deep, low mass ratio overcontact binary systems. xiv. a ... - IOPscience

The Astronomical Journal, 150:69 (6pp), 2015 September

doi:10.1088/0004-6256/150/3/69

© 2015. The American Astronomical Society. All rights reserved.

DEEP, LOW MASS RATIO OVERCONTACT BINARY SYSTEMS. XIV. A STATISTICAL ANALYSIS OF 46 SAMPLE BINARIES Yuan-Gui Yang1,3 and Sheng-Bang Qian2,3 1

School of Physics and Electronic Information/Information College, Huaibei Normal University, Huaibei 235000, Anhui Province, China; [email protected],[email protected] 2 Yunnan Astronomical Observatory/National Astronomical Observatories of China, Kunming 650011, Yunan Province, China; [email protected] 3 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650011, China Received 2015 January 31; accepted 2015 July 4; published 2015 August 10

ABSTRACT A sample of 46 deep, low mass ratio (DLMR) overcontact binaries (i.e., q ⩽ 0.25 and f ⩾ 50%) is statistically analyzed in this paper. It is found that five relations possibly exist among some physical parameters. The primary components are little-evolved main sequence stars that lie between the zero-age main sequence line and the terminal-age main sequence (TAMS) line. Meanwhile, the secondary components may be evolved stars above the TAMS line. The super-luminosities and large radii may result from energy transfer, which causes their volumes to expand. The equations of M–L and M–R for the components are also determined. The relation of P–Mtotal implies that mass may escape from the central system when the orbital period decreases. The minimum mass ratio may preliminarily be qmin = 0.044(0.007) from the relations of q–f and q–Jspin/Jorb. With mass and angular momentum loss, the orbital period decreases, which finally causes this kind of DLMR overcontact binary to merge into a rapid-rotating single star. Key words: binaries: close – binaries: eclipsing – stars: statistics binaries (7 W-subtype and 39 A-subtype). Table 1 lists the spectroscopic and photometric elements, including the light curve type, spectral type (Sp.), period P, mass ratios qph and qsp, degree of overcontact f, orbital inclination i, radial velocities K1 and K2, relative radii r1 and r2, and effective temperatures T1 and T2 for both components. The spectral type ranges from A2 to G2, the periods of which cover from 0.2710 days (J13031–0101.9; Pribulla et al. 2009b) to 0.8665 days (KN Per; Goderya et al. 1997). From this table, 24 binaries have spectroscopic elements and 5 systems (i.e., FN Cam, AW CrB, V407 Peg, GSC 5909–189, and J13031–0101.9) have no relative radii of the components, which are approximately estimated from their related values of potentials and mass ratios using the Kopal (1959) equation. For contact binaries, the spectroscopic and photometric mass ratios are almost in agreement (Maceroni & van’t Veer 1996; Pribulla et al. 2003). The photometric mass ratio for AH Cnc is much smaller than its spectroscopic one. The former may be reliable (Qian et al. 2006). Except for AW UMa (Rucinski 2015), we thus adopted the photometric mass ratios in the following statistical analyses.

1. INTRODUCTION The W Ursae Majoris (W UMa) eclipsing binary contains two components embedded in a thin common equipotential envelope surface, which was described by Lucy (1968a, 1968b). Models of the contact binary were recently investigated by many authors (e.g., Yakut & Eggleton 2005; Stȩpień 2006; Li et al. 2007). The kinematics, age, mass, and angular momentum evolution for W UMa binaries are statistically analyzed by Bilir et al. (2005), Gazeas & Stȩpień (2008), Yilidz & Doğan (2013), and Yilidz (2014). The evolutionary state of the contact binary remains unclear because the spectra cannot be analyzed for abundance due to the extreme broadening and blending of the spectral line. The famous star AW UMa may not be a contact binary but a very tight, semi-detached one based on the sequence spectroscopy (Pribulla & Rucinski 2008), which may be a challenge for the contact binary model. However, the model still requires rigorous tests on its applicability, so observations of W UMa-type binaries would be most useful (Rucinski 2015). Although some evolutionary improvements for W UMa binaries have been made, many fundamental theoretical and observational issues still remain unresolved. Therefore, it is invaluable for us to observe deep, low mass ratio (DLMR) overcontact binaries with mass ratios of q ⩽ 0.25 and degrees of contact of f ⩾ 50% (Qian et al. 2006). This kind of binary might be close to merging and evolving into a single rapid-rotating star. A statistical study of the observed DLMR binaries can help us understand the dynamical evolution of binaries and the formation of blue straggler and FK Com-type stars (Eggleton & Kiseleva-Eggleton 2001).

3. THE INFERRED MINIMUM MASS RATIO The interest in the minimum mass ratio of the W UMa binary has attracted much theoretical attention from some investigators. Rasio (1995) first derived a minimum value of q  0.09, neglecting the spin angular momentum of the less massive component due to secular tidal instability, which was identified by Rasio & Shapiro (1995). In the evolutionary process for the DLMR overcontact binary, the system’s orbit (i.e., the separation between both components) should be shrinking, and the orbital period and mass ratio will then decrease. Therefore, we preliminarily inferred a minimum mass ratio from the relations of q, f, and Jspin Jorb .

2. CHOOSING THE SAMPLE BINARIES According to the mass ratio and degree of overcontact (i.e., q ⩽ 0.25 and f ⩾ 50%), we complied a total of 46 DLMR 1


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Table 1 Spectroscopic and Photometric Elements for 46 DLMR Overcontact Binaries Star

Type

Sp.

Period (days)

qph

qsp

K1 (km s−1)

K2 (km s−1)

r1

r2

f (%)

i (°)

T1 (K)

T2 (K)

References

QX And HV Aqr V870 Ara BO Ari AH Aur AP Aur DN Aur V410 Aur CK Boo DN Boo TZ Boo XY Boo FN Cam V776 Cas AH Cnc AW CrB YY CrB V1191Cyg V345 Gem V728 Her V857 Her FG Hya V409 Hya XY LMi V710 Mon TV Mus V2388Oph V1853Ori MW Pav IK Per KN Per V407 Peg DZ Psc EM Psc Y Sex CU Tau AW UMa HV UMa BU Vel GR Vir GSC 5909–189 NSV 13890 TYC 3836–854 TYC 4157–683 J082243+1927 J13031–0101.9

AP AT WT AT AT A A AT A A A AP AP W A AT AP WT WT WP AT AT AT AT AT AT AT W AT AT AT A A WT AT AP AT AP A AT A A A A A A

F7V L F8 L F7V A2V L G0/2V F7/8V L F/G F0V A9V F2V F5V L F8V F6V F7V F3V F6 G2V F2V L L F8V F3V L A7V A2 A5 F2 F7V L F8 G0 F2V A2V A8V F7/8V G0 L L L L L

0.4122 0.3734 0.3997 0.3182 0.4941 0.5694 0.6169 0.3664 0.3552 0.4476 0.2972 0.3706 0.6771 0.4404 0.3604 0.3609 0.3766 0.3134 0.2748 0.4713 0.3822 0.3278 0.4723 0.4369 0.4052 0.4457 0.8023 0.3830 0.7950 0.6760 0.8665 0.6369 0.3661 0.3440 0.4198 0.4125 0.4387 0.7108 0.5163 0.3278 0.4486 0.3739 0.4156 0.3961 0.2800 0.2710

0.233 0.145 0.082 0.209 0.165 0.246 0.205 0.143 0.109 0.103 0.207 0.186 0.222 0.138 0.168 0.101 0.243 0.107 0.142 0.179 0.065 0.112 0.216 0.148 0.143 0.166 0.186 0.203 0.182 0.191 0.250 0.251 0.145 0.149 0.180 0.178 0.080 0.190 0.251 0.112 0.120 0.080 0.206 0.150 0.106 0.150

0.202 L 0.082 L 0.169 L L 0.144 0.111 L 0.207 0.160 0.222 0.130 0.537 L 0.243 0.107 0.142 0.179 L 0.112 L L L 0.120 0.186 L 0.228 L L 0.253 0.136 L 0.180 L 0.099 0.190 L 0.112 L L L L L L

L L 23.3 L 47.20 L L 41.14 31.66 L 57.8 39.0 59.62 31.97 103.00 L 68.07 33.68 41.54 L L 36.0 L L L 33.2 44.62 L 52.35 L L 59.2 40.39 L 38.0 L 28.37 47.0 L 37.78 L L L L L L

L L 283.5 L 279.61 L L 291.68 285.31 L 280.02 245.0 269.01 245.31 138.00 L 279.85 315.52 291.75 L L 322.4 L L L 278.0 240.22 L 229.34 L L 234.2 297.98 L 210.0 L 286.6 254.0 L 308.81 L L L L L L

0.5366 0.5713 0.6273 0.5327 0.5685 0.5238 0.5414 0.5697 0.5874 0.6000 0.5433 0.5521 0.5569 0.5947 0.5603 0.6026 0.5374 0.5967 0.5789 0.5626 0.6316 0.5982 0.5444 0.5730 0.5803 0.5678 0.5553 0.5465 0.5511 0.5482 0.5293 0.5331 0.5789 0.5853 0.5627 0.5530 0.6192 0.5209 0.5065 0.5900 0.5869 0.6622 0.5466 0.5756 0.5848 0.5665

0.2932 0.2537 0.2379 0.2803 0.2744 0.2843 0.2765 0.2496 0.2267 0.2333 0.2810 0.2724 0.3178 0.2443 0.2658 0.2375 0.3027 0.2348 0.2629 0.2785 0.2080 0.2493 0.2899 0.2672 0.2997 0.2780 0.2781 0.2810 0.2673 0.2724 0.2974 0.3032 0.2662 0.2834 0.2830 0.2652 0.2279 0.2346 0.2999 0.2519 0.2433 0.2298 0.2845 0.2692 0.2019 0.2539

55.9 56.9 96.4 50.3 75.0 64.4 53.9 52.4 65.0 64.0 52.5 55.9 88.4 77.0 58.5 75.0 63.4 68.6 73.3 71.4 83.8 85.6 60.6 74.1 62.7 74.3 65.0 50.0 50.4 52.0 54.5 61.0 79.0 95.3 64.0 50.1 84.6 61.9 61.0 78.6 62.7 76.2 59.2 76.3 72.0 50.0

56.2 79.2 70.0 85.7 76.1 75.9 76.9 78.6 64.9 60.0 85.5 69.0 71.2 52.9 90.0 82.1 77.0 80.4 72.9 68.7 85.3 82.3 89.5 81.0 79.9 77.2 76.6 83.2 85.1 77.8 83.6 87.6 80.5 88.6 76.1 74.0 78.3 57.3 84.9 83.4 73.8 90.7 78.7 79.7 75.6 88.0

6500 6460 5860 5920 6200 9016 6830 6040 6200 6095 5890 6324 6700 6700 6300 6700 6135 6500 6115 6622 8300 5900 7000 6144 6145 5980 6900 6200 7620 9070 7650 6980 6210 5300 6210 5900 7175 7300 7500 6300 6008 6510 6200 6037 5960 L

6217 6669 6210 6055 6418 8703 6750 5915 6291 6071 5873 6307 6848 6725 6265 6808 6142 6626 6365 6794 8513 6012 6730 6093 6294 5808 6505 6261 7570 7470 7288 6484 6287 4987 6093 5938 7022 7000 7448 6163 5907 6426 6186 5888 6078 L

(1), (2) (3) (4) (5) (6), (7) (8) (9) (10), (11) (7), (12) (13) (14), (15) (10), (16) (17), (18) (17), (19) (20), (21) (22) (23), (24) (25) (26) (27) (28) (29), (30) (31) (32) (33) (34) (35), (36) (37) (38), (39) (40) (41) (42) (43) (44) (45), (46) (34) (47), (48) (49) (50) (7), (51) (52) (53) (54) (54) (55) (56)

Reference. (1) Qian et al. (2007), (2) Milone et al. (1995), (3) Li & Qian (2013), (4) Szalai et al. (2007), (5) Yang et al. (2015, in preparation), (6) Gazeas et al. (2005), (7) Rucinski & Lu (1999), (8) Li et al. (2001), (9) Goderya et al. (1996), (10) Yang et al. (2005), (11) Rucinski et al. (2003), (12) Yang et al. (2012), (13) Şenavci et al. (2008), (14) Pribulla et al. (2009a), (15) Christopoulou et al. (2011), (16) McLean & Hilditch (1983), (17) Rucinski et al. (2001), (18) Pribulla et al. (2002), (19) Zoła et al. (2004), (20) Whelan et al. (1979), (21) Qian et al. (2006), (22) Broens (2013), (23) Pribulla & Vaňko (2002), (24) Rucinksi et al. (2000), (25) Zhu et al. (2011), (26) Yang et al. (2009), (27) Nelson et al. (1995), (28) Qian et al. (2005a), (29) Lu & Rucinski (1999), (30) Qian & Yang (2005), (31) Na et al. (2014), (32) Qian et al. (2011), (33) Liu et al. (2014), (34) Qian et al. (2005b), (35) Rucinski et al. (2002), (36) Yakut et al. (2004), (37) Samec et al. (2011), (38) Lapasset (1980), (39) Rucinski & Duerbeck (2006), (40) Zhu et al. (2005), (41) Goderya et al. (1997), (42) Lee et al. (2014), (43) Yang et al. (2013), (44) Qian et al. (2008), (45) McLean & Hilditch (1983), (46) Yang & Liu (2003), (47) Yang (2008), (48) Rucinski (2015), (49) Csák et al. (2000), (50) Twigg (1979), (51) Qian & Yang (2004), (52) Wadhwa (2005), (53) Wadhwa (2006), (54) Acerbi et al. (2014), (55) Kandulapati et al. (2015), (56) Pribulla et al. (2009b).

f = 100%), the system will coalesce quickly due to dynamical instability (Rasio & Shapiro 1995). For 46 sample stars, the curve of f–q is displayed in the left panel of Figure 1, where solid and open circles represent A-subtype stars and W-subtype

3.1. The Relation Between q and f For an overcontact binary, if the common fluid surface is very close to or arrives at the outer critical Roche lobe (i.e., 2


Yang & Qian

Figure 1. Relations of q–f (left panel) and q–Jorb/Jspin (right panel) for DLMR overcontact binaries. The red solid lines are plotted using Equations (1) and (4), respectively.

ones, respectively. From this figure, there is an evident parabolic curve trend. We discarded two stars, FN Cam (Pribulla et al. 2002) and EM Psc (Qian et al. 2008), which apparently deviate from the general trend. A linear leastsquares fitting method yielded the following equation,

obtained the following equation, Jspin Jorb

- 3.7738(0.0027) ´ q + 8.2817(0.0081) ´ q 2 ,

f (%) = 117.6(7.5) - 527.6(2.5) ´ q + 1164.9(7.4) ´ q . 2

(1)

3.2. The Relation Between q and Jspin /Jorb

4. POSSIBLE EVOLUTIONARY STATES

For an overcontact binary, the total angular momentum is calculated using the following formula (Pribulla 1998),

Except for three binaries (QX And, V728 Her, and AH Cnc), the absolute parameters are directly determined for the remaining 21 of 24 DLMR binaries with spectroscopic and photometric elements. Out of 24 binaries, the masses of the primaries were estimated from the spectral types for 9 stars, and from their effective temperatures for 12 stars (Harmanec 1988). Therefore, the absolute dimensions for those 24 sample stars likely have low precision. Except for the binary J13031–0101.9 (Pribulla 2009), the absolute parameters for 45 sample stars are listed in Table 2. In the following section, we derived three relations between period, mass, radius, and luminosity.

Jtotal = G ( M1 + M2 ) a (2)

2 in which r1,2 , k1,2 are the relative radii and dimensionless gyration radii for both components, while G and a are the gravitational constant and the separation between the two components, respectively. In the brackets in Equation (2), the first term is the orbital specific angular momentum Jorb and the latter two terms are the spin specific angular momentum Jspin. So, the ratio of Jspin to Jorb can written as

Jspin Jorb

=

q é ( k1r1)2 + ( k 2 r2 )2 qùúû. 1 + q êë

(4)

which is plotted as a solid line in the right panel. When a secular tidal instability occurs at Jorb = 3Jspin (Hut 1980), another minimum mass ratio of qmin2 = 0.053(0.006) can be estimated from Equation (4). A possible truncated mass ratio for the overcontact binary may be adopted for an average value of qmin = (qmin2 + qmin2 ) 2 = 0.044(0.007). This approximates to a value of 0.044–0.047 (Yang 2010), which is smaller than the value of 0.076–0.078 (Li & Zhang 2006).

Equation (1) is plotted as a solid line in the left panel of Figure 1. The minimum mass ratio can be estimated to be qmin1 = 0.035(0.008) at f = 100%, indicating that the fluid surface of the binary system reaches the outer critical Roche surface. This will result in extreme dynamical instability, then the binary may merge into a single, rapidly rotating star.

é ( k1r1)2 + q k r 2 ùú, q ´ êê + ( 2 2) ú 1+q 1+q êë (1 + q)2 úû

= 0.5104(0.0006)

4.1. The Relation Between P and M12 The orbital periods of nine stars are larger than 0.5 days according to Table 2, which may result from observational selection bias. The periods and total masses (i.e., P versus Mtotal) of 45 sample stars are shown in Figure 2. The total mass decreases with decreasing period, which implies that the orbit is shrinking and undergoing mass loss from the central system during its evolutionary process. A linear least-squares fitting method leads to the following equation,

(3)

Then we can determine the values of Jspin Jorb for 45 DLMR binaries, which are listed in Table 2 and shown in the right panel of Figure 1. Using a least-squares fitting method, we

Mtotal = 0.0.5747(0.0160) + 2.3734(0.0331) ´ P , (5)

3


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Table 2 Absolute Estimated Parameters for DLMR Overcontact Binaries Star

M1 (M )

M2 (M)

R1 (R )

R2 (R )

L1 (L  )

L2 (L  )

Jspin Jorb

QX And HV Aqr V870 Ara BO Ari AH Aur AP Aur DN Aur V410 Aur CK Boo DN Boo TZ Boo XY Boo FN Cam V776 Cas AH Cnc AW CrB YY CrB V1191Cyg V345 Gem V728 Her V857 Her FG Hya V409 Hya XY LMi V710 Mon TV Mus V2388Oph V1853Ori MW Pav IK Per KN Per V407 Peg DZ Psc EM Psc Y Sex CU Tau AW UMa HV UMa BU Vel GR Vir GSC 5909–189 NSV 13890 TYC 3836–854 TYC 4157–683 J082243+1927

1.23 1.31 1.34 1.14 1.68 2.05 1.44 1.30 1.43 1.20 0.99 0.91 2.41 1.69 1.31 1.39 1.43 1.31 1.06 1.37 1.30 1.45 1.39 1.20 1.20 1.29 1.76 1.23 1.58 2.05 1.86 1.34 1.34 0.95 1.22 1.14 1.38 2.84 1.65 1.30 1.14 1.31 1.23 1.17 1.20

0.29 0.19 0.11 0.24 0.28 0.50 0.30 0.19 0.16 0.12 0.21 0.17 0.53 0.23 0.22 0.14 0.35 0.14 0.15 0.25 0.08 0.16 0.30 0.18 0.17 0.21 0.33 0.25 0.29 0.39 0.47 0.34 0.19 0.14 0.22 0.20 0.14 0.54 0.41 0.15 0.14 0.10 0.25 0.18 0.13

1.44 1.43 1.62 1.16 1.87 2.07 1.98 1.40 1.44 1.62 1.08 1.23 2.59 1.80 1.38 1.48 1.43 1.31 1.10 1.68 1.56 1.40 1.66 1.54 1.49 1.60 2.58 1.38 2.45 2.39 2.68 1.97 1.44 1.25 1.50 1.42 1.73 2.62 1.75 1.34 1.57 1.62 1.46 1.44 1.16

0.79 0.63 0.61 0.61 0.90 1.12 1.01 0.61 0.56 0.63 0.56 0.61 1.48 0.74 0.65 0.58 0.81 0.52 0.50 0.83 0.51 0.58 0.88 0.72 0.77 0.78 1.29 0.71 1.19 1.19 1.51 1.12 0.66 0.60 0.75 0.68 0.64 1.18 1.03 0.57 0.65 0.56 0.76 0.67 0.40

3.26 3.13 2.73 1.47 4.57 25.02 7.57 2.31 2.72 3.21 1.25 2.14 11.94 5.78 2.64 3.91 2.56 2.71 1.49 4.81 10.21 2.11 5.81 3.01 2.78 2.88 13.34 2.49 17.89 34.25 21.77 8.16 2.72 1.08 2.95 2.17 7.00 17.22 8.53 2.48 2.84 4.18 2.79 2.44 1.49

0.81 0.70 0.50 0.44 1.22 6.40 1.88 0.41 0.43 0.48 0.33 0.52 4.24 0.99 0.58 0.65 0.82 0.45 0.36 1.31 1.23 0.39 1.41 0.63 0.82 0.62 2.64 0.69 4.10 3.89 5.66 1.97 0.60 0.20 0.69 0.51 0.87 2.95 2.91 0.41 0.46 0.48 0.75 0.48 0.19

0.0815 0.1326 0.2627 0.0868 0.1185 0.0745 0.0908 0.1333 0.1784 0.1958 0.0908 0.1016 0.0915 0.1492 0.1133 0.2010 0.0796 0.1872 0.1387 0.1088 0.3291 0.1811 0.0885 0.1314 0.1397 0.1177 0.1029 0.0932 0.1028 0.0981 0.0756 0.0766 0.1364 0.1367 0.1085 0.1053 0.2157 0.0882 0.0696 0.1763 0.1641 0.2988 0.0923 0.1312 0.1807

J13031–0101.9. In both figures, the zero-age main sequence (ZAMS) and terminal-age main sequence (TAMS) lines are constructed using the binary-star evolution code (i.e., Hurley et al. 2002). Relations of M–L and M–R were derived from 43 stars, all except for V857 Her (Qian et al. 2005a) and HV UMa (Csák et al. 2000), which evidently deviate from the general trends. From the M–L diagram of Figure 4, the primary components (filled circles) lie between the ZAMS and TAMS lines, while the secondary ones (open circles) are situated above the TAMS line. This implies that the primaries may be little-evolved MS stars. Meanwhile, the secondaries may be evolved TAMS stars or He-MS stars, whose super-luminosities for their masses may be attributed to energy transfer from the primaries. A linear

where Mtotal and P are in units of M and days, respectively. The computed line from the above equation is plotted in Figure 2. After differentiating Equation (5), one may obtain the mass loss rate for the DLMR contact binary from its period decrease rate. Binary mass will be lost from the system when the period decreases. In fact, removing mass and angular momentum from the binary system is inevitable. This will cause the orbit to shrink. Finally, this kind of DLMR overcontact binary will merge into a single star. 4.2. Relations of M–L and M–R Figures 3 and 4 display the luminosities and radii versus the masses of both components for the sample stars, not including 4


Yang & Qian

least-squares fitting method yielded relations between M and L as ì log (L L ï 10 1  ï ï ï ï ï ï í ï log10 (L 2 L  ï ï ï ï ï ï î

)

= 1.1282(0.0058)

)

)

)

)

+ 3.8083(0.0035 ´ log10 (M1 M

)

= 1.0618(0.0128) + 1.7179(0.0184 ´ log10 (M2 M

(6)

which are plotted as the solid and dotted lines in Figure 3. The diagrams of M–R for both components are shown in Figure 4. The radii of the primaries lie between ZAMS and TAMS lines, which agrees with their somewhat evolved MS stars. The secondaries are located on the TAMS line, which indicates that they expand due to the additional energy from the primaries. Similar to Equation (6), we obtained the following equation,

Figure 2. q–M12 curve for 45 DLMR overcontact binaries. The solid line is computed from Equation (5).

ì log (R R ï 10 1  ï ï ï ï ï ï í ï log10 (R2 R ï ï ï ï ï ï î

)

= 0.0751(0.0014)

)

)

)

)

+ 0.9513(0.0086 ´ log10 (M1 M

)

= 0.2826(0.0035) + 0.6177(0.0050 ´ log10 (M2 M . (7)

5. SUMMARY From the previous analyses, we obtained five relative relations between geometric and physical parameters for 46 DLMR overcontact binaries. The main goal here is to study their evolutionary states and to obtain the possible minimum mass ratio for this kind of binary. From the curves of q–f and q–Jspin/ Jorb, the minimum mass ratio may arrive at qmin = 0.044(0.007). Meanwhile, we derived the relations of M–R and M–L for both components. From the relation between P and M12, the mass of the DLMR overcontact binary will escape from the central system when the orbital period decreases. When the inner and outer Roche lobes shrink, the degree of contact will become f = 100% (Rasio & Shapiro 1995), or the tidal instability occurs when the spin angular momentum is larger than the orbital one (i.e., Jspin Jorb > 1/3; Hut 1980). This kind of binary will evolve into a rapid-rotating single star. Certainly, more sample stars are necessary in order to check the statistical results for DLMR overcontact binaries in the future.

Figure 3. Relations of M–L for both components of the sample binaries. The solid and dotted symbols refer to the primary stars and the secondary ones, respectively.

Many thanks are given to the anonymous referee for constructive remarks and suggestions. This work is supported partly by the National Natural Science Foundation of China (Nos. 11473009, U1231102, and 11133007), and the Open Research Program Foundation (No. OP201110). Table 1 lists new results from BO Ari, whose observations were performed using the 1.0 m telescope at Yunnan Observatory and the 60 cm telescope at Xinglong station of NAOC in China. Figure 4. Relations of M–R for the sample stars. The symbols are the same as in Figure 3.

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