1kg of electrons in a sphere

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1kg of electrons in a sphere

Postby GoC » Mon Jun 01, 2009 2:54 am UTC

What would be the result if a single kilogram of electrons suddenly appears in front of you in the form of a 1cm sphere? They aren't being forced to stay as a sphere though.
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Re: 1kg of electrons in a sphere

Postby masher » Mon Jun 01, 2009 2:59 am UTC

You'd get a severe does of beta radiation as they all fly apart...

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Re: 1kg of electrons in a sphere

Postby gmalivuk » Mon Jun 01, 2009 3:22 am UTC

The energy stored in such a configuration, if I've done the calculation correctly, would be 3.34x1034J, which is 2.8 times as much energy as the sun puts out in a year.

I'd say "severe dose of beta radiation" is a *wee* bit of an understatement...
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Re: 1kg of electrons in a sphere

Postby masher » Mon Jun 01, 2009 3:43 am UTC

Well, I was just going through a similar calculation; you were just a bit faster... :)

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Re: 1kg of electrons in a sphere

Postby GoC » Mon Jun 01, 2009 3:45 am UTC

gmalivuk wrote:The energy stored in such a configuration, if I've done the calculation correctly, would be 3.34x1034J, which is 2.8 times as much energy as the sun puts out in a year.

I'd say "severe dose of beta radiation" is a *wee* bit of an understatement...

Can you apply Coloumb's law in this situation? Mightn't you have to consider... *pulls scientific sounding words out of a hat and strings them together* electron degeneracy pressure?
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Re: 1kg of electrons in a sphere

Postby masher » Mon Jun 01, 2009 3:57 am UTC

Probably not.

A 1cm sphere is pretty big...

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Re: 1kg of electrons in a sphere

Postby BlackSails » Mon Jun 01, 2009 4:45 am UTC

You see* a brief glimpse of light as the electrons annhilate the earth.

*: You actually dont see anything, because you are destroyed before you can process the sight.

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Re: 1kg of electrons in a sphere

Postby heyitsguay » Mon Jun 01, 2009 4:48 am UTC

You headbutt it, if you're a real man.

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Re: 1kg of electrons in a sphere

Postby Vieto » Mon Jun 01, 2009 4:49 am UTC

BlackSails wrote:You see* a brief glimpse of light as the electrons annhilate the earth.

*: You actually dont see anything, because you are destroyed before you can process the sight.


Now that is a superweapon.

Now, just to convert this energy into a laser, and we have a death-star. (anyone have a charged metallic sphere, not grounded?)

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Re: 1kg of electrons in a sphere

Postby meat.paste » Mon Jun 01, 2009 4:58 am UTC

The density is ~1900 g/cc. That's well above the densities of materials at the surface of the Earth. Is the density high enough to need to account for electron degeneracy pressure? Or is that not important until you get to neutron star densities?

The induced electric field could do very bad things to other objects. I suspect (but don't have time to do the math right now) that the charge gradient across anything nearby would be enough to induce a pretty lethal current flow or simply rip matter to shreds until the charge dissipated.
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Re: 1kg of electrons in a sphere

Postby Charlie! » Mon Jun 01, 2009 5:52 am UTC

A volume roughly 1 cm^3 is big but so is 1 kg. Maybe there's some energy there after all. Let's see... If you put 10^30 particles into a box that's 0.01^3 m^3 on a side you end with little boxes 0.01^3/10^30 in volume, means the electrons are around 10^-12 m away from each other. This is about a hudredth the radius of a hydrogen atom, so I think that at most reasonable wavelengths the electrons will be indistinguishable.

But though the fermi energy for 10^30 electrons is probably considerable (well, at least a considerable number of electron volts), it's probably much less considerable than the energy already there.
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Re: 1kg of electrons in a sphere

Postby alexh123456789 » Mon Jun 01, 2009 5:57 am UTC

meat.paste wrote:The induced electric field could do very bad things to other objects. I suspect (but don't have time to do the math right now) that the charge gradient across anything nearby would be enough to induce a pretty lethal current flow or simply rip matter to shreds until the charge dissipated.


I don't think induced current would be as big a problem as electrons traveling close the the speed of light destroying everything they touch. The energy in those electrons is over 12x earth's kinetic energy, assuming I didn't mess any calculations up.

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Re: 1kg of electrons in a sphere

Postby Vieto » Tue Jun 02, 2009 1:11 am UTC

now, what would happen if 1kg of positrons appeared next to the 1kg of electrons? :kaboom:

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Re: 1kg of electrons in a sphere

Postby gmalivuk » Tue Jun 02, 2009 1:25 am UTC

The energy of annihilation is a teeny tiny fraction of the electromagnetic energy stored in the electrons to begin with, actually.

(A kilo of electrons evenly distributed throughout a sphere with a 1cm diameter already holds as much energy as 3.71x1017kg of matter.)
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Re: 1kg of electrons in a sphere

Postby Diadem » Tue Jun 02, 2009 3:24 am UTC

I actually recently derived (well, rederived obviously. It was first done in the thirties or so) the electron degenercy energy for white dwarves. It is not very hard. The energy levels of electrons in a box are:
[math]E_k=\frac{1}{2m}\left(\frac{k\pi\hbar}{L}\right)^2=\frac{\vec{p}^2_k}{2m}[/math]
Where each energy level occurs twice, once for spin up, once for spin down. So to get the total energy stored in such a configuration we need to sum [imath]2E_k[/imath] from 1 to N/2. Instead of summing, we rewrite it as an integral over momenta. This gives:

[math]E=2\left(\frac{L}{\pi\hbar}\right)^3\frac{1}{8}\int_0^{p_F}4\pi p^2E(p)dp[/math]
The factor 2 is explained above. The factor 1/8 is because we integrate over only one octet of the sphere. The other prefactor arises when we convert a the sum into an integral. The factor 4 Pi p^2 in the integral is of course the jacobian.

The solution to this is:
[math]\frac{E}{L^3}=\frac{3}{10}\left(3\pi^2\right)^{2/3}\left(\frac{\hbar^2}{m}\right)n^{5/3}[/math]
Where n is the number density.

The rest is just plugging in numbers. L = 0.01, n = N / L^3 = 10^30 / (0.01)^3 = 10^36. Google calculater gives the answer as:

1.634 × 1016 J

That is somewhat disappointing. It's the same order of magnitude as their rest mass. Considerable, but it pales compared to the coulumb repulsion.
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Re: 1kg of electrons in a sphere

Postby GoC » Tue Jun 02, 2009 3:39 am UTC

Very nice!
Still a bit disappointing though...
Lets up the mass to say... ten tons or so and see if we get a black hole! And if so, what would the black hole "look like" (it's properties)?

EDIT: Wait a minute... this isn't a white dwarf. Wouldn't the lack of neutrons and protons (and atomic structure in general) change things?
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Re: 1kg of electrons in a sphere

Postby gmalivuk » Tue Jun 02, 2009 4:30 am UTC

Since it's likely impossible to get this configuration anyway, let's bring in some science fiction.

In Niven's Known Space, the Slaver disintegrator tool suppressed the charge of either electrons or protons. If it did this for a kg of water at normal density, it would be given an energy of 8.1e26 J, or a bit more than twice the sun's total output per second.
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Re: 1kg of electrons in a sphere

Postby Cryopyre » Tue Jun 02, 2009 7:17 am UTC

how about the configuration where the proton suppressor was right next to the electron suppressor?
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Re: 1kg of electrons in a sphere

Postby Tass » Tue Jun 02, 2009 1:17 pm UTC

Vieto wrote:now, what would happen if 1kg of positrons appeared next to the 1kg of electrons? :kaboom:


You would get the energy in the form of gamma rays instead.

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Re: 1kg of electrons in a sphere

Postby gmalivuk » Tue Jun 02, 2009 4:03 pm UTC

Only it'd be far, far less energy than from either kg separately.
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Re: 1kg of electrons in a sphere

Postby tommclaughlan » Tue Jun 02, 2009 6:01 pm UTC

Assuming a uniform sphere, wouldn't they all radiate out in a sphere? In that case, depending on where you were standing, you'd get (your surface area)/4*pi*(distance to sphere) of the electrons hitting you?

The same would be true in the positron + electron sphere. Not all of them would annihilate, as some of the electrons/positrons would radiate straight up/down/left/right at a great speed. Eventually, of course, they'd attract and annihilate, in the absence of the rest of the charge/matter in the universe.
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Re: 1kg of electrons in a sphere

Postby JWalker » Tue Jun 02, 2009 6:09 pm UTC

Another thing to consider is that as the energy of a free electron increases, the probability of it striking another particle decreases (that is, the cross section of the interaction decreases). I don't know how much this will effect the outcome of the electron sphere on Earth, since 1kg is a LOT of electrons, but in general for causing damage high energy particles are not always better.

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Re: 1kg of electrons in a sphere

Postby gmalivuk » Tue Jun 02, 2009 7:44 pm UTC

Yeah, but remember that about half of them are going to hit the Earth, and nearly that many will be going through several miles of planet. If 1/3 hit anything before escaping into space, that's still the amount of energy the sun puts out in an entire year. Which means the planet is basically every bit as fucked as it would have been if all of them hit something.

(For comparison, Earth's gravitational binding energy is 2.24e32 J, so even 1% of the energy these electrons have would be enough to literally rip the planet apart completely.)
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Re: 1kg of electrons in a sphere

Postby tommclaughlan » Tue Jun 02, 2009 8:31 pm UTC

I agree, but at least we are half as fucked as we would be if someone fired a kilo of electrons at us.

The moral of the story is, if you see a kilogram ball of electrons, stand as far away from it as possible.
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Re: 1kg of electrons in a sphere

Postby JWalker » Tue Jun 02, 2009 8:47 pm UTC

Well, I'm not sure if the fraction of electrons interacting with the earth will be as high as one percent at this energy. Typical high energy interaction probabilities are extremely small, but we do have a lot of electrons and a lot of Earth, but likewise a lot of energy (or do we? I wonder what the energy per electron is for earth sized distances). Without doing more calculations, I'm not sure whether the energy deposited in the planet will be greater than the gravitational binding energy or not. I'm not saying that such a device will not be a doomsday device, just that it might not be death star like.

An additional thing to consider is that a lot of the energy in this configuration will be released in the form of photons (as well as a small contribution from other particles). Photons typically interact with matter even less than electrons do, depending on what energy they are. Its a very complicated problem to work out how devastating this thing would be. (Yet another thing, the Earth's magnetic field will tend to bend electrons away from the surface, further reducing the yeild)

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Re: 1kg of electrons in a sphere

Postby Cryopyre » Tue Jun 02, 2009 8:49 pm UTC

Won't the fast moving electrons produce a magnetic field? What would be the result of that?
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Re: 1kg of electrons in a sphere

Postby Diadem » Tue Jun 02, 2009 9:11 pm UTC

GoC wrote:Very nice!
Still a bit disappointing though...
Lets up the mass to say... ten tons or so and see if we get a black hole! And if so, what would the black hole "look like" (it's properties)?

Well calculating the required mass to create a blackhole is easy. Just use Schwarzschild's formula for the black hole radius: r = 2GM/c^2. Inverting that gives M = r c^2 / 2G. For a 1 cm radius (r = 0.01) that gives 6.8 * 10^24 kg. So rather a lot, 1 kg is not even close. But mass is energy. And energy is mass. The M in the above formula is not just rest mass, but total energy, expressed as a mass. So the coulomb energy stored in the electron configuration counts. 6.8 * 10^24 kg is about 6 * 10^41 Joules. The electrons were worth 3 * 10^34 Joules. So nope, not a black hole. But we're getting close, only 7 orders of magnitude off.

If I'm not mistaken the coulomb energy scales quadratically with mass. So ten tons would be just enough to do the trick and create a black hole.

This black hole would be charged. I'm not sure if it would have enough charge to form a naked singularity (of you pour enough charge into a black hole the event horizon disappears and weird things happen. This is most probably not possible in nature. But if you magically create electrons like that, well, who knows). It is made up purely of electrons, but most of the energy of course is potentional energy, so it's charge / mass ration might not be very high. I don't feel like doing the calculations at the moment ;)

Of course the black hole would quicky destroy the earth.


EDIT: Wait a minute... this isn't a white dwarf. Wouldn't the lack of neutrons and protons (and atomic structure in general) change things?

Nope. The above calculation was for the electron degeneracy pressure, which is all we needed for this case. For a white dwarf you'd continue by adding the rest-mass contribution (counting neutrons and protons too) and writing down an equation of state. Then plug that into the equations for relativistic hydrostatic equilibrium (which are derived from general relativity) and solve them to find properties such as radius for a given mass, core density, etc. A lot more complicated. But not needed for this calculation.
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Re: 1kg of electrons in a sphere

Postby gmalivuk » Tue Jun 02, 2009 9:33 pm UTC

tommclaughlan wrote:The moral of the story is, if you see a kilogram ball of electrons, stand as far away from it as possible.

Very very far away.

Figure a head-on cross-sectional area of something like .5m2 and a mass of 65kg, and we want our intrepid spaceman to survive.

At 1AU, a person would get nearly a GJ per kg, or 218.3 kcal/g, which would cause instant death.

At 10AU, this decreases a hundredfold, so there's only enough energy per gram to heat it up 2183K (being mostly water), which still means asploding.

At 1/100 of a light year, the energy might be survivable if it were purely thermal and was absorbed evenly by the entire body, instead of being ionizing and hitting the surface first. (If it were evenly distributed heat, it would warm a person by a bit more than half a degree C).

At 1/10 of a light year, an exposed person with those stats would still receive a dose of 22.84 Grays of beta radiation, if it's all absorbed, which would cause a coma and certain death within a day or two. (Unfortunately, I don't know how to figure out the energy individual electrons will likely have, so I don't know how much of that energy will end up being transferred to the person.)

To be down to the career radiation limit for astronauts, you'd need to be .24ly away, and even this is only slightly below the LD50 value for acute radiation.

Edit: I found a site that says high-energy electrons lose 2MeV per centimeter in water, which is what the body is made of to a first approximation. If each electron is way more energetic than that, figure it loses something like 8pJ on its trip through the body. There are about 10^30 electrons in this configuration, so now you'd only need to be 75000km away to get under 1 Gray of radiation.
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Re: 1kg of electrons in a sphere

Postby tommclaughlan » Tue Jun 02, 2009 10:18 pm UTC

We'd best start running!
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Re: 1kg of electrons in a sphere

Postby Squid Tamer » Tue Jun 02, 2009 10:25 pm UTC

I thought that I saw in some other science thread that it would take 3 kg of electrons to make a lightning bolt from here to the moon... So this gets pretty scary close.
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Re: 1kg of electrons in a sphere

Postby qbg » Tue Jun 02, 2009 11:50 pm UTC

gmalivuk wrote:The energy stored in such a configuration, if I've done the calculation correctly, would be 3.34x1034J, which is 2.8 times as much energy as the sun puts out in a year.

Wouldn't it be at most about 8.988x10^16 J (by E=M*C^2 for a mass of 1 kg)?

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Re: 1kg of electrons in a sphere

Postby You, sir, name? » Wed Jun 03, 2009 12:28 am UTC

Cryopyre wrote:Won't the fast moving electrons produce a magnetic field? What would be the result of that?


Maxwell's equations are telling me mighty trippy things about [imath]\vec{B}[/imath] when I stuff a radial current into it. But I'm too tired right now to make any serious calculations.

Did some hand waving to get a feel for the curl, and I think the current might actually cancel out the B field entirely. Consider a tiny segment of any [imath]d\rho[/imath] surface of the current: The segment segments neighboring this segment will, in the bordering region to this one, induce currents opposite to the one this segment induces... I think.
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Re: 1kg of electrons in a sphere

Postby Cynical Idealist » Wed Jun 03, 2009 12:35 am UTC

qbg wrote:
gmalivuk wrote:The energy stored in such a configuration, if I've done the calculation correctly, would be 3.34x1034J, which is 2.8 times as much energy as the sun puts out in a year.

Wouldn't it be at most about 8.988x10^16 J (by E=M*C^2 for a mass of 1 kg)?

That's if you convert the mass into energy directly. This energy is from the electrons pushing each other away.
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Re: 1kg of electrons in a sphere

Postby GoC » Wed Jun 03, 2009 1:10 am UTC

Diadem wrote:snip

Naked singularities! Woot! 8) :mrgreen:
Could you explain to me what a Cauchy Horizon is?
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Re: 1kg of electrons in a sphere

Postby Random832 » Wed Jun 03, 2009 1:29 am UTC

GoC wrote:Lets up the mass to say... ten tons or so and see if we get a black hole! And if so, what would the black hole "look like" (it's properties)?


No, you would not. The Schwarzchild radius required to get a black hole from 10 tonnes of mass is 7.43×10−24 m. To get one with a radius of 0.5cm, you would need a mass of 6.73×1024 kg. But to answer your question...

A black hole has three properties: Angular momentum, mass, and charge. I've told you the mass already; the charge of such a mass of electrons would be 1.18×1036 C. (The charge of 10 tonnes of electrons, for comparison, is 1.76×1015 C).

Diadem wrote:Well calculating the required mass to create a blackhole is easy. Just use Schwarzschild's formula for the black hole radius: r = 2GM/c^2. Inverting that gives M = r c^2 / 2G. For a 1 cm radius (r = 0.01) that gives 6.8 * 10^24 kg. So rather a lot, 1 kg is not even close. But mass is energy. And energy is mass. The M in the above formula is not just rest mass, but total energy, expressed as a mass. So the coulomb energy stored in the electron configuration counts.


Nice trick, but if that's the mass, then I'd say it's the mass of the ball of electrons, too*. Which I guess also means my estimate for the charge would have to actually be scaled down.

*Someone correct me if I'm wrong, I guess... Does energy like this have inertia, gravitational pull, etc? It's never really present in such dramatic quantities in nature.

Squid Tamer wrote:I thought that I saw in some other science thread that it would take 3 kg of electrons to make a lightning bolt from here to the moon... So this gets pretty scary close.
(Warning: I may have no idea what I am talking about)


Well - if the medium between earth and the moon were composed entirely of air at about atmospheric pressure, a lightning bolt has a potential difference of 3MV/m - so, given the distance between the surfaces of the two bodies is about 375000km, you've got about 1.13PV.

Given that the near vacuum is going to have a lot _higher_ resistance than air at atmospheric pressure, I'd guess the voltage is likely a lot higher.

For a there to be a 1.13PV potential difference between the earth and the moon, if we assume the earth to be neutral and the moon to be negatively charged... Well, that's easy. The capacitance between the earth and the moon (yes, you can find this online) is about 159 uF. So, simply multiply these together and you get 179 GC. Which amounts to a mass of [excess - or I suppose deficient if the object is positively charged] electrons of...

dun dun dunnn.....
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ONE KILOGRAM! Wow, you can't make this up.

1.02 kg, to be more precise than we can probably get away with at this point, especially after the massive handwave of assuming space to be filled with air.

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Re: 1kg of electrons in a sphere

Postby gmalivuk » Wed Jun 03, 2009 4:57 am UTC

JWalker wrote:Typical high energy interaction probabilities are extremely small, but we do have a lot of electrons and a lot of Earth, but likewise a lot of energy

Sure, but wouldn't the electrons still lose energy as they go through matter, even if it takes them a long time to slow down enough to actually get captured by anything? And that energy will still do lots of damage when there's so much of it.
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Re: 1kg of electrons in a sphere

Postby qbg » Wed Jun 03, 2009 5:29 am UTC

Cynical Idealist wrote:
qbg wrote:
gmalivuk wrote:The energy stored in such a configuration, if I've done the calculation correctly, would be 3.34x1034J, which is 2.8 times as much energy as the sun puts out in a year.

Wouldn't it be at most about 8.988x10^16 J (by E=M*C^2 for a mass of 1 kg)?

That's if you convert the mass into energy directly. This energy is from the electrons pushing each other away.

Well, here is what I'm thinking (please correct any errors):
Suppose by magic we have this structure pop into existence. This structure contains X amount of energy from the repulsion between the electrons, Y amount from the electron's rest mass equivalence, and Z amount from the other energies the electrons have. Thus the total amount of energy we popped into existence is X+Y+Z, and so we have popped into existence the equivalence of that sum in mass. Would then upper limit not by the one I gave above, or am I measuring the mass of the sphere incorrectly?

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Re: 1kg of electrons in a sphere

Postby skeptical scientist » Wed Jun 03, 2009 5:58 am UTC

I think people were taking "1 kg of electrons" to mean "electrons with rest mass 1kg".
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Re: 1kg of electrons in a sphere

Postby GoC » Wed Jun 03, 2009 11:57 am UTC

skeptical scientist wrote:I think people were taking "1 kg of electrons" to mean "electrons with rest mass 1kg".

Yep.
Belial wrote:I'm just being a dick. It happens.

Random832
Posts: 2525
Joined: Wed Oct 10, 2007 4:38 pm UTC

Re: 1kg of electrons in a sphere

Postby Random832 » Wed Jun 03, 2009 1:19 pm UTC

I think the question is one i asked earlier - does "mass" that's in the form of energy other than rest mass contribute to the inertia and/or gravitational pull of a system? (I'd think it would contribute to the gravitational pull, if it counts for whether it will form a black hole)


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