Orbital state vectors

Position vector

The position vector r describes the position of the body in the chosen frame of reference, while the velocity vector 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.

The body does not actually have to be in orbit for its state vector 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 u ^ t is the unit vector tangent to the trajectory, then

Derivation

The velocity vector v can be derived from position vector r by differentiation with respect to time:

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 ( r and v ) can be easily used to compute the angular momentum vector as h = r × v .

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 elements because they coincide with the actual orbit only at that moment.