Astronomy Page
1. Introduction
This paper describes CurveFit, a Windows program developed for analyzing the light curves of RS Canum Venaticorum binary stars and for doing theoretical fits of circular dark spots to the distortion-wave curves of these stars. However, its use is not restricted to this class of stars. It can be used to fit any eclipsing, non-contact binary stellar systems and to match starspots to any photometric distortion wave.
The CurveFit package is written in Delphi that combines the functions of an older group of DOS programs written in Fortran. The package includes: 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); a spot fitting utility developed first by Budding and Zeilik (1986), and further refined by Rhodes (1989); a number of auxiliary utilities, all developed by Rhodes (1989), to manually enter light curve data, to bin the data (if there are more than 200 data points), to add the necessary parameters that the curve fitting and spot fitting utilities require to the raw data files, to shift the data in magnitude and phase to the format expected by the curve fitting and spot fitting utilities, to make quick screen plots of the various output data from either curve fitting or spot fitting utilities to see if the curve fitting is acceptable, and to plot the position and sizes of the spots on a Mercator projection of the stellar surface.
The CurveFit utilities are robust and user-friendly, and with the universal availability of PCs, they provide an inexpensive means of analyzing eclipsing binary light curve data in a consistent way.
2. Description of Spot Fitting Algorithms
The methodology used for extracting maculation effects from the light curves of these eclipsing binary stars is based on the chi-square minimization computational algorithms developed by Budding and Najim (1980) and further refined by Budding and Zeilik (1987, see also Budding, 1993). This 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 2 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.
SV Camelopardalis Theoretical Curve 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 CurveFit
CurveFit runs on an IBM compatible
microcomputer. 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 95--it will not run on earlier versions of
Windows, but it will run on Windows 98, 2000, NT, and XP. When the program is
run, the window shown in Figure 1 opens with drop-down menus for the program
utilities and procedures.
Note that this program replaces the earlier suite of Dos light-curve fitting
programs. Those programs are still available for those who would rather use Dos.
The CurveFit program consists of the following procedures:
Add Parameters – This option has two sub-options
Curve Fitting – Allows you to open a data file and add the required parameters for the operation of the binary light curve fitting procedure.
Spot Fitting – Allows you to open a data file and add the required parameters for the operation of the spot fitting procedure.
Bin Data – Used to bin the data in a data file containing more than 200 data points to reduce the number of data points to 200.
Fold Data – Used to combine data points of two or more files into a single file.
Enter Phase/Magnitude – Allows the manual entry of phase and magnitude data. The phase can be entered in phase units, degrees, Universal Time, or Heliocentric Julian date.
Shift Data – Contains the following sub-options:
Max Delta Mag = 0 – Shifts the values of the delta magnitude of a data file so that the maximum delta magnitude is equal to zero, the form expected by the binary curve fitting procedure. Also, if the eclipse is more positive in magnitude, it inverts the values so the eclipse is more negative in magnitude.
Max Delta Mag at Phase 0 – Shifts the phase so that the maximum delta magnitude corresponds to phase zero.
Max Delta Mag = 1.0 – Shifts the values of the delta magnitude of a data file so that the maximum delta magnitude is approximately equal to 1.0, the form needed by the spot fitting procedure.
Graph – Contains two options:
Light Curve – Plots delta magnitude versus phase. Two data files can be plotted to compare for example the modal curve with the actual data. Both light curve and spot curve data can be plotted. The graph can be saved as a Windows bitmap file for further processing or pasting into other applications
Spot – Plots the positions and sizes of star spots on a Mercator projection of the star. The graph can be saved as a Windows bitmap file for further processing or pasting into other applications
Fitting – Contains two options:
Light Curve - Fits input data of phase and delta magnitude for an eclipsing binary star to a modal curve.
Spot – Fits input data of phase and delta magnitude to a modal star spot curve.
Mercator Plot of Star Spots
The CurveFit program with accompanying documentation is available free to anyone who wants to use it. You can download it from http://winsite.com/ The DOS programs are also available there. You can also send an email request to michael_rhodes@byu.edu, and I will email you a zip-file with the program and a users manual.
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 28, 2006 by Michael Rhodes
Michael D. Rhodes