Given mass, position and eccentricity, is there a way to get initial velocity in a 2-body problem?

Question

I know that the we can get the eccentricity vector like that: $\mathbf{e} = \frac{\mathbf{v} \times \mathbf{h}}{\mu} - \frac{\mathbf{r}}{\|\mathbf{r}\|}$ (source: https://astronomy.stackexchange.com/a/29008/42546 ).

I would like to know if there's a way to get one of the initial velocity vectors $\mathbf{v}$ (I believe there's 2) of the lightest of the bodies, given the mass, the position $\mathbf{r}$ and the eccentricity (not the vector) in a 2-body problem ?

What I'm trying to do is a planetary system procedural generator and I would like, when I put initial velocity for each body, for the eccentricity to be near 0 and never equal or greater than 1 (so every orbit is elliptical); if it's impossible using eccentricity but you see another way to achieve that, I'll take it.

Thank you !

Answer 1

As previously answered by Connor Garcia, the answer is "No, mass, eccentricity, and position are not enough to determine orbital velocity, even given a specific orbital plane, unless the eccentricity is zero."

One way to intuitively envision this: There is a continuum of similar elliptical orbits with the specified eccentricity $e$, that range in semi-major axis length $\frac{|\vec{r}|}{1+e} \le a \le \frac{|\vec{r}|}{1-e}$ and for any one of those, you can always rotate it around the central body is in a position where the point at the end of the radial distance vector is on the boundary of the ellipse.

Orbits of Eccentricity $e = 0.25$ that pass through Point P

The animation shows the range of orbits the chosen eccentricity in a single plane that pass through a chosen point $\mathrm{P}$ at the end of the radial distance vector $\vec{r}$, normalized to have a length of 1 unit.

The central body at $\mathrm{F_1}$ has Standard Gravitational Parameter $\mu = 1$. This allowed the velocity vector $\vec{v}$, to be calculated using the Vis-Viva equation, tangent to the ellipse, and displayed in the prograde (counterclockwise) direction, but the retrograde direction of $-\vec{v}$ would also be a valid value.

(And if you're asking, no, the shape traced out by the range of velocity vectors is not an ellipse)

And for every semimajor axis $a$ such that $\frac{|\vec{r}|}{1+e} \lt a \lt \frac{|\vec{r}|}{1-e}$, two congruent orbits in different rotations exist as valid choices, so adding the semimajor axis and orbital direction to the calculation isn't enough to uniquely specify the orbit in a designated plane.

Answer 2

No, this is an underdetermined problem if you don't have the semi-major axis $a$ and orbital inclination $i$ (or some equivalent).

However, if you want circular orbits, then the semi-major axis $a$ is the same as the distance between the bodies.

If you want prograde orbits in the x-y plane then the orbital inclination $i$ is 0.

Then you can calculate the velocity vector $\vec{v}$ in 2D on the x-y plane with magnitude derived from the vis-viva equation, perpendicular to the vector from the large to the small body, along the same direction as the large body rotation.