- #1
qraal
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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.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 Rmax ~ 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...
Pc = 6/9.π.G.ρ2.R2 (gravitational central pressure of uniform sphere)
Pj = 4/15.π.G.ρ2.R2 (...call it the Jeans Pressure)
I would think the stable scenario would be when Pc = Pj, 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?
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 Rmax ~ 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...
Pc = 6/9.π.G.ρ2.R2 (gravitational central pressure of uniform sphere)
Pj = 4/15.π.G.ρ2.R2 (...call it the Jeans Pressure)
I would think the stable scenario would be when Pc = Pj, 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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