The baryonic Tully–Fisher relation and the CL fits

The BTFR links two quantities measured independently: the baryonic mass Mbar, from 3.6 μm light and HI, and the outer rotation speed. Your CL fits provide a third quantity, the plateau speed vL, and your BTFR papers predict how it should scale: V⁴ = 9GcHz·M, with M a few percent of Mbar. This page tests three things: the relation with Vflat and with vL, what vL adds, and whether the fitted (M, R) satisfy the two steps of that prediction.

Four velocities, same masses

The fit and scatter use the galaxies that have Vflat, so all four rows compare the same sample. Scatter is the rms of log Mbar about the fitted line. Lelli et al. (2019) find 0.24 dex observed scatter with Vflat for 123 SPARC galaxies.

Rising curves: offset from Vflat BTFR using Vlast–
… using vL (calibrated to Vflat)–
vL / Vflat (median, scatter)–
ρ(BTFR residual, vL/Vflat)–

Does vL add anything beyond Vflat?

Offset of each galaxy from the Vflat BTFR against how much its vL exceeds Vflat. If vL carried extra information about the baryons, these would correlate.

Step 1: the embedding GM/R² = cH

Distribution of the fitted bulge-edge acceleration g = GM/R² relative to cHz. The papers place the population average at 0.

Step 2: M a few percent of Mbar

Distribution of the fitted M / Mbar. The dashed line is the fraction f that the observed BTFR normalization requires if Step 1 holds exactly.

The two steps trade off along the BTFR

Each galaxy's fitted g/cH against M/Mbar. Since vL⁴ ≈ 9G·M·g and vL ≈ Vflat, galaxies on the BTFR must lie near the diagonal band, where a smaller g goes with a larger M/Mbar. The star marks the point the papers predict (g = cH, M = f·Mbar).

Reading the result

log(Mbar/V⁴) = −log(9GcH) − log(g/cH) − log(M/Mbar)

This identity follows from vL⁴ ≈ 9G·M·g for HzR ≪ √(2GM/R). The observed normalization fixes only the sum of the last two terms. The paper attributes all of it to M/Mbar, with log(g/cH) = 0. The fits split it differently.

What is independent, and what that means

  • Mbar never enters the fit. It comes from SPARC photometry and HI, so a tight Mbar–vL relation is a genuine external check of the CL velocity scale, not a circular one.
  • Statistical independence is weaker. Both quantities describe the same galaxy and share its distance (Mbar ∝ D², while the velocities do not depend on D), which adds correlated scatter along the relation.
  • What would show that CL adds information is vL beating Vflat: smaller scatter, or residuals that track vL/Vflat. Where Vflat exists, it does not. Where the curve is still rising and Vflat is undefined, vL predicts the plateau. Measured against the Vflat relation, that prediction lands about as close as the last measured speed; deeper HI data that extend those curves would settle it.

Fits and assumptions

  • Uniform single-L CL fits for all 175 galaxies, Hz = 2.2×10⁻¹⁸ s⁻¹ (cHz = 6.6×10⁻¹⁰ m/s²), with vL = √1.5·(√(2GM/R) − HzR). Fits whose R runs to its lower bound are hidden by default; they are mostly massive bulge systems.
  • Mbar = Υdisk·Ldisk + Υbulge·Lbulge + 1.33·MHI at 3.6 μm. The sliders change it live; SPARC's default is 0.5 / 0.7.
  • Reference line: Mbar = 50·V⁴ (M☉, km/s; McGaugh 2005), as quoted in your 2018 paper. Theory line: Mbar = V⁴/(9GcHz·f).