Question: You mentioned in a previous post that, theoretically, orbits could have eccentricities of either 0 or 1, but "real" orbits have eccentricities between 0 and 1. Now I can easily imagine a perfectly circular orbit (e = 0), but I'm having a hard time visualizing how an orbit could be a "line" (e = 1). A little help, please? — JK, Württemberg, Austria
Answer: That's a tough question, and I'm afraid I'll need to get a bit technical to answer it. Are you ready for a basic lesson in celestial mechanics? There's hardly any math involved, so don't panic.
The first thing we need to do is imagine a highly unlikely scenario. Visualize two identical Earths with our existing Moon nearby (above animation). Since all celestial objects attract each other with gravity, you might wonder how those two Earths could stay put and not fall toward each other. In fact, they couldn't ... unless they were in orbit around each other, so just go ahead and imagine that's what they're doing. But there's no need to actually visualize that motion.
Turns out the laws of physics allow us to ignore constant motion, and analyze objects as though they really were at rest. It's not unlike the situation where, as you pour yourself a drink aboard a jet airliner, you can ignore the fact that you're moving at 600 mph. Sure, I could have done the animation with the two Earths in circular motion around each other, but you'd get dizzy watching it. And like I said, we're allowed to ignore that constant motion. But if you really want to see that complex gravitational dance, with the background stars added for a frame of reference, go here.
Now one can define the term orbit in many ways, but what works best here is: gravitationally bound motion that repeats over time in a cyclic manner. And that's exactly what's happening in this animation.
Are you with me so far? OK, so consider that gravity always pulls things toward the centers of other things. The center of mass (COM) of the two Earths is represented by the red dot. That's where the Moon is attracted to. I represent the force of gravity with those yellow arrows that pop up in the animation. If those two Earths are identical, and if the Moon starts out exactly on the line perpendicularly bisecting the line between the two Earths, you can indeed have a stable configuration that results in the Moon oscillating back and forth as shown.
We're in the realm of what's known as the Three Body Problem in celestial mechanics. This is a very difficult problem, mathematically, but if you assume there's no other bodies exerting gravity, then this 3-body linear orbit problem actually has tractable solutions. But the initial conditions would have to be perfect.
So there you have it. An "orbit" can be a straight line. Highly improbably, but theoretically possible.