Baryonic Tully-Fisher Relation

• Baryonic Tully-Fisher Relation is an empirical scaling law that links the total baryonic mass (stars and gas) of a galaxy to its characteristic flat rotation speed.
• Observations reveal a tight power-law with a slope near 4 and intrinsic scatter often below 0.13 dex, posing challenges for ΛCDM and alternative gravity theories.
• The relation aids in distance measurement and testing galaxy formation models, with deviations at the low-mass end driven by feedback and observational biases.

The Baryonic Tully-Fisher Relation (BTFR) is a fundamental empirical scaling law linking the baryonic mass of a galaxy—comprising stars and gas—to its characteristic rotation velocity. Unlike the classical Tully-Fisher relation, which links stellar luminosity to rotation speed, the BTFR incorporates a physically motivated sum of all detectable baryonic components, and has become a critical diagnostic of galaxy formation models, dark matter theory, and alternative gravity frameworks.

1. Formal Definition and Observational Determination

The BTFR is most commonly parameterized as a power-law relation

$M_{\rm b} = A\, V^{\alpha}$

where $M_{\rm b}$ is the total baryonic mass, $V$ is a characteristic rotation speed (typically the flat outer disk velocity $V_{\rm f}$), $A$ is the normalization, and $\alpha$ is the logarithmic slope. The baryonic mass is constructed as

$M_{\rm b} = M_{*} + M_{\rm gas}$

with $M_{*}$ the total stellar mass, usually derived from multi-band photometry and population-synthesis–based mass-to-light ratios, and $M_{\rm gas}$ dominated by atomic hydrogen (converted via $M_{\rm HI}=2.36\times10^{5} D^2 F_{\rm HI}$, with $D$ the distance and $F_{\rm HI}$ the 21-cm flux), a correction for helium and molecular gas, and—if available—direct measurements or corrections for $M_{\rm H_{2}}$ and warm-ionized gas (McGaugh et al., 2015, Lelli et al., 2015, Puech et al., 2011, Gnedin, 2011).

High-quality local samples consistently find

$$M_{\rm b} \simeq 47\, V_{\rm f}^4~(M_\odot,~V_{\rm f}~{\rm in~km\,s^{-1}})$$

with best-fit slopes $\alpha$ in the range $3.8-4.0$, normalizations $A$ between $45-50~M_\odot\,({\rm km\,s}^{-1})^{-4}$, and intrinsic scatter below $0.13$ dex when measured using extended HI rotation curves or spatially-resolved kinematics (McGaugh et al., 2015, Lelli et al., 2015, Lelli et al., 2019, McGaugh, 2011). For lower-resolution or integrated single-dish data, slopes can be as low as $3.3-3.4$ (Ball et al., 2022, Kourkchi et al., 2022).

Selection of the velocity metric is critical; the tightest BTFR is always obtained using the mean velocity along the flat part of the outer rotation curve ($V_{\rm f}$). Other definitions (e.g., $V_{\rm max}$, $V_{2.2}$, HI linewidths) yield shallower slopes and larger scatter (Lelli et al., 2019). At high redshift ($z\sim0.6$), the BTFR appears not to evolve in zero point or slope, indicating that much of the gas reservoir converted into stars over the past 6 Gyr was already gravitationally bound to galaxies at $z=0.6$ (Puech et al., 2011).

2. Theoretical Interpretations in ΛCDM and Beyond

In ΛCDM, the BTFR emerges as an indirect consequence of the relationship $M_{200}\propto V_{200}^{3}$ in dark matter halos, modulated by two key factors: the baryonic fraction $f_{b}=M_{\rm b}/M_{200}$ and the velocity ratio $f_{V}=V_{\rm f}/V_{200}$. The observed steep slope ($\alpha\simeq3.8-4.0$) is primarily attributed to the systematic decrease in galaxy formation efficiency $f_{\rm gal}$ at low halo mass; in APOSTLE/EAGLE cosmological hydrodynamical simulations, this causes $M_{\rm b} \propto V_{\rm out}^{3.63}$ (with $V_{\rm out}$ the circular velocity at twice the baryonic half-mass radius) (Sales et al., 2016).

A physically derived expression in ΛCDM is

$$M_{\rm b} = f_{\rm bar}\,M_{200}\,f_{\rm gal} \propto V_{200}^{3+\beta}$$

where $f_{\rm gal} \propto V_{200}^{\beta}$ with typical $\beta \simeq 0.6$ (Sales et al., 2016, Desmond, 2012).

In simulations, the BTFR is reproduced for disk galaxies with appropriately calibrated feedback and star formation histories, but matching both the slope and the exceptional small observed scatter ($\sim0.10$ dex) presents a challenge. ΛCDM-based semi-analytic models, given the underlying scatter in halo mass–concentration and galaxy formation efficiency, are expected to yield a minimum intrinsic scatter of $0.15-0.2$ dex, highlighting an outstanding fine-tuning problem (Lelli et al., 2015, G. et al., 2016).

Alternative frameworks such as MOND predict $M_{\rm b}\propto V_{\rm f}^4$ with the normalization set by a universal acceleration scale $a_0$, i.e., $V^4=G a_0 M_b$, and zero intrinsic scatter in the deep-MOND regime. The empirical BTFR—slope, normalization, tightness—is taken as a direct consequence of this modified dynamics (McGaugh, 2011).

Recent analytic work shows that the BTFR also arises as an inevitable scaling if galaxy formation occurs by collapse from a hierarchically distributed, D$\approx$2 quasi-fractal IGM, with a characteristic acceleration $a_F\equiv4\pi G\Sigma_F$ determined by the ambient mass-surface density (Roscoe, 2023). In these frameworks, the observed BTFR becomes a manifestation of either new physical laws or boundary conditions of cosmic structure.

3. The Low-Mass/Faint-End Behavior and Curvature

While the high-mass ($M_{\rm b} \gtrsim 10^{9} M_\odot$) BTFR is well-established as a single power-law, both models and observations indicate significant changes—and potential curvature—at the dwarf galaxy regime. In $\Lambda$CDM hydrodynamical simulations such as APOSTLE/EAGLE, SAMBA, and NIHAO, the BTFR steepens significantly for $V_{\rm rot} \lesssim 40$ km/s (i.e., $M_{\rm b} \lesssim 10^{8} M_\odot$), with a strong predicted downturn (up to 1.5× deviation in $V$ at fixed $M_{\rm b}$), as baryon retention efficiency drops sharply due to feedback and photoevaporation effects (Sales et al., 2016, Glowacki et al., 2020, McQuinn et al., 2022, Ruan et al., 20 Mar 2025).

However, empirical recovery of this faint-end turnover is strongly limited by observational biases:

• Many real dwarfs only exhibit rising rotation curves out to their detectable HI disk edge $r_{\rm out}$, so $V_{\rm obs}$ substantially underestimates the true $V_{\rm max}$ (Sales et al., 2016, McQuinn et al., 2022, Ruan et al., 20 Mar 2025).
• At low $V_f$, the ionized gas fraction becomes dominant and is often systematically unaccounted for; the “observed” BTFR will deviate downward by up to a factor of two in $M_{\rm b}$ at $V_f \lesssim 40$ km/s if the ionized layer is missed (Gnedin, 2011).
• Inclination errors and selection for gas-rich, well-ordered disks can further bias velocity and mass estimates, increasing the apparent scatter and flattening the observed faint-end slope (Sales et al., 2016, McQuinn et al., 2022).

Hydrodynamical simulations predict that the intrinsic BTFR will remain tight ($\sigma_{\log V} \sim 0.05$ dex) even as the slope increases at low mass, but the observed relation will appear shallower and more scattered unless rotation curves are traced to much lower HI column densities ($\Sigma_{\rm HI} \sim 0.01\,M_\odot\,{\rm pc}^{-2}$), as targeted by FEASTS and MHONGOOSE (Ruan et al., 20 Mar 2025). Empirical studies report a marked turndown at $M_{\rm b} \sim 10^{8} M_\odot$, $V_{\max} \sim 45$ km/s once cored density profiles (FIRE-calibrated) are used to infer the true $V_{\max}$ (McQuinn et al., 2022).

4. Intrinsic Scatter, Fundamental Plane, and Physical Correlates

The BTFR is distinguished among galaxy scaling laws by its extremely small intrinsic scatter. Observationally, the perpendicular (orthogonal) intrinsic scatter is $\sigma_{\perp}\sim0.026$ dex (6%) when using $V_{\rm f}$ (Lelli et al., 2019), with best-fit values $\sim0.10-0.13$ dex across different samples and methods (McGaugh et al., 2015, Lelli et al., 2015, Ball et al., 2022, Kourkchi et al., 2022). There is no detectable correlation between residuals and galaxy structural parameters (radius, surface brightness, scale length), distinguishing the BTFR from mass–size or mass–angular-momentum (“Fall relation”) scaling laws, which exhibit $\sim7\times$ larger scatter (Lelli et al., 2019, Lelli et al., 2015).

In high-resolution simulations, the predicted scatter is typically larger ($\sim0.18-0.24$ dex), as a direct result of the theoretical scatter in halo concentration, spin, and baryon fraction $f_d$ (Desmond, 2012, G. et al., 2016). The absence of “second parameter” trends in the observed BTFR remains a stringent challenge to hierarchical galaxy formation scenarios (Lelli et al., 2015).

The universality of the BTFR scatter strongly constrains the possible variation in physical drivers such as the stellar IMF and star formation histories across disk galaxies, and limits the role of stochastic feedback and “bursty” evolutionary processes in setting the global baryonic–velocity scaling relationship (McGaugh et al., 2015, McGaugh, 2011, Lelli et al., 2015).

5. Sample Size Effects, Systematics, and Measurement Methodologies

Both the recovered slope and scatter of the BTFR depend sensitively on sample size and on homogeneous measurement methodology. Simulations with large synthetic catalogs confirm that for $N<100$ galaxies, the 1σ uncertainties in recovered slope exceed 0.08 dex, and the scatter in recovered intrinsic scatter exceeds 0.16 dex; robust, bias-free estimates require $N\gtrsim200$ with coverage across the full $M_{\rm b}$–$V$ parameter space (G. et al., 2016). Choice of velocity metric can shift the BTFR slope between $3.1$ (peak or inner velocities) and $4.0$ (outer flat velocities), with linewidth-based samples from H I surveys (e.g., ALFALFA) returning shallower slopes than high-resolution spatially-resolved rotation curves (Ball et al., 2022, Kourkchi et al., 2022, McGaugh et al., 2015, Lelli et al., 2019).

Apparent tensions between BTFR slopes derived from simulations and observed samples can often be traced to sample size limitations, non-representative selection, mismatched mass/velocity distributions, or inconsistent velocity definitions. Properly-matched samples and velocity/mass measurements are required to interrogate true model–data correspondence (Ball et al., 2022, G. et al., 2016, Lelli et al., 2015).

In distance applications, the BTFR provides a secondary, redshift-independent distance indicator for large HI and photometric samples, with typical distance uncertainties of ~0.17 dex for high-quality data (Ball et al., 2022, Kourkchi et al., 2022).

6. Extensions, Generalizations, and Systematic Variants

Recent work generalizes the BTFR in several directions:

• Dimensionless and physically-based reformulations: Dimensional analysis yields

$V_{\rm circ}^4 = a_s G M_{\rm b}$

where $a_s$ is a scaling acceleration related to the global mass discrepancy and disk surface density. A fully dimensionless variant connects $V_{\rm circ}$, dynamical mass surface density, and mass discrepancy without explicit recourse to external constants, applicable to analytic and computational frameworks (Fortune, 2020).

• Inclusion of S0/E galaxies and virial-theorem–based velocities: Extended versions of the BTFR have been constructed for large samples of spirals and ellipticals, using effective rotation velocities derived from dispersion-based dynamical masses. The joint relation remains a nearly power-law, with slopes $3.2-4.0$, supporting a broad universality for both rotation- and dispersion-supported systems (Ravi et al., 21 Dec 2025).
• Dependence on the gas-phase composition: Omission of warm ionized gas systematically depresses $M_{\rm b}$ at the faint end, artificially flattening the observed BTFR for $V_f \lesssim 40$ km/s. Analytic corrections or direct measurement of this component are required to match theoretical models (Gnedin, 2011).
• Local–global connection: Consistent with the “baryonic scaling model,” the BTFR can be viewed as the global manifestation of a radius-by-radius scaling between the local baryonic mass surface density and the local rotation speed, naturally integrating to the observed global power-law relation (Swaters et al., 2012).

7. Open Theoretical and Empirical Challenges

Despite broad empirical agreement, the BTFR continues to challenge both ΛCDM and alternative gravity frameworks:

• Fine-tuning problem in ΛCDM: The combination of the steeper-than-expected slope, near-zero intrinsic scatter, and lack of secondary parameter dependence implies a high degree of self-regulation or yet-unknown coupling between baryonic and dark matter assembly, or new feedback physics, that remains to be explained by hierarchical theories (Sales et al., 2016, Lelli et al., 2015, McGaugh, 2011).
• Origin of the acceleration scale: The empirical BTFR normalization directly recovers Milgrom’s $a_0$ in MOND, but the exact physical genesis of such a universal scale in a Newtonian or ΛCDM setting is unclear, though quasi-fractal IGM assembly and emergent boundary conditions have been proposed (Roscoe, 2023).
• Characterization at low mass: Mapping the predicted BTFR turndown at $M_{\rm b} \lesssim 10^{8} M_\odot$ remains a top-tier observational challenge. Future deep HI surveys (FEASTS, MHONGOOSE, MIGHTEE) focused on low surface densities and high angular resolution are expected to provide decisive tests of feedback-regulated BTFR steepening and its variance with environment and redshift (Ruan et al., 20 Mar 2025, McQuinn et al., 2022).

The BTFR stands as both an empirical pillar for calibrating galaxy masses and a stringent test-bed for galaxy formation and cosmological models, the physical basis of which continues to motivate high-resolution observations, sophisticated simulations, and theoretical reinterpretations.