ACTA ASTRONOMICA Vol. 45 (1995) pp. 725±745 Are Rotational Axes Perpendicular to Orbital Planes in Binary Systems. III. Main Sequence and Short-Period RS CVn Stars. by R. GøeÎbocki Institute of Theoretical Physics and Astrophysics, University of GdaÂnsk, ul. Wita Stwosza 57, 80-952 GdaÂnsk, Poland e-mail: ®[email protected] and A. Stawikowski N. Copernicus Astronomical Center, ul. RabiaÂnska 8, 87-100 ToruÂn, Poland e-mail: [email protected] Received November 24, 1995 ABSTRACT Inclinations of the rotational axes, irot , are determined for 46 main sequence binaries of F, G, K and M spectral type and short period RS CVn systems. Seven binaries are asynchronous. The i inclinations irot are then compared with the orbital inclinations, orb , to test the alignment between the equatorial and orbital planes. In all 39 cases of synchronous rotators irot is equal or nearly equal to iorb . In a sample of seven asynchronous systems, at least six, and perhaps all, are non-coplanar. Key words: Stars: rotation-binaries: general-Stars: late-type 1. Introduction Chromospherically active stars with spots present a unique possibility to de- V i termine rotational period, Prot , and with independently measured rot sin rot to evaluate inclination of rotational axis, irot . In late type binaries it allows for com- parison of inclination of rotational axes to the orbital plane. Our previous analyses (Stawikowski and GøeÎbocki 1994 a,b called hereinafterPaperIandPaperII)showed that in long-period RS CVn stars an assumption of coplanarity of the equatorial ro- = i tational and orbital planes ( irot orb ) is justi®ed for synchronous systems only. In asynchronous binaries the rotational axes are not perpendicularto the orbital plane. 726 A. A. It contradicts generally accepted assumption and requires revision of time-scales of circularization and synchronization for non-coplanar systems. In the present study we analyze main sequence F, G, K and M binaries and short-period RS CVn stars with orbital periods smaller than about 10 days. The observational data were collected from all recently available publications, but in most cases they are based on the second edition of "A catalog of chromospherically active binary stars" Strassmeier et al. (1993), hereinafter called CABS. ThemethodoftheanalysisisthesameasinPaperIandII. Becausethe accuracy of determination of irot is crucialfor®nalconclusions,specialattention is paid tothe determination of stellar temperatures and absolute magnitudes in¯uencing directly error in stellar radii and in consequence error in irot . In Chapter 2 we discuss calibration and errors of our temperature and absolute magnitude scales for main sequence F, G, K and M stars. A de®nition of synchronismin binary systems is not unique. A simple criterion = P of Porb rot is useless because of differential rotation and "pseudo synchro- nism" for highly eccentric orbits. Tan Huisong et al. (1993) suggest a parameter = j(P P )P j S S orb rot orb as a measure of synchronism, where 0 01 means S synchronism and S 0 04 means asynchronism. The case of 0 01 0 04 means either a slight asynchronism or a highly differential rotation. We use this parameter for discriminating asynchronous systems. It should be stressed that we = P assume Prot phot and we include in our analysis only stars with directly deter- V mined values of Pphot (either from periodic variations of obs or from "migration waves" in eclipsing binaries). The values of Pphot show small seasonal variations due to differential rotation, but the observed changes are smaller than 4%. i In Section 3 we present the results ± our values of stellar parameters: iorb , rot = ji i j and i orb rot . The discussion of errors is made individually for each star in Comments to Tables 2±5. 2. Observational Data We have found 69 binary systems (short-period RS CVn or main sequenceF, G, V i K,Mspectraltypebinaries)withknown Pphot and rot sin rot full®ling our selection criteria. Unfortunately, only for 46 systems other stellar and orbital parameters were i known with accuracy good enough for determination of iorb and rot . They are listed in Tables 2±5. In this sample, 21 systems are eclipsing binaries providing accurate orbital inclinations, 22 non-eclipsing objects are the SB2 systems with reliable parameters for both components. For three non-eclipsing SB1 systems (EI Eri, V833 Tau and DM UMa) evaluation of orbital inclination was possible (m) i because of extremely low value of the mass function, f . In such a case orb q is low and insensitive to the adopted values of mass, M , and mass ratio, (see Paper I). Vol. 45 727 When both (or at least one) components of the binary are main sequence stars, the evaluation of iorb is made using stellar masses determined from the mass- luminosity relation. We used the relation given in Lang (1992) compendium, which is based on careful discussion of Popper (1980). Our results were always very close to that presented in CABS. Accuracies of iorb determination in most cases are better than 2 and not worse than 5 . Binary systems for which errors in iorb exceed 5 were excluded from the sample. An interesting case of II Peg for which iorb is very uncertain but spots have been observed (and modeled) by many authors is described in Section 3. Inclinations of rotational axes were determined from the ratio of the observed i V V projected rotational velocity, Vrot sin rot , to the rotational velocity, rot , with rot obtained from rotational period, Prot , using the formula = R P Vrot 50 61 rot R P V where is radius in R , rot in days and rot in km/s. As in previous papers P only Prot based on direct phot measurements (or equivalent "migration wave" observations) have been used in the analysis. We are warning again that some authors are publishing Prot based on the a priori assumption of synchronism, especially for short-period systems. It could lead to serious errors (e.g., for VY Ari = P = P = P = Porb 13 2 while phot 16 4 or LR Hya with orb 6 86 and phot 3 14). There are two main sources of errors in the determination of irot : inaccuracy i R of Vrot sin rot measurements and evaluation of . In the estimation of error of i individual Vrot sin rot values we must rely on accuracies published in original papers. Sometimes a few independent measurements are available. If our estimate i indicates that the error of Vrot sin rot exceeds 15% we exclude such star from the sample. The stellar radius, R , was obtained from the estimation of the effective T M temperature, eff , and the absolute magnitude, v . To minimize this error special attention was paid to the calibration of the temperature scale. Fortunately, most of stars in our sample are the F, G, K, M dwarfs not very distant from the Sun. For some of them very accurate trigonometric parallaxes were available. Otherwise M T M v v was estimated from the spectral type. The scales and errors in eff and are described in the following subsection. 2.1. Effective Temperature and Absolute Magnitude Calibration Most accurate estimation of Teff can be obtained from a multicolor infrared photometry. ForF,GandK typesubgiantsandgiantsformulae found by McWilliam = f (CI ) (1990) (see his Table 6), in his extensive analysis of Teff for the BVRI photometry, provide the most reliable values of Teff . We used these formulae for all stars with the luminosity class IV, IV-III and III. Using Table 12 of McWilliam = f ( ) (1990) we have constructed a relation Teff sp type . The adopted values for Teff are the average values obtained from each color index and spectral type of a star. 728 A. A. There are many relations of Teff versus spectral type or color index for main sequence stars, e.g.,Johnson (1966), Novotny (1973), Bohm-Vitense (1981) Gray (1992) and most frequently used Landolt-Bernstein (1982). When comparing these relations some discrepanciesand systematic shifts become evident. With increasing accuracy and number of infrared observations many independent determinations of Teff for individual stars have been published. We used the following sources = f ( ) T = f (CI ) for redetermination of Teff sp type and eff for late type dwarfs: Leggett et al. (1986), Blackwell et al. (1991), Buser and Kurucz (1992), Blackwell and Lynas-Gray (1994) and Taylor (1994). Additionally we included independent Teff estimations used for the [Fe/H] determinations and published in the catalogue of Cayrel de Strobel et al. (1992). We analyzed only stars within the luminosity class V and present in the Bright Star Catalogue to avoid interstellar reddening effects. If more than one determination of temperature was available for a given star, an average value was calculated with the weight equal 2 given to Taylor's (1994) data, and the ®nal weight equal to the number of independent determinations of Teff . Information about spectral types was obtained from SIMBAD, and the most = f ( ) frequently cited ones were adopted. The resulting relation of Teff sp type for F, G, K and M main sequencestars is presented in columns 1 and 2 of Table 1. The catalogue of multicolor photometry of Lanz (1986) allowed to determine relations between effective temperature and BVRI color indices in the same way. T = f (CI ) B V V R For BS stars with known Teff the best relations eff for , , I V I R and color indices were found. They are presented in columns 3, 4, 5 and 6 of Table 1. We believe that with known spectral type and infrared color indices our calibration provides Teff with the accuracy of 50 K. Unfortunately V for 18 analyzed binary systems only spectral type and B are known.