A&A 520, A41 (2010) Astronomy DOI: 10.1051/0004-6361/201014329 & c ESO 2010 Astrophysics Implications of a new triple-α nuclear reaction rate Consequences for Cepheids P. Morel 1,J.Provost1,B.Pichon1, Y. Lebreton2,3, and F. Thévenin1 1 Université de Nice-Sophia Antipolis, CNRS UMR 6202, Observatoire de la Côte d’Azur, Laboratoire Cassiopée, BP 4229, 06304 Nice Cedex 04, France e-mail: [Pierre.Morel;Janine.Provost;Bernard.Pichon;Frederic.Thevenin]@oca.eu 2 Observatoire de Paris, Laboratoire GEPI, CNRS UMR 8111, 92195 Meudon, France e-mail: [email protected] 3 IPR, Université de Rennes 1, 35042 Rennes, France e-mail: [email protected] Received 26 February 2010 / Accepted 13 June 2010 ABSTRACT Context. Recently the triple-α reaction rate has been re-evaluated. In the temperature range, 107−108 K, with respect to the NACRE rate, the new rate is enhanced by up to 20 orders-magnitude. Aims. To validate this new rate, we investigate its consequences for the evolution of Cepheid models. Methods. The stellar evolutionary tracks are calculated with the CESAM code and displayed in the domain 4−10 M. > Results. With the new rate, the first dredge-up does not occur. For masses larger than 4.5 M each evolutionary track crosses the instability strip only once. The luminosities are higher than with the previous rate, then leading to smaller theoretical masses that better agree with the pulsational mass. Conversely, and inconsistently with one century of observations of more than two hundred Cepheids, the temporal derivative of the period keeps a positive sign. Moreover the observed depletions of atmospheric lithium and C/N ratio do not occur. A slight modification of only a few percents of the new nuclear rate allows us however to restore the loops inside the instability strip and the changes of sign of the temporal derivative of periods. Conclusions. This preliminary work indicates that the new rate may solve some of the long-lasting unsolved theoretical problems of Cepheids. Yet indisputable observations argue against its pertinence. Nonetheless, with regard to its theoretical importance, the triple-α new reaction rate still needs to be confirmed or revisited. Key words. nuclear reactions – nucleosynthesis – abundances – stars: evolution – stars: variables: Cepheids 1. Introduction Recently, Ogata et al. (2009, hereafter OKK) have re-evaluated the triple-α (3α) reaction rate by directly solving the three-body Schrödinger equation. In the temperature range 107 K–2×108 K they obtained a rate enhanced by up to 20 times compared to the rate provided in the NACRE (Angulo et al. 1999) compilation. Ogata et al. (2009) give the new rate 3αOKK as a tabulated cor- rection Δ3α to be applied to the rate of NACRE 3αNACRE: α = α + Δ α . log10(3 OKK) log10(3 NACRE) log10( 3 ) (1) Figure 1 shows the 3α reaction rates computed with the NACRE and OKK requirements respectively. It is accepted up to now that the ignition of the 3α reactions occurs at temperatures T ∼> 100 MK (Kippenhahn & Weigert 1991). At these temper- atures its rate amounts to 2.38 × 10−24 cm3 mol−1 s−1 according to NACRE. With the OKK new rate, this level corresponds to a ∼ Fig. 1. (Colour online) The NACRE (full) and enhanced Ogata et al. temperature close to T 50 MK. Hereafter we will designate by (2009) (dash) triple-alpha reaction rates variations with temperature. cold helium burning (CHeB) the OKK rate in contrast to the hot With OKK, the temperature of the ignition is about halved. For CHeBm helium burning (HHeB) of the NACRE reaction rate. The 3α he- Δ α (dot-dash) the correction log10( 3 ) is arbitrarily reduced by 7% – see lium burning occurs in physical conditions unattainable for ex- Sect. 5. perimental validation except in stellar interiors. As seen in Fig. 2 left panel, the HR diagram would need to be revisited if this new 3α rate was really at work in stars. Saruwatari & Hashimoto (2008) have investigated the theo- white dwarfs in accretion binary systems, considered as progen- retical effects of the OKK new 3α rate on helium ignition for itors of type Ia supernovae. Article published by EDP Sciences Page 1 of 7 A&A 520, A41 (2010) Fig. 2. (Colour online) Left panel: evolutionary tracks computed with cold helium burning (CHeB). Right panel: evolutionary tracks computed with hot helium burning (HHeB). The left and right dashed lines respectively delimit the blue and red sides of the Cepheids instability strip. For clarity only the part of the evolutionary tracks beyond the ZAMS is displayed. Dotter & Paxton (2009) found that the OKK rate is incom- (Valle et al. 2009, and references therein). None of these patible with the evolution of late-type stars through the helium physical processes are clearly understood or quantified up flash and the existence of extended red giant branches in old stel- to now. lar systems. In these stars the 3α ignition occurs in a degenerate medium. Yet even with a temperature sensitivity ∝T 26 for CHeB, Hereafter we study how models of Cepheids computed with instead ∝T 40 for HHeB, the 3α ignition remains explosive and CHeB are modified and if these models may provide answers the numerics are unstable. to these open questions. Precisely because their pulsations occur during the helium The paper is organized as follows. In Sect. 2 we present the burning in their core, the Cepheids are promising objects for physical processes, physical data, values of parameters and nu- exhibiting quantitative tests for the new rate. Cepheids are im- merical techniques used. Section 3 is devoted to a description portant because the period of their pulsation is related to their of evolutions using CHeB and HHeB with emphasis on mod- luminosity, they have been long recognized as primary stan- els with the mass of δ Cephei (i.e. 6.6 M). A discussion dards to estimate the distances. They still are subjects of nu- of results is made in Sect. 4. Results of numerical experiences merous theoretical and observational galactic and extra galactic with changes of physical data are briefly reported in Sect. 5. works based on observations stretched over more than a cen- Conclusions are given in Sect. 6. tury. The Cepheids pulsate when they are located in a wedge- shaped area of the Hertzsprung-Russell diagram, the so-called “instability strip” (Turner 2001). Its width is of the order of 2. Methods Δ ∼ . log10 Teff 0 09. Despite the huge theoretical investment two points remain unexplained: Our goal is to investigate in a general way the consequences of the 3α new rate on Cepheid models. Fixing the physics and the 1. these stars pulsate when they describe one or several “blue numerics, we have calculated several stellar models with charac- loops” through the instability strip. But the physical condi- teristic masses of Cepheids. tions for the existence and extent of blue-loops are still de- bated (Kippenhahn & Weigert 1991; Valle et al. 2009). As an example, in our calculations with the opacities and the Calculation of models: The models were computed with the solar mixture of Asplund et al. (2005), blue loops cross the CESAM2k code (Morel 1997; Morel & Lebreton 2008). The instability strip only for masses larger than ∼7 M; evolutions were initialized with homogeneous pre-main se- 2. since the late sixties it became apparent that the masses quence models. The EFF equation of state (Eggleton et al. of Cepheids predicted from the theory of stellar evolution 1973) was used, together with opacities determined by Iglesias were larger than those predicted by the pulsation theory (Cox & Rogers (1996) using the solar mixture of Grevesse & Noels 1980). Nowadays this problem remains open (Caputo et al. (1993), complemented at low temperatures by Alexander & 2005; Valle et al. 2009, and references therein). The most Ferguson (1994) opacities. Conductive opacities were taken currently invoked physical process to explain these discrep- from Iben (1975). We used the composition X = 0.7126, ancy is the existence during the main sequence hydrogen Y = 0.270, (Z = 0.0174, Z/X = 0.0244). In the convection burning phase of a hypothetical core overshoot in the range zones the temperature gradient was computed according to the of 0.2 to 0.8 pressure scale height (Keller 2008, and refer- Canuto & Mazitelli (1991) convection theory with the mixing ences therein). The overshooting produces a larger helium length parameter set to unity as recommended by these authors. core, it increases the amount of helium nuclear fuel available The atmosphere was restored using Eddington’s atmospheric during the pulsation. The higher the convective core over- law (Mihalas 1978). The relevant nuclear reaction rates of pp shoot, the larger the luminosity for a given mass or con- chain, CNO and 3α cycles were taken from the NACRE compi- versely for a given luminosity the smaller the theoretical lation. Mass loss, rotation, microscopic and turbulent diffusion mass. Mass loss occurring just before the Cepheid’s pulsa- are not included in the models. Overshooting is only consid- tion phase is also invoked to reduce the theoretical mass ered in Sect. 5. For models computed with CHeB, the analytical Page 2 of 7 P. Morel et al.: Implications of a new triple-α nuclear reaction rate form of the 3α reaction rate from NACRE is enhanced in accor- dance with Ogata et al. (2009) data and interpolated according to Eq. (1). Period: in a first approximation the period p (in days) of Cepheids depends on the mean density (Cox 1980): R3 p = Q¯ , (2) i M here M is the mass and R the radius of the star (in solar units), Q¯i is a parameter which depends on the physics.