Baryonic Tully-Fisher Relation (BTFR)

• BTFR is an empirical scaling law that relates a galaxy's total baryonic mass (stars plus gas) to its outer flat rotation velocity.
• Empirical calibrations using resolved rotation curves yield slopes near 4 with very tight intrinsic scatter, especially in gas-rich systems.
• This relation challenges theoretical models by constraining galaxy formation, dark matter dynamics, and alternative gravity frameworks like MOND.

The Baryonic Tully-Fisher Relation (BTFR) is a fundamental empirical scaling law that relates the total baryonic mass of a rotationally supported galaxy—summing both stars and gas—to its characteristic outer rotation velocity. Originally emerging as a generalization of the classical Tully-Fisher relation (luminosity vs. velocity), the BTFR replaces luminosity with the physically motivated baryonic mass and tightly links galaxy mass assembly and dynamics across a broad mass spectrum. It serves as a stringent constraint for theoretical models of galaxy formation, dark matter, and alternative gravity, and remains pivotal for distance measurement and studies of cosmological structure.

1. Mathematical Formulation and Empirical Calibration

The BTFR is conventionally expressed as a power-law:

$M_b = A\, V^x$

where $M_b$ is the total baryonic mass (typically $M_b = M_* + 1.33\,M_{HI}$, with the factor 1.33 for helium), $V$ is a suitably defined rotation velocity, $x$ is the slope, and $A$ is the normalization. The “tightest” empirical form of the BTFR uses the mean velocity in the flat portion of a resolved rotation curve ($V_f$), where rotational motion is dominated by the total mass distribution (Lelli et al., 2019).

Direct fits to resolved galaxy samples yield best-fit relations such as:

$$\log_{10} (M_b / M_\odot) = 1.99 \pm 0.18 + (3.85 \pm 0.09)\,\log_{10} (V_f / \mathrm{km\,s}^{-1})$$

or, in “engineering units”:

$$M_b = 10^{1.99} (V_f/\mathrm{km\,s}^{-1})^{3.85} ~ M_\odot$$

Intrinsic orthogonal scatter is minimized for carefully selected samples, reaching as low as $\sigma_\perp \simeq 0.026$ dex (about 6%) when $V_f$ is adopted. Alternate velocity proxies (unresolved HI lines, velocities at inner radii) systematically yield shallower slopes (closer to 3.3) and larger scatter (Lelli et al., 2019, Bradford et al., 2016).

Gas-rich galaxies in the low-mass regime (often $M_{\rm gas}/M_* > 2.7$) provide critical anchoring of the normalization, as their baryonic mass is less sensitive to unknown stellar mass-to-light ratios or IMF choices. For such galaxies, the BTFR is measured as (McGaugh, 2011, Papastergis et al., 2016):

$$M_b = (47 \pm 6)~ V_f^4,\quad\text{with~}V_f~\text{in km s}^{-1}$$

This normalization is equivalent to a MOND acceleration $a_0 = 1.3 \times 10^{-10}\,\mathrm{m\,s}^{-2}$, and the scatter is consistent with observational uncertainties (no intrinsic scatter detected).

2. Physical Origin and Theoretical Interpretation

The BTFR has attracted attention due to both its tightness and its unexpectedly steep slope ($x$ close to 4). In the ΛCDM framework, naive virial scaling of halos gives $M_{vir} \propto V_{vir}^3$, while the observed baryonic mass fraction and the mapping from halo to galaxy introduce a steepening. State-of-the-art abundance-matching and semi-analytic models reconcile the overall slope with data by invoking mass-dependent baryon fraction $f_{\rm bar}(M_{vir})$, halo concentration scatter, angular momentum transfer, and feedback processes (Desmond, 2012, G. et al., 2016).

Recent high-resolution hydrodynamical simulations (IllustrisTNG, EAGLE, SIMBA, NIHAO) achieve broad agreement with the shape and normalization of the BTFR for massive galaxies, typically finding slopes $3.5 < x < 4$, but often predict larger intrinsic scatter ($\sim 0.15-0.20$ dex) and increased curvature at high or low mass (Ravi et al., 21 Dec 2025, Glowacki et al., 2020, G. et al., 2016).

In the alternative framework of Modified Newtonian Dynamics (MOND), the BTFR is an exact law resulting from a modified force law in the deep-MOND regime:

$V_f^4 = G\,a_0\, M_b$

This predicts $x = 4$, a fixed normalization set by the universal acceleration $a_0$, and vanishing intrinsic scatter. Empirical calibrations using gas-rich galaxies match the predicted normalization and observe minimal scatter, providing strong empirical support for this formulation at the scale of disks (McGaugh, 2011, McGaugh et al., 2015).

Alternative gravitational and cosmological models have also derived the quartic law as a natural consequence. For instance, analysis in a D~2 hierarchical Universe also produces $V^4 \propto M_b$ as a condition for Newtonian gravitational stability, with a universal acceleration scale set by the large-scale baryonic surface density (Roscoe, 2023).

3. Observational Methodology and Systematic Dependencies

Velocity Measurement and Proxy Definitions

Choice of the velocity metric is the dominant factor affecting both the measured BTFR slope and scatter. Definitions include:

• $V_f$: outer flat rotation-curve velocity (highest precision)
• $W_{20}/2$, $W_{50}/2$: half-widths of integrated HI line profiles at 20% (or 50%) of peak
• $V_{2.2}$: velocity at 2.2 times the disk scale length
• $V_{max}$: maximum observed rotation

Resolved rotation curves that extend to the flat part of the disk minimize systematics and yield steeper slopes (near 4), whereas unresolved or inner measures are biased lower (flattening the slope) and introduce larger scatter, especially in low-mass and rising-curve systems (Bradford et al., 2016, Lelli et al., 2019). Empirical conversion relations among velocity definitions are needed to homogenize diverse surveys (Lelli et al., 2019).

Mass Estimation

Stellar masses are best estimated through near-infrared photometry (e.g., Spitzer [3.6 μm]) and color–mass-to-light ratio (CMLR) calibrations, enforcing consistency across photometric bands (McGaugh et al., 2015). For gas-rich galaxies, total baryonic mass is dominated by atomic gas; in such cases, systematic uncertainties in the stellar mass-to-light ratio or IMF are subdominant (Papastergis et al., 2016, McGaugh, 2011).

The adopted correction for helium and metals in the gas mass (e.g., $M_{gas} = 1.33\,M_{HI}$) and the specific choice of IMF for $M_*$ (e.g., Chabrier, Kroupa) shift the normalization by up to 0.05–0.15 dex but have little effect on the slope if consistently applied (McGaugh et al., 2015, Ball et al., 2022).

Intrinsic Scatter and Sample Properties

The BTFR exhibits minimal intrinsic scatter—often $<0.10$ dex—when the sample is limited to galaxies with well-measured flat velocities and robust mass measurements (Lelli et al., 2015, Papastergis et al., 2016). Selection effects, sample size, demographic bias, and inclusion of non-representative velocity/mass ranges can bias both slope and scatter significantly; for $N < 50$ random uncertainties can exceed $0.1$ dex, and non-representative subsampling induces additional systematic bias (G. et al., 2016). Mass-dependent curvature and breaks may emerge at the low-mass ($M_b \lesssim 10^8\,M_\odot$) and high-mass ends, where feedback and baryon loss truncate the power-law relation (McQuinn et al., 2022, Ruan et al., 20 Mar 2025).

4. Curvature, Mass Range, and Redshift Evolution

The BTFR is extremely linear over five to six decades in mass ($10^6 \lesssim M_b \lesssim 10^{11} M_\odot$). At high baryonic mass ($M_b \gtrsim 10^{11} M_\odot$), theory and high-resolution Hi data predict a flattening or turnover due to mass-dependent decline in galaxy formation efficiency and increasing dominance of hot gas and active galactic nucleus feedback (Desmond, 2012, Glowacki et al., 2020). At low mass ($M_b \lesssim 10^8 M_\odot$), both ΛCDM simulations and carefully corrected empirical data indicate a decrease (“turndown”) in baryon retention and a break in the BTFR corresponding to decreased galaxy formation efficiency (McQuinn et al., 2022, Ruan et al., 20 Mar 2025).

Observationally, detecting this low-mass turndown requires deep Hi mapping to very low surface densities ($\Sigma_{HI} \lesssim 0.08\,M_\odot$ pc$^{-2}$), as conventional surveys are not sensitive enough for resolved rotation curves at low mass (Ruan et al., 20 Mar 2025).

With respect to redshift, simulations (e.g., SIMBA) predict the normalization of the BTFR decreases at earlier epochs (e.g., at $z = 1$), reflecting lower average baryon content at fixed velocity. The slope $x$ remains close to 4 over cosmic time, and clear detection of this evolution for disk-dominated galaxies is a key goal of upcoming large Hi surveys (e.g., LADUMA) (Glowacki et al., 2020).

5. Astrophysical Applications and Implications

The BTFR’s precision and simplicity make it the “fundamental plane” for disk galaxies, superseding multivariate “fundamental plane” relations that include scale length, surface brightness, or angular momentum. The relation is utilized as a redshift-independent distance indicator; the BTFR, when calibrated with galaxies of known distances (Cepheids, TRGB), produces local Hubble constant estimates (e.g., $H_0 = 75.1 \pm 2.8$ km s$^{-1}$ Mpc$^{-1}$) in agreement with other independent probes (Schombert et al., 2020).

In theoretical terms, the BTFR challenges galaxy formation models since semi-analytic or simulation-based ΛCDM models must tightly couple baryon retention, feedback, and halo response to reproduce both the observed slope and the exceptionally low intrinsic scatter. MOND and other alternative gravity frameworks, for which the BTFR arises naturally as a consequence of modified force laws, remain empirically competitive in the disk regime (McGaugh et al., 2015, Roscoe, 2023).

Recent parametric extensions explicitly include disk scale radius and dynamic mass discrepancy as “hidden variables,” demonstrating that a virtually scatter-free BTFR arises simply from Newtonian dynamics combined with observed galaxy scaling properties, again with the empirically inferred acceleration scale close to Milgrom’s $a_0$, but not universal (Fortune, 2021).

Theoretical advances, including models where cosmic evolution alters the BTFR normalization exponentially with cosmic time but leaves the slope fixed, reconcile the observed offset between galaxies and clusters and embed the law in quantum gravitational structure formation (Marongwe et al., 25 Nov 2025).

6. Tables: BTFR Slope and Scatter in Representative Studies

Study / Sample	Slope ($x$)	Scatter (dex)	Velocity Definition / Notes
Lelli et al. (2019) (Lelli et al., 2019)	$3.85 \pm 0.09$	0.026	$V_f$, flat part of HI rotation curve
McGaugh & Schombert (2015) (McGaugh et al., 2015)	$4.09 \pm 0.15$	0.12	$V_f$, gas-rich calibration
Papastergis et al. (2016) (Papastergis et al., 2016)	$3.75 \pm 0.11$	0.056	$W_{50}/2$, edge-on, gas-rich
Lelli, McGaugh & Schombert (2016) (Lelli et al., 2015)	$3.85 \pm 0.09$	0.10	V_f, Spitzer [3.6 μm] masses
Bradford et al. (2016) (Bradford et al., 2016)	$3.24$ (varies 2.6–3.5)	0.14–0.41	$W_{20}/2$, systematics explored
Ball et al. (2022) (Ball et al., 2022)	$3.30 \pm 0.06$	0.025	$V_{75}$, ALFALFA auto-fit
Ravi et al. (2024) (Ravi et al., 21 Dec 2025)	$3.21$–$4.01$	0.07–0.17	MaNGA/TNG100, spirals and ellipticals

7. Future Directions and Theoretical Challenges

Current and future large-scale surveys (ALFALFA, LADUMA, WALLABY, SKA pathfinders) will continue to extend the accuracy and dynamic range of the BTFR, especially at low masses and higher redshifts. Confirming or refuting the predicted turndown at low baryonic mass, mapping its evolution, and detecting potential second-parameter dependencies remain key objectives (Ruan et al., 20 Mar 2025, Glowacki et al., 2020).

In ΛCDM, matching the unity of slope and exceptionally low scatter across six decades in mass remains non-trivial and requires careful modeling of both galactic feedback and stochastic halo properties (G. et al., 2016, Lelli et al., 2015). Conversely, falsifying the BTFR’s universality or quantifying physically meaningful second-parameter residuals would challenge the MONDian view and favor complex halo–galaxy coupling.

The BTFR thus stands as a stringent, physically fundamental, and empirically robust scaling law at the center of extragalactic astronomy and galaxy formation theory.