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In astrodynamics
Astrodynamics

Orbital mechanics or astrodynamics is the application of celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft....
 or celestial mechanics
Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motion s of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data....
 a hyperbolic trajectory is a Kepler orbit
Kepler orbit

In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space....
 with the eccentricity greater than 1. Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
 a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body
Central body

In astrodynamics a central body is a body that is being orbited by an orbiting body . Under standard assumptions in astrodynamics:* it is more massive than the orbiting body by several orders of magnitude ,...
. Similarly to parabolic trajectory
Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit....
 all hyperbolic trajectories are also escape trajectories. The specific energy
Specific orbital energy

In astrodynamics the specific orbital energy of an orbiting body traveling through space under standard assumptions in astrodynamics is the sum of its potential energy and kinetic energy per unit mass....
 of a hyperbolic trajectory orbit is positive.








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In astrodynamics
Astrodynamics

Orbital mechanics or astrodynamics is the application of celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft....
 or celestial mechanics
Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motion s of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data....
 a hyperbolic trajectory is a Kepler orbit
Kepler orbit

In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space....
 with the eccentricity greater than 1. Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
 a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body
Central body

In astrodynamics a central body is a body that is being orbited by an orbiting body . Under standard assumptions in astrodynamics:* it is more massive than the orbiting body by several orders of magnitude ,...
. Similarly to parabolic trajectory
Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit....
 all hyperbolic trajectories are also escape trajectories. The specific energy
Specific orbital energy

In astrodynamics the specific orbital energy of an orbiting body traveling through space under standard assumptions in astrodynamics is the sum of its potential energy and kinetic energy per unit mass....
 of a hyperbolic trajectory orbit is positive.

Hyperbolic excess velocity


Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
 the body traveling along hyperbolic trajectory will attain in infinity an orbital velocity called hyperbolic excess velocity that can be computed as: where:
  • is standard gravitational parameter
    Standard gravitational parameter

    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...
    ,
  • is length of semi-major axis
    Semi-major axis

    In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
     of orbit
    ORBit

    ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl....
    's hyperbola
    Hyperbola

    In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
    .


The hyperbolic excess velocity is related to the specific orbital energy
Specific orbital energy

In astrodynamics the specific orbital energy of an orbiting body traveling through space under standard assumptions in astrodynamics is the sum of its potential energy and kinetic energy per unit mass....
 or characteristic energy by

Velocity

Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
 the orbital velocity of a body traveling along hyperbolic trajectory can be computed as: where:
  • is standard gravitational parameter
    Standard gravitational parameter

    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...
    ,
  • is radial distance of orbiting body from central body
    Central body

    In astrodynamics a central body is a body that is being orbited by an orbiting body . Under standard assumptions in astrodynamics:* it is more massive than the orbiting body by several orders of magnitude ,...
    ,
  • is length of semi-major axis
    Semi-major axis

    In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
    , always a negative number for hyperbolas.


Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
, at any position in the orbit the following relation holds for orbital velocity , local escape velocity
Escape velocity

In physics, escape velocity is the speed where the kinetic energy of an object is equal to the magnitude of its gravitational potential energy, as calculated by the equation,...
 and hyperbolic excess velocity :

Note that this means that a relatively small extra delta-v
Delta-v

In astrodynamics, the term delta-v, literally "change in velocity" , has a specific meaning: it is a scalar which takes units of speed that measures the amount of "effort" needed to carry out an orbital maneuver, i.e., to change from one trajectory to another....
 above that needed to accelerate to the escape speed, results in a relatively large speed at infinity.

Energy

Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
, specific orbital energy
Specific orbital energy

In astrodynamics the specific orbital energy of an orbiting body traveling through space under standard assumptions in astrodynamics is the sum of its potential energy and kinetic energy per unit mass....
  of a hyperbolic trajectory
Hyperbolic trajectory

In astrodynamics or celestial mechanics a hyperbolic trajectory is a Kepler orbit with the eccentricity greater than 1. Under standard assumptions in astrodynamics a body traveling along this trajectory will orbital coast to infinity, arriving there with hyperbolic excess velocity relative to the central body....
 is greater than zero and the orbital energy conservation equation for this kind of trajectory takes form: where:
  • is orbital velocity of orbiting body,
  • is radial distance of orbiting body from central body
    Central body

    In astrodynamics a central body is a body that is being orbited by an orbiting body . Under standard assumptions in astrodynamics:* it is more massive than the orbiting body by several orders of magnitude ,...
    ,
  • is length of semi-major axis
    Semi-major axis

    In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
    ,
  • is standard gravitational parameter
    Standard gravitational parameter

    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...
    .


See also

  • Orbit
    ORBit

    ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl....
  • Orbital equation
  • List of orbits
    List of orbits

    The following is a list of types of Orbit s:...


External links