derstand why radial monomode pulsation is preferred by stars in the upper

Cepheid instability strip.

2.6 Amplitude limitation by resonances

That resonances may play a role in stellar pulsation has been realized well

before numerical modeling became possible. This is what Rosseland (1949)

wrote about about early attempts to explain the shapes of Cepheid light

curves: The conclusion seems unavoidable that some particular property of

the star plays an active role in shaping the curves. It may be something like

a resonance between the fundamental and higher mode suggested by Woltjer

(1935, 1937). Modern investigations have con¬rmed this.

2.6.1 The 2:1 resonance

The Hertzsprung sequence of Cepheid light curves has been successfully

explained in terms of the frequency distance from the 2:1 resonance centre

between the fundamental mode (n = 1) and the second overtone (n = 3)

(Simon & Schmidt, 1976, Buchler et al., 1990).

Also Woltjer was the ¬rst to point out that the 2:1 resonance causes

amplitude limitation. Since it is a lower-order e¬ect in terms of amplitude,

one might expect that it should be more e¬cient in amplitude limitation

than saturation. However, it does not seem to be the case in Cepheids. The

amplitudes at 10-day period, which is the resonance centre, are not markedly

lowered. The point is that this resonance may be a sole amplitude-limiting

e¬ect only if the damping of modes of higher radial order is fast enough. A

stable double-mode solution exists only if 2γ1 + γ3 < 0.

A critical role for the 2:1 resonance in the amplitude limitation was found

in limiting the growth of the -mechanism-driven instability of high-mass

stars by Papaloizou (1973). He showed that the resonance between the

Dziembowski

34

unstable fundamental mode and the stable ¬rst overtone is very e¬ective

preventing a catastrophic mass loss, which was suggested by earlier investi-

gators.

2.6.2 Parametric resonance and dwarf and giant dichotomy

Three-mode coupling caused by the parametric resonance is another lowest-

order nonlinear e¬ect leading to amplitude limitation. In this case the e¬ect

is due to dissipation of energy by a pair of linearly stable (daughter) modes

for whose sum are close to frequency of an unstable (parent) mode. Denoting

with subscripts a and b the daughter modes and with c the parent mode we

have

ωc = ωb + ωa + ∆ω ,

with |∆ω| ωc and

γc > 0 , γa < 0 , γb < 0 .

An exponential growth of modes a and b occurs if the amplitude of mode

c exceeds the critical value, which (e.g. Vandakurov, 1981) is given by

2

γa γb ∆ω

Ac,crit = 1+ , (2.5)

Cabc γd

where γd = γa + γb . The coupling coe¬cient Cabc is a volume integral with

integrand containing products of eigenfunctions of the three involved modes.

The general expression is complicated (Dziembowski, 1982) but it is easy

to show that Cabc = 0 only if the azimuthal orders satisfy the condition

mc = ma + mb and the di¬erence between the two highest is not larger

than the lowest one. For instance, if the parent mode is radial then the

daughter modes must have ma = ’mb and a = b .

Freedom in choosing a and ma allows ¬ne frequency tuning. The fre-

quency distance, ∆ ,ng , decreases approximately as ’1 (see equations 2.1

and 2.2), hence considering daughter modes with ’ ∞ we may approach

∆ω = 0. This favours high- mode excitation. The opposite e¬ect is that

of damping. If the quasi-adiabatic approximation applies then we have ap-

proximately (e.g. Van Hoolst et al., 1998)

¯

N2 2

γ≈ , (2.6)

„g ω 2

where „g is the thermal time scale of the G cavity. In main-sequence stars

the instability ¬rst appears at certain intermediate though still rather high

On the diversity of stellar pulsations 35

values implying that the daughter modes are most likely undetectable.

Important observable consequences occur for the parent mode, whose am-

plitude may be reduced to the level not much exceeding Ac,crit and may be

modulated.

The character of the terminal pulsation state resulting from the interac-

tion between the parent and the daughter modes depends on the mismatch,

∆ω, and the driving (damping ) rates γ. Stable stationary solutions with

the parent-mode amplitude given by the right-hand side of equation (2.5),

but with γd replaced by γs = γc + γd exist in wide range of parameters

(see Wersinger et al., 1980, Dziembowski, 1982). Outside that range, in

particular for a close resonance, only time-dependent amplitude limitation

is possible. The solution may take a form of a single- or multi-periodic limit

cycle and, going through a series of period doubling, become chaotic. Still

equation (2.5) may be used for a crude estimate of the mean amplitude.

If γs < 0 then amplitude limitation in any form by the sole e¬ect of the

parametric resonance is not possible.

My ¬rst application of the theory of parametric resonance was to estimate

the amplitude of an = 1 g-mode in the sun. Dilke & Gough (1972) showed

that the mode may be driven by the mechanism and speculated that it

might reach high amplitude, high enough to mix the solar interior. This was

an ingenious idea invented to solve the neutrino de¬cit problem. The results

of my calculations (Dziembowski, 1983) were unfortunately discouraging.

The parametric instability was found to set in at very low amplitudes, far

lower than needed for mixing. The next application was to explain the low

pulsation amplitudes of δ Scuti stars (Dziembowski & Kr´likowska, 1985).

o

We found that the amplitudes of unstable modes were limited by the three-

mode interaction to the level of 1 - 10 mmag, which was in a rough agreement

with observations. The values are well below the ones needed to saturate

the instability.

It seemed that we were on the road toward explaining the systematic

di¬erence between giant and dwarf pulsators. In Cepheids and RR Lyrae

stars the parametric resonance does not prevent high pulsation amplitudes

for two reasons. Damping rates of daughter modes are much higher than in

¯

δ Scuti stars due to the much higher N (see Figure 2.5) and „g is shorter.

The second reason is a weaker coupling (smaller Cabc ) between the parent

radial and the potential daughter modes. The latter are trapped in the deep

interior, where the former ones have very low amplitudes. The truth is,

however, that not much happened after those works. The di¬culty is that

most likely much more than just one pair of daughter modes is excited at

the onset of the parent-mode instability. It is easy to show that constant-

Dziembowski

36

amplitude solutions do not exist if there are more than two pairs. Beyond

that, the problem is di¬cult and to my best knowledge, was never solved.

2.6.3 Higher-order parametric resonance and the Blazkho e¬ect

Resonant coupling between radial and nonradial modes of similar frequen-

cies is a third-order e¬ect in the amplitude expansion. Nonetheless, it may

have a greater impact on RR Lyrae pulsation than the lower-order resonant

coupling due to the properties of the mode-trapping pattern. We have seen

in Figure 2.4 that nonradial modes with frequencies close to those of radial

modes are also close to the maxima of the driving rates. This means rel-

atively large amplitudes in the acoustic cavity, hence stronger coupling to

radial modes and lower damping rates. Both e¬ects promote parametric in-

stability. Figure 2.4 also shows that the trapping favours excitation of = 1

modes.

Van Hoolst et al. (1998) derived the following expression for the amplitude

of the parent mode amplitude at the onset of the instability:

∆ω 2 + γ 2,N

A2 > .

0

C00

It is similar to equation (2.5), but specialized to a radial mode and it takes

into account saturation of driving by the radial mode, which makes γ ,N

negative. Our survey (Dziembowski & Cassisi, 1999) of realistic models of

RR Lyrae stars has revealed that excitation of radial modes is quite likely.

The probability ranges from 0.3 to 0.9. A study of the nonlinear development

(Nowakowski & Dziembowski, 2001) has shown that in this case there is