eaa.iop.orgDOI: 10.1888/0333750888/2622 Tully–Fisher RelationMichael Pierce FromEncyclopedia of Astronomy & AstrophysicsP. Murdin © IOP Publishing Ltd 2006 ISBN: 0333750888Downloaded on Wed Feb 01 01:07:51 GMT 2006 [127.0.0.1]Institute of Physics PublishingBristol and PhiladelphiaTerms and ConditionsTully–Fisher RelationENCYCLOPEDIA OF ASTRONOMY AND ASTROPHYSICSTully–Fisher RelationCepheid variable stars are the primary means by whichdistances are measured over the local volume of space.However, beyond about 20 megaparsecs (Mpc) Cepheidsbecome too faint, even for Hubble Space Telescope, andso alternative means of measuring distances are needed.One of the more popular secondary methods makes useof the strong correlation between the luminosity ofSPIRALGALAXIESand their rotational velocities. This is known asthe Tully–Fisher relation.The Tully–Fisher relation has become one of themost widely used methods of measuring extragalacticdistances since spiral galaxies are relatively commonand contain young, massive stars. As a result, theyalso contain Cepheid variables, making the calibration ofthe Tully–Fisher relation using nearby systems relativelystraightforward. In applying this technique the rotationalvelocity serves as a predictor of the luminosity, or absolutemagnitude, of the galaxy. The distance is then calculatedfrom the distance modulus, the difference betweenthe apparent magnitude and the predicted absolutemagnitude of the galaxy. The Tully–Fisher relationprovides an opportunity to step from the local calibratinggalaxies to the smooth Hubble expansion in a single step.Consequently, this technique is extremely valuable for themapping of large-scale structure, the Hubble flow and anyassociated peculiar (i.e. non-expansion) velocities.The origin of the Tully–Fisher relation remainssomewhat uncertain but arises from the correlationbetween mass and luminosity. A spiral galaxy rotatesunder the influence of its own gravity and so the rotationalspeed is connected with its mass. A more massive galaxyalso contains a larger number of stars and is thereforemore luminous. The result is a correlation between theluminosity and the rotational speed for spiral galaxies,the Tully–Fisher relation. There are a few complicationsto this simple explanation. First, the total luminosityof a particular spiral galaxy is a reflection of its starformation history. This might be expected to vary fromgalaxy to galaxy. Second, the fact that spiral galaxieshave considerable amounts of ‘DARK MATTER’ means thatthe mass of a galaxy is not necessarily well representedby the amount of mass contained within its populationof stars. That is, the ratio of mass to light for spiralgalaxies might be expected to vary at a given total mass.The fact that the Tully–Fisher relations show relativelysmall scatter indicates that these variations are small andthis in turn provides a strong constraint on models forthe formation and evolution of spiral galaxies. Whilethese details are still uncertain we can still measure galaxydistances through an empirical calibration of this relation.Observational data for the Tully–Fisher relationThe necessary observational data consist of apparentmagnitudes, corrected for Galactic and internal extinctionby dust, and some measure of rotational velocities,corrected for projection effects. Usually, the rotationalvelocity is measured via the Doppler broadening of theH I 21 cm line, since spiral galaxies are easily detectedin this spectral line usingRADIO TELESCOPES. However, forgalaxies withREDSHIFTS of only a few thousand km s−1,the 21 cm line is shifted into heavily used regions ofthe radio spectrum, resulting in severe interference fromterrestrial sources (radars, cell phones, etc). In addition, atthis wavelength the resolution of even the largest single-dish radio telescopes is only a few arcminutes and so forredshifts beyond about 5000 km s−1the detected signal canbe confused with that from neighboring galaxies. This hasprompted interest in using optical emission lines such asHα to measure the rotational speed of spiral galaxies.Early application of the Tully–Fisher relations usedphotographic photometry, but today CCDs are thedetectors of choice at optical wavelengths owing to theirsuperior performance. The redder Johnson bandpasses(R and I) are usually used because of their reducedsensitivity to dust extinction and because the spectralenergy distributions of spiral galaxies are dominated bylate-type giant stars at these wavelengths. Thus, thesemeasurements are less sensitive to variations in the starformation history and dust content of the individualgalaxies. Theseadvantages continue intothe near-infraredH and K bands. However, the night sky at thesewavelengths is as much as a factor of 104brighter than atoptical wavelengths, making accurate surface photometryat faint levels extremely difficult.Atypical procedure for measuring the magnitude of agalaxy from a calibrated CCD image begins by identifyingand masking contaminating regions from foregroundstars, background galaxies and cosmetic defects. Thesurface brightness profile of the galaxy is constructed bybinning pixels as a function of radius. The integratedmagnitude within some surface brightness level canthen be computed and this measurement extrapolated toinfinite radius using some model, such as an exponential,to produce a total magnitude for the galaxy. Ellipses areoften fitted to the isophotes of the galaxy to yield the best-fit axial ratio of the inclined disk as a function of radiusand surface brightness. The axial ratio can then be used toestimate the inclination of the galaxy.There are several advantages to using the Doppler-broadened 21 cm line ofHIasthemeasure of the rotationof late-type galaxies. First, spiral galaxies are H I richso the line is extremely strong. Second, the H I withinthese systems has an extended distribution such that theouter, flat portion of the rotation curve is well sampled,providing an accurate measurement of the maximumrotational velocity. However, low angular resolution andinterference become significant limitations at redshifts of∼10 000 km s−1, as described above. The use of the Hαemission line avoids these issues. In this case, the slitof a spectrograph is positioned to lie along the long axisof aDISK GALAXY such that the resulting spectrum willsample the velocity as a function of radius for the galaxy.However, since the Hα emission line is produced