We present the TESS and BVRcIc light curves solution of four low massratio contact binary systems TIC 159102550 V1068 Her MW Pav and TIC 321576458. Except MW Pav all three systems have been studied for the first time. The period analysis of TIC 159102550 show anticorrelation between primary and secondary minima and no long term variation is reported. The systems V1068 Her and MW Pav show increasing orbital period trends. Data for TIC 321576458 is too few to determine any periodic variations. The light curve analysis using WilsonDivinney model shows that systems are low massratio contact binaries. Out of four targets two systems TIC 159102550 and V1068 Her are shallow contact binary systems with fillout factor of 20 and 14 respectively. The two contact binaries MW Pav and TIC 321576458 are in deep contact state with fillout factor 63 and 61 respectively. V1068 Her shows EBtype light curve however the temperature difference between the primary secondary component is only 17K which indicates that system is in thermal contact. To understand the evolutionary status of these systems the components are plotted on the massluminosity diagram. The primary companions are in the ZAMS zone while the secondary components of all the systems are away from TAMS which indicates that secondary is more evolved than the primary components. V1068 Her and MW Pav are expected to evolve into a single rapidly rotating star provided that they meet the wellknow Huts criterion. Through statistical investigation of more than hundred low massratio contact binary systems including our targets we have found that all of the low massratio contact binaries have undergone the massratio inversion process. Based on our sample the relationship between mass ratio and spin and orbital angular momentum ratio has been updated and proposed a new value of qmin  0.0388 for Darwins stability.
['Astrophysics - Solar and Stellar Astrophysics']
Multicolor and TESS photometric investigation of four lo massratio contact binary systems
{'MULTI-COLOR AND TESS PHOTOMETRIC INVESTIGATION OF FOUR LOW MASS-RATIO CONTACT BINARY SYSTEMS': "Ahmed Waqas Zubairi, 1, 2 Zhao Ergang, 1, 3, 4 Qian Shengbang, 5 Zhou Xiao, 1, 3, 4 and Eduardo Fern'andez Laj'us 6, 7  \n1 Yunnan Observatories, Chinese Academy of Sciences (CAS), , 650216 Kunming, China  \n2 University of Chinese Academy of Sciences, No.1 Yanqihu East Rd, Huairou District, Beijing, PR China, 101408  \n3 Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, P. R.  \nChina  \n4 Key Laboratory of the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, P. O. Box 110, 650216 Kunming,  \nChina  \n5 Department of Astronomy, Yunnan University, Kunming 650091, P. R. China  \n6 Facultad de Ciencias Astron'omicas y Geof'ısicas, Universidad Nacional de La Plata, Paseo del Bosques/n, 1900 La Plata, Buenos Aires, Argentina  \n7  \nInstituto de Astrof'ısica de La Plata (CCT La Plata - CONICET/UNLP), 1900 La Plata, Argentina", 'ABSTRACT': "We present the TESS and BVR c I c light curves solution of four low mass-ratio contact binary systems TIC 159102550), V1068 Her, MW Pav and TIC 321576458. Except MW Pav, all three systems have been studied for the first time. The BVR c I c observations of southern contact binary system, MW Pav, is carried out using 60cm Helen Sawyer Hogg (HSH) telescope located at Argentina. The TIC 321576458 is observed in BVR c I c with 70cm Sino-Thai telescope at Lijiang station of Yunnan Observatories. The period analysis of TIC 159102550 show anti-correlation between primary and secondary minima and no long term variation is reported. The systems V1068 Her and MW Pav show increasing period at dP/dt = 1 . 247(9) × 10 -7 days/yr and dP/dt = 2 . 256(6) × 10 -7 days/yr , respectively. Data for TIC 321576458 is too few to determine any periodic variations. The light curve analysis using Wilson-Divinney model shows that systems are low mass-ratio contact binaries. Out of four targets, two systems TIC 159102550 and V1068 Her, are shallow contact binary systems with fill-out factor of 20% and 14%, respectively. The two contact binaries MW Pav and TIC 321576458 are in deep contact state with fill-out factor 63% and 61%, respectively. V1068 Her shows EB-type light curve, however the temperature difference between the primary secondary component is only 17 K , which indicates that system is in thermal contact. To understand the evolutionary status of these systems, the components are plotted on the mass-luminosity diagram. The primary companions are in the ZAMS zone while the secondary components of all the systems are away from TAMS which indicates that secondary is more evolved than the primary components. V1068 Her and MW Pav are expected to evolve into a single rapidly rotating star provided that they meet the well-know Hut's criterion. Through statistical investigation of more than hundred low mass-ratio contact binary systems including our targets, we have found that all of the low mass-ratio contact binaries have undergone the mass-ratio inversion process. Based on our sample, the relationship between mass ratio and spin and orbital angular momentum ratio has been updated and proposed a new value of q min = 0 . 0388 for Darwin's stability.  \nKeywords: binaries: eclipsing; -binaries: close; - stars: evolution; -stars: individual: TIC 159102550, V1068 Her, MW Pav & TIC 321576458", '1. INTRODUCTION': "The two Roche lobe filled components of an eclipsing binary star formed a common convective envelope are referred as W UMa type eclipsing binary systems or simply contact binaries (Kopal 1956; Lucy 1968). Theses binaries are further divided into two categories; A-type and W-type (Binnendijk 1970; Lucy 1968). The continuous variation in the light curve along with small difference between primary and secondary minima put them in EW-type eclipsing binary  \nsystem. The small difference (nearly equal) depths in both minima indicates that the temperature of primary and secondary star is almost equal. However, the variation in magnitude at maxima shows the magnetic activity which can be explained with the presence of star spots (O'Connell 1951). The orbital period study of contact binaries is also important. The orbital and spin angular momentum plays an important role in maintaining orbital stability (Hut 1980; Counselman 1973) and provide useful information about the rate of mass transfer and angular momentum loss and help us to identify the potential merger candidates (e.g., Ferreira et al. (2019)).  \nThe widely accepted point of view about the formation of these system is via evolution from detached binaries due to AML or Kozai cycle. The strong interaction of binary companions makes their evolutionary scenario totally different compared to single star evolution. Some marginal contact systems are believed to be newly-formed contact binaries and are at the beginning of their contact phase (Qian et al. 2017; Qian et al. 2018, 2020). Their evolutionary process can be explained by the combination of angular momentum loss (AML) and thermal relaxation oscillation (TRO). Such system oscillate around a critical mass-ratio (Lucy 1976; Flannery 1976; Robertson & Eggleton 1977; Qian 2001, 2003; Zhu et al. 2014). These systems are expected to merge sooner into FK Comae type or blue straggler stars (Qian et al. 2005b, 2008; Zhu et al. 2014; Sriram et al. 2016; Zhu et al. 2016; Gao et al. 2022). Therefore, contact binaries, especially with low mass-ratio, work as natural laboratories for studying the mass and energy transfer, the angular momentum loss of the system, and stellar evolution & merging processes.  \nIn this paper, we are presenting photometric investigation of four low mass-ratio contact binary systems. As a time of this writing, three systems (TIC 159102550, V1068 Herand TIC 321576458) have not been studied in detailed before. We are presenting the first photometric investigation for these systems. The MW Pav(MW Pavonis) is a low-mass ratio contact binary system in southern hemisphere. The first photometric solution was presented by Eggen (1968). Later on, Williamon (1971) and Lapasset (1977) also observed MW Pavbut did not present any detailed solution. The radial velocity solution of Rucinski & Duerbeck (2006) reported the spectral type to be F3/IV-V along with spectroscopic mass ratio of q sp = 0 . 228 which was not consistent with their photometric mass-ratio ( q ph = 0 . 122). Deb & Singh (2011), uses the spectroscopic mass-ratio of Rucinski & Duerbeck (2006) and face problems in light curve fitting. However, they proposed the photometric mass-ratio of q = 0 . 200 with 52% degree of contact. More recently, Alvarez et al. (2015), presented combined light and velocity curve solution. Their UBV photometric solution suggest a new mass-ratio q = 0 . 222 and a high fill-out factor of 60% for the system.", '2.1. TESS Observations': "The Transiting Exoplanet Survey Satellite (TESS) has rapidly increased the eclipsing binaries research (Ricker et al. 2014). The obtained photometric data of a number of eclipsing binaries can be downloaded from Mikulski Archive for Space Telescopes (MAST) archive. 1  \nThe TIC 159102550( α 2000 = 19 h 18 m 17 . 44 s , δ 2000 = +42 0 02 ' 47 . 39 '' ) is listed as EW type binary in VSX catalog. The system is observed by TESS from May 14, 2020 to June 08, 2020 in sector 25 and 26 respectively. The data of V1068 Her(=TIC 193580427) ( α 2000 = 17 h 43 m 23 . 08 s , δ 2000 = +47 0 51 ' 41 . 79 '' ) is available from July 24, 2021 to August 20, 2021 in sector 41.  \nBoth of these two targets (TIC 159102550 and V1068 Her) were also observed by LAMOST (Wu et al. 2011; Cui et al. 2012). The obtained atmospheric parameters are listed in Table 1.  \nTable 1. Atmospheric parameter of TIC 159102550 and V1068 Her observed by LAMOST", '2.2. BVR c I c Observations': "MW Pav(=TIC 372127422) ( α 2000 = 20 h 46 m 27 . 74 s , δ 2000 = -71 0 56 ' 58 . 46 '' , V m ag = 8 . 79) has been observed with 60cm Helen Sawyer Hogg (HSH) telescope at Complejo Astronomico El Leoncito (CASLEO), San Juan, Argentina.  \nThe new multi-color light curve is taken during three nights on August 09, 15, 20, 2019. These observations were carried out using SBIG STL1001E CCD camera with standard Jhonson's BVR c I c filters attached to the HSH telescope giving 9 . 3 ' × 9 . 3 ' field of view. MW Pavis also observed by TESS from July 05, 2020 to September 30, 2020 in sector27 using 2-minutes cadence. The differential magnitudes from HSH observations are determined using the standard aperture photometry procedure in Image Reduction and Data Analysis Facility (IRAF) software packages (Tody 1986, 1993).  \nTIC 321576458( α 2000 = 15 h 42 m 12 . 72 s , δ 2000 = +37 0 52 ' 46 . 50 '' , V m ag = 13 . 74) has been observed with 70cm SinoThai telescope at Lijiang station of Yunnan Observatories. The first BVR c I c light curve of TIC 321576458is obtained on April 13 and April 17, 2023. The telescope is equipped with Andor iKon 936 (2048x2048) CCD camera with standard Jhonson's BVR c I c filters. The system is also observed by TESS from July 05, 2020 to September 30, 2020 in sector-27 using 2-minutes cadence. The differential magnitudes from the observations of 70cm Sino-Tahi telescope are determined using the standard aperture photometry procedure in Image Reduction and Data Analysis Facility (IRAF) software packages (Tody 1986, 1993).", '3. ORBITAL PERIOD INVESTIGATION': '3.1. Period Study of TIC 159102550  \nThe period search of TIC 159102550 gives 0 . 488253 d from ROTSE All-Sky Surveys (Akerlof et al. 2000). No other database has any information on previously published period of TIC 159102550. Therefore, the O-C values are determined by using epoch MinI = BJD 2459446 . 1702 + 0 . 488253 E .  \nFigure 1. O-C diagram of TIC 159102550. The primary and secondary minimas shows anti-correlation.  \n<!-- image -->  \nThe Figure 1 shows that primary and secondary minimas are anti-correlated with each other. It further indicates that the period should be corrected from left upper panel of Figure 2 for the primary minima. The revised ephemeris and period are given in Equation 1. A sample of times of primary and secondary minima are listed in Table 3.1.  \nMinI = BJD 2459446 . 1702(7)+ 0 . 48823883(8) × E (1)  \nWe reconstruct the O-C diagram with the corrected period and ephemeris which is displayed in right panel of Figure 2. The O-C for primary minima does not displays any large-scale variation as shown in upper panel of Figure 2, which means that the revised period gives a better value than before. However, the corrected period from secondary minima  \nshows a different behavior as shown in below panel of Figure 2. The revised O-C values are also tabulated in Table 3.1.  \nFigure 2. O-C diagram of TIC 159102550. Left panel is calculated by the original period for the primary minima and display the revised one. The right panel is the same but for secondary minima.  \n<!-- image -->  \nTable 2. Times of minima for TIC 159102550. The BJD are +2400000.', '3.2. Long term Period variation of V1068 Her': 'The newly determined times of minima of V1068 Her along with values available in literature are listed in Table 3. All HJDs are converted to BJD using https://astroutils.astronomy.osu.edu/time/hjd2bjd.html. The new O-C values are calculated by using the following epoch MinI = BJD 245934 . 9772+0 . 394304 E . The trend of O-C values indicate a upward parabolic fit, as shown in the Figure 3. The least square method yields that period of V1068 Her is increasing. We obtained the new ephemeris and period of V1068 Her as shown in Equation 2.  \nFigure 3. O-C diagram of V1068 Her with all available times of minima. The parabolic fit is shown in black line. Residuals are plotted in the bottom panel.  \n<!-- image -->  \nMinI = 2459034 . 9767(6) + 0 . 39430544(2) × E +6 . 73(5) × 10 E  \n-11 2 (2)  \nThe quadratic term in Equation 2 leads us to determine the period increase at rate of dP/dt = 1 . 247(9) × 10 -7 days/yr .  \nTable 3 . Times of minima for V1068 Her. The BJD are +2400000  \nTable 3 . Times of minima for V1068 Her. The BJD are +24000003.3. Long term Period variation of MW Pav  \nThe first observations were presented by Eggen (1968) with a period of 0.562979 days. Later on, the different period of MW Pav, was reported by many authors (Williamon 1971; Lapasset 1977). The radial velocity solution of Rucinski & Duerbeck (2006) suggests that the period of system is 0 . 7949810 days . The O-C gateway 2 and the spreadsheets of Bob Nelson, maintained at AAVSO database (Bob 2011) contains most of the data which we have used during period investigation. Table 4 contains all the values that have been collected from the literature, BVR c I c filters at HSH telescope and TESS data. All HJDs are converted into BJD. The O-C diagram indicates parabolic fit  \nas shown in the Figure 4. The least-square method was applied to obtain new ephemeris and period of MW Pav as shown in Equation 3.  \nMinI = 2458704 . 61313(8)+ 0 . 79499753(5) × E +2 . 417(3) × 10 -10 E 2 (3)  \nThe quadratic term in Equation 3 helped us to determine the period increase rate of dP/dt = 2 . 220(9) × 10 -7 days/yr . If we compare this value with Alvarez et al. (2015), it can be noted that the rate of increase in period has been boosted up by 2 . 213 × 10 -7 days/yr .  \nFigure 4. The O-C diagram of MW Pav is constructed using MinI = HJD 2458704 . 6089 + 0 . 79498855 E . Black line shows parabolic fit. Residuals are plotted in the bottom panel.  \n<!-- image -->  \nTable 4 . Times of minima for MW Pav. The BJD are +2400000  \nTable 4 . Times of minima for MW Pav. The BJD are +2400000  \nTable 4 . Times of minima for MW Pav. The BJD are +24000003.4. Period Study of TIC 321576458  \nThe period of TIC 321576458was first reported to be 0.37614d by (Drake et al. 2014). No times of minimas are available for this target as of this writing. We have determine new times of primary and secondary minima from our  \nBVR c I c observations and TESS light curves using Kwee & Wan Woerden algorithm (Kwee & van Woerden 1956). The O-C diagram is shown in Figure 5 and the times of primary and secondary minima are tabulated in Table 5. The figure clearly shows no variation in the orbital period. The new period and ephemeris are shown in Equation 4.  \nMinI = 2460052 . 3995(2)+ 0 . 376144829(1) × E (4)  \nFigure 5. The O-C diagram of TIC 321576458 is constructed using MinI = BJD 2460052 . 39813 + 0 . 37614 E . Residuals are plotted in the bottom panel.  \n<!-- image -->  \nTable 5 . A sample of Times of minima for TIC 321576458. The BJD are +2400000  \nTable 5 . A sample of Times of minima for TIC 321576458. The BJD are +2400000', '4. LIGHT CURVE SOLUTIONS': "We use Mode-3 of the Wilson-Divenney (WD) code to obtain the solution of light curves (Wilson & Devinney 1971; Wilson & Van Hamme 2014). According to Lucy (1967) and Ruci'nski (1969), the gravity darkening coefficients ( g 1 , g 2 ) and albedos ( A 1 , A 2 ) should be 0 . 32 and 0 . 5, respectively as these values are suitable for binaries with convective envelopes. The limb darkening parameters were obtained from van Hamme (1993). As per Mode-3 constraint in the WD program, the fixed parameters were g 1 , g 2 , A 1 , A 2 , x 1 , x 2 , y 1 , y 2 . The gravitational potentials are kept equal (Ω 1 = Ω 2 ) for contact configuration.", '4.1. TIC 159102550': 'In the absence of spectroscopic and photometric mass-ratio of this target, we apply the well-known q-search process to estimate the mass-ratio of the system. According to Pribulla et al. (2003) a reliable mass-ratio can be obtained for totally eclipsing binary systems. The results showed in Figure 6 indicates q = 0 . 20 for TIC 159102550.  \nWe use this q-value and keep it adjustable along with T 2 , Ω 1 , i, q, and L 1 to get the final convergent solution. The temperature of primary component ( T 1 ) was opted from LAMOST data (Table 1). The modeled and observed data showed variation in amplitude of the light curve at phase 0.25 and 0.75. In order to cover this discrepancy in light curve solution, we introduced a cool spot on primary component for which parameters are listed in Table 8. The final light curve solution is shown in Figure 7.  \nThe obtained photometric parameters are listed in Table 6', '4.2. V1068 Her': 'Since no photometric and spectroscopic mass ratio of this target was avaialbel in the literature, so we employed the q-search process to determine the unique value for the mass-ratio. The results of q-search method indicate the q = 0 . 28 for V1068 Her. The results are displayed in Figure 6.  \nWe use this q-value and keep it adjustable along with T 2 , Ω 1 , i, q, and L 1 to get the final convergent solution. The temperature of primary component ( T 1 ) was opted from Table 1. The convergent solution when superimposed on the observed data shows difference in Max-I and Max-II values. Therefore, we performed a spot solution on the light curve. The obtained photometric parameters are listed in Table 6 and the spot parameters are tabulated in Table 8. The final light curve solution is shown in Figure 7.  \nFigure 6. The q-search results of TIC 159102550, (top left) V1068 Her(top right) and TIC 321576458(bottom).  \n<!-- image -->  \nFigure 7. The light curve solution of TIC 159102550 (left) and V1068 Her(right). The modeled (solid green line) is superimposed over observed light curve.  \n<!-- image -->  \n4.3. MW Pav  \nTable 6. Photometric parameters of TIC 159102550 and V1068 Her  \nThe solution of Rucinski & Duerbeck (2006) suggests photometric mass-ratio of the system be q ph = 0 . 122 and q sp = 0 . 228. They also determined the spectral type of MW Pav to be F3/IV-V. Deb & Singh (2011), uses the spectroscopic mass-ratio of Rucinski & Duerbeck (2006) and face problems in light curve fitting. However, they proposed the photometric mass-ratio of q = 0 . 200 with a fill-out factor of 52%. More recently, Alvarez et al. (2015), presented combined light and velocity curve solution. Their UBV photometric solution suggest a new mass-ratio q = 0 . 222 and a high fill-out factor of 60% for the system.  \nWe have analysed the new BVR c I c and TESS light curves of this target. The initial value of mass ratio ( q = 0 . 228) and the proposed spectral type of MW Pav by Rucinski & Duerbeck (2006) help us to set the temperature of primary component at T 1 = 6900 K from Cox (2000). We use this q-value and get the convergent solution in Mode-3 of the WD program. The difference in Max-I and Max-II values compelled us to determine the spot parameters. We, therefore, run a spot solution for both, BVR c I c and TESS, light curves. The final photometric solution is shown in Figure 8 and the parameters are listed in Table 7.  \nFigure 8. The BVR c I c light curve of MW Pav is shown in the left and the TESS curve is on right.  \n<!-- image -->  \na  \n-  \nSince this target is also observed for the first time and no other photometric and spectroscopic mass-ratio is available in the literature. Therefore, we employed the q-search process to determine the initial mass-ratio of the system. The results are shown in Figure 6 which indicates the q=0.143. By using this q-value and keep it adjustable along with T 2 , Ω 1 , i, q, and L 1 to get the final convergent solution. The temperature of primary component ( T 1 ) was taken from GAIA data. The modeled light curve fits well over the observed light curve and therefore not spot solution was performed. The fitted light curves are shown in Figure 9.', '5.1. TIC 159102550': "By combining all times of observed primary and secondary minima, we have constructed the O-C diagrams to investigate the period changes. The O-C diagram of TIC 159102550 hints to correct the period. After period correction, we determined new values of O-C and reconstructed the diagram. The O-C for primary minima displays no largescale change as shown in upper panel of Figure 2, which means that the revised period gives a better value than before. If we apply the same to secondary minima, it shows a different behavior as shown in below panel of Figure 2. The O-C of many targets has variable change, which include regular and irregular, such as long-term increasing and decreasing (Zhao et al. 2021; Li et al. 2022b; Zhao & Tian 2022). This can be explained by mass-transfer between both components, a third body, magnetic activities or apsidal motion (Er-gang et al. 2019; Li et al. 2022a). Sudden  \nTable 7. Photometric parameters of MW PavTable 8. Star spot parameters for TIC 159102550, V1068 Her and MW Pav  \na Assumed  \nFigure 9. Combined light curves of TIC 321576458. The BVR c I c light curve of is shown in the left and the TESS curves are on right side. Solid line represents the modeled solution.  \n<!-- image -->  \nPhase  \ndMag  \n<!-- image -->  \nPhase  \nTable 9. Photometric parameters of MW Pav  \nchange and even anti-correlations between the primary and secondary eclipse timing has also been reported. This anti-correlation behaviour can be explained with rapid variation in light curve due to migration of star spot over time. This phenomena can deviate the eclipse times from theoretical phase which results in different trend in O-C diagram over a small timescale (Tran et al. 2013; Zhou et al. 2019; Chang et al. 2023).  \nIn the absence of spectroscopic parameters, absolute values of binary components cannot be determined directly. However, if we assume that the primary component is a main-sequence star, then we can estimate the mass of secondary component from Cox (2000). The absolute parameters are tabulated in Table 10. The photometric solution suggest that system is an A-type low mass-ratio ( q = 0 . 211(2)) contact binary with a weak degree of contact (20 . 74%). It's secondary component is an evolved star while the more massive primary is still on the main-sequence as shown in Figure 10. We suggest continuous monitoring of this target.", '5.2. V1068 Her': 'The period investigation of V1068 Her suggest that binary is going through increase in period by dP/dt = 1 . 247(9) × 10 -7 days/yr . The continuous increase in period indicates that binary is undergoing a process of mass transfer from less massive secondary to more massive primary component. We can use the well-known equation (Singh & Chaubey 1986; Carroll & Ostlie 2017)  \ndP/dt P = 3 dM 2 dt ( 1 M 2 -1 M 1 ) (5)  \nand determine the mass transfer rate of 4 . 304(5) × 10 -8 M /circledot yr -1 at thermal time-scale of 6 . 481(5) × 10 6 yr for V1068 Her.  \nSince no absolute parameters of this target was previously published, so we have estimated the mass of primary component from Cox (2000). All the parameters are listed in Table 10. The photometric solution indicates the massratio of q = 0 . 266(3) and shallow degree of contact ( f = 14 . 11%). The system lies in A-subtype category of contact binaries. Its more massive primary component is a main-sequence star while the less massive secondary companion is an evolved star. With the transfer of mass from secondary to primary component, the mass-ratio of the system will further decrease. If the mass transfer process is conservative, the period change of V1068 Hersuggest that the system will oscillate around a critical mass-ratio. Since both component of binary are in thermal contact (∆ T = 15 K ), they may not be oscillating between contact and semi-detached state. When the mass-ratio decreases to critical value the period will decrease which indicates the reversal in mass transfer direction. Therefore, the mass-ratio will start increasing and as soon as it increases the critical value, the period of V1068 Her will start increasing again and the system will keep on oscillating. A critical AML rate will keep the V1068 Herin shallow contact stage. The larger AML rate will make the stars to reach at the outer Lagrangian point ( L 2 ) before merger into a single star (Rahunen 1981; Qian 2003; Qian et al. 2013).', '5.3. MW Pav': "The long term period analysis of MW Pav indicates the increase in period by 2 . 220(9) × 10 -7 days/yr . The continuous increase in period indicates that both systems are undergoing a process of mass transfer from less massive secondary to more massive primary component. We have used the Equation 5 and determined that system is transferring the mass from less massive secondary to more massive primary component at the rate of 2 . 307(6) × 10 -8 M /circledot yr -1 in thermal time-scale of 1 . 11(4) × 10 7 yr .  \nThe BVR c I c light curve analysis shows that system is a W-subtype low mass-ratio ( q = 0 . 154(1)) contact binary in deep contact stage ( f = 71 . 57%). However, light curve solution of TESS data put it in A-subtype category. It is possible that binary has now changed from W-type to A-type contact system. Such phenomena has also been reported by Qian & Yang (2005) and Qian et al. (2008). We have opted the radial velocity data from Rucinski & Duerbeck (2006) and combined it with our photometric solution to determine the absolute parameters of MW Pav. They are presented in Table 10. With the increase in period and the transfer of mass from secondary to primary component, the mass-ratio of the system will further decrease. The increasing period will widen the orbit, however, the strong degree of contact, (71 . 51%) for HSH observations and (63 . 20%) for TESS, will hold the binary in contact configuration for a longer duration of time until the Hut's criteria (Hut 1980) is achieved ( J orbital ≤ 3 J spin ). The MW Pav will eventually evolve into single rapidly rotating star.", '5.4. TIC 321576458': "The O-C diagram of this target (Figure 5) does not hint of any period variation during the short time span, therefore, period change and mass transfer rate cannot be determined. Assuming that the primary component is a main-sequence star, we have estimated the absolute parameter from Cox (2000) and shown in Table 10. The photometric solution suggest that system is an W-type low mass-ratio ( q = 0 . 153(1)) with moderate fill-out factor of 41 . 7%. However, TESS light curve analysis put it in A-type category with higher contact degree of 61 . 2%. It's secondary component is an evolved star while the more massive primary is still on the main-sequence as shown in Figure 10. More times of primary and secondary minima will help to study the orbital period in detail. We suggest to monitor the target for a longer duration.  \nTable 10. Absolute parameters four contact binaries  \nDuring recent years, many authors have paid much attention to the statistical studies on contact binary systems. The analysis of orbital period distribution of several authors (e.g., (Lucy 1976), (Rucinski 2007)) showed a strong and sharp peak at nearly 0.22 days which is considered as short-period limit for contact binaries. Qian et al. (2017), through investigation of more than 40,000 systems reported 0.29 days and Li et al. (2020) established 0.30 days while analyzing 446 systems. Not only this, Eggen (1967), established an important period-color relationship for contact binaries which helped to the model of thermal relaxation oscillation theory. Yang & Qian (2015) also performed statistical analysis on the evolutionary scenarios of contact binary systems but his analysis was limited to deep contact binaries. We have discussed the distribution of low mass-ratio contact binary systems on the Mass-Luminosity (ML) diagram to study their evolutionary track. We have also calculated their initial mass, age and momentum ratio of these system. We have collected absolute parameters of 122 contact binaries, including our present study, to understand the evolutionary scenario of our target binaries. The data contains 47 extreme low mass-ratio (ELMR) binaries with q ≤ 0 . 100 tabulated in Table 11 and 75 low mass-ratio (LMR) binaries with q = 0 . 10 -0 . 25 listed in Table 12.  \nTable 11. Absolute parameters of extreme low mass-ratio contact binaries  \nTable 12. Absolute parameters of low mass-ratio contact binaries  \nThe position of primary and secondary components of our targets along with other binaries listed in Table 12 is shown in Figure 10. The filled symbols are chosen to represent primary and hollow symbols represents secondary components of each contact binary system. Our targets are presented in blue color. From Figure 10, it is clear that most of the primary components, along with our observed targets, are residing around Zero-age Main-Sequence (ZAMS), whereas, the less massive secondaries are above the Terminal Age Main-Sequence (TAMS) line. This indicates that primary stars are little or un-evolved and secondaries have evolved out of main sequence zone. Our four targets are also following the same trend.  \nAccording to Yildiz & Do˘gan (2013); Yıldız (2014) the initial mass of both, primary and secondary, components also help us understand the evolutionary process of contact binaries. They proposed a new method and calculate the initial mass of primary and secondary components of a binary system. They also estimated the age of contact binary systems.  \nFigure 10. The mass-luminosity diagram of low mass ratio contact binaries. The solid symbols represents the primary components and the hollow are secondary components. Symbols with blue color shows our targets.  \n<!-- image -->  \nWe have employed the model proposed by Yildiz & Do˘gan (2013) to calculate the initial mass of both components. We have computed the initial mass of secondary component ( M 2 i ) in terms of its higher luminosity for present mass using Equation 6, with M 2 as the present mass of secondary and M L 2 as ( L 2 / 1 . 49) 1 / 4 . 216 in solar units. The initial mass of the primary component M 1 i is calculated by using γ = 0 . 6640, the same value as used by Yildiz & Do˘gan (2013). In order to calculate the age of our sample stars we have used the following equations 7 and 8. We have also compared our target with other low mass ratio binaries. The values of their initial masses are tabulated in Table 13  \nM 2 i = M 2 +2 . 50 × ( M L 2 -M 2 -0 . 07) 0 . 64 , (6)  \nM 2 mean = M 2 i + M L 2 2 (7)  \nτ MS = 10 ( M/M /circledot ) 4 . 05 × [ 0 . 00560 × ( M M /circledot +3 . 993 ) 3 . 16 +0 . 042 ] Gyr (8)  \nFor these binaries the ratio of spin angular momentum ( J s ) and orbital angular momentum ( J o ) is calculated using the equation 9 provided by Yang & Qian (2015) and is listed in Table 13  \nJ s J o = 1 + q q ( k 2 1 r 2 1 + qk 2 2 r 2 2 ) , (9)  \nThe value of gyration radii ( k 2 1 = k 2 2 = k 2 ) was opted from (Li & Zhang 2006) as 0 . 06. The ratio of spin and orbital angular momentum indicates that all LMR, including our targets, are in dynamically stable state. However, a few of the ELMR binaries have crossed the threshold value of 1 / 3. According to Li et al. (2021a), the most likely reason of this, is that the gyration radii (k) must be lower than 0.06. This means that the limit of mass ratio of contact binaries is affected by k 2 .  \nIf we keep the system V1187 Her exactly at the stability limit, then the value of k 2 is 0.03214. The relation between q and spin-orbital angular momentum ratio is shown in figure 11. The fitting function is shown in Equation 10.  \nJ s J o = 0 . 052( ± 0 . 002) + 0 . 64( ± 0 . 02) × e -21 . 15( ± 0 . 48) × q (10)  \nBased on our samples, the updated relationship between mass ratio and spin-orbital angular momentum ratio gives us a new value of q min = 0 . 0388 for Darwin's stability in contact binaries.  \nFigure 11. The relationship with mass ratio and J s /J o . The horizontal dashed line represents critical value for Darwin's stability. Left Figure shows the values with k 2 = 0 . 06 and right one with k 2 = 0 . 03214  \n<!-- image -->", '6. ACKNOWLEDGMENTS': 'This study is partly supported by the National Natural Science Foundation of China (Grant Nos. 11933008), Yunnan Natural Science Foundation (No. 202001AT070051) and Yunnan Revitalization Talent Support Program Young Talent Project. Some data of Present study is from LAMOST, which is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences. This work also used TESS and GAIA data and authors thanks a lot for the team of TESS and GAIA. This research has also made use of the International Variable Star Index (VSX) database, operated at AAVSO, Cambridge, Massachusetts, USA.  \nTable 13 . Momentum ratio, and initial mass of primary and secondary components of selected binary systems  \nContinuation of Table 13  \n*Revised value', 'REFERENCES': "Akerlof, C., Amrose, S., Balsano, R., et al. 2000, AJ, 119, 1901, doi: 10.1086/301321  \nAlvarez, G. E., Sowell, J. R., Williamon, R. M., & Lapasset, E. 2015, Publications of the Astronomical Society of the Pacific, 127, 742, doi: 10.1086/682388 Binnendijk, L. 1970, Vistas in Astronomy, 12, 217, doi: https://doi.org/10.1016/0083-6656(70)90041-3 Bob, N. 2011, Bob Nelson's database of eclipsing binary O-C files, https://www.aavso.org/bob-nelsons-o-c-files Brat, L., Trnka, J., Lehky, M., et al. 2009, Open European Journal on Variable Stars, 107, 1 Brat, L., Trnka, J., Smelcer, L., et al. 2011, Open European Journal on Variable Stars, 137, 1 Broens, E. 2013, MNRAS, 430, 3070, doi: 10.1093/mnras/stt113 Carroll, B. W., & Ostlie, D. A. 2017, An Introduction to Modern Astrophysics, 2nd edn., ed. S. F. P. 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