Much of the debate over the distance scale arises from the large distances that we need to cover to be sure we are beyond the range of peculiar velocities such as Virgocentric flow. Eventually, we find that only global galaxy properties and their correlations are usable. In the ladder of distance indicators, propagation of errors becomes dominant. See Rowan-Robinson, The Cosmological Distance Ladder (Cambridge 1987), for a full discussion. Modern methods are described in Galaxy Distances and Deviations from Universal Expansion, ed. B. Madore and R.B. Tully (NATO ASI 180). We will consider the methods in the traditionql distance ladder in turn.

Trigonometric parallax. This is useful out to a few hundred pc for individual stars if we have milliarcsecond precision, which Hipparcos delivered for tens of thousands of stars. This is the only (almost) completely foolproof technique for distances, since we know the size of the Earth's orbit well. Statistical applications can be applied to whole groups of stars, using (for example) the solar motion through the galactic disk to generate secular parallax. These still sample only a tiny region of the galaxy, and in particular do not reach to either very luminous stars or Cepheid variables (though Hipparcos delivered statistically useful parallaxes for some Cepheids).

Cluster convergent points. For nearby clusters of appreciable angular extent (like the Hyades) perspective makes the proper motions of individual stars not parallel, but directed toward a point in the sky parallel to the cluster's mean motion relative to the Sun. This gives the angle between our line of sight and the cluster's motion, and thus what fraction of the cluster's space motion is seen as proper motion and what as radial velocity. Measuring the average radial velocity then allows a distance determination, as the distance for which the radial velocity and proper motion are consistent with the angle between line-of-sight and space motion. This lets us calibrate absolute magnitudes for all the cluster members - including upper main-sequence and red giant stars. The classic example is the Hyades cluster, seen here using Hipparcos proper motions from Perryman et al. (1998 A&A 331, 81):

Main-sequence fitting. For even more distant star clusters (that might contain OB stars or Cepheids, for example) we estimate distances by assuming that main-sequence stars of identical spectral type have the same absolute magnitude. This amounts to, for example, shifting the main-sequence location of a cluster until it coincides with that of some reference cluster like the Hyades. The reddening must be reasonably well determined to make this work. This can be done for systems as distant as the Magellanic clouds, which is the easiest place to calibrate Cepheids. For this purpose, each Magellanic Cloud can be thought of as a giant cluster.

Cepheid variables. These are supergiants in the instability strip on the H-R diagram, undergoing regular pulsations that are expressed by luminosity and temperature variations. Their high optical luminosity makes them easy to pick out (though, being rather massive stars, they don't occur in elliptical galaxies). Recent data give a period-luminosity relation of the form <M_V> = -3.53 log P + 2.13 (<B₀> - <V₀>) + f where f ~ -2.25 is a zero point. P is in days here, and the brackets denote averaging over a cycle of the light curve. The relations for the SMC and LMC are shown by Mathewson, Ford and Visvanathan 1986 (ApJ 301, 664) as follows, from their Fig. 3 (courtesy of the AAS):

To use Cepheids effectively, one must deal with the following points:

Cepheids have been measured from the ground throughout the Local Group (which Hubble could do - the astronomer, not the telescope), and can be detected in the M81 and Sculptor groups, and more recently in M101 at a distance of 7 Mpc (Cook, Aaronson, and Illingworth 1986 ApJLett 301, L45), and even an amazing detection of a couple in the late-type Virgo spiral NGC 4751, when the seeing and stellar crowding all worked together (Pierce et al. 1994 BAAS 26, 1411). Note that it is traditional to quote the distance modulus m-M = 5 log D - 5 rather than the distance itself in many publications on the distance scale - for example, the DM of the LMC is close to 18.5. To date, the HST key project on the distance scale has reported detections of Cepheids to 25 Mpc, and it can in principle go well beyond Virgo. A real shame there aren't any spirals which can be shown to live in the Coma core. The best-known report of this work was for NGC 4321=M100 in Virgo by Ferrarese et al (1996 ApJ 464, 568), see also Freedman et al 1994 (Nature 371, 757). The project, using Cepheids to calibrate secondary distance indicators through common galaxy and group membership, was described by Kennicutt, Mould, and Freedman 1995 (AJ 110, 1476). Some of their Cepheid light curves are shown below -- for M100 alone, they already detect more Cepheids than are known in the LMC, so the LMC calibration becomes a weak link. The project has gotten all its data, and a recent summary (Mould et al. 2000 ApJ 529, 7867) gives a grand average value of H₀= 71 ± 6 km/s Mpc as based on HST Cepheid distances to 25 galaxies, in ridiculously close agreement with results of fitting the WMAP power spectrum of CMB fluctuation.

This plot colects the Key Project Cepheid distances. Note the large peculiar motions within Virgo; the one galaxy lying right on the mean line at that distance is NGC 7331, almost opposite Virgo in the sky.

RR Lyrae stars. These are lower-luminosity stars, where the instability strip crosses the horizontal branch. They may appear on cluster H-R diagrams by omission in the "RR Lyrae gap", since variables are usually not plotted. The absolute magnitude of all RR Lyrae variables seems to be nearly constant at <M_V = 0.75 ± 0.1. There may be some poorly-determined metallicity dependence. No period determination is needed here, just the determination that a star is of this type (which means you get the period anyway). Problems are: RR Lyraes are intrinsically about 2 magnitudes fainter than Cepheids, and similarly difficult to calibrate; only a couple are close enough for a parallax measurement with Hipparcos, so statistical parallaxes are still important.

Automated image detection has proven fruitful in finding these stars throughout the Local Group, even before HST. Saha and Hoessel (1990, AJ 99, 97) report finding 151 in the small elliptical NGC 185, as seen in their Fig. 5 courtesy of the AAS:

Most luminous (blue/red) stars. There is an empirical relation between a galaxy's absolute magnitude and that of the brightest individual stars - this amounts to assuming a constant form for the upper end of the luminosity function and letting statistics operate. Conveniently, these are the first stars to be resolved. Possible problems: confusion with compact clusters (as in 30 Doradus), unknown variation with galaxy type.

All of the stellar indicators listed above for other galaxies are easiest to use in systems with substantial population I components, and in rather open galaxies so that crowding is reduced. One therefore tries to deal with a galaxy's outer regions, and rather late-type galaxies (see the Sandage and Bedke atlas for illustrations of resolution into stars for such galaxies, which was the point of their producing this volume). There are also several temporary or indirect stellar distance indicators:

Novae. There is a relation between absolute magnitude and fading rate for novae, as best we can tell from the Local Group. They can easily be picked out as transient Ha sources, and two seem to have been detected in this way as far away as M87 (Pritchet and van den Bergh 1987 ApJLett 288, L41); as well, data series sufficient to find Cepheids may find them as continuum sources. Ciardullo et al. (1990 ApJ 356, 472) discuss 11 well-observed novae in M31. The relation between fading rate and absolute B magnitude is only partially followed by Ha, so that a combination of Ha discovery, continuum observations near maximum, and Ha observations to faint levels seems the most effective approach. Faint continuum measurements are impossible because the nova blends into the overall stellar background. This technique may be used for population II systems.

Planetary nebulae. These can also trace the population II components, since they can be produced by old stars. Their usefulness as a distance indicator relies on the fact that their luminosity function appears to be invariant, and is easily understood from stellar evolution (Jacoby 1989 ApJ 339, 39). Large numbers of planetaries can be detected in nearby galaxies by using narrow-band images around the [O III] l5007 line, which is extremely strong in planetaries but not most H II regions. Sufficient planetaries have been detected for estimates of the distance to Virgo (Jacoby et al. 1990, ApJ 356, 332). The fitting technique for an incomplete luminosity function is illustrated by Fig. 3 of Ciardullo et al. 1989 (ApJ 339, 53) for M31 (courtesy the AAS):

Supernovae. Type I (population II) supernovae can be recognized (and divided into subgroups a,b, and maybe c) based on their spectra and light curves. Available evidence is consistent with peak luminosity being roughly fixed for at at least type Ia (but watch out, new understanding of subluminous ones like 1987A may change this). Supernovae can be seen a long way off (like z=1.7 if you're looking hard), so they would make wonderful distance indicators if (1) we really know this peak luminosity, (2) it really is constant, and (3) we can account for dust obscuration (hello IR). The peak brightness is given by supernova models, but SN in galaxies nearby enough for checking are rare. For cosmologically distant SN the rate of decay is stretched by the dilation factor (1+z). These are the objects which first provided strong evidence for an acceleration of the Hubble expansion (perhaps to be identified with Einstein's cosmological constant).

A direct measure of distance for expanding or pulsating objects is in principle possible via the Baade-Wesselink method. One measures the change in bolometric luminosity and the integral (change in relative) radial velocity over this time. Then, applying either a blackbody approximation or a more realistic spectrum, the angular size difference between two epochs is derived, which gives a distance by requiring it to be consistent with the radius change from radial velocities. Problems center around just how the observed velocity is weighted across the photosphere and whether the opacity structure changes between epochs.

Surface-brightness fluctuations. Well before a galaxy is truly resolved into even its brightest stars, the image will be mottled by statistical fluctuations; for example, if the surface brightness is such that there are 100 red giants per seeing disk, one expects 10% Poisson fluctuations. These may be distinguished from photon noise because these fluctuations have the same spatial power spectrum as the seeing disk (or more generally the system response, i.e. PSF), not white noise (Tonry and Schneider 1988 AJ 96, 807). As a sample, this image shows M32 HST data resampled as if seen at progressively greater distances (each step increasing by a factor 2). The technique is surprisingly powerful as long as one can compare galaxies with similar stellar populations - basically one must assume a characteristic (well-defined) mean luminosity for stars. The method has already been extended to Virgo, giving excellent agreement with planetary-nebula determinations and first hints as to which galaxies are on the near and far sides (Tonry et al. 1989 ApJ 346, L57).

H II regions. By necessity these require active star formation and OB stars. They are luminous and measurable to very large distances. The first approach (Sandage and Tammann 1974 ApJ 190, 525) was to assume that the diameter of the brightest H II regions is related to galaxy absolute magnitude. However, Kennicutt 1979 (ApJ 228, 704) showed that seeing effects compromise visual and isophotal diameters so strongly that this cannot work as a distance indicator. More recent work has focussed on emission-line luminosities, assuming in essence that the more star formation, the brighter the galaxy, and statistically the brighter the biggest few H II regions are. This might be considered a variant on the brightest blue stars method.

The emission-line widths have also been considered, with a claim by Terlevich and Melnick (1981 MNRAS 195, 839) that an L - s⁴ relation holds for supergiant H II regions; that is that they are bound by a gravitational mass propertional to (ionizing-UV) starlight intensity. This would be useful in the same way as the Tully-Fisher relation or the analogous relation for elliptical galaxies. However, further work (Gallagher and Hunter 1983 ApJ 274, 141; Roy et al. 1986 ApJ 300, 624) has clouded the picture; for more extended samples, the correlation is much less striking, and the gas motions are largely supersonic, driven by stellar winds and SN rather than being gravitationally produced.

Galaxy structures. We may consider identifiable structure within galaxies as distance indicators, if they have constant or calibratable sizes. Candidates have included:

A refinement, including a second parameter related to surface brightness, has been used by the Seven Samurai to compile a large set of redshift-independent distances for mapping the local velocity field (Dressler et al. 1987 ApJ 313, 42; data in Faber et al. 1989 ApJSuppl 69, 763).

Global galaxy properties: These must be used for more and more distant systems, requiring extensive calibration from the techniques above. Specific indicators include:

Corrections to observed magnitudes must be applied for (1) measuring aperture size (2) passband redshifting, the so-called K-correction (3) the redshifting of both photon energy and arrival rate, and (4) any assumed evolution - at least passive evolution of the stellar population must be taking place.

All of the above methods rely on a straightforward application of the inverse-square law or the angular diameter-distance relation. There is also a range of techniques that use more involved or indirect combinations of observables. Some examples are:

The Hubble time: for simple big-bang models, ages of objects (stars, radioactive nuclei) set bounds on H₀. The age of the universe is of order the Hubble time t_H =1/H₀, to within a factor of order unity depending on the deceleration history of the expansion. For H₀=50 km/s Mpc, t_H= 2 x10¹⁰ years; for 100 km/s Mpc, 10¹⁰ years. This must be greater than the age determined from geological and stellar-evolutionary timescales, nuclear isotopic clocks like ²³⁵U/²³⁸U, and consistent with the dynamical status of galaxies and clusters. The small amount of evolution observed in elliptical galaxies to about z=1 favors smaller H₀ in simple models (Hamilton 1985 ApJ 297, 371). One should beware subtly circular arguments - globular-cluster ages were beautifully consistent with H₀=50 but had been calculated by people who know the answer they expected to get and tuned a few parametere accordingly. There was, for several years, a widely-publicized discrepancy between t_H from HST Cepheid results and globular-cluster ages, but recent calculations of effects of mixing on stellar evolution and the Hipparcos distance revisions to Cepheids both go in the direction of reducing the problem.

Gravitational lenses: we need to know the lens mass (for example through the cluster velocity dispersion) and the time delay between images (say from QSO variability). Then we can derive the lens proper distance. The differential time delay may be the hardest part here, especially in the presence of microlensing.

Light echoes: this has given an independent distance to the LMC, by using the time of illumination of a circumstellar ring (seen from IUE, Panagia et al. 1991 ApJL 380, L23) to give an absolute front-back size, and the angular size of the ring (from HST) for a transverse measurement. This example was done by, for example, Gould (1995 ApJ 452, 189). A similar approach can also be used (with polarization to tell where the ring is) for distant supernovae (Sparks 1994 ApJ 433, 19).

Emission/absorption measures: here one uses the different dependences of emission and absorption on density versus path length. An example is the IGM in clusters seen in emission via X-rays and in absorption (more precisely upward scattering) against the microwave background (the Sunyaev-Zeldovich effect). This works because on astrophysical grounds we expect the hot gas to be smoothly distributed through the cluster potential; clumping would make this more useful for probing structure than distance. So far, this isn't accurate enough for use as more than a consistency argument because the absorption is very weak, but in principle is free of many of the assumptions of other methods (10⁷ K gas should be very smoothly distributed). This technique for detecting hot cluster atmospheres is almost equally sensitive for all cluster redshifts z>0.5 because it's an area measure, so surveys are in progress to find high-redshift clusters as S-Z spots.

Proper motions: a maser in a star-forming region should be detectable with the VLBA all the way to Virgo. Its proper motion due to the rotation of a typical spiral should be of order 3 microarcseconds per year, which, it has been claimed, should be measurable in a decade or so. One then determines the distance at which this matches the disk rotational velocity at the appropriate radius. The most distant actual application so far has been to masers in the nuclear disk of NGC 4258 (Herrnstein et al. 1999 Nature 400, 539).

Distances to nearby galaxies are not in serious dispute, but the role of peculiar velocity on these scales is. Some useful distances are (in Mpc)

This means that H(Virgo) is about 60 km/s Mpc, but is this value globally applicable? Two major camps long existed: Sandage at 50 (the "long" distance scale) and de Vaucouleurs at 100 (the "short" scale). Data occasionally drown in invective on this issue. Doing a systematic error treatment, Hanes 1981 (MNRAS) and Rowan-Robinson in his book found that 80 km/s Mpc satisfies all the error bars and is what the IR T-F relation gives at large distances. This is essentially the Key Project global value as well, with the CMBR global fitting giving a value of 71. Maybe the compromise value of 75 that many people have used was actually more than fence-sitting.

Aaronson, Huchra, and Mould found evidence for systematic departures from the Hubble flow toward Virgo, so that the redshift-distance relation is nonlinear, and in some places double or triple-valued.

A first indication of such disturbances was the study by Rubin and Ford (1987 AJ 81, 719) of 96 Sc I galaxies, which showed an asymmetry on the sky in redshift-magnitude space such that we were likely to be moving at about 500 km/s with respect to the centroid of these galaxies. This eventually turned into an industry, with the 7 Samurai announcing a "Great Attractor" off in Centaurus (l=299°, b=-11°) that messes up the velocity field out to about 3000 km/s (Lynden-Bell et al. 1987 ApJLett 313, L37). We are approaching this mass at about 700 km/s; this is actually consistent with the Rubin and Ford result if Virgo infall is included. Lauer and Postman (1994 ApJ 425, 418) find yet a different motion relative to 119 Abell clusters at z < 0.05 - 561 ± 284 km/s toward l=220 °, b= -28 °, yet a different direction and certainly an unexpected magnitude. A somewhat different motion is derived with respect to the microwave background, which is the grandest average we can find - the final COBE data set gives 368 km/s toward l=264.3, b=48.1 with independent analysis of the FIRAS and DMR instruments in good agreement (Lineweaver et al. 1996 ApJ 470, 38). This has just been refined with WMAP to l=263.8, b=48.2 (Bennett et al. ApJ submitted, astro-ph/0302207). At some point one wonders about the the scale on which the cosmological principle is adequately realized. This means that the Grail itself, H₀, must be sought at even larger distances than thought before (to the extent that it would be useful in itself if the Hubble flow is really lumpy, though the tightness of the Hubble diagram for standard candles suggests that it isn't all that bad).

There are also isolated instances of galaxies flagrantly violating the Hubble flow. Perhaps the best is in the direction of NGC 1275. The main galaxy has v=5000 km/s, and has something that looks like a late-type spiral demonstrably in front of it but having v=8100 km/s. Images from Keel 1983 (AJ 88, 1579) isolate the foreground and background systems in Ha:

while the foreground system is visible in absorption by dust in this HST image:

This is too fast to be just free fall into a cluster core - and if there are many galaxies shooting around at 3000 km/s there should be huge scatter in the Hubble diagram. Thus there can't be many of these, but how far would we have gone wrong if we saw the spiral by itself?

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