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Astron. Astrophys. 326, 1001-1012 (1997)

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2. Numerical procedure

The stellar evolution code used for this study was described in several articles (Forestini 1991, 1994), and currently used opacities or atmosphere models are analyzed in detail by Siess et al. (1996b). Modifications needed to take accretion into account are the following.

2.1. Mass and composition update

SF derived the distribution f of accreted matter deposited in the star as a function of its thermal, chemical and mechanical properties. By definition [FORMULA] represents the ratio of the mass [FORMULA] accreted in shell j to the mass of this shell, [FORMULA]. Therefore, when f is computed, the new mass distribution after accretion, [FORMULA], is given by 1

[EQUATION]

Our computations reveal that the distribution of matter inside the star does not depend on the mesh point. Indeed in completely convective stars, the accreted matter distributes uniformly in the stellar interior whatever are the distribution of mass shells or time step. On the other hand if the accretion process involves only the surface layers of the star, the periodic redistribution of mass shells has very weak effect because in this region the mass interval between two shells is always very small. Moreover the time step [FORMULA] is also constrained by the accretion process and must fulfill [FORMULA]. This condition ensures small changes in function f between two consecutive models.
This modification of the mass distribution is accompanied by a modification of chemical composition, since the chemical composition of the accreted matter is not necessary the same as that of the star. This is especially the case for [FORMULA] H, as we shall see. Species conservation leads (cf Appendix A) to a new mass fraction of element i at shell j

[EQUATION]

where [FORMULA] is the stellar mass fraction of element i at shell j before accretion and [FORMULA] the mass fraction of this element in the accreted matter. In our model we assume that a turbulent element keeps its identity until it dissolves, therefore we can replace [FORMULA] by [FORMULA] in the above expression. As expected, if the chemical composition of the circumstellar environment does not differ from that of the star ([FORMULA]), there is no chemical change. Additional modifications come from the fact that [FORMULA] has to be rescaled to the new mass distribution (Appendix A). The energetic properties of the accreted matter are discussed in the next section.

2.2. Energy equation

During the accretion process, a large amount of potential energy is released. If we assume that accretion is ultimately caused by the presence of an accretion disk around the star, then half of the accretion luminosity is radiated by the disk while the other half [FORMULA] is deposited in the boundary layer

[EQUATION]

where G is the gravitational constant, [FORMULA] is the protostellar mass, [FORMULA] is the mass accretion rate and [FORMULA] is the radius of the hydrostatic core. Generally it is assumed that the accreted matter has the same specific entropy as matter on the stellar surface (Kippenhahn et al. 1977, Mercer-Smith et al. 1984, Stringfellow 1989, Braun et al. 1995) but this assumption is not necessarily satisfied. Physical conditions prevailing in the boundary layer are still subject to many uncertainties (Popham et al. 1993, Regev & Bertout 1995). In particular, the fraction of accretion luminosity that is effectively radiated away is still uncertain. We thus use in the following a parameter [FORMULA] ([FORMULA]), representing the fraction of [FORMULA] that is transfered to the accreted matter as internal energy, the fraction [FORMULA] escaping radiatively. The accretion luminosity imparted to the star is written

[EQUATION]

In a situation where [FORMULA], all the energy is radiated away in the boundary layer and the accreted matter mixes with the same properties as the local stellar matter. Conversely, if [FORMULA], [FORMULA] is available to heat up the stellar matter.

During an evolutionary time step [FORMULA], a mass [FORMULA] is accreted and a total energy [FORMULA] is given to the star. Assuming instantaneous and uniform heat repartition, each gram of accreted matter releases on average, an energy [FORMULA] per second. The distribution of [FORMULA] then corresponds to the distribution of accreted matter. In consequence, the value [FORMULA] at shell j is

[EQUATION]

where [FORMULA] is the accretion function. The factor [FORMULA] takes into account the mass modification of the shells. This relation satisfies [FORMULA].
The fact that accreted matter can radiate and lose part of its heat excess during the accretion process is explicitly taken into account by the function f. Actually, the distribution of mass deposition inside the star depends on the thermal property of the accreted material (see the discussion in SF).

Finally, the equation of energy conservation can be written

[EQUATION]

where [FORMULA] is the local luminosity and where [FORMULA] ([FORMULA]) is the nuclear (gravitational) energy production rate per unit mass.

2.3. Nucleosynthesis and accretion

During classical PMS evolution, the luminosity [FORMULA] is provided by the gravitational contraction and the star evolves on the Kelvin-Helmholtz time scale, [FORMULA] defined by

[EQUATION]

For a star of 1 [FORMULA], [FORMULA] is of the order of [FORMULA] yr. Deuterium ignition in the early evolution does not alter the general evolution; it only slows the stellar contraction. This is due to both the small amount of deuterium (the deuterium mass fraction is [FORMULA]) and its very high burning rate, which do not maintain nuclear energy production during more than [FORMULA] yr.

However, inclusion of the accretion process modifies that scheme. As long as matter is accreted (typically a few million years), the nuclear energy production is maintained by the inflow of fresh deuterium. The deuterium production due to the nuclear reactions

[EQUATION]

is negligible compared to the inflow of fresh deuterium. From now on, the structure is governed by nuclear burning and evolves on a nuclear time scale. In order to decouple the equations of nucleosynthesis from those of stellar structure we normally have to constrain the time step to be small enough to guarantee a constant deuterium abundance over the time step. I.e., the evolutionary time step reduces to a few hundred years. In other words, the evolution of deuterium abundance in the presence of accretion cannot be treated independently from the overall evolution of stellar structure.

During a time step [FORMULA], accreted material brings in the star a number of deuterium atoms [FORMULA], where [FORMULA] is the deuterium mass fraction in the accreted matter and [FORMULA] the mass of the deuterium atom. The equation governing deuterium abundance evolution is then given by 2

[EQUATION]

where [FORMULA] is the characteristic time scale of deuterium burning, [FORMULA] and [FORMULA] the number density of deuterium in the star before and after addition of accreted matter. Usually the nucleosynthesis equations are written in terms of the mole fraction [FORMULA], a quantity that is related to [FORMULA] by

[EQUATION]

where [FORMULA] is the Avogadro number, [FORMULA] the density and [FORMULA] the deuterium mass number. From Eq. (7), we have

[EQUATION]

This expression couples the structural and nucleosynthetic evolution. The evolutionary time-step we choose for each model must verify the following equation in each shell of the star

[EQUATION]

where [FORMULA] is the density difference between the two last successfully calculated models. Condition (9) allows us to neglect density time-variations in Eq. (8), so that Eq. (6) simplifies to

[EQUATION]

where [FORMULA] is the mole fraction of deuterium in the accreted matter. Mass accretion causes a dilution of the shells and to account for that, we define new variables, [FORMULA] and [FORMULA] as

[EQUATION]

Finally the general equation governing the evolution of deuterium is

[EQUATION]

the formal solution of which is given by

[EQUATION]

The first term in Eq. (14) gives the deuterium abundance after its nuclear depletion during a period [FORMULA]. The second term represents the contribution due to accretion including the partial destruction of this element during [FORMULA]. In the very early evolution, the reaction rate is very slow ([FORMULA]) and we find that

[EQUATION]

i.e. the composition remains unchanged. This equation is solved, during the convergence process, at each iteration for the stellar structure.

2.4. Treatment of convective regions

In convective regions, we assume instantaneous mixing. The chemical elements are homogenized and the resulting mean abundances are given by

[EQUATION]

where [FORMULA] is the mass fraction of element i, [FORMULA] and [FORMULA] the mass coordinates of the bottom and the top of the convective zone, respectively. Applied to Eqs. (11) and (12) this leads to

[EQUATION]

and

[EQUATION]

where [FORMULA] is the mass of the convective region, [FORMULA] the mass accreted in this zone and [FORMULA].

Furthermore, as we make the assumption of completely homogeneous convective regions, nucleosynthesis within this zone can also be calculated in one shot, in a mean fictitious shell where we consider the average destruction time scale

[EQUATION]

The average is made on [FORMULA] because it represents the mean reaction rate. Thus, inside convective regions, deuterium evolves according to

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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