\documentstyle[hyperref,epsf,twoside]{article}

\input ibvs2.sty

\begin{document}

\IBVSheadDOI{63}{6223}{23 November 2017}

\IBVStitletl{Direct Distance Estimation and Absolute}{ Parameters of Z Draconis}

\IBVSauth{Terrell, D.$^1$; Nelson, Robert H.$^{2,3}$}

\IBVSinst{Dept.\ of Space Studies, Southwest Research Institute,
1050 Walnut St., Suite 300,\\ \indent  Boulder, CO 80302, USA,
e-mail: terrell@boulder.swri.edu}
\IBVSinst{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{Z Dra}
\IBVStyp{   EB   }
\IBVSkey{photometry}
\IBVSkey{spectroscopy}
\IBVSabs{New radial velocities of the Algol-type binary Z Draconis are reported. Images of the field taken in 2005 are re-reduced using APASS standards to derive calibrated photometry. Simultaneous analysis of the radial velocities and photometry using the Direct Distance Estimation method yields the absolute parameters, as well as the distance to the system.}

\begintext

Terrell (2006) briefly discussed the early observational efforts on the Algol-type binary Z Draconis and presented differential $BVR_CI_C$ light curves,
the first published light curves obtained on a modern photometric system. Since then, the availability of the AAVSO Photometric All-Sky Survey 
(APASS; Henden, et al.\ 2012) has made it possible to place the observations on the standard (absolute) system by using APASS standards in the field of Z Dra.
We present a re-reduction of the $BV$ images of Terrell (2006) that place the data on the standard system. 
We also present new spectroscopic observations that yield radial velocities of the primary star and, for the first time, the secondary star,
thus enabling us to measure the mass ratio accurately. The combination of standard photometry (with flux calibrations, {\it viz.} Wilson, et al.\ 2010)
and radial velocities allows for the inclusion of the distance to the binary as a solution parameter, yielding a distance estimate and corresponding error that 
includes the correlations with other adjusted model parameters directly, rather than being an after-the-fact estimate with simplifying assumptions 
({\it e.g.} spherical stars). This Direct Distance Estimation (DDE) approach is described in Wilson (2008) and application examples
are found in Wilson \& Van Hamme (2009), Vaccaro, et al.\ (2010), Wilson \& Raichur (2011) and Vaccaro, et al.\ (2015).

The equipment used to make the photometric observations is described in Terrell (2006). The $BV$ images were bias/dark subtracted and 
flatfielded using the ccdproc routine in IRAF (Tody, 1993), and instrumental magnitudes were measured using PSF fitting with SExtractor (Bertin \& Arnouts, 1996) and PSFEx (Bertin, 2011).
The instrumental magnitudes were then transformed onto the standard system using the method described in Terrell, et al.\ (2016). The resulting
$BV$ magnitudes are available from the IBVS web site as file 6223-t3.txt. We chose to use the $BV$ images and not the
$R_CI_C$ images for two reasons. First APASS does not provide $R_CI_C$ magnitudes for standards directly, and transformations from the APASS passbands 
($BVg'r'i'$) to $R_CI_C$ are still preliminary. Secondly, the DDE approach is best suited to the analysis of light curves in two passbands when 
solving for the surface temperatures of both stars, as we do here ({\it viz.} Wilson, 2008). The addition of a light curve in a third passband would
allow us to add the interstellar extinction as an adjustable parameter, but the extinction towards Z Dra appears to be very small (Terrell, 2006), as 
expected for its high galactic latitude and close distance. We did perform some solutions with $BVI_C$ light curves, both adjusting the extinction directly and by doing
solutions on a grid of fixed values for the extinction, but the results were not encouraging. The likely small value of the extinction combined with
the uncertainties in the $I_C$ calibration probably play a role in the inability to measure the extinction with our data. 

In April of 2011, RHN secured a total of six medium resolution (R$\approx$10,000) spectra of Z Dra at the Dominion 
Astrophysical Observatory (DAO) in Victoria, British Columbia, Canada using the Cassegrain spectrograph attached to the 1.85 m 
Plaskett Telescope.  The 21181 configuration was employed using 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. Frame reduction 
was performed by software RaVeRe (Nelson 2013).  See Nelson (2010) and Nelson et al. (2014) for further details. Radial velocities were 
determined using the Rucinski broadening functions (Rucinski, 2004, Nelson, 2010) as implemented in software Broad (Nelson, 2013; Nelson et al.\ 2014). A log of the spectroscopic observations is given in Table 1. 


\begin{table}
\centering
\caption{Radial velocity observations of Z Dra.}
\vskip2mm
\begin{tabular}{cccccc}
\hline
DAO  & Mid time & Exposure  & Phase at  & $V_1$ & $V_2$ \\\ 
image \#  & (HJD--2400000) & (sec) & mid-exposure${^\dag}$ & ($km\ s^{-1}$) & ($km\ s^{-1}$)\\ 
\hline
11-02487 & 55666.7962 & 3600 & 0.275 & --82.9 $\pm$ 2.3 & 140.8 $\pm$ 5.4\\
11-02532 & 55668.8275 & 1097 & 0.772 & 25.6 $\pm$ 2.7 & --203.3 $\pm$ 3.2\\
11-02569 & 55670.7885 & 3600 & 0.216 & --82.0 $\pm$ 2.3 & 143.0 $\pm$ 5.9\\
11-02710 & 55676.7913 & 3600 & 0.639 &  14.4 $\pm$ 2.3 & --189.7 $\pm$ 7.6\\
11-02719 & 55676.9429 & 3600 & 0.750 &  25.9 $\pm$ 2.5 & --205.4 $\pm$ 3.0\\
11-02752 & 55678.8829 & 3600 & 0.180 & --75.8 $\pm$ 2.4 & 117.9$\pm$ 5.6\\
\hline
\multicolumn{6}{c}{\normalsize $^\dag$ Phases computed using the ephemeris parameters in Table 2.}
\end{tabular}
\end{table}

The $BV$ light curves and the new radial velocities were analyzed simultaneously with the 2013 version of the 
Wilson--Devinney program (WD; Wilson \& Devinney, 1971; Wilson, 1979; Wilson, 2008). Since Z Dra is a semi-detached
system with the lower mass secondary filling its Roche lobe (confirmed by initial experiments with the model using a detached 
configuration), we employed WD mode 5 in all of our solutions. We performed fitting experiments assuming both convective and radiative
envelopes for the primary star, but found that models assuming a convective envelope gave superior fits in all cases, thus our
best-fit model assumes a value of 0.32 for the gravity darkening exponents of both stars and a value of 0.5 for the bolometric albedoes. 
Limb darkening coefficients were automatically computed at each iteration from the Van Hamme (1993) tables. Weights for the 
various light and velocity curves were determined automatically by WD at each iteration.

\IBVSfig{10cm}{6223-f1.eps}{The fits to the $B$ and $V$ light curves of Z Dra. The residuals (observed - computed) from the fits are shown at the bottom.}
\IBVSfigKey{6223-f1.eps}{Z Dra}{light curve}
\IBVSfig{10cm}{6223-f2.eps}{The fits to the radial velocity curves curves of Z Dra. The sizes of the error bars on the primary star velocities are
 approximately the same size as the points.}
\IBVSfigKey{6223-f12eps}{Z Dra}{radial velocity curve}

In contrast to the traditional way of analyzing photometry using independently and arbitrarily scaled light curves in several passbands,
the DDE approach uses standard magnitudes and preserves the color information found in the differences between the
light curves in each passband at each point in the binary orbit. With two light curves in different passbands, it is therefore possible 
to allow the surface temperatures of both stars to adjust in the solution, as opposed to the traditional approach 
where the temperature of one star is fixed at a value derived from other sources such as spectral types or colors as, 
for example, in the solution for Z Dra of Terrell (2006). 

Table 2 shows the adjusted parameters which includes the distance to the system in addition to the expected parameters for
a semi-detached solution with light and radial velocity curves. Figure 1 shows the fits to the light curves, and Figure 2 the 
radial velocity fits. There are clearly small asymmetries present in the light curves, probably due to spots, 
and we did attempt a few fits with a single cool spot on the primary, but the improvement in the fit was marginal and the question of
the uniqueness of such solutions with modest-precision light curves led us to abandon the spot fits.
As noted in Terrell (2006), the derived value of $\dot{P}$
is not particularly informative given the complex period changes in the system, but it was included to allow for the change in 
period between the epochs of the photometric and spectroscopic observations so that they could be analyzed simultaneously. 
Previous studies of the eclipse timings ({\it viz.} Khaliullina, 2016 and 
references therein) conclude that a third star may be present in the system and we included third light as an adjustable parameter
in our solutions, but this led to physically unrealistic (negative) values.The estimated distance to the system is 276 $\pm$ 3 pc 
and that compares well to the value of 283$^{+19}_{-17}$ pc from Gaia Data Release 1 (Gaia Collaboration, et al.\ 2016).  If there were third 
light in the system that was unaccounted for in the model, the distance to the system would be understimated because the system would
be too bright for its actual distance. The good agreement with the distance from Gaia supports the argument that any third light in
Z Dra must be negligible.

\begin{table}
\centering
\caption{Parameters from the light/velocity curve solution.}
\vskip2mm
\begin{tabular}{ccc}
\hline
Parameter & Value & Std. error\\ 
\hline
$a$ & 6.29 $R_{\odot}$ & 0.08 $R_{\odot}$ \\ 
$V_\gamma$ & $-28.2$ km sec$^{-1}$ & 0.5 km sec$^{-1}$ \\ 
$i$ & 86\fdeg94 & 0\fdeg06 \\ 
$T_1$ & 6446 K & 11 K \\ 
$T_2$ & 3936 K & 14 K \\ 
$q$ & 0.304 & 0.002 \\ 
$\Omega_1$ & 4.56 & 0.02 \\ 
$\Omega_2$ & 2.475 & (lobe filling constraint) \\ 
$\mathrm{HJD}_0$ & 2453430.71668 & 0.00006 \\
$P$ & 1\fday3574226 & 0\fday00001 \\
$\dot{P}$ & $2.0 \times 10^{-8}$& $1.1 \times 10^{-8}$ \\
$log(d)^\dag$ & 2.441 & 0.005 \\
$M_1$ & 1.39 $M_\odot$ & 0.05 $M_\odot$\\
$M_2$ & 0.42 $M_\odot$ & 0.02 $M_\odot$\\
$R_1$ & 1.48 $R_\odot$ & 0.02 $R_\odot$\\
$R_2$ & 1.77 $R_\odot$ & 0.02 $R_\odot$\\
$L_{V,1}$ & 4.0 $L_\odot$ & 0.1 $L_\odot$\\
$L_{V,2}$ & 0.134 $L_\odot$ & 0.004 $L_\odot$\\
\hline
\multicolumn{3}{c}{\normalsize{$^\dag$ Distance $d$ to the binary in parsecs.}}\\
\end{tabular}
\end{table}

With a mass of 0.42 $M_\odot$ and a radius of 1.77 $R_\odot$, the secondary component is clearly evolved, making Z Dra a short-period Algol.
Still unresolved is the nature of the period changes in the system. A period increase due to mass transfer from the lobe-filling secondary
seems to be a reasonable conclusion but the somewhat periodic changes on top of that are still debated. The light time effect, the 
Applegate (1992) mechanism, or a combination of both, are plausible explanations at this point. Further observations, standardized
photometry in particular to measure luminosity changes predicted by the Applegate mechanism, will be needed to decide between the
various possibilities.

\vskip 1cm

\textit{Acknowledgements:} This research was made possible through the use of the AAVSO Photometric All-Sky Survey (APASS), 
funded by the Robert Martin Ayers Sciences Fund and U.S. National Science Foundation grant 1412587.
It is a pleasure to thank the staff members at the DAO (David Bohlender, Dmitry Monin, and the late 
Les Saddlemyer) for their usual splendid help and assistance.  Much use was made of the SIMBAD database 
during this research.

\references

Applegate, J.H., 1992, {\it ApJ}, {\bf 385}, 621 \DOI{10.1086/170967}

Bertin, E., 2011, {\it ASP Conf.\ Ser.} {\bf 442}, 435

Bertin, E., Arnouts, S., 1996, {\it A\&A Suppl. Ser.}, {\bf 117}, 393 \DOI{10.1051/aas:1996164}

Gaia Collaboration, et al., 2016, {\it A\&A}, {\bf 595}, A1 \DOI{10.1051/0004-6361/201629272}

Henden, A. A., Levine, S. E., Terrell, D., Smith, T. C., Welch, D., 2012, {\it JAAVSO}, {\bf 40}, 430 

Khaliullina, A.I., 2016, {\it Astronomy Reports}, {\bf 60}, 517 \DOI{10.1134/S1063772916030070}

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)

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

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

Rucinski, S. M., 2004, {\it IAU Symp.}, {\bf 215}, 17

Terrell, D. 2006, {\it IBVS}, {\bf 5742}

Terrell, D. 2016, {\it IBVS}, {\bf 6166}

Tody, D., 1993, {\it ASP Conf.\ Ser.}, {\bf 52}, 173

Vaccaro, T.R., Terrell, D., Wilson, R.E., 2010, {\it ASP Conf.\ Ser.} {\bf 435}, 89

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

Wilson, R.E., 1979, {\it ApJ}, {\bf 234}, 1054 \DOI{10.1086/157588}

Wilson, R.E., 2008, {\it ApJ}, {\bf 672}, 575 \DOI{10.1086/523634}

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

Wilson, R.E., Raichur, H., 2011, {\it MNRAS}, {\bf 415}, 596 \DOI{10.1111/j.1365-2966.2011.18741.x}

Wilson, R.E., Van Hamme, W., 2009, {\it ApJ}, {\bf 699}, 118 \DOI{10.1088/0004-637X/699/1/118}

Wilson, R.E., Van Hamme, W., Terrell, D., 2010, {\it ApJ}, {\bf 723}, 1469 \DOI{10.1088/0004-637X/723/2/1469}

\endreferences

\IBVSedata{6223-t3.txt} 
\IBVSdataKey{6223-t3.txt}{Z Dra}{light curve}

\end{document}
