Given
| Parameter |
Symbol |
Units |
| Position Vector |
$\vec{r}$ |
distance |
| Orbital Inclination |
$i$ |
radians |
| Orbital speed at position vector |
$v$ |
distance/time |
| Eccentricity Vector |
$\vec{e}$ |
|
Requirements for these calculations:
- Eccentricity Vector has a positive magnitude
- Eccentricity Vector is
not parallel to Position Vector.
- All angular measurements are in
radians.
1. Calculate the orbital eccentricity $e$ and the position unit vector
$\hat{r}$:
$$e = ||\vec{e}||$$
$$\hat{r} =\frac{ \vec{r}}{||\vec{r}||}$$
2. Calculate the Perifocal Unit Vectors ($\hat{p},\hat{q},\hat{w}$)
The perifocal unit vectors are the orthagonal vectors in a reference frame centered on the body being orbited. They're useful for dealing with orbits using a canonical rotation, in their own plane.
$\hat{p}$ points at the orbital periapsis, and is co-directional with the eccentricity vector. This calculation requires a non-zero eccentricity.
$$\hat{p} = \frac{\vec{e}}{e}$$
$\hat{w}$ is perpendicular to the orbital plane, co-directional with the angular momentum vector. Since $\hat{p}$ and $\hat{r}$ are coplanar unit vectors, and as long as they aren't parallel, their cross product will be perpendicular to the orbital plane, and we can use its dot product with the z-axis unit vector $\hat{k}$ and the orbital inclination to determine the correct direction of $\hat{w}$.
$$\hat{u} = \hat{p} \times \hat{r}$$
$$\hat{w} = \begin{cases}
\hat{u}, & \text{if ( $\hat{u} \cdot \hat{k} \ge 0$ and $i \le \frac{\pi}{2}$) or ( $\hat{u} \cdot \hat{k} \lt 0$ and $i \gt \frac{\pi}{2}$)} \\
-\hat{u},& \text{otherwise}
\end{cases}$$
$\hat{q}$ is in the direction of True Anomaly $\frac{\pi}{2}$, and can be found from the cross product of the other two perifocal unit vectors.
$$\hat{q} = \hat{w} \times \hat{p}$$
3. Calculate true anomaly $\theta$
True anomaly is the angle between the periapsis and the orbital position, measured in the direction of travel around the orbit, and can be determined from the arccos of the dot product of our two unit vectors, using the dot product of $\hat{r}$ and $\hat{q}$ to determine the correct angle.
$$\theta =
\begin{cases}
\arccos\left(\hat{p} \cdot \hat{r}\right), & \text{if $\hat{r} \cdot \hat{q} \ge 0$ } \\
-\arccos\left(\hat{p} \cdot \hat{r}\right), &\text{otherwise}
\end{cases}
$$
4. Calculate the perifocal velocity angle $\psi$
From there, we can determine the angle between the velocity vector and the perifocal vector $\hat{p}$
$$\psi = \theta - \mathrm{atan2}(e \sin \theta, 1 + e \cos \theta) + \frac{\pi}{2}$$
5. Calculate the velocity vector $\vec{v}$
And finally, using the perifocal unit vectors, he perifocal velocity angle, and the speed at the chosen position to find the velocity vector.
$$\vec{v} = v (\hat{p} \cos \psi + \hat{q} \sin \psi )$$
