The First Determination Of The Hubble Constant

Lundmark beat Lemaitre by two years

Historians of science tell use that it was actually Lemaitre, in 1927, who came up with the first value.

Neither of these claims is correct. The first value was determined by Lundmark, in 1925 (MNRAS, 85, 865). He didn't know it at the time. It is found in Table I on line 20. Lundmark took the same data he used in his 1924 paper (MNRAS, 84, 747) and fit a number of different models for the velocity-distance relation. The solutions all involved solving for a vertex motion (a systemic motion of the solar system relative to the external galaxies) and a "K-term" that accounts for the remaining contribution to the velocities. This procedure paralleled the one that Hubble would use in 1929. Lundmark tried 20 different combinations of data cuts and K-term formulations. Line 20, the last one in the table, is the precise formulation used by Hubble - a K-term linear in distance, with no constant offset. The coefficient of this relation is the Hubble constant.

The only quirk is that Lundmark expressed his distances in units of the distance to M31, and in order to convert the coefficient to absolute units we need to know that distance. By reverse-engineering Table III of the 1924 paper, we find that his preferred value at that time was 200 kpc. The coefficient on line 20 was 6.96. We divide by 0.2 Mpc for the distance to M31, and thus arrive at a Hubble constant of 35 km/s/Mpc. For once we have a Hubble constant from the 1920's that was actually too small compared to the modern value. Also notable, his value was the only one around that time that did not depend on Hubble's data. (The distance calibration was based on the magnitudes of novae.)

Why has Lundmark's epochal accomplishment not been recognized before? Lundmark was always his own worst enemy. Alan Sandage called him a "pure Baconian empiricist". In this case his preferred solution from Table I was line 18, a K-term that was a quadratic function of distance with a large positive constant offset and a negative 2nd-order term, which meant that velocities would reach a maximum and then decline as the distance became larger. He wrote down line 20, but attached no significance to it. All we can do now is look back and take note. It was the first-ever calculation of the Hubble constant, but it had no impact on anyone who followed.

Addenda

Lundmark's value of 35 is enormously different from Lemaitre's value of 625, even though they nominally used similar methods and nearly the same data! Why the difference? Here are the reasons that I can figure out so far.

1. Lundmark used M31 as his standard candle. Lemaitre used Hubble's mean absolute magnitude of all nearby galaxies with Cepheid distances as his standard candle. M31 is 2 magnitudes brighter than this mean. That is a factor 2.5
2. Lundmark made a huge blunder in calculating distances from apparent magnitudes - he used the wrong equation . The faintest galaxies (mostly at large distance) were assigned distance moduli 4 mag fainter than they should have been. Since Lundmark averaged apparent magnitude distances with angular diameter distances, it wasn't quite that bad. However, it may amount to another factor 2.5
3. Lundmark's most distant galaxy, NGC 1700, had the wrong redshift . It carried so much weight that it might have impacted the result by another factor 2.

In "Did Edwin Hubble Plagiarize?" (2011, arXiv:1107.0442), Shaviv recalculated the slope that Lundmark could have derived and found 71.2 km/s/Mpc, twice the value of Line 20. Shaviv used the data from Lundmark's 1924 paper verbatim and thus did not correct for the vertex motion (unlike Lundmark's Line 20 value), perhaps explaining the different values.

Hermann Weyl actually attempted to calculate the Hubble constant two years earlier (1923, Raum, Zeit, Materie), and, while he had a decent distance to M31, he assigned it a velocity that was largely invention. Thus, while his estimate of the Hubble radius was in the right ballpark, it was too dicey to give him credit ahead of Lundmark.

Lundmark came up with two other remarkable results in this paper, which also have had little impact on the field.

1. Lundmark assumed that the vertex motion of the sun represented rotation around the center of the Milky Way. By combining this velocity with Shapley's distance to the center, he derived a dynamical mass for the galaxy. He found a value of 1.e12 solar masses and noted that it was much higher than what was in the known stars, which were 0.8e10 solar masses. He attribute the discrepancy to a combination of dark matter, a too-large value for the distance to the Milky Way, and to the possibility that the vertex motion represents more than just rotation. He was correct on all counts, although he couldn't demonstrate it at the time.
2. Lundmark noted that S Andromedae, long thought to be a nova, was too bright compared to the large numbers of other novae that had been found in M31 and had similar properties to SN 1572 (Tycho's supernova). He proposed that novae split into two classes, which he callled "Upper Class" and "Lower Class" novae. His conjecture was correct, but his nomenclature did not survive; we call the former "supernovae", while the latter are ordinary novae.