62 Odessa Astronomical Publications, vol. 23 (2010) FOUR NEW VARIABLE STARS NEAR CL AURIGAE. II Chun-Hwey Kim1 , Jae Woo Lee2 , Duck Hyun Kim1 , Ivan L. Andronov3 1 2 3 Department of Astronomy and Space Science, and CBNU Observatory, Chungbuk National University, Cheongju 361-763, Korea, kimch@chungbuk.ac.kr Korea Astronomy and Space Science Institute, Daejeon 305-348, Korea, jwlee@kasi.re.kr Department ”High and Applied Mathematics”, Odessa National Maritime University, Odessa, Ukraine, il-a@mail.ru ABSTRACT. We report on a discovery of four new variable stars (USNO-B1.0 1234-0103195, 12350097170, 1236-0100293 and 1236-0100092) in the field of CL Aur. The stars are classified as eclipsing binary stars with orbital periods of 0.5137413(23) (EW type), 0.8698365(26) (EA) and 4.0055842(40) (EA with a significant orbital eccentricity), respectively. The fourth star (USNO-B1.0 1236-0100092) showed only one partial ascending branch of the light curves, although 22 nights were covered at the 61-cm telescope at the Sobaeksan Optical Astronomy Observatory (SOAO) in Korea. Fourteen minima timings for these stars are published separately. In an addition to the original discovery paper (Kim et al. 2010), we discuss methodological problems and present results of mathematical modeling of the light curves using other methods, i.e. trigonometric polynomial fits and the newly developed fit “NAV” (”New Algol Variable”). with T0 = 2453412.0000(5) and Porb = 4.d 0055842(40). Here in brackets are the error estimates in units of last decimal digit. The secondary minimum is not covered completely by observations, but significantly shifted from the phase 0.5, indicating an elliptic orbit. USNO-B1.0 1234-0103195 shows only one ascending branch typical for EA-type stars, but, for determination of elements, new observatins are needed. USNO B1.0 1236-0100293: EW -type The object is a EW-type binary system. For analysis, we used an trigonometric polynomial (TP) fit ms (t) of order s: ms (t) = Cs,1 + s  (Cs,2j · cos(jωt) + Cs,2j+1 · sin(jωt)), j=1 where ω = 2πf, f = 1/P is trial frequency. The test Key words: Variable stars: eclipsing: EA, EW function is defined as -type; Data analysis: “NAV” algorithm. N 2  σO−C,s 2 , σO−C,s = (mk − ms (tk ))2 , S(f ) = 2 σO−C,0 k=1 During a study of the eclipsing binary CL Aur, 2 we have discovered four new variable stars. The where σO−C,s is variance of deviation of observational observations were made on 22 nights from 2003 points from the s − th order fit. The value S(f ) is November to 2005 February. The CCD camera has a square of the correlation coefficient between the ob2048×2048 pixels and an FOV of about 20.′ 5 × 20.′ 5, served and calculated (for a given f ) values. Detailed The filter set is attached to the 61 cm reflector at discussion of statistical properties of this test funcSobaeksan Optical Astronomy Observatory (SOAO) tion, coefficients was presented by Andronov (1994, in Korea. The exposure times were 75∼140 s for B, 2003). For the periodogram analysis, we have used 45∼85 s for V , 33∼65 s for R, and 30∼60 s for I, the computer program ”Multi-Column View” (MCV) respectively. A 2×2 binning mode was used. The described by Andronov and Baklanov (2004). nearby stars GSC 2393-1424 and GSC 2393-1418, The periodogram (dependence of the test function imaged on the chip at the same time as the variable, S(f ) on trial frequency is shown in Fig. 1. Taking into were chosen as comparison and check stars, respec- account the EW type of variability, the first approach tively. The co-ordinates (2000.0) of the comparison is TP fit with s = 2, which corresponds to a double are 05h 13m 27.s 48, +33◦ 26’46.3”. Unfortunately, there wave and different depth of minima and (in a case of is no multicolor calibration for the comparison star, O’Connel effect). Thus the most prominent peak at so the photometry is in differences “var-comp”. The the periodogram occurs at the orbital period Porb , and discovery paper was published by Kim et al. (2010). the second one (in height) at a double frequency (halfThe star USNO-B1.0 1236-0100092 is an EA-type star period). Odessa Astronomical Publications, vol. 23 (2010) 63 Figure 4: Trigonometric polynomial fits of 7-th orFigure 1: Periodogram of VRI observations (from top der for BVRI observations of USNO-B1.0 1235-0097170 to bottom) of USNO B1.0 1236-0100293 computed us- with “1σ” corridor. ing the 2-nd order trigonometric polynomial fit Figure 2: Dependence of the r.m.s. error estimate of the smoothing curve on the degree s of trigonometric polynomial for filters VRI. Figure 5: Dependence of the ”unit weight error” on the eclipse half-width ∆φ for filters BVRI. Figure 3: 4-nd order trigonometric polynomial fit to the VRI observations of USNO B1.0 1236-0100293. The thickness of the line corresponds to the “1σ” corridor. The ephemeris for the primary minimum is Min.HJD= 2453215.5773 + 0.5137405 · E. Figure 6: ”NAV” fits to the BVRI observations of USNO-B1.0 1235-0097170 with “1σ” corridor. The ephemeris for the primary minimum is Min.HJD= 2453208.8503 + 0.8698365 · E. 64 Odessa Astronomical Publications, vol. 23 (2010) Due to statistical errors and non-orthogonality of the basic functions, the best fit values of the period and coefficients (Cs,i are generally dependent on s even for the same i. Kim et al. (2010) used the method of Scargle (1982) with a least squares approximation mc (t) = m̄ + Cs,2 · cos(ωt) + Cs,3 · sin(ωt), with Cs,1 = m̄ (sample mean of the observations) instead of a least square solution in TP-1. Moreover, the estimates of period and other coefficients are dependent on the photometric system. For the TP-2 fit, we obtained for VRI the period estimates of 0.d 5137397(68), 0.d 5137347(36), 0.d 5137469(32). The weighted mean value is Porb =0.d 5137413(23), is slightly different from that of 0.d 5137580(5) published by Kim et al. (2010). The initial epoch for the primary minimum is T0 = 2453215.57299(83). In Fig. 2, the dependence of the r.m.s. error estimate of the smoothing curve σ[xC ] (= σobs in Eq. (18) of Andronov (1994))is shown as a function of the degree s of trigonometric polynomial. For all three filters, the minimum of this function is seen at s = 2. Additional analysis using the program FDCN (FOUR-N) by Andronov (1994) had shown, that the coefficients Cs,2j are statistically significant up to j = 4. Thus we have chosen s = 4, which corresponds to physically better phase curve (i.e. more sharp minima than maxima). For this s = 4, the period estimates are 0.d 5137419(70), 0.d 5137332(37), 0.d 5137454(32). The mean weighted value is Porb = 0.5137405(23) and T0 = 2453215.5773(13). We also computed a “mean weighted” periodogram G(f ) = 3  wi · (1 − Si (f )) i=1 with weights proportional to wi = (n − 1 − 2s)/(1 − Smax,i ), where Smax,i− is maximal value of the periodogram for a given filter i. The minimum of this function occurs at P = 0.d 5137404(85). Although the period estimate is fairly close to a mean weighted value, the error estimate is significantly larger. The phase light curves and the corresponding 4-th order trigonometric polynomial fits are shown in Fig. 3. The depth of the primary minimum in different filters is ∆1 V = 0.m 382(13), ∆1 R = 0.m 415(7), and ∆1 I = 0.m 363(6), i.e. very similar. The amplitude is at small maximum in the filter R. The depth of the secondary minimum is ∆2 V = 0.m 240(13), ∆2 R = 0.m 219(8), ∆2 I = 0.m 229(6) is the same within error estimates. The phase-averaged mean brightness is C1 = 4.m 035(6), 3.m 365(3) and 2.m 836(3) for V,R,I, respectively. Although there is no calibration of the comparison star, so the color indices of the object are available only in respect to this comparison star. The differences ((V − R)var − (V − R)comp ) = 0.m 570(7) and ((R − I)var − (R − I)comp ) = 0.m 629(4) are rather large, indicating that the comparison star is a blue one, and the variable is yellow or red. This is in an agreement with expectations for W UMa - type stars (e.g. Tsessevich 1971). USNO-B1.0 1235-0097170: EA -type This star was classified as an Algol - type variable. The coefficients of the trigonometric polynomial fit aree statistically significant while s ≤ 7. The TP-7 fits are shown in Fig. 4. The mean weighted values are Porb = 0.d 8698365(26) and T0 = 2453208.8503(9). For each of these fits, 16 parameters are determined using least squares, with estimates for the r.m.s. accuracy of the fit of σ[xC ] =0.m 0214, 0.m 0154, 0.m 0105, 0.m 0085. However, one may see apparent waves at the light curve, especially when at phases badly covered by observations. Such phenomenon is a common problem for signals with very asinusoidal shape. To improve accuracy of fits for EA variables, Andronov (2010) proposed a “New Algol Variable” (NAV) fit, which was also tested by Virnina (2010). The free parameter is the eclipse half-width ∆φ. We adopted value β = 2. To determine its statistically optimal value, we computed dependence of the ”unit weight error” σ0 on ∆φ for filters BVRI, which is shown in Fig. 5. The minima of this test function appeared at ∆φ from 0.089 to 0.100 with a weighted mean of 0.094. The corresponding values σ[xC ] of 0.m 0128, 0.m 0098, 0.m 0066, 0.m 0053 are by a factor of ∼ 1.6 better than for the TP fit. The fits are shown in Fig. 6. One may see a significant ellipticity effect aruguing that the red star is tidally distorted, and (because of small depth of the secondary minimum) much larger than another component. Acknowledgements. This work was supported by the Korea Research Foundation (KRF) grant funded by the Korea government (MEST)(No. 2010-0016968) and has been done as part of a cooperative project between Chungbuk National University and the Korea Astronomy and Space Science Institute. References Andronov I.L.: 1994, OAP, 7, 49 Andronov I.L.: 2003, ASPC, 292, 391 Andronov I.L.: 2010, http://www.astrokarpaty.net/ kolos2010abstractbook.pdf Andronov I.L., Baklanov A.V.: 2004, Astronomy School Reports, 5, 264, http://uavso.pochta.ru/mcv Kim Chun-Hwey, Lee Jae Woo, Kim Duck Hyun, Andronov I.L.: 2010, OEJV, 126, 1 Scargle J.D.: 1982, ApJ, 263, 835 Tsessevich V.P. (ed.): 1971, Instationary stars and methods of their investigation. 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