arXiv:1110.1104v1 [astro-ph.SR] 5 Oct 2011 fietavrain,wihi fteodro 10 of order moment the the of of is timescale which the variations, on inertia occur of should r stellar changes that implies period This i sta conserved. of is hydrogen momentum angular mass-loss the fusing without phase, objects MS the During (MS) cores. their main-sequence angul condensed. prevailing as the they the lives of which spend their of fraction rotate. out a stars cloud all inherit Consequently, mother stars the all of mass, momentum to addition In Introduction 1. tr.Teoeaudn lmnsi hi topee r c are atmospheres their uppe in in evolution elements suit overabundant rotational most The the the studying stars. for are beds composition, test chemical able surface abnormal an this? test we do How 2000). Maeder & (Meynet e-mail: edo Send coe 0 2018 30, October srnm Astrophysics & Astronomy .O Kudrjavtsev O. D. .Mikul´aˇsekZ. h antcceial euir(C)sas hc have which stars, (mCP) peculiar chemically magnetic The upiigvrain ntertto fteceial pec chemically the of rotation the in variations Surprising eevd2 uy2011 July 29 Received 11 10 information. e words. e Key to stars s upper-main-sequence able these magnetic is in of evolution body periods and stellar structure rotation rotating of faster variability inner, alternating an with wind, yea the severa in of d timescale 1.538771 Conclusions. a of on maximum Vir a CU reaching quarter-century, in changes it period reached smaller it much when o 1968, intervals alternating the exhibit until stars shortening both gradually time, same the met novel At a using stars the Results. of variations period the analysed Methods. Aims. Context. UVradV0 r eogaogteefwmPsaswoerota whose stars mCP few a these of among model belong Ori the V901 by and explained Vir adequately CU be can that variations ecigislclmxmmo .276 nteya 05 Si 2005. year the in d 0.5207163 of maximum local its reaching ff 9 8 7 6 2 5 4 3 1 [email protected] rn eussto requests print rvt bevtr,6 ikBro od lmta,Cp T Cape Plumstead, Road, Burton Dick 61 Observatory, Private EI,Osraor ePrs NS PC Universit´e Pari UPMC, CNRS, Paris, de Italy Observatoire Catania, LESIA, di Astrofisico Osservatorio - NAF nttt o srnm fteUiest fVen,Vienna Vienna, of University the of Astronomy for Russia Institute Pulkovo, at Observatory Astronomical Central pca srpyia bevtr fRS ihi Arkhyz, Nizhnij RAS, of Observatory Astrophysical Special bevtr n lntru fJhn aia V Palisa, Johann of Planetarium and Observatory srnmclIsiueo lvkAaeyo cec,Tatra Science, of Academy Slovak of Institute Republic Astronomical Slovak 60, 059 SK 133, Tatransk´a Lomnica etro xelnei nomto ytm,TneseSta Tennessee Systems, Information in Excellence of Center eateto hoeia hsc n srpyis Masar Astrophysics, and Physics Theoretical of Department eamt td h tblt fteprosi UVradV0 O V901 and Vir CU in periods the of stability the study to aim We efudta h hpso hi hs uvswr osatd constant were curves phase their of shapes the that found We h aoiyo antcceial euir(C)sasex stars (mCP) peculiar chemically magnetic of majority The ecletdalaalberlvn rhvdobservations archived relevant available all collected We tr:ceial euir–sas aibe tr:indi stars: – variables stars: – peculiar chemically stars: 1 epooeta yaia neatosbtenati,oute thin, a between interactions dynamical that propose We , 2 .Krtiˇcka J. , 6 Zdenˇek Mikul´aˇsek,: .I Romanyuk I. I. , / cetd2 etme 2011 September 20 Accepted aucitn.Variations14 no. manuscript tr UVrii n 91Orionis V901 and Virginis CU stars 1 .W Henry W. G. , 6 .A Sokolov A. N. , 3 .Jan´ık J. , 7 − B–TcnclUiest,Otaa zc Republic Czech Ostrava, University, Technical – SB ˇ 10 9 otation ol eoetesi-ontm aao n rnsanwinsi new a brings and paradox time spin-down the remove would atof part 7 MS r tars. oainlbaigadaclrto.Tertto period rotation The acceleration. and braking rotational f 1 o htalw o h obnto fdt fdvresorts. diverse of data of combination the for allows that hod ABSTRACT .L¨uftinger T. , .Zverko J. , oa iiu f05079 .Tepro hnsatdincr started then period The d. 0.52067198 of minimum local s on- panteosre oainlvraiiy theoretical A variability. rotational observed the xplain 03 hntertto eidbgnt decrease. to began period rotation the when 2003, r ar rs Austria , iil oaigmi-eunesa ihpritn surfa persistent with main-sequence rotating rigidly c httm h oainhsbgnt ceeaeaan ea We again. accelerate to begun has rotation the time that nce kUiest,Kt´ˇs´ ,C 1 7 ro zc Republ Czech Brno, Kotl´aˇrsk´a University, 37, 611 yk CZ 2, s´ onc,Soa Republic Slovak nsk´a Lomnica, n eUiest,Nsvle ense,USA Tennessee, Nashville, University, te - Russia er.Tertto eido 91Oiwsicesn o th for increasing was Ori V901 of period rotation The years. l ieo;5paeJlsJnsn 29 ednCdx France Cedex, Meudon 92190 Janssen, Jules place 5 Diderot; s w,SuhAfrica South own, oee,teei loasalsbru fhtmPsasthat stars mCP hot of subgroup small a also is 20 Adelman there & However, (Pyper stars distorted magnetically of cession lysclrcci hne ntesaeo hi ih curve light their of shape e.g. d the stars in mCP changes few cyclic A secular decades. uppe play of most timescales of on curves constant light are and stars periods expe rotation theoretical the our that tions performe confirm been They have On-line (2011). 2007) al. Mikul´aˇsek the chemica al. et (Mikul´aˇsek magnetic in et stars of listed peculiar observations stars photometric mCP of best-monitored catalogue the of dozens unprecede curacy. with decades evolution several rotational past their the reconstruct obse over can archival collected magne and stars mCP present and of both vations spectrum, Combining observed. brightness, r are star the field the in As variations regions. spot periodic persistent large, into centrated iul UVr 91Oi–sas rotation stars: – Ori V901 Vir, CU vidual: inprosvr ntmsae fdecades. of timescales on vary periods tion ii eidclgt ai,setocpcadspectropol and spectroscopic radio, light, periodic hibit upeetdwt u e esrmnso hs tr and stars these of measurements new our with supplemented rn h atsvrldcds hl h eid eechang were periods the while decades, several last the uring aeu eidaaye fpooercosrain of observations photometric of analyses period Careful izn vkye l 00,wihcnb trbtdt h pre the to attributed be can which Ziˇzˇnovsk´y 2000), al. et ˇ 4 iuigalacsil bevtoa aacnann phas containing data observational accessible all using ri J. , 8 antclycnndevlp,bae ytestellar the by braked envelope, magnetically-confined r .Trigilio C. , Ziˇzˇnovsk´y ˇ 9 5 .Neiner C. , .Zejda M. , 1 10 .Liˇska J. , n .N eVilliers de N. S. and , fC i was Vir CU of 1 yunexpected ly estructures. ce .Zvˇeˇrina P. , ic

s found lso arimetric uliar c h into ght easing, S 2018 ESO past e ing. tdac- nted e otates, (see s 1 MS r 11 we , in d , 04). cta- lly is- tic r- - 2 Z. Mikul´aˇsek et al.: Surprising rotation variations of the mCP stars CU Vir and V901 Ori have stable light curve shapes and spectroscopic variability, but 2.5 exhibit variable rotation periods (Pyper et al. 1998; Mikul´aˇsek et 0.35 2 al. 2008; Townsend et al. 2010). Constancy of their light curves 0.3 on the scale of decades excludes precession as the cause of the 1.5 observed period changes (for details see Mikul´aˇsek et al. 2008). 1 0.25 lin This letter concentrates on the period analysis of the best- 0.5 0.2 monitored mCP stars – CU Virginis and V901 Orionis. O−C 0 P [s]

∆ 0.15 −0.5 0.1 2. The stars −1 2.1. CU Virginis −1.5 0.05 −2 CU Vir = HD 124224 = HR 5313 is one of the most enig- 0 −2.5 matic stars in the upper MS. This very fast rotating silicon mCP 1960 1980 2000 1960 1980 2000 star has a moderate, nearly dipolar magnetic field (Borra & Years Years Landstreet 1980; Pyper et al. 1998) with the pole strength of ◦ Fig. 1. Long-term variations of the rotation period and O-C values for Bp = 3.0 kG tilted towards the rotational axis by β = 74 , the ◦ the magnetic CU Vir. Left: The model curve of axis inclination being i ≃ 43 (Trigilio et al. 2000). CU Vir is the long-term rotational period changes ∆P of CU Vir in seconds with re- only known MS star that shows variable radio emission, resem- d spect to the mean period P0 = 0.52069415. The formal accuracy of the bling the radio lighthouse of pulsars (Trigilio et al. 2008, 2011; polynomial fit is comparable with the thickness of the fitted line. Right: Ravi et al. 2010), it also displays variations in light and spectral The difference of the observed and calculated times of zero phase ac- lines of He i, Si ii,H i, and other ions. The nature of its variabil- cording to a linear ephemeris in days. Each point represents the aver- ity is minutely studied in Krtiˇcka et al. (2011). CU Vir belongs age of 498 consecutive individual measurements; the weights of these to the most frequently studied mCP stars. means are indicated by their areas. Occasional rapid increases in its rotation period have been reported and discussed. Pyper et al. (1997, 1998) discovered an d d the negative value of the second derivative of the period: P¨ = abruptincrease of the period from 0.5206778to 0.52070854 that −13 −1 occurred approximately in 1984 and Pyper & Adelman (2004) −29(13) × 10 d , which indicated that braking could change discussed two possible scenarios of the explanation of the ob- into acceleration (Mikul´aˇsek et al. 2008). Nevertheless, the sig- served O-C diagram, namely a continually changing period or nificance of this conclusion was fairly low. two constant periods. After the year 1998 Trigilio et al. (2008, 2011) observed another increase in the period of radio pulses of 3. Observations ∆P = 1.12 s with respect to the period determinedby Pyper et al. (1998). However, the 2010 measurements indicate some period For the period analyses of CUVir we obtained some new pho- decrease. The rate of the deceleration can be evaluated using the tometric or spectroscopic data, specifically: 1) JK derived 210 spin-down time, τ, defined as τ = P/P˙¯, where P(t) is the instan- individual spectrophotometric magnitudes in 21 bands (201 - taneous rotation period at the time t and P˙¯ is the mean rate of 299 nm) by processing 10 IUE spectra, in 1979 January - March the rotational deceleration. The paradox of CUVir is according (for details see Krtiˇcka et al. 2011); 2) GWH obtained 374 pre- to Mikul´aˇsek et al. (2011) that its spin down time, τ ∼ 6 × 105 cise BV measurements at Fairborn Observatory Arizona, USA, years, is more than two ordersof magnitude shorter than the esti- in 2010 February - 2011 June; 3) JL obtained 38 V measure- matedage of thestar – 9×107 yr (Kochukhov & Bagnulo 2006). ments using his own microtelescope at the SAAO, South Africa and Brno, CR, in 2010April- June ; 4) JJ obtained251 vb ob- servations at Suhora Observatory, Poland, in 2011 March; and 2.2. V901 Orionis JJ+JL obtained 536 UBV measurements using the 0.5m reflector V901 Ori = HD 37776 is a young hot star (B2IV) residing in of the SAAO, South Africa, in 2011 May. the emission IC 432, with a global, extraordinarily strong We collected a total of 8965 individual measurements of CUVir obtained between 1949 and 2011 (62 years or 43662 (Bs ≈ 20 kG), and unusually complex magnetic field (Thompson & Landstreet 1985; Kochukhov et al. 2011). The observed mod- revolutions of the star) including 8270 photometric and spec- erate light variations are caused by the spots of overabundant trophotometric measurements in photometric bands from 200 to silicon and helium (Krtiˇcka et al. 2007). 753 nm as well as spectroscopic, spectropolarimetric and radio- Three decades of precise photometric and spectroscopic metric observations. The diverse observational data originated monitoring have revealed a continuous rotational deceleration from 39 sources (see the list in Appendix B). (Mikul´aˇsek et al. 2008), increasing the period of about 1d.5387 The V901Ori data listed in Mikul´aˇsek et al. (2008) were en- by a remarkable 18 s! Ruling out (a) the light-time effect in a bi- hanced with 98 precise BV measurements obtained by GWH at nary star, (b) the precession of the star’s rotational axis, and (c) Fairborn Observatory Arizona, USA from September 2009 to evolutionary effects as possible causes of the period change, we February 2011. We used a total of 2611 individualmeasurements interpreted the deceleration in terms of the rotational braking of obtained from 1976 to 2011 from 14 sources. In addition to 2409 photometric measurements, the data include 202 measurements the outer stellar layers as caused by the angular momentum loss i in the stellar . of equivalent widths of eleven selected He lines. However, this cannot explain the discrepancy between the 5 spin-down time, τ = 2.5×10 yr, and the star’s age of one million 4. The models and their results years or older (Mikul´aˇsek et al. 2008, 2010, 2011; Kochukhov & Bagnulo 2006). The interpretation of the rotation period evo- The method of data processing, outlined in the Appendix A, as- lution by simple angular motion loss was also questioned by sumes that phase curves of all monitored quantities are constant, Z. Mikul´aˇsek et al.: Surprising rotation variations of the mCP stars CU Vir and V901 Ori 3 while the ‘instant period’ of changes P(t) is variable in time t. It −3 is advantageous to introduce a ‘phase function’ ϑ(t, b), which is x 10 a continuous monotonic function of time t. The fractional part of 4 it corresponds to the common phase ϕ, the integer part is the so- 2 called (E) and b isaset of gb free parameters describing a model of the period development. Quantities P(t) and ϑ(t, b) are bound by the simple equality P(t) = 1/ϑ˙, where the dot refers to 0 the first time derivative of the quantity. Because we did not know O−C [d] the true nature of observed period changes of P(t) we chose stan- −2 dard low-orders polynomials for their description.

−4 1988 1990 1992 1994 1996 4.1. CU Virginis Years The period changes P(t) (see Fig.1) can be adequately approxi- mated by a cubic parabola with the origin at the time t = T0, Fig. 2. Medium-term variations in the rotational period of CU Vir. The modulation of the long-term rotational changes in the observed time 3 P(t) = P0 + A 3Θ − 4 Θ ; Θ = (t − T0) /Π, (1) of the light minimum minus the calculated time of light minimum is   expressed in days. Owing to uneven sampling within the observing where A and Π are parameters expressing the amplitude and seasons, we were not able to detect rotational modulation on shorter timescale of the observed period changes, and Θ is a time-like timescales. function. The instant period P(t) reaches its local extrema P0 ∓ A 1 at Θ1,2 = ∓ 2 , where P0 is the periodat thetime of t(Θ = 0) = T0. The value Π then equals the duration of the rotation deceleration 5 1 0.45 epoch that took place in the time interval t1,2 = T0 ∓ Π. 2 0.4 ˙ 0 Applying the equality ϑ = 1/P(t), we obtain the following 0.35 approximation for the phase function in the form of the fourth- −5 0.3 order polynomial of time 0.25 lin P [s] B t − M A Π ∆ −10 0.2 ϑ(t, b)  ϑ − 3 Θ2 − Θ4 ; ϑ = 0 , B = , (2) O−C 0 2 0 0.15 P0 P0 P0 −15   0.1 where ϑ is the phase function for a linear ephemeris with the 0 0.05 origin at M0 ≡ 2446730.4447 and the basic period P0. −20 0 Using the above formulated phenomenological model, we d −0.05 derive P0 = 0.52069415(8), Π = 13260(70) d = 36.29(19) 1980 1990 2000 2010 1980 1990 2000 2010 Years Years yr, T0 = 2446636(24) or 1986.56(7), A = 1.915(10) s, and B = 0.5643(29) d. The period P(t) reached its local minimum in Fig. 3. Long-term variations in the rotational period and O-C diagram d the year 1968.4, Pmin = 0.52067198(7), and its local maximum of V901 Ori. Left: Variation in the rotation period ∆P in seconds with d d in 2004.7: Pmax = 0.52071631(18). The rotational deceleration respect to the mean period P0 = 1.538756(3), derived from 2611 in- rate reached its maximum P˙ = 0.158syr−1 in the year 1986.6. dividual measurements. The 1-σ deflections from the quadratic fit are The zero phase times of CUVir can be evaluated using the rela- denoted by dashed lines. Points with error bars correspond to virtual  3 2 4 period deflections for appropriately selected groups of consecutive indi- tion JD(k) M0 + P0 k + B ( 2 Θk − Θk ), where k is an integer and Θ = (M + P k − T )/Π. vidual data items. Right: The difference of the observed zero phase time k 0 0 0 and the time calculated according to the linear ephemeris in fractions of The basic model fits the observed long-term period changes a day. Each point represents an average of 435 consecutive measure- very well and enables us to predict the zero phase times with an ments; the weights of these means are indicated by their areas. accuracy of 0.001 d. However, the analysis of the residuals from the accepted polynomial model reveals an additional variation. Between 1988 and 1998, for which there is excellent photomet- We put the origin, M0 = 2449967.969(5), at a bright- ric coverage (Adelman et al. 1992; Pyper et al. 1998), we find a ness maximum (in 1999). Our fitted parameter values include d small modulation of the period on a timescale of several years P0 = 1.538756(3), the instantaneous period at time M0, and −9 −1 (Fig. 2). P˙ 0 = 11.3(0.7) × 10 = 0.356(23)s yr and P¨ = −4.4(6) × 10−12d−1 = −1.38(19) × 10−5 syr−2, the first and second time derivatives. The negative value of P¨ is now established with 4.2. V901 Orionis 7-σ certainty. Accordingly, the star reached its longest period, d For the phase function of V901Ori we adopted polynomials Pmax = 1.5387709(15), in 2002.8 ± 1.1, and is now accelerat- in the form of the third-order Taylor-expansion, also used in ing again (see Fig.3). The zero-phase timing can be calculated  1 ˙ 2 1 ¨ 3 Mikul´aˇsek et al. (2008): by the relation JD(k) M0 + P0 (k + 2 P0 k + 6 P0Pk ), where k is an integer. There is no indication of any short-term period 1 ˙ 2 1 ¨ 3 ϑ(t, b) = ϑ0 − 2 P0ϑ0 − 6 P0 Pϑ0; ϑ0 = (t − M0)/P0; modulation analogous to that seen in CUVir. ˙  ˙ 1 ¨ 2 P(t) = 1/ϑ P0 1 + P0ϑ0 + 2 P0Pϑ0 , (3)   5. Discussion where ϑ0 is the phase function for the linear ephemeris with the origin at M0. P0 and P˙0 are the instantaneous period and its first We were able to reveal unexpected alternating lengthenings and derivative at M0. We assumed that the second derivative of the shortenings in the periods of CUVir and V901Ori owing to period, P¨, is constant throughout the interval of observations. simultaneous processing of all available observational material 4 Z. Mikul´aˇsek et al.: Surprising rotation variations of the mCP stars CU Vir and V901 Ori obtained during many decades, including our own observations period changes are not monotonic – intervals of rotational de- made in 2007–11.The period evolutions of both stars were mod- celeration alternate with intervals of rotational acceleration, all elled by polynomials of low degrees. In the case of CUVir we on the timescale of several decades. These results explain the also tested a fit with a harmonic function (Mikul´aˇsek et al. 2011) spin-down time paradox of these stars, according to which the and a Gaussian function corresponding to the transient nature of spin-down time was significantly shorter than the age of these the phenomenon. The only differences among them are notice- stars. On the other hand, this unexpected behaviour of two fairly able at the very beginning of our observations, which slightly dissimilar mCP stars poses a strong challenge for any theoretical favour the latter two alternatives. models. The possible harmonic-like changes of the observed peri- ods of both stars together with the strict constancy of the phase Acknowledgements. We thank R. von Unge, D. M. Pyper, M. Lenc, S. J. Adelman, T. Ryabchikova, and R. Arlt for valuable discussions and unpub- curves can in principle be explained by the light time effect lished observational data. This work was funded by the following grants: GAAV caused by an invisible companion orbiting the star in a nearly IAA301630901, GACRˇ 205/08/H005, MEB 061014/WTZ CZ 10-2010, MEB circular orbit. However, this orbital motion would be revealed 0810095, VEGA 2/0074/09, and MUNI/A/0968/2009. We also thank the referee by considerable variations ∆RV. In the case of D. Moss for valuable comments and inspiring suggestions. CUVir the observed change in the period ∆P of 3.8 s corre- ∆ = ∆ = −1 sponds to a variation RV c P/P 25 kms , while the References observed change in V901Ori period of ∆P ≃ 20 s predicts a ∆RV variation of even 45 km s−1! However, no such long- Abt, H. A., & Snowden, M. S. 1973, A&AS, 25, 137 term RV variations synchronised with period changes have been Abuladze, O. P. 1968, Bjul. Abustam. astrofiz. obs., 36, 43 Adelman, S. J., Dukes, R. J. Jr., & Pyper, D. M. 1992, AJ, 104, 314 found (Pyper et al. 1997; Mikul´aˇsek et al. 2008). Additionally, Blanco, C., & Catalano, F. 1971, AJ, 76, 630 the light time effect cannot explain the complex medium-term Borra, E. F., & Landstreet, J. D. 1980, ApJS, 42, 421 period variations observed in CUVir (see Sec. 4.1, Fig.2). Braithwaite, J., & Nordlund, Å. 2006, A&A, 450, 1077 We conclude that the observed period variations are caused Charbonneau, P., & MacGregor, K. 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O. 2011, A&A, may affect only the outer stellar envelope that is hardened by the 450, 763 global magnetic field, leaving the faster rotating core unaffected. Krivosheina, A. A., Ryabchikova, T. A., & Khokhlova, V. L. 1980, Nauchnii Inform. Astron. Council, Ser. “Astrofizika”, No.43, 70 (in Russian) Intervals of rotational deceleration may then alternate with inter- Kuschnig, R., Ryabchikova, T. A., Piskunov, N. E., et al. 1999, A&A, 348, 924 vals of angular momentumexchangebetween the slowly rotating Krtiˇcka, J., & Kub´at, J. 2001, A&A, 369, 222 envelope and the faster rotating core. The latter manifests itself Krtiˇcka, J., Mikul´aˇsek, Z., Zverko J., & Ziˇzˇnovsk´yJ.ˇ 2007, A&A, 470, 1089 by the rotational acceleration of the outer envelope. Krtiˇcka, J., Mikul´aˇsek, Z., L¨uftinger, T., et al. 2011, A&A – submitted Landstreet, J. D., & Borra, E. F. 1978, ApJ, 224, 5 However, this explanation is theoretically challenging. 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J. Adelman, & W. W. Weiss (Cambridge the younger star V901Ori rotates more slowly than CUVir. University Press, Cambridge), 307 Ravi, V., Hobbs, G., Wickramasinghe, D., Champion, D. J., & Keith M. 2010, All these considerations should be the subject of additional MNRAS, 408, 99 physical modelling. Nevertheless, the aim of this letter is more Ste¸pie´n, K. 1998, A&A, 337, 754 unpretentious – we only want to draw the attention to an inter- Thompson, I. B., & Landstreet, J. D. 1985, ApJ, 289, 9 esting property of the rotation of some upper MS stars. Townsend, R. H. D., Oksala, M. E., Cohen, D. H., Owocki, S. P., & ud-Doula, A. 2010, ApJ, 714, 318 Trigilio, C., Leto, P., Leone, F., Umana, G., & Buemi, C. 2000, A&A, 362, 281 Trigilio, C., Leto, P., Umana, G., Buemi, C. S., & Leone, F. 2008, MNRAS, 384, 6. Conclusions 1437 Trigilio, C., Leto, P., Umana, G., Buemi, C. S., & Leone, F. 2011, ApJ, 739, L10 Our study investigated the nature of the rotational period vari- Weiss, W. 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Appendix A: The outline of the method the instant period from a group of observations to generate the model curves in our figures. The techniques used to analyse the data are based on the Relation A.1 can also be used to determine reliably zero- rigourous application of non-linear, weighted, least-squares phase times for selected groups of observations Ok. These quan- methods used simultaneously for all relevant data that contain tities depend only marginally on the chosen model of the phase phase information. Our technique does not use an O-C diagram function ϑ(t, b). Therefore we can use these Ok in the process of as an intermediate stage of data processing; O-C diagrams are the phase function modelling. used only as a visual check on the adequacy of the models. Let us assume that all observed phase curves of a star are ade- quately described by the unique general model function F(ϑ, a), Appendix B: Brief specification of CU Vir data described here by ga parameters contained in a parameter vector a, a = (a1,..., a j,..., aga ) . In our computation we assume that the form of all the phase variations is constant and that the time Table B.1. Data with phase information about CU Vir used in the period variability of the observed quantities are given by a phase func- analysis. tion ϑ(t, b), which is a monotonic function of time t. The frac- tional part of it corresponds to the common phase, the integer season/s Data type N Reference part is the so-called epoch (E). We can express the phase func- 1949-52 EW He i 34 Deutsch (1952) tion by means of a simple model quantified by gb parameters b, 1955 UBV 162 Hardie (1958) b = (b1,..., bk,..., bgb ). The instantaneous period is given simply 1964-6 UBV 702 Abuladze (1968) by the equality P(t) = 1/ϑ˙. 1964 EW Si ii 40 Peterson (1996) i For a realistic modelling of phase variations for all data types 1964 EW He 9 Peterson (1996) for a star, we need g free parameters for the description of the 1965-6 RV Hγ 16 Abt & Snowden (1973) a 1966 B 33 Hardie et al. (1967) model function F(ϑ, a)and gb free parameters for the description 1967,76 EW He i 37 Hardorp & Megessier (1977) of the phase function ϑ(t, b) . The computationof the free param- 1967,76 EW Si ii 19 Hardorp & Megessier (1977) eters was iterative under the basic condition that the weighted 1968-9 EW Si ii 124 Krivosheina et al. (1980) sum S (a, b) of the quadrates of the difference ∆yi of the ob- 1968-9 EW Hγ 21 Krivosheina et al. (1980) served value yi and its model prediction is minimal (wi being 1968 UBV 287 Blanco & Catalano (1971) the individual weight of the i-th measurement). 1972-81 spf mag 179 Pyper & Adelman (1985) 1972 UBV 29 Winzer (1974) n 1974 β 95 Weiss et al. (1976) 2 ∆yi = yi − F(ϑi); S = ∆yi wi; δS = 0; ⇒ (A.1) 1974 uvbyW 460 Weiss et al. (1976) Xi=1 1975 spf OAO II 54 Molnar & Wu (1978) n n 1976-8 Beff 14 Borra & Landstreet (1980) ∂F(ϑi, a) ∂F ∂ϑ(ti, b) 1977 EW He i 69 Pedersen (1978) ∆yi wi = 0; ∆yi wi = 0 ∂a ∂ϑ ∂b 1979 spf IUE 210 Krtiˇcka et al. (2011) Xi=1 j Xi=1 i k 1980-3 uvby 89 Pyper & Adelman (1985) = + 1980-3 β 23 Pyper & Adelman (1985) We obtain here g ga gb equations of g unknown param- i eters. The weights of individual measurements w are inversely 1983 EW He 9 Hiesberger et al. (1995) i 1987-9 UBV 1012 Adelman et al. (1992) proportional to their expected uncertainty. The system is non- 1988 β 25 Musiełok et al. (1990) linear; we have to determine the parameters iteratively. With a 1990-3 BTVTHp 288 ESA (1998) good initial estimate of the parameter vectors a and b, the itera- 1991-7 uvby 3635 Pyper et al. (1998) tions converge very quickly. Usually we need only several tens 1994-5 EW He i 19 Kuschnig et al. (1999) of iterations to complete the iteration procedure. 1995 RV Hδ 19 Kuschnig et al. (1999) 1995 EW Si ii 19 Kuschnig et al. (1999) 1995 EW Hδ 18 Kuschnig et al. (1999) A.1. Virtual O-C diagrams. Evolution of periods. 1995 Beff 21 Pyper et al. (1998) 1998-09 radio 5 Ravi et al. (2010) The short-term modulation of the phase function was analysed 2002-9 V 353 Pojma´nski et al. (2001) by means of the residuals of the observed data ∆yi, creating in- 2010-11 BV 374 Henry - this paper dividual values of the phase shifts expressed in days (O-C) j with 2010-11 EW Si ii 59 Jan´ık - this paper adapted individual weight W j for each observed datum and aver- 2010 V 38 Liˇska - this paper ages of the phase shifts defined for arbitrarily selected groups of 2011 vb 251 Jan´ık - this paper 2011 UBV 402 Jan´ık & Liˇska - this paper measurements (O-C)i or deflection of the mean period from the instant model period ∆Pk(tk):

−1 2 ∂F ∂F The data used for the analysis of CUVir are given in Table B.1. (O-C) j = −P(t j) ∆y j ; W j = w j; (A.2) ∂ϑ ! ∂ϑ ! Here we used the following abbreviations: EW – the equivalent nk nk width, RV – radial velocity, β –Hβ photometry, UBV – Johnson j=1 (O-C) j W j j=1 (O-C) j ϑ j W j (O-C) = ; ∆Pk = . UBV photometry, uvby – Stroemgren uvby photometry, BTVTHp k P nk W P nk ϑ2 j=1 j j=1 j W j – Hipparcos photometry, Beff – the mean longitudinal magnetic P P induction, spf – magnitudes derived from spectrograms obtained by UV satellites OAO2 and IUE, and radio - timings of ra- Computations of (O-C)k and ∆Pk followed after the model parameters were found; consequently they had no influence on diopulses. the model solution. They were used only to visualise of the so- lution. Similarly, we can compute virtual ‘observed’ values of