Strange Quarks

Grant Foster’s book “Analyzing Light Curves: A Practical Guide” (section 5.5) gives an example of using polynomials to determine critical points of a light curve, in particular: a Mira maximum.

The book addresses the question of how to determine minima/maxima, especially in the presence of scatter in the data. The following figure shows a 7 degree polynomial fit for a Visual JD range around maximum for Mira.

To obtain this plot, load the data from the AAVSO International Database for the JD range shown (2451460.0764 to 2451559.539), select Polynomial Fit from the Analysis menu or toolbar, select the Visual series, then the number of degrees for the polynomial, in his case: 7.

Experiment with the degree value to see the effect upon the least squares polynomial fit. A 5 days-per-bin mean series (again, based upon the Visual series) makes a useful comparison. This can be changed via the View menu’s Plot Control dialog.

Switching to the Model and Means tabs in turn and clicking the Magnitude column to re-order it, allows the maximum value in the series to be easily found. Selecting such a row causes the cross hair in the plot to move also.

Grant’s discussion goes beyond simple polynomial fits, including a discussion of information criteria or “goodness measures” and a consideration of alternatives such as the Lowess smooth, both of which are on the roadmap for VStar. He also spoke about this in more detail at one of the Astro April Citizen Sky talks about uncertainty in determining time of minimum/maximum.

Given that it is currently near maximum, while writing this entry I also created polynomial fits for filtered ranges of R Car, another long period variable. I’ll leave that for another post though.

On a related note, I’ve been asked by a few people recently about when VStar will include a Time of Minimum/Maximum (ToM) capability such as Kwee-van Woerden for use with eclipsing binary light curves. This is working its way higher up the list.

Grant Foster’s 1995 CLEANest Fourier Spectrum paper (Foster, G., 1995, “The CLEANest Fourier Spectrum”, The Astronomical Journal, vol 109, no 4, 1889–1902­) gives a number of examples of applying the CLEANest algorithm to datasets, artificial and real. Two of these use AAVSO visual magnitude estimates: S Ori and AA Cas.

This post shows VStar’s CLEANest implementation applied to AA Cas. Foster 1995 uses an AA Cas dataset in the JD range 2447500 to 2449500. The following shows that dataset loaded from the AAVSO International Database (AID) via VStar’s file menu.

aa Cas Visual JD 2447500 to 2449500

A DCDFT with frequency range can be initiated from VStar’s Analysis menu, selecting the Visual band and specifying minimum and maximum frequencies, the range over which to scan, and frequency resolution over the range.

This results in the following power spectrum (in the Power vs Frequency pane) with the orange squares showing peaks or “top hits”.

aa Cas Visual JD 2447500 to 2449500 power spectrum

These top hits are shown in the next diagram in tabular form.

aa Cas Visual JD 2447500 to 2449500 selection for input to CLEANest(7)

In this example, seven top hits have been selected using combinations of shift-click and control-click  (Windows) or command-click (Mac). The initial input values to CLEANest are not stated in Foster 1995 (section 5, page 1900), but the rows selected above fairly closely correspond to what I think they should be.

Clicking the CLEANest button opens this dialog from the Top Hits pane.

aaCas CLEANest(7) input dialog

Clicking OK here adds seven new top hits with the same power value, shown multiply-selected in the top hits list and annotated on the power spectrum.

Now click Create Model in the Top Hits pane and the following dialog will open.

aaCas CLEANest(7) model creation dialog

Click OK and the main plot will have an additional “model” series added. Dismiss the main DCDFT dialog to return to the main VStar window.

aa Cas Visual JD 2447500 to 2449500 model from CLEANest(7)

Something on my TODO list is to make the model series continuous rather than discrete as it currently appears. The residuals for this model can be viewed by opening the Plot Control dialog from the View menu and setting it as shown, including changing the Days per Mean Series Bin (and clicking Apply).

Dismissing the dialog changes the plot to look like this.

aa Cas Residuals and binned means from Visual JD 2447500 to 2449500 CLEANest(7)

Performing a DCDFT on the residuals with the same frequency parameters as for the visual series, but by selecting the residuals gives the following power spectrum.

aa Cas power spectrum from Residuals

Looking at the Power axis suggests that there is very little discrimination between any of the frequencies.

In addition, the ANOVA value in the Info dialog (File menu) also suggests that there is unlikely to be any signal remaining to be found in the residuals, i.e. the null hypothesis that there is no significant signal present should be accepted.

aa Cas ANOVA from Residuals