The Premain Sequence Phase and the Birthlines

In the pre-MS phase, the stars are in a stage of hydrostatic equilibrium; the evolution becomes much slower proceeding at the Kelvin-Helmholtz timescale, which is about 30 million years for the Sun. In addition to contraction, the nuclear deuterium burning produces an energy which significantly influences the evolution of solar and lower mass stars and produces an inflation of the radius. The accretion of matter from the parent cloud may continue for some time, if so the rate of mass accretion is a key parameter determining the properties of pre-MS stars. This phase of evolution leads the stars to the main sequence phase, where they experience nuclear fusion reactions of hydrogen into helium.

20.1 General Properties of Non-adiabatic Contraction

The pre- and protostellar phases were treated as adiabatic, because despite the growing luminosity, the total loss of energy was limited due to the short timescales. When evolution becomes slower, the adiabatic approximation is no longer valid. For a 1M0 star, the fast adiabatic phase ends at an age of ~ 106 yr, when the internal temperature T « 105 K and the density q ~ 10~2 g cm~3. Henceforth, the rhythm of contraction is much slower, because the whole star, supported by gas pressure, is now about in hydrostatic equilibrium.

20.1.1 The Kelvin-Helmholtz Timescale

For a star in equilibrium, the Virial theorem (Sect. 1.3) indicates that the increase AU =[1/(3y- 3)] (-AQ) of the specific internal energy is related to the difference of the specific potential energy AQ. The radiated energy AUrad is the difference

*This chapter may form the matter of a basic introductory course.

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between what is liberated by gravitational contraction and the increase AU of internal energy:

3y-4

For an ionized gas y = 5/3, half of the energy is used for heating the gas, while the other half is radiated away. For an average luminosity L over a timescale tKH, one has AUrad = LtKH; thus we get tKH = AUd = ^qGM2 ^ GM , (20.2)

L 3y-3h RLRL

with a numerical coefficient of the order of unity, (i.e., 3/10 for constant density and 3/4 for a typical density distribution with a polytropic index n = 3, Sect. 24.5). Numerically, this becomes for a coefficient equal to 1

This is the Kelvin-Helmholtz timescale, expressing the time during which a star can radiate a luminosity L from the potential energy only. Currently the ratio of tKH to the MS lifetime is of the order of tKH/tMS = 10-2 (0.005 for a solar-type star and 0.02 for a massive star). For a mass-luminosity relation of the form L ~ M3 and a mass-radius relation R ~ M0 7 valid over the upper MS, one has tKH ~ M-17. We recall that tKH is also the timescale for the stellar thermal adjustments (Sect. 3.2.4).

20.2 Pre-MS Evolution at Constant Mass

For long, it was considered that when the star reaches the Hayashi phase at the end of the adiabatic phase, the whole stellar mass has been assembled and that the star then evolves keeping a constant mass. Now, more refined scenarios are considered as discussed in further sections.

20.2.1 The Hayashi Line

The Hayashi line is an important concept in stellar evolution. It is the location of fully convective stars in the HR diagram. This locus is nearly vertical (combined effect of convective transport and opacity) and depends on the mass of the star. This is close to the location of post-MS red giants, since red giants have very extended convective envelope. Stars on the Hayashi line have large radii (Sect. 19.4). The main properties of the Hayashi line are as follows:

- The Hayashi line is at about constant Teff« 3500 K (Figs. 20.1 and 20.2). This results from the high opacity below T ~ 7000 K, due mainly to the photo-ionization of ion H". At low T, the opacity k behaves like k ~ q05T9 (8.25). Thus, if T grows, k grows even faster, which in turn reduces T. This produces a feedback maintaining T about constant.

- The Hayashi line is the lower Teff limit of convective stars in the HR diagram, i.e., it applies to fully convective stars. To the right of it, there are no stable stars. If for a star model of given M and L, one would reduce Teff below the value of the Hayashi line and make a larger radius, say R', the integration of the equation for mass conservation from the stellar center up to R' would give a mass superior to M, since the density distribution of a convective body is anyhow fixed by the polytropic index n (Sect. 24.5), i.e., for a fixed density distribution, a larger stellar radius would lead to a larger mass. Therefore, there is no star in equilibrium to the right of the Hayashi line.

- The location of the Hayashi line also depends on the mixing-length ratio t/HP for convection, since this ratio influences the stellar radius. An increase of t/HP from 1.0 to 1.5 shifts log Teff by +0.06 dex, since a more efficient energy transport, as resulting from a larger t, produces a smaller stellar radius.

- The Teff of the Hayashi line also depends on metallicity Z and in particular on the abundances of metals with a low ionization potential, since they provide free electrons making ions H" which contribute a lot to the opacity at low T. A lower Z reduces the opacity and thus increases the luminosity. To provide more energy, contraction is enhanced; the radius becomes smaller and so the Teff of the Hayashi branch is higher.

- The initial model at the top of the Hayashi branch can be taken as a polytrope with index n = 1.5 (or even n = 1 with an analytical solution, Sect. 24.5). Stellar contraction makes a higher density, which in turn increases the opacity and thus the luminosity decreases: the star goes down the Hayashi line (Fig. 20.1).

- The Hayashi line terminates at a minimum luminosity and thereafter the star moves toward higher Teff being made of a growing radiative core and an external convective envelope. The minimum luminosity is proportional to some power of the mass.

- When a star after the main sequence phase becomes a red giant, it evolves back toward low Teff, it will actually settle down on a new Hayashi line according to the radiative core to convective envelope ratio.

20.2.2 Gravitational Energy Production and D Burning

During contraction, the rate of gravitational energy production is given by egrav = -Cp T + (8/q)P (Sect. 3.3.2). With Vad = P8/(Cp qT), this gives cgrav

grav

From homology relations for stellar contraction (Sect. 3.4), we have P/P = -4R/R and q/q = —3R/R and with (3.86) we get

For a perfect and monatomic gas with a = 1, 5 = 1, CP = cp and Vad = 2/5, this is grav

Contraction (R < 0) produces stellar energy. The energy production is not much concentrated near the stellar center, as for nuclear burning.

Figure 20.1 shows the pre-MS evolution at constant mass for 1 and 3 M0 models. The Hayashi phase lasts about 2 x 107 yr for a 1M0 star and 106 yr for a 3 M0 star. The radius decreases considerably during this phase, but the fusion of deuterium (noted D or 2H) slows down the contraction. Figures 20.1 and 20.2 show that D early ignites on the Hayashi line for low-mass stars. D burning occurs at central temperature Tc between 1 and 2 x 106 K for stars from 1 to 60 M0 (Table 20.1);

Fig. 20.1 Pre-MS evolutionary tracks for evolution at constant mass for 1.0 and 3.0 Mq. The places where the ignition of deuterium, lithium and hydrogen starts for the 1 Mq model are indicated, as well as the ignition of deuterium and 12C for the 3 Mq model. Adapted from L. Siess [529]

log TefT

Fig. 20.1 Pre-MS evolutionary tracks for evolution at constant mass for 1.0 and 3.0 Mq. The places where the ignition of deuterium, lithium and hydrogen starts for the 1 Mq model are indicated, as well as the ignition of deuterium and 12C for the 3 Mq model. Adapted from L. Siess [529]

Fig. 20.2 Pre-MS tracks for evolution at constant mass. The squares on the tracks show the location of D ignition. The long-broken line represents the end of the convective envelope, the dotted line the appearance of convective cores (which rapidly disappear for lower masses). From P. Bernasconi and the author [44]

Fig. 20.2 Pre-MS tracks for evolution at constant mass. The squares on the tracks show the location of D ignition. The long-broken line represents the end of the convective envelope, the dotted line the appearance of convective cores (which rapidly disappear for lower masses). From P. Bernasconi and the author [44]

it starts at higher T in more massive stars because their average density is lower. D burning plays a great role during pre-MS evolution: by contributing to the luminosity, driving convection and inflating the stellar radius. The main reactions for deuterium burning are

D(p, y) 3He D(D,p) 3H(, e" v) 3He D(D, n) 3He. (20.7)

Table 20.1 The pre-MS lifetimes for constant mass evolution (¿kh is of the same order), its ratio to the H-burning phase and the temperature when 1% of the deuterium is burnt

Mass tpre-MS ¿pre-MS T(D burning)

in units of 106 yr H in units of 106 K

60 0.028 0.0082 1.93

15 0.117 0.0101 1.39

The first one is the most important by a factor 100 with respect to the other two at the relevant T; its energy generation rate is approximated by [456]

1gcm-3 / V106K

with n = 11.7. [D/H] = 2 x 10~5 is the typical number ratio of hydrogen to deuterium. The high T sensitivity implies that during D burning the temperature does not change much and thus D burning has an efficient thermostatic effect, keeping T almost constant. The second and the third reactions are of nearly equal importance (they are often ignored in literature). In the stellar context, the ^-disintegration of tritium can be regarded as instantaneous, since its half-lifetime is i1/2 = 12.26 yr, while the survival time of 2H is much larger in pre-MS stars. D burning can contribute up to 90% of the stellar luminosity in pre-MS stars, the rest being produced by contraction. In low-M stars, D burning can nearly stop for a while the star contraction on the Hayashi line. After 3 x 105 yr, D is exhausted in a 1M0 model (105 yr for 3M0); the star continues its contraction down the Hayashi line. The importance of D burning is lower for higher masses, which are dominated by contraction. For M > 4.5 M0, D burning starts in regions which are already radiative due to the fast recession of the external convective zone, while below this mass limit it completely burns in the fully convective interior.

After D burning, nuclear burning of lithium occurs for T > 2.5 x 106 K; the reaction is very sensitive to temperature (like T20):

Li burning occurs at smaller ages for larger masses: 109, 2.0 x 107, 1.4 x 106 and 2.4 x 105 yr for 0.06,0.4, 1 and 3 M0, respectively (Sect. 20.7). Beryllium Be burns for T > 3.2 x 106 K and boron B for T > 4 x 106 K. The minimum masses for Li, Be and B-burning are 0.055, 0.085 and 0.08 M0 [106]. The fusion of these rare elements is not energetically significant; however they make the Li and Be abundances to change during pre-MS evolution. Thus, a high Li abundance is a distinctive signature of pre-MS stars (Sect. 20.7).

20.2.3 From the Hayashi Line to the ZAMS

Moving down on the Hayashi line, the star becomes more transparent, as H and He are ionized over a larger region. This reduces the radiative gradient (Vrad ~ kL) and convection disappears, first in the center, where a radiative core forms and then from almost the whole star. This produces an internal mass reorganization with steeper density gradients, corresponding to change of polytropic index n from 1.5-3 to 33.6. This transition occurs near the minimum of surface luminosity. The ages then are, respectively, 2 x 106 and 1.5 x 105 yr for 1 and 3M0.

Contraction is non-homologous, being faster in the center. A part of the released gravitational energy is used to expand the external layers, while the rest is radiated. Thus, the dip in luminosity is followed by a fast growth of the stellar radius and the luminosity. This effect is particularly visible for the higher mass models, where radiation pressure is large (Fig. 20.2). Convection completely disappears at the location indicated by a broken line. The star settles in the HR diagram on a slightly oblique long leftward ascending branch, which is the locus of radiative stars with a polytropic index n « 3. For masses lower than about 1.5 M0, a proper radiative branch is absent, since these stars maintain a convective envelope during the whole contraction to the zero-age main sequence (ZAMS).

The start of H burning is preceded by a short phase of rather intense C-burning, during which carbon is brought to equilibrium (the cosmic C abundance is much larger than that of CNO equilibrium). Only a part of the CN cycle (Fig. 25.1) is operating at this stage, namely the reactions

The above chain converts 12C to 14N and thus brings CN elements to equilibrium. The energy production rate £ by 12C(p, y)13N goes like T19, which favors the appearance of a convective core (the release of gravitational energy is also playing a role in massive stars). The resulting central expansion pumps energy and reduces L and Teff; a knee on the tracks is resulting (Fig. 20.2). We may distinguish two mass domains:

- Stars with mass above about 1.3 M0: when 12C is brought to equilibrium, T is not yet high enough to make the full CN cycle operating. The core shrinks again and contraction operates until T is high enough. This produces a second knee in the tracks. Then, the growth of the central T allows the CNO cycle to operate.

- Below 1.3M0: the exhaustion of central 12C makes the convective core to disappear; it is not revived by the pp chains, due to their low T-dependence, these chains stop contraction and the star sets on the ZAMS.

Various criteria have been proposed to fix precisely the time when a contracting star sets on the ZAMS, for example, the minimum stellar radius, the local maximum core radius, the minimum luminosity, a central H depletion of 1%. Some criteria cannot be applied to the whole range of masses. Moreover, they do not exactly coincide and thus do not lead to similar values for the pre-MS lifetimes. For example, the fourth criterion leads to much longer pre-MS lifetimes, in particular for very massive stars. The first three generally give comparable values. It seems advisable to take the first one, because it is clearly defined. The pre-MS lifetimes from the top of the Hayashi line to the ZAMS are given in Table 20.1 with this definition. They are of the order of the Kelvin-Helmholtz timescale given by (20.2). They are very short for the most massive stars (i.e., of the order of a few 104 yr) due to their high luminosities.

20.3 Pre-MS Evolution with Mass Accretion

The above models assume that the whole stellar mass is already assembled at the end of the adiabatic phase. This is not necessarily the case. We shall thus examine the evolution of an initially small accreting core in hydrostatic equilibrium from the end of the adiabatic phase to the ZAMS. In many models [456], it is considered that stars are accreting at a constant rate Maccr, typically of 10"5 M0 yr"1 for solar-type stars. The accretion continues for some time, then the evolution proceeds at constant mass. At the end of accretion, the star experiences a period of relaxation and thermal re-adjustment and it joins a track very close to the track with its actual constant mass in the HR diagram. In some models, a transition period with decreasing mass accretion rates is considered (Sect. 20.7), but it does not make great changes.

20.3.1 The Birthline and Its Timescales

We define the birthline as the track described in the HR diagram by stars accreting mass at a "substantial rate". Since accretion stops when the star leaves the birthline, it is also in general the place where the star first becomes visible. What is a "substantial rate" of mass accretion? For solar-type stars, it is typically of the order of 10"5 M0 yr"1 (Fig. 20.5). A more physical definition may be given: accretion dominates if Maccr is large enough so that

M GM2

Maccr R2 L

The star does not have the time to adjust thermally. Being not in thermal equilibrium, the star occupies a particular location in the HR diagram: the birthline, where evolution is dominated by accretion.

If the timescales are such that taccr > tKH, the star has the time to adjust thermally and the evolution proceeds in thermal equilibrium, i.e., the star is on the ZAMS. There is a continuous set of birthlines for different values of Maccr (Fig. 20.5). The birthline forms the upper envelope of the individual further tracks. (Some authors define the birthline as the equilibrium position of fully convective D-burning stars in the HR diagram.)

For stars with M < 4.5 M0 (a limit between 2.5 and 9 M0 depending on authors) taccr < tKH for the above current accretion rates. This is due to the relatively long Kelvin-Helmholtz timescale (20.1). For larger masses, we have the opposite: taccr > tKH. The thermal equilibrium is rapidly realized and, even if the accretion is not terminated, these stars generally lie on the ZAMS or close to it (Sect. 22.2).

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