EARLY PRE-MAIN SEQUENCE EVOLUTION OF LOW MASS STARS IBRAHIM KÜÇÜK Department of Physics, Erciyes University, 38039, Kayseri, Turkey; E-mail: kucuk@newton.physics.metu.edu.tr NILGÜN KIZILOG̃LU and RIKKAT CIVELEK Department of Physics, Middle East Technical University, 06531, Ankara, Turkey (Received 10 February, 1998; accepted 17 June, 1998) Abstract. Evolution of the gravitational contraction phase of stars having masses 0.3 < M/M⊙ < 1.5 were studied along with deuterium burning. The equation of state developed by Mihalas et al. (1988) and OPAL opacity tables were used in our investigation. The theoretical time lines were compared with observations. 1. Introduction The early pre-main sequence phase (PMS) evolution of low mass stars have been previously discussed by several authors (Iben, 1965; Ezer and Cameron, 1965; Grossman et al., 1974; VandenBerg et al., 1983). They showed attention to the PMS evolution phase of stars in order to learn about the ages of young galactic clusters (Ezer and Cameron, 1967; Mazzitelli and Moretti, 1980). The results of theoretical evolutionary studies of low mass stars enable due to draw theoretical lines of equal age in the Hertzsprung–Russell (H–R) diagram by joining the points corresponding to the same evolutionary time on the tracks of considered masses. One can give an upper limit for the age of a given cluster by plotting the observed positions of stars of a galactic cluster in the H–R diagram. The inspiration that led us to calculate the new PMS tracks of low mass stars were: i) the new opacity tables for the interior and atmosphere of stars (Iglesias et al., 1992 (OPAL opacities); Alexander and Ferguson, 1994); ii) the new equation of state studies, one developed by Hummer and Mihalas (1988) and Mihalas et al. (1988), and the other by Rogers et al. (1996); iii) the new data on the observation of PMS stars concerning the locations in the H–R diagram. In the present study, we have computed the PMS evolutionary models of low mass stars starting from the threshold of the energy stability. At the threshold, total released gravitational potential energy is sufficient to provide enough internal energy plus the energy required for the dissociation of hydrogen molecules and ionization of hydrogen. The considered mass range is between 0.3M⊙ < M < 1.5M⊙ . Ezer’s evolutionary code was used in the calculation, even though, it was Astrophysics and Space Science 259: 279–284, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands. 280 IBRAHIM KÜÇÜK ET AL. modified to represent recent improvements in the opacity and in the treatment of equation of state (Yıldız, 1996; Yıldız and Kızılog̃lu, 1997). Similar studies have been carried out by D’Antona and Mazzitelli (1994), and Forestini (1994). They both computed the PMS evolutionary tracks of stars whose masses were smaller than 2.5M⊙ in order to compare the lithium depletion in young clusters by using the OPAL opacities but with different equation of state treatments. Recently, Rogers and Iglesias (1996) presented some new results about low mass stars obtained by using the OPAL equation of state and opacity. In Section 2, the main features for obtaining the models are summarized. The calculated evolutionary models and the comparison of models with observations are presented in Section 3. 2. The Models The numerical integration procedure and the physical inputs are described elsewhere (Yıldız and Kızılog̃lu, 1997). Only the key features will be given here. The evolutionary study was carried out using the Henyey method which was applied from the center toward the surface. All the nuclear reaction rates are obtained from Caughlan and Fowler (1988). The deuterium burning through the sequences 2 H (p, γ )3H e and 2 H (d, n)3 H e was also included in the calculations which was known to lengthen the contraction time of the low mass stars. An initial deuterium abundance of 1.4 × 10−5 by mass was chosen as given by Hubbard et al. (1994), which is less than primordial deuterium abundance value given by Songoila et al. (1994) (D/H ∼ 1.9 − 2.5 × 10−4 ). The radiative opacities were found by quadratic interpolation using OPAL opacity tables (Iglesias et al., 1992). For any arbitrary combinations of temperature, density and chemical composition, we preferred to use low temperature opacity tables of Alexander and Ferguson (1994) in the outer layers of stars. The equation of state developed by Mihalas et al. (1988, 1990) (MHD EOS) was used in the present study which is based on minimization of free energy and includes non ideal effects, such as the Coulomb correction to pressure, pressure due to partially degenerate electrons and correction for the size of particles. The effects of non ideal terms in the equation of state are more important in lower mass stars. Fractional contributions of Coulomb and degenerate electron pressures and contributions from the derivatives of the partition function of H and H e+ increase as the mass of the star decreases toward 0.3M⊙ (Yıldız, 1996). This EOS was adopted for ρ < 10−2 gr/cc if the temperature is below 105 K. Therefore we limited the presentation of results to models whose density–temperature structures did not enter in this forbidden region. Convection was treated by the mixing length theory. The ratio of mixing length to pressure scale height was assumed to be 1.74. For the chemical composition we took the fractional hydrogen abundance X = 0.699 and heavy element abundance EARLY PRE-MAIN SEQUENCE EVOLUTION OF LOW MASS STARS 281 TABLE I Some physical parameters when deuterium burning starts M/M⊙ tdeut (yr) log Teff log L/L⊙ Tc (◦ K) ρc (g cm−3 ) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.3 1.5 1.355 ×105 1.336 ×105 1.043 ×105 8.053 ×104 7.679 ×104 6.450 ×104 6.988 ×104 5.260 ×104 4.789 ×104 3.820 ×104 3.572 3.583 3.600 3.613 3.622 3.629 3.638 3.639 3.651 3.651 0.225 0.403 0.615 0.828 0.960 1.110 1.178 1.335 1.516 1.650 7.985 ×105 9.018 ×105 9.514 ×105 9.464 ×105 9.848 ×105 9.744 ×105 1.054 ×106 9.830 ×105 1.090 ×106 1.072 ×106 9.937 ×10−2 8.179 ×10−2 6.210 ×10−2 4.276 ×10−2 3.557 ×10−2 2.650 ×10−2 2.660 ×10−2 1.752 ×10−2 1.422 ×10−2 1.022 ×10−2 Z = 0.019, by mass. When these parameters are used, the present theoretical solar model fits well to its observational values (Yıldız, 1996; Yıldız and Kızılo g̃lu, 1997). 3. Results and Discussion The theoretical evolutionary paths of low mass stars with deuterium burning were followed up to the point where the density was greater than 0.01 gr/cc for temperatures below 105 K due to the limitation in the MHD EOS. Table I gives some physical parameters of low mass stars at the deuterium main sequence (DMS). The deuterium ignition time for the masses examined is given in the second column. The remaining columns in turn, give the logarithm of effective temperature, the logarithm of luminosity, central temperature, and central density at corresponding times. Only the 0.3M⊙ star could not deplete all of its deuterium content during the time that we consider, because it is still fully convective during this period. As it can be seen from the table, deuterium burns in the temperature range, 8 × 105 K < Tc < 106 K for the mass range under consideration. The time variations of the quantities, central temperature, and luminosity for 0.4 and 1.5M⊙ are shown in Figure 1. As seen in this figure, during the deuterium burning phase, the rate of decrease in the surface luminosity and the increase in central temperature of the stars decreases in time. Deuterium begins to burn at an age of 1.3 × 105 yr and 3.8 × 104 yr for stars of 0.4 and 1.5M⊙ , respectively, while the duration of the deuterium burning phase is about 3.4 × 105 yr for 0.4M⊙ , it is 1.7 × 105 yr for the latter one (at the end of this burning phase, the deuterium contents have dropped to 10−14 percent of their initial values). Deuterium burning lengthens the contraction 282 IBRAHIM KÜÇÜK ET AL. Figure 1. Variation of luminosity and central temperature in time for 0.4 and 1.5M⊙ stars. The decrease in Tc and increase in L are noticeable during the deuterium burning phase in both cases. time of the lower mass stars particularly. When the deuterium burning reaction rates are at maximum, the nuclear energy supplies almost 95% of total energy, for these stars. We compare the present results with the previous calculations of low mass stars (Küçük, 1993) for which Cox and Stewart opacity (1970) and Ezer EOS (1965) were used. Evolutionary tracks of the present study shift toward the left part of the H–R diagram due to the increase in the value of convective parameter α to 1.74. The present models also have lower luminosities at a given age during the gravitational contraction since OPAL opacity makes the transport of energy easier from the deeper layers to the surface of the stars. This shows that the present isochrones should indicate a younger age for the star clusters. Two young galactic clusters, namely Trapezium and NGC 6611 and T Tauri stars in the Taurus Auriga region have been selected for comparison. The observational data were taken from Herbig and Terndrup (1986) for Trapezium Cluster, The et al. (1990) and de Winter et al. (1997) for NGC 6611, and from Kenyon and Hartmann (1995) for T Tauri stars, respectively. The distance modulus V–Mv are taken as 10.62, 12.50 and 5.73 for Trapezium, NGC 6611 and T Tauri stars, respectively. The observational Colour–Magnitude Diagram of samples has been transformed into theoretical H–R diagram using the transformation table presented EARLY PRE-MAIN SEQUENCE EVOLUTION OF LOW MASS STARS 283 Figure 2. Observational H–R diagram for Trapezium Cluster, NGC 6611 and T Tauri stars. The continuous line with open box represents the position of the theoretical DMS. The continuous lines are the theoretical isochrones for t = 103 , 104 , 105 , 5 × 106 , 106 yr. The early pre-main sequence evolutionary tracks of 0.3, 0.6, 0.8 and 1.5M⊙ stars are also shown. by Kenyon and Hartmann (1995). The theoretical isochrones and H–R diagram of the samples are presented in Figure 2. In the same figure, the evolutionary tracks for 0.3, 0.6, 0.8, and 1.5M⊙ were also plotted. The continuous line with open box represents the position of the theoretical DMS. The solid lines are the theoretical isochrones for the ages of 103 , 104 , 105 , 5 × 105 , 106 yr. From this figure one can assume that most of our sample stars are in gravitational contraction phase of their evolution. Although there is a significant age spread among the stars of clusters under consideration, we can assign an upper limit of 5 × 105 yr to the age of T Tauri stars and NGC 6611. As stated by de Winter et al. (1990) most of foreground stars of NGC 6611 are of late spectral type though some early spectral type and post–ZAMS stars are observed. They concluded that the cluster contains objects of a few 0.1 Myr as well as objects of about 6 Myr. 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