Dark matter and Shell Theorem by Isaac Newton

[David Levy]

Brief info on Shell theorem

http://en.wikipedia.org/wiki/Shell_theorem

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 said 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.

Brief info on Dark matter

http://en.wikipedia.org/wiki/Dark_matter

Dark matter is a hypothetical kind of matter that cannot be seen with telescopes but accounts for most of the matter in the universe. The existence and properties of dark matter are inferred from its gravitational effects on visible matter, radiation, and the large-scale structure of the universe. It has not been detected directly, making it one of the greatest mysteries in modern astrophysics.

based on the standard model of cosmology, the total mass–energy of the known universe contains 4.9% ordinary matter, 26.8% dark matter and 68.3% dark energy.^[2][3] Thus, dark matter is estimated to constitute 84.5% of the total matter in the universe, while dark energy plus dark matter constitute 95.1% of the total mass–energy content of the universe.^[

 

Let's suppose that the density of the dark matter is the same everywhere in the Universe.

In this case, we should look at the Universe as a Hollow ball full with Dark matter.

Based on Isaac Newton shell theorem [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."

It is clear that that no net gravitational force is exerted by the shell on any object inside this shell.

Therefore, Based on this explanation the net effect of the dark matter in the universe is Zero!

This is also fully correlated also with the following example:

Let's set the Sun outside the galaxy at a place without any stars or galaxies nearby, but we will keep the same density of the Dark matter.

In this case, the Sun will get the same dark matter' gravity force from all directions. Therefore, the net gravity force vector of the dark matter on the sun should be zero.

Do you agree?

Edited March 16, 2015 by David Levy

[Janus]

Dark matter is not uniformly distributed throughout the universe. it concentrates in spherical halos around galaxies and also has a filament like structure on a large scale such as shown in this image of a dark matter map:

 

[Strange]

Also, the density of dark matter is greater at the centre of the galaxy.

[swansont]

 

Let's suppose that the density of the dark matter is the same everywhere in the Universe.

In this case, we should look at the Universe as a Hollow ball full with Dark matter.

Based on Isaac Newton shell theorem %5B2%5D:

"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."

It is clear that that no net gravitational force is exerted by the shell on any object inside this shell.

Therefore, Based on this explanation the net effect of the dark matter in the universe is Zero!

This is also fully correlated also with the following example:

Let's set the Sun outside the galaxy at a place without any stars or galaxies nearby, but we will keep the same density of the Dark matter.

In this case, the Sun will get the same dark matter' gravity force from all directions. Therefore, the net gravity force vector of the dark matter on the sun should be zero.

Do you agree?

 

If the assumption is correct. But if the assumption is wrong, then so is the result.

 

If the distribution of normal matter was uniform, there would be no gravitational collapse to form galaxies. But we have galaxies, therefore we can conclude that the distribution was not. In a similar fashion, we should conclude that we can't assume what you are assuming.

[Strange]

Also, the density of dark matter is greater at the centre of the galaxy.

 

I should explain that this distribution of dark matter is required to explain the observed velocity distributions of gas and stars in galaxies.

[David Levy]

Dark matter is not uniformly distributed throughout the universe. it concentrates in spherical halos around galaxies and also has a filament like structure on a large scale such as shown in this image of a dark matter map:

 

O.K.

 

But I have asked about hypothetical uniformed distribiution.

So, do you agree that if (just if) the dark matter is uniformed distributed throughout the universe, then no net gravitational force is exerted by the shell on any object inside the Universe.

Yes or no?

[Strange]

But I have asked about hypothetical uniformed distribiution.

So, do you agree that if (just if) the dark matter is uniformed distributed throughout the universe, then no net gravitational force is exerted by the shell on any object inside the Universe.

Yes or no?

 

Yes.

 

Which is why the distribution is not uniform.

[Mordred]

We already know it's not. Dark matter is influenced by gravity, it falls into gravity wells just as normal matter does. We account for this in the BAO (baryon accoustic oscillations) in the CMB measurements. As anisotropies form DM falls into the localized anistropies. This provides a major aid in early large scale structure formation.

 

Around galaxies (spiral) your distribution profile is the Navarro Frank White NFW profile

 

http://en.m.wikipedia.org/wiki/Navarro%E2%80%93Frenk%E2%80%93White_profile

[David Levy]

 

Yes.

 

Which is why the distribution is not uniform.

 

Thanks

 

Is it correct that the Science accept the idea of shell theorem in the central bulge of spiral galaxy?

If so, why the matter distributed in the bulge is more uniform then the dark matter?

Edited March 16, 2015 by David Levy

[swansont]

 

If so, why the matter distributed in the bulge is more uniform then the dark matter?

 

Is it? I'd like to see corroboration of this claim.

[Strange]

Is it correct that the Science accept the idea of shell theorem in the central bulge of spiral galaxy?

 

Of course. It is a mathematical proof.

 

 

If so, why the matter distributed in the bulge is more uniform then the dark matter?

 

Is it?

[David Levy]

Of course. It is a mathematical proof.

 

Is it?

So, there is a mathematical proof that the central bulge of spiral galaxy meets the shell theorem, (although, the matter in the bulge isn't uniformly distributed).

Can you also please direct me to that article?

However, don't you think that the dark matter should be more uniformly distributed then the matter in the bulge.

In any case, if the science accepts the idea that the bulge could be uniform, why the same idea isn't applicable for the dark mass?

Edited March 16, 2015 by David Levy

[Strange]

So, there is a mathematical proof that the central bulge of spiral galaxy meets the shell theorem, (although, the matter in the bulge isn't uniformly distributed).

Can you also please direct me to that article?

 

No there isn't. What I meant was that the shell theorem is a mathematical proof. It can be applied to any spherical mass (such as the central bulge).

 

However, don't you think that the dark matter should be more uniformly distributed then the matter in the bulge.

In any case, if the science accepts the idea that the bulge could be uniform, why the same idea isn't applicable for the dark mass?

 

 

Is the bulge more uniformly distributed? It seems unlikely. Can you provide a reference for this? And how is that connected to the shell theorem?

 

The distribution of dark matter is calculated from the velocities of stars and from simulation of the behaviour of dark matter. I assume the same is done for the stars in the bulge. So it will be interestuing to see why it is more uniform.

Edited March 16, 2015 by Strange

[David Levy]

 

What I meant was that the shell theorem is a mathematical proof. It can be applied to any spherical mass (such as the central bulge).

 

 

Thanks

 

Sorry that I have to ask you again almost the same question.

However, I still do not understand this critical issue.

The shell theorem can be applied to any spherical mass.

So, if the central bulge is accepted as spherical mass, why the dark matter in the universe can't be also accepted as spherical mass?

Edited March 16, 2015 by David Levy

[Strange]

So, if the central bulge is accepted as spherical mass, why the dark matter in the universe can't be also accepted as spherical mass?

It is. It is a spherical mass, which extends beyond the visible galaxy and is densest at the centre.

[Mordred]

The detail your having trouble with is the distribution of DM is everywhere within and outside the galaxy . The difference is in the density at different locations.

 

This is detailed in the NFW profile

 

note the NFW uses the virial theory.

 

Shell theory is great but it's not practical when your distribution is everywhere at varying densities

Edited March 16, 2015 by Mordred

[swansont]

So, if the central bulge is accepted as spherical mass, why the dark matter in the universe can't be also accepted as spherical mass?

Spherical mass is not enough. It must have a symmetric distribution, which can only vary radially. The universe doesn't qualify.

[Mordred]

Spherical mass is not enough. It must have a symmetric distribution, which can only vary radially. The universe doesn't qualify.

In point there is a spherical symmetric treatment of the dark matter halo. The trick is defining the radius boundary.

 

A common methodology is to set the boundary at

 

[latex]R_{200}[/latex] The 200 value is 200* the critical density.

 

 

 

This is the method used in the older universal rotation curve.

 

Ned Wright has a good coverage.

 

http://ned.ipac.caltech.edu/level5/March01/Battaner/node7.html

 

I had almost forgotten that method as its more common to use NFW

(Don't ask me why 200 was chosen, I honestly do not recall why.

In regards to the question of the Op

 

The spherical symmetric DM shell has net force zero within the shell, if and only if you have uniform density,

 

which we don't. But even if we did the net force is only zero at the center of mass.

 

This line is the one you overlooked on the wiki page.

 

"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"

 

The main difference between a solid and a gas is density in terms of gravity.so assuming a perfect ball of uniform density the net force is only zero at the center of mass, not throughout the inner sphere.

 

Now here is the next problem, the assumption of a ball like spherical DM halo. The halo being defined by R_ 200 will be elliptical. Remember gravity affects dark matter the same as baryonic matter.

 

 

Now lets look at the singular Isothermal sphere. You have a sphere defined by velocity rotation. This causes a flattening, much like an asteroid belt or the the rings of Saturn. The galaxy rotation affects the dark element in a similar manner defined by R_200. We have to set the radius boundary somewhere.

 

This is where the formula

 

[latex]\rho_r=\frac{\sigma^2_v}{2\pi Gr^2}[/latex] comes into play, this is part of the NFW profile.

 

[latex]\sigma^2_v[/latex] is the velocity dispersion

 

Note [latex]R_{200}[/latex] is also used as the radius boundary of the dark sector in the NFW profile.

 

I would also suggest you look at how the virial theorem is involved.

 

Read this wiki carefully.

 

In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy, [latex]\left\langle T \right\rangle[/latex], of a stable system consisting of N particles, bound by potential forces, with that of the total potential energy,[latex] \left\langle V_\text{TOT} \right\rangle[/latex], where angle brackets represent the average over time of the enclosed quantity

 

http://en.m.wikipedia.org/wiki/Virial_theorem

Edited March 17, 2015 by Mordred

[David Levy]

It is. It is a spherical mass, which extends beyond the visible galaxy and is densest at the centre.

 

Sorry, if the dark mass is spherical mass, than it is expected by definition that it will fully obey to Isaac Newton shell theorem.

I had almost forgotten that method as its more common to use NFW

(Don't ask me why 200 was chosen, I honestly do not recall why.

In regards to the question of the Op

 

The spherical symmetric DM shell has net force zero within the shell, if and only if you have uniform density,

 

which we don't. But even if we did the net force is only zero at the center of mass.

 

 

Sorry, this is incorrect - based on Newton.

The net force is zero regardless of the object's location.

Please see again the following description from Newton:

"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."

 

However, Why it isn't 100 of 500?

Somehow, the science is making his selection based on his needs.

Now, let's agree that the dark mass is not symmetric distribution. However, it is still need to be everywhere at different density.

Therefore, why it is claimed that it is concentrated in some sections of the galaxy? How do we know that? What kind of proves do we have?

With regards to the following:

http://cdms.berkeley.edu/Education/DMpages/FAQ/question36.html

"The average density of dark matter near the solar system is approximately 1 proton-mass for every 3 cubic centimeters, which is roughly 6x10^-28 kg/cm^3"

"The radius of the sun's orbit is about 2.5x10¹⁷ km, so the total mass of dark matter within that orbit is 6x10⁴⁰ kg. This is the mass of 3x10¹⁰ (30 billion) stars like the sun! The entire galaxy only contains ~100 billion stars, so the dark matter does have a significant effect on the sun's orbit through the galaxy. For objects farther out near the edge of the galaxy, the dark matter is actually the main thing keeping them in their orbits. This is more or less how dark matter was discovered by astronomer Vera Rubin and others: the orbital speeds of galactic stars and gas clouds don't match our expectations from the visible matter."

Why all the dark mass outside the Sun's orbit has been neglected? It must have significant impact on the total Sun' gravity?

Actually if the distribution was uniform, then we all agree that the total gravity net force on the sun should be zero (regardless of the Sun location). Therefore, we shouldn't neglect all the dark mass outside the orbit ring.

This dark mass, must have negative gravity force on the Sun. So it must decrease dramatically the net gravity force on the sun. We shouldn't ignore this mass. It is a severe mistake!

Edited March 17, 2015 by David Levy

[Mordred]

No your understanding is incorrect.

 

Take the Earth itself for example. It's a spherical mass

 

Is the force of gravity zero beneath the surface?

 

Come on use some math not just descriptions. The key is NET force and vector sum.

Dark matter distribution is NOT a hollow sphere its treatment is the same mathematically as the SOLID example. Yeesh

The only difference is density. Do you not understand that term????:

 

 

Density, every formula related to rotation curves uses this term learn it.

I even gave you a galaxy modelling article. Did you bother reading it?

Why do I even bother trying to teach you the real science involved I post you references supporting EVERY statement and example I provide. You still refuse to accept it.

http://www.google.ca/url?sa=t&source=web&cd=5&ved=0CCwQFjAE&url=https%3A%2F%2Fwikis.uit.tufts.edu%2Fconfluence%2Fdownload%2Fattachments%2F9440479%2Fchemouni_bach_GE_dec07.pdf%3Fversion%3D1&rct=j&q=singular%20isothermal%20sphere%20profile%20of%20spiral%20arms&ei=xNQDVaDFLsfwoATsxoCACQ&usg=AFQjCNGm931PDgYo5WOdtIksZVMLKqwVSQ&sig2=fa628v9sgDG0sloyufaLVg&bvm=bv.88198703,d.eXY

 

Here is the galaxy modelling article again.

 

Read it

 

Sorry, if the dark mass is spherical mass, than it is expected by definition that it will fully obey to Isaac Newton shell theorem.

 

 

This dark mass, must have negative gravity force on the Sun. So it must decrease dramatically the net gravity force on the sun. We shouldn't ignore this mass. It is a severe mistake!

There is no such thing as anti gravity.

 

Gravity affects dark matter the EXACT same as baryonic matter.

 

It does obey the shell theorem, you don't understand how the shell theorem Is correctly applied.

 

Reason you don't apply it correctly is your not applying the vector sums.

 

In shell theorem when the vector sum of mass =0 is the center of mass.

 

It's also used in barycenter orbits.

and Keplers laws.

(The saving grace with all the effort I put into these answers is even if you don't study the material other readers will. )

Edited March 17, 2015 by Mordred

[Strange]

Sorry, if the dark mass is spherical mass, than it is expected by definition that it will fully obey to Isaac Newton shell theorem.

 

Of course.

 

 

Sorry, this is incorrect - based on Newton.

The net force is zero regardless of the object's location.

Please see again the following description from Newton:

"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."

 

But dark matter is not distributed as a "shell".

 

 

Now, let's agree that the dark mass is not symmetric distribution.

 

I haven't seen anything to suggest it is not symmetrical. Can you provide a reference?

 

 

Therefore, why it is claimed that it is concentrated in some sections of the galaxy? How do we know that? What kind of proves do we have?

 

1. Because that is the density pattern required to produce the observed velocities of stars and dust.

 

2. Because that is also the result of simulations of the behaviour of dark matter (based on what we know about it)

 

 

Why all the dark mass outside the Sun's orbit has been neglected?

 

Because of the shell theorem - the (symmetrically distributed) mass outside the solar system has no effect.

[swansont]

Sorry, if the dark mass is spherical mass, than it is expected by definition that it will fully obey to Isaac Newton shell theorem.

You keep dropping the symmetric part of the description. It's critical.

 

Sorry, this is incorrect - based on Newton.

The net force is zero regardless of the object's location.

Please see again the following description from Newton:

"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."

 

Sorry, but you keep getting this wrong. I've added emphasis on the part you are missing. The shell exerts no force inside of it, but the mass inside still exerts a force on anything outside of it. So there is a force inside the shell, for a symmetric spherical mass.

[Janus]

To illustrate what Swansont is saying, consider the following image of a spherical mass divided into concentric shells:

Each shell has zero gravitational effect on the shells or anything else inside of them. However, each shell does have a gravitational effect on anything outside of it. Thus anything sitting on the outer shell of this sphere will feel the sum of the gravitational effect of all the shells just like you feel the entire gravitational effect of the Earth's mass standing on its surface. If you place it further in, the shells outside of it have no effect on it, but it will still feel the sum of the gravitational effect of all the shells closer to the center than it is. It will be the same as if you had just removed the outer shells and it now rests on the surface of a smaller, less massive sphere.

Edited March 17, 2015 by Janus

[David Levy]

Thanks you all.

I do appreciate your valuable answers.

Dark matter is not uniformly distributed throughout the universe. it concentrates in spherical halos around galaxies and also has a filament like structure on a large scale such as shown in this image of a dark matter map:

 

With regards to this photo;

It looks to me like a ripple effect on sea water.

Please see the following:

http://www.fotothing.com/photos/43c/43c7519c765015dc9e4a4c1b7ea8b8c3_d34.jpg

If that is correct, than we could consider the following analogy:

The dark matter could be considered as water and the galaxy is similar to the ship which crosses the ocean.

Therefore, if it is a correct analogy, than it is expected that the dark matter is not moving with the galaxies, but the galaxies are crossing the dark matter.

From another aspect – We are using the the dark matter as a cosmology constant. Therefore, we can assume that it is a constant matter in the Universe.

Do you agree?

Edited March 18, 2015 by David Levy

[Strange]

If that is correct, than we could consider the following analogy:

The dark matter could be considered as water and the galaxy is similar to the ship which crosses the ocean.

Therefore, if it is a correct analogy, than it is expected that the dark matter is not moving with the galaxies, but the galaxies are crossing the dark matter.

 

It is not a good analogy. Just because one thing looks a bit like another thing, doesn't mean there is any conneection. You are comparing a 2D surface with a 3D structure. We know that dark matter and galaxies are located together in this large scale structure of the universe. In fact, from simulations, it seems pretty clear that the dark matter is an important factor in creating this structure.

 

From another aspect – We are using the the dark matter as a cosmology constant.

 

You seem to be confusing dark matter and dark energy.