Evolution on the Birthline

Let us follow the evolution of an accreting star on the birthline with an accretion rate of 10~5 M0 yr-1. The initial age is equal to about (M/ < M >) for an average initial rate < M >. Apart from this choice of the initial age, the evolution on the birthline is independent of the initial conditions.

Up to 1.2M0: for brown dwarfs and low-mass stars between 0.01 and 0.4M0, the luminosity is so low and the Kelvin-Helmholtz time so long that even the moderate thermostatic support of D burning maintains the star in equilibrium, making a sort of deuterium main sequence. Low-mass stars are fully convective at the end of the adiabatic phase, as a result of the high opacity, up to a limit of 2.5 M0. Due to D burning, for masses larger than about 0.2 M0 the stellar radius of the accreting star grows almost linearly with the increasing stellar mass (Fig. 20.3) up to a mass of about 1.2 M0. The reason is that the central stellar temperature behaves like Tc ~ M/R. As Tc is essentially constant during D burning, a linear relation between mass and radius is established.

From 1.2 to 2.5 M0: if accretion goes on, the star on the birthline remains fully convective up to a mass limit of about 2.5 M0. Above it, a radiative core develops; convection remains in an outer zone up to about 4-4.5 M0. After the linear part ending at 1.2 M0, the radius slightly declines as mass is growing up to about 2.5-3M0 (Figs. 20.3 and 20.4).

From 2.5 to 4 M0: stars more massive than about 2.5 M0 form an off-center radiative barrier, where D burns in shell and no longer reaches the stellar center, which has switched to radiative equilibrium and carries on with gravitational contraction. The increase of central T and the resulting increase of transparency favor the growth of the radiative core (Fig. 20.4).

Table 20.2 shows the main parameters for a star evolving on the birthline with M from (1 — 2) x 10—5 M0 yr—1 [44]. The corresponding birthline is illustrated by a dotted line of 10—5 M0 yr—1 in Fig. 20.5. The energy produced in the outer layers by D burning makes a luminosity LD (20.12) larger than the outgoing luminosity (Fig. 20.7). This defines a mass limit of 4-4.5 M0 up to which radiative equilibrium is not fast enough to dissipate the energy of D burning; this limit also corresponds to about the mass where convection disappears in the envelope.

The energy of D burning produces a fast stellar inflation with almost a doubling of the stellar radius as shown in Figs. 20.3 and 20.4. Of course, the larger the

Fig. 20.3 Relation between the radius and mass of a star accreting at a rate Maccr = 10 5 Mq yr 1. Adapted from F. Palla and S.W. Stahler [456, 457]

20.4 Evolution on the Birthline 10 i

20.4 Evolution on the Birthline 10 i

Fig. 20.4 Left: evolution of the convective zones (in gray) in a star with a growing mass. Initially the star is fully convective and then convection recedes from the center to finally disappear in the whole star. The star is then fully radiative and at higher masses a convective core appears. Right: corresponding evolution of the radius. From P. Bernasconi and the author [44]

Fig. 20.4 Left: evolution of the convective zones (in gray) in a star with a growing mass. Initially the star is fully convective and then convection recedes from the center to finally disappear in the whole star. The star is then fully radiative and at higher masses a convective core appears. Right: corresponding evolution of the radius. From P. Bernasconi and the author [44]

Table 20.2 Properties of stars on the birthline at Z = 0.02 for accretion rates (1 - 2) x 10-5 M© yr-1. The age, mass, luminosity, Teff, radius and envelope mass fraction are given at various stages on the birthline. The slight variation of the mass loss rates is due to the account of the internal turbulent pressure in the cloud [44]

Table 20.2 Properties of stars on the birthline at Z = 0.02 for accretion rates (1 - 2) x 10-5 M© yr-1. The age, mass, luminosity, Teff, radius and envelope mass fraction are given at various stages on the birthline. The slight variation of the mass loss rates is due to the account of the internal turbulent pressure in the cloud [44]

Age

Ml

Mass

Log L

Log Teff

R

Menv

yr

M© yr-1

K

Menv/M

7.923

x 104

1.10

x 10-5

0.801

0.731

3.632

4.223

1.000

8.826

104

1.11

10-5

0.901

0.781

3.640

4.311

1.000

9.718

104

1.13

10-5

1.000

0.813

3.647

4.331

1.000

1.192

105

1.16

10-5

1.250

0.856

3.662

4.247

1.000

1.406

105

1.20

10-5

1.500

0.875

3.673

4.126

1.000

1.815

105

1.26

10-5

2.000

0.903

3.692

3.905

1.000

2.201

105

1.33

10-5

2.500

0.922

3.708

3.708

0.750

2.568

105

1.40

10-5

3.000

0.962

3.723

3.623

0.291

3.250

105

1.53

10-5

4.001

1.236

3.760

4.189

0.022

3.875

105

1.66

10-5

5.001

2.781

4.062

6.173

0.000

4.457

105

1.79

10-5

6.006

3.238

4.278

3.864

0.000

4.994

105

1.92

10-5

7.001

3.265

4.315

3.362

0.000

5.507

105

2.05

10-5

8.013

3.468

4.365

3.374

0.000

Directional Trajectory

Fig. 20.5 The dashed lines show evolutionary tracks at constant mass (Fig. 20.2). The continuous lines show the birthlines for constant accretion rates Maccr = 10-6, 10-5, 10-4 [email protected] yr-1; these birthlines also extend to the upper part of the diagram in the MS stage. The dotted lines represent birthlines for accretion rates slightly growing with mass which may better fit observations for T Tauri stars (cf. Table 20.2; these three dotted birthlines correspond, respectively, to 0.1, 1.0 and 10.0 times the accretion rates considered in Table 20.2 and they start with accretion rates equal to 10-6, 10-5and10-4 M0 yr-1 , respectively. The thick continuous line is the ZAMS. Adapted from P. Norberg and the author [443]

Fig. 20.5 The dashed lines show evolutionary tracks at constant mass (Fig. 20.2). The continuous lines show the birthlines for constant accretion rates Maccr = 10-6, 10-5, 10-4 [email protected] yr-1; these birthlines also extend to the upper part of the diagram in the MS stage. The dotted lines represent birthlines for accretion rates slightly growing with mass which may better fit observations for T Tauri stars (cf. Table 20.2; these three dotted birthlines correspond, respectively, to 0.1, 1.0 and 10.0 times the accretion rates considered in Table 20.2 and they start with accretion rates equal to 10-6, 10-5and10-4 M0 yr-1 , respectively. The thick continuous line is the ZAMS. Adapted from P. Norberg and the author [443]

accretion rate, the larger the radius inflation. From the expression of the central temperature as a function of mass and radius (1.26), one has dR/R = dM/M - dTc/Tc. The temperature of the interior and of the D shell is not changing very much; thus the increase in the mass produces a corresponding fast increase in the radius. Thus, the birthline shows a steep increase in the luminosity in Fig. 20.8 due to the fact that the internal peak of luminosity has radiatively diffused toward the stellar surface. The phase of shell D burning, which inflates the radius, also delays the role of gravitational contraction as the main energy source.

Above 4M0: the radiated luminosity becomes greater than the accretion luminosity, i.e.

gmM m gm2

Lrad > -— ^ >7-i.e., taccr > tKH . (20.13)

Thus, thermal relaxation intervenes fast enough and thermal equilibrium is achieved. Accretion may still proceed, but the rate of thermal adjustment is in general faster and the stars move toward the ZAMS. Evidently, the various mass limits depend very much on the accretion rates; those given here correspond to rates of the order of 10-5 M0 yr-1.

The end of the radius inflation for a star on the birthline occurs for masses of about 4.5 M0; the convective envelope disappears at the same stage (cf. Fig. 20.4) and the star becomes fully radiative. The accretion still brings deuterium, but it stays at the stellar surface and escapes destruction. Thus, D has no effect anymore on the birthline and the energy only comes from contraction. The accreting star is fully radiative between about 4.5 and 6M0. It contracts slowly and this brings the accreting star on the ZAMS in the HR diagram (Fig. 20.5). There the CN cycle comes to equilibrium first (Sect. 20.2) and then hydrogen ignition occurs.

Higher accretion rates produce higher birthlines (Fig. 20.5). The accreting star effectively reaches the ZAMS near 4, 8 or 12 M0 for constant accretion rate of 10-6, 10-5 or 10-4 M0 yr-1, respectively. If the star continues to accrete mass when it has reached the ZAMS, it will move upward along the ZAMS since it is in thermal equilibrium due to the short Kelvin-Helmholtz timescale. Then, as H burning proceeds, the star starts moving leftward in the HR diagram at the nuclear timescale.

The birthline is an upper envelope (Fig. 20.8) of stellar tracks which leave the birthline and then evolve at constant mass. This can be shown easily. For radiative stars, one has the homology relation (Sect. 24.3)

Thus, a star on the birthline with a growing mass becomes more luminous than a star which keeps its mass constant after leaving the birthline. Higher accretion rates shift the radius vs. mass relation toward higher radius for a given mass. Figure 20.5 also shows that growing accretion rates produce, for the same reasons, slightly steeper birthlines.

Comparisons of birthlines with various accretion rates and observations of Ae/Be Herbig and T Tauri stars are shown in Fig. 20.6. As the birthline is the upper envelope of pre-MS stars in the HR diagram, we see that in the range of 2-8 M0 the rates should likely be about 1-3 times the rates of Table 20.2, i.e., of the order of 1-6 x 10-5 M0 yr-1. The comparison of birthlines and observations ofpre-MS stars in the HR diagram may provide an indication of the accretion rates Maccr, which is the leading parameter for pre-MS evolution.

Fig. 20.6 Comparison of various birthlines with accretion rates equal from bottom to top to 0.1, 0.15, 0.2, 0.3, 0.5, 0.75, 1.0, 1.25, 1.75, 2.5, 3.5 and 5.0, the case illustrated by the data of Table 20.2. Pre-MS tracks at constant mass are indicated by dot-dashed lines and post-MS tracks of high masses by dashed lines. Numbers indicate the stellar masses. The observations are Ae/Be Herbig and T Tauri stars with different values of Maccr and sources. From P. Norberg and the author [443]

Fig. 20.6 Comparison of various birthlines with accretion rates equal from bottom to top to 0.1, 0.15, 0.2, 0.3, 0.5, 0.75, 1.0, 1.25, 1.75, 2.5, 3.5 and 5.0, the case illustrated by the data of Table 20.2. Pre-MS tracks at constant mass are indicated by dot-dashed lines and post-MS tracks of high masses by dashed lines. Numbers indicate the stellar masses. The observations are Ae/Be Herbig and T Tauri stars with different values of Maccr and sources. From P. Norberg and the author [443]

20.5 Evolution from the Birthline to the ZAMS

At some stage, accretion comes to an end or becomes negligible, because for example the cloud fragment has delivered most of its mass to the central body. When this occurs, maybe after some transition with a decreasing accretion rate, the stars leave the birthline. They follow tracks which finally join the track of the same constant mass (after some re-adjustments of the internal thermal equilibrium if it is not yet achieved, i.e., for M < 4 M0). Figure 20.8 shows a grid of star models of different final masses. During the first part of their evolution, the stars are accreting with rates of about 10-5 M0 yr-1 and they are on the birthline (Table 20.2). When accretion stops, they leave the birthline on a track with a lower luminosity as shown above.

L/L0

Fig. 20.7 Evolution of the internal luminosity of a 3.5 [email protected] star after it has left the birthline. The time is counted since the birthline. There is an outward shift of the luminosity peak as the star tends to thermal equilibrium. During these phases, the surface luminosity grows strongly. Adapted from F. Palla and S.W. Stahler [456]

Fig. 20.7 Evolution of the internal luminosity of a 3.5 [email protected] star after it has left the birthline. The time is counted since the birthline. There is an outward shift of the luminosity peak as the star tends to thermal equilibrium. During these phases, the surface luminosity grows strongly. Adapted from F. Palla and S.W. Stahler [456]

Stars of different masses leave the birthline in a different way. Stars with M < 2.3 M0, which are fully convective, leave the birthline descending vertically on the Hayashi line (Fig. 20.8), because they burn fast the deuterium present, which is no longer replenished after the star has left the birthline. Contraction provides the main energy source and the stellar radius declines. A radiative core appears a little before the star reaches its minimum luminosity. Convection recedes toward the surface and the star settles on a radiative track. A convective envelope only remains in stars with M < 1.5 M0. After the minimum luminosity, the rise in brightness is very steep due to the readjustment of the internal L profile: the maximum which was deep in the star radiatively diffuses toward the surface (Fig. 20.7). For stars with M > 1.2 M0, CN burning starts slightly before the ZAMS and brings CN elements to their equilibrium values. Finally, H burning starts via the pp chains (M < 1.2 M0) or via the CNO cycle (M > 1.2 M0).

When they leave the birthline, stars with mass between 2.5 and 4M0 have a radiative core which grows until the star is fully radiative. Thus, these stars have no Hayashi phase: when they become visible, their Teff are much higher than for evolution at constant mass: a 5M0 star appears with a Tef ~ 11500 K (Fig. 20.8) instead of 4000 K at constant mass. The tracks show first a steep increase in luminosity, rather similar to that occurring for stars on the birthline for the above reasons. Non-homologous contraction supplies most of the energy: the interior, deprived of other energy source, contracts, while the outer layers expand due to the shift of the luminosity maximum toward the surface. The stars then join the tracks with constant mass and follow about the same evolution. A convective core forms when CN burning starts before the ZAMS.

log Teff

Fig. 20.8 Pre-MS tracks at solar composition with accretion from a cloud with thermal and turbulent support (cf. Table 20.2). The stellar masses on the ZAMS are indicated. The birthLine is indicated by a thick line which forms the upper envelope of the various tracks. From Bernasconi [43]

log Teff

Fig. 20.8 Pre-MS tracks at solar composition with accretion from a cloud with thermal and turbulent support (cf. Table 20.2). The stellar masses on the ZAMS are indicated. The birthLine is indicated by a thick line which forms the upper envelope of the various tracks. From Bernasconi [43]

Stars with mass above about 4.5 M0 are fully radiative when they leave the birthline and they adjust rapidly on the corresponding tracks with constant mass, following the above L ~ R-1/2 relation. The part of the radiative track, where they are observable, is short (Fig. 20.8).

Stars with mass above ~ 6 M0 already have a convective core when they leave the birthline. They contract for a short time before the ignition of the CNO cycle; then contraction stops. Stars with M > 8 M0 are studied in Chap. 22.

In the above accretion scenario, when they become visible the stars generally have smaller radii and luminosities (by a factor of about 10 and 100, respectively) than if they would appear at the top of the Hayashi line (Sect. 19.4). Compared to the tracks at constant mass of Fig. 20.2, the visible part of the tracks with accretion covers a much smaller area of the HR diagram, since stars on the birthline are still embedded in their parent material. The comparison (Fig. 20.6) of the birthline and observations of T Tauri stars and Ae/Be Herbig stars supports the above scenario.

The various mass limits given above depend on metallicity Z and accretion rates. As an example, at Z = 0.001 the core remains fully convective up to a mass of 2 M0 instead of 2.5 M0. The convective envelope disappears at about 3.1 M0 instead of about 4.5 M0; the star joins the ZAMS near 6.5 M0 instead of 7.5 M0. There is a global shift of the domain to lower masses at lower Z, because due to lower opacities the stars are brighter and hotter; they behave like slightly more massive stars with solar Z.

20.6 Lifetimes, Ages and Isochrones

There are various possible definitions of the ages of T Tauri stars (< 2 M0), of Ae/Be stars (> 2 M0) and forming stars.

- We may define the zero age when the stars first become optically visible, i.e., when the phase of heavy accretion is over and the star leaves the birthline. Let us call tpre-MS the age defined in this way. This is often the age definition in the context of accretion models [456].

- Another possibility is to define an age tform as the lifetime since the end of the isothermal collapse, i.e., from the starting point in Fig. 19.5. Such models are in their infancy and such ages are rarely available. The total formation time tform could be estimated as the sum of the initial accretion time tini.accr = Mini/ < Mjni.accr >, where Mini is the mass at the beginning of the birthline phase (typically a few tens of M0) and < Mfini.accr > an appropriate average of the initial accretion rates, plus the time tbirthl spent on the birthline, plus the age tpre-MS after the birthline:

- In general for MS stars and post-MS stars, the ages are counted since the ZAMS. The relative error is small since the formation time is of the order of 1% of the H-burning lifetime.

Table 20.3 compares these to the lifetimes tclassic of models with constant mass (Sect. 20.2). Below 2M0, tform and tpre-MS are nearly identical, since the pre-MS phase starts nearly on the Hayashi line. For larger masses more time is spent on the birthline and thus the pre-MS phase is shorter. For stars with mass close to 7M0, tpre-MS tends toward zero, since there the birthline joins the ZAMS. The differences of tform and tpre-MS with respect to tclassic are also small up to about 1.2 M0, since in both scenarios the stars have a long way to reach the ZAMS.

It is striking that above about 1.5 M0 (and up to about 7M0) the total formation time tform in the accretion scenario is shorter than the classical time. This is surprising at first, because during a part of its evolution on the birthline the accreting

Table 20.3 The lifetimes tform and ipre-MS up to the ZAMS for the models with the accretion rates of Table 20.2 compared to the lifetime tciassic of models at constant mass (Sect. 20.2). An average rate < Aiini.accr >= 10~5 M® yr-1 is assumed before the birthline. The columns give the final masses, the total lifetimes tform, the pre-MS lifetimes tpre-MS and the ratios with respect to the classic timescales of models at constant mass

Table 20.3 The lifetimes tform and ipre-MS up to the ZAMS for the models with the accretion rates of Table 20.2 compared to the lifetime tciassic of models at constant mass (Sect. 20.2). An average rate < Aiini.accr >= 10~5 M® yr-1 is assumed before the birthline. The columns give the final masses, the total lifetimes tform, the pre-MS lifetimes tpre-MS and the ratios with respect to the classic timescales of models at constant mass

Final mass

form

¿pre-MS

¿form

¿pre-MS

M0

yr

yr

¿classic

¿classic

0.8

7.154 >

< 107

7.147 >

107

1.045

1.044

1.0

3.821 >

< 107

3.805 >

107

0.982

0.978

1.5

3.095>

107

3.081 >

107

0.874

0.870

2.0

1.172 >

107

1.153 >

107

0.501

0.493

3.0

2.683 >

< 106

2.423 >

: 106

0.371

0.335

4.0

1.414 >

< 106

0.881 >

: 106

0.557

0.347

5.0

0.799 >

< 106

0.412 >

: 106

0.694

0.358

star has a lower luminosity than for constant mass evolution. However, it is a general property [529] that accreting stars have a higher central temperature and are younger than non-accreting stars. An accreting star goes through a succession of structures, which are more non-homologous than in the case of constant mass, i.e., contraction is faster in the center, where T becomes higher (the star is younger at the same central T). Also, not all the deuterium is burnt in the accretion scheme, while it is exhausted in the case of constant mass, which are fully convective initially and this makes more energy available. For M > 7 M0, the ratio iform/iclassic very much depends on the accretion rates (Sect. 22.2).

Fig. 20.9 Isochrones of various ages for pre-MS models with constant accretion rates of 10~5 [email protected] yr-1. Light gray shading shows the location of stars between 106 and 107 yr; dark area is for stars above 107 yr. The ages indicated here are ipre-MS (from the birthline). Adapted from S.W. Stahler and F. Palla [548]

log Teff

Fig. 20.9 Isochrones of various ages for pre-MS models with constant accretion rates of 10~5 [email protected] yr-1. Light gray shading shows the location of stars between 106 and 107 yr; dark area is for stars above 107 yr. The ages indicated here are ipre-MS (from the birthline). Adapted from S.W. Stahler and F. Palla [548]

Figure 20.9 provides an example of isochrones with various pre-MS ages ipre-MS from 105 to 3 x 107 yr. A few properties may be quoted:

- The smaller the ipre-MS, the farther the isochrone from the ZAMS.

- Isochrones with larger ipre-MS reach the ZAMS at lower luminosities.

- For the age estimates, the values derived from accretion tracks are a factor of about 2-3 smaller for ages lower than 5 x 105 yr (with respect to tracks at constant mass), while above 106 yr the differences are negligible.

The comparison of HR diagrams of very young clusters and isochrones gives some insight into their history of star formation. However, the results are valid only if the models correspond well to the reality.

20.7 Lithium Depletion in Pre-MS Stars

Light elements are fragile. If the models have an external convective zone which extends deep enough to reach T = 2.5 x 106 K, the abundance of 7Li at the stellar surface starts decreasing. The surface abundance of lithium depends critically on the depth of the external convective zone. The initial abundance of lithium is very low; the number ratio N(Li)/N(H) is 1.3 x 10~9 in the interstellar medium, i.e., logN(Li) = 3.1 in a scale where logN(H) = 12.0. In the Sun, the lithium abundance is about a factor 102 lower than in the interstellar medium, being logN(Li) = 1.1. Lithium is observable, thanks to a faint line Li I 6708 A, which allows lithium abundances to provide a test on the model structure for both pre-MS and MS stars.

20.7.1 Model Predictions

Pre-MS stars start contracting with a central T lower than 2.5 x 106 K; thus there is no Li depletion at the beginning of the birthline for all masses. Surface depletion appears during pre-MS evolution for M < 1.2 M0 when they reach the dip in luminosity at the bottom of their descending tracks. There, contraction makes the internal T high enough to destroy lithium. This is about the stage where a radiative core develops at the stellar center for M > 0.4 M0. If the bottom of the receding convective zone is above the critical level for Li burning, the surface abundance of lithium is not affected. This happens in stars with masses (on the ZAMS) above 1.2M0, while in stars with M < 0.8 - 0.9 M0, the external convective zone is deep enough to largely destroy lithium before the star reaches the ZAMS (Fig. 20.10).

For 1M0, after a fast depletion by ~ 10, there is a kind of a plateau (Fig. 20.10). The Li depletion goes on slowly during the whole MS phase due to internal mixing, so that the total decrease with respect to the initial Li abundance reaches a factor of 100 at the solar age. For the 0.8 M0, the initial fast destruction is larger due to a deeper convective zone; it also reaches a plateau. For 0.6M0, the destruction

Fig. 20.10 Left: the evolution of the 7Li abundance as a function of log (age) for different masses. The Li abundance is expressed as a number ratio with respect to hydrogen. The value of log H in the Sun is 12.0. The initial value adopted for the number ratio N(Li) /N(H) is 2 x 10-9. Adapted from L.T.S. Mendes, F. D'Antona and I. Mazzitelli [397]. Right: the abundance of lithium 7Li as a function of stellar masses on the ZAMS relatively to the initial abundance Li0. Adapted from L. Siess [529]

Fig. 20.10 Left: the evolution of the 7Li abundance as a function of log (age) for different masses. The Li abundance is expressed as a number ratio with respect to hydrogen. The value of log H in the Sun is 12.0. The initial value adopted for the number ratio N(Li) /N(H) is 2 x 10-9. Adapted from L.T.S. Mendes, F. D'Antona and I. Mazzitelli [397]. Right: the abundance of lithium 7Li as a function of stellar masses on the ZAMS relatively to the initial abundance Li0. Adapted from L. Siess [529]

is complete in about 107 yr. The Li depletion becomes larger as the pre-MS stars approach the ZAMS. Figure 20.11 shows the region of the HR diagram of solar and lower mass stars where lithium depletion is expected from theoretical models [548]; this provides a test about the depth of the outer convective zone in pre-MS models.

Deuterium, more fragile than Li, is destroyed in stars below about 1.2 M0 during their initial fully convective phase, while above that limit, due to the absence of a

Stellar Birthline

Fig. 20.11 Pre-MS evolutionary tracks for various stellar masses in M0. The upper and lower thick lines are, respectively, the birthline and the ZAMS; the broken line is an isochrone of 106 yr. In the white area between the two thick lines, there is no Li depletion; in the light shaded area, the depletion is down to 10% of the initial value; in the dark shaded area depletion is between 10% of the initial value and complete depletion. Adapted from S.W. Stahler and F. Palla [548]

Fig. 20.11 Pre-MS evolutionary tracks for various stellar masses in M0. The upper and lower thick lines are, respectively, the birthline and the ZAMS; the broken line is an isochrone of 106 yr. In the white area between the two thick lines, there is no Li depletion; in the light shaded area, the depletion is down to 10% of the initial value; in the dark shaded area depletion is between 10% of the initial value and complete depletion. Adapted from S.W. Stahler and F. Palla [548]

deep convective zone, deuterium keeps its initial abundance in case of accretion. The depth of the external convective zone is very sensitive to the physical inputs of the models, such as metallicity, opacity, convection, overshooting, rotation, extra mixing and residual accretion. Stars with a higher metallicity Z have a stronger Li depletion, because the opacity is higher and thus their external convective regions are deeper. Conversely lower Z stars have less Li depletion.

20.7.2 Li and D in T Tauri Stars and Residual Accretion

A residual accretion rate Maccr during the pre-MS phase affects the tracks in the HR diagram and the Li abundances; however for Maccr < 10-7M0 yr-1 there is little effect for solar-type stars. Figure 20.12 compares three different cases of evolution of mass accretion with time [529], all starting from 0.5 M0 and finishing with 1.2 M0 to a case of constant mass evolution with 1.2 M0. The higher the initial accretion, the earlier the accreting star joins the track of constant mass. In the two cases where the initial accretion is not negligible, the accretion produces a small increase in stellar radius, before contraction takes over. The higher the initial accretion, the earlier the formation of a radiative core. The mass and age estimates from the location in the HR diagram are little affected by residual accretion.

The abundances of Li and D offer tests whether there is a continuing accretion during the pre-MS phase. Above 1.5 M0, no differences in the Li abundances are expected. Around 1M0, surprisingly the abundance of Li is decreased by a factor of 4 in models with a residual accretion, due to the faster increase of the central T in accreting models, because of a more efficient contraction [529]. The best signature of late accretion (after the birthline) would be the presence of D on stars more massive than 1.2 M0 (but below about 4M0, since these stars have no or only a very thin convective zone, cf. Fig. 20.4, and T is below 106 K so that D is not destroyed). Below 1.2M0, D is anyway completely destroyed in convective envelopes.

Observations of lithium in T Tauri stars also provide [380] interesting constraints on models. For L > 0.9 L0, the mean observed value of Li abundance is in agreement with the cosmic value of logN(Li)=3.1, with a marginal trend to have less depletion than predicted. For masses lower than the Sun, models tend to predict too much lithium depletion.

On the whole, it is still uncertain whether there is a significant pre-MS accretion after the birthline, either in the form of continuous matter infall or as Jupiter-like objects [612]. Li depletion in pre-MS stars seems related to rotation, in the sense that fast rotators near 1M0 generally show less Li depletion than slow rotators. This might be an effect of rotation on the limit of the convective zone (Solberg-Hoiland criterion, Sect. 6.4.3).

IVLff

-I 1

1

11 1 -

1- i l

/

.2

s

-I 1

1

3.75

3.70

3.65

log T

3.60

Fig. 20.12 The continuous thick line shows a pre-MS evolution for a constant 1.2 M® star. The three other tracks start at the top of the Hayashi line with M = 0.5 M®. The dotted line shows a fast initial accretion; the star reaching its final mass in 106 yr. The long-broken track is for a decreasing accretion rate. The short-broken line shows the track for a negligible accretion until 106 yr and a large after that. The black squares indicate the return to radius contraction as the main energy source, after some expansion due to D burning. The ellipses mark the beginning of a radiative core; this is about the point where Li is burning. The stars indicate the end of accretion. At point marked 1.2 M®, the star reaches the ZAMS. Loci of constant radius are indicated. Adapted from Siess [529]

20.7.3 Li Depletion in Low-Mass Stars and Brown Dwarfs

The fact that the low-mass stars with masses < 0.5 M® keep a constant Teff during their descending tracks in the HR diagram (Fig. 20.11) leads to interesting developments [46] allowing us to obtain the radius, age and luminosity as a function of Li depletion. These stars are fully convective during Li burning; thus they have a polytropic structure of index n = 1.5 (Sect. 24.5). For such a structure the central density qc is proportional to the average density; as usual the central T behaves like GM/R (1.26); numerically

where ¡ieff < ^, because partial degeneracy produces free electrons and thus the effective mean molecular weight is lower than that for perfect gas (however, for simplification here we take |eff « |). Except during the initial D burning, gravitational contraction powers the stellar luminosity; thus by deriving the potential energy of the polytrope with n = 3/2 (Sect. 24.5) with respect to radius R and remembering that half of the change of potential energy is radiated, one gets

4nR2oTff

We may integrate this equation from t = 0 with R = to the present radius R at time t. This time differs from the time tpre-MS counted since the star has left the birthline. However, in low-mass stars, Li depletion occurs long after (> 10 Myr) the birthline; thus the times tform and tpreMS are not too different (Table 20.3). One has

numerically [46]

Equation (20.18) can be integrated keeping L constant; one has with (20.19)

3 GM2 1

Numerically

Teff

3000 K J

106yr

These equations show how R and L decrease with time when the star goes down the Hayashi track. The lithium depletion can be considered as achieved when the timescale tLi for the nuclear destruction of lithium (cf. 9.9) by 7Li(p,a)4He is equal to the characteristic time tcontr of the star contraction:

where X is the H mass fraction. The equality of these two times gives a relation between the central temperature Tc and (M, Teff) the time of depletion. One may eliminate Tc by (20.17) and obtain the dependence of the radius at the depletion time

Rdepl

Teff1/6 M7/813/4 (Fig. 20.13), as well as the scaling of the Li depletion time tLi eff

Fig. 20.13 Lines of constant Li depletion as a function of radius and mass. The lines of 5 and 95% depletion are shown. Adapted from L. Bildsten et al. [46]

Fig. 20.13 Lines of constant Li depletion as a function of radius and mass. The lines of 5 and 95% depletion are shown. Adapted from L. Bildsten et al. [46]

The age is 50 Myr for 0.1 M0, Teff = 3000 K, ^ = 0.6 and a depletion factor of 2 (Fig. 20.14). Expression (20.24) applies to fully convective stars when degeneracy is unimportant (from 0.2 to 0.5 M0). Degeneracy introduces correction terms in this equation [46], but the behavior is about the same: tLi ~ Te£f351M~0.715. If Teff can be estimated and mass inferred from a grid of models, the observation of Li depletion provides with (20.24) or Fig. 20.15 an age estimate for clusters with an accuracy better than 25%.

The lithium abundance is also a constraint on the models. In Fig. 20.13, models predict that at a given mass, a contracting star above a certain radius limit should show no or little Li depletion. Below another limit, Li should be depleted. The two lines are close to each other, because of the high T-sensitivity of the Li-burning reaction. Since Li burns at a nearly constant T, the relation Tc ~ M/R (20.17) imposes that the radii at these limits behave almost linearly with masses. The depletion times as a function of stellar masses are shown in Fig. 20.14 [105]. The two lines for depletion by factors 2 and 100 are close to each other for the same reasons as above. The lowest mass for Li burning is 0.06 M0 at solar composition. For masses larger than this minimum mass, the depletion time decreases fast.

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