G.A. Tammann, L. Labhardt & M. Federspiel
Astronomisches Institut der Universität Basel, Venusstr. 7, CH-4102 Binningen, Switzerland
The Observatories of the Carnegie Institution, 813 Santa Barbara Street, Pasadena, CA 91101, USA
A. Saha, F.D. Macchetto & N. Panagia
Space Telescope Science Institute 3700 San Martin Drive, Baltimore, MD 21218, USA
The influence of HST on the determination of is already enormous and it can only grow. In Table 1 a compilation is given of distance determinations with HST having some bearing on . The values scatter between 55 and 80 and the formal mean is . This, however, is not the best value, because some values shown are mutually incompatible.
There are now two self-consistent routes to to which HST has heavily contributed. The first route via the Virgo cluster is described in Section 2, the second route using supernovae of type Ia (SNe Ia) is discussed in Section 3.
Table 1: determinations from HST
Much of what follows depends on Cepheid distances. A word on the reliability of their P-L relation is, therefore, in place. The P-L relation of Madore & Freedman (1991), adopted in the following, assumes an LMC modulus of 18.50. Actual confirmation of this value to better than 0.10 mag comes from the P-L relation calibrated by Galactic Cepheids (Sandage & Tammann 1968, Feast & Walker 1987) and independently from the ring of SN 1987A (Panagia et al. 1991, Gould 1994); further support is given by RR Lyr stars and other distance determinations (cf. Tammann 1996). The zero point of the adopted P-L relation is therefore secure to in linear distance. The slope of the relation is well determined by the LMC Cepheids; it is of less importance as long as the Cepheids under consideration cover a sufficient period range, which is also needed to avoid selection effects (Sandage 1988). Metallicity effects are believed to be small (Freedman & Madore 1990, Chiosi et al. 1993).
When the first reliable Cepheid distance of a Virgo galaxy (NGC 4321) became available from HST (Freedman et al. 1994), it was precipitately hailed the Virgo cluster distance (Mould et al. 1995, Kennicutt et al. 1995) although the value of 17 Mpc was suspiciously low. The next two Virgo galaxies, NGC 4536 (Saha et al. 1996a) and NGC 4496A (Saha et al. 1996b), again had very low distances. Only the fourth galaxy, NGC 4639, revealed the important depth of the cluster (note for comparison: the spiral members span in the sky!). Its distance is 25 Mpc (Sandage et al. 1996) and yet it is a bona fide cluster member; with a small velocity of 800 kms, it cannot be assigned to the background.
It is no accident that the first three Virgo spirals with Cepheid distances lie on the near side of the cluster. They were selected from Sandage & Bedke's (1988) atlas of galaxies most suited for HST; they were thus biased to begin with. NGC 4639 looks much more difficult and would not have been selected had it not produced the archetypal SN 1990N.
It is now clear that it will take at least a dozen Cepheid distances of a randomly selected sample of Virgo members to obtain a meaningful mean cluster distance.
Meanwhile Tanvir et al. (1995) have suggested to step up their Cepheid-based Leo group distance of out to the Virgo cluster by means of relative distance determinations. The best available relative distances are compiled in Table 2. Adding the mean difference of to the above distance modulus gives a Virgo cluster modulus of . For brevity we will refer to this value in the following as the Cepheid distance of the Virgo cluster .
Table 2: The distance modulus difference between the Leo group and the Virgo cluster
There are now 11 spiral galaxies with Cepheid distances (seven of which come from HST) which are useful for the calibration of the relation between absolute magnitude and (w= inclination-corrected 21cm-line width). Two close companions of M101 bring the number of useful calibrators to 13 (cf. Table 3). Two galaxies (M101 and M100) are less inclined than the frequently adopted limit of ; yet their inclinations are so well defined by mapping their velocity field that they are still useful as calibrators.
Table 3: Galaxies with Cepheid distances for the calibration of the Tully-Fisher relation
The date in Table 3 yield the following calibration of the TF relation
(Federspiel et al. 1996), where the slope is taken from the Virgo cluster.
An objective and complete sample of Virgo spirals is defined by the 48 non-peculiar galaxies of type Sab--Sdm from the Virgo Cluster Catalog (Binggeli et al. 1985) with and lying within the isopleths of substructures A and B (see Binggeli et al. 1993) or, without changing the result, within the X-ray contour of the cluster (Böhringer et al. 1994). This sample together with equation (1) gives . The use of infrared instead of B magnitudes does not bring an advantage (Schröder 1995), nor does the application of the inverse TF relation. For the robustness of the result against variations of the input parameters the reader is referred to Federspiel et al. (1996).
The peak of the luminosity function (LF) of globular clusters (GC) has frequently been used as a standard candle. A modern calibration of the GCs in the Galaxy and in M31 combined with a compilation of published GCLFs in five Virgo ellipticals has led to a Virgo modulus of (Sandage & Tammann 1995). Meanwhile Whitmore et al. (1995) found a very bright peak magnitude in V and I for NGC 4486, which is well determined with HST and which corresponds, with our precepts, to a modulus of (Sandage & Tammann 1996). However, the GCs in NGC 4486 have a bimodal color distribution which is suggestive of age differences and possible merger effects (Fritze-von Alvensleben 1995, Elson & Santiago 1996). Turning a blind eye to this problem and averaging over all available GCLFs in Virgo we obtain . We are aware that the method may still face considerable problems.
The method, normally applied to ellipticals, was extended to the bulges of S0 and spiral galaxies by Dressler (1987). Using the bulges of the Galaxy, M31, and M81 as local calibrators, one obtains (Tammann 1988).
Novae are potentially powerful distance indicators through their luminosity-decline rate relation. Using the Galactic calibration of Cohen (1985), Capaccioli et al. (1989) have found the apparent distance modulus of M31 to be (i.e., somewhat less than indicated by Cepheids). From six novae in three Virgo ellipticals Pritchet & van den Bergh (1987) concluded that the cluster is more distant by mag than the apparent modulus of M31, implying (zero absorption is adopted for the Virgo cluster, see Section 2.6). The result carries still small weight, but is interesting because it is based on novae exclusively. HST observations, although time-consuming, of novae in the Virgo cluster could much improve this independent result.
Theoretical models of SNe Ia by various authors converge towards for ``Branch normal'' objects (Branch 1996, Höflich & Khokhlov 1996, Ruiz-Lapuente 1996). It is true that fainter, nearby SNe Ia are known, but being red and spectroscopically peculiar they can easily be singled out, and they do not contaminate distant, luminosity-selected samples of SNe Ia (cf. Section 3). Eight SNe Ia which have occurred in the Virgo cluster have mag. This value combined with the theoretical calibration gives .---Had we used instead the empirical calibration of Table 5 below, the Virgo modulus would have become larger by 0.08 mag. We refrain from using this value because the routes towards in Sections 2 and 3 are to be kept strictly apart.
The assumption that the LF of the shells of planetary nebulae (PN) in the light of the 5007Å line had a universal cutoff at mag has led to a Virgo modulus significantly lower than obtained from the six methods discussed above (Jacoby et al. 1990). Yet it was pointed out that the cutoff magnitude depends on the sample size (i.e., the absolute magnitude of the parent galaxy; Bottinelli et al. 1991, Tammann 1993). Numerically simulated LFs of the shell luminosities confirm, indeed, the dependence on sample size and population age (Méndez et al. 1993). As a consequence, the published PN distances deviate systematically from the Cepheid distances. The deviations increase with distance from M81 (Jacoby et al. 1989), NGC 5253 (Jacoby & Ciardullo 1993), and the Leo group (Ciardullo et al. 1989) to reach at the Virgo cluster (Jacoby et al. 1990) 0.74 mag !---A new method to derive PN distances, allowing for sample size and other effects, has been proposed by Soffner et al. (1995); the first result for the nearby galaxy NGC 300 is encouraging.
Surface brightness fluctuations (SBF) have also been proposed as distance indicators (Tonry & Schneider 1988). The first ``test'' has remained rather unconvincing, spreading the elliptical Virgo cluster members over an interval of 12 to 24 Mpc (Tonry et al. 1990); this interval was interpreted as real although early-type galaxies are known to be concentrated in the cores of galaxy clusters. Moreover, the individual distances correlate with the Mg index (Lorenz et al. 1993). Finally, we note that the SBF distances of NGC 5253 (Phillips et al. 1992), the Leo group (Tonry 1991), and the Virgo cluster (Tonry 1991) are smaller than Cepheid distances by as much as 0.97, 0.48, and 0.56 mag, respectively.
For the stated reasons we use neither the PN nor the SBF distances.
To obtain the cosmic recession velocity of the Virgo cluster, the observed value must still be corrected for the deceleration of the Local Group. We adopt (cf. Tammann 1996) and find for Virgo
Yet we prefer a very similar value the rational of which, however, is quite different. Many authors step up the Virgo distance out to the Coma cluster using the relative distance modulus between Virgo and Coma, and find at the distance of Coma. One can repeat that with any cluster whose distance relative to Virgo is known. In fact there are at least 14 clusters with rather good relative distances and velocities ( is the velocity in the frame of the microwave background). The best cosmic value of is then the all-sky mean over 14 different determinations. But more elegant is the reverse method: the relative distances are used to scale down the velocities of the 14 clusters and to predict a mean Virgo cluster velocity, i.e., the velocity which the cluster would have in the absence of all local deviations from an ideal expansion field. The result of this procedure is (Sandage & Tammann 1990, Jerjen & Tammann 1993, Jerjen 1995 for a more rigid error determination).
The six independent distance determinations of the Virgo cluster in Sections 2.1--2.3 are repeated in Table 4.
All distance moduli are taken to be true values, i.e., zero absorption is assumed towards the Virgo cluster. If the B-absorption implied by Burstein & Heiles (1984) is applied individually to all galaxies used for the distance determinations, the modulus becomes lower by only 0.06 mag. Even this almost negligible amount may be an overestimation as discussed by Sandage & Tammann (1996).
Table 4: Distance moduli of the Virgo cluster
If the adopted mean cluster distance of Mpc is combined with one obtains
An HST program has been mounted to determine the large-scale value of . The aim is to derive Cepheid distances (in V and I to control absorption effects) of up to ten galaxies which have produced well observed SNe Ia. So far we have calibrated the peak luminosity of six SNe Ia. A seventh object has become available through Tanvir's et al. (1995) Cepheid distance of the Leo group. (It can be assumed that the member galaxies of this compact group lie practically at the same distance.) The resulting absolute magnitudes of the seven SNe Ia are shown in Table 5. Detailed discussions of the input parameters are given elsewhere (Sandage et al. 1996, Tammann et al. 1996; negligible differences between these sources are due to a different weighting of individual sources). The agreement to within the errors between the individual luminosities supports the claim that SNe Ia are (nearly) perfect standard candles. Independent confirmation of the luminosities comes from Höflich et al. (1996) who have three SNe Ia in common with Table 5. Their model luminosities are the same to within mag. Branch's (1996) model luminosity of SN 1981B agrees fortuitously well with ours, and two SNe Ia of Ruiz-Lapuente (1996) are fainter by only mag judging from their late spectra and the inferred Ni mass.
Figure: Hubble diagrams in B and V of all non-red SNe Ia with known maximum magnitudes. Open circles and crosses are from the older archive literature. Filled circles are the modern data provided by Phillips (1993) and Hamuy et al. (1995). The very small K-corrections are applied.
Table 5: Absolute Magnitudes of SNe Ia at Maximum.
The Hubble diagram of all SNe Ia beyond 1100 kms with reasonably well determined maximum magnitudes is shown in Fig. 1 (Tammann & Sandage 1995). Their intrinsic luminosity scatter must be considerably less than 0.35 mag, because much of the scatter is expected from observational errors and peculiar motions. Indeed, the intrinsic scatter must be very small because even the most distant SNe Ia lie very close to the theoretical Hubble line of slope 0.2. The argument goes as follows. The most distant SNe Ia occupy a volume about 18000 times larger than that of the local calibrators. The large volume must contain exceptionally luminous SNe Ia---if they existed---and they have a much enhanced discovery chance for two reasons: their apparent magnitude is brighter than average and they stay longer above the detection limit. But still, there are no objects significantly above the Hubble line, not even at large distances. This means: The sample of SNe Ia shown in Fig. 1 constitutes a homogeneous class of very luminous and unabsorbed objects.
When in the following the calibration of Table 3 is applied to the SNe Ia in Fig. 1, it should be kept in mind that Branch normal SNe Ia (cf. Branch et al. 1993) are compared with the most luminous SNe Ia known. Therefore the resulting value of can only be, if anything, an upper limit.
Forcing a slope of 5 (corresponding to linear expansion) to the data in Fig. 1 gives
An easy calculation shows that the constant term in equations (3) and (4) is determined by
Inserting and from Table 5 leads directly to and , from which we adopt
Equations (3) and (4) are defined out to kms. The value of therefore represents the truly cosmic expansion rate.
At a time when only the very first calibrating SNe Ia were known, it was suggested that SN 1972E was overluminous on the basis of its light curve shape (Hamuy et al. 1995, Riess et al. 1995) and that consequently the true value of was larger. In the light of seven calibrators this argument is now impossible. From first principles of stellar statistics, it is known that seven nearby objects can on average not be more luminous than a distant, luminosity-segregated sample.
Figure: The absolute magnitude of all SNe Ia in or beyond the Virgo cluster with known versus velocity distance. Also shown are all faint red SNe Ia; they illustrate our point that no underluminous (or absorbed) SNe Ia are found at large distances. The distant objects must therefore be among the very brightest ones. of the calibrators (squares) is based on their Cepheid distances. For all other SNe Ia is calculated from the recession velocities and (upper panel) and (lower panel). Note that for the impossible case arises that the distant SNe Ia are on average less luminous than the nearby calibrators.
This point is illustrated in Fig. 2, where the absolute magnitudes of the seven calibrators are compared to the absolute magnitudes of the distant SN sample. The latter are calculated once with and once with . For the absurd situation arises that the distant SN Ia are systematically fainter than the nearby calibrators. The firm conclusion from this is that .
A more detailed discussion of all external errors is given elsewhere (Tammann et al. 1996). It yields a confidence range of .
The two independent routes towards the large-scale value of , via the Virgo cluster and SNe Ia, give 54 and 56 (internal errors), respectively. Their only interdependence is that they rely on Cepheids (predominantly observed with HST), which are the least controversial distance indicators at present. Together they make a strong case for (external error). Values of are equally unlikely as values of .
The relatively low value of is supported by additional methods, e.g., TF and other distances of field galaxies (Sandage 1994, 1996 and references therein), and the Zeldovich-Sunyaev effect (Lasenby & Hancock 1995, Rephaeli 1995). Baum's et al. (1995) HST photometry of globular clusters in the Coma cluster requires . A gravitationally lensed quasar sets (Dahle et al. 1994). Models of SNe Ia could not be understood if was 60 (Branch et al. 1996) or in no case 70 (Höflich & Khokhlov 1996, Ruiz-Lapuente 1996).
We believe that literature values significantly larger than are explained by an unwarrantedly high Virgo velocity, the unrealistic hope to fathom the depth of the Virgo cluster with only a single galaxy, the myth of a sharp, dispersionless cutoff of the luminosity function of planetary nebula shells, the reliance on the suspicious surface brightness fluctuation method, and/or simply by Malmquist bias which always artificially increases the value of .
The Basel group acknowledges the support of the Swiss National Science Foundation. A.S. and A.S. thank NASA for its support for the overall project. N.P. and F.D.M. acknowledge the support of ESA as part of its ongoing commitment to the science done with HST. We are all grateful to the people behind the scenes of HST, without whom none of the observations used here would have been obtained.
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