--- title: 'Tully-Fisher Relation: Spiral Galaxy Dynamics' url: https://www.emergentmind.com/topics/tully-fisher-relation-tfr type: topic --- # Tully-Fisher Relation: Spiral Galaxy Dynamics The Tully-Fisher relation (TFR) is an empirical scaling law linking the luminosity or mass of a spiral galaxy to its rotational velocity, usually inferred from a global H I line width or from a resolved rotation curve. In its standard calibrated form it is written as \(M = a + b \log W\), while mass-based variants relate \(V_{\rm rot}\) to stellar or baryonic mass. The relation functions both as a redshift-independent distance indicator and as a dynamical constraint on the coupling between baryons, disk structure, and dark matter halos, which is why it is used in extragalactic distance work, peculiar-velocity studies, and tests of galaxy-formation models [2310.16053, 1607.03100]. ## 1. Historical and conceptual foundations The modern TFR is usually traced to Tully and Fisher’s 1977 formulation, but the underlying idea predates it. An early use was due to Öpik in 1922, who employed a velocity–magnitude relation to estimate the distance to Andromeda, and later Balkowski and collaborators identified a correlation between H I line width and luminosity in a small sample of spiral and irregular galaxies. Tully and Fisher’s contribution was to establish the relation as a practical distance tool for inclined spiral galaxies; their 1977 work also produced an early estimate of the Hubble constant near \(80\ {\rm km\ s^{-1}\ Mpc^{-1}}\) [2310.16053]. The standard physical heuristic starts from circular motion, \(V_{\rm rot}^2 \propto M/r\). If one further assumes roughly constant mass-to-light ratio and surface brightness across spirals, then \(L \propto V_{\rm rot}^4\). The same chapter stresses, however, that real galaxies depart from this idealization because dark matter haloes, variations in stellar populations, gas fractions, and non-uniform star-formation histories alter the light–mass relation [2310.16053]. Although the TFR is often described as a one-dimensional late-type-galaxy analogue of the fundamental plane, that characterization is only approximate. One synthesis explicitly states that late-type galaxies have bulges as well as disks, and that because the surface density of disks is only standard for the more massive galaxies, additional parameters are involved. In that view, bulge kinematics, disk surface density, and central surface brightness are not mere nuisance terms but physically relevant hidden variables in the observed relation [2005.12527]. ## 2. Observational construction and kinematic measures In practice, the TFR is built from photometry and a velocity proxy. The photometric term may be an optical, near-infrared, or mid-infrared absolute magnitude, while the kinematic term may be a global H I width such as \(W_{20}\) or \(W_{50}\), an optical emission-line width, a CO line width, a characteristic rotation-curve velocity such as \(V_{\max}\) or \(V_{\rm flat}\), or a dynamical circular velocity from Jeans modeling [2310.16053, 1704.08788, 1007.4072, 1806.06272]. The velocity definition is not innocuous. A direct comparison between early-type spirals and S0 galaxies showed that global H I line widths, ionized-gas position–velocity diagrams, asymmetric-drift-corrected stellar kinematics, and circular velocities from dynamical Jeans models do not yield identical numerical velocities. That study concluded that comparison of different galaxy populations is vulnerable to systematic and uncertain biases if different rotation measures are mixed, and that the same tracer and same methodology must be used for both samples [1007.4072]. Inclination is a central correction because the observed line width gives only the projected velocity, \(w = v \sin i\). A standard optical-axis-ratio correction is \[ \cos^2 i = \frac{q^2-q_0^2}{1-q_0^2}, \] with \(q\) the observed axis ratio and \(q_0\) the intrinsic thickness [1309.6051]. Inclination errors are especially consequential because they affect both the deprojection of the velocity width and the internal-extinction correction; one methodological study found that if inclination measurements have mean errors larger than about \(10^\circ\), it is better not to use any inclinations than to treat those measurements as exact [1309.6051]. Two methodological extensions address precisely this difficulty. First, an inclination-free maximum-likelihood method marginalizes over \(\rho(i)=\sin i\) and recovers the TFR zero-point, slope, and scatter with statistical errors only about \(1.5\) times larger than the ideal case of perfectly known inclinations [1309.6051]. Second, H I stacking provides a statistical TFR without galaxy-by-galaxy linewidth measurements: for realistic galaxy populations, the width of the stacked H I profile is approximately a constant factor of the intrinsic deprojected width, \[ W_{50}^{\rm stack} \approx 0.93\,W_{50}^{\rm ref}, \] which enables TFR work at lower masses and higher redshifts than direct individual detections [1510.07785]. Spatially resolved data add a further layer of robustness. A multi-wavelength H I study compared three velocity measures—\(W^{i}_{50}\), \(V_{\max}\), and \(V_{\rm flat}\)—and found that the tightest correlation occurs for the \(3.6\,\mu{\rm m}\) band paired with \(V_{\rm flat}\), rather than with a global linewidth proxy [1704.08788]. ## 3. Classical, stellar-mass, baryonic, and generalized forms Several closely related but distinct TFR formulations are in current use. | Formulation | Luminosity or mass term | Kinematic term | |---|---|---| | Classical TFR | Absolute magnitude or luminosity | H I profile width or rotation velocity | | Stellar-mass TFR | \(M_*\) | \(V_{\rm rot}\) | | Baryonic TFR | \(M_{\rm bar}=M_*+M_{\rm gas}\) | \(V_{\rm rot}\) | | Generalized \(S_{0.5}\) TFR | \(M_*\) | \(S_{0.5}=\sqrt{0.5V_{\rm rot}^2+\sigma^2}\) | The classical relation is the one originally calibrated as a luminosity–velocity law for disk galaxies [2310.16053]. The stellar-mass TFR replaces luminosity with \(M_*\), thereby reducing sensitivity to bandpass and stellar-population effects, while the baryonic TFR replaces luminosity with total baryonic mass, \[ M_{\rm bar}=M_*+M_{\rm gas}, \] which one review describes as likely being more fundamental because baryonic mass traces the galaxy’s true mass budget better than luminosity alone [1102.2311, 2310.16053]. A generalized kinematic formulation uses \[ S_{0.5}=\sqrt{0.5V_{\rm rot}^2+\sigma^2}, \] or more generally \(S_K=\sqrt{K V_{\rm rot}^2+\sigma^2}\), to combine ordered and disordered motions. This is motivated by the fact that many high-redshift or low-mass galaxies are not cleanly rotation-dominated disks; using \(S_{0.5}\) allows one to place both rotation-supported and dispersion-supported systems on a common stellar-mass–kinematics sequence [1506.04144, 1706.02745]. The kinematic tracer itself can also be changed. A CO-based study of star-forming galaxies at \(z=0.05\)–0.3 constructed reverse \(K_s\)-band, stellar-mass, and baryonic-mass TFRs using the inclination-corrected width \(W_{50}/\sin i\), and concluded that CO is a suitable and attractive alternative to H I, provided the molecular gas reaches the flat part of the rotation curve, as indicated by double-horned or boxy line profiles [1806.06272]. ## 4. Scatter, wavelength dependence, and hidden variables A persistent empirical result is that the TFR steepens and tightens toward longer wavelengths. In the TNG50 cosmological simulation at \(z=0\), the slope changes from \(-7.46 \pm 0.14\ {\rm mag\,dex^{-1}}\) in NUV to \(-9.66 \pm 0.09\ {\rm mag\,dex^{-1}}\) in IRAC [4.5], while the perpendicular dispersion declines from \(0.116 \pm 0.006\) dex in FUV to a nearly wavelength-independent \(\sim 0.045\)–\(0.046\) dex in the red and infrared [2503.00194]. An observational analysis based on resolved H I kinematics found the tightest TFR for the \(3.6\,\mu{\rm m}\) band combined with \(V_{\rm flat}\), with \(\sigma_{\perp,{\rm obs}} = 0.036 \pm 0.010\) dex [1704.08788]. Mid-infrared work also indicates that the TFR is not always strictly linear. A WISE calibration of the W1 and W2 relations found evidence of curvature and provided quadratic fits, \[ \mathcal{M}^{b,i,k,a}_{W1} = -20.48 - 8.36 (\log W^i_{mx} - 2.5) + 3.60 (\log W^i_{mx} - 2.5)^2, \] \[ \mathcal{M}^{b,i,k,a}_{W2} = -19.91 - 8.40 (\log W^i_{mx} - 2.5) + 4.32 (\log W^i_{mx} - 2.5)^2. \] The same study showed that an I-band–WISE color correction reduces the scatter from \(0.54\) to \(0.46\) mag in W1 and from \(0.56\) to \(0.46\) mag in W2 [1407.7528]. Residuals from the TFR are physically structured rather than random. In TNG50, the strongest residual correlations in the UV and optical are with \(u-r\) color and sSFR; after a color correction, the modified TFR has an almost constant intrinsic tightness of about \(0.046\) dex across the entire UV-to-MIR range [2503.00194]. A more general structural interpretation identifies bulge fraction, bulge velocity dispersion, disk surface density or central surface brightness, gas depletion or stripping in clusters, and baryonic mass distribution as sources of TFR scatter and deviation [2005.12527]. Morphology and kinematic regularity are equally important. A morphologically blind sample at \(0.19.5\) [1711.10726]. Methodological biases can mimic astrophysical trends. For a sample of 2411 Sb–Sc galaxies, the fitted coefficients \((\alpha,\beta)\) vary systematically with the width and placement of the magnitude interval used for the fit, and a runs test rejects the null hypothesis of no underlying trend at \(98\%\)–\(99\%\) confidence in most cases [1609.08534]. By contrast, an SDSS analysis of \(25{,}698\) late spirals found no significant dependence of the fibre-based TFR on projected neighbour density once aperture effects and sample bias were modeled [1206.1662]. ## 5. Redshift evolution and theoretical interpretation Observed redshift evolution depends strongly on which TFR variant is used. In the IMAGES survey, the stellar-mass TFR for rotating disks at \(z\sim 0.6\) is offset downward by \(0.34^{+0.21}_{-0.06}\) dex relative to the local relation, meaning that at fixed \(V_{\rm rot}\) those galaxies had roughly a factor of \(\sim 2\) less stellar mass than local disks. The same work derived, for the first time, a high-redshift baryonic TFR by inverting the Schmidt-Kennicutt relation to estimate gas masses, found a median gas fraction of \(\sim 31 \pm 1\%\), and reported that the baryonic zero point shows no significant shift between \(z\sim0.6\) and \(z=0\) [1102.2311]. The stated interpretation is that most of the gas converted into stars over the past \(6\) Gyr was already gravitationally bound to galaxies at \(z\sim0.6\), although cold gas inflow at the level of roughly \(\sim 30\%\) of the present local baryonic mass remains allowed within the systematic uncertainties [1102.2311]. At higher redshift, the luminosity TFR evolves more strongly in the blue than in the infrared. A study at \(1.13\) and the existence of a correlation becomes unclear. The same analysis argued for a break or turnover near \(M_*\sim10^{10}\ M_\odot\), with a steeper low-mass slope, and obtained \(\Delta({\rm BIC})=7\) in favor of an asymptotic broken relation over a single straight line [1706.02745]. At lower lookback time, the CO TFR out to \(z=0.3\) showed no statistically significant evolution in slope or zero point in \(K_s\)-band luminosity, stellar mass, or baryonic mass [1806.06272]. Theoretical interpretation has shifted from a simple halo virial picture to a more structured galaxy–halo coupling. EAGLE-based work makes the dependence explicit through \[ M_{\rm str}=\frac{1}{10GH(z)}\frac{f_m}{f_v^3}V_{\rm rot}^3, \] where \(f_m=M_{\rm str}/M_{200}\) and \(f_v=V_{\rm rot}/V_{200}\). In that framework, the slope, zero point, and scatter of the TFR are controlled by \(f_m/f_v^3\), and galaxy size becomes critical because smaller disks increase the baryonic contribution to the measured rotation speed. EAGLE approximately matches both the abundance-matching relation and the observed size–mass relation, and therefore reproduces the observed TFR and its weak evolution out to \(z=1\) [1607.03100]. A more general abundance-matching analysis reached related but sharper constraints. Agreement with the observed stellar-mass TFR places an upper limit on abundance-matching scatter of \(\lesssim 0.35\) dex at \(2\sigma\), requires weak or reversed halo contraction, and, once combined with the mass–size relation, constrains the ratio of disc to halo specific angular momentum to approximately \(0.6\)–\(1.2\). That same framework, however, predicts too large an intrinsic scatter for the mass–size relation and too strong an anticorrelation between TFR and mass–size residuals, implying that a correlation between stellar surface density and enclosed dark matter mass is missing from the simplest one-parameter models [1506.00169]. ## 6. Distance scale, peculiar velocities, and survey applications The TFR remains one of the principal redshift-independent distance indicators for spiral galaxies. Its utility is strongest in the nearby universe, where the observed radial velocity contains both Hubble expansion and a peculiar-velocity component; one review emphasizes that spiral galaxies are abundant enough below roughly \(z \lesssim 0.07\) for TFR distances to be central to cosmography, bulk-flow studies, and three-dimensional density and velocity-field reconstruction [2310.16053]. Infrared calibrations are especially prominent in this role. A WISE W1/W2 calibration based on 310 galaxies in 13 clusters and 37 Cepheid/TRGB zero-point calibrators obtained \[ \mathcal{M}^{b,i,k,a}_{W1} = -20.35 - 9.56 (\log W^i_{mx} - 2.5) \] with \(0.54\) mag rms and \[ \mathcal{M}^{b,i,k,a}_{W2} = -19.76 - 9.74 (\log W^i_{mx} - 2.5) \] with \(0.56\) mag rms. Using curved WISE relations, color-corrected WISE relations, and an updated I-band calibration to recalibrate the UNION2 SN Ia distance scale, that study derived \[ H_0 = 74.4 \pm 1.4\ {\rm (stat)} \pm 2.4\ {\rm (sys)}\ {\rm km\ s^{-1}\ Mpc^{-1}} \] [1407.7528]. The TFR is now being implemented in large spectroscopic and H I survey pipelines. The DESI DR1 Peculiar Velocity Survey calibrated an \(r\)-band TFR for 10,262 spiral galaxies using velocities measured at \(0.4R_{26}\) and found a slope of \(-7.22 \pm 0.01\) AB mag with intrinsic scatter \(0.466 \pm 0.001\) AB mag. It also produced a peculiar-velocity catalog for all 10,262 galaxies and a clustering catalog for 6807 of them [2512.03227]. The WALLABY pilot survey, using ASKAP H I spectra with \(30\) arcsec angular resolution and \(\sim4\ {\rm km\ s^{-1}}\) velocity resolution, demonstrated that pilot-quality TFR distances can support state-of-the-art local velocity-field work and reported that a pilot-sized sample with WALLABY-like scatter is sufficient to recover a \(500\ {\rm km\ s^{-1}}\) bulk flow at \(2.5\sigma\) significance [2406.10877]. Methodological diversification has broadened the relation’s survey applicability. CO-based TFRs extend line-width work beyond the local Universe [1806.06272], inclination-free likelihood methods are directly relevant to future H I surveys by the SKA and its pathfinders [1309.6051], and stacked H I approaches are designed to push TFR measurements to lower H I masses and higher redshifts than galaxy-by-galaxy detection allows [1510.07785]. Taken together, these developments indicate that the TFR is no longer a single optical linewidth relation, but a family of calibrated dynamical scaling laws adapted to different tracers, wavelength regimes, and survey architectures. Source: https://www.emergentmind.com/topics/tully-fisher-relation-tfr