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Standard gravitational parameter

 

 
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Standard gravitational parameter
 
 
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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....
, the standard gravitational parameter of a celestial body is the product of 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....
  and the mass :

The units of the standard gravitational parameter are km3s-2
r standard assumptions in astrodynamics
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,...
 we have: where: and the relevant standard gravitational parameter is that of the larger body.For all 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....
s around a given central body: where: where: See Kepler's third 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."...
.For all parabolic trajectories
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....
  is constant and equal to ;.

For elliptic and hyperbolic orbits is twice the semi-major axis times the absolute value of 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....
.

re:

Then:

Terminology and accuracy
The value for the Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
 is called geocentric gravitational constant and equal to 398 600.441 8 ± 0.000 8 km3s-2.






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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....
, the standard gravitational parameter of a celestial body is the product of 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....
  and the mass :

The units of the standard gravitational parameter are km3s-2

Small body orbiting a central body

Under standard assumptions in astrodynamics
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,...
 we have: where:
  • is the mass 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 the mass of 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 ,...
    ,
and the relevant standard gravitational parameter is that of the larger body.For all 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....
s around a given central body: where:
  • is the orbit radius
    RADIUS

    Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
    ,
  • is the orbital speed
    Orbital speed

    The orbital speed of a body, generally a planet, a natural satellite, an satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body....
    ,
  • is the angular speed,
  • is 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....
    .The last equality has a very simple generalization to 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....
    s:
where:
  • is the semi-major axis
    Semi-major axis

    In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
    .
See Kepler's third 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."...
.For all parabolic trajectories
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....
  is constant and equal to ;.

For elliptic and hyperbolic orbits is twice the semi-major axis times the absolute value of 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....
.

Two bodies orbiting each other


In the more general case where the bodies need not be a large one and a small one, we define:
  • the vector is the position of one body relative to the other
  • , , and in the case of an 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....
    , the semi-major axis
    Semi-major axis

    In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
     , are defined accordingly (hence is the distance)
  • (the sum of the two values)
where:
  • and are the masses of the two bodies.


Then:
  • for 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....
    s
  • for 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....
    s: (with a expressed in AU and T in years, and with M the total mass relative to that of the Sun, we get )
  • for parabolic trajectories
    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....
      is constant and equal to
  • for elliptic and hyperbolic orbits is twice the semi-major axis times the absolute value of 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....
    , where the latter is defined as the total energy of the system divided by the reduced mass
    Reduced mass

    Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the Units_of_measurement of mass, which allows the two-body problem to be solved as if it were a one-body problem....
    .


Terminology and accuracy


The value for the Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
 is called geocentric gravitational constant and equal to 398 600.441 8 ± 0.000 8 km3s-2. Thus the uncertainty is 1 to 500 000 000, much smaller than the uncertainties in and separately (1 to 7000 each).

The value for the Sun
Sun

The Sun , a G V star, is the star at the center of the Solar System. The Earth and other matter orbit the Sun, which by itself accounts for about 98.6% of the Solar System's mass....
 is called heliocentric gravitational constant and equals 1.32712440018 m3s-2.