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The motion of a rigid body in orbit or near a massive object

Published by Erik Dahlén at Shakespir.

Copyright Erik Dahlén.

This copy is for your private use only, no distribution is allowed without consent from the author.

**Abstract**

p<>{color:#000;}. [Aim] To investigate whether a rigid body in motion near a massive object can be seen as a point mass or if the motion most be calculated for each part of the rigid body independently.

[Method] Investigating known equations for bodies in motion around massive objects.

[Result] Each part of a rigid body in motion near a massive object, e.g. a rigid body in orbit, will move in its own path and hence can the rigid body not be seen as a point mass. Therefore a non-rotating rigid body in orbit will show the same side toward the central mass all the time, contrary to the common belief [Ref 1].

TODAY most people believe that the Moon rotates around its own axis at the same rate as it revolves around Earth, [Ref 1]. This idea is based on the assumption that a rigid body will orbit as one object and that this object will orbit according to Newtonian motions. Below will a logic argument explain why this assumption cannot be made.

The Newtonian theory claims that a non-rotating object will orbit a central mass according to figure 1, where the triangle is a rigid object in orbit and the circle is the central mass and the mass of the triangle is negligible compare to the mass of the central object.

Figure 1. The blue circle is the central mass, the black triangle is the rigid body in orbit and the black arrow is the way of orbit.

In order to find out if this Newtonian view of rigid bodies in orbit is a good one, the first thing to look at is how non-rigid bodies orbit. In figure 2 there is 6 particles inside a gas cloud, the particles orbits the blue central mass and the mass of the gas cloud is negligible compare to the central mass.

Figure 2. The big blue circle is the central mass, the 6 small black circles with numbers are bodies in orbit and the black arrow is the way of orbit.

If these are 6 small rigid bodies they will still orbit in the same constellation. The reason for this is that every object will orbit in its own orbit, a little faster if it is closer to the central mass. In a non-relativistic system equation 1 is a good approximation, [Ref 2], for the speed of an object in orbit, where v is the speed, G the gravitational constant, M the mass of the central object and r the distance from the central mass and the object. A higher r will give a lower v.

If the 6 particles are glued together by rods, they will now become one object and not 6 individual objects. The main question now is how this one object will orbit. According to the Newtonian theory explaining figure 1, it should orbit like figure 3.

Figure 3. The big blue circle is the central mass, the 6 small black circles with numbers and rods between them is a body in orbit and the black arrow is the way of orbit. Orbiting as if the object were a point mass.

This will not happen, since every part will feel the gravitational force individually. If the object were to orbit in the orbital speed for part 3 and 4, part 5 and 6 would orbit in a too low speed for their altitude and they would try to fall into a lower orbit to gain kinetical energy from potential energy so they could start to orbit in another lower orbit, which would be an eccentric orbit. In the same time part 1 and 2 would orbit in a too high speed for their altitude and they would try to fall into a higher orbit to loss kinetical energy to potential energy so they could start to orbit in another higher orbit, which would be an eccentric orbit. Figure 4 shows how they would really orbit, if every part were to orbit in its own orbit.

Figure 4. The big blue circle is the central mass, the 6 small black circles with numbers and rods between them is a body in orbit and the black arrow is the way of orbit. Orbiting as each part were to orbit the central mass in its own path.

Figure 5 shows which orbits every part wants to orbit in, if they were to orbit in the orbital speed of part 3 and 4. The illustration is exaggerated and the perihelion precession is neglected.

Figure 5. The big blue circle is the central mass, the 6 small black circles with numbers and rods between them is a body in orbit and the 3 black transparent circles are the path of the individual part of the object in orbit. The illustration is an exaggeration and perihelion precession has been neglected.

The reason for each part not to orbit in its own elliptical orbit is of course that it is a rigid body and the force to keep the object together is stronger that the difference is gravity for each part, otherwise gravity would rip the object apart, which can happen close to black holes. But even if each part cannot follow its own elliptical orbit it will want to do so, so it will stay in its place and it will not move to another altitude which is suggested in the Newtonian view. For all massive objects with a shape, i.e. not a point mass, half of the objects mass will try to move into a higher orbit and half of its mass will try to move into a lower object, resulting in the object to orbit in the orbit for the center of mass, but the parts at lower orbit will not move into a higher orbit since they try to get to an even lower orbit and not to a higher orbit. Hence is the Newtonian view from [Ref 1] wrong.

To demonstrate that the difference of orbital speed, or the gravitational force, between two points in a small object in orbit around a large massive object is not negligible an example will be calculated.

For a man made satellite orbiting in geostationary orbit its distance to the center of earth is 4.2 * 10^7 m [Ref 2], and with the mass of Earth to 5.9722*10^24 kg [Ref 3] and the gravitational constant at 6.6741*10^-11 m^3*kg^-1*s^-2 [Ref 4]. A satellite at this orbit will orbit in a speed of about 3080 m/s, from equation 1. If this satellite were a sphere of radius 1 m, the part which is 1 m closer to earth will move in the same speed as the rest of the satellite, since it is a rigid body, but the orbital speed at 1 m closer to earth is slightly higher than the speed of the satellite. Equation 2 gives the difference in orbital speed between two different altitudes. Using the same G, gravitational constant, and M, mass of Earth, and change r with 1 m for r2 the difference between the two orbital speeds are 0.037 m/s or 0.13 m/hour, for an object orbiting Earth with an period of almost 24 h and a radius of 1 m the difference of speeds of 0.13 m/h is not negligible. So the part closes to Earth of this satellite will therefore move 0.13 m/h too slow for its altitude and hence want to move closer to Earth, and not further away from Earth which is needed if it will orbit as in figure 1.

A more visual way to illustrate that something is wrong with today’s view on how objects orbit around massive objects is to think of an object being built up. First think of a non-rotating object in orbit, as the left part of figure 6, then think of a two objects orbiting one after the other, as the right part of figure 6. If the two objects orbiting were to get closer to each other they will still orbiting one after the other, but what will happen if they were to get connected, how would they then orbit, as the left part of figure 7 or the right part of figure 7.

Figure 6. The big blue circle is the central mass, the 2 or 4 small black circles are rigid bodies in orbit and the black arrow is the way of orbit.

Figure 7. The big blue circle is the central mass, the 2 small black circles are rigid bodies in orbit and the black arrow is the way of orbit.

According to the first part of this article it is obvious that the right part of figure 7 is the correct one, and according to the Newtonian view of figure 1 the left part of figure 7 is the correct one. If the two objects were orbiting one after the other, there is no good reason why they would start to orbit as the left part of figure 7.

A more peculiar situation with the Newtonian view of figure 1 is if one object were to be inside another object, with vacuum between them, as in figure 8. Again the left part of the figure as the Newtonian view of figure 1 and the right part as described in this article. The green small object inside the larger square will start to move in relation to the larger square, if it is vacuum inside the square, according to the Newtonian view from figure 1. In the view presented in this article the small green object inside the larger square would stay put.

Figure 8. The big blue circle is the central mass, the small black square is rigid body in orbit with a small body, the green circle, inside itself and the black arrow is the way of orbit.

This article presented good arguments that the old Newtonian view in figure 1 is a wrong way to view non-rotating rigid bodies in orbit. Hence and alternative theory most be presented, regardless of the theory presented here is the correct one, a new is needed.

The new theory presented in this article claims that each part of an object is orbiting in its own path and not in the path of the central mass of the object, but since it is a rigid body it cannot fall into a lower or higher orbit and therefore the part of the object will remain in its place.

Some ideas and pictures for this article has been taken from [Ref 5].

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p<>{color:#000;}. *https://solarsystem.nasa.gov/planets/moon/indepth* *[2017-06-02]*

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p<>{color:#000;}. [_ https://en.wikipedia.org/wiki/Orbital_speed [2017-06-02]_]

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p<>{color:#000;}. [_ https://en.wikipedia.org/wiki/Earth_mass [2017-06-02]_]

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p<>{color:#000;}. [_ https://en.wikipedia.org/wiki/Gravitational_constant [2017-06-02]_]

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p<>{color:#000;}. *The Rotation Of The Moon, E. Dahlen, 2016, Shakespir*

- Author: Erik DahlÃ©n
- Published: 2017-07-25 15:50:09
- Words: 1826