We presented fundamental stellar parameters and evolutionary statuses of six solar type detached eclipsing binaries whose masses are in the range of 0.97-1.43
M
⊙
. EK Cep and FL Lyr belong to the zero age main sequence. HS Hya, IT Cas and CD Tau are on the main sequence. Their ages are 1.3, 1.9 and 2.2 Gyr, respectively. Both component stars of AI Phe evolved to sub giants and its age is 4.0 Gyr. Those ages of the detached binary systems show good agreement with the time scale for synchronization and circularization of the binary systems.
1. INTRODUCTION
Photometric observations of eclipsing binaries produce light curves whose morphologies provide relative parameters of the binary systems, while the size of semi-major axis can be determined in mks unit using radial velocity curves produced by spectroscopic observations of the binaries. Therefore the eclipsing double lined spectroscopic binaries provide fundamental parameters such as stellar masses and radii. Even luminosities and distances can be determined if temperature can be assigned to the component stars.
In this paper we investigated six solar type double lined spectroscopic eclipsing binaries for their evolutionary statuses which include absolute their dimensions, evolution tracks, isochrones, and ages. Generally the evolution models (Yi et al. 2003, Bertelli et al. 2009, Ekstr?m et al. 2012) have been used to determine the evolutionary statuses of stars in clusters. Our aim is to find the evolutionary statuses of field stars using detached eclipsing binaries. This work is expected to cross check the fundamental stellar parameters.
The absolute dimensions were determined in Section 2, the evolution tracks were calculated in Section 3. Finally we discussed and summarized our results in Sections 4 and 5, respectively
2. ABSOLUTE DIMENSIONS
We collected well-detached eclipsing binary systems which not only show no interactions between components but also provide multi color light curves and double-lined radial velocity curves for the purpose of stellar fundamental parameters. Thus far, we first have collected approximately 80 eclipsing detached binary stars. We first analyzed first six solar type detached eclipsing binary systems whose mass range between 0.9 and 1.4 solar masses. The ephemerides of six detached binary systems are listed in
Table 1
.
Double lined radial velocity curves were analyzed using the 2007 version of Wilson & Devinney (1971) differential correction computer code for spectroscopic parameters. The system velocity (V
r
), semi-major axis (a) and mass ratio (q) were adjusted to fit to the observed radial velocity curves for the circular orbits, while additional parameters, eccentricity (e), and longitude of periastron (
ω
) were adjusted for the eccentric orbits.
For the photometric parameters we estimated effective temperatures of the binary systems using their color indexes of the binary systems, and relation between color index (
B-V
) and effective temperature (Surina 2009). The estimated temperatures were compared with the temperatures of six
The ephemerides of six detached binary systems.aError was not published by authors.
The ephemerides of six detached binary systems. aError was not published by authors.
The system temperatures of six detached binary systems.
The system temperatures of six detached binary systems.
Spectroscopic and photometric solutions of six detached binary systems.
Spectroscopic and photometric solutions of six detached binary systems.
Absolute dimensions of six detached binary systems.
Absolute dimensions of six detached binary systems.
binary systems determined by spectral types. The temperatures of six detached binary systems are listed in
Table 2
.
The temperatures determined by color index of CD Tau, HS Hya, IT Cas, and FL Cep show good agreement with those of spectral types, while EK Cep and AI Phe show deviation between temperatures by color index and by spectral type. This is mainly due to the incorrect collection of color excess in the literatures. Therefore we used temperatures determined by spectral types. We assumed the temperature of the binary system to be an initial temperature of the primary star. Later the binary system temperatures were divided into temperatures of primary and second stars (Kang et al. 2004). The system temperatures are listed in
Table 2
. The mass ratio and semi major axis determined in the analysis of radial velocity curves were adopted. The initial values of limb darkening coefficients (x
1
, x
2
) were adopted from the theoretical mean values of van Hamme (1993). The gravitational darkening coefficients (g
1
, g
2
), were assumed to 0.32 for convective envelope and 1.0 for radiative envelope suggested by Lucy (1967). The bolometric albedos (A
1
, A
2
) are 0.5 and 1.0 for convective envelope and radiative envelope, respectively. These parameters were then adjusted, if necessary.
The theoretical light curves were fitted to the observed light curves by adjusting parameters which include temperature, relative radii, and other photometric parameters. The spectroscopic and photometric parameters were combined to determine absolute dimensions of the detached eclipsing binary systems. The photometric parameters and absolute dimensions are listed in
Table. 3
and
4
, respectively. Brief research histories and current interesting topics of each binary system are described below.
- 2.1 EK Cep
EK Cep was discovered to be an eclipsing binary system by Strohmeier (1959). The photometric observations were carried out by Ebbighausen (1966a), Khaliullin (1983), and Hill & Ebbighausen (1984). Spectroscopic observations were published by Ebbighausen (1966b) and Tomkin (1983). Later, the evolution of EK Cep was studied by various investigators (Popper 1987, Martin & Rebolo 1993, Claret et al. 1995, Palla & Stahler 2001, Yildiz 2003, Marques et al. 2004, Claret 2006). We analyzed the radial velocity curves of Tomkin (1983) and the
VR
light curves published by Ebbighausen & Hill (1990). EK Cep is an interesting eclipsing binary star whose secondary star is a pre main sequence star. The system shows apsidal motion with a high relativistic contribution. The more massive companion seems to be at the beginning of the hydrogen burning phase. The fitted radial velocity and light curves of EK Cep are plotted in
Figs. 1
and
2
, respectively.
- 2.2 CD Tau
CD Tau (HD 34335, HIP 24663) was found to be a double lined spectroscopic binary by Sanford (1928). Wesselink (1932) found it to be an eclipsing binary. Popper (1971) published radial velocity curves. Photometric observations were carried out by three different investigators (Srivastava 1976, Wood 1976, Gulmen et al. 1980). The light curve analyses had been carried out by Gulmen et al. (1980) and Russo et al. (1981). The radial velocity curves had been analyzed by Popper (1971) and Ribas et al. (1999), respectively. The evolution of CD Tau was studied by Ribas et al. (2000).
CD Tau is known as a possible member of a triple system (Chambliss 1992). It has a K-type close visual companion (CD Tau C). We analyzed combined radial velocity curves observed by Popper (1971) and Torres et al. (1997). The
UBV
light curves observed by Gyldenkerne et al. (1975) were analyzed. The fitted radial velocity and light curves of CD Tau plotted in
Figs. 3
and
4
, respectively.
Radial velocity curves of EK Cep with our fitted model curves.
VR light curves of EK Cep with our fitted model curves.
Radial velocity curves of CD Tau with our fitted model curves.
- 2.3 IT Cas
IT Cas was discovered to be an eclipsing binary system by Fadeeva (Parenago 1938). Lacy et al. (1997) published
UBVR
light curves and the first radial velocity curves of IT
UBV light curves of CD Tau with our fitted model curves.
Radial velocity curves of IT Cas with our fitted model curve.
UBVR light curves of IT Cas with our fitted model curves.
Cas. We analyzed radial velocity and light curves for their absolute dimension. The orbit of IT Cas is an eccentric orbit and show apsidal motion. The fitted radial velocity and light curves are plotted in
Figs. 5
and
6
, respectively.
Radial velocity curves of HS Hya with our fitted model curves.
ubvy light curves of HS Hya with our fitted model curves.
- 2.4 HS Hya
HS Hya (HD 90242) was discovered to be an eclipsing binary star by Strohmeier et al. (1965). The spectroscopic observations were published by Popper (1971) and Torres et al. (1997). Gyldenkerne et al. (1975) carried out the photometric observations. The light curves had been analyzed by Gyldenkerne et al. (1975) and Giuricin et al. (1980). HS Hya has a third star in the system (Torres et al. 1997). The third star is a M0 dwarf with 0.5 solar mass and 190 day orbit.
The combined radial velocity curves observed by Popper (1971) and Torres et al. (1997), and
ubvy
light curves of Gyldenkerne et al. (1975) were analyzed. The fitted radial velocity and light curves are plotted in
Figs. 7
and
8
, respectively.
Radial velocity curves of FL Lyr with our fitted model curves.
VR light curves of FL Lyr with our fitted model curves.
- 2.5 FL Lyr
FL Lyr was discovered to be a detached eclipsing binary system by Popper et al. (1986). The first light curves of this system were published by Cristaldi (1965).
We analyzed radial velocity curves published by Popper et al. (1986) and
VR
light curves published by Cristaldi (1965) and Popper et al. (1986). The components of FL Lyr binary system is known to close to the Sun in their properties which well describe F and G type main sequence stars. The fitted radial velocity and light curves are plotted in
Figs. 9
and
10
, respectively.
- 2.6 AI Phe
AI Phe was discovered to be an eclipsing binary system by Strohmeier (1972). The spectroscopic observations were carried out by Imbert (1979) and Andersen et al. (1988). The
UBVRI
light curves were published by Imbert (1979) and Hrivnak & Milone (1984), respectively. Both component
Radial velocity curves of AI Phe with our fitted model curves.
UBVRI light curves of AI Phe with our fitted model curves.
stars of AI Phe are known as evolved stars. We analyzed the radial velocity and light curves published by Hrivnak & Milone (1984) and by Andersen et al. (1988), respectively. The fitted radial velocity and light curves are plotted in
Figs. 11
and
12
, respectively.
3. EVOLUTION TRACKS OF 6 DETACHED SYSTEMS
The evolution tracks of six detached eclipsing systems were computed using the Yale stellar evolution code (Guenther et al. 1992). Cemical mixture is assumed to be the scaled solar value introduced by Grevesse & Sauval (1998) because targets are solar neighborhood with mass range (0.9 ~ 2.0 M
⊙
). Two opacity sets have been used in computation of transport of radiant energy. OPAL opacities (Iglesias & Rogers 1996) for interior and low temperature opacities (Ferguson et al. 2005) for outer region have been considered, respectively. These two opacity sets are in a good agreement in the overlapping region.
Evolution tracks and isochrone of EK Cep.
Evolution tracks and isochrone of CD Tau.
Evolution tracks and isochrone of IT Cas.
Evolution tracks and isochrone of HS Hya.
Evolution tracks and isochrone of FL Lyr.
Evolution tracks and isochrone of AI Phe.
Chemical compositions and ages of six detached binary systems.
Chemical compositions and ages of six detached binary systems.
We plotted the location of a given star with propagation error (box) using their luminosity and temperature on Hertzsprung-Russell diagram. The horizontal axis of the diagram is the effective temperature, Log (T
eff
) and vertical axis is luminosity, Log (L/L
⊙
). We plot evolution tracks using several parameter sets for a given star. Then we adjusted Y
⊙
and Z
⊙
values until the evolution track satisfy the primary and secondary stars and their isochrones. The evolution tracks and isochrones for six detached systems are plotted in
Figs. 13
-
18
. Each plot shows zero age main sequence (green line), evolution tracks for primary (black line) and secondary (black dotted) stars of each binary system, and isochrones (red thick line).
According to the fitted evolution tracks, EK Cep and FL Lyr belong to zero age main sequence. HS Hya, IT Cas and CD Tau are on the main sequence and their ages are 1.3, 1.9 and 2.2 Gyr, respectively. Both component stars of AI Phe evolved to sub giants and its age is 4.3 Gyr. Chemical compositions and ages of six binary systems are listed in
Table 5
.
4. DISCUSSION
We investigated evolution tracks, isochrones, and ages of the six detached binary stars. EK Cep and FL Lyr belong to the zero age main sequence. HS Hya and IT Cas are on the main sequence and their ages are 1.3, 1.9 and 2.2 Gyrs, respectively. Both component stars of AI Phe binary system evolved to sub giants and its age is 4.3 Gyrs. Those ages of the detached binary systems shows good agreement with the time for synchronization and circularization of the binary systems.
It is gerally accepted that a binary star formed with an eccentric orbit which becomes circular orbit. The rotaion of star synchronized to their orbital motion. It is due to the tidal forces in close detached binary stars. The tidal forces drive an exchange of angular momentum between the rotation of the stars and their orbital motion. The theory of orbital pertubations caused by tidal forces between the components of close binaries has been studied by Zahn (1975, 1977, 1978, 1989, 1992). The cumulative effects of such tidal interactions with time is referred to as tidal evolution. The characteristic signs of tidal evolution are
Period?age diagram for synchronization and the circularization.
1) alignment of the stellar spin axes perpendicular to the orbital plane; 2) synchronization of the rotation of the stars to the orbital motion; and 3) circularization of the orbits (Meibom et al. 2006). Hilditch (2001) summerized rough time scales of the synchronization and the circularization for convective envelope stars as below
where the orbital period is in days.
These equations show that the time scales strongly depend on the orbital period. The ages determined by evolution tracks were compared with the time scales of synchronization and circularization in
Fig. 19
. The time limits for synchronization and circularization were calculated using above two equations. and were plotted against orbital period in
Fig. 19
.
The evolution tracks of CD Tau and HS Hya determined their ages as 1.9 Gyrs and 1.3 Gyrs, respectively. Their solutions of light and radial velocity curves show both binary systems have been syncronized and their orbits are circular. The analyses of light and radial velocity curves for CD Tau and HS Hya binay systems do not show eccentricity and asynchronization in their orbits. Therefore ages determined by evolution tracks of these binary systems are good agreement with time scales of synchronization and the circularization. The evolution tracks of EK Cep shows that both components are on the zero age main sequence.The analyses of light and radial velocity curves show that EK Cep has an eccentric orbit (e = 0.10) but its rotaion is synchronized to the orbital motion. Therefore it passed synchronization period, but it is still on the circularization period. We can assume its age approximaly between 10
7
Mass?Luminosity relation for six detached binary systems.
Gyr and 10
9
Gyr. EK Cep also show good agreement with those of tital evolution theory. AI Phe is not plotted becaue of the period of AI Phe is too long compared those of other systems. The age determined by evolution tracks of AI Phe is 4.3 Gyr. It has been passed the period of tidal evolution.
The mass?lumonosity relation is plotted using component stars of six detached eclipsing binares in
Fig. 20
. The open circles and triangles stand for primary and secondary stars, repectively. All stars show good agrrement except two stars which are component stars of AI Phe whose components are evolved to sub giant stars. The component stars of five binary systems shows the mass?luminosity relation as below
This relation shows good agrement with the mass ? luminosity relation for mass range between 0.43
M⊙
and 2.0
M⊙ .
5. SUMMARY
The radial velocity and light curves collected in the literatures for six detached eclipsing binaries were analyzed using the Wilson & Devinny (1971) differential correction program. The findings for these analyses are summarized as follows:
(ⅰ) The fundamental parameters which include masses, radii, temperatures, luminosities of both component stars in the six binary systems are obtained.
(ⅱ) The evolution tracks, isochrones, ages, metal abundances of six binary systems are obtained.
(ⅲ) Ages of six binary systems were confirmed by comparing with time period of synchronization and circularization of a binary system. The ages of the binary systems show good agreement with the time period of tidal evolution period Therefore we found absolute dimensions of six stars are well matched to their evolutionary statues.
Acknowledgements
This work was supported in part by Mid-career and General Researcher Programs, Science and Technology 2011-0028001 and 2011-0010696, and Basic Science Research Program (grant no. 2010-0024954) through the National Research Foundation (NRF) funded by the Ministry of Education.
Andersen J
,
Clausen JV
,
Nordstrom B
,
Gustafsson B
,
Vandenberg DA
1988
Absolute dimensions of eclipsing binaries. XIII. AI pheonicis: a case study in stellar evolution.
A&A
196
128 -
140
Bertelli G
,
Nasi E
,
Girardi L
,
Marigo P
2009
Scaled solar tracks and isochrones in a large region of the Z-Y plane. II. From 2.5 to 20 Msun stars.
A&A
508
355 -
369
DOI : 10.1051/0004-6361/200912093
Chambliss CR
1992
Eclipsing binaries in multiple-star systems.
PASP
104
663 -
677
DOI : 10.1086/133036
Claret A
,
Gimenez A
,
Martin EL
1995
A test case of stellar evolution: the eclipsing binary EK Cephei. A system with accurate dimensions, apsidal motion rate and lithium depletion level.
A&A
302
741 -
744
Cristaldi S
1965
Curva di luce ed elementi della binaria fotometrica FL Lyrae.
MmSAI
36
77 -
97
Ebbighausen EG
1966a
Photometric elements of the eclipsing binary EK Cephei.
AJ
71
642 -
646
DOI : 10.1086/109974
Ebbighausen EG
1966b
Spectroscopic orbit of the eclipsing binary EK Cephei.
AJ
71
730 -
731
DOI : 10.1086/110178
Ebbighausen EG
,
Hill G
1990
VR photometry of the eclipsing binary EK Cephei.
A&AS
82
489 -
496
Ekström S
,
Georgy C
,
Eggenberger P
,
Meynet G
,
Mowlavi N
2012
Grids of stellar models with rotation. I. Models from 0.8 to 120 Msunat solar metallicity (Z = 0.014).
A&A
537
146 -
DOI : 10.1051/0004-6361/201117751
Ferguson JW
,
Alexander DR
,
Allard F
,
Barman T
,
Bodnarik JG
2005
Low-temperature opacities.
ApJ
623
585 -
596
DOI : 10.1086/428642
Gimenez A
,
Margrave TE
1985
Relativistic apsidal motion in the eclipsing binary systems V1143 Cygni and EK Cephei.
AJ
90
358 -
363
DOI : 10.1086/113740
Giuricin G
,
Mardirossian F
,
Mezzetti M
,
Predolin F
1980
Revised photometric elements of the detached eclipsing binaries RS Cha, RZ Cha, and HS Hya.
A&A
85
259 -
262
Guenther DB
,
Demarque P
,
Kim Y-C
,
Pinsonneault MH
1992
Standard solar model.
ApJ
387
372 -
393
DOI : 10.1086/171090
Gulmen O
,
Ibanoglu C
,
Gudur N
,
Bozkurt S
1980
Two colour photometry of the eclipsing binary CD Tauri.
A&AS
40
145 -
150
Gyldenkerne K
,
Jørgensen HE
,
Carstensen E
1975
Four-colour photometry of eclipsing binaries. I. HS Hya, light curves, photometric elements, and determination of helium content.
A&A
42
303 -
309
Hilditch RW
2001
An introduction to close binary stars
Cambridge University Press
Cambridge
Hill G
,
Ebbighausen EG
1984
Photometric elements of the eclipsing binary EK Cephei.
AJ
89
1256 -
1260
DOI : 10.1086/113620
Hrivnak BJ
,
Milone EF
1984
Observations, analysis, and absolute parameters of the evolved binary AI Phoenicis.
ApJ
282
748 -
757
DOI : 10.1086/162258
Imbert M
1979
Orbital elements and dimensions of the eclipsing binary AI Phe.
A&AS
36
453 -
456
Khaliullin KF
1983
The relativistic orbit rotation of the eclipsing binary Ek-Cephei.
SvA
27
43 -
49
Lacy CHS
,
Torres G
,
Latham DW
,
Zakirov MM
,
Arzumanyants GC
1997
Absolute dimensions and masses of IT cassiopeiae.
AJ
114
1206 -
1220
DOI : 10.1086/118553
Lucy LB
1967
Gravity-darkening for stars with convective envelopes.
ZA
65
89 -
92
Marques JP
,
Fernandes J
,
Monteiro MJPFG
2004
The role of the time step and overshooting in the modelling of PMS evolution: the case of EK Cephei.
A&A
422
239 -
245
DOI : 10.1051/0004-6361:20035693
Martin EL
,
Rebolo R
1993
EK Cephei B: a test object for pre-ZAMS models of solar-type stars.
A&A
274
274 -
278
Meibom S
,
Mathieu RD
,
Stassun KG
2006
An observational study of tidal synchronization in solar-type binary stars in the open clusters M35 and M34.
ApJ
653
621 -
635
DOI : 10.1086/508252
Palla F
,
Stahler SW
2001
Binary masses as a test for pre-main-sequence tracks.
ApJ
553
299 -
306
DOI : 10.1086/320658
Parenago PP
1938
Issledovaniia izmenenii bleska 208 peremennykh zvezd (1920-1937), Trudy Gosudarstvennogo astronomicheskogo instituta im. P. K. Sternberga t. 12, vyp. 1
Instituta
Moskva
Popper DM
1971
Six main-sequence eclipsing binaries of types A to F.
ApJ
166
361 -
372
DOI : 10.1086/150965
Popper DM
1987
A pre-main sequence star in the detached binary EK Cephei.
ApJ
313
L81 -
L83
DOI : 10.1086/184835
Popper DM
,
Lacy CH
,
Frueh ML
,
Turner AE
1986
Properties of main-sequence eclipsing binaries--into the G stars with HS Aurigae, FL Lyrae, and EW Orionis.
AJ
91
383 -
404
DOI : 10.1086/114018
Ribas I
,
Jordi C
,
Torra J
,
Giménez Á
2000
Chemical composition of eclipsing binaries: a new approach to the helium-to-metal enrichment ratio.
MNRAS
313
99 -
111
DOI : 10.1046/j.1365-8711.2000.03195.x
Russo G
,
Milano L
,
D✝Orsi A
,
Marcozzi S
1981
Revised photometric elements of the eclipsing systems Bs-Draconis and Cd-Tauri.
Ap&SS
79
359 -
366
DOI : 10.1007/BF00649432
Sanford RF
1928
The orbits of the spectroscopic binaries Struve 674A and Struve 1561B.
ApJ
68
42 -
50
DOI : 10.1086/143128
Strohmeier W
1972
Three new bright eclipsing binaries.
IBVS
665
1 -
Strohmeier W
,
Knigge R
,
Ott H
1965
Bright southern BV-stars.
IBVS
107
1 -
Surina F
2009
Absolute dimensions of twenty five standard eclipsing binaries.
Sejong University
MS Thesis
Tomkin J
1983
Secondaries of eclipsing binaries. V. EK Cephei.
ApJ
271
717 -
724
DOI : 10.1086/161238
Torres G
,
Stefanik RP
,
Andersen J
,
Nordstrom B
,
Latham DW
1997
The absolute dimensions of eclipsing binaries. XXII. The unevolved F-type system HS Hydrae.
AJ
114
2764 -
2777
DOI : 10.1086/118685
van Hamme W
1993
New limb-darkening coefficients for modeling binary star light curves.
AJ
106
2096 -
2117
DOI : 10.1086/116788
Wesselink AJ
1932
Provisional elements of the eclipsing variable star C. P. D. -23°3476.
BAN
6
185 -
Wilson RE
,
Devinney EJ
1971
Realization of accurate close-binary light curves: application to MR Cygni.
ApJ
166
605 -
DOI : 10.1086/150986
Wood DB
1976
Intermediate-band photometric analysis of the eclipsing binary star CD Tauri.
AJ
81
855 -
861
DOI : 10.1086/111965
Yi SK
,
Kim Y-C
,
Demarque P
2003
The Y2stellar evolutionary tracks.
ApJS
144
259 -
261
DOI : 10.1086/345101
Yildiz M
2003
Modeling dissimilar components of the eclipsing binary EK Cep: does the primary star have a rapidly rotating core?
A&A
409
689 -
695
DOI : 10.1051/0004-6361:20031118
Zahn J-P
1975
The dynamical tide in close binaries
A&A
41
329 -
344
Zahn J-P
1977
Tidal friction in close binary stars.
A&A
57
383 -
394
Zahn J-P
1978
Erratum; Tidal friction in close binary stars.
A&A
67
162 -
Zahn J-P
1989
Tidal evolution of close binary stars. I. Revisiting the theory of the equilibrium tide.
A&A
220
112 -
116
Zahn J-P
1992
Circulation and turbulence in rotating stars.
A&A
265
115 -
132