, Saturn, Uranus, Neptune and Pluto. Pluto s status as a planet is being reconsidered. 

The outer or giant planets - Jupiter. Saturn, Uranus, and Neptune - are massive low-density bodies with a rocky core surrounded by deep layers consisting mainly of solid, liquid, and gaseous hydrogen and helium. They are much further from the sun and therefore much cooler. All have large numbers of satellites Jupiter has at least 63 Saturn at least 61 Uranus 27 and Neptune 13. The outer planets also have ring systems composed of smaller bodies, rocks, dust, and ice particles. 

The atmospheres of the planets can be divided Into two main groups. The terrestrial planets Mars and Venus lie close to the Sun, have masses comparable to the Earth and atmospheres dominated by the heavy elements, oxygen, nitrogen and carbon. The Jovian planets Jupiter, Saturn,... 

The giant planets, Jupiter, Saturn, Uranus, and Neptune, occupy the 5th through 8th planetary orbits, eounting outward from the Sun. They seem to form pairs Jupiter and Saturn are similar in size and other properties, and so are Uranus and Neptune (see Appendix 3). All giant planets have deep atmospheres, composed mainly of... 

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