3. THE PRIMARY UNCERTAINTIES

Measurement errors in m⁰_V are no longer the limiting factor that determines the accuracy when using the GCLF for distance determinations. Typical internal uncertainties are now 0.1 - 0.2 mag, and can be further reduced by averaging several galaxies together. The uncertainties introduced by the following questions generally dominate the calculation.

1. How secure are the "local" calibrators?
2. How universal is the GCLF?
3. How important is interstellar extinction?
4. What is the appropriate velocity for the calculation of the Hubble constant?

While most of these are the same questions that were with us 20 and 30 years ago (see Sandage 1968, de Vaucouleurs 1970, Hanes 1979), the recent increase in the number of high quality GCLFs and the new capabilities provided by HST are making it possible to resolve many of these questions. For example, rather than relying on the Milky Way and M31 for our calibration it is now possible to use Cepheid measurements to directly calibrate the ellipticals in the Virgo cluster.

3.1. Local calibrators

Until recently, the globular cluster systems of the Milky Way and M31 provided the primary calibration for GCLF distance estimates. While a few other local galaxies have also been measured (M33, M81, NGC 4565, NGC 4594; see H96), the data is relatively poor and the errors roughly twice as large, so most studies have not used them. Secker (1992) and Sandage & Tammann (1995) have recently reexamined the use of M31 and the Milky Way as the local calibrator. Secker finds that M⁰_V is roughly 0.2 mag brighter in M31 (M⁰_V = -7.51 ± 0.15 mag) compared to the Milky Way (M⁰_V = -7.29 ± 0.13 mag). Sandage and Tammann (1995), adopting an RR Lyrae calibration for the Milky Way that is roughly 0.25 mag brighter than in general use, find that M⁰_V is nearly identical for M31 (M⁰_V = -7.70 ± 0.20 mag) and the Milky Way (M⁰_V = -7.60 ± 0.11 mag). The mean of all four values is M⁰_V = -7.52 ± 0.17 mag where the uncertainty is the scatter in the four values, which provides the best estimate of the accuracy in this case since systematic effects (i.e. the RR Lyrae distance scale) dominate the results.

The use of local spirals to calibrate distant ellipticals has always been one of the primary weaknesses of the GCLF method for measuring distances, as will be discussed in the section on universality (Section 3.2.2). A practical concern is that faint GCs are more difficult to identify in nearby spirals since they can be confused with open clusters, and the smaller number and larger angular extent makes them more difficult to differentiate from background galaxies. Reed et al. (1992) suggests that incompleteness at the faint end may be the reason the value of M⁰_V in M31 appears to be slightly higher, and the value of slightly lower, than the comparable values for the Milky Way.

It is now possible to replace this comparison of "apples and oranges" with a direct calibration of elliptical galaxies using measurements of Cepheid variables in the Virgo cluster galaxies. The primary uncertainty with this method is the possibility that the galaxy with Cepheid measurements might be in the front or back of the cluster. It is therefore important to observe several galaxies in the same cluster. At present there are five galaxies in the Virgo cluster with distance determinations using Cepheid variables (NGC 4321 - Ferrarese et al. 1996 find m - M = 31.04 ± 0.17 mag; NGC 4496A - Saha et al. 1996a find m - M = 31.03 ± 0.14 mag; NGC 4536 - Saha et al. 1996b find m - M = 31.23 ± 0.05 mag; NGC 4571 - Pierce et al. 1994 find m - M = 30.87 mag; and NGC 4639 - Sandage et al. 1996 find m - M = 32.00 ± 0.23 mag). The value for NGC 4639 is nearly a magnitude higher than the other values. We therefore remove the high and low values to obtain our estimate for the Virgo cluster distance modulus of m - M = 31.10 ± 0.14 mag (where the error estimate is the mean of the value for three galaxies [underestimate] and the value for five galaxies [overestimate]). Using the value of m_V = 23.89 ± 0.09 mag from Section 2.3, and an uncertainty of 0.20 mag for the Cepheid zeropoint (Freedman et al. 1994) yields our final value of M⁰_V = -7.21 ± 0.26 mag. Keeping all five Cepheid measurements would result in a value of M⁰_V which is 0.13 mag brighter. The dominant uncertainty is due to the Cepheid distance determination for the Virgo cluster and the Cepheid zeropoint, both of which should improve in the near future.

Hence, the value of M⁰_V derived from the Milky Way and M31 (i.e., M⁰_V = -7.52 mag) is 0.31 mag brighter than our estimate of M⁰_V for bright ellipticals in the Virgo cluster. This should be considered a tentative result due to the large uncertainties. It is interesting to note, however, that the value is similar to the what has been advocated by Fleming et al. (1995) and Ashman, Conti, & Zepf (1995).

3.2. Universality

In 1958, Sandage wrote "We are, therefore, forced to the conclusion that globular clusters are very poor distance indicators because, at least with the present data, the absolute magnitude apparently vary from galaxy to galaxy". This was based on a comparison between the Milky Way and M31. van den Bergh (1975) came to essentially the same conclusion. As the data improved, these discrepancies went away, and it became more and more evident that the GCLF was actually quite a good distance indicator. For example, Sandage & Tammann (1995) find M⁰_V for the Milky Way and M31 agree to within 0.1 mag, well within the statistical uncertainty, and argue that the GCLF is nearly universal.

The question of universality takes four basic forms; 1) luminosity, 2) Hubble type, 3) color, 4) environment.

3.2.1. Luminosity

Do bright and faint galaxies have the same value of M⁰_V? H96 treats this in some detail, showing that dwarf ellipticals have values of M⁰_V which are roughly 0.3 mag fainter than bright ellipticals. One possibility is that fainter clusters can survive in these systems due to the weaker tidal fields. This would predict that values of would be larger. While the value of for a composite of 8 dwarf ellipticals in Virgo cluster observed by Durrell et al. (1996) is the highest measured value in Table 1 from H96 ( = 1.48 ± 0.20) the large uncertainty due to the small number of clusters makes this very uncertain. The new HST snapshot survey of 30 dwarf ellipticals (Ferguson et al. 1996) should improve the situation.

The difference in M⁰_V for faint galaxies is not actually very important for the distance scale problem, since we are most interested in the bright ellipticals with large populations of GCs needed to provide small statistical uncertainties. The study of the GCLF in dwarf galaxies will primarily be useful for understanding the formation and evolution of GCs.

3.2.2. Hubble type

The question of whether spiral and elliptical galaxies have the same GCLF is important since, until recently, the local calibrators were all spirals while the distant galaxies were all ellipticals. This circumstance arose because a more accurate value of M⁰_V can be measured for a giant elliptical which has more GCs, fewer point-like sources to be confused with the GCs (e.g. open clusters, HII regions, bright stars), and negligible internal extinction. Unfortunately, there are no giant ellipticals in the local group. The measurement of Cepheid variables in galaxies as far as the Virgo cluster is now circumventing this problem (see Section 3.1), but at the price of making the GCLF a secondary distance indicator. The question of universality versus Hubble type is still very important in order to provide a cross check on the Cepheid scale, as well as to gain a better understanding of how GCs form and evolve.

3.2.3. Color (i.e. metallicity and age effects)

As the precision of observations has increased it has become clear that many galaxies have bi- or multi-modal color distributions. Recent improvements in spectral synthesis models (e.g. Bruzual & Charlot 1993, 1996, Worthey 1994, Fritze-v.Alvensleben & Gerhard 1994) now make it possible to interpret these effects in terms of metallicity and/or age differences. Ashman, Conti, & Zepf (1995) have employed the Worthy (1994) stellar evolution models to show that if spirals and ellipticals have the same mass function (an assumption that needs to be carefully checked, since there are a variety of physical mechanisms that could modify the GC mass function; see Section 4), then spirals should have values of M⁰_V which are roughly 0.2 mag brighter than ellipticals due to differences in the mean metallicity of the globular clusters. They also show that the I band is less sensitive to metallicity effects.

The observations of red and blue clusters in M87 (Whitmore et al. 1995, Elson & Santiago 1996a, b) provide some observational evidence for this trend, with a shift of 0.1 mag seen by Whitmore et al. (1995) and a shift of 0.3 mag seen by Elson & Santiago (1996b). These effects, if real, could affect the value of M⁰_V at the 0.1 - 0.2 mag level. Correcting for the effect offers the potential of reducing the small dispersion in M⁰_V still further.

The discovery of candidate young globular clusters in both merger remnants (e.g. Holtzman et al. 1992, Whitmore et al. 1993, Whitmore & Schweizer 1995) and in starburst galaxies (e.g. Conti & Vacca 1994, Hunter, O'Connell, & Gallagher 1994, and Meurer et al. 1995) raises the question of whether M⁰_V might be affected by the presence of young and intermediate-age clusters for some galaxies. While very young clusters are easy to identify, since they are up to 6 magnitudes brighter in M⁰_V and 0.4 to 1.0 mag bluer in V-I, Whitmore et al. (1996) show that intermediate-age clusters are more difficult to isolate. At an age of 1 - 2 Gyr the clusters have roughly the same colors as old globular clusters, based on the Bruzual & Charlot (1996) models, although they are 1 - 2 mag brighter. Older clusters are redder, and either slightly brighter (2 - 6 Gyr) or slightly fainter (> 6 Gyr) than the old population. Intermediate-age clusters may therefore lead to an estimate for M⁰_V which is either slightly too large or too small. It may be possible to minimize this potential problem by selecting galaxies that do not have evidence of recent merger activity (e.g. shells and loops), or by using only the blue clusters to determine M⁰_V.

3.2.4. Environment

There are now enough galaxies with well determined GCLFs to begin to ask whether the environment and the position within the galaxy (i.e. the local environment) can affect the GCLF.

Blakeslee and Tonry (1996), using only six galaxies, find tentative evidence for a correlation between M_V and environment. Using the group velocity dispersion as an indicator of environment they find that M⁰_V is fainter by about 0.5 mag in the Coma cluster than in the local group. However, the correlation is quite dependent on the observations of NGC 4881 in the Coma cluster (Baum et al. 1995) which are ambiguous (see Section 2.2). In addition, even if the correlation turns out to be real it is quite possible that a difference in M⁰_V for different Hubble types is the cause rather than an environmental effect, since the galaxies in the low velocity dispersion environment are all spirals while the other galaxies are ellipticals. Finally, the opposite trend appears to be present for the dwarf ellipticals in the field compared to the dwarf ellipticals in the Virgo cluster (H96).

A large number of studies have concluded that M⁰_V is not a function of position in a galaxy (e.g. Grillmair, Pritchet, & van den Bergh 1986, van den Bergh 1992, McLauglin, Harris, & Hanes 1994, Harris & Pudritz 1994, and Forbes 1996a). However, a recent paper by van den Bergh (1996) indicates that the luminosity distribution of globular clusters in the Milky Way depends on cluster size, which is known to correlate with Galactocentric distance. In addition, the clusters in the outer halo (> 10 kpc) with red horizontal branches are fainter than normal clusters by a factor of 10. Many of these are the anomalous clusters such as Pal 1, Pal 12, and Ter 7 which are both very extended and very faint, and would not be picked up by most studies of GC's in external galaxies. It will be important to examine the more recent HST datasets to search for possible trends in ellipticals, but based on earlier results the effects on M⁰_V are likely to be quite small.

3.3. Extinction

Extinction from dust in the Milky Way (galactic extinction), and dust in other galaxies (internal extinction), results in relatively minor uncertainties. For example, Sandage & Tammann (1995) advocate a value of A_V = 0.00 mag for the galactic extinction in M87 while several other studies use a value from Burstein and Heiles (1984) of A_V 0.06 mag (e.g. Whitmore et al. 1995). The situation is similar for the Fornax cluster, where values of A_V = 0.00 mag, 0.05 mag, and 0.10 mag are commonly used.

Effects of internal extinction are generally negligible since elliptical galaxies, containing very little gas and dust, are used for most GCLF distance determinations. However, for the Milky Way GCs typical values of the color excess, E(B - V), are 0.3 mag (Sandage & Tammann 1995), and 0.1 mag for M31 (Reed, Harris, & Harris 1992). Internal extinction is potentially important if these galaxies are used to set the zeropoint. If we use galaxies with Cepheid variables to calibrate our zero point the uncertainty due to extinction is quite small. For example, Ferrarese et al. (1996) estimate a total color excess E(B - V) = 0.10 mag for the Cepheid measurements in M100, and a combined uncertainty of 0.02 mag in the error budget for absorption in M100 and the LMC (used to define the Cepheid zeropoint).

3.4. Effect of "peculiar" velocities on the calculation of H₀

A determination of the Hubble constant requires an estimate of the distance and of the appropriate "expansion" velocity of a galaxy. Globular clusters share a problem common to several distance indicators, namely they only provide accurate distance estimates out to a few thousand km s^-1. Since peculiar velocities of a few hundred km s^-1 are common for clusters of galaxies (e.g. Lauer & Postman 1994), this can introduce substantial uncertainties in our estimate of the Hubble constant. For example, a typical estimate for the Virgocentric infall velocity is 200 km s^-1 (Han & Mould 1990). This represents an 15% correction to the Virgo velocity. The spread in the different measurements ( 70 km s^-1) represents a 5% uncertainty in H₀.

One approach to circumvent this problem is to use HST to push out to larger distances. Baum et al. (1995) has shown that it is difficult but not impossible to reach the turnover of the GCLF at the distance of the Coma cluster ( 7000 km s^-1). Another technique is to offset from a nearby cluster to a more distant cluster, assuming the relative distances of the two clusters are well known (e.g. Freedman et al. 1994). This appears to be the case between the Virgo and Coma clusters, where estimates of _Coma-Virgo(m - M) = 3.71 ± 0.05 mag (van den Bergh 1992, based on 12 studies), _Coma-Virgo(m - M) = 3.71 ± 0.03 mag (de Vaucouleurs 1993, based on 12 studies, with 7 overlapping van den Bergh 1992), and _Coma-Virgo(m - M) = 3.80 ± 0.17 mag (Jerjen & Tammann 1993, based on five studies) are in good agreement. A related technique is to use the relative distance between a single cluster and several more distant clusters, so that any peculiar motions of the distant clusters tend to average out. Jerjen and Tammann (1993) use this technique to estimate the "Machian" velocity of the Virgo cluster based on 15 clusters. The main shortcoming of this technique is that the number of galaxy clusters with accurate relative distance is limited, so adding more clusters may simply add noise in some cases.

It is generally assumed that all the galaxies in a cluster are at the same distance, and the systemic velocity of the cluster can therefore be used as the appropriate velocity for the calculation of the Hubble constant. However, the galaxy may actually be in the front or back of the cluster. For a system as large and ill-defined as the spiral galaxies in the Virgo cluster (i.e. with a "radius" in the range 3° - 6°), this may result in a 5 - 10% error in the true distance. In addition, for some studies there may be a tendency to select galaxies which are slightly nearer (e.g. Cepheids), since it is easier to resolve the stars in this case. It will therefore be important to observe a number of galaxies and a number of different groups and clusters.

One way to solve this problem is to observe the dominant D or cD galaxy in the cluster, since there is good evidence that these galaxies reside very near the center of the cluster. For example, the position at the center of the extended x-ray halo in the Virgo cluster is good evidence that M87 is at the bottom of the potential well. Another method is to use a very compact cluster such as Fornax. In this case the radius of the cluster is roughly 1°, implying a maximum line-of-sight excursion of about 2% if the cluster is roughly spherical.

3.5. Error estimate

As discussed in Section 2.3, the combination of the intrinsic dispersion in the value of M⁰_V and the measurement uncertainty results in internal uncertainties of 0.2 mag. Data for several galaxies can be averaged together reducing the uncertainty still further. Hence, the external errors are generally the limiting factors for distance determinations using the GCLF.

Table 3 provides an estimate of the total error budget for the following three cases: 1) using the Milky Way and M31 for calibration, 2) using Cepheids in the Virgo cluster for calibration, 3) using Cepheids in several groups and clusters to nail down the zeropoint, and ten ellipticals in the range 2,000 - 5,000 km s^-1 with good sky coverage to average out peculiar velocities. See Section 2 and Section 3 for the discussion of the values in Table 3. Values in parenthesis assume ten galaxies have been averaged together. The value of ± 0.10 mag for the calibration of Method # 3 assumes the uncertainty in the Cepheid distance scale will be reduced by roughly a factor of two in the next few years (e.g. more observations in the Virgo and Fornax cluster and a better determination of the distance to the LMC). The table shows that at present, the limiting factors are the zeropoint calibration, peculiar velocities, and for Method # 1, the question of universality. These lead to current uncertainties of 15% in H₀. The estimate for method # 3 shows that within a few years it should be possible to use the GCLF to determine distances for individual ellipticals to 10%, and to determine H₀ to about 7%.