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Numerical Model Of Solar System
A numerical model of the Solar System is a set of mathematical equations, which, when solved, give the approximate positions of the planets as a function of time. Attempts to create such a model established the more general field of celestial mechanics. The results of this simulation can be compared with past measurements to check for accuracy and then be used to predict future positions. Its main use therefore is in preparation of almanacs. Older efforts The simulations can be done in either Cartesian or in spherical coordinates. The former are easier, but extremely calculation intensive, and only practical on an electronic computer. As such only the latter was used in former times. Strictly speaking, the latter was not much less calculation intensive, but it was possible to start with some simple approximations and then to add perturbations, as much as needed to reach the wanted accuracy. In essence this mathematical simulation of the Solar System is a form of the ''N-body probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Celestial Mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data. History Modern analytic celestial mechanics started with Isaac Newton's ''Principia'' (1687). The name celestial mechanics is more recent than that. Newton wrote that the field should be called "rational mechanics". The term "dynamics" came in a little later with Gottfried Leibniz, and over a century after Newton, Pierre-Simon Laplace introduced the term ''celestial mechanics''. Prior to Kepler, there was little connection between exact, quantitative prediction of planetary positions, using geometrical or numerical techniques, and contemporary discussions of the physical causes of the planets' motion. Laws of planetary motion Johannes Kepler was the first to closely integrate the predictive geometrical a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating Point
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base. Numbers of this form are called floating-point numbers. For example, the number 2469/200 is a floating-point number in base ten with five digits: 2469/200 = 12.345 = \! \underbrace_\text \! \times \! \underbrace_\text\!\!\!\!\!\!\!\overbrace^ However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, 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 number, 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 (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Physics
Computational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline (or offshoot) of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics — an area of study which supplements both theory and experiment. Overview In physics, different theories based on mathematical models provide very precise predictions on how systems behave. Unfortunately, it is often the case that solving the mathematical model for a particular system in order to produce a useful prediction is not feasible. This can occur, for instance, when the solution does not have a closed-form expression, or is too complicated. In such cases, numerical approximations are required. Computational physics is the subject that deals with these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richmond, Virginia
Richmond ( ) is the List of capitals in the United States, capital city of the Commonwealth (U.S. state), U.S. commonwealth of Virginia. Incorporated in 1742, Richmond has been an independent city (United States), independent city since 1871. The city's population in the 2020 United States census was 226,610, up from 204,214 in 2010, making it Virginia's List of cities and counties in Virginia#Largest cities, fourth-most populous city. The Greater Richmond Region, Richmond metropolitan area, with over 1.3 million residents, is the Commonwealth's Virginia statistical areas, third-most populous. Richmond is located at the Atlantic Seaboard fall line, James River's fall line, west of Williamsburg, Virginia, Williamsburg, east of Charlottesville, Virginia, Charlottesville, east of Lynchburg, Virginia, Lynchburg and south of Washington, D.C. Surrounded by Henrico County, Virginia, Henrico and Chesterfield County, Virginia, Chesterfield counties, Richmond is at the intersection o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
VSOP (planets)
The semi-analytic planetary theory VSOP (French: ''Variations Séculaires des Orbites Planétaires'') is a mathematical model describing long-term changes ( secular variation) in the orbits of the planets Mercury to Neptune. The earliest modern scientific model considered only the gravitational attraction between the Sun and each planet, with the resulting orbits being unvarying Keplerian ellipses. In reality, all the planets exert slight forces on each other, causing slow changes in the shape and orientation of these ellipses. Increasingly complex analytical models have been made of these deviations, as well as efficient and accurate numerical approximation methods. VSOP was developed and is maintained (updated with the latest data) by the scientists at the Bureau des Longitudes in Paris. The first version, VSOP82, computed only the orbital elements at any moment. An updated version, VSOP87, computed the positions of the planets directly at any moment, as well as their orbita ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ephemeris
In astronomy and celestial navigation, an ephemeris (; ; , ) is a book with tables that gives the trajectory of naturally occurring astronomical objects and artificial satellites in the sky, i.e., the position (and possibly velocity) over time. Historically, positions were given as printed tables of values, given at regular intervals of date and time. The calculation of these tables was one of the first applications of mechanical computers. Modern ephemerides are often provided in electronic form. However, printed ephemerides are still produced, as they are useful when computational devices are not available. The astronomical position calculated from an ephemeris is often given in the spherical polar coordinate system of right ascension and declination, together with the distance from the origin if applicable. Some of the astronomical phenomena of interest to astronomers are eclipses, apparent retrograde motion/planetary stations, planetary es, sidereal time, positions for ... [...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 historically been stable as observed, and will be in the "short" term, their weak gravitational effects on one another can add up in ways that are not predictable by any simple means. For this reason (among others), the Solar System is ''chaotic'' in the technical sense defined by mathematical chaos theory, and that chaotic behavior degrades even the most precise long-term numerical or analytic models for the orbital motion in the Solar System, so they cannot be valid beyond more than a few tens of millions of years into the past or future – about 1% its present age. The Solar System is stable on the time-scale of the existence of humans, 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 (16 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction (geometry), direction and a magnitude, and both are conserved. Bicycle and motorcycle dynamics, Bicycles and motorcycles, flying discs, Rifling, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it. The three-dimensional angular momentum for a point particle is classically represented as a pseudovector , the cross product of the particle's position vector (relative to some origin) and its mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and light. Energy is a Conservation law, conserved quantity—the law of conservation of energy states that energy can be Energy transformation, converted in form, but not created or destroyed. The unit of measurement for energy in the International System of Units (SI) is the joule (J). Forms of energy include the kinetic energy of a moving object, the potential energy stored by an object (for instance due to its position in a Classical field theory, field), the elastic energy stored in a solid object, chemical energy associated with chemical reactions, the radiant energy carried by electromagnetic radiation, the internal energy contained within a thermodynamic system, and rest energy associated with an object's rest mass. These are not mutual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncation
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb_+ to be truncated and n \in \mathbb_0, the number of elements to be kept behind the decimal point, the truncated value of x is :\operatorname(x,n) = \frac. However, for negative numbers truncation does not round in the same direction as the floor function: truncation always rounds toward zero, the \operatorname function rounds towards negative infinity. For a given number x \in \mathbb_-, the function \operatorname is used instead :\operatorname(x,n) = \frac. Causes of truncation With computers, truncation can occur when a decimal number is typecast as an integer; it is truncated to zero decimal digits because integers cannot store non-integer real numbers In mathematics, a real number is a number that can be used to measurem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
