Physical properties of three short period close binaries: kic 2715417, kic 6050116 and kic 6287172
aa r X i v : . [ a s t r o - ph . S R ] J u l Research in Astronomy and Astrophysics manuscript no.(L A TEX: ms0007.tex; printed on July 3, 2018; 0:38)
PHYSICAL PROPERTIES OF THREE SHORT PERIOD CLOSEBINARIES: KIC 2715417, KIC 6050116 AND KIC 6287172 ∗ M. A. NegmEldin , A. Essam , Shahinaz M. Yousef Astronomy,Space Science and Meteorology Department, Faculty of Science, Cairo University, Cairo,12613, Egypt;
[email protected] Department of Astronomy, National Research Institute of Astronomy and Geophysics, Helwan, 11421,Egypt
Received 2018 January 11; accepted 2018 June 13
Abstract
We present the physical parameters of three short period close binaries usingthe observed data from Kepler Space Telescope. All of these observations were taken ina single bandpass (which approximates the Johnson V -band). Our three systems are KIC2715417, KIC 6050116 and KIC 6287172. The first system KIC 2715417 is considered asemi-detached system with the secondary component filling the Roche lobe. The second sys-tem KIC 6050116 is an overcontact system, while the third system KIC 6287172 belongs toellipsoidal variables (ELV) as deduced from the Roche lobe geometry. For photometric anal-ysis, we used PHOEBE software package which is based on the Wilson & Devinney code.Due to lack of spectroscopic data, the photometric mass ratios are determined from the anal-yses of the light curves using the Q -search method. The absolute parameters are determinedusing three different methods (Harmanec, Maceroni & Van’tVeer and Gazeas & Niarchos). Key words: stars:binaries:eclipsing — stars: fundamental parameters — stars: luminosityfunction, mass function — stars: individuals (KIC 2715417, KIC 6050116 and KIC 6287172)
We have many types of close binary systems as a result of close binary evolution. These types can beclassified as follows:The type of Semi-Detached systems, with the surfaces of the low mass components, are found in contactwith the inner Lagrangian surfaces (the critical potential surfaces) and the surfaces of the more massivecomponents are bounded within a separate equipotential surface. The Beta Lyrae systems, where one ofthe two components has become a giant or supergiant in the course of its evolution, the matter can beescaped from the critical potential surface at which the gravitation at their surface is weak to transfer into ∗ Discovered by Kepler Space Telescope
M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries the other component. W UMa type or late-type of contact systems have both of their components fillingthe inner Lagrangian surfaces and share the common envelope. Ellipsoidal variables can be described asvery close binaries with the two components are non-spherical shapes (ellipsoidal shapes) due to the mutualgravitation. The binary systems can be considered as close binary for the following reasons:The close binaries have short periods. The distance separating the two components is comparable to theirsize. The two components are close enough that their shapes are distorted by mutual gravitation forces tonon-spherical shapes such as in the type of ellipsoidal variables. The surface of both components overflowtheir critical Lagrangian surfaces and share a common envelope such as in the case of overcontact type.When one of its two components become giant or supergiant in the final stages of its evolution, matter mayfreely flow from one component to the other as in the case of Beta Lyrae type.
Kepler is a space telescope launched by NASA on March 7, 2009 to discover extra solar planets orbitingaround other stars in the field of Cygnus constellation. Kepler has a primary mirror equal to 1.4 metersin diameter. The FOV of Kepler spacecraft has a 105 deg . The photometer is composed of an array of42 CCDs, each CCD has 2200x1024 pixels . The Keplers catalogue ID, coordinates, V magnitude, colorexcess and the interstellar extinction of the target objects are listed in Table 1Table 1: Catalogue ID, Coordinates, and V Magnitude Kepler ′ s Catalogue ID α δ V mag E(B-V) A ν ( hh : mm : ss.ss ) ( ± dd : mm : ss.ss ) KIC 2715417 19:27:52.565 +37:55:39.97 14.070 0.063 0.427KIC 6050116 19:36:33.113 +41:20:22.56 14.258 0.059 0.369KIC 6287172 19:29:59.69 +41:37:45.0 12.714 0.114 0.339
From the Kepler data archive , we have collected the observations of the light curves for the abovethree systems. Observations of the KIC 2715417 system started at 2454964.51259 JD and ended at2456390.95883 JD. The light curve of these data is shown in Figure 1. In the case of KIC 6050116 system,observations started at 2454964.51229 JD and ended at 2456390.95881 JD and the corresponding lightcurve is shown in Figure 2. For the system KIC 6287172, the observations started at 2454953.53905 JDand ended at 2456390.95893 JD. The corresponding light curve of this system is shown in Figure 3. http://keplerebs.villanova.edu/ . A. NegmEldin et al. : Physical properties of three short period close binaries 3 Fig. 1: Light Curve of the system KIC 2715417.Fig. 2: Light Curve of the system KIC 6050116.Fig. 3: Light Curve of the system KIC 6287172.
M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
In the case of the eclipsing binary system KIC 2715417 and from the Kepler Input Catalog , we foundthat the effective temperature of the primary component is ◦ K, the Kepler magnitude is 14.070 mag ,metallicity = -0.458, log G ( log Surface gravity) = 4.830 and the color Excess reddening E ( B − V ) =0.063. It is considered a semi-detached binary with the secondary component filling the Roche lobe. Therepresented ephemeris of the system can be written as: HJD ( M inI ) = 2454964 . . × E (1)Where HJD (Min I) represents the minima epoch times in Heliocentric Julian Date and E is the integernumber of cycles.For the eclipsing binary system KIC 6050116 and from the Kepler Input Catalog, it is found that the ef-fective temperature of the primary component is ◦ K, the Kepler magnitude equals 14.25 mag , Metallicity(Solar= 0.0134) =-0.849, log G (
Log surface gravity) = 4.524 and color Excess reddening E ( B − V ) =0.059, it is an overcontact system. The represented ephemeris of the system can be written as: HJD ( M inI ) = 2454964 . . × E (2)Using the Same Kepler Input Catalog for the eclipsing binary system KIC 6287172, we also found that theeffective temperature of the primary component is ◦ K and the Kepler magnitude is equal to 12.714 mag and Metallicity (Solar=0.0134) = -0.470 and log G = 4.286 and color Excess reddening E ( B − V ) = 0.114,it is considered an ellipsoidal variable ( ELV ). The represented ephemeris of the system can be written as:
HJD ( M inI ) = 2454953 . . × E (3) For the three systems (KIC 2715417, KIC 6050116, and KIC 6287172) it appears that their temperaturesare below ◦ K which indicates that they are convective, hence the gravity darkening for the convectivestars g = g = 0 . ( ? ). The bolometric albedo A = A = 0 . ( ? ) and the limb darkening coefficientswere adopted from ? based on the linear cosine law model.With the lacking of any spectroscopic data for these three systems, it is difficult to determine the massratio accurately. Thus, we used the Q -search method as discussed in ? to determine the approximate valueof the photometric mass ratios for these systems. We have adjusted the PHOEBE software model of un-constrained binary system to estimate a set of parameters that represent the observed light curve. In orderto determine the photometric mass ratio, we used a series of mass ratios values ranging from 0.1 to 1.0in steps of 0.1 after that we took the three values of mass ratio corresponding to the minimum values ofthe residuals. Then, we used the same technique but with steps of 0.01. For each value of mass ratio, weobtained the sum of the squared deviations (residuals) from the fitting solutions of the light curve modeling.Figure 4 shows that the minimum occurs at value q= 0.29 for the system KIC 2715417. Figure 5 shows thatthe minimum occurs at value q=0.57 for the system KIC 6050116. Finally Figure 6 shows that the minimumoccurs at value q=0.61 for the system KIC 6287172 which is not to be trusted due to very small inclinationof (ELV) Type. Rather, the models offered here are the best that can be done with the available data. http://archive.stsci.edu/kepler/keplerfov/search.php . A. NegmEldin et al. : Physical properties of three short period close binaries 5 ( o - c ) q KIC 6050116
Fig. 4: The Q -search diagram for the system KIC 2715417. ( o - c ) q KIC 2715417
Fig. 5: The Q -search diagram for the system KIC 6050116. ( o - c ) q KIC 6287172
Fig. 6: The Q -search diagram for the system KIC 6287172. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Fig. 6: The Q -search diagram for the system KIC 6287172. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
The light curve of the system KIC 2715417 in the V -band as shown in Figure 1 has been analyzed using thePHOEBE package, version 0.31a ( ? ) which is based on the code of ? . From the Kepler Input Catalogue,it is found that the corresponding temperature T = 5189 ◦ K. First, from the PHOEBE software we haveused the model of unconstrained binary system to find approximate values of the parameters that representthe observed light curve. The data obtained from the PHOEBE model of unconstrained binary system,were analyzed to find the fillout factor (f) using the software Binary Maker 3 (BM3). This software is amodification of the parameter defined by ? to specify equipotentials for contact, overcontact and detachedsystems after specifying the mass ratio. In the case of detached type, the fill-out is given by formula (A).For overcontact, the fill-out is given by formula (B): f = Ω inner Ω − , f or Ω inner < Ω (
Detached ) ( A ) f = Ω inner − ΩΩ inner − Ω outer , f or Ω inner > Ω (
Overcontact ) ( B ) Thus, the fill-out factor (f) for detached stars will lie between ( − < f < . The fill-out factor (f) forovercontact systems will lie between (0 < f < . In the case of the contact system, the fill-out factors oftwo stars equals zero ( f = 0) . When (f) is near zero, the system can be described as a near contact system(e.g. the smaller star has a fill-out factor ( f ) = 0 . and the larger star has ( f ) = − . ).In our case, we found the fill-out factor of the primary component f = − . , and the secondarycomponent has fill-out factor f = 0 . . According to the above mentioned rules, the best model describ-ing the binary system KIC 2715417 is a near contact system. The best photometric fitting has been reachedafter several runs, which shows that the primary component is massive and hotter than the secondary one,with a temperature difference of about ◦ K.The orbital and physical parameters of the system KIC 2715417 are listed in Table 2. Figure 7 displaysthe observed light curve for the interval 2454964.51259 JD - 2454972.99293 JD, together with the syntheticcurve in the V -band while Figure 8 displays the light curve residual error for different phases. According tothe effective temperature of both the primary and secondary components of the system KIC 2715417 andfrom the calibration of Morgan-Keenan (MK) spectral types for the main sequence class( ? ), the spectraltypes are nearest to K and K respectively.Using the orbital and physical parameters listed in Table 2 with the Binary Maker 3 software , wepresent the shape of the system KIC 2715417 at phases 0.0, 0.25, 0.5, and 0.75 in Figure 9. Also we presentthe Roche lobe geometry of the system in Figure 10. . A. NegmEldin et al. : Physical properties of three short period close binaries 7 Table 2:
The Orbital and Physical Parameters of the system KIC 2715417Parameter Value Parameter ValueEpoch 2454964.666158 X X . ◦ ± . g g ( T ) 5189 ◦ K ( F ixed ) A ( T ) 4711 ◦ ◦ K A (Ω ) 2 . ± . L L + L . ± . PHSV (Ω ) ± L L + L ± ± Residual P ( o − c ) ( f ) - 0.1150 ( f ) r ( back ) ± r ( back ) ± r ( side ) ± r ( side ) ± r ( pole ) ± r ( pole ) ± r ( point ) ± r ( point ) ± r ) 0.4616 ± r ( Mean ) ± ± ± ◦ ± ◦ ± Longitude ( λ ) ◦ ±
12 87 ◦ ± Co-Latitude ( φ ) ◦ ± ◦ ± -1.0 -0.5 0.0 0.5 1.00.901.05 KIC 2715417
OBSERVED DATA SYNTHETIC DATA F l u x Phase
Fig. 7: Light curve in V filter showing the coherence betweenobserved and synthetic curves. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Fig. 7: Light curve in V filter showing the coherence betweenobserved and synthetic curves. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Fig. 8: Light curve residual error for different phases for thesystem KIC 2715417. (a) phase 0.25 (b) phase 0.75(c) phase 0.0 (d) phase 0.5
Fig. 9: The shape of the system KIC 2715417 at phases 0.0, 0.25, 0.5, and 0.75 .Fig. 10: Roche Lobe Geometry of the system KIC 2715417. . A. NegmEldin et al. : Physical properties of three short period close binaries 9
Because of the O’Connell effect ( ? ), we applied the cool spot on the surface of star to achieve the bestfit. We have one cool spot at each of the primary and secondary components for the system KIC 2715417as shown in Figure 9. Table 2 gives the spot parameters where the spot of the secondary component has thelargest radius and the coolest temperature than the primary component.The temperature of the primary component is ◦ K while the temperature factor of the cool spot is . ± . . This means that the primary cool spot has a temperature of ◦ ± ◦ K .On the other hand,the temperature of the secondary component is ◦ + 14 ◦ K and the temperature factor of the cool spotis . ± . . Thus the temperature of the cool spot on secondary component is ◦ ± ◦ K . The light curve of the system KIC 6050116 in the V -band as shown in Figure 2 has been analyzed usingthe PHOEBE package. It is found that the corresponding temperature T = 4569 ◦ K from the Kepler InputCatalogue. With the PHOEBE software we have used the model of unconstrained binary system to estimatea set of parameters that represent the observed light curve. The best photometric fitting has been reachedafter several runs.From the analyses of the software Binary Maker 3, the type of binary system is considered to be over-contact, since the fill-out factors of the first component is f = 0.0704 and the secondary component is f =0.0704. The fill-out factor is equal for the two components of contact and overcontact binaries as the twostars are in contact or overcontact with each other. They must have the same gravitational equipotential (Ω) otherwise gas will literally leak away from the system until it reaches equilibrium.The primary component is massive and hotter than the secondary one, with a temperature difference ofabout ◦ K.The orbital and physical parameters of the system KIC 6050116 are listed in Table 3. Figure 11displays the observed light curve for the interval 2454964.51229 JD - 2454974.48433 JD, together with thesynthetic curve in the V band while Figure 12 displays the light curve residual error for different phases.According to the effective temperature of both the primary and secondary components of the systemKIC 6050116 and from the calibration of Morgan-Keenan (MK) spectral types for the main sequence class,the spectral types are nearest to K for both components. Using the orbital and physical parameters listedin Table 3 with the BM3 program, we present the shape of the system KIC 6050116 at phases 0.0, 0.25, 0.5,and 0.75 in Figure 13. We also present the Roche lobe geometry of the system in Figure 14. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Because of the O’Connell effect ( ? ), we applied the cool spot on the surface of star to achieve the bestfit. We have one cool spot at each of the primary and secondary components for the system KIC 2715417as shown in Figure 9. Table 2 gives the spot parameters where the spot of the secondary component has thelargest radius and the coolest temperature than the primary component.The temperature of the primary component is ◦ K while the temperature factor of the cool spot is . ± . . This means that the primary cool spot has a temperature of ◦ ± ◦ K .On the other hand,the temperature of the secondary component is ◦ + 14 ◦ K and the temperature factor of the cool spotis . ± . . Thus the temperature of the cool spot on secondary component is ◦ ± ◦ K . The light curve of the system KIC 6050116 in the V -band as shown in Figure 2 has been analyzed usingthe PHOEBE package. It is found that the corresponding temperature T = 4569 ◦ K from the Kepler InputCatalogue. With the PHOEBE software we have used the model of unconstrained binary system to estimatea set of parameters that represent the observed light curve. The best photometric fitting has been reachedafter several runs.From the analyses of the software Binary Maker 3, the type of binary system is considered to be over-contact, since the fill-out factors of the first component is f = 0.0704 and the secondary component is f =0.0704. The fill-out factor is equal for the two components of contact and overcontact binaries as the twostars are in contact or overcontact with each other. They must have the same gravitational equipotential (Ω) otherwise gas will literally leak away from the system until it reaches equilibrium.The primary component is massive and hotter than the secondary one, with a temperature difference ofabout ◦ K.The orbital and physical parameters of the system KIC 6050116 are listed in Table 3. Figure 11displays the observed light curve for the interval 2454964.51229 JD - 2454974.48433 JD, together with thesynthetic curve in the V band while Figure 12 displays the light curve residual error for different phases.According to the effective temperature of both the primary and secondary components of the systemKIC 6050116 and from the calibration of Morgan-Keenan (MK) spectral types for the main sequence class,the spectral types are nearest to K for both components. Using the orbital and physical parameters listedin Table 3 with the BM3 program, we present the shape of the system KIC 6050116 at phases 0.0, 0.25, 0.5,and 0.75 in Figure 13. We also present the Roche lobe geometry of the system in Figure 14. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Table 3:
The Orbital and Physical Parameters of the system KIC 6050116Parameter Value Parameter ValueEpoch 2454964.708006 X X . ◦ ± . g g ( T ) 4569 ◦ K ( F ixed ) A ( T ) 4485 ◦ ◦ K A (Ω ) 2 . ± . L L + L . ± . PHSV (Ω ) ± L L + L ± ± P ( o − c ) ( f ) ( f ) r ( back ) ± r ( back ) ± r ( side ) ± r ( side ) ± r ( pole ) ± r ( pole ) ± r ( point ) ± r ( point ) ± r ( Mean ) ± r ( Mean ) ± ± ◦ ± Longitude ( λ ) ◦ ± Co-Latitude ( φ ) ◦ ± -1.0 -0.5 0.0 0.5 1.00.61.2 KIC 6050116
OBSERVED DATA SYNTHETIC DATA F l u x Phase
Fig. 11: Light curve in V filter showing the coherence be-tween observed and synthetic curves. . A. NegmEldin et al. : Physical properties of three short period close binaries 11 Fig. 12: Light curve residual error for different phases for thesystem KIC 6050116. (a) phase 0.25 (b) phase 0.75(c) phase 0.0 (d) phase 0.5
Fig. 13: The shape of the system KIC 6050116 at phases 0.0, 0.25, 0.5, and 0.75 .Fig. 14: Roche Lobe Geometry of the system KIC 6050116. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Fig. 13: The shape of the system KIC 6050116 at phases 0.0, 0.25, 0.5, and 0.75 .Fig. 14: Roche Lobe Geometry of the system KIC 6050116. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
We have also one cool spot at secondary component for the system KIC 6050116 as shown in Figure 13.Table 3 gives the temperature of the secondary component as ◦ ± ◦ K and the temperature factor ofthe cool spot is . ± . . Thus the temperature of the cool spot of secondary component is ◦ ± ◦ K. The light curve of the system KIC 6287172 in the V -band as shown in Figure 3 has been analyzed usingthe PHOEBE package. We found that the corresponding temperature T = 6646 ◦ K from the Kepler InputCatalogue. With the PHOEBE software we have used the model of unconstrained binary to determineapproximate values of the physical and geometrical parameters that represent the observed light curve.After the best photometric fitting has been reached, we have used the data obtained from PHOEBE softwareinserted in the other software Binary Maker 3 to find the type of binary system from the fill-out factors ofthe two components.We found that the fill-out parameters of the primary star is f = 0.432657 and for the secondary star is f = 0.432657 which is considered a non-eclipsing overcontact type or Ellipsoidal Variables ( ELV ) . Theprimary component is massive and hotter than the secondary one, with a temperature difference of about ◦ K. The orbital and physical parameters of the system KIC 6287172 are listed in Table 4. Figure 15displays the observed light curve for the interval 2454953.53905 JD - 2454967.31191 JD, together with thesynthetic curve in the V -band, while Figure 16 represents the light curve residual error for different phases.According to the effective temperature of both the primary and secondary components of the systemKIC 6287172 and from the calibration of Morgan-Keenan (MK) spectral types for the main sequence class,the spectral types are nearest to F for both components. Using the orbital and physical parameters listed inTable 4 with the BM3 program, we present the shape of the system KIC 6287172 at phases 0.0, 0.25, 0.5,and 0.75 in Figure 17. We present the Roche geometry of the system in Figure 18. . A. NegmEldin et al. : Physical properties of three short period close binaries 13 Table 4:
The Orbital and Physical Parameters of the system KIC 6287172Parameter Value Parameter ValueEpoch 2454953.651911 X X . ◦ ± . g g ( T ) 6646 ◦ K ( F ixed ) A ( T ) 6624 ◦ ◦ K A (Ω ) 2 . ± . L L + L . ± . PHSV (Ω ) ± L L + L ± ± P ( o − c ) ( f ) ( f ) r ( back ) ± r ( back ) ± r ( side ) ± r ( side ) ± r ( pole ) ± r ( pole ) ± r ( point ) ± r ( point ) ± r ( Mean ) ± r ( Mean ) ± -1.0 -0.5 0.0 0.5 1.00.9961.002 KIC 6287172
OBSERVED DATA SYNTHETIC DATA F l u x Phase
Fig. 15: Light curve in V filter showing the coherence be-tween observed and synthetic curves. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Fig. 15: Light curve in V filter showing the coherence be-tween observed and synthetic curves. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Fig. 16: Light curve residual error for different phases. (a) phase 0.25 (b) phase 0.75(c) phase 0.0 (d) phase 0.5
Fig. 17: The shape of the system KIC 6287172 at phases 0.0, 0.25, 0.5, and 0.75 . . A. NegmEldin et al. : Physical properties of three short period close binaries 15
Fig. 18: Roche Lobe Geometry of the system KIC 6287172. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Fig. 18: Roche Lobe Geometry of the system KIC 6287172. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
The physical parameters: effective temperature T eff , absolute magnitude M V , relative radius RR ⊙ , relativeluminosity LL ⊙ , and surface gravity g, were calculated using the equations of stellar structure. The adoptedconstants which we used are T eff ⊙ = 5772 ◦ K , log g ⊙ = 4 . , and M bol ⊙ = 4 . .We have three methods to determine only the primary masses for each type of binary. These threemethods are: ? , ? and ? . We compute the physical parameters according to the empirical relation derived by ? from his work aboutthe stellar masses and radii based on modern binary data. These relation obtained by the least-square fitvia Chebychev polynomials for the data introduced by ? . Harmanecs method is the mass - temperaturerelation used to determine the masses of the primary and secondary components M and M for detachedbinaries. However, in the case of overcontact binaries and any other type of binaries, it is only applicablefor determination of the primary component M which is slightly affected by thermal contact with thesecondary low mass stars. If we assumed that the effect of temperature of the secondary component on theprimary is negligible because the primary has higher temperature. Thus the primary can be deduced fromthe previous relations with good approximation. The mass of the secondary component cant accuratelybe calculated using this method because the secondary component is greatly affected by thermal contactwith the primary hotter star. The mass of primary stars according to Harmanec were calculated from thefollowing equation: log M M ⊙ = ((1 . X − . X + 88 . X − . (4)Where X = log( T eff ) Equation 4 is applicable for . ≥ log( T eff ) ≥ . The mass of secondary component is determined from the photometric mass ratio, ( q = M M ) The physical parameters can be determined from the relation between the total mass ( M T ) and total lu-minosity ( L T ) of the binary systems from ? taking the assumption that the interaction between the twocomponents of the binary system is not affecting the total luminosity of the system, therefore the commonenvelope radiates the luminosity given by the sum of the internal luminosities In other words, the total lu-minosity is the same for the binary system. Due to the pervious assumption, this method can be applicablefor any type of binary system such as detached, over contact or ellipsoidal to determine the total mass ofthe binary system. Using the value of mass ratio for each binary, we can get the individual masses M and M .for the binary system as shown in the following equations: log( L T ) = 23 log( M T ) + c (5)Where c = log h cP ( ) ( r T + r T ) i In the same paper ( ? ) at equation (2), the constant c was written as: . A. NegmEldin et al. : Physical properties of three short period close binaries 17 c = (4 π ) G σ But we note that, there is a mistake in the formula of c and it should be written as: c = ( π ) G σ Using the evolutionary tracks for non-rotating model which have been computed by ? for zero age mainsequence stars (ZAMS) with metallicity Z = 0 . . We correlate the total luminosity and total mass fromthe fitting data of the binary system of stars located on the ZAMS as follows: log( L T ) = 4 . M T ) − . (6)From the intersections between the straight lines of binary systems represented by equations (5) and(6) as shown in Figures 19, 23 and 27, we can deduced the total mass ( M T ) of binary system. Fromthe knowledge of the mass ratio ( q ) and total mass ( M T ) of the binary system, the individual masses ( M and M ) can readily be calculated from equation 7. where M = M T q + 1 , M = M × q = M T × q ( q + 1) (7) This is the period - mass relation used to determine the mass of the primary component. It is only applicablefor short period eclipsing binary systems such as overcontact or near contact types with orbital period log
P < − . and is not included Ellipsoidal variables (non-eclipsing binaries) according to ? and ? . Themass of the primary star ( M ) can be obtained from the following expression: log( M ) = (0 . ± . P ) + (0 . ± . (8)Where P is the orbital period of the binary system of W UMa-type. All of these three methods areused to calculate only the mass of the primary component of the binary system. The mass of the secondarycomponent of binary system is found from the photometric mass ratio ( q = M M ) and the primary mass.The photometric mass ratio of the contact type is more precise than undercontact type such as the detachedbinary. All of these three methods are used to calculate only the individual masses of the binary systems.From the Kepler third law, the semi-major axis of the orbit for binary system was determined using thefollowing equation: a = r G × ( M + M ) × P π (9)Where G is the gravitational constant ( G = 6 . × − kg − m s − ) . Knowing the semi-majoraxis ( a ) thus, we can calculate the radius of primary and secondary stars according to the relation R j = r j a , where j can take 1 for primary component or 2 for secondary component, r is the mean fractional radii ofthe star. The luminosity of the primary and secondary stars were calculated using the direct equation: L j L ⊙ = ( R j R ⊙ ) × ( T j T ⊙ ) (10)The bolometric magnitude of the primary and secondary components of binary systems is calculatedusing the following equation: M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
P < − . and is not included Ellipsoidal variables (non-eclipsing binaries) according to ? and ? . Themass of the primary star ( M ) can be obtained from the following expression: log( M ) = (0 . ± . P ) + (0 . ± . (8)Where P is the orbital period of the binary system of W UMa-type. All of these three methods areused to calculate only the mass of the primary component of the binary system. The mass of the secondarycomponent of binary system is found from the photometric mass ratio ( q = M M ) and the primary mass.The photometric mass ratio of the contact type is more precise than undercontact type such as the detachedbinary. All of these three methods are used to calculate only the individual masses of the binary systems.From the Kepler third law, the semi-major axis of the orbit for binary system was determined using thefollowing equation: a = r G × ( M + M ) × P π (9)Where G is the gravitational constant ( G = 6 . × − kg − m s − ) . Knowing the semi-majoraxis ( a ) thus, we can calculate the radius of primary and secondary stars according to the relation R j = r j a , where j can take 1 for primary component or 2 for secondary component, r is the mean fractional radii ofthe star. The luminosity of the primary and secondary stars were calculated using the direct equation: L j L ⊙ = ( R j R ⊙ ) × ( T j T ⊙ ) (10)The bolometric magnitude of the primary and secondary components of binary systems is calculatedusing the following equation: M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries M bolj = M bol ⊙ − . L j L ⊙ (11)The absolute magnitude, M V , is related to the bolometric magnitude, M bol , via: M V = M bol − BC (12)Where BC is the bolometric correction given by ? : BC = − . T ) − + 13 . T ) − − . T ) − − .
901 [log ( T ) − − . (13)We used the relation in equation 12 to calculate the absolute magnitude for all three systems. Using theinterstellar extinction value ( A ν ) corresponding to the equatorial coordinates J2000 obtained from ? for the V -band at effective wavelength =5517 ˚A . Finally we found that, for system KIC 2715417 ( A ν = 0 . ),for the System 6050116 ( A ν = 0 . ) and for the system 6287172 ( A ν = 0 . ). Using the distancemodulus relation to calculate the distance of the three systems as follows: D = 10 . m − M +5 − A ν ) (14)Where the distance is in parsec (pc), m represents the apparent magnitude and M V is the absolutemagnitude. All the absolute parameters (mass, semi major axis, radius, luminosity, bolometric magnitudeand the distance) of the three systems which have been calculated are listed in Tables 5, 6, and 7.Table 5: The absolute parameters of the system KIC 2715417Parameter Harmanec Maceroni Gazeas M ( M ⊙ ) ± ± ± M ( M ⊙ ) ± ± ± a ( R ⊙ ) ± ± ± R ( R ⊙ ) ± ± ± R ( R ⊙ ) ± ± ± L ( L ⊙ ) ± ± ± L ( L ⊙ ) ± ± ± M bol ± ± ± M bol ± ± ± D ( pc ) ± ± ± . A. NegmEldin et al. : Physical properties of three short period close binaries 19 Table 6:
The absolute parameters of the system KIC 6050116Parameter Harmanec Maceroni Gazeas M ( M ⊙ ) ± ± ± M ( M ⊙ ) ± ± ± a ( R ⊙ ) ± ± ± R ( R ⊙ ) ± ± ± R ( R ⊙ ) ± ± ± L ( L ⊙ ) ± ± ± L ( L ⊙ ) ± ± ± M bol ± ± ± M bol ± ± ± D ( pc ) ± ± ± Table 7:
The absolute parameters of the system KIC 6287172Parameter Harmanec Maceroni M ( M ⊙ ) ± ± M ( M ⊙ ) ± ± a ( R ⊙ ) ± ± R ( R ⊙ ) ± ± R ( R ⊙ ) ± ± L ( L ⊙ ) ± ± L ( L ⊙ ) ± ± M bol ± ± M bol ± ± D ( pc ) ±
12 387 ± Notes: The system KIC 6287172 belongs to ELV type. Gazeas & Niarchos (2006) Method is not included inTable 7 because it is only applicable to W UMa-type.
In order to study the evolutionary state of the three systems (KIC 2715417, KIC 6050116, and KIC6287172), we have plotted the physical parameters listed in Tables 5, 6 and 7 of the components of thethree systems on the H − R , M − R and M − L diagrams in Figures 19 -30. Using the evolutionary tracksfor non-rotating model which have been computed by ? for both zero age main sequence stars ( ZAM S ) and terminal age main sequence stars ( T AM S ) with metallicity Z = 0 . (Solar metallicity). As it isclear from Figures 19, 23 and 27 the straight line (Binary System) represents the relation between the totalluminosity and the total mass of the binary system as written in equation 5. We can determine the totalmass from the intersection between the two lines, Binary System and ZAM S . In Figures 20, 24 and 28 theprimary components of the three systems are located on
ZAM S , however the secondary components of thethree systems are located above
ZAM S . This can be attributed to the rise of temperature of the secondarycomponent due to thermal contact between the two components of the binary systems and consequently thevalue of luminosity will rise without any change in mass. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
ZAM S . This can be attributed to the rise of temperature of the secondarycomponent due to thermal contact between the two components of the binary systems and consequently thevalue of luminosity will rise without any change in mass. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
In Figures 21, 25 and 29 the primary components of the three systems are located on
ZAM S , but forthe secondary components of the three systems are located below
ZAM S . That may be due to thermal con-tact. In the figures 22 and 26 the primary components of the two systems (KIC 2715417, KIC 6050116) arelocated on
ZAM S , but the secondary component of these two systems are located faraway from
ZAM S .In Figure 30 we note that the primary component of the system KIC 6287172 is located above
ZAM S withadditional mass accumulated due to matter transfer from the secondary component. This system belongs tothe type of Ellipsoidal Variables ( ELV ) . The type of ELV can be described as very close non-eclipsingbinaries whose two components are of non-spherical shape due to the mutual gravitation. Their light varia-tion can change as seen from the Earth as a result of the rotation of the two components where their surfacesfacing the observers are changing. -0.4 -0.2 0.0 0.2 0.4 0.6-2-1012
Binary System Zams Tams
Log L / Lo ( Lo ) Log M/Mo (Mo)
KIC 2715417
Fig. 19: L-M of Binary System KIC 2715417. -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-3-2-10123
Primary Star Secondary Star Zams Tams
Log L / Lo ( Lo ) Log M/Mo (Mo)
KIC 2715417
Fig. 20: L-M relation of the system KIC 2715417. . A. NegmEldin et al. : Physical properties of three short period close binaries 21
Primary Star Secondary Star Zams Tams
Log L / Lo ( Lo ) Log T (K)
KIC 2715417
Fig. 21: L-T relation of the system KIC 2715417. -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6-0.4-0.20.00.20.40.60.8
Primary Star Secondary Star Zams Tams
Log M / M o ( M o ) Log R/Ro (Ro)
KIC 2715417
Fig. 22: M-R relation of the system KIC 2715417. -0.4 -0.2 0.0 0.2 0.4 0.6-2-1012
Binary System Zams Tams
Log L / Lo ( Lo ) Log M/Mo (Mo)
KIC 6050116
Fig. 23: L-M of Binary system KIC 6050116. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Fig. 23: L-M of Binary system KIC 6050116. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-3-2-10123
Primary Star Secondary Star Zams Tams
Log L / Lo ( Lo ) Log M/Mo (Mo)
KIC 6287172
Fig. 24: L-M relation of the system KIC 6050116.
Primary Star Secondary Star Zams Tams
Log L / Lo ( Lo ) Log T (K)
KIC 6050116
Fig. 25: L-T relation of the system KIC 6050116. -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6-0.4-0.20.00.20.40.60.8
Primary Star Secondary Star Zams Tams
Log M / M o ( M o ) Log R/Ro (Ro)
KIC 6050116
Fig. 26: M-R relation of the system KIC 6050116. . A. NegmEldin et al. : Physical properties of three short period close binaries 23 -0.2 0.0 0.2 0.4 0.6-101
Binary System Zams Tams
Log LT / Lo ( Lo ) Log MT/Mo (Mo)
KIC 6287172
Fig. 27: L-M of Binary system KIC 6287172. -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-3-2-10123
Primary Star Secondary Star Zams Tams
Log L / Lo ( Lo ) Log M/Mo (Mo)
KIC 6287172
Fig. 28: L-M relation of the system KIC 6287172.
Primary Star Secondary Star Zams Tams
Log L / Lo ( Lo ) Log T (K)
KIC 6287172
Fig. 29: L-T relation of the system KIC 6287172. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries
Fig. 29: L-T relation of the system KIC 6287172. M. A. NegmEldin et al. : Physical properties of three short period close binaries: Physical properties of three short period close binaries -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6-0.4-0.20.00.20.40.60.8
Primary Star Secondary Star Zams Tams
Log M / M o ( M o ) Log R/Ro (Ro)
KIC 6287172
Fig. 30: M-R relation of the system KIC 6287172. . A. NegmEldin et al. : Physical properties of three short period close binaries 25
We presented the physical properties of the systems KIC 2715417, KIC 6050116 and KIC 6287172 derivedfrom the analyses of the Kepler data according to the above results, we arrived to following conclusions:The system KIC 2715417 is considered a near contact system. The second system KIC 6050116 is anovercontact system, while the system KIC 6287172 belongs to ellipsoidal variable ( ELV ) or non-eclipsingovercontact type. The effective temperature of the primary star for all the three systems are slightly higherthan the secondary component. We found that the primary components for our systems are located on ZAM S line while the secondary components are located away from
ZAM S . This can be due to theheating of the secondary components of the close binaries by thermal contact.We used the three mentioned methods to determine the masses of the primary components of the binarysystems and calculated the masses of the secondary components from the photometric mass ratios obtainedfrom the Q -search method which gives good values compared with the spectroscopic mass ratio particularlyfor contact type. This enabled us to obtain the required comparison of the systems under considerations. Itis found that the method of ? gives the best results for our close binary systems which have thermal contactbetween the two components.For the system KIC 2715417 the primary component has a mass of . ± . M ⊙ while the sec-ondary component has a mass . ± . M ⊙ . The distance equal ± P c . Each component has acool spot while the spectral type is K and K for the primary and secondary components respectively. Thesystem KIC 6050116 consists of two components with masses . ± . M ⊙ and . ± . M ⊙ for the primary and secondary components respectively. This system is located at a distance ± P c .Only the primary component has a cool spot, while the spectral type is K for both two components of thesystem.The third binary system KIC 6287172 has larger masses . ± . M ⊙ and . ± . M ⊙ forthe primary and secondary components respectively. The system is further away than the previous two sys-tems at a distance of ± P c , while the spectral type is F for both primary and secondary componentswithout any spot for both components. The difference between two maxima in any light curve indicatesthe presence of spot on the surface of the component which is called O’Connell effect ( ? ). The presence ofcool spots on the primary and secondary components of the KIC 2715417 and the primary component ofthe system KIC 6050116 indicates the existence of magnetic field inside the spots which inhibit convectionsimilar to the case of sunspots. Acknowledgements
In this paper we collected all the data from Kepler mission. Kepler was selected as the10th mission of the Discovery Program. Also we used the Vizier database maintained at CDS, Strasbourg,France. This work is part of M.Sc. thesis submit to Cairo University by one of us (M. A. NegmEldin).
References
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