4 V401 Cyg

The eclipsing binary V401 Cyg (also BD  $+30^{\circ}$ 3592, HIP 95816, AN 88.1929, FL 2747, P1824; $\alpha_{2000} = 19^{\rm h}29^{\rm m}20.3^{\rm s}$, $\delta_{2000} = +30^{\circ}24\hbox{$^\prime$ }28.5\hbox{$^{\prime\prime}$ }$, $V_{\max}=$ 10.8 mag; Sp. F0) is a relatively well-known W UMa-type binary with period of about 0.58 days. It was discovered to be a variable by Hoffmeister (1929) and for a long time it was misclassified as RR Lyrae type variable with period 0.2895 days. Lurye (1947) found its eclipsing binary nature and derived the light elements with the correct period:

$${\rm Pri.~Min.} = {\rm HJD~24~28838.382} + 0\hbox{$.\!\!^{\rm d}$ }58271446 \cdot E.$$

Early photoelectric observations were obtained by Spinrad (1959) in June 1958 at Berkeley. He confirmed the eclipsing nature of its variability and found a colour index B - V =0.27 mag. Next B, V photoelectric photometry was obtained by Purgathofer (1964) at Lowell observatory during several nights in 1961. He found the value of B - V= 0.36 mag and revealed the remarkable increase in its period. This led him to the calculation of the new quadratic ephemeris:

$$\begin{eqnarray*}{\rm Pri.~Min.} &=& {\rm HJD~24~34215.693}\\ && + 0\hbox{$.\!\... ... }58271901 \cdot E + 2\hbox{$.\!\!^{\rm d}$ }55 \ 10^{-10} E ^2. \end{eqnarray*}$$

A period study of V401 Cyg was presented also by Herczeg (1993), who reported several photoelectric times of minima and confirmed that the period was still increasing. He also derived the corrected quadratic light elements:

$$\begin{eqnarray*}{\rm Pri.~Min.} &=& {\rm HJD~24~34215.695}\\ && + 0\hbox{$.\!\... ...$ }5827187 \cdot E + 1\hbox{$.\!\!^{\rm d}$ }10 \ 10^{-10} E ^2. \end{eqnarray*}$$

All times of minimum light given in Herczeg (1993, his Table 1) as well as new timings were incorporated in our analysis. Using the method of least squares we calculated the following light elements with a quadratic term:

$$\begin{eqnarray*}{\rm Pri.~Min.} &=& {\rm HJD~24~34215.6924}\\ && + 0\hbox{$.\!... ... }58272061 \cdot E + 7\hbox{$.\!\!^{\rm d}$ }39 \ 10^{-11} E ^2. \end{eqnarray*}$$

The O-C residuals for all times of minimum with respect to the linear ephemeris are shown in Fig. 2.

Figure 2: O-C graph for V401 Cyg. See legend for Fig. 1

The non-linear fit, corresponding to the light elements given above is plotted as a continuous curve. The period increase ${\rm d}P/{\rm d}E = 1.48 \ 10^{-10}$ day/cycle or $9.26 \ 10^{-8}$ day/year or 0.8 seconds/century resulting from these elements is relatively large for a W UMa-star. Assuming the value of mass ratio q = 0.3, resulting from our light curve analysis, and a total mass $M_1 + M_2 = 1.8~M_{\odot}$, one can obtain a mass-exchange rate for this binary ${\rm d}M/{\rm d}t = 2.03 \ 10^{-8}~M_{\odot}$/year.

The B and V light curves of V401 Cyg published by Purgathofer (1964), as well as our R light curve, have been used simultaneously for the determination of the geometric and photometric elements using the Binary Maker 2.0 reduction software (Bradstreet 1993). This program, based on the Wilson-Devinney algorithm, is used for a preliminary analysis of light curves by graphically producing models of close binaries. First, the observed points ordered in phase were combined into 177 normalized binned points in each colour. Based on the GCVS's spectral classification of the primary component, F0, and the colour index B - V=0.36 mag (Purgathofer 1964), the temperature was adopted to be T₁=6700 K. The other adopted parameters are the gravity-darkening coefficients g₁ = g₂, the albedoes A₁ = A₂ and the linear limb-darkening coeficients x₁ and x₂. We adjusted the following parameters: the mass ratio q = M₂/M₁, the potential function $\Omega_1$, $\Omega_2$, the temperature of the secondary component T₂, and the orbital inclination i. The final solution is given in Table 2; the uncertainty in the inclination is about $\pm 2$ degrees and the difference in temperature is known to about $\pm 10\%$, both dependent on the mass ratio.

 

 

Table 2: Synthetic light curve parameters for V401 Cyg

Parameter	Unit	Value
$\lambda_B$	Å	4400
$\lambda_V$	Å	5500
$\lambda_R$	Å	6900
x1B = x2B		0.74
x1V = x2V		0.60
x1R = x2R		0.48
g1 = g2		0.32
T1	K	6700
T2	K	6650
i	deg	77
$\Omega_1 = \Omega_2$		2.38
q		0.30
L3		0
r1, r2 (pole)		0.474, 0.281
r1, r2 (side)		0.515, 0.296
r1, r2 (back)		0.547, 0.348
fill out		46%

The computed light curve based on these elements is shown in Fig. 3.

Figure 3: V light curve of V401 Cyg obtained by Purgathofer (1964). The continuous line represents the light-curve solution with parameters given in Table 2. The shift of the secondary minimum from the phase 0.5 is clearly visible

This model takes into account a constant interval of brightness in the secondary minimum announced by Herczeg (1993) and confirmed by our observations ( $d_R \simeq 35$ min, see Fig. 4).

Figure 4: A plot of the differential R-magnitudes of V401 Cyg obtained during the secondary eclipse at JD 24 51738

The geometrical representation of V401 Cyg at phase 0.25 is displayed in Fig. 5.

Figure 5: Three-dimensional model of the contact system V401 Cyg at phase 0.25