Estudamos o interior das estrelas observando variações do seu brilho. Da mesma forma que estudamos o interior da Terra. Todo mundo já viu desenhos mostrando informações detalhadas do interior da Terra. Todo mundo acredita nesses desenhos. E todo mundo sabe (?) que não temos nenhuma informação direta de lugares mais profundos que 6km (minas mais profundas do mundo).

De onde sabemos tais coisas?
Vamos começar do começo.

Escute os sons:

O que essas sons têm em comum? Os dois são a nota DÓ. Mas eles são bem diferentes. E ninguém precisa ser músico e nem físico para dizer que o primeiro é um piano e o segundo um violão.

A figura abaixo mostra a onda sonora produzida pelo piano:

Onda sonora produzida por um DÓ tocado ao piano.

E esta figura mostra a onda sonora produzida pelo violão:

Onda sonora produzida por um DÓ tocado ao violão.

Como estas figuras foram feitas? - Aqui explicamos como gravar sons com o computador, converter um arquivo de áudio em um arquivo de dados e visualiza-los num gráfico.

But if both sounds have the same pitch (the same frequency), how come they have a different timbre, in a way that we can tell a piano from a guitar? The pictures above show the sound data plotted for a small range of the domain (time). If we take a look at plots for the whole length of the record, a difference becomes evident... A C on the piano:

The same C on the guitar:

What we see on the pictures is that the envelopes of the sound waves are different. That's why both sounds sound pretty different while still having the same pitch (a frequency of roughly 261 Hz). We also know that this, like any sound, is not composed of only one pitch, but by a fundamental frequency (C4 = 261 Hz) and its harmonics, found at N times the fundamental frequency. Mathematically, we expect to find the first harmonic to be at ~ 2x(261 Hz) = 522 Hz, the third harmonic at ~ 3x(261 Hz) = 783 Hz and so on. It might be much better for us to analyze our data and the relationships between these different frequencies if we could look at at a plot of FREQUENCY vs amplitude (instead of TIME vs amplitude). This means we need to go from the time domain to the frequency domain. Luckily, there's a mathematical method to do just that. We can apply a Fourier Transform (FT) to our data, which decomposes a continuous function or a discrete data set (such as ours) into the frequencies that make it up. For a practical reason (reducing the processing time) we took a sample from each audio file (one from the piano, one from the guitar), some one or two seconds after the corresponding string started to vibrate and applied a discrete Fourier Transform (DFT), with a program written in C, to each of them. We then plotted both "frequency spectra" superposed. And voilà, we have the confirmation that the same note, or "chord", consisting of roughly the same pitches was played on both instruments:

We can clearly see the peaks corresponding to the fundamental frequency and the fist two harmonics. Note: if the frequencies of the peaks we see on the FT plot are not exactly the ones expected for a C4 note (261 Hz, 522 Hz, 738 Hz...) this means our instruments were just slightly out of tune. The Fourier Transform is the best of way of checking the tuning of an instrument. What electronic tools used to tune string instruments do is in fact a "live" FT that tells the fundamental frequency produced by the string. In the frequencies spectrum, we can tell one instrument from another in a more analytical way: by the relation between the amplitudes of the peaks corresponding to the fundamental frequency and each harmonic. We can see in the picture above that this relation is different for the piano and for the guitar. In fact, each instrument, while still having peaks at the same frequencies, would show its own different pattern.

At this point you might be asking yourself: what on earth does all this have to do with stars and their internal structure? When we look at a pulsating star with a telescope for some nights and plot its flux against the time we observed it (when we measure the amount of radiation emitted by a star we say we're performing a photometry) we might get a result just like the following:

This is a light curve for a pulsating white dwarf. Now, it kind of looks kind of like our sound waves, doesn't it? It's only plotted over a larger time-range. We see the periodical nature of the star's pulsation. A vibrating string has a fundamental frequency and the corresponding harmonics; when we record its sound and plot the data we see no more than two of them. In pulsating white dwarf's light-curves there are more pulsation modes present than harmonics we see at the wave produced by a string. In the same way a frequency spectrum obtained through a Fourier Transform will help us better understand the timbre of different instruments (a result of their different constructions), we can also run a FT on the star's photometry and be able to get information about its internal structure. The frequency spectrum for a pulsating white dwarf looks like this:

On the plot above we can see at least five distinctive pulsation modes. It is through the study of these pulsation modes that we're able to determine some basic properties of the observed stars, such as their mass, density, rotation speed, density and internal composition.

some files to play with (zip)