CL Fits and the Lelli Relation

Lelli, Fraternali & Verheijen (2013) found that the central rise of a rotation curve, dRV(0), scales with the central surface brightness roughly as I00.5. Here the relation is shown for 131 SPARC galaxies with three gradient estimates: the secant through the first three data points, the CL model gradient sCL = vL/(√3 R), and the arctan control. Click any galaxy to see its rotation curve, the two fits, and the curves predicted from its baryonic data alone.

Central gradient against central surface brightness

Gradient
Surface brightness
bisector fitslope 0.5Lelli range through the sample medianbulge

One galaxy: fits and prediction

CL fitCL from Mbar, Σinarctan fitarctan from Mbar, Σin

The inner rise: secant against central slope

secant through first three pointsCL slope sCL·rarctan slope at r = 0

Prediction accuracy across the sample

Each dot is one galaxy: the rms deviation, in dex, between the observed curve and a curve. Tick marks are medians. Dashed rows are predictions from Mbar and Σin with leave-one-out calibration, so no galaxy helps to predict itself. Click a dot to open that galaxy.

Reading the panels

  • The correlation belongs to the data. The measured secant, the CL gradient and the arctan gradient give nearly the same Spearman ρ with Σin (0.76–0.78). Switching the estimator moves the cloud up or down; it hardly changes its shape.
  • CL changes the normalisation. In the median sCL/sdata = 1.01, while the arctan slope at the centre is 1.38 times the secant. For a curve that bends near the centre, a secant through finite radii underestimates the slope at r = 0, so the arctan value is not necessarily biased. The CL inner branch is straight up to R, so its central slope equals its secant.
  • The proxy matters as Lelli et al. describe. With the extrapolated disc value Σd,0, ρ falls to 0.58 and the bulge galaxies leave the relation, because their central light exceeds the disc fit.
  • Slopes depend on the regression direction. Both axes scatter, so OLS gives a lower slope and the inverse fit a higher one; the bisector is shown as a descriptive symmetric estimate.

How a curve is predicted from the light

For each galaxy, both scaling relations are refitted on the other 130 galaxies. Then

vL from Mbar (BTFR) · sCL from Σin (gradient relation) · R = vL/(√3 sCL) · M from the CL relation

and the same for the arctan curve with Vc and its central slope. The median rms is 0.097 dex for CL and 0.091 dex for arctan, a typical factor of about 1.25, against 0.037 and 0.027 dex for direct fits. The two relations recover the amplitude and shape of most curves, but not as well as the radial acceleration relation does from the full baryon distribution.

Σin is sampled at the innermost kinematic radius, and the sample of 131 was selected partly on the CL fits being constrained, so this is a prediction from baryonic quantities for the selected sample, not a fully independent photometric one. Source: de Haas (2026), code and data at doi:10.5281/zenodo.23340027. SPARC: Lelli, McGaugh & Schombert (2016).