NationMaster - Encyclopedia: Hyperbolic trajectory

In astrodynamics or celestial mechanics a hyperbolic trajectory is an orbit with the eccentricity greater than 1. Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body. Similarly to parabolic trajectory all hyperbolic trajectories are also escape trajectories. Specific energy of hyperbolic trajectory orbit is positive. Astrodynamics is the study of the motion of rockets, missiles, and space vehicles, as determined from Sir Isaac Newtons laws of motion and his law of universal gravitation. ... Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... In astrodynamics, under standard assumptions any orbit must be of conic section shape. ... For most of the problems in astrodynamics involving two bodies and standard assumptions are usually the following: A1: and are the only objects in the universe and thus influence of other objects is disregarded, A2: The orbiting body () is far smaller than central body (), i. ... In astrodynamics a central body () is a body that is being orbited by orbiting body(). Under standard assumptions in astrodynamics: it is orders of magnitude heavier than orbiting body (i. ... In astrodynamics or celestial mechanics a parabolic trajectory is an orbit with the eccentricity equal to 1. ... An escape orbit (also known as C3 = 0 orbit) is the high-energy parabolic orbit around the central body. ... In astrodynamics the specific orbital energy (or vis-viva energy) of an orbiting body traveling through space under standard assumptions is the sum of its potential energy () and kinetic energy () per unit mass. ...

1 Hyperbolic excess velocity
2 Velocity
3 Energy
4 See also
5 External links

Hyperbolic excess velocity

Under standard assumptions the body traveling along hyperbolic trajectory will attain in infinity an orbital velocity called hyperbolic excess velocity ( $v_infty,!$ ) that can be computed as: For most of the problems in astrodynamics involving two bodies and standard assumptions are usually the following: A1: and are the only objects in the universe and thus influence of other objects is disregarded, A2: The orbiting body () is far smaller than central body (), i. ... The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. ...

$v_infty=sqrt{muover{a}},!$

where:

$mu,!$ is standard gravitational parameter,
$a,!$ is length of semi-major axis of orbit's hyperbola.

The hyperbolic excess velocity is related to the specific orbital energy or characteristic energy by In astrodynamics, the standard gravitational parameter () of a celestial body is the product of the gravitational constant () and the mass : The units of the standard gravitational parameter are km3s-2 Small body orbiting a central body Under standard assumptions in astrodynamics we have: where: is the mass of the orbiting... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... A graph of a hyperbola, where h = k = 0 and a = b = 2. ... In astrodynamics the specific orbital energy (or vis-viva energy) of an orbiting body traveling through space under standard assumptions is the sum of its potential energy () and kinetic energy () per unit mass. ...

$2epsilon=C_3=v_{infty}^2,!$

Velocity

Under standard assumptions the orbital velocity ( $v,$ ) of a body traveling along hyperbolic trajctory can be computed as: For most of the problems in astrodynamics involving two bodies and standard assumptions are usually the following: A1: and are the only objects in the universe and thus influence of other objects is disregarded, A2: The orbiting body () is far smaller than central body (), i. ... The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. ...

$v=sqrt{2muleft({1over{r}}+{1over{2a}}right)}$

where:

$mu,$ is standard gravitational parameter,
$r,$ is radial distance of orbiting body from central body,
$a,!$ is length of semi-major axis.

Under standard assumptions, at any position in the orbit the following relation holds for orbital velocity ( $v,$ ), local escape velocity( ${v_{esc}},$ ) and hyperbolic excess velocity ( $v_infty,!$ ): In astrodynamics, the standard gravitational parameter () of a celestial body is the product of the gravitational constant () and the mass : The units of the standard gravitational parameter are km3s-2 Small body orbiting a central body Under standard assumptions in astrodynamics we have: where: is the mass of the orbiting... In astrodynamics a central body () is a body that is being orbited by orbiting body(). Under standard assumptions in astrodynamics: it is orders of magnitude heavier than orbiting body (i. ... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas. ... For most of the problems in astrodynamics involving two bodies and standard assumptions are usually the following: A1: and are the only objects in the universe and thus influence of other objects is disregarded, A2: The orbiting body () is far smaller than central body (), i. ... The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. ... In physics, for a given gravitational field and a given position, the escape velocity is the minimum speed an object without propulsion, at that position, needs to have to move away indefinitely from the source of the field, as opposed to falling back or staying in an orbit within a...

$v^2={v_{esc}}^2+{v_infty}^2$

Note that this means that a relatively small extra delta-v above that needed to accelerate to the escape speed, results in a relatively large speed at infinity. General In general physics delta-v is simply the change in velocity. ...

Energy

Under standard assumptions, specific orbital energy ( $epsilon,$ ) of a hyperbolic trajectory is greater than zero and the orbital energy conservation equation for this kind of trajectory takes form: For most of the problems in astrodynamics involving two bodies and standard assumptions are usually the following: A1: and are the only objects in the universe and thus influence of other objects is disregarded, A2: The orbiting body () is far smaller than central body (), i. ... In astrodynamics the specific orbital energy (or vis-viva energy) of an orbiting body traveling through space under standard assumptions is the sum of its potential energy () and kinetic energy () per unit mass. ... In astrodynamics vis-viva equation (also referred to as orbital energy conservation equation) is one of the fundamental and useful equations that govern the motion of orbiting bodies. ...

$epsilon={v^2over2}-{muover{r}}={muover{2a}}$

where:

$v,$ is orbital velocity of orbiting body,
$r,$ is radial distance of orbiting body from central body,
$a,$ is length of semi-major axis,
$mu,$ is standard gravitational parameter.

The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. ... In astrodynamics a central body () is a body that is being orbited by orbiting body(). Under standard assumptions in astrodynamics: it is orders of magnitude heavier than orbiting body (i. ... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas. ... In astrodynamics, the standard gravitational parameter () of a celestial body is the product of the gravitational constant () and the mass : The units of the standard gravitational parameter are km3s-2 Small body orbiting a central body Under standard assumptions in astrodynamics we have: where: is the mass of the orbiting...

External links

http://www.cix.co.uk/~sjbradshaw/msc/traject.html
http://www.go.ednet.ns.ca/~larry/orbits/ellipse.html

Categories: Astrodynamics | Celestial mechanics

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Velocity

Energy

See also

External links

Hyperbolic excess velocity