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Let's assume for the sake of this question that the 1x1x1 cube was scooped by unknown methods, and is instantly brought to somewhere close to Jupiter's orbit without any harm or any change to nearby planets, their conditions and their orbits. What would likely happen to this cube?

Would the colder temperatures at Jupiter's orbit cause the plasma of the cube to reorganize into atoms of Hydrogen and Helium and what not? Would it continue to stay as Plasma, and explosively spread away? Would it still continue to emit light?

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It would exapand and cool.

The sun's surface (however that is defined) is a mixture of hydrogen and helium plasma in a ratio of about 3:1. It is plasma because it is hot.

If you moved some to somewhere cold it would cool down. There is no magic in this, it is essentially the same as a hot cup of coffee cooling down.

As it cools it would return to being gas, the electrons in the plasma would reconnect with the atoms, and the hydrogen atoms would react to form $H_2$. Again there is nothing surprising about this, no more than water becoming ice when put in the freezer. If you put plasma somewhere cold, it transfers heat to its environment, and changes state.

It would be explosive, simply because a cube of plasma at pressures of $10^{-3}$ bar (the surface pressure of the sun) and temperatures of 5500 degrees would expand explosively if placed in space, but it wouldn't be very explosive, as there isn't much material, so not much energy.

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Plasma is extremely hot gas. In the sun it is compressed by the high gravitational forces. Therefore nuclear fusion is possible. If the plasma would be taken out it would expand and cool down.
It would definitely not stay plasma as there would not be enough pressure and heat.

As the nuclei cool down they would "take back" their electrons and become normal atoms again. The remains would be a cloud of hydrogen/helium

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    $\begingroup$ On the surface (however you define it) it is neither dense enough nor hot enough to sustain fusion $\endgroup$ Jul 15 at 11:41

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