Issue 
A&A
Volume 585, January 2016



Article Number  A65  
Number of page(s)  10  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201527202  
Published online  18 December 2015 
Massive star formation by accretion
I. Disc accretion
^{1}
Observatoire de Genève, Université de Genève,
chemin des Maillettes 51,
1290
Sauverny,
Switzerland
email:
lionel.haemmerle@unige.ch
^{2}
Institute für Theoretische Astrophysik, Zentrum für Astronomie der
Universität Heidelberg, AlbertUeberleStr. 2, 69120
Heidelberg,
Germany
Received: 17 August 2015
Accepted: 3 November 2015
Context. Massive stars likely form by accretion and the evolutionary track of an accreting forming star corresponds to what is called the birthline in the HertzsprungRussell (HR) diagram. The shape of this birthline is quite sensitive to the evolution of the entropy in the accreting star.
Aims. We first study the reasons why some birthlines published in past years present different behaviours for a given accretion rate. We then revisit the question of the accretion rate, which allows us to understand the distribution of the observed premainsequence (preMS) stars in the HR diagram. Finally, we identify the conditions needed to obtain a large inflation of the star along its preMS evolution that may push the birthline towards the Hayashi line in the upper part of the HR diagram.
Methods. We present new preMS models including accretion at various rates and for different initial structures of the accreting core. We compare them with previously published equivalent models. From the observed upper envelope of preMS stars in the HR diagram, we deduce the accretion law that best matches the accretion history of most of the intermediatemass stars.
Results. In the numerical computation of the time derivative of the entropy, some treatment leads to an artificial loss of entropy and thus reduces the inflation that the accreting star undergoes along the birthline. In the case of cold disc accretion, the existence of a significant swelling during the accretion phase, which leads to radii ≳ 100 R_{⊙} and brings the star back to the red part of the HR diagram, depends sensitively on the initial conditions. For an accretion rate of 10^{3}M_{⊙} yr^{1}, only models starting from a core with a significant radiative region evolve back to the red part of the HR diagram. We also obtain that, in order to reproduce the observed upper envelope of preMS stars in the HR diagram with an accretion law deduced from the observed mass outflows in ultracompact HII regions, the fraction of the mass that is accreted onto the star should represent a decreasing fraction of the mass outflows when the mass of the accreting object increases. In other words, the accretion efficiency (mass effectively accreted onto the star with respect to the total in falling matter) decreases when the mass of the star increases.
Key words: stars: formation / stars: evolution / accretion, accretion disks
© ESO, 2015
1. Introduction
Massive stars are rare, but their influence on our Universe is critical. They give the main chemical and dynamical inputs to the interstellar medium. Most of the heavy elements in our Universe were produced in the core of massive stars or during their deaths. The massloss during their lifetimes, and more dramatically their deaths in supernovae, feed the interstellar medium with these newly produced heavy elements. Such outflows and explosions, as well as the stellar radiation itself, are also thought to be the main mechanisms that trigger star formation. Thus, massive stars drive the star formation in their host galaxies and make them evolve. However, the formation of massive stars is currently the most poorly known part of their evolution.
Stars form in the gravitational collapse of cold dense clouds. Since the collapse is highly nonhomologous, hydrostatic equilibrium is first reached by a small central core while the rest of the cloud is still collapsing, nearly in freefall (e.g. McNally 1964; Bodenheimer & Sweigart 1968; Larson 1969, 1972). As the collapse proceeds, the hydrostatic core evolves as a premainsequence (preMS) star, with a mass increasing with time. This is the accretion scenario (e.g. Shu 1977; Stahler et al. 1980a,b, 1986b,a; Palla & Stahler 1990, 1991, 1992).
In the case of spherical symmetry, it has been shown that this scenario is not valid for massive stars (M ≳ 40 M_{⊙}) owing to their strong radiation field (Kahn 1974; Yorke & Kruegel 1977; Wolfire & Cassinelli 1987). But we know now that an accretion flow with a disc geometry allows this limitation to be circumvented (Nakano 1989; Yorke & Sonnhalter 2002; Krumholz et al. 2009; Peters et al. 2010; Kuiper et al. 2010, 2011). Hydrodynamic simulations for the accretion flow in massive protostellar clouds (e.g. Peters et al. 2010; Kuiper et al. 2010), as well as observations of highmass protostellar objects (e.g. Fazal et al. 2008), suggest that the typical value of the accretion rate during the formation of massive stars is ~10^{3}M_{⊙} yr^{1}. This is in contrast with the case of lowmass stars, for which values of ~10^{5}M_{⊙} yr^{1} are expected (see e.g. Stahler et al. 1980a,b; Stahler 1983).
The accretion scenario for massive star formation has been explored by many authors in a stellar evolution approach. Bernasconi & Maeder (1996), Norberg & Maeder (2000), Behrend & Maeder (2001), and Haemmerlé et al. (2013) have computed stellar models with accretion using the Geneva Stellar Evolution code. The accretion rates used in these works were increasing functions of time, i.e. of the actual stellar mass. In the lowmass range (M ≲ 2 M_{⊙}) values around 10^{5}M_{⊙} yr^{1} were considered, while in the highmass range (M ≳ 10 M_{⊙}) the values were ~10^{3}M_{⊙} yr^{1}.
Hosokawa & Omukai (2009) and Hosokawa et al. (2010) have also computed stellar models with accretion for the formation of massive stars, using their own code. The accretion rates they used were constant in time (i.e. independent of the actual stellar mass), and so already had high values (~10^{3}M_{⊙} yr^{1}) in the lowmass range. They found that when a star accretes at such high rates in the intermediatemass range (2 M_{⊙} ≲ M ≲ 10 M_{⊙}), it goes through a rapid swelling phase where the stellar radius can increase by more than one order of magnitude, leading to values as high as several 100 R_{⊙} depending on the physical conditions. They also found that above a critical accretion rate of ≃ 2−3 × 10^{3}M_{⊙} yr^{1}, a second swelling occurs before the star can contract to the zeroagemainsequence (ZAMS), leading to stellar radii near 1000 R_{⊙}. In this case, at a mass around ~50 M_{⊙}, steady accretion is no longer possible owing to the high radiation pressure that approaches the Eddington limit.
Since the works listed above were published, the treatment of accretion in the Geneva Stellar Evolution code has been improved (Haemmerlé 2014). In the original version of the code, an inconsistency in the treatment of the time derivatives was responsible for an artificial loss of entropy during the accretion phase. This inconsistency has been removed in the new version of the code. In the present paper, we show how these improvements modify preMS evolution with accretion. The paper is organized as follows. In Sect. 2, we describe the improvements brought to the code. In Sect. 3, we present models computed with the new version of the code using constant accretion rates, and we compare the results with those obtained with the original version of the code and with those of other authors. In Sect. 4, we do the same with the timedependent accretion rate used by Behrend & Maeder (2001) and Haemmerlé et al. (2013). The conclusions are summarized in Sect. 5.
2. Improvements in the code
A full description of the original treatment of accretion in the Geneva Stellar Evolution code has already been given in the previous papers (Bernasconi & Maeder 1996; Haemmerlé et al. 2013). Here we focus on the improvements implemented since the last paper. A full description of the treatment of accretion in the new version of the code is available in Haemmerlé (2014). The treatment for the burning of light elements is also described in Haemmerlé et al. (2013), and a general description of the other physical ingredients in the code is available in Eggenberger et al. (2008).
The new improvements are related to the time variation of the thermodynamic quantities. Among the four equations of structure, the equation of energy conservation is the only one that involves time, so this is the equation that makes the star evolve. The equation is obtained from the energetic balance of a given mass element dm. A fraction of the energy generated by nuclear reactions is transformed into heat, the rest is radiated, (1)where ϵ_{n} is the energy generation rate by nuclear reactions, q the heat by unit of mass, t the time, F the flux, and ρ the mass density. Since this equality is satisfied for any mass element, we have (2)where s is the specific entropy, T the temperature, L_{r} the luminosity in the shell of radius r, and M_{r} the total mass contained in the interior of this shell. We emphasize that (3)since we started from the energetic balance of a given mass element (Lagrangian coordinates).
An alternative treatment is to use the relative mass coordinate (4)which is not a Lagrangian coordinate in the case of accretion since M increases with time. With this coordinate, Eq. (2) becomes (5)Equation (5) contains one more term than Eq. (2), which is proportional to the accretion rate and the entropy gradient. Since the entropy increases outwards and Ṁ> 0 during accretion, this additional term is negative. This procedure has been used by e.g. Stahler et al. (1980a,b) and Hosokawa & Omukai (2009). These two methods are physically equivalent.
In our code, we keep the Lagrangian formulation using Eq. (2). However, in the original version, we treated the differential ds as a time variation at fixed μ instead of fixed M_{r}. In the constant mass case, such a treatment is fully consistent since a fixed μ also corresponds to a fixed M_{r}. However, in the case of accretion, it leads to an artificial loss of entropy. This inconsistency is physically equivalent to the neglect of the additional term in Eq. (5). If we neglect this negative term, this is equivalent to assuming that , while actually . Thus, we see that the neglect of the additional term implies a value for the time derivative of the entropy that is too large or in other words, an artificial loss of entropy. This inconsistency has been removed in the new version of the code: we still use Eq. (2), but the differential ds that appears in this equation is now treated consistently as a time variation at constant M_{r}.
For the accretion of entropy (i.e. the computation of ds/ dt in the layer newly accreted at each time step), we make the assumption that the entropy of the material that is accreted is the same as the entropy of the stellar surface before the material is accreted. This is the assumption of cold disc accretion (see e.g. Palla & Stahler 1992; Hosokawa et al. 2010; Haemmerlé et al. 2013). It corresponds to the case of an accretion flow with a thin disc geometry, allowing any entropy excess to be radiated away in the polar directions before it is advected in the stellar interior. Thus, we do not add any contribution from the gravitational energy liberated by the material when it is accreted, and the entropy profiles are built continuously. This assumption is generally considered as the lower limit for the accretion of entropy (e.g. Hosokawa et al. 2010). The corresponding boundary conditions are the usual photospheric boundary conditions.
We use the Schwarzschild criterion for convection without overshooting. The chemical composition is solar (Z = 0.014) with the abundances of Asplund et al. (2005) and Cunha et al. (2006). For deuterium, we take a mass fraction of 5 × 10^{5} as in Bernasconi & Maeder (1996), Norberg & Maeder (2000), Behrend & Maeder (2001), and Haemmerlé et al. (2013).
3. Models with constant accretion rates
3.1. Case with Ṁ = 10^{4}M_{⊙} yr^{1}
In order to illustrate the effect of the improvements described in Sect. 2, we first consider a preMS star accreting at a constant rate of (6)For the initial model, we consider (7)This initial model has a radius of 6.06 R_{⊙} and is fully convective. Starting from this initial model, we compute two models accreting at the rate of Eq. (6), one with the original version of the code, the other with the new version, including the improvements described in Sect. 2. The evolutionary tracks and the evolution of the radii and the internal structures of both models are shown in Figs. 1 and 2.
Fig. 1 Evolutionary tracks of two models having the same initial conditions (Eq. (7)) accreting at the same rate (10^{4}M_{⊙} yr^{1}), but computed with the two different versions of the code. The black dotted straight lines are isoradius for the indicated values of R, and the black shortdashed line is the ZAMS of Ekström et al. (2012) with a few masses indicated. From bottom to top, the filled squares on each track are the apparition of the radiative core, the end of the convective envelope, and the apparition of the convective core (see Fig. 2). 

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Fig. 2 Evolution of the stellar radius and the internal structure for the same models as in Fig. 1. The horizontal axis corresponds to the stellar mass, which is a time coordinate in the case of accretion. In each case, the upper line indicates the stellar radius, the other continuous and longdashed lines indicate the boundary between radiation and convection. The thin blue dotted lines are isomass of 1, 2, 3, ..., 21 M_{⊙} for the model computed with the new version of the code. 

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In the first stages of the evolution (lowmass range, M ≲ 2 M_{⊙}), both models are identical. In convective regions, the entropy gradient is flat, so that the time variation of entropy at fixed μ or at fixed M_{r} is the same.
The initial value of the central temperature is too low for a significant Dburning, and the star takes its energy from gravitational contraction. At M ≃ 1 M_{⊙}, the central temperature reaches ~10^{6} K and Dburning becomes significant. Owing to its high sensitivity in temperature (ϵ_{D} ~ T^{11.7}, see e.g. Maeder 2009), Dburning has a thermostatic effect, leading to a central temperature T_{c} that is nearly constant until the end of central burning. Since T_{c} ~ M/R, it produces an increase in the stellar radius, nearly linearly with respect to the stellar mass (see Fig. 2). The deuterium mass fraction decreases in the whole star, and goes to 0 at M ≃ 2−3 M_{⊙}. Dburning becomes too weak for the star and gravitational contraction starts again.
It is at this point that both models start to diverge with a smaller radius for the original model than for the new model. But while the star is still fully convective, the differences remain relatively modest. The small differences that appear are due to the nonadiabaticity of convection in the external regions.
Since central Dburning is now negligible, T_{c} can increase again, and the opacity decreases. At M ≃ 4 M_{⊙}, a radiative core appears, earlier in the original model (M = 3.8 M_{⊙}) than in the new model (M = 4.3 M_{⊙}). The huge decrease in opacity in the deep regions leads to the socalled luminosity wave (e.g. Larson 1972; Hosokawa et al. 2010): the low opacity in the radiative core imposes a flux that is too strong for the convective envelope, in which opacity is still high. The convective envelope absorbs the flux, so that the luminosity profile has a maximum near the boundary between these two regions. As the temperature increases in the star, the radiative core grows in mass and the maximum in luminosity moves outwards. The energy absorbed by the convective envelope leads to an increase in the stellar radius, which becomes particularly strong as the convective envelope recedes.
It is at this point that the differences between the two models become really significant. While the original model reaches only 14.7 R_{⊙}, the new one reaches 21.5 R_{⊙}, which is 46% higher. Because of the strong entropy gradient that develops in the radiative regions, a ds at a given μ or at a given M_{r} are significantly different.
When the whole star becomes radiative, the radius decreases again, and the star contracts towards the ZAMS. Around 11−12 M_{⊙}, the triggering of the CN cycle produces a convective core, and the energy released by the nuclear reactions replaces the gravitational contraction as the main energy source of the star. Contraction slows down until the stellar radius reaches a minimum, which can be seen as the ZAMS at masses of 13.6 M_{⊙} and 15.6 M_{⊙} for the original and the new model, respectively. At this point, the differences between the two models become small again: for the original model the minimum radius is 4.3 R_{⊙}, while it is 4.7 R_{⊙} for the new model (only 9% higher). Then, as accretion goes on the stellar radius increases nearly linearly with the stellar mass, following the ZAMS (see Fig. 1). We stopped the computation when the central mass fraction of hydrogen decreased by 0.3% with respect to its initial value.
To summarize, the differences between the two models are as follows:

the new model evolves more slowly than the original;

the new model reaches higher radii than the original, mainly during the swelling phase.
The other differences remain modest.
Why does the neglect of the additional term have the strongest effect during the swelling? Because the swelling occurs in the intermediatemass range, when the star is fully radiative, having thus a strong entropy gradient. In this case, the artificial loss of entropy in the original model becomes significant. Using the homology relations P ~ M^{2}/R^{4} and T ~ M/R the specific entropy reads (8)(Hosokawa & Omukai 2009). It follows that, for a given mass, the smaller the entropy is, the smaller the radius. As a consequence, the artificial loss of entropy leads to artificially small radii, softening the swelling. Furthermore, since the gravitational contraction – which governs preMS evolution – corresponds to a decrease of the entropy, the artificial loss of entropy accelerates preMS evolution.
3.2. Case with Ṁ = 10^{3}M_{⊙} yr^{1}
The typical value of the accretion rate for massive star formation is higher than the 10^{4}M_{⊙} yr^{1} considered in Sect. 3.1. In the present section, we make the same comparison as in Sect. 3.1, but using the rate (9)For the initial model, we first consider (10)labelled CV1, which has a radius of R = 12.1 R_{⊙} and is again fully convective^{1}. The evolutionary tracks and the evolution of the radii and the internal structures of models starting from CV1 accreting at the rate of Eq. (9) computed with both versions of the code are shown in Figs. 3 and 4.
Fig. 3 Same as Fig. 1, but for an accretion rate of 10^{3}M_{⊙} yr^{1} and the initial model CV1. 

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Fig. 4 Same as Fig. 2, but for an accretion rate of 10^{3}M_{⊙} yr^{1} and the initial model CV1. (The internal structure is shown only for the new model.) 

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The result is qualitatively similar to the case with 10^{4}M_{⊙} yr^{1}. The original and the new model are identical until the end of central Dburning. Then the new model has larger radii and evolves more slowly than the original model. Again, the main difference is the magnitude of the swelling: while the maximum value of the stellar radius is 22.0 R_{⊙} for the original model, it is 36.2 R_{⊙} for the new model, which makes a relative difference of 65%. It illustrates the fact that the artificial loss of entropy is more important when the accretion rate is higher. For a given timestep dt and a given value of μ, the difference in M_{r} between the model of age t and the model of age t + dt increases with the accretion rate since (11)In order to test the dependence of our results on the initial model, we compute two additional models with the new version of the code at the same rate of 10^{3}M_{⊙} yr^{1}, but using two different initial models, labelled CV2 and RC.
The CV2 model is given by (12)This model has a radius of 20.4 R_{⊙} and is fully convective, as is CV1. Both models are fully convective; CV2 differs from CV1 only in terms of global properties, and not in terms of internal structure. We note, however, that the CV2 model has a larger radius than the model of the same mass obtained by accretion starting from CV1. With CV1, the stellar radius reached when the mass is equal to 2 M_{⊙} is only 9.4 R_{⊙}, so about a factor of two smaller than the radius of CV2. In agreement with Eq. (8), the larger radius of the CV2 model corresponds to a higher total amount of entropy.
Finally, the RC model is given by (13)It has a radius of 2.53 R_{⊙} and is thus much more compact than CV1 and CV2. Moreover, in contrast to all the previous cases, the initial RC model is not fully convective, but is made of a radiative core (M_{r}< 1.53 M_{⊙}) and a convective envelope. We note that for both CV2 and RC, we used a mass of 2 M_{⊙}, which is higher than for CV1. Because of numerical difficulties, it is easier to compute accreting models with the structure of RC at M = 2 M_{⊙} than at M = 1 M_{⊙}. In order to allow a direct comparison, we used the same mass for CV2.
In terms of thermodynamic quantities, the main difference between the initial models CV2 and RC is the entropy profile. Both entropy profiles are shown in Fig. 5. First, we see that the global entropy is lower in the RC case than in the CV2 case, which was expected from Eq. (8). Moreover, in contrast to the entropy profile of CV2, which is flat (corresponding to a fully convective structure with adiabatic convection), the entropy profile of RC increases outwards (except in the convective envelope where it is flat). In the radiative core, the entropy profile increases nearly linearly with the Lagrangian coordinate with a gradient given by (14)In other words, the RC model differs from the CV2 model not only in terms of global properties, but also (and more importantly) in terms of internal structure.
Fig. 5 Entropy profiles of the initial models CV2 and RC. The square along the RC profile indicates the boundary between the radiative core and the convective envelope. 

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Another difference between these initial models is the deuterium abundance. The low central entropy of the RC model also corresponds to a high central temperature, which exceeds the temperature for Dburning: T_{c} = 6.1 × 10^{6} K, instead of 7.7 × 10^{5} K in the case CV2. Indeed, Eq. (8) and T_{c} ~ M/R give , so that for a given mass, the lower is the entropy, the higher is the central temperature. With such a high temperature, we expect all the deuterium contained originally in the material of this “initial” model to be destroyed in the previous stages of the evolution, and we take X_{2} = 0 in the initial model RC. However, we keep the usual mass fraction (X_{2} = 5 × 10^{5}, Sect. 2) in the accreted material, as in the CV cases.
Fig. 6 Evolutionary tracks of the models computed from the initial models CV1, CV2 and RC at Ṁ = 10^{3}M_{⊙} yr^{1}, using the new version of the code. For comparison, a series of points from the model MD3D of Hosokawa et al. (2010) is indicated with connected black squares. The black dashed curve is the ZAMS of Ekström et al. (2012) with a few masses indicated, and the black dotted straight lines are isoradius of 1, 10, 100 and 1000 R_{⊙}. The blue squares along the tracks correspond to the same masses as those indicated along the ZAMS. (Notice that, on the vertical axis, L is the intrinsic stellar luminosity, without any contribution from the accretion luminosity.) 

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The evolutionary tracks obtained from these two new initial models are shown in Fig. 6, together with the track obtained from CV1 (which is the same as the blue track in Fig. 3). The evolution obtained from CV2 is similar to the CV1 case. The star is initially fully convective, until a radiative core appears, here at a mass of 8.8 M_{⊙}. Then the convective envelope recedes and disappears, while the luminosity wave described in Sect. 3.1 produces the swelling of the star. The radius reaches its maximum value of 48.6 R_{⊙} at a mass of 12.6 M_{⊙}, before the star contracts towards the ZAMS (reached at M ~ 30 M_{⊙}). This example shows that a change in the total amount of entropy in the initial model, without any difference in the internal distribution of the entropy, does not result in a significant difference in the stellar evolution during the accretion phase. For instance, while the initial radius of the CV2 model is more than a factor of 2 larger than that of CV1, the maximum value reached by the radius during the swelling is increased by only 34%.
The evolution corresponding to the RC model shows significant differences compared to the CV cases (i.e. CV1 and CV2). In the case of RC with its early radiative core, the evolution starts with the luminosity wave and the swelling. Owing to the low stellar luminosity, the external layers radiate away the entropy that they absorb from the central regions with less efficiency than in the CV cases: the KelvinHelmholtz time is longer. As a consequence, the accretion produces a stronger swelling, leading to radii higher than 100 R_{⊙} already in the intermediatemass range. In particular, the radius reaches its maximum value of 204 R_{⊙} at a mass of 7.5 M_{⊙}. At this stage, the luminosity is sufficiently high to radiate the energy that was stored in the envelope and that was responsible for the swelling of the star. The accreting star decreases in radius and the birthline joins the ZAMS (reached again at M ~ 30 M_{⊙}).
We see that in this case the maximum value reached by the radius during the accretion phase is nearly one order of magnitude higher than in the CV cases. It shows that a change in the internal distribution of entropy in the initial model results in significant differences in the stellar evolution during the accretion phase.
The rate considered in the present section (Eq. (9)) is the same as the one used by Hosokawa et al. (2010). The model of these authors reached radii of ~100 R_{⊙} during the swelling, while our CV models reach only ~40 R_{⊙}. However, we saw with the previous examples that the maximum value reached by the radius during the swelling depends on the initial model with a high sensitivity. When we use the initial model RC, we obtain values for the maximum radius that are higher than those of Hosokawa et al. (2010) by a factor of 2.
In the fiducial model of Hosokawa et al. (2010), the initial entropy profile is not flat. It increases outwards, linearly with the Lagrangian coordinate: (15)This initial model is thus fully radiative, instead of fully convective. Their initial mass (0.1 M_{⊙}) is also lower by one order of magnitude than in our case, so that a direct comparison is not possible. However, if we use the same assumption of Eq. (15) for a 2 M_{⊙} star, corresponding to the mass of our initial models CV2 and RC, we obtain an entropy gradient of (16)Thus we can see the fiducial model of Hosokawa et al. (2010) as an intermediate case between our models CV2, with a flat entropy profile, and RC, with a gradient given essentially by Eq. (14), (17)however, the second inequality holds only in the radiative core of the model RC, i.e. for M_{r} ≲ 1.5 M_{⊙}. In Fig. 6, we added a series of points from the fiducial model of Hosokawa et al. (2010; their model labelled MD3D). We see that the HertzsprungRussell (HR) track of this model also corresponds to an intermediate case between our tracks CV2 and RC, which is consistent with Eq. (17).
Moreover, Hosokawa et al. (2010) tested the dependence of their results on the choice of the initial model by considering different initial entropy profiles. In particular, they used an initial model with a flat entropy profile (labelled MD3cv), which is thus fully convective. In this case, during the swelling they obtained a maximum radius of only 30 R_{⊙} (see their Fig. 9), which is even lower than the value we obtain in our models with a flat initial entropy profile (CV cases). We note that the total amount of entropy in their initial model is also lower than in our cases. Taking into account these differences in the initial conditions, it appears that for given physical conditions our models are in good agreement with those of Hosokawa et al. (2010).
It follows from the above examples that the maximum radius reached by a given star and the possibility for it to evolve towards the red part of the HR diagram depends sensitively on the initial conditions, namely on the initial entropy profile, which reflects the physics of the prestellar collapse, and is currently not known.
4. Models with the ChurchwellHenning accretion rate
4.1. Accretion rate
As mentioned in Sect. 1, preMS models with accretion for the formation of massive stars have been computed using an accretion rate increasing with time, instead of the constant accretion rates described in Sect. 3. In particular, Behrend & Maeder (2001) and Haemmerlé et al. (2013) used an accretion rate based on the ChurchwellHenning relation. This relation, established by Churchwell (1999) and confirmed by Henning et al. (2000), is an empirical correlation between the massloss rates through outflows Ṁ_{out} in ultracompact HII regions and the bolometric luminosity L of the corresponding central sources. A polynomial fit gives (Behrend & Maeder 2001) (18)The correlation covers 6 orders of magnitude in luminosity, from ~L_{⊙} to ~10^{6}L_{⊙}, corresponding to ZAMS luminosities on the whole intermediate and highmass range.
Such mass outflows are expected to be produced at the inner boundary of accretion discs by mechanisms such as magnetic field or radiative heating in a way that is still poorly understood (Maeder 2009). From the material coming from the disc, a fraction f is accreted by the central star, the rest (1−f) is rejected in the polar directions, producing the outflows. If we know the value of f, the ChurchwellHenning relation provides an accretion rate, given as a function of the bolometric luminosity of the accreting object. Indeed, since Ṁ = fṀ_{disc} and Ṁ_{out} = (1 − f) Ṁ_{disc}, Eq. (18) gives (19)This procedure was used first by Behrend & Maeder (2001). Since the mechanisms producing the outflows are not yet understood, the value of f is not known. With the assumption that f is constant (independent of the mass of the accreting object), Behrend & Maeder (2001) computed birthlines^{2} with different values of f. By comparing the birthlines with the location of observed Herbig Ae/Be stars on the HR diagram, they found that the best fit was given by the value f = 1/3 (i.e. (1 − f) /f = 2). With this value of f, they obtained a birthline which is in agreement with observations on the whole stellarmass range. With a constant f, Eq. (19) gives an accretion rate that increases with L and thus with M. We note that, since the relation between M and L is given by the stellar models, the accretion history Ṁ(M) also depends on the stellar models. With f = 1/3, the values of the accretion rate obtained by Behrend & Maeder (2001) were ~10^{5}M_{⊙} yr^{1} in the lowmass range, ~10^{4}M_{⊙} yr^{1} in the intermediatemass range, and ~10^{3}M_{⊙} yr^{1} in the highmass range. With this accretion rate, they were able to produce stars of 85 M_{⊙} still close to the ZAMS.
However, these results were obtained with the version of the code that overestimated the losses of entropy (see Sect. 2). It is therefore interesting to reconsider this question with the new version. This is the aim of the present section. We compute birthlines with the new version of the code, using the accretion rate given by Eq. (19) and various values of f. For L, we consider the intrinsic stellar luminosity without any contribution from the accretion luminosity, as in Behrend & Maeder (2001).
4.2. Model with f = 1/3
We first consider the ChurchwellHenning accretion rate (Eq. (19)) with the same value f = 1/3 as used by Behrend & Maeder (2001). For the initial model, we take (20)and a solar composition, as in Sect. 3. This initial model has a radius of 7.2 R_{⊙} and is again fully convective. As in Sect. 3.1 and 3.2, we compute birthlines starting with the same initial model (Eq. (20)), accreting at the same rate^{3} (Eq. (19); f = 1/3), but with the two different versions of the code (Sect. 2). The evolutionary tracks and the accretion histories of this models are shown in Figs. 7 and 8 (longdashed red line and solid blue line). The birthline obtained in this way with the original version of the code (red dashed), is the same as that of Behrend & Maeder (2001). We see that this birthline closely corresponds to the empirical upper envelope of the preMS intermediatemass stars observed by Alecian et al. (2013) and the preMS lowmass stars observed by Cohen & Kuhi (1979), indicated by the green dots.
Fig. 7 Birthlines computed with the ChurchwellHenning accretion rate (Eq. (19)) and the same initial model (Eq. (20)), but with different versions of the code (labelled “original” and “new”) and different values of the fraction f that appears in Eq. (19), f = 1/3 and 1/11. The green dots indicate observations of preMS stars (Herbig Ae/Be stars by Alecian et al. 2013, filled squares with error bars; T Tauri stars by Cohen & Kuhi 1979, empty squares). The black shortdashed line is the ZAMS of Ekström et al. (2012), with a few masses indicated, and the black dotted straight lines are isoradius of indicated values. The final mass of each model is indicated at the end of the corresponding track. 

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Fig. 8 Accretion histories of the same models as in Fig. 7. The black dotted horizontal line indicates the critical rate of Hosokawa & Omukai (2009) and Hosokawa et al. (2010). 

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As in the case with a constant accretion rate (Sect. 3), both models remain essentially identical in the lowmass range, while the star is fully convective. But again, when the star enters the intermediatemass range, becomes radiative, and starts its swelling, the two models diverge; the new model has a higher radius and luminosity than the original model (Fig. 7). However, in this case, the dependence of the accretion rate on the stellar luminosity produces a positive feedback: the rapid increase in luminosity in the new model leads to a rapid increase in its accretion rate (see Fig. 8). In this model, the accretion rate reaches 10^{3}M_{⊙} yr^{1} already in the intermediatemass range (at M = 7.3 M_{⊙}), while in the original model it is reached only in the highmass range (at M = 10.6 M_{⊙}). As a consequence, the swelling is enhanced in the new model compared to the original one, leading to radii as high as 100 R_{⊙} (Fig. 7) at a mass of 10.2 M_{⊙}. At such a mass, which corresponds roughly to the beginning of the highmass range, the swelling slows down, but before the star can contract again, the positive feedback between the stellar luminosity and the accretion rate leads to the critical value of Hosokawa (Fig. 8, at a mass of 11.5 M_{⊙}). At this point, the stellar luminosity exceeds 50% of the Eddington luminosity (Fig. 9), and the second swelling occurs (see Sect. 1 or Hosokawa & Omukai 2009; Hosokawa et al. 2010 for more details). As we can see in Fig. 7, during this second swelling, the star goes back to the red until it reaches the boundary of the forbidden region of Hayashi, as in the case of a red giant. Then, as the swelling goes on, the star moves upwards on the HR diagram, along a vertical line, similar to the red giant branch (i.e. it moves up the Hayashi line). We stopped the computation when the stellar mass was 32 M_{⊙} with a radius approaching 1000 R_{⊙} and a luminosity close to 75% of the Eddington luminosity (Fig. 9). If we stop accretion at this point, the star simply contracts towards the ZAMS in a KelvinHelmholtz time (45 000 yr for such a mass), as in the canonical scenario of preMS at constant mass. It shows that the canonical scenario corresponds to the limit of the accretion scenario for a high accretion rate.
Fig. 9 Eddington factor as a function of the stellar mass, for the model “new (1/3)” of Fig. 7, i.e. of the model computed with the accretion rate of Eq. (19) with f = 1/3 using the new version of the code (Sect. 2), starting from the initial model given by Eq. (20). 

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The internal structure of this model is shown in Fig. 10 ^{4}. Until the end of the first swelling, the internal structure is similar to the cases of constant accretion rates described in Sect. 3. But when the second swelling occurs, the convective envelope comes back, as in the case of a red giant or in the initial model of a ~30 M_{⊙} star in the canonical scenario.
Fig. 10 Evolution of the radius (upper blue curve) and the internal structure of the model “new (1/3)” of Fig. 7, i.e. of the model computed with the accretion rate of Eq. (19) with f = 1/3 using the new version of the code (Sect. 2), starting from the initial model given by Eq. (20). The dark blue areas are convective zones, the light blue areas are radiative regions, and the grey area is the envelope (see footnote 4). 

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The new birthline computed with f = 1/3, if representative of the birthline for the majority of stars, would predict a much more extended distribution than the one observed for the preMS stars. In the next section we discuss a birthline obtained with a lower value of f, which produces an upper envelope more compatible with the observations.
4.3. Model with f = 1/11
We computed a third birthline with the new version of the code, the same initial model, and the ChurchwellHenning accretion rate, but with a value of f = 1/11 (i.e. (1−f) /f = 10). The evolutionary track of this birthline is shown in Fig. 7 (blue dashed line), and the corresponding accretion history is in Fig. 8.
As we can see, thanks to the low value of f, the accretion rate of this new birthline is lower than in the other two cases by nearly one order of magnitude. This new birthline diverges rapidly from the other two already in the lowmass range, but when the star enters the intermediatemass range, the swelling brings the new birthline back to the original one (see Fig. 7). Then, the two birthlines remain close to each other until the end of the computation at masses above 85 M_{⊙}. In particular, with this new birthline, we can also reach such high masses still close to the ZAMS.
4.4. Discussion
As we can see by looking at the two new birthlines in Sects. 4.2 and 4.3 (blue tracks in Fig. 7), if we want to reproduce the original birthline, which is in agreement with the observations in the whole stellarmass range, it is necessary to consider a value of f that decreases with the stellar mass. With the original value of f = 1/3, we still obtain a satisfactory result in the lowmass range, but in the intermediate and highmass range, such a high value of f leads to luminosities that are too high compared to those of Herbig Ae/Be stars. If such an accretion history was realistic, we would expect to observe massive preMS stars contracting towards the ZAMS at such high luminosities, which is not the case^{5}. Inversely, if we consider the value f = 1/11, in the lowmass range we obtain an accretion rate that is much lower than the typical rate of 10^{5}M_{⊙} yr^{1} expected for such masses (e.g. Stahler et al. 1980a,b). With this accretion rate, we obtain luminosities that are too low compared to the observed T Tauri stars, as can be seen in Fig. 7. But in the intermediate and highmass range, this new value of f leads to a birthline that is in good agreement with the observations, fitting the upper envelope of Herbig Ae/Be stars and allowing us to produce stars of masses above 80 M_{⊙} still close to the ZAMS. Interestingly, this value is also in good agreement with the observational estimations of Churchwell (1999) for Btype stars, which corresponds to the intermediatemass range. As a consequence, our results suggest that the efficiency of accretion from a disc decreases with the stellar mass: the higher the stellar mass, the larger the fraction of the material coming from the disc that is rejected through the bipolar outflows.
The assumption of a unique accretion history is certainly an approximation, and we expect the real accretion histories of stars to depend on the final mass they are destined to reach. In this sense, each of our birthlines has to be seen as an upper envelope of the evolutionary tracks of each individual accreting star, corresponding to the main accretion phase (Maeder 2009). Even in this case, however, the value of f during the main accretion phase has to be lower for stars of intermediate and highmass than for lowmass stars in order to reproduce the upper envelope of observed preMS stars.
We note that when we compute the accretion rate from the luminosity with Eq. (19), we do not include the accretion luminosity L_{accr}. This additional contribution would increase the accretion rate, leading to a higher birthline on the HR diagram. In order to keep the birthline in agreement with the observations, we would need to reduce the value of f. Thus the value f = 1/11 has to be seen as an upper limit for the intermediate and highmass range. Moreover, L_{accr} represents a larger fraction of the total luminosity L_{tot} = L + L_{accr} in the lowmass range than in the intermediate and highmass ranges. In the lowmass range, L_{accr} ≃ 90%L_{tot} typically, while in the intermediate and highmass ranges L_{accr} ≲ 50%L_{tot}. As a consequence, taking into account the accretion luminosity would reduce the decrease of f as a function of M.
Another issue is the dependence of these results on the initial model. In Sect. 3.2 we showed that for a constant accretion rate of 10^{3}M_{⊙} yr^{1} the choice of the initial model has a critical impact on the evolution. However, in the case of the ChurchwellHenning accretion law for f = 1/11, the accretion rate remains much lower than 10^{3}M_{⊙} yr^{1} during the whole preMS phase (Fig. 8), and it turns out that the shape of this birthline is nearly unchanged when we use the initial model RC of Sect. 3.2 instead of the initial model given by Eq. (20). In other words, the choice of the initial model has no significant influence on the results of the present section.
4.5. Main sequence evolution and isochrones
Usually, MS models are computed starting from the ZAMS with initial conditions obtained from canonical preMS at constant mass. However, Bernasconi & Maeder (1996) obtained in their models that the internal structure of a MS star is not the same in the accretion scenario as in the canonical scenario. They found that for a given mass and central mass fraction of hydrogen, a MS star formed by accretion has a smaller convective core than a MS star coming from a canonical preMS, leading thus to a shorter MS lifetime^{6}. In order to see how the preMS scenario described in the previous sections modifies the structure and the evolution of MS stars compared to canonical MS models, we computed contractions^{7} at constant masses from the birthline described in Sect. 4.3 (ChurchwellHenning accretion rate with f = 1/11), and the subsequent MS evolution. We obtain that the internal structure on the ZAMS and the evolution on the MS are identical to those of canonical MS stars. The HR tracks of the contractions for M = 2, 5, 8, 20, 50, and 80 M_{⊙} and the subsequent MS tracks are shown in Fig. 11. We also added several isochrones of 2, 4, 8, 30, and 70 Myr. A more complete grid with isochrones will be available in a forthcoming paper.
Fig. 11 HR tracks of the birthline “new(1/11)” (Sect. 4.3), and the subsequent contractions and MS tracks for M = 2, 5, 8, 20, 50, and 80 M_{⊙} (solid black lines). Several isochrones are indicated with coloured solid lines. The isoradius of 1, 10, 100, and 1000 R_{⊙} are indicated with black dotted lines and the ZAMS of Ekström et al. (2012) is shown with a black dashed line. 

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5. Conclusions and perspectives
We showed that the previous versions of the code contained an inconsistency that lead to an artificial loss of entropy in the case of an accreting star dominated by the radiative transport. As a result of this artificial loss of entropy, we underestimated the magnitude of the swelling that occurs when an intermediatemass star accretes at a high rate (~10^{3}M_{⊙} yr^{1}). With the new version of the code, we obtain results that are in good agreement with those of other authors, provided that we compare models with the same physical conditions. In this context, we showed that the existence of a significant swelling that brings the star back towards the red part of the HR diagram, as described by other authors, depends critically on the initial conditions.
We looked at the effects of these improvements on the accretion history given by the ChurchwellHenning relation and considered various values of f, the fraction of the material coming from the disc which is effectively accreted by the star. Using the original value of f = 1/3, we found that the first swelling of the star leads to a radius of 100 R_{⊙}; before the star can contract again, a second swelling occurs, which leads the star to the red part of the HR diagram until the boundary of the forbidden region of Hayashi with radii near 1000 R_{⊙} and a new convective envelope. We showed that instead, with a fraction f = 1/11, the birthline remains close to the original birthline of Behrend & Maeder (2001) in the intermediate and highmass range. This new value of f is in good agreement with the estimations of Churchwell (1999) and provides a good fit to the observed upper envelope of preMS stars in the HR diagram for luminosities higher than about 2.5. At lower luminosities, a value of f equal to 1/3 gives a better fit. Therefore, in order to reproduce these observations with the new version of the code, we conclude that it is necessary to consider a fraction f that decreases with the stellar mass, which suggests that, when the stellar mass increases, the mass that is accreted represents a still smaller fraction of the mass that is in the outflows.
In the next paper in this series we will add the effects of rotation. Since preMS models simultaneously including the effects of accretion and rotation and computed with the original version of the code have been published (Haemmerlé et al. 2013), we will describe how the new improvements modify the results in the case of rotation.
The birthline is the set of the points on the HR diagram where the stars of various masses become optically visible. With the assumption that the accretion history Ṁ(M) is unique, which can be partially justified with Newton’s theorem, the birthline is also the evolutionary track of each individual accreting star. If this assumption is correct, such a track has to reproduce the upper envelope of observed preMS objects. Without this assumption, the birthline can be seen as the upper envelope of each individual track. See Maeder (2009) for more details.
I.e. the same Ṁ(L), but not the same Ṁ(M), as mentioned in Sect. 4.1.
This result was obtained with the original version of the Geneva Stellar Evolution code, see Sect. 2.
Acknowledgments
Part of this work was supported by the Swiss National Science Foundation.
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All Figures
Fig. 1 Evolutionary tracks of two models having the same initial conditions (Eq. (7)) accreting at the same rate (10^{4}M_{⊙} yr^{1}), but computed with the two different versions of the code. The black dotted straight lines are isoradius for the indicated values of R, and the black shortdashed line is the ZAMS of Ekström et al. (2012) with a few masses indicated. From bottom to top, the filled squares on each track are the apparition of the radiative core, the end of the convective envelope, and the apparition of the convective core (see Fig. 2). 

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In the text 
Fig. 2 Evolution of the stellar radius and the internal structure for the same models as in Fig. 1. The horizontal axis corresponds to the stellar mass, which is a time coordinate in the case of accretion. In each case, the upper line indicates the stellar radius, the other continuous and longdashed lines indicate the boundary between radiation and convection. The thin blue dotted lines are isomass of 1, 2, 3, ..., 21 M_{⊙} for the model computed with the new version of the code. 

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In the text 
Fig. 3 Same as Fig. 1, but for an accretion rate of 10^{3}M_{⊙} yr^{1} and the initial model CV1. 

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In the text 
Fig. 4 Same as Fig. 2, but for an accretion rate of 10^{3}M_{⊙} yr^{1} and the initial model CV1. (The internal structure is shown only for the new model.) 

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In the text 
Fig. 5 Entropy profiles of the initial models CV2 and RC. The square along the RC profile indicates the boundary between the radiative core and the convective envelope. 

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In the text 
Fig. 6 Evolutionary tracks of the models computed from the initial models CV1, CV2 and RC at Ṁ = 10^{3}M_{⊙} yr^{1}, using the new version of the code. For comparison, a series of points from the model MD3D of Hosokawa et al. (2010) is indicated with connected black squares. The black dashed curve is the ZAMS of Ekström et al. (2012) with a few masses indicated, and the black dotted straight lines are isoradius of 1, 10, 100 and 1000 R_{⊙}. The blue squares along the tracks correspond to the same masses as those indicated along the ZAMS. (Notice that, on the vertical axis, L is the intrinsic stellar luminosity, without any contribution from the accretion luminosity.) 

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In the text 
Fig. 7 Birthlines computed with the ChurchwellHenning accretion rate (Eq. (19)) and the same initial model (Eq. (20)), but with different versions of the code (labelled “original” and “new”) and different values of the fraction f that appears in Eq. (19), f = 1/3 and 1/11. The green dots indicate observations of preMS stars (Herbig Ae/Be stars by Alecian et al. 2013, filled squares with error bars; T Tauri stars by Cohen & Kuhi 1979, empty squares). The black shortdashed line is the ZAMS of Ekström et al. (2012), with a few masses indicated, and the black dotted straight lines are isoradius of indicated values. The final mass of each model is indicated at the end of the corresponding track. 

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In the text 
Fig. 8 Accretion histories of the same models as in Fig. 7. The black dotted horizontal line indicates the critical rate of Hosokawa & Omukai (2009) and Hosokawa et al. (2010). 

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In the text 
Fig. 9 Eddington factor as a function of the stellar mass, for the model “new (1/3)” of Fig. 7, i.e. of the model computed with the accretion rate of Eq. (19) with f = 1/3 using the new version of the code (Sect. 2), starting from the initial model given by Eq. (20). 

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In the text 
Fig. 10 Evolution of the radius (upper blue curve) and the internal structure of the model “new (1/3)” of Fig. 7, i.e. of the model computed with the accretion rate of Eq. (19) with f = 1/3 using the new version of the code (Sect. 2), starting from the initial model given by Eq. (20). The dark blue areas are convective zones, the light blue areas are radiative regions, and the grey area is the envelope (see footnote 4). 

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In the text 
Fig. 11 HR tracks of the birthline “new(1/11)” (Sect. 4.3), and the subsequent contractions and MS tracks for M = 2, 5, 8, 20, 50, and 80 M_{⊙} (solid black lines). Several isochrones are indicated with coloured solid lines. The isoradius of 1, 10, 100, and 1000 R_{⊙} are indicated with black dotted lines and the ZAMS of Ekström et al. (2012) is shown with a black dashed line. 

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In the text 
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