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\title

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Can spot activity solve the problem of abnormal gravity darkening in semi-detached system?



\author(s)

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H. Rovithis-Livaniou$^{\rm 1}$, G. Djura\v sevi\'c$^{\rm 2}$, N. Georgiades$^{\rm 1}$, P. Rovithis$^{\rm 3}$,\\ and  S. Erkapic$^{\rm 2}$



$^{\rm 1}${\it Section of Astrophysics, Astronomy \& Mechanics, Dept. of Physics, Athens University, Panepistimiopolis, Zografos 157 84, Greece}\\

$^{\rm 2}${\it Astronomical Observatory, Volgina 7, 11050 Belgrade, Yugoslavia}\\

$^{\rm 3}${\it Institute of Astronomy \& Astrophysics, National Observatory of Athens, P.O. Box 20048, 11810 Athens, Greece}\\


\text



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\large



\section*{Abstract}
Abnormal gravity darkening exponents, found in some semi-detached systems from their light curve analysis, can turn to normal, if a spotted model is used. To show this, an application to the semi-detached eclipsing binary TV Cas is presented.


\section{General Introductory Remarks}
According to gravity darkening law: {\it for stars in hydrostatic equilibrium, the local emergent flux is proportional to the local gravity acceleration}. That is: $F \sim g^{\tau}$, where {\it F} is the emergent flux, {\it g} the gravity acceleration, and {\it $\tau$} the gravity darkening coefficient. As a consequence, we have:                                                      
$T_{eff} \sim g^{{\tau}/4}$  or  $T_{eff} \sim g^{\beta}$, where $\beta={\tau}/4$; 
$\beta$ being known as the gravity darkening exponent.

For stars in radiative equilibrium, $\tau=1$, or $\beta=0.25$, von Zeipel (1924).
For stars with convective envelopes, $\tau=0.32$, or $\beta=0.08$, Lucy (1967);                  
while, a generalization was made by Kippenhahn (1977). Moreover, recently Claret (2000) has re-examined the gravity darkening effect, studying stellar rotation, and yield to new results for very low mass stars. 

The foregoing mentioned theoretical values of $\beta$ are usually used in the light curve analysis. On the other hand, $\beta$ can be treated as free parameter and calculated from light curves analysis of eclipsing binaries (e.g. Twigg \& Rafert 1980, Kitamura \& Nakamura 1987). A similar study has been carried out by Gjura\v sevi\'c et al. (2000a), with very important results. One of them is that abnormal gravity exponents could become normal, if a spotted model is used for the light-curve analysis. The case of the semi-detached system TV Cas is presented and used as an example to show this very interesting result.


\section{Methodology}
(Gjura\v sevi\'c et al. 2000a) analyzed many semi-detached systems to find their gravity darkening exponents. The analysis was based on good photoelectric observations, and made with our latest improved computer program (e.g. Djura\v sevi\'c et al. 2000b). The examined systems were chosen such that to consist of a main sequence primary component, being well inside its Roche lobe, and a sub-giant (or giant) secondary filling its corresponding Roche lobe. The gravity darkening exponents of the primaries were fixed to their theoretical values, while those of the secondaries were treated as free parameters, and calculated.

In most of the systems, the computed values of $\beta$ were very close to the theoretically predicted; but, there was also a number of systems for which abnormal values were found. In these particular systems that the gravity darkening exponents were found to be much higher than predicted by theory, and because there is no any physical mechanism that could explain this, a re-analysis was done.   

The re-analysis was made assuming the existence of some spot activity on their secondary components. An assumption very reliable because of the presence of a small O'Connell effect, being obvious either directly from the light curves of the systems, or in the residuals between the observed and the theoretically computed light curves. 

\section{The TV Cas case}    
TV Cas is a short period eclipsing binary, ($P \sim 1.8116$ days), consisting of a hot ($T_{eff}=10500$), main sequence primary of B9 spectral type, and a cool secondary component.  The BV light curves published by Papousek (1974), were used in the present study. The mass ratio of the two components was taken equal to 0.405 
(Khalesseh \& Hill 1992). The two stars' albedos, were assumed equal to one, and 0.5 for the hot, and the cool components, respectively; while, the gravity darkening exponent of the hot primary was fixed to 0.25.
The TV Cas system was firsty analyzed with $\beta$ treated as free parameter. In this case, $\beta$ was found to be equal to 0.15; an abnormal, and much higher value than the theoretically expected 0.08, for stars with convective envelopes. Then, the system was re-analyzed using a spotted model, because of the presence of a small O'Connell effect (Max I is less than Max II) in its light curves. The results for both cases, are given in Table 1, and presented in Fig.1.


\section{Discussion of the Results}
>From the results of the light curve analysis of the semi-detached eclipsing binary TV Cas, for the two different cases:\\
1) the gravity darkening exponent $\beta_{c}$ of the secondary cool component, to be taken as free parameter to be adjusted, and\\
2) fixing $\beta_{c}$ to its theoretical value, and using a spotted model,\\
given in Table 1, and presented in Fig.1, it is obvious that:\\
a) the fitting between observed and calculated light-curves is very good in both cases,
b) the spotted and unspotted models yield to almost identical results, as  concerns the principal elements of the two components.\\
{\bf Thus, if a spotted model is used, is not necessary to leave $\beta_{c}$ to get abnormal values.}\\  
A similar result has been also found in the LT Her case (Djura\v sevi\'c et al., 2000c). This, might mean that perhaps spot activity could be the answer and solve the problem of abnormal values of gravity darkening  exponents, found by some investigators in  many semi-detached systems; at least in some of them.
We are working towards this direction, and try to find out if this is also the case in other semi-detached systems, and under which conditions (Djura\v sevi\'c et al., 2002).

\section*{Acknowledgments}

This work was partially financial supported by Athens University, (grant No. 70/4/3305). This work has made use of ADS data base at Strasburge, France.



\section*{References}
Claret A., 2000, A\&A, 359, 289.\\  
Djura\v sevi\'c G., Rovithis-Livaniou H., Rovithis P., Milovanovic N., 2000a, JENAM-2000, ABSTRACTS, p.74.\\
Djura\v sevi\'c G., Rovithis-Livaniou H., Rovithis P., 2000b, A\&A, 364, 543.\\
Djura\v sevi\'c G., Rovithis-Livaniou H., Rovithis P, Georgiades N., Erkapi\'c, 2000c, Proc. 20th SPIG, (Petrovi\'c Z.Lj. et al. Eds), p. 501.\\
Djura\v sevi\'c G., Rovithis-Livaniou H., Rovithis P, Georgiades N., Erkapi\'c, 2002, (work in progress).\\
Ebbighausen E.G., Peyegor G., 1974, PASP, 86, 203.\\ 
Kippenhahn R., 1977, A\&A, 58, 271.\\  
Kitamura M., Nakamura  Y., 1987, Ann. Tokyo Astron. Obs., 2nd 
Series 21, No 4, 387.\\     
Lucy B.L., 1967, Zeitschrift fur Astrophysik, 65, 89.\\ 
Papousek J., 1974, BAICz, 25, 152.\\
Russo G., Milano L., 1983, A\&AS, 69, 261.\\
Twigg L.W., Rafert J.B., 1980, MNRAS, 193, 775.\\
von Zeipel H., 1924, MNRAS, 84, 665.\\



   \begin{table*}
\caption{Results of the analysis of the TV Cas light curves 
obtained by solving the inverse problem for the Roche model. 
{\bf Left} - Gravity-darkening coefficients of the cooler secondary 
component ($\beta_{\rm c}$) is free parameter ; {\bf Right} - Spot on the 
secondary component.} 
         \label{ResTVC}
      \[
        \begin{array}{lllll}
            \hline
            \noalign{\smallskip}
{\rm CB \ System} &{\rm TV \ Cas} &{\rm TV \ Cas} &{\rm TV \ Cas} &{\rm TV \ Cas}\\
{\rm Quantity}    &{\rm B - filter} &{\rm V - filter} &{\rm B - filter} &{\rm V - filter} \\
            \noalign{\smallskip}
            \hline
            \noalign{\smallskip}
      n                         & 143             & 143             & 143             & 143            \\
{\rm \Sigma(O-C)^2}             & 0.0583          & 0.1239          & 0.0506          & 0.0989         \\                                                           
{\rm \sigma}                    & 0.0203          & 0.0295          & 0.0189          & 0.0264         \\                                                           
q=m_{\rm c}/m_{\rm h}           & 0.405           &                 &                 &                \\                                                           
T_{\rm h}                       & 10500           &                 &                 &                \\                                                           
\beta_{\rm h}                   & 0.25            &                 &                 &                \\                                                           
A_{\rm h}                       & 1.0             &                 &                 &                \\                                                           
A_{\rm c}                       & 0.5             &                 &                 &                \\                                                           
f_{\rm h}=f_{\rm c}             & 1.0             &                 &                 &                \\                                                           
T_{\rm c}                       & 5404  \pm 38    & 5345  \pm 36    & 5425  \pm 29    & 5423  \pm 28   \\                                                           
F_{\rm h}                       &0.647  \pm 0.002 &0.647  \pm 0.002 &0.648  \pm 0.001 &0.649  \pm 0.003\\                                                           
F_{\rm c}                       &0.998  \pm 0.001 &0.995  \pm 0.001 &1.000  \pm 0.001 &1.000  \pm 0.001\\                                                           
     i                          &79.00  \pm 0.02  & 79.0  \pm 0.03  &79.00  \pm 0.02  &79.00  \pm 0.03 \\                                                           
A_{\rm S}=T_{\rm S}/T_{\rm c}   &                 &                 &0.65^*           &0.65^*          \\                                                                       
\theta_{\rm S}                  &                 &                 &26.1   \pm 2.0   &26.0   \pm 1.5  \\
\lambda_{\rm S}                 &                 &                 &70.2   \pm 5.3   &71.7   \pm 4.3  \\                                                           
\varphi_{\rm S}                 &                 &                 &43.7   \pm 5.8   &44.7   \pm 4.1  \\                                                           
\beta_{\rm c}                   &{\bf0.15}\pm0.02 &{\bf0.16}\pm 0.02&0.08^*           &0.08^*          \\                                                           
u_{\rm h}                       &0.57             & 0.49            & 0.57            &0.49            \\                                                           
u_{\rm c}                       &0.78             & 0.70            & 0.78            &0.70            \\                                                           
\Omega_{\rm h}                  &3.968            &3.970            &3.962            &3.961           \\                                                           
\Omega_{\rm c}                  &2.691            &2.696            &2.688            &2.688           \\                                                           
R_{\rm h} [D=1]                 &0.279            &0.279            &0.280            &0.280           \\                                                                       
R_{\rm c} [D=1]                 &0.283            &0.282            &0.283            &0.283           \\                                                                       
L_{\rm h}/(L_{\rm h}+L_{\rm c}) &0.946            &0.902            &0.947            &0.900           \\                                                                       
{\rm {\cal M}_h [M_{\odot}]}   & 3.78     & 3.78  & 3.78            & 3.78          \\                     
{\rm {\cal M}_c [M_{\odot}]}   & 1.53     & 1.53  & 1.53            & 1.53          \\                                               
{\rm {\cal R}_h [R_{\odot}]}   & 3.09     & 3.09  & 3.10            & 3.10          \\                                               
{\rm {\cal R}_c [R_{\odot}]}   & 3.30     & 3.29  & 3.31            & 3.31          \\                                               
{\rm log} \ g_{\rm h}          & 4.03     & 4.03  & 4.03            & 4.03          \\                                               
{\rm log} \ g_{\rm c}          & 3.59     & 3.59  & 3.58            & 3.58          \\                                               
M^{\rm h}_{\rm bol}            &-0.26     &-0.26  &-0.26            &-0.26          \\                                               
M^{\rm c}_{\rm bol}            & 2.48     & 2.54  & 2.46            & 2.46          \\                                               
a{\rm _{orb} [R_{\odot}]}      &10.90     &10.90  &10.90            &10.90          \\                                               
            \noalign{\smallskip}
            \hline                    
         \end{array}
      \]
\medskip                                                                         
\noindent BaSeL approximation of the stellar atmospheres 
(${\rm [Fe/H]_{h,c}=0.0}$ - accepted metallicity of the components)
\bigskip

\noindent {\bf Note:} $n$ - number of observations, $\Sigma(O-C)^2$ - final 
sum of squares of residuals between observed (LCO) and synthetic (LCC) light 
curves, $\sigma$ - standard deviation of the observations, 
$q=m_{\rm c}/m_{\rm h}$ - mass ratio of the components, $T_{\rm h}$ - 
temperature of the hotter component, $\beta_{\rm h}$ - gravity-darkening 
coefficients of the hotter component, $A_{\rm h}=A_{\rm c}=1.0$ - albedo 
coefficients of the components, $f_{\rm h}=f_{\rm c}=1.00$ - nonsynchronous 
rotation coefficients of the components, $T_{\rm c}$ - temperature of the 
cooler component, $F_{\rm h,c}$ - filling factors for critical Roche 
lobes of the hotter primary (h) and cooler secondary (c), $i$ - orbit 
inclination (in arc degrees), $\beta_{\rm c}$ - gravity-darkening coefficient 
of the cooler secondary component, $u_{\rm h,c}$ - limb-darkening coefficients 
of the components, $\Omega_{\rm h,c}$ - dimensionless surface potentials of the 
components, $R_{\rm h,c}$ - polar radii of the components in units of the 
distance between the component centres, $L_{\rm h}/(L_{\rm h}+L_{\rm c})$ - 
luminosity of the hotter star, ${\rm {\cal M}_{h,c} [{\cal M}_{\odot}]}$ - 
masses of stars in solar units, ${\rm {\cal R}_{h,c} [R_{\odot}]}$ - mean radii 
of stars in solar units, ${\rm log} \ g_{\rm {h,c}}$ - logarithm (base 10) of 
the mean surface acceleration (effective gravity) for components, 
$M^{\rm h,c}_{\rm bol}$ - absolute bolometric magnitudes of stars and 
$a{\rm _{orb} [R_{\odot}]}$ - orbital semi-major axis in units of the solar 
radius.
   \end{table*}

\begin{figure*}
   \resizebox{\hsize}{!}{\includegraphics{TVC.eps}}
%     \vspace{21cm}                                                       
\caption{Observed (LCO) and final synthetic (LCC) light curves of 
TV Cas with final O-C residuals obtained by analysing B and V observations; 
The view of the system TV Cas at orbital phase 0.25, obtained 
with the parameters estimated by analysing observations. {\bf Left} -
Gravity-darkening coefficient of the cooler secondary component is a
free parameter ; {\bf Right} - Spot on the secondary component.}
     \label{LcTVC}
    \end{figure*}
     

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