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Orbital state vectors
 
 
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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 dynamics orbital state vectors (sometimes state vectors) are vectors of position
Position

Position may refer to:* A location in a coordinate system, usually in two or more dimensions; the science of position and its generalizations is topology...
  and velocity
Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
  that together with their time (epoch
Epoch (astronomy)

In astronomy, an epoch is a moment in time used as a reference for the orbital elements of a celestial body. Typically, the epoch is either the moment an observation was made or the moment for which a prediction was calculated....
) uniquely determine the state of an orbiting body.

State vectors are excellent for pre-launch orbital predictions when combined with time (epoch) expressed as an offset to the launch time. This makes the state vectors time-independent and good general prediction for 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....
.


state vectors must be considered in a particular inertial frame of reference
Inertial frame of reference

In physics, an inertial frame of reference is a frame of reference, tied to the state of motion of an Observer , with the property that each physical law portrays itself in the same form in every inertial frame....
 setting.






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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 dynamics orbital state vectors (sometimes state vectors) are vectors of position
Position

Position may refer to:* A location in a coordinate system, usually in two or more dimensions; the science of position and its generalizations is topology...
  and velocity
Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
  that together with their time (epoch
Epoch (astronomy)

In astronomy, an epoch is a moment in time used as a reference for the orbital elements of a celestial body. Typically, the epoch is either the moment an observation was made or the moment for which a prediction was calculated....
) uniquely determine the state of an orbiting body.

State vectors are excellent for pre-launch orbital predictions when combined with time (epoch) expressed as an offset to the launch time. This makes the state vectors time-independent and good general prediction for 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....
.


Frame of reference

The state vectors must be considered in a particular inertial frame of reference
Inertial frame of reference

In physics, an inertial frame of reference is a frame of reference, tied to the state of motion of an Observer , with the property that each physical law portrays itself in the same form in every inertial frame....
 setting. For most practical applications in astrodynamics this is usually assumed to have the following properties:
  • cartesian right-handed coordinate system
    Cartesian coordinate system

    In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
    :
    • with x-axis pointing to vernal equinox,
    • with z-axis pointing upwards, meaning the x-y plane is the reference plane.


Position vector

The orbital position vector is a cartesian vector describing the position of the orbiting body in Frame of reference. Together, the orbital position vector and orbital velocity vector describe uniquely the state of an orbiting body and thus are called Orbital state vectors
Orbital state vectors

In astrodynamics or celestial dynamics orbital state vectors are vectors of position and velocity that together with their time uniquely determine the state of an orbiting body....
.

Velocity vector

Orbital velocity vector is a cartesian vector describing velocity of the orbiting body in Frame of reference. Orbital velocity vector together with orbital position vector describe uniquely state of the orbiting body and thus are called Orbital state vectors
Orbital state vectors

In astrodynamics or celestial dynamics orbital state vectors are vectors of position and velocity that together with their time uniquely determine the state of an orbiting body....
.

For any object moving through space, the velocity vector is tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 to the trajectory. If is the unit vector
Unit vector

In mathematics, a unit vector in a normed vector space is a Vector space whose Norm is 1 . A unit vector is often denoted by a lowercase letter with a superscribed caret or ?hat?, like this: ....
 tangent to the trajectory, then

Derivation

Orbital velocity vector can be derived from orbital position vector by differentiation with respect to time:

Relation to orbital elements

Orbital state vectors are equivalent to orbital elements
Orbital elements

In celestial mechanics, the elements of an orbit are the parameters needed to specify that orbit uniquely. Orbital elements are generally considered in classical mechanics two-body systems, where a Kepler orbit is used ....
 (Keplerian elements) and each can be computed with each other (and used to derive other parameters of the 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....
).

Both state vectors and orbital elements have unique advantages over the other. Computed in advance state vectors are more useful for orbital prediction. A time-independent state vector can be combined with the launch time using xxx method in order to arrive at a valid set of orbital elements whereas computed in advance orbital elements are valid only when launch occurs without the slip.

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....
 Orbital state vectors
Orbital state vectors

In astrodynamics or celestial dynamics orbital state vectors are vectors of position and velocity that together with their time uniquely determine the state of an orbiting body....
 ( and ) are used with the help of following auxiliary vector:
  • specific relative angular momentum
    Specific relative angular momentum

    In astrodynamics, the specific relative angular momentum of an orbiting body with respect to a central body is the relative angular momentum of the first body per Units of measurement mass....
     vector


Orbital state vectors
Orbital state vectors

In astrodynamics or celestial dynamics orbital state vectors are vectors of position and velocity that together with their time uniquely determine the state of an orbiting body....
 can then be used to calculate following orbital elements
Orbital elements

In celestial mechanics, the elements of an orbit are the parameters needed to specify that orbit uniquely. Orbital elements are generally considered in classical mechanics two-body systems, where a Kepler orbit is used ....
 (Keplerian elements) (see their definitions for directions):
  • Inclination
    Inclination

    Inclination in general is the angle between a reference plane and another plane or Axis_of_rotation of direction. The axial tilt is expressed as the angle made by the planet's axis and a line drawn through the planet's center perpendicular to the orbital plane....
     
  • Eccentricity
  • Longitude of ascending node
  • Argument of periapsis
    Argument of periapsis

    The argument of periapsis is the orbital element describing the angle of an orbiting body's apsis , relative to its ascending node . The angle is measured in the orbital plane and in the direction of motion....
     
  • Mean anomaly
    Mean anomaly

    In celestial mechanics, mean anomaly is one of the orbital elements that defines a Kepler orbit. It specifies the position of the orbiting objects along the ellipse defined by the other elements, but does not correspond to an actual geometric angle....
     
  • 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....
     
together with time ( epoch
Epoch (astronomy)

In astronomy, an epoch is a moment in time used as a reference for the orbital elements of a celestial body. Typically, the epoch is either the moment an observation was made or the moment for which a prediction was calculated....
) those can be used to compute other orbit's parameters:
  • 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 ....
     
  • Semi-major axis
    Semi-major axis

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

    In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis....
     
  • Linear eccentricity
    Eccentricity (mathematics)

    In mathematics, the eccentricity, denoted e or , is a parameter associated with every Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular....
     
  • Periapsis distance
  • Apoapsis distance
  • Eccentric anomaly
    Eccentric anomaly

    The definition of eccentric anomaly for an ellipse as a geometric figure directly applies for an elliptic Kepler orbit. The definitions of the true anomaly and the eccentric anomaly for an ellipse and the relations between these entities are all in Ellipse#True anomaly and Ellipse#Eccentric anomaly....
     
  • Mean longitude
    Mean longitude

    In astrodynamics or celestial dynamics mean longitude is the longitude at which an orbiting body could be found if its orbit were circular orbit and its inclination were zero....
     
  • True longitude
    True longitude

    In astrodynamics true longitude is the longitude at which an orbiting body could actually be found if its inclination were zero. Together with the inclination and the ascending node, the true longitude can tell us the precise direction from the central object at which the body would be located at a particular time....
     


Keplerian elements typically define an osculating orbit
Osculating orbit

In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space is the gravitational Kepler orbit that it would have about its central body if perturbations were not present....
 because of perturbations in the orbital path. The osculating orbit is valid only at the epoch of the original Cartesian elements.