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\begin{document}
\IBVShead{5880}{16 March 2009}

\IBVStitletl{MW UMa, A Detached Binary:}{Observations and Analysis}

\IBVSauth{Nelson, Robert H.$^{1,2}$}

\IBVSinst{1393 Garvin Street, Prince George, BC, Canada, V2M 3Z1
        e-mail: b-o-b.nelson@shaw.ca [remove dashes]}
\IBVSinst{Guest investigator, Dominion Astrophysical Observatory, Herzberg Institute of Astrophysics, National Research Council of Canada}

\SIMBADobj{MW UMa}

\begintext

While studying the optical variability of ROSAT X-ray sources, Robb et al. (2002) found MW UMa (= GSC 4153-0634 = RXJ 114302+603435) to be a detached eclipsing binary of period 1.2347 days.  They presented eight CCD times of minima, photometry, a light curve in $R_{\mathrm{C}}$ (Cousins), and a classification spectrum.  The spectrum and the photometry yielded approximate spectral types of F6V +F9V.  The full light curves in $V$, $R_{\mathrm{C}}$, and $I_{\mathrm{C}}$ were kindly forwarded to the author.

During April of 2006, the author took five high resolution (10 \AA/mm reciprocal dispersion) spectra at the Dominion Astrophysical Observatory (DAO) in Victoria, British Columbia, Canada; he then used the Rucinski broadening functions (Rucinski, 2004) to obtain radial velocity (RV) curves (see Nelson, et al., 2006a for details).  The spectral range was 5004-5267 \AA \ and the reciprocal dispersion, 10 \AA/mm.  A log of DAO observations and RV results is presented in Table 1.

\begin {center}
{\bf Table 1: } \vskip 1mm
\begin{tabular}{cccccc}
\hline
DAO&Mid Time&Exposure&Phase at&$V_1$&$V_2$\\
Image \#&(HJD-2400000)&(sec)&Mid-exp&(km/s)&(km/s)\\
\hline
3697&53847.9490&1422&0.778&\ 122.2&-138.4\\
3720&53848.9623&3600&0.599&\ \ 71.5&\ \ -84.5\\
3729&53849.7431&3600&0.231&-124.1&\ 134.5\\
3731&53849.7853&3600&0.266&-122.9&\ 138.0\\
3736&53849.8639&3600&0.329&-107.4&\ 123.1\\
\hline
\end{tabular}

\end{center}

The following elements were used for phasing throughout (see Nelson, 2006b for the O-C relation):

$$\rm{JD\,Hel\,Min\,I} = 52402.3277(1) + 1.23475(80) E$$

The author used the 2004 version of the Wilson-Devinney (WD) light curve and radial velocity analysis program with the Kurucz atmospheres (Wilson and Devinney, 1971, Wilson, 1990, Kallrath, et al., 1998) as implemented in the Windows software WDwint (Nelson, 2005) to analyze the data.  To get started, a spectral type F6V mentioned above and a temperature $T_1 = 6514 \pm 240$\,K were used; interpolated tables from Cox (2000) which gave $\log g = 4.368$ were used; an interpolation program by Terrell (1994) gave the (van Hamme, 1993) limb darkening values; and finally, a logarithmic (LD\,=\,2) law for the extinction coefficients was selected, appropriate for cooler stars (Bessell, 1979).  Convective envelopes were chosen for both stars.  (Radiative envelopes were later tried as a check but gave a poorer fit.)

Mode 2 (for detached stars) was chosen based on the general appearance of the light curves.  Mindful that, for detached systems, stellar size is not well constrained by the light curves (Terrell, 2009), the author determined the flux ratio based on the areas under the (well-defined) peaks in the Rucinski broadening function (see Fig.~1).  This is justified (Rucinski, 2008) because the Rucinski broadening functions (Rucinski, 2004) are linear.  Values were taken for all the spectra; a mean value of Flux$_2$/Flux$_1 = 0.690 \pm 0.003$ (sd of mean) was obtained.  (Note that this was for a wavelength band centred at 5136 Angstroms; it was taken to represent the $V$ band, the closest available band.)

% Fig. 1.  A Rucinski broadening function for MW UMa, image \#3697
\IBVSfig{3.7cm}{5880-f1.eps}{A Rucinski broadening function for MW~UMa, image \#3697.}

During modelling, convergence was attained through the method of adjusting sets of uncorrelated parameters (fixed parameters are given in Table 2).  In particular, the mass ratio $q = M_2/M_1$ was held fixed because this value ($0.886 \pm 0.003$) was well determined from the RV curves; in contrast, it is not well constrained from the photometric data.  Various starting values of $\Omega_1$ were used and the DC procedure followed through to a best solution.  The resultant flux ratios $= L_2/L_1$ were then found and compared to the required value of 0.690.  In this way, the optimal value for $\Omega_1$ was found and a consistent solution found.

To investigate the situation further, the author made use of a contour-plotting routine in WDwint in which the best model served as the starting point, after which the program varied two parameters -- in this case, $\Omega_1$ and $\Omega_2$ (the surface potentials of stars 1 and 2, resp.) -- over a matrix of 100 preset values.  For each value of these two parameters, it searched for a best fit varying other selected parameters (in this case, $L_1$, $T_2$, and inclination $i$) over two iterations or loops (each loop varying one parameter, making the correction, then on to the next parameter, then on to the third).   Mass ratio, $q$, was kept fixed, as mentioned above.  A long 'rift valley' in $\Omega_1$--$\Omega_2$ space resulted, in which low values of the sum of residuals ($\Sigma\omega_{\mathrm{res}}^2$ ) nearly equaled that of the best solution all the way along the trough.  (See Fig.~2.)  This result underscored the need for independent flux ratio determinations obtained from the spectra, else the solution would be truly indeterminate.  The lines of constant flux ratio (not shown) are lines of positive slope; the one for the required value neatly intersected the imaginary line along the trough for the best (spotted) solution (marked by a cross).

The author then ran a set of runs using the square root law (LD = 3), interacting with the above contour plot to get $\Omega_2$ as a function of $\Omega_1$. The best solution yielded a sum of the residues indistinguishable from the solution with LD = 2; further, there was no significant change in any of the output values.

Runs with non-zero eccentricity were attempted, but with the formal errors close to the final value of 0.001, circular orbits were adopted.  Two separate runs were made:  Model 0: no spots, and Model 1: spots of star 1.  The final results for both models are given in Table 4.  It may be seen that Model 1 (spots on star 1) gives a significantly lower residual than the unspotted model and a much better fit visually; hence it is adopted (but the unspotted results given for comparison).  Note that the quoted errors are formal errors produced by the WD program; actual errors may be larger.  Also the quoted error in $T_2$ is relative to $T_1$ -- that is, it is the error in ($T_2 - T_1$).  The absolute error in $T_1$ is much larger, corresponding to half one spectral sub-class (so obviously, the absolute error in $T_2$ will be of the same order).  Third light was tested for, but no significant differences from zero were found.

The fundamental parameters, using the results from Model 1 are given in Table 3.  Also, the interstellar coefficient, $A_V = 0.19 \pm 0.09$ was taken from Robb, et al. (2002) to yield an estimate of the distance.

\IBVSfig{8.3cm}{5880-f2.eps}{A contour plot of $\Sigma\omega_{\mathrm{res}}^2$ plotting $\Omega_1$ versus $\Omega_1$.}

\begin {center}
{\bf Table 2: }

\nopagebreak\vskip1mm
\begin{tabular}{cccc}
\hline
Quantity&\multicolumn{2}{c}{Value}&Error\\
&Star 1&Star 2&\\
\hline
$g$&0.320&0.320&[fixed]\\
$A$&0.500&0.500&[fixed]\\
$x$ (bol)&0.639&0.644&[fixed]\\
$y$ (bol)&0.241&0.225&[fixed]\\
$x$ ($V$)&0.710&0.741&[fixed]\\
$y$ ($V$)&0.275&0.257&[fixed]\\
$x$ ($R_{\mathrm{C}}$)&0.637&0.669&[fixed]\\
$y$ ($R_{\mathrm{C}}$)&0.285&0.271&[fixed]\\
$x$ ($I_{\mathrm{C}}$)&0.553&0.585&[fixed]\\
$y$ ($I_{\mathrm{C}}$)&0.276&0.264&[fixed]\\
\hline
\end{tabular}
\end{center}

\begin {center}
{\bf Table 3: } \vskip 1mm
\begin{tabular}{ccccccc}
\hline
Fundamental&Star 1&Star 1&Star 1&Star 2&Star 2&Star 2\\
Quantity&Tabular&WD&error&Tabular&WD&Error\\
\hline
Sp. Type&F6 V&--&--&F9 V&--&--\\
Mass ($M_0$)&1.32&1.28&0.10&1.11&1.14&0.09\\
Radius ($R_0$)&1.26&1.23&0.01&1.14&1.18&0.01\\
$\log g$ (cgs)&4.36&4.36&0.004&4.37&4.35&0.004\\
Luminosity ($L_0$)&2.56&2.47&0.36&1.60&1.75&0.25\\
Distance (pc)&--&159&11&--&--&--\\
\hline

\end{tabular}
\end{center}



\begin {center}
\small
{\bf Table 4: } \vskip 1mm
\begin{tabular}{cccccccc}
\hline
Quantity.&Model 0&Model 1&Error&Quantity.&Model 0&Model 1&Error\\
--&(no spot)&(spot on 2)&--&--&(no spot)&(spot on 2)&--\\
\hline
$T_1$ (K)&6514&6514&130&$L_2/L_1\,\, (V)$&0.670&0.690&-\\
$T_2$ (K)&6026&6112&22&a (solar radii)&6.48&6.49&0.008\\
$\Omega_1$&6.27&6.17&0.02&$V_\gamma$ (km/s)&-0.7&-0.7&0.3\\
$\Omega_1$&5.89&5.95&0.02&$r_1$ (pole)&0.185&0.189&0.001\\
$q = M2/M1$&0.886&0.886&0.005&$r_1$ (point)&0.188&0.192&0.001\\
$i$ (deg)&82.90&82.78&0.05&$r_1$ (side)&0.186&0.190&0.001\\
Spot Lat (deg)&na&90&30&$r_1$ (back)&0.188&0.191&0.001\\
Spot Lng (deg)&na&185&2&$r_2$ (pole)&0.183&0.180&0.001\\
Spot Rad. (deg)&na&74&5&$r_2$ (point)&0.186&0.184&0.001\\
Spot Temp fac&na&0.992&0.004&$r_2$ (side)&0.186&0.182&0.001\\
$L_1/(L_1+L_2)\,\,(V)$&0.592&0.592&0.002&$r_2$ (back)&0.186&0.183&0.001\\
$L_1/(L_1+L_2)\,\,(R)$&0.578&0.581&0.002&$\Sigma\omega_{\mathrm{res}}^2$&0.02140&0.01832&--\\
$L_1/(L_1+L_2)\,\,(I)$&0.567&0.571&0.002&&&&\\
\hline
\end{tabular}
\end{center}


The resultant $V$ light curves are displayed in Figs.~3 and 4 together with the residuals, and the radial velocity curves, in Fig.~5.  A 3-dimensional representation from Binary Maker 3 (Bradstreet, 1993) is displayed in Fig.~6.
Fits for all colours are found in the on-line version in Figs.~7-8.

The validity of spots resulting from WD analysis is, in the opinion of the author, tenuous.  Spots are added because they yield significant improvements to the fit (as happened here).  One needs to use care, however, as the unrestrained addition of spots can, in principle, fit anything!

% Figure 3. Photometric Data and WD Output (V, Rc, and Ic) -- Model 0 (no spot)
\IBVSfigNoCaption{3.5cm}{5880-f3a.eps}
\IBVSfigKey{5880-f3a.eps}{MW UMa}{light curve}
\IBVSfig{5.75cm}{5880-f3b.eps}{Photometric Data and WD Output ($V$, top panel), residual (bottom panel) -- Model 0 (no spot)}
\IBVSfigKey{5880-f3b.eps}{MW UMa}{}

% Figure 4. Photometric Data and WD Output (V, Rc, and Ic) -- Model 1 (with spot)
\IBVSfigNoCaption{3.5cm}{5880-f4a.eps}
\IBVSfigKey{5880-f4a.eps}{MW UMa}{light curve}
\IBVSfig{5.75cm}{5880-f4b.eps}{Photometric Data and WD Output ($V$, top panel), residual (bottom panel) -- Model 1 (with spot)}
\IBVSfigKey{5880-f4b.eps}{MW UMa}{}


% Figure 5. Radial Velocity Data and WD Output
\IBVSfig{7cm}{5880-f5.eps}{Radial Velocity Data and WD Output}
\IBVSfigKey{5880-f5.eps}{MW UMa}{radial velocity curve}

% Figure 6.  Binary Maker depiction of the system at phases 0.75 and 0.97 respectively.
\IBVSfig{4cm}{5880-f6.eps}{Binary Maker depiction of the system at phases 0.75 and 0.97 respectively}
\IBVSfigKey{5880-f6.eps}{MW UMa}{}

\IBVSefigure{5880-f7.eps}
\IBVSfigKey{5880-f7.eps}{MW UMa}{light curve}
\IBVSefigure{5880-f8.eps}
\IBVSfigKey{5880-f8.eps}{MW UMa}{light curve}


In conclusion, MW UMa is a detached binary with temperatures of about 6500 and 6100 K, and luminosities of about 2.6 and 2.5 solar units respectively.  Reference to the evolutionary tracks of Schaller et al. (1992) for Y = 0.300 and z = 0.020 reveals that both stars are unevolved from the main sequence.

{\it Acknowledgements.} It is a pleasure to thank Russ Robb, at the University of Victoria, for suggesting this star and for providing the photometric data.  It is also a pleasure to thank the staff members at the DAO (especially Dmitry Monin and Les Saddlemyer) for their usual splendid help and assistance.  Thanks are also due to an anonymous referee for pointing out problems with the original paper, and to Dirk Terrell for his advice in the modelling process and for reviewing the final manuscript.

\references
% References:

Bessell, M.S., 1979, PASP, 91, 589

Bradstreet, D.H.; 1993, Binary Maker 2.0, An Interactive Graphical Tool for Preliminary Light Curve Analysis, in Milone, E.F. (ed.) Light Curve Modelling of Eclipsing Binary Stars,  pp 151-166 (Springer, New York) \BIBCODE{1993IAUCB..21..151B}

Cox, A.N., ed, 2000, Allen's Astrophysical Quantities, 4th ed., (Athlone Press, London).

Kallrath, J., Milone, E.F., Terrell, D., and Young, A.T., 1998, ApJ, 508, 308.

Nelson, R.H., 2005, Software, by Bob Nelson,\\ {\tt http://members.shaw.ca/bob.nelson/software1.htm}

Nelson, R.H., Terrell, D., \& Gross, J., 2006a, IBVS, 5715

Nelson, R.H., 2006b, Eclipsing Binary O-C Files,\\ {\tt http://www.aavso.org/observing/programs/eb/omc/nelson\_omc.shtml}

Robb, R.M., Thanjavur, K., and Clem, J.L., 2002, IBVS, 5305

Rucinski, S. M., 2004, IAUS, 215, 17, in: Stellar Rotation, ed.: A. Maeder, P. Eenens, San Francisco, ASP

Rucinski, S. M. 2008, private communication

Schaller, G., Schaerer, D., Meynet, G and Maeder, A., 1992, A\&ASS, 96, 269

Terrell, D., 1994, Van Hamme Limb Darkening Tables, vers. 1.1.

Terrell, D., 2009, private communication

van Hamme, 1993, AJ, 106, 2096

Wilson, R.E., and Devinney, E.J., 1971, ApJ, 166, 605

Wilson, R.E., 1990, ApJ, 356, 613

\endreferences

\end{document}
