3 DK Cyg

The eclipsing binary DK Cyg (also BD  $+33^{\circ}$ 4304, HIP 106574, AN 1.1927, FL 3231; $\alpha_{2000} = 21^{\rm h}35^{\rm m}02.7^{\rm s}$, $\delta_{2000} = +34^{\circ}35\hbox{$^\prime$ }45.4\hbox{$^{\prime\prime}$ }$, $V_{\max}= 10.30$ mag; Sp. A8V) is a well-known W UMa-type binary with a period of about 0.4707 days. It was discovered to be a variable by Guthnick & Prager (1927). The first photoelectric observations were made by Hinderer (1960). Binnendijk (1964) in his photometric study derived linear light elements:

$${\rm Pri.~Min. = HJD~24~37999.5838} + 0\hbox{$.\!\!^{\rm d}$ }47069055 \cdot E$$

and found the change of the secondary minimum depth. The period study of DK Cyg was published by Paparo et al. (1985). They showed that the orbital period was increasing and calculated the first parabolic light elements:

$$\begin{eqnarray*}{\rm Pri.~Min.} &=& {\rm HJD~24~37999.5828} \\ &&+ 0\hbox{$.\!... ... }47069066 \cdot E + 5\hbox{$.\!\!^{\rm d}$ }39 \ 10^{-11} E ^2. \end{eqnarray*}$$

The next photoelectric study was published by Awadalla (1994) who refined the light elements:

$$\begin{eqnarray*}{\rm Pri.~Min.} &=& {\rm HJD~24~37999.58249}\\ && + 0\hbox{$.\... ...$ }47069073 \cdot E + 5\hbox{$.\!\!^{\rm d}$ }76 \ 10^{-11} E ^2 \end{eqnarray*}$$

and confirmed the light curve variability. See also that paper for further details and a historical review of other observations. The first spectroscopic study and radial velocity curve analysis was presented by Rucinski & Lu (1999) who found q = m₂/m₁ = 0.325. Recently, updated linear ephemeris of DK Cyg was presented by Kiss et al. (1999):

$${\rm Pri.~Min.} = {\rm HJD~24~51000.0999} + 0\hbox{$.\!\!^{\rm d}$ }47069290 \cdot E.$$

All photoelectric times of minimum light published in Hinderer (1960), Binnendijk (1964), Paparo et al. (1985), Agerer (1988, 1990, 1992), Awadalla (1994), Agerer & Hübscher (1995, 1996) and Hegedüs et al. (1996), as well as new timings given in Table 1 were incorporated in our analysis. One additional time of primary minimum was obtained by J. Strobl at the Jindrichuv Hradec Observatory, Czech Republic, with a CCD camera SBIG ST6 and a 16 cm Newtonian telescope. Other numerous visual and photographic estimations obtained by the AAVSO, BAV and BBSAG observers were used with less weight. A total of 95 times of minimum were incorporated in our analysis. Using the method of least squares we derived the following light elements with a quadratic term:

${\rm Pri.~Min.} = {\rm HJD~24~37999.5825}\ +$ $\pm 0.0004$ $0\hbox{$.\!\!^{\rm d}$ }47069064 \cdot E$ + $5\hbox{$.\!\!^{\rm d}$ }75 \ 10^{-11} E ^2.$

$\pm 0.00000008$    $\pm 0.03$

The O-C residuals for all times of minimum with respect to the linear ephemeris are shown in Fig. 1.

Figure 1: O-C residuals for the times of minimum of DK Cyg with respect to the linear light elements. The continuous curve represents the parabolic approximation. The individual photoelectric times are denoted by dots, photographic or visual estimations by circles

The non-linear fit, corresponding to the light elements given above, is plotted as a continuous curve. The period increase resulting from these elements is ${\rm d}P/{\rm d}E = 1.15 \ 10^{-10}$ day/cycle or $8.92 \ 10^{-8}$ day/year or 0.8 seconds/century. Because the long-term increase in orbital period is usually explained by mass transfer from the secondary to the primary component, we calculated the value of this supposed mass transfer for DK Cyg. If the mass transfer is conservative with no magnetic field we determine the mass transfer rate ${\rm d}M/{\rm d}t = 2.86 \ 10^{-8}~M_{\odot}$/year for a total mass of the system $M_1 + M_2 = 2.0~M_{\odot}$ and the mass ratio q = 0.325 (Rucinski & Lu 1999).