Drops of Jupiter

I heard a song the other day. Half the lyrics sounded like the sort of meaningless but positive-sounding things you expect out of song lyrics, and the other half were astronomy. Bad astronomy. I utterly failed to take the song seriously, because whoever it’s being sung to is very extremely dead.

“She’s back in the atmosphere with drops of Jupiter in her hair.”
Believe it or not, saying “drops of Jupiter” actually makes sense. There exists a substance that, while not unique to Jupiter, is characteristic of it. It makes up a large fraction of the planet and is in fact liquid. Perfect. That material is metallic hydrogen. So naturally the question is, what happens if a drop of metallic hydrogen from Jupiter were to appear in someone’s hair?

Not pictured: Safety.

At the shallowest relevant depth in Jupiter’s atmosphere (there are much bigger numbers available, but we want to give the lyric-writers the benefit of the doubt), the temperature is 10,000 Kelvin, and the pressure is hundreds of thousands of gigaPascals, millions of times Earth’s atmospheric pressure. We’re talking center-of-the-earth type pressure and temperature hotter than the surface of the Sun. It should go without saying that you do not want that in your hair.

Wolfram|Alpha tells me that a “metric drop” is a unit of volume. I needed a number, so I may as well use the actual one. So this person has a couple of drops of metallic hydrogen in her hair, each drop being .05 cm^3. To find the actual amount of hydrogen in each drop, PV=nRT. Yes, that’s an equation for gases not liquids, but this temperature and pressure is a transition point where it’s going between those phases, so I’m hoping it’s still valid to use it here. (Despite the term “metallic,” this is kind of like a gas and kind of like a liquid and not remotely solid.)

If this equation doesn’t work here because the material is too liquid, well, the liquid phase is obviously denser than the gas anyway, so this will be an underestimate rather than an overestimate. So just pretend I added another clause on to each paragraph saying that it’s at least this bad.

PV=nRT. (2*10^8 kPa)*(.05 cm^3)=n*(8.31×103 cm3*kPa*K-1*mol−1)*(10000 K). Solve for n, and it’s 0.12 moles of hydrogen. Thanks to hydrogen’s excessively convenient molar mass, that’s also 0.12 grams. That…doesn’t sound so bad. Until you remember the temperature and pressure it’s at.

The first thing that happens is that this small drop immediately and explosively expands until it’s at the same pressure as the air around it. Since the pressure is dropping by a factor of 2*10^8 kPa/101.3 kPa =1974000, the volume increases by the same factor. So that’s how many metric drops of volume it occupies, and that means it takes up about three and a half cubic feet. Not a huge amount, but enough that when the explosion happens right next to someone’s head, well, it does not go well for them. I might run those numbers later.

The next thing that happens is that the hydrogen sublimates and cools down. It goes from being in its metallic liquid phase to being diatomic gaseous, and then since it’s surrounded by air that’s cooler than it is, its temperature decreases. Hydrogen has a specific heat of 14.3 J/g/K, meaning it takes 14.3 Joules of energy to raise the temperature of one gram of hydrogen by one degree Kelvin. Conversely, for each degree Kelvin that the temperature decreases, there are 14.3 Joules per gram of hydrogen released. That energy has to go somewhere.

The specific heat is actually higher, because these are some pretty extreme temperatures and it’s not really a constant. But I’ll round down so the numbers are simpler, and add a third “it’s actually worse than this” disclaimer. We’ve got 0.12 grams here, so each degree of temperature change sends 1.716 Joules of energy into the air or whatever else is unfortunate enough to be close.

Since the temperature starts at a staggeringly high 10000 K, and is going down to regular Earth atmosphere temperatures like 300 K (80º F), that’s a change of 9700 K. So it’s releasing more than 16.6 kJ of energy in the form of heat to whatever happens to be nearby. That’s surprisingly small; Wikipedia tells me it’s about the energy contained in nine M16 rifle cartridges or four grams of TNT, but to be honest I was expecting something more apocalyptic. But it does happen to be centered on a human head, which is bad.

A human brain masses about 1400 g, and has a specific heat of 3.64, so if all the heat were going in to the brain and were evenly distributed, it would only raise the brain temperature a few degrees. Which, sure, could kill the person. And yes, it won’t be evenly distributed. The heat will be concentrated at the skin near the drop. All the skin on that side of the head will be instantly destroyed and I don’t know what will happen to the skull. But it’s a lot less bad than it could have been; I was expecting the heat to leave her head completely vaporized.

Well, that was less bad than expected. Let’s stop worrying about the heat and look back at the force of the explosion.
The center of the blast, of course, was close enough to the person’s head to be described as “in her hair,” so a significant fraction of the incredibly hot exploding gas is going to come into contact with her head. It’ll be less than half, but in about that ballpark.

A sphere with a volume of one drop has surface area 6.56*10^-5 square meters. The pressure in that drop was 2*10^8 kPa= two hundred billion Newtons per square meter. So the force being directed headward is about half of the product of those. (Can I use these numbers this way? It seems like I should be able to.) If that much force hits the head of the person in question, that could be millions of Newtons. Like, the kind of force like when you’re minding your own business and suddenly there’s a blue whale on top of your head. That kind of force. It’s even worse than I was expecting.

To make things even worse, the singer said drops. Plural. It is very safe to say that the subject of this song no longer has a head, and the rest of her body is probably obliterated as well. So, as a general rule of thumb, you should probably keep drops of Jupiter away from your hair.

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2 thoughts on “Drops of Jupiter

  1. Glomerulus

    This is a nice analysis! But be careful, I think you missed something: the heat shouldn’t be as bad as you say. Expanding explosively cools you down a lot, because you’re doing work on the atmosphere: specifically 3.5 cubic feet * 1 atm = 10kJ of work. So there’s ‘only’ around 6.6 kJ of heat, not 16.6 kJ. Still not something I’d want near my head!

    Reply

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