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\title{Photometric investigation of three ultrashort-period contact binaries: Two of them are in special evolutionary stages}

\author{L. Liu\altaffilmark{1,2,3,4}, S.-B. Qian\altaffilmark{1,2,3,4}, E. Fern\'{a}ndez Laj\'{u}s\altaffilmark{5,6}, A. Essam\altaffilmark{7}, M. A. El-Sadek\altaffilmark{7} and X. Xiong\altaffilmark{1,2,3,4}}\singlespace


\altaffiltext{1}{Yunnan Observatories, Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming, 650216, P. R. China (e-mail: LiuL@ynao.ac.cn)}
\altaffiltext{2}{Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming, 650216, P. R. China}
\altaffiltext{3}{Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, P. R. China}
\altaffiltext{4}{University of Chinese Academy of Sciences, Yuquan Road 19\#, Sijingshang Block, 100049 Beijing, China}
\altaffiltext{5}{Facultad de Ciencias Astron$\acute{o}$micas y Geof$\acute{i}$sicas, Universidad Nacional de La Plata, 1900 La Plata, Buenos Aires, Argentina}
\altaffiltext{6}{Instituto de Astrof$\acute{s}$ica de La Plata (CCT La Plata-CONICET, UNLP), Argentina}
\altaffiltext{7}{National Research Institute of Astronomy and Geophysics, Department of Astronomy, Helwan, Cairo, Egypt}


\begin{abstract}
We carried out high-precision photometric observations of three ultrashort-period eclipsing binaries with 2-meter-class telescopes. The light-curve solutions suggested that all targets had evolved into contact phase. The photometric results are as follows: a) 1SWASP\,J030749.87$-$365201.7, $q=0.439\pm0.003, f=0.0\pm3.6\,\%$; b) 1SWASP\,J213252.93$-$441822.6, $q=0.560\pm0.003, f=14.2\pm1.9\,\%$; c) 1SWASP\,J200059.78$+$054408.9, $q=0.436\pm0.008, f=58.4\pm1.8\,\%$. All of their light curves showed O'Connell effects, which can be modeled by assumed cool spots. The cool spots model is strongly  supported by the night-to-night variation of the I-band light curves of 1SWASP\,J030749.87$-$365201.7. For a comparing study, we collected all of well-studied ultrashort-period contact binaries (USPCBs) with P $<$ 0.24 day (27 in total), and found that most of them (16 of 27) are shallow contact ($f<20\,\%$) while only 4 of 27 USPCBs are deep fill-out factors ($f>50\,\%$). One of the four deep USPCBs is found by this paper. Generally, contact binaries with deep fill-out factors are going to merge. But the USPCBs are believed as they just evolved into contact phase. Hence, the deep USPCB, 1SWASP\,J200059.78$+$054408.9, is a contradiction, which makes it interesting. Particularly, 1SWASP\,J030749.87$-$365201.7 is a zero contact binary within a thermal equilibrium, implying that it should be a tune-off sample predicted by the thermal relaxation oscillation (TRO) theory.
\end{abstract}

\keywords{binaries : eclipsing --
Stars: individuals (1SWASP\,J030749.87$-$365201.7, 1SWASP\,J213252.93$-$441822.6, 1SWASP\,J200059.78$+$054408.9) --
Stars: evolution}

\section{Introduction}
The W UMa-type contact binaries are usually composed of two cool main sequence stars where both components are covered by a common convective envelope (CCE). The short-period limit of this type binary is an unquestioned fact. There were some ideals for explaining the limitation: (1) the components become fully convective below a certain period \citep[e.g.][]{Rucinski1992, Rucinski1997, Paczynskietal2006, Beckeretal2011} so that the system becomes unstable; (2) the timescale of the angular momentum loss (AML) is much longer so that the USPCBs cannot be produced under the current age of the universe \citep{Stepien2006, Stepien2011}. Almost fully convective structure makes USPCBs to be different from the F, G and K type contact binaries. Many USPCBs have been found just in recent years with the extraplanet searching projects \citep[One can find the names of the surveys that were introducd by][]{Koenetal2016}. \citet{Nortonetal2011} published a catalogue of 53 ultrashort-period eclipsing binary (USPEB) candidates, while \citet{Lohretal2013a} investigated the period changes of 143 USPEBs, based on the Super Wide Angle Search for Planets (SuperWASP). \citet{Koenetal2016} confirmed 29 USPEB systems being in overcontact configuration according to a preliminary Fourier decomposition analysis. A series of studies on individual USPCBs have been done in last 5 years \citep[e.g.][]{DimitrovKjurkchieva2015, Qianetal2015a, Liuetal2015a, Jiangetal2015a}. These studies showed that most USPCBs have shallower fill-out factors ($<20\,\%$). It indicates that USPCBs are in the beginning phase of the contact configuration. And, some USPCBs already have broken the known short-period limitation or low-mass limitation \citep[e.g., SDSS\,J001641$-$000925,][]{Davenportetal2013, Qianetal2015b}. The special USPCBs which are under the critical conditions are interesting and important to understand the evolutionary boundary conditions.


For this purpose, and because of the unqualified data of surveys, we observed the USPCB candidates, 1SWASP\,J030749.87$-$365201.7, 1SWASP\,J213252.93$-$441822.6 and 1SWASP\,J200059.78$+$054408.9, with 2-meter-class telescopes. The first one was discovered by \citet{Nortonetal2011}, while the second and the third one were discovered by \citet{Lohretal2013a}. The period of these three system are 19584.393\,s (0.22667122\,day), 19114.669\,s (0.22123459\,day) and 17771.663\,s (0.20569054\,day), respectively \citep{Lohretal2013a}. In the literatures, the light curves of these three USPCBs shown very large scatters. So, in this paper, we analyzed these systems with our new observed high-precision multi-colour light curves. We report here that 1SWASP\,J030749.87$-$365201.7 and 1SWASP\,J200059.78$+$054408.9, a zero contact and a deep contact, are such special samples mentioned above.


\section{Observation and data reduction}
The two south sky targets were observed by the 2.15\,m Jorge Sahade Telescope (JST) in Complejo Astronomico El Leoncito Observatory (CASLEO), San Juan, Argentina. The JST has been equipped a Versarray 2048B CCD camera at the Cassegrain focus (f/8.5) with a binned of 5$\times$5, yielding a field of view of 5$\times$5 arcmin$^2$. The standard UBVRI filter system was used. 1SWASP\,J030749.87$-$365201.7 was observed from November 24 to the December 4 in 2014 with JST, while 1SWASP\,J213252.93$-$441822.6 was observed in September 11, 2015. One can find more observational details in Table~\ref{tab:summary observation_table}. The JHK magnitudes \citep[from 2MASS catalogue,][]{Cutrietal2003} of target, comparison and check stars are listed in Table~\ref{tab:target comparison_table}. The observed data were reduced by the IRAF, with the zero and flat corrections. The differential photometry measuring errors of former are about 0.009\,mag, while those of the later are about 0.007\,mag. The phases of 1SWASP\,J030749.87$-$365201.7 were calculated according to the ephemeris of $2456994.77031+0^d.22667122{\times}E$ while those of 1SWASP\,J213252.93$-$441822.6 were computed based on the formula, $2457276.57884+0^d.22123459{\times}E$, respectively. One can find the corresponding light curves in Fig~\ref{fig:light curves_figure}.


The north sky target was observed by the 1.88\,m reflector telescope in the Kottamia Observatory, Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), 11421 Helwan, Cairo, Egypt. This telescope has been equipped a EEV CCD 42-40 camera with a format of 2048$\times$2048 pixels, cooled by liquid nitrogen attached on the Newtonian focus of the 1.88\,m Kottamia reflector telescope, yielding a field of view of 10$\times$10 arcmin$^2$. The standard UBVRI filter system was used, too. As the same as the former two, the target, comparison and check stars are listed in Table~\ref{tab:target comparison_table}, too. The same data reduction process has been done. The differential photometry measuring errors are about 0.006\,mag. The formula of $2457604.37104+0^d.20569054{\times}E$ was used to yield the light curves plotted in Fig~\ref{fig:light curves_figure}, too.


\section{Light curves solution}
To determine photometric elements and to understand the geometrical structure and evolutionary state of these three USPCBs, we used the latest version of the W-D code \citep{WilsonDevinney1971, Wilson1979, Wilson1990, Wilson2008, Wilson2012, vanHammeWilson2007, Wilsonetal2010, WilsonvanHamme2014} to analyze their multi-colour light curves.


Usually, the effective temperatures are estimated by colours. At the beginning, we estimated the $T_1$ (effective temperature of the hot component) based on the JHK colour indexes because our targets are in later spectral type (red colour). The corresponding JHK magnitudes \citep[2MASS catalogue,][]{Cutrietal2003} are listed in Table~\ref{tab:target comparison_table}. The effective temperatures are computed by the method of \citet{WortheyLee2011}. Then, we applied the $q$-search method to find an initial $q$ as the input parameter for the W-D code. Such grid method $q$-searching is just aimed to make the program to converge faster. In other words, the advantage of the $q$-search method is just for saving computed time. The results of $q$-search are displayed in Fig~\ref{fig:q search_figure}. The red crosses are the suggested values of $q$ for each system. During the solutions, the bolometric albedo $A_1=A_2=0.5$ \citep{Rucinski1969} and the values of the gravity-darkening coefficient $g_1=g_2=0.32$ \citep{Lucy1967} were used, which correspond to the common convective envelope of both components; square root limb-darkening coefficients were used, according to \citet{ClaretGimenez1990}. The adjustable parameters were: the mass ratio $q$; the orbital inclination $i$; the potential $\Omega$ (because all solution finally converged to mode\,3); the mean temperature of star\,2, $T_2$; the monochromatic luminosity of star\,1.

However, all light curves showed O'Connell effects \citep{O'Connell1951}. The presence of cool spots on the more massive component can simulate the light curves very well, and it is supported by i) later type stars have deep convective zones so that they have strong magnetic field; ii) fast rotate later type stars should have more stronger magnetic field than normals \citep{BarnesCollierCameron2001, Barnesetal2004}; iii) the more massive component has a deeper convective zone than the less massive one \citep{Mullan1975}. For the first target, the situations of the cool spots are more complicated, which is supported by its I-band night-to-night variation of light curves (Fig~\ref{fig:night-to-night variation_figure}). Three cool spots are added to simulate its light curves. One is on the less massive component and two are on the more massive component. 1SWASP\,J030749.87$-$365201.7 has so many spots suggest that it has very strong magnetic activities. Emphatically, the longitudes of spots can be limited exactly by the distortions in the light curves. The other three parameters, latitudes, radii and temperature ratios, however, are not independent. A very recent study of cool spots on M dwarfs revealed that the fractus cool spots occurred on the high latitudes with high frequency \citep{Barnesetal2015}, according to the doppler maps. Although our cool spots model is a probably result, the differential of results with and without cool spots is usually less than 10\% \citep[e.g.][]{Qianetal2011, Qianetal2013a}.

Finally, the light curve solutions are listed in Table~\ref{tab:solution_table}, and the elements of the assumptive cool spots are listed in Table~\ref{tab:cool spots_table}. The corresponding fittings of each light curve are shown in Fig~\ref{fig:light curves_figure}, while the residuals of the fittings are shown in Fig~\ref{fig:residuals_figure}. The differential colours diagrams are shown in Fig~\ref{fig:colour index_figure}. In this figure, it is clearly seen that colours of the systems varied with the phases, like their light curves. And there is a swell in colour curves of 1SWASP\,J030749.87$-$365201.7 around the phase of 0.40. The swell means a higher temperature. The amplitude of the colour curves suggest an uncertainty of 700\,K for the temperature estimation, but it do not change any uncertainty of the temperature ratio ($T_1/T_2$). At last, the geometric constructions of the three USPCBs and the present cool spots are displayed in Fig~\ref{fig:cool spots_figure}.


\section{Discussion of the solutions}
In this section, we will mainly discuss about the reliability of the photometric solutions and the evolutionary states revealed by the solutions. The resolved inclinations of these three systems are greater than 75 degree, strongly indicating that their photometric mass ratios are similar to their spectroscopic mass ratios \citep{Maceronivan1996}. Hence, the photometric mass ratios of our solution should be acceptable. The solutions show that these three systems are in contact phase, with moderate mass ratios. However, 1SWASP\,J200059.78$+$054408.9, with the shortest period among the three, have a deep contact factor of 58.4\,\%. Deep USPCBs are very few. It is supported both by the observational facts and by the theory. 27 well-studied USPCBs (P $<$ 0.24 day) are collected in Table~\ref{tab:27 well-studied USPCBs_table}. Only 4 of 27 USPCBs are deep fill-out factors ($f>50\,\%$). And 16 of 27 systems are shallow contact ($f<20\,\%$). Unlike the F and G type contact binaries, non of their mass ratios are less than 0.3. Generally, shallow contact factor implies an early phase of contact. According to the thermal relaxation oscillation (TRO) theory \citep{Lucy1976, Flannery1976, Webbink1977}, the material flow started when the primary component (donor) fully filled its Roche lobe, which made the orbit shrinking, and then, the secondary component (accretor) fully filled its Roche lobe, too. The accretor will expand continuously when it accepted masses, which made the orbit wider, until the contact configuration broke. Subsequently, the orbit will shrink again with the mass transfer from the primary to the secondary component. It forms a cycle of contact-semidetached-contact. In these cycles, the temperatures of the two components will be similar because of a thermal exchange with the mass transfer, and the fill-out fact cannot reach high when a system undergoes the TRO cycle because of the existence of the contact broken phase. Hence, TRO explained why it is such high fraction of shallow USPCBs (16 of 27 systems). And for this reason, the zero contact binary within thermal equilibrium, 1SWASP\,J030749.87$-$365201.7, should be in the turn-off phase of the TRO cycle. On the other hand, however, the TRO theory cannot explain the presence of the deep USPCBs with high or moderate mass ratios (e.g., 1SWASP\,J200059.78$+$054408.9).

If the USPCBs would reach a deep fill-out factor, a rapid orbit shrinking mechanism is required. Angular momentum loss (AML) caused by the magnetic stellar winds \citep[e.g.][]{Stepien2006, Stepien2011} may be the mechanism. He pointed out that if there is a third body or the system is located in a dense field, it can lose a lot of AM. An excessive AML also can occurred in the pre-main-sequence phase. In that time, he said these possibilities, especially for the last one, are quite rare, according to the observations \citep{Stepien2006}. However, some studies showed that the third bodies is not very rare \citep[e.g.,][]{PribullaRucinski2006, DAngeloetal2006, Rucinskietal2007, Qianetal2013b, Qianetal2015b}. Maybe the few deep USPCBs in Table~\ref{tab:27 well-studied USPCBs_table}, including 1SWASP\,J200059.78$+$054408.9, are formed by the rare AML way, e.g., interaction with third bodies.


\section{Global parameter correction of the ultrashort-period contact binaries}
The physical parameters of close binaries are sufficiently different from the singles or wide binaries because of the strong inter-gravity. For example, the surface gravity accelerations of components in contact binaries systems are observably less than those of the same masses single main-sequence stars. Hence, to obtain these parameters for close binaries independently is very important to build an improved evolutionary model for them. The USPCBs are predicted that they have a well empirical global parameter relation than that of F, G and early K type contact binaries because most of them are near the zero age main sequence and unevolved. This character is good for estimating the parameters of USPCBs when it lacks of spectroscopic data. \citet{DimitrovKjurkchieva2015} had already summarized a relationship between the period and the semi-major axis, based on the 14 well-studied binaries with P $<$ 0.27 d ($a = -1.154 + 14.633 \times P - 10.319 \times P^2$). However, one point deviated this relation, i.e., GSC\,1387-0475. It was first investigated by \citet{RucinskiPribulla2008} with yielding a spectroscopic mass ratio. Nevertheless, their photometric light curve was not quality well. \citet{Yangetal2010} observed this system again and obtained a BVR-bands light curve. They adopted the spectroscopic mass ratio yielded by \citet{RucinskiPribulla2008}, and found smaller masses of the components than the results of the former. We assumed that the inconformity of these two results is caused by  the deep fill-out factor. Consequently, we realized that the semi-major axis probably is over estimated with the relation of \citet{DimitrovKjurkchieva2015} when the fill-out factor is huge (e.g. $f>50\,\%$). According to the well known Roche potential,
\begin{equation}
{\psi}=\frac{2}{1+q}\cdot\frac{1}{r_1}+\frac{2q}{1+q}\cdot\frac{1}{r_2}+(x-\frac{q}{1+q})^2+y^2
	\label{eq:Roche potential}
\end{equation}
where ${r_1}^2=x^2+y^2+z^2$, ${r_2}^2=(1-x)^2+y^2+z^2$, $q=M_2/M_1$, $xyz$ are normalized coordinates (with semi-major axis, A), and with the definition of fill-out factor
\begin{equation}
f=\frac{{\psi}-{\psi}_{\rm{in}}}{{\psi}_{\rm{out}}-{\psi}_{\rm{in}}},
	\label{eq:fill-out factor}
\end{equation}
we compute the $r_{\rm{side}}$, $r_{\rm{back}}$ and $r_{\rm{pole}}$ with a certain fill-out factor $f$ and mass ratio $q$. The effective radius is calculated by $r_{\rm{E}}=\sqrt[3]{r_{\rm{side}}{\cdot}r_{\rm{back}}{\cdot}r_{\rm{pole}}}$.
Then, we compare this $r_{\rm{E}}$ to $r_{\rm{L}}$ which was yielded by \citet{Eggleton1983} with an error less than 1\,\%,
\begin{equation}
r_{\rm{L}}=\frac{0.49q^{-2/3}}{0.6q^{-2/3}+\ln(1+q^{-1/3})}.
	\label{eq:critical Roche radii}
\end{equation}
Finally, we obtain a ratio of these two normalized radii, and use it to correct the semi-major axis. The corresponding results are list in Table~\ref{tab:Semi-major axis corrections_table}. All errors in this table are estimated with the error propagation formula. We find that the masses of GSC\,1387-0475 are high close to the values of \citet{Yangetal2010} after the correction. We adopt the corrections to 1SWASP\,J200059.78$+$054408.9. It may be composed by two M type components with mass $0.458\pm0.066\,M_{\odot}$ and $0.199\pm0.033\,M_{\odot}$, respectively.

\section{Conclusions}
1SWASP\,J030749.87$-$365201.7 is a zero contact binary, with $q=0.439\pm0.003$ and $f=0.0\pm3.6\,\%$. It is on the turn-off phase of the TRO cycle. 1SWASP\,J213252.93$-$441822.6 is a shallow contact binary with $q=0.560\pm0.003$ and $f=14.2\pm1.9\,\%$. It is under the TRO controlling. 1SWASP\,J200059.78$+$054408.9 is a deep contact binary, with $q=0.436\pm0.008$ and $f=58.4\pm1.8\,\%$. It should be formed by a rapid AML mechanism. All targets showed strong magnetic activities. In summary, 1SWASP\,J030749.87$-$365201.7 and 1SWASP\,J200059.78$+$054408.9 are in particular evolutionary stages. They are worth to monitor in future.


%\bigskip
\acknowledgments{
We thank Allen W. Shafter for giving us constructive suggestions that improved the paper greatly.

This work is partly supported by the Yunnan Natural Science Foundation (2016FB004), the young academic and technology leaders project of Yunnan Province (No. 2015HB098), Chinese Natural Science Foundation (Nos. 11403095 and 11325315), the Key Research Programme of the Chinese Academy of Sciences (grant No. KGZD-EW-603), and Strategic Priority Research Programme ``The Emergence of Cosmological Structures'' of the Chinese Academy of Sciences (No.XDB09010202). New observations of the two south sky targets were obtained with the 2.15m telescope at Complejo Astron$\rm\acute{o}$mico El Leoncito observatory (CASLEO), San Juan, Argentina, and those of the north sky target were obtained with the 1.88 m Kottamia reflector telescope in Egypt.
}


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%\begin{references}
%\end{references}

\begin{table}
	\centering
\begin{footnotesize}
\caption{Summary of the observations.}
	\label{tab:summary observation_table}
\begin{tabular}{ccccccc}
\hline
        Target	               &  Obs Date	& Texp (s)	    & Filter	    &N Img	&Telescope	    &Seeing ($''$)\\
\hline
1SWASP\,J030749.87$-$365201.7  & 2014-11-24	& 30	        &   I	        & 471	&2.15\,m JST	&3.0-7.0   \\
	                           & 2014-11-25	& 30	        &   I	        & 440	&2.15\,m JST	&3.5-7.5   \\
	                           & 2014-12-03	& 30	        &   V	        & 669	&2.15\,m JST	&3.5-7.5   \\
	                           & 2014-12-04	& 60	        &   B	        & 325	&2.15\,m JST	&3.5-7.0   \\
1SWASP\,J213252.93$-$441822.6  & 2015-09-11	& 60	        &   R	        & 308	&2.15\,m JST	&3.4-7.4   \\
1SWASP\,J200059.78$+$054408.9  & 2016-08-01	& 200	        &   V	        & 49	&1.88\,m KRT	&3.0-6.5   \\
	                           & 2016-08-01	& 135	        &   R	        & 46	&1.88\,m KRT	&3.0-6.5   \\
	                           & 2016-08-01	& 170	        &   I	        & 46	&1.88\,m KRT	&3.0-6.5   \\
\hline
\end{tabular}
\end{footnotesize}
\end{table}


\begin{table}
	\centering
\begin{tiny}
\caption{The JHK magnitudes for the targets, comparisons and check stars. Estimated temperatures of the primary components for the three USPCBs based on the JHK colours.}
	\label{tab:target comparison_table}
\begin{tabular}{lllllll}
\hline
Name				&  J (mag) &  H (mag) & K (mag) &  J-K & Type        &$T_p$ (K)\\
\hline
1SWASP\,J030749.87$-$365201.7	&13.552    &13.007    &12.887 	&0.665  &Target	     &4750  \\
2MASS\,J03075380$-$3653319	    &12.840    &12.407    &12.359 	&0.481  &Comparison  &      \\
2MASS\,J03075601$-$3652174	    &15.165    &14.910    &14.873 	&0.292  &Check	     &      \\
1SWASP\,J213252.93$-$441822.6	&14.299    &13.667    &13.622 	&0.677  &Target	     &4700  \\
2MASS\,J21325995$-$4418032	    &12.782    &12.317    &12.228 	&0.554  &Comparison  &      \\
2MASS\,J21330002$-$4418197	    &13.652    &13.263    &13.178 	&0.474  &Check	     &      \\
1SWASP\,J200059.78$+$054408.9	&13.412    &12.875    &12.764 	&0.648  &Target	     &4800  \\
2MASS\,J20011488$+$0543026	    &13.519    &13.162    &13.048 	&0.471  &Comparison  &      \\
2MASS\,J20011491$+$0542524	    &13.420    &13.204    &13.225 	&0.195  &Check	     &      \\
\hline
\end{tabular}
\end{tiny}
\end{table}


\begin{table}
	\centering
\begin{tiny}
\caption{Photometric solutions for the three USPCBs.}
	\label{tab:solution_table}
\begin{tabular}{lclclcl}
\hline
                        &  1SWASP\,J030749.87$-$365201.7   &	 &  1SWASP\,J213252.93$-$441822.6    &       &  1SWASP\,J200059.78$+$054408.9          \\
                                                                                                                                                       \\
Parameters              &  Photometric elements  &  errors       &  Photometric elements  &  errors          &  Photometric elements  &  errors        \\
\hline                                                                                                                                                 \\
$g_1=g_2$               &     0.32               & assumed       &     0.32               & assumed          &     0.32               & assumed        \\
$A_1=A_2$               &     0.50               & assumed       &     0.50               & assumed          &     0.50               & assumed        \\
$x_{1bolo}=x_{2bolo}$   &     0.315              & assumed       &     0.315              & assumed          &     0.311              & assumed        \\
$y_{1bolo}=y_{2bolo}$   &     0.371              & assumed       &     0.370              & assumed          &     0.377              & assumed        \\
$x_{1V}=x_{2B}$         &     1.036              & assumed       &     --                 & --               &     --                 & --             \\
$y_{1V}=y_{2B}$         &   $-0.216$             & assumed       &     --                 & --               &     --                 & --             \\
$x_{1V}=x_{2V}$         &     0.682              & assumed       &     --                 & --               &     0.658              & assumed        \\
$y_{1V}=y_{2V}$         &     0.135              & assumed       &     --                 & --               &     0.162              & assumed        \\
$x_{1R}=x_{2R}$         &     0.423              & assumed       &     0.437              & assumed          &     0.407              & assumed        \\
$y_{1R}=y_{2R}$         &     0.343              & assumed       &     0.330              & assumed          &     0.360              & assumed        \\
$x_{1I}=x_{2I}$         &     0.258              & assumed       &     --                 & --               &     0.247              & assumed        \\
$y_{1I}=y_{2I}$         &     0.423              & assumed       &     --                 & --               &     0.433              & assumed        \\
Phase shift             &     --                 & --            &     --                 & --               &     0.0032             & $\pm0.0006$    \\
$T_h$ (K)               &     4750               & $\pm700$      &     4700               & $\pm400$         &     4800               & $\pm300$       \\
$T_c$ (K)               &     4697               & $\pm703$  	 &     4671               & $\pm405$         &     4528               & $\pm316$       \\
$q=M_c/M_h$             &     2.280              & $\pm0.010$	 &     1.785              & $\pm0.007$       &     2.296              & $\pm0.040$     \\
$\Omega_{in}$           &     5.6432             & --     		 &     4.9443             & --               &     6.4432             & --             \\
$\Omega_{out}$          &     5.0383             & --     		 &     4.3545             & --               &     5.8264             & --             \\
$\Omega_1=\Omega_2$     &     5.6431             & $\pm0.0220$   &     2.0763             & $\pm0.0033$	     &     5.3123             & $\pm0.0112$    \\ 	
$i (^{\circ})$          &     $78.2$             & $\pm0.1$ 	 &     81.9               & $\pm0.1$         &     $75.3$             & $\pm0.5$       \\
$L_1/(L_1+L_2)(B)$      &     0.3415             & $\pm0.0015$ 	 &     --                 & --               &     --                 & --             \\
$L_1/(L_1+L_2)(V)$      &     0.3231             & $\pm0.0012$ 	 &     --                 & --               &     0.4196             & $\pm0.0067$    \\
$L_1/(L_1+L_2)(R)$      &     0.3035             & $\pm0.0010$ 	 &     0.3789             & $\pm0.0013$      &     0.3975             & $\pm0.0054$    \\
$L_1/(L_1+L_2)(I)$      &     0.2801           	 & $\pm0.0009$	 &     --                 & --   		     &     0.3850             & $\pm0.0048$    \\
$r_1(pole)$             &     0.2951             & $\pm0.0014$	 &     0.3165             & $\pm0.0012$ 	 &     0.3200             & $\pm0.0064$    \\
$r_1(side)$             &     0.3079             & $\pm0.0017$	 &     0.3319             & $\pm0.0015$      &     0.3392             & $\pm0.0082$    \\
$r_2(back)$             &     0.3402             & $\pm0.0027$	 &     0.3692             & $\pm0.0025$      &     0.4027             & $\pm0.0195$    \\
$r_2(pole)$             &     0.4185             & $\pm0.0013$	 &     0.4124             & $\pm0.0011$      &     0.4547             & $\pm0.0049$    \\
$r_2(side)$             &     0.4449             & $\pm0.0016$   &     0.4386             & $\pm0.0014$      &     0.4924             & $\pm0.0069$    \\
$r_2(back)$             &     0.4724             & $\pm0.0021$   &     0.4704             & $\pm0.0020$      &     0.5333             & $\pm0.0102$    \\
$f$ (\%)                &     0.0                & $\pm3.6$		 &     14.2               & $\pm1.9$	     &     58.4               & $\pm1.8$	   \\
\hline
\end{tabular}
\end{tiny}
\end{table}


\begin{table}
	\centering
\begin{tiny}
\caption{Cool spot elements based on the light curve solutions.}
	\label{tab:cool spots_table}
\begin{tabular}{lccccc}
\hline
                               &          & $\theta$ ($^\circ$) &  $\psi$ ($^\circ$) &  $\Omega($sr$)$ &  $T_s/T_*$\\
\hline
1SWASP\,J030749.87$-$365201.7  & star 1   &  89.95              &  157.08            &    0.35845      &    0.700  \\
                               & star 2   &  86.51              &  235.49            &    0.22845      &    0.850  \\
                               & star 2   &  89.95              &  180.00            &    0.27845      &    0.700  \\
1SWASP\,J213252.93$-$441822.6  & star 2   &  89.67              &   77.01            &    0.20800      &    0.700  \\
1SWASP\,J200059.78$+$054408.9  & star 2   &  89.70              &  134.30            &    0.20845      &    0.700  \\
\hline
\end{tabular}
\end{tiny}
\end{table}


\clearpage
\begin{table}
\begin{tiny}
\begin{center}
\caption{27 well-studied USPCBs.}
	\label{tab:27 well-studied USPCBs_table}
\begin{tabular}{lccccccccl}
\hline
Name                            &   Period (day)&$q=M_s/M_p$& $f (\%)$& $i (^{\circ})$& $M_p^*$ & $M_s^*$ & $R_p^*$ & $R_s^*$ & reference                       \\
\hline
1SWASP\,J030749.87$-$365201.7   &   0.2266712   &   0.439  &  0.0   &  78.2 & 0.789 	&0.346 	&0.737 	&0.507 & This paper                               \\
RW Com                          &   0.2373464   &   0.471  &  6.1   &  74.9 & 0.849 	&0.400 	&0.774 	&0.549 & \citet{Djurasevicetal2011}                   \\
NSVS\,2700153                   &   0.228456    &   0.775  &  7.1   &  47.8 & 0.650 	&0.504 	&0.662 	&0.589 & \citet{DimitrovKjurkchieva2015}            \\
1SWASP\,J055416.98$+$442534.0   &   0.21825     &   0.792  &  8.6   &  70.1 & 0.582 	&0.461 	&0.618 	&0.556 & \citet{DimitrovKjurkchieva2015}            \\
1SWASP\,J200503.05$-$343726.5   &   0.2288836   &   0.934  &  9.0   &  73.8 & 0.599 	&0.560 	&0.637 	&0.617 & \citet{Zhangetal2017}                        \\
1SWASP\,J160156.04$+$202821.6   &   0.22653     &   0.670  &  10.0  &  79.5 & 0.679 	&0.455 	&0.675 	&0.563 & \citet{Lohretal2014, Essametal2014}   \\
1SWASP\,J022727.03$+$115641.7   &   0.21095     &   0.463  &  10.4  &       & 0.659 	&0.305 	&0.658 	&0.464 & \citet{Liuetal2015a}                       \\
1SWASP\,J150822.80$-$054236.9   &   0.23006     &   0.514  &  12.0  &  90.0 & 0.774 	&0.398 	&0.729 	&0.538 & \citet{Lohretal2014}                       \\
1SWASP\,J074658.62$+$224448.5   &   0.22085     &   0.395  &  12.6  &  78.6 & 0.769 	&0.304 	&0.726 	&0.476 & \citet{Jiangetal2015a}                      \\
1SWASP\,J080150.03$+$471433.8   &   0.217531    &   0.432  &  13.6  &  83.8 & 0.723 	&0.313 	&0.698 	&0.476 & \citet{DimitrovKjurkchieva2015}            \\
1SWASP\,J213252.93$-$441822.6   &   0.22123459  &   0.560  &  14.2  &  81.9 & 0.690 	&0.386 	&0.679 	&0.521 & This paper                               \\
1SWASP\,J015100.23$-$100524.2   &   0.2145001   &   0.320  &  14.6  &  79.4 & 0.760 	&0.243 	&0.724 	&0.432 & \citet{Qianetal2015b}                         \\
V1104 Her                       &   0.22788     &   0.623  &  15.0  &       & 0.707 	&0.441 	&0.692 	&0.557 & \citet{Liuetal2015b}                       \\
CC Com                          &   0.22068516  &   0.526  &  17.0  &  89.8 & 0.701 	&0.369 	&0.685 	&0.511 & \citet{Koseetal2011}                         \\
SDSS\,J012119.10�C001949.9       &   0.2052      &   0.500  &  18.9  &  83.9&  0.600 	&0.300 	&0.622 	&0.454  & \citet{Jiangetal2015b}                     \\
NSVS\,7179685                   &   0.20974     &   0.451  &  19.3  &  85.5 & 0.655 	&0.295 	&0.656 	&0.457 & \citet{DimitrovKjurkchieva2015}            \\
NSVS\,8626028                   &   0.217407    &   0.805  &  20.7  &  65.9 & 0.573 	&0.461 	&0.612 	&0.555 & \citet{DimitrovKjurkchieva2015}            \\
2MASS\,J00164102$-$0009251      &   0.198561    &   0.630  &  22.0  &  53.3 & 0.507 	&0.319 	&0.564 	&0.457 & \citet{Davenportetal2013}                    \\
1SWASP\,J024148.62$+$372848.3   &   0.21975076  &   0.813  &  23.3  &  68.7 & 0.585 	&0.475 	&0.621 	&0.565 & \citet{Jiangetal2015c}                      \\
1SWASP\,J133105.91$+$121538.0   &   0.21801     &   0.828  &  25.2  &  77.6 & 0.570 	&0.472 	&0.611 	&0.561 & \citet{Elkhateebetal2014a}                 \\
V523 Cas                        &   0.233693    &   0.516  &  29.0  &  85.4 & 0.798 	&0.412 	&0.744 	&0.550 & \citet{Samecetal2004}                        \\
1SWASP\,J210318.76$+$021002.2   &   0.22859     &   0.877  &  34.2  &  81.9 & 0.616 	&0.540 	&0.645 	&0.607 & \citet{Elkhateebetal2014b}                 \\
1SWASP\,J200059.78$+$054408.9   &   0.20569054  &   0.435  &  58.4  &  75.3 & 0.631 	&0.274 	&0.642 	&0.439 & This paper                               \\
NSVS\,925605                    &   0.217629    &   0.678  &  70.2  &  57.2 & 0.618 	&0.419 	&0.637 	&0.533 & \citet{DimitrovKjurkchieva2015}            \\
GSC\,1387$-$0475                &   0.21781128  &   0.474  &  76.3  &  49.9 & 0.705 	&0.334 	&0.687 	&0.489 & \citet{Yangetal2010}                         \\
1SWASP\,J075102.16$+$342405.3   &   0.20917224  &   0.740  &  96.0  &  76.0 & 0.543 	&0.401 	&0.590 	&0.514 & \citet{Jiangetal2015d}                      \\
1SWASP\,J234401.81$-$212229.1   &   0.21367     &   0.422  &        &  79.4 & 0.699 	&0.295 	&0.683 	&0.461 & \citet{Lohretal2013b, Koen2014}           \\
\hline
\end{tabular}
\end{center}
\noindent {$^*$ Computed by the relation of \citet{DimitrovKjurkchieva2015} and by eqution~\ref{eq:critical Roche radii}. The footnote P denotes the primary component, while S denotes the secondary component.}
\end{tiny}
\end{table}


\begin{table}
	\centering
\begin{tiny}
\caption{Semi-major axis corrections for the deep USPCBs.}
	\label{tab:Semi-major axis corrections_table}
\begin{tabular}{lcccc}
\hline
Name                                                         & 1SWASP\,J200059.78$+$054408.9  & NSVS\,925605     & GSC\,1387$-$0475       & 1SWASP\,J075102.16$+$342405.3     \\
Period (day)                                                 & 0.20569054 	                  & 0.21762900       & 0.21781128 	          & 0.20917224                        \\
$q = M_s/M_p$                                                & $0.436\pm0.008$ 	              & $0.678\pm0.003$  & $0.474\pm0.008$        & $0.740\pm0.040$                   \\
$f(\%)$                                                      & $58.4\pm1.8$ 	              & $70.2\pm2.6$ 	 & $76.3\pm2.9$ 	      & $95.0\pm4.0$                      \\
Refrence                                                     &This paper                      &\citet{DimitrovKjurkchieva2015} & \citet{Yangetal2010} & \citet{Jiangetal2015d}\\
\hline
a $(R_{\odot})$                                              & $1.419\pm0.050$ 	              & $1.542\pm0.050$  & $1.544\pm0.050$ 	      & $1.455\pm0.050$                   \\
$r_{\rm{L1}}$                                                & $0.452\pm0.005$  	          & $0.413\pm0.004$  & $0.445\pm0.004$  	  & $0.405\pm0.004$                   \\
$r_{\rm{L2}}$                                                & $0.310\pm0.003$  	          & $0.346\pm0.003$  & $0.316\pm0.003$  	  & $0.353\pm0.004$                   \\
$r_{\rm{E1}}$                                                & $0.492\pm0.011$  	          & $0.475\pm0.012$  & $0.502\pm0.012$  	  & $0.505\pm0.012$                   \\
$r_{\rm{E2}}$                                                & $0.352\pm0.010$  	          & $0.412\pm0.010$  & $0.380\pm0.011$  	  & $0.467\pm0.011$                   \\
$cor=(r_{\rm{L1}}/r_{\rm{E1}}+r_{\rm{L2}}/r_{\rm{E2}})/2$    & $0.899\pm0.013$   	          & $0.854\pm0.013$  & $0.860\pm0.015$  	  & $0.780\pm0.010$                   \\
$a'=a \cdot cor (R_{\odot})$                                 & $1.276\pm0.064$ 	              & $1.317\pm0.063$  & $1.327\pm0.066$ 	      & $1.135\pm0.054$                   \\
$M_p (M_{\odot})$                                            & $0.458\pm0.066$  	          & $0.385\pm0.055$  & $0.448\pm0.064$ 	      & $0.257\pm0.031$                   \\
$M_s (M_{\odot})$                                            & $0.199\pm0.033$  	          & $0.261\pm0.038$  & $0.212\pm0.034$ 	      & $0.190\pm0.033$                   \\
$R_p=r_{\rm{E1}} \cdot a' (R_{\odot})$                       & $0.628 \pm0.046$  	          & $0.626\pm0.046$  & $0.666\pm0.049$  	  & $0.572\pm0.041$                   \\
$R_s=r_{\rm{E2}} \cdot a' (R_{\odot})$                       & $0.449 \pm0.035$  	          & $0.543\pm0.039$  & $0.504\pm0.040$  	  & $0.530\pm0.038$                   \\
\hline
\end{tabular}
\end{tiny}
\end{table}

\clearpage
\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=1.3]{lcs.eps}
 \caption{Observed and fitted curves for the three USPCBs. The colour points denote the observed data with different filters. Blue denotes B filter; green denotes V filter; orange denotes R filter; red denotes I filter. The black solid lines are the theoretical fittings, with the modeling of dark spots.}
    \label{fig:light curves_figure}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=.85]{qs.eps}
\caption{The relation between $q$ and fitting residuals for the three USPCBs. The red cross in each diagram presents the initial value of $q$ at the beginning of the W-D program running.}
    \label{fig:q search_figure}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=1.5]{j030.eps}
\caption{The night-to-night light curve variation of 1SWASP\,J030749.87$-$365201.7. The red data points were observed on the night of 2014-11-24 with the I filter, while the blue data points were observed on the night of 2014-11-25 with the same filter. We add four times of $P_0$ (0.2266712 day) to the red points. However, this two parts of light curves do not join well. Moreover, a changing of depths of minima is clear seen. This phenomenon could be caused by variations of the presented cool spots.}
    \label{fig:night-to-night variation_figure}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=.7]{res.eps}
\caption{Fitting residuals of light curves for the three USPCBs. The colours denote the same meaning as the Fig~\ref{fig:light curves_figure}.}
    \label{fig:residuals_figure}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=.85]{colour.eps}
\caption{Observed differential colours for the three USPCBs. Blue denotes $\Delta$(B-V); orange denotes $\Delta$(V-R); red denotes $\Delta$(V-I); magenta denotes $\Delta$(R-I). The solid lines are yielded by the W-D program.}
    \label{fig:colour index_figure}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[angle=0,scale=2]{str.eps}
\caption{The geometrical structure of the three USPCBs with their present cool spots.}
    \label{fig:cool spots_figure}
\end{center}
\end{figure}


\end{document}
