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Hyperbolic Trajectory
In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one. Under simplistic assumptions a body traveling along this trajectory will coast towards infinity, settling to a final excess velocity relative to the central body. Similarly to parabolic trajectories, all hyperbolic trajectories are also escape trajectories. The specific energy of a hyperbolic trajectory orbit is positive. Planetary flybys, used for gravitational slingshots, can be described within the planet's sphere of influence using hyperbolic trajectories. Parameters describing a hyperbolic trajectory Like an elliptical orbit, a hyper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-minor Axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle. The length of the semi-major axis of an ellipse is related to the semi-minor axis's length through the eccentricity and the semi-latus rectum \ell, as follows: The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the cen ... [...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] |
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surface is made up of the ocean, dwarfing Earth's polar ice, lakes, and rivers. The remaining 29% of Earth's surface is land, consisting of continents and islands. Earth's surface layer is formed of several slowly moving tectonic plates, which interact to produce mountain ranges, volcanoes, and earthquakes. Earth's liquid outer core generates the magnetic field that shapes the magnetosphere of the Earth, deflecting destructive solar winds. The atmosphere of the Earth consists mostly of nitrogen and oxygen. Greenhouse gases in the atmosphere like carbon dioxide (CO2) trap a part of the energy from the Sun close to the surface. Water vapor is widely present in the atmosphere and forms clouds that cover most of the planet. More s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closest Approach
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elliptic orbit. The name for each apsis is created from the prefixes ''ap-'', ''apo-'' (), or ''peri-'' (), each referring to the farthest and closest point to the primary body the affixing necessary suffix that describes the primary body in the orbit. In this case, the suffix for Earth is ''-gee'', so the apsides' names are ''apogee'' and ''perigee''. For the Sun, its suffix is ''-helion'', so the names are ''aphelion'' and ''perihelion''. According to Newton's laws of motion, all periodic orbits are ellipses. The barycenter of the two bodies may lie well within the bigger body—e.g., the Earth–Moon barycenter is about 75% of the way from Earth's center to its surface. If, compared to the larger mass, the smaller mass is negligible (e.g., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impact Parameter
In physics, the impact parameter is defined as the perpendicular distance between the path of a projectile and the center of a potential field created by an object that the projectile is approaching (see diagram). It is often referred to in nuclear physics (see Rutherford scattering) and in classical mechanics. The impact parameter is related to the scattering angle by : \theta = \pi - 2b\int_^\infty \frac, where is the velocity of the projectile when it is far from the center, and is its closest distance from the center. Scattering from a hard sphere The simplest example illustrating the use of the impact parameter is in the case of scattering from a sphere. Here, the object that the projectile is approaching is a hard sphere with radius R. In the case of a hard sphere, U(r) = 0 when r > R, and U(r) = \infty for r \leq R . When b > R , the projectile misses the hard sphere. We immediately see that \theta = 0. When b \leq R, we find that b = R \cos\tfrac. Collisi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Trajectories With Different Impact Parameters
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they manifest hyperbolas, not because something about them is exaggerated. * Hyperbolic angle, an unbounded variable referring to a hyperbola instead of a circle * Hyperbolic coordinates, location by geometric mean and hyperbolic angle in quadrant I * Hyperbolic distribution, a probability distribution characterized by the logarithm of the probability density function being a hyperbola * Hyperbolic equilibrium point, a fixed point that does not have any center manifolds * Hyperbolic function, an analog of an ordinary trigonometric or circular function * Hyperbolic geometric graph, a random network generated by connecting nearby points sprinkled in a hyperbolic space * Hyperbolic geometry, a non-Euclidean geometry * Hyperbolic group, a finite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
True Anomaly
In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse (the point around which the object orbits). The true anomaly is usually denoted by the Greek letters or , or the Latin letter , and is usually restricted to the range 0–360° (0–2π). As shown in the image, the true anomaly is one of three angular parameters (''anomalies'') that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly. Formulas From state vectors For elliptic orbits, the true anomaly can be calculated from orbital state vectors as: : \nu = \arccos ::(if then replace by ) where: * v is the orbital velocity vector of the orbiting body, * e is the eccentricity vector, * r is the orbital position vector (segment ''FP'' in the figure) of the orbi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Gravitational Parameter
In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM when one body is much larger than the other. \mu=GM \ For several objects in the Solar System, the value of ''μ'' is known to greater accuracy than either ''G'' or ''M''. The SI units of the standard gravitational parameter are . However, units of are 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Energy
In astrodynamics, the characteristic energy (C_3) is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length2 time−2, i.e. velocity squared, or energy per mass. Every object in a 2-body ballistic trajectory has a constant specific orbital energy \epsilon equal to the sum of its specific kinetic and specific potential energy: \epsilon = \frac v^2 - \frac = \text = \frac C_3, where \mu = GM is the standard gravitational parameter of the massive body with mass M, and r is the radial distance from its center. As an object in an escape trajectory moves outward, its kinetic energy decreases as its potential energy (which is always negative) increases, maintaining a constant sum. Note that ''C''3 is ''twice'' the specific orbital energy \epsilon of the escaping object. Non-escape trajectory A spacecraft with insufficient energy to escape will remain in a closed orbit (unless it intersects the central body), with C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler's Laws Of Planetary Motion
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that: # The orbit of a planet is an ellipse with the Sun at one of the two foci. # A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. # The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit. The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law helps to establish that when a planet is closer to the Sun, it travels faster. The thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Specific Angular Momentum
In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative position and relative linear momentum, divided by the mass of the body in question. Specific relative angular momentum plays a pivotal role in the analysis of the two-body problem, as it remains constant for a given orbit under ideal conditions. " Specific" in this context indicates angular momentum per unit mass. The SI unit for specific relative angular momentum is square meter per second. Definition The specific relative angular momentum is defined as the cross product of the relative position vector \mathbf and the relative velocity vector \mathbf . \mathbf = \mathbf\times \mathbf = \frac where \mathbf is the angular momentum vector, defined as \mathbf \times m \mathbf. The \mathbf vector is always perpendicular to the inst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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