Sheng-Bang Qian, 1, 2 Jiajia He, 1, 2 Fuyuan Xiang, 3 Xiyang Ding, 4 and Soonthornthum Boonrucksar 5

The Astronomical Journal, 129:1686 1693, 2005 March # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A. PERIOD CHANGES OF AO CAMELOPARDALIS AND AM LEONIS AND THEIR IMPLICATIONS FOR THE PRESENCE OF TERTIARY COMPONENTS AND THE EVOLUTIONARY STATES OF THE TWO OVERCONTACT BINARY SYSTEMS Sheng-Bang Qian, 1, 2 Jiajia He, 1, 2 Fuyuan Xiang, 3 Xiyang Ding, 4 and Soonthornthum Boonrucksar 5 Receivved 2003 October 6; accepted 2004 November 30 ABSTRACT AO Cam and AM Leo are two short-period overcontact binary systems whose physical properties are nearly the same. In the present paper, three CCD times of minimum light for AO Cam and two photoelectric eclipse times for AM Leo are presented. Combining the new determined eclipse times with others compiled from the literature, the behavior of their O C variations were investigated. For AO Cam, its orbital period is revised, and it is found that the period shows a cyclic variation with a period of 20.1 yr and an amplitude of 0.0047 days. In AM Leo, a new period increase has occurred recently. Two possibilities of period variations, sudden and continuous changes, are discussed. The period changes of both systems cannot be explained by apsidal motion, since both the primary and the secondary eclipse times vary in the same O C trend. Of the three mechanisms of magnetic activity cycles, internal mass motions, and the presence of a third body, we think that the last is more plausible. Assuming that the period changes are due to third bodies revolving around the eclipsing pairs, the parameters of the third-component stars are determined. If it exists, the third body in AO Cam is revolving in a circular orbit, whereas that in AM Leo is revolving in an eccentric orbit (e 0 ¼ 0:58). No secular period changes were discovered in either overcontact binary star, which is in agreement with the recent statistical relation of Qian. This may suggest that AO Cam and AM Leo are in transition between the angular momentum loss controlled and the thermal relaxation oscillation controlled stages of the evolutionary scheme recently proposed by Qian. Key words: binaries: close binaries: eclipsing stars: evolution stars: individual (AO Camelopardalis, AM Leonis) 1. INTRODUCTION The light variability of the W UMa type system AO Cam (BD +52 0826) was discovered by Hoffmeister (1966). The first photoelectric light curves in the UBV filters were published by Milone et al. (1982). They classified the spectral type of the binary star as G5 and gave a period of 0.329917 days. Milone et al. (1982) analyzed their data with the method of Rucinski (1973, 1974). Evans et al. (1984, 1985) obtained B and V light curves and analyzed them with the Wilson-Devinney method (Wilson & Devinney 1971). However, no unique solutions were determined. The photometric studies by Cooke & Leung (1985) and Barone et al. (1993) revealed that AO Cam is a W-type system, which is confirmed by the spectroscopic investigation of Rucinski et al. (2000). However, the photometric mass ratio is very different from the spectroscopic one (q sp ¼ 0:413). This may be caused by the fact that the partial eclipse makes it very difficult to determine the mass ratio photometrically. Up to now, no O C curves of the binary system have been constructed and analyzed. The other W UMa type binary star, AM Leo (BD +10 2234), is the brighter component of the visual binary ADS 8024. Since its discovery by Worley & Eggen (1956), it has been extensively investigated. Photoelectric observations of the binary 1 National Astronomical Observatories, Yunnan Observatory, Chinese Academy of Sciences, P.O. Box 110, 650011 Kunming, China; qsb@netease.com. 2 Also at United Laboratory of Optical Astronomy, Chinese Academy of Sciences, 100012 Beijing, China. 3 Department of Physics, Xiangtan University, 411105 Xiangtan, Funan, China. 4 Bijie County Second Middle School, 551700 Bijie, Guizhou, China. 5 Department of Physics, Chiang Mai University, 50200 Chiang Mai, Thailand. 1686 system were published by Worley & Eggen (1956), Abrami (1959), Binnendijk (1969), Hoffmann & Hopp (1982), Derman et al. (1991), and Demircan et al. (1992). The light curve was observed to change from W- to A-type or vice versa (Binnendijk 1969; Hoffmann & Hopp 1982; Derman et al. 1991). A variable O Connell effect was also seen in the light curve. The binary was assigned the spectral type F5 by Hill et al. (1975), but unfortunately no radial velocity curves have been published. Photometric solutions by Hutchings & Hill (1973), Binnendijk (1984), and Hrivnak (1993) with a light-curve synthesis technique and by Binnendijk (1969) with the Russell-Merrill method ( Russell & Merrill 1952) all revealed a W-type overcontact binary system. The period variation of AM Leo was first noticed by Binnendijk (1969), who pointed out a period decrease that occurred around 1960. After this period decrease, Hoffmann & Hopp (1982) showed that no further period change could be found. The O C curve was described by Hoffmann & Hopp (1982) with two separate linear fits to the data before and after 1960 or with a second-order polynomial. They found no significant difference between the two fits, although the fit by linear sections is more likely. A period increase near 1985 was discovered by Demircan & Derman (1992), and several possible mechanisms (e.g., mass transfer, third body, magnetic activity cycles, internal mass motions) were proposed to explain the O C oscillation by them. Our two eclipse times and those times of minimum light published by Pribulla et al. (2002), Agerer & Hübscher (2003), Gürol et al. (2003), Dvorak (2004), and Hiller et al. (2004) reveal that the period of AM Leo has been increasing recently. This indicates that the period change of the binary system needs to be investigated in detail. The physical parameters of the two W UMa stars are listed in Table 1 in solar units. The values of M 1 and M 2 of AO Cam

PERIOD CHANGES OF AO CAM AND AM LEO 1687 TABLE 1 Physical Parameters of AO Camelopardalis and AM Leonis Star Name P i (deg) M 1 M 2 q Spectral Type Ref. AO Cam... 0.3299 76 1.16 0.50 0.41 G0 V 1, 2 AM Leo... 0.3658 87 1.25 0.53 0.42 F5 V 3, 4 References. (1) Rucinski et al. 2000; (2) Evans et al. 1985; (3) Binnendijk 1984; (4) Hrivnak 1993. are determined with the spectroscopic solutions of Rucinski et al. (2000) by taking i ¼ 76 (Evans et al. 1985), whereas those of AM Leo were from Maceroni & van t Veer (1996). As we can see from Table 1, the masses of the two components, the mass ratios, and the orbital periods of the two systems are nearly equal, which suggests that their physical properties are nearly the same. In order to study the period changes of the two W UMa binary stars, we intended to observe them at epochs of minimum light. 2. NEW OBSERVATIONS FOR AO CAMELOPARDALIS AND AM LEONIS The observations of AO Cam in the V band were carried out on 2002 February 5, 7, and 8 with the PI1024 TKB CCD photometric system attached to the 1.0 m reflecting telescope at the Yunnan Observatory in China. The V passband used is close to that in the Johnson UBV system (Yang & Li 1999). The effective field of view of the photometric system is 6A5 ; 6A5atthe Cassegrain focus. The integration time for each image was 120 s. The PHOT task (which measures magnitudes for a list of stars) in the IRAF aperture photometry package was used to reduce the observed images. Three times of minimum light were derived from the observations and are shown in Table 2. The other W UMa type binary star, AM Leo, was observed on 2002 March 2 with the WET high-speed three channel photoelectric photometer (Jiang & Hu 1998) attached to the 85 cm telescope at the Xinglong Station of the National Astronomical Observatories (NAO). During the observation, the integration time was 10 s, and Johnson s V filter was used. No complete light curve was obtained, but with the observed data, one primary and one secondary time of minimum light were determined by using a parabolic fitting method. They are given in Table 2, with the uncertainties in the third column. 3. ORBITAL PERIOD CHANGE OF AO CAMELOPARDALIS Since the discovery of AO Cam by Hoffmeister (1966), no O C curves have been constructed and analyzed for it. In order to investigate the period change of AO Cam, all available times of minimum light were compiled and are listed in Table 3. The (O C) 1 values in the table were calculated with the linear ephemeris given by Evans et al. (1985), Min: I ¼ HJD 2; 445; 745:6394 þ 0:329905519E; and are displayed in Figure 1, where crosses refer to visual times of minimum light, whereas filled and open circles represent the primary and secondary eclipse times observed with the methods of photoelectric and CCD (hereafter PC) photometries, respectively. As we can see from Figure 1, the period of AO Cam needs to be revised, and the period may show a cyclic variation. Thus, a sinusoidal term was added to a linear ephemeris to give a good fit to the (O C) 1 curve (Fig. 1, solid line). With weights of 1 for visual data and 20 for the PC observations, a weighted leastsquares solution led to the following equation: Min: I ¼ HJD 2; 4457; 45:6423(0:0004) þ 0:32990465(0:00000002) E þ 0:0047(4) sin ½0N0162E þ 306N4(3N9)Š: The sinusoidal term in the equation reveals a periodic variation with a period of 20.1 yr and an amplitude of 0.0047 days, which is more clearly seen in Figure 2, where the (O C) 2 values with respect to the linear part of equation (2) are shown. The errors of the data published after HJD 2,451,000 are shown in the second column of Table 3. Apart from two data points (HJD 2,451,237.397 and 2,451,597.486, whose O C values follow the general O C trend), the errors are less than 0.0003 days. Although the errors of those PC observations obtained before HJD 2,451,000 were not given in the original references, we think that they are usually less than 0.0010 days. During the analysis of the O C variation, a quadratic ephemeris was also used to fit the O C curve (Fig. 2, dashed line). The value of the sum of the weighted squares of the residuals from equation (2) is ¼ 0:0012. However, the value of from the quadratic ephemeris is ¼ 0:0016, suggesting that the sinusoidal curve fits the data better. The residuals from equation (2) are displayed in Figure 3, where no changes can be traced, indicating that the cyclic variation can describe the general O C trend satisfactorily. ð1þ ð2þ TABLE 2 New Times of Minimum Light for AO Camelopardalis and AM Leonis Star Name HJD Error Methods Min. Filter 4. VARIATIONS IN O C FOR AM LEONIS Times of minimum light of AM Leo have been compiled by Demircan & Derman (1992). Following this collection, some times of minimum light were compiled by Hiller et al. (2004) and at the Eclipsing Binaries Minima Database (EBMD). 6 After these collections, some PC data were published and are shown in Table 4. In order to compare these data with the period study by AO Cam... 2,452,311.0689 0.0003 CCD I V 2,452,313.0484 0.0002 CCD I V 2,452,314.0387 0.0003 CCD I V AM Leo... 2,452,336.0852 0.0002 pe II V 2,452,336.2705 0.0001 pe I V 6 Available at http://www.oa.uj.edu.pl/ktt/krttk_dn.html.

TABLE 3 Times of Minimum Light for AO Camelopardalis HJD (2,400,000+) Errors Min. Method E (O C) 1 (O C) 2 Ref. 44,520.8657...... II pe 3712.5 þ0.0005 0.0054 1 44,558.8054...... II pe 3597.5 þ0.0011 0.0047 1 44,559.7950...... II pe 3594.5 þ0.0010 0.0048 1 44,559.9610...... I pe 3594 þ0.0020 0.0038 1 45,732.6076...... II pe 39.5 0.0005 0.0033 2 45,738.5463...... II pe 21.5 0.0001 0.0029 2 45,745.6402...... I pe 0 þ0.0008 0.0019 2 45,752.5676...... I pe 21 þ0.0002 0.0025 2 46,006.4281...... II pe 790.5 0.0016 0.0037 3 46,022.4291...... I pe 839 0.0011 0.0031 3 46,024.7389...... I pe 846 0.0006 0.0026 4 46,036.2858...... I pe 881 0.0004 0.0024 3 46,036.4498...... II pe 881.5 0.0013 0.0033 3 46,036.6157...... I pe 882 0.0004 0.0031 3 46,100.2871...... I pe 1075 0.0007 0.0025 5 46,107.3796...... II pe 1096.5 0.0012 0.0030 5 46,429.6979...... II pe 2073.5 0.0006 0.0016 6 46,436.6260...... II pe 2094.5 0.0005 0.0014 6 47,864.7879...... II pe 6423.5 þ0.0004 þ0.0031 7 50,415.281...... II vis 14154.5 0.0061 þ0.0028 8 50,421.220...... II vis 14172.5 0.0054 þ0.0036 8 50,434.246...... I vis 14212 0.0106 0.0015 8 51,199.4585... 0.0001 II pe 16531.5 0.0140 0.0025 9 51,199.6237... 0.0001 I pe 16532 0.0137 0.0022 9 51,237.397... 0.004 II pe 16646.5 0.0146 0.0030 9 51,241.5223... 0.0002 I pe 16659 0.0131 0.0015 9 51,597.486... 0.005 I pe 17738 0.0175 0.0050 9 51,798.5674... 0.0002 II pe 18347.5 0.0135 0.0005 10 52,311.6089... 0.0003 I CCD 19901 0.0202 0.0059 11 52,313.0484... 0.0002 I CCD 19907 0.0202 0.0059 11 52,314.0387... 0.0003 I CCD 19910 0.0196 0.0052 11 52,607.6550... 0.0001 I CCD 20800 0.0192 0.0041 12 52,631.7377... 0.0001 I CCD 20873 0.0196 0.0044 13 52,943.8268... 0.00005 I CCD 21819 0.0212 0.0052 14 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) Mullis & Faulkner 1991; (8) Dalmazio 1997; (9) Ogloza et al. 2000; (10) Pribulla et al. 2001; (11) this paper; (12) Dvorak 2003; (13) Nelson 2003; (14) Nelson 2004. Fig. 1. The (O C) 1 curve of AO Cam based on the linear ephemeris given by Evans et al. (1985). Crosses refer to visual times of minimum light, whereas filled and open circles refer to the primary and secondary times observed with PC photometry, respectively. The solid line represents a combination of a new linear ephemeris and a cyclic variation, and the dashed line is the new linear ephemeris. Fig. 2. The (O C) 2 diagram for AO Cam from the linear ephemeris. The dashed line refers to a quadratic fit. Symbols are the same as in Fig. 1.

PERIOD CHANGES OF AO CAM AND AM LEO 1689 Fig. 3. Residuals for AO Cam with respect to eq. (2). Symbols are the same as in Figs. 1 and 2. The solid line refers to a linear fit to the residuals. The slope of the line is +8:8 ; 10 11, which is much smaller than the error of the orbital period of AO Cam. Demircan & Derman (1992), the (O C) 1 values of all available eclipse timings were calculated with the light elements given by Binnendijk (1969), Min: I ¼ 2; 439; 936:8337 þ 0:36579720E: The corresponding (O C) 1 diagram is plotted against epoch number in Figure 4, where crosses refer to visual and photographic times of minimum light and circles represent the PC observations (filled and open circles stand for primary and secondary minimum light, respectively). As displayed in Figure 4, the (O C) 1 values of those times of minimum light observed visually or photographically show large scatter (up to 0.028 days). These data do not contribute to form the general O C trend and tend to mislead regarding the real period change. Therefore, only PC observations were used for the present period study. AsshowninFigure4,aftertheperiodincreasereportedby Demircan & Derman (1992), another period increase occurred around HJD 2,450,800 and was also reported by Hiller et al. (2004). By assuming that the period change is not continuous, the O C curve can be divided into four portions (AB, BC, CD, ð3þ Fig. 4. The (O C) 1 diagram of the overcontact binary system AM Leo computed with the ephemeris given by Binnendijk (1969). Crosses refer to visual and photographic times of minimum light, and filled and open circles refer to the primary and secondary eclipsing times determined with PC photometric observations, respectively. The dashed lines indicate that several period jumps may occur in the system. and DE) in which the period change is discussed. Fortunately, apart from the last portion, the data in other portions are sufficiently covered. The values of T 0 and P 0 in the equation (O C) 1 ¼ T 0 þ P 0 E ð4þ were calculated for each portion by using a least-squares method and are shown in Table 5, where T 0 and P 0 are the corrections in epoch and period, respectively. As listed in the table, a period decrease of P ¼ 2:395 ; 10 6 days ¼ 0:21 s occurred near 1960, which was first noticed by Binnendijk (1969) and was later pointed out by Hoffmann & Hopp (1982) and Demircan & Derman (1992). Then a period increase (P ¼ þ0:561 ; 10 6 days ¼þ0:05 s) took place around 1982, and finally the period increased again around 1998 with the value of P ¼þ2:524 ; 10 6 days ¼þ0:22 s. After the three period jumps are removed, the residuals are as displayed in Figure 5. TABLE 4 New Available Times of Minimum Light for AM Leonis HJD (2,400,000+) Errors Min. Method E (O C) 1 (O C) 2 Ref. 52,322.3645... 0.0004 I pe 33,859 þ0.0034 0.0058 1 52,336.0852... 0.0002 II pe 33,896.5 þ0.0067 0.0025 2 52,336.2705... 0.0001 I pe 33,897 þ0.0091 0.0001 2 52,339.37402... 0.0006 II CCD 33,905.5 þ0.0034 0.0059 3 52,339.55334... 0.0002 I CCD 33,906 0.0002 0.0095 3 52,344.3147... 0.0005 I pe 33,919 þ0.0051 0.0035 1 52,373.3957... 0.0002 II pe 33,998.5 þ0.0059 0.0034 1 52,682.8617... 0.0004 II CCD 34,844.5 þ0.0075 0.0023 4 52,683.4106... 0.0004 I pe 34,846 þ0.0077 0.0021 1 52,719.4402... 0.0022 II pe 34,944.5 þ0.0062 0.0035 1 52,724.3811... 0.0004 I pe 34,958 þ0.0089 0.0009 1 52,730.4154...... II pe 34,974.5 þ0.0075 0.0023 1 52,736.4516...... I pe 34,991 þ0.0081 0.0017 1 52,750.3516... 0.0004 I pe 35,029 þ0.0078 0.0020 1 References. (1) Agerer& Hübscher 2003; (2) this paper; (3) Gürol et al. 2003; (4) Dvorak 2004.

1690 QIAN ET AL. Vol. 129 TABLE 5 Several Assumed Period Jumps for AM Leonis Portions T 0 P 0 AB... þ0.0172(0.0024) þ2.02(0.22) ; 10 6 BC... þ0.0007(0.0005) 3.75(0.66) ; 10 7 CD... 0.0073(0.0014) þ1.86(0.63) ; 10 7 DE... 0.0869(0.0234) þ2.71(0.68) ; 10 6 On the other hand, the period change may be continuous. The (O C) 2 values based on the linear ephemeris recently published by Kreiner et al. (2001), Min: I ¼ 2; 439; 936:8260 þ 0:36579770E; are displayed in Figure 6. As plotted in this figure, the (O C) 2 curve may show a cyclic variation, which may be caused by the light-time effect of a third body. As shown in this figure, the shape of the oscillation is not strictly sinusoidal, indicating that the third body is moving in an elliptical orbit (e 0 < 0:6). Thus, the following equation, (O C) 2 ¼ a 0 þ X2 i¼1 ½a i cos (ie) þ b i sin (ie)š; where a i, b i,andare well-known Fourier constants, was used to express those (O C) 2 residuals. Based on various numerical trials with different values of, the best solutions with those observations (Fig. 6, solid line) yielded a 0 ¼þ0:0008(2), a 1 ¼þ0:0091(6), b 1 ¼ 0:0035(5), a 2 ¼ 0:0003(8), b 2 ¼ 0:0028(1), and ¼ 0N0070(assumed). The residuals from equation (6) are displayed in Figure 7. With ¼ 360 P e =T, the orbital period of the eclipsing pair around the center of mass of the triple system was determined to be T ¼ 51:4 yr. The orbital parameters of the third-body orbit were computed by using the formulae given by Kopal (1959): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a12 0 sin i0 ¼ c a 2 1 þ b2 1; ð7þ ð5þ ð6þ Fig. 6. The (O C) 2 plot for AM Leo with respect to the ephemeris recently published by Kreiner et al. (2001). The solid line refers to a theoretical orbit of an assumed third body. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 0 ¼ 2 a 2 2 þ b2 2 a 2 1 þ ; b2 1 ð8þ! 0 ¼ arctan (b 2 1 a 2 1 )b 2 þ 2a 1 a 2 b 1 (a 2 1 b 2 1 )a 2 þ 2a 1 b 1 b 2 ; ð9þ 0 ¼ t 0 T 2 arctan a 1b 2 b 1 a 2 a 1 a 2 þ b 1 b 2 ; ð10þ where c is the speed of light and a 12 0, i 0, e 0,! 0,and 0 are the semimajor axis, the orbital inclination, the eccentricity, the longitude of periastron from the ascending node, and the time of the periastron passage, respectively. The results are a12 0 sin i0 ¼ 1:69(0:10) AU, e 0 ¼ 0:58(0:07),! 0 ¼ 54N0(16N6), and 0 ¼ 2;436;021(859) days. The sum of the squares of the residuals from equation (6) is ¼ 0:0016, whereas that from the dashed lines described by equation (4) is only ¼ 0:0013. This indicates that the dashed lines fit the O C curve better than does equation (6). 5. DISCUSSION AND CONCLUSION We combined our five times of minimum light with others compiled from the literature to find the period changes of two Fig. 5. Residuals from the dashed lines in Fig. 4. Symbols are the same as in Figs. 1 4. The slope of the solid line is 5:5 ; 10 11, which is much smaller than the error of the orbital period of AM Leo. Fig. 7. Residuals for AM Leo from the best fit of eq. (6). Symbols are the same as in Figs. 1 6. The solid line refers to a linear fit to the residuals. The slope of the line is 1:3 ; 10 8, which is smaller than the error of the orbital period.

No. 3, 2005 PERIOD CHANGES OF AO CAM AND AM LEO 1691 TABLE 6 Comparison between the Original Times of Minimum Light and Those Determined with the Parabolic Method Original Minima (2,400,000+) (1) Parabolic Fitting Method (2,400,000+) (2) Min. (3) Difference (4) Ref. (5) 36,988.6917... 36,988.6918(0.0008) II 0.0001 1 36,995.6413... 36,995.6415(0.0004) II 0.0002 1 39,936.8337... 39,936.8334(0.0003) I þ0.0003 2 39,937.7490... 39,937.7494(0.0002) II 0.0004 2 39,937.9310... 39,937.9315(0.0004) I 0.0005 2 39,938.8465... 39,938.8461(0.0003) II þ0.0004 2 39,939.7598... 39,939.7600(0.0003) I 0.0002 2 44,292.3766... 44,292.3772(0.0007) I 0.0006 3 44,336.4553... 44,336.4547(0.0014) II þ0.0006 3 44,343.4055... 44,343.4052(0.0006) II þ0.0003 3 47,216.5595... 47,216.5602(0.0007) I 0.0007 4 47,241.4346... 47,241.4343(0.0009) I þ0.0003 4 47,562.6039... 47,562.6037(0.0009) I þ0.0002 4 47,947.4223... 47,947.4218(0.0006) I þ0.0005 4 47,966.4454... 47,966.4451(0.0010) I þ0.0003 4 References. (1) Bookmyer 1961; (2) Binnendijk 1969; (3) Demircan & Derman 1992; (4) Demircan et al. 1992. overcontact binary stars, AO Cam and AM Leo. For AO Cam, the residuals in Figure 2 indicate that its period change is cyclic with a period of 20.1 yr. For the other system, AM Leo, it is revealed that the period was increasing around 1998. Two possibilities, sudden changes and a continuous period variation, were described in the previous section. As shown in Figures 2 and 6, both the primary and secondary times of minimum light follow the same general trends of O C variation, suggesting that the O C curves of the two overcontact binary stars cannot be explained by apsidal motion. Variations of the light curve for AM Leo have been reported by several investigators (e.g., Binnendijk 1969; Hoffmann & Hopp 1982; Derman et al. 1991). For AO Cam, it can be noted from Figures 1, 2, and 3 that the secondary times of minimum light fall systematically lower than those of the primary minimum whenever both types of minima are observed. It is possible that the observed period changes in the two binary systems are caused by the influence of the asymmetric minima via the presence of dark spots and/or the method of measuring the epochs of minima (van t Veer 1973; Maceroni & van t Veer 1994). To check whether the changes are real, we chose AM Leo as an example and have newly computed some times of minimum light with a parabolic fitting method. The original minima in Table 6 published by Demircan & Derman (1992) and Demircan et al. (1992) were calculated with the method of Kwee & van Woerden (1956), but the methods used to determine the others were not given in the references. The redetermined eclipse minima are shown in column (2) of Table 6, and the values listed in column (4) are the differences (the published minima minus those determined by the parabolic fitting method). It is shown that the differences are smaller than 0.0007 days, which is much smaller than the scatter of the residuals in Figures 5 and 7. The light curve of AM Leo published by Abrami (1959) was averaged by Binnendijk (1969), who showed that this light curve was asymmetric, especially around the primary minimum. The reflected and unreflected branches of this minimum are displayed in Figure 8. The thick solid line and the thin solid line in the figure represent parabolic fits for both cases. It is shown that the effect of the asymmetric Fig. 8. Normal light curve of Abrami s (1959) observations. Filled circles refer to the reflected branch of the minimum (assuming that the primary minimum is symmetric), and open circles refer to the observations. The thick solid line represents the parabolic fit of the reflected branches of the minimum, whereas the thin solid line represents the parabolic fit of the observations. minimum on the determination of eclipse time is relatively small. Therefore, the observed systematic changes in the orbital periods of AO Cam and AM Leo cannot be attributed to the asymmetric minima and to the different methods (e.g., the method of Kwee & Van Woerden vs. the parabolic fitting method). The alternate period change of a close binary system containing at least one late-type star can be explained by a magnetic activity cycle mechanism (e.g., Applegate 1992; Lanza et al. 1998). This mechanism expects a connection between the period change and the luminosity variation (the variation of the spot area). For AO Cam, the observed data are too insufficient to check this connection. For AM Leo, the study by Demircan & Derman (1992) showed no grounds to support any relation. Moreover, the O C curve plotted in Figure 4 suggests that, strictly speaking, the period jumps do not occur alternately. Two successive period increases occurred around 1982 and1998. Thus, we think that the period change of AM Leo is not caused by magnetic activity cycles of both components or that the magnetic activity cycles of the components have been affected by the presence of the common convective envelope, as in some Algol-type binaries (e.g., TW Dra and SW Cyg) in which the magnetic activity cycles may be affected by rapid mass loss from the cool components (Qian & Boonrucksar 2002; Qian et al. 2002). The mechanism related to internal material motions was proposed by van t Veer (1986) to explain alternating period changes of close binary stars. Van t Veer pointed out that such motions related to inhomogeneous nuclear processes may temporarily bring heavy material from the inner radiative core outward and conversely. These motions should displace the location of the center of gravity and result in period changes. Since this mechanism does not need the period to change in a strictly alternating way, compared with the magnetic activity mechanism, it is a more plausible cause for the period change of AM Leo. However, the verification of van t Veer s idea is difficult to confirm or refute and is a great weakness of the idea (van t Veer 1986). Therefore, although the dashed lines fit the O C curve of AM Leo better, we think that its O C curve is more reasonably explained as the result of the presence of a third body in the system. The sinusoidal O C variation in Figure 2 may indicate that the third body in AO Cam is moving in a circular orbit. For AM Leo, the third body is rotating in an eccentric orbit

1692 QIAN ET AL. Vol. 129 Fig. 9. Relations between the masses and the orbital inclinations (i 0 ) for the assumed third-component stars in AO Cam and AM Leo. The dashed line is for AO Cam, and the solid line is for AM Leo. (e 0 ¼ 0:58) with a period of 51.4 yr. A calculation with the following equation, f (m) ¼ (M 3 sin i 0 ) 3 (M 1 þ M 2 þ M 3 ) 2 ¼ 42 2(a 0 12 sin i 0 ) 3 ; ð11þ leads to small mass functions for the third bodies in the two W UMa type stars (see Table 6). With the parameters listed in Table 1, the masses and the orbital radii for the third bodies are computed. The correlations between the mass and the orbital inclination (i 0 ) for the assumed third bodies are displayed in Figure 9. The values of the masses and the orbital radii of the third-component stars for several different orbital inclinations (i 0 ) are shown in Table 7. As we can see from this table, the TABLE 7 Values of the Masses and Orbital Radii of the Assumed Third Component Stars in AO Camelopardalis and AM Leonis for Several Different Orbital Inclinations (i 0 ) Parameter AO Cam AM Leo A... 0.0047(0.0004) 0.0097(0.0006) T (yr)... 20.1(assumed) 51.4(assumed) e 0... 0(assumed) 0.58(0.07) a12 0 sin i0 (AU)... 0.81(0.07) 1.69(0.10) f (m) (M )... 1.30(0.34) ; 10 3 1.82(0.33) ; 10 3 m 3 (i 0 =90 )(M )... 0.16(0.02) 0.19(0.01) m 3 (i 0 =70 )(M )... 0.18(0.02) 0.21(0.01) m 3 (i 0 =50 )(M )... 0.22(0.03) 0.26(0.02) m 3 (i 0 =30 )(M )... 0.35(0.04) 0.41(0.03) m 3 (i 0 =10 )(M )... 1.31(0.16) 1.58(0.14) a 3 (i 0 =90 )(M )... 9.8(1.7) 15.6(1.4) a 3 (i 0 =70 )(M )... 9.5(1.6) 15.6(1.4) a 3 (i 0 =50 )(M )... 8.9(1.3) 15.3(1.4) a 3 (i 0 =30 )(M )... 8.0(1.1) 14.6(1.3) a 3 (i 0 =10 )(M )... 6.0(0.9) 11.0(1.2) Fig. 10. Positions of AO Cam and AM Leo ( filled circles) onthedp=dt-q diagram of the observed overcontact binary systems. assumed third bodies in the two systems are invisible unless the orbital inclination i 0 is very small (i 0 < 10 ). Recently, based on the period changes of 59 overcontact binary systems, a statistical investigation of long-period variations has been given by Qian (2003a). A statistical relation between dp=dt and mass ratio q for cooler systems (M 1 < 1:35 M ), dp=dt ¼ 1:5(0:8) ; 10 7 þ 3:7(1:7) ; 10 7 q; ð12þ has been obtained. This equation tells us that when q ¼ 0:41, dp=dt ¼ 0(1:1) in units of 10 7 days yr 1. For AO Cam and AM Leo, the mass ratios are q ¼ 0:41 and 0.42, respectively. The period analysis discussed above shows that no clear longperiod changes are found in the two systems, i.e., dp=dt 0, which is in good agreement with this statistical relation. The positions of AO Cam and AM Leo on the dp=dt q diagram are shown in Figure 10. In order to explain the secular period changes of W UMa type binary stars, an evolutionary scenario was proposed by Qian (2001a, 2001b, 2003a). This scenario is that the evolution of an overcontact binary involves both thermal relaxation oscillations (TROs) and variable angular momentum loss (AML) via the change in depth of the overcontact. According to this evolutionary scheme, systems (e.g., V417 Aql; see Qian 2003b) with secularly decreasing periods are in the AML-controlled evolutionary stage, whereas those (e.g., CE Leo; see Qian 2002 and Kang et al. 2004) showing an increasing period are in the TRO-controlled stage. AO Cam and AM Leo may be in the transition between the AML-controlled and the TRO-controlled stages. This work was supported by the Chinese Academy of Sciences (KJCXZSW-T06), the Yunnan Natural Science Foundation (2001A0026Q and 2003A0072M), the Chinese Natural Science Foundation (10003004), the Science and Technology Department of Yunnan Province (2003RC19), the National Key Fundamental Research Project through grant G1999075405, and the National Research Council of Thailand. We have used the

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