View source for Orbital state vectors

[[Image:Orbital state vectors.png|thumbnail|300px|Orbital position vector, orbital velocity vector, other [[orbital element]]s]]

In [[astrodynamics]] and [[celestial dynamics]], the '''orbital state vectors''' (sometimes '''state vectors''') of an [[orbit]] are
[[cartesian coordinate system|Cartesian]] vectors of [[position (vector)|position]] (<math>\mathbf{r}</math>) and [[velocity]] (<math>\mathbf{v}</math>) that together with their time ([[epoch (astronomy)|epoch]]) (<math>t</math>) uniquely determine the trajectory of the orbiting body in space.<ref name=om>{{cite book|url=http://www.nssc.ac.cn/wxzygx/weixin/201607/P020160718380095698873.pdf|title=Orbital Mechanics for Engineering Students |author=Howard Curtis |publisher=[[Elsevier]]|location=Embry-Riddle Aeronautical University Daytona Beach, Florida|isbn=0 7506 6169 0|date=2005-01-10}}</ref>{{rp|154}}

==Frame of reference==
State vectors are defined with respect to some [[frame of reference]], usually but not always an [[inertial reference frame]]. One of the more popular reference frames for the state vectors of bodies moving near [[Earth]] is the [[Earth-centered inertial|Earth-centered equatorial]] system defined as follows:<ref name=om/>{{rp|23}}

*The [[origin (geometry)|origin]] is Earth's [[center of mass]];
*The Z axis is coincident with Earth's rotational axis, positive northward;
*The X/Y plane coincides with Earth's equatorial plane, with the +X axis pointing toward the [[vernal equinox]] and the Y axis completing a right-handed set.

This reference frame is not truly inertial because of the slow, 26,000 year [[axial precession (astronomy)|precession of Earth's axis]], so the reference frames defined by Earth's orientation at a standard [[epoch (astronomy)|astronomical epoch]] such as B1950 or J2000 are also commonly used.<ref>{{cite book|doi=10.1007/978-3-662-50367-6_2|chapter=Coordinate and Time Systems|year=2016|title=GPS Theory, Algorithms and Applications|last1=Xu|first1=Guochang|last2=Xu|first2=Yan|isbn=978-3-662-50365-2|url=https://www.springer.com/cda/content/document/cda_downloaddocument/9783662503652-c2.pdf?SGWID=0-0-45-1581503-p179955995}}</ref>{{rp|24}}

Many other reference frames can be used to meet various application requirements, including those centered on the Sun or on other planets or moons, the one defined by the [[barycenter]] and total angular momentum of the solar system, or even a spacecraft's own orbital plane and angular momentum.

==Position and velocity vectors==<!--linked from [[Semi-major axis]]-->
The '''position vector''' <math>\mathbf{r}</math> describes the position of the body in the chosen [[#Frame of reference|frame of reference]], while the
'''velocity vector''' <math>\mathbf{v}</math> describes its velocity in the same frame at the same time. Together, these two vectors and the time at which they are valid uniquely describe the body's trajectory.

The body does not actually have to be in orbit for its state vectors to determine its trajectory; it only has to move [[ballistically]], i.e., solely under the effects of its own inertia and gravity. For example, it could be a spacecraft or missile in a [[suborbital]] trajectory. If other forces such as drag or thrust are significant, they must be added vectorially to those of gravity when performing the integration to determine future position and velocity.

For any object moving through space, the velocity vector is [[tangent]] to the trajectory. If <math> \hat{\mathbf{u}}_t</math> is the [[unit vector]] tangent to the trajectory, then
:<math>\mathbf{v} = v\hat{\mathbf{u}}_t</math>

===Derivation===
The '''velocity vector''' <math>\mathbf{v}\,</math> can be derived from '''position vector''' <math>\mathbf{r}\,</math> by [[Differentiation (mathematics)|differentiation]] with respect to time:

:<math>\mathbf{v} = {d\mathbf{r}\over{dt}}</math>

An object's state vector can be used to compute its classical or Keplerian [[orbital elements]] and vice versa. Each representation has its advantages. The elements are more descriptive of the size, shape and orientation of an orbit, and may be used to quickly and easily estimate the object's state at any arbitrary time provided its motion is accurately modeled by the [[two-body problem]] with only small perturbations.

On the other hand, the state vector is more directly useful in a [[numerical integration]] that accounts for significant, arbitrary, time-varying forces such as drag, thrust and gravitational perturbations from third bodies as well as the gravity of the primary body.

The state vectors (<math>\mathbf{r}</math> and <math>\mathbf{v}</math>) can be easily used to compute the [[angular momentum]] vector as <math>\mathbf{h}=\mathbf{r}\times\mathbf{v}</math>.

Because even satellites in low Earth orbit experience significant perturbations (primarily from Earth's non-spherical shape), the Keplerian elements computed from the state vector at any moment are only valid at that time. Such element sets are known as [[osculating orbit|osculating]] elements because they coincide with the actual orbit only at that moment.

==See also==
* [[ECEF]]
* [[Earth-centered inertial]]
* [[Orbital plane]]
* [[State vector (navigation)]]

==References==
<references />

{{orbits}}

[[Category:Orbits]]
[[Category:Vectors (mathematics and physics)]]