Cepheid variable light curves

Question

Basically, given a set of noisy observations for the apparent magnitude of a Cepheid variable, how is this fit to a curve which allows the period, and therefore distance, to be found? Cepheids' luminosity isn't sinusoidal. My first though was to use a Fourier approximation, i.e. fit using least sqaures error to $$L(t) = \sum_{k=0}^na_i\sin(k\omega t)+b_i\cos(k\omega t)$$

For some small $n$, to find the most accurate value of $\omega$, but this model would have serious problems with overfitting to the noisy data. So how is this accomplished? Two possible ideas: first would be to have some sort of addition to the error function which penalizes large coefficients as is usually done with things like logistic regression. The other would be to fit a Fourier approximation to a much less noisy set of observations and then fit just a scalar multiple and new $\omega$ to this curve. What is usually done? I think it's a pretty imporant question to get these values accurately, as we base our determination of Hubble's constant from the periods of Cepheids.

Answer 1

I'm not an expert on analysing classical pulsators, but the thing to remember is that, given a good signal-to-noise ratio, the Fourier transform will still have it's strongest amplitude at the correct period. You'll just see secondary peaks at other frequencies that are integral multiples of the main frequency. i.e. overtones. Plus, in many systems you can see the pulsation periods quite clearly by eye. e.g. the curves here.

The first thing I would add is that this does break down a bit when the star oscillates at more than one frequency. In that case, the distribution of peaks gets more complicated, although you can still try to work out what the two primary frequencies are, and which of the other peaks are beats, overtones, or combinations of beats and overtones. As you can guess, this isn't a simple process. In some systems, it's a real headache!

The second thing is that, nowadays, we can model the pulsations, and these models are able to reasonably reproduce the basic shape, which we can then fit. I don't know the details of the modelling but I've seen convincing results presented at conferences. (A quick search didn't turn up details but I'll have a closer look when I have more time.)

The third thing is that, indeed, there are many other methods for trying to find the main frequency. This is a review from a conference I attended a few months ago and this is the cited paper that compares a variety of methods.