V700 Cygni: A Dynamically Active W UMatype Binary Star II
 Author: Kim ChunHwey, Jeong Jang Hae
 Organization: Kim ChunHwey; Jeong Jang Hae
 Publish: Journal of Astronomy and Space Sciences Volume 29, Issue2, p151~161, 15 June 2012

ABSTRACT
An intensive analysis of 148 timings of V700 Cyg was performed, including our new timings and 59 timings calculated from the super wide angle search for planets (SWASP) observations, and the dynamical evidence of the W UMa W subtype binary was examined. It was found that the orbital period of the system has varied over approximately 66^{y} in two complicated cyclical components superposed on a weak upward parabolic path. The orbital period secularly increased at a rate of +8.7 (±3.4) × 10^{9} day/year, which is one order of magnitude lower than those obtained by previous investigators. The small secular period increase is interpreted as a combination of both angular momentum loss (due to magnetic braking) and masstransfer from the less massive component to the more massive component. One cyclical component had a 20.^{y}3 period with an amplitude of 0.^{d}0037, and the other had a 62.^{y}8 period with an amplitude of 0.^{d}0258. The components had an approximate 1:3 relation between their periods and a 1:7 ratio between their amplitudes. Two plausible mechanisms (i.e., the lighttime effects [LTEs] caused by the presence of additional bodies and the Applegate model) were considered as possible explanations for the cyclical components. Based on the LTE interpretation, the minimum masses of 0.29
M_{⊙} for the shorter period and 0.50M_{⊙} for the longer one were calculated. The total light contributions were within 5%, which was in agreement with the 3% thirdlight obtained from the light curve synthesis performed by Yang & Dai (2009). The Applegate model parameters show that the root mean square luminosity variations (relative to the luminosities of the eclipsing components) are 3 times smaller than the nominal value (ΔL/L_{p,s} ？ 0.1), indicating that the variations are hardly detectable from the light curves. Presently, the LTE interpretation (due to the third and fourth stars) is preferred as the possible cause of the two cycling period changes. A possible evolutionary implication for the V700 Cyg system is discussed.

KEYWORD
W UMatype star , V700 Cyg , period change , lighttime effects , magnetic activities

1. INTRODUCTION
The eclipsing nature of V700 Cyg (OV 26, 2MASS J20310525 3847006, CCDM J20311 3847A, WDS J20311 3847A) was discovered by Whitney (1952) who classified it as a W UMatype binary star with a period of 0.^{d}340048. The first photoelectric BV light curves with an O’Connell effect (Max II fainter than Max I) were secured by Niarchos et al. (1997) who studied global photometric properties of the system through light curve synthesis as well as period study. According to those authors, V700 Cyg is a W UMa Wsubtype binary star with the spectral type of G5V and an orbital inclination of 79.9° . The more massive 0.92
M _{⊙} star is 374K cooler than the less massive 0.60 M_{⊙} star with a temperature of 5770 K. The fillout factor is approximately 27% in moderate contact with one another. The OC diagram with the light elements (C = HJD 2445163.4896 0.^{d}340045602 E) by Niarchos et al. (1997) showed a shortterm period instability. A close physical companion that was approximately 1.4^{m} fainter in the V bandpass than V700 Cyg was noted by Eggen (1967). V700 Cyg was later catalogued as CCDM J20311 3847A (Dommanget & Nys 2002) and WDS J20311 3847A (Mason et al. 2001). The earlier history was well summarized by Niarchos et al. (1997).In recent years, V700 Cyg was revisited by Qian (2003), Molik & Wolf (2004), Xiang et al. (2009) and Yang & Dai (2009). Qian (2003) obtained new light elements (
C = HJD 2449999.3003 0.^{d}29063148E ) and noted that the shortterm period instability detected by Niarchos et al. (1997) is not real but is caused by the use of an incorrect longer period. They also suggested that the orbital period varied in a cyclical manner with a period of 40^{y} and an amplitude of 0.^{d}0095, which was superposed on a secular period increasing at a rate of 8.41 × 10^{8} d/yr. Molik & Wolf (2004) independently analyzed the times of minima that were available to them and found only a cyclic period change of 46^{y} without any secular change. Yang & Dai (2009) confirmed the result of Qian’s (2003) period study and obtained a cyclic period of 54^{y}, with an amplitude of 0.^{d}0212 longer than those identified by Qian (2003) and Molik & Wolf (2004). Yang & Dai (2009) derived the rate of the secular period increase as 2.89 × 10^{8} d/yr, 3 times smaller than that of Qian (2003). In addition, new symmetric VR light curves showing no O’Connell effect were secured by Yang & Dai (2009). Their analyses of the light curves with the WilsonDevinney binary model (Wilson & Devinney 1971) showed a temperature difference of Δ T = 503K between the components, a mass ratio of 0.544, an inclination of 84.o1 and a fillout factor of 15.1%. These values are larger than Δ T = 374K, smaller than 0.652, 79.^{o}9 and 27%, respectively, according to Niarchos et al. (1997). Although the reasons for the differences between the two investigators are currently unknown, it is true that the differences cannot be negligible and must be solved with new photometric and spectroscopic observations. Xiang et al. (2009) also conducted a period study by using the times of minima available to them and found a period of 39.y1 for the cyclic term with an amplitude of 0.^{d}0197; a secular period increase of 2.8 × 10^{8} d/yr was derived. Xiang et al. (2009) suggested an additional cyclic oscillation with a period of approximately 25.^{y}0 and a small amplitude of 0.^{d}0031 in the residuals from the first cyclic term.Surveying the historical period studies performed by the investigators above, we note the following: 1) even concurrent studies (Xiang et al. 2009, Yang & Dai 2009) made with nearly the same times of minima do not show consistency for the period of the seemingly cyclic change; and 2) numerous timings before and after the recent period studies of Xiang et al. (2009) and Yang & Dai (2009) have been published (e.g., see Table 1 and the references therein). In this paper, we perform an intensive reanalysis of all the published timings available to us, including our new two timings and 59 others calculated from the super wide angle search for planets (SWASP) observations. Two cyclic changes superposed on a weakly constrained upward parabolic variation are suggested. The following is a discussion of the implications of existing working hypotheses.
2. OBSERVATIONS AND TIMES OF MINIMA
The CCD photometric observations of V700 Cyg for the purpose of timing determination were made on the nights of November 4 and 12 in 2008 with the 35 cm campus station reflector at the Chungbuk National University Observatory in Korea. The telescope was equipped with an SBIG ST8 CCD imaging system and electrically cooled with a 19' × 12'field of view. No filters were used in our observations. GSC 3152527 and GSC 3153193 were chosen as the comparison and check stars, respectively. The instrumentation used and reduction method for the raw CCD frames have been described in detail by Kim et al. (2006). The resultant standard error of our observations (check minus comparison) was approximately ± 0.m009. The light curves for the observations are shown in Fig. 1.
From our observations using the conventional Kwee & van Woerden (1956, KW) method, two times of minimum light were determined. In addition, we researched whether
V700 Cyg has been observed in survey works such as the all sky automated survey (ASAS) and the SWASP. Fortunately, V700 Cyg is identified as 1SWASP J203105.27 384701.8 in the SWASP public archive that is available at the Web^{1} and managed by the SWASP community. Using the SWASP data observed from June 14 to October 25 in 2007, we calculated a total of 59 timings (27 for the primary timing and 32 for the secondary timing) by the KW method.
1http://www.wasp.le.ac.uk/public/lc/index.php
3. PERIOD STUDY
To investigate the period variation of V700 Cyg, a total of 148 (9 visual, 14 plate, 7 photographic, and 118 photoelectric and CCD) times of minimum light (including ours and those from the SWASP) were collected from a modern database (Kreiner et al. 2001) and from the recent literature. Table 1 lists only the photoelectric and CCD minima, which were not compiled by Yang & Dai (2009) or published after their study.
The (
OC ) residuals of all the timings were calculated with the linear ephemeris of Kreiner et al. (2001) as follows:The (
OC _{1}) diagram is shown in Fig. 2 where different symbols in size and shape are used according to the observational method and the type of eclipse for each of the timings. As shown in Fig. 2, it is clear that the variation pattern of the observed (OC _{1}) residuals from 1946^{y} to 2012^{y} is complex. The orbital period before about 2008y seemed to be globally sinusoidal; however, there is a large gap of about 12 years between 1982^{y} and 1994^{y}. This fact was noticed by Qian (2003), Xiang et al. (2009) and Yang & Dai (2009). These authors made intensive period studies of V700 Cyg and adopted a quadratic plus sine ephemeris to explain the (OC ) residuals constructed with their available timings. Note that the initial epoch (HJD 2449999.3003) used by Qian (2003) and Yang & Dai (2009) is not a primary epoch representing a deeper eclipse, rather it is a secondary epoch and the eclipse types of the timings in Table 3 from the Yang & Dai (2009) study were listed reversely. The primary timings in the Yang & Dai (2009) Table 3 should be changedto secondary timings and viceversa. Xiang et al. (2009) used the ephemeris (Eq. 1) from Kreiner et al. (2001) and improved it as follows:
The dashed and solid lines in Fig. 2 denote the linear (plus quadratic) and full terms, respectively, from Eq. (2). The residuals from Eq. (2) were plotted at the bottom of Fig. 2 where the recent timings (since 2007^{y}) have been clearly deviating from the ephemeris, which implies that Eq. (2) should be revised and/or the period change of V700 Cyg may be more complicated.
To understand the general period change behavior, we tried to fit all of the times of minima to a quadratic plus LTE ephemeris described as:
where τ _{3} is the lighttime due to the assumed third body and includes the five orbital parameters (
a _{12} sini _{12},e _{12}, ω _{12},P _{12},T _{12}) for the mass center of the eclipsing pair around the mass center of the triple system. The parametric and differential forms of the orbital elements for the lighttime orbit were taken from Irwin (1952, 1959). The LevenbergMarquardt method (Press et al. 1992) was used to solve Eq. (3) simultaneously. As shown in the 2^{nd} column of Table 2, the final solution was quickly derived. In this and subsequent calculations, we assigned different weights to the data according to the inverse values of the scatters observed for the observational methods: 30 to the photoelectric or CCD observations and 1 to the photographic, plate and visual data. The resultant (OC ) diagram with the linear term in Table 2 was drawn in Fig. 3, which shows dashed and solidlines representing the quadratic and LTE terms, respectively. The residuals from the full terms were plotted in the bottom of Fig. 3.
As shown in Fig. 3, Eq. (3) provides a more satisfactory fit to the observed timings than Eq. (2). The cyclic period is calculated as 73.^{y}9 (± 3.^{y}7), which is 27.^{y}9, 34.^{y}8 and 19.^{y}2 longer than those provided by Molik & Wolf (2004), Xiang et al. (2009) and Yang & Dai (2009), respectively. Note that the photoelectric and CCD residuals at the bottom of Fig. 3 are distributed in an oscillatory pattern with a small amplitude of about 0.d003 and with a short period of about 20^{y}. The standard deviation for the photoelectric and CCD residuals is σ = ± 0.^{d}0019, which is 4 times larger than the nominal error (± 0.^{d}0005) of the modern photoelectric and CCD timing observations. The coefficient of the quadratic term has a value smaller than its standard error (with a negative sign), which is contrary to the findings of Xiang et al. (2009) and Yang & Dai (2009). This result implies that the secular
variation is not well constrained by the present timing data because the quasisinusoidal variation is dominant and also because the secondary oscillatory pattern is not considered in Eq. (3).
To find the variation characteristics of the small oscillation in Fig. 3, we used two LTEs ephemeris, which are expressed by simply adding τ _{4} to Eq. (3) as follows:
where τ _{4} is the lighttime due to the assumed fourth body and also includes the five orbital parameters (a_{123} sin
i _{123},e _{123},ω _{123},P _{123},T _{123}) from the masscenter of the eclipsing pair as well as the thirdbody revolving around the mass center of the quadruple system. The LevenbergMarquardt method was used again to solve Eq. (4) simultaneously. Thirteen parameters were adjusted in Eq. (4). The results are listed in the 3^{rd} and 4^{th} columns of Table 2. The quality of the final solution represented by Eq. (4) can be judged by Fig. 4, which plots the (OC ) residuals with the linear term of Eq. (4); the dashed and solid lines represent the quadratic and full terms of Eq. (4), respectively. As shown at the bottom of Fig. 4, the standard deviation for the photoelectric and CCD residuals from the full terms of Eq. (4) is σ = ± 0.^{d}0006, which is compatible with the nominal, modern photoelectric and CCD timing accuracy. Figs. 5 and 6 show two cyclic (OC ) diagrams phased with each of their periods; the continuous lines were drawn with the solution parameters in Table 2. The residuals from the linear and full terms of Eq. (4) are listed in the fifth and sixth columns of Table 2, respectively.From the coefficient (
A ) of the quadratic term in Table 2, a secularly increasing rate of the period is calculated as 8.7 (± 4.0) × 10^{9} day/year, which is approximately one order of magnitude lower than the results obtained by Xiang et al. (2009) and Yang & Dai (2009). The secular period increase is not convincingly revealed with the present timing data.4. TWO CYCLIC PERIOD CHANGES
In the above section, we have decoupled two cyclic terms via Eq. (4) from the observed (
OC ) residuals shown in Figs. 46. One LTE orbit with the longer period of 62.^{y}8 (P _{out}) has a semiamplitude of 0.^{d}0258 (K _{out}) and a relatively small eccentricity of 0.067 (e _{out}) while the LTE orbit with the shorter period of 20.^{y}_{3} (P _{in}) has a semiamplitude of 0.^{d}0037 (K _{in}) and a remarkably large eccentricity of 0.934 (e _{in}). There is a commensurable relationship of 1:3 betweenP _{in} andP _{out}, 1:7 betweenK _{in} andK _{out} and 14:1 betweene _{in} ande _{out}.Two rival theories may explain these cyclic period
[Table 3.] The basic parameters for the hypothetical third and fourth bodies in the V700 Cyg system.
changes: 1) The LTEs are caused by additional third and fourth bodies in the V700 Cyg system; or 2) the Applegate (1992) mechanism, which was improved by Lanza et al. (1998). If the LTEs are assumed to be the mechanism to produce two cyclic period changes, then mass functions, masses and bolometric luminosities for different inclinations of two tertiary bodies were calculated by using the absolute dimension (
M _{1} = 0.92M _{⊙}, _{2} = 0.60M M _{⊙},R _{1} = 1.04R _{⊙},R _{2} = 0.86R _{⊙},L _{1} = 0.83L _{⊙} ,L _{2} = 0.74L _{⊙}) of the eclipsing pair, which was provided by Niarchos et al. (1997). The results are listed in Table 3. In Table 3, the minimum masses (i ’ = 90˚ ) of 0.29M _{⊙} for the thirdbody and 0.50M _{⊙} for the fourth body would contribute small fractions of approximately 0.7% and 5% lights, respectively, to the total luminosity (l t =l 1 +l _{2} +l _{3} +l _{4}). From their VR light curves synthesis, Yang & Dai (2009) derived a third light of about 3% that shows a good agreement with our value. If the third and fourth bodies (at least one additional body) in the V700 Cyg system are assumed to be real, the small contribution of their lights to the total light of the quadruple system wouldmean that the additional bodies revolve around the mass center of the system in planes nearly coplanar with the orbital plane of the eclipsing pair.
The Applegate (1992) model can alternatively explain the cyclical components of the period variability. According to this theory, the cyclic changes in the magnetic dynamo activity of tidally locked cool stars in binary systems can produce small cyclic orbital period modulations. Using the modulation periods and amplitudes listed in Table 2 and adopting the absolute dimensions explained above, the model parameter variations needed to change the orbital period for V700 Cyg can be obtained from the formula provided by Applegate (1992). The calculations were made with the assumption that two cyclical period changes are produced by both components. Table 4 lists the resultant Applegate parameters where the root mean square (RMS) luminosities converted to magnitude scale were calculated with the formula provided by Kim et al. (1997). The parameters in Table 4 show that the Applegate mechanism could operate well in both the primary and secondary components for both cyclic period changes, although the RMS luminosity variations relative to the luminosities of the primary and secondary stars are 3 times smaller than the nominal value (Δ
L/Lp,s ？ 0.1) that the Applegate model requires. These small RMS luminosity variations correspond to ± 0.m01, and they may be hardly detectable from any of the light curves measured during the suggested modulation periods (because of the dominant light variations caused by the spot activities involved). We made a morphology investigation of historical light curves secured at two different epochs by Niarchos et al. (1997) and Yang & Dai (2009). The BV photoelectric light curves of Niarchos et al. (1997) show an O’Connell effect for both light curves. However, the VR CCD light curves of Yang & Dai (2009) are perfectly symmetrical within their observational errors.The observation epochs of light curves by Niarchos et al. (1997) and Yang & Dai (2009) are marked as ‘a’ and ‘b’ with an arrow in Figs. 5 and 6, respectively. As shown in Fig. 5, the light curves from Niarchos et al. (1997) were obtained from a position somewhat distant from the ascending nodal point in the 63.^{y}4 sinusoidal curve, but Yang & Dai’s (2009) obtained their light curves just near the descending nodal point. Because the two epochs are near nodal points and the Applegate RMS brightness variation calculation is small, the expectation is that the light curves at the two epochs are symmetrical, but the reality described above is different. In Fig. 6, we see that the light curves of Niarchos et al. (1997) and Yang & Dai (2009) were observed near the bottom and top, respectively, of the highly quasisinusoidal curve with the 20.^{y}4 shorter period. If the Applegate mechanism is assumed to be a cause of the shorter period modulation, then any light curves at the positions should show as asymmetrical because of the highly enhanced magnetic activities, but the perfect symmetric light curves from Yang & Dai (2009) defy that explanation. Based on these discussions, the preferred explanation is that the geometrical LTEs (due to the third and fourth stars) cause the two cyclic period variations, although the Applegate mechanism cannot be ruled out based on the presently available insufficient data (light curves and timings).
5. DISCUSSION AND CONCLUSION
To understand the dynamical picture of this active W UMa binary system in detail, we analyzed a total of 148 timings of V700 Cyg, including our two timings and 59 others that were calculated from the SWASP observations. It was found that the orbital period of the system has varied over about 66^{y} in two cyclical components superposed on a weakly constrained upward parabola. Three decoupled components of the period variability were investigated.
The secular period increase of + 8.7 (± 4.0) × 10^{9} day/year is one order of magnitude lower than those obtained previously (Xiang et al. 2009, Yang & Dai 2009), indicating that the secular period increase over 66y is not convincingly revealed with the present timing data. The weak secular period increase of V700 Cyg may be a special feature in that the secular period increases (or decreases) of most of W UMa contact binaries are approximately one or two orders of magnitude larger than that of V700 Cyg (e.g., Wolf et al. 2000, Qian 2003, Kim et al. 2003, Lee et al. 2008). In general, W UMa binaries are known to suffer from a secular period decrease via angular momentum loss (AML) due to magnetic braking (van’t Veer 1979, Rucinski 1982, Bradstreet & Guinan 1994, Maceroni & van’t Veer 1991, Stepien 1995, 2006, Demircan 1999). Mass and energy exchanges through the connecting neck between the two components are also possible, resulting in a thermal relaxation oscillation (TRO; Lucy 1976, Flannery 1976, Robertson & Eggleton 1977). If we assume that the two mechanisms occur concurrently in any contact binary system and that the size of the period decreasing rate predicted by the AML is slightly smaller than that of the increase rate caused by a mass transfer from the less massive component to the more massive component, then a weak secular period increase could possibly be observed (as in the V700 system). Under this assumption, an observed secular period rate can be formulated as the sum of the period change rates of the two mechanisms as follows:
where the first and second terms in the right side represent the period variation rates due to magnetic braking (mb) and mass transfer (mt), respectively. In the case of V700 Cyg, the decreasing period rate caused by the magnetic braking was determined to be 5.8 × 10^{8} day/year with the formula provided by Bradstreet & Guinan (1994). Therefore, the secular period increase by a mass transfer from the less massive star to the more massive one is calculated with Eq. (5) as 6.7× 108 day/year, which is comparable to those of other W UMa contact binaries (e.g., Wolf et al. 2000, Qian 2003, Kim et al. 2003, Lee et al. 2008). If this picture is true of V700 Cyg, then the system is on an evolutionary phase in that the increasing period rate by a secular mass transfer from the less massive hot star to the more massive cool one is comparable to or slightly larger than the decreasing rate by a secular AML (due to magnetic braking). Future timings are important in resolving whether this conjecture is realistic.
In addition to the secular period increase, the complex period variations of V700 Cyg were decoupled into two cyclical components via two LTE ephemeris. The inner orbit has a period of 20._{y}3 (
P _{in}), a small semiamplitude of 0.^{d}0037 (K _{in}) and a remarkably large eccentricity of 0.934 (e _{in}), while the outer orbit has a longer period of 62.^{y}8 (P _{out}), a semiamplitude of 0.^{d}0258 (K _{out}) and a small eccentricity of 0.067 (e _{out}). Some commensurable relations between these elements seem to exist such asP _{in}:P _{out} ？ 1:3,K _{in}:K _{out} ？ 1:7, ande _{in}:e _{out} ？ 14:1. Our investigation shows that these types of commensurabilities, particularly between periods, exist possibly in some eclipsing binary stars with orbital periods that have been reported to suffer from two sinusoidal period changes. For example, there is a 40:11 relationship between the shorter and longer periods in the OO Aql system (Albayrak et al. 2005), 13:5 in the VW Cep system (Zasche & Wolf 2007), 7:2 in the WZ Cep system (Jeong & Kim 2011), 7:1 in the AH Cep (Kim et al. 2005), 2:1 in the SZ Her system (Lee et al. 2012), 11:3 in the T LMi system (Zasche et al. 2006), 41:7 in the SW Lyn system (Kim et al. 2010), and 5:2 in the V508 Oph system (Albayrak et al. 2005). Similar relationships are also found in other eclipsing binary systems, which are known to have substellar companions (see Lee et al. 2012). If these commensurable relationships are truly produced by additional bodies in the systems, they would be interpreted as a stable orbital resonances produced by a longterm gravitational interaction between two tertiary bodies in the systems (Peale 1976, Kley et al. 2004). If so, it is important to question their origin and evolution. Are the tertiary bodies captured dynamically during the long evolutionary path of a close binary or are they formed primordially as a multiple system? At the moment, these questions are difficult to answer. However, observational evidence supports the latter question (Tokovinin et al. 2006, Rucinski et al. 2007, Eggleton & Tokovinin 2008). Eggleton (2012) suggested a series of evolutionary sequences in a global manner, which may provide a good answer to the latter question. According to Eggleton (2012), the primordial triple systems in which the inner and outer orbits have a high mutual inclination secularly evolve through Kozai (1962) cycles and induced tidal frictions into close binaries within 10day periods. Due to magnetic braking, the late type close binaries can further evolve into contact binaries with common envelopes via AML. After suffering from many TRO processes, the contact binaries emerge into single stars because of violent Darwin instability. Finally, wide binaries consisting of emerged single stars and relatively unchanged components remain. If this scenario is correct, then the additional bodies we have suggested in the V700 Cyg system may be evidence that strongly supports the scenario. The minimum masses for the inner and outer bodies of V700 Cyg were derived as 0.29M _{⊙} and 0.50M _{⊙}, respectively, under the assumption that the additional bodies are mainsequence stars. The sum of the lights emitting from the two extrabodies corresponds to approximately 5% light relative to the total luminosity of the quadruple system, which is somewhat compatible to the 3% light derived from the light curve synthesis by Yang & Dai (2009). Dai (2009).The low luminosities of the suggested bodies could imply that they are revolving around the mass center of the system in planes nearly coplanar with the orbital plane of the eclipsing pair.
The Applegate model parameters in Table 4 indicate that the RMS luminosity variations relative to the luminosities of the primary and secondary stars are 3 times smaller than the nominal value (Δ
L/L_{p,s} ？ 0.1), which corresponds to ± 0.^{m}01 and may be hardly detectable from the light curves because of the spot activities involved. The historical light curves at different epochs do not seem to follow the mechanism, although the Applegate mechanism cannot be ruled out based on the evidence currently available.

[Fig. 1.] The observed eclipse light curves of V700 Cyg and the check star (GSC 3153193) relative to the comparison star (GSC 3152527).

[Table 1.] The CCD times of minima of V700 Cyg not listed in Yang & Dai (2009) or published after their study.

[Table 11.] (Continued)

[Fig. 2.] The (OC) diagram of V700 Cyg drawn together with the nonlinear terms (solid and dashed lines) of Eq. (2). The recent residuals from Eq. (2) at the bottom deviate remarkably from the nonlinear terms of Eq. (2).

[Table 2.] The Final solutions of Eqs. (3) and (4).

[Fig. 3.] The (OC) diagram of V700 Cyg drawn together with the nonlinear terms (solid and dashed lines) of Eq. (3) and Table 2. The recent residuals from Eq. (3) at the bottom show a small oscillatory pattern.

[Fig. 4.] The (OC) diagram of V700 Cyg drawn together with the nonlinear terms (solid and dashed lines) of Eq. (4) and Table 2.

[Fig. 5.] The (OC) diagram of V700 Cyg phased with the longer period of 63.y4 in Table 2. The arrows marked as ‘a’ and ‘b’ represent the phases that correspond to the light curves secured by Niarchos et al. (1997) and Yang & Dai (2009), respectively.

[Fig. 6.] The (OC) diagram of V700 Cyg phased with the shorter period of 20.y4 in Table 2. The arrows marked as ‘a’ and ‘b’ represent the phases corresponding to the light curves secured by Niarchos et al. (1997) and Yang & Dai (2009), respectively.

[Table 3.] The basic parameters for the hypothetical third and fourth bodies in the V700 Cyg system.

[Table 4.] The Applegate model parameters for the shorter and longer periods.