# Theoretical predictions of surface light element abundances in protostellar
and pre-Main Sequence phase.

E. Tognelli 1,2, S. Degl’Innocenti 1,3, P.G. Prada Moroni 1,3 L. Lamia4,5,6,
R.G. Pizzone5, A. Tumino5,7, C. Spitaleri4,5 and A. Chiavassa8

###### Abstract

## 1

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 $\approx 2$ to $\approx 5$ million degrees) and thus in
the early evolutionary stages of a star they are gradually destroyed at
different depths of stellar interior mainly by (p, $\alpha$) burning
reactions, in dependence on the stellar mass. Their surface abundances are
strongly influenced by the nuclear cross sections, as well as by the extension
toward the stellar interior of the convective envelope and by the temperature
at its bottom, which depend on the characteristics of the star (mass and
chemical composition) as well as on the energy transport in the convective
stellar envelope. In recent years, a great effort has been made to improve the
precision of light element burning cross sections. However, theoretical
predictions surface light element abundance are challenging because they are
also influenced by the uncertainties in the input physics adopted in the
calculations as well as the efficiency of several standard and non-standard
physical processes active in young stars (i.e. diffusion, radiative
levitation, magnetic fields, rotation). Moreover, it is still not completely
clear how much the previous protostellar evolution affects the pre-Main
Sequence characteristics and thus the light element depletion. This paper
presents the state-of-the-art of theoretical predictions for protostars and
pre-Main Sequence stars and their light element surface abundances, discussing
the role of (p, $\alpha$) nuclear reaction rates and other input physics on
the stellar evolution and on the temporal evolution of the predicted surface
abundances.

##  2 Keywords:

Nuclear reactions, nucleosynthesis, abundances, Stars: pre-main sequence,
Stars: evolution

##  3 Introduction

Light elements – lithium, beryllium and boron (hereafter Li, Be and B) – are
burned at relatively low temperatures ($T$ from $\approx 2$ to $\approx 5$
million degrees) easy to reach in stellar interiors at the bottom of the
convective envelope, even during the early pre-Main Sequence (pre-MS)
evolution. Therefore, surface light elements are depleted if the mixing
processes become efficient enough to bring them down to the destruction
region. This property makes such elements very good tracers of the mixing
efficiency in stellar envelopes whose theoretical treatment is still a
difficult task in stellar physics. Due to the different burning temperatures,
the comparison between theory and observation for Li, Be and B, if possible,
would be very useful to constrain theoretical models and in particular the
extension of the convective envelope. The most of the observations concern the
abundance of 7Li because, in most stars, surface 6Li is completely destroyed
during the pre-MS phase and Be and B isotopic measurements are very
problematic (e.g. Cunha [51], Kaufer [103], Delgado Mena et al. [64]).

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. Xiong and Deng [219], Piau and Turck-Chièze [151], Sestito et al. [174],
Deliyannis et al. [66], Jeffries [96], Pinsonneault et al. [152], Jeffries
[97], Talon [193]).

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 [151], Burke et al. [33], Richard
et al. [168, 167], Tognelli et al. [197, 201]). 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 [17], MacDonald and Mullan [123], Feiden and Chaboyer
[74], Somers and Pinsonneault [177, 178]). 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 [136], Talon and Charbonnel [194], Pace et al. [146], Charbonnel
et al. [40] 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. [18], Baraffe and Chabrier [17], Tognelli et al. [200],
Kunitomo et al. [106], Tognelli et al. [203]).

Light element burning cross sections are fundamental ingredients in the
predictions of the time behaviour of light element stellar surface abundances.
In recent years new values for (p,$\alpha$) reaction rates have been proposed,
mainly estimated using the Trojan Horse Method, greatly improving the
precision of these quantities.

The present review summarises 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 analyse 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.

##  4 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 $\sim 1$ Myr) with solar or slightly
sub-solar metallicity; among these objects, some of them show a still
detectable accretion disc, or protoplanetary disc, and very low accretion
rates (see .g. Hartmann and Kenyon [85], Muzerolle et al. [138], Calvet et al.
[36], Muzerolle et al. [139, 140], Bae et al. [16], Ingleby et al. [94]). Such
residual accretion discs show clear footprints of a previous accretion phase.

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. [199, 202].)

As an example, Fig. 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] M⊙ (Tognelli et al. [199]). Such observed stars are
fully formed, in the sense that the measured accretion rates are extremely
small, thus, they have already reached their final mass. Thus they can be
considered stars evolving as constant mass structures. This figure is intended
to qualitatively show the position of observed young stars in a HR diagram;
they are located in a region which theoretically corresponds to pre-MS models
undergoing to a gravitational contraction (we will discuss this evolutionary
stage in more details in Section 5). Standard stellar models generally agree
with data for young stars in the colour-magnitude (CM) or in the HR diagram
(see e.g. Tognelli [196], Randich et al. [166]), as qualitatively shown in
figure. We remark that stellar models should be able to populate such region
of the HR diagram where young stars are observed. Thus, the simple comparison
with observations of young associations/clusters (especially in the GAIA era)
in the HR/CM diagram can put strong constraints on stellar evolution
theoretical predictions (see e.g. Gaia Collaboration et al. [78], Randich et
al. [166], Bossini et al. [32]). This is a fundamental point especially when
accretion phases are taken into account (see Section 6), helping in
constraining free parameters adopted in model computations.

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. [132], Gennaro et al. [80]).

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. Deliyannis et al. [66], Charbonnel
et al. [41], Pinsonneault et al. [152], Piau and Turck-Chièze [151], Randich
[164], Tognelli et al. [197]). These issues will be discussed in the present
review.

##  5 General characteristics of standard pre-main sequence evolution

Figure 2: HR diagram for low-mass stars with indicated the main evolutionary
stages during the pre-MS evolution: Hayashi track (fully convective star, grey
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.

The pre-MS evolution starts at the end of the accretion phase and ends with
the Zero Age Main Sequence, or simply ZAMS, position111The 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.. 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 [86] and
Hayashi and Nakano [87]; 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] M⊙
computed using the PISA stellar evolutionary code (Degl’Innocenti et al. [62],
Dell’Omodarme et al. [67]), with the adopted input physics/parameters
described in Tognelli et al. [202, 203]); the same figure also shows a
qualitative representation of some of the main evolutionary stages
characteristics of such mass range.

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 $L$:

$\tau_{KH}=\frac{\beta}{2}\frac{GM^{2}}{RL}.$ (1)

The factor $\beta$ takes into account the density profile inside the star. For
an (unrealistic) model of homogeneous and spherical star with a constant
density, $\beta=5/3$. The gravitational contraction leads to an internal
temperature increase. We recall that for non-degenerate structures222When
talking about _degeneration_ we refer to electron quantum degeneracy. the
central temperature, $T_{c}$, depends on the stellar mass $M$, the radius $R$
and the chemical composition (mean molecular weight $\mu$) in the following
way:

$T_{c}\propto\frac{\mu M}{R}.$ (2)

From the relation above a contraction naturally leads to a rise in $T_{c}$.
Using this result, the Stephan-Boltzmann law ($L\propto
R^{2}$$T_{\mathrm{eff}}$4) and the virial theorem, it can be shown that the
luminosity of the star decreases following a simple power law, $L\propto
t^{-2/3}$.

The gravitational contraction is the only energy source until the central
temperature reaches about $10^{6}$ K, when the deuterium burning reaction
D(p,$\gamma$)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, $\epsilon_{pD}$,
($\epsilon_{pD}\propto T_{c}^{12}$); such a dependence limits the $T_{c}$
increase, halting the gravitational contraction (because of the $T_{c}\propto
1/R$ relation).

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 Fig. 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 0.1 - 1 M⊙ varies between about $0.1$-2 Myr, depending on the mass;
as an example the D-burning phase lasts about 1-2$\times 10^{6}$ yr for 0.1 M⊙
and about $10^{5}$ yr for a 1 M⊙ (see e.g. Chabrier and Baraffe [39], D’Antona
and Mazzitelli [59], Tognelli et al. [199]).

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
$X_{D}\approx 2\times 10^{-5}$ should be adopted (see e.g. the review by
Sembach [172]); such a value is smaller than that predicted by the BBN
($X_{D}\approx 4\times 10^{-5}$, see e.g. Steigman et al. [191], Pettini et
al. [150], Pitrou et al. [153], Mossa et al. [137]), as expected – e.g. by
galactic evolution models – because deuterium is destroyed in stars.

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 $L\propto
t^{-2/3}$. This second gravitational contraction increases the temperature and
density in the inner region of the star. Depending on the total mass, such a
temperature increase could lead to a drop in the radiative opacity
$\kappa_{R}$. For stars with $M\gtrsim 0.3$ M⊙, the internal opacity drop
reduces the radiative gradient leading to the formation of a central radiative
stable zone. As a consequence of this fact, the star leaves the Hayashi track
in the HR diagram, shifting towards larger temperatures as the radiative core
grows in mass, until the star efficiently ignites the central hydrogen burning
(reaching the ZAMS). This part of the stellar evolution is traditionally
called the _Henyey track_ and corresponds to the blue part of the evolutionary
track in Fig. 2. For $M<0.3$ M⊙, the temperature increase is not enough to
produce such an opacity drop and the star continues its contraction along the
Hayashi line. In this mass range, if the total mass is larger than
approximately $0.08$ M⊙, the contraction continues until the central
temperature is large enough to ignite central hydrogen burning, which becomes
the main energy source of the star (see e.g. Iben [93]). On the other hand, if
$M<0.08$ M⊙, during the contraction the star become so dense that the pressure
is dominated by the degenerate electron contribution; in such a configuration
the pressure is only very slightly dependent on the temperature. Then the
contraction slows down and the star (called brown dwarf) evolves along a
cooling sequence which, in the HR diagram, follows a precise mass-radius
relation.

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.

##  6 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. [187, 188, 189], Hartmann
et al. [83], Baraffe et al. [19]).

Protostellar accretion has been analysed in the literature starting from the
pioneering works by Stahler et al. [187], Stahler [186], Palla and Stahler
[147], Palla and Stahler [148], Palla and Stahler [149], Hartmann et al. [83]
and Siess and Livio [175], to more recent works by Baraffe et al. [18],
Hosokawa and Omukai [90], Baraffe and Chabrier [17], Tognelli [196], Kunitomo
et al. [107] and Tognelli et al. [203]. 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.

###  6.1 Cloud collapse and protostellar accretion

The main phases of the protostellar evolution are briefly described below (for
more details see Larson [113, 114, 115]).

  * •

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 0.01 M⊙ and a radius
of several AU) out of which the matter is still falling on the core. A
transition region (shock front) develops close to the core surface where the
matter settles and passes from supersonic to subsonic.

  * •

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 ($T\sim 2000~{}$K) is
reached. Then contraction energy does not warm anymore the core but it is used
to dissociate $H_{2}$, forcing the core to break the condition of hydrostatic
equilibrium. At this stage, the core density and pressure increase.

  * •

Formation of the 2nd Larson core: when $H_{2}$ is fully dissociated a further
increase of density and pressure, due to contraction, while mass is still
falling radially on the core, leads to a second, hydrostatic equilibrium for
the central core with a mass of the order of $\sim 0.001$ M⊙ ($=1$ MJ, Jupiter
mass) and a radius of about $1$ R⊙. From this moment on the central objects
maintains its hydrostatic configuration while its mass increases.

The protostellar evolutionary phases listed above are quite general (for solar
metallicity stars) and almost independent of the computation details. Larson
[113] remarked that at some stages of the cloud evolution a hydrostatic
central object (2nd 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 [131], Machida et al. [124], Tomida et al. [204], Vaytet et al.
[214]). Reasonable intervals for the mass, radius and temperature of the
stable hydrostatic core are: mass range of 1 - 20 MJ, radius values of 0.5 -
10 R⊙ and central temperature of 2-6$\times 10^{4}$ K.

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 5 M⊙ models. Figure
adapted from Larson [114].

In the HR diagram of Fig. 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 [113, 114], adopting
selected accretion parameters, followed the subsequent evolution until the the
Hayashi track, finding that low mass stars ($M<1~{}$M⊙) attain, after the
protostellar accretion, characteristics similar to that of standard evolution
along the Hayashi track. In contrast, as the mass increases ($M>2~{}$M⊙),
models skip the Hayashi line, ending the protostellar phase closer and closer
to the MS position, where they join the ”standard” track.

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.

###  6.2 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 is not needed to employ a hydrodynamic code, as the
protostar itself is in hydrostatic equilibrium after the formation of the 2nd
Larson core. In this approximation, the central object can be described using
a mono dimensional hydrostatic stellar evolutionary code (see e.g. Siess and
Livio [175], Stahler et al. [187, 188]).

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 is 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 summarised below.

  * •

Accretion rate. The accretion rate ($\dot{m}$) defines the rate at which the
star changes its mass; $\dot{m}$ can vary by orders of magnitude during the
accretion phase, passing from $\dot{m}$$<10^{-6}$-$10^{-7}~{}$M⊙/yr (quiescent
accretion) to rapid and intense episodes of mass accretion (bursts) with
$\dot{m}$$\sim 10^{-4}$-$10^{-3}~{}$M⊙/yr, e.g. as observed in FU Ori stars,
(Hartmann and Kenyon [85], Hartmann et al. [84], Audard et al. [14]).

  * •

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. [19]).

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. [187], Stahler [186], Palla and
Stahler [147, 148, 149], Hosokawa and Omukai [90]). 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. [83], Siess and Livio [175], Baraffe et al.
[18], Baraffe and Chabrier [17], Kunitomo et al. [107], Tognelli [196]). 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 [216], Baraffe
et al. [19]).

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.
[141], Meyer et al. [134], Watson et al. [218], Machida et al. [125] 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 is possible
to estimate the amount of energy retained by the accreted matter, in disc
accretion this quantity is defined by a free parameter
($\alpha_{\mathrm{acc}}$). The impact of $\alpha_{\mathrm{acc}}$ on the
evolution is discussed in the next sections.

###  6.3 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.

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 [149].

The impact of spherical accretion on the formation of pre-MS stars has been
largely analysed in the pioneering works by Larson (Larson [113, 114]),
Stahler and Palla (Stahler et al. [187], Stahler [186], Palla and Stahler
[147, 148, 149]), and more recently also by Hosokawa and collaborators
(Hosokawa and Omukai [90, 91], Hosokawa et al. [92, 89]). 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 $10^{-5}~{}$M⊙/yr, it is possible to obtain fully accreted
stars in a region of the HR diagram that corresponds to the upper envelope of
the locus where young pre-MS stars are observed (see Figure 4). This sequence
was called ” _birthline_ ”, that is the locus of stars with different masses
where the accretion ends and the stars become optically visible (Palla and
Stahler [149]).

More recent sets of birthlines can be found in Hosokawa and Omukai [90]: such
accretion models was computed for different values of the accretion rate (from
$10^{-6}$ to $10^{-3}~{}$M⊙/yr), adopting a spherical protostellar accretion
code (similar to that used by Stahler and Palla). They showed that increasing
$\dot{m}$, the birthline moves towards larger luminosities and radii, thus
still in full agreement with the observations. Moreover, since spherical
accretion models produce low-mass stars (on the birthline) in a region that
corresponds to the top of the Hayashi track of standard stellar models (see
Figure 4), the differences between standard and spherical accreting models in
pre-MS low-mass stars are negligible. This validates the results of standard
evolutionary tracks/isochrones (at least for ages higher than 1 Myr).

However, it is 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.
[83]) 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 is possible to follow also an analytic
approach to analyse the main characteristics of the accreting star. Following
the formalism presented in Hartmann et al. [83], it is possible to write a
simple equation for the temporal evolution of the accreting star radius:

$\displaystyle\frac{\dot{R}}{R}=\frac{7}{3}\frac{R}{GM^{2}}\bigg{[}\beta_{D}-L_{\mathrm{ph}}+\bigg{(}\alpha_{\mathrm{acc}}-\frac{1}{7}\bigg{)}\frac{GM^{2}}{R}\frac{\dot{m}}{M}\bigg{]}$
(3)

where $M$ and $R$ are the stellar mass and radius, $\beta_{D}$ expresses the
luminosity due to the deuterium burning (D-burning), $L_{\mathrm{ph}}$ is the
luminosity of the stellar surface, $\dot{m}$ is the mass accretion rate and
$\alpha_{\mathrm{acc}}$ represents the fraction of the accretion energy
deposed into the star (thermal energy of the accreted matter).

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
($L_{\mathrm{ph}}$) with the inclusion of D-burning energy production
$\beta_{\mathrm{D}}$, while the last term represents the accretion effect,
which is proportional to $\dot{m}$. This term accounts for the fraction of the
thermal energy of the accreted matter retained by the star,
$\alpha_{\mathrm{acc}}$. Such a parameter has to be specified as an external
free parameter, ranging from 0 (no energy acquired by the star) to about 1 (or
1/2 in case of thin disc, see e.g. Siess and Livio [175]). From the same
equation, it is also evident that
$\alpha_{\mathrm{acc}}=1/7\equiv\alpha_{\mathrm{acc,cr}}$ defines a critical
value; for $\alpha_{\mathrm{acc}}<\alpha_{\mathrm{acc,cr}}$ the third term is
negative, and it contributes to the contraction of the star. For
$\alpha_{\mathrm{acc}}>\alpha_{\mathrm{acc,cr}}$ the same term produces a
radius expansion. It is common to refer to the case $\alpha_{\mathrm{acc}}\sim
0$ (or $\alpha_{\mathrm{acc}}\ll\alpha_{\mathrm{acc,cr}}$) as _cold disc
accretion_ and $\alpha_{\mathrm{acc}}>\alpha_{\mathrm{acc,cr}}$ as _hot disc
accretion_.

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 $\beta_{D}$) or via an efficient accretion
energy transport into the protostar
($\alpha_{\mathrm{acc}}>\alpha_{\mathrm{acc,cr}}$). These two cases are
discussed separately in the next two sections.

###  6.4 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 $\alpha_{\mathrm{acc}}=0$. From eq.
(3), to produce a radius increase, D-burning has to supply the star with
enough energy to counterbalance the radiative losses at the stellar surface
plus the gravitational energy decrease caused by the mass ingestion. If this
condition is not satisfied, the protostar contracts and the resulting model at
the end of the protostellar phase has a radius much smaller than that observed
in young disk stars and expected in spherical accretion cases.

Figure 5: Effect of different deuterium original abundances (see labels),
$X_{\mathrm{D}}$, on the protostellar evolution of 1 M⊙ model. The
protostellar accretion starts with a seed mass of 10 MJ, with a constant
accretion rate of $10^{-5}$ M⊙/yr. Filled circles indicate the end of the
protostellar accretion phase and triangles the beginning of D-burning. Figure
adapted from Kunitomo et al. [107].

The dependency of the radius on original deuterium abundance $X_{\mathrm{D}}$
has been investigated in Tognelli [196] and more recently in Kunitomo et al.
[107]. In Kunitomo et al. [107] the authors assumed for the 2nd Larson core
mass the value $M_{\mathrm{seed}}=0.01$ M⊙(=10 MJ). Figure 5 shows a
comparison between birthlines obtained assuming different values of
$X_{\mathrm{D}}$, for a cold accretion scenario with $M_{\mathrm{seed}}$=10
MJ. When no deuterium is taken into account in the stellar matter, the star
inevitably contracts: in this model, the star ignites the hydrogen burning
close to the end of the protostellar accretion, thus totally skipping the pre-
MS evolution. The situation changes increasing the deuterium mass fraction
abundance in the star. To partially reproduce the standard pre-MS evolution, a
deuterium content of $X_{D}\approx 4\times 10^{-5}$ (i.e. 40 ppm) is required.
If a more reliable deuterium content is adopted, $X_{D}\approx 2\times
10^{-5}$, the models with protostellar accretion converges to standard models
only in the Henyey track; in this case, the evolution along the Hayashi track
is missed contrarily to what observed in young clusters. We want to comment
about the fact that the uncertainty on galactic deuterium mass fraction
abundance is not larger than 10 ppm (see e.g. Fig. 2 and table 1 in Sembach
[172]), thus an initial deuterium content of $X_{D}\approx 40$ ppm is an over
estimation for disk stars. This fact seems to indicate that deuterium alone is
not capable of maintaining the star bright enough to reconcile protostellar
cold accretion models and the results obtained in a standard non accreting
scenario.

###  6.5 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 $\alpha_{\mathrm{acc}}>\alpha_{\mathrm{acc,cr}}$.
In Hartmann et al. [83] it was shown that non-cold accretion models
($\alpha_{\mathrm{acc}}$$>\alpha_{\mathrm{acc,cr}}$) can attain a radius
expansion large enough to reproduce observed stars; in this case the disc
accretion mimics spherical-accretion birthline obtained by Stahler [186]. More
recently, Kunitomo et al. [107] analysed in more details the impact of
$\alpha_{\mathrm{acc}}$ on the formation of a 1 M⊙ model, finding that the
inclusion of a certain fraction of the total accretion energy (i.e.
$\alpha_{\mathrm{acc}}$$\in[0,\,1]$) in the star is capable of maintaining the
structure at large radii. Figure 6 shows the birthline computed in Tognelli
[196] – by means of the PISA stellar evolutionary code – for solar metallicity
stars using three values of $\alpha_{\mathrm{acc}}$=0 (cold case), 0.5 and 1
(hot case), for a seed mass value of 5 MJ.

Figure 6: Effect of $\alpha_{\mathrm{acc}}$ on the protostellar evolution for
three values of $\alpha_{\mathrm{acc}}$, 0 (cold accretion), 0.5 and 1 (hot
accretion). Dotted lines are standard pre-MS tracks. Figure adapted from
Tognelli [196].

From Fig. 6, it is evident that adopting a value of
$\alpha_{\mathrm{acc}}$$\geq 0.5$ models on the birthline are bright and
intersect the standard evolutionary tracks in the standard Hayashi track, for
all the selected mass range [0.1, 2.0] M⊙. Recently, Tognelli et al. [203]
have obtained similar results for metal poor models: they showed that even in
the low metallicity case, the inclusion of accretion energy produces expanse
objects that intersect the Hayashi track of standard non accreting models at
the end of the protostellar accretion stage.

###  6.6 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. [18], Tognelli et al. [203]).

Figure 7 shows the evolution in the HR diagram of cold accretion models
starting from different $M_{\mathrm{seed}}$ and ending with different final
masses, as discussed in details in Baraffe et al. [18]. It is difficult to
reproduce the Hayashi track of pre-MS stars starting from $M_{\mathrm{seed}}$
of few Jupiter masses (i.e. cases A, B, D). Moreover, the position of the 1
Myr model (filled square) in accretion models is relatively far from the
standard 1 Myr isochrone; in most of the cases, the position of pre-MS models
with the inclusion of cold protostellar accretion at 1 Myr is very close to
the standard non accreting 10 Myr isochrone, witnessing the strong impact of
cold accretion on pre-MS evolution.

Figure 7: Evolution in the HR diagram of protostellar models with different
values of $M_{\mathrm{seed}}$, compared with the standard 1 Myr and 10 Myr
isochrones (black long dashed lines). The different letters indicate models
with different seeds and final masses ($M_{fin}$), in particular: (A)
$M_{\mathrm{seed}}$= 1 MJ, $M_{fin}$ = 0.05 M⊙, (B) $M_{\mathrm{seed}}$= 1 MJ,
$M_{fin}$=0.1 M⊙, (D) $M_{\mathrm{seed}}$= 1 MJ, $M_{fin}$=0.5 M⊙, (H)
$M_{\mathrm{seed}}$= 10 MJ, $M_{fin}$ = 0.21 M⊙, (I) $M_{\mathrm{seed}}$= 50
MJ, $M_{fin}$ = 0.55 M⊙, (J) $M_{fin}$=1.05 M⊙, (K) $M_{\mathrm{seed}}$= 100
MJ, $M_{fin}$ = 1.1 M⊙, (L) $M_{\mathrm{seed}}$=100 MJ, $M_{fin}$ =1.85 M⊙.
Filled squares represent the position of the 1 Myr model. Figure adapted from
Baraffe et al. [18].

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 firstly by Hosokawa et
al. [89] to show that the protostar remains bright enough to end the
protostellar phase close to a Hayashi track. Top panel of Fig. 8 shows the
models by Hosokawa et al. [89]. 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 0.03 M⊙ (magenta dashed line), 0.1 M⊙ (magenta solid line) and at 0.3
M⊙ (magenta dotted line). It is interesting to notice that in all cases, the
birthline is still quite luminous, being very close to the 1 Myr isochrone.
Similar results have been obtained for metal poor models by Tognelli et al.
[203] (bottom panel of Fig. 8).

Figure 8: Evolution in the HR diagram of purely and partially hot models
computed with an accretion rate $\dot{m}=10^{-5}$ M⊙/yr. Top panel: comparison
between purely hot (solid purple line), purely cold (green and red solid
lines) and hot+cold birthlines (magenta lines). The two purely cold cases
differ for the seed radius, 3.7 R⊙ (green line) and 1.5 R⊙ (red line, mC5-C).
The accretion switches from hot to cold at a given value of the mass, which is
0.03 M⊙ (magenta dashed line), 0.1 M⊙ (magenta solid line), and 0.3 M⊙
(magenta dotted line). Black lines are isochrones of 1 and 10 Myr (dashed
lines) and the ZAMS (dot-dashed line) for standard non accreting models.
Squares, triangles and circles represent observations of some young pre-MS
stars. Figure adapted from Hosokawa et al. [89]. Bottom panel: models at low
metallicity (Z=0.0001) for a total final mass of 0.7 and 0.8 M⊙. The accretion
switches from hot to cold at a mass value 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6 M⊙
as indicated in the labels. The thick grey line represents the hot birhtline
($\alpha_{\mathrm{acc}}=1$). Figure adapted from Tognelli et al. [203].

Baraffe et al. [18] and Baraffe et al. [19] 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
($\alpha_{\mathrm{acc}}>\alpha_{\mathrm{acc,cr}}$, see the Appendix in Baraffe
et al. [19]), to switch back to cold accretion at the end of each burst. The
authors showed that in this case it’s still possible to produce models that
end their protostellar accretion close to the standard position of the Hayashi
track, to reproduce data (see also Tognelli et al. [203] for metal poor
protostars).

What emerges from the previous analysis is that, if one assumes masses and
radii typical of the 2nd 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.

##  7 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] M⊙, the ranges of burning
temperatures for the different elements approximately are: 2.4-3.5$\times
10^{6}$ K ($T(^{6,7}$Li)), 3.5-4.0$\times 10^{6}$ K ($T(^{9}$Be)) and
4.2-5.0$\times 10^{6}$ K ($T(^{10,11}$B)). In the literature the temperatures
given for the burning are sometimes slightly different form the values
reported here; the reason is that usually authors do not take into account
that stars with different masses ignites these elements at slightly different
temperatures because the nuclear burning rates also depends (even if at a
lower level) on the density in the region where the burning occurs. Moreover,
such temperatures can be different for MS or pre-MS stars with the same mass
because of the different time scales in which light elements are destroyed. In
MS stars the evolutionary time scale is much longer than that in pre-MS (for
the same mass and chemical composition) consequently a smaller burning rate
due to a smaller threshold temperature at the bottom of the convective
envelope, is compensated by the longer time during which that element is
destroyed. As a result the burning of light elements in MS can efficiently
occur even at thresholds temperature smaller than those required in pre-MS.

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. [111].

Due to the differences in their burning temperatures in pre-MS, Li, Be and B
are gradually destroyed at different depths inside the stellar interior and at
different ages, in dependence on the stellar mass. As an example, Fig. 9 shows
the portion along the evolutionary track where surface Li, Be and B are burnt
at the bottom of the convective envelope in a set of solar chemical
composition stars in the mass range [0.08, 1.0] M⊙. It is interesting to
notice that while Li is burnt (at the bottom of the convective envelope) in
the whole selected mass range, surface 9Be burning occurs only for masses
between about 0.08 and 0.5 M⊙, while B is burnt in an even smaller mass range
(about 0.1 - 0.3 M⊙).

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 [173], Delgado Mena et al. [65, 63], Aguilera-Gómez et al. [2] and
references therein).

6Li burns at a lower temperature with respect to 7Li, consequently it is
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. [49]). 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. [12]) and it is below the detection level also for most thick
disk and halo stars (see e.g. Spite and Spite [184]). 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.
7.5).

Beryllium and boron measurements are more problematic than 7Li observations.
9Be abundance is measured using near-UV lines, only in stars with
$T_{\mathrm{eff}}$$\gtrsim 5000$ K, which corresponds in pre-MS to a mass
range where Be is expected to be preserved and not destroyed (see e.g. Garcia
Lopez et al. [79], Santos et al. [170], Randich et al. [165], Smiljanic et al.
[176], Delgado Mena et al. [64], Lamia et al. [111]).

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 [51]). 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 [28], Lodders et al. [119], Primas [162], Tan
et al. [195], Boesgaard et al. [29]). 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 [163], Prantzos [160]). 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. [10], Asplund et al.
[12], Lodders et al. [119], Lodders [118]).

The temperatures for light elements burning can be reached in stellar
interiors during the pre-MS evolution of stars with masses larger than about
0.05-0.1 M⊙ (depending on the requested burning threshold temperature). We
recall that at the beginning of the pre-MS evolutions stars are, independently
of their mass, fully convective. Thus, if a nuclear burning occurs at this
evolutionary stage, the burning affects the chemical abundance in the whole
star, from the centre to the surface. However, as the star contracts and warms
up, the opacity decreases at the stellar centre and stars with $M\gtrsim 0.3$
M⊙ develop a radiative core. From this moment on the chemical evolution of the
surface is (during the pre-MS) decoupled from that of the centre if the bottom
of the convective envelope does not reach a region deep and hot enough to
process –via nuclear burning– the surface matter. Thus, for partially
convective pre-MS stars, a condition to have a partial surface depletion of a
specific element is that the bottom of the convective envelope reaches a
temperature high enough to make nuclear burning efficient and then recedes
toward the external parts of the star at lower temperatures.

Figure 10: Temporal evolution of the temperature at the bottom of the
convective envelope for masses in the range [0.08,1.0] M⊙. The threshold
temperature required to ignite Li, Be and B burning are indicated as coloured
horizontal lines. The ZAMS position is marked by a diamond. The not regular
behaviour of the 0.3 M⊙ model at $\log t(yr)\sim 7.5-8.3$ is caused by the
formation of a transient convective core before the ZAMS (figure adapted from
Lamia et al. [111]).

Figure 10 shows the temporal evolution of the temperature at the bottom of the
convective envelope, $T_{\mathrm{bce}}$, for stars with different masses
between 0.08 and 1.0 M⊙ at solar metallicity, with indicated the approximate
values for the Li, Be and B burning temperatures. In fully convective stars
$T_{\mathrm{bce}}$ coincides with the central temperature, Tc. When stars are
fully convective $T_{\mathrm{bce}}$ progressively increases until the star
reaches the ZAMS, while in stars that develop a radiative core
$T_{\mathrm{bce}}$ stops increasing when the radiative core forms and
$T_{\mathrm{bce}}$ slowly decreases as the radiative core grows in mass. This
has a direct impact on the interval of time during which the surface light
elements depletion occurs. Considering e.g. 7Li abundance in fully convective
stars (i.e. M$\leq 0.3$ M⊙), $T_{\mathrm{bce}}$ overcomes $T(^{7}$Li$)$ at
young ages (i.e. about 50 Myr for 0.1 M⊙ and 5 Myr for 0.3 M⊙); then surface
Li burning continues during the whole pre-MS and MS phase. Since
$T_{\mathrm{bce}}$ continuously increases, the burning efficiency increases
too. On the other hand, in partially convective stars $T_{\mathrm{bce}}$
reaches a maximum and then decreases as the star evolves towards the ZAMS. For
$M\gtrsim 0.5$ M⊙, $T_{\mathrm{bce}}$ decreases below $T(^{7}$Li$)$ at some
point during the pre-MS, thus halting the lithium burning at the bottom of the
convective envelope. From this moment on, surface lithium abundance remains
constant during the pre-MS phase. Figure 10 also shows that increasing the
mass of the star the time interval during which surface lithium is destroyed
is shorter and the maximum value of $T_{\mathrm{bce}}$ reduces too; this is
due to the fact that increasing the mass the convective envelope becomes
thinner. This indicates that, increasing the mass, surface lithium is
destroyed progressively less efficiently.

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 $0.08\lesssim M$/M⊙$\lesssim 0.5$. On the other hand B in
destroyed only in the mass range $0.1\lesssim M$/M⊙$\lesssim 0.3$.

Figure 11 gives an example of the 7Li surface abundance time behaviour
predicted for stars in the mass range [0.08, 1.0] M⊙; it is important to
notice that light element surface abundances depend not only on the capability
of $T_{\mathrm{bce}}$ to overcome the threshold temperature for the considered
element, but also on the duration of the burning and on the difference between
the threshold and $T_{\mathrm{bce}}$. In particular, this last quantity is
very important because the burning rate of light elements is proportional to
$T^{a}$ with $a\approx 20$ for lithium, $a\approx 23$ for beryllium and
$a\approx 25$ for boron.

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] M⊙and
solar chemical composition. Stellar models have been computed using the same
input parameters described in Lamia et al. [111].

Referring to Fig. 11, in fully convective stars, i.e. $M\lesssim 0.3$ M⊙, at a
fixed age, the surface lithium depletion progressively increases increasing
the stellar mass. For partially convective models, this behaviour breaks up
and, at a fixed age, the lithium depletion decreases as the mass increases.
This is clearly visible in the figure comparing e.g. the predicted lithium
abundance at about $\log t(yr)=7.5$ for 0.7, 0.8, 0.9, and 1.0 M⊙ models. The
amount of residual surface lithium increases with the mass, as the consequence
of the decrease of $T_{\mathrm{bce}}$ (in time) when the radiative core forms
and grows.

Figures 10 and 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 7.2.

###  7.1 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. [61], D’Antona and Montalbán [60], Jeffries [97],
Tognelli et al. [197] and references therein). Moreover, the presence of a
large scatter in the observed Li abundance among stars with similar
$T_{\mathrm{eff}}$ in young clusters poses questions about the possible
mechanisms producing different amounts of lithium depletion in stars with the
same mass, age and chemical composition (King et al. [104], Jeffries [96],
Clarke et al. [45], Xiong and Deng [220], King et al. [105]).

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.

It is worth noticing that, qualitatively, standard models (without accretion)
are capable to produce a pattern of lithium vs mass (or $T_{\mathrm{eff}}$)
similar to that observed in young clusters. This pattern can be divided into
three regions, and, referring to Fig. 12, it can be summarised as it follows:

  * •

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 $T_{\mathrm{eff}}$ (or the stellar mass), until it reaches
a plateau corresponding to stars that do not deplete Li (hot stars). Regions
(3)-(4) in Fig. 12 correspond to partially convective models of increasing
mass. As previously discussed, the more massive is the star, the thinner is
the convective envelope and, in turn, the less efficient is the surface Li
depletion. The plateau corresponds to stars with a convective envelope so thin
that $T_{\mathrm{bce}}$$<$ $T(^{7}$Li$)$.

  * •

For lower masses (and thus $T_{\mathrm{eff}}$), e.g. in regions (1)-(2), stars
are fully convective and lithium is burned in the whole star. At a fixed age
the lithium burning efficiency increases with the stellar mass and lithium
surface abundance rapidly changes varying the stellar mass.

  * •

In region (1), reducing the mass (or $T_{\mathrm{eff}}$), one approaches the
minimum mass that reaches the Li burning temperature in fully convective
stars. Below this minimum mass the surface lithium abundance is constant and
equal to the original value.

###  7.2 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 analysed by Baraffe and Chabrier [17], 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. [19], Tognelli
[196], Kunitomo et al. [106]). 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 ($Z=0.005$, $Z=0.001$,
and $Z=0.0001$) was discussed in Tognelli et al. [203]. We also performed some
tests to verify that what obtained for sub-solar metallicity is still valid at
solar metallicities.

The results by Tognelli et al. [203], similarly to what already obtained by
Baraffe and Chabrier [17], show that two scenarios can be found:

  * •

_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 is
only slightly affected by the other accretion parameters (accretion rates,
initial radius). After the protostellar phase, for masses larger than about
0.1-0.2 M⊙, lithium is completely destroyed in an extremely hot and fully
convective pre-MS structure. This prediction is in complete disagreement with
observations available for young clusters, where $M\approx 0.8$-1 M⊙ stars
show no relevant lithium depletion (see Section 7.4 for more details).
Moreover, such accreting models are in disagreement with the observed position
of very young disk pre-MS stars in the HR diagram. The disagreement between
data and accretion models is partially mitigated if a larger seed mass is
adopted (of the order of 10 MJ). In this case it is possible to reduce the
level of lithium depletion in very low mass stars (i.e. $M\lesssim 0.3$ M⊙),
but not for stars close to 1 M⊙ where lithium is totally depleted in pre-MS.

  * •

_Hot accretion case_. In Section 6.6 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. [203]). 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 $T_{\mathrm{eff}}$ and the same chemical composition show
different luminosities (see e.g. Hillenbrand [88], Jeffries [98], Da Rio et
al. [54, 55]). A possible dependence on the protostellar accretion of such a
luminosity spread was analysed by Baraffe et al. [18]; the adoption of a
different accretion history during the protostellar phase can strongly affect
the luminosity and $T_{\mathrm{eff}}$ of a star at the end of the protostellar
phase, as already discussed in previous sections. If this is the case, faint
stars, which experienced cold accretion, should show a clear lower lithium
content than bright ones. In other words, such a luminosity spread should
directly reflect in a surface lithium content spread. This point deserves to
be further investigated to clearly confirm or exclude the presence of a
correlation between lithium content and luminosity in star forming regions.

###  7.3 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
$T_{\mathrm{eff}}$$>5900$ K and [Fe/H] $<-1.5$, the so called Spite plateau
(Spite and Spite [183, 182]), with a constant logarithmic lithium
abundance333$A(Li)$ indicates an observational notation for abundances, where
$A($Li$)=\log N_{\mathrm{Li}}/N_{\mathrm{H}}+12$, with $N$ being the number of
particles of a given specie. $A($Li$)=2.1$-$2.4$ (Charbonnel and Primas [42],
Asplund et al. [13], Meléndez et al. [133], Sbordone et al. [171]), and
references therein). From the theoretical point of view, stars with such
temperatures and metallicities are expected to preserve their initial lithium
content, moreover galactic enrichment due to cosmic rays and spallation
processes should be totally negligible at such low metallicities. Thus the
Spite plateau is expected to represent the primordial lithium content produced
during the big bang nucleosynthesis (BBN).

However, BBN predicts a primordial lithium content of $A($Li$)=2.67$-$2.75$,
(see e.g. Cyburt et al. [53], Pitrou et al. [153]). 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 [75], Pizzone et
al. [156], Goudelis et al. [82], Coc and Vangioni [47], Damone et al. [57],
Lamia et al. [108]).

Similarly, on the stellar evolution side, the problem has been analysed 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. [167]). 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 $T_{\mathrm{eff}}$. Thus
no Spite plateau would be possible without tuning the diffusion efficiency.
Also turbulent mixing could produce an effect similar to that of pure
diffusion, on similar time scales, but also in this case an ad hoc tuning is
required (see e.g. Richard et al. [167], Spite et al. [185] and references
therein). Also mass loss coupled to diffusion and turbulent mixing can be
tuned to produce a constant lithium depletion along the Spite plateau (Swenson
[192], Vauclair and Charbonnel [213]), but, again, there is the need for a
fine tuning of the parameters.

Another possibility is that lithium depletion occurs during the pre-MS. In
this case, Fu et al. [77] suggested that a certain level of
undershooting444The 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. 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 $T_{\mathrm{eff}}$ (or mass) dependent depletion on MS
time scales.

Recently Tognelli et al. [203] analysed 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 6.6 and
7.2, 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 is
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 the stellar mass accretion (see e.g. Fu et al.
[77], Tognelli et al. [203]). 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.

###  7.4 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 play 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 [151], Burke et al. [33], Tognelli et al.
[197, 201]). The quantities that mainly affects lithium, as analysed in the
quoted papers, are the following (see e.g. Chapter 3 in Tognelli [196]):

  * •

_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
($\nabla_{rad}\equiv(d\log T/d\log P)_{rad}$) is larger than the adiabatic
one. The radiative gradient is proportional to the Rosseland mean radiative
opacity $\kappa_{R}$, thus change in $\kappa_{R}$ directly affects the
position of the convective unstable boundary. Generally an uncertainty on
$\kappa_{R}$ of about $\pm 5\%$ is assumed in the computations (Badnell et al.
[15], Blancard et al. [27], Mondet et al. [135]). Similarly an uncertainty on
the adiabatic gradient, $\nabla_{ad}$, thus in the equation of state, modifies
the position of the bottom of the convective envelope. An increase in
$\kappa_{R}$ or a decrease of $\nabla_{ad}$ lead to an extension of the
convective envelope that can reach deeper and hotter layers, increasing the
efficiency of surface lithium burning (see e.g. Tognelli [196]). The variation
of surface lithium abundance, due to the change in the equation of state or
radiative opacity, strongly depends on the selected mass range and age.
However, in those models that efficiently deplete lithium (e.g. for 0.7 and
0.8 M⊙) a variation in lithium abundance of approximately 0.1-0.2 dex (due to
equation of state uncertainty) and 0.4-0.5 dex (due to opacity error) can be
obtained. In the worst cases (i.e. $M\sim$0.6 M⊙) a variation of $5\%$ of the
radiative opacity can lead to a difference of $\sim 0.8$ dex in the predicted
lithium content. The effect on surface lithium of the equation of state or
radiative opacity reduces as the mass increases.

  * •

_Outer boundary conditions_. The outer boundary conditions are the pressure
$P_{atm}$ and temperature $T_{atm}$ at the bottom of the atmosphere. These
quantity deeply affect the temperature profile in the convective envelope thus
modifying also its depth. The uncertainty on $(P_{atm},\,T_{atm})$ it is not
provided by stellar atmospheric calculations, but one can test the effect on
the stellar characteristics of the adoption of different atmospheric models
available in the literature (see e.g. Tognelli et al. [199]). As said above,
the effect on lithium depends on the mass/age of the models; a typical
variation of 0.3-0.5 dex is expected, which reduces as the mass increases.

  * •

_Mixing length parameter_. The convection efficiency in super-adiabatic
regimes in 1D stellar evolution codes commonly relies on mixing-length theory
(MLT) Böhm-Vitense [30]. In this formalism, the scale on which the heat is
efficiently transported by convection is defined as
$\ell=\alpha_{\mathrm{ML}}\times H_{p}$, where $H_{p}$ is the local pressure
scale height and $\alpha_{\mathrm{ML}}$ is the mixing length parameter, a free
parameter to be calibrated. The extension of the convective envelope and thus
the temperature at its bottom and the the surface lithium abundance are
strongly affected by the adopted $\alpha_{\mathrm{ML}}$. The adoption of
different values of this quantity within plausible ranges can produce a
variation of surface lithium abundance as large as 1 order of magnitude in
those stars where the external envelope is largely super adiabatic and where
lithium is efficiently destroyed at the bottom of the convective envelope
(i.e. for masses in the range [0.5, 1.0] M⊙ (see e.g. Piau and Turck-Chièze
[151] and Tognelli et al. [197]).

  * •

_Nuclear cross section_. The error on the cross section for the 7Li(p,
$\alpha)^{4}$He reaction directly affects the rate at which lithium is
destroyed and thus its temporal evolution. Since the energy released by such
reactions is inconsequential for the stellar structure, the only effect is on
the surface lithium content at a fixed age. In Section 8 we will discuss this
point in more detail.

  * •

_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
introduced 555With _alpha enhancement_ one indicates an enhancement of the
relative abundance of $\alpha$ elements (C, O, Ne, Mg, Si, S, Ar and Ca) with
respect to the solar composition. It is generally expressed as:
[$\alpha$/Fe]=$\log($Nα/N${}_{Fe})_{\mathrm{star}}$ \-
$\log($Nα/N${}_{Fe})_{\odot}$.. 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 ($Y_{p}$), 2) the metal-to-helium enrichment ratio
($\Delta Y/\Delta Z$), 3) the metal-to-hydrogen ratio in the Sun
($(Z/X)_{\odot}$), 4) and the [$\alpha$/Fe] (alpha-enhancement) for metal poor
stars (Gennaro et al. [80], Tognelli et al. [197, 201]). The observational
error on [Fe/H] has thus to be combined with the uncertainties of such
quantities, to estimate the final global uncertainty on the initial helium and
metal mass fraction to be used in the computation of stellar models (see e.g.
Gennaro et al. [80], Tognelli et al. [197, 201]); for solar chemical
composition, the uncertainty on $Y$ and $Z$ are estimated to be of the order
of 4-5% for $Y$ and about 20% for $Z$ (Tognelli et al. [201]). The variations
of $Y$ and $Z$ have a strong impact on the lithium burning because they change
both the extension of the convective envelope and the temperature inside a
star (see e.g. Piau and Turck-Chièze [151], Tognelli [196] and Tognelli et al.
[201]); the uncertainty on the chemical composition can produce a variation of
the surface lithium abundance up to one order of magnitude, especially in
stars with $M\lesssim 0.7$ M⊙. The effect reduces at larger masses.

Tognelli [196] 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. [197] and Tognelli [196]). 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
1 order of magnitude, especially for stars with $T_{\mathrm{eff}}$$\lesssim
4700$ K.

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 [196].

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. [144], Collet et al. [48], Freytag et al.
[76], Magic et al. [127], Trampedach et al. [206], Trampedach et al. [208],
Trampedach et al. [207], Pratt et al. [161], 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. [122], Ludwig et al. [121],
Arnett et al. [8, 7]). 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 [30], where convection efficiency
depends on the free parameter $\alpha_{\mathrm{ML}}$. A variation of such a
parameter can produce a large effect on the surface lithium abundance at a
given age in stars with a super-adiabatic envelope. This effect is
particularly important in stars with masses larger than about 0.5-0.6 M⊙. It
has been shown in the literature that models with a reduced convection
efficiency ($\alpha_{\mathrm{ML}}$$<$ $\alpha_{\mathrm{ML}}$⊙, solar
calibrated value) attains a better agreement with data for both young clusters
and binary stars (Piau and Turck-Chièze [151], D’Antona and Montalbán [60],
Tognelli et al. [197]).

Figure 14 shows the results obtained by Tognelli et al. [197] where the
observed surface lithium abundance in five young open clusters (IC2602,
$\alpha$ 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 colours 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 $T_{\mathrm{eff}}$ pattern. The
availability of young clusters to perform such an analysis (ages below 100-150
Myr) is mandatory to avoid possible effects due to MS non-standard mixing
processes which act on timescales of the order of $\sim$ Gyr.

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 ($\alpha_{\mathrm{ML}}$=1) in
pre-MS are shown as continuous lines. In the case of NGC2516, the models were
computed using two different values of [Fe/H], [Fe/H] = -0.10 (bottom left
panel) and [Fe/H] = $+0.07$ (bottom right panel), as reported in the
literature. Figure adapted from Tognelli et al. [197].

Referring to Fig. 14, it is evident that in all cases, the adoption of a
constant value of $\alpha_{\mathrm{ML}}$ (calibrated on MS stars) produces a
lithium depletion much larger than observed. On the other hand it is possible
to tune $\alpha_{\mathrm{ML}}$ during the pre-MS to reproduce the observed
lithium pattern and the most important result is that the derived
$\alpha_{\mathrm{ML}}$ in pre-MS is independent of the cluster age and on the
stellar mass. The authors derived a value of
$\alpha_{\mathrm{ML}}$${}_{\mathrm{,PMS}}=1$. We mention that the reduction of
the efficiency of superadiabatic convection in stellar envelope has been put
forward as a plausible mechanism to explain the discrepancies in the radius
observed and predicted in pre-MS binary systems. A value of
$\alpha_{\mathrm{ML}}$$\sim 1$ has been suggested to explain the radius in
young binary systems (see e.g. Gennaro et al. [80]).

As said, $\alpha_{\mathrm{ML}}$ is a free parameter which reflects the present
not precise knowledge of external convection efficiency, thus one should find
a physical reason for its variation during the evolution of a given star. A
reduced convection efficiency could be motivated by the attempt in 1D
evolutionary codes to mimic the main effects of some non-standard mechanisms
active in young stars, such as the presence of a not negligible magnetic field
(especially in the convective region, see e.g. Ventura et al. [215]). To this
regard Feiden and Chaboyer [74] found that the inclusion of a magnetic field
in partially convective stars produces a radius expansion caused by the
inhibition of convection efficiency in the convective envelope (see also
Feiden and Chaboyer [73]). Figure 15 shows the results of their work on
evolutionary models computed with and without the inclusion of a magnetic
field. The models are compared to the characteristics of YY Gem binary system
(both stars have masses of about 0.6 M⊙, Torres and Ribas [205]). Such a
system exhibit evidences of a relatively strong magnetic field (surface spots,
X-ray, gyrosynchrotron radio emissions, flaring events). Standard models
underestimate the radius of the components by about 8%, a difference that can
be erased if a magnetic field of 4-5 KG is included in the computations. The
stronger is the magnetic field the larger the radius of the star at the same
age (left panel). In the radius vs $T_{\mathrm{eff}}$ plane, the inclusion of
a magnetic field produces a cooler and brighter star (see right panel of Fig.
15). Feiden and Chaboyer [74] also showed that it is possible to reproduce the
main effects of a magnetic field in 1D non-magnetic stellar models by using a
properly tuned value of the mixing length parameter. To do this an
$\alpha_{ML}$ value lower than the solar calibrated one (i.e. close to unity)
should be adopted.

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 [72]). Figure 16 shows the
comparison between the expected surface lithium abundance as a function of
$T_{\mathrm{eff}}$ in standard non magnetic models of 5 and 10 Myr compared to
a model of 10 Myr in which a magnetic field is included (the magnetic field
strength Beq shown in figure varies with the mass but in the mass range 0.1-1
M⊙ it is of the order of 2-2.6 KG). The inclusion of the magnetic field has a
strong impact on the resulting lithium abundance, drastically reducing the
level of depletion and thus pointing in the direction to improve the agreement
between data and model predictions for pre-MS stars.

Figure 15: Effect of magnetic fields (for the labelled surface magnetic field
strength $<$B$f>$) on the radius evolution of a partially convective pre-MS
star ($M=0.599$ M⊙). The radius is compared to that of the observed YY Gem
binary system (horizontal and vertical stripes). Left panel: radius vs age.
Right panel: stellar radius vs $T_{\mathrm{eff}}$ for magnetic and non
magnetic models. Figure adapted from Feiden and Chaboyer [74]. Figure 16:
Effect on the surface lithium abundance of the presence of a magnetic field
(Beq, see text). The surface lithium abundance behaviour as a function of the
effective temperature is shown for standard (non magnetic) models of 5 and 10
Myr compared with a model of 10 Myr in which a magnetic field is included.
Figure adapted from Feiden [72].

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 [95] and Somers and Pinsonneault [178]). 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. Figure 17 shows an application of
stellar models with surface spots to the surface lithium abundance observed in
the Pleiades cluster (see Somers and Pinsonneault [178]). 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.

Figure 17: Comparison between Pleiades data and predicted surface lithium
abundance obtained in models that account for surface spots for several values
of the spot coverage fraction $f_{\mathrm{spot}}$. Dotted lines connect models
with the same mass and different values of $f_{\mathrm{spot}}$. The dimension
of the circles is proportional to the rotation velocity of the star. Figure
adapted from Somers and Pinsonneault [178].

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 [220]). 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 analysed 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 $\gtrsim 500$ Myr). However, such mechanisms are not relevant for
the evolution of young pre-MS stars and thus we will not discuss them in this
context (see e.g. Charbonnel et al. [40], Eggenberger et al. [71], Landin et
al. [112])

###  7.5 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 modelling 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 modelling realistic stellar atmospheres, the
interested reader can find more details in other reviews (see e.g. Jofré et
al. [101]).

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. [144]). 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 (Collet et al. [48], Nordlund et al. [144]) and CO5BOLD (Freytag
et al. [76]). 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. [120], Magic et
al. [127], Trampedach et al. [206]) have proven that the convection-related
surface structures have different size, depth, and temporal variations,
depending on the stellar type (Tremblay et al. [209], Beeck et al. [22], Magic
and Asplund [126]). 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. [25], Creevey et al. [50],
Chiavassa et al. [43]), radial velocity (Allende Prieto et al. [3], Chiavassa
et al. [44]), chemical abundances estimates (Asplund et al. [12], Caffau et
al. [34]), and photometric colours (Chiavassa et al. [44], Bonifacio et al.
[31]).

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 idealised LTE (Asplund [9]). 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 [143]).
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 (Asplund et al. [e.g., 11], Caffau et al. [e.g., 35], Lind et al.
[e.g., 116], Nordlander et al. [e.g., 142], Magic et al. [e.g., 127], Amarsi
et al. [e.g., 4]). 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. [23]). Using this method, Wang et
al. [217] derived a new 3D NLTE solar abundance of A(Li) = 0.96 $\pm$ 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. [5], Bergemann et al.
[24]). 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.7 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. [37], Asplund et al. [13]). Cayrel et al. [38]
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 (Fig. 18), because they have a preferential influence over the blue-
and red-shifted part of the line profile, respectively (Lind et al. [117]). To
investigate this aspect, Steffen et al. [190] and Lind et al. [117] used a 3D
NLTE treatment with 3D RHD simulation snapshots carried out with CO5BOLD and
Stagger Code , respectively. They re-analysed 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 characterise the calibration
lines, to decrease the observational error. Eventually, a very recent
publication by González Hernández et al. [81], confirms the non detection of
6Li for a very metal poor binary star ([Fe/H]$\sim-3.7$ dex), finding an upper
limit for the isotopic ratio of 6Li/7Li$~{}<~{}10\%$.

Figure 18: Example of synthetic stellar surface profiles of the 7Li doublet
line at 6708.0 Å, computed from 3D RHD simulation of metal-poor stars with an
LTE (dashed) and NLTE (solid) approach. The 7Li-abundance is the same. The LTE
line profile interpolated to meet the same equivalent width as the NLTE line
is also displayed (dotted). An additional absorption appears in the red wing
relative to the blue for NLTE models due to strongly differential effects in
granules and inter-granular regions (Lind et al. [117]).

##  8 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 is 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 [169]),

$N_{A}<\sigma
v>_{b}=\sqrt{\frac{8}{\pi\mu}}\frac{N_{A}}{(K_{B}T)^{\frac{3}{2}}}\int_{0}^{+\infty}\sigma(E)_{b}\,E\,e^{-\frac{E}{K_{B}T}}\,dE\,\,(\mathrm{cm}^{3}\,\,\mathrm{mol}^{-1}\,\mathrm{s}^{-1})$
(4)

where $\sigma(E)$ is the cross section of the process, the subscript $b$ means
that the reaction rate is for two bare nuclei (i.e. without any electron
screening effect), $T$ is the temperature in Kelvin (K). In stellar evolution
calculations, the energy at which the process occurs are generally so small
that it is convenient to write the cross section in terms of another quantity
called the astrophysical factor $S(E)$ defined as it follows,

$S(E)_{b}=E\,\sigma(E)_{b}\,e^{2\pi\eta(E)}$ (5)

with $\eta(E)$ the Sommerfeld parameter related to the tunnel effect of two
interacting charged particles, that can be written as:

$2\pi\eta(E)=\frac{Z_{1}\,Z_{2}\,e^{2}}{2\varepsilon_{0}\hbar}\sqrt{\frac{\mu}{2E_{\mathrm{cm}}}}=31.3998\,Z_{1}\,Z_{2}\,\sqrt{\frac{A_{\mu}}{E_{\mathrm{cm}}(\mathrm{KeV})}}$
(6)

where $\mu=m_{1}m_{2}/(m_{1}+m_{2})$ is the reduced mass and $A_{\mu}$ is the
same quantity but expressed in atomic mass units (amu),
$E_{\mathrm{cm}}(\mathrm{KeV})$ is the energy in the centre of mass expressed
in KeV. Using this quantity, eq. (4) assumes the following form,

$N_{A}<\sigma
v>_{b}=\sqrt{\frac{8}{\pi\mu}}\frac{N_{A}}{K_{B}T^{\frac{3}{2}}}\int_{0}^{+\infty}S(E)_{b}\,e^{-2\pi\eta(E)-\frac{E}{K_{B}T}}\,dE\,\,(\mathrm{cm}^{3}\,\,\mathrm{mol}^{-1}\,\mathrm{s}^{-1})$
(7)

For many application in stellar astrophysics, it is possible to expand the
astrophysical factor around a specific value of the energy, thus obtaining,

$\displaystyle S(E)\approx
S(E_{0})\bigg{(}1+\frac{S^{\prime}(E_{0})}{S(E_{0})}(E-E_{0})+\frac{1}{2}\frac{S^{\prime\prime}(E_{0})}{S(E_{0})}(E-E_{0})^{2}+\dots\bigg{)}$
(8)

The quantity $E_{0}$ is also known as the Gamow peak energy, and it
corresponds to the energy where the exponential quantity inside the integral
in eq. (7) has its maximum value. $E_{0}$ is defined in the following way
(Rolfs and Rodney [169]),

$E_{0}=1.22(A_{\mu}Z_{1}^{2}Z_{2}^{2}T_{6}^{2})^{1/3}\,\,(\mathrm{KeV})$ (9)

with $T_{6}$ the temperature expressed in million kelvin.

The expansion of $S(E)$ given above depends on the temperature at which the
considered reaction occurs (thorough $E_{0}$), which in turn depends on the
stellar mass. However, at low energy typical of reactions in stars, $S(E)$
varies slowly with the energy, thus it is convenient to expand $S(E)$ around
$E\approx 0$: in this case the reaction rate can be evaluated knowing $S(0)$
and its derivatives (usually it is enough to have $S^{\prime}(0)$ and
$S^{\prime\prime}(0)$).

Light element (p,$\alpha$) reaction rates have been recently revised through
the indirect Trojan Horse Method (THM, see e.g. Baur [21], Spitaleri et al.
[179, 180, 181] 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-noise 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
[154, 108] 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 [155] and
references therein). THM was also applied to reactions between heavier
systems, which are relevant in the late stellar evolutionary stages (Tumino et
al. [211]). 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.

###  8.1 Effects of deuterium burning cross sections on pre-MS evolution

As discussed in Sec.s 5, 6.3 and 6.4, 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,$\gamma$)3He process in stellar conditions has been
reported by several authors (see Adelberger et al. [1] for a review) both from
measurements and theoretical calculations along with its uncertainty.
Adelberger et al. [1] redetermined the best value for the astrophysical factor
$S(E)$ for such a reaction at zero energy ($S(0)$) and the uncertainty on it,
concluding that the current uncertainty on $S(0)$ for such burning reaction is
$\approx 7\%$. Recently, Mossa et al. [137] redetermined the D+p cross
sections at energies typical of the BBN (between 32-263 KeV) – thus larger
than those used in stellar calculations – estimating an uncertainty of about
$3\%$.

We tested the impact on pre-MS evolution of the D+p cross section variation,
using the uncertainty given by Adelberger et al. [1] at stellar energies,
which is $\pm 7\%$. Such a variation of the deuterium burning reaction rate
produces a negligible effect on the stellar structure evolution. The
negligible effect is related to the large dependency on the temperature of the
p(D,$\gamma$)3He burning channel (about $T^{12}$); if $S(0)$ is artificially
varied (e.g. reduced) by 7% (independently of the temperature), to obtain the
same energy production rate, which sustains the star, an increase of the
burning temperature is required. However, given the high temperature
dependency of the rate, a very small temperature variation is enough to
restore the energy production. Thus the models are essentially unaffected by
the current uncertainty on the p(D,$\gamma$)3He reaction rate. From this
analysis we can conclude that the main uncertainty source on the D-burning
phase in stellar models is the error on the initial deuterium abundance which
can be as large as 50% as discussed in Section 6.4.

Recently Tumino et al. [210] (see also Tumino et al. [212]) 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.

Figure 19: Comparison of the THM rates for the D(D,p)3H (left panels) and
D(D,n)3He (right panels) reactions with the NACRE (Angulo et al. [6]), JINA
REACLIB (Cyburt [52]) and Descouvemont et al. [68] rates. The stripe
corresponds to the THM estimated uncertainty. Figure adapted from Tumino et
al. [210].

Figure 19 shows, for the quoted reactions, the THM rates compared to the ones
of still widely used NACRE compilation (Angulo et al. [6]), of the JINA
REACLIB (Cyburt [52]) and to the Descouvemont et al. [68] rates. The estimated
uncertainty on the analysed burning channels (of about 5%) are also shown. At
temperatures typical of stellar deuterium burning ($\sim 10^{6}$ 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 [52]. The differences sensitively
reduce at larger temperatures, more important for cosmological calculations.
If the Descouvemont et al. [68] 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 $T\sim 10^{9}$ K.

The new THM rate for the D(D,n)3He reaction is $\sim$10% larger than the
others (NACRE, JINA and Descouvemont et al. [68]) for temperatures smaller
than about $5\times 10^{7}$ K. At larger temperatures the differences reduce
and for $T\gtrsim 2\times 10^{8}$ K the THM rate is smaller than the others by
about 5-10%. Tumino et al. [210] evaluated the effect of the new rates in
stellar models, showing that the change in the cross sections does not produce
any effect on the stellar structure. The result was expected because such
burning channels are quite negligible in stellar models, where D is mainly
destroyed via the p(D,$\gamma$)3He channel. On the contrary, these reactions
could be more important for BBN (Coc and Vangioni [46], Cyburt [52]). Tumino
et al. [210] estimated that the new reaction rates could result in a variation
of the primordial deuterium abundance inferred from the BBN by about $2\%$,
while an impact on the 7Li abundance up to about 10% is expected.

###  8.2 Stellar surface abundance of light elements and updated (p,$\alpha$)
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.
The different fragility of Li, Be and B against (p,$\alpha$) destruction
reactions potentially allows to investigate the characteristics of different
depths of the stellar interior.

In Fig.s 20 and 25 the reaction rates for the most relevant light element
burning (p,$\alpha$) 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.

####  8.2.1 6Li and 7Li surface abundance and (p,$\alpha$) reaction rates
efficiency

Figure 20: Left panel: ratio between the THM 6Li(p,$\alpha$)3He reaction rate
and the [159] (pt05) reported in the JINA REACLIB (solid blue line) together
with THM upper and lower limits (red dashed line). The black dashed area
represents the upper and lower limits of the JINA REACLIB rate assuming the
same uncertainties given in the NACRE compilation. $T9$ indicates the
temperature in units of 109 K. Figure adapted from Lamia et al. [110]. Right
panel: Ratio of the adopted THM 7Li(p, $\alpha$)4He reaction rate to that
evaluated by the still widely used NACRE compilation (solid blue line). The
THM rate upper and lower limits (red dashed lines) are compared with the upper
and lower values recommended by NACRE (black dashed lines). Figure adapted
from Lamia et al. [109].

The first attempts to apply THM (p,$\alpha$) reaction rates to pre-MS lithium
surface abundance calculations were performed by Pizzone et al. [158] and
[159] (hereafter pt05) and successively updated, after re-normalization to
recently released direct data, in Lamia et al. [109, 110]. The left panel of
Fig. 20 shows the 6Li(p,$\alpha$)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
T$\approx~{}10^{6}$ K, typical of 6Li burning in the pre-MS phase, a value
that is larger than the current estimated uncertainty (about 10%) on the rate
itself.

Lamia et al. [110] 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 $-2.0$). Figure 21 shows the
time evolution of the surface 6Li abundance – normalised to the original
value– obtained adopting the three different 6Li(p,$\alpha$)3He reaction rates
– THM, JINA (pt05), and NACRE. From Fig. 21 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.,
$M=1.2$ M⊙), which have the thinnest external convective envelopes, only that
with the highest metallicity (i.e., [Fe/H]$=-0.5$) efficiently depletes
surface 6Li. In the selected [Fe/H] range, the difference in the 6Li depletion
between the THM and NACRE models ranges from about 13% (for $M=1.2$ M⊙) to
about 60% (for $M=1.0$ M⊙). The difference reduces if JINA rate is adopted, as
expected due to the smaller differences between the two rates.

Figure 21: Time evolution of the surface 6Li abundance, when the three
different labelled 6Li(p,$\gamma$)3He reaction rates are adopted in the
models. Each panel corresponds, as indicated, to a different chemical
composition for which four different masses are evolved. See text for details.
Figure adapted from Lamia et al. [110].

Right panel of Fig. 20 shows the comparison between the THM and NACRE 7Li$+$p
reaction rate; the difference is of about 13% at a temperature of
T$\approx~{}10^{6}$ K, not much larger than the current uncertainty on the
rate (about 10%). Figure 22 shows the time evolution of the surface 7Li
abundance for different masses when THM and NACRE 7Li$+$p reaction rates are
adopted. The differences between the two calculations range from about 7% (for
$M=1.0~{}$M⊙) to about 25% (for $M=0.6$ M⊙).

Figure 22: Time evolution of surface 7Li abundance for PISA models with solar
chemical composition and the labelled masses computed with THM, and NACRE
reaction rates (figure adapted from Tognelli et al. [198]).

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. 7.4 and the discussions in Pizzone et
al. [157] and Tognelli et al. [197]). 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. [109, 110]).

####  8.2.2 Lithium Depletion Boundary

In the mass range $M\approx$0.06-0.4 M⊙ (the exact values depending on the
chemical composition), 7Li is completely destroyed in pre-MS in fully
convective structures. The larger is the mass the higher is the temperature
inside the star, and consequently the earlier is the onset of lithium burning.
The larger temperature in more massive stars produces also a more efficient
lithium burning, and consequently the age at which lithium is fully depleted
in such convective stars strongly depends on the stellar mass. In a coeval
population of stars with ages between about 15 and 350 Myr, one would expect
to observe a sharp transition in very low-mass regime between stars with and
without surface lithium at a given mass (corresponding to the higher stellar
mass that, at the cluster age, fully destroy lithium). Such a transition,
usually called the Lithium Depletion Boundary (LDB), is a powerful age
indicator (see e.g. D’Antona and Mazzitelli [58]) thanks to the connection
between luminosity, mass and age. Figure 23 shows an example of the age vs
luminosity relation for stars located at the LDB in the mass range [0.06
$\div$ 0.4] M⊙.

Figure 23: Age vs luminosity relation for stars at the Lithium Depletion
Boundary for solar chemical composition and $\alpha_{\mathrm{ML}}$=1. The
labels along the sequence indicate the stellar mass in M⊙. Figure adapted from
Tognelli et al. [201].

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. [20], Oliveira et al. [145], Jeffries and Oliveira [100],
Manzi et al. [128], Dobbie et al. [69], Jeffries et al. [99], Binks and
Jeffries [26], Juarez et al. [102], Dahm [56], Dupuy et al. [70], Martín et
al. [130, 129] and references therein). The uncertainties on age determination
through the LDB technique have been analysed in Burke et al. [33] and, more
recently, in Tognelli et al. [201]; the main uncertainties that potentially
affect the LDB age determination are those already discussed in Section 7.4.
Figure 24 shows the relative age uncertainty on LDB age determination obtained
by Tognelli et al. [201] 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 ($0.06\lesssim
M$/M⊙$\lesssim 0.1$) have errors of the order of about 5%, an uncertainty that
increases increasing the stellar luminosity at the LDB and thus the derived
age. Tognelli et al. [201] showed that a large part of the uncertainty on the
LDB ($\sim 40\%$) comes from the chemical composition indetermination while
the lithium burning rate causes a variation of the LDB age of about 1%.

Figure 24: Relative uncertainty on the LDB age as a function of the luminosity
of the star at the LDB. The shaded area represents the cumulative error stripe
(see text). The lines are estimations of the uncertainty obtained by a linear
or quadratic combination of the individual effect of the variation of the
input physics and chemical composition. Figure adapted from Tognelli et al.
[201].

####  8.2.3 9Be and 10B surface abundance and (p,$\alpha$) reaction rates
efficiency

Figure 25: Left panel: ratio between the 9Be(p,$\alpha$)6Li THM reaction rate
and the one reported in the NACRE compilation. The red filled area marks the
THM reaction rate uncertainty compared to the NACRE one (grey region). Right
panel: ratio between the 10B(p,$\alpha$)7Be THM reaction rate and that
obtained by the NACRE compilation (blue line). The red filled area marks the
THM reaction rate uncertainty compared to the NACRE one (grey region). Figure
adapted from Lamia et al. [111].

Figure 25 shows the comparison between the recent THM reaction rates and other
reaction rates used in the literature. The THM 9Be(p,$\alpha$)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. [221], even if the THM uncertainty region is sensibly smaller than
that of the NACRE II rate, see Lamia et al. [111]. Left panel of Fig. 26 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 M $\lesssim 0.5$ M⊙).

We recall that in stars, 9Be is destroyed through two channels:
9Be(p,$\alpha$)6Li (R1, the rate analysed by Lamia et al. [111]) and 9Be(p,
2$\alpha$)2H (R2). The ratio R1/R${}_{2}\approx 1.2$, for stellar conditions
at the temperature of interest; thus the R2 contribution to beryllium
destruction is not negligible. Changing only R1, as done in Lamia et al. [111]
affects the final beryllium abundance by a factor that is given by the R1
reaction rate change (about 25%) multiplied by the probability that the 9Be
burning occurs in that channel, i.e., 25% $\times$ R1/(R1 \+ R2 ) $\approx$
14%, thus leading to a change in the predicted Beryllium abundance lower than
what expected if only R1 channel was active in stars. The effect of the THM
uncertainty on the 9Be(p,$\alpha$)6Li rate – approximately 25% which is equal
to the difference with respect to NACRE rate – is expected to produce a
variation of the predicted 9Be surface abundance of the same order of
magnitude discussed above.

Figure 26: Temporal evolution of surface 9Be (left panel) and 10B (right
panel) abundances (normalised to one) for solar chemical composition models
with the labelled stellar mass. Models computed using the THM (thick solid
line) and the NACRE (dashed line) reaction rates are shown. Figure adapted
from Lamia et al. [111]. Figure 27: Surface logarithmic abundances of 9Be
(upper panel) and 10 B (lower panel) as a function of the star effective
temperature for different stellar masses in the range [0.06, 0.8] M⊙ and for
three ages typical of young/intermediate open clusters. Models are calculated
for [Fe/H]$=+0.0$ adopting both the THM (solid blue line) and the NACRE
(dashed red line) reaction rates. Figure adapted from Lamia et al. [111].

Right panel of Fig. 25 shows the comparison between the THM and the NACRE rate
for the 10B(p,$\alpha$)7Be reaction: at temperatures of a few millions of
kelvin, the THM rate is $\sim$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. [221],
the differences are significantly reduced, see Lamia et al. [111].

The effect of the different 10B(p,$\alpha$)7Be reaction rates on surface B
abundance in low mass stars (at solar metallicity) is shown in the right panel
of Fig. 26. The lower THM 10B(p,$\alpha$)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 $M~{}\lesssim 0.4$ M⊙.
Also notice that the typical timescale at a fixed mass where 10B is destroyed
is longer than that of 9Be.

For completeness in Fig. 26 we point out for ages typical of MS evolution
(e.g. log t$\gtrsim 8.5$ for $M=0.5$ M⊙ for the 9Be abundance behaviour and
for $M=0.4$ M⊙ for 10B) the effect of microscopic diffusion, which leads to
the settling of light elements toward the stellar interior.

Figure 27 shows the surface logarithmic abundances of 9Be and 10B as a
function of the effective temperature, $T_{\mathrm{eff}}$. Although
observational 9Be and 10B abundances are still not available for the low
temperature/mass regimes typical of efficient 9Be and/or 10B burning, it is
worth to estimate the role of the improvements in nuclear physics in surface
abundance predictions. The models are computed using the THM and NACRE
reaction rates discussed above; we remind that $T_{\mathrm{eff}}$ is not
affected by the change of the (p,$\alpha$) rates. Each curve represents the
abundance isochrone, i.e., the locus of models with the same age but different
masses. For those models where 9Be (10B) burns the differences between the
adoption of the NACRE and THM reaction rates can be as large as about 0.2-0.3
dex for 9Be and almost 1 dex for 10B.

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 is 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,$\alpha$) Be and B burning reaction rates is not the dominant error source
for the surface abundance predictions of such elements.

##  9 Summary and conclusions

Surface light elements abundances prediction in stellar models is a difficult
task because it is 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 towards 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