Minute-cadence Observations of the LAMOST Fields with the TMTS. VI. Physical Parameters of Contact Binaries

The TMTS is located at the Xinglong Station of the National Astronomical Observatory of China (NAOC), using a multitube telescope system consisting of four 40 cm optical telescopes and a large field of view (FoV) of approximately 18 deg² (J.-C. Zhang et al. 2020). Luminous filter (L band hereafter) was conducted a wide wavelength range from 390 nm to about 900 nm (J.-C. Zhang et al. 2020), similar to Gaia G band (330–1050 nm; Gaia Collaboration et al. 2018). During the first 2 yr survey, TMTS has monitored 449 LAMOST/TMTS plates by the end of 2022. Machine learning classification was performed on the periodic variable sources obtained from uninterrupted light curves, confirming 5698 W UMa type contact binaries, namely EW contact binaries (F. Guo et al. 2024).

The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) is a 4 m Schmidt telescope with an FOV of 5 square degrees, and is equipped with 4000 fibers (X.-Q. Cui et al. 2012; G. Zhao et al. 2012; A. L. Luo et al. 2015). LAMOST medium resolution survey (MRS) Data Release (DR) 10⁸ was used as the source of radial velocities (RVs) for the W UMa contact binaries with a resolution of 7500. For each observed object, two spectra are obtained within a single exposure, which include a blue (B) side spectrum with a wavelength range of 4950–5350 Å, and a red (R) side spectrum with a wavelength range of 6300–6800 Å. The blue arm contains more absorption lines than the red arm, allowing us to measure the RVs with a precision up to 1 km s⁻¹ for most stars (N. Liu et al. 2019).

First, we cross-matched the EW catalog of TMTS with the LAMOST MRS catalog and filtered according to the following criteria:

1. 1  
   Select targets within an angular radius of 3″.
2. 2  
   SNRs of B-band spectra greater than 10.
3. 3  
   Number of consecutive exposures greater than or equal to 3.

After the above selection process, we obtained 57 targets that met the criteria. Then, using the period and times of minimum of the TMTS light curves of each target, we calculated the phase corresponding to the observed times for each target. We chose targets with at least three phases between 0.2 and 0.3 or 0.7 and 0.8. After that, we visually inspected their light curves (LCs) again, excluding LCs with insufficient SNR and those targets that might be ellipsoidal variables. Finally 10 targets were identified from the LAMOST MRS spectra for subsequent RV calculations, which are listed in Table 1. Figure 1 shows the phase-folded light curves of the 10 W UMa contact binaries from the TMTS.

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We also cross-matched these targets with VSX (C. L. Watson et al. 2006), Gaia (Gaia Collaboration et al. 2016, 2018), and the LAMOST low-resolution (LR) spectra. The information obtained is listed in Tables 1 and 2. Figure 2 shows the LR spectra, which are typical FGK-type spectra and characterized by apparent Balmer absorption lines.

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Table 1 provides information on these targets, including their TMTS catalog identifier (Source ID), Name (in our work), R.A. in decimal degrees, decl. in decimal degrees, observation date (Obs. Date), orbital period from TMTS (P_TMTS), orbital period from VSX (P_{V SX}), median magnitude from TMTS (L₀), absolute magnitude derived from the Gaia DR2 database (G_abs), color index derived from the Gaia DR2 database (B_P − R_P), exposure times of LAMOST LR spectra (N_LRS), exposure times of LAMOST MR spectra (N_MRS), and VSX variability type (VSX type). Table 2 lists the spectral information of the 10 targets from LAMOST DR10, including Name, Obs. Date, spectral type (Spec.type), effective temperature (T_eff), surface gravity (log g), metallicity (Fe/H), heliocentric RV, and SNR of the g band (SNR_g).

All of these targets have been identified as contact binaries or eclipsing binary candidates by the Asteroid Terrestrial-impact Last Alert System (A. N. Heinze et al. 2018) or Gaia Data Release 3 (A. Panchal & Y. C. Joshi 2021), and their orbital periods have been determined, respectively. The periods, which were used for the phase-folded and orbital period analysis, are listed in Table 5. Among the 10 targets, J0132, J0305, J1300, and J1402 have undergone the analysis of multiband photometric solution and studies of period. Detailed explanations of these targets are provided in the following.

• (i)  
  J0132 (V471 Cas) was discovered photographically by C. Hoffmeister (1966), and classified as a W UMa type binary by H. Gessner & I. Meinunger (1973). The photometric solution was first applied to J0132 by X. Liu & H. Tan (1991), in which the mass ratio was determined as 0.5947(±0.0149) and the overcontact factor is 0.19. D. P. Kjurkchieva et al. (2019) reanalyzed the new light curves and obtained the mass ratio and fill-out factor with 0.635 and 0.078. Through the analysis of orbital period, they revised the period of J0132 to 0.400937 days and confirmed that within J0132 there may exist a third body with a sinusoidal period of 12.8 yr.
• (ii)  
  J0305 (NSVS 6599082) was first classified as a W UMa type binary by D. I. Hoffman et al. (2009). A. Panchal & Y. C. Joshi (2021) presented the photometric and spectroscopic analysis, which indicates that J0305 shows a long-term increase in orbital period, with dp/dt = 1.78(±1.52) × 10⁻⁶ day yr⁻¹. The mass ratio and fill-out factor are 0.31(±0.01) and 0.105, respectively, the equivalent width (EW) of Hα is measured as 1.031 ± 0.018 Å.
• (iii)  
  J1300 (MM Com) was discovered as a contact binary with a period of 0.30199999 days from Robotic Optical Transient Search Experiment (ROTSE) all-sky surveys (C. Akerlof et al. 2000). D. P. Kjurkchieva et al. (2018) obtained light curves in the and bands and derived the photometric solution. J0913 is a W-type contact binary, with a mass ratio, orbital inclination, and fill-out factor of 4.66(±0.02), 806(±0.03), and 0.2388, respectively. Then, Y. Yang et al. (2023) analyzed new photometric and spectroscopic data for J1300, determining a mass ratio of 4.747(±0.005), an orbital inclination of 7976(±0.16), and a fill-out factor of 0.32(±0.04). The study of O − C suggests that J0913 may be in a triple system with a period of 20.02(±0.43) yr.
• (iv)  
  J1402 (EI CVn) was discovered as an eclipsing binary system by ROTSE with a period of 0.260775 days (C. Akerlof et al. 2000). Then it was carried out as a photometric orbital solution by Y.-G. Yang (2011), which was found to be a W-type contact binary with a mass ratio of 0.461(±0.003) and a fill-out factor of 0.21(±0.07). The O − C analysis indicates that the orbital period of J1402 is decreasing at a long-term rate of dp/dt = −3.11(±0.03) × 10⁻⁷ day yr⁻¹. K. B. Alton & K. Stepień (2021) obtained the new CCD photometric data of J1402 and performed a W-D analysis, resulting in a mass ratio of 0.443(±0.001) and a fill-out factor of 0.15. The study of O − C suggests a long-term period decrease rate of dp/dt = −1.35(±0.01) × 10⁻⁷ day yr⁻¹, with a periodic modulation of 10.14 ± 1.13 days.

The other six targets, J0047, J0638, J0759, J0913, J1042, and J2236, have not been systematically analyzed since their discovery.

In order to obtain the RVs of these 10 targets, we first used the Python package laspec (B. Zhang et al. 2020, 2021) to process the LAMOST MR spectra and measured the cross-correlation functions (CCFs) for each component. In this process, PHOENIX (P. H. Hauschildt 1993; P. H. Hauschildt & E. Baron 2006; E. Baron & P. H. Hauschildt 2007) was used to construct template spectra, whose resolution was degraded to match that of the LAMOST MR spectra. Simultaneously, since radial velocity zero-points (RVZPs) of MR spectra may vary with time and different fibers, we used the method of B. Zhang et al. (2021) to calibrate RVZPs. Then, to determine the positions of the two peaks of the CCF, we used the Python tool GaussPy to fit the RVs of primary and secondary components, which can implement the Autonomous Gaussian Decomposition algorithm (R. R. Lindner et al. 2015). The RVs obtained for all targets are listed in Table 3.

We used the 2013 version of the W-D method to analyze the TMTS light curves and the RVs of these 10 systems to obtain their orbital and absolute parameters. The initial epoch and period used to convert BJD to phase of each target are the corrected epoch and period in Table 5. During the W-D analysis, we analyzed every light curve with RV for each target simultaneously.

For targets with LAMOST LR spectra, we used the temperature or the average temperature of multiple spectra provided by LAMOST as the effective temperature of the primary star (T₁, the hotter component) during the W-D process. For targets without LAMOST LR spectra, we examined their LAMOST MR spectra. Since the MR spectra of J0132 did not yield a temperature and the temperature uncertainties for J0913 exceeded 50 K, the temperatures provided by Gaia were adopted as T₁ for these two targets. The specific values T₁ are listed in Table 7. Then we fixed the effective temperature of the primary (T₁) and adjusted that for the secondary (T₂, the cooler component). The gravity-darkening and bolometric albedo coefficients were set to g_1,2 = 0.32 and A_1,2 = 0.5 following L. B. Lucy (1967) and S. M. Ruciński (1969). The bolometric limb-darkening and bandpass limb-darkening coefficients were internally computed, which were obtained from W. van Hamme (1993), and the limb-darkening law is the square root law. The weights of the light curves and the radial velocities were set according to the observation errors, using the reciprocal of the square of the error. During the modeling, the fixed parameters were as follows: the effective temperatures of the primary star T₁, the orbital period P, and the orbital eccentricity e. The orbital eccentricity e was fixed to zero, consistent with the assumption that contact binaries typically have circular orbits due to strong tidal interactions. The adjustable parameters were as follows: the orbital semimajor axis a, the systemic RV V_γ, the orbital inclination i, the mass ratio q = M₂/M₁, the effective temperature of the secondary star T₂, the monochromatic luminosity of the primary star L₁, and the dimensionless potential of the two components Ω₁ = Ω₂. The fill-out factor was calculated by f = (Ω − Ω_in)/(Ω_out − Ω_in), where Ω, Ω_in, and Ω_out represent potentials for the common photosphere, the inner and outer contact surfaces, respectively. When f = 0, the two components just fill the inner contact surface and begin to contact, when f = 1, the two components have already filled the outer contact surface. After running the automatic iteration, the convergence criterion was established as the condition in which the standard error of the adjustable parameters is greater than their respective correction values, and the difference between the input mean residual and the predicted mean residual does not exceed 1%.

From their TMTS light curves, it can be seen that the two maxima at phases 0.25 and 0.75 exhibit some difference, a phenomenon known as the O’Connell effect, which is typically attributed to the magnetic activity of the components (e.g., Y.-N. Guo et al. 2020; K. Li et al. 2021a; A. Papageorgiou et al. 2023; A. Čeki et al. 2024). Meanwhile, Doppler boosting also contributes to the observed flux variations (A. Loeb & B. S. Gaudi 2003; S. Zucker et al. 2007). Therefore, a dark spot or a hot spot model was employed during the W-D analysis to obtain a better fit first. All targets used models with a dark spot added to either the primary or the secondary star. The latitudes were both fixed at 90°, because the most reliable parameter of a spot is the longitude of the spot, which can be used to study the evolution of the spot and stellar activity cycles (Z. Eker 1999; S. V. Berdyugina 2005). Subsequently, we calculated the observed magnitude variations, Δm (m_0.25–m_0.75), in phases 0.25 and 0.75 from the light curves derived from the W-D fitting and listed in Table 7. For each target, considering the temperatures (T_eff) of the components and the observational band (λ ≈ 650 nm), the beaming factors were determined (S. Zucker et al. 2007). Using the radial velocities of each component at phases 0.25 and 0.75, we calculated the beaming-magnitude variations for each target. All of these values are listed in Table 7. The observed magnitude variations are on the order of percent of a magnitude, while the beaming-magnitude variations are on the order of thousandths. Therefore, Doppler boosting is unlikely to explain the O’Connell effect. The spot model was adopted to account for the O’Connell effect.

The orbital and spot parameters determined by the light curves and the radial velocities are all tabulated in Table 7. The uncertainties of the adjustable parameters are only internal uncertainties of the final step of the light curve analysis determined by the W-D code. To account for the uncertainty of T₁ derived from spectroscopy to T₂, we conducted the following discussion. Based on photometric data, the W-D code can precisely determine T₂/T₁. Therefore, assuming T₁ is fixed, the uncertainty of T₂ derived from W-D can be used to determine the uncertainty of T₂/T₁. Subsequently, combining the uncertainties of T₂/T₁ and T₁, the accurate uncertainty of T₂ can be obtained. The final uncertainties of T₁ and T₂ are also listed in Table 7. Theoretical light curves and radial velocities for the 10 targets are shown in Figure 5. The smooth residuals of the light curves indicate that the model with the added spot resulted in a good fit for all 10 targets. Figure 6 shows the geometric structures of each target, we can clearly see the location and angular radius of the spots. Table 7 also provides absolute physical parameters from the W-D process, with detailed descriptions presented in Section 5.

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When the photosphere or chromosphere of a star exhibits magnetic activity, the atmosphere of the star is heated nonthermally by the magnetic field. The spectral subtraction technique is usually used to investigate chromospheric activity (S. C. Barden 1985). The activity is quantified by the EW of the emission line, such as the Balmer series (Hα, Hβ, Hγ) and the Ca ii infrared triplet (IRT), which serve as useful indicators of chromospheric activity in many contact binaries (Y. V. Pavlenko et al. 2018; K. Li et al. 2022; X.-Y. Liu et al. 2023; L.-Z. Li et al. 2024).

We analyzed the LAMOST LR spectra for these 10 targets to assess their chromospheric activities. Excluding J0132 and J0913, there are a total of 14 spectra for eight targets. The analysis process follows three fundamental steps. First, based on the temperatures of the two components obtained from the W-D solution for each target, we selected template spectra for the primary and secondary stars from the RV standard star catalog of Y. Huang et al. (2018). The temperatures of the inactive template spectra were chosen to closely match the components of binary, with a difference not exceeding 200 K. Second, we downloaded these template spectra from LAMOST DR10, removing the cosmic rays and normalizing the spectra. Third, the Fortran code STARMOD was used to construct the synthetic spectra of the binaries, taking into account the RV, the rotationally broadening, and the relative weight of the two components. The values for the RV and rotationally broadening were referenced from the LAMOST LR spectra, while the weights of the two components were derived from the W-D solution. These parameters were also treated as free during the STARMOD process. Finally, the subtracted spectra between the LAMOST observed spectra and the synthetic spectra were obtained and displayed in Figure 7, showing only the Hα region.

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The blue lines in Figure 7 represent the subtracted spectra, which show the varying intensities of the emission lines. We used the Python package PySpecKit to calculate the EWs of the Hα emission lines, which are listed in Table 2. In previous studies, stars with EW of Hα greater than 0.75 Å were considered magnetically active (A. A. West et al. 2011, 2015). Therefore, we infer that J0047, J0305, J0638, and J1402 exhibit strong magnetic activities.

According to the W-D code DC and LC program, the orbital and absolute physical parameters of the 10 targets were determined, including the mass ratio q, inclination i, fill-out factor f, semimajor axis a, masses of the two components (M₁, M₂), radii of the two components (R₁, R₂). The luminosities of two components (L₁, L₂) are determined by the Stefan–Boltzmann law. All orbital and absolute physical parameters are tabulated in Table 7. The RV data for these 10 targets have been obtained for the first time. Note that four of them have been systematically analyzed in previous studies, with their physical parameters being estimated. Each of these four targets will be discussed below.

• (i)  
  For J0132, X. Liu & H. Tan (1991) reported a mass ratio of 0.5947 (M₂/M₁), and D. P. Kjurkchieva et al. (2019) gave a value of 0.635 (M₂/M₁). In this work, the mass ratio was determined as 0.676 (M₁/M₂).
• (ii)  
  For J0305, A. Panchal & Y. C. Joshi (2021) reported a mass ratio of 0.31 (M₂/M₁), and our analysis gives a value of 0.35 (M₁/M₂).
• (iii)  
  For J1300, the mass ratios previously determined by D. P. Kjurkchieva et al. (2018) and Y. Yang et al. (2023) were 4.66 (M₂/M₁) and 4.747 (M₂/M₁), respectively, which are approximately the same as the value of 4.51 (M₂/M₁) obtained in this work.
• (iv)  
  For J1402, Y.-G. Yang (2011) and K. B. Alton & K. Stepień (2021) reported mass ratios of 0.461 (M₁/M₂) and 0.443 (M₁/M₂), respectively. The mass ratio of 0.457 (M₁/M₂) determined in this study is also consistent with their results.

Therefore, the mass ratios obtained for these four targets in our study are generally consistent with those reported in previous works. For J0132 and J0305, the temperatures of the two components may have been assigned to different components due to their small temperature differences.

With the LC minima collected from superWASP, TESS, ZTF, ASAS, CSS, ASAS-SN, BRNO, TMTS, and other literature, we analyzed orbital period variations for the targets. The final analysis revealed that all targets provided the corrected initial epochs and orbital periods. Except for J1042, the periods of the other nine targets exhibit a long-term increasing or decreasing trend, with three targets also showing cyclic period variations.

The long-term period decrease or increase can probably be explained by the mass transfer between two components. Assuming the conservation of mass and angular momentum, we used the following equation to calculate the rate of mass transfer,

By combining the period change (dp/dt) of each target and the masses of the two components provided in Tables 5 and 7, we calculated the (dM₁/dt, rate of mass transfer) for each target, which are listed in Table 5. The positive value indicates that the primary star M₁ is receiving mass, while the negative value indicates that the primary star M₁ is losing mass.

A long-term decreasing period is caused by mass transfer from the more massive star to the less massive star. This explanation is applied to J0047, J0638, and J1402. Therefore, we calculated the timescale of mass transfer () and the thermal timescale () for these three targets, and listed in Table 8.

Table 8. The Mass Transfer Timescale and Thermal Timescale for J0047, J0638, and J1402

Name	× 107 yr	× 107 yr
J0047	0.21	5.20
J0638	0.10	9.85
J1402	0.43	26.18

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The two timescales for these three targets differ significantly, with the mass transfer timescale being 4.00%, 1.04%, and 1.64% of the thermal timescale, respectively. Therefore, mass transfer cannot explain the long-term decrease in period for J0047, J0638, and J1402. The long-term period decrease in these targets is probably due to AML. Furthermore, the possibility that O − C of J0047 and J0638 is part of a cyclic variation cannot be ruled out.

For the other seven targets, the long-term increasing trend of the period may be due to mass transfer from the less massive star to the more massive one. Due to the conservation of angular momentum, the distance between the two components increases as their mass transfer, leading to a decrease in the degree of contact. The systems will evolve from the current contact state to a semidetached or detached state. According to stellar evolution theory, the more massive star will fill its Roche lobe first. As a result, the mass will be transferred from the more massive star to the less massive one. The mass and energy are then exchanged between the two components, leading to the evolution toward a contact state again. This is called the thermal relaxation oscillation model (TRO) of contact binaries (B. P. Flannery 1976; L. B. Lucy 1976; J. A. Robertson & P. P. Eggleton 1977). Therefore, long-term monitoring of these targets is necessary in the future.

The cyclic variation of the O − C parameter can be explained by the light travel-time effect (LTTE) due to the presence of a companion star or by a magnetic activity cycle. The existence of a third body is often used to explain the cyclic variation of orbital period, such as BK Vul (S. Adalalı & E. Soydugan 2024); CW Aqr (A. Vijaya & K. Sriram 2023); CSS_J154915.7+375506 (J.-F. Wu et al. 2024). If J0132, J1300, and J1402 are in triple systems, we can apply the function below to describe them:

where G and P₃ are the gravitational constant and the period of the (O − C)₂ oscillation, respectively. The amplitude of the oscillation is , where is the distance between binary and barycenter of the triple body system , c is the speed of light and is the inclination of the orbit of the third component. M₁ and M₂ represent the mass of the two components, and M₃ is the mass of the third body. We determined f(m) of the additional component for the three targets. The orbital distance between the third body and the central binary can be estimated as a₃ = (M₁ + M₂) × a₁₂/M₃. If the orbital inclination of the third body is the same as the binaries ( = i) for the three targets, respectively, the mass and the distance of the tertiary companion are calculated. All parameters are listed in Table 6. Therefore, if the third body is a main sequence star, the spectral type and the luminosity were determined according to the relation of A. N. Cox (2000). Finally, the M₃ of J0132 is too small for it to be a main sequence star and may be a brown dwarf. The spectral type and the proportion of the third body to the total luminosity for J1300 and J1402 are determined and listed in Table 6.

Another possible mechanism is magnetic activity. The Applegate mechanism (J. H. Applegate 1992) involves magnetic activity that produces a change in the variation of the quadrupole moment and, finally, the change in the orbital period. This can be described by using the following equation (A. F. Lanza & M. Rodonò 2002)

where ΔP is the period of cyclic variation, P is the orbital period, M is the mass for each component of the binary system, and a is the semimajor axis of binary system. The quadrupole momenta of the required variation of both components (ΔQ₁ and ΔQ₂) were determined and tabulated in Table 6. The typical value is usually 10⁵¹–10⁵² g cm² for close binaries (A. F. Lanza & M. Rodonò 1999), and ΔQ = 10⁴⁹ g cm² for cataclysmic variables (A. F. Lanza & M. Rodonò 1999). Although the quadrupole momenta of many binaries are not in agreement with the typical value, such as DZ Psc (Y. G. Yang et al. 2013), V1101 Her (Q.-f. Pi et al. 2017), and V0474 Cam (D. F. Guo et al. 2018). They are all similar with the value of 10⁴⁹ g cm². Thus, we cannot exclude the possibility of magnetic activity, especially for J1402. The light curve of J1402 exhibits the O’Connell effect and its spectra show strong magnetic activity.

Therefore, the periodic change for J1300 and J1402 may be caused by the existence of a dim third body, but the Applegate mechanism cannot be excluded because both show evidence of magnetic activity. The cyclic variation of O − C for J0132 may be due to magnetic activity. More observational data are needed in the future to confirm this.

With the absolute physical parameters, the positions of binaries on the mass–luminosity (M–L) and mass–radius (M–R) diagrams can be determined. In Figure 8, the solid and dotted lines show the zero age main sequence (ZAMS) and the terminal age main sequence (TAMS), constructed with the help of the binary star evolution code (BSE; J. R. Hurley et al. 2002).

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The more massive stars of all binary samples except J0132 and J0913 are located between the ZAMS line and the TAMS line, while the less massive stars are located above the TAMS line. This indicates that the currently more massive star is a main sequence star, and the currently less massive star is overluminous and oversized relative to main sequence counterparts with the same mass. This excess in luminosity and radius may be explicable by developments in the early evolution of the system. For J0132, both components are located in the main sequence, while for J0913, both components are above the TAMS. This suggests that these two targets may have evolved more slowly or more rapidly compared to the other eight targets.

Following P. E. Christopoulou & A. Papageorgiou (2013), we used the equation below to describe the relationship between mass and angular momentum J_orb of contact binaries:

where J_orb, M_T, P, and q represent the angular momentum, total mass of the binary system, orbital period, and the mass ratio, respectively. Combining the values of these parameters, the angular momentum of these 10 targets was determined and listed in Table 7. The relationship of M_T and J_orb for detached binaries and overcontact binaries is shown in Figure 9, where the boundary line separates detached binaries and overcontact binaries. Note that the data for the detached binaries in the sample were collected from Z. Eker et al. (2006), while the data for overcontact binaries were collected from Z. Eker et al. (2006) and K. Yakut & P. P. Eggleton (2005). The positions of 10 targets are marked in Figure 9. It can be seen that all targets are below the boundary line, and J0132 and J0913 are very close to it. Many researchers have proposed that the W UMa binaries may be formed from short-period detached binaries by AML (e.g., K. Stepien 2006; K. Stepień 2011; M. Yildiz & T. Doğan 2013; M. Yıldız 2014; S.-B. Qian et al. 2017). Considering their positions on the M–L and M–R diagrams in Figure 8, along with the overall trends and their low fill-out factors, we infer that these two targets may be newly formed contact binaries.

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