Cpto e o Po s

Equations (15.19), (15.24), (15.30) and (15.32) are the four linear homogeneous differential equations for the four unknowns r', p', t' ,i' as functions of Mr. The coefficients are those from the equilibrium models.

The quantities r',... or 8r(Mr, t),... in (15.11) are developed as, sr(Mr, t) = sr(Mr) eiat = sr(Mr) e-ajtei(Rt .

(0 is in general complex, (Or and (Oj represent the real and imaginary parts of the frequency. In a non-adiabatic study, if Oj is positive, the amplitude is decreasing, if negative it is increasing. One also has -jj = i o. When the above four differential equations are completed by appropriate boundary conditions (Sect. 15.2.2), the solutions r', p', t'as functions of Mr can be determined. However, solutions of the equations with their boundary conditions only exist for some specific values al, with n = 0,1, 2,3 The (On are ordered by increasing values of n. The value Oo, the shortest frequency defines the fundamental mode, while o1, o2, ... are the first, the second overtone, etc. To these various modes, the corresponding solutions are r0 (Mr), r'1 (Mr), r2 (Mr),... with similar notations for the other variables p' (Mr), t'(Mr) and £'(Mr).

Figure 15.1 shows the behavior of SL/L = (! in a Cepheid model of 7 M0 in relation with the opacity and the cumulated work provided be the pulsation (i.e., the sum of the work, with its sign from the interior to the surface). We see that in the region where the opacity grows with depth, i.e., kt > 0 and kp > 0 the work is

Zone

Fig. 15.1 The relative excess of luminosity 5L/L, the cumulative work and the opacity in a Cepheid model of 7 [email protected] as a function of the zoning of the outer layers. The stellar surface is on the left, the white area is the atmosphere with optical depth < 2/3, the gray area is the optically thick envelope. The cumulative work is the sum of the work, positive or negative, from the inner zone (600) up to the surface. As the integrated work up to the surface is positive, the star is unstable. Adapted from Schaller [512]

Zone

Fig. 15.1 The relative excess of luminosity 5L/L, the cumulative work and the opacity in a Cepheid model of 7 [email protected] as a function of the zoning of the outer layers. The stellar surface is on the left, the white area is the atmosphere with optical depth < 2/3, the gray area is the optically thick envelope. The cumulative work is the sum of the work, positive or negative, from the inner zone (600) up to the surface. As the integrated work up to the surface is positive, the star is unstable. Adapted from Schaller [512]

positive. There, some luminosity is subtracted from the emergent flux to feed the pulsation. The cumulated work at the surface (on the left of the figure) is positive, i.e., the integral work from the interior to the surface is positive and thus the star is unstable. If there would only be the k mechanism at work, we would expect the work to grow, starting from the interior, only in regions above the maximum of k, i.e., where k decreases outward. However, there is also the effect of the partial ionization on the adiabatic exponents r to be accounted for (cf. Fig. 7.4). There is a minimum of r3 deep in the zone 550 and this has a destabilizing effect which makes the driving zone to start a bit deeper than as given by the location of the maximum of the opacity. Figure 15.4 below further shows in a simplified model which effects contribute to stability and instability.

The above equations can be solved numerically by finite differences and by application of the Henyey method. It is preferable [96] to first discretize the equations in finite differences and then to linearize them. These equations are used for a simple analytical model below (Sect. 15.3).

15.2.1 Convection

Convection in cores and envelopes introduces an additional difficulty. The first and simplest approach is to assume that convection does not vary with pulsation. In fact, this means that at all times the convective flux would be the same as in the equilibrium model. A second approach is to develop to the first order the expressions for the convective flux, assuming that convection adapts instantaneously to the new state created by pulsation.

The first approach is acceptable if the turnover time of convective motions (5.49) is much longer than the pulsation period P, while the second is acceptable in the opposite case,

WnoTCr > P constant convection, WnoTCr C P convection adapts itself.

For the solar envelope as well as for the convective cores of massive stars, one rather has the second case. In the first case, there is almost no effect of convection. In the second case, the instability is moderately favored [59]. However, often the reality corresponds to an intermediate situation, with turnover times of the same order as the pulsation periods (e.g., for red giants) and thus appropriate developments have to be made [196, 222].

15.2.2 Boundary Conditions and Eigenvalue Problem

Four boundary conditions are needed and various degrees of refinement are possible, often leading to an increased complexity. The solutions are evidently not insensitive to the boundary conditions and great care has to be taken. We summarize a basic choice of boundary conditions. At the outer boundary, a first condition can be obtained by assuming dp'

at the base of the atmosphere, which means that no force is applied at the surface. However, if there is mass loss and running wave in the atmosphere, this condition does not apply. Running waves produce a considerable damping of the pulsation.

A second condition at the surface is obtained by assuming that the photosphere (with given mass and pressure at its basis) is floating above the inner pulsating layers. A relation connecting (!, r', t' at the surface is easily obtained by linearizing the temperature-optical depth T(t) relation (24.19; see for example[34]). In the center, the conditions are r' = 0 and £' = 0 , (15.36)

which imply no displacement and light fluctuations (some of the early models had only one condition in the deep interior, e.g., a condition of adiabacy like, C (dt'/dt) _ (dp'/dt) = 0; if so, the problem is not completely defined and a continuous range of solutions for co is possible). Through the boundary conditions, the proper frequencies are a property of the whole star. Nevertheless, the outer layers play a dominant role since they harbor the non-adiabatic effects and also determine the pulsation period, since the sound speed is smaller there.

Now, we have fourEqs. (15.19), (15.24), (15.30) and (15.32) with four boundary equations, the problem is in principle determined for any value of The point is that the equations are linear and homogeneous, so that the scaling of the solutions is not yet fixed. A scaling must be adopted and one usually takes r' = Sr/r = 1 at the stellar surface as a normalization. This makes one more, i.e., five constraints, to be satisfied by the system of four equations. The consequence is that the system only accepts solutions for some specific values of the frequency, called eigenfunctions ( with n = 0,1,2,3... as mentioned above. Practically, if the integration of the system of equations is started from a couple of boundary conditions, it will match the other couple of boundary conditions only for some values (n of the frequency. For each (n, corresponding to the fundamental or to the overtones, there are specific solutions, r'n(r), p'n(Mr), t'n(Mr) and £'n(Mr), called eigenfunctions. The fundamental oscillation with a frequency ( 0 has no node between the center and the surface. The first overtone has one node between the center and the surface, the second overtone has two, etc.

Physically, the eigenvalue problem appears because the star may have stationary oscillations only for some suitable oscillation frequencies which permit an integer number of half wavelength to be fitted within the stellar resonant cavity. The real part ( of the frequency determines the period Pn = 2k/ \ (\ and the imaginary part ( gives the damping or the driving of the pulsation. Mathematically, the problem is a so-called Sturm-Liouville problem. In practice the choice of a good numerical method for finding the eigenfunctions is essential [96, 512].

Figure 15.2 illustrates the eigenfunctions r'0, r'1 and r2 for a ¡5 Cephei variable star, an about 11 M0 star in the MS stage. The pulsation amplitudes are large at the surface (where they are normalized to unity) and very small in the deep layers, particularly in the interior with respect to the nodes of the overtones. The square of the amplitudes at some point in the star vary like the inverse of the density, consistently with an even distribution of the pulsation energy in the star. The higher overtones with more internal modes have shorter oscillation periods, since the wavelengths are shorter. The values of the periods and period ratios of the modes are discussed in term of the pulsation constant Qn associated to each mode (Sect. 15.5.1).

Since the basic equations (15.10) contain all the physics of the problem, it is also possible to study the pulsations by integrating numerically these equations which include the hydrodynamic term d2r/dt2. The procedure is started from an equilibrium solution, which is perturbed in some way, and the evolution of the system is followed with very short time steps, i.e, small fractions of the expected pulsation period. Similar boundary conditions as those mentioned above can be adopted. After some transient harmonics have dissipated, the system either converges toward the initial equilibrium and is thus stable or stationary oscillations progressively emerge with some limiting amplitudes since the non-linear effects are accounted for by the full set of equations.

Fig. 15.2 The perturbations of radius for the fundamental mode, the first and second overtones as functions of radius in a Vir, all [email protected] Main Sequence star. Each mode is normalized to 1.0 at the surface. Adapted from A.N.Cox [144]

Fig. 15.2 The perturbations of radius for the fundamental mode, the first and second overtones as functions of radius in a Vir, all [email protected] Main Sequence star. Each mode is normalized to 1.0 at the surface. Adapted from A.N.Cox [144]

15.3 Baker's One-Zone Analytical Model

The previous developments form a basis for numerical models, which allow us to study the stability of a particular stellar model. It is, however, useful to have an analytical model for understanding the essential physical effects of pulsation. The one-zone model by Baker [32] is a clever approach, which was the starting point of further developments [146].

The coefficients of the four Eqs. (15.19), (15.24), (15.30) and (15.32) are time independent, thus time and space variables can be separated. The one-zone model keeps the time dependence, but in order to get rid of the space dependence of the coefficients and variables, it considers only one single thin spherical layer of mass m located somewhere in the star (cf. Fig. 15.3). The equilibrium state of the thin shell is defined by r0, p0, t0, l0. The fluctuations r', p', t' are considered as constant through the shell, d - d' -0. (15.37)

dMr dMr dMr

The same equation for t would mean adiabacy and thus no driving or damping of the pulsation, thus an essential property would be lost. Therefore, we need to calculate the gain or loss of luminosity through the shell. We assume that there is a luminosity fluctuation ¿'L at the lower limit of the shell and tu at the upper limit. The average and gradient of t are thus f' + f' d f' f' — f' f = + h and d- = . (15.38)

2 dMr m

Fig. 15.3 Schematic representation of the one-zone model

We then assume that there is no fluctuation at the bottom of the shell. Thus, the variations of the emerging flux result only from effects in the shell, the interactions of the shell with the surroundings are ignored. We have i'L = o and thus i'

We keep the essential properties of the interaction of the gas layer with the radiation passing through it. The gas modulates with a certain frequency the constant flux entering at the bottom (i'L = 0) and produces a variable flux at the top. This happens because radiative energy may be removed from the flux and fed into mechanical pulsation energy, alternatively radiative energy may be radiated away at the expense of the mechanical oscillation energy. Thus, we have either a driving of the pulsation or a radiative damping. The oscillation frequency is a characteristic of the physical conditions of the system. With conditions (15.37) and (15.39), the four equations (15.19), (15.24), (15.30) and (15.32) become

dr = °o (4 r + p'), i' - 4r' + Kpp' + (kt - 4)t' = 0 , d t' d p' ' Cd - d = -K°0i with

The various quantities are taken at the level considered. With (3.76) to express P0 S, we see that K scales as the ratio of the luminosity L0 by the product of the heat content of the shell times the frequency a0. Thus K « i0/(a0Eh) is the ratio of the luminosity to the heat content Eh of the shell times the pulsation period (cf. also 15.76). As some of this content may be radiated away during one pulsation cycle, K plays an essential role for the non-adiabatic effects. If K = 0, we have C (dt'/dt) _ (dp'/dt) = 0 which implies adiabacy according to (3.63). If so, the mechanical and thermal parts of the pulsations would essentially be uncoupled.

Now we combine the four equations into a single one, without making yet the hypothesis of adiabacy. First, i' is eliminated between the third and the fourth equation,

C _ dp- = _4Ko0 r + Ko0kpp' + Ko0(kT _ 4)t'. (15.45) at at

The first equation gives

which allows one to eliminate t',

= _4K00r + Ko0Kpp' + +KO0(KT _4)^ + KO0(KT _4)ap-. (15.47)

Now, p' is eliminated with the second equation,

+KO0KP—^—t + KO0(KT _4W _Ko0KP4r' O(2 at2 o a 1 d2r' a

This is a third-order equation of the form, d3r- d2r- dr'

with

From the definition of C (15.33), from (3.76) and (7.57), we have r2 1 C

One also has from (7.58)

and with (7.66), we can eliminate r3 — 1 and obtain, r = and r — 1 = —L— . (15.53)

This means that the coefficients A, B, D can be expressed in terms of r, a, 5 and

We now suppose that the relative fluctuations r' of the radius have the following form r'(Mr) = %{Mr) est . (15.54)

This expression introduced in the differential equation (15.49) leads to s3 + KaoAs2 + Cq Bs + Ka03D = 0 . (15.55)

This cubic equation defines the complex frequencies s of the pulsations of the model. This is the eigenvalue equation for the one-zone model. We now examine the solutions and constraints resulting from this relation, both for the adiabatic and non-adiabatic pulsations.

15.3.1 Adiabatic Pulsations

Let us consider the case where the pulsations do not exchange heat with their surroundings. As a consequence, one cannot say whether the pulsations are sustained or damped, i.e., whether the star is stable or not. Nevertheless, a pulsation period can be determined, which is always very close to the period obtained in the non-adiabatic case (see Sect. 15.4.2). From (3.64), dq = 0 implies Cp(dT/dt) — (8/g)(dP/dt) = 0 and thus with (15.33), this gives dt' dp'

dt dt

Following the steps from (15.45) (15.46) (15.47) (15.48) and (15.49), we have

The adiabatic solution for s is

with B given by (15.5o) with relations (15.51) (15.52) and (15.53), we get

There are two possibilities for sad:

- If — > 4/3: the solutions for r' (15.54) are sinusoidal, without damping since this is the adiabatic case.

- If — < 4/3: r' grows exponentially and we have a dynamical instability, consistently with the result (1.69) from the Virial theorem.

The above expression (15.6o) of the pulsation frequency also shows that the ion-ization in the envelope, which decreases —, contributes to increase the pulsation period. The above expression leads to the well-known period-luminosity-color relation for Cepheids discussed in Sect. 15.5.1.

15.4 Non-adiabatic Effects in Pulsations

Non-adiabatic effects produce the driving and the damping of pulsations. Their study allows us to understand the physical effects playing a role in stellar pulsations. Large uncertainties still concern the interactions of pulsations with processes such as convection and stellar winds.

15.4.1 The k and y Mechanisms

Equation (15.55) with K = 0 determines the stability. For the star to be stable, the three roots s1, s2, s3 of the equation must have negative real parts, so that the amplitude of the oscillations decrease in time according to (15.54). As shown below, the conditions for negative roots are [32]

and sad becomes

Go2 B > o , K Go D > o , K Go (AB-D) > o •

This results from the properties of the solutions of a polynomial equation of the form s3 + a1 s2 + a2 s + a3 = 0 . (15.64)

The three solutions s1; s2, s3 of a cubic equation follow the properties [69], s1 + s2 + s3 = -a1 , (15.65)

- Condition (15.61) results from the fact that the sum of the products in (15.67) must be positive, thus a2 has to be positive. This gives condition (15.61), which implies r1 > 4/3, which is the condition for dynamic stability, as mentioned just above. If not satisfied, the star implodes or explodes on the dynamical timescale.

- Condition (15.62) results from the fact that the product (15.66) must be negative, if the three roots are negative. Thus a3 is positive, which gives condition (15.62). It expresses the secular stability (i.e., over a time of the order of the Kelvin-Helmholtz timescale) of the system discussed in Sect. 15.4.3.

- Condition (15.63) expresses the pulsational stability. The reason and meaning of this condition are derived later (see 15.75). Let us just verify here that it is self-consistent. This condition implies that the difference a1 a2 - a3 > 0. Since a2 and a3 are both positive, the previous inequality necessarily implies that a1 is positive, which is consistent with condition (15.65) if the three roots are negative. One also verifies that a1 a2 - a3 > 0 gives

- s2s2 - s1s3 - s1s2s3 - s1s2 - s1s2s3 - s2s3 - s1s3 - s2s2 > 0 (15.68) a relation which is evidently satisfied if all roots are negative.

Let us examine the pulsation stability. One can express (15.63) more explicitly with (15.50). After simplification of identical terms, one has

AB -D = -3CkpS - 4saC - 3KTS + 12s + 4s = 3Cs kt -4(aC - s) + 12

C 3C

With (15.51) to (15.53), this gives finally the stability condition

The instabilities, when present, are vibrational instabilities, i.e., the star pulsates periodically. The examination of this relation shows the factors which favor the stability and instabilities:

0 0

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