## Radial Pulsations of Stars

Variable stars have always fascinated mankind, showing non-immutable objects on the celestial sphere thought to be the domain of gods. Indeed, oscillatory phenomena are frequent in natural systems: the level of the sea shows tidal oscillations and waves, the blowing winds produce oscillating noise, the clouds form waves on the side of mountains, some geysers are periodic, etc. Stars do not escape to this rule of Nature.

What makes the beautifully self-controlled stellar nuclear reactors oscillating periodically, like is the case for the famous Cepheids? This question was answered by Eddington in 1926 [168]. If at an appropriate depth, neither too superficial nor too deep, in a stellar envelope, the opacity increases with temperature T (contrarily to the general behavior of the opacity), a small compression induces a higher T , thus a higher opacity. More heat is retained and as it cannot easily go out due to the higher opacity, it produces an expansion which goes beyond the equilibrium point. Then, T becoming lower, the opacity declines, the energy goes out and gravity recalls the system backward, which again produces a compression, etc. In this way the star engine is working cyclically.

There are several kinds of pulsations. They may be radial, meaning that the star inflates and contracts with purely radial motions, keeping the spherical symmetry during a pulsation cycle. Radial pulsations can produce large changes of radius and luminosity. Pulsations may be non-radial, with motions also having a horizontal component. The amplitudes of non-radial oscillations are generally small. Stellar pulsations offer tools for a better understanding of the internal stellar physics. Cepheids are standard candles for the calibration of the distances in the Universe. Some fundamental references about the theory of radial pulsations are Ledoux & Walraven [317], Ledoux [319], Cox [146], Gautschy & Saio [203]; see also Bono et al. [56].

### 15.1 Thermodynamics of the Pulsations

There is a similarity between the thermodynamics of an engine and of a star. In an engine, there are some driving and damping effects which tend respectively to inA. Maeder, Physics, Formation and Evolution of Rotating Stars, Astronomy and 371 Astrophysics Library, DOI 10.1007/978-3-540-76949-1_15, " © Springer-Verlag Berlin Heidelberg 2009

crease or reduce the work production. Some basic conditions have to be fulfilled for an engine or a star to produce work to the outside or to sustain pulsations against dissipation effects. Let us consider a mass element in the stellar gas. The First Principle says that the energy provided to the system goes into the increase of the internal energy SU and into the work ST^ provided to the outside, i.e., SQ^ = SU + ST^. This also applies over a pulsation cycle, j>SQ/ = j SU + y ST/ . (15.1)

At the end of a cycle, the internal energy is back to its initial value. The work toward the exterior is positive (i.e., the pulsation is sustained) if the mass element is absorbing some heat from the pulsation cycle. Let us consider the entropy SS of the mass element, SS = SQ:/T. It is a function of the state of the medium, so that over a cycle, one has

This shows that a medium keeping T constant during a cycle cannot produce any work since f SQ: = 0. Let us suppose that the temperature T(t) at a time t is equal to an average T0 plus a small fluctuation AT(t),

The entropy over a cycle can be written to the first order

which is also the work provided to the outside,

In order a positive work ST^ > 0 to be produced, S Q: and A T must have the same sign, i.e., the heat must be provided when the temperature is high and released when the temperature is low,

SQ: > 0 with AT > 0 and/or SQ: < 0 with AT < 0. (15.7)

It may be sufficient that only the first or the second condition is satisfied. In a pulsating star, some regions may absorb heat and other parts lose it. The condition for pulsation is that the total work over the star is positive,

In stars, two possible mechanisms able to be the driving engine of pulsations have been identified:

- The e mechanism. During compression by a pulsation, T increases so the rate £ of nuclear energy production is also increased, cf. (9.34). This satisfies the first part of condition (15.7). Then the system expands, goes beyond its equilibrium point and comes back under the recall force of gravitation.

- The k mechanism. Normally in a non-pulsating star, a compression produces an increase of q and T so the opacity k decreases. The medium becomes more transparent (e.g., 8.43) and the energy escapes faster. This means SQ/ < 0 with AT > 0, i.e., the opposite of the first condition of (15.7). Thus, there is no pulsation in general.

If, however, k increases with T, as it occurs in a region of partial hydrogen and helium ionization, or as due to the increase of the opacity by the ion (cf. Sect. 8.6.1), a compression makes T higher, thus a higher k. This means that at higher T, more heat is retained, i.e., SQ/ > 0 and the first of conditions (15.7) is satisfied. The second is also satisfied, because the retained energy makes an expansion, T and k decrease, thus the energy escapes more easily from the star, i.e., one has SQ/ < 0 for AT < 0. The star is pulsating.

These mechanisms will appear in the study of the non-adiabatic pulsations. The k mechanism is the dominant instability mechanism of pulsating stars. The Cepheids are the best example with pulsations driven by the zone of partial second ioniza-tion of helium, see also Sect. 15.4. The e mechanism is in general not working in stars, because the damping dominates, however, it is possible that Wolf-Rayet stars are unstable, due to both the e and k mechanisms [145, 342]. Let us note that the damping is not the viscous damping, but the "radiative damping": radiation escapes from the compressed hot layers, this removes energy from the pulsations and tends to damp them.

### 15.2 Linear Analysis of Radial Oscillations

There are several ways to study the radial stellar instabilities. Here, we examine the linear theory with developments which may be used for both numerical models and analytical studies. Oscillations of small amplitudes around an equilibrium situation are considered. A linear theory does not provide the amplitudes which are determined by non-linear effects. However, it allows us to determine the pulsation periods and the various overtones. If the non-adiabatic effects are accounted for, it is also possible to determine whether a star is stable or not, the linear theory showing which are the driving and damping regions.

Generally, the pulsation properties are mainly determined by the envelope, which harbors the non-adiabatic effects and also determines the pulsation period, since the sound speed (C.27) is lower there. We start from the four basic equations (1.12), (1.4), (3.17) and (3.40) of stellar structure expressed in Lagrangian form, dr 1 dP 1 ( GMr d2r dMr 4nr2Q ' dMr 4nr2 \ r2 dt2

The acceleration term is included in the second equation, but not rotation, otherwise spherical symmetry would be broken. We consider the case of radiative transfer. Convection introduces, in some cases, an additional complexity. We comment on it below (Sect. 15.2.1). The heat egrav brought by contraction or removed by expansion is expressed by (3.64). We consider here stellar envelopes and thus ignore the nuclear term e.

Let us consider small Lagrangian perturbations around the equilibrium values, denoted with a subscript "o". For a given mass element, one has r(Mr, t ) = ro(Mr) +Sr(Mr, t)

Now, we introduce these last expressions in the equilibrium Eq. (15.1o) keeping only the first-order terms. In the first equation, the first and the second members become, respectively,

15.2 Linear Analysis of Radial Oscillations 375

1 dr

4rcr2 Q0 (1 + r')2(1 + q') dM, It remains with account of the corresponding equilibrium equation re= P0- (_3 r'_ q') . (15.16)

We want to eliminate the fluctuations of density q ' with the equation of state (3.60), in order to only have the four dependent variables r, P, T and Lr,

where a and S are given by (3.60), the variations of ^ due to ionization are accounted for by a and S . Finally, one has dr' 1

In the second equation for the motions, the first and second members give, respectively, dP- + p,)1-(> + p') & + P&, -.5.20)

2nd order

The last term leads to a second order and we ignore it. We define

2 GMr

Ob is a frequency, it is the fundamental frequency, which is of the order of the inverse of the dynamical timescale (1.28). We are left with dp0 ,dpo dp' 1 2 , „ A 1 d2r'

dM + P'dM + P0^ = - ÏT °° ^ - 4r') - ÏT • (15-23)

The first terms on the left and the right cancel each other and we have with account that (dpo/dMr) = -a-02/(4rcro), dp' 1

In the third equation for heat transfer, we have for the first member,

dMr dMr dMr dMr dMr '

Now, in the second member we express the perturbation of the opacity,

We write the opacity as follows, k = K0PKpTKt , (15.27)

where the logarithmic derivatives kp and kt are exponents slowly variable with temperature and density. We have also, ln k = kp ln P + kt ln T , dln k = KPdlnP + ktdlnT , k' = Kpp' + KTt' . (15.28)

The second member of the equation of transfer gives

Now, the perturbed equation of transfer becomes dto dt' , dto 3ko£o , . . . . „ ,

T--- + to^— +1'^ =--o " , (1 + / + Kpp' + KTt' -4r' - 3t'

dMr dMr dMr 4ac 16n2tor4 v PV

The first terms on the left and the right cancel each other and the perturbed equation of transfer is finally dt ' dMr

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