Orbital state vectors

In astrodynamics or celestial dynamics orbital state vectors (sometimes state vectors) are vectors of position ($\mathbf{r}$) and velocity ($\mathbf{v}$) that together with their time (epoch) ($t\,$) 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.

Orbital position vector and orbital velocity vector and other orbit's elements

Frame of reference

The state vectors must be considered in a particular inertial frame of reference setting. For most practical applications in astrodynamics this is usually assumed to have the following properties:

• cartesian right-handed coordinate system:
  • 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 $\mathbf{r}$ 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.

Velocity vector

Orbital velocity vector $\mathbf{v}$ 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.

For any object moving through space, the velocity vector is tangent to the trajectory. If $\hat{\mathbf{u}}_t$ is the unit vector tangent to the trajectory, then

$\mathbf{v} = v\hat{\mathbf{u}}_t$

Derivation

Orbital velocity vector $\mathbf{v}\,$ can be derived from orbital position vector $\mathbf{r}\,$ by differentiation with respect to time:

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

Orbital position is when a planet rotates another planet.

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 orbital state vectors ($\mathbf{r}$ and $\mathbf{v}$) are used with the help of following auxiliary vector:

• specific relative angular momentum vector $\mathbf{h}=\mathbf{r}\times\mathbf{v}$

Orbital state vectors can then be used to calculate following orbital elements (Keplerian elements) (see their definitions for directions):

• Inclination ($i\,$)
• Eccentricity ($e\,$)
• Longitude of ascending node ($\Omega\,$)
• Argument of periapsis ($\omega\,$)
• Mean anomaly ($M\,$)
• Orbital period ($T\,$)

together with time ($t\,$) (epoch) those can be used to compute other orbit's parameters:

• True anomaly ($\nu\,$)
• Semi-major axis ($a\,$)
• Semi-minor axis ($b\,$)
• Beta Angle ($\boldsymbol{\beta}$)
• Linear eccentricity ($\epsilon\,$)
• Periapsis distance ($d_p\,$)
• Apoapsis distance ($d_a\,$)
• Eccentric anomaly ($E\,$)
• Mean longitude ($L\,$)
• True longitude ($l\,$)

Keplerian elements typically define an osculating orbit because of perturbations in the orbital path. The osculating orbit is valid only at the epoch of the original Cartesian elements.