Tully Fisher RelationEdit
The Tully-Fisher relation is an empirical link between the intrinsic brightness of a spiral galaxy and the speed at which its stars and gas rotate. First identified in the late 1970s by R. Brent Tully and J. Richard Fisher, the relation provides a practical bridge between what a galaxy looks like in the sky (its luminosity) and how it moves (its rotation curve). As a result, it has become a central tool in the extragalactic distance ladder, helping astronomers estimate distances to nearby galaxies when direct geometric measurements are not possible.
At its core, the Tully-Fisher relation expresses a tight correlation: more luminous disk galaxies tend to rotate faster. In practice, observers often use the relation in magnitude form (a linear relation between absolute magnitude and the logarithm of the rotation velocity) or in terms of log-luminosity versus log-velocity. Because rotation velocities can be inferred from the broadening of spectral lines, particularly the 21-cm line of neutral hydrogen, the relation provides a way to convert a shift in a galaxy’s spectrum into a measure of its distance. For background context, this sits alongside other distance indicators in the Cosmic distance ladder and complements the use of standard candles like Cepheid variables and Type Ia supernovae.
The physical interpretation of the Tully-Fisher relation touches on several pillars of modern galaxy formation theory. The rotation velocity of a disk is governed by the total mass contained within the galaxy, including both luminous (stars and gas) and dark matter components. The observed tightness of the relation implies a fairly regular coupling between a galaxy’s baryonic content and its dark matter halo, reflecting how baryons settle into disks within halos shaped by gravity and feedback processes. In recent years, a version of the relation that uses total baryonic mass rather than luminosity—the Baryonic Tully-Fisher relation—has gained prominence as it more directly traces the mass that participates in rotation. For the standard luminosity-based form, researchers must account for how light traces mass in different bands and how dust attenuation and stellar populations affect that tracing.
Observational implementation of the Tully-Fisher relation rests on several practical choices. The rotation velocity can be estimated from the velocity width of the HI 21-cm line or from resolved rotation curves obtained in optical emission lines such as H-alpha. Each method carries its own uncertainties, including the inclination of the disk relative to the line of sight, which must be corrected to recover the true rotational speed. Observers also select a photometric band in which to measure luminosity; near-infrared bands (for example the K-band and other infrared astronomy bands) are favored because they are less affected by dust and better trace the grown-up stellar mass, reducing scatter in the relation. The relation’s zero point and slope can vary with bandpass and with the sample selection, so careful calibration against galaxies with independently measured distances is essential. For discussions of biases that can affect distance estimates, see Malmquist bias.
The TF relation is widely used to estimate distances to galaxies and clusters, which in turn informs measurements of the Hubble constant and maps of peculiar velocities. When calibrated with nearby standard candles or geometric distance anchors, the relation contributes to a consistent picture of the local and intermediate-scale expansion of the universe. Beyond distance measurements, the relation offers a diagnostic of galaxy formation and evolution: its slope, scatter, and any systematic deviations across galaxy types illuminate how efficiently galaxies convert baryons into stars and how their dark matter halos influence or constrain disk growth. The relation is therefore discussed in tandem with broader topics such as dark matter halos, disk stability, and the overall framework of Lambda-CDM model cosmology.
There is ongoing debate about the universality and exact form of the Tully-Fisher relation across environments and galaxy types. While the relation is tight for many late-type, rotationally supported disks, dwarfs and low-surface-brightness galaxies can exhibit larger scatter or systematic departures from the canonical slope. The baryonic version of the relation helps address some of these concerns by focusing on total baryonic mass rather than light alone, but it too faces challenges in accurately measuring gas content and stellar mass in diverse systems. Debates also center on how best to treat aging stellar populations, metallicity effects, and the influence of star formation history on mass-to-light ratios in different bands. For more on related stellar and galactic properties, see stellar population theory and galaxy morphology analyses.
In practice, researchers use the TF relation as one piece of a larger toolkit for understanding the cosmos. It interacts with other distance indicators, tests of galaxy formation physics, and constraints on the distribution of matter in the local universe. Studies continually refine calibration methods, assess selection effects, and explore the relation’s behavior in various galaxy populations to improve its accuracy and reliability as a distance tool. The Tully-Fisher relation thus remains a foundational concept in observational cosmology and galaxy evolution, linking how galaxies shine to how they spin and how the cosmos is expanding.
History and discovery
The relationship was first established in the late 1970s by R. Brent Tully and J. Richard Fisher through observations of spiral galaxies that showed brighter systems rotate more quickly. Early work demonstrated a clear correlation between a galaxy’s rotational velocity and its luminosity, opening a practical route to estimate distances to galaxies beyond the reach of direct parallax measurements. Over the decades, the idea has been refined, extended to multiple photometric bands, and reframed through the lens of the total baryonic content of galaxies.
Physical basis and formulations
The TF relation can be expressed in terms of luminosity L or absolute magnitude M as a function of rotation velocity v, typically written in a form like L ∝ v^α or M ∝ −β log v, with α and β determined empirically from calibrations. In the baryonic version, the total baryonic mass M_b scales with v^4, reflecting a deeper link between dark matter halo properties and the disk’s baryonic content. The approximate universality of the relation across many spirals points to regularities in how gas, stars, and dark matter co-evolve in disk galaxies.
Observational considerations and accuracy
Accurate application of the TF relation requires careful measurement and correction: rotation speeds must be inferred from well-resolved kinematic data or calibrated line widths, disk inclinations must be estimated, extinction and stellar population effects must be accounted for in the luminosity, and the chosen photometric band should minimize scatter. The relation is most precise in the near-infrared, where dust effects are reduced and stellar mass is traced more faithfully. Recognizing and mitigating selection biases, such as Malmquist bias, is essential when using the TF relation to infer distances or cosmic expansion properties. See Malmquist bias for discussions of these biases in astronomical samples.
Applications and implications
Beyond measuring distances, the TF relation serves as a testbed for theories of galaxy formation and the coupling between baryons and dark matter halos. It provides empirical constraints on how efficiently galaxies convert gas into stars and how feedback processes shape disk growth within halos predicted by the Lambda-CDM model. The relation also underpins practical distance-scale work, contributing to estimates of the Hubble constant in combination with other distance indicators and velocity fields. For related distance methods and calibrations, see Cosmic distance ladder and Cepheid variables.