## The distance to M31

To compare my results for the distance to M31 directly to Hubble’s, I needed to use Shapley’s flawed calibration of the period-luminosity relationship. Curiously, Hubble adjusted this graph to yield absolute magnitude at maximum, rather than the average absolute magnitude as originally suggested by Leavitt and used in virtually all other calibrations. Hubble indicated he believed his maximum magnitude readings were more reliable than those obtained during dimmer portions of the cycle. Based on this graph, the logarithm of my 31.91-day period yielded an absolute magnitude for M31-V1’s maximum of –3.6.

Once you have an object’s absolute magnitude and corresponding apparent magnitude, it is simple to calculate its distance using an equation called the distance modulus: m – M = 5[log10(d/10)], where m is the apparent magnitude, M is the absolute magnitude, and d is the distance in parsecs. (One parsec is equal to 3.26 light-years.) Solving this equation for d gives d = 10(m-M+5)/5.

My values for m and M yielded a distance of 275,423 parsecs, or 897,879 light-years — very close to Hubble’s published value of 900,000. With this, I had accomplished my goal. Despite the flawed calibration, I had also proven M31 is so far away it must be a separate galaxy.

I was extremely pleased that my result was so close to Hubble’s. The difference was only 2,121 light-years. Then, while reading Hubble’s 1929 publication again, I noticed something remarkable. Although Hubble did not show his values for maximum apparent and absolute magnitude, he gave their difference: m – M = 22.2. That was precisely the value I had obtained! That could only mean Hubble had obtained exactly the same distance: 897,879 light-years. He simply rounded to 900,000 light-years in his paper. Now I was thrilled. I had done in my backyard what Hubble did at Mount Wilson, with precisely the same result.

Still, that result is incorrect because it is based on an incorrect calibration of the period-luminosity relationship. Since 1929, as technology has improved, the calibration has been revised. This has greatly increased the calculated distance to M31. With the advent of HST, that number is now 2.537 million light-years.

## A memorable endeavor

With that, I declared the project a great success. Following in Hubble’s footsteps was an exhilarating experience I will certainly remember for the rest of my life. Most of all, I am amazed that I, a mere amateur astronomer using equipment in my backyard, was able to reproduce a feat that less than a century ago was accomplished by the world’s greatest astronomer using the world’s largest telescope.

This is a testament to amateur astronomy as a hobby. Want to be an amateur archaeologist or paleontologist? Good luck accessing an Egyptian tomb or a T. rex fossil bed to conduct your own research projects. Such valuable materials are reserved exclusively for professionals.

Not so with amateur astronomy. All astronomers have unrestricted access to the same crucial resource: the entire sky above us. And with that, the sky is truly the limit of what we amateurs can do.

## Up to Date

One of HST’s chief goals was to precisely determine distances to 10 Milky Way Cepheids by measuring their trigonometric stellar parallaxes — which can only be done from space — to produce a calibration of unprecedented accuracy for the period-luminosity relationship.

The equation for the period-luminosity relationship using the HST calibration is: M = (–2.43 + 0.12)[log10(P)-1.0] – (4.05 + 0.02), where M is absolute magnitude and P is the period.

Using my data with the HST calibration, how close could I come to the currently accepted distance of M31? My period of 31.91 days yields an absolute magnitude of –5.27 using this calibration. Then the distance modulus gives a distance of 776,247 parsecs or 2.531 million light-years.

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