Time Series Analysis of Long-Term Photometry of the RS Cvn Star IM Pegasi

Master’s thesis International Master’s Programme in Space Science

Time series analysis of long-term photometry of the RS CVn star IM Pegasi

Victor S, olea

May 2013

Tutor: Doc. Lauri Jetsu Censors: Prof. Alexis Finoguenov Doc. Lauri Jetsu

University of Helsinki Department of Physics P.O. Box 64 (Gustaf Hallstr¨ omin¨ katu 2a) FIN-00014 University of Helsinki Helsingin yliopisto — Helsingfors universitet — University of Helsinki Tiedekunta/Osasto — Fakultet/Sektion — Faculty Laitos — Institution — Department

Faculty of Science Department of Physics Tekij¨a — F¨orfattare — Author

Victor S, olea Ty¨on nimi — Arbetets titel — Title Time series analysis of long-term photometry of the RS CVn star IM Pegasi

Oppiaine — Läroämne — Subject International Master’s Programme in Space Science Työn laji — Arbetets art — Level Aika — Datum — Month and year Sivumäärä — Sidoantal — Number of pages Master’s thesis May 2013 31 Tiivistelmä — Referat — Abstract

We applied the Continuous Period Search (CPS) method to 23 years of V-band photometric data of the spectroscopic binary star IM Pegasi (primary: K2-class giant; secondary: G– K-class dwarf). We studied the short and long-term activity changes of the light curve. Our modelling gave the mean magnitude, amplitude, period and the minima of the light curve, as well as their error estimates. There was not enough data to establish whether the long-term changes of the spot distribution followed an activity cycle. We also studied the differential rotation and detected that it was significantly stronger than expected, k ≥ 0.093. This result is based on the assumption that the law of solar differential rotation is valid also for IM Peg. We detected two long-lived active longitudes, rotating with a mean d d period of 24 .59, which is slightly slower that the rotation period of the star PW = 24 .6488. These two active longitudes were present during the whole observation period. There were also three abrupt activity shifts, in 1989, 1992 and 1993.

Avainsanat — Nyckelord — Keywords Methods: data analysis; Stars: activity, individual (IM Pegasi), rotation, starspots Säilytyspaikka — Förvaringsställe — Where deposited Kumpula Science Library Muita tietoja — övriga uppgifter — Additional information Contents

1 Introduction 1 1.1 The activity of the Sun ...... 1 1.2 Variable stars of type RS CVn ...... 2 1.3 The star IM Pegasi ...... 2

2 Observations 5

3 The continuous period search 8

4 Long-term activity changes 11

5 Differential rotation 16

6 Active longitudes 19

7 Conclusions 22

References 23

A The data and its format 27

B CPS test results 28

i Chapter 1

Introduction

1.1 The activity of the Sun

The Sun, being the closest star to us, has been observed extensively for centuries. Sunspots have been observed since the invention of the telescope. Schwabe began in 1826 the first systematic observations of sunspots. He determined later the length of the sunspot cycle to be approximately 11 years long (Schwabe 1843). However, the lengths of the observed cycles have varied between 8 and 15 years. Gleissberg (1945) noticed that the number of sunspots during these cycles oscillates with a period of approximately 80 years (the so-called Gleissberg cycle). The activity has sometimes been very low or even ceased altogether: one such period was the Maunder Minimum, lasting approximately between 1645 and 1715. This coincided with an unusually cold weather on Earth, the so-called "Little Ice Age". The cause of sunspots was discovered in the beginning of the 20th century: Hale (1908) proposed a solar magnetic field as an explanation. Solar magnetic activity is also the cause of several other phenomena, such as flares or the very high temperature of the chromosphere and corona above the visible photosphere. At the beginning of each activity cycle, pairs of sunspots of opposing magnetic polarity form at higher latitudes on both hemispheres (but usually not more than 45◦ from the equator). As the cycle proceeds, they form at lower and lower latitudes. The plot of the latitudinal positions of these pairs as a function of time gives the well- known "butterfly diagram". The magnetic field polarity of the leading and trailing sunspot pairs changes at the end of each cycle. It has been observed that the solar luminosity varies by 0.1% during the sunspot cycle. Willson and Hudson (1991) showed that the maximum luminosity is reached at the sunspot maximum. There are also similar variations in the intensity of several spectral emission lines, for example in the chromospheric Ca II H & K emission lines. It is thus an obvious question to ask whether other stars also exhibit similar cycles and whether spots can be observed (directly or indirectly) on other stellar surfaces. It

1 CHAPTER 1. INTRODUCTION 2 is not an impossible task, since detecting luminosity changes, like those in the Sun, is within the capabilities of modern instruments.

1.2 Variable stars of type RS CVn

Hall (1976) deﬁned a new class of variable stars, which he named RS CVn stars. These stars, like their prototype star RS Canum Venaticorum (HD 114519), are close binaries with a massive F–K-class giant and a G–M-class subgiant or dwarf. The variability in the V-band usually has an amplitude of lower than 0m.6. The main characteristics of this class are:

• Photometrical variability caused by starspots

• Active chromosphere: presence of chromospheric Ca II H & K and Hα emission lines, as well as X-ray and UV emission

• Fast synchronous rotation and orbital motion, with variations in the rotation period

• Orbital periods between 1 and 14 days. Those with periods over two weeks are referred to as "the long-period class".

The members of these binaries are close to each other, tidally locked and therefore rotate fast. As a consequence of their fast rotation, they display strong magnetic activity, which is observed e.g. as large starspots. These characteristics make RS CVn stars interesting targets for light curve modelling.

1.3 The star IM Pegasi

IM Peg (HR 8703, HD 216489) is a long-period spectroscopic binary of RS CVn type. It was already included in the ﬁrst survey of Hall (1976). It has a visual magnitude V = 5m.6 (Strassmeier et al. 1997). The K2 III primary exhibits strong starspot activity (Berdyugina et al. 1999; Berdyugin et al. 2006; Marsden et al. 2005). It has an estimated mass of 1.8 0.2M⊙, an effective surface temperature of about 4500 K and a radius of at least 12 R⊙ (Strassmeier et al. 1997). There is, however, disagreement about the spectral class of the unseen secondary. Berdyugina et al. (1999) considered it to be a K0 dwarf, while Berdyugin et al. (2006) suggested that it is a G2 dwarf. No eclipses have been observed, i.e. the inclination of the orbital plane is not close ◦ to 90 . Strassmeier et al. (1997) reported an orbital period Porb = 24.65 days and a photometric period Pphot = 24.4 days. Additionally, they observed some seasonal variations in the photometric period: for the 1993-94 observing season its mean was CHAPTER 1. INTRODUCTION 3

25.19 0.39 days, for the 94-95 season 24.108 0.081 days, and for the 95-96 season

24.498 0.030 days. This gave a long-term average of Pphot = 24.4936 0.0024 days between 1978-1996. However, they did not ﬁnd signs of regular long-term Pphot changes. IM Peg has a long observational history. This means that the orbital parameters and several other physical characteristics have been determined accurately, e.g. by Marsden et al. (2005). We have collected the following physical parameters of IM

Peg into Table 1.1: effective surface temperature (Teﬀ ), rotational velocity (v sin i), photometric period (Pphot), light curve amplitude (A), the radius (R) and spectral class of the primary. There is a good agreement between the results of the different authors in Table 1.1. For example, the following photometric observations of IM Peg have been made:

• Between October 1983 and March 1987 with the 10 inch Automatic Photoelec- tric Telescope in Phoenix, Arizona and Mt. Hopkins, Arizona, 275 observations in standard Johnson UVB photometry (Strassmeier et al. 1989)

• 218 observations between November 1993 and June 1996 with the 0.75 m Ama- deus APT at Mt. Hopkins, Arizona, and 111 observations with the 0.8 m Catania APT on Mt. Etna, Italy (Strassmeier et al. 1997)

• Between 1996 and 2001 with the 0,75 m Amadeus Automatic Photoelectric Telescope at Mt. Hopkins, Arizona (Ribárik et al. 2003)

In this thesis, we will analyze 23 years of photometric observations of IM Peg made with the 40 cm Vanderbilt/Tennessee State Automatic Photoelectric Telescope (APT) at Fairborn Observatory, Arizona. CHAPTER 1. INTRODUCTION 4

Reference Teﬀ V sin i Pphot Amplitude R Spectral Class −1 [K] [km s ] [d] [mag] [R⊙] 1 24.643 2 4265 3 24.4248 0.16 K1 III-IV 4 24.6 K0 5 4250 24.65 K1 III 6 24.39 K1 III-IV 7 24 24.4 12 K1 III 8 4200 24.547 9 36 24.6 10 24.43 0.156 11 4420 24 24.349 K1 III 12 28.1 13.6 13 28.2 24.45 0.22-0.4 >13.6 K2 II-III 14 4450 26.5 24.6488 13.3 K2 III Table 1.1: Physical parameters of IM Peg: The references are: 1. Lucy and Sweeney (1971), 2. Gottlieb and Bell (1972), 3. Herbst (1973), 4. Spangler et al. (1977), 5. Glebocki and Stawikowski (1979), 6. Eaton et al. (1983), 7. Fekel et al. (1986), 8. Basri (1987), 9. Huisong and Xuefu (1987), 10. Nelson and Zeilik (1990), 11. Dempsey et al. (1993), 12. Fekel (1997), 13. Strassmeier et al. (1999), 14. Berdyugina et al. (1999) Chapter 2

Observations

We studied photometric observations made between November 1987 and July 2010 with the 40 cm Vanderbilt/Tennessee State Automatic Photoelectric Telescope (APT) at Fairborn Observatory. The telescope, which is mainly used for the observation of chromospherically active stars, commenced operations in November 1987. The data are standard Johnson UBV differential photometry. We analyzed only the V-band data. For each observation, the ﬁxed measurement sequence K – S – C – V – C – V – C – V – C – S – K was performed, where the notations are: K = check star, S = sky background, C = comparison star, V = variable star. Each individual measurement had a 10-second integration time. The reported values for one observation are the means of the three measurements. Observations with an error larger than 0m.02, were discarded. Henry (1995) and Fekel and Henry (2005) have described the instrument and the data reduction process. Young et al. (1991) have reviewed the precision of this instrument in detail and their estimate for the internal accuracy was 0m.006 in the V-band. We obtained 2663 observations for ∆VS−C and 2373 observations for ∆VK−C . The data was divided into 41 observing seasons (Table 2.1). Figure 2.1 shows the data. As can be seen, there was no signiﬁcant seasonal variation in the differential photometry of comparison star (K = HD 218235, V = 6m.13) minus check star (C = HD 216635, V = 6m.612). However, there were strong variations in the differential photometry of the target star (V = IM Peg) minus the comparison star. m The observed standard deviation of ∆VK−C was 0 .0065, which is close to the value 0m.006 determined by Young et al. (1991). A useful test for checking the 2 goodness of a model gi = g(ti) is the χ test statistic

∑n ϵ 2 χ2 = i , σ2 i=1 i where σi is the accuracy of the yi data and ϵi = yi − gi are the residuals. For a good 2 2 model, one should have |ϵi| ≈ σi ⇒ χ ≈ n. A value χ ≫ n indicates that the model is unreliable. We tested the model gi = my, where my is the mean of all data

5 CHAPTER 2. OBSERVATIONS 6

Season Begins Ends n Season Begins Ends n 1 16.11.1987 15.01.1988 19 22 8.09.1999 28.01.2000 99 2 20.05.1988 17.06.1988 23 23 18.09.2000 26.01.2001 71 3 11.09.1988 20.01.1989 103 24 30.04.2001 2.07.2001 61 4 15.05.1989 25.06.1989 34 25 24.09.2001 19.01.2002 116 5 10.09.1989 18.01.1990 99 26 4.05.2002 1.07.2002 58 6 16.05.1990 25.06.1990 22 27 20.09.2002 7.02.2003 164 7 6.10.1990 16.01.1991 35 28 22.04.2003 10.07.2003 63 8 26.04.1991 2.07.1991 50 29 12.09.2003 30.01.2004 242 9 7.09.1992 22.01.1993 54 30 20.04.2004 8.07.2004 131 10 26.04.1993 28.06.1993 46 31 7.09.2004 4.02.2005 286 11 4.09.1993 22.01.1994 62 32 19.04.2005 16.07.2005 99 12 13.10.1994 14.01.1995 43 33 14.09.2005 20.01.2006 116 13 24.05.1995 11.07.1995 28 34 7.05.2006 21.06.2006 29 14 23.09.1995 16.01.1996 87 35 17.09.2006 15.01.2007 68 15 27.05.1996 25.06.1996 19 36 9.05.2007 4.07.2007 29 16 4.11.1996 25.01.1997 46 37 28.09.2007 20.01.2008 62 17 30.04.1997 13.07.1997 35 38 26.05.2008 1.07.2008 12 18 23.09.1997 28.01.1998 84 39 19.09.2008 16.01.2009 65 19 31.05.1998 3.07.1998 16 40 5.10.2009 15.01.2010 46 20 29.09.1998 28.01.1999 84 41 26.05.2010 6.07.2010 20 21 1.05.1999 1.07.1999 37 Table 2.1: The observing seasons

points yi = ∆VS−C (ti). We used σi = 0.006 from Young et al. (1991). The result was 2 χ S−C = 534756 for n = 2663 observations. This suggested that the variability of 2 the target star IM Peg was real. On the other hand, the χ for ∆VK−C was 5412 for n = 2362 observations. This result, that the two quantities, n and χ2, have nearly the same order of magnitude, suggested that the variations of the comparison and check star were very small. CHAPTER 2. OBSERVATIONS 7

Figure 2.1: The differential photometry ∆VS−C (top, n = 2663 observations) and ∆VC−K (bottom, n = 2373 observations). The mean magnitudes for the observing seasons are marked with diamonds and connected by dashed lines. The magnitude scale is the same in both panels. Chapter 3

The continuous period search

The continuous period search (hereafter CPS) ﬁts the periodic model

∑K ¯ g(t) = g(t; β) = M + [Bk cos(2kπft) + Ck sin(2kπft)] k=1 to the data y¯ = [y(t1), y(t2), . . . , y(tn)] = [y1, y2, . . . , yn] (Lehtinen et al. 2011). The ¯ free parameters of the model are β = [M,B1,...,Bn,C1,...,Cn, f], i.e. the mean

M, the frequency f, and the individual sine and cosine amplitudes Bk and Ck. The model residuals are ϵi = yi − gi = y(ti) − g(ti).

The mean of all time points ti of a modelled dataset is denoted with τ. The physically meaningful light curve parameters are: the mean M(τ), the period P (τ), the total amplitude A(τ) = max[g(t)] − min[g(t)] and the epochs of the primary and secondary light curve minima tmin,1(τ) and tmin,2(τ). Before modelling, the data was divided into segments and datasets. Each dataset was modelled separately. The segments are disjoint, while the datasets may overlap. However, each dataset must contain at least one data point which does not belong to the preceding or the following dataset. In our case, the segments were the 41 observing seasons specified in Table 2.1. The length of a dataset was chosen in such a way that there was a sufficient number of data points for the light curve modelling. A dataset covering a too long timespan would be insensitive to rapid light curve changes. There- fore, we used a one-month maximum length (∆Tmax = 30 days) and we imposed a threshold of n ≥ 10 data points for an accepted dataset. This means that any succes- sion of at least 10 consecutive observations within a 30-day period gave an acceptable dataset. Observations not included into any dataset were not modelled. The main advantage of having partly overlapping datasets is a better time reso- lution. This can however result in an undesired correlation between the models for overlapping datasets. Therefore it is necessary to keep a record of the datasets which do not overlap. The first analyzed dataset is automatically labelled as "independent". Subsequent datasets which have no common data points with the previous indepen-

8 CHAPTER 3. THE CONTINUOUS PERIOD SEARCH 9

IND(τ) = 1 IND(τ) = 0 IND(τ) = 0 R(τ) = 0 R(τ) = 0 R(τ) = 1 M(τ) 87 (■) 719 (×) 35 (×) A(τ) 87 (■) 719 (×) 33 (×) P (τ) 87 (■) 719 (×) 33 (×) tmin,1(τ) 87 (■) 719 (×) 33 (×) tmin,2(τ) 84 (▲) 697 (×) 33 (×) Table 3.1: The symbols used in our ﬁgures for the different types of light curve parameters (IND(τ) = 1, IND(τ) = 0) and for the reliability of these parameter estimates (R(τ) = 1, R(τ) = 0). Error bars are plotted only for independent (IND = 1) model parameter estimates. dent dataset are also "independent". All other datasets are "not independent". The notation IND(τ) = 1 is used for independent datasets, and IND(τ) = 0 for the other datasets. The best model order K was determined for each dataset. We tested the cases K = 0, K = 1 and K = 2. The best order was selected with the Bayesian information criterion formulated in (Lehtinen et al. 2011, Eq. 6). From the total of 842 modelled datasets, no periodicity was detected in two datasets (K = 0). For 23 datasets, the best order was K = 1, and for the remaining 817 datasets, it was K = 2. The physically meaningful parameters P (τ), A(τ) and tmin,1(τ) exist only if the model order is K ≥ 1. The secondary minimum epoch tmin,2(τ) can exist only for the K = 2 models. The light curves of all independent datasets are shown in Fig. 3.1. We used the bootstrap method to test the reliability of the model and to obtain error estimates for the model parameters M(τ), A(τ), P (τ), tmin,1(τ) and tmin,2(τ). The error of each parameter is the standard deviation of 200 bootstrap estimates. The bootstrap distributions for every model parameter and the residuals were tested against a Gaussian hypothesis with the Kolmogorov-Smirnov test using a signiﬁcance level γ = 0.01. A model was considered reliable if all the above mentioned bootstrap parameter estimate distributions were Gaussian. We used the notation R(τ) = 0 for the reliable model parameter estimates and R(τ) = 1 for the unreliable ones, as in Lehtinen et al. (2011).

Another quantity of interest is the time scale of change, TC , which gives an estimate of the time that the light curve stays unchanged. It was computed by determining how long the model for a dataset still ﬁts the subsequent datasets in the same segment.

Obviously, this parameter TC is meaningful only in the case of reliable models. This is why the default value TC (τ) = −1 was used for unreliable models. A value of

TC (τ) = −2 denotes the models that ﬁt the data of all subsequent datasets in the same segment. There were a total of 806 reliable datasets. Out of these, no change was detected for 411 datasets (TC (τ) = −2). The remaining 395 datasets had a mean

TC (τ) = 29.346 days, a median TC (τ) = 27.287 days, a minimum TC (τ) = 5.5 days CHAPTER 3. THE CONTINUOUS PERIOD SEARCH 10

Figure 3.1: Light curves of all independent datasets (IND(τ) = 1). The light curve phases were computed from ϕi = FRAC[(ti − tmin,1(τ))/P (τ)], where FRAC[x] removed the integer part of x.

and a maximum TC (τ) = 74.76 days. The mean and median TC (τ) values were very d close to the maximum dataset length ∆Tmax = 30 . Hence, the results of our modelling could be considered reliable, because the mean and the median of TC indicated d that the light curves did not change during ∆Tmax = 30 . Chapter 4

Long-term activity changes

In this section, we discuss the long-term activity changes of IM Peg. Photometric variability is caused by stellar magnetic activity, analogous to the 11-year-long solar cycle. Oláh et al. (2000) suggested that there was a long-term cycle of 48.3 years, but noted that it was twice as long as their data. Oláh et al. (2009) also detected an approximately 9-year-long cycle in the mean brightness of IM Peg. They did not exclude the presence of other cycles. However, since the observational record extends only to the 1970s, establishing cycles longer than about 40 years is not yet possible.

The frequently applied method by Horne and Baliunas (1986) (hereafter called the HB-method) was used to search for activity cycles in M(τ), A(τ), M(τ) + A(τ)/2, M(τ) − A(τ)/2. We analysed only the n = 87 reliable values of the independent data sets, i.e. IND(τ) = 1, REL(τ) = 0.

The amplitude A(τ) measures the longitudinally nonuniform spot distribution. This parameter is large, for example, in the case of one large spot close to the stellar equator. As the star rotates, both the spotted and the spot-free surface come into view. This determines the amplitude A(τ) of the observed light curve. The larger the spot is, the higher the amplitude (Fig. 4.1, top). The mean M(τ) of the light curve measures the longitudinally uniform spot distribution. For example, there could be a sector of spots throughout the whole equatorial area. It would cause little or no brightness changes during one rotation of the star. Such a sector would determine the mean M(τ) of the light curve (Fig. 4.1, bottom). A greater brightness corresponds to a smaller spotted area. Hence, M(τ) − A(τ)/2 indicates the least spotted area. The parameter M(τ) + A(τ)/2 indicates the most spotted area.

The HB-method, which is based on a sinusoidal model, resembles the power spec- trum method formulated by Scargle (1982). The HB-method is used for period detec- tion in noisy and unevenly spaced time series. The HB-method periodogram at each

11 CHAPTER 4. LONG-TERM ACTIVITY CHANGES 12

Figure 4.1: The qualitative light curve in case of a single large starspot close to the stellar equator (top) and in case of numerous small spots distributed uniformly in the equatorial area of the star (bottom). tested frequency f is

 [ ] [ ]  ∑n 2 ∑n 2  y′ cos 2πf(t − τ) y′ sin 2πf(t − τ)   i i i i  1 z (f) =  i=1 + i=1  (4.1) HB  ∑n ∑n  2 ′ − 2 ′ − 2 σ 2 [yi cos 2πf(ti τ)] 2 [yi sin 2πf(ti τ)] i=1 i=1 ∑ −1 ′ − where yi = yi(ti) are the data points, my = n yi is the mean, yi = yi my and 2 σ is the variance of yi. For each tested frequency, the parameter τ is computed from

( )( )− ∑n ∑n 1 tan (4πfτ) = sin 4πfti cos 4πfti . i=1 i=1

The connection between Scargle’s periodogram (zLS) and the HB-method periodogram (zHB) is that the latter is normalized with the variance σ of the data. The 2 relation is zHB = zLS/σ . CHAPTER 4. LONG-TERM ACTIVITY CHANGES 13

Figure 4.2: The Horne-Baliunas periodograms zHB(f) for M(τ), A(τ), M(τ) − −1 A(τ)/2 and M(τ) + A(τ)/2. The units of τ are years. The best frequency [fC ] = y is marked with a diamond. The numerical PC = 1/fC values are given in Table 4.1.

2 Parameter PC [years] F χ M(τ) 14.68 0.18 1.31 × 10−11 22929 A(τ) 8.01 0.16 9.00 × 10−3 23004 M(τ) + A(τ)/2 12.24 0.16 5.99 × 10−10 18012 M(τ) − A(τ)/2 15.73 0.29 3.00 × 10−5 27463

Table 4.1: Activity cycle periods PC obtained with the Horne-Baliunas method

For IM Peg, the range of tested frequencies was between

1 f = ≡ P −1 ≈ 22.026 years−1 min ∆T max and n f = ≡ P −1 ≈ 0.498 years−1, max 2∆T min where ∆T = tn − t1 ≈ 22.026 years was the length of the time series. The false alarm probability F is an estimate for the signiﬁcance of a peak having a height z0 in the periodogram zHB(f). To compute F , one has to ﬁrst determine the number of independent tested frequencies, which is

m = −6.362 + 1.193n + 0.00098n2. CHAPTER 4. LONG-TERM ACTIVITY CHANGES 14

Let us consider a periodogram peak z0. The probability that zHB exceeds z0 for m = 1 −z0 independent tested frequency is Q = P (zHB > z0) = e (Horne and Baliunas

1986). The probability for the complementary event is P (z0 ≤ zHB) = 1 − Q. For m > 1 independent tested frequencies, the probability that zHB does not exceed z0 is m (1 − Q) . Finally, the complementary probability that zHB exceeds z0 at least once − is 1 − (1 − e z0 )m. The false alarm probability F is thus the probability that at least one periodogram value of zHB exceeds the level of the peak z0 when m independent frequencies are tested. This probability is

− m F = 1 − [1 − e z0 ] . (4.2)

This is the probability that there exists a frequency value f1 such that zHB(f1) = z1 > −1 − z0. Therefore, the probability that the period PC = f1 is real, is 1 F .

The error estimate for the cycle period PC detected with the HB-method is

3πσ σ = N , C 2am1/2∆T where a is the amplitude of the sine ﬁt g¯ to y¯ and σN is the variance of the residuals ϵ¯ =y ¯ − g¯ (Horne and Baliunas 1986). To check the goodness of these sine ﬁts, we also computed the χ2 test statistic. The data y¯ were M(τ), A(τ), M(τ) + A(τ)/2 and

M(τ) − A(τ)/2 and the models were their respective sine ﬁts g¯. The errors σM and

σA of M(τ) and A(τ) were determined directly by the CPS. For y = M(τ) A(τ)/2, we used ( ) ( ) ∂y 2 ∂y 2 1 σ2 = σ2 + σ2 = σ2 + σ2 . y ∂M M ∂A A M 4 A The sinusoidal ﬁts g¯ to the light curve parameters M(τ), A(τ), M(τ) + A(τ)/2 and M(τ) − A(τ)/2 are shown in Fig. 4.3 (continuous lines). The best cycle periods

PC , their error estimates σPC and their false alarm probabilities F are given in Table 4.1, as well as the χ2 values of the sinusoidal ﬁts g¯. The false alarm probabilities for

M(τ) and M(τ) + A(τ)/2 indicate that their cycle periods PC = 14.68 and 12.24 years could be real. On the other hand, the very large χ2 ≫ n = 87 values show that a sinusoidal model is not suitable for our data. In this sense, the presence of activity cycles could not be indisputably established. Oláh et al. (2009), who used a different method of periodicity determination, mentioned both a mean magnitude M(τ) cycle length of 9-10 years and one of 18.9 years. Our PC = 14.68 years value for M(τ) was different. The sinusoidal ﬁt with the smallest χ2 value was the one for M + A/2, which is an approximation of the maximum spotted area. CHAPTER 4. LONG-TERM ACTIVITY CHANGES 15

Figure 4.3: Long-term changes of M(τ), A(τ), M(τ) + A(τ)/2 and M(τ) − A(τ)/2 (from top to bottom). A sinusoidal ﬁt has been made for the model parameters of the independent datasets (IND = 1). Symbols are explained in Table 3.1 Chapter 5

Differential rotation

If there were surface differential rotation in IM Peg, it may be detected as changes of the photometric rotation period P (τ). We used a parameter Z for estimating these changes. It was computed from

6∆P Z = W , PW √ ( ) ∑ ∑ ∑ √∑ −1 −1 2 where PW = ( wiPi)( wi) , ∆PW = wi(Pi − PW ) wi and w = σ−2 were the weights of P (τ). This parameter gives the variability of the i Pi i period P (τ) within the weighed 3∆PW limits (Jetsu et al. 2000, Sect. 3.3). The re- d d liable P (τ) estimates for the independent datasets gave PW ∆PW = 24 .59 0 .38 d and Z = 0.094 ≡ 9.4%. This value of PW was very close to Porb = 24 .6488 (Fig. 5.1).

We denote the real and spurious period variations with Zphys and Zsp, respectively. The accuracy of the data was σ = 0.m006 and the light curve amplitudes were typically

Figure 5.1: The long-term changes of the photometric period P (τ) of IM Peg. The symbols are explained in Table 3.1. The orbital period Porb is marked with a dotted line, PW with a solid line, and PW 3∆PW with dashed lines.

16 CHAPTER 5. DIFFERENTIAL ROTATION 17

A ≈ 0.m2. This gave an amplitude to noise ratio A/σ ≈ 33. The spurious period variations induced by this ratio should be small, Zsp ≈ 0.016 ≡ 1.6% (Lehtinen et al. 2 2 − 2 ≡ 2011, Table 2). From Zphys = Z Zsp, we obtained Zphys = 0.093 9.3%. This means that the contribution of Zsp is insigniﬁcant. We also tested the hypothesis that the observed period variations were not real and that the period was constant. This hypothesis was rejected, because the χ2 statistic for the n = 87 independent and reliable P (τ) estimates was 93145. This indicated the presence of surface differential rotation, i.e. the period variations were real.

We assumed that the law of solar differential rotation holds also for IM Peg. A frequently used approximation for the Sun and other active stars is

P P (θ) = eq , (5.1) 1 − k sin2 θ where θ is the latitude, P (θ) denotes the latitudinal rotation period, Peq is the rotation period at the equator and k is the differential rotation coefﬁcient (Hall and Busby 1990). For the Sun, it is k = 0.189. This value was observed directly from the movement of the sunspots (Hall and Busby 1990). The relation between k and Z is 2 2 |k| ≈ Z/h, where h = sin θmax − sin θmin. The angles θmax and θmin are the maximum and minimum latitudes of starspot formation (Jetsu et al. 2000). If we assume ◦ ◦ θmin = 0 and θmax = 90 , then k = Zphys = 0.093. This value of k is the lower limit, because the latitudinal range of spot activity in IM Peg is most probably less that 90 degrees (i.e. h < 1). Assuming a solar-like distribution, i.e. 0◦ ≤ θ ≤ 30◦, would give h = 0.25 and |k| = 0.372, which is signiﬁcantly larger than in the Sun.

Like in most RS CVn binaries, IM Peg shows synchronous rotation, i.e. Porb ≈

Prot (Hall 1976). If Eq. 5.1 holds, then the synchronously rotating latitude can not d d be θmin or θmax, because PW − 3∆PW = 22 .13 < Porb < PW + 3∆PW = 27 .05 (Figure 5.1). The relation in Eq. 5.1 requires that the maximum and minimum values of P (τ) occur at the latitudes θmin or θmax, depending on the sign of k. Henry et al. (1995) published a relation for the differential rotation coefﬁcient. Their equation was

log k = −2.12 + 0.76 log Prot − 0.57H, (5.2)

where H = Rstar/rRoche is the Roche lobe ﬁlling factor, and Prot was expressed in days. The parameter rRoche can be approximated as ( ) 2/3 0.49 M1 r M2 Roche = ( ) ( ( ) ) a 2/3 1/3 0.6 M1 + ln 1 + M1 M2 M2 CHAPTER 5. DIFFERENTIAL ROTATION 18

Parameter Unit Value Mass M⊙ 1.8 0.2 (primary), 1.0 0.1 (secondary) Radius R⊙ 13.3 0.6 (primary), 1.0 0.07 (secondary) a sin i R⊙ 16.70 0.02 (primary), 30.34 0.03 (secondary) Porb days 24.64877 i degrees 65 . . . 80 Tconj HJD 2 450 342.905 0.004 Teﬀ K 4550 50 (primary), 5650 200 (secondary) Table 5.1: Parameters for IM Peg from Ransom et al. (2012).

(Eggleton 1983), where a is the semimajor axis of the system. For the subsequent calculations we used the values from Ransom et al. (2012) (Ta- ble 5.1). The value of a was easily determined from Kepler’s 3rd law: M1 + M2 = 3 2 a /P , with the masses M1 and M2 in units of M⊙, a in astronomical units, and the period P in years. This gave a = 0.24 AU. We then obtained rRoche = 0.10 AU =

21.62R⊙ and H = 0.62. Inserting the values F = 0.62 and Prot = 24.6488 in Eq. 5.2 gave k = 0.039, which was significantly below our previously determined lower limit k ≥ 0.093. Henry et al. (1995, Fig. 28) also published a relationship between the rotation period Prot and the differential rotation coefficient k. We have log k = −0.429 and log Prot = 1.392, which means that IM Peg would be an outlier in their sample, dis- playing much stronger differential rotation than expected. Collier Cameron (2007) gave a relationship between the effective temperature and the differential rotation rate of the star: ( ) T 8.6 ∆Ω = 0.053 eff . 5130

−1 Using the the Teﬀ value from Table 5.1, we obtain ∆Ω = 0.019 rad d . Considering that ∆Ω = 2πk/P (Lehtinen et al. 2011), we have k = 0.074. This value is larger than the one detected above (k = 0.039), but still below our minimum physically possible differential rotation coefﬁcient k ≥ 0.093. Generally, differential rotation should be much weaker in rapidly rotating giants (Henry et al. 1995). Our results for IM Peg contradicted this idea. Chapter 6

Active longitudes

There seems to be a tendency for spots on chromospherically active binary stars to form at preferred longitudes with respect to the companion star (Eaton and Hall 1979). If these active longitudes are long-lived, they can be detected with nonparametric methods, like the Kuiper method. The Kuiper test is closely related to the Kolmogorov-Smirnov test (Jetsu and Pelt 1996). It has been applied in searching for active longitudes (Jetsu 1996). It ﬁnds out whether the distribution of phases for n time points, t1, . . . , tn is regular with d some period P . We performed the Kuiper test between Pmin = 0.9PW = 22 .13 and d Pmax = 1.1PW = 27 .05. The analyzed time points ti were the reliable primary and the secondary light curve minima tmin,1 and tmin,2 of the independent datasets. Their phases were obtained from ϕi = FRAC[(ti − t0)/P ], where t0 was an arbitrary epoch point before t1. The results of the Kuiper test do not depend on the chosen value of t0. The phases were then sorted into ascending order. The lowest ϕi value was denoted with x1, and the highest one with xn. The Kuiper test cumulative probability distribution function is a step function   0 if ϕ < x1 −1 Fn(ϕ) = in if x ≤ ϕ < x , for 1 ≤ i ≤ n − 1.  i i+1  1 if ϕ ≥ xn

The null hypothesis H0 is that the phases ϕi are uniformly distributed between 0 and 1, i.e. their cumulative probability density distribution function is F (ϕ) = ϕ. The + − + − Kuiper test statistic is Vn = D − D , where D and D are the maximum values of Fn(ϕ) − F (ϕ) and F (ϕ) − Fn(ϕ). The Kuiper test periodogram Vn(f) is shown in Fig. 6.1. The best periods maximize Vn(f), i.e. they are found at the peaks of d d Vn(f). The best period detected for the tmin,1 epochs was P = 24 .453 0 .012, −11 with QK = 1.7 · 10 , where QK is the critical level, i.e. the probability that Vn reaches or exceeds the computed ﬁxed value. The best period for tmin,1 and tmin,2 was

19 CHAPTER 6. ACTIVE LONGITUDES 20

Figure 6.1: Kuiper test periodogram Vn(f) for the reliable and independent tmin,1 (top), and the same periodogram for the reliable and independent tmin,1 and tmin,2 (bottom). The best periods P = 24d.453 0d.012 and P = 24d.466 0d.017 are marked with a vertical line.

d d −6 P = 24 .466 0 .017, with QK = 8.7 · 10 . These two periods were the same within their error limits. The critical levels QK were computed as in (Jetsu and Pelt

1996, Eq. 24). The error estimates σP were obtained with the bootstrap method, as described in the same paper.

We chose the zero epoch t0 = 2 450 342.883 − 130Porb = 2 447 138.539, where d Porb = 24 .6488, was the value published by Berdyugina et al. (1999). The phases of tmin,1 and tmin,1 calculated with the ephemeris HJD = 2 447 138.539 + 24.466E are shown in Fig. 6.2. The main result was that the active longitudes on the surface of the d primary star did not corotate with the orbital period Porb = 24 .6488 of the binary. For most of the time, activity was present at both active longitudes. These structures were very stable during the whole observed time interval. There were also activity shifts in October 1989, September 1992 and March 1993. CHAPTER 6. ACTIVE LONGITUDES 21

Figure 6.2: The phases ϕmin,1 and ϕmin,1 of the light curve minima with the best period obtained from the Kuiper test HJD = 2 447 138.539 + 24.466E. The symbols are explained in Table 3.1. Chapter 7

Conclusions

In this work, we analysed photometric data of the RS CVn-class star IM Peg collected between November 1987 and July 2010. We performed an analysis with the continuous period search method (CPS). We studied both the short and long-term variations of the light curve parameters. The mean of the time scale of the change of these light d curves, TC = 29 .346, was slightly shorter than the length of the modelled datasets,

∆Tmax = 30 days. This confirmed that our CPS models were reliable. There was not enough data available to either confirm or refute previous detections of long-term activity cycles (e.g. by Oláh et al. (2009)). A simple sinusoidal fit does not describe the long-term changes well, as could be seen from the large χ2 values. The best fit, while still poor, was the one for M + A/2, approximating the maximum starspot area. There exists also a possibility, as suggested by Strassmeier et al. (1997) that, while a long-term variation does exist and it is clearly detectable, it is not periodic. The star displayed strong differential rotation, |k| ≥ 0.093. This minimum value is correct only if the law of solar differential rotation holds also for IM Peg and if the range of starspot distribution extends from the equator to the poles. Most likely, however, the latitudinal range of starspot activity is narrower. This means that the differential rotation of IM Peg is much stronger than in the Sun, and very much stronger than expected for a rapidly rotating K-class giant (Hall and Henry 1990). We also detected two long-lived and stable active longitudes, rotating with a period d −6 of PAL = 24 .466 (QK = 8.7 · 10 ). They existed for the whole observation period. We also noticed a few activity shifts, when the stronger activity switched from one active longitude to the other one. These active longitudes did not corotate with the d orbital motion, Porb = 24 .6488. The long-term mean photometric period PW = 24d.59 was slightly shorter than the orbital period.

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The data and its format

The ﬁrst 46 observations are presented in table A. The ﬁrst column contains the helio- centric Julian date (HJD) of the observation. The next three columns contain the differential photometry in the U, B and V bands for IM Peg minus HD 218235. The last three columns contain the differential photometry of HD 218235 minus HD 216635. When there was no observation, the default value of 99.999 was used. We analyzed only the V-band data (columns 1, 4 and 7).

HJD ∆US−C ∆BS−C ∆VS−C ∆UK−C ∆BK−C ∆VK−C 2447163.6123 99.999 99.999 -0.790 99.999 99.999 99.999 2447173.5849 -0.420 -0.636 -0.766 -1.694 -0.986 -0.437 2447174.5821 -0.458 -0.661 -0.784 -1.694 -0.981 -0.428 2447175.5782 -0.471 -0.677 -0.798 -1.701 -0.988 -0.431 2447301.9664 99.999 -0.634 -0.755 99.999 99.999 99.999 2447301.9813 99.999 99.999 -0.762 99.999 99.999 99.999 2447302.9643 99.999 -0.646 -0.783 99.999 -0.981 -0.454 2447302.9738 99.999 -0.657 -0.787 99.999 99.999 99.999 2447303.9637 99.999 -0.686 -0.814 99.999 99.999 99.999 2447304.9695 99.999 -0.751 -0.880 99.999 99.999 99.999 2447306.9752 99.999 99.999 99.999 99.999 -1.105 -0.506 2447307.9516 99.999 -0.792 -0.902 99.999 -0.980 -0.452 2447307.9615 99.999 -0.797 -0.902 99.999 99.999 99.999 2447308.9508 99.999 -0.755 -0.877 99.999 -0.980 -0.450 2447308.9718 99.999 -0.753 -0.872 99.999 -0.987 -0.451 2447309.9495 99.999 -0.699 -0.828 99.999 -0.984 -0.456 2447310.9452 99.999 -0.626 -0.762 99.999 -0.985 -0.457 2447312.9446 99.999 -0.508 -0.651 99.999 -0.985 -0.458 2447313.9394 99.999 -0.470 -0.623 99.999 -0.990 -0.460 2447315.9339 99.999 -0.466 -0.624 99.999 -0.979 -0.451 2447316.9300 99.999 -0.476 -0.669 99.999 -0.956 -0.474 2447317.9277 99.999 -0.524 -0.670 99.999 -0.966 -0.446 2447319.9229 99.999 -0.598 -0.730 99.999 99.999 99.999 2447320.9213 99.999 -0.618 -0.750 99.999 -0.979 -0.455 2447324.9067 99.999 -0.592 -0.722 99.999 -0.971 -0.432 2447328.8975 99.999 -0.692 -0.822 99.999 -0.968 -0.462 2447329.9622 99.999 -0.749 -0.861 99.999 -0.981 -0.452 2447415.8244 99.999 -0.545 -0.684 99.999 -0.984 -0.452 2447316.9300 99.999 -0.476 -0.669 99.999 -0.956 -0.474 2447317.9277 99.999 -0.524 -0.670 99.999 -0.966 -0.446 2447319.9229 99.999 -0.598 -0.730 99.999 99.999 99.999 2447320.9213 99.999 -0.618 -0.750 99.999 -0.979 -0.455 2447324.9067 99.999 -0.592 -0.722 99.999 -0.971 -0.432 2447328.8975 99.999 -0.692 -0.822 99.999 -0.968 -0.462 2447329.9622 99.999 -0.749 -0.861 99.999 -0.981 -0.452 2447415.8244 99.999 -0.545 -0.684 99.999 -0.984 -0.452 2447417.7490 99.999 -0.607 -0.736 99.999 -0.989 -0.457 2447417.8855 99.999 -0.603 -0.733 99.999 -0.979 -0.447 2447418.7246 99.999 -0.621 -0.751 99.999 -0.978 -0.445 2447421.7457 99.999 -0.606 -0.732 99.999 -0.981 -0.444 2447421.8952 99.999 -0.600 -0.727 99.999 -0.981 -0.443 2447422.7184 99.999 -0.582 -0.723 99.999 -0.983 -0.453 2447422.8592 99.999 -0.584 -0.716 99.999 -0.981 -0.456 2447423.7168 99.999 -0.618 -0.740 99.999 -0.987 -0.454 2447423.8727 99.999 -0.627 -0.741 99.999 -0.984 -0.451 2447424.7162 99.999 -0.612 -0.741 99.999 -0.985 -0.450 2447424.8748 99.999 -0.606 -0.751 99.999 -0.984 -0.455 Table A.1: The format of our data

27 Appendix B

CPS test results

Here are our results for the ﬁrst 12 datasets. The format is the same as described in (Lehtinen et al. 2011, Appendix A).

3 47415.8244 5 1 47115.7264 1 47430.7051 47461.6731 47442.1650 0 47115.7264 47141.6760 47127.5316 0 32 -0.7148 0.0708 12 -0.6963 0.0528 2 0.0086 50.8278 0 0.0528 -2.0000 -0.7397 0.0021 0 -0.6963 0.0153 0 24.1825 0.1724 0 -1.0000 -1.0000 1 0.2387 0.0072 0 -1.0000 -1.0000 1 47437.2634 0.0737 0 -1.0000 -1.0000 1 47447.6522 0.1552 0 -1.0000 -1.0000 1 3 47415.8244 6 2 47301.9664 1 47431.7039 47462.7923 47444.1628 0 47301.9664 47329.9622 47312.5380 1 32 -0.6997 0.0623 22 -0.7748 0.0892 2 0.0087 70.5228 2 0.0072 -2.0000 -0.7357 0.0029 0 -0.7532 0.0019 0 24.2338 0.1535 0 25.1209 0.1501 0 0.2289 0.0103 0 0.2975 0.0061 0 47437.3552 0.0834 0 47314.9338 0.0596 0 47447.9308 0.2000 0 47324.9822 0.1482 0 3 47415.8244 7 3 47415.8244 1 47432.7028 47463.6856 47445.5917 0 47415.8244 47446.8100 47432.1107 1 31 -0.6898 0.0545 42 -0.7491 0.0734 2 0.0090 69.0939 2 0.0110 27.2868 -0.7338 0.0038 0 -0.7497 0.0018 0 24.2758 0.1625 0 24.3425 0.3508 0 0.2229 0.0139 0 0.2600 0.0066 0 47437.3880 0.0916 0 47437.1387 0.0969 0 47448.0548 0.2467 0 47422.6500 0.1886 0 3 47415.8244 8 3 47415.8244 2 47434.7038 47465.6686 47448.0692 0 47417.7490 47448.8073 47433.2631 0 28 -0.6818 0.0510 43 -0.7493 0.0720 2 0.0087 41.5556 2 0.0111 33.7486 -0.7426 0.0058 0 -0.7489 0.0017 0 24.2342 0.1375 0 25.1575 0.3660 0 0.2573 0.0224 0 0.2576 0.0061 0 47437.4277 0.0863 0 47437.2259 0.0936 0 47447.7709 0.2277 0 47422.4031 0.1713 0 3 47415.8244 9 3 47415.8244 3 47436.6984 47467.7428 47450.3864 0 47428.7300 47459.7927 47439.1706 0 28 -0.6859 0.0540 32 -0.7411 0.0818 2 0.0073 42.6064 2 0.0128 27.8410 -0.7337 0.0065 0 -0.7475 0.0024 0 24.5038 0.1743 0 24.5396 0.2314 0 0.2262 0.0237 0 0.2625 0.0073 0 47437.1493 0.1448 0 47437.0833 0.0822 0 47447.9113 0.2522 0 47447.2966 0.2076 0 3 47415.8244 10 3 47415.8244 4 47437.6968 47468.6687 47452.0723 0 47429.7135 47460.7737 47441.1663 0 27 -0.6928 0.0555 32 -0.7250 0.0780 2 0.0065 37.5525 2 0.0131 42.2351 -0.7327 0.0053 0 -0.7472 0.0027 0 24.7179 0.1490 0 24.4881 0.2272 0 0.2301 0.0198 0 0.2655 0.0100 0 47461.6880 0.0795 0 47437.0746 0.0977 0 47447.9102 0.2045 0 47447.2764 0.1792 0