# Calculate large and small frequency separation for the Sun

I want to determine the big and small frequency seperation from timeseries data for the sun. An excerpt of the data (timeseries and power series) is plotted below.

The power series is calculated in MATLAB like this:

n = length(t_obs);
dt = diff(t_obs(1:2));
y = fft(d_obs, n);
P = y .* conj(y)/n;
f = (0:n/2)/(n*dt);

f = f(1:(n/2));
P = P(1:(n/2));
plot(f, P);


What I can't understand is this:

• How can I get the big frequency separation $\Delta \nu$ and the small separation $\delta \nu$ for $l=0$ and $l=1$ from the powerseries without having to read it manually from the plot (there are many datasets). If you should want to show an example of implementing this (although an explanation or a hint will suffice), I'm fluent in Python as well as MATLAB.
• What is the unit of the y-axis on the powerseries?

Timeseries plot: Power series plot: For reference: $\delta\nu_l = \nu_{nl}-\nu_{n-1l+2}$

I hope you can help.

• @MichaelBrown: Ok, how do I move it, then? I've just created a user there on this account. – user4452 May 30 '13 at 20:24

Frequency separations

The easiest way to estimate the large separation (without fitting the individual frequencies) is to take the autocorrelation of the power spectrum and find the maximum. That's a start. To find the small separation, you'd be looking for a second peak. Anyway, have a look at the autocorrelation of the pwer spectrum and see if you can see the separations.

You could try to find the maxima of the autocorrelation, but I suspect the numerical derivatives will make it hard to find the correct zero-point. And, if you actually fit the frequencies, then you can take the average of the differences directly.

Units of the power spectrum

You can follow the units through the calculation of the objects you're working with. Suppose your time-series has units $x$. The Fourier transform is a sum of products of the time-series with something like $e^{i\omega}$, which is dimensionless. So the elements of the Fourier transform have units $x$ too. Finally, the power series is like the transform squared, so it has units $x^2$.

If you end up with weird units this way, note that in asteroseismology, people usually divide their signal my the mean (or median) and subtract one, so that have a dimensionless number. They usually express the power series in units of parts per million (ppm).

• I've never tried to fit the frequencies, could you tell me what equation I should fit to (or link to a page where it's explained)? – user4452 May 31 '13 at 8:34
• I'd recommend having a look at the autocorrelation of the power spectrum first. As for fitting, I don't know of the best original sources, but you have a look at Chapter 2 of Thorsten Stahn's PhD thesis and references therein: solar-system-school.de/alumni/stahn.pdf – Warrick May 31 '13 at 10:59