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Author Topic: Where is the point of maximum gravity inside a planet or star?  (Read 9063 times)

Offline a_dark_knight

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We always treat gravity as attracting things to the *center* (of gravity) of a planet or star, for instance let's consider one called Plar. But if you were *inside* Plar, it would be different. At the center, you would be pulled out in every direction right? So do those forces simply cancel out and you feel nothing or do they create a tension that pulls things apart? I lean towards the first one.

In that case, where is the net force due to gravity maximum? How far into the center? And also, where is the "pressure" maximum from all the stuff sitting on top of you? Is it still the center because it all adds up from the higher material?

You can assume Plar is homogenous if that helps, surely it can be figured out with some calculus, but I'd also be interested in Earth. Finally, does this phenomenon have any noticeable effects, like on the nuclear reactions in stars or can a hole form from material being pulled out?
« Last Edit: 19/12/2012 15:16:52 by chris »


 

Offline Pmb

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #1 on: 05/12/2012 06:46:53 »
We always treat gravity as attracting things to the *center* (of gravity) of a planet or star, for instance let's consider one called Plar.
To be precise, that holds only between objects which are point particles. For arbitrary bodies one has to integrate over each body. If Plar is a sphere with a uniform distribution of mass then a point particle inside Plar would be attracted to geometric center of the body, which would also be the center of mass too.

Quote from: a_dark_knight
But if you were *inside* Plar, it would be different. At the center, you would be pulled out in every direction right?
Since Plar has a uniform distribution of mass the the force on any point particle inside Plar would be directed to its center. In this case the center of mass of the object would be attracted to the center of the body. The body would be crushed rather than pulled apart. However if the center of the body was not center of the center of Plar then the body would be pulled part in Plar's radial direction and squeezed in a direction which is tangengential to a radial line eminnating from Plar's center.
 

Offline a_dark_knight

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #2 on: 05/12/2012 12:26:37 »
Okay, so the net force at any point is always toward the center, but near the center, that net force becomes small (or even zero). Does that have any significant implications? And is the pressure maximum at the very center or somewhere further out where the net force due to gravity is larger (or even maximum)?

And yes, I figured out that technically gravity should be integrated over all the bodies involved but planets and stars were always simplified to a point when I was taught (in high school), which is perfectly accurate for most applications. But when is the difference between the two versions no longer negligible? How close do two bodies have to be before their sizes become relevant?
 

Offline Phractality

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #3 on: 05/12/2012 18:16:41 »
Newton did the integration which proves, for a sphere of uniform density, the strength of gravity is zero at the center and increases linearly toward the surface; beyond the surface, it decreases as the inverse square of distance. At every point, the direction of gravity is toward the center. This is called Newton's shell theorem; he calculated the gravity for a hollow shell and divided the solid sphere into concentric hollow shells.

Stars are not uniformly dense; the density increases toward the middle; so the gravity increases more rapidly as you move away from the center and more slowly as you near the surface. The slope of the graph of gravity is proportional to the density at any given distance from the center. Still, the direction of gravity is always toward the center, and therefore, the pressure is greatest at the center.
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #4 on: 08/12/2012 23:04:36 »
You do get a little headache thinking of it, we're so used to it being gravity 'drawing us down' :)
Pressure doesn't need gravity, I think? I'm not totally sure as you could imagine molecules flowing in a very weak, or undetectable, gravitational 'field'. Would they be able to create a pressure? What about one molecule in that field, is that a pressure?

Pressure is about kinetic energy as I think of it, thingies 'moving around' relative each other, and as they do, and collide, they transfer energy to each other, with gravity defining a preferred direction for the kinetic energy's 'build up'. So we get a increased pressure the further down we get, gravitationally seen, all the way to the center. But it is also so that inside that very center you will have 'pillars of pressure acting' on you from all points of the perfect sphere (earth) so (as a weird thought example) if the earth was made of hollow cheese :) you should be crushed, although weightlessly so.

If I'm thinking right that is?

What about a gas cloud in space, ahh, there will be gravity as soon as there is matter, right :)
so it will act as a having a preferred direction then too, to the gas clouds center.
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #5 on: 08/12/2012 23:19:02 »
the one about one molecule, or particle, is just me trying to see what 'kinetic energy' you could define to it. In fact you can define any or no energy at all to it, if it only is one particle existing in field, otherwise devoid of matter. It's not a very good question as with one particle there is no way to define any 'motion', except possibly quantum as in HUP giving it some sort of intrinsic probability of a 'motion'. But normally all motion must be relative something else, and that should be earth for us :) except accelerations then, that according to relativity also is equivalent to a 'gravity', if you accept me including all forms of accelerations in that statement.
==

Keep picking the wrong words here :)

And btw, 'pressure' definitely seems to be needing a preferred direction it seems? To build a pillar of pressure. Or else being confined inside something, then adding heat which inside a sphere should become a outward pressure equal in all points on that balls walls. What more ways to define pressure?

(Is there a 'minimum space' for each air molecule relative another? Thinking of filling a balloon with air, how does it keep its 'shape' and that kinetic energy acting on the walls, after cooling down, without such a requirement? )

Eh, that should be temperature/heat, but also a factor related to a 'minimum distance' between molecules, adapting to the temperature surrounding the balloon, if I'm thinking right that is? What makes that distance?
=

You can of course relate it to the mass energy of the amount of molecules you have inside that balloon, relative the ones outside, but still? Ahh, they have more molecules to collide with which, in the end, must keep imparting more kinetic energy to the walls, maybe :)

Not maybe, it better be so or I'm gonna get a bigger headache than I already have..
Been playing too much WOW lately :)
« Last Edit: 08/12/2012 23:58:58 by yor_on »
 

Offline Phractality

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #6 on: 09/12/2012 06:23:58 »
Pressure is just force spread over an area. For a spherical surface concentric with the mass of a gaseous planet, the pressure is constant. Multiply that pressure by the area of the sphere and you get the total weight of all the gas outside that sphere. The gas inside the sphere pushes outward with equal and opposite force of the outside gas trying to fall in.
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #7 on: 09/12/2012 15:09:36 »
Yes but I'm still trying to see the mechanical properties of the particles interacting inside the balloon.
And I'm not too happy with my last explanation either :) Why would the particles closest to the walls have a larger momentum than the ones, say in the middle? When you blow a balloon up you heat that air, and impart 'energy'. But after a while that energy/heat must disappear. What you then have is a larger amount of particles inside the balloon than outside, all of them in motion. They are closer to each other than on the outside, but should now have the approximate same energy/motion per particle. How do they then keep the balloon inflated?

Collisions?
=

Alternative, how do they keep a larger 'energy' per particle after the heat has dissipated?
« Last Edit: 09/12/2012 15:11:18 by yor_on »
 

Offline Phractality

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #8 on: 10/12/2012 04:22:03 »
Yes but I'm still trying to see the mechanical properties of the particles interacting inside the balloon.
And I'm not too happy with my last explanation either :) Why would the particles closest to the walls have a larger momentum than the ones, say in the middle? When you blow a balloon up you heat that air, and impart 'energy'. But after a while that energy/heat must disappear. What you then have is a larger amount of particles inside the balloon than outside, all of them in motion. They are closer to each other than on the outside, but should now have the approximate same energy/motion per particle. How do they then keep the balloon inflated?

Collisions?
=

Alternative, how do they keep a larger 'energy' per particle after the heat has dissipated?
Inflating the balloon increases the number of molecules inside the balloon. Initially those molecules are heated by compression, but conduction of heat thru the membrane equalizes the temperature. Equalizing the temperature equalizes the energy per molecule, but you still have more molecules inside the balloon. The balloon needs to be bigger to contain more molecules, and stretching the balloon puts it in tension. The tension in the stretched rubber is exactly matched by the pressure difference between the inside and outside.

You can calculate the tension in the stretched rubber by considering an imaginary plane that bisects the balloon. The pressure difference times the cross section is the total force pushing the two halves apart from inside. That is equal and opposite to the tension times the circumference of the cross section times the thickness of the membrane. This works even if the plane does not bisect the balloon, because both the circumference and area are smaller.
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #9 on: 11/12/2012 16:04:20 »
Yes, but if the balloon need to expand due to a higher concentration of molecules inside it than outside, temperature being equivalent, doesn't that imply that molecules need a certain space around them?
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #10 on: 11/12/2012 20:07:01 »
You should by weighting be able to tell the amount of molecules, relative 'normal unpressurized' air surrounding it, and then by measuring the volume decide where the limit goes for the molecules becoming 'pressuring' on the balloons walls, right? And it's a balance relative what pressure there exist outside it naturally. What happens to a balloon in space? It burst due to pressure differences but what does it mean?
==

You could see it as two questions, the first assuming that there must be a 'rest space' (for lack of a better expression) for a molecule, wondering what that would be? The other wondering what mechanism make those molecules bursting the balloon (in space). They go into each other it seems :)
« Last Edit: 11/12/2012 20:14:54 by yor_on »
 

Offline Phractality

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #11 on: 11/12/2012 20:11:24 »
Yor_on,

Avogadro's gas law:
PV = kNT \,whereP is the absolute pressureV is the volumeN is the number of gas moleculesk is the Boltzmann constant (1.381◊10−23 J∑K−1 in SI units)T is the temperature (K)

If you need to understand the math of how molecules bouncing off one another maintain the spacing between them, that is covered by Einsteinís Quantum Theory of the Monatomic Ideal Gas:
That paper is sometime called "worthless", but it lays the foundation for his more important
Atoms, entropy, quanta: Einsteinís miraculous argument of 1905.

If you want to avoid high-power math, just accept the fact that higher temperature makes the molecules move faster and bounce off one another harder and more often, so they need more volume to maintain a constant pressure, or more pressure to maintain a constant volume. If you give them more space they get cooler; crowd them together, they get hotter. Bouncing off the walls of a container heats the walls, which then transfer the heat to the surroundings, either by conduction or by radiation.
 
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #12 on: 11/12/2012 20:16:21 »
Seems we wrote past each other there, take a look at what I added to the question. Liked your pdf:s btw :)
« Last Edit: 11/12/2012 20:21:40 by yor_on »
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #13 on: 11/12/2012 21:09:35 »
A molecule represent a certain energy, it takes up a space, it also have a intrinsic motion. We cant ignore its 'relative motion' (as an added 'energy') even when uniform, as I think? Introduce more molecules and we get collisions which are representations of the combined 'energies' of the particles colliding, including relative (uniform) motion. In relativity 'energy' always seems to become a expression of a 'system' consisting of two, or more, objects, and the question of where it stores the 'energy' or 'potential energy' becomes meaningless as there is no way I see to define it, it's all about relations.

But assuming that a particle takes up a space, according to the Pauli exclusion principle, then so does a molecule :) Why do they separate in space (burst the balloon)? And could you define a 'rest space' from that? As if there was some minimalistic definition of 'no pressure', which then could be said to leave the molecule/particles in a 'rest space', as we use 'rest mass'?
=

Crazy, isn't it :)
« Last Edit: 11/12/2012 21:59:47 by yor_on »
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #14 on: 11/12/2012 22:15:33 »
I'm thinking of it this way. The rest mass of a particle is defined to be the same, everywhere. At a black hole or on earth, is there something governing a minimal 'rest space' for particles too? And if it is, how do you get to be 'at rest' with such a space? If pressure is what makes the balloon burst, and it is what we define it as, right? Could you track that to a 'rest space'? Maybe you need some more degrees of freedom for it to make sense though, that we don't notice? The momentum(s) inside that balloon hasn't changed, neither has their energies/rest mass.

Can there be a 'rest space'?
 

Offline Phractality

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #15 on: 12/12/2012 03:15:45 »
I'm thinking of it this way. The rest mass of a particle is defined to be the same, everywhere. At a black hole or on earth, is there something governing a minimal 'rest space' for particles too? And if it is, how do you get to be 'at rest' with such a space? If pressure is what makes the balloon burst, and it is what we define it as, right? Could you track that to a 'rest space'? Maybe you need some more degrees of freedom for it to make sense though, that we don't notice? The momentum(s) inside that balloon hasn't changed, neither has their energies/rest mass.

Can there be a 'rest space'?
The point of Einstein's SR is that every reference frame is equivalent (provided there are no FTL phenomena). You can take the center of mass of a solid object, or the center of a balloon full of gas, as the origin of a reference frame. You could assign a velocity to each molecule in the object or the gas and apply SR to get a ratio of kinetic mass to rest mass. As the temperature rises, the sum of all the kinetic masses will increase.

Then, you could consider each molecule's kinetic mass in a reference frame which is moving relative to the first reference frame, and add them up to get the kinetic mass of the whole in that reference frame. Turns out, you get the same mass by applying SR to each molecule and adding them up, or by applying SR to the whole group of molecules.

The mass increase due to added heat can be used to determine an average change of speed of each molecule, which will be different in different reference frames. This average is not simply the sum divided by the number of molecules, even if all the molecules are identical. Instead, you determine what group speed would yield the same mass increase as the sum of all the individual mass increases.

(If you could capture two photons in an endless loop, their radiant energy would become a rest mass. SR could be applied to each photon or to the pair with the same result, the same as if the photons were particles. But two-photon whirls have been moved to New Theories.)
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #16 on: 12/12/2012 22:31:49 »
Yes :)

That's mass and energy, but I'm thinking space. Matter takes up space and I'm sort of wondering how it does it? A rest mass is a very good idea, especially as we can follow it to a defined energy, being 'at rest' with it. As you say, it all becomes relative though,  yourself or whatever other theoretical frame of reference you use as your anchor, defining that  'energy' or mass-energy' of what you measure on.

What I'm wondering about is the concept itself, a 'rest mass', wondering if you could assume a 'rest space' being reserved as well? If there is such a thing that is :) Point particles (elementary particles) as leptons photons etc are said to be pointlike, meaning that you either can define them as taking up 'no place' or if you like a field 'no specific place' in that field, until measured. But both you and me are more than fields, we consist of combinations of all those particles, and they obey our least thought, my fingers (consisting of those particles) move as I write etc. To put such a behavior into a 'field concept' should be impossible, at least today. Fields are easier to understand when we speak about spontaneous pair productions, there and gone, than matter although we have examples on particles surviving that pair experience. But going to living matter and using fields?

Why I'm discussing fields is because from such an idea you don't need any reserved 'space' before as I see it, it could be that the field create the space, but thinking of living matter? There should be reserved 'spaces' for the particles making me up, and if there is, is the idea of a rest space possible?
 

Offline evan_au

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #17 on: 15/12/2012 09:51:48 »
My conclusion for this hypothetical situation: If you had an incredibly strong and well-insulated spaceship which you managed to pilot to the exact center of the star Plar (or the center of the Earth):
  • The spaceship would need to be incredibly strong, because the pressure is at a maximum at the exact center of the body.
  • You would float around inside the spaceship, just as if you were in free-fall, in space.
  • You would not be pulled apart by the gravity pulling your arms and legs in opposite directions - the gravity in opposite directions cancels out at the center
  • But if you were foolish enough to open the incredibly strong door door of your incredibly strong spaceship, you would be immediately crushed by matter rushing into the spaceship, propelled by the immense pressure of the overlying layers of matter, each trying to get just a little closer to the center of gravity.
  • ... and just for a moment, succeeding!
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #18 on: 15/12/2012 22:46:29 »
Don't you think the point where gravity 'equals out' should be rather small? I would expect that point to be like a 'point like particle' myself, gravity(ies) surrounding it. Although it/they should be weaker the closer to that point you get, so yeah. One would hold together I guess.
==

But there should be some ratio to it in where, if we imagine us inside something very massive as a neutronstar, we would be 'spagettified', I think? :)
« Last Edit: 15/12/2012 22:52:45 by yor_on »
 

Offline CliffordK

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #19 on: 15/12/2012 23:23:01 »
You would not be pulled apart by the gravity pulling your arms and legs in opposite directions - the gravity in opposite directions cancels out at the center

The gravity vector is always towards the center of mass of the body...  thus no pulling...  just pushing.  However, near the center, the gravity vector will be very small.

I.E.  if your star is a million miles in diameter...  No matter where you are in the star, the differences in the gravity in any cubic mile would be essentially undetectable.  Although, if you were in a cubic mile sphere at the exact center, and managed to hang weights, then they would all be extremely light, but would generally hang towards the middle.

Keep in mind that the center of gravity may not be exactly at the middle of the body.

I'm seeing notes that the center of gravity of the Earth is about 4600 km from the middle...  And, since the Earth is spinning, it would be essentially be moving around all the time.
 

Offline Phractality

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #20 on: 16/12/2012 00:54:33 »
I'm seeing notes that the center of gravity of the Earth is about 4600 km from the middle...  And, since the Earth is spinning, it would be essentially be moving around all the time.

The point you are referring to, 4641 km from the center of the Earth is the location of the Earth-moon system's barycenter. If you were orbiting the Earth-moon system at a point far beyond the moon you would orbit around that barycenter, with wobbles as you pass near the line between the Earth and moon. In low Earth orbit, you orbit the center of Earth, but the Earth happens to be orbiting the barycenter. In low lunar orbit, you orbit the center of the moon, which is also orbiting the barycenter. So your choice of reference frame is important. Near Earth, we usually name the center of Earth as the origin; near the moon, we name the center of the moon as the origin.

If you droped something inside a bubble at the barycenter, it would accelerate toward the center of the Earth. The barycenter is roughly 3/4 of the way out from the center to the surface, but the denser parts are in the middle, so your dropped object would accelerate toward the center at roughly 1/2 g. That's if your reference frame is centered on the Earth and rotating with the Earth.
 

Offline CliffordK

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #21 on: 16/12/2012 03:34:36 »
Thanks,
So, if you had two objects of equal mass, then the center of mass of the two object system would be between them.

But, on either one object, the majority of the gravity would be towards the center of the object...

But, can one entirely ignore the second object?  The tides, of course. are about 6 feet, but are rather symmetrical around the center of the planet.

Or, is the symmetry of the tides the answer.  So, the gravitational center is offset by a few feet, but well balanced by centrifugal force.
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #22 on: 16/12/2012 04:41:49 »
Not as I think Clifford. Mass and gravity goes together, when you're at the 'center' Earths mass is surrounding you, negating its 'pull outwards'. But if we magically cut away three quarters of the earth you would find 'gravity' again, pointing to where you came from, and no weightlessness. It doesn't matter which side of the earth you are, the direction of gravity is approximately inwards to the center but the deeper you get, the more mass you leave behind (above, and around) you.
==

Pressure will point inwards in a perfect non rotating sphere, undisturbed by other heavenly objects gravity. But pressure will also 'line itself up' relative gravity's direction, assuming kinetic energy to define a pressure. So if we cut away 3/4 of the Earth (starting at that center) you would find yourself on a top instead of a center, and the pressure would act the other way, following gravity's new direction. And from that we can see that energy obey gravity :)
« Last Edit: 16/12/2012 05:07:46 by yor_on »
 

Offline Phractality

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #23 on: 16/12/2012 06:08:48 »
This is all covered by Newton's shell theorem. Given uniform distribution of mass in a hollow shell, the gravity inside the shell is zero. The mass outside of the shell doesn't pull things apart; the gravity from opposite direction merely cancels out. It doesn't pull different directions on different parts of an object; instead, it doesn't pull at all on any part of the object.

Outside the shell, gravity is the same as if all the mass of the shell were concentrated at the center. So outside of the shell, gravity is proportional to the inverse square of distance from the center.

Inside a solid sphere, at distance r from the center, you can divide the problem into two parts. The part closer than r to the center acts like all of its mass is concentrated at the center. The part farther than r from the center is just a hollow shell, so it contributes nothing to the gravity at radius r.

As r increases, the mass inside r increases as r3, so the gravity at r increases in direct proportion to r3/r2 = r.



That graph assumes uniform density. In fact the interior is denser, so the slope is steeper near the center.
 

Offline yor_on

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #24 on: 16/12/2012 15:19:26 »
Da*, you're right :)

It was me getting confused thinking about it. Gravity could be treated as a pressure of a 'density' I guess and then you just have to imagine something moving inside that. The direction is inwards everywhere, so my spaghettifcation can only come to be due to tidal forces as something (neutronstar) rotates. It was me misleading meself here :)
==

Reminder to myself: Have to stop playing wow, it influences my physics :) I sort of forgot to count that other side of the sphere counteracting my 'possible' pull. But the rest, I insist, is correct, ahem...
« Last Edit: 16/12/2012 15:26:34 by yor_on »
 

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Re: Point of maximum gravity/pressure inside a planet or star
« Reply #24 on: 16/12/2012 15:19:26 »

 

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