Questions about the inverse square law and dark matter

Hello Gentlemen,

I have some questions about the inverse square law.

For systems of particles, I think that applying that law is a bit more
complex than for surfaces. Unluckily, reading Wikipedia, other sites
and parts of graduate papers, I have not found reference to this.

I use ^ for exponentiation, space for multiplication, and the second
letter of a variable as a sub-index.

For one particle with respect to an observation point, we can have (L
= luminosity or power, I = intensity or brightness and r = distance
from
the emitting point to the observation point):

L = I 4 pi r^2

If r is an hypotenuse in three dimensions, then we have:

r^2 = x^2 + y^2 + z^2

L = I 4 pi (x^2 + y^2 + z^2)

Lets assume that x and y form the observation plane (e.g. CCD from
camera) and z is the distance to the parallel plane where the emitting
point source resides.

For a system of particles, we then have (S = Sigma or sum for all
particles with j from 1 to number of particles):

S(Lj) = 4 pi S(Ij (xj^2 + yj^2 + zj^2))

If the system is isotropic having: an average distance d to the
particles, in z, and an average intensity Ia, then for simplification
we could have (where p (positive) and n (negative) are sub-indexes
representing each half the particles in j, i.e. from 1 to n/2):

for each zp = d + z'p, another zn = d - z'n and z'p = z'n

S(zj^2) = S(zp^2) + S(zn^2)= S((d + z'p)^2 + (d - z'p)^2)

S(zj^2) = 2 S(d^2 + z'p^2)

S(Lj) = 4 pi Ia (2 S(xp^2 + yp^2) + n d^2 + 2 S(z'p^2))

Again simplifying, we can imagine that each sum to the right (for each
dimension) can be simplified because the particle distribution is
isotropic, so there is an average separation t between particles. So
we apply the formula for square pyramidal numbers:

we have particles in x at distances:
-ex, ... -3 t, -2 t, -t, 0, t, 2 t, 3 t, ... , ex

So we use the square pyramidal number sum with i from 1 to ex/t (where
ex is the extent of the particles in the x dimension from the origin
to any side; it can be a radius depending on the geometry of the
particle system):

S(xp^2) = S((i t)^2) = t^2 S(i^2) = t^2 P(ex/t)

Replacing that in the last S(Lj), we get:

S(Lj) =
8 pi Ia t^2 (P(ex/t) + P(ey/t) + P(ez'/t) + n/2 d^2/t^2)

The last term is then (where I = Ia n; and d = r, the distance from
the observing point to the center of the particle system):

I 4 pi d^2

That is the formula used at the beginning for simple surfaces. With
respect to that, in the first 3 terms, we get an excess of:

8 pi Ia t^2 (P(ex/t) + P(ey/t) + P(ez'/t))

Which is average intensity per particle proportional to a metric of
the extent of the particle system in the three dimensions.

The square pyramidal number for the x dimension is:

P(ex/t) = (2 (ex/t)^3 + 3 (ex/t)^2 + ex/t)/6

and t^2 P(ex/t) = (2 ex^3/t + 3 ex^2 + t ex)/6

So that the 3D metric of the particle system is proportional to the
sum of the cubes of the extent of the system in each dimension.

If the separation between particles is very big (very low density)
then
we will have the cube term reduced:

2 ex^2 ex/t

So that the 3D metric will be characterized somewhere between the
squares and the cubes of the extent of the system in each dimension.

So my questions a

Could the excess term here be related to dark matter in galaxies
(where separation between stars makes the excess term proportional to
somewhere between the square and the cube of the galaxy size)?

Is it possible that for the gas surrounding the galaxies and the gas
at the center of galaxy clusters that the term can explain some of the
dark matter?

Maybe at least a small outer part of galaxies seems dark because we
should not imagine the incoming rays as collimated (but this depends
on the obsevation method, so here the answer is most probably no)?

I know that the rays of a galaxy would be for all practical purposes
collimated but as in the classic experiment to calculate G with small
masses, big distances can really make a difference.

Thanks in advance for your replies.

Gustavo Broos wrote in news:8a7be8e4-2915-45ff-988d-
:

[snip nonsense]

So my questions a

Could the excess term here be related to dark matter in galaxies
(where separation between stars makes the excess term proportional to
somewhere between the square and the cube of the galaxy size)?

It'd help if you knew why dark matter came about. Galactic rotation
curves are obtained by the measurements of invidual components of stars
within the galaxy which then come together to show the rotation curve of
the galaxy.

The next method is through gravitational lensing.

Neither method has a single thing to do with anything you wrote and I
haven't the faintest idea what it is supposed to mean.


Is it possible that for the gas surrounding the galaxies and the gas
at the center of galaxy clusters that the term can explain some of the
dark matter?

To the same degree that neutrinos are relevant to dark matter. In
principle when doing the accounting, yes. In practice, no.


Maybe at least a small outer part of galaxies seems dark because we
should not imagine the incoming rays as collimated (but this depends
on the obsevation method, so here the answer is most probably no)?

I know that the rays of a galaxy would be for all practical purposes
collimated but as in the classic experiment to calculate G with small
masses, big distances can really make a difference.

Thanks in advance for your replies.


Collimation is irrelevant to dark matter.

"Gustavo Broos" wrote in message
...
| Hello Gentlemen,
|
| I have some questions about the inverse square law.
|
| For systems of particles, I think that applying that law is a bit more
| complex than for surfaces. Unluckily, reading Wikipedia, other sites
| and parts of graduate papers, I have not found reference to this.
|
| I use ^ for exponentiation, space for multiplication, and the second
| letter of a variable as a sub-index.
|
| For one particle with respect to an observation point, we can have (L
| = luminosity or power, I = intensity or brightness and r = distance
| from
| the emitting point to the observation point):
|
| L = I 4 pi r^2
|
| If r is an hypotenuse in three dimensions, then we have:
|
| r^2 = x^2 + y^2 + z^2
|
| L = I 4 pi (x^2 + y^2 + z^2)
|
| Lets assume that x and y form the observation plane (e.g. CCD from
| camera) and z is the distance to the parallel plane where the emitting
| point source resides.
|
| For a system of particles, we then have (S = Sigma or sum for all
| particles with j from 1 to number of particles):
|
| S(Lj) = 4 pi S(Ij (xj^2 + yj^2 + zj^2))
|
| If the system is isotropic having: an average distance d to the
| particles, in z, and an average intensity Ia, then for simplification
| we could have (where p (positive) and n (negative) are sub-indexes
| representing each half the particles in j, i.e. from 1 to n/2):
|
| for each zp = d + z'p, another zn = d - z'n and z'p = z'n
|
| S(zj^2) = S(zp^2) + S(zn^2)= S((d + z'p)^2 + (d - z'p)^2)
|
| S(zj^2) = 2 S(d^2 + z'p^2)
|
| S(Lj) = 4 pi Ia (2 S(xp^2 + yp^2) + n d^2 + 2 S(z'p^2))
|
| Again simplifying, we can imagine that each sum to the right (for each
| dimension) can be simplified because the particle distribution is
| isotropic, so there is an average separation t between particles. So
| we apply the formula for square pyramidal numbers:
|
| we have particles in x at distances:
| -ex, ... -3 t, -2 t, -t, 0, t, 2 t, 3 t, ... , ex
|
| So we use the square pyramidal number sum with i from 1 to ex/t (where
| ex is the extent of the particles in the x dimension from the origin
| to any side; it can be a radius depending on the geometry of the
| particle system):
|
| S(xp^2) = S((i t)^2) = t^2 S(i^2) = t^2 P(ex/t)
|
| Replacing that in the last S(Lj), we get:
|
| S(Lj) =
| 8 pi Ia t^2 (P(ex/t) + P(ey/t) + P(ez'/t) + n/2 d^2/t^2)
|
| The last term is then (where I = Ia n; and d = r, the distance from
| the observing point to the center of the particle system):
|
| I 4 pi d^2
|
| That is the formula used at the beginning for simple surfaces. With
| respect to that, in the first 3 terms, we get an excess of:
|
| 8 pi Ia t^2 (P(ex/t) + P(ey/t) + P(ez'/t))
|
| Which is average intensity per particle proportional to a metric of
| the extent of the particle system in the three dimensions.
|
| The square pyramidal number for the x dimension is:
|
| P(ex/t) = (2 (ex/t)^3 + 3 (ex/t)^2 + ex/t)/6
|
| and t^2 P(ex/t) = (2 ex^3/t + 3 ex^2 + t ex)/6
|
| So that the 3D metric of the particle system is proportional to the
| sum of the cubes of the extent of the system in each dimension.
|
| If the separation between particles is very big (very low density)
| then
| we will have the cube term reduced:
|
| 2 ex^2 ex/t
|
| So that the 3D metric will be characterized somewhere between the
| squares and the cubes of the extent of the system in each dimension.
|
| So my questions a
|
| Could the excess term here be related to dark matter in galaxies
| (where separation between stars makes the excess term proportional to
| somewhere between the square and the cube of the galaxy size)?
|
| Is it possible that for the gas surrounding the galaxies and the gas
| at the center of galaxy clusters that the term can explain some of the
| dark matter?
|
| Maybe at least a small outer part of galaxies seems dark because we
| should not imagine the incoming rays as collimated (but this depends
| on the obsevation method, so here the answer is most probably no)?
|
| I know that the rays of a galaxy would be for all practical purposes
| collimated but as in the classic experiment to calculate G with small
| masses, big distances can really make a difference.
|
| Thanks in advance for your replies.

If one photon arrives in one square eyeball every second,
then at twice the distance one photon will arrive in an area
of four square eyeballs every second or in an areas of one
square eyeball every four seconds. That's why cameras have
shutter speeds and dorks have dork matter.

Dear Gustavo Broos:

On Jul 12, 2:06*am, Gustavo Broos wrote:
....
So my questions a
....
Is it possible that for the gas surrounding the galaxies
and the gas at the center of galaxy clusters that the
term can explain some of the dark matter?

There is little / no gas (more importantly dust) at the center of
spiral galaxies, and there is more farther out. And yes, all of Dark
Matter can be explained by just our accounting / guessing at normal
mass distributions, but other problems arise if you "twist the titty"
that hard.

Maybe at least a small outer part of galaxies seems
dark because we should not imagine the incoming
rays as collimated (but this depends on the obsevation
method, so here the answer is most probably no)?

We found a large amount of ionized gas between galaxies. Only visible
because we had x-ray sources that could still interact with not-fully-
ionized oxygen. It is unlikely that there is enough ionized hydrogen
to be all of Dark Matter. Which leaves still several candidates.

I know that the rays of a galaxy would be for all
practical purposes collimated

I don't think that is the word you want

but as in the classic experiment to calculate G with
small masses, big distances can really make a
difference.

Thanks in advance for your replies.

There are more than 1000 papers, in the last year alone, on
arxiv.org
.... that talk about Dark Matter. Most are approximations,
estimations, and models. Some are actual observations, and talk about
methods. I'd recommend you learn about why Dark Matter came up, the
assumptions that created so much of it. Neutrinos still are required
as part of Dark Matter.

David A. Smith

dlzc wrote in news:e48f5307-46c6-49ff-8184-2087a1aeae58
@q34g2000prf.googlegroups.com:

[...]

There are more than 1000 papers, in the last year alone, on
arxiv.org
... that talk about Dark Matter. Most are approximations,
estimations, and models. Some are actual observations, and talk about
methods. I'd recommend you learn about why Dark Matter came up, the
assumptions that created so much of it. Neutrinos still are required
as part of Dark Matter.

David A. Smith


Last I checked, neutrino's (collective, summed) mass was a factor of two or
so smaller than would be required to handle dark matter. Which doesn't
begin to tackle the fact that it would only solve the dark matter of today
problem as opposed to the imprint left on the CMB.

That's the problem with invoking things like black holes, dust, etc. It has
to have existed at decoupling, while true for the case of neutrinos it
doesn't work because they were relativistic back then.

On Jul 12, 11:06*am, Gustavo Broos wrote:
Hello Gentlemen,

I have some questions about the inverse square law.


How sweet,the inverse square law which is assigned to Kepler which
correlates orbital periods with distance from the Sun never worked -

"The proportion existing between the periodic times of any two planets
is exactly the sesquiplicate proportion of the mean distances of the
orbits, or as generally given,the squares of the periodic times are
proportional to the cubes of the mean distances." Kepler

Gentlemen indeed !,how I wish that were true as it takes an
interesting detour to the Fomahaut system to determine whether it
works or not -

http://apod.nasa.gov/apod/image/0811...ut_hst_lab.jpg

Any other era before the toxic strain of empiricism emerged and it
would be a lively discussion but this is no normal era,the contact,if
any has been either hostile or non existent even when direct imaging
is brought to bear on planetary dynamics.The idea of moving directly
from a human level to a macroscopic level blows me away as though
reader believe no information is lost .The thing about it ,as I have
discovered through many years of experience,is not only do followers
of Newton do not care what he tried to do,they couldn't tell you the
method Kepler used to correlate the orbital period of a planet with
its orbital radius !.

Dear eric gisse:

On Jul 12, 9:52*am, eric gisse wrote:
dlzc wrote in news:e48f5307-46c6-49ff-8184-2087a1aeae58
@q34g2000prf.googlegroups.com:

[...]

There are more than 1000 papers, in the last year alone, on
arxiv.org
... that talk about Dark Matter. *Most are approximations,
estimations, and models. *Some are actual observations,
and talk about methods. *I'd recommend you learn about
why Dark Matter came up, the assumptions that created
so much of it. *Neutrinos still are required as part of Dark
Matter.

Last I checked, neutrino's (collective, summed) mass was
a factor of two or so smaller than would be required to
handle dark matter. Which doesn't begin to tackle the fact
that it would only solve the dark matter of today problem
as opposed to the imprint left on the CMB.

We've been making neutrinos by the boatload since the CMB quenched.

That's the problem with invoking things like black holes,
dust, etc. It has to have existed at decoupling, while true
for the case of neutrinos

And helium, and metals missing lots of electrons...

it doesn't work because they were relativistic back then.

Presumably "they" is "black holes, dust, etc.", or are the neutrinos
relativistic? Your response is not clear here.

The most of the "initial" neutrinos would have cooled by now, so it
would be possible to gravitationally capture them. There would be
"newer" ones that were still too hot to capture...

David A. Smith

dlzc wrote in news:f2d49c48-57a9-4ed0-aa3e-eb5191ecd116
@j14g2000prn.googlegroups.com:

Dear eric gisse:

On Jul 12, 9:52*am, eric gisse wrote:
dlzc wrote in news:e48f5307-46c6-49ff-8184-
2087a1aeae58
@q34g2000prf.googlegroups.com:

[...]

There are more than 1000 papers, in the last year alone, on
arxiv.org
... that talk about Dark Matter. *Most are approximations,
estimations, and models. *Some are actual observations,
and talk about methods. *I'd recommend you learn about
why Dark Matter came up, the assumptions that created
so much of it. *Neutrinos still are required as part of Dark
Matter.

Last I checked, neutrino's (collective, summed) mass was
a factor of two or so smaller than would be required to
handle dark matter. Which doesn't begin to tackle the fact
that it would only solve the dark matter of today problem
as opposed to the imprint left on the CMB.

We've been making neutrinos by the boatload since the CMB quenched.

A largely irrelevant contribution. The absolute number and mass of the
neutrinos is impressive if you integrate over the lifetime of the
universe, but they do not make a meaningful contribution to anything.


That's the problem with invoking things like black holes,
dust, etc. It has to have existed at decoupling, while true
for the case of neutrinos

And helium, and metals missing lots of electrons...

it doesn't work because they were relativistic back then.

Presumably "they" is "black holes, dust, etc.", or are the neutrinos
relativistic? Your response is not clear here.

Neutrinos.



The most of the "initial" neutrinos would have cooled by now, so it
would be possible to gravitationally capture them. There would be
"newer" ones that were still too hot to capture...

David A. Smith


By now - sure. Not recently enough to have been meaningful WRT galactic
formation, nor massive enough. Remember that baryonic matter is
outmassed by a significant margin. Neutrinos just can't cut it.

If the dark matter to baryonic matter ratio were inverted then neutrinos
would be far more likely.

Dear eric gisse:

On Jul 13, 7:45*am, eric gisse wrote:
dlzc wrote in news:f2d49c48-57a9-4ed0-aa3e-eb5191ecd116
@j14g2000prn.googlegroups.com:
On Jul 12, 9:52*am, eric gisse wrote:
dlzc wrote in news:e48f5307-46c6-49ff-8184-
2087a1aeae58
@q34g2000prf.googlegroups.com:

[...]

There are more than 1000 papers, in the last year alone, on
arxiv.org
... that talk about Dark Matter. *Most are approximations,
estimations, and models. *Some are actual observations,
and talk about methods. *I'd recommend you learn about
why Dark Matter came up, the assumptions that created
so much of it. *Neutrinos still are required as part of Dark
Matter.

Last I checked, neutrino's (collective, summed) mass was
a factor of two or so smaller than would be required to
handle dark matter. Which doesn't begin to tackle the fact
that it would only solve the dark matter of today problem
as opposed to the imprint left on the CMB.

We've been making neutrinos by the boatload since the
CMB quenched.

A largely irrelevant contribution. The absolute number and mass of the
neutrinos is impressive if you integrate over the lifetime of the
universe, but they do not make a meaningful contribution to anything.

Maybe you have seen this:
http://arxiv.org/abs/1107.2166

That's the problem with invoking things like black holes,
dust, etc. It has to have existed at decoupling, while true
for the case of neutrinos

And helium, and metals missing lots of electrons...

it doesn't work because they were relativistic back then.

Presumably "they" is "black holes, dust, etc.", or are the
neutrinos relativistic? *Your response is not clear here.

Neutrinos.



The most of the "initial" neutrinos would have cooled by
now, so it would be possible to gravitationally capture
them. *There would be "newer" ones that were still too
hot to capture...

By now - sure. Not recently enough to have been meaningful
WRT galactic formation, nor massive enough. Remember that
baryonic matter is outmassed by a significant margin.

Based on *assumptions* of baryonic matter distribution, at a galactic
core we only imagine.

Neutrinos just can't cut it.

http://arxiv.org/abs/1106.2543
.... maybe they can look exactly like they can cut it. At least at the
quench of the CMB.

If the dark matter to baryonic matter ratio were inverted
then neutrinos would be far more likely.

David A. Smith

dlzc wrote in news:fd67e2fc-24c9-4ec3-9806-d94bad9fda57@
28g2000pry.googlegroups.com:

Dear eric gisse:

On Jul 13, 7:45*am, eric gisse wrote:
dlzc wrote in news:f2d49c48-57a9-4ed0-aa3e-
eb5191ecd116
@j14g2000prn.googlegroups.com:
On Jul 12, 9:52*am, eric gisse wrote:
dlzc wrote in news:e48f5307-46c6-49ff-8184-
2087a1aeae58
@q34g2000prf.googlegroups.com:

[...]

There are more than 1000 papers, in the last year alone, on
arxiv.org
... that talk about Dark Matter. *Most are approximations,
estimations, and models. *Some are actual observations,
and talk about methods. *I'd recommend you learn about
why Dark Matter came up, the assumptions that created
so much of it. *Neutrinos still are required as part of Dark
Matter.

Last I checked, neutrino's (collective, summed) mass was
a factor of two or so smaller than would be required to
handle dark matter. Which doesn't begin to tackle the fact
that it would only solve the dark matter of today problem
as opposed to the imprint left on the CMB.

We've been making neutrinos by the boatload since the
CMB quenched.

A largely irrelevant contribution. The absolute number and mass of
the
neutrinos is impressive if you integrate over the lifetime of the
universe, but they do not make a meaningful contribution to anything.

Maybe you have seen this:
http://arxiv.org/abs/1107.2166

10^+/-0.46? That's a pretty goddamn awkward figure.

At any rate, I skimmed because it didn't actually seem to attempt to
distinguish between that model and the other mentioned models thus my
interest was minimal. I recall there have been attempts at fitting other
profiles to galaxies but the NFW profile seems to work the best thus
far.


That's the problem with invoking things like black holes,
dust, etc. It has to have existed at decoupling, while true
for the case of neutrinos

And helium, and metals missing lots of electrons...

it doesn't work because they were relativistic back then.

Presumably "they" is "black holes, dust, etc.", or are the
neutrinos relativistic? *Your response is not clear here.

Neutrinos.



The most of the "initial" neutrinos would have cooled by
now, so it would be possible to gravitationally capture
them. *There would be "newer" ones that were still too
hot to capture...

By now - sure. Not recently enough to have been meaningful
WRT galactic formation, nor massive enough. Remember that
baryonic matter is outmassed by a significant margin.

Based on *assumptions* of baryonic matter distribution, at a galactic
core we only imagine.

Neutrinos just can't cut it.

http://arxiv.org/abs/1106.2543

Far more interesting. Sum of neutrino masses has an upper bound of 0.17
eV? Yikes. It was ****-all before, but 0.17 is about 3x smaller than the
previous bound.

... maybe they can look exactly like they can cut it. At least at the
quench of the CMB.

Not so much. The CMB is irrelevant here - this is quasar spectra.

"Naturally, neutrinos have only a mild effect on dark matter halos
[13,15], since their large velocity dispersion prevents their
clusteringon small scales. In contrast, we find that the impact of
neutrinos on void properties is much stronger."

Have to say that it would not have come to mind to study the effects of
neutrinos on the void.

My grasp is that the mass of the neutrinos alters the distribution of
voids, in the form of increasing/decreasing the amount of voids that pop
up. Which in turn is visible in quasar spectra, presumably via the size
of the gaps in the spectra.


If the dark matter to baryonic matter ratio were inverted
then neutrinos would be far more likely.

David A. Smith