3 Collapse of Bonnor-Ebert spheres

Figure 2: The relation between age and luminosity for the collapse of Bonnor-Ebert spheres ( full lines) and classical, fully hydrostatic models of pre-main sequence evolution. These relations are relevant for the determination of PMS-ages. For the comparison we use the quasi-hydrostatic results by D'Antona & Mazzitelli (1994) ("Alexander + MLT'' case) for 0.1, 0.5, 1 and $2~ M_\odot$ ( dotted lines from left to right). Classical PMS-models do not accrete and evolve at constant mass. The new PMS-ages calculated from the collapse of cloud fragments are larger than the classical values by up to a Myr, depending on luminosity. This is due to an age offset because the formation process is modeled and due to a different PMS-contraction behaviour caused by the difference between the assumed initial thermal structure in classical tracks and the thermal structure calculated from the collapse and accretion process. Note that collapse luminosities show an initial rise, close to zero age, as the protostars are "switched'' on after core formation. Ages inferred using collapse PMS-tracks will be larger than those inferred from classical tracks by up to a Myr at ages below 2 Myr. The dashed and dash-dotted line, close to the $1~{M_\odot}$-relation, are obtained by changing the mixing length parameter $\alpha _{\rm ML}$ to 1.2 and 1.4, respectively, for the same fragment mass, see text.

In this section we discuss the results for the collapse of critical Bonnor-Ebert spheres with a temperature of $10 ~ \rm K$ and masses of 0.05 to $10~{M_\odot}$ within a constant volume, as obtained by the methods described above. For a description of the method of solution we refer to Appendix B.

3.1 Structure of results and definitions

The results described below are quantities derived from the flows which are obtained as the solution of equations given in Table .1, for the Bonnor-Ebert collapses. Since there is no a priori hydrostatic part and no a priori photospheric model in our model (they result as special properties of the solutions of our flow equations) we describe how the stellar parameters age, effective temperature, radii and masses are obtained from the radiation hydrodynamical flows. We restrict our discussion mostly to these basic parameters to give an overview over the collapse solutions. We start our discussion with a proposal for the definition of zero stellar age, because as it turns out that most our definitions of stellar parameters will only be valid after this instant of time.

3.2 Stellar zero age

The modeling of the star formation process together with the PMS-contraction opens up the possibility to define a stellar zero age. We require the instant of age zero to be (1), well defined, (2), a unique clock for an individual star; it should be reset by a destruction of the star and stellar mergers, and (3), the gravitationally weakly bound, tenuous states of the cloud should not contribute to stellar age - a long-lasting quasi-static cloud phase would otherwise induce almost arbitrary age additions without any consequences for the subsequent evolution. To compare the luminosities of objects of various masses at a given age a general definition of zero age is necessary. As the thermally controlled Kelvin-Helmholtz contraction is the dominating evolutionary process in young gas spheres, whether they are giant planets, brown dwarfs or stars, we propose to use as age zero the instant of time, when the interior of the gas spheres is thermally enclosed for the first time. The enclosure means that photons cannot escape from the entire object directly but are radiated from a photosphere. Energy transfer to the photospheric bottleneck then delays the cooling and determines how the thermal reservoir in the interior is emptied. The cooling history can then be used to define an age since the heat reservoir was formed. As a practical definition for age zero we propose to use the instant of time when the Rosseland mean optical depth of a gaseous object equals 2/3, i.e.,

$$\mbox{Age} = 0 \!:\ \tau_{\rm Ross} (t) = \int^R_{r^\prime} \kappa_{\rm Ross} \varrho ~ {\rm d}r = \frac{2}{3} \cdot$$ (1)

We use 2/3 here because of the advantage for, e.g., the $T_{\rm eff}$ definition in the comparison to D'Antona & Mazzitelli (1994) but recommend to use $\mbox{Age} = 0$ at $\tau_{\rm Ross} (t) = 1$, in general. $\tau=2/3$ results from an averaging of optical depth over the stellar disk that is specific to the grey Eddington approximation.

For a protostar this instant of time is practically at the end of the isothermal phase of the collapse. Furthermore, with definition (1) the luminosities of the protoplanets can be compared to stars and brown dwarfs of similar age. The proposed definition of zero age has the following properties: (1), it preceeds the short phase of the "final hydrostatic core formation'' for protostars that was used in earlier work (Appenzeller & Tscharnuter 1975; Winkler & Newman 1980a) by only $\sim$ $2000 ~{\rm yr}$. It corresponds to the sharp initial luminosity rise (within 1000 yr for 0.05 to 10 $M_\odot$) that occurs when the first (molecular hydrogen) protostellar cores form. Because this happens during this rapid dynamical density enhancement due to early cloud collapse, the proposed definition is very sharp, in the sense that the properties of the cloud vary rapidly in the vicinity of zero age; $\mbox{Age} = 0$ is reached during a phase of rapid changes in the observables. As a consequence it will be observationally well defined; (2), it is "close'' to the formation time of the oldest "rocks'' in the solar system. They presumably form during the process of planetesimal formation that occurs according to present understanding within $\sim$ $10^4 ~{\rm yr}$ after pre-planetary nebula formation. The pre-planetary nebula forms roughly at the time when the final hydrostatic protostellar core settles into hydrostatic equilibrium (Tscharnuter 1987). Hence, together with the $\sim$ $2000 ~{\rm yr}$ for final core formation, the ages of the oldest meteorites and meteoritic components that are determined by absolute radioactive dating should be only $\sim$ $10^4 ~{\rm yr}$ smaller than the ages defined here. This results in a conceptual consistency of our ages and the meteoritic ages to within probably < $10^5 ~{\rm yr}$. It is a useful property because solar system ages of $4.566\pm 0.002$ Gyr (Allègre et al. 1995; for Allende CAIs) are used to calibrate stellar structure theory by assuming that the age of the Sun is equal to the age of the meteorites and a comparison of stellar-evolution models of the Sun at theoretical ages with properties of the present Sun; (3), relative ages of different objects in a star formation region are physically well defined; (4), the thermal clock which is based on the dominating dynamical process in early stellar evolution, namely gravitational contraction and cooling is (re)set at the proposed zero age; (5), the age spread in young clusters is conceptually accessible without reference to the ZAMS, and knowledge about it. Concepts involving the ZAMS rely on the assumption that it will be reached. This is not guaranteed as it does not exist for brown dwarfs and planets. Hence such concepts would be inapplicable for those objects. Our proposed zero age is a well defined concept for all masses at least from an earth mass up to the largest stellar masses; (6), questions about coevolution of very young binaries can be addressed because individual component ages are well defined; (7), with this zero age we use the earliest instant when an effective temperature can be defined, based on a radius defined by using optical depth. With age definition (1) we can study the luminosity emitted by our cloud volume as a function of age. In the beginning the luminosity comes from the still almost isothermal cloud fragment, in the end from the pre-main sequence star that has formed in the centre. We will use this age to discuss the effect of the cloud collapse on pre-main sequence ages. We can expect two main effects relative to hydrostatic models of early stellar evolution: (i) an age offset that is due to the fact that the collapse phase which is not modeled in fully hydrostatic PMS-studies adds contributions of the order of a free-fall time to the ages usually given (see Tables 1, 2), and (ii) an effect due to the different PMS-contraction behaviour that is due to the difference between the assumed initial thermal structure of classical PMS-calculations and the thermal structure as calculated from the collapse and accretion process in this work.

In our discussion, we use the relation between luminosity and age for two reasons: (1) because of the uncertainties (both theoretical and observational) in the determination of the effective temperature of objects in the vicinity of the Hayashi-tracks, and (2) because these tracks are very steep in the HRD during the early PMS evolution - the luminosity decreases at almost constant effective temperature. These relations for the collapse of Bonnor-Ebert spheres of 0.1, 0.5, 1 and $2~ M_\odot$ are depicted in Fig. 2. The relations determined from collapse (full lines) originate from zero luminosity, i.e., before the cloud fragment starts to radiate significantly. The fully hydrostatic relations (dotted lines) begin at an assumed initial age and at a luminosity that is the consequence of the assumed (usually fully convective, or n=3/2 polytrope) initial thermal structure.

Notice the age offset of about $10^5 ~{\rm yr}$, about a free-fall time, between the beginning of the classical and collapse curves in all cases. This is due to the assumed initial ages of fully hydrostatic calculations. Next we look at the possibility of attributing all the differences to an age offset. This could be done by shifting the classical curves upward to higher ages until the luminosities match the new curves. Such a procedure could be considered successful if the rates of change in luminosity of classical and collapse L(t)-relations match, too. This would be indicated by the luminosity difference between corresponding old and new relations of the same mass to be constant. For the $0.1~{M_\odot}$ case this is fairly successful and the collapse may be essentially viewed as producing an age offset of about $300~ {\rm kyr}$. For the other masses this procedure does not hold. Even for the $0.5 ~{M_\odot}$ relations an addition of $400~ {\rm kyr}$ to the classical relation might just bring the luminosities in accord, but the relations still diverge because of the different contraction behaviour.

The consequences for the inferred ages at given luminosity are already considerable for $0.5 ~{M_\odot}$: at the end of the collapse calculation, a PMS-object with solar luminosity has a new age of $1.5 ~{\rm Myr}$ while following down the $1~ L_\odot$-ordinate to the classical tracks gives $0.5~\rm Myr$. The situation is very similar for the 1 and $2~ M_\odot$ clouds, but the differences in luminosity evolution to the classical relations are even more pronounced. Changing the mixing length, a procedure that probably produces the largest changes in classical models does not significantly change this conclusion. This can be seen by comparing the dashed and dash-dotted lines, for $\alpha_{\rm ML}=1.2$ and 1.4, respectively, that are very close to the standard 1 $M_\odot$ collapse relation for $\alpha _{\rm ML} = 1.5$.

Next the adopted age definition allows us to define the "stellar'' properties, i.e., the properties of the optically thick, ultimately hydrostatic parts of the flow for any instant after zero age. This is described in the next section.

3.3 Radii

Here we focus on the observables of the collapsing cloud fragments and the resulting young stars, so we define our stellar radii to be the optical depth radii, $R_\tau$, obtained by determining the radius r' at which

$$\tau = \int^R_{r^\prime} \kappa_{\rm Ross} \varrho ~ {\rm d}r = 2/3 .$$ (2)

Because of our definition of zero age such a radius exists. This is not necessarily close to the density radius, $R_\varrho$, i.e., the location of the maximum absolute value of the density gradient, that usually defines the stellar radius (density gradients in shocks are ignored for this procedure cf. Winkler & Newman 1980a,b; Baschek et al. 1991). Note that Balluch (1991a-c) used the radius of the accretion shock as an estimate for $R_\varrho$ and the stellar radius, which is a good estimate due to the small stand-off distance of the shock.

The optical depth radius gives a measure of the radius from which the photons escape and which parts of the flow are thermally shielded from the outside. It is the length scale a (bolometric) observer would see. It is very close to the density radius once the protostellar cocoon has been sufficiently diluted by accretion to make it transparent down to the stellar photosphere. The two radii converge when $R_\tau$ "jumps'' from the dust photosphere to the stellar photosphere and intercepts $R_\varrho$.

3.4 Luminosity and effective temperature

With the luminosity at the outer boundary and the optical depth radius we determine an effective temperature by

$$L = 4 \pi R_\tau^2 \sigma T_{\rm eff}^4 .$$ (3)

The so defined $T_{\rm eff}$-values are used in our Hertzsprung-Russell diagrams. To give an estimate for the uncertainty involved when placing a protostar into the Hertzsprung-Russell diagram we additionally plot two alternative temperature definitions in Fig. 1: (1) a temperature T(R_T)based on the temperature radius, R_T, that is governed by

$$r=R_T: L(r) = 4 \pi r^2 \sigma T(r)^4 ,$$ (4)

cf. Baschek et al. (1991). This is the temperature at a distinct point in the flow where the luminosity corresponds to the value that would be obtained by a blackbody radiating at the local flow matter-temperature. (2) a photospheric temperature defined by

$$T_{\rm phot} = T(R_\tau) ,$$ (5)

i.e., the local flow temperature at the optical depth radius. It is a measure of the temperature at the location from which the photons typically escape towards an observer.

In a static, compact, radiative photosphere in Eddington approximation, the definitions result in the same temperatures (cf. Baschek et al. 1991).

3.5 Collapse of a solar mass

The resulting theoretical Hertzsprung-Russell diagram for the collapse of a $1~{M_\odot}$ Bonnor-Ebert sphere is shown in Fig. 1. In this case the mixing length parameter $\alpha _{\rm ML}$ is chosen to be 1.4 to facilitate the comparison with the fully hydrostatic pre-main sequence calculation. The thick line shows the evolutionary track in the Hertzsprung-Russell diagram by using the effective temperature based on the optical depth radius $R_\tau$ (as used in earlier collapse studies by Appenzeller & Tscharnuter 1975 and Winkler & Newman 1980a). The track starts at the instant of zero age, indicated by the diamond symbol. As above noted the adopted definition of zero age coincides with the first instant at which the effective temperature concept can be used.

The initial decrease in $T_{\rm eff}$ is due to the fact that by definition $R_\tau$ starts out very small but rapidly increases during the first core formation. This occurs because the core's density and optical depth increase rapidly during this settling into hydrostatic equilibrium. $R_\tau$ increases at approximately constant luminosity and hence $T_{\rm eff}$ drops. Temperatures given before zero age (dash-dotted part) are central temperatures of the still transparent, almost isothermal cloud. With the onset of accretion onto the core the luminosity rises sharply with a moderate increase in $T_{\rm eff}$. The "mouse-head'' is the hall mark of the formation of the second (inner) atomic hydrogen core that forms after the (second) collapse of the dissociating molecular hydrogen core. At the cusp after the first luminosity maximum the final accretion phase is entered. As the luminosity rises towards the second maximum, the first core is accreted onto the second one and the Mach numbers of the flow increase above ten. The small luminosity decrease after the second maximum accompanies the adjustment of the accretion flow and the mass accretion rate onto the final core to the substantial luminosity now radiated from the accretion shock.

When the luminosity starts rising again the quasi-steady main-accretion phase commences. Shortly after the flat top part of the track is reached, the luminosity, due to D burning, becomes significant and reaches a maximum of 60% at the global luminosity maximum. When the volume is significantly depleted by the accretion process the luminosity starts decreasing and $T_{\rm eff}$ rises as parts of the flow closer to the young star become visible. When the dust cocoon becomes transparent the optical depth radius "jumps'' into the "stellar'' photosphere the effective temperature sharply increases, crosses the classical PMS-track, the Hayashi line and finally reaches the "corner'' of the collapse track, at maximum $T_{\rm eff}$. From this point on $T_{\rm eff}$ coincides with the temperature definition based on the temperature radius (thick and dashed line overlap).

Hence the radiation hydrodynamical results for temperature based on R_T and $R_\tau$ converge to the equality valid in classical, hydrostatic photospheres. The photospheric temperature (dotted) at $\tau=2/3$ passes the shock at this instant and produces a sharp spike to the left as $\tau=2/3$ is at the peak temperature just downstream from the accretion shock, and still upstream from the radiative relaxation layer. As the accretion luminosity becomes negligibly small and after "D burnout'' the collapse track turns almost parallel to the classical track, slightly pointing towards the ZAMS location of the Sun on the classical track. The photospheric temperature remains different from the other two temperatures, and slightly cooler than the $T_{\rm eff}$ value of the fully hydrostatic classical track. Note that the photosphere of the collapse calculation is convective all along the Hayashi-part of the track and $T_{\rm phot}=T_{\rm eff}$, valid for a compact, hydrostatic, radiative photosphere in Eddington approximation is not guaranteed under these conditions.

3.6 Collapse PMS-tracks and isochrones

Figure 3: Evolutionary tracks in the Hertzsprung-Russell diagram for the collapse and early pre-main sequence evolution of 0.05, 0.1, 0.5, 1, 2 and $10~{M_\odot}$ cloud fragments ( full lines). The dashed lines indicate isochrones for the collapse tracks, labeled with the respective ages. Zero age is defined here as the moment when the respective cloud fragment becomes optically thick and the interior is thermally locked as the first photosphere forms (Rosseland mean optical depth reaches 2/3). Quasi-hydrostatic tracks for 0.1, 0.5, 1 and $2~ M_\odot$ from D'Antona & Mazzitelli (1994), "Alexander MLT''-case, $\alpha _{\rm ML}=1.4$ ( dotted lines) are plotted for comparison.

Keeping in mind the above discussion concerning temperatures we now proceed to a discussion of evolutionary tracks for cloud fragments of various masses and the respective isochrones in the Hertzsprung-Russell diagram, based on the effective temperature with the optical depth-radius definition (see Baschek et al. 1991 for a discussion, including the protostellar application). It is important to note that the isochrones are based on the chosen zero age and hence the assumption that the first photospheres of all stars have formed at the same time. Different collapse- and accretion times of fragments of different masses are included. This is in contrast to hydrostatic studies that count their ages from an arbitrary zero point related to there initial condition.

In Fig. 3 an overview over the collapse and PMS evolution of Bonnor-Ebert-sphere cloud fragments with masses of 0.05, 0.1, 0.5, 1, 2 and $10~{M_\odot}$ is given. They are compared to classical, non accreting, fully hydrostatic tracks for 0.1, 0.5, 1, and $2~ M_\odot$ after D'Antona & Mazzitelli (1994). For the comparison an electronic version of their "Alexander, MLT''-tracks for $\alpha _{\rm ML}=1.4$ was used.

The early evolution dominated by the initial cloud collapse is very similar for all masses. At an age of 1000 yr the first hydrostatic cores form, with an age spread of a few hundred years. All fragments enter the main accretion phase at an age slightly above 8 kyr. All tracks remain below effective temperatures of $2000~\rm K$ up to $30~\rm kyr$. The protostars still remain in their dust cocoons and radiate from dust photospheres. The luminosity decline has already started at this age for the lowest mass fragment. At $60~\rm kyr$ the first object has arrived on the pre-main sequence, close to the Hayashi track. It has formed from the lowest mass cloud fragment of $0.05~ M_{\odot}$ and is a young brown dwarf. The higher mass fragments follow until the $2~ M_\odot$ fragment arrives on its PMS-track at $0.5~\rm Myr$. The $10~{M_\odot}$ fragment is still accreting at $0.7~{\rm Myr}$, with a hydrostatic central stellar embryo of $4.5 ~ M_\odot$.

Figure 3 illustrates that star formation is a gravitational heating event that increases temperature by about two orders of magnitude. Due to the negative, so-called gravothermal specific heat of common, thermally supported self-gravitating systems the heating slows down when energy losses slow-down because matter becomes opaque due to the cross-sections of atoms. Ultimately nuclear energy is added which acts as a long-term thermostatic effect.

The early PMS evolution of the collapsed cloud fragments shows a relation to the classical tracks that changes with mass. The $2~ M_\odot$ fragment arrives close to the radiative, Henyey part of the corresponding classical track. The $1~{M_\odot}$ and the $0.5 ~{M_\odot}$ case arrive on the PMS with a luminosity corresponding to about half way down on the convective, Hayashi parts of the classical tracks. The $0.1~{M_\odot}$ fragment arrives with a luminosity corresponding to the top of the classical track. The $0.05~ M_{\odot}$ fragment (a proto brown dwarf) is shown to give an indication of the further mass dependence of the relation to the $0.1~{M_\odot}$ classical track. For the two lowest masses the age difference can be attributed to an offset in zero age. Furthermore, D burning has not begun for the $0.1~{M_\odot}$ case-as might be expected so close to the brown dwarf limit. It follows that, by using the appropriate initial thermal structure, this mass probably will show a quasi-classical PMS evolution starting near the top of a Hayashi track and a somewhat higher effective temperature, but, as found in previous studies, with the slow-down of contraction due to D burning.

On the other hand the results of the collapse calculations indicate substantial corrections in PMS stellar properties for masses of $0.5 ~{M_\odot}$ and higher. PMS objects of these masses already are half-way in the direction of the Henyey-track when they become visible.

3.7 Lines of constant mass-isopleths

Due to our radiation hydrodynamical approach only the masses of the cloud fragments are prescribed. The stellar mass and the corresponding mass accretion rate are calculated from the flow. Hence, quantities like stellar mass and mass accretion rate are determined during the evolution of the flow.

For our discussion it is useful to use the optically thick mass, $M_\tau$, for further considerations. It follows naturally by determining the mass inside the optical depth radius, $R_\tau$ that we have already used in the definition of $T_{\rm eff}$. This corresponds to the hydrostatic mass, defined, e.g., via the density radius (cf. Baschek et al. 1991) for most of the time but has to be distinguished during the very early accretion when calculating the accretion-luminosity from $\dot{M_*} GM_*/R_*$. We use $M_\tau$ for the remainder.

In order to show the stellar properties as accretion proceeds we display lines of constant optically thick mass, $M_\tau$ - which we may call isopleths - in the Hertzsprung-Russell diagram in Fig. 4. It is the actual stellar mass that determines the stellar properties and not the fragment mass that includes the entire circumstellar environment under consideration. The $10~{M_\odot}$ fragment, for example, starts as the other fragments with $M_\tau=0.04~{M_\odot}$ and produces the corresponding luminosities and effective temperatures despite of its $10~{M_\odot}$protostellar envelope.

Figure 4: Optically thick mass, $M_\tau$, in the Hertzsprung-Russell diagram for the collapse and early pre-main sequence evolution of 0.05, 0.1, 0.5, 1, 2 and $10~{M_\odot}$ cloud fragments. $M_\tau$ is the mass interior to the optical depth radius, $R_\tau$. Dash-dotted lines are locations of constant $M_\tau$, isopleths, with values labelled along the $10~{M_\odot}$-track. The locations where $M_\tau$ corresponds to 80, 90 and 99% of the final mass are connected by the thin lines. The thin line for 99% is used as an estimate for the end of significant mass accretion. The lines of constant mass are essentially tangential to the tracks beyond this point. Quasi-hydrostatic tracks for 0.1, 0.5, 1 and $2~ M_\odot$. (D'Antona & Mazzitelli 1994, "Alexander MLT''-case, $\alpha _{\rm ML}=1.4$) are shown for comparison.

The lowest isopleth at $0.01~ {M_\odot}$ illustrates the fact that hydrostatic structures of that mass become optically thick (opacity limit of fragmentation) and, as a consequence, all initial protostellar cores typically form with a few percent of a solar mass. At $M_\tau=0.04~{M_\odot}$ the lowest mass fragment has almost terminated its main accretion phase while the higher mass fragments still grow their cores and accelerate their accretion flows. As $M_\tau$ increases to the final mass the isopleths become tangential to the evolutionary tracks and at constant mass the two curves are identical. Note the decrease in luminosity when the track of a fragment approaches an isopleth close to its final mass. The beginning exhaustion of the mass reservoir becomes then noticeable as a drop in accretion luminosity. To mark the end of significant mass accretion, the point at which 99% of the final mass has been accreted is indicated by a thin line. For the two smallest masses this happens well before they reach the Hayashi phase. Accretion, as a consequence, does not play a role during their PMS-contraction as calculated here. However accretion does play a role in defining their thermal structure at the "top of the track''. For masses from $0.5 ~{M_\odot}$ upward the end of significant mass accretion is terminated after the Hayashi line has been crossed.