\documentstyle[twoside,epsf,hyperref]{article}
\input ibvs2.sty

\begin{document}
\IBVSheadDOI{63}{6226}{11 December 2017}

\IBVStitle{V736 Cephei -- An A-Type Overcontact Binary}

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

\IBVSinst{Mountain Ash Observatory, 1393 Garvin Street, Prince George, BC, Canada, V2M 3Z1 \\ email: bob.nelson@shaw.ca}
\IBVSinst{Guest investigator, Dominion Astrophysical Observatory, Herzberg Institute of Astrophysics, National Research Council of Canada}

\SIMBADobj{V736 Cephei}


\begintext

The discoverer of the variability of V736 Cep (NSV 13635, NSVS 3275157, HD 235475. SAO 33275, TYC 3957-12-1) appears to be undocumented.  As part of the HD catalogue, it was classified presumably by Cannon and Pickering (1993) as F8.  The first relevant reference is to Otero et al. (2005) who provided coordinates, elements (epoch and period), apparent reference to the above classification, and an eclipse type (EA).  Since then, there have been numerous eclipse timings published, but no light curve or analysis.

In order to rectify this lack, the author first secured, in September of 2011, 2013, 2014, and 2015,  a total of 14 medium resolution ($R\sim10000$ on average) spectra of V736 Cep at the Dominion Astrophysical Observatory (DAO) in Victoria, British Columbia, Canada using the Cassegrain spectrograph attached to the 1.85 m Plaskett Telescope.  He used the 21181 configuration and a grating with 1800 lines/mm, blazed at 5000 \AA, and giving a reciprocal linear dispersion of 10 \AA/mm in the first order.  The wavelengths ranged from 5000 to 5260 \AA, approximately.  A log of observations is given in Table 1 and an eclipse timing diagram, in Figure 9 later in the paper.  The following elements were used for both radial velocity (RV) and photometric phasing:

\begin{equation}
{\rm JD~(Hel)~Min~I} = 24 57619.7380+ 0.8578464 E
\end{equation}

\begin{table}
\centering
\caption{Log of DAO observations.}
\begin{tabular}{lccccc}
\hline
DAO &	Mid Time &	Exposure &	Phase at &	$V_1$	& $V_2$ \\
Image\# & (HJD$-$2400000) & (sec) & Mid-exp & (km/s) & (km/s) \\
\hline
11-08068 & 55815.9293 & 3600 & 0.282 & $-$71.5 (5.8) & 224.5 (5.8) \\ 
11-08086 & 55816.7831 & 3600 & 0.278 & $-$57.1 (7.3) & 236.9 (13.4) \\ 
11-08169 & 55823.9181 & 3600 & 0.595 & 13.1 (5.0) & --- \\ 
11-08200 & 55824.9408 & 3600 & 0.787 & 38.8 (5.7) & $-$239.6 (4.0) \\ 
11-08211 & 55825.7162 & 3600 & 0.691 & 29.3 (4.3) & $-$242.3 (4.1) \\ 
11-08214 & 55825.7657 & 3600 & 0.749 & 11.7 (4.8) & $-$268.6 (3.8) \\ 
13-09667 & 56545.9245 & 3600 & 0.245 & $-$60.2 (5.2) & 249.2 (10.7) \\ 
14-24341 & 56906.7363 & 1200 & 0.847 & 26.5 (5.8) & $-$224.1 (5.4) \\ 
14-24403 & 56908.7899 & 1200 & 0.241 & $-$70.5 (4.1) & --- \\ 
14-24415 & 56908.8968 & 1200 & 0.365 & $-$43.7 (9.1) & 176.2 (12.6) \\ 
15-13014 & 57291.8559 & 1200 & 0.785 & 39.7 (5.6) & $-$247.1 (3.5) \\ 
15-13015 & 57291.8736 & 1800 & 0.805 & 43.2 (8.0) & $-$250.9 (4.8) \\ 
15-13128 & 57295.6376 & 3600 & 0.193 & $-$68.8 (6.1) & 238.4 (7.0) \\ 
15-12130 & 57295.6697 & 1800 & 0.230 & $-$64.3 (6.0) & 237.2 (8.2) \\ 
\hline
\end{tabular}
\end{table}

Frame reduction was performed by software RaVeRe (Nelson 2013).  See Nelson (2010) and Nelson et al. (2014) for further details.  The normalized spectra are reproduced in Fig.~1, sorted by phase.  Note towards the right the strong neutral iron lines (at 5167.487 and 5171.595 \AA) and the strong neutral magnesium triplet (at 5167.33, 5172.68, and 5183.61 \AA). 

\IBVSfig{5.5cm}{6226-f1.eps}{V736 Cep spectra at phases 0.193, 0.230, 0.245, 0.278, 0.282, 0.365, 0.691, 0.749, 0.785, 0.787, 0.805, 0.847 (from top to bottom).}
\IBVSfigKey{6226-f1.eps}{V376 Cep}{spectrum}

Radial velocities were determined using the Rucinski broadening functions (Rucinski, 2004, Nelson, 2010) as implemented in software Broad25 (Nelson, 2013).  See Nelson et al. (2014) for further details. An Excel worksheet with built-in macros (written by him) was used to do the necessary radial velocity conversions to geocentric and back to heliocentric values (Nelson 2014).  The resulting RV determinations are also presented in Table 1.  The mean rms errors for RV$_1$ and RV$_2$ are 5.9 and 6.9 km/s, respectively, and the overall rms deviation from the (sinusoidal) curves of best fit is 9.1 km/s. The best fit yielded the values $K_1 = 49.8$ (1.7) km/s, $K_2 = 251.1$ (2.3) km/s and $V_{\gamma} = -12.4$ (1.4) km/s, and thus a mass ratio $q_{\rm sp} = K_1/K_2 = M_2/M_1 = 0.198$ (7).

Representative broadening functions, at phases 0.232 and 0.778 are depicted in Figs. 2 and 3, respectively.  Smoothing by a Gaussian filter is routinely done in order to centroid the peak values for determining the radial velocities.

\IBVSfig{7cm}{6226-f2.eps}{Broadening functions at phase 0.230--smoothed and unsmoothed.}
\IBVSfigKey{6226-f2.eps}{V376 Cep}{other}


\IBVSfig{7cm}{6226-f3.eps}{ Broadening functions at phase 0.785--smoothed and unsmoothed.}
\IBVSfigKey{6226-f3.eps}{V376 Cep}{other}

In the autumn months of 2015 and 2016 the author took a total of 269 frames in $V$, 277 in $R_{\rm C}$ (Cousins) and 277 in the $I_{\rm C}$ (Cousins) band at his private observatory in Prince George, B.C., Canada.  Renamed Mountain Ash Observatory, it is the former Sylvester Robotic Observatory described in Nelson (2009).  A finder chart is included as Fig. 10 at the end of the paper.

Standard reductions were then applied (see Nelson et al., 2014 for more details). The variable, comparison and check stars are listed in Table 2.  The coordinates and magnitudes for V736 Cep, the comparison, and check stars are from the Tycho Catalogue, Hog, et al., (2000), and converted to standard Johnson values using relations due to Henden (2001).


\begin{table}
\centering
\caption{Details of variable, comparison and check stars.}
\begin{tabular}{lccccc}
\hline
Object & GSC & RA (J2000) & Dec (J2000) & $V$ (mag) & $B-V$ (mag) \\ 
\hline
Variable & 3957-0012 & 21\hr 16\mm 29\fsec1133 & +55\deg 23\arcm 10\farcs236 & 9.82 (3) & +0.40 (4) \\ 
Comparison & 3957-0898 & 21\hr 17\mm 29\fsec8881 & +55\deg 33\arcm 32\farcs048 & 10.10 (3) & +0.87 (7) \\ 
Check & 3957-0310 & 21\hr 17\mm 07\fsec2846 & +55\deg 23\arcm 03\farcs045 & 10.66 (5) & +0.34 (7) \\ 
\hline
\end{tabular}
\end{table}

The author used the 2003 version of the Wilson-Devinney (WD) light curve and radial velocity analysis program with the Kurucz atmospheres (Wilson and Devinney, 1971, Wilson, 1990, Kallrath and Milone, 1998, Wilson, 1998) as implemented in the Windows front-end software WDwint (Nelson, 2013) to analyse the data.  To get started, the spectral type F8 (taken from SIMBAD, no reference given; but there is an implied reference to Cannon and Pickering (1993) in Otero et al. (2005).  Interpolated tables from Flower (1996) gave a temperature $T_1 = 6199 \pm 120$ K and $\log g = 4.367 \pm 0.004$.  (The quoted errors refer to one spectral sub-class.)  An interpolation program by Terrell (1994, available from Nelson 2013) gave the Van Hamme (1993) limb darkening values; and finally, a logarithmic (LD=2) law for the limb darkening coefficients was selected, appropriate for temperatures $< 8500$ K (ibid.).  The limb darkening coefficients are listed  in Table~3.  (The values for the second star are based on the later-determined temperature of 6101 K and assumed spectral type of F8-9.)  Convective envelopes for both stars were used, appropriate for cooler stars (hence values gravity exponent $g = 0.32$ and albedo $A = 0.500$ were used for each).

\begin{table}
\centering
\caption{Limb darkening values from Van Hamme (1993).}
\begin{tabular}{ccccc}
\hline
Band & $x_1$ & $x_2$ & $y_1$ & $y_2$  \\ 
\hline
Bol & 0.645 & 0.644 & 0.227 & 0.226 \\ 
$V$ & 0.735 & 0.739 & 0.263 & 0.259 \\ 
$R_{\rm C}$ & 0.662 & 0.667 & 0.274 & 0.272 \\ 
$I_{\rm C}$ & 0.579 & 0.583 & 0.265 & 0.264 \\
\hline
\end{tabular}
\end{table}

From the GCVS 4 designation (EW) and from the shape of the light curve, mode 3 (overcontact binary) mode was used. 

Convergence was attempted by the method of multiple subsets. The subsets were: ($a$, $V_{\gamma}$, $i$, $L_1$), ($T_2$, $\Omega_1$), and ($q$, $L_1$). However, no reasonable fit could be obtained until a spot was placed on the back side of star 1 (visible during secondary minimum).  Thereafter, the fitting proceeded smoothly.

Detailed reflections were tried, with $n_{\rm ref} = 3$, but there was little--if any--difference in the fit from the simple treatment.  There are certain uncertainties in the process (see Csizmadia et al., 2013, Kurucz, 2000).  On the other hand, the solution is very weakly dependent on the exact values used.

The model is presented in Table 4.  For the most part, the error estimates are those provided by the WD routines and are known to be low; however, it is a common practice to quote these values and we do so here.  Also, estimating the uncertainties in temperatures $T_1$ and $T_2$ is somewhat problematic.  A common practice is to quote the temperature difference over--say--one spectral sub-class (assuming that the classification is good to one spectral sub-class, the precision being unknown in this case).  In addition, various different calibrations have been made (Cox, 2000, page 388-390 and references therein, and Flower, 1996), and the variations between the various calibrations can be significant.  If the classification is $\pm$ one sub-class, an uncertainty of $\pm 120$ K to the absolute temperatures of each, would be reasonable.  The modelling error in temperature $T_2$, relative to $T_1$, is indicated by the WD output to be much smaller, around 7 K.


\IBVSfig{6cm}{6226-f4.eps}{$V$ light curves for V736 Cep  -- data, WD fit, and residuals.}
\IBVSfigKey{6226-f4.eps}{V376 Cep}{light curve}


\IBVSfig{6cm}{6226-f5.eps}{$R$ light curves for V736 Cep  -- data, WD fit, and residuals.}
\IBVSfigKey{6226-f5.eps}{V376 Cep}{light curve}

\IBVSfig{6cm}{6226-f6.eps}{$I$ light curves for V736 Cep  -- data, WD fit, and residuals.}
\IBVSfigKey{6226-f6.eps}{V376 Cep}{light curve}

\begin{table}
\centering
\caption{Wilson-Devinney parameters.}
\begin{tabular}{lccc}
\hline
WD Quantity & Value & error & Unit \\ 
\hline
Temperature, $T_1$ & 6199 & [fixed] & K \\ 
Temperature, $T_2$ & 6101 & 120 & K \\ 
$q = m_2/m_1$ & 0.189 & 0.001 & --- \\ 
Potential, $\Omega_1 = \Omega_2$ & 2.152 & 0.002 & --- \\ 
Inclination, $i$ & 80.68 & 0.17 & degrees \\ 
Semi-maj. axis, $a$ & 5.23 & 0.06 & solar radii. \\ 
$V_\gamma$ & $-$13.4 & 1.8 & km/s \\ 
Fill-out, $f_1$ & 0.431 & 0.024 &  \\ 
Spot co-latitude & 70 & 5 & degrees \\ 
Spot longitude & 171 & 2 & degrees \\ 
Spot radius & 17.2 & 0.5 & degrees \\ 
Spot temp. factor & 0.948 & 0.004 &  \\ 
$L_1/(L_1+L_2)$ (V) & 0.8203 & 0.0002 & --- \\ 
$L_1/(L_1+L_2)$ (R$_{\rm C}c$) & 0.8188 & 0.0001 & --- \\ 
$L_1/(L_1+L_2)$ (I$_{\rm C}c$) & 0.8174 & 0.0001 & --- \\ 
$r_1$ (pole) & 0.5041 & 0.0004 & orbital radii \\ 
$r_1$ (side) & 0.5541 & 0.0007 & orbital radii. \\ 
$r_1$ (back) & 0.5803 & 0.0009 & orbital radii \\ 
$r_2$ (pole) & 0.2434 & 0.0011 & orbital radii \\ 
$r_2$ (side) & 0.2553 & 0.0014 & orbital radii \\ 
$r_2$ (back) & 0.3038 & 0.0033 & orbital radii. \\ 
Phase shift & 0.0004 & 0.0001 & --- \\ 
$\Sigma\omega_{\rm res}^2$ & 0.07958 & --- & --- \\ 
\hline
\end{tabular}
\end{table}

The light curve data and the fitted curves are depicted in Figures 4-6.  The residuals (in the sense observed-calculated) are also plotted, shifted upwards by 0.65 units.




The radial velocities are shown in Fig. 7.  A three-dimensional representation from Binary Maker 3 (Bradstreet, 1993) is shown in Fig. 8.


The WD output fundamental parameters and errors are listed in Table 5.  Most of the errors are output or derived estimates from the WD routines.  From Kallrath \& Milone (1998), the fill-out factor is $f = (\Omega_{\rm I}-\Omega)/ (\Omega_{\rm I}-\Omega_{\rm O})$, where $\Omega$ is the modified Kopal potential of the system, $\Omega_{\rm I}$ is that of the inner Lagrangian surface, and $\Omega_{\rm O}$, that of the outer Lagrangian surface, was also calculated.

\begin{table}
\centering
\caption{Fundamental parameters.}
\begin{tabular}{lccc}
\hline
Quantity & Value & Error & unit \\ 
\hline
Temperature, $T_1$ & 6199 & 120 & K \\ 
Temperature, $T_2$ & 6101 & 120 & K \\ 
Mass, $m_1$ & 2.20 & 0.06 & M0 \\ 
Mass, $m_2$ & 0.41 & 0.02 & M0 \\ 
Radius, $R_1$ & 2.86 & 0.01 & R0 \\ 
Radius, $R_2$ & 1.40 & 0.01 & R0 \\ 
$M_{\rm bol,1}$ & 2.20 & 0.02 & mag \\ 
$M_{\rm bol,2}$ & 3.82 & 0.02 & mag \\ 
$\log g_1$ & 3.87 & 0.01 & cgs \\ 
$\log g_2$ & 3.76 & 0.01 & cgs \\ 
Luminosity, $L_1$ & 10.9 & 0.2 & L0 \\ 
Luminosity, $L_2$ & 2.44 & 0.05 & L0 \\ 
Fill-out factor 1,2 & 0.43 & 0.02 & --- \\ 
Distance, $r$ & 316 & 6 & pc \\ 
\hline
\end{tabular}
\end{table}

To determine the distance, the analysis proceeded as follows: First the WD routine gave the absolute bolometric magnitudes of each component; these were then converted to the absolute visual ($V$) magnitudes of both, $M_{V,1}$ and $M_{V,2}$, using the bolometric corrections BC = $-$0.160 and $-$0.17 for stars 1 and 2 respectively.  The latter were taken from interpolated tables constructed from Cox (2000).  The absolute $V$ magnitude was then computed in the usual way, getting $M_{V} = 2.14 \pm 0.02$ magnitudes.  The apparent magnitude in the $V$ passband was $V = 9.854 \pm 0.03$, taken from the Tycho values (Hog et al. 2000) and converted to the Johnson magnitude $9.816 \pm 0.03$ using relations due to Henden (2001).


\IBVSfig{7cm}{6226-f7.eps}{Radial velocity curves for V736 Cep -- data and WD fit.}
\IBVSfigKey{6226-f7.eps}{V376 Cep}{radial velocity curve}

\IBVSfig{3cm}{6226-f8.eps}{Binary Maker 3 representation of the system -- at phases 0.03, 0.43 and 0.75.}
\IBVSfigKey{6226-f8.eps}{V376 Cep}{other}

Ignoring interstellar absorption (setting $A_V = 0$), we calculated a preliminary value for the distance $r = 343$ pc from the standard relation:	

\begin{equation}
r = 10^{0.2(V - M_{\rm V} -  A_{\rm V} +  5)} {\rm~parsecs}
\end{equation}

Galactic extinction was obtained from a model by Am\^{o}res \& L\'epine (2005).  The code (available in IDL and converted by the author to a Visual Basic routine) assumes that the interstellar dust is well mixed with the dust, that the galaxy is axi-symmetric, that the gas density in the disk is a function of the Galactic radius and of the distance from the Galactic plane, and that extinction is proportional to the column density of the gas,  Using Galactic coordinates of $l = 95\fdeg5859$ and $b = +4\fdeg3732$ (SIMBAD), and the initial distance estimate of $d = 0.343$ kpc, a value of $A_V = 0.175$ magnitude was determined,  A further iteration revealed little change in $A_V$.  Substitution into (2) gave $r = 316$ pc.

The errors were assigned as follows: $\delta M_{\rm bol,1} = \delta M_{\rm bol,2} = 0.02$, $\delta {\rm BC}_1 = \delta {\rm BC}_2 = 0.009$ (the variation of 1 spectral sub-class), $\delta$V = 0.03, $\delta A_V = 0.01$, all in magnitudes.  Combining the errors rigorously (i.e., by adding the variances) yielded an estimated error in $r$ of 6 pc which is probably far too low.

Some comments regarding the period variation are in order.  An eclipse timing difference (O--C) plot using timings from 1999 is depicted in Fig. 9.  Although there is considerable scatter, a linear relation over the interval, cycle 4400 (in 2009) to cycle 7380 (in 2016) was determined.  This yielded a best-fit linear solution and ephemeris of Equation (1) above.

\IBVSfig{6cm}{6226-f9.eps}{V736 Cep -- eclipse timing (O--C) diagram with a linear fit for points after cycle 4000.}
\IBVSfigKey{6226-f9.eps}{V376 Cep}{O-C diagram}

In conclusion, all the fundamental parameters for V736 Cephei have been determined.  It will be interesting to monitor this system photometrically in the coming years to observe the evolution of the spot.

The Excel file (and many others) are available at Nelson (2016).  The 8000+ files are updated annually.

\vskip 1cm


\IBVSfig{10cm}{6226-f10.eps}{Sample CCD frame of the field of view showing the stars of interest.}
\IBVSfigKey{6226-f10.eps}{V376 Cep}{finding chart}

{\bf Acknowledgements:}
It is a pleasure to thank the staff members at the DAO (Dmitry Monin, David Bohlender, and the late Les Saddlmyer) for their usual splendid help and assistance.  Much use was made of the SIMBAD database during this research.



\references 

Am\^{o}res, E.B., L\'epine, J.R.D., 2005, {\it AJ}, {\bf 130}, 659 \DOI{10.1086/430957}

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, N.Y.) \DOI{10.1007/978-1-4612-2742-7}

Cannon, A.J., Pickering, E.C., 1993, Harv. Ann., 91-100 (1918-1924; ADC 1989), Henry Draper Catalogue and Extension 1 (HD, HDE)

Cox, A.N., ed, 2000, Allen's Astrophysical Quantities, 4th ed., (Springer, New York, NY) \DOI{10.1007/978-1-4612-1186-0}

Csizmadia, S., Pasternacki, T., Dreyer, C., Cabrera, A., Erikson, A., Rauer, H., 2013, {\it A\&A}, {\bf 549}, A9 \DOI{10.1051/0004-6361/201219888}

Flower, P.J., 1996, {\it ApJ}, {\bf 469}, 355 \DOI{10.1086/177785}

Henden, A., 2001, {\tt http://www.tass-survey.org/tass/catalogs/tycho.old.html}

Hog, E., et al., 2000, {\it A\&A}, {\bf 355}, L27

Kallrath, J., Milone, E.F., 1998, Eclipsing Binary Stars--Modeling and Analysis (Springer-Verlag). \DOI{10.1007/978-1-4419-0699-1}

Kurucz, R.L., 2000, {\it BaltA}, {\bf 11}, 101

Nelson, R.H., 2009, {\it IBVS}, {\bf 5884}

Nelson, R.H., 2010, ``Spectroscopy for Eclipsing Binary Analysis'' in The Alt-Az Initiative, Telescope Mirror \& Instrument Developments (Collins Foundation Press, Santa Margarita, CA), R.M. Genet, J.M. Johnson and V. Wallen (eds) [available on ResearchGate]

Nelson, R.H., 2013, Software by Bob Nelson, \\ {\tt https://www.variablestarssouth.org/bob-nelson/}

Nelson, R.H., 2014, Spreadsheets, by Bob Nelson, \\ {\tt https://www.variablestarssouth.org/bob-nelson/}

Nelson, R.H., 2016, Bob Nelson's $O-C$ Files, {\tt http://www.aavso.org/bob-nelsons-o-c-files}

Nelson, R.H., \c{S}enavci, H.V., Ba\c{s}t\"{u}rk, \"{O}, and Bahar, E., 2014, {\it NewA}, {\bf 29}, 57 \DOI{ 10.1016/j.newast.2013.11.006}

Otero, S.A., Wils, P, and Dubovsky, P.A., 2005, {\it IBVS}, {\bf 5586}

Rucinski, S. M., 2004, ``Advantages of the Broadening Function (BF) over the Cross-Correlation Function (CCF)'', in Stellar Rotation, {\it Proc. IAU Symp.}, {\bf 215}, 17

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

Van Hamme, W., 1993, {\it AJ}, {\bf 106}, 2096 \DOI{10.1086/116788}

Wilson, R.E., 1990, {\it ApJ}, {\bf 356}, 613 \DOI{10.1086/168867}

Wilson, R.E., 1998, Documentation of Eclipsing Binary Computer Model (available from the author)

Wilson, R.E., and Devinney, E.J., 1971, {\it ApJ}, {\bf 166}, 605 \DOI{10.1086/150986}

\endreferences 

\end{document}
