J is for Jeans Mass and Jeans Length

Essentially, the Universe is a bunch of hydrogen floating about and occasionally clumping together.

You're like, "Dude! No way!" And I'm like, "Way!" And you're all, "Nuh-uh!" And I'm all, "Yuh-huh!" And you go, "But it can't be just hydrogen. There's clumps of stuff!" And Sir Issac Newton goes, "Why's there clumps of stuff? There shouldn't be (but there is! I see clumps of stuff!)" and James Jeans said, "Strewth, there's clumps, because science!"

And he went on to explain how the clumps came to be.

See, our buddy Issac knew just enough about science to question the issues regarding stuff (in this case, hydrogen) clumping together.

So, we've got a whole lotta hydrogen floating about out there in a giant molecular cloud of the stuff. As you know, as soon as a hydrogen atom (or molecule) gets close enough to another one, gravity draws them together.  Get a whole lotta atoms/molecules together, and gravity draws them in. Ah, but here was Newton's quandary: Get enough of them together in a dense enough clump, and the gas pressure would force them away from each other (like how a balloon gets puffy when you fill it). Yet, we've got lots of lumps of stuff. How did that happen?

Our buddy Jimmy figured that if there were clumps, there had to be a reason. He worked out that there was a certain point where, if you got enough atoms togethers, their combined gravity would be stronger than gas pressure, and clumps would happen.

When gravity is equal to or lesser than the gas pressure, nothing happens. When the gravity is stronger, then you get clumps of stuff like stars. The greater the mass of the cloud, the smaller its size, and the colder its temperature, the more unable it will be to resist gravitational collapse. (Temperature? Sure Temperature causes pressure, which is why a boiling kettle whistles while a cold one does not. If your cloud's got a fever, it's gonna need more mass to collapse.)

There was a certain point that this swapover happens. We call that the Jeans Mass (after our buddy Jimmy).

Jeans Length is the radius of a cloud where this collapse will start to happen.

And this is how baby stars are born. As we already know, stars are powered under fusion, which drives nucleosynthesis, and when the stars die and go boom (for various kinds of boom), it scatters all its soot and ash, which then clumps together into rocks and stuff through a process called accretion.

Hey nerd! Go hardcore and do the math:

Jeans Mass:

kB = Boltzmann's constant
T = temperature of the cloud (Kelvin)
r = radius of the cloud
μ = mass of your hydrogen atom
G = gravitational constant
ρ = cloud's mass density (cloud mass divided by cloud volume)

(Wanna cheat?)

Jeans Length:

Same variables as above, with k being the Boltzmann's constant and ρm being the cloud's mass density.

Does math make things easier or harder for you?

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Her Grace does not like math, but she will do it all the same.