Telescope Reviews: Period Luminosity relationship and Cepheids

Sorry to be replying so late, but just in case you (or other readers) are still interested ...

The "statistical parallax" method is a classical technique for trying to estimate the distances to a set of (ideally) identical stars. The regular parallax technique involves taking pictures of a region at intervals of six months; as the Earth moves around in its orbit, the changing point of view causes nearby stars to shift relative to more distant ones. Unfortunately, stars must be very close for this method to work. The limit of the technique at the time of Shapley's work was probably around 100 parsecs, and very few or no Cepheids are that close.

So, what is "statistical parallax?" The basic idea goes something like this: first, identify a set of stars which are

a) nearly identical in their physical properties --
same color, same apparent magnitude, same spectral type.

If you do this right, and all the stars really are similar in their intrinsic properties, AND in the apparent magnitude, then they should all be at roughly the same distance from you. Ideally, they will make a sort of hollow shell which surrounds you. You want to find the radius of this shell ... but how?

If you watch these stars over several decades, they will all move slightly in the sky relative to more distant stars. This "proper motion" is due to each star's particular motion as it orbits the center of the Milky Way, and due to the Earth's motion in its orbit. Now, if you have a large enough sample, and if the stars really are at the same distance from the Earth ... then their proper motions, considered as a set, should almost cancel out. That is, there should be roughly equal numbers of them moving left, and moving right; moving up, and moving down. If the Earth were stationary relative to the center of the shell of stars, you'd expect all the motions to cancel exactly.

But -- the Sun is moving in its own orbit through the galaxy. That means that the Sun is NOT motionless relative to this shell of stars. And that means that when you add up all the proper motions, you'll find a small left-over motion. This motion is really a reflection of the Sun's motion. If you know the direction and speed of the Sun's motion through the galaxy, relative to other stars -- which we do know, to some degree -- then you can use the apparent angular residual motion to determine the distance to the shell of stars.

A bunch of really, really luminous stars of apparent magnitude 10 would be very far away from us ... and so the Sun's motion through space would cause almost no residual motion. We'd measure almost no residual -- the motions of the different stars would cancel exactly.

A bunch of low-luminosity stars of apparent magnitude 10 might be within a few parsecs of us. The sun's motion relative to this group of stars would leave a very large residual when we average all the proper motions ... which would tell us that the stars in this group must be very close.

That's the basic idea of statistical parallax. You can find more detailed descriptions, with the math, in graduate textbooks like Mihalas and Binney or Binney and Tremaine.

I just checked the ADS, and don't see any papers written by Shapley during the 1910s or 1920s which deal with statistical parallax. Hmmm.