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Lambert's Problem
In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination. Suppose a body under the influence of a central gravitational force is observed to travel from point ''P''1 on its conic trajectory, to a point ''P''2 in a time ''T''. The time of flight is related to other variables by Lambert's theorem, which states: :''The transfer time of a body moving between two points on a conic trajectory is a function only of the sum of the distances of the two points from the origin of the force, the linear distance between the points, and the semimajor axis of the conic.'' Stated another way, Lambert's problem is the boundary value problem for the differential equation \ddot = - ... [...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 of 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 arithmetical techniques, and contemporary discussions of the physical causes of the planets' motion. Johannes Kepler Johannes Kepler (1571–1630) was the first to closely integrate the predictive geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambert Fig3
Lambert may refer to People *Lambert (name), a given name and surname * Lambert, Bishop of Ostia (c. 1036–1130), became Pope Honorius II *Lambert, Margrave of Tuscany ( fl. 929–931), also count and duke of Lucca *Lambert (pianist), stage-name of German pianist and composer Paul Lambert Places United States *Lambert, Mississippi, a town *Lambert, Missouri, a village *St. Louis Lambert International Airport, St. Louis, Missouri *Lambert, Montana, a rural town in Montana *Lambert, Oklahoma, a town *Lambert Township, Red Lake County, Minnesota *Lambert Castle, a mansion in Paterson, New Jersey *Lambert Creek, San Mateo County, California Elsewhere * Lambert Gravitational Centre, the geographical centre of Australia *Lambert (lunar crater), named after Johann Heinrich Lambert *Lambert (Martian crater), named after Johann Heinrich Lambert Transportation *Lambert (automobile), a defunct American automobile brand *Lambert (cyclecar), British three-wheeled cyclecar *''Lambert'', one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbits
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbital ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbit Determination
Orbit determination is the estimation of orbits of objects such as moons, planets, and spacecraft. One major application is to allow tracking newly observed asteroids and verify that they have not been previously discovered. The basic methods were discovered in the 17th century and have been continuously refined. ''Observations'' are the raw data fed into orbit determination algorithms. Observations made by a ground-based observer typically consist of time-tagged azimuth, elevation, range, and/or range rate values. Telescopes or radar apparatus are used, because naked-eye observations are inadequate for precise orbit determination. With more or better observations, the accuracy of the orbit determination process also improves, and fewer " false alarms" result. After orbits are determined, mathematical propagation techniques can be used to predict the future positions of orbiting objects. As time goes by, the actual path of an orbiting object tends to diverge from the predicte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patched Conic Approximation
In astrodynamics, the patched conic approximation or patched two-body approximation is a method to simplify trajectory calculations for spacecraft in a multiple-body environment. Method The simplification is achieved by dividing space into various parts by assigning each of the ''n'' bodies (e.g. the Sun, planets, moons) its own sphere of influence. When the spacecraft is within the sphere of influence of a smaller body, only the gravitational force between the spacecraft and that smaller body is considered, otherwise the gravitational force between the spacecraft and the larger body is used. This reduces a complicated n-body problem to multiple two-body problems, for which the solutions are the well-known conic sections of the Kepler orbits. Although this method gives a good approximation of trajectories for interplanetary spacecraft missions, there are missions for which this approximation does not provide sufficiently accurate results. Notably, it does not model Lagrangian po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambert Fig4
Lambert may refer to People *Lambert (name), a given name and surname * Lambert, Bishop of Ostia (c. 1036–1130), became Pope Honorius II *Lambert, Margrave of Tuscany ( fl. 929–931), also count and duke of Lucca *Lambert (pianist), stage-name of German pianist and composer Paul Lambert Places United States *Lambert, Mississippi, a town *Lambert, Missouri, a village *St. Louis Lambert International Airport, St. Louis, Missouri *Lambert, Montana, a rural town in Montana *Lambert, Oklahoma, a town *Lambert Township, Red Lake County, Minnesota *Lambert Castle, a mansion in Paterson, New Jersey *Lambert Creek, San Mateo County, California Elsewhere * Lambert Gravitational Centre, the geographical centre of Australia *Lambert (lunar crater), named after Johann Heinrich Lambert *Lambert (Martian crater), named after Johann Heinrich Lambert Transportation *Lambert (automobile), a defunct American automobile brand *Lambert (cyclecar), British three-wheeled cyclecar *''Lambert'', one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbital Pole
An orbital pole is either point at the ends of an imaginary line segment that runs through the center of an orbit (of a revolving body like a planet, Natural satellite, moon or satellite) and is perpendicular to the orbital plane. Projected onto the celestial sphere, orbital poles are similar in concept to celestial poles, but are based on the body's orbit instead of its celestial equator, equator. The north orbital pole of a revolving body is defined by the right-hand rule. If the fingers of the right hand are curved along the retrograde and prograde motion, direction of orbital motion, with the thumb extended and oriented to be parallel to the orbital axis of rotation, axis, then the direction the thumb points is defined to be the orbital north. The poles of Earth's orbit are referred to as the ecliptic poles. For the remaining planets, the orbital pole in ecliptic coordinates is given by the longitude of the ascending node (☊) and inclination (''i''): ''l'' = ☊ - 90°, '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the reciprocal function y(x) = 1/x in the Cartesian plane, * as the path followed by the shadow of the tip of a sundial, * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity (mathematics), eccentricity e, a number ranging from e = 0 (the Limiting case (mathematics), limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution. Analytic geometry, Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: : \frac+\frac = 1 . Assuming a \ge b, the foci are (\pm c, 0) for c = \sqrt. The standard parametric e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambert Fig2
Lambert may refer to People *Lambert (name), a given name and surname * Lambert, Bishop of Ostia (c. 1036–1130), became Pope Honorius II *Lambert, Margrave of Tuscany ( fl. 929–931), also count and duke of Lucca *Lambert (pianist), stage-name of German pianist and composer Paul Lambert Places United States *Lambert, Mississippi, a town *Lambert, Missouri, a village *St. Louis Lambert International Airport, St. Louis, Missouri *Lambert, Montana, a rural town in Montana *Lambert, Oklahoma, a town *Lambert Township, Red Lake County, Minnesota *Lambert Castle, a mansion in Paterson, New Jersey *Lambert Creek, San Mateo County, California Elsewhere * Lambert Gravitational Centre, the geographical centre of Australia *Lambert (lunar crater), named after Johann Heinrich Lambert *Lambert (Martian crater), named after Johann Heinrich Lambert Transportation *Lambert (automobile), a defunct American automobile brand *Lambert (cyclecar), British three-wheeled cyclecar *''Lambert'', one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Heinrich Lambert
Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections. Biography Lambert was born in 1728 into a Huguenot family in the city of Mulhouse (now in Alsace, France), at that time a city-state allied to Switzerland. Some sources give 26 August as his birth date and others 28 August. Leaving school at 12, he continued to study in his free time while undertaking a series of jobs. These included assistant to his father (a tailor), a clerk at a nearby iron works, a private tutor, secretary to the editor of ''Basler Zeitung'' and, at the age of 20, private tutor to the sons of Count Salis in Chur. Travelling Europe with his charges (1756–1758) allowed him to meet established mathematicians in the German ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambert Fig1
Lambert may refer to People *Lambert (name), a given name and surname * Lambert, Bishop of Ostia (c. 1036–1130), became Pope Honorius II *Lambert, Margrave of Tuscany ( fl. 929–931), also count and duke of Lucca *Lambert (pianist), stage-name of German pianist and composer Paul Lambert Places United States *Lambert, Mississippi, a town *Lambert, Missouri, a village *St. Louis Lambert International Airport, St. Louis, Missouri *Lambert, Montana, a rural town in Montana *Lambert, Oklahoma, a town *Lambert Township, Red Lake County, Minnesota *Lambert Castle, a mansion in Paterson, New Jersey *Lambert Creek, San Mateo County, California Elsewhere * Lambert Gravitational Centre, the geographical centre of Australia *Lambert (lunar crater), named after Johann Heinrich Lambert *Lambert (Martian crater), named after Johann Heinrich Lambert Transportation *Lambert (automobile), a defunct American automobile brand *Lambert (cyclecar), British three-wheeled cyclecar *''Lambert'', one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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