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Resonant Orbits
In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relationship is found between a pair of objects (binary resonance). The physical principle behind orbital resonance is similar in concept to pushing a child on a swing, whereby the orbit and the swing both have a natural frequency, and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies (i.e., their ability to alter or constrain each other's orbits). In most cases, this results in an ''unstable'' interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be self-correcting and thus stable. Examples are the 1:2:4 resonance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galilean Moon Laplace Resonance Animation 2
Generically, a Galilean (; he, גלילי; grc, Γαλιλαίων; la, Galilaeos) is an inhabitant of Galilee, a region of Israel surrounding the Sea of Galilee (Kinneret). The New Testament notes that the Apostle Peter's accent gave him away as a Galilean (Matthew 26:73 and Mark 14:70). The Galilean dialect referred to in the New Testament was a form of Jewish Palestinian Aramaic spoken by people in Galilee from the late Second Temple period through the Apostolic Age . Later the term was used to refer to the early Christians by Roman emperors Julian and Marcus Aurelius, among others. History Biblical narrative According to the Biblical history of the Twelve Tribes, the region of Galilee was allotted to the tribes of Naphtali and Dan, at points overlapping with the domain of the Tribe of Asher and neighboring the region of Issachar. In the First Book of Kings, the Phoenician ruler King Hiram I of Sidon was awarded twenty cities in the region of Galilee, given to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planetary System
A planetary system is a set of gravitationally bound non- stellar objects in or out of orbit around a star or star system. Generally speaking, systems with one or more planets constitute a planetary system, although such systems may also consist of bodies such as dwarf planets, asteroids, natural satellites, meteoroids, comets, planetesimals and circumstellar disks. The Sun together with the planetary system revolving around it, including Earth, forms the Solar System. The term exoplanetary system is sometimes used in reference to other planetary systems. Debris disks are also known to be common, though other objects are more difficult to observe. Of particular interest to astrobiology is the habitable zone of planetary systems where planets could have surface liquid water, and thus the capacity to support Earth-like life. History Heliocentrism Historically, heliocentrism (the doctrine that the Sun is at the centre of the universe) was opposed to geocentrism (placing Earth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pandora (moon)
Pandora is an inner satellite of Saturn. It was discovered in 1980 from photos taken by the ''Voyager 1'' probe, and was provisionally designated . In late 1985 it was officially named after Pandora from Greek mythology. It is also designated . Pandora was thought to be an outer shepherd satellite of the F Ring. However, recent studies indicate that it does not play such a role, and that only Prometheus, the inner shepherd, contributes to the confinement of the narrow ring. It is more heavily cratered than nearby Prometheus, and has at least two large craters in diameter. The majority of craters on Pandora are shallow as a result of being filled with debris. Ridges and grooves are also present on moon's surface. The orbit of Pandora appears to be chaotic, as a consequence of a series of four 118:121 mean-motion resonances with Prometheus. The most appreciable changes in their orbits occur approximately every 6.2 years, when the periapsis of Pandora lines up with the apoapsi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Tongue
In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or other related invariant property thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form geometric shapes that resemble tongues, in which case they are called Arnold tongues. Arnold tongues are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes and cardiac electric waves. Sometimes the frequency of oscillation depends on, or is constrained (i.e., ''phase-locked'' or ''mode-locked'', in some contexts) based on some quantity, and it is often of interest to study this relation. For instance, the outset of a tumor trigger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resonant Interaction
In nonlinear systems, a resonant interaction is the interaction of three or more waves, usually but not always of small amplitude. Resonant interactions occur when a simple set of criteria coupling wave-vectors and the dispersion equation are met. The simplicity of the criteria make technique popular in multiple fields. Its most prominent and well-developed forms appear in the study of gravity waves, but also finds numerous applications from astrophysics and biology to engineering and medicine. Theoretical work on partial differential equations provides insights into chaos theory; there are curious links to number theory. Resonant interactions allow waves to (elastically) scatter, diffuse or to become unstable. Diffusion processes are responsible for the eventual thermalization of most nonlinear systems; instabilities offer insight into high-dimensional chaos and turbulence. Discussion The underlying concept is that when the sum total of the energy and momentum of several vib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Musica Universalis
The ''musica universalis'' (literally universal music), also called music of the spheres or harmony of the spheres, is a philosophical concept that regards proportions in the movements of celestial bodies – the Sun, Moon, and planets – as a form of music. The theory, originating in ancient Greece, was a tenet of Pythagoreanism, and was later developed by 16th-century astronomer Johannes Kepler. Kepler did not believe this "music" to be audible, but felt that it could nevertheless be heard by the soul. The idea continued to appeal to scholars until the end of the Renaissance, influencing many schools of thought, including humanism. History The concept of the "music of the spheres" incorporates the metaphysical principle that mathematical relationships express qualities or "tones" of energy which manifest in numbers, visual angles, shapes and sounds – all connected within a pattern of proportion. Pythagoras first identified that the pitch of a musical note is i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galilean Moon
The Galilean moons (), or Galilean satellites, are the four largest moons of Jupiter: Io, Europa, Ganymede, and Callisto. They were first seen by Galileo Galilei in December 1609 or January 1610, and recognized by him as satellites of Jupiter in March 1610. They were the first objects found to orbit a planet other than the Earth. They are among the largest objects in the Solar System with the exception of the Sun and the eight planets, with a radius larger than any of the dwarf planets. Ganymede is the largest moon in the Solar System, and is even bigger than the planet Mercury, though only around half as massive. The three inner moons— Io, Europa, and Ganymede—are in a 4:2:1 orbital resonance with each other. While the Galilean moons are spherical, all of Jupiter's much smaller remaining moons have irregular forms because of their weaker self-gravitation. The Galilean moons were observed in either 1609 or 1610 when Galileo made improvements to his telescope, which e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solar System
The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar System" and "solar system" structures in theinaming guidelines document. The name is commonly rendered in lower case ('solar system'), as, for example, in the ''Oxford English Dictionary'' an''Merriam-Webster's 11th Collegiate Dictionary''. is the gravitationally bound system of the Sun and the objects that orbit it. It formed 4.6 billion years ago from the gravitational collapse of a giant interstellar molecular cloud. The vast majority (99.86%) of the system's mass is in the Sun, with most of the remaining mass contained in the planet Jupiter. The four inner system planets— Mercury, Venus, Earth and Mars—are terrestrial planets, being composed primarily of rock and metal. The four giant planets of the outer system are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-body Problem
In physics, the -body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the -body problem, especially Ms. Kovalevskaya's 1868–1888 twenty-year complex-variables approach, failure; Section 1: "The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics" (Chapter 1, "The motion of a rigid body about a fixed point (Euler and Poisson equations)"; Chapter 2, "Mathematical Exterior Ballistics"), good precursor background to the -body problem; Section 2: "Celestial Mechanics" (Chapter 1, "The Uniformization of the Three-body Problem (Restricted Three-body Problem)"; Chapter 2, "Capture in the Three-Body Problem"; Chapter 3, "Generalized -body Problem"). Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars. In the 20th century, und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized and extended the work of his predecessors in his five-volume Traité de mécanique céleste, ''Mécanique céleste'' (''Celestial Mechanics'') (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian probability, Bayesian interpretation of probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplace operator, Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Of The Solar System
The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways. For this reason (among others), the Solar System is chaotic in the technical sense of mathematical chaos theory, and even the most precise long-term models for the orbital motion of the Solar System are not valid over more than a few tens of millions of years. The Solar System is stable in human terms, and far beyond, given that it is unlikely any of the planets will collide with each other or be ejected from the system in the next few billion years, and that Earth's orbit will be relatively stable. Since Newton's law of gravitation (1687), mathematicians and astronomers (such as Pierre-Simon Laplace, Joseph Louis Lagrange, Carl Friedrich Gauss, Henri Poincaré, Andrey Kolmogorov, Vladimir Arnold, and Jürgen Moser) have search ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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