Orbital state vectors

Orbital state vectors - in astrodynamics and celestial mechanics, Cartesian position vectors ( $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ $\ mathbf {r}$ ) and speed ( $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ $\ mathbf {v}$ ), together with a moment in time ( era ) ( $t\,$ ${\ displaystyle t \,}$ $t \,$ ) uniquely determine the trajectory of the circulating body.

Position vectors in orbit (indicated by blue), velocity (indicated by green) and other elements of the orbit

Content

1 reference system
2 Position vector
- 2.1 Conclusion
3 notes

Reference system

State vectors are determined relative to some reference frame , usually inertial . One of the most used reference systems for bodies moving near the Earth is the geocentric equatorial reference system, defined as follows:

the origin is at the center of mass of the earth;
the Z axis coincides with the axis of rotation of the Earth, the positive coordinates correspond to the northern hemisphere;
the X / Y plane coincides with the plane of the earth's equator, the X axis is directed to the vernal equinox , the Y axis forms the right triple of vectors with the X and Z axes.

This reference frame is actually not inertial due to the slow precession of the axis of rotation of the Earth, so the reference system is usually determined by the position of the Earth in a standard astronomical era, such as B1950 or J2000.

You can use other reference systems for various tasks, for example, heliocentric or planetocentric, reference systems with a beginning in the barycenter of the solar system, reference systems with the orbital plane of the spacecraft as a reference plane.

Position Vector

Position Vector $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ describes the position of the body in the selected reference frame, the velocity vector $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ shows the speed in the same reference frame at the same time. These vectors, together with the point in time at which they are indicated, describe the trajectory of the body.

The body does not have to be in orbit so that state vectors describe the trajectory of its movement: the body should move only under the influence of inertia and gravity. For example, a spacecraft on a suborbital trajectory can be considered as a body. If other forces are significant, they can be vectorially added to the force of attraction during integration to determine position and speed in the future.

For any object moving in outer space, the velocity vector will be tangent to the trajectory. If ${\hat {\mathbf {u} }}_{t}$ ${\ displaystyle {\ hat {\ mathbf {u}}} _ {t}}$ is the unit vector tangent to the trajectory, then

\mathbf {v} =v{\hat {\mathbf {u} }}_{t}.

{\ displaystyle \ mathbf {v} = v {\ hat {\ mathbf {u}}} _ {t}.}

Conclusion

Speed vector $\mathbf {v} \,$ ${\ displaystyle \ mathbf {v} \,}$ can be obtained from the position vector $\mathbf {r} \,$ ${\ displaystyle \ mathbf {r} \,}$ with time differentiation:

\mathbf {v} ={d\mathbf {r}  \over {dt}}.

{\ displaystyle \ mathbf {v} = {d \ mathbf {r} \ over {dt}}.}

The position vector of an object can be used to calculate classical or Kepler orbital elements. Keplerian elements show the size, shape and orientation of the orbit and can be used to quickly calculate the state of an object at a given point in time if the movement of the object is exactly simulated by a two-body problem with very small perturbations.

On the other hand, state vectors are more useful for numerical integration , which takes into account significant additional forms of forces that vary with time and gravitational perturbations from other bodies.

State vectors ( $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ and $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ ) can be used in calculating the angular momentum vector in the form $\mathbf {h} =\mathbf {r} \times \mathbf {v}$ ${\ displaystyle \ mathbf {h} = \ mathbf {r} \ times \ mathbf {v}}$ .

Since even satellites in low Earth orbit experience significant disturbances (especially due to the non-sphericity of the Earth), Keplerian elements calculated by the state vector at a given time are valid only for this moment in time. Such elements are called osculating .

Orbital state vectors

Content

Reference system

Position Vector

Conclusion

Notes

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