Skip to main content
Sandbox Physics
Molecular cloud collapse and magnetic support

Molecular clouds · L3

When a grid can no longer resolve contraction

Sandbox Physics ·

As density rises, gravity can amplify shorter scales, but a fixed grid keeps the same cell width. A sharpening image needs comparisons with finer grids and an independent method. Brightness alone does not determine whether the concentration is resolved.

A cell stores an average amount of gas

Divide a periodic square into cells. Each stores mean volume density ρ\rho and momentum densities mx=ρvxm_x=\rho v_x and my=ρvym_y=\rho v_y. They represent area integrals divided by cell area, with the same state at every unit of invariant depth. A function evaluated at the cell center generally differs from its cell average.

A finite-volume method shares one interface flux between neighboring cells. Mass leaving one enters the other. Sum every cell and internal interface contributions cancel in pairs; periodic edges cancel too. Gravity enters momentum, with no direct mass source.

Let Δx\Delta x be cell width and Δt\Delta t the time step. The mass fluxes FF and HH cross the two directions. Higher-order reconstruction and interface solvers change local errors, but the shared-flux structure must remain.

ρˉij n+1=ρˉij n−ΔtΔx(Fi+1/2,j−Fi−1/2,j+Hi,j+1/2−Hi,j−1/2)\bar\rho_{ij}^{\,n+1}=\bar\rho_{ij}^{\,n}-\frac{\Delta t}{\Delta x}(F_{i+1/2,j}-F_{i-1/2,j}+H_{i,j+1/2}-H_{i,j-1/2})
Mass balance for equal-area square cells. Superscripts identify successive time levels.

Reading: Molecular-cloud Jeans standalone scientific reproduction

Periodic gravity responds to departures from the mean

The integral of a periodic Laplacian is zero, so Poisson’s right-hand side must also have zero mean. Use ρ−⟨ρ⟩\rho-\langle\rho\rangle as the source. Nonzero Fourier modes give the potential directly. Setting the uniform potential to zero fixes an arbitrary constant; the mean density’s global gravity has already been removed from this model.

These are three-dimensional gas equations restricted to fields invariant in the third direction. An infinitely thin gravitating sheet would have a different gravity kernel. The periodic patch also has no outer boundary that falls inward as an isolated sphere.

The same density field determines acceleration at each momentum update. Density, velocity and gravity diagnostics follow that state. Nonpositive or nonfinite states fail the calculation; adding a density floor would alter the mass balance.

∇xy2Φ=ρ−⟨ρ⟩,Φ^k=−ρ−⟨ρ⟩^k∣k∣2(k≠0)\nabla_{xy}^2\Phi=\rho-\langle\rho\rangle,\qquad \widehat\Phi_{\mathbf k}=-\frac{\widehat{\rho-\langle\rho\rangle}_{\mathbf k}}{|\mathbf k|^2}\quad(\mathbf k\ne0)
Units here set 4πG=14\pi G=1. Hats indicate Fourier coefficients; acceleration is −∇Φ-\nabla\Phi.

Reading: Molecular-cloud Jeans standalone scientific reproduction · Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field

Time refinement and grid refinement test different errors

Fix J=1.5J=1.5, initial contrast 0.20.2 and zero initial momentum, then evolve to t=2.5t=2.5. Grids of 32232^2, 64264^2 and 1282128^2 cells give peak densities about 2.131092.13109, 2.152142.15214 and 2.156062.15606. Before comparing complete fields, average neighboring fine subcells into the same coarse cell.

The adjacent-grid three-component state RMS differences fall from 0.0011180.001118 to 0.0001910.000191. RMS means the square root of the mean squared difference. It summarizes the field, whereas peak density examines one cell. A small shift in a sharp peak can produce a large local difference.

Now hold the spatial grid fixed and halve the time step. This isolates a different discretization error. Smaller time steps do not add spatial information to a concentration occupying only one or two cells.

An actual early failure illustrates why two numbers can mislead. The maximum linear-wave amplitude error divided by its initial amplitude was about 0.0008270.000827 at 32 cells and 0.0009500.000950 at 64 cells. Coarse effects happened to cancel. The original failure is retained; the same test at 64, 128 and 256 cells shows a steadily decreasing error.

Reading: Molecular-cloud Jeans standalone scientific reproduction

An independent method needs its own spatial checks

The second method represents gas as Fourier waves, evaluates nonlinear fluxes on a grid and projects back to retained waves. It shares no finite-volume reconstruction or interface flux. Its point values are converted through the exact Fourier cell-average window before comparison. The original point-versus-average comparison remains in the failure history.

This pseudospectral reference has errors too. Momentum flux contains m2/ρm^2/\rho. Division can generate infinitely many harmonics even when the inputs contain finitely many. Coarse sampling can mistake short waves for longer ones, an effect called aliasing. Keeping only two thirds of the output band does not certify an alias-free nonpolynomial flux.

The corrected reference evaluates flux on twofold and fourfold quadrature grids before projection. It separately increases retained spatial modes and halves the time step. All ten original support cases and the fixed longer case have completed these three comparisons.

The largest twofold/fourfold density RMS difference is 8.22×10−168.22\times10^{-16}. Increasing retained modes gives maximum density RMS difference 6.64×10−136.64\times10^{-13}; the separate half-step maximum complete-state difference is 4.78×10−114.78\times10^{-11}. These are measured differences for specified cases, not a uniform error theorem over continuous controls.

A smooth positive density 1+0.999cos⁡(2πx)1+0.999\cos(2\pi x) with constant momentum supplies an analytic algebraic counterexample. At 64 points with only output masking, the retained flux-derivative coefficient error is about 526.9526.9. Sixteenfold quadrature reduces it to about 4.69×10−104.69\times10^{-10}. This near-zero-density example lies outside the operating domain and exposes the limitation of the masking assumption.

Reading: Molecular-cloud Jeans standalone scientific reproduction

Jeans cells provide a stopping indicator

The local isothermal Jeans length scales as ρ−1/2\rho^{-1/2}. With NN cells per side and mean density one, its cell count at the peak is NJ=N/(Jρmax⁡)N_J=N/(J\sqrt{\rho_{\max}}). The producer stops before accepting a candidate below eight cells, retaining the last accepted field.

A 64-cell, J=1.5J=1.5 stress run accepts t=3.89153t=3.89153, peak density 28.331828.3318 and NJ=8.01589N_J=8.01589. Its next candidate at 3.901383.90138 has only 7.843087.84308 cells and is rejected. An intentionally underresolved 16-cell, J=2.5J=2.5 initial state stops before its first step.

The late accepted field still has eight cells, but it has no same-time spatial convergence evidence. Its smooth isothermal energy diagnostic changes by about 40.6%40.6\% of the sum of the initial absolute energy components. The diagnostic combines kinetic, isothermal compression and gravitational terms to examine numerical evolution; it is not a radiative cooling-energy calculation. This state demonstrates stopping, rather than supplying an accurate stellar-fragment mass.

The usable time window follows the comparisons actually completed. The continuous operating rectangle is tested only through time two. The fixed example has additional evidence through two and a half. Continuing a movie does not extend that evidence automatically.

NJ=λJ,localΔx=NJρmax⁡/ρ0N_J=\frac{\lambda_{J,\rm local}}{\Delta x}=\frac{N}{J\sqrt{\rho_{\max}/\rho_0}}
A local thermal-Jeans indicator based on peak cell-average density. Other physical scales and magnetic geometry can require finer resolution.

Reading: Molecular-cloud Jeans standalone scientific reproduction · Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1

Keep comparisons that another reader can repeat

Save initial conditions, actual time, grid, method and the complete numerical state alongside an image. Another reader can then give the same physical case to a different grid or algorithm and inspect both the position of a concentration and its mass balance.

The reproduction retains rebound, growth, coarse-error cancellation, the representation mismatch and resolution stops. Each failure identifies a concrete next check: initial velocity, spatial representation, flux quadrature or insufficient resolution. New physics can be assessed against the same chain as the model develops.

Reading: Molecular-cloud Jeans standalone scientific reproduction

Exercises and explanations

  1. Does reducing the time step by four prove that a sharp peak has sufficient spatial resolution?

    Read the explanation

    It checks time discretization on the same cells. Increase spatial resolution, compare cell-average fields at the same physical time, and check the independent reference’s spatial error. A small time error does not create missing spatial information.

  2. A concentration has 8.028.02 local Jeans cells. Can it be treated as an accurate physical fragment?

    Read the explanation

    It passes only the stated stopping indicator. Same-time spatial convergence, numerical dissipation and physical approximations still matter. The stress record demonstrates this: its last field retains eight cells without late-time accuracy certification.

  3. Why can halving the pseudospectral time step not replace increasing the flux quadrature grid?

    Read the explanation

    A rational flux can generate short waves that the sampling grid misrepresents. Smaller steps can repeat the same spatial projection error more often. Change quadrature and retained spatial modes separately, and compare low modes and actual cell averages.

Try the prediction

Watch pressure and gravity change the same gas

  1. Save an operating-window field and compare 64 and 128 cells at the same time. Average the fine cells first, then examine both peak and full-field differences.
  2. Open the standalone long-time stress record and locate the accepted and rejected Jeans-cell values. Explain why it demonstrates stopping.
  3. Write grid error, time error and reference quadrature difference on separate lines. State which numerical setting changed in each, keeping physical initial conditions fixed.

Sources and calculation scope

  1. Chandrasekhar & Fermi 1953, Problems of Gravitational Stability in the Presence of a Magnetic Field

    Section V, pages 138–141, equations 158–173. The original uses an adiabatic sound derivative and Gaussian magnetic units; the chapters state their isothermal derivative and SI normalization. Original paper linked, not redistributed.

  2. Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1

    Pinned v1, sections 2–4, Figures 8–10 and equations 8–11/17/22–23. Read the mass-weighted percentiles and the number distribution separately. The Figure 8 model-name pairing and approximate dimensional example are preserved in context; no author correction is assumed.

  3. Molecular-cloud Jeans standalone scientific reproduction

    Independent finite-volume and oversampled pseudospectral producers; analytic Jeans and full linear MHD references; all ten support cases, a fixed longer case, original official inputs and retained failures. Default replay actually recomputes its scientific steps offline after ordinary Python dependencies are installed.

FeedbackSomething broken, confusing, or missing on this page?Tell us
What kind of feedback?

Sent with this page’s address, its Lab ID, the site language and your device type. Please leave out personal details. Feedback is kept for up to 12 months.Privacy policy