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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.M

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.R

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/m

R

so R

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

3/(8π).G.M

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

P

I would think the stable scenario would be when P

So can anyone explain it to me?

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.M

^{2}/Rand, 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.R

^{3}.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/m

^{3}), that means we can computeR

^{2}= 15/(4πG).P/ρ^{2}so R

_{max}~ 34,750 kmHere's the thing. The central pressure of a uniform ball, from its gravity alone, is

3/(8π).G.M

^{2}/R^{4}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.ρ^{2}.R^{2}(gravitational central pressure of uniform sphere)P

_{j}= 4/15.π.G.ρ^{2}.R^{2}(...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?

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