I have drones in my game that need to approach and orbit a node and shoot at it. Problem is I want to stay away from a real physics simulation, meaning I don't want to give the node and drone a mass and the drone's thrusters' a force. I just want to find the best way to approach and then enter orbit. There was a pretty good answer about using bezier curves and doing it that way, but that is essentially a tween between two fixed points. The nodes are also moving as the drones enter orbit.
Remember that, at least in terms of gravitational pull (and not, say, collision), mass is only necessary to determine gravitational strength, and that is only necessary to determine acceleration of other objects toward the mass. And force is only necessary to determine acceleration, which is already known/calculated in this situation.
Thus, it's fairly straightforward to "fake" physics without doing a complicated simulation. You only need to alter acceleration of the drone to "fake" the gravity of the node.
The drone's acceleration (that is self-applied) should be altered by an acceleration vector before being applied to the velocity/position according to the direction of the node and the distance to the node. The direction of the acceleration vector is the direction of the node, and the magnitude is the inverse of the distance. This would simulate the gravitational pull of the node, as drones would be pulled faster towards nodes the closer they are.
The drone must, of course, change its orientation (and thus, acceleration) towards the orbit it wishes to establish (but this is a tangential AI concern). However, if no orbit was desired, this system would even fake the gravitational pull nodes would have on drones that are trying to fly away from them.
The specific algorithm for the acceleration depends on the implementation specifics, such as the drone accelerations and velocities. Testing with a simple run-time input can allow you to quickly determine the acceleration needed to get the right "feel".
This would even result in an eventual circular orbit, as long as the drone continually accelerates itself in the direction of the orbit's tangent, and the node continually "pulls" it inward.