I've found this essay among my old emails (in PDF), and I thought why not publish it here?
On the gravitational effects of third bodies on two specified bodies in space, and their consequences
Abstract:
In a system of many objects or bodies, the movements of two bodies close to each other are determined not only by their own masses, locations and velocities, but also by the masses, locations and velocities of all the other (third) bodies in the system. In some cases the gravitational fields of these third bodies act to increase attraction between the two specified bodies, but in some other cases they act as if to cause some degree of repulsion between the two bodies, like a tidal force. The author argues that this might be the real cause of many flat structures in the Universe, like the rings of Saturn, the planets of the Solar System, or the stars of the Milky Way.
Content:
Gravity is not as simple as calculations with the center of mass. Even in case of three bodies, the resulting problem of calculating their motions turned out to be very difficult, called the three-body-problem [1]. For the same reason, and also because I am not a Physicist (etc), I cannot come up with final calculations on the subject in the meantime. I just have one simplified example as follows: Let A and B be two bodies (actually point masses) of mass m, and their distance be 2s. Let O be the midpoint of the AB segment. Let c be a circle with midpoint O and radius 3s. For the sake of simplicity, our third bodies will be on the circle c. If the circle had a larger radius, our results would be similar, just different in scale. Let C1 and C2 be the intersections of the line AB and the circle c. Let e be a line through O perpendicular to the line AB, and let C3 and C4 be the intersections of the line e and the circle c. Let C1, C2, C3, and C4 also be point masses, or third bodies of mass m, and of course, let the initial velocities of A, B, C1, C2, C3 and C4 all be zero. Here is the figure:
Now we could apply the Newtonian equation of universal gravitation [3] to (approximately) get the forces which would move our bodies or point masses (F=G*m^2/r^2). C1 attracts A by F_A=G*m^2/(2s)^2, while C1 attracts B by only F_B= G*m^2/(4s)^2. As a result, C1 will cause A and B to increase their distance (of course, this will not be realized as A and B also attract each other, but still, their distance will be greater with C1 than without it). C2 will behave the same way as C1 did. However, C3 and C4 are in another position, and they will make A and B to be closer to each other than without C3 and C4. As for other possible bodies, on the circle c or farther, their effect on A and B will be somewhere in-between the repulsive (tidal [2]) effect of C1 and C2, and the attractive effect of C3 and C4. What does this mean? To me, it probably means that three-dimensional structures (with alike sizes in each dimension) tend to collapse, but flat structures tend to be preserved more. This could be the cause of why many structures in the Universe are (somewhat) flat: the Milky Way, the Solar System, and the rings of Saturn. Of course, this is harder to see in three dimensions with all the bodies having initial velocities (in different directions) and different masses, so I may let the evaluation of this to my Physicist readers. (Note that the WikiPedia article on „tidal force” [2] also mentions that the ring system of Saturn is caused by the tidal force, however, another WikiPedia article says that there is still no consensus as to how the rings of Saturn were formed [7].) Why is this important? Because otherwise the flatness of the celestial structures may be explained by other theories, like that which I have watched in a YouTube video by „minutephysics” entitled „Why is the Solar System Flat?” [4]. It says that every three-dimensional cloud of matter has a plane which defines its total angular momentum, which is preserved due to the law of conservation of angular momentum [5]… that's OK... but it also says that the matter not on this plane gets closer to it due to collisions between celestial bodies (maybe coming from different sides of this plane). Here I would argue that no collisions are necessary, but celestial bodies might arrange themselves near a plane for reasons I described above, i. e. the gravity of third bodies (but afterwards, they might collide as well, of course, especially when they enter into the stronger gravitational field of one another). Also, the „collision theory” might not adequately explain why the rings of Saturn are composed of so small objects, or why the asteroid belt [6] is there between the orbits of Mars and Jupiter. Well, I have come up with this idea of mine in 2014 or 2015, and included it in my math blog, and I might also have shared it on UseNet. Since then, I deleted that math blog of mine, and it was offline for about 5 years. Recently I have put some parts of it on the net again, with a different address, and some edition, so you could read my brief mentioning of this topic there [8]. I am not a Physicist, so I think it would be good if this topic were reviewed by some Physicists competent on this area.
References:
[1] Three-body problem, WikiPedia
https://en.wikipedia.org/wiki/Three-body_problem
[2] Tidal force, WikiPedia
https://en.wikipedia.org/wiki/Tidal_force
[3] Newton's law of universal gravitation, WikiPedia
https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
[4] „Why is the Solar System Flat?” by „minutephysics”, YouTube
https://www.youtube.com/watch?v=tmNXKqeUtJM
[5] Conservation of angular momentum, WikiPedia
https://en.wikipedia.org/wiki/Angular_momentum#Conservation_of_angular_momentum
[6] Asteroid belt, WikiPedia
https://en.wikipedia.org/wiki/Asteroid_belt
[7] Rings of Saturn, WikiPedia
https://en.wikipedia.org/wiki/Rings_of_Saturn
[8] Noble challenge: search for an error in Mathematics (my blog post mentioning this idea) from the blog „Recreation in Mathematics by Árpád Fekete” (dead link removed)
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