## Cp q T

In the case of a perfect mono-atomic gas, the additional term is just equal to V,. One could expect that the above expressions have important consequences for the criteria of convective stability. This is however not the case, because even in a medium of variable ¡ , one follows the internal gradient of a fluid element in which ¡ does not vary (except possibly for ionization, but this is accounted for by a and 8). Thus, the adiabatic gradient is just that given by (3.76). There are however cases, in particular for meridional circulation, where the changes of ¡ have to be accounted for in the variations of entropy.

3.4 Changes of T and q for Non-adiabatic Contraction

In case of non-adiabatic contraction (e.g., as it occurs in the pre-main sequence phase or between the H- and He-burning phases), there is a relation between the internal density and temperature. Let us consider a star which contracts from a radius R to R',

For contraction, CR < 1. We consider a homologous contraction, i.e., we assume a uniform contraction: at each level in the star, one has the same relation r' = CR r with the same value of CR. The mass does not change, so that SMr = 0 and CM = 1. We have seen above that Pc ~ GM2/R4 ((1.20); see also Sect. 24.3), so that

SP 1

which gives in the linear approximation

SP Sr

Similarly for the density, one has Ce = Cm/CR and Sq/q = -3Sr/r. With (3.84) one has a relation between the variations of pressure and density

With the general form of the equation of state (3.60) written here for a constant mean molecular weight p, one gets

This is an important expression relating the changes of temperature and density during a slow (non-adiabatic) contraction. This relation depends on the equation of state through the coefficients a and S (3.60).

### 3.4.1 Major Consequences for Evolution

The above expression (3.86) has remarkable consequences both for star formation and evolution. For a perfect gas a = S = 1, thus one has dln T = 3dln q . (3.87)

The slope is 1/3 in the non-adiabatic case, i.e., the half of the slope in the adiabatic case (19.30, the slope 2/3 applies only in very short phases, i.e., which occur with the dynamical timescale). The smaller slope here is understandable, since half of the energy from contraction is radiated away in the non-adiabatic case (Sect. 1.3.3).

log Tc

Perlect gas ^^

4 !H «He

/

Degenerate

<* /

Fig. 3.2 Schematic representation of the central evolution of a star with indication of the perfect gas and degenerate domains separated by a slope of 2/3 (continuous line). The track (dashed line) with an arrow indicates the path with slope 1/3 followed by a contracting star. The dotted line indicates the T limit where H ignition occurs. The star marks the crossing of the dotted and continuous lines, which determines the brown dwarf upper limit log pc

Fig. 3.2 Schematic representation of the central evolution of a star with indication of the perfect gas and degenerate domains separated by a slope of 2/3 (continuous line). The track (dashed line) with an arrow indicates the path with slope 1/3 followed by a contracting star. The dotted line indicates the T limit where H ignition occurs. The star marks the crossing of the dotted and continuous lines, which determines the brown dwarf upper limit

In the pre-main sequence phase, a slope of ~1/3 applies when the evolution proceeds with the Kelvin-Helmholtz timescale (i.e., from point C in Fig. 19.4 until the star sets on the main sequence). It also approximately applies in the post-MS stages, when contraction supplies energy between the main phases of nuclear burning. Thus, the slope 1/3 defines the main trend of stellar evolution in the diagram log Tc vs. log Qc, which represents the evolution of the central temperature as a function of the central density and is a fundamental diagram of stellar evolution (Fig. 3.2).

In the plane log Tc vs. log Qc, the separation between the domains of the perfect gas and of the degenerate electron gas has a slope 2/3 (Fig. 3.2, see Sect. 7.8).

The evolution of stellar centers proceeds with a flatter slope, implying that during evolution a star unavoidably moves toward the domain of degenerate gas. This is the basic reason why stars evolve to degenerate end points.

When the star enters the domain of complete electron degeneracy, the coefficients 3 5

a ^ 5 and 5 ^ 0 (Sect. 7.8). This implies that at some stage during evolution a becomes smaller than |, while 5 is not yet equal to zero. The ratio (4a - 3)/3 5 may thus become negative, meaning that for a star entering the degenerate domain, contraction does not produce a T increase.

The slope in the plane log Tc vs. log qc also intervenes for the lower mass limit of stars having hydrogen nuclear fusion. If a star enters the degenerate domain with a temperature below the limit for H burning of about 6 x 106 K, its further contraction does not produce an increase of T and H ignition will never occur (Fig. 3.2). The lower mass limit for H burning is about 0.08 M0 at solar metallicity. The objects between about 0.01 and 0.08 M0 resulting from the contraction-fragmentation process and below the limit for H ignition are the brown dwarfs.

### 3.5 Secular Stability of Nuclear Burning

We have seen above (Sect. 3.2.1) the main effects which allow stellar equilibrium to exist. Nuclear burning can exist in different conditions of state, it may occur in shell or in the presence of strong neutrino emissions. We now have the necessary basis to examine the conditions for the long-term (secular) stellar stability [608]. Let us consider a small sphere of radius r and mass Mr around the center and apply to it Eq. (3.40) of energetic equilibrium using (3.64) for the gravitational energy production,

Let us suppose a small temperature perturbation T1 around the equilibrium value T0, so that the temperature can be written as T = T0 + T1. It results similarly in perturbations P = P0 + P1, q = qo + q1 , £ = £o + £1. The equilibrium values imply T0 = 0 and P0 = 0. We assume the perturbations to be fast enough so that the adiabatic approximation is valid, i.e., Lr 1 = 0. From (3.88), we get the relation between the changes of P, T and the perturbation of the nuclear energy production rate

From (3.86) and (3.85), we can relate the change of P and T,

which also yields P1/P0 = Z T1/T0 and P1/P0 = Z T1(/T0. Thus, we can express the perturbations of P in terms of those of T in (3.89) and get

£i = CpTi - — Po Z T = CpTo (1 - Vad Z) T , (3.91)

Q0 to to where (3.76) is used. This relates the changes of input energy and temperature of the central regions. Now, we write the nuclear reaction rate in the simplified form £ = £o q Tv (9.34). We also have

Finally, the relative perturbation of temperature is related to its time variation by

A < 0 implies stability and A > 0 instability since the perturbation will grow. This expression can be applied to several situations.

Nuclear reactions in MS stars: the law of perfect gas dominates with a = 8 = 1, thus Z = 4. One has Vad = 0.4, thus 1 - VadZ = -0.6 and v is positive. Thus, A is negative and nuclear burning in MS stars is stable.

Neutrino losses: there one has negative £ (3.38). Thus, the conclusions are opposite to the above ones. Significant neutrino losses generally occur in degenerate conditions producing T < 0 and the star cools down, which reduces the neutrino emissions and leads to a stable situation. For a perfect gas, a strong (very hypothetical) neutrino emission would be destabilizing favoring central collapse.

Completely degenerate gas: in this case, a = 3/5 and 8 = 0, which gives Z = 0. Thus, one gets A > 0 (the value of Vad for complete non-relativistic degeneracy is also 0.4, cf. Schatzman [517]). Nuclear burning in a degenerate region is highly unstable. The additional energy £1 does not lead to an expansion, but the excess energy is stored thermally which increases T, which increases £1, etc. (Sect. 3.2.1). From (3.86), we get 3 8 d ln Tc = (4 a- 3) d ln q c. Since 8 = 0 and (4a - 3) = 0, one gets dln qc = 0 and the star moves vertically in the log Tc vs. log qc diagram (Fig. 26.9) during the event, until either the degeneracy is lifted or the star is disrupted. In stars with M < 2.2 M0 at standard Z, which ignite helium in degenerate conditions at the top of the red giant branch, this instability gives the "He flash". For stars in the mass range of 8-9 M0, which ignites carbon in degenerate conditions (Sect. 26.4.1), this gives the "C detonation".

### 3.5.1 Shell Source Instability

Very thin nuclear-burning shells may also produce nuclear instabilities. Let us consider shell source burning at a distance r0 from the center in a very thin shell of thickness D C r0, as it occurs for example in the AGB stars. The mass in the shell is m = 4nr^Q d. If the shell extends, due to some perturbation keeping m and r0 « constant, the relative change of density in the shell is

0 0