arXiv:astro-ph/0203477v1 27 Mar 2002
ASP Conference Series, Vol. **VOLUME**, **PUBLICATION YEAR**
Problems with the Pulsation Mode Selection Mechanism
in the Lower Instability Strip (Observations and Theory)
M. Breger1, A. A. Pamyatnykh1,2,3
1Institut f¨ ur Astronomie, Universit¨ at Wien, T¨ urkenschanzstr. 17,
A–1180 Wien, Austria;email: email@example.com
2Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw,
Poland; email: firstname.lastname@example.org
3Institute of Astronomy, Russian Academy of Sciences,
Pyatnitskaya 48, 109017 Moscow, Russia
predicted and observed pulsation modes for δ Scuti stars. The selection
of nonradial modes trapped in the outer envelope is considered on the
basis of kinetic energy arguments. The trapped ℓ = 1 modes for the star
4 CVn are in good, but not perfect agreement with the observations. The
trapping of the ℓ = 2 modes is weaker, so that this simple rule of mode
selection may apply to ℓ = 1, and possibly not to ℓ = 2 modes.
We examine the severe disagreement between the number of
Delta Scuti star models predict pulsational instability in many radial and non-
radial modes. The observed number of low-degree modes is much lower than
the predicted number. The problem of mode selection is most severe for post-
main-sequence δ Scuti stars, which comprise about 40 percent of the observed
δ Scuti stars. The theoretical frequency spectrum of unstable modes is very
dense. Most modes are of mixed character: they behave like p-modes in the
envelope and like g-modes in the interior. For example, for a model of 4 CVn we
predict 554 unstable modes of ℓ = 0 to 2: 6 for ℓ = 0, 168 for ℓ = 1, and 380 for
ℓ = 2. However, only 18 (and an additional 16 combination frequencies) were
observed (Breger et al., 1999). This is demonstrated in Fig.1. The problem for
other δ Scuti stars, such as BI CMi, is similar.
Consequently, either the theoretical predictions or the observational tech-
niques are imperfect. On the observational side, only modes with photometric
amplitudes in excess of 0.5 mmag are observed. However, this does not solve the
dilemma, since the problem could then be rephrased to why only some favored
modes are excited to observable amplitudes, which usually are much higher than
the observational limit.
A number of different scenarios ranging from chaotic mode selection to
energy transfer between the modes through nonlinear mode coupling can be in-
voked to explain the disagreement. Some of these hypotheses are very promising
and are amenable to observational tests:
Breger & Pamyatnykh
2. Scenario: random mode selection
Here the selection of the star’s observationally visible modes from the multitudes
of possible modes is random. A consequence of this scenario is that we will not
be able to make use of nonradial-mode frequencies for seismic probing.
This hypotheses leads to several observational predictions: (i) there are no
regular frequency patterns of the photometrically visisble modes and, (ii) there
is no similarity of the frequencies of the visible modes from star to star with
similar masses, temperature and ages.
3.Scenario: temporary growth of randomly selected modes to ob-
servable amplitudes by mode interaction
In this scenario, most or all of the predicted modes are indeed unstable, but with
amplitudes too small to observe photometrically. However, due to mode inter-
action, power is transferred between the modes. A few modes can temporarily
grow to high amplitudes and are observed. After several years, which represent
the typical growth cycle of δ Scuti stars, these modes will decay again.
The hypothesis is supported by the fact that δ Scuti stars show time-variable
amplitudes as well as linear mode combinations of the modes with high ampli-
tudes. The amplitude variability may be so large that over a decade the star
may look like a completely different star (e.g. compare the power spectra of
4 CVn during the 1966-1970, 1974-1978, 1983-1984 and 1996-1997 time peri-
ods). However, a detailed investigation of the data shows that modes do not
disappear to be replaced by others, but are still present at millimag amplitudes.
This argues against the hypothesis, at least on time scales of decades.
4. Scenario: trapped envelope modes
In this scenario, the mode selection mechanism is related to modes trapped in
the envelope (Dziembowski & Kr´ olikowska, 1990). It may be easier to excite
these modes with lower kinetic energy to a given amplitude on the surface. Such
a mode selection rule would be very simple and the computations rely only on
linear theory. However, the strong proof must come from the nonlinear theory.
Fig.2 illustrates the complexity of the oscillation frequency spectrum for
a model of 4 CVn, which oscillates with at least 18 frequencies in the range
4.7 - 9.7 c/d. The parameters of the model are within the range allowed by
observational data: 2.4 M⊙, logL/L⊙= 1.76,
Teff= 6800K, logg = 3.32, Vrot= 82 km/s. The chemical composition X =
0.70, Z = 0.02 was assumed. The ordinate gives the oscillation kinetic energy,
which is evaluated assuming the same radial displacement at the surface for each
mode. The high density of the oscillation spectrum of nonradial modes is caused
by very large values of the Brunt-V¨ ais¨ al¨ a frequency in the deep interior. This
diagram is similar to one given by Dziembowski (1997), with the observational
mode identification and selected rotational splitting added.
The most trapped modes are analogs of pure acoustic modes. It can be seen
that the trapping is much more effective for dipole (ℓ = 1) than for quadrupole
modes (ℓ = 2). The trapping effect for ℓ = 2 modes is especially weak at low
Pulsation Mode Selection Mechanism
frequencies and may not apply here. Therefore, the selection rule might work
for only ℓ = 1 modes (see also Dziembowski & Kr´ olikowska, 1990).
The observed spacing between identified ℓ = 1 modes is about 1.2 c/d,
corresponding to the theoretical spacing between trapped ℓ = 1 modes. Also,
the rotational splitting for the trapped ℓ = 1 and also for ℓ = 2 modes is similar
to the observed splitting for identified modes. Note, however, that the conditions
for the excitation of modes of different azimuthal order m may differ. Therefore,
only some components of multiplets may be excited to observed amplitudes.
For the model considered the theoretical frequencies of the best trapped
modes of ℓ = 1 fit the identified observed frequencies quite well. However, we
still consider this model to be only an illustrative one. To fit theoretical and
observed frequencies quantitatively, we must also take the effect of the rotational
coupling of modes of different ℓ values into account. Due to this effect, close
modes with the same m and whose ℓ degrees differ by 2, are pushed away in
frequency. This was demonstrated for a model of XX Pyx (Pamyatnykh et al.
In evolved δ Scuti models the trapping of nonradial modes in the envelope
can be more effective than in RR Lyrae models. For example, for the most-
trapped modes of ℓ = 1 in 4 CVn up to 67 percent of kinetic energy can be
contributed by the envelope – in comparison with less than 20 percent in the
RR Lyrae variables (see Dziembowski & Cassisi, 1999).
It was shown that there is a high probability of a resonant excitation of
nonradial modes in radially pulsating RR Lyrae star models (see Nowakowski,
these proceedings and references therein). Very recently, nonradial modes were
detected in these stars (see Kov´ acs, these proceedings and references therein).
The theoretical frequency spectra of evolved δ Scuti stars are similar to those of
RR Lyrae stars, so resonances may be important for these stars too.
Breger, M., Handler, G., Garrido, R., et al. 1999, A&A, 349, 225
Dziembowski, W.A. 1997, in Sounding solar and stellar interiors, eds.J. Provost
& F.-X. Schmieder, (Kluwer, Dordrecht), 317
Dziembowski, W.A. & Cassisi, S. 1999, Acta Astron., 49, 371
Dziembowski, W. & Kr´ olikowska, M. 1990, Acta Astron., 40, 19
Pamyatnykh, A.A., Dziembowski, W.A., Handler, G., & Pikall, H. 1998, A&A,
Breger & Pamyatnykh
4 CVn: Frequency spectrum and probable mode identification
Amplitude in millimag
Frequency in c/d
(Main frequency range only, no combination frequencies)
Observed pulsation modes and probable mode identifica-
tions for the evolved δ Scuti star 4 CVn.
Pulsation Mode Selection Mechanism Download full-text
of the evolved δ Scuti star 4 CVn. The trapped modes are the modes
with low kinetic energy values. Only the m = 0 modes are shown.
Empty symbols denote stable modes. For the most trapped modes all
components of the multiplets are shown separately and compared to
the observed frequencies with probable mode identifications.
Kinetic energy values associated with the unstable modes