Tully-Fisher Relation

Tully-Fisher relation (TFR) is an empirical astrophysical scaling law for rotationally supported spiral galaxies (and irregular disk galaxies), demonstrating that a galaxy's intrinsic luminosity L (or total baryonic mass M_b) is proportional to roughly the fourth power of its maximum or asymptotic circular rotational velocity (L ∝ V_rot^α, with exponent α ≈ 3.0 to 4.0 depending on photometric bandpass). Discovered in 1977 by the astronomers R. Brent Tully and J. Richard Fisher using neutral hydrogen (H I) 21-centimeter line-width observations at the National Radio Astronomy Observatory (NRAO)—as the disk-galaxy counterpart to the Faber-Jackson relation for elliptical spheroids—the Tully–Fisher relation serves two key roles in modern astrophysics: (1) it acts as one of the primary secondary standardizable candles on the extragalactic distance ladder (measuring distances to tens of thousands of spiral galaxies out to ~100–200 megaparsecs to map local peculiar velocity flows, the Laniakea Supercluster, and the Hubble constant H₀); and (2) in its generalized mass form—the Baryonic Tully–Fisher Relation (BTFR, M_b ∝ V_flat⁴)—it imposes a tight empirical constraint on the coupling between luminous baryons and dark matter halos across five orders of magnitude in galaxy mass.

How did Tully and Fisher measure the intrinsic luminosity of a distant spiral galaxy solely from its radio spectrum without knowing its distance d?

1. Neutral Hydrogen (H I) 21-cm Global Profile (W₂₀ and W₅₀): Cold atomic hydrogen gas (H I) in a spiral galaxy disk extends far beyond the optical stellar disk into the flat outer region of the galaxy's rotation curve, emitting the hyperfine spin-flip transition at rest frequency ν₀ = 1420.40575 MHz (λ₀ = 21.106 cm). When a single-dish radio telescope (such as Green Bank, Arecibo, or Parkes) observes an inclined rotating spiral galaxy, gas on the side of the disk rotating toward the observer is Doppler-blueshifted (+V_rot sin i), while gas on the receding side is Doppler-redshifted (−V_rot sin i). This produces a classic double-horned 21-cm velocity profile whose full width at 20% or 50% of peak flux (W₂₀ or W₅₀, corrected for instrumental broadening and internal turbulent dispersion σ_gas ≈ 8–10 km/s) equals twice the projected rotation speed:
   • W_corr = 2 V_max sin i.
2. Optical/Near-Infrared Inclination Correction (i): Because a face-on galaxy (i = 0°) shows zero line-of-sight rotational Doppler shift (sin i = 0), Tully–Fisher measurements are restricted to spiral galaxies inclined at 45° ≤ i ≤ 80°. The inclination angle i is determined from the observed minor-to-major axial ratio (b / a) of the galaxy's outer optical or near-infrared isophotes using the Hubble oblateness formula:
   • cos² i = [ (b / a)² − q₀² ] / [ 1 − q₀² ] (where q₀ ≈ 0.13 to 0.20 is the intrinsic edge-on thickness ratio of a spiral disk).
3. Distance Modulus Extraction (μ = m − M): Once the true de-projected rotation velocity V_max = W_corr / (2 sin i) is computed, plugging V_max into the calibrated Tully–Fisher linear relation:
   • M_λ = a_λ + b_λ [ log₁₀(2 V_max) − 2.5 ]

yields the galaxy's absolute magnitude M_λ. Comparing M_λ with the extinction-corrected apparent magnitude m_λ directly yields the luminosity distance d = 10^{(m_λ − M_λ − 25)/5} megaparsecs with a distance precision of ~15–20% per galaxy (0.30–0.40 mag scatter).

In Tully and Fisher's original 1977 blue-band (B-band, λ ≈ 440 nm) photographic measurements, the slope of the relation was relatively shallow (L_B ∝ V_max^2.5–3.0) and exhibited substantial scatter (~0.5–0.7 magnitudes) for two physical reasons: (a) blue light is heavily attenuated by internal interstellar dust extinction in inclined spiral disks, and (b) blue luminosity is dominated by short-lived, massive O and B main-sequence stars formed in stochastic recent starbursts rather than the long-lived low-mass stellar population that dominates the galaxy's stellar mass M_*.

When astronomers (Marc Aaronson, Jeremy Mould, and John Huchra, 1979) shifted observations to the near-infrared (H, K_s, and Spitzer 3.6 μm bands), where dust extinction is negligible (< 0.05 mag) and light traces old K and M giant/dwarf stars proportional to total stellar mass M_*, the slope steepened to L_NIR ∝ V_flat^3.8–4.0 and the intrinsic scatter dropped below 0.25–0.35 magnitudes.

The Baryonic Tully–Fisher Relation (BTFR) in Gas-Rich Dwarfs

A notable puzzle emerged in the 1990s: when astronomers plotted the Stellar-Mass Tully–Fisher Relation (M_* vs. V_flat) down into the regime of low-mass, gas-rich dwarf irregular galaxies (V_flat ≈ 20 to 80 km/s), the power law broke down! Dwarf galaxies appeared under-luminous for their rotation speeds.

In 2000, Stacy S. McGaugh, James M. Schombert, Greg D. Bothun, and W. J. G. de Blok resolved this anomaly by recognizing that in low-mass dwarf disks, the majority of the luminous baryonic mass (50% to > 90%) has never been converted into stars—it remains in the form of neutral atomic hydrogen (H I) and primordial helium gas (M_gas ≈ 1.33 M_{H I}). When the ordinate is replaced by the total baryonic mass (M_b = M_* + 1.33 M_{H I}), the downward curvature vanishes: dwarf irregulars (V_flat = 20 km/s) and giant spirals like the Andromeda Galaxy (V_flat = 250 km/s) fall onto a single, unbroken power law (M_b = A · V_flat⁴, where normalization A ≈ 47–50 M_☉ km⁻⁴ s⁴) with minimal residual dependence on galaxy radius R_d or surface brightness Σ₀.

Why does a galaxy's baryonic mass scale with the fourth power of its rotation velocity? Starting from Newtonian gravity for a disk of mass M_* and characteristic radius R rotating at circular speed V² = G M_tot(R) / R:

1. Squaring the circular velocity gives V⁴ = G² M_tot² / R².
2. Substituting the mean surface brightness I₀ = L / (π R²) and total mass-to-light ratio (M_tot / L) eliminates R², yielding:
   • L = [ 1 / (π G²) ] · [ I₀ ( M_tot / L )² ]⁻¹ · V⁴.

This simple derivation exposes the deep physical puzzle known as the Fine-Tuning Conspiracy of Low-Surface-Brightness (LSB) Galaxies: according to the equation above, two galaxies with the same V_flat but a 100-fold difference in surface brightness I₀ should be offset from each other on the Tully–Fisher diagram by 5 magnitudes unless the product [I₀ (M_tot / L)²] remains strictly constant (meaning lower-surface-brightness galaxies must be systematically more dark-matter dominated by the exact factor required to cancel their larger radius R).

• In the Standard ΛCDM Cosmological Paradigm: A spherical cold dark matter halo virializing at cosmic overdensity Δ_vir ≈ 200 satisfies the virial relation M₂₀₀ ∝ H(z)⁻¹ V₂₀₀³ (a slope of 3, not 4!). To transform the primordial halo relation M₂₀₀ ∝ V₂₀₀³ into the observed baryonic relation M_b ∝ V_flat⁴, hydrodynamic galaxy-formation simulations (such as EAGLE, IllustrisTNG, and FIRE) invoke mass-dependent stellar feedback: in shallower dwarf-galaxy potential wells (V₂₀₀ < 100 km/s), core-collapse supernova winds expel a progressively larger fraction of the primordial baryons from the disk, steepening the retained baryon fraction f_b(V₂₀₀) ∝ V₂₀₀^1.0 so that M_b = f_b M₂₀₀ ∝ V_flat⁴.
• In Modified Newtonian Dynamics (MOND, Mordehai Milgrom 1983): By contrast, if effective gravitational acceleration below a universal acceleration scale a₀ ≈ 1.2 × 10⁻¹⁰ m s⁻² transitions from the Newtonian regime (g_N = G M_b / r²) to the deep-MOND regime (g = √(g_N a₀) = √(G M_b a₀) / r), setting the centripetal acceleration V_flat² / r equal to g cancels the radius r identically and yields an exact, zero-scatter fourth-power law (V_flat⁴ = G a₀ M_b) independent of surface brightness I₀. Reproducing the ultra-small intrinsic scatter (< 0.11 dex) and surface-brightness independence of the SPARC (Spitzer Photometry and Accurate Rotation Curves) BTFR dataset remains a primary benchmark for both self-interacting dark matter models and baryonic feedback simulations.

1. Tully, R. B., & Fisher, J. R. (1977). A new method of determining distances to galaxies. Astronomy and Astrophysics, 54(3), 661–673. https://ui.adsabs.harvard.edu/abs/1977A%26A....54..661T
2. Aaronson, M., Huchra, J., & Mould, J. (1979). The infrared luminosity/H I velocity-width relation and its application to the distance scale. The Astrophysical Journal, 229, 1–13. https://doi.org/10.1086/156923
3. McGaugh, S. S., Schombert, J. M., Bothun, G. D., & de Blok, W. J. G. (2000). The Baryonic Tully-Fisher Relation. The Astrophysical Journal, 533(2), L99–L102. https://doi.org/10.1086/312628
4. Lelli, F., McGaugh, S. S., & Schombert, J. M. (2016). The small scatter of the baryonic Tully-Fisher relation. The Astrophysical Journal Letters, 816(1), L14. https://doi.org/10.3847/2041-8205/816/1/L14