Research Paper J. Astron. Space Sci. 32(2), 127-136 (2015) http://dx.doi.org/10.5140/JASS.2015.32.2.127

Phenomenological Modeling of Newly Discovered Eclipsing Binary 2MASS J18024395 + 4003309 = VSX J180243.9+400331

Ivan L. Andronov1, Yonggi Kim2,3†, Young-Hee Kim2,3, Joh-Na Yoon2,3, Lidia L. Chinarova4, Mariia G. Tkachenko1 1Department of High and Applied Mathematics, Odessa National Maritime University, Odessa 65029, Ukraine 2Chungbuk National University Observatory, Cheongju 361-763, Korea 3Dept. of Astronomy and Space Science, Chungbuk National University, Cheongju 361-763, Korea 4Astronomical Observatory, Odessa National University, Odessa 65014, Ukraine

We present a by-product of our long term photometric monitoring of cataclysmic variables. 2MASS J18024395 +4003309 = VSX J180243.9 +400331 was discovered in the field of the intermediate polar V1323 Her observed using the Korean 1-m telescope located at Mt. Lemmon, USA. An analysis of the two-color VR CCD observations of this variable covers all the phase intervals for the first time. The light curves show this object can be classified as an Algol-type variable with tidally distorted components, and an asymmetry of the maxima (the O’Connell effect). The periodogram analysis confirms the cycle numbering of Andronov et al. (2012) and for the initial approximation, the ephemeris is used as follows: Min I. BJD = 2456074.4904+0.3348837E . For phenomenological modeling, we used the trigonometric polynomial approximation of statistically optimal degree, and a recent method “NAV” (“New Algol Variable”) using local specific shapes for the eclipse. Methodological aspects and estimates of the physical parameters based on analysis of phenomenological parameters are presented. As results of our phenomenological model, we obtained for the inclination i=90°, M1=0.745M◉, M2=0.854M◉, 9 M=M1+M2=1.599M◉, the orbital separation a=1.65·10 m=2.37R◉ and relative radii r1=R1/a=0.314 and r2=R2/a=0.360. These estimates may be used as preliminary starting values for further modeling using extended physical models based on the Wilson & Devinney (1971) code and it's extensions

Keywords: 2MASS J18024395+4003309 = VSX J180243.9+400331, eclipsing binary, CCD photometry

1. INTRODUCTION et al. 2011; Kim et al. 2004, 2005). 1RXS J180340.0+401214 is an intermediate polar, subclass of magnetic cataclysmic Chungbuk National University Observatory (CBNUO) variables, which has a magnetic white dwarf accreting is monitoring many cataclysmic variables as a part of an from its secondary. This object recently got an official “Inter-Longitude Astronomy (ILA)” campaign (Andronov name V1323 Her in the “General Catalogue of Variable et al. 2010) in order to study how the physical properties Stars” (GCVS) (Samus et al. 2014). One eclipsing variable, of cataclysmic variables depend on time and luminosity 2MASS J18024395+4003309, was discovered in the vicinity state. CBNUO is also involved in developing an automatic of this intermediate polar, 1RXS J1803, by Breus (2012). observation system and an analysis program for monitoring This object was registered in the “Variable Stars index” the cataclysmic variables (Yoon et al. 2012, 2013). We have (VSX, http://aavso.org/vsx) and received the name VSX published some results from our monitoring data (Andronov J180243.9+400331 (hereafter called VSX1802 for brevity)

This is an Open Access article distributed under the terms of the Received Mar 19, 2015 Revised May 28, 2015 Accepted May 28, 2015 Creative Commons Attribution Non-Commercial License (http:// †Corresponding Author creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, E-mail: : [email protected], ORCID: 0000-0002-9532-1653 provided the original work is properly cited. Tel: +82-43-261-3202 , Fax: +82-43-274-2312

Copyright © The Korean Space Science Society 127 http://janss.kr plSSN: 2093-5587 elSSN: 2093-1409 J. Astron. Space Sci. 32(2), 127-136 (2015)

and an object identifier 282837. Unfortunately, no GCVS have used the star C1 of Andronov et al. (2011), for which name has yet been given to this object. The study of the Henden (2005) published the magnitudes of V and R filters: main target V1323 Her was presented by Andronov et al. V=14.807m, Rc=14.436m. The original observations (HJD, (2011). magnitude) are available upon request. Fig. 1 is the finding Due to a relatively large angular distance (14’) from chart of V1323 Her with VSX1802. V1323 Her, VSX1802 is seen in the field of V1323 Her only in CCD images with a focal reducer. This was the case for one night reported by Andronov et al. (2012), who noted 3. PERIODOGRAMFig. 1. Finding ANALYSIS chart of V1323 Her (marked with v) and reference stars (marked by a sharp profile of the minimum. Additional observations numbers). Reference star 22 is VSX 1802. The size of the field is 22’x22’. from the Catalina survey (Drake et al. 2009) allowed them For the periodogram analysis, we have used the trigonometric to determine photometric elements T =2456074.4904, polynomial fit ofPe a degreeriodogram s: analysis 0 𝑡𝑡 − 𝑇𝑇0 𝑡𝑡 − 𝑇𝑇0 P=0.3348837±0.0000002d. This profile is typical characteristic 𝜙𝜙 =For the− periodogramint( ) analysis, we have used the trigonometric polynomial fit 𝑃𝑃 𝑃𝑃 of systems with nearly equal radii and an inclination of of a 𝑠𝑠degree s: (1) i≈90° (i.e., both eclipses are nearly total). Due to incomplete 𝑐𝑐(𝑡𝑡) 1 2𝑗𝑗 2𝑗𝑗+1 𝑥𝑥 = 𝐶𝐶 + ∑(𝐶𝐶 ∙ cos2𝜋𝜋𝑓𝑓𝑓𝑓 + 𝐶𝐶 ∙ sin2𝜋𝜋𝜋𝜋𝜋𝜋) (1) coverage of the phases in the discovery paper by Andronov where coefficients𝑗𝑗= C1 ,α=1..m,m=1+2s are computed using α 𝑠𝑠 . . et al. (2012), more detailed study was needed for VSX1802. the least squares𝑥𝑥where𝐶𝐶( 𝑡𝑡method) =coefficients𝐶𝐶1 + 𝑚𝑚(cf.∑𝑗𝑗 =Anderson1 (𝐶𝐶2𝑗𝑗 cos 2003;2𝜋𝜋𝑓𝑓𝑓𝑓 Andronov+ 𝐶𝐶2𝑗𝑗+1 sin are2𝜋𝜋 𝑓𝑓computed𝑓𝑓), using the least squares Such work can serve as one of the good by-products of the 1994, 2003), andmethod f is𝑥𝑥𝑐𝑐 frequency(𝑡𝑡 )(cf.= ∑ Anderson𝐶𝐶𝛼𝛼 (in∙ 𝑓𝑓𝛼𝛼 (cycles/day)𝑡𝑡) 2003; Andronov. For fitting, 1994, 2003), and f is frequency (in 𝛼𝛼=1 𝐶𝐶𝛼𝛼, 𝛼𝛼 = 1. . 𝑚𝑚, 𝑚𝑚 = 1 + 2𝑠𝑠 ILA campaign. we used the statisticalcycles/day) test function. For fitting, S( f )we as used the statistical test function as  𝑛𝑛 The purpose of this study is to analyse the photometry 2 𝑘𝑘 𝑘𝑘 𝑐𝑐 𝑘𝑘 𝑆𝑆(𝑓𝑓) data with a complete coverage of the phase as well 𝛷𝛷 = ∑ 𝑤𝑤2 ∙ (𝑥𝑥 − 𝑥𝑥 2(𝑡𝑡 ,) )  (2) (2) 𝑘𝑘=1𝜎𝜎𝐶𝐶 𝜎𝜎𝑂𝑂−𝐶𝐶 as to carry out phenomenological modeling. The 2 2  𝑂𝑂 𝑂𝑂 w𝑆𝑆(here𝑓𝑓) = 𝜎𝜎 is= the1 −r.m.s. 𝜎𝜎 deviation of the observations (O) from the sample mean, C phenomenological model is based on the theory of close where σO is the r.m.s. deviation of the observations (O) binary systems e.g. presented in the classical monographs from the samplecorresponds mean, C corresponds to calculated to values calculated and Ovalues-C to the deviation of the observed values 𝜎𝜎𝑂𝑂 by Kopal (1959) and Tsessevich (1971). Because no detailed and O-C to thefrom deviation the calculated of the observed ones. values(see Andronov from the 1994, 2003 for more details). The photometric or spectroscopic studies have been reported calculated onesperiodograms (see Andronov are 1994, shown 2003 in forFig. more 2 for details). the filter R and s=1 and s=2. One can see that yet, our rough phenomenological model will help to study The periodogramsthe numerous are shown peaks in Fig. for 2 sfor=1 theare muchfilter Rlower and than those for s=2, indicating that the this object with more detail in the future. s=1 and s=2. Onemain can signal see corresponds that the numerous not to the peaks true frequency,for but to its harmonic, which is s=1 are much lower than those for s=2, indicating that the

2. OBSERVATIONS

The Korean 1-m telescope at Mt. Lemmon in Arizona, USA (LOAO), is equipped with a focal reducer for 2×2 K CCD. The field-of-view is 22.2 square arcminutes, which is very effective for studies not only of the main targets (typically intermediate polars), but also of other variable stars in the field. It should also be noted that another eclipsing binary GSC 04370-00206 (now called V442 Cam, Samus et al. 2014) has been discovered in the field of MU Cam = 1RXS J062518.2+733433 with this telescope (Kim et al. 2005). In total, we obtained 196 observations in V (range 16.51m –17.51m) and 242 observations in R (range 15.88m–16.77m) between 2012 and 2014. The total duration of observations was 45.5 hours during 11 nights in R and 8 nights in the alternatively changing filters VR. The time interval of the observations was JD 2455998–2456722. To improve calibration accuracy, we have used the method of “artificial comparison star” (Andronov & Baklanov 2004; Kim et Fig. 1. Finding chart of V1323 Her (marked with v) and reference stars (marked by numbers). Reference star 22 is VSX 1802. The size of the field is al. 2004). As the object is in the field of V1323 Her, we 22’x22’. http://dx.doi.org/10.5140/JASS.2015.32.2.127 128 Ivan L. Andronov et al. Phenomenological Modeling of VSX J1802

main signal corresponds not to the true frequency, but to 70 its harmonic, which is characteristic for eclipsing binaries V 60 R with nearly equal minima (as was shown by Andronov et al. Limit 50

(2012) from the photometry from the Catalina survey). The )

P 40 A most prominent peak at the periodogram corresponds to F ( g

l 30 f=2.986107 cycles/day. A close value of the best frequency, - f =2.986114 cycles/day is also seen for the observations in 20 the filter V. The frequencies in the filter V and R are close 10 within error estimates to the value f =2.986111±0.000002 0 0 5 10 15 cycles/day (Andronov et al. 2012), but differ by a value of 0.5/ s year from the estimate based on 3 minima by Parimucha et al. (2012). Since our periodogram analysis confirms the Fig. 2. Dependence of the parameter Ls= – lg FAP on the degree of the trigonometric polynomial s. The horizontal dashed line corresponds to a cycle numbering of Andronov et al. (2012), for the initial critical value L=3. approximation, we use their ephemeris:

Min I. BJD = 2456074.4904 + 0.3348837.E (3) 0.04

V 0.03 R ) c _ x 4. PHENOMENOLOGICAL MODELING. MULTI – ( 0.02 g i HARMONIC APPROXIMATION s 0.01

The degree of the trigonometric polynomial s is often 0 determined by visual comparison of the light curve with its 0 5 10 15 s approximation (cf. Parenago & Kukarkin 1936). To determine the statistically optimal degree s of approximation (1), Fig. 3. Dependence of the r.m.s. value of the accuracy of the smoothing function σ[x ] on the degree of the trigonometric polynomial s. one may use different criteria (see Andronov 1994, 2003 c for detailed discussion). The classical approach is to use

Fischer’s criterion. In Fig. 2, the value Ls = –lg FAP is plotted (Fischer’s criterion) are shown in Fig. 4. Comparison of vs s. It is clearly seen that L is large for even values of s=2k, the trigonometric polynomial fits with different degrees and L≈0 for s=2k+1; k=1, 2,.. This can be explained by the s shows that, for smaller s, the depth of the eclipses is good symmetry of the light curve. Adopting the critical value underestimated and there are formal waves at the “out-of- -3 Lcrit=3 (i.e. the false alarm probability FAP=10 ), one may eclipse” part. These waves are apparently present for larger s, suggest s=14 for the filter V and s=8 for R. where the accuracy of the smoothing function is larger, but Another criterion proposed by Andronov (1994) is based the central part of the eclipses is approximated better. on the r.m.s. estimate of the accuracy of the smoothing curve

σ[xc] at the moments of the observations. The corresponding dependence is shown in Fig. 3. For both filters, the minimum of 5. PHENOMENOLOGICAL MODELING. THE NAV

σ[xc] corresponds to s=6. In this approximation, the frequency ALGORITHM f is a free parameter determined using differential corrections. The best fit estimate of periods is P=0.3348842d ±0.0000005d Phenomenological modeling of the light curves of (V) , 0.3348845d ± 0.0000004d (R), the difference from the eclipsing binary stars with relatively narrow eclipses was value in (1) is not statistically significant. The corresponding described in detail by Andronov (2012). The method was moments of the primary minima are 2456238.9186 ± 0.0005d called “NAV” (”New Algol Variable”) and applied to a few (V), 24 56320.9663±0.0007d (R). The difference between these Algol-type variables (e.g. Kim et al. 2010b). 𝑡𝑡 − 𝑇𝑇0 𝑡𝑡 − 𝑇𝑇0 values is due to different cycle numbers close to the mean In short, the method𝜙𝜙 = may− beint (described) as follows. The time of observations in V and R (as required for best accuracy smoothing function is defined𝑃𝑃 as usual𝑃𝑃 in the linear least  𝑠𝑠 estimates, see Andronov 1994 for details). squares method 𝑥𝑥𝑐𝑐(𝑡𝑡) = 𝐶𝐶1 + ∑(𝐶𝐶2𝑗𝑗 ∙ cos2𝜋𝜋𝑓𝑓𝑓𝑓 + 𝐶𝐶2𝑗𝑗+1 ∙ sin2𝜋𝜋𝜋𝜋𝜋𝜋) The phase curves in the V and R filters for statistically 𝑗𝑗=1 optimal degrees of trigonometric polynomial fit s =6 𝑚𝑚 (4) (best accuracy of the smoothing function) and s=14 𝑥𝑥𝑐𝑐(𝑡𝑡) = ∑ 𝐶𝐶𝛼𝛼 ∙ 𝑓𝑓𝛼𝛼(𝑡𝑡) 𝛼𝛼=1

 𝑛𝑛 129 2 http://janss.kr 𝛷𝛷 = ∑ 𝑤𝑤𝑘𝑘 ∙ (𝑥𝑥𝑘𝑘 − 𝑥𝑥𝑐𝑐(𝑡𝑡𝑘𝑘))  𝑘𝑘=1   (5) 𝛽𝛽 3/2 if |𝑧𝑧| < 1 𝐻𝐻(𝑧𝑧, 𝛽𝛽) = {(1 − |𝑧𝑧| ) , 0 else For =0, the shape is narrow and is physically unrealistic; for =1, the shape at the center of eclipse is triangular; for =2, it is parabolic and when approaches , the shape tends to be a rectangle. The dimensionless variable z is related to phase 𝛽𝛽 ∞  as

     (6) ( − 0)−int( − 0+0.5) 𝑧𝑧 = J. Astron. Space Sci. 32(2), 127-136 (2015)

where phase is defined typically for a given initial epoch T0 and period P:

 (7) (7) 0 0 𝑡𝑡(𝑡𝑡−−𝑇𝑇𝑇𝑇0)−int(𝑡𝑡−𝑡𝑡𝑇𝑇−) 𝑇𝑇0 𝜙𝜙 = −𝑃𝑃 int( ) A discussion of= the 𝑃𝑃computation𝑃𝑃 of phase for the case of variable period may be found in Andronov & Chinarova A discussion of the computation of phase for the case of variable period may (2013). be found in Andronov & Chinarova (2013). Another free parameter is the eclipse half-width Δϕ=D/2, where D – the fullAnother width of free the parameter eclipse, in is the the GCVS eclipse (Samus half-width =D/2, where D – the full et al. 2014)width is expressed of the ineclipse, percent in ofthe the GCVS period. (Samus et al. 2014) is expressed in percent of Other shapesthe period. used for determining of the parameters are based on a Gaussian function and its modifications Other shapes used for determining of the parameters are based on a Gaussian (Mikulasek et al. 2012), which are formally of infinite width. function and its modifications (Mikulasek et al. 2012), which are formally of An opposite approach is a splitting of the phase interval and infinite width. An opposite approach is a splitting of the phase interval and approximationapproximation of the “out-of-eclipse” of the “out-of parts-eclipse” by aparts constant by a constant and of the eclipses by a and of the parabolaeclipses by(Papageorgiou a parabola (Papageorgiou et al. 2014) .et The al. 2014).disadvantages of this model are: a) the The disadvantagesdiscontinuity of this of themodel smoothin are: a)g thefunction, discontinuity b) the underestimat ion of the depth of the of the smoothingminima function,because itsb) thewidt underestimationh is set to a large of the constant and c) bad fitting of the depth of theascending minima and because descending its width branches. is set However, to a large the number of parameters is only 5 constant and(as c)the bad width fitting of theof the eclipse ascending and the and phase descending shifts are set to constants). branches. However, the number of parameters is only 5 (as the width of Ourthe eclipseprevious and approach the phase was shifts to use are 5 setparameters to for the “constant+parabola” constants).fit (brightness out of eclipse, at the primary and secondary minimum; half-width Our previous and approachshift  ), in was order to useto avoid 5 parameters discontinuity for of the smoothing function. Other types of functions were discussed by Andronov (2005) and Andronov & the “constant+parabola” 0 fit (brightness out of eclipse, at the primaryMarsakova and secondary (2006). minimum; half-width Δϕ and

shift ϕ0), in order to avoid discontinuity of the smoothing Fig. 4. Phase curves in the V and R filters for statistically optimal degrees function. Other types of functions were discussed by of the trigonometric polynomial fit s=6 (best accuracy of the smoothing Andronov (2005) and Andronov & Marsakova (2006). function) and s=14 (Fischer’s criterion). Filled circles are original observations, the smoothing functions xc(φ) are shown with 1σ and 2σ error corridors. In the NAV algorithm, the basic functions are:

where Cα are called “coefficients” and fα(t) - “basic functions”. f1(1)=1 Contrary to trigonometric polynomials with sines and cosines only, the “New Algol Variable” (NAV) algorithm combines a f2(ϕ)=cos(φ), φ=2πϕ low – order trigonometric polynomial with a special shape for the eclipses. Andronov (2012) and Tkachenko & Andronov f3(ϕ)=cos(2φ) (2014) compared a few approaches and chose a local shape dependent on a single parameter called β: f4(ϕ)=sin(φ) (5) (5) 𝛽𝛽 3/2 f (ϕ)=sin(2 φ ) (5) (1 − |𝑧𝑧𝛽𝛽| )3/2 , if |𝑧𝑧| < 1 5 𝐻𝐻(𝑧𝑧, 𝛽𝛽) = { if |𝑧𝑧| < 1 𝐻𝐻 (𝑧𝑧, 𝛽𝛽) = {(1 − |𝑧𝑧| ) , For β=0, the shape is narrow0 and is physically else unrealistic; 0 else f6(ϕ)=H((ϕ-ϕ0)/C8, C9) for β=1, theFor shape =0, atthe the shape center is narrowof eclipse and is is triangular; physically for unrealistic ; for =1, the shape at the For =0, the shape is narrow and is physically unrealistic; for =1, the shape at the β=2, it is paraboliccenter of eclipseand when is triangular β approaches; for ∞=2, the, it shapeis parabolic andf ( when)=H(( - approaches-0.5 )/C , C ,) center of eclipse is triangular; for =2, it is parabolic and7 whenϕ ϕ approachesϕ0 8 ,10 tends to bethe a shape rectangle. tends toThe be dimensionless a rectangle. The variable dimensionless z is variable z is related to phase the shape tends to be a rectangle. The dimensionless variable z is 𝛽𝛽related to phase∞ related to phase as ϕ as 𝛽𝛽 ∞  as The coefficient C corresponds to mean stellar magnitude     1     (6) after reducing the observations for the eclipse, effects of  (6) ( − 0)−int( − 0+0.5 ) (6) ( − 0)−int( − 0+0.5) reflection, ellipticity and asymmetry. Although the effects where phase is defined𝑧𝑧 = typically for a given initial epoch T of reflection, ellipticity and asymmetry are not strictly 𝑧𝑧 = 0 and period P: sinusoidal, their amplitudes are typically much smaller than where phase is defined typically for a given initial epoch T0 and period P: where phase is defined typically for a given initial epoch T0 and period P:   (7) http://dx.doi.org/10.5140/JASS.2015.32.2.127 (𝑡𝑡−𝑇𝑇0)−int(𝑡𝑡−𝑇𝑇0 ) 130 (7) (𝑡𝑡−𝑇𝑇0)−int(𝑡𝑡−𝑇𝑇0) = 𝑃𝑃 = 𝑃𝑃 A discussion of the computation of phase for the case of variable period may A discussion of the computation of phase for the case of variable period may be found in Andronov & Chinarova (2013). be found in Andronov & Chinarova (2013). Another free parameter is the eclipse half-width =D/2, where D – the full Another free parameter is the eclipse half-width =D/2, where D – the full width of the eclipse, in the GCVS (Samus et al. 2014) is expressed in percent of width of the eclipse, in the GCVS (Samus et al. 2014) is expressed in percent of the period. the period. Other shapes used for determining of the parameters are based on a Gaussian Other shapes used for determining of the parameters are based on a Gaussian function and its modifications (Mikulasek et al. 2012), which are formally of function and its modifications (Mikulasek et al. 2012), which are formally of infinite width. An opposite approach is a splitting of the phase interval and infinite width. An opposite approach is a splitting of the phase interval and approximation of the “out-of-eclipse” parts by a constant and of the eclipses by a approximation of the “out-of-eclipse” parts by a constant and of the eclipses by a parabola (Papageorgiou et al. 2014) . The disadvantages of this model are: a) the parabola (Papageorgiou et al. 2014) . The disadvantages of this model are: a) the discontinuity of the smoothing function, b) the underestimation of the depth of the discontinuity of the smoothing function, b) the underestimation of the depth of the minima because its width is set to a large constant and c) bad fitting of the minima because its width is set to a large constant and c) bad fitting of the ascending and descending branches. However, the number of parameters is only 5 ascending and descending branches. However, the number of parameters is only 5 (as the width of the eclipse and the phase shifts are set to constants). (as the width of the eclipse and the phase shifts are set to constants). Our previous approach was to use 5 parameters for the “constant+parabola” Our previous approach was to use 5 parameters for the “constant+parabola” fit (brightness out of eclipse, at the primary and secondary minimum; half-width fit (brightness out of eclipse, at the primary and secondary minimum; half-width  and shift  ), in order to avoid discontinuity of the smoothing function. Other  and shift  ), in order to avoid discontinuity of the smoothing function. Other types of functions were discussed by Andronov (2005) and Andronov & types of functions0 were discussed by Andronov (2005) and Andronov & Marsakova (2006).0 Marsakova (2006). Ivan L. Andronov et al. Phenomenological Modeling of VSX J1802

those of the eclipses, thus, we use first-order approximation;

C2 corresponds to semi-amplitude of the reflection effect; C3 corresponds to the semi-amplitude of the ellipticity effect; the asymmetry (O'Connel effect, usually interpreted by

spots) is approximated by terms with C4 and C5. The depths of the primary and secondary minimum are described by

the coefficients C6 and C7, respectively. To continue numeration of the coefficients, we introduce

C8=Δϕ, C9 = β1 (describing the profile of the primary

minimum) and C10=β2 for the secondary minimum. The

phase shift C11=ϕ0 may also be added to the set of unknown

variables, but we have used a fixed value ϕ0=0 for the analysis of the present star. Typically the period and the initial epoch determined from long-term observations from different authors are more accurate than those from the smaller set of observations used for the analysis.

For each set of trial values of C8, C9, C10, the first 7 coefficients were computed using the method of least squares (Anderson 2003; Press et al 2007). For the rest of 𝑡𝑡 − 𝑇𝑇0 𝑡𝑡 − 𝑇𝑇0 the coefficients, one𝜙𝜙 = may choose− int( among) various methods (Cherepashchuk 1993;𝑃𝑃 Andronov𝑃𝑃 & Tkachenko 2013a;  𝑠𝑠 Marquardt 1963). For our analysis, we used the method of 𝑥𝑥𝑐𝑐(𝑡𝑡) = 𝐶𝐶1 + ∑(𝐶𝐶2𝑗𝑗 ∙ cos2𝜋𝜋𝑓𝑓𝑓𝑓 + 𝐶𝐶2𝑗𝑗+1 ∙ sin2𝜋𝜋𝜋𝜋𝜋𝜋) minimizing the test𝑗𝑗= function1 at a grid.

 Usually, the test function𝑚𝑚 for one filter may be written in this way: 𝑥𝑥𝑐𝑐(𝑡𝑡) = ∑ 𝐶𝐶𝛼𝛼 ∙ 𝑓𝑓𝛼𝛼(𝑡𝑡) 𝛼𝛼=1 The “NAV” fit for the phase curve and a corresponding “1σ” corridor,  𝑛𝑛 (8) Fig. 5. 2 for the filters V (up) and R (bottom). At the phases of eclipses, an additional 𝛷𝛷 = ∑ 𝑤𝑤𝑘𝑘 ∙ (𝑥𝑥𝑘𝑘 − 𝑥𝑥𝑐𝑐(𝑡𝑡𝑘𝑘))  curve corresponding to a continuation of the “out of eclipse” parts is also where x , t are signal𝑘𝑘= 1values and times, respectively, and shown. k k of are slightly different due to statistical errors. To get a single value, as  wk are possible weight coefficients. The complete theory  theoretically8 expected, we made a scaled sum of statistical properties of the smoothing function in a case 𝛷𝛷(𝐶𝐶 ) of arbitrary functions and (wavelet-like) time- and scale- (9) (9) dependent weight functions was presented by Andronov 𝛷𝛷𝑉𝑉(𝐶𝐶8) 𝛷𝛷𝑅𝑅(𝐶𝐶8) 𝛷𝛷(𝐶𝐶8) = 𝛷𝛷𝑉𝑉,𝑚𝑚𝑚𝑚𝑚𝑚 + 𝛷𝛷𝑅𝑅,𝑚𝑚𝑖𝑖𝑖𝑖 (1997). The adopted value of the filter half-width C8=0.1177,

We have computed a dependence Φ (C )=minC ,C Φ (C ,C ,C ) corresponds to the minimum of Φ(C ). Other parameters V 8 8 9 V 8 9 10 The adopted value of the8 filter half-width =0.1177, corresponds to the minimum for the filter V and, similarly, R. The minimum of Φ(C ) are determined by the phenomenological modeling are listed in 8 of . Other parameters determined by the phenomenological modeling are slightly different due to statistical errors. To get a single Table 1. 𝐶𝐶8 listed in Table 1. value, as theoretically expected, we made a scaled sum The corresponding𝛷𝛷(𝐶𝐶 light8) curves for the filters V and R are shown in Fig. 5.Table 1. Coefficients of the phenomenological model for our observations in the Using the smoothingfilters V, R functions and that forfor Catalina the two (filters,the latter the published by Andronov et al. 2012). smoothing function of the color index V-R was computed. It Table 1. Coefficients of the phenomenological model for our observations filter V filter R Catalina is shown in Fig. 6. in the filters V, R and that for Catalina (the latter published by Andronov et al. 2012).

α filter V filter R Catalina 𝜶𝜶1 16.6742𝜶𝜶 0.0036𝜶𝜶 4592.10𝜶𝜶 𝜶𝜶 16.1052𝜶𝜶 0.0047𝜶𝜶 3411.09𝜶𝜶 𝜶𝜶 16.236𝜶𝜶 Cα σα Cα /σα Cα σα Cα /σα Cα 𝑪𝑪 𝝈𝝈 𝑪𝑪 /𝝈𝝈 𝑪𝑪 𝝈𝝈 𝑪𝑪 /𝝈𝝈 𝑪𝑪 1 16.6742 0.0036 4592.10 16.1052 0.0047 3411.09 16.236 6. DISCUSSION2 0.0309 0.0052 5.92 0.0090 0.0068 1.33 - 2 0.0309 0.0052 5.92 0.0090 0.0068 1.33 - 3 0.0921 0.0060 15.41 0.0815 0.0077 10.61 0.078 3 0.0921 0.0060 15.41 0.0815 0.0077 10.61 0.078 From these coefficients4 -0.0100 of the0.0037 “NAV” approximation,-2.73 0.0078 the 0.0047 1.68 - 4 -0.0100 0.0037 -2.73 0.0078 0.0047 1.68 - classical phenomenological parameters listed in the GCVS 5 0.0158 0.0034 4.60 0.0204 0.0043 4.72 - 5 0.0158 0.0034 4.60 0.0204 0.0043 4.72 - m m 6 0.6954 0.0224 31.04 0.6633 0.0327 20.29 0.568 (Samus et al. 2014)6 were0.6954 determined: 0.0224 Max I=16.56731.04 ±0.0060.6633, 0.0327 20.29 0.568 7 0.5457 0.0143 38.22 0.4947 0.0170 29.11 0.486 Max II=16.5927 m ±0.0060.5457 m, Min0.0143 I=17.493 38.22m ±0.0140.4947m, 0.0170 29.11 0.486

131 The corresponding light curves forhttp://janss.kr the filters V and R are shown in Fig. 5. Fig. 4. If one uses them as “depths” of the minima, the corresponding point at the Fig. 4. If one uses them as “depths” of the minima, the corresponding point at the diagram by Malkov et al. (2007) is located in the physically reliable region, diagram by Malkov et al. (2007) is located in the physically reliable region, relatively close to the line of the equal radii. relatively close to the line of the equal radii. m m m m The mean color index, (V-R)=16.674m-16.105m=0.569m 0.069 m was The mean color index, (V-R)=16.674 -16.105 =0.569 0.069 was computed from the values of C1 for two filters. In the phase curve of the color computed from the values of C1 for twom filters.m In the phase curve± of the color index (Fig. 5), variations from 0.54m 0.01m (at Max I at ±phase 0.25) to indexm (Fig. m 5), variations from 0.54 0.01 (at Max I at phase 0.25) to 0.63m 0.03m (at Min I at phase 0.00) are present. At Min II at phase 0.50, V- 0.63 m0.03 (atm Min I at phase 0.00) are± present. At Min II at phase 0.50, V- R=0.61m 0.01m. The asymmetry of the± maxima indicates the presence of the R=0.61± 0.01 . The asymmetry of the maxima indicates the presence of the O’Connell± effect. O’Connell± effect. ± These values of the color index are in the instrumental system, so, before These values of the color index are in the instrumental system, so, before taking into account the color transformation coefficients and determining (V-R) in taking into account the color transformation coefficients and determining (V-R) in the standard system, we can't estimate more precisely the "color temperature" of the standard system, we can't estimate more precisely the "color temperature" of the "star1+star2 system" (i.e. weighted mean temperature of two components) and the "star1+star2 system" (i.e. weighted mean temperature of two components) and corresponding "spectral class" (also intermediate between that of the components). J. Astron. Space Sci. 32(2), 127-136 (2015) corresponding "spectral class" (also intermediate between that of the components). Following Andronov (2012), we introduce residual relative intensities Following Andronov (2012), we introduce residual relative intensities (primary minimum) (10) (primary (primary minimum) minimum) (10) (10) −0.4𝐶𝐶6 𝐼𝐼1 = 10−0.4𝐶𝐶6 𝐼𝐼1 = 10 (secondary (secondary minimum)minimum) (11) (11) (secondary minimum) (11) −0.4𝐶𝐶7 𝐼𝐼 2 = 10−0.4𝐶𝐶7 The intensities𝐼𝐼 2 = 10 are in units of sum of intensities of two stars (out of eclipse,The intensitiesbut "reduced" are in for units the of three sum effectsof intensities of two stars (out of eclipse, The intensities are in units of sum of intensities of two stars (out of eclipse, mentionedbut above) "reduced" and the for “intensity the three depth”effects ofmentioned minimum above) and the “intensity depth” of but "reduced" for the three effects mentioned above) and the “intensity depth” of (how muchminimum light is eclipsed): (how much d =1- lightI , d is=1- eclipsed)I . : , minimum (how much1 light1 is2 eclipsed)2 : , Andronov (2012) introduced a phenomenological Andronov (2012) introduced a 𝑑𝑑phenomenological1 = 1 − 𝐼𝐼1 𝑑𝑑2 = 1 para− 𝐼𝐼2meter. defined as parameter definedAndronov as Y = d + d(2012). The caseintroduced Y = 0 corresponds a 𝑑𝑑phenomenological1 = 1 − 𝐼𝐼1 𝑑𝑑2 = 1 para− 𝐼𝐼2meter. defined as The1 2case corresponds to "no eclipse", 1 to "both eclipses are The case corresponds to "no eclipse", 1 to "both eclipses are to "no eclipse",total" 1, and,to "both for the eclipses majority are of total", cases, and, for the. For this star, we get values of majority of 𝑌𝑌total"cases,= 𝑑𝑑, 10≤and,+Y𝑑𝑑 ≤1.for2. Forthe thismajority star,𝑌𝑌 = weof0 cases,get values of . For this star, we get values of 𝑌𝑌 = 𝑑𝑑1 + 𝑑𝑑2. 𝑌𝑌 = 0 Fig. 6. The smoothing curve for the color index using the “NAV” fits for the =0.868 0.014 (filter V), =0.823 00.019≤ 𝑌𝑌 ≤(R)1. (12) phase curve in the filters V and R and a corresponding “1σ” corridor. =0.868 0.014 (filter V), =0.823 00.019≤ 𝑌𝑌 ≤(R)1. (12) Y = 0.868 ± 0.014 (filter V), Y = 0.823 ± 0.019 (R) (12) 𝑌𝑌 ± 𝑌𝑌 ± 𝑌𝑌 ± 𝑌𝑌 ± In the model of "spherical stars with no limb darkening" (Shulberg 1971; In the model ofIn "spherical the model stars of with"spherical no limb stars darkening" with no limb darkening" (Shulberg 1971; m m m Malkov et al. 2005) recently discussed by Andronov & Tkachenko (2013b) as a MinII=17.281 ±0.008 , Min I-Max I=0.926 , MinII- (Shulberg Malkov1971; Malkov et al. 2005)et al. 2005)recently recently discussed discussed by Andronov & Tkachenko (2013b) as a m "zero-order" approximation for determination of physical parameters), the eclipsed MaxI=0.714 (filter V). The corresponding point at the “depth by Andronov"zero &- order"Tkachenko approximation (2013b) for as determinationa "zero-order" of physical parameters), the eclipsed part of the light is proportional to the surface of projection at the eclipse S; the – depth” diagram (Fig.1 in Malkov et al. 2007) lies close to approximationpart forof thedetermination light is proportional of physical to parameters), the surface of projection at the eclipse S; the the line “R” of equal radii, but slightly outside the allowed the eclipsed part of the light is proportional to the surface of region. This is caused by the “simplified” model of spherical projection at the eclipse S; the surface of projection of stars components without effects of ellipticity, reflection and limb S = πr 2, S = πr 2 Here, we use dimensionless parameters, 1 surface1 2 of projection2 of stars Here, we use dimensionless darkening, which Malkov et al. (2007) have used. r =surfaceR /a, r =ofR /projectiona, where a ofis thestars orbital separation. Here, we use dimensionless 1 1 2 2 2 2 parameters, r1=R1/a, r2=R2/a, where a is2 the orbital separation.2 parameters, r1=R1/a, r2=R2/a, where1 a is12 the orbital2 separation.22 The parameters C6, and C7 correspond to the "reduced Let’sparameters introduce, r1= RL11/Va=1-, r2=LR2V2 /asa, thewhere𝑆𝑆1 relative= a𝜋𝜋 is𝑟𝑟1 the, contribution 𝑆𝑆 orbital2 = 𝜋𝜋𝑟𝑟 separation.2 . of 𝑆𝑆1 = 𝜋𝜋𝑟𝑟1 , 𝑆𝑆2 = 𝜋𝜋𝑟𝑟2 . depth" of the minimum, i.e., not to the maximum (as in the the emission fromntroduce the first star in V, similaras theto other relative filters. contribution of the emission surface of projectionntroduce of stars as the relative Herecontribution, we use of dimensionless the emission "General Catalogue of the Variable Stars"), with respect to Let’sfromsurface define the of thatfirst projection the star first in starofV , stars similareclipses tothe other second filters star. atLet’s theHere define, we usethat dimensionless the first star from the first star in V1,𝑉𝑉 similar to2𝑉𝑉 other2 filters. 2Let’s define that the first star parametersLet’s, r i1=R1/a, r2=𝐿𝐿R1𝑉𝑉2/a=, where1 −1 𝐿𝐿2 a𝑉𝑉 is 12 the orbital2 separation.22 the continuation of the "out-of-eclipse" curve, as shown primaryparameterseclipses (moreLet’s the, rsecond ideep)1=R1/a ,minimum.star r2=𝐿𝐿 atR2 /thea=, where1primary Under𝑆𝑆− 𝐿𝐿= a𝜋𝜋 isthis 𝑟𝑟(more the, definition,𝑆𝑆 orbital deep)= 𝜋𝜋𝑟𝑟 separation.minimum. the . Under this definition, eclipses the second star at the primary𝑆𝑆1 = 𝜋𝜋 𝑟𝑟(more1 , 𝑆𝑆2 deep)= 𝜋𝜋𝑟𝑟 2minimum. . Under this definition, in Fig. 4. If one uses them as “depths” of the minima, the secondthe second star has star largerntroduce has largersurface surface brightness brightness (temperature).as the (temperature). relative contribution of the emission the second starntroduce has larger surface brightnessas the (temperature). relative contribution of the emission corresponding point at the diagram by Malkov et al. (2007) Sofrom one the may first write star a systemin V, ofsimilar 4 equations to other filters. Let’s define that the first star from theSo onefirst may star write in V 1a,𝑉𝑉 systemsimilar of to2 4𝑉𝑉 otherequations filters . Let’s define that the first star is located in the physically reliable region, relatively close to eclipsesLet’sSo the one second i may write star𝐿𝐿 at1a𝑉𝑉 systemthe= 1primary− of𝐿𝐿2 4𝑉𝑉 equations (more deep) minimum. Under this definition, eclipsesLet’s the second i star𝐿𝐿 at the= 1primary− 𝐿𝐿 (more deep) minimum. Under this definition, the line of the equal radii. (13) the second star has larger surface brightness (temperature). (13) m m m the second star has larger surface brightness (temperature). (13) The mean color index, (V-R)=16.674 -16.105 =0.569 . . . 1𝑉𝑉 1𝑉𝑉 . 2 2𝑉𝑉. 2 2𝑉𝑉. 𝑑𝑑1𝑉𝑉 =So(1 one− 𝐿𝐿 1may𝑉𝑉) 𝑆𝑆 write/𝑆𝑆2 = a system𝐿𝐿2𝑉𝑉 𝑆𝑆 /𝑆𝑆 of2 =4 equations𝐹𝐹2𝑉𝑉 𝑆𝑆 (14) ±0.069m was computed from the values of C for two filters. 𝑑𝑑1𝑉𝑉 =So(1 one− 𝐿𝐿 1may𝑉𝑉) 𝑆𝑆 write/𝑆𝑆2 = a system𝐿𝐿2𝑉𝑉 𝑆𝑆 /𝑆𝑆 of2 =4 equations𝐹𝐹2𝑉𝑉 𝑆𝑆 (14)(14) 1 𝑑𝑑 = (1 −. 𝐿𝐿 ) 𝑆𝑆/𝑆𝑆 =. 𝐿𝐿 𝑆𝑆 /𝑆𝑆 = 𝐹𝐹 𝑆𝑆 (14) . . In the phase curve of the color index (Fig. 5), variations 2𝑉𝑉 1𝑉𝑉. 1 1𝑉𝑉. (13)(15) 𝑑𝑑2𝑉𝑉 = 𝐿𝐿1𝑉𝑉 𝑆𝑆/𝑆𝑆1 = 𝐹𝐹1𝑉𝑉 𝑆𝑆 (15) (13)(15) m m m m 𝑑𝑑 = 𝐿𝐿 𝑆𝑆/𝑆𝑆 = 𝐹𝐹 𝑆𝑆 . . (15) from 0.54 ± 0.01 (at Max I at phase 0.25) to 0.63 ± 0.03 . . . 1𝑅𝑅𝑉𝑉 1𝑅𝑅𝑉𝑉 . 2 2𝑉𝑉𝑅𝑅. 2 2𝑉𝑉𝑅𝑅. 𝑑𝑑1𝑅𝑅 = (1 − 𝐿𝐿1𝑅𝑅) 𝑆𝑆/𝑆𝑆2 = 𝐿𝐿2𝑅𝑅 𝑆𝑆 /𝑆𝑆2 = 𝐹𝐹2𝑅𝑅 𝑆𝑆 (14)(16) (at Min I at phase 0.00) are present. At Min II at phase 0.50, 𝑑𝑑1𝑅𝑅𝑉𝑉 = (1 − 𝐿𝐿1𝑅𝑅𝑉𝑉) 𝑆𝑆/𝑆𝑆2 = 𝐿𝐿2𝑉𝑉𝑅𝑅 𝑆𝑆 /𝑆𝑆2 = 𝐹𝐹2𝑉𝑉𝑅𝑅 𝑆𝑆 (14)(16)(16) 𝑑𝑑 = (1 −. 𝐿𝐿 ) 𝑆𝑆/𝑆𝑆 =. 𝐿𝐿 𝑆𝑆 /𝑆𝑆 = 𝐹𝐹 𝑆𝑆 (16) m m 2𝑅𝑅𝑉𝑉 1𝑉𝑉𝑅𝑅. 1 1𝑉𝑉𝑅𝑅. V-R=0.61 ± 0.01 . The asymmetry of the maxima indicates 𝑑𝑑with2𝑅𝑅 =5 𝐿𝐿u1𝑅𝑅nknowns𝑆𝑆/𝑆𝑆1 =, 𝐹𝐹1𝑅𝑅 𝑆𝑆 . Four parameters – the relative(15) surface 𝑑𝑑with2𝑅𝑅𝑉𝑉 =5 𝐿𝐿u1𝑉𝑉𝑅𝑅nknowns𝑆𝑆/𝑆𝑆1 =, 𝐹𝐹1𝑉𝑉𝑅𝑅 𝑆𝑆 . Four parameters – the relative(15) surface the presence of the O’Connell effect. with 5 unknowns, L. , L , S, S ,. S . Four parameters. – the brightness1𝑅𝑅 1𝑅𝑅 . 1V 121𝑉𝑉R and12𝑅𝑅𝑅𝑅1 .similarly2 21 2 2for𝑅𝑅. the star 1 and filter V – are related to 𝑑𝑑brightness= (1 − 𝐿𝐿 ) 𝑆𝑆/𝐿𝐿𝑆𝑆1𝑉𝑉=, 𝐿𝐿and 𝐿𝐿1𝑅𝑅, similarly𝑆𝑆𝑆𝑆 /,𝑆𝑆𝑆𝑆1=, 𝑆𝑆 𝐹𝐹2 for𝑆𝑆 the star(16) 1 and filter V – are related to These values of the color index are in the instrumental relativethat1𝑅𝑅 mentioned surface brightness1𝑅𝑅 above𝐿𝐿12.𝑉𝑉 So, F 𝐿𝐿 1it2𝑅𝑅= 𝑅𝑅,isL𝑆𝑆 not,/S𝑆𝑆2 1 possible,and 𝑆𝑆 22 similarly𝑅𝑅 to determine for(16) the all 5 parameters without 𝑑𝑑that =mentioned(1 −. 2𝐿𝐿𝑅𝑅 above) 𝑆𝑆/2𝐿𝐿𝑅𝑅𝑆𝑆 . =.So,2 𝐿𝐿 2𝐿𝐿Rit ,is𝑆𝑆𝑆𝑆 2notR/,𝑆𝑆𝑆𝑆 2possible=, 𝑆𝑆𝐹𝐹 𝑆𝑆 to determine all 5 parameters without that mentioned𝐹𝐹2𝑅𝑅 =above𝐿𝐿2𝑅𝑅/. 𝑆𝑆So2 it is not possible to determine all 5 parameters without system, so, before taking into account the color star𝑑𝑑withadditional 21𝑅𝑅 and =5 𝐿𝐿 u1filter𝑅𝑅nknowns information.𝐹𝐹𝑆𝑆2/𝑅𝑅 𝑆𝑆V1= –= 𝐿𝐿are, 2 𝐹𝐹𝑅𝑅1/ 𝑅𝑅related 𝑆𝑆(e.g..𝑆𝑆2 color to that index mentioned-. luminosityFour parameters above. (or a bsolute – the m agnitude)relative surface-radius 𝑑𝑑withadditional2𝑅𝑅 =5 𝐿𝐿u1𝑅𝑅nknowns information𝑆𝑆/𝑆𝑆1 =, 𝐹𝐹1𝑅𝑅 (e.g.𝑆𝑆 color index-. luminosityFour parameters (or absolute – the m agnitude)relative surface-radius transformation coefficients and determining (V-R) in the So brightnessrelation,it is not assumingpossible to Main determine Sequence and similarly all) 5 - parametersth usfor we the need star without an 1 "extra"and filter equation. V – are related to brightnessrelation, assuming Main𝐿𝐿1𝑉𝑉, Sequence 𝐿𝐿and1𝑅𝑅, similarly𝑆𝑆, 𝑆𝑆)1 -, th𝑆𝑆2 usfor we the need star an 1 "extra"and filter equation. V – are related to standard system, we can't estimate more precisely the "color additionalthat mentioned information above𝐿𝐿1. 𝑉𝑉 (e.g.So, 𝐿𝐿 1it 𝑅𝑅color ,is𝑆𝑆 not, 𝑆𝑆index- 1possible, 𝑆𝑆2 luminosity to determine (or all 5 parameters without thatHowever mentioned, we2𝑅𝑅 can above estimate2𝑅𝑅 . So2 it is not possible to determine all 5 parameters without temperature" of the "star1+star2 system" (i.e. weighted absoluteadditionalHowever magnitude)-radius, weinformation𝐹𝐹2𝑅𝑅 can= 𝐿𝐿estimate2𝑅𝑅/ 𝑆𝑆(e.g.2 color relation, index -assuming luminosity Main (or a bsolute magnitude)-radius additional information𝐹𝐹 = 𝐿𝐿 / 𝑆𝑆(e.g. color index- luminosity (or absolute magnitude)-radius mean temperature of two components) and corresponding Sequence)relation, -assuming thus we need Main an Sequence "extra" equation.) - th us we need an "extra" equation.(17) relation, assuming Main Sequence) - th us we need an "extra" equation.(17) "spectral class" (also intermediate between that of the However,1𝑉𝑉 1𝑅𝑅 we can2𝑉𝑉 estimate2𝑅𝑅 However𝐿𝐿1𝑉𝑉/𝐿𝐿1𝑅𝑅 =, we𝑑𝑑2 can𝑉𝑉/𝑑𝑑 estimate2𝑅𝑅 = 1.080 ± 0.037 (18) components). However𝐿𝐿1𝑉𝑉/𝐿𝐿1𝑅𝑅 =, we𝑑𝑑2 can𝑉𝑉/𝑑𝑑 estimate2𝑅𝑅 = 1.080 ± 0.037 (18) 2𝑉𝑉 2𝑅𝑅 1𝑉𝑉 1𝑅𝑅 Following Andronov (2012), we introduce residual 𝐿𝐿From2𝑉𝑉/𝐿𝐿 this,2𝑅𝑅 = we𝑑𝑑1 𝑉𝑉can/𝑑𝑑 1derive𝑅𝑅 = 1 .another035 ± 0 combination.046 (17) (17) 𝐿𝐿From2𝑉𝑉/𝐿𝐿 this,2𝑅𝑅 = we𝑑𝑑1 𝑉𝑉can/𝑑𝑑 1derive𝑅𝑅 = 1 .another035 ± 0 combination.046 (17) relative intensities 𝐿𝐿1𝑉𝑉/𝐿𝐿1𝑅𝑅 = 𝑑𝑑2𝑉𝑉/𝑑𝑑2𝑅𝑅 = 1.080 ± 0.037 (18) (19) 𝐿𝐿1𝑉𝑉/𝐿𝐿1𝑅𝑅 = 𝑑𝑑2𝑉𝑉/𝑑𝑑2𝑅𝑅 = 1.080 ± 0.037 (18) (18) (19) 2𝑉𝑉1 2𝑅𝑅𝑉𝑉 1𝑉𝑉 21𝑅𝑅 2𝑉𝑉 2𝑅𝑅 1𝑉𝑉 1𝑅𝑅 𝐿𝐿From(i.e.𝑑𝑑1 /equal𝑑𝑑𝐿𝐿 this,2)𝑉𝑉 =/to we( 𝑑𝑑unity1 can//𝑑𝑑𝑑𝑑2 within)derive𝑅𝑅 = 1( 𝐿𝐿.erroranother0352𝑉𝑉/𝐿𝐿 ±estimates.2𝑅𝑅0 combination)./046(𝐿𝐿1𝑉𝑉/ A𝐿𝐿1 s𝑅𝑅mall) = 0differe.958 ±nce0. 053(even if smaller error), 𝐿𝐿From(i.e.𝑑𝑑2𝑉𝑉1 /equal𝑑𝑑𝐿𝐿 this,2𝑅𝑅)𝑉𝑉 =/to we( 𝑑𝑑unity1 𝑉𝑉can//𝑑𝑑𝑑𝑑2 within1)derive𝑅𝑅 = 1( 𝐿𝐿.erroranother0352𝑉𝑉/𝐿𝐿 ±estimates.2𝑅𝑅0 combination)./046(𝐿𝐿1𝑉𝑉/ A𝐿𝐿1 s𝑅𝑅mall) = 0differe.958 ±nce0. 053(even if smaller error), could be related to limb darkening. could be related to limb darkening. (19) http://dx.doi.org/10.5140/JASS.2015.32.2.127 132 (19) Other combinations are for the surface brightness (i.e.𝑑𝑑1 /equal𝑑𝑑2Other)𝑉𝑉 /to( 𝑑𝑑 unitycombinations1/𝑑𝑑2 within)𝑅𝑅 = ( 𝐿𝐿error 2are𝑉𝑉/ 𝐿𝐿 forestimates.2𝑅𝑅 )the/(𝐿𝐿 surface1𝑉𝑉/ A𝐿𝐿1 s𝑅𝑅 mall)brightness= 0differe.958 ±nce0. 053(even if smaller error), (i.e.𝑑𝑑1 /equal𝑑𝑑2)𝑉𝑉 /to( 𝑑𝑑unity1/𝑑𝑑2 within)𝑅𝑅 = ( 𝐿𝐿error2𝑉𝑉/𝐿𝐿 estimates.2𝑅𝑅)/(𝐿𝐿1𝑉𝑉/ A𝐿𝐿1 s𝑅𝑅mall) = 0differe.958 ±nce0. 053(even if smaller error), could be related to limb darkening. (20) could be related to limb darkening. (20) 1𝑉𝑉 2𝑉𝑉 2𝑉𝑉 1𝑉𝑉 𝑑𝑑1𝑉𝑉/𝑑𝑑Other2𝑉𝑉 = 𝐹𝐹combinations2𝑉𝑉/𝐹𝐹1𝑉𝑉 = 1.197 are ±for0 the.037 surface brightness (21) 𝑑𝑑1𝑉𝑉/𝑑𝑑Other2𝑉𝑉 = 𝐹𝐹combinations2𝑉𝑉/𝐹𝐹1𝑉𝑉 = 1.197 are ±for0 the.037 surface brightness (21) 1𝑅𝑅 2𝑅𝑅 2𝑅𝑅 1𝑅𝑅 𝑑𝑑1𝑅𝑅/𝑑𝑑T2𝑅𝑅hese= 𝐹𝐹 phenomenological2𝑅𝑅/𝐹𝐹1𝑅𝑅 = 1.249 ± val0.057ues may be used to estimate (20) temperature ratios, 𝑑𝑑1𝑅𝑅/𝑑𝑑T2𝑅𝑅hese= 𝐹𝐹 phenomenological2𝑅𝑅/𝐹𝐹1𝑅𝑅 = 1.249 ± val0.057ues may be used to estimate (20) temperature ratios, e.g. using "Main Sequence" (MS) relations". From these MS assumptions, one may 𝑑𝑑e.g.1𝑉𝑉 /using𝑑𝑑2𝑉𝑉 = "Main𝐹𝐹2𝑉𝑉/ 𝐹𝐹Sequence"1𝑉𝑉 = 1.197 (MS)± 0 .relations"037 . From these MS assumptions,(21) one may 𝑑𝑑make1𝑉𝑉/𝑑𝑑 a2 𝑉𝑉sequence= 𝐹𝐹2𝑉𝑉/𝐹𝐹 of1𝑉𝑉 suggestions= 1.197 ±, 0which.037 will be discussed below(21). make a sequence of suggestions, which will be discussed below. 𝑑𝑑1𝑅𝑅/𝑑𝑑T2𝑅𝑅hese= 𝐹𝐹 phenomenological2𝑅𝑅/𝐹𝐹1𝑅𝑅 = 1.249 ± val0.057ues may be used to estimate temperature ratios, 𝑑𝑑1𝑅𝑅/𝑑𝑑T2𝑅𝑅hese= 𝐹𝐹 phenomenological2𝑅𝑅/𝐹𝐹1𝑅𝑅 = 1.249 ± val0.057ues may be used to estimate temperature ratios, e.g. using "Main Sequence" (MS) relations". From these MS assumptions, one may e.g. using "Main Sequence" (MS) relations". From these MS assumptions, one may make a sequence of suggestions, which will be discussed below. make a sequence of suggestions, which will be discussed below. The second star (eclipsed at a primary minimum) has a larger surface brightness, consequently a larger radius and luminosity than the first star.

Taking into account Y≈1, one may suggest that the secondary eclipse is total surface of projection of stars Here, we use dimensionless(or close to total). Thus, , and the system of equations may be solved 2 2 completely: surface of projection of parametersstars , r1=R1/a, r2=R2/a,Here where1, we a isuse the dimensionlessorbital2 separation. 1 surface of projection of stars 𝑆𝑆 = 𝜋𝜋𝑟𝑟1 , 𝑆𝑆 = 𝜋𝜋𝑟𝑟2 .Here, we use dimensionless 𝑆𝑆 = 𝑆𝑆 surface of projection of stars Here, we use dimensionless parameters, r1=R1/a, r2=R2/a, where a is2 the orbital separation.2 parameters1 , r1ntroduce1=R1/a2, r2 =R22/a, where a isas2 the the orbital relative separation.2 contribution of the emission (22) 𝑆𝑆 = 𝜋𝜋𝑟𝑟 , 𝑆𝑆 = 𝜋𝜋𝑟𝑟 . 1 12 2 22 parameters, r1=R1/a, r2=R2/a, where𝑆𝑆1 = a𝜋𝜋 is𝑟𝑟1 the, 𝑆𝑆 orbital2 = 𝜋𝜋𝑟𝑟 separation.2 . ntroduce from the firstas star the inrelative V1,𝑉𝑉 similar contribution𝑆𝑆 =to2𝑉𝑉 𝜋𝜋other𝑟𝑟 , of  𝑆𝑆 filtersthe= 𝜋𝜋emission.𝑟𝑟 Let’s. define that the first1𝑉𝑉 star2𝑉𝑉 Let’s introduce 𝐿𝐿 = 1 − 𝐿𝐿 as the relative contribution of the 𝐿𝐿emission= 𝑑𝑑 = 0.395 ± 0.008 (23) eclipses the secondntroduce star at the primary (moreas the deep)relative minimum contribution. Under of this the definition,emission from the first star in V1,𝑉𝑉from similar the to2first𝑉𝑉 other star filters in V., Let’ssimilar define to other that filtersthe first. Let’s star define that the first star Let’s i 𝐿𝐿 fromthe= second1 the− 𝐿𝐿 first star starhas largerin V1,𝑉𝑉 surfacesimilar brightness to2𝑉𝑉 other filters(temperature).. Let’s define that the first1𝑅𝑅 star2𝑅𝑅 (24) eclipses the second star ateclipses the primaryLet’s the second i(more deep)star𝐿𝐿 at1𝑉𝑉 minimumthe= 1primary− 𝐿𝐿2. 𝑉𝑉Under (more this deep) definition, minimum . Under this definition,𝐿𝐿 = 𝑑𝑑 = 0.366 ± 0.010 eclipsesLet’s the second i star𝐿𝐿 at the= 1primary− 𝐿𝐿 (more deep) minimum. Under this definition, the second star has larger thesurface secondSo brightness one star may has write (temperature).larger a surfacesystem brightnessof 4 equations (temperature). 1 2 𝑉𝑉 1𝑉𝑉 2𝑉𝑉 the second star has larger surface brightness (temperature). (𝑆𝑆 /𝑆𝑆 ) = 𝑑𝑑 /(1 − 𝑑𝑑 ) = 0.782 ± 0.021 (25) So one may write a systemSo of one 4 equations may write a system of 4 equations (13) 1 2 𝑅𝑅 1𝑅𝑅 2𝑅𝑅 So one may write a system of 4 equations (𝑆𝑆 /𝑆𝑆 These) = 𝑑𝑑values/(1 −slightly𝑑𝑑 ) = differ0.721 by± 0a.029 value of 0.061 0.035=1.7, and the . . . 1𝑉𝑉 1𝑉𝑉 2 2𝑉𝑉 2 2𝑉𝑉 (13) difference is not statistically significant. The mean weighted value of the ratio of 𝑑𝑑 = (1 − 𝐿𝐿 ) 𝑆𝑆/𝑆𝑆 = 𝐿𝐿 𝑆𝑆 /𝑆𝑆 = 𝐹𝐹 𝑆𝑆 (14) (13) ± . . . (13) cross-sections of the two stars is and the ratio of radii 1𝑉𝑉 1𝑉𝑉 2 2𝑉𝑉 2 . 2𝑉𝑉 . . . . 𝑑𝑑 = (1 − 𝐿𝐿 ) 𝑆𝑆/𝑆𝑆 = 𝐿𝐿12𝑉𝑉 𝑆𝑆 /𝑆𝑆1𝑉𝑉= 𝐹𝐹1𝑉𝑉1 .𝑆𝑆 12𝑉𝑉 (14)2𝑉𝑉 . 2 2𝑉𝑉. 𝑑𝑑1𝑉𝑉 = (𝐿𝐿1 −𝑆𝑆𝐿𝐿/1𝑆𝑆𝑉𝑉)=𝑆𝑆/ 𝐹𝐹𝑆𝑆2 =𝑆𝑆 𝐿𝐿2𝑉𝑉 𝑆𝑆 /𝑆𝑆2 = 𝐹𝐹2𝑉𝑉 𝑆𝑆 (14) (15) is (radius is in units of the orbital separation). For the . . 𝑑𝑑 = (1 − 𝐿𝐿 ) 𝑆𝑆/𝑆𝑆 = 𝐿𝐿 𝑆𝑆 /𝑆𝑆 = 𝐹𝐹 𝑆𝑆 (14) 1 2 2𝑉𝑉 1𝑉𝑉 1 1𝑉𝑉 . . . . . (𝑆𝑆 /𝑆𝑆 ) = 0.761 ± 0.017, 𝑑𝑑 = 𝐿𝐿 𝑆𝑆/𝑆𝑆 = 𝐹𝐹 𝑆𝑆 12𝑅𝑅𝑉𝑉 1𝑉𝑉. 1𝑅𝑅1 12𝑉𝑉. 2𝑅𝑅 2 2𝑅𝑅 (15) low-mass1 2 part of the Main Sequence, R~M (e.g. Faulkner 1971) so this value may 𝑑𝑑2𝑉𝑉 = (𝐿𝐿1𝑉𝑉−𝑆𝑆𝐿𝐿/𝑆𝑆1)=𝑆𝑆/ 𝐹𝐹𝑆𝑆1𝑉𝑉=𝑆𝑆 𝐿𝐿 𝑆𝑆 /𝑆𝑆 = 𝐹𝐹 𝑆𝑆 (16) (15) (𝑅𝑅 /𝑅𝑅 ) = 0.872 ± 0.010 . 𝑑𝑑 .= 𝐿𝐿 𝑆𝑆/𝑆𝑆 .= 𝐹𝐹 𝑆𝑆 (15) be an estimate of the mass ratio, . From the table of 1𝑅𝑅 1𝑅𝑅 2 2𝑅𝑅 2 . 2𝑅𝑅 . . . . 𝑑𝑑 = (1 − 𝐿𝐿 ) 𝑆𝑆/𝑆𝑆 = 𝐿𝐿12𝑅𝑅 𝑆𝑆 /𝑆𝑆1𝑅𝑅= 𝐹𝐹1𝑅𝑅1 .𝑆𝑆 12𝑅𝑅 (16)2𝑅𝑅 . 2 2𝑅𝑅. 0.72 𝑑𝑑with1𝑅𝑅 =5 (𝐿𝐿1unknowns−𝑆𝑆𝐿𝐿/1𝑆𝑆𝑅𝑅)=𝑆𝑆,/ 𝐹𝐹𝑆𝑆2 =𝑆𝑆 𝐿𝐿2𝑅𝑅 𝑆𝑆 /𝑆𝑆2 = 𝐹𝐹2.𝑅𝑅 F𝑆𝑆our parameters(16) – the relativeAllen surface (1973) for the Main Sequence one may obtain a statistical relation R≈M . . 𝑑𝑑 = (1 − 𝐿𝐿 ) 𝑆𝑆/𝑆𝑆 = 𝐿𝐿 𝑆𝑆 /𝑆𝑆 = 𝐹𝐹 𝑆𝑆 (16) 1 2 2𝑅𝑅 1𝑅𝑅 1 1𝑅𝑅 . . 𝑞𝑞 = 𝑀𝑀 /𝑀𝑀 = 0.872 ± 0.010 𝑑𝑑with =5 𝐿𝐿unknowns𝑆𝑆/𝑆𝑆 =, 𝐹𝐹 𝑆𝑆 brightness2𝑅𝑅 1𝑅𝑅 . .1 Four11 𝑅𝑅𝑉𝑉parameters. and1𝑅𝑅 similarly –1 the2 for relative the star surface 1 and filter V – are relatedfor a wide to range of spectral classes: from A0 to M0. In this case, we get a slightly 𝑑𝑑with2𝑅𝑅 =5 𝐿𝐿u1𝑅𝑅nknowns𝑆𝑆/𝑆𝑆1 =, 𝐹𝐹𝐿𝐿1𝑅𝑅 ,𝑆𝑆𝐿𝐿 , 𝑆𝑆, 𝑆𝑆 , 𝑆𝑆 . Four parameters – the relative surface brightness 𝑑𝑑withthatand =similarlymentioned5 𝐿𝐿unknowns𝑆𝑆2/𝑅𝑅 𝑆𝑆for above= the, 2 𝐹𝐹𝑅𝑅 .star So𝑆𝑆2 1it andis not filter possible V. –F our areto determinerelatedparameters to all – 5 theparamet relativeersdifferent withoutsurface estimate . From more 1𝑉𝑉 brightness1𝑅𝑅 1 𝐹𝐹 2 = 𝐿𝐿 /𝑆𝑆 and similarly for the star 1 and filter V – are related to 𝐿𝐿 , 𝐿𝐿 , 𝑆𝑆, 𝑆𝑆 , 𝑆𝑆 1𝑉𝑉 1𝑅𝑅 1 2 0.80 1/0.72 that mentioned above. Soadditionalbrightness it is not possible information to𝐿𝐿 1determine𝑉𝑉 (e.g., 𝐿𝐿and1 𝑅𝑅color, similarly𝑆𝑆 ,all 𝑆𝑆index1 5, 𝑆𝑆paramet2 -for luminosity theers star without 1(or and absolute filter Vm agnitude)– are relatedrecent-radius tablesto of Cox (2000), R≈1.01M , thus 2𝑅𝑅 2𝑅𝑅 2 that mentioned above𝐿𝐿 . So, 𝐿𝐿 it ,is𝑆𝑆 not, 𝑆𝑆 possible, 𝑆𝑆 to determine all 5 parameters without 1 2 1 2 𝐹𝐹 = 𝐿𝐿 /𝑆𝑆 thatrelation, mentioned assuming2𝑅𝑅 above Main2𝑅𝑅 . So2 Sequence it is not) possible- thus we to need determine an "extra" all equation.5 paramet ers without 𝑞𝑞 = 𝑀𝑀 /𝑀𝑀 ≈ (𝑅𝑅 /𝑅𝑅 ) = 0.826 ± 0.013 additional information (e.g.additional color index information𝐹𝐹2𝑅𝑅- =luminosity𝐿𝐿2𝑅𝑅/ 𝑆𝑆(e.g.2 ( orcolor absolute index m- luminosityagnitude)-radius (or absolute magnitude)-radius additional information𝐹𝐹 = 𝐿𝐿 / 𝑆𝑆(e.g. color index- luminosity (or absolute magnitude)-radius relation, assuming Main SequenceHoweverrelation, )assuming, -we th uscan we estimate Main need Sequencean "extra") -equation. thus we need an "extra" equation. 1 (26) relation, assuming Main Sequence) - thus we need an "extra" equation. 𝑅𝑅1 0.80 Ivan L. Andronov1 et al.2 Phenomenological Modeling of VSX J1802 However, we can estimateHowever , we can estimate (17) 𝑞𝑞 = 𝑀𝑀 /𝑀𝑀 ≈ (𝑅𝑅2) = 0.842 ± 0.012 However, we can estimate However, the (unknown) deviations from the mass-radius relation may 1𝑉𝑉 1𝑅𝑅 2 𝑉𝑉 2 𝑅𝑅 (17) exceed this maximal 5 percent difference in estimates using different coefficients 𝐿𝐿 /𝐿𝐿 = 𝑑𝑑 /𝑑𝑑 = 1.080 ± 0.037 (18)(17) (17) of the statistical dependence. From the duration of eclipse, one may suggest an 1𝑉𝑉 1𝑅𝑅 2𝑉𝑉 2𝑅𝑅 From this, we can derive another combination From the duration of eclipse, one may suggest an inequality 12𝑉𝑉 12𝑅𝑅 21 𝑉𝑉 12 𝑅𝑅 (18) 𝐿𝐿 /𝐿𝐿 = 𝑑𝑑 /𝑑𝑑 = 1.𝐿𝐿From080/𝐿𝐿± this,0.037= we𝑑𝑑 can/𝑑𝑑 derive= 1 .another035080 ± 0 combination.046037 (18) inequality 𝐿𝐿1𝑉𝑉/𝐿𝐿1𝑅𝑅 = 𝑑𝑑2𝑉𝑉/𝑑𝑑2𝑅𝑅 = 1.080 ± 0.037 (18) 2𝑉𝑉 2𝑅𝑅 1𝑉𝑉 1𝑅𝑅 𝐿𝐿From/𝐿𝐿 this,= we𝑑𝑑 can/𝑑𝑑 derive= 1 .another𝐿𝐿From0352𝑉𝑉/±𝐿𝐿 this,2𝑅𝑅0 combination.046= we𝑑𝑑1 𝑉𝑉can/𝑑𝑑 1derive𝑅𝑅 = 1 .another035 ± 0 combination.046 (19) (19) (27) (27) 𝐿𝐿From2𝑉𝑉/𝐿𝐿 this,2𝑅𝑅 = we𝑑𝑑1 𝑉𝑉can/𝑑𝑑 1derive𝑅𝑅 = 1 .another035 ± 0 combination.046

1 2 𝑉𝑉 1 2 𝑅𝑅 2𝑉𝑉 2𝑅𝑅 1 𝑉𝑉 1(19)𝑅𝑅 (i.e.𝑑𝑑 /equal𝑑𝑑 ) /to( 𝑑𝑑unity/𝑑𝑑 within) = ( 𝐿𝐿error/𝐿𝐿 estimates.)/(𝐿𝐿 / A𝐿𝐿 small) = 0differe.958 ±nce0. 053(eve n if smaller(19) For1 error such2 ), large ( values,8) the stars are distorted (and we see this also from large values For (19)such 𝑟𝑟 large+ 𝑟𝑟 ≥values,sin 2 the𝜋𝜋𝐶𝐶 stars= 0 .are6739 distorted (and we 1 2 𝑉𝑉 1 2 𝑅𝑅 could2𝑉𝑉 2be𝑅𝑅 related1𝑉𝑉 to1𝑅𝑅 limb darkening. of ) and thus close to their Roche lobes. (i.e.𝑑𝑑 /equal𝑑𝑑 ) /to( 𝑑𝑑unity/𝑑𝑑 within) =i.e.,( 𝐿𝐿error(i.e. 𝑑𝑑equal1 //equal𝐿𝐿 𝑑𝑑estimates.2 )to)𝑉𝑉/ /tounity((𝐿𝐿 𝑑𝑑unity1// 𝑑𝑑Awithin𝐿𝐿2 withins)mall𝑅𝑅) = error (0differe 𝐿𝐿error.9582𝑉𝑉 estimates./𝐿𝐿 ±estimates.nce2𝑅𝑅0). /053(eve(𝐿𝐿1 A𝑉𝑉n/ smallifA𝐿𝐿 1 smallers𝑅𝑅mall) =difference 0differe .error958 ±),nce 0. 053(eveseen ifthis smaller also from error large), values of C ) and thus close to their (i.e.𝑑𝑑1 /equal𝑑𝑑2)𝑉𝑉 /to( 𝑑𝑑unity1/𝑑𝑑2 within)𝑅𝑅 = ( 𝐿𝐿error2𝑉𝑉/𝐿𝐿 estimates.2𝑅𝑅)/(𝐿𝐿1𝑉𝑉/ A𝐿𝐿1 s𝑅𝑅mall) = 0differe.958 ±nce0. 053(even if smaller error), 3 could be related to limb(even darkening.could if smaller beOther related combinationserror), to limb could darkening. beare related for the to surface limb darkening. brightness Roche lobes. 3 Although a possible W UMa – type classification may not be completely could be related to limb darkening. 𝐶𝐶 Other combinations are for the surface brightness Althoughrejected a possible, these W UMa stars – aretype not classification yet in thermal may not contact (Lucy 1976) based on the Other combinations are forOther the surface combinations brightness are for the surface brightness (20) Other combinations are for the surface brightness be completelydifferen rejected,ce of mean these brightness stars are notand yet we insuggest thermal that they are not in contact. Thus, we 1𝑉𝑉 2𝑉𝑉 2 𝑉𝑉 1 𝑉𝑉 (20) 𝑑𝑑 /𝑑𝑑 = 𝐹𝐹 /𝐹𝐹 = 1.197 ± 0.037 (20) contact(20)(21) (Lucyprefer 1976) an EA based – type on classification the difference with ellipticof mean component(s) . (20) 1𝑉𝑉 2𝑉𝑉 2𝑉𝑉 1𝑉𝑉 brightness and we suggest that they are not in contact. 1𝑅𝑅𝑉𝑉 2𝑉𝑉𝑅𝑅 2 𝑅𝑅𝑉𝑉 1 𝑉𝑉𝑅𝑅 (21) 𝑑𝑑 /𝑑𝑑 = 𝐹𝐹 /𝐹𝐹 = 1.197𝑑𝑑 /±𝑑𝑑T0hese.037= 𝐹𝐹 phenomenological/𝐹𝐹 = 1.197249 ± val0.037057ues may be used to estimate (21) temperature ratios, 𝑑𝑑1𝑉𝑉/𝑑𝑑2𝑉𝑉 = 𝐹𝐹2𝑉𝑉/𝐹𝐹1𝑉𝑉 = 1.197 ± 0.037 (21) Thus,(21) we prefer an EA – type classification with elliptic 1𝑅𝑅 2𝑅𝑅 2𝑅𝑅 1𝑅𝑅 e.g. using "Main Sequence" (MS) relations". From these MS assumptions, one may 𝑑𝑑 /𝑑𝑑These= 𝐹𝐹 phenomenological/𝐹𝐹 = 1.𝑑𝑑2491𝑅𝑅/±𝑑𝑑 T2val0𝑅𝑅hese.057=ues𝐹𝐹 phenomenological2may𝑅𝑅/𝐹𝐹 1be𝑅𝑅 = used1.249 to estimate± val0.057ues temperaturemay be used ratios, to estimate component(s). temperature ratios, 𝑑𝑑make1𝑅𝑅/𝑑𝑑 aT2 𝑅𝑅sequencehese= 𝐹𝐹 phenomenological2𝑅𝑅/𝐹𝐹 of1𝑅𝑅 suggestions= 1.249 ± ,val 0which.057ues maywill be discussedused to estimate below. temperature ratios, e.g. using "Main Sequence"Thesee.g. (MS) using phenomenological relations" "Main Sequence". From thesevalues (MS) MS may relations" assumptions, be used. From to estimate one these may MS assumptions,Using the one oversimplified may form of the MS mass-radius e.g. using "Main Sequence" (MS) relations". From these MS assumptions, one may make a sequence of suggestionstemperaturemake a, whichsequence ratios, will of be e.g.suggestions discussed using "Main,below which. Sequence" will be discussed (MS) belowrelation. make a sequence of suggestions, which will be discussed below. Using the oversimplified form of the MS mass-radius relation relations". From these MS assumptions, one may make a

sequence of suggestions, which will be discussed below.   (28) (28) The second star (eclipsed at a primary minimum) has a . and𝑅𝑅 =, 𝑅𝑅combining𝑀𝑀/𝑀𝑀 , with Kepler's third law, we obtain the inclination, o, larger surfaceThe secondbrightness, star consequently(eclipsed at aa largerprimary radius minimum) and hasand, a combining larger surfa withce Kepler's third law, we obtain the inclination, The second star (eclipsed at a primary minimum) has a larger surface    the orbital luminositybrightness, than consequently the first star. a larger radius and luminosity than thei = first 90°, star.M = 0.745M , M = 0.854M , M = M. +9M =1.599M the 𝑖𝑖 = 90 brightness, consequently a larger radius and luminosity than the first star.1 ◉ separation2 ◉a=1.65 101 m=2.372 R ◉ and relative radii r =R /a=0.314 and 1 9 2  1 2 1 1 Taking into account Y≈1, one may suggest that the orbital separation a= 1.65∙10𝑀𝑀 = 0m.745 = 2.37 𝑀𝑀R◉, and 𝑀𝑀 relative= 0.854 radii 𝑀𝑀 r1,=R𝑀𝑀1/= 𝑀𝑀 + 𝑀𝑀 = 1.599 𝑀𝑀 , Taking into account Y≈1, one may suggest that the secondary eclipse is totalr 2=R2/a=0.360. secondaryTaking eclipse into is total account (or close Y≈1, to one total). may Thus, suggest S= Sthat, and the secondarya = 0.314 eclipse and r is= Rtotal/a = 0.360. (or close to total). Thus, , and the system1 of equations may be 2solved2 the(or system close of The toequations total).second Thusmay star ,be (eclipsed solved, completely:and at thea primary system minimum)of equations hasGenerally, may a largerbe thesesolved surfa are ce minimalGenerally, estimates these are of theminimal radiuses estimates of the radiuses and masses, since for completely: o completely:brightness, consequently a larger1 radius and luminosity than the first star. completely: 𝑆𝑆 = 𝑆𝑆1 and masses, since foran aninclination inclination,, , differing i, differing from from 90 90°,, we ’ll get larger estimated values. With the 𝑆𝑆 = 𝑆𝑆1 (22)(22) we’ll get larger estimatedassumption values. of one With total the eclipse, assumption one may write cos(i) < |r2 - r1| = 0.046, thus Taking into account Y ≈1, one may suggest(22) that the secondary eclipse is total o 𝑖𝑖o of one total eclipse,87.4 one ≤may i ≤write 90 , cos(0.999≤sini)<|r (-i)r ≤1,| = 0.046, and for VSX 1802, the effects of inclination on 1𝑉𝑉(or close2𝑉𝑉 to total). Thus, , and the system of equations may be solved 2 1 𝐿𝐿1𝑉𝑉 = 𝑑𝑑2𝑉𝑉 = 0.395 ± 0.008 (23)(23) estimates of radii and masses are negligible. 𝐿𝐿 completely:= 𝑑𝑑 = 0 .395 ± 0.008 (23) thus 87.4° ≤ i ≤ 90°, 0.999 ≤ sin(i) ≤ 1, and for VSX 1802, the 1𝑅𝑅 2𝑅𝑅 1 𝐿𝐿1𝑅𝑅 = 𝑑𝑑2𝑅𝑅 = 0.366 ± 0.010 𝑆𝑆 = 𝑆𝑆 effects(24) of inclination on estimates of radii and masses are 𝐿𝐿1𝑅𝑅 = 𝑑𝑑2𝑅𝑅 = 0.366 ± 0.010 (24) (24) Even though the spectral classes of this system have not been reported yet, (22) negligible. 1 2 𝑉𝑉 1𝑉𝑉 2𝑉𝑉 we can suggest the corresponding spectral classes of these stars to be G8 and K2 (𝑆𝑆1/𝑆𝑆2)𝑉𝑉 = 𝑑𝑑1𝑉𝑉/(1 − 𝑑𝑑2𝑉𝑉) = 0.782 ± 0.021 (25) (𝑆𝑆 /1𝑆𝑆𝑉𝑉 ) =2𝑉𝑉𝑑𝑑 /(1 − 𝑑𝑑 ) = 0.782 ± 0.021 (25) (25)Even though the spectral classes of this system have (𝑆𝑆 𝐿𝐿/𝑆𝑆 )= =𝑑𝑑 𝑑𝑑 =/0(.1395− 𝑑𝑑± 0).008= 0 . 782 ± 0 .021 (23) according to Cox (2000) and the possible Roche model of these VSX 1802 system 1 2 𝑅𝑅 1𝑅𝑅 2𝑅𝑅 1 2 𝑅𝑅 1𝑅𝑅 2𝑅𝑅 not been reported yet,can webe estimatedcan suggest as inthe Fig. corresponding 7. (𝑆𝑆1/𝑆𝑆2These)𝑅𝑅 = 𝑑𝑑values1𝑅𝑅/(1 −slightly𝑑𝑑2𝑅𝑅) = differ0.721 by± 0a.029 value of 0.061 0.035=1.7, and the (𝑆𝑆 𝐿𝐿/1𝑆𝑆𝑅𝑅These)= =𝑑𝑑2 𝑅𝑅𝑑𝑑values=/0(.1366 −slightly𝑑𝑑± 0).010= differ0.721 by± 0a.029 value of 0.061 0.035=1.7(24) , and the Thesedifference values is slightlynot statistically differ by asignificant. value of 0.061±0.035=1.7 The mean weightedσ, spectralvalue of classes the ratio of theseof stars to be G8 and K2 according to difference is not statistically significant. The mean weighted± value of the ratio of andcross the-1sections difference2 𝑉𝑉 of 1the𝑉𝑉 is nottwo statisticallystars2𝑉𝑉 is significant. The ± andCox the(2000) ratio and of theradii possible Roche model of these VSX 1802 cross(𝑆𝑆-sections/𝑆𝑆 ) = of𝑑𝑑 the/( 1two− 𝑑𝑑stars) =is 0.782 ± 0.021 and (25)the ratio of radii meanis weighted value of the ratio (radius of cross-sections is in units of of the the orbital systemseparation). can be For estimated the as in Fig. 7. is (radius1 is 2in units of the orbital separation). For the 1 2 𝑅𝑅 1𝑅𝑅 2𝑅𝑅 (𝑆𝑆1/𝑆𝑆2) = 0.761 ± 0.017,  twolow stars(-𝑆𝑆mass/ 𝑆𝑆isThese )part(S1=/ Sof 𝑑𝑑2)values =the/ 0.761±0.017,( Main1 −slightly𝑑𝑑 Sequence,) = differ (0and𝑆𝑆.721/ 𝑆𝑆R the~by±)M =0ratio a(e.g..029 0value.761 ofFaulkner ±radii of0 .0170.061 1971), 0.035=1.7 so this value, andmay the low-mass1 2 part of the Main Sequence, R~M (e.g. Faulkner 1971) so this value may is (R(/difference𝑅𝑅R1/)𝑅𝑅 =2 0.872±0.010) = is0. 872not statistically± (radius0.010 is significant. in units of Thethe meanorbital weighted value of the ratio of be1( an𝑅𝑅 2/e𝑅𝑅stimate) = 0 of.872 the± mass0.010 ratio, . From the table of be an estimate of the mass ratio, ±. From the table 0.72of separation).cross-sections For the oflow-mass the two partstars of is the Main Sequence, 7. andCONCLUSIONS the ratio of0.72 radii Allen (1973) for the Main Sequence one may obtain a statistical relation R≈M0.72Fig . 7. The model of the system VSX 1802: The Roche lobes, the line of centers Allen (1973) for the Main Sequence on1e may2 obtain a statistical relation R≈M R~forM (e.g. isa wide Faulkner range 1971)of spectral so this classes value (radius𝑞𝑞 := mayfrom𝑀𝑀 1is /be A0𝑀𝑀in 2an unitsto= estimate M0.0.872 of In the± this 0orbital.010 case, separation).we get a slightly For andthe circles corresponding to estimated radii of the stars in a spherically-symmetric for a wide range of spectral classes𝑞𝑞 := from(𝑆𝑆𝑀𝑀1//𝑆𝑆 A0𝑀𝑀2) =to=0 M0.0.761.872 In±± this00.017.010 case,, we get a slightly low-mass part of the Main Sequence, R~M (e.g. Faulkner 1971) so this value mayapproximation . of differentthe mass ratio,estim ateq = M1/M2 = 0.872±0.010. From the table In this. From study, more we estimated the parameters only from different(𝑅𝑅1 /𝑅𝑅estim2) =ate0. 872 ± 0.010 . From more 0.80 of recentAllenbe an(1973)tables estimate of for Cox theof (2000),the Main mass RSequence≈ ratio1.01,M 0. 80one, thus 1may/0 .72 obtain . From the table of recent tables of Cox (2000),1 R≈21.01M1 , 2thus1/0 .72 0.72 recent tables of Cox 𝑞𝑞(2000),=0.72𝑀𝑀1/ R𝑀𝑀≈21.01≈ (M𝑅𝑅1/𝑅𝑅, 2thus) = 0.826 ± 0.013 a statisticalAllen (1973)relation for R 𝑞𝑞≈theM= Main𝑀𝑀 for/𝑀𝑀 Saequence ≈wide(𝑅𝑅 range/𝑅𝑅 on)1e ofmay 2spectral =obtain0.826 a ±statistical0.013 relation R≈M for a wide range of spectral classes𝑞𝑞 := from𝑀𝑀 / A0𝑀𝑀 to= M0.0.872 In± this0.010 case, we get a slightly classes: from A0 to M0. In1 this case, we get a slightly different 1 (26) 0.180 1/0.72 (26) estimatedifferent q = M /estimM ≈(𝑅𝑅1ateR0./80 R ) = 0.826± 0.013. From more . From more 1 2 𝑅𝑅1 01.80 2 1 2 𝑅𝑅 0.80 Conclusions 1 2 𝑅𝑅2 0.80 1/0.72 recent𝑞𝑞 =recent tables𝑀𝑀 However,1/ 𝑀𝑀tables of2 Cox≈ (of 𝑅𝑅(2000),the2 Cox) (unknown) (2000),= R ≈1.010.842 RM≈± 1.01d0e, .viationsthus012M , thusfrom the mass-radius relation may 𝑞𝑞 = 𝑀𝑀 However,/𝑀𝑀 ≈ ( 𝑅𝑅the) (unknown)𝑞𝑞==0𝑀𝑀.8421/𝑀𝑀± 2d0e≈.viations012(𝑅𝑅1/𝑅𝑅 2)from the= 0mass.826-±radius0.013 relation may In this study, we estimated the parameters only from phenomenological exceed this maximal 5 percent difference in estimates using different coefficients exceed this maximal 5 percent difference in estimates using different coefficientsmodeling of the phase light curve. The main unjustified suggestions were "no limb 1 (26) of the statistical dependence. From the duration of eclipse,(26) one may suggest andarkening" and "similar to MS dependencies" (no real quantitative relations, just of the statistical dependence𝑅𝑅1 0.80 . From the duration of eclipse, one may suggest an inequality "larger stars are brighter"). There are no detailed photometric or spectroscopic inequality 1 2 𝑅𝑅2 However,𝑞𝑞 = 𝑀𝑀 theHowever,/𝑀𝑀 (unknown)≈ ( the) deviations(unknown)= 0.842 from± d0e.viations 012the mass-radius from the mass-radius relation may (27) studies at present, therefore the true physical parameters of this system cannot be relationexc mayeed thisexceed maximal this maximal 5 percent 5 percentdifference difference in estimates(27) in usingFig. different 7. The model coefficients of the system VSX 1802: The Roche lobes, the line (27) of centers and circles correspondingestimated for to estimatednow. radii of the stars in a estimates1 of 2 theusing statistical different8 coefficientsdependence of .the From statistical the durationdependence. of eclipse,spherically-symmetric one may suggest approximation. an For𝑟𝑟1 + such𝑟𝑟2 ≥ largesin( 2values,𝜋𝜋𝐶𝐶8) = the0. 6739stars are distorted (and we see this also from large values For𝑟𝑟 +inequality such𝑟𝑟 ≥ largesin ( 2values,𝜋𝜋𝐶𝐶 ) = the0. 6739stars are distorted (and we see this also from large values Our phenomenological modeling could allow rough estimates to make the of ) and thus close to their Roche lobes. of ) and thus close to their Roche lobes. physical parameters. Although the errors of the phenomenological parameters are 3 (27) 𝐶𝐶3 Although a possible W UMa – type classification may not be completelynot very large (up to few percent), there may be systematic errors up to a dozen 𝐶𝐶3 Although a possible W UMa – type classification133 may not be completely http://janss.kr rejected, these stars are not yet in thermal contact (Lucy 1976) based on the rejectedFor𝑟𝑟1 + such,𝑟𝑟 2these≥ largesin stars( 2values,𝜋𝜋𝐶𝐶 8are) = thenot0. 6739stars yet arine thermaldistorted contact (and we (Lucy see this 1976) also frombased large on valuesthe difference of mean brightness and we suggest that they are not in contact. Thus, we differenof ce) and of meanthus close brightness to their and Roche we suggest lobes. that they are not in contact. Thus, we prefer an EA – type classification with elliptic component(s). prefer an EA – type classification with elliptic component(s). 𝐶𝐶3 Although a possible W UMa – type classification may not be completely rejected, these stars are not yet in thermal contact (Lucy 1976) based on the difference of mean brightness and we suggest that they are not in contact. Thus, we prefer an EA – type classification with elliptic component(s). J. Astron. Space Sci. 32(2), 127-136 (2015)

phenomenological modeling of the phase light curve. The Odessa, 1994), 49-54. main unjustified suggestions were "no limb darkening" Andronov IL. Method of running parabolae: Spectral and and "similar to MS dependencies" (no real quantitative statistical properties of the smoothing function, Astron. relations, just "larger stars are brighter"). There are no Astrophys. Suppl. 125, 207-217 (1997). http://dx.doi. detailed photometric or spectroscopic studies at present, org/10.1051/aas:1997217 therefore the true physical parameters of this system cannot Andronov IL, Multiperiodic versus Noise Variations: be estimated for now. Mathematical Methods, in ASP Conference Series, ed. Our phenomenological modeling could allow rough Sterken C (Astronomical Society of the Pacific, San estimates to make the physical parameters. Although the Francisco, 2003), 391-400. errors of the phenomenological parameters are not very Andronov IL, Advanced Methods for Determination of large (up to few percent), there may be systematic errors up Arguments of Characteristic Events, in ASP Conference to a dozen percent due to the simplicity of the model. These Series, ed. Sterken C (Astronomical Society of the Pacific, estimates may be used as preliminary values for further San Francisco, 2005), 37-53. modeling using extended physical models based on the Andronov IL, Phenomenological Modeling of the Light Curves Wilson & Devinney (1971) code and it’s extensions (Wilson of Algol-Type Eclipsing Binary Stars, Astrophys. 55, 536- 1979, 1994, 2012, 2014; Zola et al. 1997, 2010; Bradstreet 550 (2012). http://dx.doi.org/10.1007/s10511-012-9259-0 2005; Kallrath et al, 2009; Linnel 2012; Reed 2012; Rucinski Andronov IL, Kim YK, Yoon J-N, Breus VV, Smecker-Hane TA, 2010; Prsa et al 2012). Such study of eclipsing binaries et al., Two-Color CCD Photometry of the Intermediate with the CBNU observations are reported by Jeong & Kim Polar 1RXS J180340.0+401214, J. Korean Astron. Soc. 44, (2013); Kim & Jeong(2012); Jeong & Kim (2011) and Kim 89-96 (2011). http://dx.doi.org/10.5303/JKAS.2011.44.3.89 et al. (2010a). It should be noted that the lack of spectral Andronov IL, Antoniuk KA, Baklanov AV, Breus VV, Burwitz data doesn’t allow to study various combination of stars in V, et al., Inter-Longitude Astronomy (ILA) Project: different evolutionary stages. Some discussions with crude Current Highlights and Perspectives.I.Magnetic vs.Non- assumptions about the age and metallicity of the system Magnetic Interacting Binary Stars (Odessa Astronomical are therefore out of the scope of this study. More detailed Publications, Odessa, 2010), 8-12. photometrical and spectroscopic information will enable Andronov IL, Baklanov AV, Algorithm of the artificial more thorough studies. comparison star for the CCD Photometry, Astron School Rep. 5, 264-272 (2004). Andronov IL, Breus VV, Zola S, Determination of Characteristics ACKNOWLEDGEMENTS of Newly Discovered Eclipsing Binary 2MASS J18024395 +4003309 VSX J180243.9+400331 (Odessa Astronomical This work was supported by the research grant of Publications, Odessa, 2012), 145-147 (2012). Chungbuk National University in 2012. The data acquisition Andronov IL, Chinarova LL, Method of Running Sines: and analysis were partially supported by Basic Science Modeling Variability in Long-Period Variables, in Research Program through the National Research Czestochowski Kalendarz Astronomiczny 2014, ed. Foundation of Korea (NRF) funded by the Ministry of Wszolek B (Astronomia Nova Association, Poland, 2013), Education, Science and Technology (2011-0014954). It is 139-144. also a part of the “Inter-Longitude Astronomy” campaign Andronov IL, Marsakova VI, Variability of long-period pulsating (Andronov et al. 2010) and the project “Ukrainian Virtual stars. I. Methods for analyzing observations, Astrophys. Observatory (Vavilova et al. 2012). 49, 370-385 (2006). http://dx.doi.org/10.1007/s10511-006- 0037-8 Andronov IL, Kim Y, Yoon J-N, Breus VV, Smecker-Hane TA, et REFERENCES al., Two-Color CCD Photometry of the Intermediate Ploar 1RXS J180340.0+401214, J. Korean Astron. Soc. 44, 89-96 Allen CW, Astrophysical quantities, 3rd Ed (Athlone Press, (2011). http://dx.doi.org/10.5303/JKAS.2011.44.3.89 London, 1973). Andronov IL, Tkachenko MG, Comparative Analysis of Anderson TW, An Introduction to Multivariate Statistical Numerical Methods for Parameter Determination, Analysis, 3rd Ed (Wiley, New York, 2003), 721. Czestochowski Kalendarz Astronomiczny 2014, ed. Andronov IL, (Multi-) Frequency Variations of Stars. Some Wszolek B (Astronomia Nova Association, Poland, 2013a), Methods and Results (Odessa Astronomical Publications, 173-180. http://dx.doi.org/10.5140/JASS.2015.32.2.127 134 Ivan L. Andronov et al. Phenomenological Modeling of VSX J1802

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