I'm wondering what would be a good geometric method to compute orbital elements that fix the orbit in space, given that one is given the position vector $\vec{r}$ and the velocity vector $\vec{v}$ for a Keplerian orbit.
I know that from vis-visa equation, one gets the semi-major axis a and uses the $\vec{v} \times \vec{h} $, where $\vec{h}$ is the specific angular momentum ($\vec{r}\times\vec{v} $) of the orbiting body, to compute the eccentricity vector $\vec{e}$ using the following equation, where $\mu$ is the standard gravitational parameter.
$$
\vec{e}=\frac{\vec{v} \times \vec{h}}{\mu} – \frac{\vec{r}}{r}
$$
What other elements we need to compute in order to fix the orbit or is it enough ? I'd like to hear if anyone has some ideas with possibly a geometric visualization.
Thank you in advance !
Best Answer
Let me show, first of all, a construction for the radius of curvature of a conic (see here for a proof).
Let $P$ be a point on a conic with foci $S$ and $S'$. The bisectors of the angles formed by lines $PS$ and $PS'$ are the tangent and normal at $P$ (for a parabola these are the bisectors of the angles formed by $PS$ with the parallel to the axis passing through $P$). From $S$ and $S'$ construct the perpendiculars to $PS$ and $PS'$ respectively, intersecting the normal at $N$ and $N'$ (see figure below). The radius of curvature $\rho=PC$ can be then computed from: $$ \tag{1} {1\over \rho}={1\over2}\left({1\over PN}\pm{1\over PN'}\right), $$ where one has to take sign $+$ for an ellipse and $-$ for a hyperbola (in the case of a hyperbola $S$ must be the focus of the branch where $P$ lies). The special case of a parabola can be obtained by letting $S'\to\infty$, that is: $\rho = 2PN$.
Suppose now we have a point mass $P$ subject to gravitational attraction from a fixed point $S$. We know at a certain instant of time the position of $P$, its velocity $v$ (magnitude and direction) and from Newton's law of gravitation we also know its acceleration $a$ (directed towards $S$).
The line through $P$ perpendicular to the velocity is the normal to the orbit at $P$. We can construct the line through $S$ perpendicular to $PS$, intersecting the normal at $N$. And we can construct the normal acceleration $a_n$ by projecting $a$ onto the normal (see figure below).
$a_n$ is connected to the radius of curvature $\rho$ at $P$ by the well-known kinematical relation $$ a_n={v^2\over \rho}, $$ hence we can compute $\rho$ as $$ \rho={v^2\over a_n}. $$ We can then use $(1)$ to find $PN'$: $$ {1\over PN'}=\pm\left({2\over \rho}-{1\over PN}\right), $$ where the sign of $2/\rho-1/PN$ (which is the same as the sign of $2PN-\rho$) determines the shape of the orbit:
For an ellipse or hyperbola we can now construct $N'$ and project it on the reflection of line PS about the normal, to find the second focus S'. For a parabola constructing the directrix is easy and is left to the reader.