Ideal Newtonian model of a sphere with uniform density

[quiet]

This thread is simply an application of Newton's gravitational law and of the two corresponding shell theorems. Why have I decided to place the subject in Speculations? Because in this section, if  a debate is initiated, there is more freedom to expose opinions.

In physics we can take the laws and theorems individually. In that way we obtain pieces of information, comparable with the pieces of a puzzle.

Sometimes a problem requires applying laws and / or theorems jointly. This minimizes the amount of free parameters and, for that reason, reveals details that do not appear in individual applications, or in cases that are resolved by applying a less numerous set of laws and / or theorems.

Regarding gravitation, Newton left a law of force and two theorems, called shell theorems. The law of force does not need comment. Let's go to the shell theorems.

- Theorem 1

Shell material perfectly spherical. The density may be uniform, or it may be a function of the radius with spherical symmetry. In both cases the same theorem is viritated, which establishes the following. Let's abbreviate the word shell with the letter C. The gravitational force between C and an object located outside of C, has the same value that we would obtain supposing that all the mass of C is accumulated in the center of C.

Example.

To calculate the force between two shells of the type mentioned, the distance in the denominator of the force formula is equal to the distance between the centers of the shells. That is to say

[math] F= G \ \dfrac{ m_{_1} \ m_{_2} }{r^2}[/math]

- Theorem 2

The net gravitational force, exerted on a point object located inside the spherical shell, is equal to zero. It is fulfilled regardless of the location of the inner object.

- Let's start now the topic that interests us.

* Sphere with uniform density.

* Validity of Newton's gravitational equation and the shell theorems within the body of the sphere.

* All variables are continuous. That is, we can derive and integrate.

Why uniform density? Think of a sphere made of foam rubber with uniform density. To alter that and achieve non-uniform density, work is needed. Then, with non-uniform density, the sphere has more energy than with uniform density. When conditions permit, the sphere will minimize its energy by releasing the excess, that is, it will return to the condition of uniformity. The law of force and Newton's shell theorems admit some density distributions in the sphere and others do not. Uniform density is a supported condition.

Think of a spherical shell of infinitesimal thickness, defined within the sphere. That shell has an infinitesimal volume dV, expressed in the following way.

[math]dV = 4 \ \pi \ r^2 \ dr \ \ \ \ \ \ \ \ \ (01)[/math]

[math]r \ \rightarrow[/math] radius of the infinitesimal shell

Symbolize P to a point on that shell. We want to calculate the gravitational force at that point.

Theorem 2 informs that the part of the sphere located between [math]r[/math] and [math]R_{_T}[/math] exerts a net force equal to zero over P. Then only the part between the center and [math]r[/math] .

Theorem 1 informs that we can assume accumulated in the center all the mass of the part that interests. This applies to all points of the infinitesimal shell. For that reason we can refer the calculation to the mass [math]dm[/math] of the whole infinitesimal shell. In that case, Newton's formula expresses the sum of all the modules. Which modules? There are infinite points, [math]P_{_1}[/math],[math]P_{_2}[/math],[math]P_{_3}[/math]...[math]P_{_n}[/math], all in the same condition as P. Although the force vectors of all those points have different directions, all the modules are equal. If we put in Newton's formula the mass [math]dm[/math] of the whole shell, the result is the summation of all the modules. The individual module at a point is equal to that summation, divided by the area of the spherical surface of radius [math]r[/math] and multiplied by the infinitesimal area corresponding to the point. If we started from the individual point, we would need to do an integral, to get the same result that Newton's formula expresses when we put the mass [math]dm[/math] of the whole shell. Es decir, podemos evitar una integral y lo haremos.

The sum [math]dF[/math] of the módules is

[math] dF = G \ \dfrac{m_i \ dm}{r^2}  \ \ \ \ \ \ \ \ \ (02) [/math]

[math] m_i \ \rightarrow [/math] mass of the part enclosed by the shell of radius [math]r[/math]

[math] dm \ \rightarrow [/math] infinitesimal mass of the shell of radius [math]r[/math]

The shell of radius [math]r[/math] is in direct contact with the shell located an infinitesimally farther from the center, that is to say situated in [math]r+dr[/math]. Contact involves force transmission in the radial direction. The force points towards the center. That means that the shell located in [math]r+dr[/math] transmits force to the shell located in [math]r[/math]. On the shell located in [math]r[/math] There is an infinitesimal gravitational force, which is added to the infinitesimal from [math]r+dr[/math]. The sum of both is transmitted to [math]r-dr[/math], whose infinitesimal gravitational force is added to the sum of the other two.

That means that strength [math]F[/math], exerted on the shell of radius [math]r[/math], it is calculated integrating [math]dF[/math] between [math]r[/math] and [math]R_{_T}[/math] .

[math] F= \int_r^{R_{_T}} dF \ \ \ \ \ \ \ \ \ (03) [/math]

We apply (02) in (03).

[math] F= \int_r^{ R_{_T} } G \ \ \dfrac{m_i \ dm}{r^2} \ \ \ \ \ \ \ \ \ (04) [/math]

The density [math]\delta[/math] is uniform. Then we have the following.

[math]m_i = \delta \ \dfrac{4}{3} \ \pi \ r^3  \ \ \ \ \ \ \ \ \ (05) [/math]

[math]dm = \delta \ 4 \ \pi \ r^2 \ dr  \ \ \ \ \ \ \ \ \ (06) [/math]

We apply (05) and (06) in (04).

[math] F= \int_r^{ R_{_T} } G \ \dfrac{ \delta \ \dfrac{4}{3} \ \pi \ r^3 \ \ \delta \ 4 \ \pi \ r^2 \ dr }{r^2} [/math]

[math] F= \dfrac{\left( 4 \ \pi \right)^2 }{3} \ G\ \delta^2 \int_r^{ R_{_T} } \ r^3 \ dr [/math]

We solve the integral.

[math] F= \dfrac{\left( 4 \ \pi \right)^2 }{3} \ G\ \delta^2 \left[ \dfrac{R_{_T}^4}{4} - \dfrac{r^4}{4} \right] [/math]

[math] F= \dfrac{\left( 4 \ \pi \right)^2 }{3} \ G\ \delta^2 \left[ \dfrac{R_{_T}^4}{4} - \dfrac{r^4}{4} \right] [/math]

[math] F= \dfrac{4}{3} \ \pi^2 \ G\ \delta^2 \left( R_{_T}^4 - r^4 \right)  \ \ \ \ \ \ \ \ \ (07) [/math]

The force [math]F[/math] is distributed over the entire spherical surface of radius [math]r[/math] , as previously explained. Then the pressure [math] p [/ math] corresponding to that surface is expressed as follows.

[math] p= \dfrac{ \dfrac{4}{3} \ \pi^2 \ G\ \delta^2 \left( R_{_T}^4 - r^4 \right) }{4 \ \pi \ r^2} [/math]

[math] p= \dfrac{ \dfrac{4}{3} \ \pi^2 \ G\ \delta^2 \left( R_{_T}^4 - r^4 \right) }{4 \ \pi \ r^2} [/math]

[math] p = \dfrac{\pi}{3} \ G \ \delta^2 \ \left( \dfrac{R_{_T}^4}{r^2} - r^2\right)  \ \ \ \ \ \ \ \ \ (08) [/math]

In (08) we notice the following. If a point B is closer to the center than another point A, that is to say [math]r_{_B} < r_{_A}[/math], then the pressure in B is greater than the pressure in A. That property is independent of the density and the total size of the sphere.

In the equation (08) [math] p \rightarrow \infty [/math] when [math]r \rightarrow 0 [/math] . Is that why an absurd or incoherent equation?

When you put a model, you are not expressing all the properties of reality. Example. No real cone can have the vertex sufficiently sharp to end in a single point, which meets the ideal of the abstract geometric point, of infinitesimal size or, in practical language, of size equal to zero. The geometric model of the cone assumes that it does. That does not create incoherence in geometric theory.

Something similar is found in the ideal model of the sphere. There is a critical region, not suitable for the usual aggregation states. There is no inconsistency. The ideal model exhibits a detail that we can not ignore. The theorems of physics are as abstract as the theoretical geometry. When we are ready to interpret the results well, an infinite tendency in an ideal model means, simply, that there is a border and, in some way, the physical laws avoid going beyond that boundary. Many times the same laws determine the properties of the border.

Think of the following ideal model.

A material cone is free to move without friction along a straight tube. You place the tube vertically, with the apex of the cone facing down. Although the mass of the cone is very small, say a microgram, the ideal model gives the vertex an infinitesimal surface, equal to zero in practical terms. The weight of a microgram, divided by an infinitesimal surface, gives a pressure tending to infinite. Is the result inconsistent? No. Well interpreted, the model informs that no material cone can have the vertex perfectly sharp, until equaling the abstract geometric point, independently of the mass of the cone. 

Think of the three usual states of aggregation, gaseous, liquid and solid. Can any of those states exist near the center, where the pressure is too great? Note that [math]p \rightarrow \infty[/math] when [math]r \rightarrow 0[/math]. Too close to the center, the usual aggregation states can not subsist. What kind of state could resist? It would need to be, at the elementary level, a highly dynamic state. We will not advance in this matter, because it exceeds the scope of Newtonian physics.

If we knew the maximum pressure that the most resistant of the usual aggregation states can withstand, we could calculate the radius of the central region that neither of these states can occupy. We can do it in the following way.

In (08) we can make the following variable change.

[math] r^2=u  \ \ \ \ \ \ \ \ \ (09) [/math]

If in (08) we apply (09) a second degree equation is formed with respect to [math]x[/math].

[math] p = \dfrac{\pi}{3} \ G \ \delta^2 \ \left( \dfrac{R_{_T}^4}{u} - u\right) [/math]

We multiply by [math]u[/math] both members.

[math] p \ u = \dfrac{\pi}{3} \ G \ \delta^2 \ \left( R_{_T}^4 - u^2 \right) [/math]

We divide both members by [math] \dfrac{\pi}{3} \ G \ \delta^2 [/math] .

[math] \dfrac{ 3 \ p }{ \pi \ G \ \delta^2 } \ u = R_{_T}^4 - u^2 [/math]

We order.

[math] u^2+ \dfrac{ 3 \ p }{ \pi \ G \ \delta^2 } \ u - R_{_T}^4 = 0  \ \ \ \ \ \ \ \ \ (10) [/math]

The maximum pressure allowed by the most resistant habitual aggregation state is a data independent of the size and density of the sphere. This data could be included in tables, as the constant G . If that data were known, it would suffice to know the density of the sphere to calculate the radius of the critical region, that central region where there can be no gas, liquid, or solid.

On the internet there is data from trials that subject substances to increasing pressures, to reach disintegration. When I did the search, a synthetic substance appeared as a substance more resistant to pressure, more resistant than all the known natural substances. A sphere made of that artificial substance, with mass and radius equal to the mass and radius of the Earth, would have a surprisingly large critical region, thousands of kilometers around the center. If the sphere is made of something less resistant than the artificial substance, equation (10) gives a larger critical region.

Edited September 9, 2018 by quiet

[studiot]

Forgive me but I am struggling with the idea of a sphere of uniform density that has internal pressure increasing radially inwards.

Why will the density not increase  radially inwards as well?

[Strange]

38 minutes ago, studiot said:

Forgive me but I am struggling with the idea of a sphere of uniform density that has internal pressure increasing radially inwards.

Why will the density not increase  radially inwards as well?

For the same reason that cows are spherical 

[quiet]

2 hours ago, studiot said:

Forgive me but I am struggling with the idea of a sphere of uniform density that has internal pressure increasing radially inwards.

Why will the density not increase  radially inwards as well?

Thanks studiot for the question, which serves to reflect. We know that the density of a gas, approximately, increases with pressure, as long as the volume does not approach the value of the co-volume. Assuming constant density implies supposing that it is not a gaseous sphere. The constituent substance of the sphere is incompressible. In reality it seems difficult to find something that is strictly incompressible, maintaining that property in all conditions. In an ideal model, like this one that we have here, we can suppose perfect incompressibility in all conditions of pressure and temperature. And the model reports that ideal conditions do not prevent the existence of a critical region.

1 hour ago, Strange said:

For the same reason that cows are spherical 

The Chinese philosopher Confucius inverted the phrase, because he said that true understanding is given by the pair of symmetrical phrases that you get with the original phrase plus the inverted phrase. In your case we would have the next pair.

For the same reason that cows are spherical

For the same cows that reason is spherical

Edited September 9, 2018 by quiet

[studiot]

If you are considering the material incompressible, you should say so at the beginning.

You might like to read this paper on the stress distributions in rotating spheres growing by accretion (ie planets)

Note carefully that near the beginning the authors point out that the internal stress distribution is markedly different from a solid body that is made stress free in one go and then set rotating.

The article give much useful data pertinent to bodies in our solar system.

https://www.sciencedirect.com/science/article/pii/S0020768304006286

 

 

[quiet]

1 hour ago, studiot said:

If you are considering the material incompressible, you should say so at the beginning.

You might like to read this paper

https://www.sciencedirect.com/science/article/pii/S0020768304006286

It is true what you point out. The thread would have been better warning in the beginning that the model corresponds to incompressible material. Before your question, I had not noticed that detail. That's why I did not mention it in the beginning. I appreciate your intervention.

Thanks also for the link. I have not followed the mathematical development. I have only read the concepts that I can, in some way, understand. It seemed to me that the objective of the advanced model shown in this publication is to explain, first, how the astronomical objects of the analyzed class are formed, to study later the evolution towards a condition with more or less stable averages in the magnitudes that describe the state. As I said, that seemed to me. Maybe you can confirm or correct my interpretation.

Edited September 9, 2018 by quiet

[studiot]

13 hours ago, quiet said:

Thanks also for the link. I have not followed the mathematical development. I have only read the concepts that I can, in some way, understand. It seemed to me that the objective of the advanced model shown in this publication is to explain, first, how the astronomical objects of the analyzed class are formed, to study later the evolution towards a condition with more or less stable averages in the magnitudes that describe the state. As I said, that seemed to me. Maybe you can confirm or correct my interpretation

Yes you have the train of thought and the ideas correct.

I am not sure about your mathematical developments though.

I agree the referred article is tough mathematically, you might find this Newtonian treatment easier.

 

 

 

[quiet]

I would like the written communication to convey how much I appreciate the gravitation chapter of that book. With an outstanding didactic quality, with beautiful and elegant style, it exposes the tools that allow to pose very diverse situations and access to specific bibliography. I have saved the images. It does not appear the author's reference in the  sheets. I would like to know the data, if possible.

I must confess that regarding gravitational interaction and other interactions, little by little a very different conviction has been installed in me than what appears in the bibliography. So different that seem to me  it's an abuse to think about a gravitational field and apply the same kind of mathematical treatment that we use for the electromagnetic field. The issue exceeds the context of this thread. That is why I will not continue the comment.

[studiot]

The General Properties of Matter (1965 edition; 5th ed I think)

Newman and Searle

First published 1929

[quiet]

Thank you very much.