The Solar-Type Contact Binary BX Pegasi Revisited
aa r X i v : . [ a s t r o - ph . S R ] O c t The Solar-Type Contact Binary BX Pegasi Revisited
Jae Woo Lee, Seung-Lee Kim, Chung-Uk Lee, and Jae-Hyuck Youn
Korea Astronomy and Space Science Institute, Daejeon 305-348, Korea [email protected], [email protected], [email protected], jhyoon @ kasi.re.kr ABSTRACT
We present the results of new CCD photometry for the contact binary BX Peg,made during three successive months beginning on September 2008. As do historicallight curves, our observations display an O’Connell effect and the November data bythemselves indicate clear evidence for very short-time brightness disturbance. For thesevariations, model spots are applied separately to the two data set of Group I (Sep.–Oct.) and Group II (Nov.). The former is described by a single cool spot on thesecondary photosphere and the latter by a two-spot model with a cool spot on thecool star and a hot one on either star. These are generalized manifestations of themagnetic activity of the binary system. Twenty light-curve timings calculated fromWilson-Devinney code were used for a period study, together with all other minimumepochs. The complex period changes of BX Peg can be sorted into a secular perioddecrease caused dominantly by angular momentum loss due to magnetic stellar windbraking, a light-travel-time (LTT) effect due to the orbit of a low-mass third companion,and a previously unknown short-term oscillation. This last period modulation could beproduced either by a second LTT orbit with a period of about 16 yr due to the existenceof a fourth body or by the effect of magnetic activity with a cycle length of about 12yr.
Subject headings:
Stars
1. INTRODUCTION
For BX Peg, Lee at al. (2004a, hereafter Lee04a) remains the most recent comprehensivestudy. These authors showed that historical light curves of BX Peg, displaying year-to-year lightvariability, can all be explained by introducing a single dark spot on the more massive secondary starand found that the orbital period has varied due to a periodic oscillation overlaid on a continuousperiod decrease. They concluded that the periodic O – C residuals could not be produced by spotactivity (Maceroni & van’t Veer 1994) and were not locked into the light variation as required byApplegate (1992). Thus, the only phenomenon that could be responsible was a third body witha period of 52.4 yr and a limiting mass of M sin i =0.26 M ⊙ and the hypothetical companionbecame the default explanation. 2 –Lee04a also suggested that the timing residuals indicated an additional short-term oscillationwith a period of about 12 yr and a semi-amplitude of about 0.002 d. Since then, many newphotoelectric and CCD timings of minimum light have been reported and should now be sufficientlynumerous to test this possibility meaningfully. In this article, we present a detailed study of the O – C diagram of BX Peg together with a new light-curve synthesis.
2. NEW PHOTOMETIC OBSERVATIONS
New CCD photometry of BX Peg was performed on 9 nights from 26 September through15 November 2008. The observations were taken with a 2K CCD camera and a
BV R filter setattached to the 1.0-m reflector at Mt. Lemmon Optical Astronomy Observatory (LOAO) in Ari-zona, USA. The instrument and reduction method have been described by Lee et al. (2009). Thecomparison star (C) was chosen to be BD + 25 o B ,412 in V , and 412 in R ) and a sample of them is listed in Table 1. The light curves of BX Peg areplotted in the upper panel of Figure 1 and the (K–C) magnitude differences are shown in the lowerpanel. Measurements of the check star indicate that, on average, the comparison star remainedconstant within the 1 σ -value of about ±
3. LIGHT-CURVE SYNTHESIS
Our light curves of BX Peg are asymmetrical and continue to display season-to-season lightvariability as they have in previous years. From the analysis of historical data, Lee04a showedthat the asymmetries can be interpreted as spot activity on the secondary component presumablyproduced by a magnetic dynamo, and the variations of the asymmetries most likely arise fromthe variability of the spots with time. As shown in Figure 1, the new light curves still indicate anO’Connell effect (Max I fainter than Max II) and cycle-to-cycle intrinsic variability. Specifically, theNovember light curves are very different from the other data, especially at the first quadrature. Wesolved the new light curves in a manner almost identical to that used by Lee04a. For this synthesis,the contact mode 3 of the 2003 version of the Wilson-Devinney code (Wilson & Devinney 1971;Wilson 1979, 1990; hereafter WD) was applied separately to the two data sets of Group I (Sep.–Oct.) and Group II (Nov.) and a cool spot on the secondary photosphere was adopted for both ftp://ftp.astro.ufl.edu/pub/wilson/ ℓ ) but found that the code always returned negative values for this parameter.In Lee04a it was shown that, within errors, 10 independent light curves obtained from 1978through 2000 could all be represented by a unique geometry and by wavelength-consistent photo-metric parameters. That conclusion can now be extended to encompass the three 2008 light curvesreported here. One must understand that the formal errors returned by the WD code are lowerlimits to realistic uncertainties but, even so, the agreement over 31 years is strong enough thatthe component stars of BX Peg are well known even within the formal errors. Perhaps this is nottoo surprising since the eclipses are complete and thus light curve determinacy is high, but theimplication of the statement is that further photometric interest in this binary may be limited tomapping its intrinsic variability.With this advantage, it is possible to ask whether there has been stability to the cool spotsthat have been described over the same time interval? Within assigned errors, these spots havemigrated to larger longitudes over almost a hemisphere and (non-monotonically) have moved closerto the stellar equator. The spot sizes reached a maximum around 2000 and they have cooledprogressively over the interval of monitoring. Of course, these conclusions depend on light curvesthat under-sample the time interval over which they were accumulated. It is also clear that a hotspot can emerge in a very brief time as happened over only a month in 2008. Almost certainly,this means that the hot spot is not a signature of impact from streaming gas since the over-contactcondition has been documented since 1978. Rather, the hot disturbance and the variability of thecool spots are generalized manifestations of the magnetic activity of the BX Peg system. 4 –
4. FITTING THE O – C VARIATION
We calculated minimum epochs for each of our eclipses with the WD code by means of adjustingonly the ephemeris epoch ( T ). Ten such timings of minimum light are given in Table 4, togetherwith all photoelectric and CCD timings since the compilation of Lee04a. For further ephemerisimprovement, we used the light-curve timings given in Table 3 of Lee04a, rather than the originalminima of Samec (1990). The following standard deviations were assigned to timing residualsbased on observational technique and the method of measuring the epochs: ± ± ± plus LTT ephemeris: C = T + P E + AE + τ , (1)where τ symbolizes the LTT due to a third companion physically bound to the eclipsing pair(Irwin 1952, 1959) and includes five parameters ( a sin i , e , ω , n and T ). In order to improvethe coefficients of the former ephemeris, we fitted all times of minimum light to equation (1) usingthe Levenberg-Marquart (LM) algorithm (Press et al. 1992). The results are given in the secondcolumn of Table 5, together with related parameters. The absolute dimensions of Samec & Hube(1991) have been used for these and subsequent calculations. The value for the third-companionperiod is different from the evaluation in Lee04a because the intervening years have added muchweight to the period history. The other Irwin parameters have not changed significantly if onerecognizes that ω is poorly determined even now.The top panel of Figure 3 shows the O – C residuals constructed with the linear term ofthe ephemeris; the solid curve and the dashed parabola represent the full non-linear terms andthe quadratic term, respectively. The middle panel displays the residuals τ from the linear andquadratic terms of the equation and the bottom panel the residuals from the complete ephemeris.These appear as O – C in the third column of Table 4. In all panels, error bars are shown for onlythe timings with known errors. In the bottom panel of Figure 3, the timing residuals indicate, asbefore, a possible additional short-term oscillation. Accordingly, the period variability of the systemmust be more complicated than the form of equation (1). To get a more generalized description ofthe period variability, we introduced the times of minimum light into a different ephemeris form: C = T + P E + AE + τ + S , S = K ′ sin( ω ′ E + ω ′ ) . (2)The LM technique was again applied to solve for the eleven parameters of the ephemeris whichare listed in the third column of Table 5. Adding the generalized sine modulation decreased thethird-body period to a value close to that in Lee04a but nothing else has changed. The O – C residuals from the linear light elements are plotted in the top panel of Figure 4. The second andthird panels display the LTT orbit ( τ ) and a 12-yr period modulation ( S ), respectively, and thelowest panel the residuals from the full equation (2). These appear as O – C in the fourth columnof Table 4. It is clear that the residuals in the third panel of Figure 4 skew across the sine curve 5 –and the fit is not so good as could be wished. Consequently, the lowest panel of the figure does notshow the final residuals as randomized as they should be.The sine oscillation can be replaced in favor of a second LTT orbit ascribed to a fourthcomponent of the BX Peg system. For such a case, it is necessary to use a quadratic plus two-LTTephemeris instead of equation (2): C = T + P E + AE + τ + τ . (3)These calculations converged quickly to yield the entries listed in Table 6 where we see that notmuch has changed for the orbit of the supposed third object of the system. Figure 5, derived fromthe Table 6 parameters, is plotted in the same sense as Figure 4. The second and third panelsrefer to the τ and τ orbits, respectively, and are possibly marginally better fits to the data. Ifthe hypothetical objects are on the main sequence, the minimum masses for the putative thirdand fourth bodies correspond to spectral types of M6 and M8, respectively, and their bolometricluminosities would contribute only 0.6 % to the total luminosity of the quadruple system. Also,the semi-amplitude of the systemic radial velocity variation of the eclipsing pair due to the twosupposed companions would be only about 1 km s − . These two limits indicate that it will be noteasy to detect companions orbiting the eclipsing binary independently.As predicted by Maceroni & van’t Veer (1994) and confirmed by Lee et al. (2009), times ofminimum light may be systematically shifted by light asymmetries due to starspot activity. Thelight curve synthesis method gives more precise timings than do other techniques (e.g. Kwee &van Woerden 1956) based on the observations during a minimum alone. Clear evidence for thisassertion appears in Figures 3–5 which show almost no noise for the WD timings compared to thosefrom other methods.
5. DISCUSSION
The negative coefficients of the quadratic terms in equations (2) and (3) yield a continuousperiod decrease with a rate of − × − d yr − , which can be explained either by conservativemass transfer from the more massive cool star to its less massive hot component or by angularmomentum loss (AML) due to a magnetic stellar wind. Under the assumption of conservative masstransfer, the transfer rate is about 7.0 × − M ⊙ yr − . This value is 3.5 times greater than a rateof 2.0 × − M ⊙ yr − calculated by assuming that the cool secondary transfers mass to the hotprimary component on a thermal time scale. Therefore, the alternative mechanism, AML caused bymagnetic braking rooted in the convective zone, seems more likely. From an approximate formulagiven by Guinan & Bradstreet (1988), the period decrease rate is calculated to − − − − d yr − ) for k =0.07, 0.10, and 0.15, respectively. The last value might bea good approximation to the gyration constant k of BX Peg. It is also possible that the correctexplanation of the secular period change is some combination of non-conservative mass transferand AML but, at present, it seems that AML should be the dominant contributor. 6 –This last cautionary remark actually has some independent support. Consider the pair of W-type systems defined observationally by Binnendijk (1970), V829 Her (Erdem & ¨Ozkarde¸s 2006) andV781 Tau (Yakut et al. 2005). Their individual masses and radii and the photospheric temperaturedifference as well as the systemic mass ratio and period are the same values within 1 % yet thealgebraic signs of the secular period changes are different. This is not a unique case. The samesituation exists for V417 Aql (Qian 2003; Lee et al. 2004b) and BB Peg (Kalomeni et al. 2007).If there is little mechanical and radiometric difference between two such binaries which behavedifferently dynamically, there must be a particular evolutionary mechanism that distinguishes themor else the seeming secular period changes are not really unbounded and are just long-term cyclicaleffects. The rigorous association of a positive period change with a particular sense of mass transferand the mirror association of a negative period change with the other sense of mass transfer andwith AML appear to be too didactic.The 12-yr period modulation, shown in the third panel of Figure 4, could operate in at least onecomponent since each has a convective envelope and thus may be magnetically active (Applegate1992; Lanza et al. 1998). With the values of K ′ and P ′ , the parameters of an Applegate modelwere calculated for both components and appear in Table 7, where ∆ m rms denotes a bolometricmagnitude difference relative to the mean light level of BX Peg converted to magnitude scale withequation (4) in the paper of Kim et al. (1997). The variations of the gravitational quadrupolemoment ∆ Q correspond to typical values for contact binaries and the required light variationassociated with each component is within the value (∆ L/L p , s ∼ .
1) proposed by Applegate.This consistency indicates that Applegate’s mechanism could possibly function in both componentstars. There is no a priori reason to require a sine-type behavior for a magntic cycle in a star. IfApplegate’s mechanism is the main cause of the cyclical variation, his model requires the brightnessvariation to vary in phase with the period modulation of Figure 4.In order to check this possibility and to study the long-term light variations of BX Peg, wemeasured the light levels at four different phases (Max I, Min I, Max II and Min II) for our newlight curves and for the archival measurements of Zhai & Zhang (1979), Samec (1990) and Lee04a.The latter are taken from Table 6 of Lee04a and the comparisons are given in Table 8. All datasets are referred to the same comparison star (BD + 25 o B and ∆ V are plotted in Figure 6, where the seasonal means were subtracted from the grand meanfor all seasons giving the mean seasonal differences in the natural systems. The year and epoch foreach data set were calculated by averaging the starting and ending HJDs of the observations. InFigure 6, the third and fourth panels represent the sine term of the C ephemeris and the τ orbitof the C ephemeris, respectively, averaged over each observing season and the arrows indicate themean epoch of each seasonal light curve. There are only 5 epochs for light curves but, even as fewas they are, they do not conform to this prediction of the Applegate mechanism.This binary presents a conflict for observers. Without the minimum monitoring which hasbeen so fruitful, we would be ignorant of the time-scales and semi-amplitudes of the 50-year cycleand of the shorter one as well. Perusal of the present Figures 3 through 5 and of Figures 4 and 5 7 –in Lee04a shows that the 50-year cycle has become much better delineated over only 5 years. Thesame conclusion cannot be drawn for the shorter cycle. We have already remarked that neither thesinusoid nor a fourth-companion waveform track the residuals well. Many of the minimum timingsare much less precise than those from LOAO and their noisy appearance is readily evident in allthe figures. Residuals as large as − REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
10 – (cid:39) B Sep. 26-28Oct. 19-22Nov. 14-15-0.20.00.20.7 (cid:39) V (cid:39) R Phase -0.2 0.0 0.2 0.4 0.6 0.8 1.0 ( K - C ) BV-0.4R-0.5
Fig. 1.— The upper panel displays
BV R light curves of BX Peg defined by individual observations.The solid curves represent the photometric solutions obtained from the measurements betweenSeptember and October 2008. The magnitude differences (K–C) between the check and comparisonstars are plotted in the lower panel. 11 –
Phase
Phase ( O - C ) B -0.050.000.05 ( O - C ) V ( O - C ) R Phase
Sep.-Oct. (cool spot) Nov. (cool spot) Nov. (cool & hot spots)
Fig. 2.— Light residuals from the models for two data sets (Group I and Group II). See the textfor details. 12 – O - C ( d ) -0.3-0.2-0.10.0 CCPEPGVI (cid:87) (cid:22) ( d ) -0.020.000.02 Cycle -80000 -60000 -40000 -20000 0 20000-0.0050.0000.005 O - C ,f u ll ( d ) Fig. 3.— The O – C diagram of BX Peg constructed with the linear terms of the quadratic plus LTT ephemeris. In the top panel, the continuous curve and the dashed, parabolic one representthe full contribution and the quadratic term of the equation, respectively. Diamond symbols referto the minimum times obtained with the WD code. The middle panel displays the residuals fromthe linear and quadratic terms and the bottom panel the residuals from the full ephemeris. Anadditional short-term oscillation seems to exist in the final residuals in the bottom panel. 13 – O - C ( d ) -0.3-0.2-0.10.0 CCPEPGVI (cid:87) (cid:22) ( d ) -0.020.000.02 Cycle -80000 -60000 -40000 -20000 0 20000 O - C ,f u ll ( d ) -0.0050.0000.005 ( d ) -0.0050.0000.005 Fig. 4.— The O – C residuals of BX Peg from the linear ephemeris of equation (2). These are drawnin the top panel with the continuous curves due to the full non-linear terms and the dashed paraboladue to the quadratic term of the equation. The second and third panels display a LTT orbit and a12-yr period oscillation, respectively, and the lowest panel the residuals from the complete equation. 14 – O - C ( d ) -0.3-0.2-0.10.0 CCPEPGVI (cid:87) (cid:22) ( d ) -0.020.000.02 Cycle -80000 -60000 -40000 -20000 0 20000 O - C ,f u ll ( d ) -0.0050.0000.005 (cid:87) (cid:23) ( d ) -0.0050.0000.005 Fig. 5.— The uper panel is constructed in the same sense as Figure 4 with the linear terms ofTable 6. The second and third panels refer to the τ and τ orbits, respectively. 15 – (cid:39) B -0.20.00.2 Max IMin IMax IIMin II-0.20.00.2 (cid:39) V Max IMin IMax IIMin II ( d ) -0.0050.0000.005 Cycle -20000 -10000 0 10000 20000 (cid:87) (cid:23) ( d ) -0.0050.0000.005 Fig. 6.— The mean seasonal variations of ∆ B and ∆ V for BX Peg at four characteristic phases(Max I, Min I, Max II and Min II). The third and fourth panels represent the sine term and the τ orbit of the C and C ephemeris forms, respectively, averaged over each observing season andthe plotted points were determined by combining all eclipse timings for a given year. Open circlesrefer to seasons with only one minimum timing. An errors bar gives the standard deviation of eachdata set. The arrows indicate the mean epoch of each seasonal light curve. 16 –Table 1. CCD photometric observations of BX Peg.HJD ∆ B HJD ∆ V HJD ∆ R − − − − − − − − − T (HJD) 2,453,208.3118 ± P (d) 0.28041759 ± q ± i (deg) 87.693 ± T (K) 5532 ±
20 5300Ω 6.135 ± A g X x B ± ± x V ± ± x R ± ± L/ ( L + L ) B ± L/ ( L + L ) V ± L/ ( L + L ) R ± r (pole) 0.2820 ± ± r (side) 0.2946 ± ± r (back) 0.3310 ± ± r (volume) a a Mean volume radius.Table 3. Spot parameters for BX Peg. a Parameter Group I Group IICool 2 Cool 2 and Hot 2 Cool 2 and Hot 1Colatitude (deg) 66.6 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± T spot / T local ± ± ± ± ± a Cool 2: a cool spot on the secondary; Hot 2: a hot spot on the secondary; Hot 1: a hot spoton the primary. 18 –Table 4. Observed photoelectric and CCD times of minimum light for BX Peg since thecompilation of Lee04a.
HJD Epoch O – C O – C O – C Min References(2,450,000+)1,899.3225 13283.0 0.00186 0.00140 0.00164 I Br´at et al. (2007)2,145.5289 14161.0 0.00129 0.00056 0.00057 I Zejda (2004)2,521.4288 15501.5 0.00095 0.00006 -0.00056 II Zejda (2004)2,521.5697 15502.0 0.00164 0.00075 0.00013 I Zejda (2004)2,878.4018 16774.5 0.00197 0.00122 0.00012 II H¨ubscher et al. (2005)2,878.5425 16775.0 0.00246 0.00171 0.00061 I H¨ubscher et al. (2005)2,886.3935 16803.0 0.00176 0.00102 -0.00008 I Krajci (2005)2,887.3756 16806.5 0.00240 0.00165 0.00056 II H¨ubscher (2005)2,887.5149 16807.0 0.00149 0.00074 -0.00035 I H¨ubscher (2005)2,887.515 16807.0 0.00159 0.00084 -0.00025 I Diethelm (2004)2,902.3758 16860.0 0.00024 -0.00049 -0.00157 I H¨ubscher (2005)2,929.4367 16956.5 0.00082 0.00011 -0.00092 II H¨ubscher (2005)2,929.5793 16957.0 0.00321 0.00250 0.00147 I H¨ubscher (2005)2,956.3592 17052.5 0.00320 0.00251 0.00156 II Br´at et al. (2007)3,208.4530 17951.5 0.00137 0.00092 0.00086 II Pribulla et al. (2005)3,209.4332 17955.0 0.00010 -0.00034 -0.00040 I H¨ubscher et al. (2005)3,209.4335 17955.0 0.00040 -0.00004 -0.00010 I Pribulla et al. (2005)3,209.5743 17955.5 0.00099 0.00055 0.00049 II Pribulla et al. (2005)3,212.5180 17966.0 0.00031 -0.00013 -0.00018 I Pribulla et al. (2005)3,217.4262 17983.5 0.00120 0.00076 0.00072 II H¨ubscher et al. (2005)3,220.3701 17994.0 0.00071 0.00028 0.00025 I Pribulla et al. (2005)3,220.5099 17994.5 0.00030 -0.00013 -0.00016 II Pribulla et al. (2005)3,220.5112 17994.5 0.00160 0.00117 0.00114 II H¨ubscher et al. (2005)3,221.4928 17998.0 0.00174 0.00131 0.00128 I H¨ubscher et al. (2005)3,224.4358 18008.5 0.00035 -0.00008 -0.00010 II Pribulla et al. (2005)3,226.5392 18016.0 0.00062 0.00019 0.00018 I Pribulla et al. (2005)3,226.5392 18016.0 0.00062 0.00019 0.00018 I H¨ubscher et al. (2005)3,228.3612 18022.5 -0.00010 -0.00052 -0.00053 II Br´at et al. (2007)3,233.4095 18040.5 0.00068 0.00026 0.00026 II H¨ubscher et al. (2005)3,233.5488 18041.0 -0.00023 -0.00065 -0.00065 I H¨ubscher et al. (2005)3,236.3538 18051.0 0.00059 0.00018 0.00019 I Pribulla et al. (2005)3,236.4931 18051.5 -0.00032 -0.00073 -0.00072 II Pribulla et al. (2005)3,240.4180 18065.5 -0.00127 -0.00168 -0.00166 II Pribulla et al. (2005)3,240.5597 18066.0 0.00023 -0.00019 -0.00017 I Pribulla et al. (2005)3,250.3747 18101.0 0.00060 0.00020 0.00024 I H¨ubscher et al. (2005)3,250.5140 18101.5 -0.00031 -0.00071 -0.00067 II H¨ubscher et al. (2005)3,255.4219 18119.0 0.00028 -0.00011 -0.00006 I H¨ubscher et al. (2005)3,255.5628 18119.5 0.00097 0.00058 0.00063 II H¨ubscher et al. (2005)3,257.3845 18126.0 -0.00004 -0.00044 -0.00038 I H¨ubscher et al. (2005)3,257.5226 18126.5 -0.00215 -0.00254 -0.00249 II H¨ubscher et al. (2005)3,282.3417 18215.0 -0.00003 -0.00039 -0.00029 I H¨ubscher et al. (2005)3,282.4834 18215.5 0.00146 0.00110 0.00120 II H¨ubscher et al. (2005)3,341.2285 18425.0 -0.00097 -0.00126 -0.00106 I Diethelm (2005)3,360.2974 18493.0 -0.00048 -0.00075 -0.00053 I Zejda et al. (2006)3,601.4569 19353.0 -0.00027 -0.00027 0.00010 I H¨ubscher et al. (2006)
19 –Table 4—Continued
HJD Epoch O – C O – C O – C Min References(2,450,000+)3,613.3741 19395.5 -0.00083 -0.00081 -0.00044 II H¨ubscher et al. (2006)3,613.5150 19396.0 -0.00013 -0.00012 0.00025 I Zejda et al. (2006)3,613.5159 19396.0 0.00077 0.00078 0.00115 I H¨ubscher et al. (2006)3,614.3566 19399.0 0.00021 0.00023 0.00059 I Br´at et al. (2007)3,614.4970 19399.5 0.00040 0.00042 0.00079 II H¨ubscher et al. (2006)3,616.4573 19406.5 -0.00222 -0.00220 -0.00184 II Br´at et al. (2007)3,617.4407 19410.0 -0.00028 -0.00026 0.00010 I Br´at et al. (2007)3,632.0216 19462.0 -0.00111 -0.00107 -0.00071 I Nagai (2006)3,632.1623 19462.5 -0.00062 -0.00058 -0.00021 II Nagai (2006)3,648.4272 19520.5 0.00005 0.00011 0.00047 II H¨ubscher et al. (2006)3,651.3711 19531.0 -0.00043 -0.00038 -0.00001 I Br´at et al. (2007)3,651.3714 19531.0 -0.00013 -0.00008 0.00029 I H¨ubscher et al. (2006)3,651.5111 19531.5 -0.00064 -0.00059 -0.00022 II H¨ubscher et al. (2006)3,659.3629 19559.5 -0.00054 -0.00047 -0.00011 II H¨ubscher et al. (2006)3,659.5033 19560.0 -0.00035 -0.00028 0.00008 I H¨ubscher et al. (2006)3,663.9891 19576.0 -0.00123 -0.00116 -0.00080 I Nagai (2006)3,951.1360 20600.0 -0.00209 -0.00177 -0.00152 I Nagai (2007)3,951.2770 20600.5 -0.00129 -0.00098 -0.00073 II Nagai (2007)3,966.4203 20654.5 -0.00055 -0.00022 0.00002 II H¨ubscher & Walter (2007)3,966.5597 20655.0 -0.00136 -0.00103 -0.00079 I H¨ubscher & Walter (2007)3,985.4890 20722.5 -0.00025 0.00009 0.00031 II Do˘gru et al. (2007)3,989.6960 20737.5 0.00048 0.00082 0.00105 II Ogloza et al. (2008)3,992.3574 20747.0 -0.00209 -0.00174 -0.00152 I H¨ubscher & Walter (2007)4,000.3487 20775.5 -0.00269 -0.00234 -0.00213 II Br´at et al. (2007)4,002.4524 20783.0 -0.00212 -0.00178 -0.00156 I H¨ubscher & Walter (2007)4,279.647 21771.5 -0.00040 0.00003 0.00008 II Paschke (2007)4,327.4581 21942.0 -0.00050 -0.00008 -0.00006 I Br´at et al. (2007)4,328.4386 21945.5 -0.00147 -0.00105 -0.00102 II Br´at et al. (2007)4,328.4386 21945.5 -0.00147 -0.00105 -0.00102 II Br´at et al. (2007)4,328.4393 21945.5 -0.00077 -0.00035 -0.00032 II Br´at et al. (2007)4,330.5428 21953.0 -0.00040 0.00002 0.00004 I Br´at et al. (2007)4,330.5431 21953.0 -0.00010 0.00032 0.00034 I Br´at et al. (2007)4,359.9874 22058.0 0.00035 0.00076 0.00077 I Nagai (2008)4,386.6281 22153.0 0.00138 0.00178 0.00178 I Samolyk (2008a)4,420.5567 22274.0 -0.00056 -0.00017 -0.00018 I Samolyk (2008a)4,650.7799 23095.0 -0.00021 0.00002 -0.00006 I Samolyk (2008b)4,702.6579 23280.0 0.00054 0.00071 0.00062 I Samolyk (2008b)4,736.72819 23401.5 0.00009 0.00022 0.00014 II this article4,736.86805 23402.0 -0.00026 -0.00012 -0.00021 I this article4,737.70921 23405.0 -0.00035 -0.00022 -0.00031 I this article4,737.84996 23405.5 0.00019 0.00032 0.00024 II this article4,738.69117 23408.5 0.00015 0.00028 0.00019 II this article4,759.72254 23483.5 0.00020 0.00030 0.00022 II this article4,760.70325 23487.0 -0.00055 -0.00045 -0.00053 I this article4,761.68556 23490.5 0.00030 0.00040 0.00031 II this article
20 –Table 4—Continued
HJD Epoch O – C O – C O – C Min References(2,450,000+)4,785.66083 23576.0 -0.00013 -0.00006 -0.00015 I this article4,786.64256 23579.5 0.00014 0.00020 0.00012 II this article