A&A 375, 770-780 (2001)
P. Fouqué 1,2 - J. M. Solanes 3 - T. Sanchis 4 - C. Balkowski 5
1 - Observatoire de Paris-Meudon DESPA, 92195 Meudon Cedex, France
2 - European Southern Observatory, Casilla 19001, Santiago 19, Chile
3 - Departament d'Enginyeria Informàtica i Matemàtiques, Escola Tècnica Superior d'Enginyeria, Universitat Rovira i Virgili, Carretera de
Salou, s/n; 43006 Tarragona, Spain
4 - Departament d'Astronomia i Meteorologia, Facultat de Físiques, Universitat de Barcelona C/Martí i Franqués 1, 08028 Barcelona, Spain
5 - Observatoire de Paris-Meudon DAEC, 92195 Meudon Cedex, France
Received 3 April 2001 / Accepted 31 May 2001
We have applied a relativistic Tolman-Bondi model of the Virgo cluster to a sample of 183 galaxies with measured distances within a radius of 8 degrees from M 87. We find that the sample is significantly contaminated by background galaxies which lead to too large a cluster mean distance if not excluded. The Tolman-Bondi model predictions, together with the HI deficiency of spiral galaxies, allows one to identify these background galaxies. One such galaxy is clearly identified among the 6 calibrating galaxies with Cepheid distances. As the Tolman-Bondi model predicts the expected distance ratio to the Virgo distance, this galaxy can still be used to estimate the Virgo distance, and the average value over the 6 galaxies is Mpc. Well-known background groups of galaxies are clearly recovered, together with filaments of galaxies which link these groups to the main cluster, and are falling into it. No foreground galaxy is clearly detected in our sample. Applying the B-band Tully-Fisher method to a sample of 51 true members of the Virgo cluster according to our classification gives a cluster distance of Mpc, larger than the mean Cepheid distance. Finally, the same model is used to estimate the Virgo cluster mass, which is within 8 degrees from the cluster center (2.2 Mpc radius), and amounts to 1.7 virial mass.
Key words: galaxies: clusters: individual: Virgo - galaxies: distances and redshifts - galaxies: elliptical and lenticular - galaxies: fundamental parameters - galaxies: spiral - cosmology: distance scale
In 1990, Fouqué et al. (1990) derived an unbiased distance to the Virgo cluster, based on a complete sample of 178 spiral galaxies, using the B-band Tully-Fisher relation (Tully & Fisher 1977). A previous similar study based on 110 spiral galaxies (Sa-Sm) was published by Kraan-Korteweg et al. (1988). Soon after, Teerikorpi et al. (1992, hereafter T92) suggested that this distance determination may have been contaminated by the inclusion of background galaxies into the complete sample, although we a priori excluded galaxies generally attributed to the background M group and W cloud.
The study of the structure of the Virgo cluster starts with de Vaucouleurs (1961), who identifies the southern extension (Virgo cloud X), and the wing (Virgo cloud W). This study is extended in de Vaucouleurs & de Vaucouleurs (1973) who separate the Virgo I cluster (Virgo E, S and S') from the Virgo II cloud complex (composed of Virgo V, X and Y) and the background W cloud (composed of Virgo Wa, Wb and W' sub-groups). Paturel (1979) applies taxonomy to disentangle these various components. Then, Tully (1982) discusses the separation between the Virgo cluster and its southern extension in the frame of the Local Supercluster. Later, Ftaclas et al. (1984) identify the M group and the N group. Pierce & Tully (1988) use the Tully & Shaya infall model (Tully & Shaya 1984) to study the velocity-distance diagram of bright Virgo spiral galaxies. Binggeli et al. (1993) clarify the situation by listing galaxies from the VCC catalogue (1985, hereafter VCC) belonging to each of these clouds (W, W', M and southern extension). T92 list several regions in the velocity-distance diagram: A, B, C1, C2, and D. Yasuda et al. (1997) also study the 3-D structure of the cluster. Finally, Gavazzi et al. (1999, hereafter G99) use distance determinations to identify new groups, named B, E, N, and S.
Please note that a clarification of the nomenclature is highly desirable, as the same letters refer to totally different groups: A can mean the main Virgo cluster or refer to galaxies with high velocities in front of Virgo and falling into the cluster (T92); B can mean the concentration around M 49, or a background group in the same region (G99), or even a foreground expanding component (T92); E can refer to the elliptical component of the Virgo cluster or an eastern group (G99); N can be a group identified by Ftaclas et al. (1984) or a northern group in G99; S refers to the spiral component of the Virgo cluster or a southern group in G99; X can mean that the galaxy lies within the X-ray contours (Federspiel et al. 1998, hereafter F98) or belongs to the Virgo X cloud (de Vaucouleurs 1961)!
To investigate the structure of the Virgo cluster and determine the mean distance of the main cluster, we have used the Tolman-Bondi model of the cluster defined in Ekholm et al. (1999, hereafter E99) in the spirit of T92, for spiral galaxies whose distances were given by the Tully-Fisher relation in B-band by Ekholm et al. (2000, hereafter E00) or in H-band by G99, but adding the information given by the HI deficiency of Virgo spiral galaxies. We also used the same model for early-type galaxies whose distances were determined by the fundamental plane method in G99, or the recent Tonry's compilation of Surface Brightness Fluctuations distances (Tonry et al. 2001, hereafter T01).
Section 2 defines the observable parameters we will use for this study. Section 3 introduces the Tolman-Bondi model and lists its adopted parameters. Section 4 investigates how the discrepant Cepheid distance to NGC 4639 is well explained by our model, and computes the average Cepheid distance to Virgo. Section 5 applies our model to the determination of the Virgo cluster mass, and compares it to other determinations. Section 6 investigates the structure of the Virgo cluster and lists galaxies which do not belong to the main cluster according to our model. Section 7 compares the average distances to Virgo adopted in the various references we used to what our model predicts. Finally, Sect. 8 summarizes the main results of this paper.
The selected sample has been extracted from the LEDA database. It consists of 584 galaxies within a radius of 8 degrees (2.25 Mpc at 16 Mpc) from M 87 ( , ) and known recession velocities smaller than 3000 kms-1. Unfortunately, not all these galaxies have distance estimates, and we therefore reduced the working sample to 125 late-type and 67 early-type galaxies, with distance measurements from G99, E00, F98 and T01. Let us describe in more detail some of the observable parameters collected for these galaxies.
These parameters are directly extracted from the LEDA database. Coordinates
are given in the J2000.0 equinox, and helocientric velocities are corrected to
the Local Group centroid using the Yahil et al. (1977) formula. A histogram of the corrected velocities for the full sample of 584 galaxies is
presented in Fig. 1.
|Figure 1: Histogram of the recession velocities, in the Local Group reference frame, for the 584 galaxies within 8 degrees from M 87. The dotted histogram corresponds to galaxies with distance measurements.|
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An accurate estimate of morphological types is important at least for two reasons: the HI deficiency parameter (see below) is calculated by comparing the HI content of a galaxy to the average HI content of an isolated galaxy of the same morphological type. Any error in the type therefore translates into a corresponding error in the HI deficiency estimate. Similarly, distances determined by E00 use the calibration of Theureau et al. (1997), where the intercept of the Tully-Fisher relation varies with the type. Any error in the type changes the galaxy distance value accordingly.
We used three sources of morphological type determinations: de Vaucouleurs et al. (1991, hereafter RC3), the VCC catalogue and van den Bergh et al. (1990). The quality of RC3 types depends upon its source, coded according to Table 3 of vol. 1 of the catalogue. If the morphological type has been measured on large reflector plates (code R), it is adopted; if it comes from PSS prints (codes P or U), we adopt an eye average with the two other sources. The adopted morphological types differ therefore from those given in LEDA and used by E00.
We adopted a numeric morphological type according to the RC3 coding scheme. For distance measurements, we decided to restrict application of the fundamental plane and SBF methods to types between -5 and -1 (E to L+) and of the Tully-Fisher method to types between 1 and 9 (Sa to Sm). Therefore, 7 galaxies were excluded from the late-type sample (types -1, 0 and 10), and 2 galaxies were excluded from the early-type sample (NGC 4440 and NGC 4531, both of type 1). The final sample thus contains 118 spiral, 43 lenticular and 22 elliptical galaxies.
For early-type galaxies, we have used the G99 compilation of distances,
obtained from the fundamental plane method in H-band. According to these
authors, their accuracy should be about 21%. These distances are based on an
assumed Virgo distance of 16 Mpc. To this compilation of 55 elliptical and
lenticular galaxies, we added SBF distances from T01. It contains 35
early-type galaxies in the Virgo cluster area, of which 25 are common with
G99. Tonry's distances are calibrated independently of any assumed Virgo
distance, and the mean shift compared to G99 amounts to:
For late-type galaxies, we also started from the G99 compilation (59 galaxies retained), obtained from the H-band Tully-Fisher method, with a claimed accuracy of about 16%. We complemented this list with galaxies with B-band Tully-Fisher distances from E00 (41) and F98 (109). We do not use galaxies from these lists outside of 8 degrees from M 87 or with morphological types outside of our adopted range (1-9).
To convert E00 distances into the G99 system, two corrections are done:
first, we correct the distances to our adopted morphological type, in the 20 cases where it differs from the LEDA type adopted in E00. Indeed, the
intercept b of the B-band Tully-Fisher relation derived by Theureau et al.
(1997) and adopted in E00 depends upon the morphological type. The
correction is given by:
Additionally, E00 distances are based on the Theureau et al.
(1997) calibration, which leads to
as E00 adopt a cosmic velocity
of Virgo of
this implies a Virgo distance of 21.8 Mpc.
Therefore, the conversion factor to the G99 assumed Virgo distance of 16 Mpc
All these shifts are significant. This means that our first guess of the
conversion factors to the G99 system was not as successful as we could hope.
We therefore repeated it with new conversion factors, until we reach on
average negligible shifts of the corrected systems to the mean one. This leads
to the following adopted conversion factors, after rejection of two galaxies
with discrepant distance measurements (NGC 4180 and NGC 4591):
|Figure 2: Histogram of the adopted distances for the 118 spiral galaxies (left) and the 65 early-type galaxies (right).|
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As we are concerned with the distance determination of individual
galaxies in the Virgo cluster, we quantify HI deficiency by means of a
distance-independent parameter DEF, based on the difference between the
expected and observed logarithm of the mean (hybrid) HI surface density,
Among our sample of 118 spiral galaxies, we have HI deficiency estimates for 106 galaxies, or 90% of the sample.
It has been suggested that Tully-Fisher distances may be underestimated for highly HI deficient galaxies (T92; Fukugita et al. 1993). For this reason, we have investigated possible differences in the rotation velocities of HI deficient and HI normal galaxies in the 12 HI deficient clusters identified by Solanes et al. (2001), as well as the influence of the HI content on the Tully-Fisher relationship of the galaxies in our sample. The results obtained allow us to conclude that HI deficiency does not affect our Tully-Fisher distances.
We use the Tolman-Bondi model of the Virgo cluster as defined in E99. Let us recall that the Tolman-Bondi model gives an analytical solution to Einstein's field equations for a spherically symmetric pressure-free density excess, embedded in an otherwise homogeneous universe. The parameters of the model are: the observed Virgo cluster velocity in the Local Group reference frame ( ), the Virgocentric infall velocity of the Local Group ( ), the Virgocentric density profile slope ( ), and the deceleration parameter of the background homogeneous universe, taken as an Einstein - de Sitter universe for simplicity ( q0 = 0.5). Although there is a general agreement about the values of 3 of these parameters, the Virgocentric density profile slope is not well known, as it should reflect the distribution of the mass around the cluster center, not only the galaxy (light) distribution. In E99, it has been constrained using 32 galaxies whose distances were known using their Cepheids, generally measured with the HST, through the PL-relation. For details about the model, the adjustment of its parameters and the influence of changing these values, the reader is referred to T92, E99 and E00.
For each galaxy, the angular distance to the Virgo center of mass (taken to
be the position of M 87) completely defines the exact shape of the
Tolman-Bondi S-curve in the velocity distance diagram, for a given choice of
model parameters. Then, the observed recession velocity of the galaxy (in the
Local Group reference frame) gives one to three possible distance ratios of the
galaxy to Virgo (see Fig. 3 for examples).
|Figure 3: Tolman-Bondi model for the six galaxies with Cepheid distances. The dotted line corresponds to the recession velocity of the galaxy, while the arrows mark the expected distance ratio if (solid arrow) or (dotted arrow).|
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To do this, we must select the most plausible one among the possible distance ratios for a given galaxy, when the model gives more than one value. Here, we made use of two criteria: we try to choose the value closest to the assumed Virgo distance (16 Mpc), and, for spiral galaxies, we take into account the HI deficiency, assuming that a true member of the cluster has a higher probability to be deficient than a galaxy falling for the first time into the cluster. We therefore attribute to each galaxy a "class number'', which is 1 if we adopt the first value of the distance ratio to Virgo (galaxies falling from in front into the cluster), 2 if we adopt the second value (true cluster members), and 3 if we adopt the third value (galaxies falling from behind into the cluster).
There are 6 galaxies in or close to the Virgo cluster whose distance is known thanks to the HST observations in V and I bands, using the Cepheid period-luminosity relation. We use the distances published in Freedman et al. (2001) (without the uncertain metallicity correction). Five of the six distances cluster about a mean distance of 14.6 Mpc, while the sixth one (NGC 4639) gives a larger distance (21 Mpc). This is perfectly explained by the Tolman-Bondi model, and this was in fact the initial motivation of this study. Figure 3 gives the result of the application of the Tolman-Bondi model to these galaxies, with a dotted line showing the recession velocity of the galaxy, and an arrow showing the distance ratio to Virgo, for an adopted distance to Virgo of 15.4 Mpc (see below). Two galaxies belong to the Virgo southern extension (NGC 4496A and NGC 4536) and lie at more than 8 degrees from M 87. However, their position in the velocity-distance diagram should be explained by our model, if we assume that no other cluster perturbes them. However, we find that the maximum velocity explained by our model for NGC 4536 is at , too small compared to its observed velocity of . For this galaxy, our model predicts it to be in the class 3, with a distance of . This in turn would lead to an inacceptable Virgo distance of , using the observed galaxy distance of 14.45 Mpc. In fact, it is well known that random velocities about 80 exist for all galaxies and explain the velocity dispersion of small groups of galaxies (Gourgoulhon et al. 1992). We therefore assume that this is the origin of the small discrepancy observed for NGC 4536.
Table 1 gives the adopted distances to the six galaxies,
|NGC 4496A||14.52||0.75||0.92||1.76||2 (1)||15.78||-0.09|
|NGC 4535||14.79||0.77||0.96||1.91||2 (1)||15.41||0.19|
|NGC 4536||14.45||(0.83)||(0.83)||1.79||1 or 2||(17.41)||0.25|
At the referee's request, we have investigated what happens if a Virgo distance of 21.5 Mpc is preferred from independent arguments. Then, four of the five galaxies classified in class 2 now fall into class 1 (infalling galaxies), NGC 4639 is the only true member of the Virgo cluster among the six, and NGC 4548 position cannot be explained by the model. The high HI deficiency of NGC 4321 is contradictory. This alternative distance is clearly less probable according to these six galaxies with Cepheid distances. A definitive answer will only become available with a larger sample of Virgo galaxies with accurate distances.
Derivation of the Virgo cluster mass from the Tolman-Bondi model, and its
comparison with the virial mass estimate, have been discussed in T92, E99 and
E00. Here, we give a summary of the useful formulae and derive the Virgo mass
for our adopted model parameters. Let
be the Virgo distance in
the cosmic recession velocity of Virgo in
and d the radius normalized to the Virgo distance. The mass
enclosed within d is the product of the "Einstein - de Sitter mass'' within
the same radius by the mass excess due to the cluster. It is given in solar
mass units by:
mass of Virgo as given by Tully & Shaya (1984) is:
By comparison, Böhringer et al. (1994) estimate the mass of the M 87 sub-cluster from X-ray emission measured by ROSAT to (1.5- within a radius of 1.8 Mpc at 20 Mpc (5 degrees). Contributions from the M 49 and M 86 sub-clusters are negligible, at about (1-3 . These values are confirmed by Schindler et al. (1999) who derive within 1.5 Mpc around M 87 (at 20 Mpc) and within 0.75 Mpc around M 49. Our Tolman-Bondi mass for the same distance would be , almost one order of magnitude larger. We do not have any explanation to offer for this discrepancy, but we note that our large mass estimate and steep density profile are supported by Tully & Shaya (1998), who find a Virgo mass of from a modeling of the velocity field of the Local Supercluster, assuming a mass-to-light ratio of in the field, but 1000 for the Virgo cluster, and .
Disentangling the different components of the Virgo cluster is difficult and
somewhat subjective. However, the use of the Tolman-Bondi model allows us to
classify each galaxy into one of the three classes defined previously. We
did that independently for early-type and spiral galaxies. In each case, we
have built several diagrams: the first one is the histogram of the "Virgo
derived distances'', and it is shown in Fig. 4.
|Figure 4: Histogram of the "Virgo derived distances'' for spiral galaxies (left) and early-type galaxies (right).|
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The second diagram gives the velocity repartition among the different
classes, and is displayed in Fig. 5.
|Figure 5: Distribution of the recession velocities among the different classes for spiral galaxies (left) and early-type galaxies (right).|
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A third diagram gives the repartition of HI deficiency among the classes,
obviously only for spiral galaxies. It is displayed in Fig. 6.
|Figure 6: Distribution of HI deficiency among the different classes for spiral galaxies.|
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Finally, Fig. 7 displays the repartition on the sky of spirals
and early-type galaxies with different symbols for different classes.
|Figure 7: Distribution on the sky of the spiral galaxies (left) and early-type galaxies (right). Filled squares correspond to class 1, crosses to class 2, open squares to class 3 and open triangles to class 4.|
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In Table 2,
|NGC 4168||2.13||M group||NGC 4168, NGC 4189, NGC 4193, NGC 4200, IC 769, IC 3061, IC 3099|
|2.07||M halo||NGC 4067, NGC 4152, IC 3074|
|NGC 4222||1.24||M infall||NGC 4222, IC 3033, IC 3066, IC 3105, UGC 7249|
|NGC 4261||2.11||W group||NGC 4180, NGC 4197, NGC 4215, NGC 4233, NGC 4235, NGC 4259, NGC 4260,|
|NGC 4261, NGC 4273, NGC 4281, IC 3225, CGCG 42-1, CGCG 42-36|
|2.17||W halo||IC 776, UGC 7579|
|NGC 4343||1.46||W infall||NGC 4252, NGC 4316, NGC 4318, NGC 4343, NGC 4353, NGC 4376, NGC 4390,|
|NGC 4411A, NGC 4434, NGC 4451, NGC 4466, IC 3115, IC 3322A, UGC 7423|
|NGC 4639||1.41||NGC 4620, NGC 4633, NGC 4639, NGC 4651, IC 3742|
|NGC 4746||1.92||NGC 4746, UGC 8085, UGC 8114|
After excluding all the galaxies which do not belong to the main cluster according to our classification, we are left with 71 spiral and 52 early-type galaxies. 67% of the original sample with distances are therefore considered as true members of the cluster; not surprisingly, the spiral sample exhibits a larger contamination (40%) than the early-type sample (20%). Among our 5 references, the proportion of true members varies from 56% (E00) and 61% (F98) to 83% (T01).
Now that we have classified all the galaxies of our sample into the different classes, we can return to the different distances adopted by the various authors to the Virgo cluster.
We will first derive an average ratio of distances from a given reference to
the predicted distance given by the Tolman-Bondi model and an adopted Virgo
distance of 15.4 Mpc. For the
galaxy measured by reference j, we
Table 4 gives the results. We can see that for G99 and T01, the agreement between the resulting mean Virgo distances from both methods (Cols. 3 and 4, respectively) and the value adopted by these authors (Col. 5) is satisfying. On the contrary, for F98 the resulting mean Virgo distances are consistent from both methods, but smaller than their adopted value: this is certainly due to inclusion of background galaxies into their sample; indeed, among the 49 galaxies of their "fiducial sample'', 14 are classified by us as non-members of the Virgo cluster. For E00 the result is inverse and the mean error is large: it seems therefore that there is a discrepancy between their adopted Hubble constant (from Theureau et al. 1997) and the Virgo distance we derive from their data.
For these two references, we adopt as the mean Virgo distance the average over the two determinations (Cols. 3 and 4). This gives 18.9 and 23.2 Mpc for F98 and E00, respectively. If we apply to these distances the conversion factors adopted in Sect. 2.3.2 to reduce F98 and E00 to a mean Virgo distance of 16 Mpc, namely 0.832 and 0.678 respectively, we now get values close to 16 Mpc (15.7 and 15.8 Mpc, respectively).
However, it is important to keep in mind that the clean samples of true
members may still be affected by the incompleteness bias (Fouqué et al.
1990 for the specific case of the Virgo cluster; Teerikorpi
1997 for a general discussion), which may lead to average cluster
distances which are too short. To estimate the amount of this bias, we
have calibrated once more the B-band Tully-Fisher relation using the 21 calibrators from Freedman et al. (2001), and the 51 true members of
Virgo according to our classification, which best match the calibrator
properties (thus excluding peculiar, interacting, HI-truncated galaxies, and
restricting to morphological types between 2 and 8, inclinations between 37 and
larger than 1.7),
|Reference (Galaxy type)||Ratio||()|
|G99 (E)||(36, 2.94)||16.0|
|T01 (E)||(29, 1.99)|
|G99 (S)||(41, 3.10)||16.0|
|F98 (S)||(67, 4.27)|
|E00 (S)||(23, 5.51)||21.8|
The 0.3 mag difference between the Cepheid and Tully-Fisher distances is worrying. Tenants of the long distance scale will see evidence that the Tully-Fisher distance is the correct one, while most galaxies with Cepheid distances are infalling on the front side of the cluster.
In this paper, we have investigated how the relativistic Tolman-Bondi model as applied in E99 gives constraints on the Virgo cluster mass and distance, and allows one to disentangle its quite intricate structure. Distances to 183 Virgo galaxies from 5 references have been used and averaged, together with HI deficiency parameters for spirals, to classify the galaxies into 4 different distance classes: an infalling component in front of Virgo, the Virgo cluster itself, an infalling component behind the main cluster, and background groups. The main results of the present study are:
We have made use of the LEDA database (http://leda.univ-lyon1.fr), supplied by the LEDA team at the CRAL - Observatoire de Lyon (France). We warmly thank all the LEDA team members for their effort. We also thank Riccardo Giovanelli and Martha Haynes for making their Arecibo General Catalog, from which we have extracted the HI data used in this study, available to us. Finally, we wish to thank the referee, Pekka Teerikorpi, for his very constructive comments. T. S. acknowledges support from a fellowship of the Ministerio de Educación, Cultura y Deporte of Spain. C. B. acknowledges ESO for a visiting position in Santiago during which this work was started.