What are the requirements for $ɣ$ shape flyby?

Question

Understanding Kepler's laws, a body with speed exceeding escape speed of a major body will move in a hyperbolic trajectory, entry and exit vector, a "less than ellipsis" curve. You don't get closed (self-intersecting) trajectories other than ellipses.

Am I correct that an inert body will not naturally pass by a major body along a route shaped like the Roman gamma letter $ɣ$?

If so, what powered flight prerequisites would the body need to fulfill to achieve exit along such a trajectory (and preferably exit with more speed than it had while entering, and possibly profit from gravitational assist and/or Oberth maneuver)?

Answer 1

Yes, but for the unpowered flight it will take a bit of cheating and it will be a three body orbit. We could argue that any rosetta orbit (a.k.a. rosette orbit, like e.g. Klemperer rosette) makes one such intersecting loop on each flyby:

                                 

                                  ^{Apsidal precession of a rosette pattern orbit (Credit: David Pratt)}

Could this be used to gain momentum with one gravity assist flyby, and not merely maintain orbital equilibrium? Yes, I believe so. Imagine a retrograde gravity assist slingshot from the direction of the assisting body's barycenter (or parallel to it). If the gravity assisting body of which we want to gain additional momentum from during the time of the full u-turn slingshot moves in orbit for more than its own breadth (diameter), then the slingshooting body would have prescribed a gamma-shaped loop, theoretically intersecting at the gravity assisting body's barycenter it orbits around. Hopefully that wouldn't be in the middle of another body, but in theory it would work. If loop entering and exiting vectors are only slightly shallower than parallel to the imaginary line between the gravity assisting body and barycenter it orbits around, then you should gain a small portion of this body's orbital momentum.

For powered flight performing Oberth maneuver, this is somewhat easier. The vehicle doing such gravity assisting flyby should enter into steep trajectory that would technically result in orbital precession and eventually hit the assisting body, and delay the Oberth burn for a bit longer, then using its own additional thrust to regain momentum and escape the gravitational grip of the assisting body. With such maneuver, it wouldn't matter if it's prograde or retrograde, but it would be crucial that the thrusters work and fire exactly when they need to. You also wouldn't achieve maximum efficiency of such a maneuver, but it is possible to exit the flyby faster than you entered it.

Answer 2

Can you make a loop? Yes and no. It depends on your point of view.

The only inertial trajectories possible are an ellipse, parabola, and hyperbola. None of those can make a loop like a $\gamma$. (An ellipse closes, but does not "cross" itself.) From the point of view of the gravitational body.

If that body is orbiting another, e.g. our Moon about the Earth, then from the point of view of Earth, the trajectory of an object about the Moon can cross itself and look like a $\gamma$ for a hyperbolic trajectory, or look like curly cues for an elliptical orbit.

Similarly, the only way for an object to leave a body with higher hyperbolic velocity than it arrives with is to measure that velocity in a frame of reference in which the body is moving. In the frame of reference of the body, the departing velocity always has exactly the same magnitude as the arriving velocity -- just different directions.

As for powered flight, with sufficient thrust you can, of course, have the object follow any trajectory you like.