Total Magnitudes of Virgo Galaxies. II.
An Investigation into the mp Scale of Volume I of Zwicky et al.'s

Catalog of Galaxies and Clusters of Galaxies

Christopher Ke-shih Young and Zheng-yi Shao, PASA, 18 (2), in press.
Next Section: Conclusions
Title/Abstract Page: Total Magnitudes of Virgo
Previous Section: Estimating Bt from mp
Contents Page: Volume 18, Number 2

Estimating mp from Bt

Should one wish to estimate mp values for would-be CGCG objects not actually listed in the catalogue, it would first be necessary to ascertain whether the mp scale suffers from a genuine faint-end scale error. This is not a trivial problem because, as is evident from Fig. 3, the hard faint-end limit of the CGCG makes it difficult to deduce how much of the skewness in the faint-end distribution of data points (with respect to the equality line) is due to noise and how much is due to scale errors. If one now considers the case shown in Fig. 5 in which mp is the dependent variable plotted as a function of Bt, we would need information about ``missing'' points above the dashed-dotted line representing the hard faint-end limit of the mp scale, in order to be able to pursue our investigation to the right of the dotted line, representing the $B_t \sim 14.7$ mag. measured completeness limit of the CGCG.

Figure 5: mp as a function of Bt for the 92 Sample MI and MII galaxies, which are plotted as `+' symbols if cluster members or ` x ' symbols if background objects. The `$\star $' symbols represent the measured (when Bt < 14.5) or estimated (when $B_t \geq 14.5$) mean mp values for bins of 0.5 mag. width (in Bt space) as tabulated in Table 2. Faintward of Bt = 14.5, they take into account ``missing'' points (for which Bt but not mp are known) that must lie above the dashed-dotted line, which represents the CGCG's

$m_{p} \leq 15.6$ cut-off, and to the right of the dotted line, which represents the measured completeness limit of Volume I in Bt space ($B_t \sim 14.7$). The solid curve represents a weighted cubic fit to these data points and is only intended as an approximate Bt-to-mp transformation equation. The dashed straight line on the other hand represents Gaztanaga & Dalton's transformation based on a sample of mainly Volume V objects.

\begin{figure} \begin{center} \psfig{file=p2f5.ps,height=7.5cm,angle=-90} \end{center} \end{figure}

One approach to the problem is that adopted by Gaztanaga & Dalton (2000), in which one assumes that the data points define a Gaussian distribution about a straight line defining the magnitude scale at the faint end on a graph analogous to Fig. 3. The main limitation with this method is that it is difficult to know a priori whether a straight line is a good approximation or not to the magnitude scale. Also, one needs very large samples of CGCG objects and even then, it is difficult to probe much deeper than

$B_t \sim 15.75$, beyond which the locus of data points becomes too detached from the equality line for one to be able to get a good handle on the parameters of the Gaussian distribution.

Table 2: Measured (when Bt < 14.5) or estimated (when $B_t \geq 14.5$) mean mp and associated scatter,

$\sigma _{m_{p}}$, as a function of Bt based on a maximum-likelihood analysis of the 88 Sample MI objects for which Bt < 16.5
Bt range sample CGCG

$\sigma _{m_{p}}$

 mean mp
(mag.) size complete- (mag.) (mag.)
  (objects) ness    
Bt < 10.0 5 1.000 0.55 ($\pm$0.17) 10.82 ($\pm$0.25)

$10.0 \leq B_t < 10.5$

 3 1.000 0.22 ($\pm$0.09) 11.20 ($\pm$0.12)

$10.5 \leq B_t < 11.0$

 0 1.000 N/A N/A

$11.0 \leq B_t < 11.5$

 3 1.000 0.33 ($\pm$0.13) 11.60 ($\pm$0.19)

$11.5 \leq B_t < 12.0$

 2 1.000 0.20 ($\pm$0.10) 12.00 ($\pm$0.14)

$12.0 \leq B_t < 12.5$

 8 1.000 0.35 ($\pm$0.09) 12.50 ($\pm$0.12)

$12.5 \leq B_t < 13.0$

 5 1.000 0.29 ($\pm$0.09) 13.04 ($\pm$0.13)

$13.0 \leq B_t < 13.5$

 11 1.000 0.32 ($\pm$0.09) 13.45 ($\pm$0.10)

$13.5 \leq B_t < 14.0$

 13 1.000 0.37 ($\pm$0.07) 14.08 ($\pm$0.10)

$14.0 \leq B_t < 14.5$

 3 1.000 0.21 ($\pm$0.08) 14.53 ($\pm$0.12)

$14.5 \leq B_t < 15.0$

 5 0.750 0.49 ($\pm$0.09) 15.32 ($\pm$0.09)

$15.0 \leq B_t < 15.5$

 9 0.714 0.28 ($\pm$0.03) 15.49 ($\pm$0.03)

$15.5 \leq B_t < 16.0$

 14 0.680 0.28 ($\pm$0.02) 15.52 ($\pm$0.02)

$16.0 \leq B_t < 16.5$

 7 0.244 0.29 ($\pm$0.06) 15.85 ($\pm$0.06)

Although our faint-galaxy sample is considerably smaller than that of Gaztanaga & Dalton, thus ruling out the use of their method here, we were able to probe as deep as Bt = 16.0 without the need for a straight-line approximation. This was possible because, within the VPC survey area at least, we knew precisely the CGCG's degree of completeness as a function of magnitude. We therefore adopted the following procedure. The 90 Sample MI objects were first sorted into bins of 0.5 mag. width (in Bt space). Although this sample is smaller than the VPC galaxy sample used to measure the completeness of the CGCG, we were still able to use the completeness ratios derived from the much larger Sample CI, as summarised in Table 1 and Table 2. Assuming that for each bin separately, the mp and missing mp values collectively observed a Gaussian distribution about a mean mp value with a sample standard deviation of

$\sigma _{m_{p}}$, we then had enough information to recover an estimate of the mean mp value and its corresponding

$\sigma _{m_{p}}$ by means of a maximum-likelihood analysis. Our results are tabulated in Table 2 and plotted on Fig. 5. The latter figure appears to confirm that there is indeed a serious scale error faintward of $B_{t} \sim 14.5$ if mp values are treated as total magnitudes. We have computed the best fitting weighted cubic curve in order to provide an approximate transformation equation for obtaining mp from Bt for objects missing from the CGCG over the range Bt < 16.5:

mp= 67.69 - 14.626 Bt + 1.2015 Bt2 - 0.030640 Bt3. (8)

Although the associated scatter is only 0.32 mag. for Bt < 14.7, polynomial fits such as this have their limitations-mainly at the faint end in this case. Therefore, when higher precision is required we recommend interpolation of the mean mp values listed in Table 2 (which also gives a detailed breakdown of the measured scatter,

$\sigma _{m_{p}}$, as a function of Bt). As is evident from Fig. 5, we find that faintward of $B_t \sim 14.7$, the Volume I magnitude scale deviates even more from a Pogson one than does the Volume V scale, as investigated by Gaztanaga & Dalton.

Next Section: Conclusions
Title/Abstract Page: Total Magnitudes of Virgo
Previous Section: Estimating Bt from mp
Contents Page: Volume 18, Number 2

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