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Vis-viva Equation
In astrodynamics, the ''vis-viva'' equation is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight which is the gravitational force determined by the product of the mass of the object and the strength of the surrounding gravitational field. '' Vis viva'' (Latin for "living force") is a term from the history of mechanics and this name is given to the orbital equation originally derived by Isaac Newton. It represents the principle that the difference between the total work of the accelerating forces of a system and that of the retarding forces is equal to one half the ''vis viva'' accumulated or lost in the system while the work is being done. Formulation For any Keplerian orbit ( elliptic, parabolic, hyperbolic, or radial), the ''vis-viva'' equation is as follows: v^2 = GM \left( - \right) where: * is the relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Astrodynamics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the Newton's law of universal gravitation, law of universal gravitation. Astrodynamics is a core discipline within space exploration, space-mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including both spacecraft and natural astronomical object, astronomical bodies such as star systems, planets, Natural satellite, moons, and comets. Orbital mechanics focuses on spacecraft trajectory, trajectories, including orbital maneuvers, orbital plane (astronomy), orbital plane changes, and interplanetary transfers, and is used by mission planners to predict the results of spacecraft propulsion, propulsive maneuvers. General relativity is a more exact theory than Newton's laws for calculati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Radial Trajectory
In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line. Classification There are three types of radial trajectories (orbits). * Radial elliptic trajectory: an orbit corresponding to the part of a degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. The relative speed of the two objects is less than the escape velocity. This is an elliptic orbit with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1, this is not a parabolic orbit. If the coefficient of restitution of the two bodies is 1 (perfectly elastic) this orbit is periodic. If the coefficient of restitution is less than 1 (inelastic) this orbit is non-periodic. * Radial parabolic trajectory, a non-periodic orbit where the relative speed of the two objects is always equal to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Orbits
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, 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 barycenter, 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 Newton's law of universal gravitation, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Specific Orbital Energy
In the gravitational two-body problem, the specific orbital energy \varepsilon (or specific ''vis-viva'' energy) of two orbiting bodies is the constant quotient of their mechanical energy (the sum of their mutual potential energy, \varepsilon_p, and their kinetic energy, \varepsilon_k) to their reduced mass. According to the orbital energy conservation equation (also referred to as ''vis-viva'' equation), it does not vary with time: \begin \varepsilon &= \varepsilon_k + \varepsilon_p \\ &= \frac - \frac = -\frac \frac \left(1 - e^2\right) = -\frac \end where *v is the relative orbital speed; *r is the orbital distance between the bodies; *\mu = (m_1 + m_2) is the sum of the standard gravitational parameters of the bodies; *h is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass; *e is the orbital eccentricity; *a is the semi-major axis. It is a kind of specific energy, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Escape Velocity
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming: * Ballistic trajectory – no other forces are acting on the object, such as propulsion and friction * No other gravity-producing objects exist. Although the term ''escape velocity'' is common, it is more accurately described as a speed than as a velocity because it is independent of direction. Because gravitational force between two objects depends on their combined mass, the escape speed also depends on mass. For artificial satellites and small natural objects, the mass of the object makes a negligible contribution to the combined mass, and so is often ignored. Escape speed varies with distance from the center of the primary body, as does the velocity of an object traveling under the gravitational influence of the primary. If an object is in a circular or elliptical orbit, its speed is always less than the es ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Suborbital
A sub-orbital spaceflight is a spaceflight in which the spacecraft reaches outer space, but its trajectory intersects the surface of the gravitating body from which it was launched. Hence, it will not complete one orbital revolution, will not become an artificial satellite nor will it reach escape velocity. For example, the path of an object launched from Earth that reaches the Kármán line (about – above sea level), and then falls back to Earth, is considered a sub-orbital spaceflight. Some sub-orbital flights have been undertaken to test spacecraft and launch vehicles later intended for orbital spaceflight. Other vehicles are specifically designed only for sub-orbital flight; examples include crewed vehicles, such as the X-15 and SpaceShipTwo, and uncrewed ones, such as ICBMs and sounding rockets. Flights which attain sufficient velocity to go into low Earth orbit, and then de-orbit before completing their first full orbit, are not considered sub-orbital. Examples o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Specific Orbital Energy
In the gravitational two-body problem, the specific orbital energy \varepsilon (or specific ''vis-viva'' energy) of two orbiting bodies is the constant quotient of their mechanical energy (the sum of their mutual potential energy, \varepsilon_p, and their kinetic energy, \varepsilon_k) to their reduced mass. According to the orbital energy conservation equation (also referred to as ''vis-viva'' equation), it does not vary with time: \begin \varepsilon &= \varepsilon_k + \varepsilon_p \\ &= \frac - \frac = -\frac \frac \left(1 - e^2\right) = -\frac \end where *v is the relative orbital speed; *r is the orbital distance between the bodies; *\mu = (m_1 + m_2) is the sum of the standard gravitational parameters of the bodies; *h is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass; *e is the orbital eccentricity; *a is the semi-major axis. It is a kind of specific energy, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Standard Gravitational Parameter
The standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of that body. For two bodies, the parameter may be expressed as , or as when one body is much larger than the other: \mu=G(M+m)\approx GM . For several objects in the Solar System, the value of ''μ'' is known to greater accuracy than either ''G'' or ''M''. The SI unit of the standard gravitational parameter is . However, the unit is frequently used in the scientific literature and in spacecraft navigation. Definition Small body orbiting a central body The central body in an orbital system can be defined as the one whose mass (''M'') is much larger than the mass of the orbiting body (''m''), or . This approximation is standard for planets orbiting the Sun or most moons and greatly simplifies equations. Under Newton's law of universal gravitation, if the distance between the bodies is ''r'', the force exerted on the smaller body is: F = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gravitational Constant
The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It is also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant, denoted by the capital letter . In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their masses and the inverse-square law, inverse square of their distance. In the Einstein field equations, it quantifies the relation between the geometry of spacetime and the energy–momentum tensor (also referred to as the stress–energy tensor). The measured value of the constant is known with some certainty to four significant digits. In SI units, its value is approximately The modern notation of Newton's law involving was introduced i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hyperbola
In mathematics, a hyperbola is a type of smooth function, smooth plane curve, 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 component (topology), connected components or branches, that are mirror images of each other and resemble two infinite bow (weapon), bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane (mathematics), plane and a double cone (geometry), 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. Besides being a conic section, a hyperbola can arise as the locus (mathematics), locus of points whose difference of distances to two fixed focus (geometry), foci is constant, as a curve for each point of which the rays to two fix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a Point (geometry), point (the Focus (geometry), focus) and a Line (geometry), line (the Directrix (conic section), directrix). The focus does not lie on the directrix. The parabola is the locus (mathematics), locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane (geometry), plane Parallel (geometry), parallel to another plane that is tangential to the conical surface. The graph of a function, graph of a quadratic function y=ax^2+bx+ c (with a\neq 0 ) is a parabola with its axis parallel to the -axis. Conversely, every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 for Perimeter of an ellipse, its perimeter (also known as circumference), Integral, integration is required to obtain an exact solution. The largest and smallest diameters of an ellipse, also known as its width and height, are typically denoted and . An ellipse has four extreme points: two ''Vertex (geometry), vertices'' at the endpoints of the major axis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
