in the RR Lyrae model than in the δ Scuti model. Spectra of nonradial

modes are actually denser than shown in Figures 2.3 and 2.4, as each of the

modes is split into 2 + 1 components by rotation. The much greater mode

density in the RR Lyrae model is a consequence of the much larger values

Dziembowski

30

Fig. 2.5. The ratio of the Brunt-V¨is¨l¨ to the fundamental radial mode frequency

a aa

in the models used in Figures. 2.3, 2.4 and in a model ZAMS star. In the ZAMS

model the central hydrogen abundance Xc = 0.7. The maximum value of the ratio

in the RR Lyrae model is about 300 and it is reached at r/R = 0.007.

of the Brunt-V¨is¨l¨ frequency N in the interior of this star. In Figure 2.5

a aa

we see the behaviour of N (r) in the two models discussed and in a ZAMS

star of the same mass as our selected δ Scuti star. The e¬ect of evolution

is a growth of N in the interior and a development of a gravity wave (G)

cavity there.

The radial order ng associated with the G cavity is approximately given

by

2

( + 1) N dr

ng ≈ ’1 (2.1)

π ω r

(e.g. Van Hoolst et al., 1998), where ω = 2πν is the angular frequency of

the mode. The integral should be taken over the G cavity. The frequency

distance between consecutive modes of the same degree is thus estimated as

ω

∆ ,ng ≈ . (2.2)

ng

In the RR Lyrae star model, at = 1 and a frequency corresponding to the

fundamental radial mode, we get ng = 180.

Already in the evolved main-sequence star, some additional nonradial

modes result from the growth of the G cavity around the shrinking convective

core. Also in this model we see structures in the γ(ν) dependence re¬‚ecting

mode trapping e¬ect. Local minima correspond to modes partially trapped

in the G cavity. The e¬ect is much more dramatic in the RR Lyrae star.

For all nonradial modes most of the oscillation energy is con¬ned to the

On the diversity of stellar pulsations 31

G cavity. The relative contribution from the acoustic cavity is only about

10% for the = 1 modes corresponding to local maxima. At this degree

still most of contribution to damping and driving arises in the outer layers

so that the di¬erence between in driving rates between the = 0 and = 1

modes is mainly due to the di¬erence in inertia. At = 2 and 3, damping

in the G cavity results in mode stability in certain frequency ranges.

Even for the most strongly trapped nonradial modes, the driving rates,

γ, are signi¬cantly lower than those for the closest radial modes, which

is not true in the δ Scuti star. We may be tempted to take this fact as

the explanation why RR Lyrae and other evolved stars exhibit preference

for radial pulsation. However, more detailed comparison with observations

warns us against such inference. The modes detected in FG Virginis at

ν ≈ 10d’1 have their driving rates lower by six orders of magnitude than

those at ν ≈ 30d’1 . The dependence of amplitude on frequency shown in

Figure 2.2 bears no resemblance to the γ(ν) dependence shown in Figure 2.3.

Clearly, we cannot rely on the driving rates for predicting amplitudes of

modes surviving in the nonlinear development.

2.5 Saturation of the linear instability

Christy (1964) was the ¬rst to construct fully nonlinear models of Cepheids

and RR Lyrae stars. His models converged to a periodic constant ampli-

tude pulsation state, which “ according to his interpretation “ was reached

through a saturation, that is, through modi¬cation induced by pulsation in

the mean structure, leading to zeroing driving rate. Depending on mean

value of L and Te¬ , the terminal state was either fundamental or ¬rst-

overtone pulsation. Later on Stobie (1969) found also second-overtone pul-

sation in his Cepheid models. It took 30 years to ¬nd such a form of pul-

sation in real objects. Stellingwerf (1975) with his novel method was able

to determine exclusive domains in the H-R diagram of ¬rst overtone and

fundamental mode pulsation and an intermediate domain, which he named

the EO (either-or), where both modes were possible depending which one

was excited ¬rst.

The origin of double-mode pulsation was not understood for a long time

and even now the problem is not fully clari¬ed. It has been approached

in a number of works by means of numerical solution of the full nonlinear

problem and with the amplitude equation formalism. Nonlinear saturation

shows up at the cubic order in pulsation amplitudes. With our cubic order

formalism we (Dziembowski & Kov´cs, 1984) derived a simple criterion for

a

double-mode pulsation. Here I outline our analysis.

Dziembowski

32

The nonlinear driving rates were written in the form

±jk A2

γj,N = γj 1+ . (2.3)

k

k

Only the case of two linearly unstable modes was considered. It was assumed

that all saturation coe¬cients ±jk are negative, which is necessary for sat-

uration and con¬rmed by subsequent numerical calculations. The terminal

amplitudes are determined by the set of two equations (2.3) with γj,N = 0.

For monomode solutions, which with our assumptions always exist, we have

1

A2 = ’ . (2.4)

j

±jj

It is stable if sk ≡ ±kj /±jj ’ 1 > 0, where k = j, which means that the

mode is more e¬ectively saturating instability of the competing mode than

that of its own. If s1 > 0 and s2 > 0 then we are in the EO domain. The

double-mode solution

1 s1 1 s2

A2 = ’ , A2 = ’

±11 s1 + s2 ’ s1 s2 ±22 s1 + s2 ’ s1 s2

1 2

exists if s1 s2 > 0. However, it is stable only if s1 < 0 and s2 < 0. Thus,

monomode and double-mode pulsations are mutually exclusive.

The fact that that double-mode pulsations are so rare may be interpreted

in two ways. Either the range of parameters leading to s1 < 0 and s2 < 0

is very narrow or we have s1 > 0 and s2 > 0 but the amplitude of mode 1

cannot reach its saturation value given in equation (2.4) due to an accidental

resonance with a damped mode. We preferred the second way and suggested

that there is a 2:1 resonance with a higher-order radial mode. The problem

with this idea, which was realized later, was lack of required resonance in

realistic models of double-mode pulsators.

Successful numerical simulations of double-mode pulsation in Cepheids

(Koll´th et al., 1998) and RR Lyrae stars (Feuchtinger, 1998) were obtained

a

only after e¬ects of convection were included. The resonance played no

role in this models. In a recent paper Koll´th et al. (2002) interpreted

a

these results with the amplitude equations. They found that the condition

s1 < 0 and s2 < 0 was satis¬ed in their double-mode pulsators. This

was never the case in purely radiative models. Unfortunately, they did not

identify the speci¬c e¬ect of convection responsible for the enhancement of

self-saturation, causing stabilization of the double-mode pulsation.

An unpleasant aspect of this solution is the fact that it rests on a crude

description of convection in which there are four adjustable parameters. Fur-

thermore, there is a problem of nonradial modes, whose presence has been

On the diversity of stellar pulsations 33

ignored in all numerical models so far. We will see later that a resonant cou-

pling with those modes may play a role for the properties of radial pulsation.

There is also a question regarding the role of strongly unstable high-degree

modes. Many such modes, having driving rates similar to the radial modes,

exist in all models of RR Lyrae star and Cepheids (e.g. Van Hoolst et al.,

1998). If the saturation is the dominant amplitude-limiting e¬ect, then we

should expect that often one of such modes wins the competition. The

resulting pulsation would be undetectable by means of photometry. Obser-

vations do not indicate that it may be the case. The RR Lyrae strip seems