Shell theorem

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In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.

Isaac Newton proved the shell theorem[1] and stated that:

  1. A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre.
  2. If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object’s location within the shell.

A corollary is that inside a solid sphere of constant density, the gravitational force varies linearly with distance from the centre, becoming zero by symmetry at the centre of mass. This can be seen as follows: take a point within such a sphere, at a distance r {\displaystyle r} r from the centre of the sphere. Then you can ignore all the shells of greater radius, according to the shell theorem. So, the remaining mass  m is proportional to r^{3}, and the gravitational force exerted on it is proportional to m/r^2, so to  r^3/r^2 =r, so is linear in r  r.

These results were important to Newton’s analysis of planetary motion; they are not immediately obvious, but they can be proven with calculus. (Alternatively, Gauss’s law for gravity offers a much simpler way to prove the same results.)

In addition to gravity, the shell theorem can also be used to describe the electric field generated by a static spherically symmetric charge density, or similarly for any other phenomenon that follows an inverse square law. The derivations below focus on gravity, but the results can easily be generalized to the electrostatic force. Moreover, the results can be generalized to the case of general ellipsoidal bodies.[2]

 

The addition of the volume to the shell is how I built my toy-model. It connects heat flow and gravity exactly. The bold text is an analogy of the heat current and the temperature gradient. The gravitational force acts  exactly  balanced towards the source of heat flow, which is the source of gravity. The force declines at a rate equal to the emissive power of heat. Gravitational energy seems to be exactly equal but acts in opposite direction to heat, which flows towards the ultimate heat sink in the vacuum.

If units of gravity are dimensioned as heat flow in a volume, their behaviour is identical.

Gravity then doesn´t depend on mass, it is equal to heat flow. Their center is ironically the same.

  1. ”If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object’s location within the shell.”

I replace ”no net gravitational force” with gravity and heat flow =0 in a spherical non interacting cavity. It also happens to be mass in the cavity, but it seems to make no difference.

I just fill out the unknown with the simplest ideal model of thermodynamic heat engines.

I added nothing, I used only geometry and proven thermodynamic principles. I introduce nothing new. It is built entirely on observation. The model produced solutions from the start.

Annonser