CHAPTER 1
INTRODUCTION

The analytical results presented in this dissertation attempt to constrain the models used to describe RS Canum Venaticorum (RS CVn) binary star systems. Recently developed techniques for the power spectrum estimation of unevenly-spaced data are utilized in the analysis of UBV photometric observations of 36 RS CVn binaries taken with the Fairborn Observatory's Automatic Photoelectric Telescope (APT) from the last quarter of 1983 to the first quarter of 1985. The content of the power spectrum for each source is extracted and models are described that are consistent with the observed characteristics within the spectra. An interactive data analysis environment was written to facilitate the investigation of the analysis techniques and the reduction of the APT data.

1.1 Background

RS Canum Venaticorum (RS CVn) type star systems are a class of chromospherically-active binaries, whose defining characteristics have evolved since Struve (1946) first called attention to the group. Oliver (1974) was the first to formally propose a set of observational characteristics to define the RS CVn criteria, but the working definition, as it is used today, was that set down by Hall (1976). He divided the RS CVn systems into five separate subgroups:

I. Regular Systems:

1. Orbital periods between 1 and 14 days.
2. The hotter component is of the spectral type F or G and luminosity class V or IV.
3. Strong Ca II H and K emission is seen outside of eclipse.

II. Short Period Systems:

1. Components are detached.
2. Orbital periods less than 1 day.
3. The hotter component is of the spectral type F or G and luminosity class V or IV.
4. Ca II H and K emission is displayed in one or both components.

III. Long Period Systems:

1. Orbital periods greater than 14 days.
2. Either component is of the spectral type G through K and luminosity class II through IV.
3. Strong Ca II H and K emission is seen outside of eclipse.

IV. Flare Star Systems

1. The hotter component is of the spectral type dKe or dMe, where the emission refers to strong Ca II H and K.

V. V471 Tau Type Systems:

1. The hotter component is a white dwarf.
2. The cooler component, spectral class G through K, displays strong Ca II H and K emission.

In creating his working definition, Hall evaluated a list of 24 systems that he considered candidates for the RS CVn criteria. He examined the observational characteristics of each system and noted which characteristics were exhibited by all 24 systems. The list of RS CVn type systems has grown to over 100 since Hall's defining paper, and the availability of the informally published Catalog of RS CVn Binary Star Systems has been announced by Nelson and Zeilik (1984). This catalog, and several others, have recently been incorporated into the Catalog of Chromospherically Active Stars (Strassmeier et al. 1988) which will serve as a prime reference for the elements of the systems under study.

The light curves of RS CVn type systems show a peculiar semiperiodic structure outside of eclipse. This structure has been referred to as a distortion wave in the light curve. Catalano and Rodono (1969, 1974, 1976) attributed the source of the distortion wave to circumstellar material around the hotter component, while Popper (1977) considered non-radial standing waves in the cooler stars outer envelope. Eaton and Hall (1979) concluded, however, that these mechanisms for generating the light variations were not viable and that the simplest mechanism was "starspots", which, in analogy to sunspots, are large, cool active regions on the photosphere.

Chromospheric activity is signaled by the presence of emission cores in the Ca II H and K resonance lines. Balmer emission, or Ha, is also associated with active chromospheres. X-ray emission is known as a tracer for active coronal regions, and ultraviolet (UV) emission and flaring are, by solar analogy, known to be associated with stellar active and transition regions. These areas on the Sun are associated with intense magnetic fields, and sunspot activity is enhanced in and around these magnetically active regions.

RS CVn's are known X-ray (Walter et al. 1980; Caillault 1982; Rengarajan 1983; Schrijver 1983; Schrijver et al. 1984) and radio emitters (cf. Mutel and Lestrade 1985). The radio emission is nonthermal in origin (gyrosynchrotron) and is one of the few direct indicators of magnetic fields. The X-ray luminosities are on the order of L_x » 10²⁴ watts. This emission has been interpreted, in solar analogy, as being caused by a hot, T ~ 10⁷ K, corona. Walter et al. (1980) have proposed a coronal model that differs from that of the Sun only in terms of scale. Schrijver et al. (1984) have shown a correlation between X-ray emission and rotational period, which can be interpreted in terms of the coronal structure visible on the sun. Majer et al. (1986), however, found no correlation between X-ray luminosities and the orbital period, rotational velocity or any other orbital parameter as is normally the case for late type main-sequence and giant stars.

Generally, RS CVn's show all the characteristic features indicative of chromospheric activity, but while the qualitative interpretations of the observations in terms of the solar-stellar connection and magnetic activity cycles are well described, the quantitative interpretations are not (Bopp 1983). Starspots are considered a logical extension of the observations from the solar-stellar connection for magnetic active regions and, when properly characterized, may be used as tracers of magnetic activity cycles in cool stars. The problem is in trying to constrain the models while introducing a minimum number of assumptions.

The starspot model was originally proposed by Kron (1947) to account for the light variations of AR Lac. It was later refined by Hall (1972) and Oliver (1974). Hall extended the model in subsequent papers (Hall 1975, 1976; Eaton and Hall 1979). The fundamental problem with the starspot model is that a unique solution to the photometric light curve cannot be determined. Kopal (1982) has pointed out that Bruns (1882) proved early on that the photometric problem is indeterminate for a star with an arbitrary surface brightness distribution, yet solutions for the distorted light curves of RS CVn's are still carried out in the classical manner (Blanco et al. 1983; Busso et al. 1984, 1985, 1986; Poe and Eaton 1985; Caton 1986). Budding and Zeilik (1987), however, have recently addressed the indeterminacy problem in terms of a rigorous treatment of a c² optimization as described by Budding and Najim (1980). Their solution is well defined, but the method is model dependent and can only be applied to high signal to noise data (~100) from eclipsing systems, and as with all forms of multivariate analysis, a global minimum in c ² can never be guaranteed.

More recently, Lines et al. (1987), have published the spectral decomposition of HD 185151, but this work was done in one color (V-band) and their method of least squares fitting of sine waves to the data was not well described in the paper. For the two periods identified, the shorter was attributed to ellipticity (P = 20.0665^d ± 0.0044) and the longer (P = 39.878^d ± 0.036) to starspots.

Several years ago Vogt and Penrod (1982, 1983) described a technique, Doppler imaging, for obtaining resolved images of the surface features of rapidly rotating stars. The technique exploits the correspondence between wavelength position across a rotationally broadened spectral line and the spatial position across the stellar disk. The difficulties in inverting the spectral line profiles into an image of the stellar disk are currently being addressed by Vogt, Penrod and Hatzes (1987).

Doppler imaging has provided one of the few direct pieces of evidence for the existence of starspots, or localized plages, but the technique is severely restricted. The data must be of very high quality (S/N > 400) and the method can only be applied to rapidly rotating stars (v sin i » 20 km/s) up to a limit where the signal is swamped by terrestrial and stellar line blends (v sin i » 100 km/s). The stellar inclination is limited to a band of 20° £ i £ 70°, and the physical properties of the atmosphere must also be known, although the process is fairly insensitive to the detailed physics of spectral line formation.

The problem still remains, however, as to the characterization of the phenomena from the light curve. If the light curve can be used as a tracer of stellar surface activity, then the photometric distortions can be used to characterize magnetic activity cycles and add to the understanding of stellar dynamics (cf. Baliunas and Vaughan 1985).

1.2 Analysis Techniques

Analysis of even the simplest eclipsing binary system is difficult owning to the complexity of the phenomena involved (limb darkening, gravity brightening, reflection, geometrical distortions, tidal effects and atmospheric eclipses). Table 1-1 summarizes the observations needed for the general characterization of a binary system. Computer programs have been developed that attempt to "solve" the light curve in both direct (i.e., Wood 1971a, 1971b, 1972, 1973a, 1973b) and indirect (i.e., Budding 1973) fashions to extract the physical parameters that describe the known physics of the system. Direct methods attempt to fit the observations by creating a computer model of the system, while the indirect methods rely on the frequency domain behavior (Kopal 1979). The difficulty with these methods is that the solutions are model dependent and they can only be applied to eclipsing systems. The problem is exacerbated by the introduction of starspot models.

The fundamental problem is to extract the information content of the light curve. For well-behaved eclipsing systems, the standard methods are well proven (Russel 1912; Kopal 1959, 1979; Sobieski 1965; Lucy 1968; Napier 1968; Hill and Hutchings 1970; Horak 1970; Wood 1971a, 1971b, 1972, 1973a, 1973b, 1976; Wilson and Devinney 1971; Mochnacki and Doughty 1972; Budding 1973; Smith and Theokas 1980; Linnell 1984; Popper 1984) even though the integrity of an individual solution is a function of the numerical techniques involved. For non-eclipsing systems, however, these methods are not readily applicable, and we must turn to extracting the spectral content of the light curve.

Photometric observation are, by their very nature, unevenly-spaced and gapped, caused by both weather and the seasonality of the objects. The estimation of the power spectrum of a time series formed by unequally-spaced observations is one of the most important problems in time series analysis (Jones 1977). The discrete Fourier transform (DFT), and its digital counterpart the fast Fourier transform (FFT), are the methods of choice when dealing with regularly-spaced data. Theoretically, the use of the DFT for unequally-spaced data is invalid since an Abelian group cannot be found such that the process is homogeneous, unless the process is Gaussian (white noise). Theoretical formalities aside, the analysis of simulated data (Ferraz-Mello 1981; Kovacs 1981) shows that the DFT gives results in good agreement with the true frequency for a wide range of situations with errors rarely exceeding 0.5%. Where the transform fails is in its inability to determine the amplitude and phase of the signal, giving errors that can be as high as 50% of the true values. If one is only interested in a coarse (0.5%) estimation of the frequency within a signal, then the discrete Fourier transform is adequate for unevenly-spaced data.

Horne and Baliunas (1986) have described a method for the accurate period determination of irregularly-spaced data based on the modified periodogram (Scargle 1982). This method not only gives an excellent estimation of the frequency of the periodic signal, but it also provides for: the estimation of the statistical error in the frequency determination (Kovacs 1981; Baliunas et al. 1985), an estimation of the expected noise peak value (Scargle 1982), an estimate of the significance of the height of a peak in the power spectrum (Horne and Baliunas 1986) and an estimate of the resolution of the periodogram.

Periodic processes may be formed by several waves, not necessarily harmonic. Although one period may be selected from inspection of the spectrum, little can be gleaned about other periodicities from the same spectrum. Other visible peaks may just be the result of leakage from the primary signal, or weaker signals may simply be swamped by the strength of the primary (Kovacs 1981). The simplest, and most direct, method to search for multiple periodic components is through harmonic filtering (Ferraz-Mello 1981) and will be detailed in Chapter 2. Once the primary frequency has been filtered, the data may be re-analyzed for secondary periodicities. This process may be repeated until all periodic structure has been determined.

The problem now arises as to how to perform harmonic filtering on an unevenly-spaced data set, since the finite Fourier transform is inadequate for this purpose. Ferraz-Mello (1981) has proposed the use of the data-compensated discrete Fourier transform (DCDFT) to overcome the limitations of the finite Fourier transform and allow for the correct filtering of a signal. The DCDFT is based on the orthonormalization of the sine and cosine phases, using the classical Gram-Schmidt formulas, over the domain. The next chapter details Fourier transforms, the modified Scargle periodogram and the DCDFT.

1.3 Approach

Rather than trying to fit data to a preconceived physical model, I extract the spectral information from the light curves using a combination of the methods outlined in Chapter 2. Most of the analysis is performed on non-eclipsing systems where the light curve is not complicated by the effects of an eclipse, but eight eclipsing systems are examined.

The data for this work was supplied by the Fairborn Observatory's automatic photoelectric telescope (Boyd, Genet and Hall 1985; Genet and Baliunas 1986; Hall 1986). Their automated system has the potential to build up a massive data base on a wide range of objects and will be described in Chapter 4. Currently, only the data from the last quarter of 1983 to the first quarter of 1985 (which contains 36 RS CVn sources) is available, but this data was sufficient for the analysis.

One difficulty in analyzing the APT data is to distinguish between the effects of data spacing and noise, and the true spectral content of the data. I modified the WINK light curve solving program (Wood 1973, 1973-1982; Vaz 1984) in order to assist in the phenomenological investigation. Light curves are generated, using the timing information from the actual data, that reflect the possible, classical, physical state of the systems. Noise is added to the synthetic data sets, and they are analyzed in the same manner as the real data in order to uncover spectral content correlations.

This type of deductive modeling allows constraints to be placed on the mechanisms that are responsible for the observed phenomenology while avoiding the indeterminacy problem addressed by Bruns (1882). It is not my intent to "solve" the light curves but rather to define the classes of models that can reproduce the observed spectral content of the systems, in effect providing a "proof of concept" for the models and eliminating their uncertainty in terms of the information content of the light curve.

1.4 Analysis Environment

One difficulty in analyzing large volumes of data, such as that produced by the automatic photoelectric telescopes (APTs), lies in the usual analytical approach. Each object is viewed as separate from the entire data base, so processing has a tendency to proceed in a serial, rather than parallel, manner. Programs are written to perform specific tasks (Fourier transforms, plotting, etc.), but they are not integrated in such a way as to allow for simple manipulation, analysis and display of the data. Integrated business packages, such as Symphony and Innovative Software's SMART system, do allow for easy manipulation of individual data elements and make it relatively painless to manipulate the data for such tasks as sorting, editing and tapering. However, these business packages are limited in their use for scientific applications.

It is possible to program the spreadsheets, for example, to perform almost any needed set of calculations, but problems such as data display capabilities and speed are an important factor and can only be addressed by the creation of specialized application programs, so it was necessary to develop a user interactive time series analysis package in order to facilitate the investigation of the numerical methods.

The analysis package, referred to as TISAN (TIme Series Analysis eNvironment), is written in Microsoft C 4.0 for the 8086/80286/80386 based work stations, with an optional math coprocessor, that have the needed speed, mass storage and low cpu-hour cost requirements. The package is based on an architecture similar to the Astronomical Image Processing System (AIPS) produced by the National Radio Astronomy Observatory (NRAO) and will be detailed in Chapter 3.

The analysis in this dissertation has not been previously addressed for at least the following reasons:

1. Large photometric data bases taken with a dedicated instrument are difficult to come by. The APT data is ideal for investigating this type of analysis since it provides a large number of sources from a single instrument over an extended period of time.
2. The numerical methods for the spectral analysis of unevenly-spaced data have only recently been described in the literature and so have yet to gain wide use.
3. The 8086/80286/80386 based microcomputers have made available the computing speed necessary for the implementation of the numerical methods, at an affordable cost, for the volume of calculations that must be performed.
4. For the investigation of the analysis techniques, it is essential that all the needed programs be written in a user interactive environment, but integrated, scientific, specialized software is difficult to find and so it must be written.

In summary, recently developed techniques for the spectral estimation of unevenly-spaced data are described in Chapter 2. These techniques are used in the analysis of UBV photometric observations of 36 RS CVn binaries taken with the Fairborn Observatory's Automatic Photoelectric Telescope, described in Chapter 4, from the last quarter of 1983 to the first quarter of 1985. The entire spectral content of each source is extracted and models are described that are consistent with the observed characteristics within the spectra. An interactive data analysis environment was written to facilitate the investigation of the proposed analysis techniques and is discussed in Chapter 3. Chapter 5 discusses the history of the sources and Chapters 6 and 7 present the results of the spectral analysis for the individual systems and the physical interpretations, respectively.

Table 1-1 Summary of Information Obtainable from Binary Systems

Table 1-1 outlines the information obtainable from binary star systems and is primarily taken from Batten (1973), where the interested reader is referred for a detailed description of the elements. The subscripts (1 and 2) refer to the primary and secondary star, respectively.

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Copyright (c) 1988-1997, Eric R. Nelson, Ph.D.