How do you determine the period?

For rare systems, like Algol, the variation is observable to the naked-eye. Algol was seen to brighten and dim over the course of almost three days and was one of the first variable stars detected. For most variables, the fluctuations are too dim to be seen by eye.

There is a huge range of variability timescales. Some variables fluctuate on timescales of a few seconds (e.g., X-ray bursters), while others may take several years (eruptive T Tauri stars). It's important to know your period range of interest, which is dependent upon two things (a) the integration time --- you cannot search for periods less than the integration time and (b) the total observation window.\[P_{\left\{\max\right\}}\le\frac{1}{3}\tau\]

In order to look for periodic signals, there are several mathematical tools. A periodogram is one of them. In signal processing, a periodogram is an estimate of the spectral density of a signal. Periodograms are super handy in determining the optimal period of a time series.

A periodogram is similar to the Fourier Transform but is optimized for unevenly time-sampled data, and for different shapes in periodic signals. This is handy, as unevenly sampled data is particularly common in astronomy. For instance, your target might rise and set over several nights or you have to stop observing with your spacecraft to download the data. Moreover, time-series data of astrophysical objects are inherently noisy measurements -- photon noise, atmospheric conditions and other factors can introduce random variation into the magnitude of the observations.

Like a Fourier Transform, a periodogram calculates the significance of different frequencies in time-series data to identify any intrinsic periodic signals. A periodogram is a brute-force tool. Many different frequencies and candidate periodic signals are evaluated for a given light curve. The statistical significance of each frequency is computed based upon a series of algebraic operations that depend on the particular algorithm used and periodic signal shape assumed. The brute-force nature of periodograms makes their computation time intensive on single-CPU systems, particularly as the number of observations and periods sampled grow. For example, on my Macbook workstation a periodogram for a 30-day light curve can take up to 20 minutes on a single CPU.