The uniqueness problem in starspot models; a critical review

New Astronomy 4 (1999) 365–376 www.elsevier.nl / locate / newast

The uniqueness problem in starspot models; a critical review Zeki Eker 1 ,2 King Saud University, College of Science, Department of Astronomy, PO Box 2455, 11451 Riyadh, Saudi Arabia Received 29 October 1998; accepted 26 January 1999 Communicated by Peter S. Conti

Abstract Within the scope of the uniqueness problem, the scientific credibility of the starspot hypothesis is examined. According to the two analyzing criticisms in this study (general and specific approaches), the starspot hypothesis is a consistent physical problem. That is, there is no uniqueness problem which discredits the starspot hypothesis. Confusions among the different meanings of the word ‘‘uniqueness’’ act as a prime source of the ambiguities. Nevertheless, the empirical evidences implying non-uniqueness of spot solutions, such as indeterminacy of spot latitudes, and therefore latitude fixing, and dissimilar solutions of models, are actually caused by insufficient accuracy of observational data not because of an intrinsic problem of uniqueness. Therefore, the starspot hypothesis must be attributed at least the same level credibility as the other light curve analyzing techniques of eclipsing binaries or cepheids since non-uniqueness because of observational data with limited accuracy is also a problem for them. Studies proving unique spot effects on the light curves and on the line profiles are reviewed. Confusing comments and misleading statements about the uniqueness problem in some leading literature are criticized.  1999 Elsevier Science B.V. All rights reserved. PACS: 97.10.pg; 97.30.b Keywords: Stars: activity; Stars: variables: other

1. Introduction For about three decades now, modeling techniques of starspots fulfilled their goal to explain distortions observed in light curves and spectral line profiles of spotted stars. However, the current aim of those models has developed along different lines: primarily to generate information about the discrete surface features that eventually will allow understanding of magnetic activity and dynamo mechanism. Consequently, in addition to generate synthetic curves 1

¨ Visiting Astronomer, Solar Observatory Kanzelhohe, Treffen, Austria. 2 E-mail: [email protected]

fitting the observations, current spot models are also expected to produce reliable spot parameters such as sizes, shapes, locations (distributions) and temperatures of virtually existing spots. But today’s most developed Doppler imaging and light curve modeling techniques, which aim to map spatially unresolved stellar surfaces, have not yet matured enough to be reliable in the predictions of spot parameters. Speaking about the reliability of predicted spot maps, the uniqueness problem certainly comes first to one’s attention. On the other hand, the problem of uniqueness is not new for the models developed in the last three decades. The history of starspot hypotheses actually goes back more than 300 years (Kopal, 1982; Hall, 1994). During the early centuries

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(17th, 18th and 19th) starspots were invoked to explain all types of variability, even novas and supernovas were no exceptions. The uniqueness problem has been noticed by the turn of the last century. After the famous article of Russell (1906) stating that a given light curve can be produced by an arbitrary distribution of brightness on the surface of a rotating body, starspots were totally rejected by the astronomers for nearly half a century. The starspot hypotheses were rejected in this period because any light curve shape, regular or irregular, could also be explained by eclipses, radial or nonradial pulsations, obscurations by circumstellar material or by the combination of similar physical events. Along with the fact that no spotted star had been found in this period, according to Hall (1994), the uniqueness problem contributed to the starspots fall from grace. Even quite recently (Kopal, 1982) the uniqueness problem has been invoked to argue that starspots do not exist. According to Vogt (1981), the uniqueness problem is one of the reasons why starspot hypotheses have been slow to achieve widespread acceptance. Despite the widespread acceptance of the starspot hypothesis today and the existence of new methods giving unique results such as the maximum entropy (Vogt et al., 1987) and ILOT (Information Limit Optimization Technique, Budding & Zeilik, 1987) the problem of uniqueness still remains unsolved. This is because, as it is usually explained, the unique solutions offered by these techniques give us the simplest, smoothest solution or a solution with the optimum number of parameters but not the true solution. Therefore many authors (Bopp & Noah, 1980; Vogt, 1981; Kopal, 1982; Vogt & Penrod, 1983; Huenemoerder & Ramsey, 1987; Neff, 1990; Sarma et al., 1991; Anders et al., 1992; Dempsey et al., 1992; Huenemoerder et al., 1993; Maceroni & van’t Veer, 1993; Hall, 1994; Patkos & Hempelmann, 1994; Elias et al., 1995; Byrne et al., 1995; Raveendran & Mohin, 1995; Hempelmann et al., 1997; Olah et al., 1997) clearly declared that an infinite number of solutions may exist as well. The problem is serious because if there are infinite solutions because of infinite possibilities of different combinations of the sizes, locations, shapes and temperatures, in reality there cannot exist any solution.

On the other hand, the concept of uniqueness appears to be different among the different research groups. Since there is no consensus, there could exist misconceptions. Eker (1996) has already argued that Russell (1906) was understood wrongly as favoring non-uniqueness. Confusing comments and unclear statements in the leading literature of the last thirty years about the problem of uniqueness show that there exist indeed misconceptions about the uniqueness problem of starspots. Criticisms of such comments with a healthy analysis of the uniqueness problem are now urgently needed. The misconceptions or the unclear ideas of the uniqueness exist not only because Russell (1906) was understood wrongly, but also because the issue of uniqueness is not simple, but complex. The word ‘‘uniqueness’’ does not seem to have a unique meaning as it is discussed in the literature. With a simple thinking, one can easily notice that there could be at least four types of uniqueness which are: (i) Uniqueness of a final solution which is obtained by the best fitting curve to observed data; (ii) Uniqueness of spot signatures on the light curves (or line profiles); (iii) Uniqueness of the true solution (recovered image being identical to the original image); (iv) The uniqueness which is not disturbed by the limitations of image reconstruction problem like the uniqueness of photographs. Therefore, an analysis of the uniqueness problem of the starspot hypothesis in a wider context will be considered to restore a proper scientific respect to it. The resolution of the issue is certainly critical for the further development and believability of the starspot hypothesis.

2. Analysis of the uniqueness problem

2.1. General approach There can be two approaches to analyze the uniqueness problem of starspot the hypothesis; (i) general approach, and (ii) specific approach. The first approach is named general because it can be applied to any physical problem. If a method involves a curve fitting to a limited number of data with a limited signal-to-noise (S / N) ratio and with

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unknown external errors, following types of uniqueness can be investigated by the questions presented.

a spot (size, shape, location and temperature) from a light curve? (or from a line profile?)

• Type 1. The uniqueness of the function to generate a fitting curve. Choosing a function in most physical problems does not pose a problem because usually there are certain mathematical expressions for certain physical problems, for example the Planck curve to represent the spectral energy distribution of a radiating object. But, if the data is not known to be associated with a specific function, usually the shape of the data plot can be used as a clue to choose a function to fit from the list of mathematical functions. In such cases, there can be many functions (curves) to fit a same data set. Unless the answer to the following question is yes, there must be a uniqueness problem of type 1. Do we have only one function to generate the theoretical curve to fit the data? • Type 2. The uniqueness of the fit. Suppose there is only one function to create a curve to fit, that is, our problem is engaged with a special function to express the observed data. Again, unless the answer to the following question is ‘‘yes’’, there must be a uniqueness problem of type 2. Is a unique fit possible with a given function (curve) for the data? • Type 3. The uniqueness of the parameter space. The uniqueness problem cannot be avoided even though one answers ‘‘yes’’ to both questions above without answering ‘‘yes’’ to the third question. Are the parameters producing the curve unique?

Certainly, both questions require the answer ‘‘yes’’ to make sure that there is no uniqueness problem. The specific approach is sufficient to indicate if there is a uniqueness problem involving specifically the starspot hypothesis. That is, it implicitly contains the answer ‘‘yes’’ to the first question in the general approach. This is because without a function, that is, without a mathematical expression for computing spot effects from the physical parameters of the spots, it would be meaningless to discuss the uniqueness of the parameters. On the other hand, one may prefer to use non-physical functions or a series containing harmonic functions to represent spots. However, the specific approach refuses to use such functions because such functions may not be good enough to represent the physical parameters of real spots. Nevertheless, there should be no need for such functions if one is able to compute spot effects on the light curves (or line profiles) by using physical parameters of real spots. Moreover, the specific approach makes it possible to analyze the uniqueness problem without involving ambiguities caused by insufficient accuracy of observational data. That is, if such ambiguities exist, they must exist also for the light variations of eclipsing or pulsating type. Therefore, answering ‘‘yes’’ to the above questions at least will show that the starspot hypothesis is not different from other types of light variations with physical credibility. On the other hand, it is commonly accepted that there are curve fitting techniques such as the least squares method or maximum entropy technique which allow one to achieve a unique solution from a limited data set with estimated standard error. Thus, answering ‘‘yes’’ to the above questions in the specific approach will be sufficient to clarify a century old problem of uniqueness in the starspot models. However, it will be shown in the following criticisms that, actually, the uniqueness problem is not that simple. Nevertheless, there can be confusions between different literal meanings of the word ‘‘uniqueness’’. Therefore, a deeper analysis of the uniqueness problem will be presented in the following sections together with criticisms of the confusing

2.2. Specific approach In order to clarify the uniqueness problem of the starspot hypothesis, clarifying the uniqueness type 3 is of crucial importance. Actually this is the problem that concerns the starspot hypothesis specifically. The following questions should be sufficient to clarify it. 1. Does a spot with specific parameters such as size, shape, location and temperature have a unique signature on the light curves? (or on the line profiles?) 2. Is it possible to recover the physical parameters of

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comments and misleading statements in some published articles.

3. Criticisms

3.1. General According to Sir Karl Popper (1972), who is one of the most distinguished philosophers of science in the 20th century, if a theory explains everything, it has no scientific value. Long before Popper developed these philosophical ideas, Russell (1906) was aware of the problem. Expressing surface brightness inhomogeneities with harmonic functions, his prime concern was that there could be discontinuities, like that of a Fourier series over the surface, but the light curve and its first derivative must be finite and continuous everywhere. Thus, Russell (1906) was proving that some types of light curves (e.g. eclipsing) cannot be reproduced by the starspot hypothesis. On the other hand when Russell (1906) said ‘‘infinite variety of ways’’, he meant mathematical possibilities to express the shape of an asteroid and then the distribution of brightness on its surface. He said also that ‘‘the problem is in general still indeterminate. In certain cases, it may not be so. For a planet, the form of the expression [B(q,w ) is an harmonic function containing truncated series of sine and cosine functions to represent surface brightness distribution] will be completely determined by the observations only when i 5 p / 2; that is, when the planet’s axis of rotation lies in the plane of its orbit’’. Finally, he ends his paper saying that ‘‘considerations of the light curve of a planet at phases from opposition may aid in determining the markings on its surface, but cannot help us to find its shape’’. Long before radiation transfer and model atmospheres developed the achievements of Russell are remarkable. Today it is clear that spherical harmonics are not good expressing spots because overlapping of harmonic functions is natural, but spots are not permitted to overlap. Moreover, the total effect of harmonic functions, that is, the value of B(q,w ), although, is finite and positive (or zero), there can be individual sub-functions in B(q,w ) with negative values. However, physical spots can have

only positive effects. Today, spherical harmonics are applied to study solar and stellar seismology where various modes of radial and non-radial pulsations may well coexist. According to the analysis of Russell, first, the uniqueness problem of type 1 was in question. Thus, He was drawing attention to the function that expresses the brightness distribution on the surface, then its derivative to represent the observed light curve. Although, it is in a form of series, a single function B(q,w ) was used. Therefore, there is no uniqueness problem of type 1. He did not discuss the second kind of non-uniqueness. Probably, it was already clear to him that it is always possible to achieve a unique fit by methods like the least squares. Actually, it was the third kind of uniqueness which he was not sure of. ‘‘The form of the expression’’ he says ‘‘will be completely determined by the observations’’ by implying the function B(q,w ). Of course, such unphysical functions may not have the ability to indicate all physical parameters of real spots. Thus, he said ‘‘the light curve of a planet can determine the markings on its surface, but not its shape’’. After 76 years, during the revival of the starspot hypothesis, Kopal (1982) was arguing ‘‘Starspot hypothesis is too simple, unphysical to explain light variability.’’ This was because, according to him ‘‘Unless an alternative check on their existence and distribution independent of the light curves, observed light curves cannot produce unique information.’’ There, Kopal was reminding of the failures of the starspot hypothesis at the turn of this century to explain other types of variations such as eclipses and pulsations etc. He interpreted this as an argument against the starspot hypothesis. Actually, he was stressing mainly the uniqueness problem of type 1. According to Kopal, a physical cause modulating the starlight is always unknown unless there is an independent acceptable evidence. So, there could be many functions (curves) to fit the observed data. A cepheid or an eclipsing type variability has independent evidences which are supplied by spectroscopic observations of correlated radial velocity variations implying binarity or surface fluctuations. Perhaps, Kopal did not accept correlated color variations of the starlight, i.e. being redder at the light minimum, as an independent evidence to cool

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surface spots. But soon after his article had been published Doppler imaging aiming to map spotted stellar surfaces was introduced (Vogt & Penrod, 1983; Vogt et al., 1987). This was an independent spectroscopic method to search for starspots. Therefore, in addition to correlated light and color variations, correlated line profile distortions (Doppler images) due to cool surface spots and correlated chromospheric activity which is indicated by Ca II H and K and Ha emissions, and magnetic activity which is supported by X-Ray and radio observations, certainly support the starspot hypothesis. On the other hand, although there is no characteristic shape of the light variation due to cool surface spots, a synthetic light curve or a line profile including spot effects can be computed from the physical parameters of spots. The function to generate theoretical curves was already named as maculation wave function (Budding & Zeilik, 1990; Budding & Zeilik, 1994). There cannot be a uniqueness problem of the first kind as Kopal (1982) suspected. Moreover, there is no need to express the surface brightness distribution by harmonic functions like Russell (1906) did and it is also meaningless to compose an unphysical function like the Hall group (Hall et al., 1990b; Hall et al., 1990a; Hall et al., 1991; Hall et al., 1995; Hall & Henry, 1992; Fekel et al., 1993; Henry et al., 1995; Crews et al., 1995; Eaton et al., 1996; Kaye et al., 1996) does by using half sine curves to fit the observed data. In the final analysis now, it should be clear that unless the parameter space is not unique, that is, physical parameters such as size, shape, location and temperature are not unique to produce a synthetic curve, just because of the second type of non-uniqueness which is caused by unknown external errors and limited signal-to-noise of the data, it is not fair to disgrace the starspot hypothesis. Limited signal-to-noise ratio and unknown probable errors also exist for a cepheid or an eclipsing type variation. Not only light variations, actually any physical problem to which solutions are sought by curve fitting is subject to face this kind of problems.

3.2. Doppler maps of spots, unique or not unique? If one starts from a limited number of data per phase bin, finite signal-to-noise (S / N) ratio per data

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point with unknown external errors, without additional constrains, it is not possible to achieve a unique solution by a simple trial and error method. One likely criterion is to choose the simplest or smoothest solution among the possible solutions as suggested by Vogt et al. (1987), who introduced the maximum entropy method into Doppler imaging. Image reconstruction by maximum entropy involves finding an image with the greatest configuration entropy. ‘‘The maximum entropy image is the one with the least amount of spatial information, and is thus the smoothest or simplest image’’ Vogt et al. (1987). A reconstructed image as being simplest or smoothest is unique by definition. Thus, even with such a simple logic, the maximum entropy allows one to obtain a unique solution. However, uniqueness associated with such a solution is often confused with the uniqueness of the true solution, which is the image identical to the original image. Therefore, despite the fact that the maximum entropy method supplies a unique solution, Vogt et al. (1987) claim that there is, in principle, no unique solution because this simplest or smoothest solution is not guaranteed to be the true solution. After explaining why the maximum entropy method gives a unique solution, Vogt et al. (1987) try to explain why there is no unique solution without noticing their verbal contradictions. They say, ‘‘casting the problem in matrix form, IR 5 D, where I and D are the image and data vectors and R the transfer matrix between the two quantities. Mathematically, this equation can be solved directly for the image vector by finding the inverse of matrix R. Then, I 5 DR 21 .’’ In practice, they claim, this cannot be done. This is because, R to be invertable, not only must the matrix R be square, but also the rows of R must be independent. Since the inverse of R is not available, guessing the image vector I relies on iterations, trials and errors between the image and data space. Their explanation appears convincing if one is not aware of logical fallacies. First, a solution found by trials and errors cannot be trusted as unique. The second is more striking because according to vector algebra it is known that such transfer matrices of size m 3 n, where m ± n, do not support a unique transfer. At this point the word uniqueness implies a one to one transformation between the

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image space and the data space. Thus, according to Vogt et al. (1987), there cannot be a unique transfer between these two spaces. However, even themselves are not sure about this since they have argued that even if there is one, the reconstructed image always inherits an uncertainty from the data with finite signal-to-noise ratio which means that one should not hope to recover the true image. Thus, recovered spot maps cannot be unique. However, the opposite was proven by their ‘‘vogtstar’’ test. The test image, which are the letters V-O-G-T as being artificial spot images on the hypothetical ‘‘vogtstar’’, has been recovered successfully and uniquely from the pre-generated spectral line profiles, certainly well enough to recognize the letter shaped spots and even some details to differentiate between the ‘‘O’’ and the ‘‘G’’ letters. It seems that the least of their expectations (invertable R) came true. Finally, they said ‘‘Unlike the traditional light curve modeling approach, which provides almost no unique information about spot shapes and locations, Doppler imaging determines both rather accurately’’. However, according to them, the ‘‘nonuniqueness’’ still exists but within the uncertainty inherited from the data. Non-uniqueness cannot be denied if there is no unique transformation between the data and image vectors. But it would be a logical fallacy to accept this blindly. Theoretically, there can be an infinite number of independent data to construct a data vector. On the other hand, an infinite number of pixels is required to construct an image even on a limited surface if the pixel size is infinitely small. In practice, however, the number of data is limited when constructing a data vector. Nevertheless, this does not mean the transformation matrix R may not be square. Actually, it should mean that there is always a limited resolution to construct an image from a limited data set. It is a well known rule that one cannot ask for more information than the data can supply. Therefore, R must be a square matrix of size n 3 n, where the value of n can be estimated from the independent data or otherwise it is a free choice. It is a logical trap to claim the transformation matrix R as being rectangular with a size m 3 n, where m ± n, because the size of R is arbitrary, i.e. it can be chosen freely. But still, the authors (Vogt et al., 1987) are not satisfied with R being rectangular

so that they continued and said ‘‘Even if R were directly invertable, this approach may be problematical since the data are always of finite signal-tonoise ratio and thus contain inherent uncertainty’’. These words are definitely misleading because such uncertainties exist even for direct imaging such as photography and cartography; but no-one claims that photographic images are not unique since a captured image lacks the details due to limited resolution. Otherwise, it would be a great mistake to use photographs on identification cards. For a reconstructed image non-uniqueness can be a problem only if distortions and fuzziness, which are caused by limited resolution and reconstruction mechanisms, are at a level that unique features of the original image are totally lost. But, this kind of non-uniqueness which is caused by the uncertainties of the input data not only endangers the solutions of Doppler imaging, physical problems of any kind which relay on curve fitting are subject to face it. Therefore, should we claim that such problems of science never provide unique solutions? Then, how can the uniqueness provided by the least squares fitting technique be explained? Despite the confusing comments and the misleading statements, Vogt et al. (1987) make a remarkable contribution to the uniqueness study of the starspot hypothesis. In fact, according to their ‘‘Vogt star’’ test and its results, the two questions in the specific approach can be answered ‘‘yes’’, ‘‘yes’’. Therefore, in spite of annoying deformations caused by the uncertainty inherited from the data, the ‘‘vogtstar’’ test provides convincing evidence that unique spot maps of spotted stars are attainable by Doppler imaging. To achieve an image quality with a resolution comparable to a photograph may a dream now, but it is not impossible theoretically. Consequently, it can be concluded that starspots have unique effects on the rotationally broadened spectral lines. This is because without the uniqueness of these effects the ‘‘vogtstar’’ test would have failed. Recovery of the original image is an undeniable evidence to the uniqueness of spot effects on the line profiles.

3.3. Criticisms of the uniqueness problem in photometric imaging ‘‘Photometric imaging’’ is a term introduced by

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Eker & Al-Malki (1999) to express starspot modeling techniques using photometric data. First quantitative formulations of photometric imaging appeared in the early seventies which is about a decade earlier than Doppler imaging. Unlike rotationally broadened line profiles, which are able to indicate numerous discrete cool surface regions, the observed light curves usually show one, occasionally two minima in a single period. Therefore, a simplest model (image) was established by a single spot with a uniform temperature lower than the photosphere. Long before the maximum entropy method introduced into Doppler imaging, its principles were already applied by the early applications of photometric imaging. A simplest spot image is circular (or rectangular for the solutions with numerical approach, Strassmeier, 1988). A uniform temperature was assumed to make it smoothest. So, there were classical one-spot modeling techniques (Bopp & Evans, 1973; Torrez et al., 1973; Budding, 1977; Vogt, 1981). Early formulations of a single spot failed to explain commonly observed asymmetric light curves. Therefore, two-spot modeling techniques became standard (Bopp & Noah, 1980; Dorren et al., 1981; Poe & Eaton, 1985; Lodenquai & McTavish, 1988; Zeilik et al., 1990; Hall et al., 1990b). Models with three spots appeared only rarely (Strassmeier & Olah, 1992; Zhai et al., 1994; Strassmeier, 1994). Although, those classical light curve modeling techniques with a fixed number of spots successfully produced the synthetic curves fitting the observations within the error limits of observational data, they definitely have not yet proved their credibility. To improve light curve solutions for a better imaging, new methods like the least squares (Kang & Wilson, 1989; Samec et al., 1993; Wilson, 1994) and ILOT (Information Limit Optimization Technique; Budding & Zeilik, 1987) have been developed. Unlike traditional techniques with one or two spots, ILOT aims to determine the optimum number of parameters in computing synthetic light curves. Nevertheless, failures of the photometric imaging continued. This is because there were complaints like indeterminacy of spot latitudes (Torrez et al., 1973; Bopp & Noah, 1980; Poe & Eaton, 1985; Stauffer et al., 1986; Dorren, 1987; Budding & Zeilik, 1987; Budding & Zeilik, 1995; Kang & Wilson, 1989; Zeilik et al., 1989; Zeilik et al., 1990; Zeilik et al., 1994; Strassmeier, 1990; Neff, 1990; Banks &

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Budding, 1990; Banks, 1991; Banks et al., 1991; Strassmeier & Olah, 1992; Anders et al., 1992; Eaton et al., 1993; Olah et al., 1994; Patkos & Hempelmann, 1994; Heckert & Ordway, 1995; Raveendran & Mohin, 1995; Summers & Heckert, 1997; Hempelmann et al., 1997) and, therefore, latitude fixing (Torrez et al., 1973; Stauffer et al., 1986; Dorren, 1987; Kang & Wilson, 1989; Zeilik et al., 1989; Eaton et al., 1993; Heckert & Ordway, 1995; Summers & Heckert, 1997). Among them, models produced by the least squares and ILOT also exist. Moreover, the non-uniqueness of spot solutions was often declared – as if a tradition (Bopp & Noah, 1980; Vogt, 1981; Kopal, 1982; Vogt et al., 1987; Huenemoerder & Ramsey, 1987; Neff, 1990; Sarma et al., 1991; Anders et al., 1992; Dempsey et al., 1992; Huenemoerder et al., 1993; Maceroni & van’t Veer, 1993; Hall, 1994; Patkos & Hempelmann, 1994; Elias et al., 1995; Byrne et al., 1995; Raveendran & Mohin, 1995; Hempelmann et al., 1997; Olah et al., 1997). In fact, those failures contributed to the nonuniqueness ambiguities. For instance, latitude fixing implies non-uniqueness because a fixed latitude is a free choice among all or a range of possible solutions. On the other hand, in some cases the nonuniqueness of photometric solutions appeared obvious. Completely different predicted solutions for HD 12545 (XX Tri) by different investigators (Strassmeier & Olah, 1992; Eker, 1995; Hampton et al., 1996), who were using the same data, were interpreted as an evidence to this kind of non-uniqueness. On the other hand, according to Maceroni & van’t Veer (1993), non-physical spurious solutions in some W UMa systems could also be attributed to the non-uniqueness associated to the starspot hypothesis. Moreover, a non-uniqueness of spot solutions was naturally suspected because it is generally imagined that a smaller but cooler spot produces the same effect on a light curve as a larger but warmer spot. The trade-off between the spot area (or size) and the spot temperature is, therefore, considered as a frustrating ambiguity causing non-uniqueness (Vogt, 1981; Bopp & Noah, 1980; Huenemoerder et al., 1993). A similar trade-off between the spot latitude and the spot size is believed to be the main source of non-uniqueness by Anders et al. (1992). Hempelmann et al. (1997) blame the triple trade-off (temperature-size-latitude) for the non-uniqueness.

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In contrast, Eker’s (1996) uniqueness study indicates that a change in one parameter cannot be fully compensated by the changes of the other parameters. In other words, a synthetic light curve of a spot cannot be reproduced by another spot that is cooler but smaller, or warmer but larger, or smaller pieces unless they are natural components of the original spot itself at the same location. Thus, Eker (1996) has shown that a spot has a unique signature on the light curves. The Uniqueness of spot effects has been proved by Eker (1996) by deriving original spot parameters (size, location and temperature) analytically from pre-generated synthetic light curves. Nevertheless, the conclusion is that one can produce the same amount of light loss at a certain phase by using these trade-offs, but not identical light curves covering all phases. The misconceptions about the non-uniqueness, apparently, are caused by not noticing the crucial difference between the non-uniqueness of a light-loss at a single phase and the nonuniqueness of the whole fitting curve. The two questions in the specific approach, therefore, can be answered ‘‘yes’’, ‘‘yes’’ according to the studies of Eker (1994); Eker (1996). A spot with specific parameters (size, shape, location and temperature) must have a unique signature on the light curves. Otherwise, recovering original parameters from the synthetic curves would have been impossible. Therefore, it is clear that the starspot hypothesis does not have a uniqueness problem also in the field of photometry according to an analysis of the specific approach. Nevertheless, the uniqueness problem would also be tested according to the rules of the general approach. First, there should not be a uniqueness problem of type 3 because the uniqueness of the parameters producing a synthetic light curve (maculation wave) has been shown by Eker (1996). On the other hand, there should not be a uniqueness problem of type 1 because, like in most physical problems, the starspot hypothesis is represented by a single function which is named ‘‘maculation wave function’’. Although it may appear differently in various models, it can be proven that it is actually a same function. Eker’s (1994) formulation is not different from Dorren’s (1987) formulation except including additional forms of limb darkening (no limb darkening, linear and quadratic limb darkening). (Dorren,

1987) has already shown that his formulation is not different from Budding’s (1977) maculation wave function. Moreover, it has also been shown by Budding & Zeilik (1990); Budding & Zeilik (1995) that some suitably weighted derivative of the photometric maculation wave function could even imitate the profile function which is used by Doppler imaging. Essentially, the final curve fitting problem, whether one is dealing with a well-defined light curve or a well defined line profile, would be amenable to quite the same methodology. Since a single profile scans effectively one complete hemisphere, one might expect the information content to be comparable to that of half a complete light curve on average, for a given level of probable error in the data set (Budding & Zeilik, 1990; Budding & Zeilik, 1995). Consequently, it can be concluded that if still there is a uniqueness problem associated with the starspot hypothesis, it must have been caused by insufficient phase coverage of low quality data. This logical deduction has been supported by a new study (Eker, 1999). The analytical relations presented by Eker (1996) to derive original spot parameters from the synthetic light curves, also allowed Eker (1999) to study the propagation of observational uncertainties to the recovered parameters. According to the propagated uncertainties, at least 60.0001 mag or better accuracy in the observed curves is needed for a successful modeling. Unfortunately, the best accuracy available from ground-based observations is not better than 60.005 mag. According to Eker’s models with a maculation wave having about 0.15 mag amplitude with 60.005 mag accuracy, the uncertainty of the size is about the original size of the spots which are used when generating the synthetic curves of the test models. The spot latitudes, on the other hand, are the most uncertain with an amount bigger than physically meaningful limits. Even for the models with known i (inclination of rotation axis), the uncertainty of latitudes is greater than 6908. That is, it exceeds the allowed limits ( 2 908 , b , 908) for a latitude. Thus, Eker’s (1999) study indicates that all the troubles and failures for determining spot latitudes can now be attributed to the insufficient accuracy of the observed data. Neither Russell (1906) nor Kopal (1982) dis-

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cussed observational uncertainties as a source of non-uniqueness ambiguities. This must be because: first, the problem of achieving a unique fit for a given function does not pose a problem to the starspot hypothesis only. Second, it is commonly known that there are methods which are designed to obtain unique fits from a data set with a limited accuracy. Actually, one should be cautious to use the word ‘‘non-unique’’ referring to the solutions which are obtained by these methods because methods like the least squares, or x 2 (or rms – root mean square) minimization guarantee a unique solution for a fitting problem. As was stated by Vogt et al. (1987), for a linear image reconstruction problem surfaces of constant x 2 are convex ellipsoids in N-dimensional image space. Since entropy surfaces are strictly convex, the maximum entropy reconstruction is thus unique (Skilling & Bryan, 1984; Vogt et al., 1987; Lyon et al., 1997). A solution relying on a best fit, which is a state of fitting at which the deviation of the data from the curve is minimum, must be unique by definition anyhow. However, one may still argue and ask why there are inconsistent results among the published models even though the methods like the least squares or x 2 minimization were used. Like the least squares method, the ILOT technique must also produce unique results. But there are published models produced by ILOT with fixed latitudes. This, however, implies non-uniqueness of the solutions because a fixed latitude represents very different latitudes as possible solutions. Unfortunately such problems exist. However, first, it would be a great mistake to disgrace the starspot hypothesis because of such problems. Second, these problems exist because it is customary to stop iterations at the stage when the rms deviation of the data from the fitting curve becomes smaller than the standard error of a single datum. This is a kind of precaution not to overinterpret the data. Therefore, if one stops iterations before reaching the best fitting conditions, there can be always non-unique solutions if the accuracy of the data is not sufficient to differentiate among the different solutions (curves or models). One way to eliminate spurious solutions of this kind is to increase the accuracy of the data. Imaging must have a limited resolution. Resolution of photometric images are thus limited by the

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uncertainties inherited by the data. If the quality and the quantity of the data allows to resolve n number of surface spots, one does not have the right to ask for one more, which is a well known rule in mathematics. For a S / N ratio value of around 100, a maculation wave can generally determine at least 3, sometimes 4 or 5 independent parameter values (Budding & Zeilik, 1990; Budding & Zeilik, 1994).

4. Discussion

4.1. Model dependence of photometric solutions According to certain entropy conditions enforced on a model, a unique solution with proper parameters can be obtained by a best fitting. The best fitting conditions are achieved when the deviation of observed data from the computed curve becomes a minimum. Unfortunately, special entropy conditions make photometric solutions model dependent. Let us consider examples. Suppose the same data set is used for classical one-spot models, standard two-spot models, three-spot solutions, etc. Each modeling technique produces its own best fitting curve and corresponding parameters. It is not only the image implied by each modeling technique, but also the corresponding light curves will not be identical. It is possible to eliminate some modeling techniques (models) if their curves cannot adequately explain the observed data. That was the reason for abandoning classical one-spot models because they were not successful in producing asymmetric light curves. On the other hand, rms, x 2 or any other parameter, which is used to choose the best fitting curve for each model, can again be used to choose the best modeling technique. In this way the model dependence of photometric solutions can be reduced. One may argue that non-uniqueness is unavoidable because the uncertainty of the data may not permit us to differentiate between the synthetic curves of some models. However, non-uniqueness associated with this argument is like the non-uniqueness confined to the fine structure of the ‘‘Vogt star’’ test by Vogt et al. (1987). It is very clear that this kind of nonuniqueness does not require the non-uniqueness of the recovered image and does not falsify the uniqueness of the solutions offered by the best fitting

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techniques. Even with fuzzy and distorted images, some unique features of the original image can be determined.

of many solutions with different spot images are required.

4.2. Unambiguous determination of spot temperatures

5. Conclusions

Since synthetic light curves are unique functions of the involved parameters (Eker, 1996), an unambiguous determination of the spot temperature without knowing the detailed spot map, as Vogt (1981) claimed, cannot be achieved. Unique solutions do not permit a trade-off between the parameters. According to Eker (1996), synthetic curves are unique functions as long as non-circular spot shapes or at least two spots or more with any shape are used to compute them. Since model dependence would deduce different solution parameters (different images) from the same data, unique spot temperatures are impossible. This thesis seems to be confirmed by the models applied to HD 12545 (XX Tri). The spot temperatures for HD 12545, which are predicted by various authors (Strassmeier & Olah, 1992; Eker, 1995; Hampton et al., 1996) are very different even though the same data were analyzed. The differences are bigger than the estimated uncertainties of each model. Since the configurations of spots on XX Tri are very different, different temperatures have been computed. (Hampton et al., 1996) noticed that the derived temperatures depend rather strongly on the inclination assumed. In fact, it is better to emphasize the dependence not only on the inclination but also on the other parameters as well. The temperature alone cannot be unique unless other values in the parameter set are also unique. If one is able to claim an unambiguous determination of the spot temperature as Vogt (1981) regardless of the sizes, shapes and distributions of discrete spots, there is no choice but to claim nonuniqueness of spot effects on the light curves. This is like the models with fixed latitude which cannot be unique. Fixing a spot latitude, since it is a notoriously hard parameter to attain, implies non-uniqueness because in such cases all or many latitudes are equally likely to be a possible solution. Similarly, if it is determined independently from shapes, orientations, sizes, locations, a single temperature makes a unique solution impossible. Otherwise, the existence

• There is no doubt about the existence of starspots. This is not only because of the solar analogy but also because there are independent photometric and spectroscopic evidences which indicate that starspots exist. Following evidences can be counted: 1.1. Photometric evidence: Correlated color variations, the star being redder at the light minimum. Cool spots contribute more at longer wavelengths. 1.2. Spectroscopic evidences: (a) Doppler imaging, that is, correlated line profile variations due to the low continuum of cooler spots. (b) Modeling technique with TiO bands. Due to cooler temperatures, molecular lines are stronger over the spot regions (Saar & Neff, 1990; Neff et al., 1995). 1.3. Auxiliary evidences: (a) Correlated chromospheric activity indicated by Ca H and K and Ha emissions. (b) Activities at the transition region and the corona indicated by UV, Xray and radio observations. • The starspot hypothesis is a consistent physical problem. This is because: 2.1. It is possible to generate synthetic light curves and synthetic line profiles from the physical parameters of spots with a given size, shape, location and temperature. 2.2. It is possible to recover those parameters from the spot effects on light curves and line profiles. The uniqueness of spot effects on the line profiles has been proven by the ‘‘Vogt star’’ test (Vogt et al., 1987) by recovering original spot images from the pre-generated line profiles. On the other hand, Eker (1996) has shown that spot effects on the light curves are also unique. Therefore, it is possible to recover the physical parameters of the spots from the line profiles or from the light curves. • Also, according to the test questions in the general approach in this study, it can be shown

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that the starspot hypothesis is consistent and does not have a uniqueness problem. 3.1. Like most physical problems (e.g. spectral energy distribution of a hot object) having a special function, the starspot hypothesis also has a special function which generates synthetic curves from the real physical spot parameters. The name of the function is ‘‘maculation wave function’’. Therefore, there should not be a uniqueness problem of type 1. 3.2. With special entropy conditions, that is, with a reasonable number of parameters specified, a unique solution relying on a best fit is always possible. There are methods (like the least squares and x 2 minimization) which are designed to produce unique results. Therefore, there cannot be a uniqueness problem of type 2. 3.3. The maculation wave function is a unique function of the involved parameters. That is, there are no two or more different spot images (distribution) which can produce identical light curves. Thus, there cannot be a uniqueness problem of type 3. • However, non-uniqueness ambiguities still exist in practical applications of the starspot hypothesis. The reasons are: 4.1. The accuracy of photometric data is insufficient to differentiate between the various models. Thus, iterations never reach the best fitting stage or they are cut by an intention as precaution not to overinterpret the data. 4.2. The non-uniqueness of light loss has been confused with the non-uniqueness of synthetic curves. However, the first is true, while the second is not. 4.3. There are confusions between different meanings of the word ‘‘uniqueness’’. • Achieving a true solution or an ideal image, which are unique in another sense, is impossible. But, it is a mistake to identify this as a uniqueness problem and thus to disgrace the starspot hypothesis. Such problems exist also for the eclipsing or cepheid type variability. Not only light variations, many physical problems may face such ambiguities. • Reliable mapping of spotted surfaces requires

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high quality, high precision observations covering sufficient rotation phases. For the photometric solutions, Eker (1999) estimates 60.0001 mag or better accuracy to be needed for a reliable photometric imaging. The resolution of the photometric images depends on the accuracy and phase coverage of the photometric observations.

Acknowledgements The author acknowledges Peter N. Brandt for his critical reading of the text and for his useful suggestions. The author also thanks Professor Mahjoob O. Tara for his criticism.

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