Is there truth about a pinhead size of the core of the Sun?

by webboffin
 P: 24 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.
 Sci Advisor P: 2,713 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...
P: 24
 Quote by 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...
Thanks and as the only reply and a scientific one it points to a real result.

Mentor
P: 11,592
Is there truth about a pinhead size of the core of the Sun?

 Quote by webboffin 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.
P: 24
 Quote by Drakkith 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.
Mentor
P: 11,592
 Quote by webboffin 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.
P: 2,713
 Quote by webboffin 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!
 P: 24 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.
 P: 89 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
Emeritus
P: 7,599
 Quote by 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...
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.
P: 2,713
 Quote by pervect 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.

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