[Supervolant (OP)]

Hello everyone,
Yesterday I thought about the giants of our past and with that of Isaac Newton. All I knew about him was that he found out about gravity by seeing an apple falling down a tree next to him and he decided to lay down some laws pretty important to modern physics. But that's about it, so I decided to lay down in bed and watch an easy to comprehend documentary on youtube about him and his life. It was well made and interesting, sometimes a bit spooky but after watching it I now have a fair amount of respect for him and his work, even though he was, as I mentioned before a bit spooky!
Watch it yourself here:

I was immensely impressed by the thought experiment he created. A cannon shot it's cannonball from a mountain. The ball, as usual, will fall down after a few meters. Shoot it faster and it will go further. Shot faster and faster and faster, it might go around half the world and then touching down. When shot fast enough and with the right amount of precision it will go around the world and turn into an orbit, which is nothing else then a balance act between speed and gravitation. By the way the same principle lays behind the International Space Station's orbit. Pretty cool explanation right?

After watching it or somewhere near the end of the documentary (I don't know) I fell to sleep and beside some weird short dreams there was this one which really struck my thoughtprocess the whole day after...

What I saw was two planets next to each other, pretty close. These two planets could only be so close because of a rotation of these two planets in a chain providing the centrifugal force high enough so the gravitation of the two planets won't make the planets collide.
Suddenly I saw Newtons thought experiment happening again. The cannon shot and the ball started doing it's orbit around the first planet. After some time it went between the two planets moving directly into the most extreme proximity midpoint of these planets. The cannonball "decided" for the other planet and it's gravitation making the turn now go around that planet. The procedure repeated and went for the first planet again. All this, it seemed to me, went literally to infinity! Seeing this thought experiment in pictures you might notice the pattern the cannonball followed, a laid down figure eight, in modern language and mathematics representing infinity. ∞ This infinity symbol really shined in a bright yellow. I woke up directly afterward and couldn't sleep for a while after... Is this an cosmological perpetuum mobile?

Is what I saw possible? Will it move into infinity?

What does Newton's first law say in this context? Newton's first law of motion is often stated as. An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

The force's involved seem pretty balanced in the case I described!

Centrifugal force is clearly on my side. The cannonball decides for the other planet because its centrifugal force is pointing for the other planet and it's gravitational field.
On the other hand, centripetal force is not! This force would like to shoot the cannonball straight away after passing the two planets, right?

I just don't know and would love to hear your thoughts about this, afterwards (hopefully) understanding the universe a tiny bit better!

Hope I can stimulate your thinking,
- Robert

[chris]

What's your question? Can you phrase the thread title as one, please.

[Supervolant (OP)]

How will the cannonball, out of the thought experiment I have described, behave?

[Kryptid]

If I am understanding your setup correctly, you start off with the (correct) assumption that a cannonball at the right speed will remain at a constant altitude above the surface of a planet despite being in constant free fall (assuming frictional losses from the atmosphere are ignored). That is, of course, what we call an orbit. Throughout the course of the cannonball's orbit, the force acting on it will be constant towards the planet's center. With this in mind, we can safely assume that it will stay in orbit. If the planet is uniform in density and the cannonball fired precisely enough, the orbit can be very nearly circular.

Things change when you add a second planet. We will consider that this new planet is placed at a distance such that the cannonball's original orbit will bring it midway between the two planets at some point. The cannonball orbits Planet 1, but the gravitational influence from Planet 2 will have a very large impact on the force that the cannonball experiences as it moves. Before Planet 2 was added, there was an equilibrium between the centripetal force pulling the cannonball towards Planet 1 and the centrifugal force trying to pull it away. Now, consider what happens when this cannonball begins to enter the part of its orbit where it travels between the planets. The amount of force acting to pull the cannonball away from Planet 1 is now higher than it was before. This will bring it closer to Planet 2 than it was to Planet 1, bring it out of Planet 1's Hill sphere: https://en.wikipedia.org/wiki/Hill_sphere

I don't believe this will result in a stable orbit around Planet 2, however. Based on my intuition, the resulting orbit around Planet 2 will be eccentric (oval-shaped). If the cannonball manages to survive the first orbit around Planet 2 without either colliding with it or being thrown out of orbit, its orbit will be even more greatly perturbed by its close pass to Planet 1 again. My best guess is that its orbit will eventually become so disturbed that it will be thrown away from the binary planets or collide with one of them. The eternal exchange between the two planets as you describe is probably not possible in reality.

[Supervolant (OP)]

Thanks for your answer Kryptid.

Your explanation of the forces involved make complete sense to me. I have made no thought about the gravity of planet 2 acting on the cannonball even when the furthest away (still orbiting planet 1).

Dream physics are just not real physics! Still, the dream was cool enough...

Again, thanks for clearing things up.

- Robert

[Tomassci]

Theoretically, the dream scenario is possible. In real life, there is that pesky slowing because of air, and that few atoms in universe. Not to mind, if there was third object, it could have been out of the way.