Physical properties of three short period close binaries: KIC 2715417, KIC 6050116 and KIC 6287172^*

Kepler is a space telescope launched by NASA on 2009 March 7 to discover extrasolar planets orbiting around other stars in the field of the constellation Cygnus. Kepler has a primary mirror 1.4 meters in diameter. The field of view of the Kepler spacecraft is 105 deg². The photometer is composed of an array of 42 CCDs and each CCD has 2200 × 1024 pixels¹. The Kepler catalog ID, coordinates, V magnitude, color excess and interstellar extinction of the target objects are listed in Table 1.

From the Kepler data archive², we have collected observations of the light curves for the above three systems. Observations of the KIC 2715417 system started at 2454964.51259 JD and ended at 2456390.95883 JD. The light curve of these data is shown in Figure 1.

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In the case of the KIC 6050116 system, observations started at 2454964.51229 JD and ended at 2456390.95881 JD, and the corresponding light curve is shown in Figure 2. For the system KIC 6287172, the observations started at 2454953.53905 JD and ended at 2456390.95893 JD. The corresponding light curve of this system is shown in Figure 3.

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In the case of the eclipsing binary system KIC 2715417 and from the Kepler Input Catalog³, we found that the effective temperature of the primary component is 5189 K, the Kepler magnitude is 14.070 mag, metallicity = −0.458, log₁₀ g (log₁₀ surface gravity) = 4.830 and color excess reddening E(B − V) = 0.063. It is considered a semi-detached binary with the secondary component filling its Roche lobe. The associated ephemeris of the system can be written as

$$\begin{eqnarray}&&{\rm{HJD}}\,({\rm{MinI}})=2454964.666158+0.2364399\times E,\end{eqnarray} \tag{ 1 }$$

where HJD (Min I) represents the minima epoch times in Heliocentric Julian Date and E is the integer number of cycles.

For the eclipsing binary system KIC 6050116 and from the Kepler Input Catalog, it is found that the effective temperature of the primary component is 4569K, the Kepler magnitude equals 14.25 mag, metallicity (solar = 0.0134) = −0.849, log₁₀ g (log₁₀ surface gravity) = 4.524 and color excess reddening E(B − V) = 0.059, indicating it is an overcontact system. The associated ephemeris of the system can be written as

$$\begin{eqnarray}&&{\rm{HJD}}\,({\rm{MinI}})=2454964.708006+0.2399081\times E.\end{eqnarray} \tag{ 2 }$$

Using the same Kepler Input Catalog for the eclipsing binary system KIC 6287172, we also found that the effective temperature of the primary component is 6646 K, the Kepler magnitude is equal to 12.714 mag, metallicity (solar = 0.0134) = −0.470, log₁₀ g = 4.286 and color excess reddening E(B − V) = 0.114. It can be considered an ELV. The associated ephemeris of the system can be written as

$$\begin{eqnarray}&&{\rm{HJD}}\,({\rm{MinI}})=2454953.651911+0.2038732\times E.\end{eqnarray} \tag{ 3 }$$

The light curve of the system KIC 2715417 in the V-band as shown in Figure 1 has been analyzed using the PHOEBE package, version 0.31a (Prsa & Zwitter 2005) which is based on the code of Wilson & Devinney (1971). From the Kepler Input Catalog, it is found that the corresponding temperature T₁ = 5189 K. First, from the PHOEBE software, we have used the model of the unconstrained binary system to find approximate values of the parameters that represent the observed light curve. Data obtained from the PHOEBE model of the unconstrained binary system were analyzed to find the fill-out factor (f) using the software Binary Maker 3 (BM3). This software is a modification of the parameter defined by Lucy & Wilson (1979) to specify equipotentials for contact, overcontact and detached systems after specifying themass ratio. In case of the detached type, the fill-out is given by formula (A). For overcontact, the fill-out is given by formula (B)

$$\begin{eqnarray*}&&\begin{array}{l}f=\frac{{\Omega }_{{\rm{inner}}}}{\Omega }-1,\,\,{\rm{for}}\,\,{\Omega }_{{\rm{inner}}}\lt \Omega \,({\rm{Detached}})\,\,({\rm{A}}),\\ f{\rm{=}}\frac{{\Omega }_{{\rm{inner}}}{\rm{-}}\Omega }{{\Omega }_{{\rm{inner}}}{\rm{-}}{\Omega }_{{\rm{outer}}}},\,\,{\rm{for}}\,\,{\Omega }_{{\rm{inner}}}\geqslant \Omega \,({\rm{Overcontact}})\,({\rm{B}}).\end{array}\end{eqnarray*}$$

Thus, the fill-out factor (f) for detached stars will lie between (−1 < f < 0). The fill-out factor (f) for overcontact systems will lie between (0 < f < 1). In the case of a contact system, the fill-out factors of two stars equal 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 (f₂) = 0.00 and the larger star has (f₁) = −0.02).

In our case, we found the fill-out factor of the primary component f₁ = − 0.1150 and the secondary component has fill-out factor f₂ = 0.0013. According to the above mentioned rules, the best model describing the binary system KIC 2715417 is a near contact system. The best photometric fitting has been reached after several runs, which shows that the primary component is more massive and hotter than the secondary one, with a temperature difference of about 478 K.

The orbital and physical parameters of the system KIC 2715417 are listed in Table 2. Figure 7 displays the observed light curve for the interval 2454964.51259 JD – 2454972.99293 JD, together with the synthetic curve in the V-band while Figure 8 displays the light curve residual error for different phases. According to the effective temperature of both the primary and secondary components of the system KIC 2715417 and from the calibration of Morgan-Keenan (MK) spectral types for main sequence stars (Drilling & Landolt 2002), the spectral types are nearest to K0 and K2 respectively.

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Using the orbital and physical parameters listed in Table 2 with the BM3 software⁴, we present the shape of the system KIC 2715417 at phases 0.0, 0.25, 0.5 and 0.75 in Figure 9. Also we present the Roche lobe geometry of the system in Figure 10.

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Because of the O'Connell effect (O'Connell 1951), we included cool spots on the surface of the star to achieve the best fit. We have one cool spot on both the primary and secondary components for the system KIC 2715417 as shown in Figure 9. Table 2 gives the spot parameters where the spot of the secondary component has a larger radius and cooler temperature than the primary component.

Table 2.  Orbital and Physical Parameters of the System KIC 2715417

Parameter	Value	Parameter	Value
Epoch	2454964.666158	X1	0.655
Period (d)	0.2364399	X2	0.711
Inclination (i)	56.56° ± 0.04°	g1	0.32
Mass ratio (q)	0.289 ± 0.002	g2	0.32
TAVH (T1)	5189 K (Fixed)	A1	0.50
TAVC (T2)	4711 ± 14K	A2	0.50
PHSV (Ω1)	2.7635 ± 0.0061	$\frac{{L}_{1}}{{L}_{1}+{L}_{2}}$	0.7909 ± 0.0016
PHSV (Ω2)	2.4416 ± 0.0032	$\frac{{L}_{1}}{{L}_{1}+{L}_{2}}$	0.2091 ± 0.0030
Phase Shift	0.00402 ± 0.00082	Residual ∑(O − C)2	0.0401
Fill-out factor (f1)	–0.1150	(f2)	0.0013
r1 (back)	0.4716 ± 0.0005	r2 (back)	0.3109 ± 0.0024
r1 (side)	0.4625 ± 0.0004	r2 (side)	0.2932 ± 0.0017
r1 (pole)	0.4505 ± 0.0003	r2 (pole)	0.2764 ± 0.0015
r1 (point)	0.4859 ± 0.0008	r2 (point)	0.3256 ± 0.0029
Mean fractional radii (r1)	0.4616 ± 0.0004	r2 (mean)	0.2935 ± 0.0019

Spot Parameters	Spot 1 (primary)	Spot 2 (secondary)
Temp Factor	0.86 ± 0.04	0.80 ± 0.05
Spot Radius	7° ± 2°	14° ± 3°
Longitude (λ)	74° ± 12°	87° ± 11°
Co-Latitude (φ)	122° ± 8°	46° ± 9°

The temperature of the primary component is 5189 K while the temperature factor of the cool spot is 0.86 ± 0.04. This means that the primary cool spot has a temperature of 4462 ± 208 K. On the other hand, the temperature of the secondary component is 4711 ± 14 K and the temperature factor of the cool spot is 0.80 ± 0.05. Thus, the temperature of the cool spot on the secondary component is 3769 ± 236 K.

The light curve of the system KIC 6050116 in the V-band as shown in Figure 2 has been analyzed using the PHOEBE package. It is found that the corresponding temperature T₁ = 4569 K from the Kepler Input Catalog. With the PHOEBE software we have used the model of an unconstrained binary system to estimate a set of parameters that represent the observed light curve. The best photometric fitting has been reached after several runs.

From the analyses of the BM3 software, the type of binary system is considered to be overcontact, since the fill-out factor of the first component is f₁ = 0.0704 and that of the secondary component is f₂ = 0.0704. The fillout factor is equal for the two components in contact and overcontact binaries as the two stars 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 more massive and hotter than the secondary one, with a temperature difference of about 84K. The orbital and physical parameters of the system KIC 6050116 are listed in Table 3.

Figure 11 displays the observed light curve for the interval 2454964.51229 JD – 2454974.48433 JD, together with the synthetic curve in the V band while Figure 12 depicts the light curve residual error for different phases.

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According to the effective temperatures of both the primary and secondary components of the system KIC 6050116 and from the calibration ofMK spectral types for main sequence stars, the spectral types are nearest to K5 for both components. Using the orbital and physical parameters listed in 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.

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Table 3.  Orbital and Physical Parameters of the System KIC 6050116

Parameter	Value	Parameter	Value
Epoch	2454964.708006	X1	0.787
Period (d)	0.2399081	X2	0.795
Inclination (i)	69.2° ± 0.7°	g1	0.32
Mass ratio (q)	0.573 ± 0.012	g2	0.32
TAVH (T1)	4569 K (Fixed)	A1	0.50
TAVC (T2)	4485 ± 14 K	A2	0.50
PHSV (Ω1)	2.9899 ± 0.0016	$\frac{{L}_{1}}{{L}_{1}+{L}_{2}}$	0.6504 ± 0.0020
PHSV (Ω2)	2.9899 ± 0.0016	$\frac{{L}_{1}}{{L}_{1}+{L}_{2}}$	0.3496 ± 0.0037
Phase Shift	0.00857 ± 0.00035	∑ (O − C)2	0.107112
(f1)	0.0704	(f2)	0.0704
r1 (back)	0.4618 ± 0.0040	r2 (back)	0.3637 ± 0.0084
r1 (side)	0.4315 ± 0.0025	r2 (side)	0.3291 ± 0.0062
r1 (pole)	0.4067 ± 0.0018	r2 (pole)	0.3144 ± 0.0053
r1 (point)	0.5570 ± 0.0021	r2 (point)	0.4430 ± 0.0021
r1 (mean)	0.4245 ± 0.0028	r2 (mean)	0.3337 ± 00045

Spot Parameters	Spot (secondary)
Temp Factor	0.76 ± 0.04
Spot Radius	10° ± 3°
Longitude (λ)	62° ± 7°
Co-Latitude (φ)	117° ± 9°

We also have one cool spot on the secondary component for the system KIC 6050116 as shown in Figure 13. Table 3 gives the temperature of the secondary component as 4485 ± 17K and the temperature factor of the cool spot is 0.76 ± 0.04. Thus, the temperature of the cool spot on secondary component is 3409 ± 179K.

The light curve of the system KIC 6287172 in the V- band as shown in Figure 3 has been analyzed using the PHOEBE package. We found that the corresponding temperature T₁ = 6646 K from the Kepler Input Catalog. With the PHOEBE software we have used the model of an unconstrained binary to determine approximate 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 the PHOEBE software input into the other software BM3 to find the type of binary system from the fill-out factors of the two components.

We found that the fill-out parameter of the primary star is f₁ = 0.432657 and that for the secondary star is f₂ = 0.432657, so this system is considered a noneclipsing overcontact type or ELV. The primary component is more massive and hotter than the secondary one, with a temperature difference of about 22K. The orbital and physical parameters of the system KIC 6287172 are listed in Table 4.

Figure 15 displays the observed light curve for the interval 2454953.53905 JD – 2454967.31191 JD, together with the synthetic curve in the V-band, while Figure 16 presents the light curve residual error for different phases.

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According to the effective temperature of both the primary and secondary components of the system KIC 6287172 and from the calibration ofMK spectral types for main sequence stars, the spectral types are nearest to F5 for both components. Using the orbital and physical parameters listed in Table 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 display the Roche geometry of the system in Figure 18.

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Table 4.  Orbital and Physical Parameters of the System KIC 6287172

Parameter	Value	Parameter	Value
Epoch	2454953.651911	X1	0.507
Period (d)	0.2038732	X2	0.509
Inclination (i)	7.1° ± 0.2°	g1	0.32
Mass ratio (q)	0.606 ± 0.001	g2	0.32
TAVH (T1)	6646 K (Fixed)	A1	0.50
TAVC (T2)	6624 ± 11 K	A2	0.50
PHSV (Ω1)	2.9213 ± 0.0037	$\frac{{L}_{1}}{{L}_{1}+{L}_{2}}$	0.6137 ± 0.0062
PHSV (Ω2)	2.9213 ± 0.0037	$\frac{{L}_{1}}{{L}_{1}+{L}_{2}}$	0.3863 ± 0.0062
Phase Shift	0.02983 ± 0.00031	∑ (O − C)2	0.475962
(f1)	0.4327	(f2)	0.4327
r1 (back)	0.4954 ± 0.0004	r2 (back)	0.4133 ± 0.0008
r1 (side)	0.4537 ± 0.0002	r2 (side)	0.3609 ± 0.0005
r1 (pole)	0.4232 ± 0.0002	r2 (pole)	0.3409 ± 0.0004
r1 (point)	0.5513 ± 0.0002	r2 (point)	0.4487 ± 0.0002
r1 (mean)	0.4245 ± 0.0028	r2 (mean)	0.3337 ± 00045

We compute the physical parameters according to the empirical relation derived by Harmanec (1998) from his work about stellar masses and radii based on modern binary data. These relations are obtained by a least-squares fit via Chebyshev polynomials for the data introduced by Popper (1980). Harmanec's method is the mass − temperature relation used to determine the masses of the primary and secondary components, M₁ and M₂ respectively, for detached binaries. However, in the case of overcontact binaries and any other type of binaries, it is only applicable for determination of the primary component M1, which is slightly affected by thermal contact with the secondary low mass stars. If we assume that the effect of temperature of the secondary component on the primary is negligible because the primary has higher temperature, the mass of the primary can be deduced fromthe previous relations with good approximation. Themass of the secondary component cannot accurately be calculated using this method because the secondary component is greatly affected by thermal contact with the primary hotter star. The masses of primary stars according to Harmanec were calculated from the following equation

$$\begin{eqnarray}\begin{array}{lll}\mathrm{log}\frac{{M}_{1}}{{M}_{\odot }} & = & \left((1.771141X-21.46965)X\right.\\ & & +88.05700)X-121.6782,\end{array}\end{eqnarray} \tag{ 4 }$$

where X = log (T_eff1). Equation (4) is applicable for 4.62 ≥ log (T_eff) ≥ 3.71. The masses of secondary components are determined from the photometric mass ratio, $(q=\frac{{M}_{2}}{{M}_{1}})$.

The physical parameters can be determined from the relation between total mass (M_T) and total luminosity (L_T) in the binary systems fromMaceroni & Van'tVeer (1996), under the assumption that interaction between the two components of the binary system does not affect the total luminosity of the system. Therefore the common envelope radiates the luminosity given by the sum of the internal luminosities. In other words, the total luminosity is the same for the binary system. Due to the previous assumption, this method can be applicable for any type of binary system such as detached, overcontact or ellipsoidal configurations to determine the total mass of the binary system. Using the value of mass ratio for each binary, we can get the individual masses M₁ and M₂ for the binary system as shown in the following equations

$$\begin{eqnarray}&&\mathrm{log}({L}_{T})=\frac{2}{3}\mathrm{log}({M}_{T})+{c}_{1},\end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray*}&&{c}_{1}=\mathrm{log}[c{P}^{(\frac{4}{3})}({r}_{1}^{2}{T}_{1}^{4}+{r}_{2}^{2}{T}_{2}^{4})].\end{eqnarray*}$$

In the same paper (Maceroni & Van'tVeer 1996) in equation (2), the constant c was written as

$$\begin{eqnarray*}&&c={(4\pi )}^{\frac{1}{3}}{G}^{\frac{2}{3}}\sigma .\end{eqnarray*}$$

However, we note that there is a mistake in the formula for {c} and it should be written as

$$\begin{eqnarray*}&&c={\left(\frac{4}{\pi }\right)}^{\frac{1}{3}}{G}^{\frac{2}{3}}\sigma .\end{eqnarray*}$$

Using the evolutionary tracks for a non-rotating model which have been computed by Mowlavi et al. (2012) for zero age main sequence (ZAMS) stars with metallicity Z = 0.014,we correlate the total luminosity and totalmass from fitting data of the binary system of stars located on the ZAMS as follows

$$\begin{eqnarray}&&\mathrm{log}({L}_{T})=4.3\mathrm{log}({M}_{T})-\mathrm{0.539.}\end{eqnarray}$$

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 deduce the total mass (M_T) of the binary system. From knowledge of the mass ratio (q) and total mass (M_T) of the binary system, the individual masses (M₁ and M₂) can readily be calculated from Equation (7)

$$\begin{eqnarray}&&{M}_{1}=\frac{{M}_{T}}{q+1},\,{M}_{2}={M}_{1}\times q=\frac{{M}_{T}\times q}{(q+1)}.\end{eqnarray} \tag{ 7 }$$

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This is the period − mass relation used to determine the mass of the primary component. It is only applicable for short period eclipsing binary systems such as overcontact or near contact types with orbital period log P < −0.25 and does not include ELVs (non-eclipsing binaries) according to Gazeas & Niarchos (2006) and Gazeas & Stepien (2008). The mass of the primary star (M₁) can be obtained from the following expression

$$\begin{eqnarray}\begin{array}{lll}\mathrm{log}({M}_{1}) & = & (0.755\pm 0.059)\mathrm{log}(P)\\ & & +(0.416\pm 0.024),\end{array}\end{eqnarray} \tag{ 8 }$$

where P is the orbital period of a W UMa type binary system. All of these three methods are used to calculate only the mass of the primary component in the binary system. The mass of the secondary component in the binary system is found from the photometric mass ratio $(q=\frac{{M}_{2}}{{M}_{1}})$ and the primary mass. The photometric mass ratio of the contact type is more precise than undercontact type such as the detached binary. All of these three methods are used to calculate only the individual masses of the binary systems. From Kepler's third law, the semi-major axis of the orbit for the binary system can be determined using the following equation

$$\begin{eqnarray}&&a=\sqrt[3]{\frac{G\times ({M}_{1}+{M}_{2})\times {P}^{2}}{4{\pi }^{2}}},\end{eqnarray} \tag{ 9 }$$

where G is the gravitational constant (G = 6.67428 × 10^-11 kg^-1 m³ s^-2). Knowing the semi-major axis (a), thus we can calculate the radius of primary and secondary stars according to the relation R_j = r_ja, where j can take values 1 for the primary component or 2 for the secondary component and r is the mean fractional radii of the stars. The luminosities of the primary and secondary stars were calculated using the direct equation

$$\begin{eqnarray}&&\frac{{L}_{j}}{{L}_{\odot }}={\left(\frac{{R}_{j}}{{R}_{\odot }}\right)}^{2}\times {\left(\frac{{T}_{j}}{{T}_{\odot }}\right)}^{4}.\end{eqnarray} \tag{ 10 }$$

The bolometric magnitudes of the primary and secondary components of binary systems can be calculated using the following equation

$$\begin{eqnarray}&&{M}_{{\rm{bol}}j}={M}_{{\rm{bol}}\odot }-2.5{\mathrm{log}}_{10}\frac{{L}_{j}}{{L}_{\odot }}.\end{eqnarray} \tag{ 11 }$$

The absolute magnitude, M_V, is related to the bolometric magnitude, M_bol, via

$$\begin{eqnarray}&&{M}_{V}={M}_{{\rm{bol}}}-BC,\end{eqnarray} \tag{ 12 }$$

where BC is the bolometric correction given by Reed (1998)

$$\begin{eqnarray}\begin{array}{lll}BC & = & -8.499{[\mathrm{log}(T)-4]}^{4}\\ & & +13.421{[\mathrm{log}(T)-4]}^{3}-8.131{[\mathrm{log}(T)-4]}^{2}\\ & & -3.901[\mathrm{log}(T)-4]-\mathrm{0.438.}\end{array}\end{eqnarray} \tag{ 13 }$$

We used the relation in Equation (12) to calculate the absolute magnitude for all three systems. Using the interstellar extinction value (A_ν ) corresponding to the equatorial coordinates J2000 obtained from Schlafly & Finkbeiner (2011) for the V-band at effective wavelength = 5517 Å, we finally found that, for system KIC 2715417 (A_ν = 0.427), for system 6050116 (A_ν = 0.369) and for system 6287172 (A_ν = 0.339). We can apply the distance modulus relation to calculate the distance of the three systems as follows

$$\begin{eqnarray}&&D={10}^{0.2(m-M+5-{A}_{\nu })},\end{eqnarray} \tag{ 14 }$$

where the distance is in parsec (pc), m represents the apparent magnitude and M_V is the absolute magnitude. All the absolute parameters (mass, semi-major axis, radius, luminosity, bolometric magnitude and distance) of the three systems which have been calculated are listed in Tables 5, 6 and 7.