The 15 Million Degree Pinhead: The Sun's Core Materialized on Earth

[webboffin]

Is it true that a pinhead size if the Sun's core if materialised on the Earth would be too dangerous to go within 90 miles of it?

I read from Wikipedia that the energy produced in the Sun's core is about 270 watts/m3. About the same as an active compost heap. I know the pinhead of core matter would be 15 million degrees but be only be around a cube around 1mm.

 

[Matterwave]

A pinhead sized piece of the Sun's core would quickly blow apart if removed from the Sun (assuming it kept its temperature, as I'm guessing is the case in your scenario). It's far too hot for the small amount of gravitational force to keep together. But apart from it's temperature, there's nothing super exotic about the Sun's core, unlike, for example, white dwarf material or neutron star material.

We can estimate the energy contained in a 1mm cube of the Sun's core, to see how big of an explosion it would make. The Sun's core is at ~15 million Kelvin, and a density of ~100g/cm^3. Given that the average thermal energy is ~1.5NkT, then:

$$E\approx\frac{3}{2}N_Ak(15,000,000K)(100g/cm^3)(1mm^3)\approx 19,000 J$$

Which is not much. It's equivalent to about 5 grams of TNT. I expect this tiny amount of material to basically make a pop and just fizzle up in the air. Safe distance is maybe 10 meters or maybe 100 meters just to be extra safe...probably not the 90 miles though...

 

[webboffin]

A pinhead sized piece of the Sun's core would quickly blow apart if removed from the Sun (assuming it kept its temperature, as I'm guessing is the case in your scenario). It's far too hot for the small amount of gravitational force to keep together. But apart from it's temperature, there's nothing super exotic about the Sun's core, unlike, for example, white dwarf material or neutron star material.

We can estimate the energy contained in a 1mm cube of the Sun's core, to see how big of an explosion it would make. The Sun's core is at ~15 million Kelvin, and a density of ~100g/cm^3. Given that the average thermal energy is ~1.5NkT, then:

$$E\approx\frac{3}{2}N_Ak(15,000,000K)(100g/cm^3)(1mm^3)\approx 19,000 J$$

Which is not much. It's equivalent to about 5 grams of TNT. I expect this tiny amount of material to basically make a pop and just fizzle up in the air. Safe distance is maybe 10 meters or maybe 100 meters just to be extra safe...probably not the 90 miles though...

Thanks and as the only reply and a scientific one it points to a real result.

 

[Drakkith]

I read from Wikipedia that the energy produced in the Sun's core is about 270 watts/m3. About the same as an active compost heap. I know the pinhead of core matter would be 15 million degrees but be only be around a cube around 1mm.

This only refers to the energy produced by nuclear fusion. As Matterwave said, the material is very dense and very hot, and would have a lot of thermal energy that would be released.

 

[webboffin]

This only refers to the energy produced by nuclear fusion. As Matterwave said, the material is very dense and very hot, and would have a lot of thermal energy that would be released.

But would the energy of a 1mm cube of matter really be dangerous even if very hot since it is very small volume and E=MC2 would make this assumption unlikely. If a single 1mm cube of matter would potentially cause such devastation then how would the Sun's gravity contain many thousand of Earth masses of Sun's core without blowing itself apart. 15 million kelvin has been achieved in the lab without need to evacuate cities.

What I want really from this question is accurate explanation in physics to debunk this "urban legend" that has been going around as fact in several publications.

 

[Drakkith]

But would the energy of a 1mm cube of matter really be dangerous even if very hot since it is very small volume and E=MC2 would make this assumption unlikely.

No, the energy content of a single cubic mm would not be very dangerous, as matterwave said. We don't need to invoke e=mc², we just need to find out how much energy is contained as heat, which Matterwave has already done.

If a single 1mm cube of matter would potentially cause such devastation then how would the Sun's gravity contain many thousand of Earth masses of Sun's core without blowing itself apart. 15 million kelvin has been achieved in the lab without need to evacuate cities.

The core is under immense pressure from the outer layers of the star pressing down on it. This immense pressure compresses and heats the core to 15 million k, resulting in fusion. But you are correct. Just because we can heat something up to 15 million k doesn't mean we need to evacuate the surrounding area. Most terrestrial fusion reactors are burning, at most, a few grams of material at once, if not less. In addition, we have to actively work to keep the plasma at 15 million k since it wants to cool off. Getting fusion power to work here on Earth is very difficult.

What I want really from this question is accurate explanation in physics to debunk this "urban legend" that has been going around as fact in several publications.

What urban legend? The "don't get within 90 miles of a 1mm³ section of the Sun's core"? I believe Matterwave already gave you the answer. There's just not that much energy in a 1 mm³ section of the Sun's core. Even accounting for the heat and the violent decompression the material would undergo, you're still looking at a very small amount of material, only 0.1 grams.

 

[Matterwave]

But would the energy of a 1mm cube of matter really be dangerous even if very hot since it is very small volume and E=MC2 would make this assumption unlikely. If a single 1mm cube of matter would potentially cause such devastation then how would the Sun's gravity contain many thousand of Earth masses of Sun's core without blowing itself apart. 15 million kelvin has been achieved in the lab without need to evacuate cities.

What I want really from this question is accurate explanation in physics to debunk this "urban legend" that has been going around as fact in several publications.

In fact a 1mm cube of matter CAN be really dangerous, if the matter is exotic enough! The Sun's core is NOT exotic enough, but something like a neutron star IS! A neutron star is basically at nuclear densities: $\rho\approx 10^{14}g/cm^3$. A 1mm cube of neutron star matter would have mass:

$m=\rho(1mm^3)\approx 10^8 kg$

That's 100,000 tons of material! Let's look at the energy content of this material. A neutron star is entirely degenerate matter at a temperature of ~1 million Kelvin or higher. Because the neutron star is totally degenerate, it's actual energy content will NOT be $3/2NkT$ but will be $3/2NkT_F$ where $T_F$ is the "Fermi temperature" (rather than the actual temperature). But, we know that $T_F>>T$ for highly degenerate matter, so we can use ~1 million Kelvin as a LOWER BOUND for the energy calculation. Assuming Neutron star matter is made entirely of neutrons, then the lower bound energy contained in 1 cubic mm of neutron star matter would be:

$E>\frac{3}{2}k(1,000,000K)(\frac{10^8 kg}{1 amu})\approx 1.3\cdot10^{18}J$

This gave us a LOWER BOUND of the energy (probably several orders of magnitude lower!) contained within this pin drop sized piece of neutron star matter to be 10^18 Joules! That's equivalent to 240 MEGATONS of TNT! This LOWER BOUND is 5 times bigger than the Tsar Bomba's (largest fusion bomb ever created) yield. I would expect the ACTUAL energy to be at least 10 times higher. THAT's devastation right there!

 

[webboffin]

Studying the responses of Matterwave and Drakkith has brought together what I wanted overall. I understand that the Sun's core itself would by a pinhead size worth on Earth not cause much damage but by Matterwave's contribution about exotic matter densities would be more disastrous, I can accept a neutron star is certainly of exotic properties. Importantly, I can safely say the equivalent of 5g of TNT wouldn't bring a garden wood shed down never mind causing devastation over a 90 mile radius.

 

[Damo ET]

Just to bring a laypersons thought to the OP. There is no doubt that matterwave and Drak are all over the correct happenings with this, and their conclusions are accurate, but one thing to keep in mind webboffin, the people that speculate these 'what ifs' aren't scientifically minded and pose the question/fact in a way which violates multiple laws in the process. I too remember a factoid based on your OP, and I think it was meant to imply that a grain of sand or (insert small sized object here) at the temperate of the core of the sun (that doesn't follow any of the laws which should govern it) would burn anything within (set distance) by its thermal radiation alone.


Damo

 

[pervect]

A pinhead sized piece of the Sun's core would quickly blow apart if removed from the Sun (assuming it kept its temperature, as I'm guessing is the case in your scenario). It's far too hot for the small amount of gravitational force to keep together. But apart from it's temperature, there's nothing super exotic about the Sun's core, unlike, for example, white dwarf material or neutron star material.

We can estimate the energy contained in a 1mm cube of the Sun's core, to see how big of an explosion it would make. The Sun's core is at ~15 million Kelvin, and a density of ~100g/cm^3. Given that the average thermal energy is ~1.5NkT, then:

$$E\approx\frac{3}{2}N_Ak(15,000,000K)(100g/cm^3)(1mm^3)\approx 19,000 J$$

Which is not much. It's equivalent to about 5 grams of TNT. I expect this tiny amount of material to basically make a pop and just fizzle up in the air. Safe distance is maybe 10 meters or maybe 100 meters just to be extra safe...probably not the 90 miles though...

I think there is more energy than that. There should also be the energy of compression, like that stored in a spring, as well as the thermal energy. It should be proportional to P*V. I don't know what the constant of proportionality is, though, you'd have to integrate Pdv over the equation of state.

If P = 250 billion bars, and V = 1 mm^3 (looked up on the internet, please correct if they're too far off), PV = 25 megajoules, so it's probably more than the thermal energy (but I"m not sure what the constant factor is).

However, assuming the constant is 1 or less, I don't think you'd have to stay 90 miles away to be safe, even so.

 

[Matterwave]

I think there is more energy than that. There should also be the energy of compression, like that stored in a spring, as well as the thermal energy. It should be proportional to P*V. I don't know what the constant of proportionality is, though, you'd have to integrate Pdv over the equation of state.

If P = 250 billion bars, and V = 1 mm^3 (looked up on the internet, please correct if they're too far off), PV = 25 megajoules, so it's probably more than the thermal energy (but I"m not sure what the constant factor is).

However, assuming the constant is 1 or less, I don't think you'd have to stay 90 miles away to be safe, even so.

Hmm...by the perfect gas law PV=NkT, so shouldn't this be in the same order? How did it get 3 orders of magnitude higher? PV is the (Landau? Gibbs?) free energy (I can never remember which is which), I wanted to calculate the actual energy of the system, which, for an ideal gas should be entirely encompassed in E=3/2NkT.

But maybe I did my calculation wrong...

Oh, you know what, you're right, I messed up and multiplied by Avagadro's number instead of dividing by 1 amu. I guess I got confused.

Using the right formula:

$$E=\frac{3}{2}\frac{100 g/cm^3}{1 amu}k(15,000,000K)(1 mm^3)\approx 19,000,000J$$

Off by a factor of 1000! Thanks for catching my mistake! This is equivalent to 4.5kg of TNT, which although much more than the previous estimate of 5 grams, is still not Earth shattering.

 

[Jonathan Scott]

A similar point has just turned up in the Physics Forum facebook posts, which led me to search the forums and find this thread. It appears that the original intent of the idea has been missed in this thread, although it was not very clearly described.

I presume that the original point is based on the Stefan-Boltzmann law, in that that the black-body radiation from a 1mm cube at a temperature of 16 million Kelvin would certainly fry a human at a very great distance. The idea does NOT take into account how the energy would be supplied to maintain this temperature, which is a completely different matter. Certainly, the idea of a "grain of sand" or "pinhead" at that temperature is just silly.

If I assume a temperature of 16 million K and a cube (with six square faces) of size 1mm, the total power output according to the Stefan-Boltzmann law is approximately as follows: $$ \sigma T^4 \times \mbox{area of cube} =
5.67 \times 10^{-8} * (16\times 10^6 )^4 * (10^{-3})^2 * 6$$ which Google calculator tells me is about $2.2 \times 10^{16} W$. Dividing by the solid angle, I make it that you'd need to be somewhat more than 1000km away for the power per square metre to be merely equivalent to direct sunlight, at about $1400W/m^2$. I don't know what the power would need to be to kill a human, but I can quite believe that it could do so at 150km or 90 miles.

 

[Ken G]

And to add to the excellent and correct posts above, some readers might be wondering how a pinhead of solar core could both produce about the amount of heat of a similar amount of compost heap (i.e., next to nothing), and this vast number of watts of radiative energy! The answer is that light takes a very long time to escape from the core of the sun (about a hundred thousand years), so a compost-heap kind of heat source can build up a very bright radiation field. However, this means that virtually all of the watts of radiated energy coming from that pinhead in Jonathan Scott's analysis are simply re-radiated from the super-bright radiation field inside the core of the Sun. Like he said, it's all an issue of whether or not there is any way to maintain that temperature-- the pinhead in the core of the Sun easily maintains that temperature because it is being irradiated from all sides by a super-bright radiation field that takes a hundred thousand years to build up at the compost-heap rate of 270 W per cubic meter, but if you took that pinhead to Earth, where it is not so irradiated, it would cool down so fast that it would end up only releasing the energy in Matterwave's analysis, and due to expansion most of that would probably not go into radiation it would go into the explosion. So the bottom line is, although the urban legend that a pinhead of solar core seems dangerous on Earth is not completely untrue because 5 kg of TNT is nothing to scoff at, to get really super-dangerous results requires a confusion between the very effect that makes the radiation field in the Sun so bright (that it takes so long to escape) and the claimed effect that makes the pinhead dangerous (that this energy can be radiated out very quickly). You can't have it both ways, so the real truth is that the pinhead had only the explosive danger of its thermal energy content-- what it is current radiating inside the Sun is not relevant in any other environment.

(Of course, there is another calculation we can do-- the fusion energy that can be released by a cubic mm of such matter. We can estimate that by knowing that the Sun will fuse this material for about 10 billion years, at a rate not much different from its current 270 W per cubic m, so if we have a billionth of a cubic meter, times 10 billion years (and a year is 30 million seconds), it comes out about 10¹¹ Joules, which dwarfs its thermal energy, no pun intended. However, to get that energy requires putting it in a very different environment from Earth, it would need to be placed in an environment where the fusion is greatly speeded up, whereas on Earth it would expand and cease fusing almost instantly. So if that much hydrogen were part of an H-bomb, yes, you might want to keep your distance, but if H-bombs are dropping, I'm not sure where you go.)

 

[Drakkith]

Dividing by the solid angle, I make it that you'd need to be somewhat more than 1000km away for the power per square metre to be merely equivalent to direct sunlight, at about 1400W/m21400W/m^2. I don't know what the power would need to be to kill a human, but I can quite believe that it could do so at 150km or 90 miles.

Well, at 15 million K most of the radiation certainly isn't anywhere close to the visible light range, but in very high energy x-rays. I'm not sure how well those propagate through the atmosphere. Plus there's also the fact that this cube of hot plasma is going to cool off very quickly, so even though it radiates 2.2x10¹⁶ watts, this will only happen for a tiny fraction of a second. I think you'd need to find the total energy emitted to know if someone would survive or not.

 

[snorkack]

Over how long distance is Sun´s core transparent?
If it is transparent over pinhead size then the light flash will last the time it takes for light to travel through pinhead... like 3 ps.

 

[Chronos]

Assuming you could instantaneously teleport 1cc of the sun's core to earth, it would have devastating consequences. Fortunately, we don't live in that universe. As soon as it was liberated from its gravity coffin, so would all that pent up gravitational energy. The sun is plasma, it takes thousands of years for a photon emitted by the core to reach the surface of the sun. Boom goes the dynamite is the short story.

 

[Matterwave]

Well, at 15 million K most of the radiation certainly isn't anywhere close to the visible light range, but in very high energy x-rays. I'm not sure how well those propagate through the atmosphere. Plus there's also the fact that this cube of hot plasma is going to cool off very quickly, so even though it radiates 2.2x10¹⁶ watts, this will only happen for a tiny fraction of a second. I think you'd need to find the total energy emitted to know if someone would survive or not.

The total energy emitted could only roughly be the thermal energy in my post #11 above. The tiny cube of matter would cool off far too quickly to release any significant amount of fusion energy. The rate of the p-p process, as Ken alluded to in his post, is a very slow one. Thus, the time that this cube could possibly irradiate at the level of 10^16 watts would be something like 10^7/10^16=10^-9 seconds. Of course, what would actually happen is the cube would blow apart and release most of the energy, as Ken said, in the explosion and not in the radiation. If you could keep the 1mm hot via some external energy source then yes, it would be very dangerous. But in order to do so you would basically have to be pumping in the entire 10^16 watts of power as the energy reserve in that core is a tiny fraction of that. It is then this machine that is able to output 10^16 watts that is the dangerous article, and not this pin-prick sized version of the Sun's core.

 

[Ken G]

Yes that's the point exactly-- it would require a real bait-and-switch, like a con man's perpetual motion machine, to make that thing more dangerous than a small bomb.

 

[Drakkith]

The total energy emitted could only roughly be the thermal energy in my post #11 above.

Ahhh, thanks for that. I'd forgotten that you'd already calculated the energy.