Friday, September 7, 2018

An Astronomy Math Puzzle

It's pretty easy to measure the radial velocities of distant stars or galaxies. (The "radial" velocity is the motion of an object directly toward or directly away from us). Astronomers use the Doppler shift, the stretching or compression of light from the moving object. Light from stars or galaxies moving toward us gets "bunched up," and the wavelength shrinks, making the light bluer, while light from objects moving away from us gets stretched out and becomes redder. The Doppler effect works so well that we can pin down the radial velocities of the most distant galaxies in the universe -- these observations provided the first evidence for the expanding universe.

But it's a lot harder to measure the sideways, or "transverse" velocity. The only way to measure a transverse velocity is to keep careful watch on an object and wait... and wait... and wait... We've been able to measure the transverse motions of stars for a long time -- the star with the fastest transverse motion is Barnard's Star, named after Vanderbilt's E.E. Barnard. But even Barnard's star is moving at a glacial rate across the sky -- 10 arcseconds a year.  At that rate, it will take almost 200 years to move the width of a full moon.

Galaxies are much farther away than stars, so their transverse motions are minuscule in comparison and have never been observed. But recently the Gaia satellite has allowed astronomers to pinpoint the locations of distant objects with unprecedented accuracy. So there's speculation that if we could monitor distant galaxies for a long enough time (10 years? 20 years?) we might be able to measure their transverse motions.

All of which leads to a puzzle. We don't want to sit around for 10 years, only to discover that we've been watching a slowpoke galaxy that's hardly moving sideways at all. If we could monitor only a handful of distant galaxies, which ones are likely to have the biggest transverse velocities? Are they the galaxies with the largest radial velocities? That makes sense -- a galaxy with a large radial velocity is more likely to have a large total velocity, so the component of its velocity in the transverse direction is also likely to be large. But here's an argument in the opposite direction: if all the galaxies are moving at about the same speed, then a large radial velocity means that the galaxy is likely to be moving almost directly toward or away from us, so it will have a small transverse velocity, while a galaxy with almost no radial motion is likely to be moving perpendicular to our viewing direction and will have a very large transverse velocity. In that case, we should monitor the galaxies with the smallest radial velocities.

So what's the answer?