E.P.J. de Haas · Kandinsky College, Nijmegen

Galaxies and jets as flows of space

Four interactive models of the constant-Lagrangian metric-inflow galaxy model and the three-rapidity Painlevé–Gullstrand jet geometry, which run the model equations live in your browser — drag to rotate, scroll or pinch to zoom, and change the parameters. Below them, three data explorers test the CL fits against the SPARC galaxies.

The models

Spiral galaxy with polar jets

A metric-inflow disk whose arms are streamlines of the flow, fed by broad inflow cones, with surplus inflow redirected into narrow double-helix jets.

bulge mass, radius, redshiftjet cone & helix pitchsurplus fraction
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E0E5S0 SaSbSc SBaSBbSBc

The Hubble sequence from cone occupation

Every tracer stays on its cone θ₀. How the flow occupies the cones turns one model into ellipticals, lenticulars, spirals and barred spirals; arms open as bulge mass and radius grow.

clickable tuning forkmorphing tourtwo-Lagrangian bars
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The M 87 jet in a single PG geometry

Zoom across seven decades in radius: inflow on broad cones, a stagnation surface at r₀ where the helix winds tight and reverses, and the outward helix on the 6° jet cone out to HST-1 and kiloparsecs.

log-radius zoomθ₀, r₀, ε∞guided tour
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The riverbed of space

Space cells created at random give Hubble’s law; absorption only in matter drives a slow replacement chain; the active riverbed carries matter into constant-speed CL spirals that turn into a circle at rc.

step-by-step build-upbulge radius & absorption ratelimit circle
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The vacuum flow at the bulge border

Why the constant-Lagrangian flow follows from one postulate: the vacuum is an ideal, barotropic, non-dissipative flow. From the same border value, CL is compared with GR’s Lense–Thirring frame dragging and a Kepler-like rotation, by Bernoulli, Rayleigh stability and GR’s vacuum condition.

CL: L keptLense–ThirringKepler-like same border value, three outward laws

CL vacuum flow: Bernoulli and Rayleigh

A rotating bulge drags space; at its border the flow is continuous, beyond it the vacuum cannot dissipate the energy it receives. Bernoulli along the spiral streamlines then gives vφ² = L − 2GM/r, the CL flow, which also passes Rayleigh’s stability test. GR’s Lense–Thirring continuation passes neither, but is GR’s vacuum. With a 27 s motion model.

border rotationlive top viewBernoulli · Rayleigh · GR test
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Galaxy growth through cosmic time

One CL galaxy grown from its earliest onset to today in a flat ΛCDM background, in two steps: first the derived kinematic background, then the same galaxy with core resets as an exploratory foreground.

z ≈ 11z ≈ 3z ≈ 1todayrc ∝ a·M1/3

CL galaxy through cosmic time

The baseline. The CL mass grows by a standard inflow history, H(t) and a(t) follow ΛCDM, and the critical radius rc = (2GM/H²)1/3 opens up with the expansion. Rotation curve and spiral at every epoch, with ΛCDM disc and halo edges behind a switch.

derived baselinerc against a(t)earliest onsetΛCDM radii switch
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active core Rfossil stages fadeover Nint orbitsresets whenMdisc = k·Mcore

CL galaxy growth with core resets

The exploratory extension. A fixed core collects part of the inflow while a disc builds up outside it; when the disc reaches k times the core mass, the core resets to a size set by GM/R² = ε·cH(t). Fossil stages fade over Nint orbits, and the page reports whether the galaxy looks nested or single-L.

exploratoryfd, k, ε, NintBTFR slope 4 at every epochnested or single-L
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Testing the fits against SPARC

Three data explorers that set the CL fits of the SPARC galaxies against quantities the fits never used: baryonic mass, photometry and central surface brightness. Click any galaxy to follow it through the plots.

Vflat and vLMbarV⁴ ∝ Mbar

CL fits and the BTFR

The baryonic Tully–Fisher relation drawn with four velocity measures, including the observed flat speed and the CL plateau speed vL. Tests the embedding condition GM/R² = cH step by step and shows what is independent of what.

Vflat vs vLembedding GM/R² = cHresiduals
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M*, Mbul, Rd (photometry)M, R (fit)SaSbScSdIm

CL fits vs photometry

The fitted M and R against what the fits never saw: stellar, bulge and baryonic mass and the disc scale length, with Hubble type and surface brightness as colour, and controls for correlations built into the fit.

choose both axescolour by type or SBfit category vs morphology
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central surface brightnessdRV(0)fit and prediction

CL fits and the Lelli relation

The central velocity gradient against central surface brightness (Lelli et al. 2013) for 131 galaxies, with the measured secant, the CL gradient and an arctan control, and for each galaxy the rotation curve predicted from its baryonic data alone.

three gradients, three proxiesprediction from Mbar and Σininner-rise zoom
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About these models

The galaxy models use the constant-Lagrangian metric-inflow picture: the bulge sets an inflow of space whose orbital and radial channels add up to a constant vL², and spiral arms are streamlines of that flow. The jet models use the three-rapidity Painlevé–Gullstrand metric, in which flow lines lie on cones of fixed polar angle. The riverbed model shows how space cells are created and absorbed, and how the active riverbed moves matter.

Each model separates what is imposed by observation, calibrated, derived from the geometry, or interpretive, and states its display conventions (for example, jet cones drawn wider than their true angle so the helices are visible).

The two growth models share one galaxy and one cosmology. The baseline derives rc(t), the plateau speed and the curve at every epoch from M(t), R(t) and H(t); the reset model adds a core + disc layer with four parameters, of which only ε is calibrated (from the BTFR zero point). It is exploratory and labelled as such.

The SPARC explorers use the SPARC database (Lelli, McGaugh & Schombert 2016) and the single-Lagrangian CL fits. Their numbers can be reproduced with the code and data in SPARC-CL-scaling, archived at doi:10.5281/zenodo.23340027.

Papers