Determine whether A is orbiting B

Main Question or Discussion Point

Hello,

I'm looking for a way to determine wether one object is in a closed elliptical orbit around another based on their mass and state vectors. For instance, if looking at the state vectors of the distant irreguar Jupiter-moon Sinope and of an asteroid passing by Jupiter at the roughly same distance and velocity, it wouldn't be obvious that one was just barely orbiting the planet and the other one wasn't. How do I calculate this from their mass, position and velocity vectors?

Cheers,
Mike

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You can calculate the total mechanical energy (kinetic + potential) of the two-body system:
$$E = \frac{1}{2}m_1|\vec{v_1}|^2 + \frac{1}{2}m_2|\vec{v_2}|^2 - G\frac{m_1m_2}{|\vec{r_2} - \vec{r_1}|}$$ If $E < 0$ then the system is bound, and the two bodies orbits about the center of mass in elliptical trajectories.

Note: sometimes it might be more convenient to solve the problem from the reference frame of the center of mass. In such case the concept of reduced mass can be applied and the two body-problem reduces to one-body problem (total mass of the system $M$ is fixed at the origin and the reduced mass $\mu$ is orbiting around). In such case, the total energy can be calculated as:
$$E = \frac{1}{2} \mu|\vec{v}|^2 - G\frac{M \mu}{|\vec{r}|}$$

Ken G
Gold Member
And note you actually have to use the center of mass frame, or else the total mechanical energy won't mean much.