I recently heard a segment of the Canadian science program Quirks and Quarks in which an astronomer discussed how they found rotation periods of stars by timing periodic darkening (presumably from “sunspots”, although he didn’t say). It occurred to me to wonder whether the spreading of spectral lines due to one limb of the star approaching and the opposite limb receding ought to give that information unless the rotation pole is pointing at us. But he never mentioned that possibility. Is the effect simply too small or is there some other reason this is not used?
It’s done (see “Rotational velocity of a star”).
Lots of interesting stuff can be done via spectroscopy. For instance, the very slight velocities imparted on a star by exoplanets will show up as a Doppler shift in the spectrum.
The difference in the doppler shifts from the two edges can tell you the speed at which the edges are moving, but that doesn’t give the rotational period unless you know the radius of the star.
True, but for main sequence stars there is a pretty good correlation between color and size. What they were trying to do was estimate the age of main sequence stars by how much of their angular momentum they had lost. This is inherently imprecise since they have to guess the original rotation speed so that imprecision in the radius could not have mattered much.
The relative Doppler shift won’t tell you the rotational speeds directly, since you don’t know the inclination of the star. You already mentioned the possibility that the axis is pointed directly at us, but what if the axis is just pointed sort of close to us? In that case, we could still see a relative Doppler shift, but it’d be smaller than it would be if we were looking from the side.
This sort of thing is fairly typical of astronomy. Usually, any given measurement won’t actually give you the quantity you’re interested in, but a combination of several different quantities. Sometimes, if you have enough different measurements of the right sorts, you can combine those measurements to get out some of the things you actually want. Sometimes you can make reasonable guesses or approximations about some of the things, to get reasonable guesses about the things you want. But you almost never get everything you want.