Best way to procedurally generate an orbit given mass and eccentricity

Question

As stated here, I've badly expressed my problem and have made what we call a XY problem (the question was well answered nonetheless) so I restate the question. Sorry for the inconvenience

I have a 2D orbit simulator using Euler-Cromer method for now; What I'm trying to do is to generate a central body with a given mass and a body orbiting this body with the orbit having:

• a procedurally generated eccentricity ranged in $\mathbf{0≤e<1}$
• a procedurally generated semi-major axis

What I'm trying to achieve is to get the initial velocity for the simulator to make any orbit that correspond this mass, eccentricity and semi-major axis - no matter the plane of the orbit nor direction. For simplification, let's state everything appends on the same plane and goes clockwise.

So far, thanks to @ConnorGarcia 's answer, I may guess that I can:

• use the eccentricity to find $\mathbf{c}$ - the distance between the focal and the center of the ellipse - like that: $\mathbf{c = a \cdot e}$
• place the body at any random distance to the orbited body equal to the distance between one focal and apogee using $\mathbf{a + c}$
• use the vis-viva equation to have speed, and
• set velocity vector using speed as magnitude and making its direction perpendicular to the semi-major axis.

Does it feels right? If I was building this generator in 3D, should I just have to add the plane to make it correct?

Answer 1

ANSWER

I would put the massive object at the origin. Then, I would calculate the velocity at periapsis, when $r=a(1-e)$ as $$v = \sqrt{GM\frac{1}{a}\left(\frac{1+e}{1-e}\right)}$$ (stolen from Uhoh's answer to Calculating object velocity at perihelion ).

Put the periapsis on the positive y axis. Then the velocity vector at periapsis is $\vec{v}=[v,0]$. If you want the periapsis at a counter-clockwise angle $\omega$ from the positive y-axis, then multiply by your standard angle rotation matrix.

EXAMPLE

Let the mass of the central body be twice a solar mass $m=2M_{\odot}$ with a planet with semi-major axis $a=1$ AU and highly eccentric with $e=0.5$. Then periapsis is at $r=0.5$ AU.

Here, the gravitational constant is $$G \approx 887.352 \frac{AU}{M_{\odot}}(km/s)^2$$ Plugging these values into Uhoh's equation for velocity above, we get $|\vec{v}| \approx 72.97$ km/s. So, when the orbiting body's position at periapsis is at $\vec{p} = [0,0.5]$ AU, its velocity is $\vec{v} = [72.97,0]$ km/s.

Want to do this in 3D?

Then $\vec{p} = [0,0.5,0]$, $\vec{v} = [72.97,0,0]$, with an argument of periapsis $\omega$, inclination $i$, and longitude of ascending node $\Omega$, then we can calculate new position and velocity vectors as $$\vec{p}' = R_z(\Omega)R_x(i)R_z(\omega)\vec{p}$$ and $$\vec{v}' = R_z(\Omega)R_x(i)R_z(\omega)\vec{v}$$ where $R_x(\theta)$ and $R_z(\theta)$ are the 3D rotation matrices around the x and z axes respectively. Note that if $i=\omega=\Omega=0$ then all of the rotation matrices are the identity matrix so no rotations take place.