# Mass Transfer and Intrinsic Light Variability in the Contact Binary MT Cas

Authors: Haifeng Dai; Huiyu Yuan; Yuangui Yang
Journal: Advances in Astronomy
 (2019)

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## Abstract

First CCD photometry for the contact binary MT Cas is performed in 2013 in December. The spectral type of F8V is determined from the low-precision spectrum observed on 2018 Oct 22. With Wilson-Devinney code, the photometric solutions are deduced fromVRc light curves (LCs) and AAVSO’s and ASAS-SN’s data, respectively. The results imply that MT Cas is a W-type weak-contact binary with a mass ratio of q=2.365(±0.005) and a fill-out factor of f=16.6(±1.2)%, respectively. The asymmetric LCs in 2013 are modeled by a dark spot on the more massive component. By analyzing the (O-C) curve, it is discovered that the orbital period may be undergoing a secular increase at a rate of dP/dt=1.12(±0.09)×10-8dyr-1, which may result from mass transfer from the less massive component to the more massive one. With mass transferring, MT Cas may evolve into a broken-contact configuration as predicted by TRO theory.


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## Body

## 1. Introduction

W Ursae Majoris binary contains two components, which are embedded in a common envelope [1, 2]. Models for contact binary have been recently constructed by several investigators (e.g., see [3–5]). However, their evolutionary status still remains unclear because the spectra cannot be analyzed for abundances due to the extreme broadening and blending of spectral lines. Therefore, it is crucially important to observe contact binaries, which may provide some special phenomena and processes, such as magnetic activity [6], third body [7], angular momentum evolution [3], flare [8], and stellar coalescence [9]. It is helpful for us to understand their formation, structure, and evolution of contact binaries.MT Cas (=SV∗ SON 4671) was found by Götz & Wenzel [10] as a W UMa-type eclipsing binary. Its visual magnitude is 13.3 mag, and the depths of both eclipses are 0.83 mag and 0.72 mag, respectively [11]. Hoffmann [12] photoelectrically observed this binary. Unfortunately, the BV light curves did not cover the complete period. Pribulla et al. [13] derived a linear ephemeris with a period of 0.31387775 days, which was updated to be 0.3138789 days [14]. Except for some photometric data performed by several amateur observers of AAVSO (https://www.aavso.org/data-download) and ASAS-SN database [15] (https://asas-sn.osu.edu/database/light_curves/335960), no additional observations for this binary have been presented up to now.In this paper, the neglected binary MT Cas was studied photometrically and spectroscopically in Section2. The orbital period variation is analyzed in Section 3, and three sets of light curves (i.e., VRc LCs in 2013, AAVSO’s LC, and ASAS-SN LC) are modeled in Section 4. In Section 5, we estimated the absolute parameters and discuss mass transfer between two components and its evolutionary status.


## 2. New Observations

### 2.1. Photometry

CCD photometry for MT Cas was first carried out at six nights of 2013, with the 85-cm telescope [16] at the Xinglong station (XLs) of National Astronomical Observatories of China (NAOC). The standard Johnson-Cousins UBVRcIc systems were mounted onto this telescope. During the observation, VRc files are used in order to provide the enough high time resolution. The image reductions are done by using the Image Reduction (IMRED) and Aperture Photometry (APPHOT) packages in the Image Reduction and Analysis Facility (IRAF) in a standard mode. Differential magnitudes were then determined by aperture photometry.In the observing process, we chose TYC 3657-1637-1 (αJ2000=00h14m59.s4 and δJ2000=+54∘39′53.′′1) and TYC 3657-1245-1 (αJ2000=00h14m32.s6 and δJ2000=+54∘29′16.′′1) as the comparison and check stars, respectively. Typical exposure times are adopted to be 60s in V band and 50s in Rc band, respectively. In total, we obtained 383 and 373 images in V and Rc bands. The standard uncertainties are ±0.005 mag in V band and ±0.004 mag in Rc band. All individual observations (i.e., HJD and Δm) are listed in Table 1. The differential magnitudesversus orbital phases are displayed in Figure 1, where phases are computed by a period of 0.d3138789 (Kreiner et al. 2004). The VRc LCs in 2013 imply that MT Cas is an UMa-type eclipsing binary, whose amplitudes of variable light are 0.69 mag and 0.66 mag in V and Rc bands, respectively. There exists an unequal height between both maxima, i.e., O’Connell effect [17, 18]. Max.II at phase 0.75 is brighter than Max.I at phase 0.25 up to 0.025 mag and 0.029 mag in V and Rc bands, respectively. This kind of stellar activity occurs on other W UMa-type binaries, such as VW Cep [19], DV CVn [20], V532 Mon [21], and BB Peg (Kalomeni et al. 2007), and DZ Psc [22].Table 1
CCD photometric observations inV and Rc bands.
V band R c band HJD Δ m HJD Δ m HJD Δ m HJD Δ m  2456629.9265 + 0.066 2456632.9847 - 0.273 2456629.9275 + 0.269 2456632.9895 - 0.076 2456629.9285 + 0.106 2456632.9861 - 0.280 2456629.9294 + 0.322 2456632.9962 - 0.091 2456629.9304 + 0.133 2456632.9874 - 0.273 2456629.9314 + 0.349 2456632.9976 - 0.113 2456629.9323 + 0.176 2456632.9901 - 0.289 2456629.9333 + 0.384 2456632.9989 - 0.099 2456629.9343 + 0.225 2456632.9915 - 0.300 2456629.9352 + 0.413 2456633.0003 - 0.116 ...... 2456632.9766 - 0.235 2456636.0355 - 0.077 2456632.9814 - 0.057 2456632.9780 - 0.238 2456636.0373 - 0.041 2456632.9827 - 0.069 2456632.9793 - 0.263 2456636.0391 - 0.041 2456632.9841 - 0.069 2456632.9807 - 0.267 2456636.0409 + 0.007 2456632.9854 - 0.063 2456632.9820 - 0.270 2456636.0427 + 0.065 2456632.9868 - 0.081 2456632.9834 - 0.261   2456632.9881 - 0.076 Note. The entire table is available only on the online journal.Figure 1
V R c light curves of the contact binary MT Cas, which are observed in 2013 in December by using the 85-cm telescope. The dotted and solid lines are plotted by photometric solutions without or with a dark spot, respectively.


### 2.2. Low-Precision Spectrum

The low-precision spectrum for MT Cas was obtained by using theYunnan Faint Object Spectrograph and Camera (YFOSC), which is attached to the 2.4-m telescope at Lijiang station (LJs) of Yunnan Astronomical Observatory of China (YNAO) at UT 15:50:13 of 2018 October 22. During the observing process, we chose a 140-mm-length slit and aGrism-3 with a wavelength range from 3200Å to 9200Å [23]. The exposure time is 10 minutes. The phase of 0.98 almost corresponds to the observed middle time HJD 245414.1634. Reduction of the spectra was performed by using IRAF packages, including bias subtraction, flat-fielding, and cosmic-ray removal. Finally, the one-dimensional spectrum was extracted. With thewinmk software (http://www.appstate.edu/~grayro/MK/winmk.htm), we obtained a normalized spectrum, which is displayed in Figure 2. By comparing the spectra of standard stars [24], the spectral type is determined to be F8V for the primary (i.e., more massive component) of this binary because the secondary is eclipsed by the primary around phase 0.0 for the W-type contact binary (see Section 4).Figure 2
Spectroscopic observation for MT Cas, observed by the 2.4-m telescope at Lijiang station of YNAO on October 22, 2018.


## 2.1. Photometry

CCD photometry for MT Cas was first carried out at six nights of 2013, with the 85-cm telescope [16] at the Xinglong station (XLs) of National Astronomical Observatories of China (NAOC). The standard Johnson-Cousins UBVRcIc systems were mounted onto this telescope. During the observation, VRc files are used in order to provide the enough high time resolution. The image reductions are done by using the Image Reduction (IMRED) and Aperture Photometry (APPHOT) packages in the Image Reduction and Analysis Facility (IRAF) in a standard mode. Differential magnitudes were then determined by aperture photometry.In the observing process, we chose TYC 3657-1637-1 (αJ2000=00h14m59.s4 and δJ2000=+54∘39′53.′′1) and TYC 3657-1245-1 (αJ2000=00h14m32.s6 and δJ2000=+54∘29′16.′′1) as the comparison and check stars, respectively. Typical exposure times are adopted to be 60s in V band and 50s in Rc band, respectively. In total, we obtained 383 and 373 images in V and Rc bands. The standard uncertainties are ±0.005 mag in V band and ±0.004 mag in Rc band. All individual observations (i.e., HJD and Δm) are listed in Table 1. The differential magnitudesversus orbital phases are displayed in Figure 1, where phases are computed by a period of 0.d3138789 (Kreiner et al. 2004). The VRc LCs in 2013 imply that MT Cas is an UMa-type eclipsing binary, whose amplitudes of variable light are 0.69 mag and 0.66 mag in V and Rc bands, respectively. There exists an unequal height between both maxima, i.e., O’Connell effect [17, 18]. Max.II at phase 0.75 is brighter than Max.I at phase 0.25 up to 0.025 mag and 0.029 mag in V and Rc bands, respectively. This kind of stellar activity occurs on other W UMa-type binaries, such as VW Cep [19], DV CVn [20], V532 Mon [21], and BB Peg (Kalomeni et al. 2007), and DZ Psc [22].Table 1
CCD photometric observations inV and Rc bands.
V band R c band HJD Δ m HJD Δ m HJD Δ m HJD Δ m  2456629.9265 + 0.066 2456632.9847 - 0.273 2456629.9275 + 0.269 2456632.9895 - 0.076 2456629.9285 + 0.106 2456632.9861 - 0.280 2456629.9294 + 0.322 2456632.9962 - 0.091 2456629.9304 + 0.133 2456632.9874 - 0.273 2456629.9314 + 0.349 2456632.9976 - 0.113 2456629.9323 + 0.176 2456632.9901 - 0.289 2456629.9333 + 0.384 2456632.9989 - 0.099 2456629.9343 + 0.225 2456632.9915 - 0.300 2456629.9352 + 0.413 2456633.0003 - 0.116 ...... 2456632.9766 - 0.235 2456636.0355 - 0.077 2456632.9814 - 0.057 2456632.9780 - 0.238 2456636.0373 - 0.041 2456632.9827 - 0.069 2456632.9793 - 0.263 2456636.0391 - 0.041 2456632.9841 - 0.069 2456632.9807 - 0.267 2456636.0409 + 0.007 2456632.9854 - 0.063 2456632.9820 - 0.270 2456636.0427 + 0.065 2456632.9868 - 0.081 2456632.9834 - 0.261   2456632.9881 - 0.076 Note. The entire table is available only on the online journal.Figure 1
V R c light curves of the contact binary MT Cas, which are observed in 2013 in December by using the 85-cm telescope. The dotted and solid lines are plotted by photometric solutions without or with a dark spot, respectively.


## 2.2. Low-Precision Spectrum

The low-precision spectrum for MT Cas was obtained by using theYunnan Faint Object Spectrograph and Camera (YFOSC), which is attached to the 2.4-m telescope at Lijiang station (LJs) of Yunnan Astronomical Observatory of China (YNAO) at UT 15:50:13 of 2018 October 22. During the observing process, we chose a 140-mm-length slit and aGrism-3 with a wavelength range from 3200Å to 9200Å [23]. The exposure time is 10 minutes. The phase of 0.98 almost corresponds to the observed middle time HJD 245414.1634. Reduction of the spectra was performed by using IRAF packages, including bias subtraction, flat-fielding, and cosmic-ray removal. Finally, the one-dimensional spectrum was extracted. With thewinmk software (http://www.appstate.edu/~grayro/MK/winmk.htm), we obtained a normalized spectrum, which is displayed in Figure 2. By comparing the spectra of standard stars [24], the spectral type is determined to be F8V for the primary (i.e., more massive component) of this binary because the secondary is eclipsed by the primary around phase 0.0 for the W-type contact binary (see Section 4).Figure 2
Spectroscopic observation for MT Cas, observed by the 2.4-m telescope at Lijiang station of YNAO on October 22, 2018.


## 3. Eclipse Times and Period Analysis

From our new observations and AAVSO’s data, several times of primary and secondary minima are generally determined by using the method of Kwee & van Woerden [25]. From ASAS-SN database, we downloaded 134 data types inV band for this binary. With the Period04 package [26], we obtained the power spectrum, which is shown in Figure 3. The searched frequency is f1=6.37192d-1, corresponding to 0.156937 days (i.e., half of an orbital period). The derived epoch HJD 2457010.9922 (i.e., the secondary eclipse time) a bit differs from the given rough epoch HJD 2457011.76746 from the ASAS-SN database. The individual single-color minimum timings with their errors are listed in Table 2.Table 2
New observed light minimum times.
JD(Hel.) Error Min Band Telescope/Source  2456629.94355 ± 0.00025 II V 85-cm (NAOC) 2456629.94379 ± 0.00022 II R c 85-cm (NAOC) 2456631.98406 ± 0.00015 I V 85-cm (NAOC) 2456631.98401 ± 0.00014 I R c 85-cm (NAOC) 2456634.02521 ± 0.00010 II V 85-cm (NAOC) 2456634.02392 ± 0.00027 II R c 85-cm (NAOC) 2456569.20623 ± 0.00022 I V AAVSO 2456569.36322 ± 0.00007 II V AAVSO 2456573.28617 ± 0.00025 I V AAVSO 2456573.60003 ± 0.00011 I V AAVSO 2456576.42499 ± 0.00018 I V AAVSO 2456576.58236 ± 0.00010 II V AAVSO 2456579.24973 ± 0.00008 I V AAVSO 2456579.56341 ± 0.00007 I V AAVSO 2457011.76746  II V ASAS-SNFigure 3
The power spectrum for 134 ASAS-SN data types by Fourier analysis.In order to construct the(O-C) curve (i.e., observed values minus calculated ones), we collected all available light minimum times. From the O-C gateway (http://var2.astro.cz/ocgate) and TIDAK [14] (http://www.as.up.krakow.pl/minicalc/CASMT.HTM), we accumulated 10 “pg” (i.e., photographic), 21 “pe” (i.e., photoelectric), and 34 CCD measurements. Table 3 lists all those eclipsing times, whose errors are not given for 10 pg, 4 pe, and 1 CCD from literature. The standard derivations for all pe and CCD data are averaged to be 0.00137 days. For light minimum times without errors, we adopted the errors of 0.0137 for 10 pg data and 0.0014 days for 4 pe and 2 CCD ones. Therefore, the used weights depend on their errors while fitting the (O-C) curve.Table 3
All light minimum times for MT Cas.
JD(Hel.) Error Epoch Method Min ( O - C ) i ( O - C ) f Ref. (days) (days) 2429113.517  - 74508.5 pg II + 0.0160 - 0.0055 ( 1 ) 2429129.388  - 74458.0 pg I + 0.0362 + 0.0147 ( 1 ) 2429144.431  - 74410.0 pg I + 0.0131 - 0.0083 ( 1 ) 2429170.336  - 74327.5 pg II + 0.0231 + 0.0017 ( 1 ) 2429229.342  - 74139.5 pg II + 0.0200 - 0.0013 ( 1 ) 2429249.431  - 74075.5 pg II + 0.0208 - 0.0004 ( 1 ) 2429464.441  - 73390.5 pg II + 0.0243 + 0.0035 ( 1 ) 2429491.425  - 73304.5 pg II + 0.0148 - 0.0059 ( 1 ) 2429498.493  - 73282.0 pg I + 0.0205 - 0.0002 ( 1 ) 2430024.400  - 71606.5 pg II + 0.0247 + 0.0050 ( 1 ) 2444901.5705  - 24208.5 pe II - 0.0004 - 0.0015 ( 2 ) 2444941.5925  - 24081.0 pe I + 0.0021 + 0.0010 ( 2 ) 2444955.2420  - 24037.5 pe II - 0.0021 - 0.0032 ( 2 ) 2451550.2943 ± 0.0030 - 3026.0 CCD I - 0.0001 + 0.0000 ( 2 ) 2451780.3698 ± 0.0021 - 2293.0 CCD I + 0.0028 + 0.0029 ( 3 ) 2451807.3638 ± 0.0002 - 2207.0 pe I + 0.0032 + 0.0033 ( 2 ) 2451924.2811 ± 0.0016 - 1834.5 CCD II + 0.0009 + 0.0010 ( 4 ) 2452096.4442 ± 0.0031 - 1286.0 CCD I + 0.0019 + 0.0019 ( 3 ) 2452151.3705 ± 0.0025 - 1111.0 CCD I - 0.0005 - 0.0004 ( 3 ) 2452190.2867 ± 0.0016 - 987.0 CCD I - 0.0052 - 0.0052 ( 5 ) 2452205.3590 ± 0.0031 - 939.0 CCD I + 0.0010 + 0.0010 ( 3 ) 2452205.5152 ± 0.0016 - 938.5 CCD II + 0.0002 + 0.0002 ( 3 ) 2452219.4813 ± 0.0028 - 894.0 CCD I - 0.0013 - 0.0013 ( 3 ) 2452574.321 ± 0.002 236.5 CCD II - 0.0008 - 0.0008 ( 2 ) 2452684.3376 ± 0.0056 587.0 CCD I + 0.0015 + 0.0014 ( 2 ) 2452859.4828  1145.0 CCD I + 0.0027 + 0.0026 ( 3 ) 2452879.5725 ± 0.0033 1209.0 CCD I + 0.0042 + 0.0041 ( 2 ) 2453302.6749 ± 0.0004 2557.0 CCD I - 0.0011 - 0.0013 ( 2 ) 2453329.6671 ± 0.0002 2643.0 CCD I - 0.0024 - 0.0027 ( 2 ) 2453353.3663 ± 0.0007 2718.5 CCD II - 0.0010 - 0.0013 ( 2 ) 2453738.6533 ± 0.0001 3946.0 CCD I + 0.0006 + 0.0002 ( 2 ) 2453759.3698 ± 0.0022 4012.0 pe I + 0.0011 + 0.0007 ( 2 ) 2453767.5305 ± 0.0001 4038.0 CCD I + 0.0010 + 0.0006 ( 2 ) 2454205.3900 ± 0.0014 5433.0 pe I + 0.0005 - 0.0001 ( 2 ) 2454432.3199 ± 0.0003 6156.0 pe I - 0.0035 - 0.0041 ( 2 ) 2454432.4779 ± 0.0002 6156.5 pe II - 0.0024 - 0.0030 ( 2 ) 2454792.6557 ± 0.0004 7304.0 CCD I + 0.0002 - 0.0006 ( 2 ) 2455067.4622 ± 0.0014 8179.5 pe II + 0.0064 + 0.0055 ( 2 ) 2455074.3675 ± 0.0006 8201.5 pe II + 0.0064 + 0.0055 ( 2 ) 2455074.5247 ± 0.0004 8202.0 pe I + 0.0067 + 0.0058 ( 2 ) 2455473.3027 ± 0.0009 9472.5 pe II + 0.0025 + 0.0014 ( 2 ) 2455473.4600 ± 0.0056 9473.0 pe I + 0.0029 + 0.0018 ( 2 ) 2455473.6167 ± 0.0008 9473.5 pe II + 0.0026 + 0.0015 ( 2 ) 2455517.7157 ± 0.0003 9614.0 CCD I + 0.0018 + 0.0006 ( 2 ) 2455797.5418 ± 0.0087 10505.5 pe II + 0.0055 + 0.0042 ( 2 ) 2456254.7090 ± 0.0002 11962.0 CCD I + 0.0092 + 0.0076 ( 2 ) 2456569.2062 ± 0.0002 12964.0 CCD I + 0.0006 - 0.0011 (6) 2456569.3632 ± 0.0001 12964.5 CCD II + 0.0006 - 0.0011 (6) 2456573.2862 ± 0.0003 12977.0 CCD I + 0.0001 - 0.0017 (6) 2456573.6000 ± 0.0001 12978.0 CCD I + 0.0001 - 0.0017 (6) 2456576.4250 ± 0.0002 12987.0 CCD I + 0.0001 - 0.0017 (6) 2456576.5824 ± 0.0001 12987.5 CCD II + 0.0005 - 0.0013 (6) 2456579.2497 ± 0.0001 12996.0 CCD I + 0.0000 - 0.0018 (6) 2456579.5634 ± 0.0001 12997.0 CCD I - 0.0002 - 0.0019 (6) 2456629.9437 ± 0.0002 13157.5 CCD II + 0.0026 + 0.0008 (6) 2456631.9840 ± 0.0001 13164.0 CCD I + 0.0027 + 0.0009 (6) 2456634.0246 ± 0.0002 13170.5 CCD II + 0.0031 + 0.0013 (6) 2456950.2602 ± 0.0013 14178.0 pe I + 0.0065 + 0.0045 ( 2 ) 2456950.4193 ± 0.0015 14178.5 pe II + 0.0086 + 0.0066 ( 2 ) 2456950.5733 ± 0.0005 14179.0 pe I + 0.0057 + 0.0037 ( 2 ) 2457010.9918  14371.5 CCD II + 0.0037 - 0.0017 ( 6 ) 2457297.4095  15284.0 pe I + 0.0066 + 0.0044 ( 2 ) 2457322.3625 ± 0.0006 15363.5 pe II + 0.0062 + 0.0040 ( 2 ) 2457364.2629 ± 0.0004 15497.0 pe I + 0.0039 + 0.0016 ( 2 ) 2457364.4204 ± 0.0018 15497.5 pe II + 0.0045 + 0.0022 ( 2 ) Ref. (1) Götz & Wenzel 1956; (2) http://www.konkoly.hu/IBVS/IBVS.html; (3) [41]; (4) [42]; (5) [43]; (6) present work.By using the linear ephemeris [14],(1)Min.I=HJD2452500.086+0.3138789×E,we compute the initial residuals, (O-C)i, which are listed in Table 3 and shown in Figure 4(a). From this figure, there exists a long gap between HJD 2430024.400 [10] and HJD 2451550.2943 [27], except for three data types [12]. However, the general trend of (O-C)i may be evidently described by an upward parabolic curve. A linear least-squares fitting method with weights leads to the following ephemeris:(2)Min.  I=HJD2452500.0896±0.0011+0.31387820±0.00000002×E+4.82±0.04×10-12×E2After being removed by Eq. (2), we obtained the final residuals of (O-C)f, which are listed in Table 3 and shown in Figure 4(b). If we neglected low-precision photographic data, we could find no regularity, such as sinusoidal variation, only from pe and CCD data, which can be directly seen from the right part of Figure 4(b) except for a bit scatter.Figure 4
The(O-C) curve for MT Cas. The crossing, open, and solid circles represent photographic data and photoelectric and CCD measurements, respectively. The continuous line is plotted by Eq. (2).


## 4. Modeling Light Curves

From the AAVSO and ASAS-SN databases, two additionalV-band LCs for MT Cas are available and are displayed in Figure 5. Therefore three sets LCs are applied to derive the photometric elements with the updated Wilson-Devinney binary star modeling program [28, 29] (W-D program can be accessed from the site of ftp://ftp.astro.ufl.edu/pub/wilson/lcdc2015.). Kurucz’s [30] stellar atmosphere model was applied. Based on the spectral type of F8V with a subtype error, we adopted a mean effective temperature for the more massive component (i.e., the primary) to be Tp=5750(±210)K  (the subscripts “p” and “s” refer to the primary component and the secondary one, respectively). We fixed the gravity-darkening coefficients of gp,s=0.32 [31] and bolometric albedo coefficients of Ap,s=0.5 [32], which are appropriate for stars with convective envelopes. The logarithmic limb-darkening coefficients are interpolate from van Hamme’s [33] tables. Other adjustable parameters are i, q, Ts, Ωp=Ωs, and Ls.Figure 5
AAVSO’s (a) and ASAS’ (b) light curves. The solid lines are the theoretical LCs, which are computed by the photometric solutions.
(a) (b)Due to lack of radial velocity curves, the “q-search” process is carried out by obtaining a series of tried solutions fromVRc light curves. Modeling calculations start to mode 2 (i.e., detached configuration) and always converge to mode 3 (i.e., contact one). For several fixed inclinations from 70∘ to 90∘ with a step of 5∘, we first performed a series of solutions. Five computed curves for q-i are shown in Figure 6(a), in which a minimum squared residual, Σ(o-c)i2, occurs around q=2.4 and i=85∘. As a free parameter for the orbital inclination, we obtained the relations of q-Σ and q-f (the fill-out factor for contact binary is defined by f=(Ωin-Ω)/(Ωin-Ωout), in which Ωin and Ωout are inner and outer critical potentials, respectively), which are displayed in Figure 6(b). The mass-ratio of q=2.4 with a minimum value of Σ(o-c)i2 indicates that MT Cas is a W-type contact binary (i.e., the less massive component eclipsed by the more massive component at the primary minima; see [34]). After q is considered to be a free parameter, we deduced the photometric solution, which is listed in Table 4. The calculated light curves as dotted lines are shown in Figure 1. Because VRc LCs in 2013 cause unequal heights between both light maxima up to Max.I-Max.II~0.03 mag, a dark spot is added in modeling the asymmetric LCs. Assuming a cool spot on the equator of the more massive component (i.e., colatitude, ϕ=90∘), we obtained the final photometric elements, which are given in Table 4, including other three parameters of spot (i.e., longitude θ, angular radius γ, and temperature factor Tspot/Tp). The mass ratio and fill-out factor for MT Cas are q=2.365(±0.005) and f=16.6(±1.2)%, respectively. The corresponding theoretical light curves as solid lines are plotted in Figure 1. The O’Connell effect for this binary can be attributed the activity of stellar spot. Its area is up to 1.7% of the area of the more massive component. Therefore, the spotted solution is accepted to be the final solution due to its small value of Σ(o-c)i2=0.2223. Additionally, we analyzed AAVSO’s and ASAS’ LCs, respectively. The derived photometric solutions are given in Table 4. The mass ratio and fill-out factor almost agree with the previous result from our VRc light curves. The calculated LCs are plotted as continuous lines in Figure 6. Their mass-ratios approximate to the result of the photometric solution with a cool spot.Table 4
Photometric solutions for MT Cas.
Parameters LCs in 2013 LC of AAVSO LC of ASAS No Spot With Spot (2878 data types) (134 data types)  i(∘) 84.7 ± 0.3 85.2 ± 0.3 83.6 ± 0.1 80.3 ± 0.2 q=Mp/Ms 2.370 ± 0.002 2.365 ± 0.005 2.487 ± 0.025 2.415 ± 0.013 Tp(K) 5750 ± 210 5750 ± 210 Xp,Yp 0.650, 0.207 0.650, 0.207 xsV,ysV 0.774, 0.232 0.774, 0.232 xsR,ysR 0.694, 0.251 0.694, 0.251 Ts(K)a 601 0 - 238 + 145 602 6 - 249 + 231 596 6 - 232 + 232 608 0 - 137 + 236 Xs,Ys 0.649, 0.221 0.649, 0.222 0.649, 0.219 0.648, 0.225 xsV,ysV 0.746, 0.260 0.745, 0.261 0.749, 0.256 0.742, 0.265 xsR,ysR 0.675, 0.271 0.674, 0.271 - Ωp=Ωs 5.678 ± 0.006 5.660 ± 0.007 5.857 ± 0.004 5.767 ± 0.031 Ls/(Lp+Ls)V 0.3602 ± 0.0006 0.3641 ± 0.0006 0.3422 ± 0.0002 0.3681 ± 0.0049 Ls/(Lp+Ls)R 0.3544 ± 0.0005 0.3559 ± 0.0005 - r p(pole) 0.4342 ± 0.0013 0.4373 ± 0.0012 0.4375 ± 0.0005 0.4345 ± 0.0019 r p(side) 0.4643 ± 0.0018 0.4683 ± 0.0016 0.4681 ± 0.0006 0.4644 ± 0.0025 r p(back) 0.4937 ± 0.0024 0.4988 ± 0.0021 0.4868 ± 0.0008 0.4931 ± 0.0029 r s(pole) 0.2932 ± 0.0016 0.2946 ± 0.0015 0.2884 ± 0.0005 0.2900 ± 0.0018 r s(side) 0.3067 ± 0.0019 0.3083 ± 0.0018 0.3015 ± 0.0006 0.3030 ± 0.0022 r s(back) 0.3438 ± 0.0032 0.3469 ± 0.0031 0.3380 ± 0.0009 0.3390 ± 0.0023 θ ( a r c ) - 1.61 ± 0.07 - γ ( a r c ) - 0.26 ± 0.04 - Tspot/Tp - 0.85 ± 0.05 - Σ(o-c)i2 0.3777 0.2223 0.2762 1.1616 f(%) 14.8 ± 1.1 16.6 ± 1.2 11.5 ± 0.6 10.2 ± 0.5 Note. aThe uncertainty of Ts is determined from the input error for Tp.Figure 6
The relations ofq-i (a), q-Σ and q-f (b) derived from VRc light curves in 2013.
(a) (b)


## 5. Results and Discussions

Up to now no spectroscopic elements have been published, and the absolute parameters of this binary cannot be directly determined. Assuming the spectral type of F8V with a subtype uncertainty, the mass of the primary was roughly estimated to beMp=1.19(±0.07)M⊙ [35]. Using the mass ratio and orbital period, the separation between components and the mass of the secondary are estimated to be a=2.32(±0.05)R⊙ and Ms=0.50(±0.03)M⊙, respectively. Other absolute parameters for MT Cas are as follows: Rp=1.08(±0.05)R⊙, Rs=0.73(±0.04)R⊙, Lp=1.15(±0.11)R⊙, and Ls=0.63(±0.07)R⊙.The mass-luminosity diagram is displayed in Figure7, in which the zero-age main sequence (ZAMS) and the terminal-age main sequence (TAMS) are constructed by the binary-star evolution code [36]. The W-type low-temperature contact binaries (LTCBs; see [3]) are plotted as black open and filled circles. From this figure, the primary component of MT Cas is close to the TAMS line, implying that it is a normal main-sequence star. Meanwhile, the secondary component lies above the TAMS line, indicating that it may be a small helium star in an advanced evolutionary stage [4].Figure 7
The mass-luminosity diagram for MT Cas. The filled and open circles represent the primary and secondary components for the W-type LTCBs [3].From Eq. (2), the orbital period of MT Cas may be undergoing a secular increase at a rate of dP/dt=+1.04(±0.09)×10-8dyr-1. This case occurs in other weak-contact binaries, which are listed in Table 5. Therefore, the period increase rate for MT Cas may be a small value for this kind of binaries. The long-term period increases can be generally interpreted by mass transfer from the less massive component to the more massive one. Under conserved mass transfer assumption, the mass transfer rate can be computed by the following formula [37]:(3)P˙P=3MpMs-1M˙pMp.Inserting the values of P˙, P, Mp, and Ms into Eq. (3), the mass transfer rate is estimated to be M˙p=+1.04(±0.09)×10-8M⊙yr-1. With the period increasing, the separation between two components will increase. Meanwhile, mass transfer from the secondary one to the primary one also causes the mass ratio to decrease. This may result in the inner and outer Roche lobes shrinking, which will cause the fill-out factor to decrease. Finally, the weak-contact configuration may be broken. Therefore, MT Cas may evolve into a broken-contact configuration, as predicted by TRO theory [3, 38–40]. In the future, it necessitates to obtain high-precision photometry and spectroscopy to identify the orbital period variations and to determine its absolute parameters.Table 5
Several weak-contact binaries with secular period increasing.
Star Spectral Period q d P / d t f Ref. type (day)  ( × 1 0 - 7 d   y r - 1) (%)  44i Boo K2V 0.26782 0.487 + 1.49 5.0 ( 1 ) ( 2 ) CW Cas G8V 0.31886 0.448 - 0.34 6.5 ( 3 ) BE Cep - 0.42439 0.427 - 0.48 6.9 (4) DD Com - 0.26921 0.271 + 3.4 8.8 (5) AD Cnc K0V 0.28274 0.770 + 4.94 8.3 (6) XY Leo K2V 0.28410 0.610 + 0.27 6.7 (7) MT Cas F8V 0.31388 0.423 a + 0.11 16.6 (8) Note. aThe mass ratio is flipping, i.e., 1/2.365. References: (1) [44]; (2) [45]; (3) [46]; (4) [47]; (5) [48]; (6) [49]; (7) [50]; (8) present paper.



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