Skin a Cat

Before I begin let me make it clear that this is not about abusing cats. I love cats. We have a cat. We treat him very well. He treats us as though it’s our duty to worship him. He’s a cat.

This is about the old adage that “there’s more than one way to skin a cat.”

Mira-type variable stars are pulsating red giants. They’re old, and late in their lives they’ve exhausted most of the hydrogen fuel in their cores so they start burning helium in the core, and hydrogen in shells around the core. They expand to great size, and start to pulse, changing brightness in periodic fashion. But their cycles aren’t perfectly regular, each is a bit different from all the others. They’re named for the prototype of the class, omicron Ceti, which the ancients called Mira (“the wonderful”).

Our own sun, 5 billion years or so from now, will expand to red giant size and possibly turn into a Mira-type variable. It will get so large that it will engulf the earth and fry it — so we have less than 5 billion years to find another place to live.

All Miras show irregular cycles, each with a slightly different period, amplitude, and cycle shape. But for some, the period doesn’t just jitter about an average value, it actually shows a trend, a secular change. Not too long ago Templeton and Willson (2005) studied a large sample of Miras to find out which were showing secular change in period. They adopted a conservative test in order to minimize the number of “false alarms,” which means they may have missed some but the ones they did identify are “for real.” Out of 547 Mira-type variables studied, 57 showed secular trend in period, so such change may not be the norm but it’s not rare either. Their study, a major result in astronomical research, was based entirely on data collected by amateur astronomers, mostly using visual photometry (the eye) to estimate brightness.

Now to the topic of this post (finally!). How can we tell that a Mira-type variable (or any kind of data) is changing its period with a secular trend? It turns out there are many methods — more than one way to skin a cat. Let’s look at some different techniques applied to one of the stars which Templeton and Willson identified as showing a large period change, BH Crucis, or simply BH Cru.

The most obvious method is to estimate each individual period. This can be done in many ways, including finding the time at which each cycle reaches its maximum, then estimating the period as the time from one maximum to the next. This can sometimes be difficult because we can’t always see the star (especially if the sun is too close) so we might miss a maximum entirely. For BH Cru, obscuration by the sun is only a minor problem. As the light curve shows, most of the maxima are sufficiently well-observed to estimate their timing:

The earliest cycles aren’t as well-observed as later ones, but we can estimate most of the times of peak brightness, which gives these estimates of the period on a cycle-by-cycle basis:

The average period is about 525 days, but it jitters around a lot. There are also two values that are very different from the others early in the series. This is because one of the maximum timings is, in a sense, “too early” so it makes the period estimate for the preceding cycle too short and that for the following cycle too long. The maximum timing isn’t actually wrong, but it fails to represent the main period of the star — for a reason we’ll see soon. In any case, this method — estimating individual periods — doesn’t really reveal a secular trend in the period of BH Cru.

Another period analysis method is Fourier analysis. The Fourier spectrum of the star reveals its periodic nature immediately:

There’s a tall peak at frequency just less than 0.002 cycle/day, corresponding to period of 523 days. But Fourier analysis won’t tell us about period change. Or … will it?

Notice that there’s a second peak just to the right of the main peak. Sometimes such a peak is an alias of the main peak, just a ghost image. But in this case it’s real. It’s not a separate period, it mixes with, and interferes with, the main peak to produce frequency modulation and amplitude modulation of the main oscillation. There are ways to disentangle the frequencies that cause such modulation, giving this:

Notice that there’s another peak in this spectrum, a small one, at about frequency 0.0057 cycle/day (period about 175 days).

That means there’s another, smaller (and faster) oscillation of this star. That’s what caused one of the maximum timings to be “too early” to represent the main period. Most of the cycles have a clearly defined maximum, like this one:

However, if that smaller oscillation occurs near the time of maximum and is strong enough, it can make the maximum “lean to the left” and happen earlier than the peak of the main cycle, like here:

It may also be timed just right to make a maximum “lean to the right” and happen later than the peak of the main cycle, like here:

Therefore that secondary period, although weak, is strong enough to perturb the timing of maximum brightness, and make it more difficult to estimate individual periods. That’s one of the reasons the individual-periods method didn’t detect a secular trend in period, but it’s not the main reason (which we’ll see later).

More to the point, there’s a very clever way (in my opinion) to use the frequencies near the main one, the ones which combine and interfere to create frequency/amplitude modulation, to reconstruct the period changes of the star. It gives this rough estimate of the changing period, but definitely indicates the existence of a secular (but not linear) trend:

It suggests a decline, followed by a rapid increase, followed by a more gradual decrease.

There’s another technique based on Fourier analysis, in which we “window” the data, looking only at a limited time slice to estimate the average period during that window. Then we let the window slide through time to get a measure of how the period changes over time. The method is called, not surprisingly, windowed Fourier analysis. One of its advantages is that we can use the standard methods to estimate the uncertainty in each period estimate. It gives this:

Now, there’s little doubt that the period of BH Cru shows a secular trend. But again, the trend itself is rather complicated, certainly not linear, showing a decline, followed by a rapid increase, followed by a more gradual decline.

Finally, we can use the method applied by Templeton and Willson, wavelet analysis. For data unevenly spaced in time, a good method is the weighted wavelet Z-transform, or WWZ. Among its many virtues is the fact that it produces pretty pictures:

The band of reddish color indicates the main oscillation. It’s weaker earlier, but that’s mainly because it’s statistically weaker simply because there’s less data. Also, the range of colors isn’t sufficient to reveal the large range of numerical values. We can compress the numerical range simply by taking logarithms, which gives this plot:

showing that the main oscillation exists throughout the entire time span, although it’s (statistically) weaker in the early part of the data.

The WWZ also enables us to compute the period as a function of time, which looks like this:

Again, there’s a decrease, followed by a rapid increase, followed by a more gradual decline.

There you have it — four different ways to look for secular trend in the period of a data set. Let’s plot the results of the last three methods for comparison:

Clearly they’re in excellent agreement! Method 2, combining interfering frequencies, gives a rougher estimate, but it also makes it easy to detect secular period change if you didn’t already know it was there.

We can also add the result of the individual-periods method:

Now we see the reasons it didn’t reveal the secular trend right away. For one thing, it shows a lot more “jitter.” For another, the “too early” and “too late” maximum timings due to the secondary oscillation threw off some of the period estimates, by a lot (the early ones are “off the scale” on this plot). Finally, the sparsity of early data caused me not even to estimate the individual periods until after the rapid period increase was well under way, so we simply missed it.

No doubt about it, there really is more than one way to skin a cat. The general agreement of multiple methods means that there’s also no doubt about the secular trend in the period of BH Cru — a decrease followed by a rapid increase followed by a more gradual decline.

And in case I haven’t emphasized it often enough, not just this star but all 547 in the study by Templeton and Willson proves the extreme high quality, and tremendous scientific value, of the Herculean efforts of amateur astronomers.