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Orbital speed
 
 
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The orbital speed of a body, generally a planet
Planet

A planet , as 2006 definition of planet by the International Astronomical Union , is a celestial body orbiting a star or Stellar evolution#Stellar remnants that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared the neighbourhood of planetesimals....
, a natural satellite
Natural satellite

A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called the primary. Technically, the term natural satellite could refer to a planet orbiting a star, or a dwarf galaxy orbiting a major galaxy, but it is normally synonymous with moon and used to identify non-artificial satellites...
, an artificial satellite
Satellite

In the context of spaceflight, a satellite is an Physical body which has been placed into orbit by human endeavor. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....
, or a multiple star
Multiple star

A multiple star consists of three or more stars which appear from the Earth to be close to one another in the sky. This may result from the stars being physically close and gravity bound to each other, in which case it is physical, or this closeness may be merely apparent, in which case the multiple star is optical....
, is the speed at which it 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 around the barycenter of a system, usually around a more mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
ive body. It can be used to refer to either the mean orbital speed, the average speed as it completes an orbit, or instantaneous orbital speed, the speed at a particular point in its orbit.

The orbital speed at any position in the orbit can be computed from the distance to the central body at that position, and 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....
, which is independent of position: the kinetic energy
Kinetic energy

The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the mechanical work needed to accelerate a body of a given mass from rest to its current velocity....
 is the total energy minus the potential energy
Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
.

Thus, 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 speed is:

where: Note:


Transverse orbital speed
The transverse orbital speed is inversely proportional to the distance to the central body because of the law of conservation of angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
, or equivalently, Kepler
Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
's second law
Kepler's laws of planetary motion

In astronomy, Kepler's three laws of planetary motion are*"The orbit of every planet is an ellipse with the sun at a Focus ."*"A line joining a planet and the sun sweeps out equal areas during equal intervals of time."...
.






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The orbital speed of a body, generally a planet
Planet

A planet , as 2006 definition of planet by the International Astronomical Union , is a celestial body orbiting a star or Stellar evolution#Stellar remnants that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared the neighbourhood of planetesimals....
, a natural satellite
Natural satellite

A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called the primary. Technically, the term natural satellite could refer to a planet orbiting a star, or a dwarf galaxy orbiting a major galaxy, but it is normally synonymous with moon and used to identify non-artificial satellites...
, an artificial satellite
Satellite

In the context of spaceflight, a satellite is an Physical body which has been placed into orbit by human endeavor. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....
, or a multiple star
Multiple star

A multiple star consists of three or more stars which appear from the Earth to be close to one another in the sky. This may result from the stars being physically close and gravity bound to each other, in which case it is physical, or this closeness may be merely apparent, in which case the multiple star is optical....
, is the speed at which it 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 around the barycenter of a system, usually around a more mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
ive body. It can be used to refer to either the mean orbital speed, the average speed as it completes an orbit, or instantaneous orbital speed, the speed at a particular point in its orbit.

The orbital speed at any position in the orbit can be computed from the distance to the central body at that position, and 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....
, which is independent of position: the kinetic energy
Kinetic energy

The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the mechanical work needed to accelerate a body of a given mass from rest to its current velocity....
 is the total energy minus the potential energy
Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
.

Thus, 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 speed is:

  • in general:
    • 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....
      :
    • 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....
      :
    • 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....
      :
where:
  • is the 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 the distance between 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 ....
     and 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 ,...
  • is 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....
  • is the semi-major axis
    Semi-major axis

    In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
Note:
  • Velocity does not explicitly depend on eccentricity but is determined by length of semi-major axis
    Semi-major axis

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


Radial trajectories


In the case of radial motion:
  • if the energy is non-negative: the motion is either for the whole trajectory away from the central body, or for the whole trajectory towards it. For the zero-energy case, see 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....
     and 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....
    .
  • if the energy is negative: the motion can be first away from the central body, up to r=µ/|e|, then falling back. This is the limit case of an orbit which is part of an ellipse with eccentricity tending to 1, and the other end of the ellipse tending to the center of the central body. See also free-fall time
    Free-fall time

    The free-fall time is the characteristic time that would take a body to collapse under its own gravity, if no other forces existed to oppose the collapse....
    .


Transverse orbital speed


The transverse orbital speed is inversely proportional to the distance to the central body because of the law of conservation of angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
, or equivalently, Kepler
Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
's second law
Kepler's laws of planetary motion

In astronomy, Kepler's three laws of planetary motion are*"The orbit of every planet is an ellipse with the sun at a Focus ."*"A line joining a planet and the sun sweeps out equal areas during equal intervals of time."...
. This states that as a body moves around its orbit during a fixed amount of time, the line from the barycenter to the body sweeps a constant area of the orbital plane, regardless of which part of its orbit the body traces during that period of time. This means that the body moves faster near its periapsis than near its apoapsis, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."

Mean orbital speed


For orbits with small eccentricity, the length of the orbit is close to that of a circular one, and the mean orbital speed can be approximated either from observations of the orbital period
Orbital period

The orbital Periodicity is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars....
 and the semimajor axis of its orbit, or from knowledge of the mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
es of the two bodies and the semimajor axis.

where is the orbital velocity, is the length
Length

Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end....
 of the semimajor axis, is the orbital period, and is the 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...
. Note that this is only an approximation that holds true when the orbiting body is of considerably lesser mass than the central one, and eccentricity is close to zero.

Taking into account the mass of the orbiting body,

where is now the mass of the body under consideration, is the mass of the body being orbited, is specifically the distance between the two bodies (which is the sum of the distances from each to the center of mass), and is the gravitational constant
Gravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitation between objects with mass....
. This is still a simplified version; it doesn't allow for elliptical
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
 orbits, but it does at least allow for bodies of similar masses.

For an object in an eccentric orbit orbiting a much larger body, the length of the orbit decreases with eccentricity , and is given at ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
. This can be used to obtain a more accurate estimate of the average orbital speed:

The mean orbital speed decreases with eccentricity.

See also

  • 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,...
  • Delta-v budget
    Delta-v budget

    Delta-v budget is a term used in astrodynamics and aerospace industry for velocity change requirements for the various Spacecraft propulsion tasks and orbital maneuvers over phases of the space mission....
  • Hohmann transfer orbit
    Hohmann transfer orbit

    In orbital mechanics, the Hohmann transfer orbit is an orbital maneuver using two engine impulses which, under Standard assumptions in astrodynamics, move a spacecraft between two coplanar circular orbits....
  • Bi-elliptic transfer
    Bi-elliptic transfer

    In astronautics and aerospace engineering, the bi-elliptic transfer is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer orbit....