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

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3. Results and discussion

3.1. Initial model and accretion rates

Our initial models result from a polytropic model with index n =1.5. After convergence with our evolution code, we obtain a 0.5 [FORMULA] star, with a radius [FORMULA], a luminosity [FORMULA], an effective temperature [FORMULA] K and a central temperature [FORMULA] K. It is completely convective, surrounded by a thin radiative atmosphere. It is also located above the birthline defined by Stahler (1988). In the future, a more consistent approach would be to combine both the protostellar (spherical) and PMS (axisymetric) accretion phases. For self-consistent computations, these two consecutive phases will require the same constitutive physics.

Concerning the accretion model, default parameters are [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. We remind the readers that [FORMULA] represents the fraction of the Keplerian angular momentum at the stellar surface that is injected into the star, [FORMULA] the value of the Richardson number in the convective region and [FORMULA] characterizes the thermal behavior of the accreted matter. As explained in SF, a large negative value of [FORMULA] depicts a situation close to free convection where the mass distribution remains unchanged. If the accreted globule rapidly thermalizes, i.e., if [FORMULA], the accretion process does not disturb the convective motions and matter distributes uniformly. In the following, we will show the influence of changing these parameters. The chemical composition of the accreted matter is assumed identical to the initial composition of the star. Initial abundances are [FORMULA], [FORMULA] and [FORMULA].

In order to illustrate the effects of accretion on stellar evolution, we study three time variable accretion rates typical of extreme T Tauri stars (Adams et al. 1990) and limit the computations to a maximum mass of 1.2 [FORMULA]. The time accretion laws, in the unit of [FORMULA], are given by


with [FORMULA] equal to [FORMULA] and [FORMULA] yr, respectively. These functions are depicted in Fig. 1. The accretion law (R3) accounts for a large accretion phase at the beginning of the evolution followed by a slowly decreasing accretion activity. We first study the accretion phase, i.e., ages lower than [FORMULA] yr.

[FIGURE] Fig. 1. Accretion rates (above) and mass evolution (below). The accretion end corresponds a final mass of 1.2 [FORMULA]. The solid line refers to the exponential law (R3), while the dotted and dashed curves correspond to the (R1) and (R2) prescriptions, respectively.

3.2. Energetic evolution of accreting stars

At the beginning of their evolution, stars are completely convective and lie on the Hayashi portion of their PMS track. Their internal pressure cannot balance gravity and they contract. When the central temperature reaches [FORMULA] K, deuterium ignites through the reaction


This event marks the beginning of a different evolutionary phase for accreting stars.

For completely convective stars, the central temperature can be approximated by


where µ is the mean molecular weight (Cox & Giuli, 1968). Eq. (21) indicates that the rate of increase of [FORMULA] depends on the contraction and mass accretion rates. Prior to deuterium ignition, the radius evolution is not very sensitive to [FORMULA] (Fig. 4) so deuterium first ignites inside the more massive stars, i.e., those accreting at the largest rates (Fig. 2). When nuclear energy production begins, the amount of deuterium is larger in more massive stars and the generated luminosity more important.
Changes of initial conditions will not affect this result. Indeed the more distended is the star in its initial configuration, the faster it contracts and by the time deuterium ignites, a very short time elapses (typically a few thousand years). Therefore, the temperature of deuterium ignition is reached with a mass close to its initial value, almost independent of the initial conditions.

[FIGURE] Fig. 2. The nuclear luminosity [FORMULA] of our accreting stars for the three accretion rates as a function of their increasing total mass (upper panel) and as a function of time (lower panel). Default parameters are [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Curves are plotted with the same conventions as Fig. 1.

Once [FORMULA] decreases significantly due to the increasing central temperature, the luminosity drops, reaching its equilibrium value defined by


where [FORMULA] is the energy generated per reaction ([FORMULA] Mev) and [FORMULA] the constant mole mass fraction in the accretion disk. For illustration, we show in Fig. 3 the deuterium mass fraction as a function of time. When [FORMULA] reaches its maximum, most of the deuterium initially present in the star has been burned. The increasing reaction rate contributes to deplete deuterium abundance which now settles to an equilibrium value [FORMULA], defined by


[FORMULA] represents the solution of Eq. (14) in the limit of very fast reaction rate ([FORMULA]). The further drop of 2 H seen in Fig. 3 is due to either the end of accretion for the dotted line (R1), or to the appearance of a radiative core that prevents deuterium from reaching the center for (R2, R3). We see that high accretion rates can sustain a high deuterium abundance despite the increasing reaction rate. The tremendous energy generated by deuterium burning during the accretion process is reflected in the stellar structure.

[FIGURE] Fig. 3. Central deuterium mass fraction as a function of mass (upper panel) and as a function of time (lower panel) for the different accretion rates. Curves are plotted with the same conventions as Fig. 1.

3.3. Internal structure

The nuclear luminosity due to deuterium burning during the accretion phase, is so strong that it exceeds the stellar luminosity. The overall stellar gravitational energy becomes negative (Fig. 4) and the star undergoes a global swelling. In the non-accreting case or for low accretion rates 3, this phenomenon is not present because deuterium is exhausted too rapidly to cause an increase in [FORMULA]. Expansion lasts as long as [FORMULA], i.e., as long as the star has not reached thermal equilibrium. For large accretion rates ([FORMULA]) or early in the evolution when the accretion time scale [FORMULA] is smaller than the Kelvin-Helmholtz time scale, the nuclear energy production induced by the accretion process dominates the structural evolution and the star expands. When sufficient mass has been accreted, the increased gravitational potential finally limits the swelling.

[FIGURE] Fig. 4. Influence of the accretion rate on the stellar radius and luminosities. From the top to bottom panels, are depicted the stellar radius, the total, nuclear and gravitational luminosities, respectively. Curves are plotted with the same conventions as Fig. 1.

The dilatation of the star provokes a drop of central density and halts temporarily the central temperature increase as indicated in Fig. 5a-d. The large energy production at the onset of deuterium burning warms up the central regions but the great sensitivity of the reaction rate on temperature (Caughlan & Fowler 1988) forces deuterium to burn thermostatically, preventing the temperature from climbing too much. This is particularly marked in Fig. 5a-d.d for high accretion rates. However when a significant fraction of the deuterium is depleted, the nuclear energy production can only be maintained by an increase of the central temperature. At that time, the star is still completely convective and its gradient [FORMULA] is nearly adiabatic, so that [FORMULA]. For an ideal gas with a constant mean molecular weight µ (true even during deuterium burning because of the relatively low 2 H abundance level), one obtains [FORMULA]. Therefore temperature variations modify in the same way the other thermodynamic quantities [FORMULA] and [FORMULA].

[FIGURE] Fig. 5. Central density and temperature as a function of time [a and b ] and as a function of mass [c and d ] for the different accretion rates. We note the thermostatic effect of deuterium burning that maintains [FORMULA] to an almost constant value as long as [FORMULA]. Curves are plotted with the same conventions as Fig. 1.

Another interesting property of these models is the influence of accretion on the formation of the radiative core. In Fig. 6, we see that the formation of the radiative core depends on the accretion scenario. During PMS evolution, the opacity [FORMULA] continuously drops in the deep interior with rising central temperatures until the radiative temperature gradient becomes smaller than the adiabatic gradient. However, the maintenance of an important nuclear energy production at the star's center forces the radiative gradient to remain larger than the adiabatic one, so that the star remains convectively unstable. Finally, with decreasing accretion rate and nuclear burning rate, the star turns radiative in its center. The radiative core grows rapidly in mass and forms a radiative barrier that stops the freshly accreted matter. The rate of core growth depends on the central temperature at the time of its formation.

[FIGURE] Fig. 6. Mass at the base of the convective envelope. This graph shows the development of the radiative core for the different accretion rates. Initially the star is completely convective and due to the drop in opacity the radiative core suddenly appears. Curves are plotted with the same conventions as Fig. 1.

3.4. Influence of the chemical composition

We present here the effect of varying the deuterium abundance since it may vary from different regions of the interstellar medium. The evolutionary influence of deuterium can be gauged by artificially varying its interstellar concentration. For the purpose of the study, we have computed for a constant accretion rate given by relation (R1), three different evolutionary models where deuterium mass fraction was set to [FORMULA] and [FORMULA] instead of [FORMULA].

We notice from Fig. 7 that a lower abundance reduces the nuclear luminosity and postpones deuterium ignition. Indeed, the rate of nuclear energy production per unit mass [FORMULA], can be written in the form


Therefore, a higher deuterium mass fraction is able to produce a larger [FORMULA] at a given time than in a standard case.

[FIGURE] Fig. 7. Effect of deuterium abundance on the generated nuclear luminosity. For an accretion prescription given by relation (R1), the evolution of the abundance and the nuclear luminosity are plotted as a function of time. The dotted lines refers to the interstellar value of [FORMULA], the solid and dashed curves to [FORMULA] equal [FORMULA] and [FORMULA], respectively.

The flat deuterium mass fraction profile observed for the largest initial abundances is due to the great temperature sensibility of reaction (20); the star reacts to the increased mass fraction and generated luminosity by burning deuterium at a slower rate and thus for a longer time. The mechanical consequence is a substantial swelling of the entire star, an effect not present for low abundances (Fig. 8).

[FIGURE] Fig. 8. Influence of deuterium abundance on the radius, central density and temperature. Curves are plotted with the same conventions as Fig. 7.

Although a deuterium abundance ten times greater than its interstellar value would provide the same equilibrium luminosity than an accretion rate ten times larger [Eq. (22)], the behavior of the star is completely different. During all the accretion process, it is a combination of both [FORMULA] and [FORMULA] that governs the evolution of the structure, and depending on these values the star can experience either a swelling or a contraction.

3.5. Effect of accretion energy deposition

We study the role of accretion energy deposition in the star. The control parameter [FORMULA], defined in Sect. 2.2, allows us to treat accretion not only as a mechanical constraint to the structure, but also as an energetic process.

Faced with the arrival of accretion energy, the star converts the thermal energy of accreted matter principally into work against compression rather than into internal energy and consequently expands. This results from the thermostatic effect of deuterium burning which constraints temperature to weak variations. Fig. 9

[FIGURE] Fig. 9. Influence of accretion energy deposition on the central temperature and density, on the deuterium mass fraction and nuclear luminosity. The solid, dotted, short- and long-dashed lines correspond to a value of [FORMULA] equal to 0.5, 0.1, 0.01 and 0, respectively. The accretion rate is [FORMULA] [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

displays the effect of this additional source term on the central density and temperature. When a large amount of accretion energy is released, the temperature enhancement is less pronounced and consequently deuterium is depleted more slowly (Fig. 9). The difference in the deuterium profile for [FORMULA] and [FORMULA] (short- and long-dashed lines, respectively) reflects the temperature sensitivity of the reaction rate.

The larger [FORMULA], the later deuterium is depleted and, in extreme cases where most of the accretion luminosity is driven into the star ([FORMULA] close to unity), the burning rate is so slow that deuterium abundance remains constant for a longer period, up to one third of the total age of a star. The resulting swelling of the star lowers the inflow of accretion energy ([FORMULA]) and decreases the thermal relaxation time scale ([FORMULA]). Consequently radius and luminosity will become larger for greater [FORMULA] (Fig. 10). Once accretion ends, the star is already in thermal equilibrium and contraction proceeds.

[FIGURE] Fig. 10. Influence of accretion energy deposition on the radius (top panel), the stellar (mid panel) and accretion luminosity (lower panel). Curves are plotted with the same conventions as Fig. 9.

3.6. Influence of mass accretion profile

To see how changes in various parameters ([FORMULA], [FORMULA]) modify the stellar structure, we will first analyze two models with the same accretion law (R1), [FORMULA] and [FORMULA] but with two different values for [FORMULA], namely [FORMULA] and [FORMULA]. For [FORMULA], convection transport is efficient and accreted matter reaches the central regions while for [FORMULA], it mainly accumulates in the outer layers of the star (Fig. 11a-f.f). For both values of [FORMULA], we study two models one with and one without accretion energy supply ([FORMULA] and [FORMULA]).

[FIGURE] Fig. 11. Comparative study of relevant variables for two different distributions of accreted matter. The solid line refers to a model with [FORMULA] while the dotted line corresponds to [FORMULA]. [FORMULA], [FORMULA] and [FORMULA] are fixed to 0.5, 0.01 and 0, respectively, and the accretion law is given by the (R1) prescription. From the top to the bottom and from the left to the right, panels depict respectively the evolution of the stellar radius [FORMULA], stellar luminosity [FORMULA], gravitational energy at the center [FORMULA], central temperature [FORMULA], central deuterium abundance X(2 H), and at the right bottom the depth of accretion [FORMULA]. [FORMULA] equal zero means that accreted matter reaches the center of the star.

In the absence of accretion energy inflow ([FORMULA]), changes in the parameters have little influence on the evolution of the star because nuclear energy production is not modified. Indeed, for completely convective stars, Eqs. (16, 17, 19) are independent on the distribution of accreted material. Therefore, if the same amount of matter is accreted, neither [FORMULA] nor the [FORMULA] profiles given by Eq. (24) will differ in both configurations. Consequently, changes of parameters will not change the role played by deuterium burning and the stellar structure will be almost insensitive to the accreted mass distribution.
As matter penetrates deeper the gravitational potential is reinforced in the central regions and locally the star undergoes a more efficient contraction (Figs. 11a-f). The sharp growth of [FORMULA] at the surface results from the rise of f in the photospheric layers and to the boundary conditions. The star is now somewhat more compact and its higher gravity provides a larger central temperature leading to earlier deuterium depletion. As the effective temperature is almost constant, the luminosity and outer radius decrease.

When mass deposition is accompanied by energy deposition ([FORMULA]), temperature and nuclear energy rate distributions are modified. If accretion energy is preferentially deposited in the central region ([FORMULA]), conversion of [FORMULA] into work against compression must be more efficient to prevent a temperature enhancement. Consequently the star has lower average gravitational pull and its radius and luminosity increase (Fig. 12a-f). Conversely, if [FORMULA] is concentrated in the outer layers of the star, a significant fraction of the accretion energy can be radiated from the surface. As a consequence, the specific entropy is lower and the star contracts more efficiently, yielding a higher central temperature and smaller luminosity.

[FIGURE] Fig. 12. Same curve as Fig. 11a-f, but for [FORMULA]

3.7. Hertzsprung-Russell diagram

The evolution of accreting stars in the HRD is given in Fig. 13. Symbols indicate the most interesting events occurring in these accreting stars. At the beginning of its evolution, an accreting star is located on the Hayashi track corresponding to its initial mass. Contraction proceeds until the central temperature reaches roughly [FORMULA] K, i.e., when deuterium ignites. Within a few thousand years, the nuclear luminosity becomes higher than the surface luminosity (triangle) and forces the star to swell. However depletion of initial deuterium causes a drop in the nuclear energy production; the star undergoes contraction again (square). The next important step in the evolution of the structure is the appearance of the radiative core (pentagon). Finally accretion ends (open circle) and the star relaxes to the standard track corresponding to final mass 1.2 [FORMULA].

[FIGURE] Fig. 13. The evolutionary tracks in the HRD of Z = 0.02 accreting stars. The solid, dotted and short-dashed curves correspond to three different accretion rates: bi-exponential (R3), constant at [FORMULA] [FORMULA] (R1) and constant at [FORMULA] [FORMULA] (R2), respectively. In our scenario, the end of the accretion phase for the (R1) and (R2) prescriptions is modeled by an exponential decrease. The initial mass is 0.5 [FORMULA] and the accretion is stopped when the stellar mass reaches 1.2 [FORMULA]. The long-dashed line represents the standard PMS track of a [FORMULA] star. Triangles mark the beginning of the star expansion due to deuterium ignition and squares the return to contraction. The radiative core appears in the pentagon location and the open circles mark the end of accretion. In the case of small accretion rate (R2 prescription), triangles and squares are merged and represent the minimum in gravitational luminosity. Stars indicate the time from which the accretion rate falls below [FORMULA] [FORMULA].

Comparing tracks corresponding to various accretion rates, some general conclusions emerge: [FORMULA] the motion of accreting stars in the HRD is always directed to higher effective temperature 4, [FORMULA] the locus of these stars in the HRD is bounded by the two standard PMS tracks corresponding to their initial and final mass and [FORMULA] all accreting stars converge to the same standard track defined by their final mass.

We emphasize on the fact that the luminosity given in Fig. 13 represents the photospheric luminosity only. We do not consider here the circumstellar environment of these objects nor the large amount of energy released by the disk or through the boundary layer. The total luminosity [FORMULA] seen by an observer is in fact the composition of the different sources


where [FORMULA] is the luminosity emanating from the disk. All these components could also potentially modify the effective temperature and luminosity of the global system, star plus circumstellar surrounding (Kenyon & Hartmann 1990). A confrontation with observations will be presented in Siess et al. (1997).

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998