Lundmark blunders calculating galaxy distances

Lundmark (1924, MNRAS, 84, 747) was the first to create a "Hubble Diagram" correlating distance with velocity. His distances were computed by assuming that galaxies are standard rulers and candles. Thus, he obtained two distance estimates for each galaxy, one from the diameter and one from the apparent magnitude. These estimates were then averaged together. He adopted M31 as his standard galaxy. Table III in the paper gives a list of distances (expressed as parallaxes.) Regarding these distances, it has been stated that, "Lundmark's extragalactic distance estimates were far more accurate than Hubble's, consistent with an expansion rate (Hubble constant) that was within 1% of the best measurements today." Let's see if that's true.

Lundmark did not actually tabulate the data that he used to derive the distances estimates. However, his data sources were the same as or similar to those used by Hubble (1926, ApJ, 64, 321), so we should be able to use the parameters tabulated in Hubble's paper to see if we can reproduce Lundmark's distance estimates.

I have done so for both the diameter- and apparent magnitude-based distances. The diameter-based distances all look good: distance estimates computed from Hubble's diameters correspond closely to the estimates from Lundmark. However, there is a serious problem trying to reproduce the distance estimates derived from apparent magnitudes. The problem is illustrated in the following figures.

I have reverse-engineered Lundmark's parallaxes using Lundmark's explanation of his procedures in order to derive the original diameters and apparent magnitudes.

The figure on the left shows Lundmark's diameters v. Hubble's diameters. The green line has a slope of 1 and is a good match to the measurements. The figure in the middle shows Lundmark's magnitudes v. Hubble's magnitudes. Here, something is terribly wrong. The green line again has a slope of 1, but it does not match the points at all. The red line matches the points but has a slope of 2. [On the right, Hubble's magnitudes are compared with B(T) magnitudes from NED along with a line of slope 1, showing that Hubble's magnitudes are sensible.] The most straightfoward explanation is that Lundmark made a blunder in converting magnitudes into distances. The proper equation is:

log10 [D/D(M31)] = [mag - mag(M31)]/5.
where D is the distance and mag is the apparent magnitude. The trend in the middle figure can be reproduced if one assumes that Lundmark divided the distance modulus on the right hand side by 2.5, not 5. (We must additionally assume that Lundmark adjusted the zero-point. Lundmark basically admitted that that is what he did: "The parallaxes derived from the total magnitudes have been reduced to the scale of those derived from the apparent diameters." Translation: "I screwed up.")

I guess we should really call him "Blundmark".

Addendum

Both Lundmark and Hubble obtained galaxy apparent magnitudes from Holetschek (1907, Annalen der Wiener Sternwarte, 20), whose instrumentation consisted of a 6 inch telescope and his eyeball. The right hand figure above shows that his eyeball did a pretty good job!