All this I know. But if a lesser body orbits (falls) around the the gravity of a greater body how do you explain highly eliptical orbits? Is it the original escape velocity, constantly dragged back by gravity? There must be other forces than the gravity of the central body acting upon an eliptically orbiting body? Well, there are, it happens, it's a fact, but I don't get the expanation?
Nice, simple universe with just two bodies in it, right? They're actually both orbiting their common centre of mass, but we'll make one of them a star, and the other a comet so said centre is well within the larger body, so we can say the comet is orbiting the star and ignore all distractions.
The energy of the comet at any point in its orbit is the sum of its potential energy (mass times the total integral of the gravitational force from the 'height' at which it is) and its kinetic energy (mass times the square of the speed at which it is travelling - that's right, speed, not velocity, at this point. But the equation looks the same).
Mass cancels out, so it doesn't matter if we're doing the experiment with a frozen pea or a medium sized planet.
We'll put our object at 50 AU from the star, with a relatively minor lateral velocity; well below the escape velocity for that star at that distance. Then wait a few centuries.
The force on our body is relatively small at that distance, but continuous, and it starts to accelerate – vector of acceleration directly towards the star. The closer it gets, the faster it gets, and the faster it gets faster, as gravity drops off as the square of the distance. Now, if it were a real comet, by now it would be outgassing and complicating the equations, so we'll make the star a black dwarf, producing no heat and with no atmosphere, and no mountains over a mm in height. Furthermore, the approaching object is utterly unaffected by magnetic fields. No eddy braking or solar wind friction.
If our object hits the star, the experiment has a trivial solution, much like the green hand grenades. If, however, it misses, due to its original velocity, by more than a couple of millimetres, the force on it is directly towards the star, while its velocity (constantly changing) becomes lateral to the surface; it rushes past, being dragged around the gravitational centre and its trajectory becomes a continuous curve , until it is flung away, all of that lovely kinetic energy carrying it further and further, the gravitational force getting less and less but dragging speed of of it all the time, the energy sum always constant until it reaches the distance it started from, and its original speed, starts falling again and the whole cycle repeats. If you map the forces, velocities and distances onto a piece of graph paper, the result is a perfect ellipse, with the centre of gravity at one focus and the other one way out in the cold dark where there's nothing much going on. A short, close, very fast bit, and a long, slow, leisurely winter, but exactly the same geometrical shape.
Obviously, our universe has some complicating factors:- the tidal forces would tear it apart, it's orbit's perturbed by the planets kicking around, there's friction from solar wind and its own tail loses it mass. But it's still close enough that we can predict when any given comet will be back (several thousand years {that is, terrestrial years. Exactly one cometary year, evidently} in some cases).
Yes, at first sight it does look as if that second focus should contain something to pull the object back round, doesn't it? But actually, it's just run out of oomph, and, like a dropped superball at the top of its trajectory, falls back down.
