**1. Introduction**

This article describes **WinFitter**, a
Windows program developed for analyzing the light curves of eclipsing binary stars and providing theoretical fits of circular dark spots to the distortion-wave curves of these stars. It has also been expanded to allow the analysis of transiting planet light curves collected by the Kepler Space Telescope, replacing the previous separate program **WinKepler**.

The **WinFitter** package was developed using **Microsoft Visual Studio 2010**, utilizing both C# and Fortan 95. The package includes: **Fitter**, a binary curve fitting utility derived from Budding's Standard Eclipsing Binary Model Optimization Procedure (Budding, 1973,
1977, 1993), with corrections and enhancements to the program made by Rhodes
(1989 and 2004); **SpotFitter**, a spot fitting utility developed first by Budding and Zeilik
(1986), and further refined by Rhodes (1989); **Plot Spot Location**, a plotting procedure to plot the position and sizes of the spots on
a Mercator projection of the stellar surface, and **Binner**, a simple binning program to reduce the size of light curve data sets. It replaces an earlier suite of **Dos** light-curve fitting
programs.

The **WinFitter** applications are robust and
user-friendly, and with the universal availability of PCs, they provide an
inexpensive means of analyzing eclipsing binary light curve data of both eclipsing binaries and transiting planets in a consistent
way.

**2.** **Description of the Light Curve and Spot Fitting
Algorithms**

The **Fitter** application uses an optimized chi-square fit of the light curve to the physical parameters of the system such as mass, luminosity, period and separation. It also includes the relevant proximity effects such as radiative interaction (reflection effects), tidal and rotational distortions, gravity brightening (ellipticity effects), limb darkening, Doppler beaming effect, and orbital eccentricity (if present). It can also be used to analyze the light curves of stars with transiting planets, such as those produced by the Kepler Space Telescope. **Fitter** also produces a distortion-wave light curve that is the difference between the observed and model light curves. This “difference curve” can then be analyzed by **SpotFitter**, a starspot modeling application, which uses a circular, dark starspot model to find the latitude, longitude, size and temperature of one or two spot groups on the active primary star.

A key aspect of the both the **Fitter** and **SpotFitter** applications is that the information content of the data is evaluated by calculating the curvature Hessian (the matrix whose components are the second partial derivative matrix of the function) and its inverse, the error matrix, in the region of chi-square space surrounding the optimal solution. This makes it possible to determine which of the adjustable parameters has a determinate solution as well as providing an error range for each of the parameterized values.

The spot fitting analysis assumes that the maculation effects can be separated from the light curve arising from the geometry and physical properties of the system. The process uses basically two separate curve fitting programs—an eclipsing binary curve fitting program and a starspot fitting program. These algorithms parameterize the photometric light curve in a way that rigorously extracts information about the latitude, longitude, size, and relative flux of spotted regions on these binary systems. What is especially important, however, is that the algorithms provide an objective evaluation of the information content of the best-fit solutions, so that we know how much information can be reliably extracted from the data.

The spot fitting algorithms model the maculation effects by one or more large, dark, circular regions on the photosphere (Budding, 1977, 1993). Circular regions have the advantage of having the minimum possible area for a given maculation effect. In essence, a circular spot is the smallest size the spot group could have—any other shape would be larger. In the spot fitting analysis it is important to distinguish between the "AC" and "DC" components of the activity. The AC component is the maculation effects produced by the starspot(s). The DC component, on the other hand, is the shift of the light curve up or down in magnitude due to an overall increase or decrease in light output of the star system. This model only addresses the AC component.

This circular spot model has a very concise algebraic form, lending itself to quick calculations while thoroughly searching parameter space for optimal solutions (Vogt, 1981). Moreover, it parameterizes the properties of the starspot in such a way as to represent the actual situation of a large active region. The derived parameter sets are then suitable measures of the starspot activity. In addition, it is possible to effectively evaluate the information content of the solution by reference to the curvature Hessian (the matrix whose components are the second partial derivative matrix of the function) in the vicinity of the optimal solution in the chi-square parameter hyperspace (Bevington, 1969; Adby and Dempster, 1974; Budding and Najim, 1980, Budding, 1993). The "best fit" is determined by minimizing the value of the chi-square variate. The program includes an optimizing routine that makes it possible to follow and control the approach to the optimum, and then calculates the curvature Hessian, and its inverse, the error matrix, in the vicinity of this optimum. Negative eigenvalues to the Hessian signal a breakdown of determinacy (also shown by negative elements on the central diagonal of the error matrix), and the corresponding eigenvectors indicate, by their relative orientations to the various parameter axes, which parameters are intrinsically well determined.

In summary, the starspot model for the RS CVn activity implies that the photometric maculation effects contain information about the net surface areas, longitudes, latitudes, and temperatures of localized active regions (the AC component), together with a more uniform background (DC) component. The parameters we can derive from sufficiently detailed and accurate photometry parallel aspects of the sun's "11-year" sunspot cycle, which is a fundamental reflection of the action of the solar dynamo.

The precision with which one can estimate optimal parameter values is clearly dependent on the inherent observational accuracy. However, the numerical details of the curvature Hessian, and its related quantities, make it possible to see when effects like the correlation between spot latitude and size prove troublesome to the determinacy. In short, the procedures make the analyst keenly aware of the information limit in the data, so he or she can be on guard against going beyond this limit in the parameterization. Hence, with due caution, we can use historical data, which are valuable in extending the time span of our knowledge of stellar magnetic activity.

Hall (1981), Vogt (1983), Rodonò
(1986), and Olah (1986) have all discussed the shortcomings of starspot modeling
of photometric data. The essence of these criticisms is that there are generally
more parameters to adjust in the model than available information to fix them.
That is certainly true for non-eclipsing systems where the orbital inclination
may not be known or known well. But *eclipsing* systems are a different
matter. First, the eclipses provide a well-defined longitude reference. Second,
the geometry of the eclipses fixes the orbital inclination in a reliable way.
Third, eclipses of the *spots*, if the geometry brings this about, should
improve resolution of their longitudinal extent, their shapes, and even their
latitudes. The Budding and Zeilik starspot modelling procedure used here does
include a *formal* treatment of this contingency; that is, it can model the
eclipse of starspots. As a result, it can establish reliable basic parameters
for these sunlike stars themselves by separating the maculation and eclipse
effects (Budding and Zeilik, 1987, Budding 1993).

With the orbital inclination known, the starspot parameter set includes: mean light level; net area; and the longitudes, latitudes, and temperatures of one or two active regions responsible for the maculation wave. Justification for using only one or two active regions comes from an analysis of the photometric light curves of 31 RS CVn systems, by Nelson and Zeilik (1988), in which they found that one or two active regions could account for the periodicities in the light curves; larger numbers of active regions were not required to model the photometric behavior.

To reach reasonable solutions, the temperature should be separated from the geometry (Vogt, 1981). That can be accomplished from simultaneous visual (V) and red (R or I) photometry using the Planck blackbody flux equation or from infrared observations (Busso et al., 1984). Hence, one requirement for complete observations is that they include a red band filter.

Longitudes would normally be readily
established from the minima in the maculation waves, which are easily visible in
high-quality photometry. The widths of the minima contain information about the
area of the active region. Finally, the *shapes* of the minima provide
information about active region latitudes, at least in an approximate way.
However, Budding and Zeilik (1987) found from their analyses that such
information can be reliably derived *only* from photometry that is 1% or
better, but latitude values *can* sometimes be extracted. There *is*
an unresolvable ambiguity as to which hemisphere (northern or southern) the
latitude location refers to, but a corresponding ambiguity in the sign of
orbital inclination holds, and is an accepted limitation for the photometric
analyses of all eclipsing binary systems.

**3. Analysis Procedure**

The eclipsing binary curve fitting utility is first used to produce a theoretical light curve which "best" represents the initial data set. It uses a Roche model for the system, and the curve fitting algorithms include the relevant proximity effects such as radiative interactions (reflection effects), tidal and rotational distortions, gravity brightening (ellipticity effects) and limb darkening. If there is orbital eccentricity involved, the model can deal with this as well. These all must be properly dealt with in order to leave behind the effect of the starspots alone on the light curve.

Using a set of standard parameters for an initial fit (Budding and Zeilik, 1987, Budding 1993), the eclipsing binary fitting utility generates an optimal curve fit that is separated from the original data, leaving the maculation wave. Figure 1 is an example of a graph of an initial theoretical curve with the observed points it is trying to fit. The difference between this theoretical curve and the observed light curve defines the maculation wave associated with the effects of starspots.

Figure 1. Theoretical Curve of SV Camelopardalis Fit to Data

This maculation wave (or difference curve) is then input into the starspot fitting procedure, which uses a circular dark spot model to find an optimal fit to the maculation effects (maculation wave + net mean depression) and calculates the size, relative fluxes, and positions of the spot groups on the maculated star.

In addition, the starspot fitting procedure separates out the maculation effects from the light curve and puts them in a correction data file. The eclipsing binary procedure can then be run with this correction data. With the maculation effects removed, we can then find optimized solutions for the physical parameters of the system. This provides a check on the spot fitting procedure, since if it is properly accounting for the maculation effects, the optimized physical parameters should remain consistent through all observed light curves.

In practice, this process is repeated several times, varying first some, and then others of the system parameters until definite solutions with a reasonable chi-square have been found.

**4. Description of WinFitter**

WinFitter runs on an PC. The minimum suggested hardware configuration is at least a 486
system with 8 MB of RAM and a hard disk—a Pentium with 16 MB of RAM is
recommended.

The program requires at least Windows XP--it will not run on earlier versions of
Windows, but does run on Windows Vista, Windows 7 and Windows 8.. It also requires .NET Framework 4.0 Client. If you do not have this on your computer, the InstallWinFitter program will install it for you. When the program is
run, the window shown in Figure 1 opens with drop-down menus for the program
utilities and procedures.

Figure 2. WinFitter Main Window

The WinFitter program consists of the following procedures:

Run Fitter– Opens a directory window to choose a data file as input to theFitterapplication and then runs the procedure.

Run SpotFitter– Opens a directory window to choose a data file as input to theSpotFitter application and then runs the application

PlotSpot Location– Plots the positions and sizes of star spots on a Mercator projection of the star.

Bin Data- Opens a separate window that allows the user to reduce the number of data points in a data file.

Limb-darkening Coefficient- Opens a separate window that allows the user to calculate limb-darkening coefficients based on the Van Hamme tables.

Help– Has two submenus:

Help Manual– Opens a pdf file of the help manual.

About– Gives information about the program, version number, etc.

Figure 3. Mercator Plot of Star Spots

You can download the WinFitter user's manual by clicking the following link: WinFitter user's manual.

The WinFitter program with accompanying documentation is available free to anyone who wants to use it. A zip file containing the installation file, the user's manual and example input and output files can be downloaded from DropBox by clicking the following link: WinFitter 2.3.zip.

**References**

Adby, P. R. and Dempster, M. A. M., 1974, *Introduction
to Optimization Methods* (London: Chapman and Hall).

Bevington, P. R., 1969, *Data Reduction and Error
Analysis forthe Physical Sciences* (New York: McGraw-Hill).

Budding, E., 1973, *Astrophys. Space Sci.,* **22**,
87.

________, 1977, *Astrophys. Space Sci*., **46**,
407.

________, 1993,* An Introduction to Astronomical
Photometry* (Cambridge: Cambridge University Press).

Budding, E., and Najim, N. N., 1980, *Astrophys. Space
Sci*., **72**, 369.

Budding, E. and Zeilik, M., 1986, in *Cool Stars,
Stellar Systems, and the Sun*, ed. M. Zeilik and D. M. Gibson (Heidelberg:
Springer-Verlag), p. 290.

Budding, E. and Zeilik, M., 1987, *Ap. J*., **319**,
827.

Busso M. et al., 1984, *Astron. Ap*., **135**,
255.

Hall, D. S., 1981, in *Solar Phenomena in Stars and
Stellar Systems*, ed. R. M. Bonett and A. K. Dupree (Dordrecht: Reidel), p.
431.

Olah, K., Hall, D. S., and Gesztelyi, L., 1986, preprint.

Rodonò, M., 1986, in *Cool Stars, Stellar Systems, and
the Sun*, ed. M. Zeilik and D. M. Gibson (Heidelberg: Springer-Verlag), p.
470.

Vogt, S. S., 1981, *Ap. J*., **250**, 327.

Vogt, S. S., 1983, in *Activity in Red Dwarf Stars*,
ed P. B. Byrne and M. Rodonò (Dordrecht: Reidel), p. 137.

*Last
updated: February 8, 2014 by Michael Rhodes*

Michael_Rhodes@byu.edu