Baryonic Tully-Fisher Relation (bTFR)

• The baryonic Tully-Fisher Relation is a power-law scaling between a galaxy’s baryonic mass and its flat rotation velocity, exhibiting a slope of ~3.95 with minimal intrinsic scatter.
• High-precision calibrations using Cepheid/TRGB distances and resolved HI rotation curves yield robust estimates of the normalization and slope through detailed regression methods.
• Its application as a secondary distance indicator provides an independent measurement of the Hubble constant, challenging conventional ΛCDM predictions and supporting alternative theories like MOND.

The Baryonic Tully-Fisher Relation (bTFR) is a power-law scaling between a galaxy’s total baryonic mass and its asymptotic rotation velocity. Empirically, it is among the tightest known dynamical relations in extragalactic astrophysics, linking photometric and kinematic observables across five decades in mass. The bTFR has profound implications for galaxy formation models, theories of gravity, and cosmological distance scaling. This article synthesizes foundational results, state-of-the-art calibrations, methodological protocols, theoretical context, and principal systematics, primarily as delineated in "Using The Baryonic Tully-Fisher Relation to Measure $H_0$" (Schombert et al., 2020), while integrating corroborating results from key calibration and theory papers.

1. Mathematical Formulation and Empirical Calibration

The bTFR is conventionally expressed as a power law,

$M_b = A V_f^x$

where $M_b$ is total baryonic mass (stars, gas), $V_f$ is the flat circular velocity, $A$ is a normalization, and $x$ is the slope. In high-precision calibrations using galaxies with Cepheid and TRGB distances, the relation is tightly constrained:

• Slope: $x=3.95 \pm 0.16$
• Zero-point: $\log A = 1.79 \pm 0.34$ (i.e., $A \simeq 62\,M_\odot\,(\mathrm{km\,s}^{-1})^{-x}$)
• Scatter: Orthogonal scatter $\sigma_\perp \simeq 0.048$ dex in $\log M_b$ These parameters are stable across well-selected calibrator samples (Schombert et al., 2020), confirming and refining earlier gas-rich galaxy measurements: $x=3.98 \pm 0.06$, $A=47\pm 6\,M_\odot\,(\mathrm{km\,s}^{-1})^{-4}$ (McGaugh, 2011). The intrinsic scatter is consistent with being due entirely to observational uncertainties, with no compelling evidence for additional physical scatter.

2. Observable Construction: Mass and Velocity

Precise measurement of $M_b$ and $V_f$ critically determines the tightness and calibration of the relation:

• Stellar Mass ($M_*$): Derived from deep Spitzer $3.6\,\mu\mathrm{m}$ imaging, with a fixed near-IR mass-to-light ratio ($\Upsilon_* = 0.50$ for disks, $0.70$ for bulges), rooted in stellar population models matching observed colors and star-formation histories (Schombert et al., 2020).
• Gas Mass ($M_g$): Obtained via HI $21\,\mathrm{cm}$ line flux for atomic hydrogen ($M_{\mathrm{HI}}$), corrected by a factor $\eta \simeq 1.33-1.40$ for helium and metals. A mean $5\%$ contribution from $H_2$ is empirically added (Schombert et al., 2020, McGaugh, 2011).
• Rotation Velocity ($V_f$): Defined as the mean speed over the flat, outer portion of resolved HI rotation curves, constructed from interferometric data to minimize inclination, non-circular motions, and asymmetries (typical $8\%$ error, $0.02$ dex) (Schombert et al., 2020, Lelli et al., 2019).

The final baryonic mass estimator is $M_b = M_* + \eta\,M_{\mathrm{HI}} + M_{H_2}$, yielding uncertainties of $0.06$ dex in $\log M_b$.

3. Calibration Protocols, Regression, and Uncertainty

Calibration employs galaxies with direct Cepheid and/or TRGB distances, high-quality rotation curves, and adequate inclination ($i>30^\circ$):

• Sample: 30 SPARC galaxies plus 20 from Ponomareva et al. (2018), spanning $V_f \sim 20\text{--}300\,\mathrm{km\,s}^{-1}$ (Schombert et al., 2020).
• Fitting: Maximum-likelihood orthogonal regression (BayesLineFit), treating both axes' errors and intrinsic scatter. Combined distance, flux, and velocity errors yield total axes uncertainties of $0.06$ dex ($M_b$) and $0.02$ dex ($V_f$).
• Systematic Tests: Variation in $\Upsilon_*$ ($\pm 0.1-0.15$), gas correction $\eta$ ($\pm 0.03$), or TRGB zero-point ($\pm 0.05$ mag) shifts $H_0$ by less than $0.4\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$—smaller than flow model uncertainties (Schombert et al., 2020).

The combination of photometric and kinematic orthogonality yields an empirical scale free of $H_0$ assumptions.

4. Application: Distance Scale and Hubble Constant Measurement

Using the calibrated bTFR as a secondary distance indicator:

• Sample: 95 independent SPARC “flow” galaxies, matching the calibrator selection on rotation curve quality and inclination.
• Velocity Correction: CosmicFlows-3 multi-attractor flow model corrects heliocentric redshifts for local large-scale flows (Virgo, Great Attractor).
• Method: For a trial value of $H_0$, convert flow-corrected velocities to distances ($D=v/H_0$), recompute $M_b \propto D^2$, and fit the flow sample’s zero-point offset relative to the calibrator bTFR. Minimize the offset to solve for $H_0$.
• Result: $$H_0 = 75.1 \pm 2.3 (\mathrm{stat}) \pm 1.5 (\mathrm{sys})\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$$, with the error budget dominated by large-scale velocity flow model uncertainties (Schombert et al., 2020).

This bTFR-based $H_0$ is in mild tension with CMB-derived values (e.g., Planck: $$H_0 \approx 67.4\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$$), anchoring an independent local expansion scale.

5. Theoretical Context: ΛCDM, MOND, and Fine Tuning

• ΛCDM: Simple virial-theory arguments ($M_b \propto V^3$) underpredict the empirical slope ($x>3.9$). Realistic models must invoke systematic variations in baryon fraction, feedback efficacy, and concentration to match the observed slope and normalization (McGaugh, 2011, Desmond, 2012). The minuscule intrinsic scatter ($\sigma_\perp \lesssim 0.05$ dex) requires narrow coupling between disk baryon fraction and halo properties, presenting a fine-tuning challenge for hierarchical formation models (Lelli et al., 2015).
• MOND: Predicts a unique relation $M_b = (G a_0)^{-1} V_f^4$; recent gas-rich samples yield $$a_0 = (1.3 \pm 0.3)\times 10^{-10}\, \mathrm{m\,s}^{-2}$$, matching canonical values (McGaugh, 2011). The bTFR’s empirical slope and negligible scatter are direct predictions of MOND, with no adjustable parameters.
• Alternative Newtonian derivations: Fractal models of the intergalactic medium (IGM, $D\approx 2$) reproduce $V^4 \sim M$ scaling from Newtonian equilibrium conditions contextualized by hierarchical cosmic structure (Roscoe, 2023).

6. Systematics, Limitations, and Precision Prospects

• Sample Selection and Velocity Definition: The velocity indicator ($V_f$, $W_{20}$, $W_{50}$, etc.), gas fraction, and radius of measurement all systematically impact the fitted slope, zero-point, and scatter. Unresolved HI linewidths typically yield lower slopes ($\sim 3.3$), whereas spatially resolved $V_f$ delivers steeper values ($\sim 3.9$) (Bradford et al., 2016, Lelli et al., 2019).
• Dominant Sources of Uncertainty: The largest systematic is the local flow correction model. Other systematics—stellar population assumptions, gas mass calibration, regression method, and photometric zero-point—are subdominant and largely cancel through matched calibration and application.
• Future Sensitivity: Intrinsic scatter ($\sim 0.05$ dex) is now observationally limited; deeper HI mapping, higher-precision near-IR photometry, and increased numbers of calibrators promise few-percent-level distance precision. Expanding spatial coverage will reduce sensitivity to peculiar velocities and flow model uncertainties.
• Physical Limitation: At the low-mass end ($M_b\lesssim 10^8\,M_\odot$), the bTFR may “turn down” from the extrapolated power law due to the combined effects of feedback and reionization suppressing baryon retention and measurable rotation velocities (McQuinn et al., 2022, Ruan et al., 20 Mar 2025).

7. Astrophysical and Cosmological Applications

• Distance Indicator: The bTFR is now a competitive, redshift-independent secondary distance indicator for galaxies in the nearby universe, with systematic uncertainties quantifiable and dominated by large-scale flow corrections (Schombert et al., 2020).
• Fundamental Scaling Law: Its empirical tightness and independence from collisional baryonic physics makes it the “fundamental plane” of disk galaxy dynamics, more fundamental than mass-size or angular momentum relations (Lelli et al., 2019).
• Constraints on Galaxy Formation: The bTFR’s scatter and slope provide stringent constraints on feedback, baryon fraction calibration, and dark matter halo response in theoretical models.
• Hubble Constant Tension: The bTFR calibration yields $H_0$ consistently around $75\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$, in mild tension with the lower values favored by CMB cosmology.
• Integrated Cosmic Structure: Recent quantum-gravity frameworks and fractal cosmologies interpret the bTFR normalization and slope as reflecting deeper cosmic structure and epoch dependence (Marongwe et al., 25 Nov 2025, Roscoe, 2023).

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This synthesis demonstrates that the baryonic Tully-Fisher relation is a robust empirical law uniting independent observations and theory. Its calibration and application continue to sharpen constraints on both local cosmological parameters and the small-scale physics of galaxy formation.