Jeans Mass and Central pressure

[qraal]

Hello Forumites

In Karl Schroeder's Virga series, the Virga he describes as a vast air-filled space in which various "steam-punk" societies exist. Electronics are artificially stopped from working inside Virga, to preserve a low-tech sample of humanity. In pondering such a construct I wondered just how big it could get before running into physical limits. The most obvious is the Jeans Mass - when the gravitational self-attraction overwhelms sound waves restoring pressure/density perturbations away from equilibrium. The Virial Theorem says an equilibrium system is on the verge of self-collapse when...

2 Kinetic Energy = Ω

...Ω being the gravitational binding energy, in the case of a uniform density ball that's

3/5.G.M²/R

and, by the ideal gas law, the 2.KE = 3.N.k.T, N being the total number of particles. Assuming a constant mix of particles through out, N = M/μ, where M is the total mass and μ is the mean molecular mass. So 2.KE = 3M.(kT/μ) and the mass is 4/3.π.R³.ρ, where ρ is the density.

According to the ideal gas law

P.V = N.k.T

and N = M/μ, and since ρ = M/V, we can derive P/ρ = k.T/μ

which means 2.KE = 3M.P/ρ

For the Virga if we take P and ρ to be their sea-level standards (101,325 Pa and 1.225 kg/m³), that means we can compute

R² = 15/(4πG).P/ρ²

so R_max ~ 34,750 km

Here's the thing. The central pressure of a uniform ball, from its gravity alone, is

3/(8π).G.M²/R⁴

which gives us the central pressure as 253,312 Pa for the figures we just derived. How can that be right? The ratio is 5/2. To compare...

P_c = 6/9.π.G.ρ².R² (gravitational central pressure of uniform sphere)

P_j = 4/15.π.G.ρ².R² (...call it the Jeans Pressure)

I would think the stable scenario would be when P_c = P_j, but I suspect I am missing something subtle. I have assumed the system is isothermal (i.e. the interior is being heated to match radiative losses from the surface.)

So can anyone explain it to me?

 

[cepheid]

I know that for something to be supported against gravitational collapse (in hydrostatic equilibrium), there has to be a pressure *gradient*, meaning that the pressure increases as you move from the outside to the centre. You have assumed an ideal gas, but you have also assumed a constant density. Problem: the equation of state for an ideal gas says that if T = const, then P ~ ρ, and hence constant ρ ==> constant P. But we just said that P cannot be constant in hydrostatic equilibrium. Conclusion: an isothermal ball of (ideal) gas in hydrostatic equilibrium cannot have a constant density throughout. Could this be the issue?

 

[qraal]

I know that for something to be supported against gravitational collapse (in hydrostatic equilibrium), there has to be a pressure *gradient*, meaning that the pressure increases as you move from the outside to the centre. You have assumed an ideal gas, but you have also assumed a constant density. Problem: the equation of state for an ideal gas says that if T = const, then P ~ ρ, and hence constant ρ ==> constant P. But we just said that P cannot be constant in hydrostatic equilibrium. Conclusion: an isothermal ball of (ideal) gas in hydrostatic equilibrium cannot have a constant density throughout. Could this be the issue?

I had wondered the same. I think the problem is that I assumed the Jeans Mass is stable - it's not. Numerical simulations (using SPH) of Jeans Mass gas blobs will collapse inexorably, as will slightly sub-Jeans Mass blobs, but at about ~60% Jeans Mass the blobs disperse. Thus I think the maximum stable size is probably when Pc = P(average), so that the speed of sound can dampen out any perturbations from equilibrium. That means M(max) is roughly at (2/5)^(3/2) of Mj, but that's a conjecture, not a proof.

 

[harcel]

What simulations show to collapse or not is not very relevant for the question. The idea is that the Jeans mass is indeed the border between stability and collapse.

Cepheid was right: a sphere of gas is not in hydrostatic equilibrium if the density is constant.

 

[qraal]

Putting numbers to my conjecture, that means a Virga style air-enclosure is 44,000 km across, using a standard N2/O2 mix to breathe. That's quite immense - Karl Schroeder settles for just 5,000 km in the books. I had wondered about just how strong the structure had to be to avoid bursting and derived some figures on using gravitational pressure of the enclosure itself to counterbalance the air-pressure - it works, but the material required is immense. Yet assuming we could ramscoop the Sun and sift for metals (astrophysical metals) there's ~20 Jupiter masses, to make stuff, available.