Theoretical Predictions of Surface Light Element Abundances in Protostellar and Pre-Main Sequence Phase
- 1Universitá di Pisa, Dipartimento di Fisica “Enrico Fermi”, Pisa, Italy
- 2INAF, Osservatorio Astronomico d’Abruzzo, Teramo, Italy
- 3INFN, Sezione di Pisa, Pisa, Italy
- 4Dipartimento di Fisica e Astronomia, Università di Catania, Catania, Italy
- 5INFN - Laboratori Nazionali del Sud, Catania,Italy
- 6CSFNSM-Centro Siciliano di Fisica Nucleare e Struttura della Materia, Catania, Italy
- 7Dipartimento di Ingegneria e Architettura, Universitá di Enna, Enna, Italy
- 8Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Nice, France
Theoretical prediction of surface stellar abundances of light elements–lithium, beryllium, and boron–represents one of the most interesting open problems in astrophysics. As well known, several measurements of 7Li abundances in stellar atmospheres point out a disagreement between predictions and observations in different stellar evolutionary phases, rising doubts about the capability of present stellar models to precisely reproduce stellar envelope characteristics. The problem takes different aspects in the various evolutionary phases; the present analysis is restricted to protostellar and pre-Main Sequence phases. Light elements are burned at relatively low temperatures (T from
Light elements – lithium, beryllium and boron (hereafter Li, Be and B) – are burned at relatively low temperatures (T from
A huge amount of data for surface 7Li abundances are available both for disk, thick disk and halo field stars and for open clusters; however, the well known discrepancy between predictions and observations of this quantity in clusters or in the Sun (the so-called “lithium-problem”) is still an open question (see e.g., Charbonnel et al., 2000; Jeffries, 2000; Piau and Turck-Chièze, 2002; Xiong and Deng, 2002; Sestito et al., 2003; Jeffries, 2006; Talon, 2008).
The theoretical prediction of surface light element abundances is complex because they are sensitive to both the input physics (i.e., equation of state, reaction rates, opacity, etc…) and chemical element abundances (i.e., initial abundance of deuterium, helium, metals, etc…) adopted in stellar models, together with the assumed efficiency of microscopic diffusion and radiative acceleration (see e.g., Piau and Turck-Chièze, 2002; Richard et al., 2002; Burke et al., 2004; Richard et al., 2005; Tognelli et al., 2012; Tognelli et al., 2015). The situation is even more complicated because surface light element abundances seem to be affected by additional “non standard” physical processes, not routinely included in stellar evolutionary codes, as the possible presence of relevant magnetic fields and mass accretion processes in some young pre-MS stars (see e.g., Baraffe and Chabrier, 2010; MacDonald and Mullan, 2012; Feiden and Chaboyer, 2013; Somers and Pinsonneault, 2014; Somers and Pinsonneault, 2015). Moreover, rotation-induced mixing, turbulent mixing, gravity waves and mass loss processes could play a role, though mainly for Main Sequence and more evolved stars (see e.g., Montalbán and Schatzman, 2000; Talon and Charbonnel, 2010; Pace et al., 2012; Charbonnel et al., 2013 and references therein).
The pre-MS is the first stellar phase where the star evolves as a fully formed object. To reach this evolutionary stage, the future star has to accrete mass, until its final value, in the previous “protostellar phase.” The details of this phase, when matter of the protostellar cloud is still falling on the surface of the protostar, are complex and uncertain. The full understanding of how the protostellar accreting phase affects the predictions for pre-MS characteristics (and thus light element abundances) is still an open problem. The inclusion of the protostellar accretion phase in evolutionary codes produces stars in the early pre-MS phase different from what expected in standard non accreting models, in which stars essentially contract along the Hayashi track. This eventually results in differences between standard and accreting models still visible during the whole pre-MS or the MS phase, with effects on both the structure and chemical composition of the stellar models (Baraffe et al., 2009; Baraffe and Chabrier, 2010; Tognelli et al., 2013; Kunitomo et al., 2018; Tognelli et al., 2020).
Light element burning cross sections are fundamental ingredients in the predictions of the time behavior of light element stellar surface abundances. In recent years new values for (p,α) reaction rates have been proposed, mainly estimated using the Trojan Horse Method, greatly improving the precision of these quantities.
The present review summarizes the state-of-the-art of theoretical predictions for protostars and pre-MS stars and their light element surface abundances, in the light of recent improvements in the adopted input physics, updated reaction rates and description of the formation and evolution of pre-MS stars.
The paper is structured as it follows. In Section 2 we qualitatively show the location of observed young pre-MS stars in the HR diagram and we compared it to the predictions of standard non accreting models. In Section 3 we give a brief overview of the main characteristics and evolutionary stages of a pre-MS model without protostellar accretion. In Section 4 we introduce the protostellar accretion phase, discussing the differences between spherical and disc accretion, along with the main parameters the determine the structure of an accreting protostar. In Section 5 we analyze the burning of light elements (Li, Be and B) in pre-MS stars and the predicted surface abundances during the pre-MS for stellar models of different masses without or with protostellar accretion phase. In Section 6 we review the impact of updated cross sections for the burning of light elements and their impact in the predictions of surface abundances in pre-MS stellar models. In Section 7 we summarize the main aspects highlighted in the review.
2 Observational Data of Pre-Main Sequence Stars as Test of Theoretical Models
This review is focused on theoretical predictions for protostellar/pre-MS models, which can be validated only through comparison with observational data. Given the difficulty in directly observing the stellar formation and the protostellar phase, only an investigation of the characteristics of very young pre-MS stars, close to the end of the protostellar phase, can indirectly give information on the previous accretion period. The availability of observations for very young pre-MS stars is thus fundamental. A great number of data is available for young pre-MS stars (ages
As an example, Figure 1 shows a sample of young pre-MS stars compared to standard isochrones from 0.1 to 10 Myr and evolutionary tracks for masses in the range [0.01, 1.0]
FIGURE 1. HR diagram for young pre-MS stars extracted from the literature, compared with a set of standard evolutionary models and isochrones from the PISA database (Tognelli et al., 2011; Tognelli et al., 2018).
Other constraints are provided from pre-MS stars in double-lined eclipsing binary (EB) systems, whose masses and radii can be determined with high precision. In recent years, an increasing number of EB systems have been studied in detail, giving the possibility to check pre-MS model predictions against data (see e.g., Mathieu et al., 2007; Gennaro et al., 2012).
Further constraints come from the measurements in low-mass stars of the surface abundance of lithium-7, which, being an element whose destruction rate is extremely sensitive to the temperature, can be used to test the temporal evolution of the pre-MS stellar structures (Charbonnel et al., 2000; Piau and Turck-Chièze, 2002; Randich, 2010; Tognelli et al., 2012). These issues will be discussed in the present rewiew.
3 General Characteristics of Standard Pre-Main Sequence Evolution
The pre-MS evolution starts at the end of the accretion phase and ends with the Zero Age Main Sequence, or simply ZAMS, position1. The star is totally formed, and the mass can be considered constant at least for the whole pre-MS and Main Sequence (MS) evolution. The first consistent description of the pre-MS evolution was given by Hayashi (1961) and Hayashi and Nakano (1963); the basic idea is that a pre-MS star starts from a cold, expanse and luminous model. Due to the low temperatures, the opacity of the stellar matter is large and thus the radiative temperature gradient in the whole structure is larger than the adiabatic one. This leads to convective motions extended within the entire stellar structure; thus the star is fully mixed and chemically homogeneous. Figure 2 shows an example of the evolution of pre-MS solar metallicity low mass stars in the mass range [0.1, 1.0]
FIGURE 2. HR diagram for low-mass stars with indicated the main evolutionary stages during the pre-MS evolution: Hayashi track (fully convective star, gray line), partially convective star (blue line), locus of the Zero Age Main Sequence (ZAMS, magenta line). The locus corresponding to the deuterium burning is indicated by the red stripe.
Due to their low temperatures, during these first stages of the standard pre-MS evolution, stars cannot produce the nuclear energy required to balance the surface energy losses by radiation and their evolution essentially consists in a gravitational contraction. The evolution time scale is thus given by the thermal (Kelvin-Helmholtz) time scale, which is the time of energy transport throughout the star. It is common to define the Kelvin-Helmholtz time scale as the ratio between the total gravitational energy of the star and its luminosity L:
The factor β takes into account the density profile inside the star. For an (unrealistic) model of homogeneous and spherical star with a constant density,
From the relation above a contraction naturally leads to a rise in
The gravitational contraction is the only energy source until the central temperature reaches about 106 K, when the deuterium burning reaction D(p,γ)3He (D-burning) becomes efficient. Such a reaction generates the energy required to maintain the star stable on nuclear time scales, longer than the thermal one. This is guaranteed also by the steep dependence on the temperature of the energy generation rate,
The ignition of the D-burning, due to the produced energy flux, maintains the star fully convective and deuterium is burnt in the whole star. The D-burning phase is shown in Figure 2 as a the red stripe, which indicates the part of the Hayashi track where D-burning provides more than 10% of the total stellar luminosity, for stars with different masses.
The nuclear time scale of D-burning depends on the characteristics of the star, mainly on the mass. The luminosity of a star at the beginning D-burning phase increases with the stellar mass; this means that increasing the stellar mass the D-burning increases its efficiency to balance the higher energy losses at the stellar surface. Thus the rate of deuterium destruction increases with mass. The typical nuclear D-burning time scale for masses in the range
The duration of the D-burning phase in pre-MS depends not only on the stellar mass but it is also proportional to the original stellar deuterium mass fraction abundance. Observations suggest that for disc stars a value of
Once deuterium has been completely exhausted in the whole star a pure gravitational contraction phase starts again. As for the previous evolution the stellar luminosity is well approximated by the power law
This general picture describes the evolution of a pre-MS star in the standard case; theoretical calculations are started when the star is a fully formed object, chemically homogeneous at high luminosity (large radius) on the Hayashi line. However, it is well known that stars undergo a formation phase, the protostellar phase, during which the mass is accreted from the protostellar cloud and/or from a disk to reach the final stellar mass. The inclusion of such a phase could, at least in principle, modify the standard theoretical picture.
4 Protostellar Accretion Phase
The stellar formation process starts with the collapse and the fragmentation of a molecular cloud that contracts forming denser cores which eventually become protostars and then stars. During this process, the protostellar mass progressively increases as the matter in the cloud falls onto the central dense object. The cloud collapse is a complex hydrodynamic problem, in which one has also to take into account cooling processes by molecules and dust. At a given time during the collapse a stable hydrostatic core forms, on which mass continues to fall, so that the accretion treatment does not require anymore hydrodinamical models (e.g., Stahler et al., 1980a; Stahler et al., 1980b; Stahler et al., 1981; Hartmann et al., 1997; Baraffe et al., 2012).
Protostellar accretion has been analyzed in the literature starting from the pioneering works by Stahler et al. (1980a), Stahler (1988), Palla and Stahler (1991), Palla and Stahler (1992), Palla and Stahler (1993), Hartmann et al. (1997), and Siess and Livio (1997), to more recent works by Baraffe et al. (2009), Hosokawa and Omukai (2009a), Baraffe and Chabrier (2010), Tognelli (2013), Kunitomo et al. (2017) and Tognelli et al. (2020). Depending on the characteristics of the accretion assumed in the computations (chemical composition, magnetic fields, rotation, geometry…) the collapse of the cloud and the stellar formation can produce different evolution whose footprints are still visible in pre-MS stars.
4.1 Cloud Collapse and Protostellar Accretion
• Isothermal collapse: The protostellar cloud, during its first collapse (until the central density is lower than about 10−13 g·cm−3) does not warm, because its density is too low to trap the energy produced by the contraction. When the density further increases above this limit the radiation can be partially trapped.
• Formation of the first Larson core: the energy trapped inside the denser regions of the cloud prevent a further collapse of this region. A first temporarily hydrostatic core forms (with a mass of about
• Second collapse: the hydrostatic core contracts as long as it radiates energy from its surface. So, although its mass is increasing due to mass accretion, its radius shrinks. The contraction of the core leads to a temperature rise, until the temperature of molecular hydrogen dissociation
• Formation of the second Larson core: when
The protostellar evolutionary phases listed above are quite general (for solar metallicity stars) and almost independent of the computation details. Larson (1969) remarked that at some stages of the cloud evolution a hydrostatic central object (second Larson core) forms that can be considered the first protostellar core. The characteristics of this core (i.e., mass, radius, density and central temperature) appear to be barely sensitive to the adopted cloud initial conditions or to the adopted input physics (see e.g., Masunaga and Inutsuka, 2000; Machida et al., 2008; Tomida et al., 2013; Vaytet et al., 2013). Reasonable intervals for the mass, radius and temperature of the stable hydrostatic core are: mass range of 1–20 MJ, radius values of
In the HR diagram of Figure 3 the sequence that identifies the end of the protostellar accretion (when the star becomes visible) is compared to standard isochrones. Interestingly, the end of the protostellar evolution is very close to the position of the 1 Myr standard isochrone. Larson (1969) and Larson (1972) adopting selected accretion parameters, followed the subsequent evolution until the Hayashi track, finding that low mass stars
FIGURE 3. Comparison between standard isochrones (thin solid lines) and the loci of the end of protostellar accreting sequence for two different initial temperatures of the cloud (10 K thick solid and 20 K thick dashed line). Circles mark the position of 0.25, 0.5, 1, 1.5, 2, 3, and
It is worth to remark that theoretical models for the protostellar evolution cannot be easily checked with observations, as these accreting phases occur when the star is still embedded inside the cloud and thus the central core is largely masked by the matter around it.
4.2 Protostellar Accretion in Hydrostatic Stellar Evolution Codes
As already discussed, hydrodynamic evolution of accreting stars is still a challenging task from the computational point of view. However, concerning the central protostar, it’s not needed to employ a hydrodynamic code, as the protostar itself is in hydrostatic equilibrium after the formation of the second Larson core. In this approximation, the central object can be described using a mono dimensional hydrostatic stellar evolutionary code (see e.g. Stahler et al., 1980a; Stahler et al., 1980b; Siess and Livio, 1997).
On the other hand, hydrodynamic models are needed to predict the structure of the envelope surrounding the protostar, which does not satisfy the hydrostatic conditions but it’s essential to determine the characteristics of the accretion flow. More precisely, the envelope gives information about the accretion rate, the percentage of the energy of the falling matter transferred to the star and the accretion geometry. Information about these quantities are needed inputs for hydrostatic protostellar models. Due to the still present uncertainty on hydrodynamic calculations, all the previous accretion parameters are affected by not negligible theoretical indeterminacy, as briefly summarized below.
• Accretion rate. The accretion rate
• Energy transferred by the accreted matter. The matter falling to the star before reaching the stellar surface has a kinetic energy that can be estimated approximating the matter velocity to the free fall one. However, when it settles on the stellar surface, the kinetic energy has become equal to zero, so the kinetic energy has been converted into another energy form. It can be thermal energy carried inside the star (the accreted matter is hot), or the energy can be partially or totally radiated (photons) before the matter reaches the stellar surface (at the shock front). The fraction of the kinetic energy transferred to the protostar depends on the characteristics of the accretion flow (i.e., density, accretion geometry, accretion rate, see e.g., Baraffe et al., 2012).
The difficulty in treating simultaneously the protostar and the envelope evolution requires some simplifications, which mainly concern the geometry of the accreting protostar-envelope system:
• Spherical accretion (see e.g., Stahler et al., 1980a; Stahler, 1988; Palla and Stahler, 1991; Palla and Stahler, 1992; Palla and Stahler, 1993; Hosokawa and Omukai, 2009a). The star is supposed to be deeply embedded into the parental cloud and the matter falls on it almost radially. The whole stellar surface is subjected to the accretion and the energy radiated by the star can be reabsorbed by the envelope. The whole protostellar accretion occurs as a radial infall from a cloud that has mass enough to generate the star, at a fixed value of the accretion rate.
• Disc accretion (Hartmann et al., 1997; Siess and Livio, 1997; Baraffe et al., 2009; Baraffe and Chabrier, 2010; Tognelli, 2013; Kunitomo et al., 2017). The matter falls from a boundary layer of a circumstellar disc and reaches the star via accretion streams. Most of the stellar surface is not subjected by the accretion and the star is free to radiate its energy, most of which is lost in space. The disc is assumed to be totally decoupled from the central star and it is not treated in the stellar evolution codes. The parameters that define the accretion (accretion rate, disk lifetime, accretion energy) are considered as external free parameters which can be obtained from detailed accretion disc evolution calculations (e.g., Vorobyov and Basu, 2010; Baraffe et al., 2012).
The spherical accretion scenario is likely to describe the first stages of the formation of the protostar when it is still embedded within the cloud that retains an approximate spherical geometry. However, observations suggest that at some stage of protostellar evolution, the cloud collapses to a disc – because of angular momentum conservation – and that it is during the disc accretion that the star gains most of its final mass (see e.g., Natta et al., 2000; Meyer et al., 2007; Watson et al., 2007; Machida et al., 2010 and references therein). So, both scenarios are interesting and describe a part of the protostellar accretion.
The most important difference between spherical and disc accretion, which deeply affects the protostellar evolution, is the amount of energy retained by the accreted matter. Indeed, while in the spherical accretion it’s possible to estimate the amount of energy retained by the accreted matter, in disc accretion this quantity is defined by a free parameter
4.3 Spherical and Disc Protostellar Accretion
The spherical accretion scenario applies to a star that is deeply embedded in a gas cloud. In this case, the evolution of the star and of the envelope have to be treated simultaneously. This allows (at least in principle) to have a consistent evaluation of the accretion rate and the amount of thermal energy that the accretion flows bring inside the star. Qualitatively, the energy emitted from the stellar surface is not free to escape into space since it has to interact with the matter around the star. Thus such an energy is partially reabsorbed by the matter in the envelope, and it eventually reaches the star. The effect of this process is that the star has a kind of external energy source that warms up the stellar surface. The injection of thermal energy from the accreted mass forces the star to expand or at least to compensate for the radius decrease caused by injection of mass.
The impact of spherical accretion on the formation of pre-MS stars has been largely analyzed in the pioneering works by Larson (Larson, 1969; Larson, 1972; Stahler et al., 1980a; Stahler, 1988; Palla and Stahler, 1991; Palla and Stahler, 1992; Palla and Stahler, 1993), and more recently also by Hosokawa and collaborators (Hosokawa and Omukai, 2009a; Hosokawa and Omukai, 2009b; Hosokawa et al., 2010; Hosokawa et al., 2011). One of the main results of such a spherical accretion scenario is that stars during the accretion phase remain bright and with large radii. Using a mild and constant accretion rate of
FIGURE 4. Comparison between standard tracks (solid lines) and the birthline (dotted line). Dots represent T Tauri and Herbig Ae/Be stars. Figure adapted from Palla and Stahler (1993).
More recent sets of birthlines can be found in Hosokawa and Omukai (2009a): such accretion models was computed for different values of the accretion rate (from
However, it’s commonly accepted that stars do not accrete mass spherically during their entire protostellar phase; on the contrary they gain most of their mass from an accretion disc. This motivates the detailed study of protostellar accretion from a disk geometry. Differently from the spherical accretion, in the disc geometry the accretion streams cover only a very limited part of the stellar surface (few percent, see e.g., Hartmann et al., 1997) and almost the whole star is free to radiate its energy into space. Another difference is that all the accretion parameters (i.e., accretion rate, fraction of energy inside the accreted matter, etc..) are treated as external parameters in disc accretion models.
In the disc accretion geometry, it’s possible to follow also an analytic approach to analyze the main characteristics of the accreting star. Following the formalism presented in Hartmann et al. (1997), it’s possible to write a simple equation for the temporal evolution of the accreting star radius:
where M and R are the stellar mass and radius,
Equation 3 contains three terms: the first and second are the normal terms that define the evolution of the star with a surface radiative loss
Looking at Eq. 3 it is clear that a radius expansion requires a positive value of the right side of the equation, which can be obtained or via an efficient deuterium burning (large
4.4 D-Burning During Protostellar Accretion
To check if the D-burning alone can produce a protostar with a large radius, in agreement with observations, we assume
FIGURE 5. Effect of different deuterium original abundances (see labels),
4.5 Accretion Energy
There is another natural way to obtain a radius expansion in protostars, which is assuming that the ingested matter retains part of its internal energy; this means to assume a value of
FIGURE 6. Effect of
4.6 Connecting the Standard pre-MS and the Protostellar Accretion Phase
From the previous discussion, it emerges that, depending on the characteristics of the protostellar accretion, the protostar could end its first evolution with a structure similar or in some cases profoundly different from that obtained in a normal gravitational contraction along the Hayashi track. The largest discrepancy with standard pre-MS evolution occurs in the case of cold accretion starting from a seed of the order of few Jupiter masses, as in that case the classical Hayashi track is almost completely skipped (see e.g., Baraffe et al., 2009; Tognelli et al., 2020).
Figure 7 shows the evolution in the HR diagram of cold accretion models starting from different
FIGURE 7. Evolution in the HR diagram of protostellar models with different values of
As discussed, it is likely that stars first accrete in a spherical hot scenario and then, at a given stage, switch to a disk-like accretion. In this case the transition from hot to cold accretion occurs for some value of the protostellar mass (possibly dependent on the amount of mass available in the cloud/disk). This mixed scenario has been investigated by Hosokawa et al. (2011) to show that the protostar remains bright enough to end the protostellar phase close to a Hayashi track. Top panel of Figure 8 shows the models by Hosokawa et al. (2011). The purely hot accretion scenario (purple solid line), which corresponds to a hot birthline obtained assuming a spherical accretion, attains large luminosities and radii well above the standard 1 Myr isochrone. Figure shows also the results of models where the accretion switches from hot to cold at a given value of the stellar mass, namely
FIGURE 8. Evolution in the HR diagram of purely and partially hot models computed with an accretion rate
Baraffe et al. (2009) and Baraffe et al. (2012) investigated also the possibility to produce bright objects using an episodic accretion. The basic idea behind the models is that during intense bursts mass accretion phases the protostar can accrete matter in the hot-accretion configuration [
What emerges from the previous analysis is that, if one assumes masses and radii typical of the second Larson core, cold models cannot produce the observed bright stars in young clusters, but it is required the presence of hot accretion phases. Thus, the results seem quite comfortable: in most hot disk or spherical geometry, the protostellar accretion leads to pre-MS models with characteristics similar to those predicted in standard pre-MS evolution. More importantly, the position in the HR diagram of such models is in agreement with observational data. On the contrary, for the accretion parameters leading to a final mass model different to that of the standard one, as in the cold accretion scenario, the position in the HR diagram is in disagreement with disk star observations, rising doubts about the validity of such models.
5 Light Elements Surface Abundances and Nuclear Burning During the pre-MS Phase
Lithium, together with beryllium and boron, belong to the class of light elements burnt in pre-MS, because of their relatively low nuclear destruction temperature (between 2–5 million degree). The threshold values for the burning temperature depend mainly on the considered element, on the stellar mass (density and evolutionary stage) and slightly on the chemical composition of the star (in particular on helium and metals abundances). For pre-MS solar metallicity stars in the mass interval [0.08, 1.0]
FIGURE 9. Evolutionary tracks for solar metallicity in the HR diagram with indicated the regions where surface light element abundances decrease due to burning (where the temperature at the bottom of the convective envelope is higher than the burning temperature of Li, Be and B). The stellar models have been computed using the PISA evolutionary code with the same input parameters described in Lamia et al. (2015).
The abundance of light elements at the stellar surface are strongly influenced by the nuclear burning as well as by the inwards extension of the convective envelope and by the temperature at its bottom. Consequently, the comparison between theory and observation for Li, Be and B surface abundances are useful to constrain theoretical models and in particular the convective envelope depth.
From the observational point of view, most of the data for light elements concern the abundance of 7Li whose line (at 670.779 nm) can be safely resolved even in cold stars, as witnessed by the huge amount of data for stars in clusters or isolated stars at different metallicities (see e.g., Sestito and Randich, 2005; Delgado Mena et al., 2014; Delgado Mena et al., 2015; Aguilera-Gómez et al., 2018 and references therein).
6Li burns at a lower temperature with respect to 7Li, consequently it’s almost completely destroyed when 7Li burning becomes efficient. Thus a potential detection of observable amounts of 6Li in stellar atmospheres would constrain the destruction of the less fragile 7Li (Copi et al., 1997). Since the depth of the convective zone increases with metallicity, 6Li is almost completely depleted in high metallicity disk stars, as in the Sun (see e.g., Asplund et al., 2009) and it is below the detection level also for most thick disk and halo stars (see e.g., Spite and Spite, 2010). The possible abundance of 6Li below the limit of detection also for halo stars could be explained by the fact that the amount of 6Li formed by the standard Big Bang and by the cosmic rays is supposed to be very low. Moreover, a very small 6Li abundance in these stars would be very difficult to detect, in particular because the lines (doublets) of 6Li and 7Li are overlapping (see also discussion in Sec. 5.5).
Beryllium and boron measurements are more problematic than 7Li observations. 9Be abundance is measured using near-UV lines, only in stars with
The abundance of the boron isotopes is even more difficult to measure than Be because the boron lines fall mainly in the UV part of the spectra where the Earth atmosphere is not transparent. In addition, for disk metallicity stars, B lines suffer strong blending problems (see, e.g., Cunha, 2010). Similarly to Be, B abundance are available in a mass range where B is expected to be not burnt in standard models. Despite the observational difficulties Be and B surface abundances data are available for some stars even at low metallicities (see e.g., Boesgaard and Novicki, 2005; Lodders et al., 2009; Tan et al., 2009; Primas, 2010; Boesgaard et al., 2011). In the observed stars, the ratio 11B/10B seems to be of the order of four, in agreement with solar values and meteorite results, even if it is very difficult to spectroscopically discriminate among the boron isotopes (see, e.g., Proffitt and Quigley, 1999; Prantzos, 2012). Be and B surface abundances have been also measured in the Sun where, as expected, they are not burned (see e.g., Asplund et al., 2005; Asplund et al., 2009; Lodders et al., 2009; Lodders, 2010).
The temperatures for light elements burning can be reached in stellar interiors during the pre-MS evolution of stars with masses larger than about 0.05–
Figure 10 shows the temporal evolution of the temperature at the bottom of the convective envelope,
FIGURE 10. Temporal evolution of the temperature at the bottom of the convective envelope for masses in the range [0.08,1.0]
The situation is similar for the other light elements; clearly one has to take into account the different burning temperatures, so that the mass range in which Be and B are destroyed at the base of the convective envelope is different from that in which lithium is burned. As an example, for solar composition models, Be can be burnt at the bottom of the convective envelope in the mass interval
Figure 11 gives an example of the 7Li surface abundance time behavior predicted for stars in the mass range [0.08, 1.0]
FIGURE 11. Temporal evolution of surface lithium abundance (normalized to the initial one) during the pre-MS for stars in the mass range [0.08, 1.0]
Referring to Figure 11, in fully convective stars, i.e.,
Figures 10, 11 refer to a standard evolution, where the star is fully formed at the top of the Hayashi track. The situation can be different if protostellar accretion is taken into account, in particular in those cases where the star at the end of the protostellar phase is compact and faint, which corresponds essentially to the case of cold accretion models. This could affect light element burning in two different ways: 1) in principle for some possible values of the accretion parameters it could be possible the burning of light elements (most likely lithium) during the protostellar phase 2) accretion can change the pre-MS stellar characteristics with respect to those already predicted by the standard scenario so that the light element burning efficiency is changed too. We will discuss the effect of the protostellar accretion on the surface chemical composition in Section 5.2.
5.1 Surface Lithium Abundance in Open Clusters
Many questions are still open about the large discrepancies between the predicted and observed surface lithium abundance in young clusters, where standard models tend to underestimate the surface abundance at a given age (see e.g., D’Antona et al., 2000; D’Antona and Montalbán, 2003; Jeffries, 2006; Tognelli et al., 2012 and references therein). Moreover the presence of a large scatter in the observed Li abundance among stars with similar
It is worth noticing that, qualitatively, standard models (without accretion) are capable to produce a pattern of lithium vs. mass (or
FIGURE 12. Theoretical surface lithium abundance predicted for solar metallicity stars at 10 Myr, as obtained in standard evolutionary models using the PISA evolutionary code.
• Starting from a certain value of the effective temperature (that depends on the cluster age) the surface lithium content, at a given cluster age, increases with the
• For lower masses (and thus
• In region (1), reducing the mass (or
5.2 Lithium Abundance Evolution During Protostellar Accretion
As discussed in the previous section, the inclusion of the protostellar accretion phase could (in dependence of the adopted accretion parameters) drastically alter the evolution of a pre-MS star. In this section we briefly review the main effects of the protostellar accretion phase on the surface lithium abundance during the protostellar and pre-MS evolution as a function of the different possible accretion parameters.
This problem was first analyzed by Baraffe and Chabrier (2010), who showed that the inclusion of protostellar accretion in solar metallicity stars with different input parameters can lead to a variety of cases for which the resulting lithium abundance (in pre-MS or in MS) is different from what expected in standard pre-MS evolution (see also Baraffe et al., 2012; Tognelli, 2013; Kunitomo et al., 2018). We recall that accretion models depend on many parameters, but the main quantities that strongly affect the pre-MS evolution are the seed mass and the accretion energy deposed into the star. The general picture that emerges is that in cold accretion models lithium is efficiently destroyed during the protostellar accretion or at the very beginning of the pre-MS phase. Thus these stars should show a very low surface lithium content.
A detailed analysis of the effect of the protostellar accretion on surface lithium abundance for different subsolar metallicities (
• Pure cold accretion case. The accretion leads to stellar structures at the end of the protostellar phase different with respect to standard non accreting models, affecting also the lithium burning efficiency. If the seed mass is of the order of few Jupiter masses, the models result to be so compact and hot that start to efficiently deplete lithium before the end of the accretion phase. The level of depletion is mainly determined by the seed mass and it’s only slightly affected by the other accretion parameters (accretion rates, initial radius). After the protostellar phase, for masses larger than about 0.1–
• Hot accretion case. In Section 4.6 we showed that if stars accrete part of their mass during an hot accretion phase (during which the protostar is maintained at large radius by the accretion energy), the star at the end of the accretion phase is more similar to a standard evolutionary models. In this case, protostars are relatively cold and they do not deplete an appreciable amount of Li. Then, when the star enters the pre-MS the residual lithium is essentially equal to the original one, as predicted by models without accretion and from this moment on the lithium evolution proceeds as in standard stellar evolutionary models.
These two scenarios embrace many other possible solutions, obtained by modify/tuning the accretion parameters and the accretion history to produce, at least in principle, intermediate scenarios. However, a fine tuning of the accretion parameters that depends also on the stellar mass is unlikely and could produce artificial results (Tognelli et al., 2020). The two extreme scenarios highlight an important point. The expected Li abundance is strictly connected to the protostellar evolution. Stars that due to the inclusion of the protostellar accretion skip the Hayashi track (i.e., pure cold accretion) undergo to an efficient lithium burning during the protostellar phase, in disagreement with standard predictions. This kind of models are excluded, at least for disk metallicities, by observational data.
The possible effects of accretion on stellar characteristics and Li temporal evolution could also be linked to the question of the luminosity spread observed in star forming regions. The problem consists in the fact that stars with the same
5.3 Lithium in Old Metal Poor Stars
An interesting aspect to be discussed about lithium evolution is the cosmological lithium problem. Halo stars show a lithium plateau for
However, BBN predicts a primordial lithium content of A(Li)-2.75, (see e.g., Cyburt et al., 2016; Pitrou et al., 2018). This estimate depends on the density of baryons, which is related to the fluctuations of the cosmic microwave background measured with WMAP and Planck satellites. The BBN predictions for the primordial lithium abundance are thus 0.3–0.6 dex larger than the Spite plateau value. This discrepancy constitutes the so called “cosmological lithium problem.” Several attempts to introduce new physics (exotic particles) or to review the reaction rates during the BBN have been performed, but without any firm conclusion (see e.g., Fields, 2011; Pizzone et al., 2014; Goudelis et al., 2016; Coc and Vangioni, 2017; Damone et al., 2018; Lamia et al., 2019).
Similarly, on the stellar evolution side, the problem has been analyzed to find a possible mechanism to deplete the same lithium amount for the stars with different masses and metallicities which populate the Spite plateau. Diffusion has been investigated as a possible solution, as it slowly brings surface lithium below the convective region (Richard et al., 2005). This process acts on timescales typical of the MS evolution, but its efficiency depends on the stellar mass and thus in the mass range corresponding to the Spite Plateau the effect of diffusion increases with
Another possibility is that lithium depletion occurs during the pre-MS. In this case, Fu et al. (2015) suggested that a certain level of undershooting4 at the bottom of the convective envelope of pre-MS stars could increase the depletion of surface lithium. In addition, a residual matter accretion, regulated by the stellar luminosity, could provide gas with pristine chemical composition (and thus lithium to the star), obtaining in pre-MS the depletion level required to produce the Spite plateau. However, in such models, MS diffusion must be inhibited to avoid a
Recently Tognelli et al. (2020) analyzed the possibility to produce a constant lithium depletion in pre-MS taking into account in the models the protostellar evolution with different accretion parameters. As discussed in Sec. s 4.6 and 5.2, depending on the scenario adopted during the protostellar evolution, stars at the beginning of the pre-MS can be profoundly different from the ones evolved starting from the Hayashi track. The reason is that the protostellar phase can deeply affect the thermal structure of a star. As a result, it’s possible to induce a lithium depletion in pre-MS or even during the protostellar phase, but it requires the adoption of a fine tuning of the parameters that govern the stellar mass acretion (see e.g., Fu et al., 2015; Tognelli et al., 2020). Moreover, as already discussed, the models that show a significant Li depletion, follow a pre-MS evolution in the HR diagram that is different to that observed for high metallicity pre-MS stars. The lack of Galactic very young and metal poor stellar systems, in which one could observe pre-MS stars with Spite plateau metallicities, avoid the possibility to restrict the range of valid accretion parameters and reach firm conclusions.
To conclude, the proposed mechanisms could in principle alleviate the cosmological lithium problem, but the weakness of all these suggested solutions is that a fine tuning of the parameters is still required to produce a constant lithium depletion reproducing the Spite plateau.
5.4 Uncertainties on Predicted Surface Lithium Abundance
The predicted depletion of surface lithium abundance (and, in general, of the light element surface abundances) is affected by the uncertainties on the input physics adopted in stellar models and on the assumed chemical composition, that influence the extension of convective envelope and temperature structure of the star. In particular, also the uncertainty on the nuclear burning cross sections plays a role in the light element abundance predictions. In the literature there are several attempts to estimate the impact of the uncertainties on the parameters/input physics adopted in stellar models on the predictions of lithium abundance in low and very-low mass stars (see e.g., Piau and Turck-Chièze, 2002; Burke et al., 2004; Tognelli et al., 2012; Tognelli et al., 2015). The quantities that mainly affects lithium, as analyzed in the quoted papers, are the following (see e.g., Chapter 3 in Tognelli, 2013):
• Radiative opacity and equation of state. The extension of the convective envelope is determined by the Schwarzschild criterion which simply states that a region is convective if the radiative temperature gradient
• Outer boundary conditions. The outer boundary conditions are the pressure
• Mixing length parameter. The convection efficiency in super-adiabatic regimes in 1D stellar evolution codes commonly relies on mixing-length theory (MLT) (Böhm-Vitense, 1958). In this formalism, the scale on which the heat is efficiently transported by convection is defined as
• Nuclear cross section. The error on the cross section for the 7Li(p,
• Chemical composition. The initial abundance of helium (Y) and metals (Z) in the star is not known, but it can be estimated from the observed [Fe/H], assuming for metal rich stars the same relative abundance of metals of the Sun, while for metal poor galactic stars a suitable alpha-enhancement must be introduced5. The conversion of [Fe/H] into Y and Z depends on the assumed values of: 1) the primordial helium mass fraction produced in the BBN
Tognelli (2013) quantitatively evaluated the impact on the predicted surface lithium abundance of the uncertainties in the input physics and in the initial chemical composition discussed above, calculating upper and lower limits of surface 7Li in stellar models. Figure 13 shows the estimated upper/lower limits (plotted as error bars) of surface lithium abundance and effective temperature, due to the contribution of the input physics uncertainties (top panel) and chemical composition indeterminacy (bottom panel). Stars with different masses at different ages typical of young clusters are shown (for more details on the procedure adopted to obtain these limits see Tognelli et al., 2012; Tognelli, 2013). The errors on the present input physics and the typical uncertainties on the adopted chemical composition have a drastic impact on the predicted surface lithium abundance, which can vary by more than one order of magnitude, especially for stars with
FIGURE 13. Uncertainties on surface lithium abundance and effective temperature due to the errors on adopted input physics (left panel) and chemical composition (right panel). Figure adapted from Tognelli (2013).
In standard models, the only possibility to deplete surface lithium in pre-MS is via convective mixing. If the bottom of the convective envelope reaches a region hot enough to burn lithium, then the surface lithium decreases in time. The level of depletion depends on a key parameter in convective stars, which is the efficiency of convective energy transport. A more efficient convective transport produces hotter stars that consequently experience a more efficient lithium burning. The opposite occurs if the convection efficiency reduces.
A precise physical treatment of external convection would require three-dimensional hydrodynamic models which have been improved in recent years, but only for limited regions of the star corresponding mainly to the atmospheric regions (see e.g., Nordlund et al., 2009; Collet et al., 2011; Freytag et al., 2012; Magic et al., 2013; Trampedach et al., 2013; Trampedach et al., 2014; Trampedach et al., 2015; Pratt et al., 2016, and references therein). These codes are state-of-the-art (magneto) hydrodynamic code that solves the time-dependent hydrodynamic equations for mass, momentum, and energy conservation, coupled with the 3D radiative transfer equation, in order to correctly account for the interaction between the radiation field and the plasma. However, hydrodynamic calculations still cannot cover the wide range of physics quantities needed to model the Galactic stellar populations. Moreover, their results cannot be easily adopted in stellar evolutionary codes, although attempts to implement approximations directly based on 3D simulations in 1D stellar models exist in the literature (e.g., Lydon et al., 1992; Ludwig et al., 1999; Arnett et al., 2015; Arnett et al., 2018). The commonly adopted procedure to treat the convection efficiency in super-adiabatic regimes in 1D stellar evolution codes relies on mixing-length theory (MLT) (Böhm-Vitense, 1958), where convection efficiency depends on the free parameter
Figure 14 shows the results obtained by Tognelli et al. (2012) where the observed surface lithium abundance in five young open clusters (IC2602, α Per, Blanco1, Pleiades and NGC2516) has been compared to theoretical predictions obtained adopting two different values for the mixing length parameter during the Pre-MS phase: one calibrated on MS stars to reproduce their colors and the other corresponding to a much less efficient convective energy transport. The best value to be used in pre-MS has been estimated in order to reproduce the A(Li) vs.
FIGURE 14. Comparison between data and theoretical model predictions for surface lithium in young clusters. Models with the same mixing length parameter in pre-MS and MS phases, calibrated with MS stars, are shown as dashed lines while models with calibrated convection efficiency in MS and artificially reduced mixing length parameter
FIGURE 15. Effect of magnetic fields (for the labeled surface magnetic field strength <B f>) on the radius evolution of a partially convective pre-MS star
The presence of a magnetic field makes the star cooler and modifies the temperature stratification inside the star: this has a direct impact on surface lithium burning efficiency (Feiden, 2016). Supplementary Figure S1 shows the comparison between the expected surface lithium abundance as a function of
Another aspect related to the presence of magnetic field in the star is the possibility to include in stellar models a surface spots coverage fraction (see e.g., Jackson and Jeffries, 2014; Somers and Pinsonneault, 2015). The effect of the spots is to reduce the outgoing flux at the stellar surface producing a radius inflation and a decrease of the stellar effective temperature. Such an effect goes in the same direction of an artificially decreased convection efficiency and, as expected, leads to a cooler envelope and to a less efficient lithium burning. Supplementary Figure S2 shows an application of stellar models with surface spots to the surface lithium abundance observed in the Pleiades cluster (see Somers and Pinsonneault, 2015). Standard models predict a level of pre-MS lithium depletion larger than that observed. The agreement can be restored assuming that a certain fraction of the stellar surface is covered by spots; increasing the coverage fraction, the models are cooler and thus the surface lithium depletion is reduced. It’s important to notice that the presence of magnetic fields of different strength or a different spot coverage fraction in stars with similar mass can introduce a star-to-star scatter in the surface lithium abundance. This could partially answer another important open question about young clusters, i.e., which is the cause of a spread in the lithium abundance, measured in stars with similar effective temperature. The extent of such spread is much larger than the quoted uncertainties, so it represents a real spread (see e.g., Xiong and Deng, 2006). The inclusion of a different surface spots coverage or different magnetic fields strength could produce stars with similar effective temperatures (but different total masses) thus leading to an apparent dispersion in the lithium abundance.
Additional mechanisms that can alter the level of lithium burning in stars have been analyzed in the literature. An induced extra mixing due to the presence of rotation, gravity waves, diffusion, or mass loss has been put forward to reproduce the surface lithium abundance pattern typical of older stars (ages
5.5 Uncertainties on Atmospheric Models for Surface Lithium Abundance Determination
The determination of surface element abundances involves the interpretation of the observed absorption lines through atmospheric models as accurate as possible. However modeling stellar atmospheres is a difficult task and the uncertainties on the measurements of surface element abundance clearly affects the comparison between theory and observations. Here we limit to briefly discuss the main difficulties in modeling realistic stellar atmospheres, the interested reader can find more details in other reviews (see e.g., Jofré et al., 2019).
The photosphere of low mass stars is covered with a complex and stochastic pattern – associated with convective heat transport – of downflowing cooler plasma and bright areas where hot plasma rises, the so called granulation (Nordlund et al., 2009). As already discussed, convection is a difficult process to understand, because it is non-local and three-dimensional, involving non-linear interactions over many disparate length scales. In recent years it has become possible to use numerical three-dimensional (3D) radiative hydrodynamical (RHD) codes to study stellar convection in atmosphere such as Stagger Code (Nordlund et al., 2009; Collet et al., 2011) and CO5BOLD (Freytag et al., 2012). Nowadays, the use of large grids of simulations covering a substantial range of values of effective temperature and surface gravity for stars in different regions of the HR diagram (Ludwig et al., 2009; Magic et al., 2013; Trampedach et al., 2013) have proven that the convection-related surface structures have different size, depth, and temporal variations, depending on the stellar type (Beeck et al., 2013; Tremblay et al., 2013; Magic and Asplund, 2014). Moreover, the related activity (in addition to other phenomena such as magnetic spots, rotation, dust, etc.) has an impact in stellar parameter determination (Bigot et al., 2011; Chiavassa et al., 2012; Creevey et al., 2012), radial velocity (Allende Prieto et al., 2013; Chiavassa et al., 2018), chemical abundances estimates (Asplund et al., 2009; Caffau et al., 2011), and photometric colors (Bonifacio et al., 2018; Chiavassa et al., 2018).
Chemical abundance ratios inferred from spectra of cool stars is based on the understanding of limitations and uncertainties of spectroscopic analyses. In this context, radiation transfer in the atmospheres of late-type stars generally takes place under non-local thermodynamic equilibrium (NLTE) conditions, rather than the idealized LTE (Asplund, 2005). The full 3D NLTE treatment would require to compute NLTE radiative transfer inside radiative hydrodynamical simulations and coupling it to the equations of gas movements. In these simulations the computational cost is dominated by the radiative transfer calculations which can be greatly reduced by adopting an approximated solution based on the opacity binning or multi-group method (Nordlund, 1982). However, introducing NLTE calculations at this stage would largely increase the computation time making very complicated to obtain a relaxed simulation. This is why 3D NLTE radiative transfer calculations are only affordable in a post-processing manner, i.e., each 3D RHD simulation is computed in LTE and then the so called <3D> models are computed averaging multiple snapshots of 3D RHD simulations over regions of equal optical depth and over the time series (e.g., Asplund et al., 2004; Caffau et al., 2009; Magic et al., 2013; Lind et al., 2017; Nordlander et al., 2017; Amarsi et al., 2018). This approach offers a middle-ground between full 3D NLTE and 1D NLTE, by accounting for NLTE in model atoms of arbitrary size, and through the use of time-independent 1D structures derived from the full 3D hydrodynamic simulations (Bergemann et al., 2017). Using this method, Wang et al. (2021) derived a new 3D NLTE solar abundance of A(Li) = 0.96 ± 0.05, which is 0.09 dex lower than the commonly used value and provided a grids of synthetic spectra and abundance corrections publicly available. Eventually, it has also become possible to undertake large samples of observations from disk and halo stars with this 3D NLTE analysis (Amarsi et al., 2019; Bergemann et al., 2019). Unluckily at present 3D atmospheric calculations are not still available for Pre-Main Sequence atmospheres.
The measurement of surface lithium abundances constitutes an important example of efforts undertaken in this field. In Sec. 5 we mentioned that the stellar surface abundance of 6Li is expected to be negligible, moreover its identification is very difficult. The presence of 6Li in metal-poor halo stars can only be derived from the asymmetry in the red wing of the 7Li doublet line at 670.8 nm. Several authors attempted to detect 6Li using 1D hydrostatic models and assuming LTE for a number of metal-poor stars with [Fe/H], lower than −2 dex (Cayrel et al., 1999; Asplund et al., 2006). Cayrel et al. (2007) pointed out that the intrinsic line asymmetry – due to the stellar surface convection – in the 7Li doublet would be almost indistinguishable from the asymmetry produced by a weak 6Li blend on a (presumed) symmetric 7Li profile.
The total line strength of the Li resonance line determines the 7Li-abundance and the shape of the line profile determines the isotopic ratio due to the shift between 6Li and 7Li isotopic components. Thus it’s critical to resolve the strongly differential NLTE effects on the granules and inter-granular regions (Supplementary Figure S3), because they have a preferential influence over the blue- and red-shifted part of the line profile, respectively (Lind et al., 2013). To investigate this aspect, Steffen et al. (2012) and Lind et al. (2013) used a 3D NLTE treatment with 3D RHD simulation snapshots carried out with CO5BOLD and Stagger Code, respectively. They re-analyzed the Li feature in some metal-poor stars and were not able to confirm the previous claimed detection of 6Li. However, they pointed out that a full understanding of 3D NLTE line formation is necessary to make correct measurements of 6Li, even though from their studies they could give only upper limits for the isotopic ratio 6Li/7Li. In particular, the 3D NLTE approach is important to characterize the calibration lines, to decrease the observational error. Eventually, a very recent publication by González Hernández et al. (2019), confirms the non detection of 6Li for a very metal poor binary star [(Fe/H) ∼−3.7 dex], finding an upper limit for the isotopic ratio of 6Li/7Li
6 Effects of Light Element Burning Cross Sections on Pre-Main Sequence Characteristics and on Light Element Surface Abundances
The predicted temporal evolution of light elements is affected by the stellar evolutionary stage and by the model structure, which depends on the input physics adopted in the computations. One of the key ingredients is the adopted light element burning cross section, as derived from measurements of indirect/direct processes. Thus it’s worth discussing how the recent determination of such cross sections at energies relevant for stellar evolution have changed the prediction of surface light element abundances in low mass stars.
For stellar calculation the reaction rate of a two body process can be written in the following way (see e.g., Rolfs and Rodney, 1988),
For many application in stellar astrophysics, it is possible to expand the astrophysical factor around a specific value of the energy, thus obtaining,
The expansion of
Light element (p,α) reaction rates have been recently revised through the indirect Trojan Horse Method (THM, see e.g., Baur, 1986; Spitaleri et al., 2003; Spitaleri et al., 2016; Spitaleri et al., 2019 and references therein), which is particularly useful to measure cross sections at astrophysical energies by-passing extrapolation procedure, often affected by systematic uncertainties due, for instance, to electron screening effects. THM allows to measure the astrophysically relevant cross sections in correspondence, or very close, to the Gamow peak without experiencing the lowering of the signal-to-noize ratio due to the presence of the Coulomb barrier between the interacting charged particles. Moreover in the last years THM was successfully applied to reactions induced by unstable beams Pizzone et al. (2016) and Lamia et al. (2019) as well as neutron induced reactions which may play a role also in the context of light element nucleosynthesis and BBN. In particular the 3He(n,p)3H was studied at the astrophysically relevant energies (see Pizzone et al., 2020 and references therein). THM was also applied to reactions between heavier systems, which are relevant in the late stellar evolutionary stages (Tumino et al., 2018). We will discuss the effects of these improvements and of the present errors on nuclear cross sections on the light elements surface abundance calculations in pre-Main Sequence stars.
6.1 Effects of Deuterium Burning Cross Sections on pre-MS Evolution
As discussed in Sec. s 3, 4.3 and 4.4, deuterium burning plays a crucial role in the first stages of pre-MS or protostellar evolution. The value of the cross section of the p(D,γ)3He process in stellar conditions has been reported by several authors (see Adelberger et al., 2011 for a review) both from measurements and theoretical calculations along with its uncertainty. Adelberger et al. (2011) redetermined the best value for the astrophysical factor
We tested the impact on pre-MS evolution of the D+p cross section variation, using the uncertainty given by Adelberger et al. (2011) at stellar energies, which is
Recently Tumino et al. (2014) (see also Tumino et al., 2011) measured the reaction rate for two additional channels involving the D-burning, namely the D(D,p)3H and the D(D,n)3He processes, using the THM; such burning channels could potentially contribute to the D-burning in stars. Supplementary Figure S4 shows, for the quoted reactions, the THM rates compared to the ones of still widely used NACRE compilation (Angulo et al., 1999), of the JINA REACLIB (Cyburt, 2004) and to the (Descouvemont et al., 2004) rates. The estimated uncertainty on the analyzed burning channels (of about 5%) are also shown. At temperatures typical of stellar deuterium burning (∼106 K) the D(D,p)3H is about 5% larger than the NACRE, while it is much larger (about 15%) than the value reported in Cyburt (2004). The differences sensitively reduce at larger temperatures, more important for cosmological calculations. If the Descouvemont et al. (2004) rate is considered, the difference with THM is very small at stellar temperatures (below 1%), while it increases at larger temperatures, reaching about 5% at
The new THM rate for the D(D,n)3He reaction is ∼10% larger than the others (NACRE, JINA and (Descouvemont et al., 2004)) for temperatures smaller than about
6.2 Stellar Surface Abundance of Light Elements and Updated (p,α) Reaction Rates
The energy produced in the Li, Be and B nuclear reactions is negligible and such reactions do not affect stellar structures evolution. However, the surface abundances of light elements strongly depend on the nuclear burning (Tognelli et al., 2019). The different fragility of Li, Be and B against (p,α) destruction reactions potentially allows to investigate the characteristics of different depths of the stellar interior.
In Supplementary Figures S5, S10 the reaction rates for the most relevant light element burning (p,α) reactions calculated with the THM are shown and compared with the JINA REACLIB and the less recent, but still widely used, NACRE rates. The results are discussed below.
6.2.1 6Li and 7Li Surface Abundance and (p,α) Reaction Rates Efficiency
The first attempts to apply THM (p,α) reaction rates to pre-MS lithium surface abundance calculations were performed by Pizzone et al. (2003) and Pizzone et al. (2005) (hereafter pt05) and successively updated, after re-normalization to recently released direct data, in Lamia et al. (2012) and Lamia et al. (2013). The left panel of Supplementary Figure S5 shows the 6Li(p,α)3He reaction rate obtained using the THM compared to the pt05 rate available on the JINA REACLIB page. The THM estimated rate deviates from the pt05 by about 15% at a temperature of
Lamia et al. (2013) evaluated the effect on the surface 6Li abundance of the update of the 6Li+p reaction rate for a range of stellar masses at three different metallicities [(Fe/H) = −0.5, −1.0, and −2.0]. Supplementary Figure S6 shows the time evolution of the surface 6Li abundance – normalized to the original value– obtained adopting the three different 6Li(p,α)3He reaction rates – THM, JINA (pt05), and NACRE. From Supplementary Figure S6 it is evident that 6Li depletion, at fixed burning reaction rate, varies significantly for different masses and metallicities. This can be understood recalling that the higher the metallicity (or the lower the stellar mass) the deeper and hotter the base of the convective envelope. Note that among the most massive models (i.e.,
Right panel of Supplementary Figure S5 shows the comparison between the THM and NACRE 7Li + p reaction rate; the difference is of about 13% at a temperature of
In general, the effect of adopting different 6Li and 7Li burning reaction rates, although not negligible, is less important than the effects due to errors in other quantities used in the computation of a stellar model, such as the original chemical composition, external convection efficiency, or the uncertainties in some other physical inputs adopted in the calculations (e.g., opacity and equation of state, see Sec. 5.4 and the discussions in Pizzone et al., 2010 and Tognelli et al., 2012). Thus, at the moment an uncertainty on the burning reaction rate of the order of 10% is not the main error source in the determination of the surface lithium abundance in stellar models (Lamia et al., 2012; Lamia et al., 2013).
6.2.2 Lithium Depletion Boundary
In the mass range
The method of LDB has been successfully adopted to assign ages to young clusters as a competitive method to the use of isochrone fitting (e.g., Barrado y Navascués et al., 1999; Oliveira et al., 2003; Jeffries and Oliveira, 2005; Manzi et al., 2008; Dobbie et al., 2010; Jeffries et al., 2013; Binks and Jeffries, 2014; Juarez et al., 2014; Dahm, 2015; Dupuy et al., 2016; Martín et al., 2018; Martín et al., 2020 and references therein). The uncertainties on age determination through the LDB technique have been analyzed in Burke et al. (2004) and, more recently, in Tognelli et al. (2015); the main uncertainties that potentially affect the LDB age determination are those already discussed in Section 5.4. Supplementary Figure S9 shows the relative age uncertainty on LDB age determination obtained by Tognelli et al. (2015) taking into account the errors on the adopted input physics and chemical composition. The shaded area has been obtained by a cumulative variation of all the considered input physics and chemical abundances within their uncertainties (going into more detail would require a too long discussion out of the purposes of this review, see the quoted paper for additional information). The uncertainty of the method depends on the stellar luminosity (or mass) at the LDB, which, in turn, translates in an age, but it is in any case lower than about 10%. As a general comment, faint stars that correspond to LDB ages of the order of 50–60 Myr
6.2.3 9Be and 10B Surface Abundance and (p,α) Reaction Rates Efficiency
Supplementary Figure S10 shows the comparison between the recent THM reaction rates and other reaction rates used in the literature. The THM 9Be(p,α)6Li rate (at temperatures of few million degrees) is about 25% larger than NACRE and the uncertainty on the THM 9Be burning rate is about 25% (blue dashed area in the figure). The THM reaction rate is quite similar to that in the recent NACRE II (Xu et al., 2013), even if the THM uncertainty region is sensibly smaller than that of the NACRE II rate, see (Lamia et al., 2015). Left panel of Supplementary Figure S11 shows the comparison between the predicted surface Be abundances computed using the THM reaction rate and the NACRE reaction rate, for solar metallicity stars. The higher THM rate leads to a faster 9Be destruction consequently, at the same age, models with the THM rate show a lower 9Be surface abundance with respect to models with the NACRE rate. The differences in the predicted surface abundances are significant in those models where 9Be is efficiently destroyed (i.e., for
We recall that in stars, 9Be is destroyed through two channels: 9Be(p,α)6Li (R1, the rate analyzed by Lamia et al., 2015) and 9Be(p, 2α)2H (R2). The ratio R1/
Right panel of Supplementary Figure S10 shows the comparison between the THM and the NACRE rate for the 10B(p,α)7Be reaction: at temperatures of a few millions of kelvin, the THM rate is ∼30% lower than the NACRE one. The error of the THM rate at the temperatures of interest is about 20%. However if the THM rate is compared with the one of the updated NACRE II Compilation (Xu et al., 2013), the differences are significantly reduced, see (Lamia et al., 2015).
The effect of the different 10B(p,α)7Be reaction rates on surface B abundance in low mass stars (at solar metallicity) is shown in the right panel of Supplementary Figure S11. The lower THM 10B(p,α)7Be cross section leads to a smaller 10B destruction and thus to a larger surface 10B abundance at a fixed age. Due to the higher 10B burning temperature with respect to 9Be, the effect of reaction rate change is significant only for masses
For completeness in Supplementary Figure S11 we point out for ages typical of MS evolution [e.g.,
Supplementary Figure S12 shows the surface logarithmic abundances of 9Be and 10B as a function of the effective temperature,
To our knowledge, an analysis of the dependence of surface 9Be or 10B abundances on the errors in input physics and chemical composition adopted in the calculations is not available in the literature. However, it’s a good approximation assuming a Be and B burning sensitivity to the input physics similar to that obtained for 7Li, as the burning temperatures are not much different. Under this assumption, the effects of the uncertainty on (p,α) Be and B burning reaction rates is not the dominant error source for the surface abundance predictions of such elements.
7 Summary and Conclusion
Surface light elements abundances prediction in stellar models is a difficult task because it’s affected by several errors in the adopted input physics and uncertainties in the efficiency of some physical mechanisms as e.g., convection in the stellar envelope. Moreover, pre-MS characteristics and surface light element abundances depend on the previous protostellar phase, which is the phase when the star forms. Analysis of the effects of different choices of accretion parameters (accretion rate, radius and mass of the first stable hydrostatic protostar, accretion history, accretion geometry, the amount of energy transferred from the accreted matter to the accreting star, etc..) on the subsequent pre-MS evolution have been performed in the literature. The results show that if the accretion process leads to bright and extended structures at the end of the protostellar phase the stellar characteristics (including the surface light element abundances) are very similar to those of standard (non-accreting) pre-MS models with the same final mass. The structure of a pre-MS star at the end of the accretion phase is affected by the inclusion of the protostellar accretion only for a restricted range of accretion parameters, mainly in the so called “cold accretion scenario.” In these cases a significant reduction of the surface light element abundances during the protostellar phase (in contrast to standard models) has been obtained; however the position of the stars in the HR diagram is in disagreement with observations for disk stars, rising doubts about the validity of the adopted accretion parameters.
Protostellar accretion in low mass halo stars was suggested in the literature as one of the possible solutions for the cosmological lithium problem. However, theoretical calculations show that the reproduction of the Spite Plateau would require a fine tuning of the parameters that govern the protostellar phase and, more important, the models with the required Li depletion follow a pre-MS evolution in the HR diagram which is quite different to the one observed for high metallicity pre-MS stars. Comparison between theoretical predictions and observations for surface lithium abundance in young open clusters still show discrepancies. During the pre-MS phase surface Li abundance is strongly influenced by the nuclear burning as well as by the extension toward the interior of the convective envelope and by the temperature at its bottom. These last two quantities depend on the input physics adopted in the calculations (radiative opacity, atmospheric models etc..), on the assumed stellar chemical composition and on the convection efficiency in superadiabatic regions, whose precise physical treatment is not still fully available.
Comparison between predictions and observations for pre-MS stars in open clusters suggest a less efficient convection during the pre-MS phase with respect to the Main Sequence. This is true even if one takes into account the uncertainties on the results due to the errors in the adopted input physics and assumed chemical composition. A possible explanation of this result could be the fact that in 1D evolutionary codes a reduced convection efficiency could mimic the main effects of some non-standard mechanisms active in young stars, such as the presence of a not negligible magnetic field and/or surface spot coverage.
The energy produced by the Li, Be, B burning reactions is negligible, thus their effects on stellar structures are irrelevant. However, the surface abundances of light elements strongly depends on the nuclear burning and thus on the adopted reaction rates. The only nuclear burning that during the pre-MS or protostellar accretion phase affects stellar evolution is the deuterium burning. The impact on pre-MS evolution of a variation of the p(D,γ)3He reaction rate by its present uncertainty (±7%) has been analyzed in the literature, finding a negligible effect on stellar models. Two other D-burning channels have been considered, namely the D(D,p)3H and the D(D,n)3He. However, as expected, the inclusion of such channels does not produce relevant effects on pre-MS evolution as the largest part of deuterium is destroyed via p(D,γ)3He.
The effects on the other light elements surface abundance predictions of the still present (even if greatly reduced) uncertainty on (p,α) cross sections have been evaluated in detail and compared to the influence on the results of the errors in the other physics ingredients and in the stellar chemical composition. Light element (p,α) reaction rates have been recently revised through the indirect Trojan Horse method (THM), sensibly reducing their estimated uncertainty and finding differences with previous estimates at the energies of astrophysical interest. In general, differences in the predicted surface Li, Be, B abundances if the THM or the less recent but still widely used NACRE reaction rates are adopted are significant for stars in which light elements are efficiently burned.
The current uncertainty on the 6Li and 7Li proton capture reaction rates is of the order of 10%. Numerical calculations show that the effects on the 6Li and 7Li surface abundances due to this uncertainty, although not negligible, are less important than the influence of errors in other quantities used in the computation of a stellar model. The present errors on the 9Be(p,α)6Li and 10B(p,α)7Be rates, at the temperatures of interest, are, respectively, of about 25 and 20%. Due to the higher 9Be and 10B burning temperature, with respect to the Li burning, the effects of the reaction rate change/uncertainty are significant only for masses lower than
In conclusion, recent cross section measurements for light element (p,α) burning reactions sensibly reduced their uncertainty, even if it is still not negligible. Pre-Main Sequence theoretical calculations and consequently prediction for light element surface abundances are affected by several uncertainty sources: the not precise knowledge of the protostellar evolution and the efficiency of superadiabatic convection, the still present errors on the input physics adopted in the calculations and in the assumed stellar chemical composition. On the other hand, the errors on light element nuclear cross sections do not constitute the main uncertainty source for the prediction of light elements surface abundances.
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
This work has been partially supported by INFN (Iniziativa specifica TAsP) and the Pisa University project PRA 2018/2019 “Le stelle come laboratori cosmici di fisica fondamentale.”
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The reviewer OS declared a past co-authorship with one of the authors RP to the handling editor.
AC acknowledges the support of the “visiting fellow” program (BVF 2019) of the Pisa University. ET acknowledges INAF-OAAb for the fellowship “Analisi dell’influenza dell’evoluzione protostellare sull’abbondanza superficiale di elementi leggeri in stelle di piccola massa in fase di pre-sequenza principale.” LL acknowledges the program “Starting Grant 2020” by University of Catania.
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fspas.2021.604872/full#supplementary-material
1The ZAMS corresponds to the phase when central hydrogen begins to be burned into helium with secondary burning elements at equilibrium and nuclear energy production fully supporting the star.
2When talking about degeneration we refer to electron quantum degeneracy.
4The term “undershooting” indicates a sink of the external convection toward the stellar interior larger than the one predicted in classical models i.e., by the Schwarzschild criterion.
5With alpha enhancement one indicates an enhancement of the relative abundance of α elements (C, O, Ne, Mg, Si, S, Ar and Ca) with respect to the solar composition. It is generally expressed as: [α/Fe] =
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Keywords: Nuclear reactions, nucleosynthesis, abundances, stars:evolution, stars:pre-main sequence
Citation: Tognelli E, Degl’Innocenti S, Prada Moroni PG, Lamia L, Pizzone RG, Tumino A, Spitaleri C and Chiavassa A (2021) Theoretical Predictions of Surface Light Element Abundances in Protostellar and Pre-Main Sequence Phase. Front. Astron. Space Sci. 8:604872. doi: 10.3389/fspas.2021.604872
Received: 10 September 2020; Accepted: 02 February 2021;
Published: 13 May 2021.
Edited by:Paul Denis Stevenson, University of Surrey, United Kingdom
Reviewed by:Maria Lugaro, Hungarian Academy of Sciences (MTA), Hungary
Oscar Straniero, Astronomical Observatory of Abruzzo (INAF), Italy
Copyright © 2021 Tognelli, Degl’Innocenti, Prada Moroni, Lamia, Pizzone, Tumino, Spitaleri and Chiavassa. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: E. Tognelli, firstname.lastname@example.org