ORBITAL PERIOD CHANGES AND THEIR EVOLUTIONARY STATUS FOR THE WEAK-CONTACT BINARIES. III. AO CAMELOPARDALIS AND AH TAURI

The Astronomical Journal, 139:195 204, 2010 January C 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi:10.1088/0004-6256/139/1/195 ORBITAL PERIOD CHANGES AND THEIR EVOLUTIONARY STATUS FOR THE WEAK-CONTACT BINARIES. III. AO CAMELOPARDALIS AND AH TAURI Y.-G. Yang 1,2, J.-Y. Wei 2,J.M.Kreiner 3,andH.-L.Li 2 1 School of Physics and Electric Information, Huaibei Coal Industry Teachers College, 235000 Huaibei, Anhui Province, China; yygcn@163.com 2 National Astronomical Observatories, Chinese Academy of Sciences, 100012 Beijing, China 3 Mt. Suhora Astronomical Observatory, Cracow Pedagogical University, ul.podchor ażych 2, 30-084 Cracow, Poland; sfkreine@cyf-kr.edu.pl Received 2009 July 20; accepted 2009 October 28; published 2009 December 10 ABSTRACT In this paper, we presented multicolor photometric observations for two eclipsing binaries, AO Camelopardalis and AH Tauri, obtained on 2008 December 16 and 17. Using the Wilson Devinney Code, the photometric solution of AH Tau was determined from our new CCD data. The mass ratio and the fill-out factor are q = 0.503(±0.003) and f = 10.8%(±0.1%), respectively. This indicates that AH Tau is in weak contact. For the weak-contact binary AO Cam, BVI light curves clearly show a difference in the heights of the maxima (i.e., the O Connell effect), which may be explained by spot activity. By analyzing the O C curves for AO Cam and AH Tau, it is found that the orbital periods appear to show a secular period decrease with a cyclic variation. The observed period modulation is ΔP/P 10 6. For AO Cam, the cyclic oscillation with a short period of 7.63(±0.07) yr and a low amplitude of 0 ḍ 0019(±0 ḍ 0003) may be preferably attributed to the cyclic magnetic activity. The period and amplitude of the cyclic variation for AH Tau are 45.8(±1.1) yr and 0 ḍ 0171(±0 ḍ 0005), which may more likely result from the light-time effect via a third body. The secular period decrease rates are dp /dt = 1.26(±0.04) 10 7 days yr 1 for AO Cam and dp /dt = 6.98(±0.07) 10 8 days yr 1 for AH Tau. This kind of period decrease can be plausibly explained by the mass transfer from the primary to the secondary, and may result in the system evolving into a deep contact configuration. Key words: binaries: close binaries: eclipsing stars: individual (AH Tauri, AO Camelopardalis) Online-only material: machine-readable and VO tables 1. INTRODUCTION AO Cam (BD+52 826; α 2000.0 = 04 h 28 m 13 ṣ 647, δ 2000.0 = +53 02 44. 60) is a W Ursae Majoris type eclipsing binary, with a short period of 0 ḍ 3299024 (Kreiner 2004). Its light variability was discovered by Hoffmeister (1966). Milone et al. (1982) first analyzed the photoelectric UBV light curves using the methods of Rucinski (1973, 1974). They estimated the spectral type of G5 with a mass ratio of 0.7 q 0.8. Subsequently, Evans et al. (1984, 1985) obtained new BV light curves. However, they were unable to find a unique solution with the method of Wilson & Devinney (1971). The observations of Milone et al. (1982) were analyzed by Cooke & Leung (1985), who found a very shallow contact degree of 1.4% and a mass ratio of q = 1.3. Barone et al. (1993) pointed out that AO Cam is definitely a W-type contact binary whose primary minimum is an occultation; that is, the smaller and less massive component is eclipsed during the deeper minimum. Their photometric results show that the mass ratio, the fill-out factor, and the temperature difference between both components are q = 1.71(±0.04) or 1/q = 0.585, f = 10.5%, and ΔT = 327 K, respectively. They noticed that the asymmetries of the light curves due to the presence of spots are of little relevance. Therefore, there exists contradictory data for the derived mass ratios. Subsequently, the spectroscopic observations of this binary were performed by Rucinski et al. (2000), who assigned to AO Cam the spectral type of G0V, and confirmed it to be a W-type contact binary. The spectroscopic mass ratio and the mass function are q sp = 2.420(±0.011) or 1/q sp = 0.413(±0.011) and f (m) = (M 1 + M 2 )sin 3 i = 1.520(±0.030) M, respectively. Baran et al. (2004) published the VRI light curves, which they analyzed to obtain the absolute parameters of AO Cam; that they obtained during a long winter night. The asymmetric light curves (i.e., ΔM 0.02 mag) were modeled by the dark spot on the more luminous component. The fill-out factor is f = 12%, indicating that this binary is in weak contact. The orbital period changes for AO Cam was studied by Qian et al. (2005), who proposed that there exists a cyclic variation. The period and amplitude of this cyclic oscillation are 20.1 yr and 0 ḍ 0047, respectively, which may be caused by the presence of a third body. Recently, Rucinski et al. (2007) detected a visual companion to AO Cam at a separation of 1. 309(5), proving the hypothesis that contact binaries require a third star to become as close as observed. The estimated spectral type for this visual companion is M5V, with a period P vis = 1630 yr. AH Tauri (HV 6187; α 2000.0 = 03 h 47 m 11 ṣ 964, δ 2000.0 = +25 06 59. 31) was discovered by Shapley & Hughes (1934), who classified this star as an RRc-type variable with a period of 0 ḍ 163334. Photographic observations were performed by Binnendijk (1950) and Romano (1962). The former classified the system as a W UMa type binary with a period of 0 ḍ 33267447, a spectral type of G1p, and a color index of 0.50, while the latter classified it to be of β Lyrae-type binary. Further photoelectric observations were subsequently made by Bookmyer (1971), Magalashvili & Kumsishvili (1980), and Liu et al. (1991). Bookmyer (1971) revised AH Tauri s period to 0 ḍ 33267557, gave it a color index of B V = 0.66, and from this color index deduced a spectral type of about G5. Magalashvili & Kumsishvili (1980) noticed that the light curves were asymmetrical and the secondary mid-eclipse occurred at a phase of 0.47. Liu et al. (1991) published BV normal observations and determined the photometric solution. The results show that AH Tau is an A-type W UMa binary with a mass ratio of 0.502, an inclination of 84. 29, and a fill-out factor of 9.1%. However, Byboth 195

196 YANG ET AL. Vol. 139 et al. (2004) gave a photometric mass ratio of 0.773 based on an analysis of their VRI observations, with an overcontact degree of 9.3%. These two mass ratios are very different. The orbital period analysis was performed by Yang & Liu (2002), who gave a complicated period change, including a continuous period decrease at a rate of ΔP/P = 1.4 10 11 between 1944 and 1976, a sudden decrease by 0.33 s around 1976, and a secular period increase at a rate of ΔP/P = +1.5 10 10 after 1976. They interpreted those changes as due to the magnetic activity, the mass loss, and the radius swelling of the components. Recently, Zasche et al. (2009) described the O C curve of AH Tau by the light-time effect. The derived semiamplitude, eccentricity, and period of the assumed third body are A = 0 ḍ 0319(±0 ḍ 0023), e = 0.31(±0.12), and P 3 = 77.6(±7.3) yr, respectively. However, the observing light minimum times of AH Tau since 1943 (i.e., HJD 2431062.5106) only covered 86% of the complete orbital period for the third body. The orbital period of a close binary is one of the main parameters for understanding the dynamics of binaries and possibly their stellar structure and evolution. In previous papers, three weak-contact binaries, TY Boo (Yang et al. 2007, hereafter Paper I), CC Com, and BV Dra (Yang et al. 2009, hereafter Paper II), were studied in detail. The continuous period increase or decrease may result from mass transfer between components, accompanied by angular momentum loss. This kind of mass transfer may cause the weak-contact binary to evolve into either a deeper-contact configuration or a broken-contact configuration as predicted by thermal relaxation oscillation (TRO) models (Lucy 1976; Flannery 1976; Robertson & Eggleton 1977). Therefore, this kind of weak-contact binary is a key evolutionary stage for the formation and evolution of a close binary. In this paper, we presented new CCD observations for AO Cam and AH Tau in Section 2. Their orbital period changes were analyzed in Section 3, and the photometric solution for AH Tau was derived from new observations in Section 4. Finally, their properties of orbital period changes and evolutionary states were discussed. 2. NEW OBSERVATIONS FOR AO CAM AND AH TAU CCD photometry for two eclipsing binaries, AO Cam and AH Tau, was performed using the 85 cm telescope and the 80 cm telescope at the Xinglong Station of the National Astronomical Observatories of China (NAOC). For the 85 cm telescope, a standard Johnson Cousin Bessel multicolor CCD photometric system was mounted on the primary focus (Zhou et al. 2009). The PI 1024 BFT camera has 1024 1024 square pixels, each subtending a projected angle on the sky of 0. 96 and resulting in a field of view of 16. 5 16. 5. The 80 cm telescope was equipped with a PI 1340 1300 CCD, giving a field of view of 11 11. Both of the CCD cameras are equipped with a standard Johnson Cousin Bessel set of UBVRI filters. The reductions of observations were done using the IMRED and APPHOT packages in IRAF. Zero and flat-fielding corrections were applied to the images. Extinction corrections were small and were not made to the observations. Then magnitudes were determined by aperture photometry. The sizes of the apertures were 3. 8 for AO Cam and 2. 0 for AH Tau. Photometric observations for AO Cam were carried out on 2008 December 17. The comparison and check stars are BD+52 824 (α J 2000.0 = 04 h 27 m 26 ṣ 817, δ J 2000.0 = 52 55 06. 21) and GSC 3732 0706 (α J 2000.0 = 04 h 26 m 58 ṣ 14, δ J 2000.0 = 52 58 06. 84), respectively. The exposure times in the BVI bands were adopted to be 10 s, 7 s, and 5 s, respectively. The magnitude differences in the sense of the variable minus the Table 1 BVI Observations for the Eclipsing Binary AO Cam Observed on 2008 December 17 B Band V Band I Band HJD Δm HJD Δm HJD Δm 2454818.0082 +1.218 2454818.0085 +0.767 2454818.0087 +0.143 2454818.0090 +1.213 2454818.0093 +0.755 2454818.0096 +0.133 2454818.0099 +1.199 2454818.0102 +0.743 2454818.0105 +0.114 2454818.0108 +1.195 2454818.0111 +0.740 2454818.0114 +0.108 2454818.0117 +1.176 2454818.0120 +0.719 2454818.0123 +0.101 2454818.0126 +1.171 2454818.0129 +0.719 2454818.0131 +0.099 2454818.0135 +1.155 2454818.0137 +0.694 2454818.0140 +0.075 2454818.0143 +1.133 2454818.0146 +0.686 2454818.0149 +0.069 2454818.0152 +1.122 2454818.0155 +0.672 2454818.0158 +0.062 2454818.0161 +1.116 2454818.0164 +0.664 2454818.0167 +0.038 2454818.0170 +1.098 2454818.0173 +0.639 2454818.0176 +0.037 2454818.0179 +1.076 2454818.0182 +0.641 2454818.0184 +0.025 (This table is available in its entirety in machine-readable and Virtual Observatory (VO) forms in the online journal. A portion is shown here for guidance regarding its form and content.) comparison star together with their heliocentric Julian dates are listed in the online Table 1. All those individual observations (i.e., 431 for the B band, 432 for the V band, and 430 for the I band) are displayed in the left panel of Figure 1. From this figure, the amplitudes of variable light are 0 ṃ 64, 0 ṃ 59, and 0 ṃ 54 for the B, V, and I bands, respectively. The light curves possess unequal height between the two maxima, i.e., the O Connell effect (Milone 1968; Davidge & Milone 1984). Max. II at phase 0.75 is brighter than Max. I at phase 0.25. The values of Max. I Max. II are 0 ṃ 031, 0 ṃ 022, and 0 ṃ 010 for the B, V, and I bands, while Baran et al. (2004) had Max. I brighter than Max. II by 0 ṃ 02. Therefore, the light curves for AO Cam show a noticeable variation from 2004 to 2008, which may be explained by dark-spot activity on the common convective envelope. In order to assess the quality of the differential light curves, the mean variation in the differences between the magnitude of the check star and that of the comparison one were calculated to be ±0.007 mag, ±0.008 mag, and ±0.007 mag for the B, V, and I bands, respectively. AH Tau was observed on 2008 December 16. The typical exposure times for the V and R bands were 30 s and 20 s, respectively. The comparison star and check star are HD 282955 (α J 2000.0 = 03 h 46 m 52 ṣ 04, δ J 2000.0 = 25 07 36. 39) and GSC 1804 2470 (α J 2000.0 = 03 h 47 m 00 ṣ 14, δ J 2000.0 = 25 05 29. 10), respectively. The magnitude differences versus heliocentric Julian dates are listed in the online Table 2. The complete light curves, including 305 individual observations in the V band and 307 individual observations in the R band, are shown in the right panel of Figure 1. As can be seen from this figure, AH Tau is a typical A-type contact binary, whose secondary eclipse is an occultation. The duration time of the occultation is about 18 minutes. The amplitudes of variable light are 0 ṃ 70 and 0 ṃ 68 for the V and R bands, respectively. Based on the differential comparison and check star observations, the precision of the individual differential magnitudes is estimated to be ±0.004 mag for the V band and ±0.003 mag for the R band. From the observations of AO Cam, two light minimum times were determined using the parabolic fitting method. Moreover, four light minimum times were calculated from the observations of Baran et al. (2004). 4 For AH Tau, we obtained two light 4 The observations for AO Cam are available at http://www.as.up.krakow.pl/.

No. 1, 2010 ORBITAL PERIOD CHANGES FOR WEAK-CONTACT BINARIES. III. 197 Figure 1. Left panel: BVI observations of AO Cam, observed on 2008 December 17. Right panel: VR observations of AH Tau, obtained on 2008 December 16. The theoretical light curves from our new photometric solution for AH Tau are shown with solid lines. Table 2 VR Observations for the Eclipsing Binary AH Tau Observed on 2008 December 16 V Band R Band HJD Δm HJD Δm HJD Δm HJD Δm 2454816.9370 0.464 2454817.1097 0.537 2454816.9376 0.463 2454817.1092 0.513 2454816.9381 0.472 2454817.1109 0.537 2454816.9387 0.473 2454817.1103 0.521 2454816.9392 0.485 2454817.1120 0.557 2454816.9398 0.479 2454817.1114 0.531 2454816.9403 0.496 2454817.1131 0.558 2454816.9409 0.495 2454817.1125 0.549 2454816.9414 0.511 2454817.1142 0.564 2454816.9420 0.501 2454817.1136 0.546 2454816.9425 0.521 2454817.1153 0.564 2454816.9431 0.507 2454817.1147 0.550 2454816.9436 0.522 2454817.1164 0.570 2454816.9442 0.520 2454817.1158 0.566 2454816.9448 0.531 2454817.1175 0.579 2454816.9453 0.525 2454817.1170 0.556 2454816.9459 0.538 2454817.1186 0.575 2454816.9464 0.530 2454817.1181 0.571 2454816.9470 0.545 2454817.1197 0.583 2454816.9475 0.548 2454817.1192 0.573 2454816.9481 0.553 2454817.1208 0.591 2454816.9486 0.544 2454817.1203 0.580 2454816.9492 0.557 2454817.1219 0.592 2454816.9497 0.560 2454817.1214 0.586 (This table is available in its entirety in machine-readable and Virtual Observatory (VO) forms in the online journal. A portion is shown here for guidance regarding its form and content.) minimum times from our new VR observations. Additionally, one light minimum time each for AO Cam and AH Tau was measured on 2007 January 29 using the 85 cm telescope. The individual times of minimum light, together with their errors, are listed in Table 3. 3. STUDYING ORBITAL PERIOD CHANGES FOR AO CAM AND AH TAU 3.1. Period Analysis of AO Camelopardalis For the eclipsing binary AO Cam, Qian et al. (2005) showed that it included a cyclic variation. This kind of cyclic oscillation may be part of a parabolic curve based on the O C curve. Therefore, it is necessary to reanalyze the period changes for this binary. We collected a total of 77 light minimum timings, consisting of 11 visual, 39 photoelectric, and 27 charge-coupled device observations. The compiled observations are listed in the first column of Table 4. In the third column of this table, vi, pe, and CCD refer to visual observations, photoelectric measurements, and charge-coupled device, respectively. Using a linear ephemeris (Evans et al. 1985), Min. I = HJD 2445745.6394 + 0 ḍ 329905519 E, (1) we can calculated the residuals (O C) of the eclipse timings, listed in the fifth column of Table 4. TheO C curve was constructed in the left panel of Figure 2. The general O C trend can be described by a downward parabolic curve superimposed with fluctuations, indicating that there exists a continuous period decrease with a cyclic variation. During the course of the fitting procedure, weights of 1, 10, and 10 were attributed to the visual, photoelectric, and CCD light minimum times, respectively. The Levenberg Marquardt technique (Press et al. 1992) was applied to solve for several fitting parameters. A weighted nonlinear least-squares fitting yielded the following equation: O C = +0.0006(±0.0002) + 1.7(±0.4) 10 7 E 5.67(±0.16) 10 11 E 2 +0.0019(±0.0003) sin[7.44(±0.07) 10 4 E +3.7439(±0.1432)]. (2) The corresponding residuals of those light minimum times are shown in the right panel of Figure 2 and are listed in the sixth column of Table 4. In the left panel of Figure 2, the solid and dotted lines are plotted to represent the entire and the parabolic parts of Equation (2), respectively. The coefficient of the quadratic term of the equation leads to a continuous

198 YANG ET AL. Vol. 139 Table 3 New Light Minimum Times for AO Cam and AH Tau Stars JD(Hel.) Type Error Filter AO Cam 2453025.1472 II 0.0002 R 2453025.3125 I 0.0002 V 2453025.3123 I 0.0002 R 2453025.3128 I 0.0002 I 2453025.4786 II 0.0001 V 2453025.4784 II 0.0002 R 2453025.4784 II 0.0002 I 2453025.6435 I 0.0002 V 2453025.6430 I 0.0002 R 2453025.6435 I 0.0002 I 2454129.9916 II 0.0006 V 2454818.1662 II 0.0002 B 2454818.1662 II 0.0002 V 2454818.1664 II 0.0001 R 2454818.3317 I 0.0001 B 2454818.3318 I 0.0001 V 2454818.3314 I 0.0002 R AH Tau 2454130.1004 I 0.0003 V 2454817.0700 I 0.0001 V 2454817.0703 I 0.0002 R 2454817.2356 II 0.0001 V 2454817.2356 II 0.0002 R Note. The first 10 light minimum times for AO Cam were calculated from the observations of Baran et al. (2004). period increase rate dp /dt = 1.26(±0.04) 10 7 dyr 1 = 1.09(±0.03) s/century. The sinusoidal term reveals a cyclic oscillation with an amplitude of A = 0 ḍ 0019(±0 ḍ 0003). The period of this variation is determined to be P 3 = 7.63(±0.07) yr. 3.2. Period Analysis of AH Tauri For AH Tau, many light minimum times have been published in the literature. Early eclipse times with 173 data entries were collected by one of the authors, J. M. Kreiner (Kreiner 2004). After 2004, another 24 timings were subsequently published in IBVS 5643, 5657, 5668, 5694, 5736, 5746, 5814, and 5870, OEJVS 3 (Locher 2005), and VSOLJ 46 (Nagai 2008). Together with our three new CCD times, we have compiled a total of 189 light minimum times, spreading over 66 years from 1943 to 2009. Table 5 tabulates all those light minimum times, consisting of 9 plate, 108 visual, 14 photographic, 13 photoelectric, and 45 CCD observations. In the third column of this table, p and pg represent plate and photographic measurements; 5 other symbols are the same as in Table 4. The O C values of those eclipse timings were computed with the linear ephemeris (Kreiner et al. 2001) Min. I = HJD2431822.3653 + 0 ḍ 33267368 E. (3) Those values of O C are listed in the fifth column of Table 5, and are displayed in the left panel of Figure 3. The s refer to visual, plate, or photographic observations, while the filled circles represent either photoelectric or CCD ones. From the left panel of Figure 3, the orbital period of AH Tau seems to show a continuous variable. Except for eight photometric points 5 Plate represents the time of the mid-exposure of a photographic plate, taken accidentally when the star was near minimum, while photographic refers to the time of minimum light from a series of photographic observations (Kreiner et al. 2001). Table 4 All Available Light Minimum Times for the Eclipsing Binary AO Cam JD(Hel.) Epoch Method Min (O C) Residuals Ref. (1) (2) (3) (4) (5) (6) (7) 2444520.7006 3713.0 pe I +0.0004 0.0004 (1) 2444520.8653 3712.5 pe II +0.0001 0.0007 (1) 2444556.8257 3603.5 pe II +0.0008 0.0001 (1) 2444558.8057 3597.5 pe II +0.0014 +0.0005 (1) 2444559.9607 3594.0 pe I +0.0017 +0.0008 (1) 2445732.6076 39.5 pe II 0.0005 0.0001 (2) 2454510.3667 +26567.5 CCD II 0.0376 0.0008 (23) 2454818.1662 +27500.5 CCD II 0.0399 0.0007 (21) 2454818.3317 +27501.0 CCD I 0.0394 0.0002 (21) 2454830.7036 +27538.5 CCD II 0.0389 +0.0003 (31) 2454842.4143 +27574.0 pe I 0.0399 0.0006 (32) 2454843.4044 +27577.0 pe I 0.0395 0.0002 (32) References. (1) Milone et al. 1982; (2) Evans et al. 1984; (3) Pohl et al. 1985; (4) Evans et al. 1985; (5) Pohl et al. 1987; (6) Faulkner 1986; (7) Pickard 1986; (8) Mullis & Faulkner 1991; (9) Dalmazio 1997; (10) Rucinski et al. 2000; (11) Ogłoza et al. 2000; (12) Hübscher et al. 2000; (13) Tikkanen 2000; (14) Hübscher et al. 2001; (15) Pribulla et al. 2001; (16) Dróżdż & Ogłoza 2005; (17) Qian et al. 2005; (18) Dvorak 2003; (19) Nelson 2003; (20) Nelson 2004; (21) Present work; (22) Krajci 2005; (23) Hübscher et al. 2009a; (24) Hübscher et al. 2005; (25) Nelson 2006; (26) Dvorak 2008; (27) Parimucha et al. 2007; (28) Hübscher et al. 2006; (29) Nelson 2008; (30) Diethelm 2007; (31) Diethelm 2009; (32) Hübscher et al. 2009b. (This table is available in its entirety in machine-readable and Virtual Observatory (VO) forms in the online journal. A portion is shown here for guidance regarding its form and content.) (Bookmyer 1971; Magalashvili & Kumsishvili 1980; Liu et al. 1991), only the plate, visual, or photographic times of minimum light were observed before 1997. Those measurements have relatively less accuracy. Disregarding those observations, however, would result in a total loss of information needed to understand orbital period changes. Therefore, the plate, visual, or photographic data were assigned a weight of 1 and the photoelectric or CCD ones a weight of 10. The shape of the O C curve may suggest a downward parabolic trend with a sinusoidal variation. Using the weighted nonlinear fitting, one can obtain the following equation: O C = 0.0111(±0.0005) + 1.80(±0.04) 10 6 E 3.18(±0.05) 10 11 E 2 +0.0171(±0.0005) sin[1.25(±0.03) 10 4 E +4.1911(±0.0320)]. (4) The residuals computed from Equation (4) are shown in the right panel of Figure 3, and are listed in the sixth column of Table 5. With the coefficient of the quadratic term of the equation, a continuous period increase rate dp /dt = 6.98(±0.07) 10 8 dyr 1 (i.e., 0.60 ± 0.01) s/century) can be derived. The sinusoidal term reveals a cyclic oscillation with an amplitude of A = 0 ḍ 0171(±0 ḍ 0005) and a period of P 3 = 45.8(±1.1) yr, which is much smaller than the previously published value of P 3 = 77.6 yr (Zasche et al. 2009). 4. LIGHT-CURVE SYNTHESIS FOR AH TAU For AH Tauri, Liu et al. (1991) derived the photometric mass ratio of q = 0.502, while Byboth et al. (2004) found q = 0.773.

No. 1, 2010 ORBITAL PERIOD CHANGES FOR WEAK-CONTACT BINARIES. III. 199 Figure 2. O C curve (left panel) and the corresponding residuals from Equation (2) (right panel) for the eclipsing binary AO Cam. The s refer to visual observations, while the filled circles represent photoelectric or CCD data. The solid and dotted lines were constructed based on Equation (2) using the entire equation and the parabolic part only, respectively. Figure 3. O C curve (left panel) and the corresponding residuals from Equation (4) (right panel) for the eclipsing binary AH Tau. The crossings refer to visual, plate, or photographic observations, while the filled circles represent photoelectric or CCD ones. The solid and dotted lines were constructed based on Equation (4) using the entire equation and the parabolic part only, respectively. There exists a large difference between those two values. Therefore, VR light curves were simultaneously analyzed to deduce a new photometric solution. Photometric elements of AH Tau were obtained using the 2003 version of the W D program (Wilson & Devinney 1971; Wilson 1979; Wilson & van Hamme 2003). During the calculation, the commonly adjustable parameters employed were the orbital inclination, i; the mean temperature of Star 2, T 2 ; the potential of the components, Ω 1 = Ω 2 ; the monochromatic luminosity of Star 1, L 1 ; and the third light, l 3. The reflection effect was computed with the detailed model of Wilson (1990). The stellar atmosphere model of Kurucz (1993) was used. The mean effective temperature for Star 1 (the more massive component eclipsed at the transit minimum) was fixed at T 1 = 5900 K, which is the same as the value from Liu et al. (1991). The logarithmic bolometric (X and Y) and monochromatic (x and y) limb-darkening coefficients were interpolated from the values of van Hamme (1993). Following Lucy (1967) and Rucinski (1973), the gravity darkening exponents of both components and their bolometric albedo coefficients were set at the values of g 1,2 = 4β 1,2 = 0.32 and A 1,2 = 0.5, respectively, which are appropriate for stars with convective envelopes. To search for a reliable mass ratio, solutions are obtained for a series of fixed values of the mass ratio from q = 0.40 to 0.90 in increments of 0.05. For each value of the mass ratio, the calculation started at mode 2 (i.e., detached mode), but the solution always converged to mode 3 (i.e., overcontact mode). The sum of the squared residuals, Σ(O C) 2 i, for the corresponding mass ratios is plotted in Figure 4, where a minimum of Σ(O C) 2 i = 0.1306 is achieved at q = 0.50. At this point, the set of adjustable parameters was expanded to include q. After some iterations, the set of parameters (i.e., i, q, T 2, Ω, and L 1 ) is finally derived, and listed in Table 6. The corresponding theoretical light curves, shown as solid lines in the right panel of Figure 1, fit the new observations fairly well. The photometric mass ratio and the inclination are q = 0.490(±0.001) and i = 85. 95(±0. 22), which approximately agree with the values obtained by Liu et al. (1991). The fill-out factor is f = 10.8%(±0.1%), implying that AH Tau is in weak contact.

200 YANG ET AL. Vol. 139 Table 5 All Available Light Minimum Times for the Eclipsing Binary AH Tau JD(Hel.) Epoch Method Min (O C) Residuals Ref. (1) (2) (3) (4) (5) (6) (7) 2431062.5106 2284.0 pg I 0.0280 0.0007 (1) 2431142.3516 2044.0 pg I 0.0287 0.0015 (1) 2431143.3511 2041.0 pg I 0.0272 +0.0000 (1) 2431144.3497 2038.0 pg I 0.0266 +0.0006 (1) 2431145.3483 2035.0 pg I 0.0261 +0.0011 (1) 2431822.3390 +0.0 pg I 0.0263 0.0002 (1) 2454475.5807 68094.5 CCD II 0.0325 +0.0008 (90) 2454475.7472 68095.0 CCD I 0.0323 +0.0010 (90) 2454505.3542 68184.0 CCD I 0.0333 +0.0001 (89) 2454736.8952 68880.0 CCD I 0.0332 +0.0004 (90) 2454817.0700 69121.0 CCD I 0.0327 +0.0011 (86) 2454817.2356 69121.5 CCD II 0.0335 +0.0003 (86) References. (1) Binnendijk 1950; (2) Romano 1962; (3) Bookmyer 1971; (4) Locher 1968; (5) Diethelm 1975; (6) Locher & Diethelm 1975a; (7) Locher & Diethelm 1975b; (8) Locher 1975; (9) Magalashvili & Kumsishvili 1980; (10) Locher 1976; (11) Poretti 1984; (12) Locher 1978; (13) Locher 1979; (14) Diethelm 1979; (15) Diethelm 1980a; (16) Diethelm 1980b; (17) Diethelm 1981; (18) Locher 1982; (19) Germann 1983a; (20) Germann 1983b; (21) Locher 1984; (22) Germann 1985; (23) Germann 1986; (24) Locher 1986; (25) Liu et al. 1991; (26) Locher 1987a; (27) Blättler 1987; (28) Locher 1987b; (29) Locher 1988a; (30) Blättler et al. 1988; (31) Blättler & Peter 1988; (32) Locher 1988b; (33) Peter 1989; (34) Peter & Locher 1989; (35) Locher 1989; (36) Locher 1990; (37) Peter & Locher 1990; (38) Locher & Peter 1990; (39) Peter & Locher 1991; (40) Locher 1991; (41) Locher 1992a; (42) Peter 1992; (43) Locher 1992b; (44) Locher & Peter 1993; (45) Locher 1993; (46) Locher 1994a; (47) Peter & Locher 1994; (48) Locher 1994b; (49) Peter & Locher 1995; (50) Peter & Locher 1996; (51) Locher & Peter 1996; (52) Locher & Peter 1997; (53) Locher 1997; (54) Paschke & Locher 1998; (55) Locher 1998; (56) Peter 1998; (57) Locher 1999; (58) Locher 2000a; (59) Yang & Liu 2002; (60) Locher 2000b; (61) Locher 2000c; (62) Nelson 2001; (63) Locher et al. 2001; (64) Agerer & Hübscher 2002; (65) Pribulla et al. 2001; (66) Zejda 2004; (67) Locher 2001; (68) T. Pribulla 2004, private communication; (69) Locher 2002a; (70) Csizmadia et al. 2002; (71) Locher 2002b; (72) Diethelm 2003; (73) Bakiş et al. 2003; (74) Nagai 2003; (75) Agerer & Hübscher 2003; (76) Diethelm 2004; (77) Nelson 2004; (78) Pribulla et al. 2005; (79) Hübscher 2005; (80) Kim et al. 2006; (81) Byboth et al. 2004; (82) Locher 2005; (83) Hübscher et al. 2005; (84) Csizmadia et al. 2006; (85) Doǧru et al. 2007; (86) Present work; (87) Dvorak 2008; (88) Nagai 2008; (89) Hübscher et al. 2009b; (90) Dvorak 2009. (This table is available in its entirety in machine-readable and Virtual Observatory (VO) forms in the online journal. A portion is shown here for guidance regarding its form and content.) 5. DISCUSSIONS Through modeling light curves of AH Tau, we derived a photometric solution indicating that AH Tauri is a weak-contact binary (i.e., f = 10.8% for AH Tau). As seen from the O C diagrams, the orbital periods of AO Cam and AH Tau appear to possess long-term period decreases with cyclic variations. These kinds of period changes may occur in many weak-contact binaries, such as BI CVn (Qian et al. 2008), V700 Cyg (Yang &Dai2009), AR Dra (Lee et al. 2009), and other stars listed in Table 1 of Paper II (Yang et al. 2009). Gazeas et al. (2005) determined the absolute parameters of AO Cam (i.e., mass, radii, and luminosity), which are listed in Table 7. Since there are no published spectroscopic elements, the absolute parameters of AH Tau cannot be determined directly. According to the spectral type of G1p, the mass of the more massive component, M 1 = 1.04 M, was adopted (Cox 2000). Based on this, the other parameters of AH Tauri are calculated and also listed in Table 7. Figure 4. Relation between the sum residuals (O C) 2 ; and the mass ratio of q for AH Tauri. Table 6 Photometric Elements for the Eclipsing Binary AH Tau Parameters Value Standard Error i( ) 85.95 ±0.22 T 1 (K) 5900 T 2 (K) 5887 ±4 X, Y 0.649, 0.221 x 1,2V, y 1,2V 0.747, 0.259 x 1,2R, y 1,2R 0.655, 0.268 q = M 2 /M 1 0.490 ±0.001 Ω 1 = Ω 2 2.8250 ±0.0025 f 10.8% ±0.1% L 1 /(L 1 + L 2 ) V 0.6585 ±0.0010 L 1 /(L 1 + L 2 ) R 0.6581 ±0.0009 r 1 (pole) 0.4213 ±0.0005 r 1 (side) 0.4489 ±0.0006 r 1 (back) 0.4789 ±0.0009 r 2 (pole) 0.3038 ±0.0006 r 2 (side) 0.3178 ±0.0008 r 2 (back) 0.3540 ±0.0013 (O C) 2 i 0.1228 5.1. Interpreting the Cyclic Variations for AO Cam and AH Tau For weak-contact binaries AO Cam and AH Tau, cyclic variations exist in their orbital period changes. The periods and semiamplitudes are 7.63(±0.07) yr and 0 ḍ 0019(±0 ḍ 0003) for AO Cam, and 45.8(±1.1) yr and 0 ḍ 0171(±0 ḍ 0005) for AH Tau. The possible mechanism for the kind of cyclic oscillation is generally attributed to either the magnetic activity cycles in one or both components or the light-time effect due to a third body. Applegate (1992) proposed that the cyclic magnetic activity may result in the observed cyclic variation by gravitational coupling mechanism. This magnetic modulation was subsequently developed by Lanza et al. (1998), Lanza & Rodonò(2004), and Lanza (2006). Using the fitted parameters of cyclic variations in Equations (2) and (4), the values of ΔP/P can be calculated by the equation (Rovithis-Livaniou et al. 2000) ΔP = 2[1 cos (2πP/P 3 )], (5) where P and P 3 are the orbital period of the eclipsing binary and the period of cyclic variation, respectively. We can then

No. 1, 2010 ORBITAL PERIOD CHANGES FOR WEAK-CONTACT BINARIES. III. 201 Table 7 Some Derived Parameters for the Two Weak-contact Binaries AO Cam and AH Tau Parameters AO Camelopardalis AH Tauri M 1 (M ) 1.119(±0.007) 1.04 M 2 (M ) 0.486(±0.005) 0.52 R 1 (R ) 1.092(±0.005) 1.05 R 2 (R ) 0.732(±0.004) 0.77 L 1 (L ) 1.529(±0.069) 1.19 L 2 (L ) 0.574(±0.006) 0.64 a(r ) 2.346 2.35 P (d yr 1 ) 1.26(±0.04) 10 7 6.98(±0.07) 10 8 Ṁ 1 (M yr 1 ) 1.09(±0.03) 10 7 0.70(±0.01) 10 7 A(d) 0.0019(±0.0003) 0.0171(±0.0005) P 3 (yr) 7.63(±0.07) 45.8(±1.1) ΔP/P 4.38(±0.63) 10 6 6.83(±0.21) 10 6 ΔQ 1 ( 10 49 gcm 2 ) 2.90(±0.42) 3.91(±0.16) ΔQ 2 ( 10 49 gcm 2 ) 1.26(±0.18) 1.91(±0.61) ΔJ 1 ( 10 47 gcm 2 s 1 ) 1.34(±0.19) 1.89(±0.60) ΔJ 2 ( 10 47 gcm 2 s 1 ) 0.84(±0.12) 1.20(±0.38) ΔE 1 ( 10 41 erg) 4.17(±1.19) 0.96(±0.06) ΔE 2 ( 10 41 erg) 8.37(±2.40) 1.51(±0.10) (ΔΩ/Ω) 1 ( 10 3 ) 0.71(±0.10) 1.17(±0.04) (ΔΩ/Ω) 2 ( 10 3 ) 2.27(±0.33) 2.89(±0.09) ΔL 1 ( 10 32 erg) 5.44(±1.56) 2.09(±0.13) ΔL 2 ( 10 32 erg) 10.9(±3.13) 3.29(±0.21) B 1 (kg) 15.4(±1.1) 7.92(±0.13) B 2 (kg) 22.3(±1.6) 1.03(±0.16) a 12 sin i (AU) 0.3362(±0.0481) 2.9548(±0.0945) f (m)(m ) 6.53(±2.81) 10 4 1.23(±0.12) 10 2 M 3 (M )(i = 90 ) 0.125(±0.017) 0.36(±0.01) M 3 (M )(i = 70 ) 0.133(±0.018) 0.38(±0.01) M 3 (M )(i = 50 ) 0.166(±0.022) 0.48(±0.01) M 3 (M )(i = 30 ) 0.263(±0.034) 0.82(±0.02) a 12 (AU)(i = 90 ) 4.32(±1.21) 12.91(±0.78) a 12 (AU)(i = 70 ) 4.30(±1.20) 12.79(±0.77) a 12 (AU)(i = 50 ) 4.25(±1.18) 12.36(±0.74) a 12 (AU)(i = 30 ) 4.10(±1.12) 11.15(±0.65) Note. The absolute parameters for AO Cam are adopted from Gazeas et al. (2005). compute the Applegate model parameters (i.e., the variation of the quadrupole moment ΔQ, the angular momentum transfer ΔJ, the transfer energy ΔE, the variation of the differential rotation ΔΩ/Ω, the rms luminosity change ΔL, and the mean subsurface magnetic field strength B). The parameters of both components of AO Cam and AH Tau, which could explain the observed period modulation of ΔP/P 10 6 are listed in Table 7. As an alternative explanation for the cyclic variation, the light-time effect via the presence of a third body may result in the observed cyclic variation (Irwin 1952). By considering that the assumed third body moves in a circular orbit, we can calculate the value of a 12 sin i = A c, where A is the semiamplitude of the cyclic oscillation and c is the speed of light. With the equation (M 3 sin i ) 3 f (m) = (M 1 + M 2 + M 3 ) = 4π 2 2 GP3 2 (a 12 sin i ) 3, (6) we can calculate the values of the mass function for the third body. Therefore, for some orbital inclination i, the unknown mass M 3 can be derived by the iteration method. With the absolute parameters of AO Cam and AH Tau, the values of the masses and the orbital radii of the third body for several different values of i (i.e., i = 90, 70, 50, and 30 ) were calculated, which are listed in Table 7. Assuming a coplanar orbit with the binary (i.e., i = 76 ), the mass and the orbital radius of the third body for AO Cam should be M 3 = 0.13(±0.02) M and a 12 = 4.31(±1.21)AU, respectively. For AH Tau, the mass and the orbital radius of the third body are M 3 = 0.36(±0.01) M and a 12 = 12.9(±0.8)AU in the coplanar orbit with the binary (i.e., i = 85. 95). According to Allen s table (Cox 2000), the spectral types for the assumed third bodies of AO Cam and AH Tau based on the coplanar masses can be estimated to be M6 M7 and M3 M4, which may be brown or yellow dwarfs with extremely low luminosities. This is the cause why the third body was not detected by spectroscopic and photometric observations. For the stability of the triple system, Harrington (1977) suggested the following sufficient condition as follows: a log 3/2 <K (1 e )a 12 log [ 1+ M ], (7) 3 M 1 +M 2 where a and (1 e )a 12 are the semimajor axis of the binary orbit and the periastron distance of the tertiary, respectively. For corevolving systems, K is approximately 0.28, while for counterrevolving systems it is approximately 0.36. When the orbit of the third body is circular, the equation of light-time τ (Irwin 1952) then reduces to a sinusoidal curve. From Equations (2) and (4), the value of e is zero. In a coplanar orbit, inserting the approximate values of a, M 1, M 2, and M 3 into Equation (7)show that those two triple systems should be dynamically stable. For the cyclic variation, the applegate mechanism requires period changes to be in phase with the magnetic activity cycles of the active component and light-level changes. For AO Cam, the light curve changes from one cycle to another, indicating that there may exist an intrinsic variability (Baran et al. 2004). Additionally, the light curves display the variable O Connell effect from 2004 to 2008. Rucinski et al. (2007) detected a visual companion from AO Cam with a period of 1630 yr. Therefore, the magnetic activity cycles may be preferably accepted to explain the cyclic variation of AO Cam. For AH Tau, there is some evidence of the third light in the photometric solution although the value of l 3 is very low. The previous light curves (Liu et al. 1991; Byboth et al. 2004) and our VR light curves are all symmetric with no evidence for magnetic spots. Moreover, there is evidence that most contact binary stars exist in multiple systems (D Angelo et al. 2006; Pribulla & Rucinski 2006; Rucinski et al. 2007), although this hypothesis needs to be observationally confirmed with further research. Therefore, the cyclic variation of AH Tau may more likely be attributed to the light-time effect via the presence of the third body, making it a triple star system. 5.2. Long-term Period Decreases and Evolutionary Status of AO Cam and AH Tau According to the binary-star evolution code (i.e., BSE; Hurley et al. 2002), we constructed the zero-age main-sequence (ZAMS) diagrams, which are plotted in log M log L and log M log R in Figure 5, where we have also plotted the parameters of the binary stars. From both panels of this figure, it is found that the more massive components are close to the ZAMS line, while less massive components lie above the ZAMS line. This implies that the secondary components of AO Cam and AH Tau are overly large and overluminous due to the mass and energy transfer from the more massive component to the less massive one during the evolutionary process.

202 YANG ET AL. Vol. 139 Figure 5. Positions of both components of AO Cam and AH Tau in the log M log L (left panel) and log M log R diagrams (right panel). The solid line refers to the ZAMS line (Hurley et al. 2002). The filled and open symbols represent the more massive component and the less massive one, respectively. As seen from the coefficients of Equations (2) and (4), the orbital periods of AO Cam and AH Tau are continuously decreasing. These kind of secular period changes may result from the mass transferring from the more massive component to the less massive one. The continuous period decrease rates are dp /dt = 1.26(±0.04) 10 7 days yr 1 for AO Cam and dp /dt = 6.98(±0.07) 10 8 days yr 1 for AH Tau. Assuming conservative mass transfer, the mass transfer rate can be estimated by the following equation (Singh & Chaubey, 1986): 3(1 q) M 1 =, (8) q M 1 where M 1 and q = M 2 /M 1 indicate the mass of the more massive component and the mass ratio of the binary, respectively. Inserting the values of P, P, M 1, and M 2 for those two binaries into Equation (8), we can calculate the mass transfer rates of dm 1 /dt = 1.04(±0.03) 10 7 M yr 1 for AO Cam and dm 1 /dt = 0.70(±0.01) 10 7 M yr 1 for AH Tau. These are listed in Table 7. For the weak-contact binary with decreasing orbital period, the mass transfers from the more massive component to the less massive one. Its decreasing orbital period will result in the shrinking of the inner and outer critical Lagrangian surfaces, causing the contact degree to increase. Therefore, this kind of weak-contact binary with a decreasing period will evolve into a deep contact configuration. P P 6. CONCLUSIONS The results of our detailed investigations on two weak-contact binaries, AO Cam and AH Tau, are summarized as follows. 1. The photometric solution for AH Tau was derived from our new CCD observations. The results show that AH Tau is a weak-contact binary (i.e., f = 10.8%). The mass ratio of 0.503(±0.004) is very close to the derived value in Liu et al. (1991) of 0.502. For the weak-contact binary AO Cam, BVI light curves clearly show a difference in the heights of the maxima (i.e., the O Connell effect), usually explained by spot activity. 2. From the O C curves for AO Cam and AH Tau, the orbital periods show a long-term period decrease with a cyclic variation. For the cyclic variations, the periods and semiamplitudes are 7.63(±0.07) yr and 0 ḍ 0019(±0 ḍ 0003) for AO Cam, and 45.8(±1.1) yr and 0 ḍ 0171(±0 ḍ 0005) for AH Tau. Based on the asymmetric light curves, the cyclic oscillation for AO Cam may be attributed to the cyclic magnetic activity in one or both components, while the cyclic oscillation of AH Tau may more likely result from the light-time effect due to a third body. This additional component may extract angular momentum from the central system. 3. Based on the absolute physical parameters for AO Cam and AH Tau, the less massive components are overluminous compared with the ZAMS. The secular period decrease rates are dp /dt = 1.26(±0.04) 10 7 days yr 1 for AO Cam and dp /dt = 6.98(±0.07) 10 8 days yr 1 for AH Tau, indicating that the mass transfers from the more massive component to the less massive component. With mass transferring, this kind of weak-contact binary with a decreasing period will evolve into a deep contact configuration. 4. Future observations would be helpful to obtain more light minimum times with high precision in order to check the nature of their orbital period changes and evolutionary status for two weak-contact binaries AO Cam and AH Tau. It would be especially important to carry out spectroscopic observations for AH Tau to obtain radial velocity curves to confirm the mass ratio and to determine the absolute physical parameters. Many thanks are expressed to the anonymous referee for the helpful suggestions and comments, including the improvement of the author s English. This work is supported partly by the Joint Fund of Astronomy of the National Natural Science Foundation of China (NSFC) and the Chinese Academy of Sciences (CAS) grant 10778707, the Special Foundation of the President and West Light Foundation of the Chinese Academy of Sciences, and the Yunnan Natural Science Foundation (2008CD157). New observations of AO Cam and AH Tau were performed using the 85 cm telescope and the 80 cm telescope at the Xinglong Station of the NAOC, the Chinese Academic of Sciences. This research has made use of the Simbad Database, operated at CDS, Strasbourg, France.

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