01 · core
Cold molecular gas
A weakly rotating, gravitationally bound core begins on interstellar scales.
~10,000 AU · ~10⁵ yr
/ Teaching Tools / Cloud Core to Planetesimal
Live equations of motion for all ten tracked paths
Large-format numerical states update below the animation. Hatted vectors use the normalized display basis; the full reduced physical law retains gravity, potential, pressure, drag, dissipation, force, torque, and angular momentum.
Ten live numerical states · normalized display basisτ = 0.00000
Focused equation of motionA · collapse clock
focused path · G0675 · derivatives advance with the visible clock τ
LIVE CHECK · waiting for the numerical state
LIVE BASIS: normalized render coordinates.
Ten-object compact telemetry
| ID | R̂ | vR | vφ | |v̂| | |â| | ĵz | fall | land |
|---|
Declare the reduced formation model
These inputs are carried through the complete chain. The first four describe the prestellar core; the remaining six describe a representative patch of the resulting disk. “Compile” recomputes every derived time, scale, threshold, ledger, and verification test.
Initial state and coupled-model parameters
uniform-core anchor · isothermal vertical disk · local dust patchsolar-analog chain · ready
—mean core density · kg m⁻³
—thermal virial parameter
—rotational / binding energy
—disk temperature · K
—radial drift time
—surrogate birth diameter
Eight explanations of one continuous history
These are diagnostic views—not eight stages and not eight separate animations. The clock determines what the material is doing. A view determines which relationship is emphasized: initial binding, simultaneous infall and rotation, landing, vertical flattening, angular-momentum transport, dust drift, pebble concentration, or binding.
Tracked-object state, velocity, and momentum ledger
Every plotted point is a persistent representative tracer, not an individual molecule. Indices preserve identity through the shared clock. Views A–F show the same global state with different measurements; G and H explicitly zoom to a reduced local patch and clump scale.
A · CORE STATEρ̄, tff, support
B · INFALL FIELDgravity + rotation
C · DISK BIRTHj, rc, landing
D · VERTICAL STATEinfall, shocks, H/R
E · TRANSPORTmass inward, J outward
F · DUST COUPLINGSt, settling, drift
G · CONCENTRATIONfeedback, filaments
H · BINDINGρpebble / ρR
Numerical and analytic verification
The live engine tests the identities it relies on
The values below are evaluated from the same compiled parameters and functions used by the animation. They are not hard-coded status badges.
| Test | computed here | analytic target | status |
|---|
Scale and time telescope
The visual camera crosses more than fifteen orders of magnitude in length. Screen size and orbital phase are compressed when the display zooms from the global cloud to a disk patch and then to a clump, while the global clock stays continuous. The live panel explicitly separates normalized render-state derivatives from physical analytic quantities in SI or astronomical units.
02 · disk
Protostar and disk
Infall stalls at the centrifugal barrier; dissipation removes vertical and random energy.
~10–100 AU · 10⁴–10⁶ yr
03 · dust patch
Settled pebble layer
Aerodynamic coupling is organized by the Stokes number and the local orbital clock.
~0.01 AU · 10–10³ orbits
04 · planetesimal
Bound solid seed
A pebble cloud that exceeds its tidal density collapses on approximately a dynamical time.
~10–100 km · days to years
Validity and honesty ledger
Direct analytic layer
Free-fall time, energy ratios, centrifugal balance, vertical scale height, drift speed, and Roche density are evaluated directly.
TIER A · COMPUTED
Reduced evolution layer
Persistent tracers undergo concurrent infall, spin-up, landing, and damping. Cyan velocity and orange acceleration are ninth-order finite derivatives of the same normalized position history; gold is the corresponding equal-weight net force F = m̂a with m̂ = 1, so force and acceleration are necessarily collinear and the gold arrow is drawn on its own longer display scale purely so both remain legible. The dashed violet / green / magenta arrows on the focused path are the exact cylindrical split a = a_R R̂ + a_φ φ̂ + a_z ẑ, drawn tip-to-tail on one common linear scale so they close on the orange arrowhead. Arrow scales are fixed per diagnostic view and never masquerade as SI units. Before landing the parcel follows the exact Ulrich (1976) infall streamline through its own initial point, μ(r) = μ₀(1 − r_c/r)/(1 − r_c/r₀) with r_c = j_z²/GM ∝ (r₀ sin θ₀)⁴, so μ → 0 and every parcel arrives in the midplane — that is the flattening, and it is solved, not asserted. The azimuth is integrated from exact j_z conservation, dφ/dt = j_z/R_cyl², along that same streamline with a per-parcel Simpson table. Only the rate along the streamline is a reduced clock; the streamline shape is the analytic solution. The landed disk's vertical seed is normalised so its measured ⟨z²⟩½/⟨R²⟩½ reproduces the declared H/R. The translucent nebular contours are evolving isodensity guides, not hard material surfaces. The G→D handoff preserves each selected material tag through a deterministic gas-to-solid provenance mapping; it is not a literal one-to-one condensation calculation. Landing, damping, transport, and drag remain declared reduced forces rather than a radiative-MHD solution.
TIER B · KINEMATICALLY CLOSED REDUCTION
Streaming-instability layer
The local clumping response is a transparent surrogate informed by simulation regimes. It is not a live hydrodynamic solution or a universal threshold.
TIER C · SURROGATE
Missing star-formation physics
No radiative MHD, nuclear reaction network, chemistry, turbulence cascade, magnetic braking calculation, feedback, fragmentation, or multiplicity is solved self-consistently. The optional element colors are inherited provenance tags from earlier stellar generations—not nucleosynthesis inside the young central object.
BOUNDARY DECLARED
Angular momentum ledger
Parcel-specific j is conserved only before torques. View E tracks angular momentum redistributed through the disk or exported by the braking/outflow term.
GLOBAL BUDGET · EXPLICIT
Flattening mechanism
Solved, not asserted. Infall follows the analytic Ulrich (1976) streamline μ(r) = μ₀(1 − r_c/r)/(1 − r_c/r₀), so μ → 0 and every parcel arrives in the midplane; the azimuth is integrated from exact j_z conservation along that same streamline. Vertical gravity plus the declared dissipation then relax the landed gas to the hydrostatic H/R, and the rendered disk is normalised to reproduce that H/R (checked to 5%). The instrument measures ⟨z²⟩½/⟨R²⟩½ live from the particle states: it starts at 1/√2 = 0.7071 for the isotropic core and ends near 0.005. With rotation OFF the same engine collapses isotropically and the ratio does not move — the counterfactual is a test, not a caption.
CONCURRENT CAUSAL MODEL
Birth size is illustrative
The displayed planetesimal diameter comes from one declared clump-scale prescription. Real initial mass functions depend on box size, pressure structure, and numerical physics.
NOT A UNIQUE PREDICTION
End of scope
The model stops at the first bound solid body. Pebble accretion, runaway growth, embryos, migration, and impacts belong to the Solar System Formation chain.
HANDOFF · DELIBERATE
Bottom line. The page is trustworthy where it evaluates a printed equation or integrates a declared reduction. At the streaming-instability handoff it changes honesty tier instead of disguising a simulation-informed morphology as first-principles hydrodynamics.
Primary references and model anchors
- Shu (1977). Self-similar collapse of isothermal spheres and star formation. ADS ↗
- Terebey, Shu & Cassen (1984). Collapse of slowly rotating isothermal cloud cores. DOI ↗
- Ulrich (1976). An infall model for the T Tauri phenomenon: rotating streamlines feed a centrifugal disk. ADS ↗
- Banerjee, Pudritz & Holmes (2004). Three-dimensional collapse of rotating Bonnor–Ebert spheres through protostellar disk formation. DOI ↗
- Lynden-Bell & Pringle (1974). Evolution of viscous discs and nebular variables. ADS ↗
- Weidenschilling (1977). Aerodynamics of solid bodies in the solar nebula. DOI ↗
- Youdin & Goodman (2005). Streaming instabilities in protoplanetary disks. ADS ↗
- Johansen et al. (2007). Rapid planetesimal formation in turbulent circumstellar disks. DOI ↗
- Rucska & Wadsley (2023). Planetesimal formation via streaming instability with multiple grain sizes. DOI ↗
- Magnan, Heinemann & Latter (2024). The physical mechanism of the streaming instability. DOI ↗