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 star, with a radius , a luminosity , an effective temperature K and a central temperature 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 , , and . We remind the readers that represents the fraction of the Keplerian angular momentum at the stellar surface that is injected into the star, the value of the Richardson number in the convective region and characterizes the thermal behavior of the accreted matter. As explained in SF, a large negative value of depicts a situation close to free convection where the mass distribution remains unchanged. If the accreted globule rapidly thermalizes, i.e., if , 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 , and .
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 . The time accretion laws, in the unit of , are given by
with equal to and 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 yr.
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 K, deuterium ignites through the reaction
This event marks the beginning of a different evolutionary phase for accreting stars.
where µ is the mean molecular weight (Cox & Giuli,
1968). Eq. (21) indicates that the rate of increase of
depends on the contraction and mass accretion
rates. Prior to deuterium ignition, the radius evolution is not very
sensitive to (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.
where is the energy generated per reaction ( Mev) and 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 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 , defined by
represents the solution of Eq. (14) in the limit of very fast reaction rate (). 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.
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 . Expansion lasts as long as , i.e., as long as the star has not reached thermal equilibrium. For large accretion rates () or early in the evolution when the accretion time scale 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.
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 is nearly adiabatic, so that . 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 . Therefore temperature variations modify in the same way the other thermodynamic quantities and .
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 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.
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 and instead of .
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 , can be written in the form
Therefore, a higher deuterium mass fraction is able to produce a larger at a given time than in a standard case.
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).
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 and 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 , 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
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 and (short- and long-dashed lines, respectively) reflects the temperature sensitivity of the reaction rate.
The larger , the later deuterium is depleted and, in extreme cases where most of the accretion luminosity is driven into the star ( 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 () and decreases the thermal relaxation time scale (). Consequently radius and luminosity will become larger for greater (Fig. 10). Once accretion ends, the star is already in thermal equilibrium and contraction proceeds.
3.6. Influence of mass accretion profile
To see how changes in various parameters (, ) modify the stellar structure, we will first analyze two models with the same accretion law (R1), and but with two different values for , namely and . For , convection transport is efficient and accreted matter reaches the central regions while for , it mainly accumulates in the outer layers of the star (Fig. 11a-f.f). For both values of , we study two models one with and one without accretion energy supply ( and ).
In the absence of accretion energy inflow
(), 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
nor the 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
When mass deposition is accompanied by energy deposition (), temperature and nuclear energy rate distributions are modified. If accretion energy is preferentially deposited in the central region (), conversion of 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 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.
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 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 .
Comparing tracks corresponding to various accretion rates, some general conclusions emerge: the motion of accreting stars in the HRD is always directed to higher effective temperature 4, the locus of these stars in the HRD is bounded by the two standard PMS tracks corresponding to their initial and final mass and 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 seen by an observer is in fact the composition of the different sources
where 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).
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998