Ever since reading Grant Foster’s description of modelling in Analyzing Light Curves: A Practical Approach, I’ve been keen to incorporate this functionality into VStar. In recent SourceForge commits I’ve implemented a first cut of this.
If you are happy to “live on the bleeding edge”, you are welcome to try it now by checking out from the SourceForge Subversion repository or downloading a tarball. Otherwise, it will be available in the next VStar release.
I want to give an example from chapter 10 in Grant’s book: A day in the life of BZ UMa. BZ UMa is a cataclysmic variable that has undergone numerous outbursts; see this Slacker Astronomy video page re: a BZ UMa poster by Aaron Price.
Grant works through an analysis of just one day of BZ UMa data that shows short-term periodic changes. In talking with Grant, it turns out that the JD range used for the example is 2,454,205.3 to 2,454,205.9 not 2,445,205.3 to 2,445,205.9 as shown in the book.
Obtaining data in the range 2,454,205.3 to 2,454,205.9 from the AAVSO International Database with VStar, and looking at just the V band data gives this light curve:
BZ UMa V band for the JD range 2,454,205.3 to 2,454,205.9
Applying a Date Compensated Discrete Fourier Transform (DC DFT) gives this periodogram:
BZ UMa V band DC DFT
Switching to the Top Hits tab shows that the tallest peak corresponds to a frequency of around 14.18 cycles per day (depending upon exactly what frequency scan parameters you provide; I changed the High Frequency to 100 and the Resolution to 0.01 in the period analysis parameters dialog). In chapter 10, Grant suggests creating a model based upon this main frequency. In VStar, clicking the “Create Model” button in the Top Hits pane will create this and plot it against the V band data as follows:
BZ UMa V Model with main frequency of 14.18 cycles per day.
Apart from the model series, VStar also generates a series called “Residuals” which results from subtraction of the model data from the model source series (V band in this case) data that looks like this:
BZ UMa residuals for model of 14.18 cycles per day
This so-called pre-whitened data has the main frequency removed from the raw V band data. Section 8.6 of Grant’s book is about pre-whitening.
The question then is whether period analysis of the residuals would reveal further signal. A DC DFT of the BZ UMa residuals above gives this periodogram:
BZ UMa DC DFT of residuals from 14.18 cycles per day model
This power spectrum shows a Top Hit of around 28.36. In the light of this additional frequency, Grant suggests creating a model based upon two frequencies, in this case: 14.18 and 28.36. You can of course create a model based upon just this new frequency:
BZ UMa Residuals Model
Now, one of the current limitations of VStar’s implementation of modelling is that you cannot incrementally add to an existing model or combine two or more. I definitely intend to permit both however, in addition (a little later) to the construction of arbitrary models. For now, the best we can do in VStar is to return to the initial V band DC DFT and select two frequencies that are close to these (in this case: 14.18 and about 28.58) giving this model plotted against the V band data:
BZ UMa V band two-frequency model
VStar’s Analysis menu now has a Models item that opens a dialog when selected containing a list of created models. They can be selected for re-plotting or deletion. Only one model-residuals series pair can be viewed at a time. Note also that new tabs in the main VStar window contain tables of the model and residual data. Polynomial fits and their residuals are now also included in the models dialog box.
Of course, model and residual data can also be included in a phase plot. In order for a model to make sense for some stars (e.g. del Cep), a phase plot is pretty much mandatory; not so for the current BZ UMa example.
Grant goes on to use the residuals from this model to identify additional harmonics beyond the fundamental frequency and first harmonic mentioned above to further refine the model. His analysis also adjusts for two groups of observer bias to compensate for consistently different estimates by two observers. VStar’s filter feature can be used to illustrate one such bias group:
BZ UMa Observer aBias
As mentioned already, VStar’s current modelling capability does not yet permit such refinements as observer bias to be included in the model. In any case, I hope this example has provided some insight into where VStar’s modelling functionality is heading.