Astron. Astrophys. 322, 730-746 (1997)

7. Discussion and conclusions

In this section we shall now discuss the legitimacy of our derived value: or according to diameter or magnitude relation respectively. We study the influence of the velocity field model used to define the kinematical distance scale, comment on the adopted galactic absorption, and discuss the choice of the calibrator sample. We finally compare our results to those of the supernova method reaching larger redshifts.

7.1. Influence of the velocity field model

We have used a linear velocity model (Peebles 1976) with Local Group infall velocity toward Virgo 150 km s^-1 and observed Virgo radial velocity 980 km s^-1. As noted in Sect. 5, the linear Peebles's model is a good approximation of the non-linear or Tolman-Bondi solution (e.g. Kraan-Korteweg, 1986) outside the triple-value region.

Particularly in the methods which use directly the Virgo cluster, changing the velocity model has a crucial effect on the derived . In Table 7 we list our values of obtained when different velocity parameters are used. The choise of and do not seem to affect significantly the results, all models are compatible with km s^-1.

Table 7. Values for with different LG Virgo infall and observed Virgo recession velocities.

Some understanding on the small sensitivity of is provided by numerical experiments. We tested how accurately the Peebles model gives kinematical distances and individual values when we assume an underlying exact Tolman-Bondi velocity field and use somewhat erroneous value for the Local Group infall velocity.

7.1.1. Sensitivity to errors in the model

We conducted a theoretical experiment asking how large an error emerges in the kinematical distances if the adopted value for is larger (smaller) than the "true" value. Results from this experiment are shown in Fig. 14.

Fig. 14. Upper panel: the relative errors of kinematical distances obtained with the Peebles's model, for objects with different angular distances from Virgo and different observed radial velocities . On the left we show the error contours when the true infall velocity is 50 km s-1 lower, and on the right 50 km s-1 higher than the used value 150 km s-1. Lower panel: now the error contours refer to the error in . Dashed lines represent the zero-error curves

A uniform grid was created. We assume the true distance in Virgo units and the angular distance with respect to Virgo to be known for each grid point. For each -pair the corresponding observed velocity is calculated from the Tolman-Bondi solution. For details of the model cf. Ekholm (1996). Two possibilities were considered: either km s^-1 or km s^-1. The common parameters used were , and km s^-1. Using and we then solve from the Peebles model the kinematical distance (Eq. (2) in BGPT86), presuming an infall velocity km s^-1. In order to get a regular, uniform grid for the solutions we interpolate for each the corresponding value of . The relative error is shown as contours of equal error in the upper panels of Fig. 14. The zero-error curves are plotted as dashed lines, positive errors as solid lines and negative errors as dotted lines. Each curve reflects an increase (decrease) in the error by one percent.

The lower zero-error line represents the tangential points where the infall velocity is perpendicular to the line-of-sight thus cancelling out. On the upper zero-error line are the points which with the LG and Virgo form an isosceles triangle. Now the line-of-sight components of the infall velocity of LG and each point cancel out as they are equal but of opposite signs.

We also note how the linearized model of Peebles collapses as we approach Virgo. Following tendencies are seen from Fig. 14 (upper panels). An overestimation of yields too small kinematical distances () and an underestimation too large distances (). An inaccuracy of km s^-1 yields errors of order of a few percent. Tests also show that decrease in accuracy rapidly increases the errors.

The value of is determined, however, using the corrected velocity . As the error in induces an error in of opposite sign, we suspect the error in to be smaller than in . The lower panels in Fig. 14 show that this indeed is the case. Now the error-curves are based on . In particular, in regions used for the plateau the errors are now of order of one percent or smaller. Though being rather restricted, this experiment helps one to understand why the value of is so insensitive to .

7.1.2. The effect of the Local Group infall velocity on the plateau dispersion

The results above showed the insensitivity of on the velocity model used. However, for the model to have some validity, its influence should be seen somewhere. From previous subsection we expect the dispersion of in the plateau to be larger when incorrect velocities for Local Group infall is used. In Fig. 15 the dispersion is plotted as a function of for the adopted Virgo observed velocity km s^-1. The value of is derived from the real plateau data, choosing the plateau limit strictly enough so that, regardless of the value of , all the data points still lie in the unbiased region.

Fig. 15. Plateau dispersion with different values of . Diameter relation has lower values of (triangles). We have also marked the minimum of the curve and the limits.

The best value of the LG infall is then given by the minimum of the observed -curve. From Fig. 15 we get km s^-1 and km s^-1 using diameter and magnitude TF relations respectively. The -limits have been calculated by multiplying the minimum dispersion by . Our standard value km s^-1 is within of these values. Agreement with the infall velocity prefered by Tammann et al. (1996) ( km s^-1, see Table 7) is even better.

7.2. Influence of the adopted galactic absorption

Changing the value of the galactic extinction affects the derived Hubble constant. Smaller values lead to larger derived distances, and one obtains finally a lower value for . It has been shown in Paturel et al. (1996), that the RC2 and the RC3 (Burstein-Heiles) systems for correcting galactic extinction differ essentially by a simple shift of roughly 0.18 mag reflecting the assumed Galactic Pole extinction (). In the present paper, is calculated using the RC2 system as it is available for the whole sky. Also, there is evidence from stellar reddening and interstellar polarization, that the Pole extinction may approach the RC2 value (Teerikorpi, 1990, Berdyugin, Snare, and Teerikorpi, 1995, Berdyugin and Teerikorpi, 1996). Using B-H corrections, the Hubble constant is 8% (with magnitude, 4% using the diameter TF relation) smaller than when the RC2 system is used, hence, becomes 49.2 with magnitudes, and 54.5 with diameters.

7.3. Influence of the absolute calibration

Special attention should be paid to the completeness of the calibrator sample. This sample, fixing the absolute TF zero-points and thus leading to a definite value of the Hubble constant, should, as for field galaxies, provide unbiased or , and be equivalent to a volume-limited sample. We studied this completeness in several ways, according to absolute diameter, absolute magnitude, and values. Also TF residuals in both diameter and magnitude TF relations were investigated. Fig. 16 shows these parameters against the "true" Cepheid distance scale. No evidence was found for any correlation of any of these parameters with distance. We computed also a normalized distance diagram for the calibrator sample (Fig. 17), and it appears that there is no deviation from a plateau region. All of this allows us to consider the calibrator sample close to volume-limited in its statistical properties. This is not so unexpected because the galaxies have been selected for Cepheid-detection and measurement primarily on the basis of their resolution into stars.

Fig. 16. Different parameters of the calibrator galaxies vs. their distance in Mpc. Primary (Cepheid) calibrators are plotted with squares, group calibrators with crosses. Absolute diameters and magnitudes are shown in two top panels, following them are the corresponding residuals with respect to values obtained from the TF relations. In the bottom panel the rotational velocities are plotted.

Fig. 17. "Plateau diagrams" (cf. Figs. 2, and 6) for calibrators. Primary and group calibrators are denoted similarly as in Fig. 16, except for Virgo galaxies, which are plotted with circles (as in Fig. 13).

However, group members corresponding to Table 2 seem to have a slightly different behaviour than objects of the pure Cepheid sample (Table 1). On average, their luminosity or size appears a little smaller than the mean value derived from the pure Cepheid sample, and tends to make a few per cent greater. This is indeed expected from the way we generally collect group members, a method suffering from a volume effect: if one selects the objects within a certain solid angle, one collects naturally more points from the back side of the group than from its front side. The resulting average distance over all the selected group members is then greater than the true distance of the group. That is just what happens when we assign to a putative group member the Cepheid distance of another group member. Consequently, the distances for these objects being slightly under-estimated (on average), the resulting absolute calibration provides a slightly too large value of . Using only the group selected calibrators we obtained and with diameters and magnitudes respectively. One can but hope, that in the future more HST time is allocated to field spirals. Direct Cepheid distances for these putative group members would be of great interest in order to complete the present study.

7.4. Concluding remarks

One of the main purposes of the KLUN-project has been to collect a large sample of several thousand galaxies which can be analyzed by the method of normalized distances. The resulting unbiased plateau set of 400 galaxies is now ten times larger than in 1986. At the same time it is one of the largest samples of field galaxies used to derive . It also reaches much greater redshifts, being well populated up to 2000 - 3000 km s^-1.

We report bellow some other novel features of the present analysis:

• for the first time, a type-dependent TF relation was used, which improves the accuracy of derived B-band TF distances
• both diameter and magnitude TF relations were used in parallel
• a new method was presented for estimating the Local Group infall from the dispersion of in the plateau
• simulations were used to show that the Hubble constant is not sensitive to the Virgo cluster velocity parameters
• the sensitivity of to different infall models was checked
• we calculated analytically what portion of the total sample is the useful unbiased subsample in a normalized distance diagram
• we checked and studied the statistical properties of the primary calibrator sample, and discussed its influence on the derived Hubble constant.

It could be argued that the velocity range of the KLUN sample again provides only a local value for the Hubble constant. However, results from farther reaching standard candles also calibrated using Cepheid galaxies, the type Ia supernovae, confirm this value for larger redshifts.

In Fig. 18 we have plotted our unbiased plateau galaxies together with SNIa from Sandage et al. (1996) in an vs. diagram. The agreement between the two samples is good: In the lower radial velocities both have a larger dispersion, partly due to the larger relative importance of the peculiar velocities. At larger the data points become fewer, but converge quite well to a mutual value of ().

Fig. 18. Plateau galaxies (dots) and type Ia supernovae (stars) in a vs. diagram. The line corresponds to average value of plateau galaxies. Diameter relation used on the top, magnitudes on the bottom.

The value of given by the SNe is of course dependent on the calibration of the method. The SNIa calibration is also based on Cepheid measurements, so we expect no systematical differences between SN and TF results from there. Our results are also in agreement with the conclusions of the "Triple-Entry-Correction" method presented for field galaxies in Sandage (1988b, 1994b), and for clusters in Sandage, Tammann, and Federspiel (1995).

Together with the recent Cepheid distances, Tully-Fisher and relations and the normalized distance method provide a firm unbiased value of the Hubble constant from field galaxies: km s^-1 Mpc^-1.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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