One constant-Lagrangian galaxy, grown from the earliest moment it can exist to today in a flat ΛCDM background. The CL mass M grows by a standard cosmological inflow (mass-accretion) history, H(t) and a(t) follow ΛCDM, and the critical radius rc = (2GM/H²)1/3, the radius where the radial river stalls, opens up with the expansion. Press play or drag the time slider. This page adds an exploratory foreground to the baseline model: a fixed core collects part of the inflow, a disc builds up outside it, and when the disc reaches k times the core mass the core resets. Earlier stages remain as fossil rings that merge over Nint orbits.
rc and a(t). For fixed M, rc ∝ a exactly while matter dominates, and freezes at (2GM/H₀²ΩΛ)1/3 once Λ takes over. With mass growth, rc ∝ a·M1/3 in the matter era.
rc as a density radius. The mean density inside rc is 3M/4πrc³ = 3H²/8πG, exactly the critical density. The ΛCDM virial radius R200 is the radius of 200 times that density, so R200 = rc/2001/3 ≈ rc/5.85 for the same mass.
The CL curve at each epoch. X = √(2GM/R) − HR, vL² = ³⁄₂X²; v² = ½X²r²/R² inside R and vL² − (√(2GM/r) − Hr)² outside, with the epoch's own M, R and H. The spiral streamlines follow tan α = vr,eff/vφ and wind up into the circle rc; the view draws them from 0.7 rc inward, because closer to rc they wind so tightly that the turns merge.
Core + disc with resets, sized by the embedding condition. The core radius follows your size law GMcore/R² = ε·cH(t) (de Haas 2025c): it is set at onset and again at each reset, with the epoch's own H(t), and stays fixed in between. Inflowing mass is split: a share 1 − fd crosses R and joins the core mass Mcore, which alone drives the CL flow; the share fd stays outside R as a disc of test masses. When the disc reaches k·Mcore it is no longer a test mass: the disc joins the core (all mass inside the new R, M2 = everything), the spiral of the ending stage freezes as a fossil between the old and new radius, and the new radius follows from the embedding condition with the new mass. No ΛCDM size relation enters this layer. Because R ∝ (M/H)1/2, the plateau obeys vL⁴ ≈ 9εGcH·Mcore: a baryonic Tully–Fisher slope of 4 at every epoch, with a zero point that scales with H(t). ε = 0.02 reproduces the observed BTFR (Mbar = 47 M☉ (km/s)−4 V⁴, Vflat ≈ vL/1.065) for M₀ taken as the total baryonic mass, and leaves today's core at GM/R² ≈ 10−1.2 cH₀, close to the 10−1.0 found in the SPARC CL fits. The disc is drawn out to 3.5 R, the SPARC median ratio of the optical edge to R.
Integration of earlier stages. Each fossil stage counts the orbits N completed inside the core since its reset (solid-body Ω = v(R)/R of the active core). Its ring and spiral fade as w = exp(−N/Nint), and inside R the rotation curve shifts with the same weight from the frozen curve of the stage that built each band (L₁ remnants) to the solid-body interior of the current Lagrangian: v² = w·vstage² + (1 − w)·vnow². The pink solid line is this blend; the dashed line is the fully merged interior. A stage counts as still distinguishable while w > 0.1, and the readout calls the galaxy nested or single-L accordingly. Inner rings complete hundreds to thousands of orbits and merge; only a recent reset stays visible, so the model predicts that galaxies with clean two-L fits had their last reset late.
Fossil orientation. A frozen spiral keeps the pattern angle it had at its reset, accumulated as ∫Ω dt over the core's history. After many orbits that angle is effectively random; the alternative setting draws a seeded random angle per reset. Neither carries information, so consecutive fossils do not line up into a spiral of their own.
Few resets. Resets need fd > k/(1+k), and each one raises the core mass by M₂/M₁ = 1 + k/(fd − k(1 − fd)). Two to four resets since tstart, with jumps of ×5–20, come from fd about 0.03–0.15 above the threshold: k = 1.5 with fd 0.65–0.75, k = 2 with fd 0.70–0.85, k = 3 with fd 0.80–0.90, k = 4 with fd 0.85–0.95. Far above the threshold resets become frequent and approach continuous growth R(t) = (GM/εcH)1/2; below it there is no reset and the disc grows without bound. The sliders are limited to this neighbourhood, and the regime readout shows the jump and its class.
Optical disc and dark-matter edges (switch, off by default). The halo mass today follows from the plateau speed, V200 = vL/(Vmax/V200) and M200 = V200³/(10GH₀), and grows with the same accretion history. The dark-matter edge is RΔ = (2GMh/ΔH²)1/3, with Δvir(z) from Bryan & Norman (1998) or Δ = 200. The optical edge is Ropt = 3.2 Rd, with Rd from r1/2 = 0.015 R200c, so Ropt ≈ 0.029 R200c. Note that rc has the same form with Δ = 1 and the CL mass M in place of the halo mass.
Earliest possible onset. The galaxy starts at the latest of three conditions: M(t) ≥ the minimum galaxy mass, R(t) < rc(t) (otherwise X ≤ 0 and no CL structure exists), and t ≥ tstart, the chosen start time (default 0.4 Gyr, z ≈ 11).
What is assumed
imposed Flat ΛCDM with Ωm = 0.31, H₀ = 67.9 km/s/Mpc (= 2.2×10⁻¹⁸ s⁻¹, the value used in the SPARC CL fits); radiation neglected, so the earliest ~1 Myr are not exact.
calibrated The growth laws are halo mass-accretion histories from ΛCDM simulations (McBride et al. 2009; Wechsler et al. 2002) or the power law of de Haas (2026a). Applying them to the CL mass M, which tracks the inner (bulge-like) mass rather than the halo, is an assumption of this model.
calibrated Average size relations: Ropt = 3.2 Rd (Persic & Salucci 1991), r1/2 ≈ 0.015 R200c (Kravtsov 2013), Δvir(z) (Bryan & Norman 1998), Vmax/V200 ≈ 1.2–1.4 for Milky-Way-mass haloes. These are population averages with about 0.2 dex scatter, applied here to one galaxy. Both edges are the standard dark-matter picture, drawn for comparison with rc; the CL model itself has no halo.
calibrated Four parameters of the core + disc layer: the disc capture fraction fd (0.75), the reset threshold k (2), the embedding factor ε (0.02, from the BTFR zero point) and the integration time Nint (30 orbits, a plausible range is 20–100; not calibrated). The size law itself, GM/R² = ε·cH(t), is your embedding condition; only its factor is calibrated.
assumed The core radius R follows one of three simple histories; none is derived from the CL theory yet.
derived rc(t), vL(t), the rotation curve and the spiral at each epoch follow from M(t), R(t) and H(t) with no further freedom.
not included Bars, mergers, virial zones and the self-gravity of the disc. The core + disc layer is a one-way kinematic foreground on the clean CL background, and the angular momentum that merging fossil stages must lose is not tracked.