RZ Cas, KO Aql and S Equ: a piece of cake of case A RLOF?

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1 The 8th Pacific Rim Conference on Stellar Astrophysics ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** B. Soonthornthum, S. Komonjinda, K. S. Cheng and K. C. Leung RZ Cas, KO Aql and S Equ: a piece of cake of case A RLOF? J.-P. De Greve (1), N. Mennekens (1), W. Van Rensbergen (1), L. Yungelson (2) (1) Astrophysical Institute, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium (2) Institute of Astronomy of the Russian Academy of Sciences, Pyatnitskaya 48, Moscow, Russia Abstract. We determine the present evolutionary state and the restrictions on the initial mass ratios of RZ Cas, KO Aql and S Equ. The gainers are in an early main sequence stage (X c > 0.5), with KO Aql being almost unevolved. The initial donor/gainer mass ratios M di /M gi must be larger than three to obtain the present mass and luminosity of the gainers. However, conservative mass transfer leads to considerable overflow of the gainer s radius over the Roche radius. 1. Setting the scene In a recent paper Soydugan et al. (2007) reported new absolute dimensions and parameters for the semi-detached eclipsing Algol-type binaries KO Aql and S Equ. The systems bear similarities to RZ Cas. For RZ Cas we use the orbital and physical parameters compiled by Maxted et al. (1994) and the mass-loss rate given by Lehmann & Mkrtichian (2007, private communication). The mentioned authors conclude from their analysis that the three systems are showing prominent features of mass transfer. The three systems form an interesting set from an evolutionary point of view. Table 1 shows the system parameters for all three. The total mass and period of RZ Cas are the smallest, Table 1. Orbital and physical parameters of RZ Cas, KO Aql and S Equ Parameter RZ Cas KO Aql S Equ gainer donor gainer donor gainer donor P (d) M (M ) R (R ) log T (K) log L/L Ṁ (M /yr) those of S Equ the largest. However, the mass ratio behaves opposite, going from 0.13 for RZ Cas (over 0.22 for KO Aql) to 0.33 for S Equ. Comparison of the system parameters with a canonical period - mass ratio function, showing the changes during Roche Lobe Overflow (RLOF), indicates that the three systems 1

2 2 J.-P. De Greve, N. Mennekens, W. Van Rensbergen, L. Yungelson Figure 1. Mass transfer rate during RLOF as a function of time for the system 2.38M M, P i = 1.67d. The arrows indicate the positions of the three systems. might be in subsequent stages of (conservative) case A mass transfer. However, appropriate model computations for these systems are lacking (Soydugan et al., 2007). To confirm the suggestion of subsequent stages we intended to model the three systems, using conservative assumptions during RLOF. Contrary to most modeling studies, we concentrated on the gainer, rather than on the donor. To avoid any confusion, we will further label the hotter, mass-gaining component (commonly called the primary) as gainer, while the cooler, mass-losing component (the secondary) is labeled as donor. The mass ratio is defined as q = M d /M g. 2. The luminosity problem First, we wanted to know if the gainers are close to thermal equilibrium. The low mass ratios point towards an advanced state of mass transfer. RLOF can be divided into a first, fast phase on a thermal timescale and a subsequent slow phase on a nuclear timescale. In that phase both components return to quasithermal equilibrium, and thus the gainers should resemble single stars with the same mass. Through comparison with a mass loss rate versus mass ratio function of a case A model calculation, we can derive the present phase of the systems. This is shown in Figure 1. As already expected from the well reversed mass ratios, the three systems are clearly beyond the phase of fast mass transfer. Comparison with single star models with masses equal to the observed gainer masses (respectively 2.21M, 2.53M and 3.21M for RZ Cas, KO Aql and S Equ) is shown in Figure 2. The comparison teaches us two things: a. The three systems are underluminous with respect to single stars, and the underluminosity increases with increasing mass, as previously noted by Soydugan et al. (2007). This indicates that either they are not close to thermal equilib-

3 RZ Cas, KO Aql and S Equ 3 Figure 2. Tracks of single star evolution in the HRD, for the mass values of Table 1. The positions of the gainers of S Equ, KO Aql and RZ Cas are given, and the central hydrogen abundance by mass, as interpolated from the models, is indicated next to the gainers. rium or have a different internal structure resulting from the previous RLOF. b. Two of the systems are close to (RZ Cas) or even on the ZAMS (KO Aql), whereas the third system (S Equ) has evolved about one third into the main sequence. If their position reflects their nuclear state, then the central hydrogen content by mass fraction X c would respectively be 0.6, 0.7 and Initial conditions for conservative case A RLOF of KO Aql In the rest of the paper we will concentrate on the evolution of KO Aql, the middle system. Adopting a conservative case A RLOF as possible history, it is easy to calculate all possible initial sets of donor mass M di, gainer mass M gi and period P i from the presently observed state. From three sets of initial conditions (set 1: M gi = 1.18M and P i = 0.68d; set 2: M gi = 1.32M and P i = 0.62d; set 3: M gi = 1.53M and P i = 0.58d) we calculated the evolution of both components simultaneously. We used the same binary code as De Loore & Van Rensbergen (2005). In the code, the structure of the gainer is calculated assuming that the added matter has the same entropy as the gainer and all effects of interaction are neglected. An initial Population I composition of X = 0.70, Z = 0.02 was adopted. 4. Not a piece of cake... The donors started losing mass early in the main sequence: respectively at X c = 0.51, 0.56 and 0.58 for set 1, 2 and 3. The systems went through a phase of rapid mass transfer and then turned into a slow phase on a nuclear timescale. Figure 3 shows the tracks for the gainers in the HRD. They climb upwards along the ZAMS. However, none of them reaches the observed position of KO Aql. Instead, around log L/L = 1, internal evolution takes over from accretion and they start to evolve away from the ZAMS. Moreover, the masses at

4 4 J.-P. De Greve, N. Mennekens, W. Van Rensbergen, L. Yungelson Figure 3. HRD with accretion tracks for the gainer of KO Aql. The initial masses are indicated next to the tracks. that point (1.80 to 1.86M ) are still far from the observed value of 2.53M. The highest initial mass ratio (1.61) results in the highest luminosity and the highest mass of the gainer at the end of the fast phase. Hence, a larger initial mass ratio is needed. Extrapolating from the previous results, we found that the following two values (with corresponding system parameters) could do the job: q i = 3.40 (set 4: M gi = 0.70M and P i = 1.67d) and q i = 3.46 (set 5: M gi = 0.69M and P i = 1.72d). The results are also shown in Figure 3. The two tracks now reach the luminosity of the gainer of KO Aql. The track of set 4 even goes through the observed position. But again, it is not a success. At the moment that the gainer in the two sets reaches the mass of KO Aql (2.53M ) it is in the wrong place in the HRD. For set 4, this is at the tip of the track, when the model is evolving away in the main sequence, its position almost coinciding with that of a single star of 2.53M with X c = For set 5 this is at log L/L = 1.62 (X c = 0.70), after which it starts slowly moving upwards again. Also the mass loss rate at that moment turns out to be wrong: for set 4 it is M /yr and for set two M /yr, too low compared to the observed M /yr. But there is more wrong with these two computations. Whereas the gainers of the first three sets remained neatly near the ZAMS, those in set 4 and 5 move almost vertically upward in the HRD during the fast phase, as a result of the large difference in thermal timescales between the two components. The different behavior of the two groups of systems can be understood in terms of the accretion timescale versus the thermal timescale t KH. This ratio is always larger than one for sets 1 to 3 with initial mass ratios between one and two (at maximum mass loss rate it reaches 6). However, for large initial mass ratios, the ratio of the timescales becomes as small as 0.01 at maximum. This behavior was already found in the 1970s by Webbink (1976, 1977). As a result, the radius of the gainer sharply increases, and becomes larger than its Roche radius. For set 4 the radius exceeds the Roche radius even up to a factor of 5. This is shown in Figure 4. The system thus represents a neat common envelope system and its evolution from R > R Roche on should be considered as formal. Note however

5 RZ Cas, KO Aql and S Equ 5 Figure 4. Evolution of the radius and the Roche radius of the gainer as a function of time during the fast phase of mass transfer for set 4: 2.38M M, P i = 1.67d. that in our computation this stage lasts only years, or only 5.6% of the fast phase (and 0.2% of the total mass transfer lifetime up to 2.53M ). 5. Conclusions We reached similar conclusions for the systems RZ Cas and S Equ. For RZ Cas an initial mass ratio of 2.94 is needed to reach both luminosity and mass. For S Equ the required initial mass ratio is minimally 3.3. In both cases, similar large excesses of the radius over the Roche radius occur. Hence, to obtain the present mass and luminosity of the gainers, large initial mass ratios are necessary, typically > 3. But under conservative conditions these result in radii that are larger than the Roche radii, implying a non-conservative approach. Moreover, the present models do not account for the observed underluminosity of the gainers. A non-conservative approach is needed, in which processes are considered that enhance the period without losing too much mass, or internal structural processes must be included that keep both luminosity and radius low. Acknowledgments. The authors thank Ron Taam for a fruitful discussion during the conference, resulting in the conclusion that a possible solution for the underluminosity should be sought through the internal redistribution of the added angular momentum during RLOF. References De Loore, C., & Van Rensbergen, W. 2005, ApSS, 296, 353 Lehmann, H., & Mkrtichian, D. E. 2007, private communication Maxted, P. F. L., Hill, G., & Hilditch, R. W. 1994, A&A, 282, 821 Soydugan, F., Frasca, A., Soydugan, E. et al. 2007, MNRAS, 379, 1533 Webbink, R. F. 1976, ApJS, 32, 583 Webbink, R. F. 1977, ApJ, 211, 486