Physics Reference

Gravitational constant (simulation)

Tuned so a planet at the habitable-zone mid-radius (~40 px) of a mass-100 star orbits in ~600 ticks.

Simulation calibration constant (no external reference) — derived from Kepler's third law.

Speed of light (simulation)

Keeps post-Newtonian corrections subtle for ordinary orbits but dramatic near black holes. Velocities are clamped so they never exceed it.

Simulation calibration constant (no external reference) — models the speed of light as an enforced limit.

Cosmological time mapping

Maps simulation ticks to cosmological years. With the default, the primordial-black-hole window (~400 ticks) sits at ~400 Myr and a mature run (~13,800 ticks) at ~13.8 Gyr.

Reference: Age of the universe (calibrated so a mature run reaches ~13.8 Gyr).

ΛCDM density parameters (Ω)

Fractions of the total mass-energy budget. The energy budget for a total is split as (total·Ω).

Reference: ΛCDM model; values from the Planck 2018 results.

Hubble expansion (Hubble's law)

A subtle expansion drift applied per tick that grows recession with distance.

Reference: Hubble's law.

Dark-energy driven acceleration (cosmological constant)

Extra Hubble-flow rate from vacuum energy, accumulating with cosmic age. The cosmic fate modulates it: a Big Rip applies phantom (×6) acceleration; a Big Crunch zeroes Λ as gravity wins.

Reference: Cosmological constant and dark energy; fates from the Big Rip / Big Crunch scenarios.

Cosmic inflation & primordial power spectrum

An exponential early expansion that freezes in quantum density fluctuations. The fluctuations follow a red-tilted, near-scale-invariant power spectrum P(k) ∝ k^(n_s−1) and seed where galaxies later form.

Reference: Cosmic inflation and the scalar spectral index n_s (Planck-measured).

Newtonian gravitation

The baseline inverse-square attraction between masses.

Reference: Newton's law of universal gravitation.

1PN perihelion precession (post-Newtonian)

First post-Newtonian correction that makes orbits precess, like Mercury's perihelion advance.

Reference: Post-Newtonian expansion; see relativistic apsidal precession.

Gravitational-wave radiation reaction (Peters)

Energy loss to gravitational waves that shrinks tight binaries and drives inspirals.

Reference: power radiated by orbiting bodies (Peters & Mathews quadrupole formula).

Schwarzschild geodesic (curved spacetime)

Weak-field equatorial expansion of motion around a non-rotating black hole, plus its event-horizon (Schwarzschild) radius.

Reference: Schwarzschild metric and Schwarzschild radius.

Kerr frame-dragging (Lense–Thirring)

A spinning black hole drags spacetime around it, adding a velocity-perpendicular cross term. The same Kerr spin also powers a magnetic Blandford–Znajek jet when the hole is threaded by the MHD field.

Reference: Frame-dragging (Lense–Thirring effect) around a Kerr black hole.

Gravitational time dilation

Clocks run slower deep in a gravity well; reaches zero at the horizon.

Reference: Gravitational time dilation.

Tidal acceleration

Differential gravity across a body of half-size δ; drives spaghettification and disruption.

Reference: Tidal force (and spaghettification).

Gravitational lensing

Light bending around a (possibly moving) mass, with velocity-dependent aberration that breaks the symmetry for a moving black hole.

Reference: Gravitational lens (light deflection 4GM/c²r).

Keplerian angular velocity

Angular speed of a circular orbit; the basis of orbital periods and the tidal mean motion n.

Reference: Kepler's third law (and mean motion).

Galaxy merging threshold

Minimum separation before two galaxies are allowed to merge.

Simulation tuning value (no external reference) — models galaxy mergers.

Degeneracy pressure

Qualitative non-relativistic degeneracy pressure supporting white dwarfs and neutron stars.

Reference: Electron degeneracy pressure.

Chandrasekhar limit

A white dwarf above this mass cannot be supported by electron degeneracy and detonates as a Type Ia supernova.

Reference: Chandrasekhar limit (≈ 1.4 M☉ in reality; scaled here to sim mass units).

Tolman–Oppenheimer–Volkoff (TOV) limit

A neutron star above this mass overcomes neutron degeneracy and collapses into a black hole.

Reference: Tolman–Oppenheimer–Volkoff limit (≈ 2.2–2.9 M☉ in reality; scaled here to sim mass units).

Quantum entanglement entropy & decoherence

Entangled black-hole pairs (used for wormholes) lose coherence exponentially each tick; the entropy is the von-Neumann form, maximal (ln 2) at p = 0.5.

Reference: von Neumann entropy and quantum decoherence.

Quantum random number generator (QRNG)

Randomness is seeded from genuine simulated-qubit sampling rather than a classical PRNG. Each draw prepares a register, mixes the amplitudes (Hadamard / phase rotations), and measures it to collapse out an integer; wider values are assembled from several of these small draws one bit at a time.

Reference: quantum random number generation and qubit measurement (Born rule). Register size is kept small because the state-vector simulator is exponential in qubit count.

Gamow-factor ignition probability

A logistic turn-on around the burning minimum decides whether a protostar ignites hydrogen fusion.

Reference: Gamow factor (quantum-tunnelling fusion) and the hydrogen-burning minimum mass.

Eddington limit & radiation coupling to the gas grid

A star is not an isolated light bulb — it is coupled to the continuum radiation transport. Each tick every living star samples the local Rosseland-mean opacity κ at its position (the same opacity model the gas / FLD solver uses) and its nominal luminosity L ∝ M^3.5 is capped at the Eddington luminosity L_Edd, above which radiation pressure overwhelms gravity and the star sheds its outer layers. The capped effective luminosity is what the rest of the simulation uses (habitable zone, planet irradiation, snapshots), and a fraction of it is injected back into the gas as a radiation-pressure source, so stars genuinely drive the radiation field — photon diffusion, ionization fronts and the build-up to reionization — exactly like the AGN / supernova sources. With no gas grid attached (or no opacity model) the cap and injection vanish and stars shine at their nominal luminosity, so behaviour is unchanged.

Reference: Eddington luminosity (radiation-pressure limit), mass–luminosity relation, and radiation pressure.

Radiative-equilibrium temperature (airless baseline)

Surface temperature of an airless grey body, scaled so a mid-HZ world (~40 px, L≈1) reads ~1.0. Albedo reflects flux; a greenhouse multiplier warms it further. Used as the fallback when no gas grid is attached — otherwise the radiation-coupled model below applies.

Reference: Planetary equilibrium temperature.

Radiation-coupled surface temperature (opacity greenhouse & ambient bath)

When the continuum is attached, the planet's surface temperature is computed with real radiation-transport-aware terms layered on the grey-body baseline. (1) Optical-depth greenhouse: the planet's atmosphere has a Rosseland optical depth τ = κ·Σ (local opacity × a mass-scaled column), and a grey atmosphere warms the surface by (1 + ¾τ)^1/4 — thicker, more opaque air runs hotter, using the same opacity model the gas solver uses. (2) Ambient radiation bath: a world immersed in the continuum radiation field (energy density E_r sampled at the planet) is heated by it in addition to its star's inverse-square flux, via T⁴ ∝ flux + w·E_r — negligible in empty space, but dominant inside an HII region or reionised gas. Planets also use each star's effective (Eddington-limited) luminosity. With τ→0 and E_r→0 this reduces exactly to the airless baseline above.

Reference: grey / idealised greenhouse model, optical depth, and radiative transfer.

Tidal heating rate

Per-tick internal heating that spikes close to the star and is amplified on eccentric orbits (drives volcanism, e.g. an Io-like world).

Reference: Tidal heating (e.g. tidal heating of Io).

Roche limit

Distance inside which tidal forces overcome a body's self-gravity and shred it into a debris/ring stream.

Reference: Roche limit.

Tidal torque

Torque on the planet's spin ω, driving it toward the mean motion n; vanishes when ω = n (tidally locked). Q is the tidal dissipation quality factor. Molten / strongly tidally-heated worlds have a soft, lossy interior, so their torque is boosted (up to MoltenBoost×, ~3e-4 effective) and they lock fastest.

Reference: Tidal locking and tidal acceleration (Q = tidal quality factor).

Dissipation Q, eccentricity damping & sub-stepping

Q is the inverse efficiency of tidal dissipation (torque ∝ 1/Q): small Q (molten) locks fast, huge Q (gas giants) barely changes spin. The body type is inferred from mass/state, Q is suppressed when young (hot, soft interior) and relaxes upward as the planet ages and cools, and far-out rocky worlds get an extra Q multiplier so locking effectively switches off beyond the close-in zone (matching the 1/r⁶ reality). On an eccentric orbit there is no single synchronous spin, so the orbit must circularize first — damping is stronger close-in and at high e. The spin update is integrated in sub-steps so a large tick approaches synchronous smoothly without overshooting.

Reference: tidal circularization and the tidal quality factor Q by body type.

Eight sinks + remnant

Sinks 1–4 stay as baryons (return to the interstellar medium); sinks 5–7 leave forever as mass-energy.

Reference: Core-collapse supernova and conservation of mass-energy.

Casimir exotic energy & traversability

Negative vacuum energy between throat walls a distance d apart props the throat open; the wormhole is traversable while the exotic-energy budget stays above the threshold.

Reference: ER=EPR conjecture, the Casimir effect, and traversable wormholes.

World & population parameters

Simulation tuning parameters (no external reference) — engine bounds and population caps.

Runtime ownership & app-config seeding

One process-wide SimulationRuntime owns the single server-authoritative universe: it holds the Universe aggregate (all mass, galaxies, stars, planets and the conserved MassTotal), its quantum-seeded RNG and event log, advances the world on the hosted tick loop, and serialises every interaction (tick, snapshot, and client commands) through one lock. Its SimConfig is bound from the Simulation section of appsettings.json at startup, so two settings are established before the first frame: Max_Mass is stamped onto Universe.MassTotal at the Big Bang and split into the ΛCDM dark-energy / dark-matter / baryon budget (only baryons form objects), and SimulationProfile is applied via SetSimulationProfile so a LongRun24x7 selection also enables the public-performance preset and hides the profile selector in the sidebar. Because the mass budget is fixed once at the Big Bang and never changes, mass is conserved for the whole lifetime of a universe.

Engine composition / configuration wiring (no external reference) — SimulationRuntime + Universe + SimConfig.

Pipeline execution order & shared EOS

Every module operates on the shared ContinuumState grid and reads one ideal-gas equation of state, so all consumers see a consistent thermodynamic state. Modules are dependency-injected as IPhysicsModule and ordered by PhysicsOrder.

Reference: ideal-gas EOS and grid-based computational astrophysics.

Two-way coupling: gravity ↔ gas

The gas grid is not a passive backdrop — it is driven directly from the N-body engine every tick and feeds back into it, so structure formation is genuinely hydrodynamic. Inside CosmosSimulation.Tick() (under the simulation lock) the engine first deposits the entity mass distribution onto the gas density field, so the solver sees where the baryons actually are; cooling, shocks, turbulence and angular-momentum transport then act on those overdensities. Stellar deaths and accreting black holes register point feedback sources (supernovae, AGN) and core-collapse events that the feedback and neutrino modules consume. The coupling is fully two-way for magnetism too: each black hole samples the local MHD field the gas solver evolves, so magnetorotational stresses boost its accretion and a spinning, magnetised hole drives a Blandford–Znajek jet whose power is injected back as extra AGN feedback. When the grid cools gas past the star-formation threshold it raises a request that the engine realises as a lightweight collapsing protogalaxy at that site, funded conservatively from the shared baryon budget; gas the grid returns (ejecta, evaporated clouds) is credited back. The whole loop is mass-conserving and tuned to evolve gradually.

The pipeline is driven exactly once per tick by the engine itself (the legacy reflection-based auto-driver is retired), guaranteeing the feedback / core-collapse / star-formation channels are populated from live engine state rather than left empty.

Reference: galaxy formation & evolution and cooling-driven / triggered star formation.

Momentum & shock heating

Gas accelerates down its pressure gradient with a viscous diffusion term; convergent flow (∇·v < 0) is irreversibly heated by von Neumann–Richtmyer artificial viscosity, reproducing SN blast shells and accretion shocks.

Reference: Navier–Stokes equations and artificial viscosity (von Neumann–Richtmyer shock capturing).

Radiative cooling curve & Jeans instability

A three-channel cooling curve Λ(ρ,T,Z) — molecular (low-T), atomic (Lyman-α bump ~10⁴) and metal-line (∝ Z) — removes thermal energy (∝ n²). Cooling lowers pressure support until a cell exceeds its Jeans length and collapses, feeding star formation.

Reference: radiative cooling function, the Jeans instability, and free-fall time.

Flux freezing, magnetic pressure & reconnection

In the ideal limit field lines are frozen into the flow; a finite resistivity η lets them diffuse and reconnect when |∇×B| exceeds a threshold, converting magnetic energy to heat. Magnetic pressure P_B = B²/2 resists compression, and magnetic braking removes angular momentum.

Reference: magnetohydrodynamics, flux freezing, magnetic reconnection, and magnetic braking.

Magnetic coupling to black holes: Blandford–Znajek jets & MRI accretion

The field the gas solver evolves is not confined to the grid — the N-body black holes feel it too. Each tick a hole samples the local magnetic energy density u_B = ½|B|² at its position (bilinear sampling, matching the flux-frozen advection). Two effects follow. (1) Magnetorotational instability (MRI): magnetic stresses transport angular momentum outward in the disk, removing the centrifugal barrier so gas accretes faster — a small, saturating boost to the accretion rate that can tip a hole into the active/quasar regime. (2) Blandford–Znajek jets: a spinning, magnetised Kerr hole extracts rotational energy electromagnetically, with luminosity P_BZ ∝ a²·B². The jet power is stored on the hole (exposed in snapshots) and injected on top of the AGN feedback. With no field — or a non-spinning (Schwarzschild) hole — both terms vanish and behaviour is unchanged.

Reference: Blandford–Znajek process (spin-powered relativistic jets) and the magnetorotational instability (disk angular-momentum transport).

Photon diffusion, opacity & ionization fronts

Radiation is transported in the diffusion (flux-limited) approximation with opacity κ = electron-scattering (grey) + Kramers (free-free / bound-free, ∝ ρT⁻³.⁵). Where ionizing flux beats recombination, an ionization front advances; radiation pressure pushes on the gas.

Reference: radiative transfer, Kramers’ opacity law, radiation pressure, and the reionization epoch.

Stellar & planetary coupling to the radiation field

The radiation field is two-way coupled to the N-body entities, not just the gas. Stars feed it: each tick a star injects a fraction of its Eddington-limited luminosity as a radiation-pressure source (see Eddington limit & radiation coupling), so populations of stars build up the diffuse radiation background, push ionization fronts outward and contribute to reionization — alongside the AGN and supernova sources. Planets read it: a planet samples the local radiation energy density E_r and Rosseland opacity κ at its position (bilinear sampling, matching the diffusion solver) to drive an opacity-based atmospheric greenhouse and an ambient-radiation heating term (see radiation-coupled surface temperature). The deposit happens before the radiation module runs, so sources are consumed the same tick; planets read the previous tick's field (one-tick lag), matching the explicit coupling used elsewhere. With no gas grid attached every coupling cleanly degrades and entity behaviour is unchanged.

Reference: H II regions (stellar ionizing radiation), reionization, and stellar feedback.

CR pressure, diffusion, winds, heating & SF regulation

Cosmic rays scatter off magnetic irregularities and diffuse (anisotropically when MHD is active). Their pressure P_cr = (γ_cr−1)E_cr adds a non-thermal force −(1/ρ)∇P_cr that is applied to the gas momentum every tick (driving smooth large-scale winds without radiating away like heat), they deposit energy as the dominant heating channel deep inside molecular clouds, and that same P_cr feeds into the star-formation support term (CrSupport·P_cr/ρ) to suppress runaway collapse.

Reference: cosmic rays and cosmic-ray feedback / transport (diffusion, pressure, heating).

Metallicity tracking & supernova yields

Each cell carries a composition (H, He, metals). When stars die, a fraction of the processed mass is injected as metals, raising Z. Higher Z strengthens the metal-line branch of the cooling curve consumed by the hydro module.

Reference: metallicity, stellar nucleosynthesis (supernova yields), and galactic chemical evolution.

Entity metal cycle (stars ↔ gas)

The metal field above is wired all the way through the entity layer, so the loop closes both ways. Stars carry a metallicity Z: gas the grid has enriched seeds new clouds, those clouds promote into galaxies and birth stars that inherit the local Z, and when those stars die their supernova feedback reports their real progenitor Z to the yield table (no longer 0). While a star lives, it also mixes a small fraction of the surrounding enriched gas into its envelope each tick, so its Z drifts toward the local interstellar value rather than staying frozen at birth. Z is clamped to a super-solar ceiling so a single hot cell can't spike it.

Reference: galactic chemical evolution and stellar populations (Pop III → II → I enrichment sequence).

Phase classification & dust

The five phases — cold molecular, warm neutral, hot ionized, coronal, and the dust component — relax toward a multi-phase pressure equilibrium. Dust tracks (and lags) the metal field, catalyses H₂ formation, and blocks radiation (shielding ∝ 1/(1+ρ_dust)).

Reference: phases of the interstellar medium and interstellar dust (grain growth & sputtering).

Phase transitions & dust evolution

Phases are not static labels — they evolve. Dust grains grow toward a metal-set equilibrium and are thermally sputtered in hot gas, while a condensation operator drives dense warm gas down onto the cold molecular phase. Because dust shields the gas, dusty cells cool and condense faster, so the warm→cold transition feeds the cold molecular reservoir that the star-formation module turns into stars — closing the dust–cooling–SF loop.

Reference: molecular-cloud formation via dust-shielded condensation, feeding star formation.

α-disk viscosity & spin evolution

The effective kinematic viscosity ν = α·c_s·H drives radial drift and sets the viscous time r²/ν. On the gas grid it acts as a shear-diffusion of angular momentum; the SpinEvolutionModel / AccretionDiskModel helpers behind it are wired into the live engine in CosmosSimulation.AngularMomentum.

• Stars — every tick a Skumanich-like magnetic-braking term spins the star down (dΩ/dt ∝ −k·Ω³), additive to the tidal coupling; accreted mass spins it back up by conserving L = I·Ω + δM·j.
• Protogalaxies — disc collapse spins the cloud up by the L-conserving contraction law Ω′ = Ω₀·(R₀/R)² (replacing the old ad-hoc factor), so the seed spin and the collapse ratio set the final disc rotation.
• Galaxies — a formed disc loses a tiny fraction of its spin per tick to winds / magnetic braking (floored so it never fully stops), with mergers still the dominant spin change.

Reference: the Shakura–Sunyaev α-disk model and accretion-disk viscosity / angular-momentum transport.

Density-threshold collapse & SFR

A cell forms stars when its density exceeds a threshold, it is colder than the ceiling, and self-gravity beats its total non-thermal support. The effective stabilising speed adds thermal, turbulent, magnetic and cosmic-ray pressure in quadrature, so the Jeans test reflects every pressure that resists collapse. The star-formation rate scales as ε_ff·ρ/t_ff; formed mass is requested back from the entity engine as a star-forming cloud.

Reference: the Kennicutt–Schmidt star-formation law and the physics of star-forming regions (Jeans collapse + free-fall efficiency, with magnetic & cosmic-ray pressure support).

Prompt IMF fragmentation (grid → entity)

When the grid flags a star-forming cell, the cooled cloud doesn't become a single object: a fraction of it is fragmented at the formation site that same tick into a cluster of stars whose masses follow a bottom-heavy initial mass function (the same Salpeter-like law the engine uses everywhere), and only the leftover gas keeps collapsing as a protogalaxy. Each draw's upper bound is the remaining cloud mass, so the cluster self-limits — early massive stars leave less behind, naturally yielding many low-mass and few high-mass stars. Stars are placed at or above the hydrogen-burning minimum (sub-stellar fragments can't ignite, so their mass stays in the gas), and mass is conserved exactly: cloud = stars + protogalaxy gas + refunded remainder.

Reference: the initial mass function (Salpeter / Kroupa) and star-cluster formation from fragmenting molecular clouds.

Stellar, supernova & AGN feedback channels

Each feedback source deposits a mix of thermal and kinetic energy over a small radius. Supernovae drive roughly spherical blast shells; AGN couple a fraction of accretion power into a narrow bipolar jet that launches outflows and bubbles. Together they regulate the gas reservoir (blowout → quenching → starburst suppression).

Reference: stellar feedback, AGN feedback, and galactic winds / outflows (galaxy self-regulation & quenching).

AGN jet → gas coupling (accretion power + Blandford–Znajek)

The power an accreting hole hands to the gas each tick is not just its accretion luminosity — the magnetically-powered jet is coupled in explicitly. When a hole is classified active or quasar, the engine emits a single AGN FeedbackSource whose power is the sum of two terms: (1) an accretion term proportional to the feed rate Ẑ_acc, scaled up 2.5× in the brighter quasar state; and (2) the Blandford–Znajek jet luminosity P_BZ the MHD coupling computed from the hole's spin and the local magnetic energy. That combined power is then split by the AGN channel into a thermal halo (heats a small sphere, suppressing cooling) and a bipolar kinetic jet (launches outflows/bubbles along the axis). So the jet you already had is now a first-class regulator of the surrounding gas, not just a visual on the hole.

Reference: AGN feedback (quasar & radio/jet modes) and the Blandford–Znajek process (spin-powered jets dumping energy into the medium).

Burst energy, ν-driven wind & cooling

The burst radiates over a few ticks (so the wind is not instantaneous) with a ~10–20 MeV thermal spectrum. A fraction (HeatingFraction) of the ν luminosity couples into the gas within a small radius above the proto-neutron star as ν heating — depositing thermal energy and a radial kinetic push that unbinds the overlying gas and drives the outflow (the shock-revival mechanism). Neutrino cooling follows a steep pair-process scaling (∝ T⁹) above a temperature threshold, the dominant cooling channel for proto-neutron stars.

Reference: supernova neutrinos, the neutrino-driven wind, and neutrino cooling of proto-neutron stars.

Pipeline execution order & disc lifecycle

A per-system ProtoplanetaryDisc (a radial array of annuli) is created lazily and retained in a ConditionalWeakTable keyed on the SolarSystem, so it is garbage-collected with the system. The formation and migration stages require live gas and naturally fall idle once the disc photoevaporates below its dispersal floor; the resonance and multi-planet stages then take over and shape the gas-free system for the rest of its life. The pipeline runs after the entity planet/tidal update (so masses and orbits are current) and before the continuum drive, inside CosmosSimulation.Tick() under the simulation lock.

All stages update UniverseState in place: planet core/envelope/total mass, orbital radius & speed, eccentricity, inclination, ascending node, migration rate/regime, gap depth, and resonance state (partner, p:q, resonant argument, libration amplitude). Throttled diagnostics log resonance capture/breaking and the onset of dynamical chaos to the event log.

Reference: nebular hypothesis / planet formation and N-body simulation of planetary systems.

Mass-budget regulation pipeline (new)

The planetary layer now runs an explicit PlanetFormationRegulationPipeline around formation to prevent runaway mass creation. Before growth, disc mass and lifetime are clamped to realistic protoplanetary ranges; during growth, pebble/planetesimal conversion is throttled; after growth, per-disc caps, destruction channels, and conservation checks run. This is why discs cannot produce unlimited planets: every stage is constrained by finite source reservoirs and tracked sinks.

Reference: observational protoplanetary-disc demographics, pebble-accretion limits, and mass-conservation constraints in coupled N-body + disc models.

Surface density, temperature & α-viscosity

Gas surface density and mid-plane temperature follow power laws in radius; the scale height, sound speed and orbital frequency set the disc thermodynamics. The turbulent α converts to a kinematic viscosity ν = α c_s H that drives viscous spreading (Lynden-Bell & Pringle) and exponential photoevaporative dispersal.

Reference: the minimum-mass solar nebula, the Shakura–Sunyaev α-disc, and viscous disc evolution.

Dust growth & radial drift

Grains coagulate toward a fragmentation/drift barrier; their aerodynamic coupling is measured by the Stokes number. Because the gas is pressure-supported and sub-Keplerian, solids feel a headwind and spiral inward fastest at St ~ 1 — the radial-drift barrier — unless a local pressure maximum (bump) cancels the gradient and traps them.

Reference: dust-grain growth and the radial-drift barrier in protoplanetary discs.

Planetesimals, pebbles & isolation mass

Below the oligarchic mass, gravitational focusing makes the biggest body grow fastest (runaway); above it, embryos grow in lockstep (oligarchic) until they sweep their feeding zone and reach the isolation mass. Pebble accretion is much faster: pebbles settling through the Hill sphere are captured efficiently until the core reaches the pebble-isolation mass and carves a pressure bump that shuts off the flux.

Reference: planetary accretion, pebble accretion, and the isolation mass.

Pebble/planetesimal regulation & destruction channels (new)

Pebble growth is now capped by flux limits, turbulence damping, pressure-trap bottlenecks, sublimation zones and fragmentation losses; planetesimal growth is gated by streaming-instability thresholds and low conversion efficiency. After accretion, unstable planets are removed via migration infall, scattering/ejection, resonant-chain collapse, tidal/Kozai-assisted infall, and mergers. These sinks are essential to avoid over-retaining mass in long-lived packed systems.

Reference: streaming instability, radial drift / fragmentation limits, and migration-driven loss channels.

Gas-envelope accretion & the giant-planet threshold

While subcritical, a core binds a slowly-contracting gas envelope whose growth is throttled by Kelvin–Helmholtz cooling. Once the core (plus envelope) passes the critical core mass, the envelope can no longer stay in hydrostatic balance and gas pours on in a runaway, building a gas giant up to the gap-limited supply.

Reference: the core-accretion model and giant-planet formation (runaway gas accretion).

Type I: Lindblad + corotation torques

The differential Lindblad torque is typically negative (inward) and scales with planet mass squared and the local disc surface density. The corotation torque (from gas executing horseshoe turns) is positive and depends on the surface-density/temperature gradients; it can reverse the net migration in regions of steep gradients, but saturates unless viscosity refreshes the horseshoe region.

Reference: Type I migration and Lindblad / corotation torques.

Type II (gap-opening) & Type III (runaway)

A planet opens a gap when its Hill radius rivals the disc scale height and its tidal torque beats viscous refilling (Crida criterion). It then drifts with the viscous accretion flow, slowed when it dominates the local disc mass. If a partial gap leaves a large co-orbital mass deficit, the feedback between drift and the asymmetric horseshoe flow runs away — fast Type III migration, inward or outward.

Reference: Type II and Type III migration, and the gap-opening (Crida) criterion.

Mean-motion resonance: capture, libration & breaking

When two planets' periods approach a ratio of small integers, the resonant argument φ stops circulating and oscillates (librates) about a fixed centre — the pair is locked and migrates together. Resonant forcing pumps the eccentricities; if the libration amplitude or eccentricity grows too large (overlap, perturbation), the lock breaks and φ circulates again.

Reference: mean-motion resonance and resonance capture by convergent migration.

Secular precession & Kozai–Lidov cycles

Away from resonance, the orbit-averaged (secular) interaction precesses the apsides and nodes and exchanges eccentricity/inclination among planets (Laplace–Lagrange theory). A distant, highly-inclined perturber (a wide binary companion or a far giant) drives Kozai–Lidov cycles: above the critical inclination the torque conserves L_z = √(1−e²)·cos i, so falling inclination is traded for rising eccentricity — a channel to extreme e (hot-Jupiter migration, tidal capture, collisions).

Reference: secular dynamics, Laplace–Lagrange theory, and the Kozai–Lidov mechanism.

Integrator choice, resonance overlap, Hill stability & chaos verdict

The long-term integrator is selectable: the default Wisdom–Holman (WHFast) symplectic scheme conserves energy/angular momentum over millions of orbits (ideal for resonances and slow chaos), while a cheaper velocity-Verlet leapfrog is available for faster runs. Each pair of planets is surrounded by a web of resonances whose widths grow with planet mass; when neighbouring resonances overlap (Chirikov), orbits wander chaotically and the elements random-walk (chaotic diffusion), ultimately driving orbital crossing and ejection. The overlap parameter scales as the mass ratio to the 2/7 power over the separation in mutual Hill radii; closely-spaced or massive planets are chaotic, widely-spaced low-mass ones are stable. A separation below ~2√3 mutual Hill radii is Hill-unstable. From the overlap the analyzer derives a Lyapunov time (the e-folding time of trajectory divergence; infinite when regular) and a coarse stable / metastable / chaotic verdict.

Reference: the Chirikov resonance-overlap criterion, Hill stability, the Wisdom–Holman symplectic integrator, and the Lyapunov time.

Detection, classification & hierarchical stability

Each pair of stars sharing a SolarSystem within the bound-separation limit becomes a BinarySystem; a nearby single star can be attached as the outer tertiary of a TripleSystem. Because detection re-runs every tick, a per-universe BinaryRegistry caches each pair (keyed on the unordered Star references) so its evolving orbit and accumulated state survive re-discovery. A triple is only retained if it is hierarchical — the inner and outer orbits must be well separated for the secular approximation to hold — tested with the Mardling–Aarseth stability criterion.

Reference: binary stars and the Mardling–Aarseth triple-stability criterion.

Tides — synchronisation, circularisation & spin–orbit coupling

In a close pair each star raises a tidal bulge on the other. Equilibrium-tide (weak-friction) dissipation drives the stellar spins toward the orbital mean motion (synchronisation), damps the eccentricity toward zero (circularisation), and exchanges angular momentum between spin and orbit. The torque scales steeply with the radius-to-separation ratio, so tides matter only for tight binaries — exactly the ones that go on to transfer mass or merge.

Reference: tidal circularization and equilibrium-tide theory (Hut 1981).

Roche lobe & mass transfer — how mass flows between stars

Each star is bounded by its Roche lobe, the teardrop equipotential around it; the Eggleton formula gives the lobe radius from the mass ratio. When an evolving (expanding) star fills its lobe at pericentre, gas spills through the inner Lagrange point onto the companion — Roche-lobe overflow (RLOF). The transfer is stable if the donor shrinks inside its lobe at least as fast as the lobe shrinks (compared via the mass-radius exponents ζ_star vs ζ_L), giving steady, long-lived transfer; it is unstable (runaway) if the lobe shrinks faster, which triggers a common envelope. Transfer is non-conservative: the accretor takes only a fraction β of the offered mass (capped near an Eddington-like rate for compact objects), the rest leaves the system carrying away angular momentum, so the orbit responds (widening or shrinking) depending on the mass ratio.

Reference: Roche lobe (Eggleton 1983) and binary mass transfer.

Common-envelope evolution — the α–λ formalism

When mass transfer is dynamically unstable, the companion is engulfed by the donor's giant envelope and the two cores orbit inside a shared envelope. Drag makes them spiral inward, and the released orbital energy is deposited into the envelope. In the energy (α–λ) formalism the envelope's binding energy is set by λ (its structure), and a fraction α of the orbital energy released during the inspiral is available to unbind it. If that energy exceeds the binding energy the envelope is ejected, leaving a much tighter detached binary (often a compact-object progenitor pair); if not, the cores merge before the envelope clears. This is the key step that hardens wide binaries into the close double-compact systems that later inspiral via gravitational waves.

Reference: common-envelope evolution and the α–λ energy formalism (Webbink 1984).

Mergers & merger products — how coalescence happens

A pair merges when its pericentre falls below the sum of the stellar radii (driven there by tides, mass transfer, a failed common envelope, gravitational-wave decay, or a Kozai–Lidov eccentricity spike). The two stars collapse into one object, conserving momentum and shedding a small mass fraction to the ISM. The product is classified from the components' kinds: two main-sequence stars make a rejuvenated, over-massive blue straggler; a compact object sinking into a giant makes a Thorne–Žytkow object; two white dwarfs make a heavier WD (or detonate, see below); two neutron stars make a kilonova leaving a heavy NS or black hole; two black holes make a single larger black hole.

Reference: blue stragglers, Thorne–Žytkow objects and neutron-star mergers.

Binary-specific supernovae — how they differ from single-star SNe

An isolated star's fate is fixed by its birth mass; in a binary the companion changes the outcome. Type Ia is a thermonuclear detonation of a carbon-oxygen white dwarf driven to the Chandrasekhar mass — either single-degenerate (the WD accretes from a companion) or double-degenerate (two WDs merge); a lone WD could never explode this way, and Type Ia dominate the universe's iron enrichment. X-ray binaries are not explosions but persistent states: a compact object accreting from a companion radiates its infalling gas as X-rays, launching relativistic jets (a microquasar) at the highest rates. Stripped-envelope (Type Ib/Ic) core-collapse occurs when a companion removes the hydrogen (Ib) and helium (Ic) envelope before the explosion, so the spectral absence of hydrogen is a direct fingerprint of the binary interaction.

Reference: Type Ia, X-ray binaries and stripped-envelope (Ib/Ic) supernovae.

Triple dynamics — how Kozai–Lidov flips induce mergers

In a hierarchical triple the distant tertiary torques the inner binary. When the mutual inclination exceeds ~39.2°, the inner eccentricity and inclination undergo large, anti-correlated Kozai–Lidov oscillations at fixed inner semi-major axis: as the inclination drops toward the orbital plane the eccentricity is pumped up toward 1, then they swap back (a “flip”). The conserved quantity L_z = √(1−e²)·cos i fixes this trade-off. The eccentricity spikes drive the inner pair to a tiny pericentre where enhanced tides or gravitational-wave emission shrink the orbit rapidly — and at the extreme the inner stars merge. A third star that never touches the inner pair is thus a leading trigger for close-binary mergers, blue stragglers, Type Ia progenitors and compact-object coalescences.

Reference: the Kozai–Lidov mechanism in hierarchical triples.

Gravitational-wave inspiral — how compact binaries coalesce

Two compact objects (NS/NS, NS/BH, BH/BH, or close WD/WD) radiate orbital energy and angular momentum as gravitational waves, so the orbit shrinks and circularises until the stars coalesce, following the Peters (1964) equations. Two features fall straight out: the orbit circularises as it decays (de/dt drives e→0), and the decay runs away at small separation (da/dt ∝ a^−3), so the final orbits are by far the fastest — the chirp. The circular merger time scales as a⁴, so a wide pair effectively never merges while a tight one (such as the hardened double-compact remnant left by a common envelope) coalesces within the simulation's lifetime; the final merger then builds a heavier, rapidly-spinning Kerr remnant.

Reference: the gravitational-wave inspiral of compact binaries (Peters 1964).

Collapse regime from progenitor mass & metallicity

The decision a star's death makes is fixed first, from its mass at collapse (with metallicity breaking the tie in the transition band). Low-mass stars never reach iron-core collapse: they shed their envelopes as a planetary nebula and leave an electron-degenerate white dwarf. Iron-core collapse of intermediate massive stars halts at nuclear density as a neutron star (Type II supernova). Above the neutron-star maximum the core cannot be supported and collapses to a black hole, either through fallback after a weak explosion or by direct (failed-SN) collapse. At extreme masses electron–positron pair-instability can unbind the entire star, leaving no bound remnant, and beyond that pulsational pair-instability sheds mass in pulses before a final collapse. These bands are exactly the rule table the collapse engine enforces.

Reference: stellar evolution, the Chandrasekhar and Tolman–Oppenheimer–Volkoff limits, and pair-instability supernovae.

Remnant-mass prescription — fallback & the NS–BH mass gap

Once the regime is chosen, a remnant-mass model turns the pre-collapse (core) mass into the bound remnant mass. The default is Fryer et al. 2012: the explosion is a proto-neutron-star plus mass-dependent fallback — light cores explode cleanly (neutron stars), heavy cores fail to explode and fall back to black holes. Its delayed engine ramps fallback smoothly (a continuous NS→BH spectrum that fills the “mass gap”), while its rapid engine keeps fallback negligible until a sharp threshold, reproducing the observed ~2–5 M☉ gap between neutron stars and black holes. Two optional modes are provided: Spera et al. 2015 (PARSEC-based, metallicity-sensitive — metal-poor stars keep a heavier core and form black holes more readily), and Belczynski et al. 2016 (a two-regime direct-collapse-above-threshold model common in binary-population synthesis).

Reference: Fryer et al. 2012 fallback prescription, Spera et al. 2015 and Belczynski et al. 2016.

Resolution pipeline & exact mass conservation

The pipeline is pure — it computes an outcome and never mutates universe state — and runs six stages in physical order so each feeds the next: (1) metallicity winds (Vink / Wolf–Rayet / AGB / rotation / binary-stripping) shrink the progenitor to a pre-collapse core; (2) binary effects let accretion or a merger raise the core, and neutron-star accretion-induced collapse can force a black hole; (3) the collapse engine selects the regime above from the original mass and metallicity; (4) the remnant-mass model applies fallback; (5) classification finalises NS vs BH vs none; (6) mass accounting partitions the progenitor into the bound remnant, radiated mass and ISM return. The three shares always sum back to the progenitor mass, so a stellar death is exactly mass-conserving before it is handed to the engine adapter.

Reference: compact remnants and core-collapse mass accounting; cross-links to the supernova mass-budget ledger and stellar winds.

Line-driven winds — how hot stars blow themselves apart

Hot, luminous OB stars and luminous blue variables drive winds by scattering their own ultraviolet photons in millions of metal absorption lines (the CAK / Vink mechanism). Because the driving is line opacity, the mass-loss rate climbs steeply with luminosity and scales with metallicity as roughly Z^0.85 — metal-poor stars at high redshift lose far less mass and keep more of their birth mass to the end. A bi-stability jump near the recombination of iron sharply increases the rate (and slows the wind) as the star cools below it, so a star crossing that temperature suddenly sheds mass faster.

Reference: line-driven stellar winds (Castor–Abbott–Klein; Vink et al.).

Dust-driven winds — the AGB superwind

Cool, pulsating asymptotic-giant-branch stars levitate their outer layers with shock waves, let the gas cool until dust grains condense, and then push that dust (and the gas dragged along with it) out by radiation pressure. Pulsation greatly enhances the loss, and carbon-rich stars (C/O > 1) form soot more readily and drive an even stronger, dustier wind. This slow, heavy superwind is how intermediate-mass stars return carbon- and dust-enriched gas to the interstellar medium before becoming white dwarfs.

Reference: AGB dust-driven mass loss and the superwind.

Cool-supergiant winds — Reimers & Blöcker

Red giants and red supergiants lose mass through a cool, dust- and chromosphere-assisted wind described empirically by the Reimers law (dM/dt ∝ LR/M). As a star climbs the giant branch its luminosity soars and the loss accelerates super-linearly — the Blöcker enhancement (∝ L^2.7) — turning the gentle red-giant breeze into the powerful wind that strips a red supergiant before core collapse. Chromospheric activity strengthens the wind further in the coolest stars.

Reference: the Reimers and Blöcker cool-giant mass-loss laws.

Magnetised winds & spin-down — why Sun-like stars slow down

Cool, low-mass stars with convective envelopes run magnetic dynamos and lose a tiny, ionised wind along open field lines. The mass loss is negligible for the budget, but the wind co-rotates out to the Alfvén radius and carries off angular momentum with a long lever arm, so the star magnetically brakes and spins down over its life (the Skumanich law). Below a saturation rotation rate the braking torque flattens off, capping how fast rapid rotators lose spin.

Reference: magnetic braking of solar-type stars.

Rotational mass loss & Be decretion discs

A rapidly rotating star is supported partly by centrifugal force, so material near the equator is much easier to eject; as the spin approaches the critical (break-up) rate the equatorial effective gravity falls toward zero and the wind is strongly enhanced there. Past a threshold fraction of critical rotation the star can no longer hold onto its equator and sheds a thin, dense decretion disc — the classical Be-star phenomenon — rather than a fast wind.

Reference: Be-star decretion discs and near-critical rotation.

Wolf–Rayet evolution & winds — stripping the envelope

A very massive star that loses its hydrogen envelope — to winds, rotation, or a binary companion — exposes its hot, helium-burning core as a Wolf–Rayet star with an extremely dense, fast, metal-enriched wind. As successive layers are peeled away the surface chemistry marches through the sequence WN (nitrogen) → WC (carbon) → WO (oxygen), and the stripped star is the progenitor of a Type Ib/Ic supernova. The formation mass threshold itself depends on metallicity, since metal-rich stars reach the Wolf–Rayet stage at lower initial mass.

Reference: Wolf–Rayet stars and stripped-envelope evolution.

Binary-wind coupling — capture, X-ray binaries & widening

In a binary the donor's wind does not just escape: the companion sweeps up part of it by Bondi–Hoyle–Lyttleton capture, and when the wind is slow it can be funnelled through the inner Lagrange point as wind Roche-lobe overflow for much more efficient transfer. A compact companion (white dwarf, neutron star, or black hole) accreting a captured wind lights up as a wind-fed X-ray binary. Because mass leaves the system, the orbit responds — isotropic (Jeans-mode) wind loss widens it as a · M_total stays roughly constant.

Reference: Bondi–Hoyle–Lyttleton accretion and wind Roche-lobe overflow.

Circumstellar medium & wind feedback

Every wind returns gas, dust, and freshly-synthesised metals to the surroundings and sweeps the ambient medium into a shell, building the circumstellar medium that a later supernova will slam into. The fast winds of hot and Wolf–Rayet stars also inflate hot, low-density wind-blown bubbles. In the simulation the shed mass (split into gas, dust, and metals) is credited back to the universe's free-mass budget so total mass is conserved, while a fraction is retained as a growing circumstellar shell around the star.

Reference: circumstellar medium and wind-blown bubbles.

Feeding-zone starvation — seeding solids into small bodies

Embryos that never clear their feeding zone leave their local solids behind as belts, comets, dwarf-planet cores and captured moons rather than sweeping them into a planet. Each tick a small fraction of the disc's current solid mass is diverted out of the disc and split across the small-body classes (the moon share is only taken when the system has a planet to host moons; otherwise it is redistributed so the seeded total is exactly preserved). This is the mechanism that removes fuel from the planet-formation pipeline — the seeded solids are conserved every step of the way but are no longer available to grow planets.

Reference: planetary feeding zones and oligarchic isolation mass.

Dwarf-planet regulation — starved embryos, rarely promoted

A dwarf planet is massive enough to be rounded by self-gravity but has failed to clear its zone, so it is fragile. Promotion to a full planet is allowed only when the shared planet mass-budget has room (the same ceiling the planet-destruction layer enforces); otherwise the dwarf is regulated by destruction — collisionally shattered into an asteroid belt, gravitationally scattered out of the system to the ISM, or tidally stripped into dust on a star-grazing orbit. The more starved the population, the more fragile it is, so destruction runs faster. This diverts solids off the “grow into a planet” track and onto the “ground to dust and removed” track.

Reference: dwarf planets and zone-clearing.

Asteroid collisional cascade — grinding the belt down

Asteroids collide at speeds set by their velocity dispersion; a belt embedded in a planetary system is stirred by mean-motion and secular resonances (the mechanism behind the Kirkwood gaps), so it grinds itself down far faster than an isolated one. Collisions come in two regimes: catastrophic disruptions pulverise both bodies mostly into fine dust, while cratering impacts merely chip off coarser meteoroid ejecta. Over many ticks this is a true Dohnanyi cascade — belt → meteoroids → dust — and the belt's size distribution shifts to the small end and depletes, so its solids cannot re-coagulate into planets.

Reference: collisional cascade and the Dohnanyi size distribution.

Comet sublimation — ices to gas, refractories to dust

Comets are volatile-rich icy bodies. When heated by the star their ices sublimate directly to gas, dragging embedded refractory grains off with them — the physical origin of a comet's gas and dust tails. The mass-loss rate rises steeply with insolation, so it scales with stellar luminosity: bright stars and close-in comets lose mass fastest. Thermal stress, spin-up from asymmetric outgassing, and tidal stress also split comets into meteoroid streams (the parents of meteor showers). Because sublimation permanently returns volatiles to the ISM as gas, comets are a net sink of solids: comet → gas + dust, and comet → meteoroids → dust.

Reference: cometary sublimation and outgassing.

Meteoroid fragmentation — the road to dust

Meteoroids (cm–m fragments) are the intermediate rung between coarse bodies and terminal dust. Mutual collisions comminute them, and near-star heating sublimates them — both paths feed the dust reservoir. A small fraction of the flux intersects a planet with an appreciable atmosphere and is ablated by ram-pressure heating, depositing its mass onto the world as micrometeoritic infall (the one meteoroid channel that delivers solids back to a planet rather than to dust). Whole-body tidal shredding near planets or compact objects is handled for macroscopic bodies by the planet-destruction layer; for the aggregate meteoroid population it appears as the same comminution-to-dust captured here.

Reference: meteoroids and atmospheric ablation.

Dust blowout & Poynting–Robertson drag — the terminal exit

Dust is the end of the cascade and the single most important stage for keeping the planet population realistic, because it is where solids permanently leave the system. For a grain the ratio of radiation pressure to gravity, β = F_rad/F_grav, is independent of distance; grains with β > 0.5 are on unbound orbits the instant they are released and are blown out to the ISM (a larger size range for more luminous stars). Bound grains still absorb and re-emit starlight, and the forward-beamed re-emission robs them of angular momentum — Poynting–Robertson drag — so they spiral inward until they are accreted by the star or sublimate to gas at the dust-destruction radius. Because seeding continuously feeds the cascade and the cascade continuously feeds dust, this stage continuously removes solids (to star + ISM), and that removal rate is the throttle on how much solid mass can remain locked in planets.

Reference: Poynting–Robertson drag and radiation-pressure blowout.

Moon tidal decay — Roche-limit destruction

A moon raises a tidal bulge on its planet; if it orbits below the synchronous radius the lagging bulge torques it inward until it crosses the Roche limit (the fate awaiting Phobos and Triton). A moon that reaches the Roche limit above the surface is torn into a Saturn-ring-like debris disc; one whose decayed orbit meets the surface first impacts the planet and delivers its whole mass. Resonant scattering can instead eject a moon to the ISM. The debris ring is not permanent: mutual collisions and viscous spreading regrind it into dust, which the dust pipeline then removes — so a destroyed moon flows moon → debris → dust → removed and never becomes a solids store. If a system has no planet to receive an impact, that mass is routed to debris so none is stranded.

Reference: the Roche limit and tidal orbital decay.

Mass-budget enforcement & conservation — keeping it honest

The small-body reservoir is not allowed to become an unphysical hoard of solids. Its total is anchored to the peak disc solids the system ever had; any excess above the allowed fraction is recondensed back to disc solids in a fixed trim order (debris → dust → meteoroids → comets → asteroids → dwarf planets → moons), so the coarsest, longest-lived bodies are preserved and the finest, most transient material is shed first. After every pass a conservation validator confirms the reservoir change equals exactly the mass that entered from, minus the mass that left to, external reservoirs; any violation is logged. Together with the shared planet mass-budget (which caps dwarf→planet promotion), this guarantees that destruction genuinely removes mass from the planet-formation pipeline, that dust removal reduces the solids available for planets, and that mass is conserved across every conversion — which is what keeps the net planet population realistic for the mass allotted.

Reference: mass-conserving debris bookkeeping; mirrors the engine's planet-destruction mass budget.