Tully-Fisher Relation

A scaling law hidden in 21-centimetre line widths

In 1977 R. Brent Tully and J. Richard Fisher, working with single-dish 21-cm radio observations of nearby spirals, plotted absolute magnitude against the width of the neutral-hydrogen line. They found a stunningly tight linear relation. The 21-cm line of HI is broadened by rotation: a face-on disk shows a narrow feature, an edge-on disk shows a double-horned profile whose width is twice the maximum rotational velocity. Tully and Fisher's key step was to use the line width as a velocity proxy that does not require resolving the disk — a property crucial for distant galaxies — and then to read off luminosity. The relation worked.

The functional form they reported was a power law,

L ∝ V_circ^α      with   α ≈ 4   (typical for IR/I-band)

or equivalently a linear relation in log L versus log V with slope ≈ 4. In the original B-band paper the slope was closer to 2.5–3, reflecting the strong dust extinction in blue; in later infrared and HI calibrations the slope converged toward 4. The scatter is remarkable: about 0.3 magnitudes (10 percent in L) at fixed V in the best modern infrared samples, an order of magnitude tighter than naive expectations from the spread in galaxy types and feedback histories.

Why the fourth power

The exponent ≈ 4 is not arbitrary. A simple dimensional sketch goes as follows. For a virially supported disk,

GM / R ~ V_circ²     →     M ~ V² R / G

If disk galaxies share a roughly constant surface brightness Σ — the Freeman law (1970) — then luminosity L = Σ π R². And if the mass-to-light ratio M/L is also roughly constant,

M = (M/L) L     →     V² R / G ∝ Σ R²     →     R ∝ V² / (G Σ)

Substituting back gives

L = Σ π R² ∝ Σ × V⁴ / (G² Σ²) ∝ V⁴ / Σ

This argument requires two regularities — constant surface brightness and constant M/L — neither of which is automatic. Both happen to hold in spirals to fair accuracy, which is why TF works at all. The deeper question, then, is not "why slope four?" but "why are spirals so uniform in surface brightness and stellar mass-to-light ratio?" That remains an unresolved tension in galaxy-formation theory.

The Baryonic Tully-Fisher relation

Stellar-luminosity TF starts to break down at low V_circ. Dwarfs and irregulars scatter widely, and many fall systematically below the line traced by giant spirals. Stacy McGaugh and collaborators showed in 2000 that the problem is not the rotational velocity — it is the choice of luminosity, which ignores gas. In dwarfs, neutral hydrogen often outweighs stars; their baryonic content is mostly gas, not light. Replacing L with total baryonic mass

M_bar = M_star + M_gas      (gas mass = 1.4 M_HI to include helium)

restores a single power law from giant spirals down to gas-rich dwarfs:

M_bar = A V_circ^4        A ≈ 50 M☉ (km/s)⁻⁴

The BTF spans more than five decades in baryonic mass with scatter consistent with the observational error budget — arguably the tightest empirical scaling relation in extragalactic astronomy. The slope is statistically indistinguishable from exactly four in IR samples, a fact that distinguishes the BTF sharply from the stellar TF and that fuels much of the MOND-vs-ΛCDM debate.

Tully-Fisher as a distance indicator

The distance use of TF rests on a simple identity. If L is the luminosity inferred from V, and F is the observed flux, then

F = L / (4π d²)     →     d = √(L / 4π F)

So measuring V (from HI line width or Hα rotation curve) and apparent magnitude (from imaging) gives a distance — provided the L–V relation has been calibrated absolutely. The calibration is set by primary distance indicators applied to nearby spirals: Cepheid variables, themselves anchored by Gaia parallaxes in the Milky Way and detached eclipsing binaries in the LMC, give precise distances to ~25 galaxies in which TF parameters can also be measured. Those galaxies fix the zero point. Once calibrated, TF reaches out to ~150 Mpc, a regime where peculiar velocities are small relative to the Hubble flow and the redshift-distance comparison directly measures H₀.

The SH0ES collaboration uses TF as one of several secondary rungs; the Cosmicflows project (Tully and collaborators) has assembled TF distances for tens of thousands of galaxies to map the local Universe's velocity field, recovering the dipole that points toward the Great Attractor and the Shapley supercluster.

Bandpass matters: IR over B

The slope and scatter of the TF relation depend strongly on the photometric band. In B (blue), dust extinction is severe and varies with inclination, type and metallicity, smearing the relation and reducing the apparent slope to ~2.5. In the near-infrared (I, J, H, K_s, Spitzer 3.6 μm), extinction is an order of magnitude smaller and old stars — which dominate the stellar mass — emit most of their light. The relation tightens and steepens; modern 3.6 μm Spitzer calibrations from the SPARC sample give slope very close to 4 and scatter ~0.3 mag.

Band	Typical slope α (L ∝ V^α)	Scatter (mag)	Strengths	Caveats
B (440 nm)	~2.5–3.0	0.5–0.6	Common archival data	Heavy, inclination-dependent extinction
R (640 nm)	~3.0–3.4	0.4–0.5	Compromise; widely available	Moderate extinction
I (806 nm)	~3.5–3.8	~0.4	Hubble WFPC2/ACS era TF	Some dust residual
K_s (2.16 μm)	~4.0	~0.35	2MASS coverage; old-star light	Surface-brightness limits
Spitzer 3.6 μm	~3.9–4.0	0.3	Cleanest stellar-mass tracer	Limited sample (S⁴G)
BTF (baryonic mass)	4.0 ± 0.1	~0.13 dex	Extends to gas-rich dwarfs	Requires HI mass measurement

Worked example: a TF distance

Suppose a face-on spiral has 21-cm HI line width W_50 = 240 km/s (after inclination correction), and an observed Spitzer 3.6 μm apparent magnitude m_3.6 = 11.2. Adopt a 3.6 μm Tully-Fisher calibration:

M_3.6 = −9.5 (log V − 2.5) − 21.5     (V in km/s; M in absolute mag)

With V = W_50 / 2 = 120 km/s:

log V = log 120 = 2.079
M_3.6 = −9.5 × (2.079 − 2.5) − 21.5
      = −9.5 × (−0.421) − 21.5
      = +4.00 − 21.5
      = −17.5

The distance modulus μ = m − M = 11.2 − (−17.5) = 28.7, so

d = 10^((μ + 5) / 5) pc = 10^(33.7/5) pc = 10^6.74 pc ≈ 5.5 Mpc

A galaxy at 5.5 Mpc. The same procedure applied to a sample of cluster galaxies at z ~ 0.05 (~200 Mpc) gives a flow-corrected distance against which the redshift can be compared, yielding H₀. Real applications carry covariances between inclination correction, line-width corrections, and Malmquist bias; the SH0ES and Cosmicflows pipelines deal with these systematically.

Theoretical interpretations

Stellar feedback in ΛCDM

In the standard cosmological framework, every spiral sits at the centre of a much more massive dark-matter halo. V_circ at large radii is set by the halo, not the disk; M_bar is set by the baryons that managed to cool, settle and form stars without being blown away by feedback. The fact that M_bar tracks V⁴ with tiny scatter therefore requires that the baryon fraction of haloes be a sharp function of halo mass (or V). Modern semi-analytic and hydrodynamic models — EAGLE, IllustrisTNG, FIRE — reproduce the BTF by tuning supernova-driven winds and AGN feedback so that low-mass haloes retain a small, regulated fraction of baryons and massive haloes lose less. The tight scatter remains a benchmark: any model whose feedback prescription introduces large halo-by-halo variance in M_bar at fixed V fails to match observation.

Modified Newtonian Dynamics

Milgrom's MOND (1983) postulates that for accelerations below a₀ ≈ 1.2 × 10⁻¹⁰ m/s², Newton's law modifies so that effective acceleration scales as √(g_N a₀) rather than g_N. In this deep-MOND regime, the asymptotic circular velocity of a galaxy at large radii satisfies

V⁴ = G M_bar a₀

This is exactly the Baryonic Tully-Fisher relation with slope 4 and a zero-point set by the universal acceleration a₀. From a MOND standpoint, BTF is a theorem, not an empirical scaling. The tightness of the observed BTF — particularly the lack of scatter, which is hard to reproduce in ΛCDM without fine-tuned feedback — is a long-standing argument for MOND in some communities. Distinguishing the two paradigms requires going beyond the BTF to the shapes of rotation curves (the radial acceleration relation, McGaugh, Lelli & Schombert 2016) and to systems where MOND and ΛCDM make different predictions (e.g. tidal dwarf galaxies, ultra-diffuse galaxies, bullet-cluster lensing).

Variants and extensions

• Stellar Tully-Fisher. The original L–V_circ relation in optical/IR bands. Slope depends on band; scatter floor ~0.3 mag at K/3.6 μm. Workhorse for distance measurements.
• Baryonic Tully-Fisher (BTF). Replace L with M_bar = M_star + M_gas. McGaugh 2000. Restores slope exactly 4; extends to dwarfs.
• Stellar-mass TF. Replace L with M_star (inferred from IR photometry and a stellar M/L). Cleaner than L but breaks down for gas-rich galaxies; included as a stepping stone to BTF.
• Faber-Jackson relation. The TF analogue for elliptical galaxies: L ∝ σ^4 where σ is the central stellar velocity dispersion. Same slope, different kinematics; both subsumed in the Fundamental Plane (Djorgovski-Davis 1987).
• Stellar-mass / halo-mass TF. Use V_circ as a halo-mass proxy and study M_star vs M_halo. Reveals the well-known double-power-law efficiency curve that peaks near Milky-Way-mass haloes (~10¹² M☉).
• High-redshift TF. KMOS and SINFONI integral-field surveys extend the relation to z ≈ 1–2, where star-forming disks are turbulent and gas-rich. The slope holds; the zero-point evolves modestly, consistent with disk growth in ΛCDM.
• Radial Acceleration Relation (RAR). A point-by-point generalisation: at each radius in each spiral, the observed acceleration g_obs is a fixed function of the baryonic g_bar, with crossover near a₀. BTF is the integrated version. McGaugh, Lelli & Schombert (2016).

Systematics that limit precision

• Inclination correction. Both line widths and apparent magnitudes need correction for the disk's tilt to the line of sight. The line-width correction divides W by sin i; the photometric correction is a band-dependent extinction term. Errors in i of ±2° translate into several percent in V and substantial mag-level corrections in B.
• Internal extinction. Dust within the target galaxy reddens and dims the apparent magnitude, particularly at high inclination. Tully and others have developed inclination-dependent extinction corrections (the TF–extinction-correction loop is itself a calibration sub-problem).
• Malmquist bias. Magnitude-limited samples preferentially detect bright galaxies at large distance; the inferred mean L at fixed V is biased high unless the sample selection is folded into the likelihood.
• Line-width measurement. W_20 (20 percent of peak), W_50 (half-peak), W_R (Tully's rotation-curve form): different definitions give different zero-points; consistency across literature requires care.
• Cluster vs field calibration. Cluster TF samples have small peculiar-velocity uncertainties (use cluster mean) but environmental effects on gas and morphology; field samples have the opposite trade-off. Modern templates merge both.
• Profile shape and asymmetry. Lopsided disks, warps, and tidal interactions broaden line widths beyond pure rotation. Detailed rotation-curve TF (Hα or HI 2-D) is preferred when available.

Where Tully-Fisher matters today

• The Hubble constant. SH0ES (Riess et al.) and Cosmicflows (Tully et al.) use TF as a secondary indicator for galaxies between 30 Mpc and 150 Mpc; the resulting H₀ feeds into the current Hubble tension between local (≈73 km/s/Mpc) and CMB-anchored (≈67.4 km/s/Mpc) values.
• Peculiar-velocity surveys. Cosmicflows-4 (2022) compiled TF distances and other indicators for 56,000 galaxies, reconstructing the gravitational potential of the local Universe out to ~100 Mpc/h.
• Bulk-flow tests. Comparing TF-inferred peculiar velocities to predictions from the redshift density field tests ΛCDM at z ≈ 0; current results are consistent with ΛCDM but with modest tensions at the largest scales.
• Galaxy-formation calibration. Tightness and slope of BTF benchmark hydrodynamic simulations (EAGLE, IllustrisTNG, FIRE). A simulation that fails to reproduce the BTF slope and scatter is widely considered to have a feedback problem.
• MOND tests. The radial acceleration relation, derived from spatially resolved rotation curves, is a stronger MOND test than BTF alone. Both come from the same SPARC dataset of ~175 disks.
• High-redshift evolution. KMOS/SINFONI/JWST surveys of z ≈ 1–2 disks track how BTF parameters evolve, constraining the growth of disks across cosmic time.

Common pitfalls

• Mixing bandpasses without re-calibration. The TF zero point and slope depend on band. Using a B-band V–L relation with K-band apparent magnitudes will produce a systematic distance error of tens of percent.
• Forgetting the inclination correction. An uncorrected line width is 2 V sin i, not 2 V. Distances scale as V², so a 10 percent error in V is a 20 percent distance error and a 40 percent error in inferred Hubble flow.
• Conflating stellar TF with baryonic TF. Dwarf galaxies appear to fall below the stellar TF line; this is not a violation of TF, it is a reminder that gas contributes most of the baryonic mass in those systems.
• Treating slope = 4 as a derivation. The Newtonian derivation requires assumptions about surface brightness and M/L that are observationally true but not theoretically guaranteed. In MOND, slope = 4 is a theorem. The two pictures predict different scatter and different deviations at extreme V.
• Reading too much into the small scatter. The 0.13 dex scatter of BTF is incredibly small but not zero, and a fraction of it is real (galaxy-to-galaxy variation in feedback efficiency). Comparison to model scatter must be apples-to-apples, including measurement uncertainty.
• Ignoring Malmquist bias at the survey edge. At the magnitude limit, TF distance estimates are systematically biased; bias-correction templates (Schechter, inverse TF) must be applied to recover unbiased H₀.