Spot activity of the RS CVn star σ Geminorum
aa r X i v : . [ a s t r o - ph . S R ] N ov Astronomy & Astrophysicsmanuscript no. sigmagem c (cid:13)
ESO 2018May 6, 2018
Spot activity of the RS CVn star σ Geminorum ⋆ P. Kajatkari , T. Hackman , L. Jetsu , J. Lehtinen , and G.W. Henry Department of Physics, P.O.Box 64, FIN-00014 University of Helsinki, Finland Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Väisäläntie 20, FI-21500 Piikkiö, Finland Center of Excellence in Information Systems, Tennessee State University,3500 John A. Merritt Blvd., Box 9501, Nashville, TN 37209, USAReceived date / Accepted date
ABSTRACT
Aims.
We model the photometry of RS CVn star σ Geminorum to obtain new information on the changes of the surface starspotdistribution, i.e., activity cycles, di ff erential rotation and active longitudes. Methods.
We use the previously published Continuous Periods Search-method (CPS) to analyse V-band di ff erential photometryobtained between the years 1987 and 2010 with the T3 0.4 m Automated Telescope at the Fairborn Observatory. The CPS-methoddivides data into short subsets and then models the light curves with Fourier-models of variable orders and provides estimates ofthe mean magnitude, amplitude, period and light curve minima. These light curve parameters are then analysed for signs of activitycycles, di ff erential rotation and active longitudes. Results.
We confirm the presence of two previously found stable active longitudes, synchronised with the orbital period P orb = d . P min , = d .
47 than the orbital motion. If the variations in the photometric rotation period were to be caused by di ff erentialrotation, this would give a di ff erential rotation coe ffi cient of α ≥ . Conclusions.
The presence of two slightly di ff erent periods of active regions may indicate a superposition of two dynamo modes,one stationary in the orbital frame and the other one propagating in the azimuthal direction. Our estimate of the di ff erential rotationis much higher than previous results. However, simulations show that this can be caused by insu ffi cient sampling in our data. Key words. stars: activity - starspots - stars: individual: σ Geminorum
1. Introduction
The RS CVn-type star σ Geminorum is a bright (V ≈ . P orb = d . ff ect on the spectrum of the binary. The secondary is mostlikely a cool, low-mass main-sequence star or possibly a neutronstar (Duemmler et al. 1997; Ayres et al. 1984). The inclinationof the rotational axis of the primary is roughly 60 ◦ (Eker 1986).The photometric variability of σ Gem was first detected byHall et al. (1977). Since 1983, intensive and continuous photo-metric observations have been made with automated photometrictelescopes (APT). The light curves acquired in this fashion havebeen studied in detail, e.g. by Fried et al. (1983), Henry et al.(1995), Jetsu (1996) and Zhang & Zhang (1999).Doppler imaging has been used to construct surface temper-ature maps of σ Gem (Hatzes 1993; Kovári et al. 2001). Thesesurface images had no polar spots, a feature often reported inother active stars. Instead, the spot activity appears to be con-strained into a latitude band between 30 ◦ and 60 ◦ .In most late-type stars, no unique, regular and persistentactivity cycle has been found. In the case of σ Gem, vari-ous analyses have yielded a wide range of di ff erent possiblequasi-periodicities, which are assumed to be an indication of ⋆ The analysed photometry and numerical results of theanalysis are both published electronically at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp: // cdsarc.u-strasbg.fr / vizbin / qcat?J / A + A / yyy / Axxx a possible stellar cycle, similar to the 11-year sunspot cycle.Strassmeier et al. (1988) suggested a possible 2.7 year period inthe spotted area of σ Gem. Henry et al. (1995), who found acycle of 8.5 years instead, suggested that the 2.7 year period isrelated to the lifetime of individual spot regions and hence thisshorter period would not represent a true spot cycle. They alsoattributed the 5.8 year cycle found by Maceroni et al. (1990) tothe spot migration rate determined by Fried et al. (1983).The light curve minima of σ Gem have shown remarkablestability in phase over time span of years, or even decades.This indicates a presence of active longitudes, a phenomenonoften seen in chromospherically active stars. Active longi-tudes are longitudinally concentrated areas that show persis-tent activity, manifesting as starspots. Active longitudes on σ Gem have previously been studied by Jetsu (1996) andBerdyugina & Tuominen (1998). The results indicate that theactive longitudes are synchronised with the orbital period, witha preference to the line connecting the binary components.Berdyugina & Tuominen (1998) also suggested that there is apossible 14.9 year activity cycle in the star.Di ff erential rotation has been studied using photometric spotmodels and Doppler-imaging techniques. Henry et al. (1995)used spot modelling to determine the migration rate of thestarspots and arrived at a value for the di ff erential rotation co-e ffi cient. Kovári et al. (2007b) analysed Doppler images, usingthe Local correlation technique (LCT). Their analysis indicatedanti-solar di ff erential rotation with α = − . ± . ff erent technique for the same data, they also got anothervalue α = − . ± .
005 (Kovári et al. 2007a).
Article number, page 1 of 10 . Observations
The observations in this paper are di ff erential photometry inthe Johnson V passband obtained at Fairborn Observatory inArizona using the 0.4 m T3 Automated Photometric Telescope(APT). Each observation is a sequence of measurements thatwere taken in the following order: K, sky, C, V, C, V, C, V,C, sky and K, where K is the check star, C is the comparisonstar and V the program star. The comparison star was HR 2896and the check star was υ Gem. Until 1992, the precision of themeasurements was 0.012 mag. Then a new precision photometerwas installed and the precision of the subsequent measurementshas been ∼ . − .
005 mag (Fekel et al. 2005). A thoroughdescription of the APT observing procedures has been given byHenry (1999).The whole time series consists of 2683 observationsand spans from JD 2447121.0481 (21 November 1987) to2455311.6556 (25 April 2010). The V-C and K-C di ff erentialmagnitudes are shown in Fig. 1. The numbers displayed inthe upper panel refer to the segment division and correspondto di ff erent observing seasons. We decided against includingpreviously published data from other sources. The continuousperiod search method (hereafter CPS)is best suited for tempo-rally continuous data of homogeneous quality and inclusion oftemporally sparse data may induce unreliable results. This isalso the approach taken in earlier studies utilising the CPS, i.e.,Lehtinen et al. (2012); Hackman et al. (2013).
3. Data analysis
Here we give a short introduction to the CPS-method and howwe used it in the time series analysis of our paper. A completedescription of the method can be found in Lehtinen et al. (2011).The CPS-method has been developed from the Three Stage Pe-riod Analysis (TSPA) by Jetsu & Pelt (1999). The CPS uses asliding window to divide the data into shorter datasets and thendetermines local models using a variable K th-order Fourier se-ries:ˆ y ( t i ) = ˆ y ( t i , ¯ β ) = M + K X k = [ B k cos ( k π f t i ) + C k sin ( k π f t i )] . (1)The optimal model order K used for each dataset is determinedby the Bayesian information criterion. The highest modellingorder used in this study was K =
2. The possibility of a constantmodel K = y i = y ( t i ) in the dataset.The first step of the CPS-analysis is to divide the data intodatasets. The datasets are composed using a rectangular win-dow function with a predetermined length ∆ T max that is movedforward through the data one night at a time. A new dataset iscreated each time when the dataset candidate determined by thewindow function includes at least one new data point that wasnot included in the previous dataset. Each modelled dataset mustalso include at least n min data points to be valid. We used values n min =
14 and which is roughly two and half times the averagephotometric period.The first dataset with a reliable model is called an indepen-dent dataset. The next independent datasets are selected withthe following two criteria. Firstly, this next independent datasetmust not share any common data with the previous independentdataset. Secondly, the model for this next independent datasetneeds to be reliable. In other words, these independent datasetsdo not overlap and their models are always reliable. With this definition, the correlations between the model parameters of in-dependent datasets represent real physical correlations, i.e. thesecorrelation are not due to bias caused by common data.The datasets are combined into segments, each representinga di ff erent observing season. The segment division does not di-rectly a ff ect the analysis because each dataset is still analysedseparately. The segment division of this analysis is given in Ta-ble 1. For each segment, the length of the segment is given, alongwith the total number of data points, the number of datasets andnumber of independent datasets.The parameters obtained from the light curve model, as afunction of the mean epoch of the dataset, τ , are: M ( τ ) = mean magnitude A ( τ ) = peak to peak light curve amplitude P ( τ ) = photometric period t min , ( τ ) = epoch of the primary minimum t min , ( τ ) = epoch of the secondary minimum T C ( τ ) = time scale of change.The CPS also provides graphical representation of the resultsfor each segment. An example of this is given later in Fig.6. Thatfigure contains the following panels:(a) standard deviation of residuals σ ( τ );(b) modelling order K ( τ ) (squares, units on the left y -axis);and number of observations per dataset n (crosses, units onthe right y -axis);(c) mean di ff erential V -magnitude M ( τ );(d) time scale of change T C ( τ )(e) amplitude A ( τ );(f) period P ( τ );(g) primary (squares) and secondary (triangles) minimumphases φ min , ( τ ) and φ min , ( τ );(h) M ( τ ) versus P ( τ );(i) A ( τ ) versus P ( τ );(j) M ( τ ) versus A ( τ );Reliable models are denoted with closed symbols and unre-liable with open symbols.The numerical results of the CPS analysis can be accessedelectronically at the CDS. The light curves and the best-fit mod-els of the independent datasets are shown in Fig. 2. The lightcurves are plotted as a function of phase φ = φ i + φ ′ i . For eachdataset, the phases φ i were first calculated using the best-fit pe-riods P ( τ ) and the epochs of the primary minima t min , ( τ ). Thephases of each dataset were then adjusted by φ ′ = φ orb , − . φ orb , are the phases of the primary minimum epochs t min , of each dataset, calculated using the orbital ephemerisJD conj = d . + d . ∆ T max = d .
2, but the result was a large number of unreliable models, thusthe longer window ∆ T max = d .
4. Results
The photometry of σ Gem has been studied before with intentionof searching for long-term activity cycles, but so far, none of the
Article number, page 2 of 10. Kajatkari et al.: Spot activity of the RS CVn star σ Geminorum (cid:0) (cid:1) (cid:2) (cid:3) (cid:4) V (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) V Fig. 1.
The di ff erential magnitudes between σ Gem and HR 2896, and the check star υ Gem and HR 2896 between the years 1987 and 2010.Di ff erent observing seasons are denoted by their corresponding segment number. Table 1.
Segments of the σ Gem photometry. Columns from left to rightare: Segment number, observing time interval, number of data points,total number of datasets and number of independent datasets.
SEG Interval n sets ind. sets1 21. 11. 1987 - 11. 3. 1988 47 13 22 13. 10. 1988 - 13. 5. 1989 182 49 33 6. 10. 1989 - 15. 5. 1990 155 51 34 22. 10. 1990 - 16. 5. 1991 77 18 35 14. 3. 1992 - 6. 5. 1992 17 3 06 5. 10. 1992 - 13. 5. 1993 97 27 47 5. 9. 1993 - 13. 5. 1994 142 59 48 12. 10. 1994 - 21. 5. 1995 119 47 49 21. 9. 1995 - 19. 5. 1996 170 59 410 3. 11. 1996 - 21. 5. 1997 138 46 311 26. 9. 1997 - 15. 5. 1998 132 55 412 29. 9. 1998 - 20. 5. 1999 152 65 413 28. 9. 1999 - 17. 5. 2000 119 49 414 12. 11. 2000 - 11. 5. 2001 72 23 315 26. 11. 2001 - 14. 5. 2002 63 27 316 24. 10. 2002 - 14. 5. 2003 82 27 317 3. 12. 2003 - 13. 5. 2004 68 24 318 19. 10. 2004 - 9. 5. 2005 84 39 319 13. 9. 2005 - 11. 5. 2006 114 48 420 16. 10. 2006 - 17. 5. 2007 82 39 321 30. 9. 2007 - 18. 5. 2008 114 56 422 11. 11. 2008 - 9. 5. 2009 69 26 323 29. 9. 2009 - 24. 4. 2010 78 28 3findings has been conclusive. We applied the CPS to the M , A and P estimates of independent datasets, using a first-ordermodel ( K = M , we find the best period P M = . ± .
21 yr. For A the best period was P A = . ± .
25 yr and for P , the best period was P P = . ± . A , M and P from indepen-dent datasets for correlations. One might expect a correlationbetween the mean brightness and starspot amplitude, simply be-cause when larger parts of the star are covered by starspots, thestar should appear dimmer. The starspot amplitude and periodmight also correlate, with changing latitude the e ff ective cov-ered area seen by the observer changes and due to di ff erentialrotation, if present, the period might also change.As the dependencies between these parameters are not nec-essarily linear, we calculated the Spearman’s rank correlation co-e ffi cient and the corresponding p-value for each pair of param-eters. Unsurprisingly, we find the strongest correlation between M and A , with a correlation coe ffi cient ρ = .
33 and a p-value p = . M and P we get ρ = − .
07 and p = .
58, andfor P and A , ρ = − .
18 and p = .
13, none of which is statisti-cally significant. Although there is relatively strong correlationbetween A and M , the periods P A and P M are di ff erent. This isat least partly explained by the spot coverage being sometimesmore axisymmetric, like in the Doppler images by Kovári et al.(2001). In the light curves obtained during similar spot configu-rations, the peak-to-peak amplitude of the light curve can be low,even though the star would appear to be faint. In order to estimate the stellar di ff erential rotation, starspots havebeen used as markers that are assumed to rotate across the vis-ible stellar disc with varying angular velocities, determined bytheir respective latitudes. To estimate the amount of surface dif-ferential rotation present in the star, we use the dimensionless Article number, page 3 of 10 (cid:11) (cid:12) (cid:13) (cid:14) (cid:15) (cid:16) (cid:17) (cid:18) (cid:19) (cid:20) (cid:21) (cid:22) (cid:23) (cid:24) (cid:25) (cid:26) (cid:27) (cid:28) (cid:29) (cid:30) (cid:31) ! " $ % & ’ ( ) * + , - . / : ; < = SEG:1 SET:2
SEG:1 SET:12
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SEG:2 SET:17
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Fig. 2.
Light curves and the best-fit models of independent datasets. The procedure used to calculate the phases is explained at the end of Section3. parameter Z = ∆ P W P W , (2)where P w ± ∆ P w is the weighted average of periods from theindependent datasets P i , given by P W = ( Σ w i P i ) / Σ P i , ∆ P w = p Σ w i ( P i − P w ) / Σ w i and w i = σ − (Jetsu 1993). The parameter Z gives the ± ∆ P w upper limit for the variation of the photomet-ric period P phot .Using only the period estimates from the independentdatasets, we get the weighted mean of the photometric period P w ± ∆ P w = d . ± d .
37, which gives Z = . A ( τ ) = . Z caused by noise were not significant (Lehtinen et al. 2011, Ta-ble 2). We can use the parameter Z to derive the di ff erential rotationprofile of a star (assuming Solar-like di ff erential rotation) Ω ( l ) = Ω (1 − α sin ( l )) (3)where l is the latitude, Ω is the rotation rate at the equator and α the di ff erential rotation coe ffi cient. The value of α can be es-timated with the relation | α | ≈ Z / h (Jetsu et al. 2000), where h = sin ( l max ) − sin ( l min ), and the parameters l min and l max arethe minimum and maximum latitudes between which the spotactivity is confined.Doppler imaging results by Hatzes (1993) and Kovári et al.(2001) indicate that the most of the spot activity on σ Gem isconstrained to latitudes between 30 ◦ and 60 ◦ , with some activityon lower latitudes, ± ◦ from the equator. This would give val-ues 0 . < ∼ h < ∼ .
75 yielding an α in the range . < ∼ α < ∼ . ,or in terms of rotational shear, 0.05 rad d − < ∼ ∆Ω < ∼ − Article number, page 4 of 10. Kajatkari et al.: Spot activity of the RS CVn star σ Geminorum > ? @ A B C D M A P Fig. 3.
The long-term changes of mean ( M ), amplitude ( A ) and period ( P ). The segment numbers are shown above the data. Table 2.
The strength of the di ff erential rotation from di ff erent papers. Paper α Henry et al. (1995) ± . ± . Kovári et al. (2007a) − . ± . − . ± . . < ∼ α < ∼ . For comparison, we have listed the previously derived values ofthe di ff erential rotation coe ffi cient α together with our estimatein Table 2.It is also of interest, how the amount of di ff erential rotationrelates to other stellar parameters in spotted stars. Henry et al.(1995) give a relation for the rotation period and the di ff eren-tial rotation. In comparison to their result, even our quite largeestimate for the di ff erential rotation is in the expected range forthis photometric period. Collier Cameron (2007) give a relationbetween the e ff ective temperature of the star and di ff erential ro-tation rate ∆Ω , ∆Ω = . T e ff ! . , (4)where T e ff is the e ff ective temperature of the star in Kelvins and ∆Ω is given as radians per day. Using the e ff ective temperature T e ff = ∆Ω ≈ . ff erential rotation rate. On the other hand, we also note that some of the di ff erentialrotation estimates for similar stars, which were used to derive theabove relation, have values comparable to our estimate.In order to estimate the reliability of our result, we calculatedsynthetic photometry using the spot-model by Budding (1977).We used a two-spot model with no di ff erential rotation, i.e. thespots rotated with a constant period P = P orb . We sampled thesynthetic light curve at the same observation times as in the orig-inal data and added normally distributed noise with zero meanand standard deviation σ N = . ǫ = y ( t i ) − ˆ y ( t i ) of our CPS model.The spot model parameters were calculated to correspond to thelocation of the active longitudes and the spot-modelling resultsfrom Kovári et al. (2001) as closely as possible. For the spot lat-itudes λ i , longitudes β i and radii r i we used values λ = ◦ , λ = ◦ , r = ◦ , r = ◦ , β = ◦ and β = ◦ .For the inclination of the star, we used value i = ◦ .The values we used for the linear limb-darkening coe ffi cient u and spot darkening fraction κ were u = .
79 and κ = . T phot = T spot = ffi cient for theJohnson V-band was calculated using the results by Claret (2000)and bilinear interpolation. The parameters used were T = T phot ,log g = .
5, microturbulence v micro = . / s and solar metal-licity. All these parameters are the same as used in Kovári et al.(2001). Article number, page 5 of 10 nalysing this synthetic photometry, we got the value Z synth = . ff ected by the long rotation periodcombined with poor sampling. More complex models, such asaddition of a third spot, rotating with a period of P = d .
47 didnot change this result, neither did addition of artificial ff -events. Active longitudes are longitudes on the surface of a star thatexhibit persistent spot activity. They can appear in pairs, situ-ated on the opposite sides of the star (Henry et al. 1995; Jetsu1996; Berdyugina & Tuominen 1998). The presence of activelongitudes in observational data is well established in many ac-tive stars and they are thought to be manifestations of non-axisymmetric dynamo modes. σ Gem has shown very persis-tent active longitudes throughout its whole observational history(Jetsu 1996). Moreover, the active longitudes on σ Gem aresynchronised with the orbital period of the tidally locked binarycomponents, whereas on many other RS CVn stars, the activelongitudes have been reported to migrate linearly in relation tothe orbital reference frame (e.g., Berdyugina & Tuominen 1998;Lindborg et al. 2011).The phase diagram of the light curve minima t min , and t min , (Fig. 4) shows clearly the two active longitudes, that are, witha few exceptions, present throughout the length of the wholetime series. The phases φ orb were calculated with the orbitalperiod, using the ephemeris JD conj = d . + d . φ = φ orb − .
2. Thus, phase φ = . t min , and t min , , were simultaneously analysed using thenon-weighted Kuiper-test. The Kuiper-test is a non-parametrictest that is suited for searching for periodicity in a series oftime points t i , when the phases, calculated using period P , φ P , i = t i P mod 1 have a bimodal (or even multimodal) distribu-tion. We used the same formulation as in Jetsu (1996). Weused a null hypothesis ( H ), that the phases φ P , i are uniformlydistributed within the interval [0 , . P W < P < . P W , where P W is the weighted aver-age of the photometric periods. The resulting best period was P min , , = d . ± d . Q = . × − .In Fig. 4 can be seen that during segments 8 −
13 and 15 − t min , using the Kuiper-test and got thebest period P min , = d . ± d . Q = . × − .If the drift of the primary minima was present only duringsingle segments, this e ff ect could be introduced by evolving spotpattern or di ff erential rotation. However, the primary minimatrace a clearly identifiable path, that can be seen for many years.The aforementioned e ff ects would only apply to single spots, notwhole active areas such as active longitudes.The main contribution to P min , comes from segments SEG9,SEG14 and SEG17. During these segments the secondary min-ima vanish altogether and the primary minimum is shifted ∼ . = y . + y . Flip-flops are a name coined by Jetsu et al. (1993) in their anal-ysis of the active giant FK Com photometry. A flip-flop is anevent where the primary and secondary minima suddenly switchtheir places in a phase diagram. We refer to these type of eventsas ff -events. An obvious interpretation is that during a ff -event,the activity jumps from one active longitude to another.In the σ Gem data, several jumps in the phase diagram canbe seen. Since all of these flip-flop candidates are not necessar-ily similar, physically related phenomena, we use the followingcriteria to distinguish true ff -events from apparent ones:C I : The region of main activity shifts about 180 degrees fromthe old active longitude and then stays on the new active lon-gitude. andC II : The primary and secondary minima are first separated byabout 180 degrees. Then the secondary minimum evolvesinto a long–lived primary minimum, and vice versa.
Although multiple phase shifts can be found in the data, thereis only one activity shift, between segments one and two, thatfulfils these two criteria. In addition, there are multiple eventssimilar to ff -events, that are abrupt, but not persistent, or are as-sociated with gradual migration of the primary minima that takeyears to complete. These events are called ab-events and gr-events, respectively. The only ff -event in the data that fulfils our criteria C I and C II oc-curs between segments SEG1 and SEG2 and has been identifiedin previous papers that used contemporaneous data, i.e., Jetsu(1996) and Berdyugina & Tuominen (1998). The first signs ofa coming ff -event can be seen at the end of SEG1, where thelight curve shows a gradual deepening of the secondary mini-mum. The switch could have actually occurred even before theend of SEG1; in the last few models of the segment, the minimahave already switched places, but the models are not reliable dueto the small number of observations. At the beginning of SEG2,the previous secondary minimum has become the new primaryminimum. After the ff -event, the primary and secondary mini-mum stay stable for several years. Article number, page 6 of 10. Kajatkari et al.: Spot activity of the RS CVn star σ Geminorum E Fig. 4.
The phases of the light curve minima of Sigma Geminorum in the orbital frame of reference. Phase φ = . = y . + y . Signs of a coming phase shift can be seen already at the endof SEG8. In this case, however, the phase shift is gradual andcaused by weakening of the primary minimum, rather than bydeepening of the secondary i.e., the activity does not "jump"from an active longitude to another, rather it diminishes on oneand stays constant on the other. During this segment, the star isat its brightest and correspondingly the light curve amplitude isvery low, decreasing throughout segments SEG8–SEG10.During SEG9 the secondary minimum vanishes altogether,giving way to a relatively unstable primary, situated halfway be-tween the previous two active longitudes.As seen in the light curves (Fig. 2), the new minimum is alsoextremely wide, most likely consisting of several large starspots.The minimum persists until the beginning of SEG10. There areseveral unreliable datasets in the beginning of this segment, dur-ing which two separate minima emerge from the previous singleminimum. This kind of phase diagram is expected when twoclose starspots rotate with di ff erent periods, gradually movingaway from each other (Lehtinen et al. 2011, Fig. 3).The Doppler images by Kovári et al. (2001) taken between1 November 1996 – 9 January 1997, and overlapping with thefirst part of segment 10, show three large starspots distributedalmost at equal distances in longitude on a latitude band situatedbetween 0 ◦ and 60 ◦ . The photometric spot models in the samepaper also show three spots (Spots 1-3), of which Spots 1 and2 correspond to the persistent active longitudes. This view doesnot support the idea of the two large spots rotating at di ff erentrates.What we consider more likely, is that the previously almostband-like spot distribution is vanishing, or the third spot (Spot3) slowly migrates and merges with the other active longitude(Spot 1). In our analysis, the beginning of SEG10 shows twominima, with the primary minima linearly migrating from φ = . φ = .
3. This supportsthe idea of the first and third spot merging. After this, the stableactive longitudes again start to dominate the light curve, onlythis time, the primary and secondary minima have switched theirplaces. During 2000-2002 and 2003-2005, the star shows again sim-ilar behaviour. During SEG13 the two active longitudes arepresent, but they disappear in SEG14 and are replaced by a sin-gle minimum, located halfway between the active longitudes. InSEG15, the two active longitudes are present again, and the min-ima have switched their places.In SEG17 the active longitudes disappear once again and arerecovered in the following segment, now with switched primaryand secondary minima. Unlike during the 1996-1997 event, theamplitude of the light curve and the mean brightness of the stardo not show any changes or patterns during the time span be-tween 2000 and 2005. There is a small linearly growing trendin the mean brightness, but it does not correlate in any way withamplitude or period during that time.
The area of main activity jumps multiple times from one activelongitude to another during the last five years of the time se-ries. Unlike before, these phase shifts happen in brief succes-sion, with intervals of only a year or two. The first ab-event,during SEG19, can be seen in the light curves and persists un-til the following SEG20, where the primary minimum switchesback. In SEG20 and SEG21 the same happens again: the activityjumps from one active longitude to another and back again in rel-atively short time. Although the phase shifts are not persistent,the primary minima are quite deep in each case and the shifts arelikely to be real and not just random fluctuation or observationalerrors.
We determined epochs for each ff -, ab-, and gr-event. In the caseof ff - and ab-events, we used the mean epoch of the two primaryminima between which the phase shift occurred. For gr-events,we determined the epoch from the moment when the primaryminimum reached and stayed on the active longitude in question.Jetsu (1996) determined the width of the active longitudes to be0.2 in phase, therefore we required that the primary minimum Article number, page 7 of 10 .10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Frequency(1.0/yr) n Fig. 5.
Kuiper-test periodogram for the ff - and gr-event epochs inTable 3. Table 3.
The ff -, gr- and ab-events found in the data. The first columnis the number of the event, second the segment or the range of segmentsduring which the event occurred. The next two columns give the esti-mated epochs of the event in JD and years. The fourth column givesthe phase calculated with the ephemeris J ff = . + . ff= flip-flop, gr = gradual phase shift,ab = abrupt phase shift. Event Segment HJD year φ ff Event type1 – – 1981.0 0.00 ff ff | φ min , − φ al , i | < .
1, where φ al , i is the meanphase of an active longitude.The mean phases of both of the active longitudes were cal-culated from independent datasets so that each minimum con-tributes only to the mean of the active longitude it is closest to inphase, and only those datasets which have two minima are used.The phases we get for the two active longitudes are φ al , = . φ al , = . ff - and gr-events, we analysed theirrespective epochs using the Kuiper-test with the period interval P min = . P max = . P ff , = . , V n = .
81. We, however, consider the secondbest period P ff , = .
99 yr, V n = .
74 to be more plausible, partlybecause the shorter period would imply that there are multipleunobserved flip-flop epochs, and partly because the 2.67 year pe-riod is an integer part of the 7.99 year period. The periodogramis plotted in Fig. 5. The small number of time points makesit impossible to calculate any meaningful significance estimatesfor the Kuiper-test statistics V n , for this, more events would berequired.
5. Discussion
The value we get for di ff erential rotation is an order of a mag-nitude greater than the values found in previous studies. Themain culprit is most likely the long rotation period. In somecases, this leads to poor phase coverage which in turn leads tolarge uncertainties in the period estimates. Another problem thelong rotation period causes is the possibility that the spot struc-ture changes on the surface of the star. In the phase diagramsof some datasets this can be seen as a superposition of two lightcurves with noticeably di ff erent shapes. In a worst case scenariothese two e ff ects appear simultaneously, i.e., the spot structurechanges between successive rotations, but this is not noticeabledue to poor phase coverage. Thus the phase diagram can createan illusion of a unique, continuous light curve, while in fact, itwas created by two di ff erent spot configurations, introducing anerror to the period estimate.Henry et al. (1995) used some of the same data analysed inthis paper. The discrepancy between our and their di ff erentialrotation estimates can be easily attributed to the di ff erent ap-proaches that were used. Henry et al. (1995) used spot mod-elling and the di ff erential rotation estimate was derived from themaximum and minimum periods determined from the spot mi-gration curve of each spot.In our analysis even slight changes in the spot structure, oc-curring faster than the rotational period, could lead to a change inthe period estimate, which we then interpret as di ff erential rota-tion. In Lehtinen et al. (2011), only the signal-to-amplitude ratiowas considered when the amount of spurious period change wasestimated. Our simulated data indicated that also the samplinge ff ects are a considerable source of spurious period changes andcan result in overestimates of the di ff erential rotation.When comparisons are made to other stars with compara-ble period, the range of di ff erential rotation these stars exhibitis greater than the di ff erence between measurement techniques.There is also considerable doubt if starspots are even reliableproxies of di ff erential rotation. Even spots on the same lati-tude might have di ff erent migration rates due to di ff erent anchordepths. This is shown by recent numerical simulations indicat-ing that if observed starspots are caused by a large-scale dynamofield, their movement is not necessarily tracing the surface dif-ferential rotation, but the movement of the magnetic field itself(Korhonen & Elstner 2011). Finally, observed spots are not nec-essary even stable or might consist of many small rapidly evolv-ing starspots instead of one large spot. In any case, it is clear thatthe use of the CPS-method might greatly overestimate di ff eren-tial rotation, if the rotation period is long. We find three kinds of events in which the activity moves froman active longitude to another. In addition to the single flip-flop fulfilling the criteria C I and C II ( ff -events), we find abrupt(ab-events) and gradual phase shifts (gr-events). It is not clearwhether or not these di ff erent kind of events are caused by thesame phenomenon or not. A similar kind of two-fold behaviourof ab- and gr-events has been reported in FK Com, by e.g.Oláh et al. (2006) and Hackman et al. (2013).The ab-events are similar to the flip-flops discovered byJetsu et al. (1993): sudden shifts of primary minima from oneactive longitude to another. The only di ff erence is that in ab-events, the phase-shift is not persistent and the area of the main Article number, page 8 of 10. Kajatkari et al.: Spot activity of the RS CVn star σ Geminorum A (e) 18.0 19.5 21.0P F G H M (h)0 20 40 60 80 100 120 140 160 180t I J K L M N O M (c) 0 20 40 60 80 100 120 140 160 180t14161820222426 P (f) 18.0 19.5 21.0P0.020.040.06 A (i)0 20 40 60 80 100 120 140 160 180t P t c (d) 0 20 40 60 80 100 120 140 160 180t0.00.20.40.60.81.0 Q m i n (g) 0.02 0.04 0.06A R S T M (j)0102030405060 U t max =49.0t=10.1989-5.1990 Sigma Gem, segment 3
Fig. 6.
CPS-analysis of segment SEG3. The contents of the panels are explained at the end of Sect. 3. active region shifts back to the original active longitude, roughlyafter a year or so. Unfortunately, it is almost impossible to de-termine which ab-events are fundamentally di ff erent from "real" ff -events, i.e., sudden, but persistent phase shifts. Some of theab-events could fail to fulfil the criteria for ff -events, simply be-cause the event is followed by an unrelated ab-event, creating anappearance of a non-persistent phase-shift.During the gr-events, the location of the main active regionshifts by ∼ ◦ in longitude and the photometry shows only onewide minimum. The disappearance of the two long-lived activelongitudes and their replacement with only one minimum couldbe illusory at least in SEG9. The overlapping Doppler imagesdated to the beginning of SEG10 show that there is a large spotarea near the longitude the single minimum was located at.The appearance of spots between the active longitudes re-sembles what Oláh et al. (2006) found in photometry of FKCom, and called phase-jumps . In a phase-jump , old active areasdisappear and then new ones emerge, with an o ff set of roughly90 ◦ in respect to the original active longitudes. In FK Com, thephase jumps cause the active longitudes to stay displaced for amuch longer time (Hackman et al. 2013). It could be that the bi-nary nature of σ Gem a ff ects the preferred location of the activelongitudes and this displacement is not long lasting.In segments SEG10, SEG14, SEG17 and SEG18 the primaryminimum traces a path towards the active longitude situated at φ al , = .
3. This may imply that in addition to the stable non-axisymmetric dynamo mode, there is also a possible azimuthaldynamo wave present, rotating faster than the star itself. Thisis also suggested by the period of the primary minima, which is shorter than the orbital period of the tidally locked binary sys-tem.If present and rotating at a constant rate, the wave wouldreturn to a same active longitude every 7.9 years. This period isremarkably close to the 7.99 year period we find from the epochsof the gr- and ff -events. It is possible that at least some of thoseobserved events occur when a spot structure corresponding toa moving dynamo wave interferes with the stable active longi-tudes, either strengthening or weakening the minima as it passesby. This does not prevent the ab-events from also being causedby this mechanism. In this model, the time between successiveflip-flops is equal only in the case that the apparent spot cover-age on both of the active longitudes is equal. If the spot cov-erage on either of the active longitudes is greater, the primaryminimum will stay on this active longitude for a longer time. Tofurther complicate things, the spot coverage on the active longi-tudes may also change independently from the migrating activearea, and this can prevent the observation of the gr-events.If there is an active region migrating in the frame of the or-bital period, we should see periodic variation in A . The interfer-ence between the stationary active longitudes and the migratingregion should modulate the light curve with an amplitude enve-lope with the same period as the flip-flop event cycle. We detectno clear sign of such modulation, although events 3 and 4 inTable 3 are associated with relatively low values of A .It is possible that this amplitude e ff ect is masked by shortterm spot evolution. This seems plausible, since the variationin A between independent datasets within one segment is quitelarge. Intriguingly, there are also disturbances in the light curves Article number, page 9 of 10 t multiple occasions, at the same epochs when the presumeddynamo wave passes an active longitude. An example of thiscan be seen in SEG3 (Fig. 6), where these abrupt changes inthe light curve even prevent reliable CPS modelling. Similar be-haviour can be seen in segments SEG11 and SEG19. As for theother CPS-parameters, M and P , there seems to be no obviousconnection between them and the ff -, gr- and ab-events. Alsothe periods found from these parameters are di ff erent from theKuiper-test periods. On the other hand, one may speculate thatthe 2.67 and 8.5 year periods found by Strassmeier et al. (1988)and Henry et al. (1995), respectively, are somehow connected tothe 2.7 and 8.5 year Kuiper-test periods. If either of those peri-ods is real and due to a migrating spot area, this could very wellbe reflected in the mean brightness of the star.
6. Summary and conclusions
By applying the CPS method to photometry of σ Gem we havebeen able to study in detail the long-term evolution of the meanbrightness ( M ), light curve amplitude ( A ) and, photometric min-ima ( t min , , t min , ) and photometric rotation period ( P ) of the star.The best periodicities in M , A and P were: P M = . ± . P A = . ± .
25 yr and P P = . ± . P could be explained by di ff erential rotation, for which weestimated a coe ffi cient of 0 . < ∼ α < ∼ .
21. From the com-bined time point series of the both primary and secondary min-ima, we found a period P min , , = d . ± d . P min , = d . ± d . A , M and P . The only obvious connectionbetween P and the other model parameters could be due to dif-ferential rotation. The di ff erential rotation estimate we get fromthe period changes is extremely large when compared to previ-ous analyses (Kovári et al. 2007a,b). Using synthetic photom-etry, we demonstrate that this is at least partly due to the longrotation period of the star, which sometimes leads to sparse anduneven phase coverage. This can cause large fluctuations in theperiod estimates, and thus, an unreasonably large di ff erential ro-tation estimate.We also confirm the presence of previously found persistentactive longitudes, which are tied to the orbital reference frameof the binary system. The most interesting result we present inthis paper is the possible connection between the flip-flop likeevents and the drift of the primary minima. This may implythat there is a superposition of two dynamo modes operating inthe star. One would be tied to the orbital period of the binarysystem, while the other one could manifest itself as an azimuthaldynamo wave, rotating faster than the star. Signs of such dynamowaves have been observed in other stars (Lindborg et al. 2011;Hackman et al. 2011, 2013, e.g.) and have also been reproducedin numerical MHD-simulations (Cole et al. 2013).Such an azimuthal dynamo could disturb the stable activelongitudes present in the star, creating the ff -, gr- and ab-events.In order to find supporting evidence for the presence of a prop-agating dynamo wave, it would be necessary to obtain newDoppler images of the star, preferably at least two sets takenat di ff erent times, and check, whether or not there are star spotsin areas indicated by the ephemeris Yr = y . + y . Acknowledgements.
This work has made use of the SIMBAD data base at CDS,Strasbourg, France and NASA’s Astrophysics Data System (ADS) bibliographicservices. The work by PK and JL was supported by Vilho, Yrjö and Kalle VäisäläFoundation. The work of TH was financed by the project "Active stars" at theUniversity of Helsinki. The automated astronomy program at Tennessee StateUniversity has been supported by NASA, NSF, TSU, and the State of Tennesseethrough the Centers of Excellence program. We thank the referee for valuablecomments on the manuscripts.
References