The W UMa-type binary star, WZ Cep (AN 244.1928,2MASS J23222421+7254566, 1RXS J232216.6+725505)was discovered by Schneller (1928), as an RR Lyr type variable star with a period of 0.^d260984. Nine years later, a photographic light curve was obtained by Balazs (1937), who correctly classified WZ Cep as a W UMa binary star with an orbital period of 0.^d41745. Detre (1940) also made photographic observations of the system, confirmed Balazs'result, and gave basic system parameters from the analysis of his light curve using Russell's classical model.The first photoelectric BV light curves were secured by Hoffmann (1984) who detected a strong O'Connell effect in his light curves, as the brightness in the 0.25 phase (Max I) was significantly higher than that in the 0.75 phase(Max II). The spectral type was assigned as F5 in the General Catalogue of Variable Stars (Kholopov et al. 1987). Hoffmann's light curves were reanalyzed by Kału？ny(1986) with the Wilson Devinney binary model (WD;Wilson & Devinney 1971). His solutions showed that WZ Cep is a contact binary with the components of unequal surface temperatures (ΔT ？ 1,000 K). The O'Connell effect in the light curve was ascribed to a small hot spot on the less massive component near the neck between stars. These interpretations were questioned by Djura？evi？ et al. (1998), who obtained new BVR light curves that were almost symmetrical. Their analysis of both their own light curves and Hoffmann's using their own Roche computer model (Djura？evi？ 1992a, b) showed that the WZ Cep system is a shallow contact binary (f = 9~14%) with the components of slightly different temperatures (ΔT ？ 140 K) and with two large dark spots on the surface of the massive, hotter primary star. Recently, WZ Cep was revisited by Lee et al. (2008) and Zhu & Qian (2009). Lee et al. (2008) presented symmetrical BVR light curves in the observing season in 2005, while Zhu & Qian (2009) obtained an asymmetrical V light curve with a strong O'Connell effect. Both authors analyzed their light curves using the WD method and confirmed the photometric results of Djura？evi？ et al. (1998). At the same time, Zhu & Qian (2009) made the first intensive period study of WZ Cep with all timings available to them, and found that the period variation consists of both a periodic and secularly decreasing terms. The periodic term has a period of 34.^y2 with amplitude of 0.^d013, while the secular period decrease rate is about -8.8 × 10^-8 day/year.

Numerous timings just before and after the period study of Zhu & Qian (2009) have been published in the literature. In this paper, all published and newly observed timings have been reanalyzed. It is noticed that the ephemeris suggested by Zhu & Qian (2009) is not compatible with recent timing residuals. New ephemeris is derived with a detection of a secondary cyclic modulation of the O-C residuals by a period of about 11.8 year. A discussion of the implications on the existing working hypotheses follows.

Charge-coupled device (CCD) photometric observations of WZ Cep for the purpose of timing determination were made on the three nights of May 28 and October 9, 2005 and September 23, 2006 with the 35-cm reflector of the campus station of Chungbuk National University Observatory in Korea. The telescope was equipped with an SBIG ST-8 CCD imaging system, electrically cooled, with a 19' × 12' field of view. No filters were used in our observations. GSC 4486-1402 and GSC 4486-1352 were chosen as the comparison and check stars, respectively. Our comparison is the same one used by Lee et al. (2008). Camera exposure time ranged between 40 seconds and 60 seconds according to the quality of the night.

The instrumentation used and the reduction method for the raw CCD frames have been described in detail by Kim et al. (2006). The resultant standard errors of our observations in terms of comparison minus check star were about ±0.^m02. From our observations, three primary times of minimum light were determined using the conventional Kwee & van Woerden (1956) method, and are listed in Table 1.

To investigate the period variation of WZ Cep, a total of 185 (91 visual, 4 photographic, and 90 photoelectric and CCD) times of minimum light, including ours, have been collected from a modern database (Kreiner et al. 2001) and from the recent literature. Table 1 lists only photoelectric and CCD minima, which were not compiled by Zhu & Qian (2009) or published after their study.

The (O-C) residuals of all timings were calculated with the linear ephemeris of Kreiner et al. (2001), as follows:

The (O-C₁) diagram is shown in Fig. 1, where the times of minimum are marked by assorted symbols that differ in size and shape according to observational method and type of eclipse. The (O-C₁) residuals were listed in the sixth column of Table 1. As can be seen in the figure, it

is clear that the variation patterns of the residuals before and after 1995^y are quite different from each other. The orbital period before 1995^y seemed to be nearly constant,

although there is a large gap of about 25 years between 1939^y and 1964^y. However, the period from 1995^y to the present has dramatically decreased in a continuous way.

This fact was noticed by Zhu & Qian (2009), who adopted a quadratic plus sine ephemeris to explain the (O-C₁) residuals constructed with the timings available to their time, as follows:

In Eq. (2) the phase angle among the arguments in sine function should be read as +103.^o5 rather than -76.^o5, which must be a typo. The value of +103.^o5 was calculated by us with the (O-C)₁ and (O-C)₂ values listed in the fifth and six columns of Table 2 in the paper of Zhu & Qian (2009). The dashed and solid lines in Fig. 1 denote the linear plus quadratic and full terms in Eq. (2), respectively. In the bottom of Fig. 1, the residuals from Eq. (2) where the recent timings since 1995y have been clearly deviating from the ephemeris are plotted, implying that Eq. (2) should be revised and/or the period change of WZ Cep may be more complicated. The standard deviation (σ) for the photoelectric and CCD residuals was calculated to be σ = ±0.^d0029. Such a value cannot be neglected because of the accuracy of modern photoelectric and CCD timings (Kim et al. 2005).

As a first step in understanding the general behavior of period change of the WZ Cep system, we tried to fit all (O-C₁) residuals to a quadratic plus sine ephemeris by using the Levenberg-Marquardt method (Press et al. 1992). The final solution was quickly converged as follows:

In this and subsequent calculations, we assigned different weights to each of the data according to the observational method: 10 to photoelectric or CCD observations,3 to photographic, and 1 to visual data, which were the same weights as those used by Zhu & Qian (2009). The linear plus quadratic and full terms in Eq. (3) were drawn with the (O-C₁) residuals as the dashed and solid lines in Fig. 2, respectively. The residuals from full terms of Eq. (3) were plotted in the bottom of Fig. 2. As shown in Fig. 2, Eq. (3) gives a more satisfactory fit to the observed timings than Eq. (2). From the argument of 0.o0089 in the sine term of Eq. (3), the cyclic period is calculated as 46.y4 (±0.^y2), which is 12.^y2 longer than that of Eq. (2) given by Zhu & Qian (2009). It is interesting to note that the residuals from Eq. (3) show an oscillatory pattern, especially in the photoelectric and CCD timings, with small amplitude of about 0.^d005 and with a short period of about 10^y.

To find the variation characteristics of the small oscillation in Fig. 2, several non-linear terms in addition to a linear plus quadratic ephemeris were considered, such as: 1) a light-time effect (LTE), 2) two sine curves, 3) a sine curve plus an LTE, 4) two LTEs. Next, attempts were made to fit all timings to each of the four ephemeris models. The result showed that the fits with the ephemeris models of both 3) and 4) were superior to those with 1) and 2). In addition, the solution with the ephemeris 4) gave zero eccentricity for the LTE orbit, with a shorter period and smaller amplitude in two LTE orbits. This means that the ephemeris 3) is best for our purpose. The ephemeris 3) is expressed as:

where τ₃ is the light-time due to the assumed third body, and includes five parameters (a₁₂ sin i₁₂, e₁₂, ω₁₂, P₁₂, T₁₂) which are the orbital parameters of the eclipsing pair around the mass center of the triple system. The parametric and differential forms of the orbital elements for the light-time orbit were taken from Irwin (1952, 1959). The Levenberg-Marquardt method was used again to solve Eq. (4) simultaneously. In this case, there are 11 parameters to be adjusted in Eq. (4). The calculations converged quickly to yield the solution listed in Table 2. The quality of the final solution represented by Eq. (4) is very satisfactory, and may be judged from Fig. 3, in which the (O-C₁) residuals were plotted with Eq. (1), and the dashed

and solid lines represent the linear plus quadratic and full terms of Eq. (4), respectively. The residuals from all terms of Eq. (4) were listed in the seventh column of Table 1, and were plotted in the bottom of Fig. 3. The standard deviation for the photoelectric and CCD residuals in the bottom of Fig. 3 is σ = ±0.^d0002, which is excellently compatible with that of the modern photoelectric and CCD timing accuracy. Figs. 4 and 5 show two cyclic (O-C) diagrams phased with each of their periods, in which the continuous lines were drawn with the solution parameters in Table 2. Summing up our analysis of all timings of WZ Cep, the period of the system has varied in two cyclical components superposed on a global downward parabola over about 80^y.

From the coefficient (A) of the quadratic term in Table 2, the secularly decreasing rate of the period of WZ Cep is calculated to be -9.^d97 (±0.14) × 10^-8 y^-1. In principle, there are two working mechanisms for the secular period decrease observed in a close binary star such as the WZ Cep system: 1) secular mass transfer from the more massive(“primary” hereinafter) to the less massive (“secondary”hereinafter) stars, 2) angular momentum loss (AML) by magnetic braking via a magnetically controlled stellar wind from a magnetically active star in a binary. WZ Cep has been known as a shallow contact system with a fill-out factor of about 9-24% (Djura？evi？ et al. 1998, Lee et al. 2008, Zhu & Qian 2009) and with the massive primary hotter than the secondary. Moreover, the primary star has a large and variable dark spot(s) (Djura？evi？ et al. 1998, Lee et al. 2008, Zhu & Qian 2009), indicating that it is a magnetically active star. This would mean that both a secular transfer of a gaseous mass stream from the primary to the secondary star and the AML from the magnetically active primary star are occurring concurrently in the WZ Cep system. On this basis, the observed secular period decrease is expressed as follows:

where the subscripts mt and mb denote mass transfer and magnetic braking, respectively. The period change by conservative mass transfer is expressed with the following well-known formula,

where M₁ and M₂ denote the masses of the primary and secondary stars, respectively. The period change due to the AML by magnetic braking is approximately expressed by the following formula (Bradstreet & Guinan 1994):

where R₁ and R₂ are the radii of the components in solar units, q is the mass ratio (q = M₂ / M₁), k² is the gy-

ration constant typically ranging from 0.07 to 0.15 for solar-type stars, and P is the orbital period in days. Before solving Eqs. (5)-(7) above, we have to estimate the absolute parameters of WZ Cep, because no reliably published absolute parameters currently exist. Thus, the absolute parameters of WZ Cep were estimated with the photometric solution of Djura？evi？ et al. (1998) and with Harmanec's (1988) empirical relation between mass and temperature. Table 3 lists the resultant absolute parameters.By using the parameters in Table. 2 and 3 and k2 = 0.1 (Webbink 1976), Eq. (7) was solved to determine the rate of decrease of the period due to magnetic braking as -6.^d72 × 10^-8 y^-1, which contributes about 67% to the observed period decrease. Therefore, we note that magnetic braking may be a major mechanism contributing to the observed secular period change. Inserting these values into Eqs. (5) and (6), the mass transfer rate from the primary to the secondary star was calculated as 1.69 × 10^-8 M_⊙ y^-1. If the observed decreasing rate of period is assumed to occur purely by a continuous mass transfer, the mass flow rate is calculated with Eq. (6) to yield about 5.16 × 10^-8 M_⊙ y^-1, which is about three times larger than that obtained when the magnetic braking and the mass transfer are occurring simultaneously. From our discussions above, one may conclude that 1) the secular period decrease of WZ Cep has resulted from the combination of AML by magnetic braking and mass transfer from the primary to the secondary components, and 2) AML is about two times more effective than mass transfer in terms of its contribution to the observed secular period decrease.

In Section 3 we have decoupled two cyclic terms via Eq. (4) from the observed (O-C₁) residuals, shown in Fig. 3 and listed in Table 2: One is a sine term with a semi-amplitude of 0.^d0054 and a period (P_sine) of 11.^y8, while the other an LTE term with a semi-amplitude of 0.^d0178, a period (P_LTE) of 41.^y3, and an eccentricity of 0.54. It is very interesting to note that there exists a commensurable 2:7 relation between P_sine and P_LTE; namely, 7 P_sine = 2 P_LTE.

There have been two rival theories to explain these cyclic period changes: One explanation is the LTEs due to additional third- and fourth-bodies in the WZ Cep system,and the other is the Applegate (1992) mechanism. Firstly, if the LTEs are assumed to be the mechanism to produce two cyclic period changes, then mass functions, masses and bolometric luminosities of two tertiary bodies were calculated for different inclinations by using the absolute parameters in Table 3. The results were listed in Table 4. In the calculation, the bolometric luminosities were obtained by using Harmanec's (1988) empirical formula for main-sequence stars. As listed in Table 4, the minimum masses (i' = 90^o) of 0.30M_⊙ for the third body and of 0.49M_⊙ for the fourth body would contribute very small fractions of about 0.1% and 2% lights, respectively, to the total luminosity (l_t = l₁ + l₂ + l₃ + l₄). Therefore, l3 and l4 would be hardly detectable in the light curve syntheses by the previous investigators (Djura？evi？ et al. 1998, Lee et al. 2008, Zhu & Qian 2009).

Secondly, as an alternative explanation for the cyclical components of the period variability of WZ Cep, Applegate's (1992) model deserves attention. According to this theory, the orbital period can be modulated from changes in the distribution of angular momentum (and corresponding changes in stellar oblateness) as the magnetically active star undergoes changes in magnetic activity.

Using the modulation periods and amplitudes listed in Table 2 and adopting the absolute dimensions provided in Table 3, the variations of parameters needed to change the orbital period of WZ Cep by specified amounts can be obtained from the formulae given by Applegate (1992). The calculations were made with the assumption that two cyclical changes of period are produced by both components. The resultant parameters were listed in Table 5, and show that the Applegate mechanism could operate only in the primary star for both two cyclic changes of period, but cannot in the secondary because the calculated luminosity change (ΔL_rms) is about 11 and 3 times larger for the shorter and longer periods, respectively, than the values required by the model. Our determination that the primary may be the magnetically active star could be supported by Mullan's (1975) conclusion that it is the more massive components of W UMa systems that should preferentially manifest star spots, and thus, magnetic activity (Eaton 1986). Previous investigators (Djura？evi？ et al. 1998, Lee et al. 2008, Zhu & Qian 2009) also invoked a cool spot (s) on the surface of the more massive primary component to explain the light curve asymmetry of WZ Cep.

Since the Applegate model predicts that the rms luminosity of the active primary star would be variable with the same period as that of the orbital period modulation,the relative lights (e.g., Max II [0.^p75] - Max I [0.^p25]) from four V bandpass light curves published so far were deduced, and their time variability was carefully investigated.But no meaningful relation between the light variation and the period variability was found, although the light asymmetry at a given epoch is remarkable and variable with different epochs. Two major reasons for this finding might be stressed: 1) The number of the light curves is only four, which is too few to test the Applegate model in detail, and 2) strong light variations produced by the localized spot activity (activities) may reside concurrently with the rms luminosity variations, or may overwhelm the rms luminosity itself. This negative conclusion somewhat strengthens the interpretation of the mechanical perturbation from the third- and fourth stars that was developed above.

We analyzed a total of 185 timings of WZ Cep, including our new three timings, to understand the dynamical picture of this very active W UMa binary system in detail. It was found that the orbital period of the system has varied in two cyclical components superposed on a global downward parabola over about 80y. Three decoupled components of the period variability were intensively investigated. For further investigations of these components,the absolute dimensions of WZ Cep were estimated with the photometric solutions of Djura？evi？ et al. (1998) and with Harmanec's (1988) empirical relation between mass and temperature.

The secular period decrease of -9.^d97 (±0.14) × 10^-8 y^-1 was calculated and interpreted as a combination of both AML due to magnetic braking and mass-transfer from the massive primary component to the secondary. The decreasing rate of period due to AML was obtained as -6.^d72 × 10^-8 y^-1, which contributes about 67% to the observed period decrease. The remaining 33% contribution results from the mass flow of about 5.16 × 10^-8 M_⊙ y^-1 from the massive primary component to the secondary. In this picture, the major driven contributor to the observed period decrease is AML by magnetic braking, which is about two times larger than mass-transfer. At the moment, WZ Cep may be considered as on an evolutionary track from the present shallow contact state of about 15% to a more deep contact state during a thermal time scale, which is expected by the magnetic braking theories (van't Veer 1979, Rucinski 1982, Maceroni & van't Veer 1991, Bradstreet& Guinan 1994, St？pie？ 1995, 2006, Demircan 1999) and the thermal relaxation oscillation theories (Flannery 1976, Lucy 1976, Robertson & Eggleton 1977).

In addition to the secular period decrease, WZ Cep has suffered from two cyclical period changes consisting of a sine curve with a shorter period of 11.^y8 and a semi-amplitude of 0.^d0054, and an LTE orbit with a longer period of 41.^y3, a semi-amplitude of 0.^d0178, and an eccentricity of 0.54. Very interestingly, there seem to exist in a commensurable 7:2 relation between their mean motion and a 33:10 relation between their amplitudes. As the possible causes for the cyclical period changes, two rival interpretations(LTE and Applegate model) were considered, and the parameters related with these were derived in Table. 4 and 5, respectively. In the LTE interpretation, the obtained minimum masses of 0.30M_⊙ and 0.49M_⊙ for the third and fourth bodies, respectively, would contribute very small fractions of about 0.1% and 2% lights, respectively,to the total luminosity (l_t = l₁ + l₂ + l₃ + l₄), if they are assumed to be main-sequence stars. This explains why previous investigators (Djura？evi？ et al. 1998, Lee et al. 2008, Zhu & Qian 2009) could not detect the third light (l₃) in their light curve syntheses. In the LTEs explanation, the 7:2 relation found between their mean motions would be interpreted as a stable 7:2 orbit resonance produced by a long-term gravitational interaction between two tertiary bodies (Peale 1976, Kley et al. 2004)

In the interpretation with the Applegate model, the model parameters in Table 5 indicate that the Applegate mechanism could work only in the primary component for both two cyclic changes of period, and cannot in the secondary. This means that the primary may be the magnetically active star, which is consistent with the studies of Mullan (1975) and previous investigators (Djura？evi？ et al. 1998, Lee et al. 2008, Zhu & Qian 2009). However, we couldn't find any meaningful relation between the light variation and the period variability from the historical light curve data observed at four different epochs, although the light asymmetry at a given epoch is remarkable and variable with different epochs. We wait for future observations to resolve these unsolved and veiled matters. At the present, we somewhat prefer the interpretation of mechanical perturbation from the third- and fourth stars.