Abstract
A comprehensive photometric study and an investigation of the orbital period variation of V53 in the globular cluster M 4 are presented. The photometric study reveals that the mass ratio and the contact degree of V53 are q ∼ 0.078 and f ∼ 69%, respectively. The observed variation in the light curve can be explained by adjusting the spot parameters. V53 belongs to extreme mass ratio (q ≤ 0.25), deep contact (f ≥ 50%) binaries, and its mass ratio is close to the minimum mass ratio predicted by theoretical studies, making it a potential object for studying the evolution of binaries and the formation of blue stragglers and FK Com-type stars. The orbital period of V53 shows a long-term decrease at a rate of dp/dt = 5.89(±0.02) × 10−8 d yr−1. This secular period decrease may be caused by the combination of mass transfer from the more massive component to the less massive component and an angular momentum loss via magnetic braking. As this mass transfer and angular momentum loss continues, V53 will ultimately evolve into a single fast-rotation star. By studying the statistics of all the contact binaries in globular clusters that have been analyzed, we found a possible correlation between the contact degree and whether or not a contact binary is a blue straggler. A contact binary is likely to become a blue straggler when its fill-out factor is more than 46.25(±2.05)%. More samples should be introduced to confirm this preliminary result in the future.
1 Introduction
Eclipsing binaries were once thought to be absent in globular clusters, but this idea has been thoroughly overthrown by new discoveries in the last three decades. At present, more than 300 eclipsing binaries have been identified in globular clusters. W UMa-type contact binaries are short-period eclipsing binaries in which both component stars are in contact or overflowing their Roche lobes, and whose EW-type light-curves vary continuously and show nearly equal primary and secondary minima. The spectral types of the two components usually range from late A to K. Through photometric and spectroscopic investigations, one can determine the absolute physical parameters of such systems. Models of W UMa-type contact binaries have been analyzed by many investigators (e.g., Lucy 1968; Flannery 1976; Webbink 1977; Robertson & Eggleton 1977; Mochnacki 1981; Yakut & Eggleton 2005; Stepien 2006), but the origins, structures, and evolution states of these systems remain unclear. Extreme mass ratio, deep contact binaries are binaries with mass ratios of q ≤ 0.25 and fill-out factors of f ≥ 50%. This kind of binary has been extensively studied during the last decade (e.g. Qian et al. 2005, 2007, 2008, 2011; Samec et al. 2011, 2012, 2013; Yang et al. 2012, 2013; Yang & Qian 2015; Zhu et al. 2005, 2011). Such systems are possible progenitors of blue stragglers and FK Com-type stars (Qian et al. 2006), and studying them could help us to clarify the evolution of binaries and the formation theory of blue stragglers and FK Com-type stars.
The globular cluster M 4 (NGC 6121) is the nearest globular cluster to the Sun at 1.8 kpc (Kaluzny et al. 2013a), and is one of the most studied globular clusters. The [Fe/H] of M 4 is −1.16 (Harris 2010), and the intermediate value of its interstellar reddening was determined to be E(B − V) = 0.392 mag by Kaluzny et al. (2013b). M 4 is located near the equator and at a short distance, making it a good target for detailed investigations, particularly in searches for variable stars. According to the online catalog compiled by C. M. Clement et al.,1 111 variables have been identified in the field of M 4, of which 31 are eclipsing binaries, making M 4 a binary-rich globular cluster. Liu et al. (2011) analyzed the light curves of two contact binaries named V47 and V53 in M 4 and found that V47 is an extremely low mass ratio binary with a small degree of contact and that V53 is a high mass ratio, deep contact binary and is a blue straggler. Recently, three detached binaries (V65, V66, and V69 in the field of M 4) were analyzed by Kaluzny et al. (2013a). By combining the photometric and spectroscopic observations, they determined the masses, radii, and luminosities of their components. In addition, they derived the distance and the age of M 4 from the detached binaries.
In this paper, we present a detailed photometric study and orbital period investigation of the W UMa-type contact binary V53 in the M 4 field. This binary was identified by Kaluzny, Thompson, and Krzeminski (1997) during a photometric survey for faint variables and shows an EW-type light curve. After the discovery, many observations were carried out. Between 1995 and 2009, Kaluzny et al. (2013b) observed this system on the 2.5 m du Pont telescope. Lovisi et al. (2010) derived the radial and rotational velocities of V53 to be Vrad = 76.4 km s−1 and Irot = 150–200 km s−1, based on the spectroscopic observations of the Very Large Telescope, and they derived an effective temperature of T = 7350 K. Liu et al. (2011) obtained BVRI light curves of V53 in 2010 by using the 2.15 m Jorge Sahade telescope at the Complejo Astronómico El Leoncito Observatory. Recently, Nascimbeni et al. (2014) presented the photometric results of V53 by using the data of deep Hubble Space Telescope WFC3 images. This binary system was first analyzed by Liu et al. (2011). They determined that V53 has a mass ratio of q = 1.23 and a contact degree of f = 79.7%, and they claimed that V53 is a blue straggler. However, in the preliminary study of Kaluzny et al. (2013b), the mass ratio of V53 was estimated to be q = 0.060. The dramatic difference in results between the two papers and the fact that V53 is a blue straggler make this binary deserving of further investigation.
2 Light-curve modeling
Kaluzny et al. (2013b) obtained the B and V light curves for V53 on 51 nights between 1995 and 2009. These data can be publicly downloaded from the CASE website.2 We analyzed these light curves using the 2013 version of the Wilson–Devinney (W–D) program (Wilson & Devinney 1971; Wilson 1979, 1990, 2008, 2012; Van Hamme & Wilson 2007; Wilson et al. 2010). Because the light curves of V53 show season-to-season variations, we divided the light curves into four segments for investigation. The details of the four segments are listed in table 1, and the four sets of light curves are displayed in the right-hand panels of figures 1 to 4. As seen in these figures, each set of light curves is close in time and different from each other. In addition, a total secondary eclipse can be seen in all figures, indicating that V53 is an A-subtype W UMa system. Kaluzny et al. (2013b) measured the color index of V53 to be (B − V) = 0.594 and the interstellar reddening to be E(B − V) = 0.385, resulting in (B − V)0 = 0.209. With a metallicity of [Fe/H] = −1.16 (Harris 2010), and using the expectation that log g lies between 4.0 and 4.5, we determined the effective temperature of the primary component of V53 to lie between T1 = 7360 K and 7470 K using the code written by Worthey and Lee (2011). An average value can be determined to be T1 = 7415 ± 55 K, which is consistent with the value T1 = 7350 K obtain by Lovisi et al. (2010). Therefore, T1 = 7415 K was used during the analysis. According to von Zeipel (1924), the radiative outer envelope of a primary component is fixed (Teff > 7200 K). The bolometric albedo A1 = 1.0 and the gravity-darkening coefficient g1 = 1.0 of the primary were assumed. For a detailed treatment of the bolometric and bandpass limb-darkening coefficients of the primary, logarithmic functions were used. The relevant coefficients were taken from van Hamme (1993). These coefficients for the secondary component can be derived due to the value of the effective temperature of the secondary component, T2.
Fig. 1.
The left-hand panel shows the relation between the resulting sum Σ of weighted square deviations and q. The upper right-hand panel shows observed and theoretical light curves in B and V bands for LC1 of V53. The crosses represent the B light curve, while the open circles display the V light curve. The black dashed lines show the theoretical light curves with no spot, the solid lines refer to the theoretical light curves with a cool spot on the primary component, and the red dashed lines represent the theoretical light curves with a hot spot on the secondary component. The lower right-hand panel displays the O − C residuals (observed light curves minus theoretical light curves) from the photometric solution, the crosses refer to the residuals with no spot, the solid circles represent the residuals with a dark spot, and the open circles show the residuals with a hot spot. (Color online)
Fig. 2.
The left-hand panel shows the relation between the resulting sum Σ of weighted square deviations and q. The right-hand panel shows observed and theoretical light curves and O − C residuals from the photometric solution in B and V bands for LC2 of V53. The symbols and the lines are the same as in figure 1. (Color online)
Fig. 3.
The left-hand panel shows the relation between the resulting sum Σ of weighted square deviations and q. The right-hand panel shows observed and theoretical light curves and O − C residuals from the photometric solution in B and V bands for LC3 of V53. The symbols and the lines are the same as in figure 1. (Color online)
Fig. 4.
The left-hand panel shows the relation between the resulting sum Σ of weighted square deviations and q. The right-hand panel shows observed and theoretical light curves and O − C residuals from the photometric solution in B and V bands for LC4 of V53. The symbols and the lines are the same as in figure 1.
During our calculations, we found that the solution was always convergent at mode 3 and the effective temperature of the secondary component, T2, is less than 7200 K. The secondary was confirmed to have a convective envelope. The adjustable parameters were the following: the effective temperature of the secondary component T2, the mass ratio q(=M2/M1), the monochromatic luminosity of the primary component in the B and V bands L1, the orbital inclination i, and the dimensionless potentials of the primary component Ω1. Because the studies of Liu et al. (2011) and Kaluzny et al. (2013b) have yielded very different mass ratios for V53, and no spectroscopic mass ratio has been obtained, the q-search method was used to determine the accurate photometric elements of V53. This means that we calculated a series of models with assumed values of mass ratios from 0.05 to 10 (when the mass ratio is more than 0.1, the step size is 0.1, whereas when the mass ratio is less than 0.1, the step size is 0.01). The sums of the weighted square deviations |$\Sigma W_i(O-C)_i^2$| for all the assumed values of q of the four sets of light curves are plotted in the left-hand panels of figures 1 to 4. As seen in those four figures, the smallest values of Σ for the four sets of light curves are all obtained at q = 0.08, indicating that the true mass ratio of V53 is around 0.08. Therefore, we adopted q = 0.08 as an initial value and an adjustable parameter, then started a new solution. When the solution is converged, the final result was extracted. The photometric elements are shown in the Appendix (table 6) and the corresponding theoretical light curves are plotted in figures 1 to 4.
Because LC1, LC2, and LC3 are asymmetric, a spotted model was used to fit each light curve. It should be noticed that in the absence of e.g., Doppler tomography to pin down spot shapes and locations, any light curve variation can in principle be reproduced by a sufficient number of spots. A cool spot on the primary component and a hot spot on the secondary component were respectively introduced to determine the final solutions, which resulted in a decrease in the residuals [ΣW(O − C)2]. The parameters with spots are shown in the Appendix (table 7), and the corresponding theoretical light curves are plotted in figures 1 to 3. In these figures, the crosses represent the B light curves and the open circles display the V light curves. The black dashed lines show the theoretical light curves with no spot, the solid lines refer to the theoretical light curves with a cool spot on the primary component, and the red dashed lines represent the theoretical light curves with a hot spot on the secondary component. As is seen in the right-hand panel of figure 1, the hot-spot model cannot fit the observed light curves very well. Therefore, for LC1, a cool spot on the primary component provides a better result. For LC2 and LC3, both a cool spot on the primary component and a hot spot on the secondary component can provide very good results. LC4 is almost symmetric; no spot is needed. In order to determine the final solution, we combined and averaged all four seasons’ photometric data to 200 normal points (one point per 0.05 phases), and the corresponding light curves (LC5) are plotted in the right-hand panel of figure 5. A q-search method was also used to derive the mass ratio, and the relationship between Σ and q is shown in the left-hand panel of figure 5. LC5 is asymmetric as well, and again a spotted model was fitted to the light curve. The model fits without a spot. The photometric results without spot, with a cool spot on the primary component, and with a hot spot on the secondary component are listed in table 2. In table 2, the filling factor f is defined as f = (Ωin − Ω)/(Ωin − Ωout). It should be noticed that the errors of the parameters in table 2 (including tables 6 and 7 in the Appendix) are unrealistically small because they are formal errors and only calculated by mathematical fitting. The synthetic light curves are displayed in the right-hand panel of figure 5. Comparing the residuals [ΣW(O − C)2] of the three modes, a cool spot on the primary component has the smallest. Photometric results with a cool spot on the primary component determined by LC5 were therefore chosen as the final solution.
Fig. 5.
The left-hand panel shows the relation between the resulting sum Σ of weighted square deviations and q. The right-hand panel shows observed and theoretical light curves and O − C residuals from the photometric solution in B and V bands for LC5 of V53. The symbols and the lines are the same as in figure 1. (Color online)
Table 1.Four sets of the light curves of V53.
Light
. | Start time
. | End time
. | Number of data
. |
---|
curves
. | (JD)
. | (JD)
. | B
. | V
. |
---|
1 (LC1) | 2449869 | 2450966 | 121 | 73 |
2 (LC2) | 2452033 | 2452033 | 72 | 111 |
3 (LC3) | 2452395 | 2452767 | 89 | 485 |
4 (LC4) | 2453857 | 2455012 | 156 | 338 |
Light
. | Start time
. | End time
. | Number of data
. |
---|
curves
. | (JD)
. | (JD)
. | B
. | V
. |
---|
1 (LC1) | 2449869 | 2450966 | 121 | 73 |
2 (LC2) | 2452033 | 2452033 | 72 | 111 |
3 (LC3) | 2452395 | 2452767 | 89 | 485 |
4 (LC4) | 2453857 | 2455012 | 156 | 338 |
Table 1.Four sets of the light curves of V53.
Light
. | Start time
. | End time
. | Number of data
. |
---|
curves
. | (JD)
. | (JD)
. | B
. | V
. |
---|
1 (LC1) | 2449869 | 2450966 | 121 | 73 |
2 (LC2) | 2452033 | 2452033 | 72 | 111 |
3 (LC3) | 2452395 | 2452767 | 89 | 485 |
4 (LC4) | 2453857 | 2455012 | 156 | 338 |
Light
. | Start time
. | End time
. | Number of data
. |
---|
curves
. | (JD)
. | (JD)
. | B
. | V
. |
---|
1 (LC1) | 2449869 | 2450966 | 121 | 73 |
2 (LC2) | 2452033 | 2452033 | 72 | 111 |
3 (LC3) | 2452395 | 2452767 | 89 | 485 |
4 (LC4) | 2453857 | 2455012 | 156 | 338 |
Table 2.Photometric results of LC5 for V53 in the globular cluster M 4.
Parameters
. | No spot
. | Cool spot
. | Hot spot
. |
---|
T 2 (K) | 6813 ± 20 | 6611 ± 28 | 6791 ± 24 |
q | 0.079 ± 0.003 | 0.078 ± 0.003 | 0.079 ± 0.003 |
i | 74.5 ± 0.6 | 74.4 ± 0.6 | 74.8 ± 0.6 |
L 1/(L1 + L2)(B) | 0.9303 ± 0.0002 | 0.9397 ± 0.0002 | 0.9333 ± 0.0002 |
L 1/(L1 + L2)(V) | 0.9246 ± 0.0002 | 0.9328 ± 0.0002 | 0.9275 ± 0.0002 |
Ωin | 1.8929 | 1.8898 | 1.8906 |
Ωout | 1.8409 | 1.8384 | 1.8391 |
Ω1 = Ω2 | 1.8544 ± 0.0044 | 1.8543 ± 0.0021 | 1.8595 ± 0.0022 |
r 1 (pole) | 0.5596 ± 0.0008 | 0.5608 ± 0.0007 | 0.5578 ± 0.0007 |
r 1 (side) | 0.6367 ± 0.0014 | 0.6389 ± 0.0012 | 0.6334 ± 0.0012 |
r 1 (back) | 0.6562 ± 0.0015 | 0.6584 ± 0.0014 | 0.6522 ± 0.0014 |
r 2 (pole) | 0.1924 ± 0.0018 | 0.1905 ± 0.0009 | 0.1879 ± 0.0009 |
r 2 (side) | 0.2023 ± 0.0033 | 0.2001 ± 0.0026 | 0.1970 ± 0.0011 |
r 2 (back) | 0.2619 ± 0.0120 | 0.2559 ± 0.0066 | 0.2456 ± 0.0033 |
f | 74.1 ± 8.5% | 69.1 ± 4.1% | 60.4 ± 4.3% |
θ (radian) | − | 0.254 ± 0.022 | 1.561 ± 0.209 |
ϕ (radian) | − | 3.974 ± 0.073 | 4.339 ± 0.092 |
r (radian) | − | 0.258 ± 0.010 | 0.304 ± 0.014 |
T f(Td/T0) | − | 0.817 ± 0.018 | 1.258 ± 0.022 |
ΣW(O − C)2 | 1.29 × 10−8 | 1.13 × 10−8 | 1.18 × 10−8 |
Parameters
. | No spot
. | Cool spot
. | Hot spot
. |
---|
T 2 (K) | 6813 ± 20 | 6611 ± 28 | 6791 ± 24 |
q | 0.079 ± 0.003 | 0.078 ± 0.003 | 0.079 ± 0.003 |
i | 74.5 ± 0.6 | 74.4 ± 0.6 | 74.8 ± 0.6 |
L 1/(L1 + L2)(B) | 0.9303 ± 0.0002 | 0.9397 ± 0.0002 | 0.9333 ± 0.0002 |
L 1/(L1 + L2)(V) | 0.9246 ± 0.0002 | 0.9328 ± 0.0002 | 0.9275 ± 0.0002 |
Ωin | 1.8929 | 1.8898 | 1.8906 |
Ωout | 1.8409 | 1.8384 | 1.8391 |
Ω1 = Ω2 | 1.8544 ± 0.0044 | 1.8543 ± 0.0021 | 1.8595 ± 0.0022 |
r 1 (pole) | 0.5596 ± 0.0008 | 0.5608 ± 0.0007 | 0.5578 ± 0.0007 |
r 1 (side) | 0.6367 ± 0.0014 | 0.6389 ± 0.0012 | 0.6334 ± 0.0012 |
r 1 (back) | 0.6562 ± 0.0015 | 0.6584 ± 0.0014 | 0.6522 ± 0.0014 |
r 2 (pole) | 0.1924 ± 0.0018 | 0.1905 ± 0.0009 | 0.1879 ± 0.0009 |
r 2 (side) | 0.2023 ± 0.0033 | 0.2001 ± 0.0026 | 0.1970 ± 0.0011 |
r 2 (back) | 0.2619 ± 0.0120 | 0.2559 ± 0.0066 | 0.2456 ± 0.0033 |
f | 74.1 ± 8.5% | 69.1 ± 4.1% | 60.4 ± 4.3% |
θ (radian) | − | 0.254 ± 0.022 | 1.561 ± 0.209 |
ϕ (radian) | − | 3.974 ± 0.073 | 4.339 ± 0.092 |
r (radian) | − | 0.258 ± 0.010 | 0.304 ± 0.014 |
T f(Td/T0) | − | 0.817 ± 0.018 | 1.258 ± 0.022 |
ΣW(O − C)2 | 1.29 × 10−8 | 1.13 × 10−8 | 1.18 × 10−8 |
Table 2.Photometric results of LC5 for V53 in the globular cluster M 4.
Parameters
. | No spot
. | Cool spot
. | Hot spot
. |
---|
T 2 (K) | 6813 ± 20 | 6611 ± 28 | 6791 ± 24 |
q | 0.079 ± 0.003 | 0.078 ± 0.003 | 0.079 ± 0.003 |
i | 74.5 ± 0.6 | 74.4 ± 0.6 | 74.8 ± 0.6 |
L 1/(L1 + L2)(B) | 0.9303 ± 0.0002 | 0.9397 ± 0.0002 | 0.9333 ± 0.0002 |
L 1/(L1 + L2)(V) | 0.9246 ± 0.0002 | 0.9328 ± 0.0002 | 0.9275 ± 0.0002 |
Ωin | 1.8929 | 1.8898 | 1.8906 |
Ωout | 1.8409 | 1.8384 | 1.8391 |
Ω1 = Ω2 | 1.8544 ± 0.0044 | 1.8543 ± 0.0021 | 1.8595 ± 0.0022 |
r 1 (pole) | 0.5596 ± 0.0008 | 0.5608 ± 0.0007 | 0.5578 ± 0.0007 |
r 1 (side) | 0.6367 ± 0.0014 | 0.6389 ± 0.0012 | 0.6334 ± 0.0012 |
r 1 (back) | 0.6562 ± 0.0015 | 0.6584 ± 0.0014 | 0.6522 ± 0.0014 |
r 2 (pole) | 0.1924 ± 0.0018 | 0.1905 ± 0.0009 | 0.1879 ± 0.0009 |
r 2 (side) | 0.2023 ± 0.0033 | 0.2001 ± 0.0026 | 0.1970 ± 0.0011 |
r 2 (back) | 0.2619 ± 0.0120 | 0.2559 ± 0.0066 | 0.2456 ± 0.0033 |
f | 74.1 ± 8.5% | 69.1 ± 4.1% | 60.4 ± 4.3% |
θ (radian) | − | 0.254 ± 0.022 | 1.561 ± 0.209 |
ϕ (radian) | − | 3.974 ± 0.073 | 4.339 ± 0.092 |
r (radian) | − | 0.258 ± 0.010 | 0.304 ± 0.014 |
T f(Td/T0) | − | 0.817 ± 0.018 | 1.258 ± 0.022 |
ΣW(O − C)2 | 1.29 × 10−8 | 1.13 × 10−8 | 1.18 × 10−8 |
Parameters
. | No spot
. | Cool spot
. | Hot spot
. |
---|
T 2 (K) | 6813 ± 20 | 6611 ± 28 | 6791 ± 24 |
q | 0.079 ± 0.003 | 0.078 ± 0.003 | 0.079 ± 0.003 |
i | 74.5 ± 0.6 | 74.4 ± 0.6 | 74.8 ± 0.6 |
L 1/(L1 + L2)(B) | 0.9303 ± 0.0002 | 0.9397 ± 0.0002 | 0.9333 ± 0.0002 |
L 1/(L1 + L2)(V) | 0.9246 ± 0.0002 | 0.9328 ± 0.0002 | 0.9275 ± 0.0002 |
Ωin | 1.8929 | 1.8898 | 1.8906 |
Ωout | 1.8409 | 1.8384 | 1.8391 |
Ω1 = Ω2 | 1.8544 ± 0.0044 | 1.8543 ± 0.0021 | 1.8595 ± 0.0022 |
r 1 (pole) | 0.5596 ± 0.0008 | 0.5608 ± 0.0007 | 0.5578 ± 0.0007 |
r 1 (side) | 0.6367 ± 0.0014 | 0.6389 ± 0.0012 | 0.6334 ± 0.0012 |
r 1 (back) | 0.6562 ± 0.0015 | 0.6584 ± 0.0014 | 0.6522 ± 0.0014 |
r 2 (pole) | 0.1924 ± 0.0018 | 0.1905 ± 0.0009 | 0.1879 ± 0.0009 |
r 2 (side) | 0.2023 ± 0.0033 | 0.2001 ± 0.0026 | 0.1970 ± 0.0011 |
r 2 (back) | 0.2619 ± 0.0120 | 0.2559 ± 0.0066 | 0.2456 ± 0.0033 |
f | 74.1 ± 8.5% | 69.1 ± 4.1% | 60.4 ± 4.3% |
θ (radian) | − | 0.254 ± 0.022 | 1.561 ± 0.209 |
ϕ (radian) | − | 3.974 ± 0.073 | 4.339 ± 0.092 |
r (radian) | − | 0.258 ± 0.010 | 0.304 ± 0.014 |
T f(Td/T0) | − | 0.817 ± 0.018 | 1.258 ± 0.022 |
ΣW(O − C)2 | 1.29 × 10−8 | 1.13 × 10−8 | 1.18 × 10−8 |
3 Period variation study
V53 has been observed over a long period of time, allowing us to analyze orbital period changes. We calculated the times of minimum light of V53 using the observational data of Kaluzny et al. (
2013b) and Nascimbeni et al. (
2014) are listed in table
3. We also collected the eclipse times from the literature, and these are also listed in table
3. As the observations from Nascimbeni et al. (
2014) are provided in BJD time, we converted all other eclipse times from HJD to BJD time based on the procedure of Eastman, Siverd, and Gaudi (
2010). Using the linear ephemeris
we calculated the
O −
C values of V53, and list them in table
3. In equation (
1), the BJD
0 is the first primary eclipse in table
3 and the period is taken from Kaluzny et al. (
2013b). The
O −
C curve is displayed in the upper panel of figure
6. As is seen in this figure, the general trend of the
O −
C curve shows a downward parabolic change, and one point marked by an open circle deviates from the general trend very much. During the analysis of the period variation, this point was discarded. Using the least-squares method, we derived the equation
The quadratic term of this equation indicates that the orbital period of V53 is undergoing a secular decrease at a rate of
dp/
dt = 5.89(±0.02) × 10
−8 d yr
−1. The residuals are shown in the lower panel of figure
6 with the continuous period decrease removed.
Fig. 6.
O − C Curve of V53 in M 54. The upper panel shows the O − C curve determined by the linear ephemeris of equation (1); the point marked by an open circle is abandoned. The residuals are shown in the lower panel with the continuous period decrease removed.
Table 3.Times of minimum light of V53 in M 4.
HJD
. | BJD
. | Errors
. | Type
. | E
. | O − C
. | Residuals
. | Sources
. |
---|
2449869.58570 | 2449869.58637 | 0.00050 | s | − 0.5 | 0.00032 | 0.00017 | Kaluzny, Thompson, and Krzeminski (1997) |
2449869.73960 | 2449869.74027 | 0.00040 | p | 0 | 0.00000 | − 0.00014 | Kaluzny, Thompson, and Krzeminski (1997) |
2449870.66500 | 2449870.66567 | 0.00020 | p | 3 | 0.00005 | − 0.00009 | Kaluzny, Thompson, and Krzeminski (1997) |
2449872.82400 | 2449872.82467 | 0.00054 | p | 10 | − 0.00009 | − 0.00024 | Kaluzny et al. (2013b)* |
2449874.67533 | 2449874.67600 | 0.00047 | p | 16 | 0.00055 | 0.00039 | Kaluzny et al. (2013b)* |
2452033.66285 | 2452033.66363 | 0.00062 | s | 7015.5 | 0.00147 | − 0.00077 | Kaluzny et al. (2013b)* |
2452033.81790 | 2452033.81868 | 0.00036 | p | 7016 | 0.00229 | 0.00004 | Kaluzny et al. (2013b)* |
2452402.72290 | 2452402.72367 | 0.00028 | p | 8212 | 0.00264 | 0.00028 | Kaluzny et al. (2013b)* |
2452402.87649 | 2452402.87726 | 0.00055 | s | 8212.5 | 0.00200 | − 0.00035 | Kaluzny et al. (2013b)* |
2452763.76136 | 2452763.76212 | 0.00053 | s | 9382.5 | 0.00187 | − 0.00052 | Kaluzny et al. (2013b)* |
2452763.91612 | 2452763.91688 | 0.00042 | p | 9383 | 0.00241 | 0.00001 | Kaluzny et al. (2013b)* |
2452767.77214 | 2452767.77290 | 0.00057 | s | 9395.5 | 0.00282 | 0.00042 | Kaluzny et al. (2013b)* |
2453107.83704 | 2453107.83779 | 0.00055 | p | 10498 | 0.00301 | 0.00063 | Kaluzny et al. (2013b)* |
2453146.70225 | 2453146.70300 | 0.00058 | p | 10624 | 0.00368 | 0.00131 | Kaluzny et al. (2013b)* |
2453154.72091 | 2453154.72166 | 0.00050 | p | 10650 | 0.00267 | 0.00030 | Kaluzny et al. (2013b)* |
2453205.61426 | 2453205.61500 | 0.00050 | p | 10815 | 0.00199 | − 0.00036 | Kaluzny et al. (2013b)* |
2453859.83483 | 2453859.83556 | 0.00029 | p | 12936 | 0.00284 | 0.00073 | Kaluzny et al. (2013b)* |
2453887.59388 | 2453887.59461 | 0.00108 | p | 13026 | 0.00151 | − 0.00057 | Kaluzny et al. (2013b)* |
2453911.65251 | 2453911.65324 | 0.00040 | p | 13104 | 0.00114 | − 0.00093 | Kaluzny et al. (2013b)* |
2454624.63204 | 2454624.63277 | 0.00042 | s | 15415.5 | 0.00149 | − 0.00003 | Kaluzny et al. (2013b)* |
2454969.63135 | 2454969.63210 | 0.00029 | p | 16534 | 0.00094 | − 0.00022 | Kaluzny et al. (2013b)* |
2455352.72445 | 2455352.72521 | 0.00060 | p | 17776 | 0.00076 | 0.00007 | Liu et al. (2011) |
2455353.64981 | 2455353.65057 | 0.00069 | p | 17779 | 0.00078 | 0.00008 | Liu et al. (2011) |
2455354.57463 | 2455354.57539 | 0.00059 | p | 17782 | 0.00025 | − 0.00043 | Liu et al. (2011) |
2455354.73894 | 2455354.73970 | 0.00098 | s | 17782.5 | 0.01034 | — | Liu et al. (2011) |
2455355.65446 | 2455355.65522 | 0.00064 | s | 17785.5 | 0.00051 | − 0.00017 | Liu et al. (2011) |
| 2456209.74945 | 0.00088 | s | 20554.5 | 0.00027 | 0.00091 | Nascimbeni et al. (2014)* |
| 2456384.33024 | 0.00101 | s | 21120.5 | − 0.00091 | 0.00006 | Nascimbeni et al. (2014)* |
| 2456455.11880 | 0.00155 | p | 21350 | − 0.00132 | − 0.00021 | Nascimbeni et al. (2014)* |
| 2456455.89103 | 0.00087 | s | 21352.5 | − 0.00022 | 0.00088 | Nascimbeni et al. (2014)* |
| 2456479.02311 | 0.00147 | p | 21427.5 | − 0.00179 | − 0.00064 | Nascimbeni et al. (2014)* |
| 2456551.04595 | 0.00140 | p | 21661 | − 0.00172 | − 0.00043 | Nascimbeni et al. (2014)* |
HJD
. | BJD
. | Errors
. | Type
. | E
. | O − C
. | Residuals
. | Sources
. |
---|
2449869.58570 | 2449869.58637 | 0.00050 | s | − 0.5 | 0.00032 | 0.00017 | Kaluzny, Thompson, and Krzeminski (1997) |
2449869.73960 | 2449869.74027 | 0.00040 | p | 0 | 0.00000 | − 0.00014 | Kaluzny, Thompson, and Krzeminski (1997) |
2449870.66500 | 2449870.66567 | 0.00020 | p | 3 | 0.00005 | − 0.00009 | Kaluzny, Thompson, and Krzeminski (1997) |
2449872.82400 | 2449872.82467 | 0.00054 | p | 10 | − 0.00009 | − 0.00024 | Kaluzny et al. (2013b)* |
2449874.67533 | 2449874.67600 | 0.00047 | p | 16 | 0.00055 | 0.00039 | Kaluzny et al. (2013b)* |
2452033.66285 | 2452033.66363 | 0.00062 | s | 7015.5 | 0.00147 | − 0.00077 | Kaluzny et al. (2013b)* |
2452033.81790 | 2452033.81868 | 0.00036 | p | 7016 | 0.00229 | 0.00004 | Kaluzny et al. (2013b)* |
2452402.72290 | 2452402.72367 | 0.00028 | p | 8212 | 0.00264 | 0.00028 | Kaluzny et al. (2013b)* |
2452402.87649 | 2452402.87726 | 0.00055 | s | 8212.5 | 0.00200 | − 0.00035 | Kaluzny et al. (2013b)* |
2452763.76136 | 2452763.76212 | 0.00053 | s | 9382.5 | 0.00187 | − 0.00052 | Kaluzny et al. (2013b)* |
2452763.91612 | 2452763.91688 | 0.00042 | p | 9383 | 0.00241 | 0.00001 | Kaluzny et al. (2013b)* |
2452767.77214 | 2452767.77290 | 0.00057 | s | 9395.5 | 0.00282 | 0.00042 | Kaluzny et al. (2013b)* |
2453107.83704 | 2453107.83779 | 0.00055 | p | 10498 | 0.00301 | 0.00063 | Kaluzny et al. (2013b)* |
2453146.70225 | 2453146.70300 | 0.00058 | p | 10624 | 0.00368 | 0.00131 | Kaluzny et al. (2013b)* |
2453154.72091 | 2453154.72166 | 0.00050 | p | 10650 | 0.00267 | 0.00030 | Kaluzny et al. (2013b)* |
2453205.61426 | 2453205.61500 | 0.00050 | p | 10815 | 0.00199 | − 0.00036 | Kaluzny et al. (2013b)* |
2453859.83483 | 2453859.83556 | 0.00029 | p | 12936 | 0.00284 | 0.00073 | Kaluzny et al. (2013b)* |
2453887.59388 | 2453887.59461 | 0.00108 | p | 13026 | 0.00151 | − 0.00057 | Kaluzny et al. (2013b)* |
2453911.65251 | 2453911.65324 | 0.00040 | p | 13104 | 0.00114 | − 0.00093 | Kaluzny et al. (2013b)* |
2454624.63204 | 2454624.63277 | 0.00042 | s | 15415.5 | 0.00149 | − 0.00003 | Kaluzny et al. (2013b)* |
2454969.63135 | 2454969.63210 | 0.00029 | p | 16534 | 0.00094 | − 0.00022 | Kaluzny et al. (2013b)* |
2455352.72445 | 2455352.72521 | 0.00060 | p | 17776 | 0.00076 | 0.00007 | Liu et al. (2011) |
2455353.64981 | 2455353.65057 | 0.00069 | p | 17779 | 0.00078 | 0.00008 | Liu et al. (2011) |
2455354.57463 | 2455354.57539 | 0.00059 | p | 17782 | 0.00025 | − 0.00043 | Liu et al. (2011) |
2455354.73894 | 2455354.73970 | 0.00098 | s | 17782.5 | 0.01034 | — | Liu et al. (2011) |
2455355.65446 | 2455355.65522 | 0.00064 | s | 17785.5 | 0.00051 | − 0.00017 | Liu et al. (2011) |
| 2456209.74945 | 0.00088 | s | 20554.5 | 0.00027 | 0.00091 | Nascimbeni et al. (2014)* |
| 2456384.33024 | 0.00101 | s | 21120.5 | − 0.00091 | 0.00006 | Nascimbeni et al. (2014)* |
| 2456455.11880 | 0.00155 | p | 21350 | − 0.00132 | − 0.00021 | Nascimbeni et al. (2014)* |
| 2456455.89103 | 0.00087 | s | 21352.5 | − 0.00022 | 0.00088 | Nascimbeni et al. (2014)* |
| 2456479.02311 | 0.00147 | p | 21427.5 | − 0.00179 | − 0.00064 | Nascimbeni et al. (2014)* |
| 2456551.04595 | 0.00140 | p | 21661 | − 0.00172 | − 0.00043 | Nascimbeni et al. (2014)* |
Table 3.Times of minimum light of V53 in M 4.
HJD
. | BJD
. | Errors
. | Type
. | E
. | O − C
. | Residuals
. | Sources
. |
---|
2449869.58570 | 2449869.58637 | 0.00050 | s | − 0.5 | 0.00032 | 0.00017 | Kaluzny, Thompson, and Krzeminski (1997) |
2449869.73960 | 2449869.74027 | 0.00040 | p | 0 | 0.00000 | − 0.00014 | Kaluzny, Thompson, and Krzeminski (1997) |
2449870.66500 | 2449870.66567 | 0.00020 | p | 3 | 0.00005 | − 0.00009 | Kaluzny, Thompson, and Krzeminski (1997) |
2449872.82400 | 2449872.82467 | 0.00054 | p | 10 | − 0.00009 | − 0.00024 | Kaluzny et al. (2013b)* |
2449874.67533 | 2449874.67600 | 0.00047 | p | 16 | 0.00055 | 0.00039 | Kaluzny et al. (2013b)* |
2452033.66285 | 2452033.66363 | 0.00062 | s | 7015.5 | 0.00147 | − 0.00077 | Kaluzny et al. (2013b)* |
2452033.81790 | 2452033.81868 | 0.00036 | p | 7016 | 0.00229 | 0.00004 | Kaluzny et al. (2013b)* |
2452402.72290 | 2452402.72367 | 0.00028 | p | 8212 | 0.00264 | 0.00028 | Kaluzny et al. (2013b)* |
2452402.87649 | 2452402.87726 | 0.00055 | s | 8212.5 | 0.00200 | − 0.00035 | Kaluzny et al. (2013b)* |
2452763.76136 | 2452763.76212 | 0.00053 | s | 9382.5 | 0.00187 | − 0.00052 | Kaluzny et al. (2013b)* |
2452763.91612 | 2452763.91688 | 0.00042 | p | 9383 | 0.00241 | 0.00001 | Kaluzny et al. (2013b)* |
2452767.77214 | 2452767.77290 | 0.00057 | s | 9395.5 | 0.00282 | 0.00042 | Kaluzny et al. (2013b)* |
2453107.83704 | 2453107.83779 | 0.00055 | p | 10498 | 0.00301 | 0.00063 | Kaluzny et al. (2013b)* |
2453146.70225 | 2453146.70300 | 0.00058 | p | 10624 | 0.00368 | 0.00131 | Kaluzny et al. (2013b)* |
2453154.72091 | 2453154.72166 | 0.00050 | p | 10650 | 0.00267 | 0.00030 | Kaluzny et al. (2013b)* |
2453205.61426 | 2453205.61500 | 0.00050 | p | 10815 | 0.00199 | − 0.00036 | Kaluzny et al. (2013b)* |
2453859.83483 | 2453859.83556 | 0.00029 | p | 12936 | 0.00284 | 0.00073 | Kaluzny et al. (2013b)* |
2453887.59388 | 2453887.59461 | 0.00108 | p | 13026 | 0.00151 | − 0.00057 | Kaluzny et al. (2013b)* |
2453911.65251 | 2453911.65324 | 0.00040 | p | 13104 | 0.00114 | − 0.00093 | Kaluzny et al. (2013b)* |
2454624.63204 | 2454624.63277 | 0.00042 | s | 15415.5 | 0.00149 | − 0.00003 | Kaluzny et al. (2013b)* |
2454969.63135 | 2454969.63210 | 0.00029 | p | 16534 | 0.00094 | − 0.00022 | Kaluzny et al. (2013b)* |
2455352.72445 | 2455352.72521 | 0.00060 | p | 17776 | 0.00076 | 0.00007 | Liu et al. (2011) |
2455353.64981 | 2455353.65057 | 0.00069 | p | 17779 | 0.00078 | 0.00008 | Liu et al. (2011) |
2455354.57463 | 2455354.57539 | 0.00059 | p | 17782 | 0.00025 | − 0.00043 | Liu et al. (2011) |
2455354.73894 | 2455354.73970 | 0.00098 | s | 17782.5 | 0.01034 | — | Liu et al. (2011) |
2455355.65446 | 2455355.65522 | 0.00064 | s | 17785.5 | 0.00051 | − 0.00017 | Liu et al. (2011) |
| 2456209.74945 | 0.00088 | s | 20554.5 | 0.00027 | 0.00091 | Nascimbeni et al. (2014)* |
| 2456384.33024 | 0.00101 | s | 21120.5 | − 0.00091 | 0.00006 | Nascimbeni et al. (2014)* |
| 2456455.11880 | 0.00155 | p | 21350 | − 0.00132 | − 0.00021 | Nascimbeni et al. (2014)* |
| 2456455.89103 | 0.00087 | s | 21352.5 | − 0.00022 | 0.00088 | Nascimbeni et al. (2014)* |
| 2456479.02311 | 0.00147 | p | 21427.5 | − 0.00179 | − 0.00064 | Nascimbeni et al. (2014)* |
| 2456551.04595 | 0.00140 | p | 21661 | − 0.00172 | − 0.00043 | Nascimbeni et al. (2014)* |
HJD
. | BJD
. | Errors
. | Type
. | E
. | O − C
. | Residuals
. | Sources
. |
---|
2449869.58570 | 2449869.58637 | 0.00050 | s | − 0.5 | 0.00032 | 0.00017 | Kaluzny, Thompson, and Krzeminski (1997) |
2449869.73960 | 2449869.74027 | 0.00040 | p | 0 | 0.00000 | − 0.00014 | Kaluzny, Thompson, and Krzeminski (1997) |
2449870.66500 | 2449870.66567 | 0.00020 | p | 3 | 0.00005 | − 0.00009 | Kaluzny, Thompson, and Krzeminski (1997) |
2449872.82400 | 2449872.82467 | 0.00054 | p | 10 | − 0.00009 | − 0.00024 | Kaluzny et al. (2013b)* |
2449874.67533 | 2449874.67600 | 0.00047 | p | 16 | 0.00055 | 0.00039 | Kaluzny et al. (2013b)* |
2452033.66285 | 2452033.66363 | 0.00062 | s | 7015.5 | 0.00147 | − 0.00077 | Kaluzny et al. (2013b)* |
2452033.81790 | 2452033.81868 | 0.00036 | p | 7016 | 0.00229 | 0.00004 | Kaluzny et al. (2013b)* |
2452402.72290 | 2452402.72367 | 0.00028 | p | 8212 | 0.00264 | 0.00028 | Kaluzny et al. (2013b)* |
2452402.87649 | 2452402.87726 | 0.00055 | s | 8212.5 | 0.00200 | − 0.00035 | Kaluzny et al. (2013b)* |
2452763.76136 | 2452763.76212 | 0.00053 | s | 9382.5 | 0.00187 | − 0.00052 | Kaluzny et al. (2013b)* |
2452763.91612 | 2452763.91688 | 0.00042 | p | 9383 | 0.00241 | 0.00001 | Kaluzny et al. (2013b)* |
2452767.77214 | 2452767.77290 | 0.00057 | s | 9395.5 | 0.00282 | 0.00042 | Kaluzny et al. (2013b)* |
2453107.83704 | 2453107.83779 | 0.00055 | p | 10498 | 0.00301 | 0.00063 | Kaluzny et al. (2013b)* |
2453146.70225 | 2453146.70300 | 0.00058 | p | 10624 | 0.00368 | 0.00131 | Kaluzny et al. (2013b)* |
2453154.72091 | 2453154.72166 | 0.00050 | p | 10650 | 0.00267 | 0.00030 | Kaluzny et al. (2013b)* |
2453205.61426 | 2453205.61500 | 0.00050 | p | 10815 | 0.00199 | − 0.00036 | Kaluzny et al. (2013b)* |
2453859.83483 | 2453859.83556 | 0.00029 | p | 12936 | 0.00284 | 0.00073 | Kaluzny et al. (2013b)* |
2453887.59388 | 2453887.59461 | 0.00108 | p | 13026 | 0.00151 | − 0.00057 | Kaluzny et al. (2013b)* |
2453911.65251 | 2453911.65324 | 0.00040 | p | 13104 | 0.00114 | − 0.00093 | Kaluzny et al. (2013b)* |
2454624.63204 | 2454624.63277 | 0.00042 | s | 15415.5 | 0.00149 | − 0.00003 | Kaluzny et al. (2013b)* |
2454969.63135 | 2454969.63210 | 0.00029 | p | 16534 | 0.00094 | − 0.00022 | Kaluzny et al. (2013b)* |
2455352.72445 | 2455352.72521 | 0.00060 | p | 17776 | 0.00076 | 0.00007 | Liu et al. (2011) |
2455353.64981 | 2455353.65057 | 0.00069 | p | 17779 | 0.00078 | 0.00008 | Liu et al. (2011) |
2455354.57463 | 2455354.57539 | 0.00059 | p | 17782 | 0.00025 | − 0.00043 | Liu et al. (2011) |
2455354.73894 | 2455354.73970 | 0.00098 | s | 17782.5 | 0.01034 | — | Liu et al. (2011) |
2455355.65446 | 2455355.65522 | 0.00064 | s | 17785.5 | 0.00051 | − 0.00017 | Liu et al. (2011) |
| 2456209.74945 | 0.00088 | s | 20554.5 | 0.00027 | 0.00091 | Nascimbeni et al. (2014)* |
| 2456384.33024 | 0.00101 | s | 21120.5 | − 0.00091 | 0.00006 | Nascimbeni et al. (2014)* |
| 2456455.11880 | 0.00155 | p | 21350 | − 0.00132 | − 0.00021 | Nascimbeni et al. (2014)* |
| 2456455.89103 | 0.00087 | s | 21352.5 | − 0.00022 | 0.00088 | Nascimbeni et al. (2014)* |
| 2456479.02311 | 0.00147 | p | 21427.5 | − 0.00179 | − 0.00064 | Nascimbeni et al. (2014)* |
| 2456551.04595 | 0.00140 | p | 21661 | − 0.00172 | − 0.00043 | Nascimbeni et al. (2014)* |
4 Discussions and conclusions
Based on the investigation of B and V light curves, we deduced that V53 in the globular cluster M 4 is an extremely low mass ratio (q ∼ 0.078), deep contact (f ∼ 69%) binary, indicating that it is at the final evolutionary stage of short-period binaries. Changes in the shape of the light curves can be explained by variations of the spot parameters. Our derived photometric results for V53 are very different from those determined by Liu et al. (2011). Kaluzny et al. (2013b) only determined a preliminary mass ratio, without showing any other parameters. We believe that our results are more reliable because: (1) the light curves that we used are more accurate; (2) the inclination we determined is i ∼ 74°, which is much larger than that determined by Liu et al. (2011) (i ∼ 39°), suggesting more reliable results; (3) the light curves that we used show a total eclipse at the secondary minimum, so the mass ratio should be less than 1.0; and (4) as is seen in the left-hand panels of figures 1 to 5, the q-search results are highly consistent with each other, and the minimum residuals are all at q = 0.08. In order to verify our results, we used the q-search method to analyze the photometric data of Liu et al. (2011). A series of values from 0.04 to 10 were applied to determine the real mass ratio. The sums of the weighted square deviations |$\Sigma W_i(O-C)_i^2$| versus mass ratio q is displayed in figure 7. The smallest value of Σ was found at q ∼ 0.08, which is in accordance with our result.
Fig. 7.
Relation between the sums of the weighted square deviations |$\Sigma W_{\,i}(O-C)_i^2$| and the mass ratio q based on the photometric data of Liu et al. (2011).
Using the apparent distance modulus in the V band of M 4, (m − M)v = 12.83 (Gratton et al. 2010), we estimated the absolute physical parameters for V53 using the same method as Liu et al. (2011). From Kaluzny et al. (2013b), the maximum V-band visual magnitude of V53 is mv = 15.757 ± 0.004 mag, then the V-band absolute magnitude of V53 can be determined to be Mv = mv − (m − M)v = 2.927 ± 0.004 mag. Using the code by Worthey and Lee (2011), we calculated the bolometric correction to lie between BCV = −0.041 mag and −0.062 mag, corresponding to values of log g = 4.0 and 4.5. Adopting an average value of BCV = −0.052 ± 0.011 mag, the absolute bolometric magnitude of V53, Mbol = 2.875 ± 0.012 mag was obtained. Then, we used the light-curve program of the W–D program to determine the relation between the semimajor axis and the combined absolute bolometric magnitude of the two components of V53, and the corresponding relation curve is shown in figure 8. The step size of the semimajor axis is 0.5 from 1.0 to 6.0, and the step size is 0.01 between 2.0 and 2.5. As one can see in this figure, when the semimajor axis is 2.24 R⊙, the combined absolute bolometric magnitude of the two components is Mbol = 2.880 mag, which is in agreement with the observed value within errors. The absolute physical parameters of V53 we calculated to be the following: |$a=2.24_{-0.01}^{+0.02}\,R_{\odot }$|, |$M_1=1.472_{-0.020}^{+0.039}\,M_{\odot }$|, |$M_2=0.115_{-0.001}^{+0.003}\,M_{\odot }$|, |$R_1=1.383_{-0.007}^{+0.012}\,R_{\odot }$|, |$R_2=0.481_{-0.001}^{+0.004}\,R_{\odot }$|, |$L_1=7.306_{-0.063}^{+0.129}\,L_\odot$|, and |$L_2=0.465_{-0.004}^{+0.008}\,L_\odot$|.
Fig. 8.
Relation between the semi-major axis and the combined absolute bolometric magnitude of the two components of V53.
Using all the available times of minimum light, we analyzed the orbital period changes. It is found that the orbital period of V53 is undergoing a secular decrease at a rate of
dp/
dt = 5.89(±0.02) × 10
−8 d yr
−1. The rate of the period decrease is very small, so it has a very long associated timescale, greater than the thermal timescale but less than the nuclear timescale. This long-term period decrease may be caused by the mass transfer from the primary to the secondary as well as an angular momentum loss via magnetic braking. Considering a conservative mass transfer from the more massive primary to the less massive secondary, and using the following equation taken from Kwee (
1958),
we can determine that the mass transfer rate is
dM1/
dt = 7.94(±0.03) × 10
−9 M⊙ yr
−1. The timescale of this conservative mass transfer is τ ∼
M1/
|$\dot{M_1}$| ∼ 1.85 × 10
8 yr, which is much larger than the thermal timescale of the primary component (τ
th ∼ 2 × 10
7M2/
RL ∼ 4.29 × 10
6 yr). This indicates that the mass transfer from the primary to the secondary is insufficient to provide the observed orbital period decrease and angular momentum loss via magnetic braking contributes an integral part. Recently, a statistic study by Liao and Qian (
2010) concluded that most W UMa binaries show cyclic changes in the
O −
C diagram. Therefore, the possibility that the secular period decrease is only a portion of a very long period (longer than the span of the data, 18.3 yr) cyclic oscillation which may be caused by the light-time effect due to a third body or the Applegate mechanism (Applegate
1992) cannot be excluded.
The cut-off mass ratio of contact binaries has been previously analyzed several times (Rasio 1995; Li & Zhang 2006; Arbutina 2007, 2009; Jiang et al. 2010). Rasio (1995) derived the minimum mass ratio of contact binaries to be about qmin ∼ 0.09 by neglecting the rotation of the secondary components. On the contrary, Li and Zhang (2006) determined a lower value of qmin ∼ 0.076. By assuming a radiative main-sequence primary and a fully convective secondary, the theoretical minimum mass ratio can be found to be in the range from 0.094 to 0.109 (Arbutina 2007), and when considering the differential rotation of the primary component, qmin decreases to 0.070–0.074 (Arbutina 2009). Jiang et al. (2010) suggested that the minimum ratio can be lowered to 0.05 based on the mass and structure of the primary. The mass ratio of V53 is determined as q ∼ 0.078, which is very close to the cut-off mass ratio of contact binaries. The statistics of all contact binaries that have mass ratios of less than 0.09 are shown in table 4. These systems are very important because their mass ratios are around the cut-off mass ratio predicted by theoretical analysis. Such systems are very good targets to challenge and constrain theoretical evolutionary models of binary mergers.
Table 4.Contact binaries with mass ratios less than 0.09.
. | Period
. |
. |
. | Parameters
. |
. |
. | dP/dt
. | Reference
. |
---|
. |
. | T1
. | T2
. | i
. | q
. | f
. |
. |
. |
---|
. | (d)
. | (K)
. | (K)
. | (°)
. |
. |
. | (d yr−1)
. |
. |
---|
V857 Her | 0.382230 | 8300 | 8513 | 85.4 | 0.065 | 83.8% | +2.90 × 10−7 | Qian et al. (2005) |
SX Crv | 0.316599 | 6340 | 6160 | 61.2 | 0.072 | 27.0% | − 1.05 × 10−6 | Zola et al. (2004) |
V870 Ara | 0.399722 | 5860 | 6210 | 70.0 | 0.082 | 96.4% | — | Szalai et al. (2007) |
ASAS J083241+2332.4 | 0.311321 | 6300 | 6672 | 82.7 | 0.068 | 69.2% | +8.80 × 10−7 | Sriram et al. (2016) |
NSV 13890 | 0.373880 | 6510 | 6426 | 76.2 | 0.080 | 90.0% | — | Wadhwa (2006) |
V53 | 0.308449 | 7415 | 6611 | 74.4 | 0.078 | 69.1% | − 5.89 × 10−8 | This paper |
. | Period
. |
. |
. | Parameters
. |
. |
. | dP/dt
. | Reference
. |
---|
. |
. | T1
. | T2
. | i
. | q
. | f
. |
. |
. |
---|
. | (d)
. | (K)
. | (K)
. | (°)
. |
. |
. | (d yr−1)
. |
. |
---|
V857 Her | 0.382230 | 8300 | 8513 | 85.4 | 0.065 | 83.8% | +2.90 × 10−7 | Qian et al. (2005) |
SX Crv | 0.316599 | 6340 | 6160 | 61.2 | 0.072 | 27.0% | − 1.05 × 10−6 | Zola et al. (2004) |
V870 Ara | 0.399722 | 5860 | 6210 | 70.0 | 0.082 | 96.4% | — | Szalai et al. (2007) |
ASAS J083241+2332.4 | 0.311321 | 6300 | 6672 | 82.7 | 0.068 | 69.2% | +8.80 × 10−7 | Sriram et al. (2016) |
NSV 13890 | 0.373880 | 6510 | 6426 | 76.2 | 0.080 | 90.0% | — | Wadhwa (2006) |
V53 | 0.308449 | 7415 | 6611 | 74.4 | 0.078 | 69.1% | − 5.89 × 10−8 | This paper |
Table 4.Contact binaries with mass ratios less than 0.09.
. | Period
. |
. |
. | Parameters
. |
. |
. | dP/dt
. | Reference
. |
---|
. |
. | T1
. | T2
. | i
. | q
. | f
. |
. |
. |
---|
. | (d)
. | (K)
. | (K)
. | (°)
. |
. |
. | (d yr−1)
. |
. |
---|
V857 Her | 0.382230 | 8300 | 8513 | 85.4 | 0.065 | 83.8% | +2.90 × 10−7 | Qian et al. (2005) |
SX Crv | 0.316599 | 6340 | 6160 | 61.2 | 0.072 | 27.0% | − 1.05 × 10−6 | Zola et al. (2004) |
V870 Ara | 0.399722 | 5860 | 6210 | 70.0 | 0.082 | 96.4% | — | Szalai et al. (2007) |
ASAS J083241+2332.4 | 0.311321 | 6300 | 6672 | 82.7 | 0.068 | 69.2% | +8.80 × 10−7 | Sriram et al. (2016) |
NSV 13890 | 0.373880 | 6510 | 6426 | 76.2 | 0.080 | 90.0% | — | Wadhwa (2006) |
V53 | 0.308449 | 7415 | 6611 | 74.4 | 0.078 | 69.1% | − 5.89 × 10−8 | This paper |
. | Period
. |
. |
. | Parameters
. |
. |
. | dP/dt
. | Reference
. |
---|
. |
. | T1
. | T2
. | i
. | q
. | f
. |
. |
. |
---|
. | (d)
. | (K)
. | (K)
. | (°)
. |
. |
. | (d yr−1)
. |
. |
---|
V857 Her | 0.382230 | 8300 | 8513 | 85.4 | 0.065 | 83.8% | +2.90 × 10−7 | Qian et al. (2005) |
SX Crv | 0.316599 | 6340 | 6160 | 61.2 | 0.072 | 27.0% | − 1.05 × 10−6 | Zola et al. (2004) |
V870 Ara | 0.399722 | 5860 | 6210 | 70.0 | 0.082 | 96.4% | — | Szalai et al. (2007) |
ASAS J083241+2332.4 | 0.311321 | 6300 | 6672 | 82.7 | 0.068 | 69.2% | +8.80 × 10−7 | Sriram et al. (2016) |
NSV 13890 | 0.373880 | 6510 | 6426 | 76.2 | 0.080 | 90.0% | — | Wadhwa (2006) |
V53 | 0.308449 | 7415 | 6611 | 74.4 | 0.078 | 69.1% | − 5.89 × 10−8 | This paper |
Table 5.Contact binaries in globular clusters.
Name
. | Cluster
. | Period
. | T1
. | q
. | f
. | Blue
. | Reference
. |
---|
. |
. | (d)
. | (K)
. |
. |
. | straggler
. |
. |
---|
NH19 | NGC 5466 | 0.34214 | 8200 | 0.495 | 94.7% | Yes | Kallrath, Milone, and Stagg (1992) |
NH30 | NGC 5466 | 0.29754 | 8130 | 0.366 | 92.0% | Yes | Kallrath, Milone, and Stagg (1992) |
V1 | M 71 | 0.34890 | 5460 | 0.330 | 14.8% | No | McVean et al. (1997) |
V2 | M 71 | 0.36719 | 6050 | 0.380 | 9.0% | No | McVean et al. (1997) |
V5 | M 71 | 0.40450 | 5900 | 0.371 | 27.0% | No | McVean et al. (1997) |
V7 | NGC 6397 | 0.26986 | 6077 | 2.690 | 2.5% | No | Li & Qian (2013b) |
V8 | NGC 6397 | 0.27124 | 6928 | 0.159 | 46.1% | Yes | Li & Qian (2013b) |
V134 | M 54 | 0.90952 | 6802 | 0.530 | 50.6% | Yes | Li & Qian (2013a) |
V144 | M 54 | 0.72159 | 7120 | 0.160 | 19.5% | Yes | Li & Qian (2013a) |
V47 | M 4 | 0.26997 | 6238 | 0.123 | 32.0% | No | Liu et al. (2011) |
V53 | M 4 | 0.30845 | 7415 | 0.078 | 69.1% | Yes | This paper |
Name
. | Cluster
. | Period
. | T1
. | q
. | f
. | Blue
. | Reference
. |
---|
. |
. | (d)
. | (K)
. |
. |
. | straggler
. |
. |
---|
NH19 | NGC 5466 | 0.34214 | 8200 | 0.495 | 94.7% | Yes | Kallrath, Milone, and Stagg (1992) |
NH30 | NGC 5466 | 0.29754 | 8130 | 0.366 | 92.0% | Yes | Kallrath, Milone, and Stagg (1992) |
V1 | M 71 | 0.34890 | 5460 | 0.330 | 14.8% | No | McVean et al. (1997) |
V2 | M 71 | 0.36719 | 6050 | 0.380 | 9.0% | No | McVean et al. (1997) |
V5 | M 71 | 0.40450 | 5900 | 0.371 | 27.0% | No | McVean et al. (1997) |
V7 | NGC 6397 | 0.26986 | 6077 | 2.690 | 2.5% | No | Li & Qian (2013b) |
V8 | NGC 6397 | 0.27124 | 6928 | 0.159 | 46.1% | Yes | Li & Qian (2013b) |
V134 | M 54 | 0.90952 | 6802 | 0.530 | 50.6% | Yes | Li & Qian (2013a) |
V144 | M 54 | 0.72159 | 7120 | 0.160 | 19.5% | Yes | Li & Qian (2013a) |
V47 | M 4 | 0.26997 | 6238 | 0.123 | 32.0% | No | Liu et al. (2011) |
V53 | M 4 | 0.30845 | 7415 | 0.078 | 69.1% | Yes | This paper |
Table 5.Contact binaries in globular clusters.
Name
. | Cluster
. | Period
. | T1
. | q
. | f
. | Blue
. | Reference
. |
---|
. |
. | (d)
. | (K)
. |
. |
. | straggler
. |
. |
---|
NH19 | NGC 5466 | 0.34214 | 8200 | 0.495 | 94.7% | Yes | Kallrath, Milone, and Stagg (1992) |
NH30 | NGC 5466 | 0.29754 | 8130 | 0.366 | 92.0% | Yes | Kallrath, Milone, and Stagg (1992) |
V1 | M 71 | 0.34890 | 5460 | 0.330 | 14.8% | No | McVean et al. (1997) |
V2 | M 71 | 0.36719 | 6050 | 0.380 | 9.0% | No | McVean et al. (1997) |
V5 | M 71 | 0.40450 | 5900 | 0.371 | 27.0% | No | McVean et al. (1997) |
V7 | NGC 6397 | 0.26986 | 6077 | 2.690 | 2.5% | No | Li & Qian (2013b) |
V8 | NGC 6397 | 0.27124 | 6928 | 0.159 | 46.1% | Yes | Li & Qian (2013b) |
V134 | M 54 | 0.90952 | 6802 | 0.530 | 50.6% | Yes | Li & Qian (2013a) |
V144 | M 54 | 0.72159 | 7120 | 0.160 | 19.5% | Yes | Li & Qian (2013a) |
V47 | M 4 | 0.26997 | 6238 | 0.123 | 32.0% | No | Liu et al. (2011) |
V53 | M 4 | 0.30845 | 7415 | 0.078 | 69.1% | Yes | This paper |
Name
. | Cluster
. | Period
. | T1
. | q
. | f
. | Blue
. | Reference
. |
---|
. |
. | (d)
. | (K)
. |
. |
. | straggler
. |
. |
---|
NH19 | NGC 5466 | 0.34214 | 8200 | 0.495 | 94.7% | Yes | Kallrath, Milone, and Stagg (1992) |
NH30 | NGC 5466 | 0.29754 | 8130 | 0.366 | 92.0% | Yes | Kallrath, Milone, and Stagg (1992) |
V1 | M 71 | 0.34890 | 5460 | 0.330 | 14.8% | No | McVean et al. (1997) |
V2 | M 71 | 0.36719 | 6050 | 0.380 | 9.0% | No | McVean et al. (1997) |
V5 | M 71 | 0.40450 | 5900 | 0.371 | 27.0% | No | McVean et al. (1997) |
V7 | NGC 6397 | 0.26986 | 6077 | 2.690 | 2.5% | No | Li & Qian (2013b) |
V8 | NGC 6397 | 0.27124 | 6928 | 0.159 | 46.1% | Yes | Li & Qian (2013b) |
V134 | M 54 | 0.90952 | 6802 | 0.530 | 50.6% | Yes | Li & Qian (2013a) |
V144 | M 54 | 0.72159 | 7120 | 0.160 | 19.5% | Yes | Li & Qian (2013a) |
V47 | M 4 | 0.26997 | 6238 | 0.123 | 32.0% | No | Liu et al. (2011) |
V53 | M 4 | 0.30845 | 7415 | 0.078 | 69.1% | Yes | This paper |
Table 6.Photometric solutions without spot for V53 in the globular cluster M 4.
Parameters
. | LC1
. | LC2
. | LC3
. | LC4
. |
---|
g 1, g2 | | 1.0, 0.32 | |
A 1, A2 | | 1.0, 0.5 | |
T 1 (K) | | 7415 ± 55 | |
T 2 (K) | 6800 ± 126 | 6437 ± 61 | 6866 ± 33 | 7026 ± 32 |
q | 0.077 ± 0.005 | 0.079 ± 0.006 | 0.078 ± 0.004 | 0.075 ± 0.003 |
i | 74.9 ± 1.5 | 73.9 ± 0.8 | 75.1 ± 0.4 | 76.2 ± 0.8 |
L 1/(L1 + L2)(B) | 0.9336 ± 0.0005 | 0.9454 ± 0.0003 | 0.9298 ± 0.0002 | 0.9245 ± 0.0003 |
L 1/(L1 + L2)(V) | 0.9276 ± 0.0005 | 0.9378 ± 0.0004 | 0.9242 ± 0.0002 | 0.9204 ± 0.0003 |
Ωin | 1.8857 | 1.8917 | 1.8875 | 1.8776 |
Ωout | 1.8351 | 1.8299 | 1.8366 | 1.8285 |
Ω1 = Ω2 | 1.8527 ± 0.0096 | 1.8436 ± 0.0062 | 1.8529 ± 0.0059 | 1.8481 ± 0.0052 |
r 1 (pole) | 0.5584 ± 0.0045 | 0.5629 ± 0.0010 | 0.5589 ± 0.0008 | 0.5599 ± 0.0023 |
r 1 (side) | 0.6347 ± 0.0075 | 0.6426 ± 0.0018 | 0.6355 ± 0.0011 | 0.6368 ± 0.0041 |
r 1 (back) | 0.6539 ± 0.0074 | 0.6629 ± 0.0021 | 0.6549 ± 0.0014 | 0.6552 ± 0.0043 |
r 2 (pole) | 0.1934 ± 0.0243 | 0.1964 ± 0.0014 | 0.1943 ± 0.0042 | 0.1868 ± 0.0096 |
r 2 (side) | 0.2036 ± 0.0325 | 0.2072 ± 0.0018 | 0.2046 ± 0.0052 | 0.1960 ± 0.0118 |
r 2 (back) | 0.2666 ± 0.0796 | 0.2881 ± 0.0131 | 0.2698 ± 0.0248 | 0.2479 ± 0.0311 |
f | 65.2 ± 19.0% | 92.9 ± 11.9% | 67.9 ± 11.6% | 60.2 ± 10.6% |
ΣW(O − C)2 | 1.90 × 10−8 | 2.12 × 10−8 | 1.40 × 10−8 | 1.67 × 10−8 |
Parameters
. | LC1
. | LC2
. | LC3
. | LC4
. |
---|
g 1, g2 | | 1.0, 0.32 | |
A 1, A2 | | 1.0, 0.5 | |
T 1 (K) | | 7415 ± 55 | |
T 2 (K) | 6800 ± 126 | 6437 ± 61 | 6866 ± 33 | 7026 ± 32 |
q | 0.077 ± 0.005 | 0.079 ± 0.006 | 0.078 ± 0.004 | 0.075 ± 0.003 |
i | 74.9 ± 1.5 | 73.9 ± 0.8 | 75.1 ± 0.4 | 76.2 ± 0.8 |
L 1/(L1 + L2)(B) | 0.9336 ± 0.0005 | 0.9454 ± 0.0003 | 0.9298 ± 0.0002 | 0.9245 ± 0.0003 |
L 1/(L1 + L2)(V) | 0.9276 ± 0.0005 | 0.9378 ± 0.0004 | 0.9242 ± 0.0002 | 0.9204 ± 0.0003 |
Ωin | 1.8857 | 1.8917 | 1.8875 | 1.8776 |
Ωout | 1.8351 | 1.8299 | 1.8366 | 1.8285 |
Ω1 = Ω2 | 1.8527 ± 0.0096 | 1.8436 ± 0.0062 | 1.8529 ± 0.0059 | 1.8481 ± 0.0052 |
r 1 (pole) | 0.5584 ± 0.0045 | 0.5629 ± 0.0010 | 0.5589 ± 0.0008 | 0.5599 ± 0.0023 |
r 1 (side) | 0.6347 ± 0.0075 | 0.6426 ± 0.0018 | 0.6355 ± 0.0011 | 0.6368 ± 0.0041 |
r 1 (back) | 0.6539 ± 0.0074 | 0.6629 ± 0.0021 | 0.6549 ± 0.0014 | 0.6552 ± 0.0043 |
r 2 (pole) | 0.1934 ± 0.0243 | 0.1964 ± 0.0014 | 0.1943 ± 0.0042 | 0.1868 ± 0.0096 |
r 2 (side) | 0.2036 ± 0.0325 | 0.2072 ± 0.0018 | 0.2046 ± 0.0052 | 0.1960 ± 0.0118 |
r 2 (back) | 0.2666 ± 0.0796 | 0.2881 ± 0.0131 | 0.2698 ± 0.0248 | 0.2479 ± 0.0311 |
f | 65.2 ± 19.0% | 92.9 ± 11.9% | 67.9 ± 11.6% | 60.2 ± 10.6% |
ΣW(O − C)2 | 1.90 × 10−8 | 2.12 × 10−8 | 1.40 × 10−8 | 1.67 × 10−8 |
Table 6.Photometric solutions without spot for V53 in the globular cluster M 4.
Parameters
. | LC1
. | LC2
. | LC3
. | LC4
. |
---|
g 1, g2 | | 1.0, 0.32 | |
A 1, A2 | | 1.0, 0.5 | |
T 1 (K) | | 7415 ± 55 | |
T 2 (K) | 6800 ± 126 | 6437 ± 61 | 6866 ± 33 | 7026 ± 32 |
q | 0.077 ± 0.005 | 0.079 ± 0.006 | 0.078 ± 0.004 | 0.075 ± 0.003 |
i | 74.9 ± 1.5 | 73.9 ± 0.8 | 75.1 ± 0.4 | 76.2 ± 0.8 |
L 1/(L1 + L2)(B) | 0.9336 ± 0.0005 | 0.9454 ± 0.0003 | 0.9298 ± 0.0002 | 0.9245 ± 0.0003 |
L 1/(L1 + L2)(V) | 0.9276 ± 0.0005 | 0.9378 ± 0.0004 | 0.9242 ± 0.0002 | 0.9204 ± 0.0003 |
Ωin | 1.8857 | 1.8917 | 1.8875 | 1.8776 |
Ωout | 1.8351 | 1.8299 | 1.8366 | 1.8285 |
Ω1 = Ω2 | 1.8527 ± 0.0096 | 1.8436 ± 0.0062 | 1.8529 ± 0.0059 | 1.8481 ± 0.0052 |
r 1 (pole) | 0.5584 ± 0.0045 | 0.5629 ± 0.0010 | 0.5589 ± 0.0008 | 0.5599 ± 0.0023 |
r 1 (side) | 0.6347 ± 0.0075 | 0.6426 ± 0.0018 | 0.6355 ± 0.0011 | 0.6368 ± 0.0041 |
r 1 (back) | 0.6539 ± 0.0074 | 0.6629 ± 0.0021 | 0.6549 ± 0.0014 | 0.6552 ± 0.0043 |
r 2 (pole) | 0.1934 ± 0.0243 | 0.1964 ± 0.0014 | 0.1943 ± 0.0042 | 0.1868 ± 0.0096 |
r 2 (side) | 0.2036 ± 0.0325 | 0.2072 ± 0.0018 | 0.2046 ± 0.0052 | 0.1960 ± 0.0118 |
r 2 (back) | 0.2666 ± 0.0796 | 0.2881 ± 0.0131 | 0.2698 ± 0.0248 | 0.2479 ± 0.0311 |
f | 65.2 ± 19.0% | 92.9 ± 11.9% | 67.9 ± 11.6% | 60.2 ± 10.6% |
ΣW(O − C)2 | 1.90 × 10−8 | 2.12 × 10−8 | 1.40 × 10−8 | 1.67 × 10−8 |
Parameters
. | LC1
. | LC2
. | LC3
. | LC4
. |
---|
g 1, g2 | | 1.0, 0.32 | |
A 1, A2 | | 1.0, 0.5 | |
T 1 (K) | | 7415 ± 55 | |
T 2 (K) | 6800 ± 126 | 6437 ± 61 | 6866 ± 33 | 7026 ± 32 |
q | 0.077 ± 0.005 | 0.079 ± 0.006 | 0.078 ± 0.004 | 0.075 ± 0.003 |
i | 74.9 ± 1.5 | 73.9 ± 0.8 | 75.1 ± 0.4 | 76.2 ± 0.8 |
L 1/(L1 + L2)(B) | 0.9336 ± 0.0005 | 0.9454 ± 0.0003 | 0.9298 ± 0.0002 | 0.9245 ± 0.0003 |
L 1/(L1 + L2)(V) | 0.9276 ± 0.0005 | 0.9378 ± 0.0004 | 0.9242 ± 0.0002 | 0.9204 ± 0.0003 |
Ωin | 1.8857 | 1.8917 | 1.8875 | 1.8776 |
Ωout | 1.8351 | 1.8299 | 1.8366 | 1.8285 |
Ω1 = Ω2 | 1.8527 ± 0.0096 | 1.8436 ± 0.0062 | 1.8529 ± 0.0059 | 1.8481 ± 0.0052 |
r 1 (pole) | 0.5584 ± 0.0045 | 0.5629 ± 0.0010 | 0.5589 ± 0.0008 | 0.5599 ± 0.0023 |
r 1 (side) | 0.6347 ± 0.0075 | 0.6426 ± 0.0018 | 0.6355 ± 0.0011 | 0.6368 ± 0.0041 |
r 1 (back) | 0.6539 ± 0.0074 | 0.6629 ± 0.0021 | 0.6549 ± 0.0014 | 0.6552 ± 0.0043 |
r 2 (pole) | 0.1934 ± 0.0243 | 0.1964 ± 0.0014 | 0.1943 ± 0.0042 | 0.1868 ± 0.0096 |
r 2 (side) | 0.2036 ± 0.0325 | 0.2072 ± 0.0018 | 0.2046 ± 0.0052 | 0.1960 ± 0.0118 |
r 2 (back) | 0.2666 ± 0.0796 | 0.2881 ± 0.0131 | 0.2698 ± 0.0248 | 0.2479 ± 0.0311 |
f | 65.2 ± 19.0% | 92.9 ± 11.9% | 67.9 ± 11.6% | 60.2 ± 10.6% |
ΣW(O − C)2 | 1.90 × 10−8 | 2.12 × 10−8 | 1.40 × 10−8 | 1.67 × 10−8 |
At present, only a few tens of extreme mass ratio (q ≤ 0.25), deep contact (f ≥ 50%) binaries have been identified. These binaries are in their final evolutionary stages and will probably merge into single fast rotating stars (FK Com-type stars) or blue stragglers (Qian et al. 2006). The mass ratio and the contact degree of V53 are q ∼ 0.078 and f ∼ 69%, respectively. V53 is definitely an extreme mass ratio, deep contact binary. As seen in figure 6 of Kaluzny et al. (2013b), V53 is above the turn-off point, located in the blue straggler region, and a member star of M 4. Therefore, V53 is a blue straggler. The orbital period of V53 is continuously decreasing. By the decrease of the orbital period, the orbit will tighten and the inner and outer critical Roche lobes will shrink, resulting in an increase of the degree of overcontact. Like the systems of GR Vir (Qian & Yang 2004), XY Leo (Qian et al. 2011), and TZ Boo (Christopoulou et al. 2011), V53 will evolve from this present high fill-out, extreme mass ratio overcontact configuration to a single rapid-rotation star. So far only one contact binary, V1309 Sco, has been observed to evolve to a merger of the components (Tylenda et al. 2011). Similar to V53, V1309 Sco was a contact binary showing a clear decrease of its orbital period before the merger.
Up to now, a few contact binaries in globular clusters have been analyzed. Table
5 lists the physical parameters of all the contact binaries in globular clusters that have been investigated. It is found that these binary systems have different contact degrees, indicating different evolutionary states. Among the 11 contact binaries, six are blue stragglers. Of these blue stragglers, all except V144 in M 54 have fill-out factors greater than 46%. There may be some correlation between the contact degree and whether or not a contact binary is a blue straggler. In order to determine the correlation, we plotted the color index difference between the five eclipsing blue stragglers (without V144 in M 54) and turn-off point versus the fill-out factor and the magnitude relative to turn-off point versus the fill-out factor in figure
9. As seen in the left-hand panel of figure
9, an exponential formula can fit the relation very well. A least-squares method yields the equation
where Δ is the relative color index to the turn-off point. From this equation, we can derive that when the fill-out factor of a contact binary exceeds 46.25(±2.05)%, it becomes a blue straggler. This result should be considered preliminary given the size of the sample.
Fig. 9.
The left-hand panel shows the relation of color index difference between the five eclipsing blue stragglers and turn-off point versus the fill-out factor. The right-hand panel shows the magnitude relative to turn-off point versus the fill-out factor.
In summary, we have presented a comprehensive photometric analysis and an investigation of the period change of V53 in the globular cluster M 4. The photometric analysis shows that V53 is an extremely low mass ratio (q ∼ 0.078) binary system with a high fill-out factor of f ∼ 69%, suggesting that V53 has the potential to challenge and constrain the theory of binary mergers. Variations in the light curves can be fitted by varying spot parameters. The O − C investigation indicates that the orbital period of V53 is continuously decreasing at a rate of dp/dt = 5.89(±0.02) × 10−8 d yr−1. In addition, V53 is a blue straggler and will evolve into a single rapid-rotating star in the future, making it a very good target for understanding the formation scenarios of FK Com-type stars and blue stragglers. These properties of V53 make it a very interesting object of study. In future, long-term photometric and spectroscopic observations are needed for V53 to confirm the orbital period variation and determine the spectroscopic mass ratio.
Acknowledgements
The authors thank Janusz Kaluzny and his collaborators very much for the seminal series of papers on binaries in globular clusters within the CASE observational program. This has been a monumental program requiring very hard observational work and considerable data-processing and interpretative effort. We also thank the CASE team for making all the observational data publicly available. This work is supported by the Joint Research Fund in Astronomy (No. U1431105) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and the Chinese Academy of Sciences (CAS), and by the Natural Science Foundation of Shandong Province (No. ZR2014AQ019), and by Young Scholars Program of Shandong University, Weihai (No. 2016WHWLJH07), and by the Open Research Program of Key Laboratory for the Structure and Evolution of Celestial Objects (No. OP201303). The authors also thank the reviewer very much for their valuable and very helpful comments to improve this paper.
Appendix. Photometric results for studying the four sets of light curves with and without spot
The photometric results of analyzing LC1 through LC4 without a spot are listed in table 6. The photometric results of analyzing LC1 through LC3 with a cool or hot spot are listed in table 7.
Table 7.Photometric solutions with spots for V53 in the globular cluster M 4.
Parameters
. | LC1
. | LC2
. | LC3
. |
---|
. | Cool spot
. | Hot spot
. | Cool spot
. | Hot spot
. | Cool spot
. | Hot spot
. |
---|
T 2 (K) | 6394 ± 65 | 6831 ± 64 | 6320 ± 34 | 6349 ± 36 | 6548 ± 18 | 6859 ± 17 |
q | 0.079 ± 0.004 | 0.078 ± 0.005 | 0.080 ± 0.004 | 0.080 ± 0.004 | 0.078 ± 0.003 | 0.077 ± 0.003 |
i | 80.3 ± 0.7 | 75.9 ± 1.6 | 79.4 ± 0.5 | 74.5 ± 0.9 | 75.0 ± 0.3 | 74.8 ± 0.4 |
L 1/(L1 + L2)(B) | 0.9506 ± 0.0002 | 0.9335 ± 0.0005 | 0.9500 ± 0.0001 | 0.9528 ± 0.0002 | 0.9432 ± 0.0001 | 0.9303 ± 0.0002 |
L 1/(L1 + L2)(V) | 0.9429 ± 0.0002 | 0.9281 ± 0.0004 | 0.9415 ± 0.0001 | 0.9447 ± 0.0002 | 0.9360 ± 0.0001 | 0.9250 ± 0.0001 |
Ωin | 1.8914 | 1.8880 | 1.8937 | 1.8968 | 1.8900 | 1.8856 |
Ωout | 1.8397 | 1.8370 | 1.8415 | 1.8441 | 1.8386 | 1.8350 |
Ω1 = Ω2 | 1.8627 ± 0.0033 | 1.8681 ± 0.0055 | 1.8478 ± 0.0028 | 1.8657 ± 0.0033 | 1.8516 ± 0.0008 | 1.8531 ± 0.0013 |
r 1 (pole) | 0.5577 ± 0.0010 | 0.5551 ± 0.0017 | 0.5622 ± 0.0006 | 0.5572 ± 0.0010 | 0.5601 ± 0.0003 | 0.5600 ± 0.0004 |
r 1 (side) | 0.6333 ± 0.0018 | 0.6292 ± 0.0029 | 0.6415 ± 0.0010 | 0.6325 ± 0.0018 | 0.6376 ± 0.0005 | 0.6373 ± 0.0007 |
r 1 (back) | 0.6521 ± 0.0020 | 0.6471 ± 0.0034 | 0.6617 ± 0.0012 | 0.6517 ± 0.0021 | 0.6570 ± 0.0005 | 0.6562 ± 0.0009 |
r 2 (pole) | 0.1880 ± 0.0014 | 0.1837 ± 0.0022 | 0.1957 ± 0.0008 | 0.1898 ± 0.0014 | 0.1918 ± 0.0004 | 0.1884 ± 0.0006 |
r 2 (side) | 0.1971 ± 0.0016 | 0.1921 ± 0.0026 | 0.2063 ± 0.0010 | 0.1993 ± 0.0017 | 0.2018 ± 0.0004 | 0.1978 ± 0.0007 |
r 2 (back) | 0.2454 ± 0.0047 | 0.2332 ± 0.0066 | 0.2797 ± 0.0056 | 0.2495 ± 0.0050 | 0.2617 ± 0.0017 | 0.2504 ± 0.0023 |
f | 55.5 ± 6.3% | 39.0 ± 10.8% | 88.1 ± 5.5% | 59.1 ± 6.3% | 74.7 ± 1.6% | 64.1 ± 2.7% |
θ (radian) | 1.052 ± 0.411 | 1.565 ± 0.198 | 0.734 ± 0.062 | 1.426 ± 0.236 | 0.347 ± 0.015 | 0.887 ± 0.052 |
ϕ (radian) | 3.903 ± 0.077 | 4.954 ± 0.274 | 4.309 ± 0.079 | 4.929 ± 0.091 | 3.616 ± 0.038 | 4.198 ± 0.059 |
r (radian) | 0.236 ± 0.015 | 0.303 ± 0.021 | 0.244 ± 0.012 | 0.510 ± 0.035 | 0.278 ± 0.065 | 0.310 ± 0.021 |
T f(Td/T0) | 0.849 ± 0.024 | 01.321 ± 0.085 | 0.806 ± 0.032 | 1.268 ± 0.093 | 0.849 ± 0.016 | 1.284 ± 0.057 |
ΣW(O − C)2 | 1.67 × 10−8 | 1.88 × 10−8 | 1.20 × 10−8 | 1.16 × 10−8 | 1.30 × 10−8 | 1.34 × 10−8 |
Parameters
. | LC1
. | LC2
. | LC3
. |
---|
. | Cool spot
. | Hot spot
. | Cool spot
. | Hot spot
. | Cool spot
. | Hot spot
. |
---|
T 2 (K) | 6394 ± 65 | 6831 ± 64 | 6320 ± 34 | 6349 ± 36 | 6548 ± 18 | 6859 ± 17 |
q | 0.079 ± 0.004 | 0.078 ± 0.005 | 0.080 ± 0.004 | 0.080 ± 0.004 | 0.078 ± 0.003 | 0.077 ± 0.003 |
i | 80.3 ± 0.7 | 75.9 ± 1.6 | 79.4 ± 0.5 | 74.5 ± 0.9 | 75.0 ± 0.3 | 74.8 ± 0.4 |
L 1/(L1 + L2)(B) | 0.9506 ± 0.0002 | 0.9335 ± 0.0005 | 0.9500 ± 0.0001 | 0.9528 ± 0.0002 | 0.9432 ± 0.0001 | 0.9303 ± 0.0002 |
L 1/(L1 + L2)(V) | 0.9429 ± 0.0002 | 0.9281 ± 0.0004 | 0.9415 ± 0.0001 | 0.9447 ± 0.0002 | 0.9360 ± 0.0001 | 0.9250 ± 0.0001 |
Ωin | 1.8914 | 1.8880 | 1.8937 | 1.8968 | 1.8900 | 1.8856 |
Ωout | 1.8397 | 1.8370 | 1.8415 | 1.8441 | 1.8386 | 1.8350 |
Ω1 = Ω2 | 1.8627 ± 0.0033 | 1.8681 ± 0.0055 | 1.8478 ± 0.0028 | 1.8657 ± 0.0033 | 1.8516 ± 0.0008 | 1.8531 ± 0.0013 |
r 1 (pole) | 0.5577 ± 0.0010 | 0.5551 ± 0.0017 | 0.5622 ± 0.0006 | 0.5572 ± 0.0010 | 0.5601 ± 0.0003 | 0.5600 ± 0.0004 |
r 1 (side) | 0.6333 ± 0.0018 | 0.6292 ± 0.0029 | 0.6415 ± 0.0010 | 0.6325 ± 0.0018 | 0.6376 ± 0.0005 | 0.6373 ± 0.0007 |
r 1 (back) | 0.6521 ± 0.0020 | 0.6471 ± 0.0034 | 0.6617 ± 0.0012 | 0.6517 ± 0.0021 | 0.6570 ± 0.0005 | 0.6562 ± 0.0009 |
r 2 (pole) | 0.1880 ± 0.0014 | 0.1837 ± 0.0022 | 0.1957 ± 0.0008 | 0.1898 ± 0.0014 | 0.1918 ± 0.0004 | 0.1884 ± 0.0006 |
r 2 (side) | 0.1971 ± 0.0016 | 0.1921 ± 0.0026 | 0.2063 ± 0.0010 | 0.1993 ± 0.0017 | 0.2018 ± 0.0004 | 0.1978 ± 0.0007 |
r 2 (back) | 0.2454 ± 0.0047 | 0.2332 ± 0.0066 | 0.2797 ± 0.0056 | 0.2495 ± 0.0050 | 0.2617 ± 0.0017 | 0.2504 ± 0.0023 |
f | 55.5 ± 6.3% | 39.0 ± 10.8% | 88.1 ± 5.5% | 59.1 ± 6.3% | 74.7 ± 1.6% | 64.1 ± 2.7% |
θ (radian) | 1.052 ± 0.411 | 1.565 ± 0.198 | 0.734 ± 0.062 | 1.426 ± 0.236 | 0.347 ± 0.015 | 0.887 ± 0.052 |
ϕ (radian) | 3.903 ± 0.077 | 4.954 ± 0.274 | 4.309 ± 0.079 | 4.929 ± 0.091 | 3.616 ± 0.038 | 4.198 ± 0.059 |
r (radian) | 0.236 ± 0.015 | 0.303 ± 0.021 | 0.244 ± 0.012 | 0.510 ± 0.035 | 0.278 ± 0.065 | 0.310 ± 0.021 |
T f(Td/T0) | 0.849 ± 0.024 | 01.321 ± 0.085 | 0.806 ± 0.032 | 1.268 ± 0.093 | 0.849 ± 0.016 | 1.284 ± 0.057 |
ΣW(O − C)2 | 1.67 × 10−8 | 1.88 × 10−8 | 1.20 × 10−8 | 1.16 × 10−8 | 1.30 × 10−8 | 1.34 × 10−8 |
Table 7.Photometric solutions with spots for V53 in the globular cluster M 4.
Parameters
. | LC1
. | LC2
. | LC3
. |
---|
. | Cool spot
. | Hot spot
. | Cool spot
. | Hot spot
. | Cool spot
. | Hot spot
. |
---|
T 2 (K) | 6394 ± 65 | 6831 ± 64 | 6320 ± 34 | 6349 ± 36 | 6548 ± 18 | 6859 ± 17 |
q | 0.079 ± 0.004 | 0.078 ± 0.005 | 0.080 ± 0.004 | 0.080 ± 0.004 | 0.078 ± 0.003 | 0.077 ± 0.003 |
i | 80.3 ± 0.7 | 75.9 ± 1.6 | 79.4 ± 0.5 | 74.5 ± 0.9 | 75.0 ± 0.3 | 74.8 ± 0.4 |
L 1/(L1 + L2)(B) | 0.9506 ± 0.0002 | 0.9335 ± 0.0005 | 0.9500 ± 0.0001 | 0.9528 ± 0.0002 | 0.9432 ± 0.0001 | 0.9303 ± 0.0002 |
L 1/(L1 + L2)(V) | 0.9429 ± 0.0002 | 0.9281 ± 0.0004 | 0.9415 ± 0.0001 | 0.9447 ± 0.0002 | 0.9360 ± 0.0001 | 0.9250 ± 0.0001 |
Ωin | 1.8914 | 1.8880 | 1.8937 | 1.8968 | 1.8900 | 1.8856 |
Ωout | 1.8397 | 1.8370 | 1.8415 | 1.8441 | 1.8386 | 1.8350 |
Ω1 = Ω2 | 1.8627 ± 0.0033 | 1.8681 ± 0.0055 | 1.8478 ± 0.0028 | 1.8657 ± 0.0033 | 1.8516 ± 0.0008 | 1.8531 ± 0.0013 |
r 1 (pole) | 0.5577 ± 0.0010 | 0.5551 ± 0.0017 | 0.5622 ± 0.0006 | 0.5572 ± 0.0010 | 0.5601 ± 0.0003 | 0.5600 ± 0.0004 |
r 1 (side) | 0.6333 ± 0.0018 | 0.6292 ± 0.0029 | 0.6415 ± 0.0010 | 0.6325 ± 0.0018 | 0.6376 ± 0.0005 | 0.6373 ± 0.0007 |
r 1 (back) | 0.6521 ± 0.0020 | 0.6471 ± 0.0034 | 0.6617 ± 0.0012 | 0.6517 ± 0.0021 | 0.6570 ± 0.0005 | 0.6562 ± 0.0009 |
r 2 (pole) | 0.1880 ± 0.0014 | 0.1837 ± 0.0022 | 0.1957 ± 0.0008 | 0.1898 ± 0.0014 | 0.1918 ± 0.0004 | 0.1884 ± 0.0006 |
r 2 (side) | 0.1971 ± 0.0016 | 0.1921 ± 0.0026 | 0.2063 ± 0.0010 | 0.1993 ± 0.0017 | 0.2018 ± 0.0004 | 0.1978 ± 0.0007 |
r 2 (back) | 0.2454 ± 0.0047 | 0.2332 ± 0.0066 | 0.2797 ± 0.0056 | 0.2495 ± 0.0050 | 0.2617 ± 0.0017 | 0.2504 ± 0.0023 |
f | 55.5 ± 6.3% | 39.0 ± 10.8% | 88.1 ± 5.5% | 59.1 ± 6.3% | 74.7 ± 1.6% | 64.1 ± 2.7% |
θ (radian) | 1.052 ± 0.411 | 1.565 ± 0.198 | 0.734 ± 0.062 | 1.426 ± 0.236 | 0.347 ± 0.015 | 0.887 ± 0.052 |
ϕ (radian) | 3.903 ± 0.077 | 4.954 ± 0.274 | 4.309 ± 0.079 | 4.929 ± 0.091 | 3.616 ± 0.038 | 4.198 ± 0.059 |
r (radian) | 0.236 ± 0.015 | 0.303 ± 0.021 | 0.244 ± 0.012 | 0.510 ± 0.035 | 0.278 ± 0.065 | 0.310 ± 0.021 |
T f(Td/T0) | 0.849 ± 0.024 | 01.321 ± 0.085 | 0.806 ± 0.032 | 1.268 ± 0.093 | 0.849 ± 0.016 | 1.284 ± 0.057 |
ΣW(O − C)2 | 1.67 × 10−8 | 1.88 × 10−8 | 1.20 × 10−8 | 1.16 × 10−8 | 1.30 × 10−8 | 1.34 × 10−8 |
Parameters
. | LC1
. | LC2
. | LC3
. |
---|
. | Cool spot
. | Hot spot
. | Cool spot
. | Hot spot
. | Cool spot
. | Hot spot
. |
---|
T 2 (K) | 6394 ± 65 | 6831 ± 64 | 6320 ± 34 | 6349 ± 36 | 6548 ± 18 | 6859 ± 17 |
q | 0.079 ± 0.004 | 0.078 ± 0.005 | 0.080 ± 0.004 | 0.080 ± 0.004 | 0.078 ± 0.003 | 0.077 ± 0.003 |
i | 80.3 ± 0.7 | 75.9 ± 1.6 | 79.4 ± 0.5 | 74.5 ± 0.9 | 75.0 ± 0.3 | 74.8 ± 0.4 |
L 1/(L1 + L2)(B) | 0.9506 ± 0.0002 | 0.9335 ± 0.0005 | 0.9500 ± 0.0001 | 0.9528 ± 0.0002 | 0.9432 ± 0.0001 | 0.9303 ± 0.0002 |
L 1/(L1 + L2)(V) | 0.9429 ± 0.0002 | 0.9281 ± 0.0004 | 0.9415 ± 0.0001 | 0.9447 ± 0.0002 | 0.9360 ± 0.0001 | 0.9250 ± 0.0001 |
Ωin | 1.8914 | 1.8880 | 1.8937 | 1.8968 | 1.8900 | 1.8856 |
Ωout | 1.8397 | 1.8370 | 1.8415 | 1.8441 | 1.8386 | 1.8350 |
Ω1 = Ω2 | 1.8627 ± 0.0033 | 1.8681 ± 0.0055 | 1.8478 ± 0.0028 | 1.8657 ± 0.0033 | 1.8516 ± 0.0008 | 1.8531 ± 0.0013 |
r 1 (pole) | 0.5577 ± 0.0010 | 0.5551 ± 0.0017 | 0.5622 ± 0.0006 | 0.5572 ± 0.0010 | 0.5601 ± 0.0003 | 0.5600 ± 0.0004 |
r 1 (side) | 0.6333 ± 0.0018 | 0.6292 ± 0.0029 | 0.6415 ± 0.0010 | 0.6325 ± 0.0018 | 0.6376 ± 0.0005 | 0.6373 ± 0.0007 |
r 1 (back) | 0.6521 ± 0.0020 | 0.6471 ± 0.0034 | 0.6617 ± 0.0012 | 0.6517 ± 0.0021 | 0.6570 ± 0.0005 | 0.6562 ± 0.0009 |
r 2 (pole) | 0.1880 ± 0.0014 | 0.1837 ± 0.0022 | 0.1957 ± 0.0008 | 0.1898 ± 0.0014 | 0.1918 ± 0.0004 | 0.1884 ± 0.0006 |
r 2 (side) | 0.1971 ± 0.0016 | 0.1921 ± 0.0026 | 0.2063 ± 0.0010 | 0.1993 ± 0.0017 | 0.2018 ± 0.0004 | 0.1978 ± 0.0007 |
r 2 (back) | 0.2454 ± 0.0047 | 0.2332 ± 0.0066 | 0.2797 ± 0.0056 | 0.2495 ± 0.0050 | 0.2617 ± 0.0017 | 0.2504 ± 0.0023 |
f | 55.5 ± 6.3% | 39.0 ± 10.8% | 88.1 ± 5.5% | 59.1 ± 6.3% | 74.7 ± 1.6% | 64.1 ± 2.7% |
θ (radian) | 1.052 ± 0.411 | 1.565 ± 0.198 | 0.734 ± 0.062 | 1.426 ± 0.236 | 0.347 ± 0.015 | 0.887 ± 0.052 |
ϕ (radian) | 3.903 ± 0.077 | 4.954 ± 0.274 | 4.309 ± 0.079 | 4.929 ± 0.091 | 3.616 ± 0.038 | 4.198 ± 0.059 |
r (radian) | 0.236 ± 0.015 | 0.303 ± 0.021 | 0.244 ± 0.012 | 0.510 ± 0.035 | 0.278 ± 0.065 | 0.310 ± 0.021 |
T f(Td/T0) | 0.849 ± 0.024 | 01.321 ± 0.085 | 0.806 ± 0.032 | 1.268 ± 0.093 | 0.849 ± 0.016 | 1.284 ± 0.057 |
ΣW(O − C)2 | 1.67 × 10−8 | 1.88 × 10−8 | 1.20 × 10−8 | 1.16 × 10−8 | 1.30 × 10−8 | 1.34 × 10−8 |
References
Applegate
J. H.
1992
,
ApJ
,
385
,
621
Arbutina
B.
2007
,
MNRAS
,
377
,
1635
Arbutina
B.
2009
,
MNRAS
,
394
,
501
Christopoulou
P.-E.
Parageorgiou
A.
Chrysopoulos
I.
2011
,
AJ
,
142
,
99
Clement
C. M.
et al.
2001
,
AJ
,
122
,
2587
Eastman
J.
Siverd
R.
Gaudi
B. S.
2010
,
PASP
,
122
,
935
Flannery
B. P.
1976
,
ApJ
,
205
,
217
Gratton
R. G.
Carretta
E.
Bragaglia
A.
Lucatello
S.
D’Orazi
V.
2010
,
A&A
,
517
,
A81
Jiang
D.
Han
Z.
Wang
J.
Jiang
T.
Li
L.
2010
,
MNRAS
,
405
,
2485
Kallrath
J.
Milone
E. F.
Stagg
C. R.
1992
,
ApJ
,
389
,
590
Kaluzny
J.
Thompson
I. B.
Krzeminski
W.
1997
,
AJ
,
113
,
2219
Kaluzny
J.
et al.
2013a
,
AJ
,
145
,
43
Kaluzny
J.
Thompson
I. B.
Rozyczka
M.
Krzeminski
W.
2013b
,
Acta Astron.
,
63
,
181
Kwee
K. K.
1958
,
BAN
,
14
,
131
Li
K.
Qian
S.-B.
2013a
,
New Astron.
,
22
,
57
Li
K.
Qian
S.-B.
2013b
,
New Astron.
,
25
,
12
Li
L.
Zhang
F.
2006
,
MNRAS
,
369
,
2001
Liao
W.-P.
Qian
S.-B.
2010
,
MNRAS
,
405
,
1930
Liu
L.
Qian
S.-B.
Fernández-Lajús
E.
2011
,
MNRAS
,
415
,
1509
Lovisi
L.
et al.
2010
,
ApJ
,
719
,
L121
Lucy
L. B.
1968
,
ApJ
,
151
,
1123
McVean
J. R.
Milone
E. F.
Mateo
M.
Yan
L.
1997
,
ApJ
,
481
,
782
Mochnacki
S. W.
1981
,
ApJ
,
245
,
650
Nascimbeni
V.
et al.
2014
,
MNRAS
,
442
,
2381
Qian
S.
Yang
Y.
Zhu
L.
He
J.
Yuan
J.
2006
,
Ap&SS
,
304
,
25
Qian
S.-B.
He
J.-J.
Soonthornthum
B.
Liu
L.
Zhu
L.-Y.
Li
L.-J.
Liao
W. P.
Dai
Z.-B.
2008
,
AJ
,
136
,
1940
Qian
S.-B.
Liu
L.
Soonthornthum
B.
Zhu
L.-Y.
He
J.-J.
2007
,
AJ
,
134
,
1475
Qian
S.-B.
Liu
L.
Zhu
L.-Y.
He
J.-J.
Yang
Y.-G.
Bernasconi
L.
2011
,
AJ
,
141
,
151
Qian
S.-B.
Yang
Y.-G.
2004
,
AJ
,
128
,
2430
Qian
S.-B.
Zhu
L.-Y.
Soonthornthum
B.
Yuan
J.-Z.
Yang
Y.-G.
He
J.-J.
2005
,
AJ
,
130
,
1206
Rasio
F. A.
1995
,
ApJ
,
444
,
L41
Robertson
J. A.
Eggleton
P. P.
1977
,
MNRAS
,
179
,
359
Samec
R. G.
Flaaten
D.
Jaso
A.
Oliver
B.
Rehn
T.
Faulkner
D. R.
Van Hamme
W.
2012
,
PASP
,
124
,
1025
Samec
R. G.
Labadorf
C. M.
Hawkins
N. C.
Faulkner
D. R.
Van Hamme
W.
2011
,
AJ
,
142
,
117
Samec
R. G.
Kring
J. D.
Faulkner
D. R.
Van Hamme
W.
2013
,
PASP
,
125
,
1200
Sriram
K.
Malu
S.
Choi
C. S.
Vivekananda Rao
P.
2016
,
AJ
,
151
,
69
Stepien
K.
2006
,
Acta Astron.
,
56
,
347
Szalai
T.
Kiss
L. L.
Mészáros
S.
Vinkó
J.
Csizmadia
S.
2007
,
A&A
,
465
,
943
Tylenda
R.
et al.
2011
,
A&A
,
528
,
A114
van Hamme
W.
1993
,
AJ
,
106
,
2096
Van Hamme
W.
Wilson
R. E.
2007
,
ApJ
,
661
,
1129
von Zeipel
H.
1924
,
MNRAS
,
84
,
665
Wadhwa
S. S.
2006
,
Ap&SS
,
301
,
195
Webbink
R. F.
1977
,
ApJ
,
215
,
851
Wilson
R. E.
1979
,
ApJ
,
234
,
1054
Wilson
R. E.
1990
,
ApJ
,
356
,
613
Wilson
R. E.
2008
,
ApJ
,
672
,
575-589
Wilson
R. E.
2012
,
AJ
,
144
,
73
Wilson
R. E.
Devinney
E. J.
1971
,
ApJ
,
166
,
605
Wilson
R. E.
Van Hamme
W.
Terrell
D.
2010
,
ApJ
,
723
,
1469
Worthey
G.
Lee
H.-c.
2011
,
ApJS
,
193
,
1
Yakut
K.
Eggleton
P. P.
2005
,
ApJ
,
629
,
1055
Yang
Y.-G.
Qian
S.-B.
2015
,
AJ
,
150
,
69
Yang
Y.-G.
Qian
S.-B.
Soonthornthum
B.
2012
,
AJ
,
143
,
122
Yang
Y.-G.
Qian
S.-B.
Zhang
L.-Y.
Dai
H.-F.
Soonthornthum
B.
2013
,
AJ
,
146
,
35
Zhu
L. Y.
Qian
S. B.
Soonthornthum
B.
He
J. J.
Liu
L.
2011
,
AJ
,
142
,
124
Zhu
L.-Y.
Qian
S.-B.
Soonthornthum
B.
Yang
Y.-G.
2005
,
AJ
,
129
,
2806
Zola
S.
et al.
2004
,
Acta Astron.
,
54
,
299
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