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Say you have an object that starts at a distance r, from a massive point-mass m that attracts it. This object has an initial velocity v tangential to said distance.

For initial velocity = zero, the trajectory is simple: it's a straight line. For any velocity larger than what is required to stay in orbit, the trajectory will be a circle, parabola, or hyperbola depending on eccentricity

What would be the trajectory of an object who's speed is too low to stay in orbit? How would you mathematically and precisely define said trajectory?

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Have a look at the Wiki article on Kepler orbits, notably the section on the simplified two-body problem.

The article has this useful picture of the different types of orbit followed by an object interacting with a substantial point mass, and provides the corresponding equations of motion.

Note that in general, if the object remains in orbit then it follows an elliptical path (accepting a circle as an ellipse with zero eccentricity) that has the point mass at one focus.

There is no case where the object has insufficient initial speed to remain in orbit - it simply moves to a 'tighter' ellipse.

illustration of Kepler orbits

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Given a point mass, no drag, and any non-zero tangential velocity it will stay in orbit.

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  • $\begingroup$ Right, I see my sign mistake in the eccentricity vector magnitude calculation. Long day, tired brain. Thanks $\endgroup$
    – MrStealYourFrog
    Commented Jan 13, 2022 at 14:55

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