Inastrodynamics andcelestial dynamics, theorbital state vectors (sometimesstate vectors) of anorbit areCartesian vectors ofposition (${\displaystyle \mathbf{r}}$) andvelocity (${\displaystyle \mathbf{v}}$) that together with their time (epoch) (${\displaystyle t}$) uniquely determine the trajectory of the orbiting body in space.^[1]: 154

Orbital state vectors come in many forms including the traditional Position-Velocity vectors,Two-line element set (TLE), and Vector Covariance Matrix (VCM).

State vectors are defined with respect to someframe of reference, usually but not always aninertial reference frame. One of the more popular reference frames for the state vectors of bodies moving nearEarth is theEarth-centered inertial (ECI) system defined as follows:^[1]: 23

• Theorigin is Earth'scenter 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 thevernal equinox and the Y axis completing a right-handed set.

The ECI reference frame is not truly inertial because of the slow, 26,000 yearprecession of Earth's axis, so the reference frames defined by Earth's orientation at a standardastronomical epoch such as B1950 or J2000 are also commonly used.^[2]: 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 thebarycenter and total angular momentum of theSolar System (in particular theICRF), or even a spacecraft's own orbital plane and angular momentum.

Theposition vector${\displaystyle \mathbf{r}}$ describes the position of the body in the chosenframe of reference, while thevelocity vector${\displaystyle \mathbf{v}}$ 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 as detailed inOrbit determination. The principal reasoning is that Newton's law of gravitation yields an acceleration${\displaystyle \ddot{\mathbf{r}}=-GM/r^2}$; if the product${\displaystyle GM}$ of gravitational constant and attractive mass at the center of the orbit are known, position and velocity are the initial values for that second order differential equation for${\displaystyle \mathbf{r}(t)}$ which has a unique solution.

The body does not actually have to be in orbit for its state vectors to determine its trajectory; it only has to moveballistically, i.e., solely under the effects of its own inertia and gravity. For example, it could be a spacecraft or missile in asuborbital 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 istangent to the trajectory. If${\displaystyle {\hat{\mathbf{u}}}_t}$ is theunit vector tangent to the trajectory, then$${\displaystyle \mathbf{v}=v{\hat{\mathbf{u}}}_t}$$

The velocity vector${\displaystyle \mathbf{v}}$ can be derived from position vector${\displaystyle \mathbf{r}}$ bydifferentiation with respect to time:$${\displaystyle \mathbf{v}=\frac{d\mathbf{r}}{dt}}$$

An object's state vector can be used to compute its classical or Keplerianorbital 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 thetwo-body problem with only small perturbations.

On the other hand, the state vector is more directly useful in anumerical 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 (${\displaystyle \mathbf{r}}$ and${\displaystyle \mathbf{v}}$) can be easily used to compute thespecific angular momentum vector as

${\displaystyle \mathbf{h}=\mathbf{r}\times \mathbf{v}}$.

Because even satellites in low Earth orbit experience significant perturbations from non-sphericalEarth's figure,solar radiation pressure, lunartide, andatmospheric drag, the Keplerian elements computed from the state vector at any moment are only valid for a short period of time and need to be recomputed often to determine a valid object state. Such element sets are known asosculating elements because they coincide with the actual orbit only at that moment.

• ECEF
• Earth-centered inertial
• Orbital plane
• Orbit determination
• State vector (navigation)
• Radial, transverse, normal

• Orbits
• Vectors (mathematics and physics)

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