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Parabolic trajectory
 
 
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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 parabolic 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 equal to 1. When moving away from the source it is called an escape orbit
Escape orbit

An escape orbit is a high-energy parabolic orbit around the central body. A body in this orbit has at each position the escape velocity with respect to this central body, for this position....
, otherwise a capture orbit
Capture orbit

A capture orbit is a reverse escape orbit. It is a parabolic orbit with as special case a straight line in the direction of the center of the central body....
.

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 an escape orbit will coast to infinity, with 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 ,...
 tending to zero, and therefore will never return. Parabolic trajectory is a minimum-energy escape trajectory.








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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 parabolic 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 equal to 1. When moving away from the source it is called an escape orbit
Escape orbit

An escape orbit is a high-energy parabolic orbit around the central body. A body in this orbit has at each position the escape velocity with respect to this central body, for this position....
, otherwise a capture orbit
Capture orbit

A capture orbit is a reverse escape orbit. It is a parabolic orbit with as special case a straight line in the direction of the center of the central body....
.

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 an escape orbit will coast to infinity, with 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 ,...
 tending to zero, and therefore will never return. Parabolic trajectory is a minimum-energy escape trajectory.

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 parabolic trajectory can be computed as: where:
  • 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 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...
    .


At any position the orbiting body has the 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,...
 for that position.

If the body has the escape velocity with respect to the Earth, this is not enough to escape the Solar System, so near the Earth the orbit resembles a parabola, but further away it bends into an elliptical orbit around the Sun.

This velocity is closely related to the orbital velocity of a body in a circular orbit
Circular orbit

In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. It is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion....
 of the radius equal to the radial position of orbiting body on the parabolic trajectory: where:
  • is orbital velocity of a body in circular orbit
    Circular orbit

    In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. It is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion....
    .


Equation of motion

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,...
, for a body moving along this kind of trajectory an orbital equation becomes: where:
  • 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 specific angular momentum of the orbiting body
    Orbiting body

    In astrodynamics, an orbiting body is a body that orbits central body . Under standard assumptions in astrodynamics:* it is less massive than the central body by several orders of magnitude ....
    ,
  • is a true anomaly
    True anomaly

    In astronomy, the true anomaly is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse ....
     of the orbiting body,
  • 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...
    .


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 parabolic trajectory is zero, so the orbital energy conservation equation for this 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 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

  • 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....
  • 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....
  • Circular orbit
    Circular orbit

    In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. It is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion....
  • Elliptic orbit
    Elliptic orbit

    In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a gravitational two-body problem with the eccentricity in this range both bodies follow Similarity elliptic orbits with the same orbital period around their common barycenter....
  • Orbital equation
  • List of orbits
    List of orbits

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