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Astron. Astrophys. 333, 795-802 (1998)

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5. Dependence of L -n and R -n distances

Binggeli & Jerjen make a big issue of the mutual dependence between the residuals in magnitude space with respect to their [FORMULA] versus n correlation and the residuals in angular-distance space with respect to their [FORMULA] versus n correlation. They plot these residuals in their Fig. 9. While they are correct in pointing out that there must be some dependence between the two sets of residuals, whether this dependence is significant enough to affect our previous findings is another matter.

We have re-plotted their Fig. 9 here as Fig. 6, this time using equal axis scales. They claim that in the absence of any dependence between the residuals, Fig. 6 should be devoid of any correlation. However, their test for dependence is fatally flawed because it is based on the prior assumption of negligible depth-as illustrated by the following example.

[FIGURE] Fig. 6. Binggeli & Jerjen cited the strength of this correlation they found using their own dataset as evidence that the analysis of YC95 was flawed. Their reasoning was that this correlation must be the product of dependence between distance estimates derived by different methods, rather than due to genuine depth in the spatial distribution of Virgo galaxies.

Imagine that we have five galaxies, which collectively constitute a complete sample of galaxies devoid of any Malmquist bias. The nearest galaxy is at [FORMULA] while the farthest is at [FORMULA], and the spatial separation between each object is [FORMULA]. The mean distance modulus of these five galaxies [in log(distance) space] is therefore [FORMULA]. Now, let us imagine that we have two perfect distance indicators based on two completely independent scaling laws which we shall denote SL1 and SL2. Both indicators can measure the distances of these objects precisely because both methods are perfect. If we were now to construct a diagram analogous to Fig. 6, we would end up with a plot like Fig. 7. The rms scatter in the residuals with respect to SL1 would be identical to that with respect to SL2, and both of these quantities would be equal to [FORMULA] mag. Now, according the Binggeli & Jerjen, for two such `independent but equivalent' measurements, we would expect the scatter with respect to the equality line on Fig. 7 to be 2 mag. However, because the distance indicators are perfect, the actual scatter with respect to the equality line is zero [regardless of which axis it is measured parallel to]. The reason for this is that while the two different measurements for an individual galaxy are `equivalent'; the measurements for different objects are not, simply because each object is at a different distance.


[FIGURE] Fig. 7. This plot is analogous to Fig. 6, but invokes hypothetical galaxy data ([FORMULA] symbols) and two hypothetical distance indicators that are based on mutually independent scaling laws, denoted SL1 and SL2. Both indicators are capable of yielding precise distance measurements. For each indicator, the lengths of the arrows represent the value of the rms scatter in the distance residuals with respect to the mean distance obtained for the galaxy sample. According to Binggeli & Jerjen, the rms scatters with respect to the equality line should both be 2 mag. when in fact they are zero!

In spite of the above, we accept that in Sect. 4 of YC95, we did indeed under-estimate our formal internal distance errors, because there must be some [non-distance related] dependence between the distance estimates based on the different scaling laws. However, even if our formal internal errors [including both intrinsic scatter and photometric errors] were as high as, say, 0.6 mag. [cf. 0.47 mag. as quoted in YC95], that would still leave room for a cluster depth of 0.54 mag., as the observed scatter per relationship in YC95 was 0.81 mag. Note that a cluster with a depth of 0.54 mag. would be half as deep as it is distant, with a further one-third of its objects lying even further out from its centroid.

We should also like to emphasize that our `independent information on the intrinsic scatter' was not `seized...by applying both the n -M and n - [FORMULA] relation at the same time'. The independent information was in fact, the much smaller scatter found in our samples of Fornax and Local Group galaxies. As long as our Fornax and Local-Group samples are representative and as long as they contain galaxies structurally similar to their counterparts in Virgo, the depth interpretation still holds.

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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