1. Jeans instability
Thermal pressure resists gravity on small scales; on large scales gravity wins. A perturbation in an isothermal gas grows when it is larger than the Jeans length, with sound speed cs = sqrt(kT / (μ mH)), μ = 2.33 for molecular gas:
lambda_J = c_s * sqrt(pi / (G rho))
M_J = (4 pi / 3) rho (lambda_J / 2)^3 (mass in a sphere of diameter lambda_J)
t_ff = sqrt(3 pi / (32 G rho)) (collapse time with no pressure)
Cold, dense gas has a small MJ (T = 10 K, n = 104 cm-3: 2.9 M☉ with this definition). Other textbooks use factors up to ~2 different, so only the scaling MJ ∝ T3/2 ρ-1/2 is universal.
2. Why the simulation threshold is not exactly MJ
A finite cloud held by outside pressure collapses below the infinite-medium MJ. Bonnor–Ebert (1955/56) gave the exact critical mass of a pressure-bounded isothermal sphere:
M_BE = 1.18 c_s^4 / (G^(3/2) P_ext^(1/2))
The experiment button measures the hydrodynamic threshold of a uniform sphere with Pext = ρ cs2 and prints it next to MBE, MJ and the virial-equilibrium limit.
3. The hydrodynamics code
The cloud is split into 160 Lagrangian mass shells (fine near the centre). Each shell edge obeys
d2r/dt2 = -G M(<r)/r^2 + j^2/r^3 - 4 pi r^2 (dP/dm)
P = rho c_s^2 [1 + (rho/rho_c)^(2/3)], rho_c = 1e-13 g/cm^3
The bracket is the standard barotropic equation of state: isothermal while the gas radiates freely, then heating (γ = 5/3) once it turns opaque (first Larson core). A von Neumann–Richtmyer artificial viscosity handles shocks; the time step obeys a CFL condition. Specific angular momentum j of every shell is conserved; a shell stops falling where gravity equals the centrifugal force, at the centrifugal radius rc = j2/(G M) — that is how the disk grows from inside out. Mass crossing 10 AU (or a density of 3×10-10 g/cm3, where H₂ dissociates and the second collapse begins) is handed to the protostar/disk.
4. Disk and jets
Material landed on the disk drains onto the star at the viscous rate tvisc = 1/(α h2 ΩK) with h = H/r = 0.033 (r/AU)1/4, α = 0.05 (+0.25 when the disk is more than half the system mass, standing in for gravitational instability). 30% of accreted mass is expelled in a magnetised wind (X-wind estimate; observed 10–40%). Jet speed is taken as 0.5 vesc, vesc = sqrt(2 G M*/R*), i.e. 100–400 km/s for 0.5–3 M☉.
5. Protostar to main sequence
L_acc = 0.5 G M* Mdot / R*
dR/dt = - L R^2 / (0.4286 G M*^2) (Kelvin-Helmholtz contraction, n = 1.5 polytrope)
T_c = 1.57e7 K * (M*/Msun) / (R*/Rsun) (solar-calibrated scaling)
While accreting faster than 10-7 M☉/yr the radius is held near 2.5 M0.27 R☉ (cold accretion). Afterwards the star contracts almost vertically in the HR diagram along the Hayashi track (Teff ≈ 4100 K M0.17). When Tc reaches 107 K the p–p chain switches on, contraction stalls at the zero-age main sequence (L ∝ M4 for 0.43–2 M☉). Below 0.075 M☉ electron degeneracy halts contraction before 107 K: a brown dwarf.
What is simplified
- 1-D (spherical) hydrodynamics with sphere-averaged rotation: no turbulence, magnetic fields, fragmentation (binary stars are common) or a true 2-D disk. Jets and cavity are drawn, not computed.
- The 10 AU sink hides the first core's internal structure; deuterium burning, the full stellar-structure code and radiative transfer are replaced by the scalings above. Pre-main-sequence times are good to a factor of ~2.
- Real cores are rarely isolated or uniform; the uniform preset is the textbook case, the peaked preset is closer to observed cores.
- The nebula uses Blender-rendered volumetric sprites placed on the actual shell positions; picture scale is logarithmic in time and re-zoomed automatically (the star is drawn enlarged in the cloud view).