Nonradial oscillations in classical Cepheids: the problem revisited

0.24 MB·English

by  C. Mulet-Marquis

Astronomy & Astrophysics manuscript no. nonrad10 (cid:13)c ESO 2008
February 4, 2008
Nonradial oscillations in classical Cepheids: the problem revisited
C. Mulet-Marquis1, W. Glatzel2, I. Baraffe1,2, C. Winisdoerffer1
1 C.R.A.L, CNRS, UMR5574 E´cole normale sup´erieure, 46 all´ee d’Italie, 69007 Lyon, France (cedric.mulet-marquis,
ibaraffe, [email protected])
2 Institut fu¨r Astrophysik, Georg-August-Universit¨at G¨ottingen, Friedrich-Hund-Platz 1, D-37077 G¨ottingen
([email protected])
7
0 Received /Accepted
0
2 ABSTRACT

n Context. We analyse the presence of nonradial oscillations in Cepheids, a problem which has not been theoretically
a
revised since the work of Dziembowsky (1977) and Osaki (1977). Our analysis is motivated by a work of Moskalik et
J
al. (2004) which reports the detection of low amplitude periodicities in a few Cepheids of the large Magellanic cloud.

2 These newly discovered periodicities were interpreted as nonradial modes.
1 Aims. Based on linear nonadiabatic stability analysis, our goal is to reanalyse the presence and stability of nonradial
modes, taking into account improvement in the main inputphysics required for the modelling of Cepheids.

1 Methods. We compare the results obtained from two different numerical methods used to solve the set of differential
v equations: a matrix method and theRicatti method.
1 Results.Weshowthelimitationofthematrixmethodtofindloworderp-modes(l<6),becauseoftheirdualcharacter
7 inevolvedstarssuchasCepheids.Forhigherorderp-modes,wefindanexcellentagreement betweenthetwomethods.
3 Conclusions. No nonradial instability is found below l = 5, whereas many unstable nonradial modes exist for higher
1 orders.Wealsofindthatnonradialmodesremainunstable,evenathottereffectivetemperaturesthantheblueedgeof
0 theCepheid instability strip, where no radial pulsations are expected.
7
0 Key words. Nonradial oscillations — stars: Cepheids
/
h
p
1. Introduction linearly unstable radial and nonradial modes with similar
-
o frequencies,orasanonlinearinteractionbetweenalinearly
r Cepheids are well known radial pulsators, oscillating in unstableradialmodeandalinearlystable,lowdegree,low-
t the fundamental mode or the low overtones. Despite a
s order nonradial mode (Van Hoolst & Waelkens 1995; Van
a very abundant literature analysing theoretically and ob-
Hoolstet al.1998).The latter hypothesis wastheoretically
: servationally their radial pulsation properties, only very
v investigated by Van Hoolst & Waelkens (1995), based on
i few studies have been devoted to nonradial oscillations in frequencies and linear growthrates of the nonradialmodes
X thesestars.Dziembowski(1971)wasthefirsttostudynon-
calculated previously by Osaki (1977).
r radial oscillations in Cepheids, based on a quasiadiabatic
a More recently, Moskalik et al. (2004) found low ampli-
linear stability analysis, and found that low-order modes
tudesecondaryperiodicitiesinafewfirstovertoneCepheids
(l ≤ 2) are stable. Following this study, Osaki (1977) and
oftheLargeMagellanicCloud,withfrequenciesclosetothe
Dziembowski (1977) showed that higher degree nonradial
unstable, radial,first overtone.These newly discoveredpe-
modes (l ≥ 4) can become unstable. These analysis high-
riodicities were interpreted as nonradial modes. They pro-
lightedthedifficultytocalculatelow-orderp-modesinsuch
vide, to our knowledge, the first direct evidence for the
evolved stars. These modes show a dual character as they
presence of nonradialmodes in classicalCepheids. But un-
can also propagate in the gravity-wave region, behaving
fortunately,theseobservationshavenotbeenconfirmedyet.
ashigh-ordergravitymodesinthe centralregionsandthus
TheobservationsofMoskaliketal.(2004)arebasedonpho-
havingseveralthousandnodesthere.Sincenoobservational
tometrical data, implying that if nonradial modes are in-
evidence for the presence of nonradial modes in Cepheids
deed detected, they must be low-degree modes. According
was available at the time of these studies, no further the-
to Osaki (1977), however, such modes should be stable.
oretical analysis of the stability of nonradial modes was
MotivatedbytheobservationsofMoskaliketal.(2004)and
performed.
the fact that since the work of Osaki (1977), no updated
Later on, however, due to advances in spectroscopy
stability analysis of nonradial modes has been performed,
and radial velocity measurement techniques, the presence
we reinvestigate this problem, taking into account the im-
of nonradial modes in classical Cepheids was more con-
provementinthemaininputphysics(opacities,equationof
vincingly considered to explain anomalous behaviors such
state) required for the modelling of Cepheids.
as amplitude modulations (Hatzes & Cochran 1995; Van
Hoolst&Waelkens1995;Koen2001;Kovtyukhetal.2003). We haveusedtwodifferentnumericalmethods for solv-
These have been interpreted either as the beating of two ing the set of linear pulsation equations. The first one is
based on a Henyey-type relaxation scheme, which is the
Send offprint requests to: C. Mulet-Marquis most commonly used in the community; the second one
2 Mulet-Marquis et al.: Nonradial oscillations in classical Cepheids
is based on the Riccati method (see Gautschy & Glatzel, δS δLrad
y = andy = ,withξ theradialandξ thehor-
5 6 r h
1990,and references therein). The two methods are briefly Cp Lrad
described in §2 and §3 respectively. Because of the above- izontal component of the displacement respectively, g the
mentioned dual character of low-degree p-modes in highly gravitational acceleration, φ the gravitational potential, p′
evolved stars, the first method requires an extremely large andφ′ theEulerianperturbationsofthepressureandgrav-
numberofgrid-pointsfortheCepheidmodelbecauseofthe itationalpotential,Lrad andδLrad the radiativeluminosity
large number of nodes in the central region. Such method anditsLagrangianperturbation,δStheLagrangianpertur-
thus suffers from resolution problems and a lack of accu- bation of the entropy, and Cp the specific heat at constant
racy. One of our purposes is to analyse and highlight such pressure. The inner boundary conditions are:
limitations and is done in §2. To overcome the problems,
c σ2
severalideashavebeensuggested,suchasasymptoticmeth- 1
y −y =0 (1)
1 2
ods providing analytical solutions (Dziembowski 1977; Lee l
1985). Osaki (1977) found that the stellar envelope could
and
beregardedasauniquepulsatingunit,withanappropriate
ly −y =0, (2)
3 4
boundary condition at the bottom of the envelope which
in principle, should take into account the enery leakage with c = m r3 and m is the mass inside the sphere of
into the core. As a more attractive alternative, the Riccati 1 mrR3 r
radius r. The outer boundary conditions at r=R are:
method as used by Gautschy & Glatzel (1990), avoids the
extra-complexity of asymptotic methods and can take into (l+1)y +y =0, (3)
3 4
accountthe entire stellar structure, providinga correctde-
scriptionoftheenergyleakageintothecorewithoutresort- (2−4∇adV)y1+4∇adV(y2−y3)+4y5−y6 =0, (4)
ing to any approximation.
y =1. (5)
1
In section §4 we present the outcome of our stability
analysis and compare the results obtained with both nu- d lnP
with∇ theadiabaticgradientandV =− .Finally,
merical methods for low-degree p-modes. ad d ln r
thesystemoflinearisedequationsisnumericallysolvedwith
a Henyey-type relaxation method. It requires a guess for
2. Linear stability analysis based on the the eigenfrequency which is derived from the solution of
Henyey-method the adiabatic problem.
2.1. Linear pulsation equations
2.2. Modal classification
The pulsation calculations are performed with a nonradial
codeoriginallydevelopedbyLee(1985)andpreviouslyused The standard modal classification of nonradial modes,
byoneoftheauthorsforanextensivestudyofradialpulsa- based on the determination of the number of nodes Ng in
tions in classicalCepheids (Alibert et al. 1999). We briefly thegravity-waveandNp intheacoustic-wavezonesrespec-
describe the method below. The system of linearisedequa- tively (see e.g Unno et al. 1989) usually fails for evolved
tions describing small amplitude stellar pulsations can be stars,asunderlinedbyDziembowski(1971),becauseofthe
foundinUnnoetal.(1989).TheeigenfunctionsX(r,θ,φ,t), dual character of the modes. Modes with p-mode charac-
solutionsofthissystemofequations,areexpressedinterms ter in the envelope do rapidly oscillate, like g-modes, in
ofsphericalharmonicsasX =x(r)Yl(θ,φ)eiσt,wherelin- the central region. This behavior is illustrated in Fig. 1
m
dicates the degree of the eigenmode and m its azimuthal which displays the radial displacement of a p-mode of de-
number. σ = σ +iσ is the eigenfrequency of the mode, gree l=10 found in a typical Cepheid model. In this case
r i
with P = 2π/σ the pulsation period and σ character- the number of nodes calculatedwith the presentnumerical
r i
ising the stability of the eigenmode. Positive values of σi method is inaccurate and meaningless, with Np=117 and
indicate stable modes. In the following, all frequencies are Ng=143. Another method must be adopted to select and
Gm classify p-modes, which are the most interesting ones since
given in units of σ0 = r R3 , with R the radius and m they can propagate to the surface and are thus potentially
detectable.
themassofthestar.Convectionisfrozenin,assumingthat
Also,ahugenumberofsolutionsisfound,amongwhich
the perturbationoftheconvectiveflux inthe lineariseden-
some are unphysical and result from numerical noise. For
ergyconservationequationisneglected.Thoughcrude,this
the selection of physical p-modes, we have used a criterion
approximation can be justified as we will be mainly inter-
based on the modal kinetic energy, E , which reads for a
ested in Cepheids close to the blue edge of the instabil- c
ity strip, where convection plays a minor role. As shown layer between radii r and r+dr:
in Alibert et al. (1999), frozen-in convection provides a
1
theoretical blue edge in good agreement with observed ra- E = σ2ρr2|δr|2,
c
2
dialfundamentalandfirstovertoneCepheidpulsators.The
boundaryconditionsarethoseimposedbytheregularityof with |δr|2 =|ξ |2+l(l+1)|ξ |2.
r h
the eigenfunctions (see Unno et al. 1989). A normalisation If the kinetic energy of an eigenmode reaches its max-
condition is added : the radial component of the displace- imum value in the acoustic-wave zone (σ > N,L, with N
ment is set to one at the surface of the star. We adopt, and L being the Brunt-Va¨is¨al¨a and the Lamb frequencies
ξr respectively), it is selected as a correct solution, which is
as in Unno et al. (1989), the following variables y = ,
1 r illustratedinFig.2.Eachselectedeigenmode,foragivenl,
1 p′ σ2rξ 1 1dφ′ is then classified as fundamental mode F or overtone (1H,
y = +φ′ = h, y = φ′, y = ,
2 gr (cid:18)ρ (cid:19) g r 3 gr 4 g dr 2H, etc) with increasingvalue of σr. This selectionmethod
Mulet-Marquis et al.: Nonradial oscillations in classical Cepheids 3
Fig.1. Radial displacement as a function of temperature Fig.2. Kinetic energy E (solid line, arbitrary units) for
c
of a p-mode with l=10,frequency σ =3.62 in units of σ , the same eigenmode with l=10 as in Fig. 1. E reaches its
r 0 c
for a typical Cepheid model (m = 5M⊙, logL/L⊙ = 3.1, maximum value in the p-zone, where the eigenfrequency
T =5930K). (dotted line) is greater than the Brunt-Va¨is¨al¨a frequency
eff
N (dashed line) and greater than the Lamb frequency L
(dash-dotted line). All frequencies are in s−1 (right-hand
y-axis). This eigenmode fullfills the selection criterion of a
allows us to find p-modes with high-degree,l ≥6, but fails
p-mode (see text §2.2).
for lower degrees due to the limitation of the numerical
method. Although we find numerous solutions for l < 6,
many of them have very close values of σ but very dif-
r is approximately the same. If the spatial resolution of en-
ferent values of σ . Moreover, they all have pathological
i velope models is too low, this implies adopting arbitrary
eigenfunctions anda selectionofthe correctsolutionbased
inner boundary conditions, leading to strongly uncertain
on the shape of the eigenfunctions or the kinetic energy is
results for the values and the sign of σ . The analysis of
not possible. i
non-radial pulsations in Cepheids based on envelope mod-
Since the problems above mentioned stem from the
els should thus be used with caution. Finally, note that all
rapid oscillations of the eigenfunctions close to the cen-
resultspresentedinthispaperwiththeHenyey-methodare
ter of the star, one could think of using the quite common
obtained with complete stellar models and a typical num-
“envelope model” approach, namely solving the dispersion
ber of grid-points of 5000, in agreement with the required
equation using envelope models rather than complete stel-
resolution estimate based on a WKB approach.
lar models. This approach, however, is not useful in the
presentcontextsinceoneneedstotransfertheunambiguous
inner boundaryconditionstakenatr =0 tothe inner edge 3. The Riccati method
oftheenvelope.Sincethereisnozonewheretheevanescent
oscillationwouldberapidlyswitchedoff,itisquitedifficult Because of the difficulties encountered with the above-
to choosethe correctinnerboundary conditionswith enve- mentioned numerical method, an independent check of the
lope models. In order to get rid of this problem, the inner resultsbothforhighandlowdegreep-modesismandatory.
boundary conditions should be taken deep enough, if not, Thereby it will be of particular interest to check whether
they will affect the solution. We found that using enve- the solutions selected with the previously described ma-
lope models requiresthe same typicalresolutionasthe one trix method are correct and no physical solution has been
requiredwithcompletestellarmodels.Thereisthusnoad- missed.
vantageofusinganenvelopemodelratherthanacomplete A method which does not suffer from resolution prob-
one since the size of the matrix which needs to be inverted lemsandprovideseigenvaluesandeigenfunctionswithpre-
4 Mulet-Marquis et al.: Nonradial oscillations in classical Cepheids
scribed accuracy irrespective of the order of an eigenfunc- Table 1. Linear stability analysis results for a Cepheid
tion is the Riccati method. Its application to stellar sta- model with mass m = 5M⊙, metallicity (in mass fraction)
bility problems is described in Gautschy & Glatzel (1990) Z = 0.01, log L/L⊙ = 3.1, Teff = 5930 K. The real part
for radial perturbations and in Glatzel & Gautschy (1992) σ and imaginary part σ of the eigenfrequencies are nor-
r i
for nonradial perturbations. We shall only briefly summa- malised to σ ∼ 1.42105 cgs. Results obtained with the
0
rize the basic essentials of the approach here and refer the Riccati (Ric.) and the Henyey (Heny.) methods are given.
reader to these publications for details. Positive values of σ indicate stable modes.
i
The rapidly oscillating character in the central regions
of the eigensolutions considered here implies severe reso- l mode σr σi σr σi Period
lution problems due to the limited size of matrices, if the (Ric.) (Ric.) (Heny.) (Heny.) (days)
eigenvalueproblemissolvedonthe basisofamatrixeigen- 0 F 2.99 -3.8 10−4 3.02 -4.4 10−4 3.43
value problem. This basic difficulty, which always occurs 1H 4.23 -6.3 10−3 4.25 -7.5 10−3 2.43
for high order eigenfunctions can in principle be overcome 2H 5.40 -2.7 10−3 5.40 -4.2 10−3 1.92
by shooting methods, which do not rely on the inversion
of large matrices. Moreover, using a shooting method, the 6 F 3.13 2.0 10−2 3.20 1.8 10−2 3.24
localstepsizecanbeadjustedtomatchprescribedaccuracy 1H 4.57 -3.9 10−3 4.58 -5.0 10−4 2.26
requirements. However, for higher order boundary value 2H 5.98 -1.1 10−3 5.96 7.5 10−3 1.74
problems(thenonadiabaticnonradialstabilityanalysisim-
pliesthesolutionofasixthordercomplexdifferentialequa- 10 F 3.60 -2.5 10−3 3.61 -1.9 10−3 2.87
tion) shooting methods tend to suffer from the parasitic 1H 5.23 -9.8 10−3 5.22 -5.2 10−3 1.98
growth problem and are numerically unstable. Moreover, 2H 6.88 2.4 10−2 6.83 3.8 10−2 1.52
withincreasingorderthenumberofparameterswhichneed
tobeiteratedintheshootingprocessbecomesprohibitively
large. Therefore shooting methods are usually not consid-
served Cepheid, LMC-SC2-208897, is a first overtone pul-
eredforthesolutionofhighorderboundaryvalueproblems.
sator with a period of P =2.42 days and absolute magni-
One way to overcome the parasitic growth problem 1
tude M ∼ −3.2. Adopting the same input physics and
and to reduce the parameters to be iterated to the com- V
evolutionarycode asinAlibertetal.(1999),wefoundthat
plex eigenvalue while still preserving the basic advantages
of a shooting method consists of transforming the linear a model with mass m = 5M⊙, metallicity (in mass frac-
boundary eigenvalue problem to a nonlinear one imply- tion)Z =0.01,log L/L⊙ =3.1,Teff =5930Kandcloseto
the end of central He burning (central mass fraction of He
ing a Riccati type equation. For the shooting process we
Y ∼ 4.10−2) provides a good fit for the observed magni-
thus obtainunambiguous initial conditions atboth bound- c
tude and period of LMC-SC2-208897. The calculated first
aries and the only parameter which needs to be iterated
overtoneperiodisunstablewithaperiodP =2.43daysand
is the eigenfrequency. In addition, integration of the non- 1
the absolutemagnitude isM =-3,inexcellentagreement
linear Riccati equation as an initial value problem is nu- V
with the observed values (see Alibert et al. 1999 for the
merically stable. As a result, the integration of the Riccati
determination of absolute magnitudes).
equation provides a complex (determinant) function of the
Excellent agreement is found between the two meth-
complexfrequency.Zerosofthisdeterminantcorrespondto
ods for the periods and growth rates of radial modes (see
the eigenvalues sought.
Table 1). For nonradial modes with degree l ≥ 6, Fig. 3
ThelatterpropertyoftheRiccatimethodoffersanaddi-
comparesthevaluesofσ andσ obtainedwithbothmeth-
tionaladvantage:eigensolutionscanbedeterminedwithout r i
ods, up to l=20. The values of σ agree within less than
resorting to any initial guess.Using matrix methods, these r
2%. Differences are found for the values of σ for over-
are usually obtained on the basis of approximations (e.g., i
tones, whereas the agreement is excellent for the funda-
the solutionofthe adiabaticproblem)which may leadto a
mental mode (see Fig. 3). Values of σ and σ for a few
missing of unexpected solutions. Using the Riccati proce- r i
nonradial modes are also given in Table 1.
dure, always the full set of equations without any approx-
For the above-mentioned effective temperature and lu-
imation is integrated to provide the determinant function.
minosity, a variation of the mixing length, between one
Bytabulatingthedeterminantasafunctionofthecomplex
pressure scale height H and 2×H or of the metallicity
frequency,a coarsedeterminationofthe eigensolutionscan P P
between Z=0.01 and Z=0.02 have only a slight influence:
be done whichsubsequently maybe improvedby iteration.
σ changes by less than 2.5% and σ by less than 8 % (if
Due to the existence of the determinant function there is r i
|σ |>5.10−3). Larger differences can appear, however, for
no problem of spurious eigenvalues when using the Riccati i ∼
method. very small values of σi (|σi|∼<5.10−3), when using the ma-
trix method described in §2.
We have also compared the results obtained with both
4. Results
methods for different effective temperatures, adopting the
samemassandluminosityastheabove-mentionedCepheid
4.1. Comparison of the results between the two methods
model. Fig. 4 shows the results for the fundamental mode,
We have compared results obtained with the two methods 1H and 2H with degree l=10. Here again, excellent agree-
described respectively in §2 and §3 for a specific Cepheid mentisfoundforthevaluesofσ ,butdifferencesappearfor
r
model. Since our work is motivated by the observations of thevaluesofσ fortheovertones.Ingeneral,bothmethods
i
Moskalik et al. (2004), we selected a model which could agreeonthe signofσ ,providingthe sameresultsconcern-
i
describe the properties of one of the observed LMC tar- ing the stability properties of modes, except for very small
gets where secondary periodicities were detected. The ob- values of σ (|σ |<5.10−3).
i i ∼
Mulet-Marquis et al.: Nonradial oscillations in classical Cepheids 5
Fig.3. Comparison, for our selected Cepheid model (m = Fig.4. Variation of σ and σ , in units of σ , as a func-
r i 0
5M⊙, logL/L⊙ = 3.1, Teff = 5930K) between the results tion of effective temperature for models with m = 5M⊙,
obtainedwiththeHenyeymethod(see§2,dottedlines)and logL/L⊙ =3.1andZ=0.01.Differentp-modeswithdegree
the Riccati method (see §3, solid lines). The upper panel l= 10 are displayed, as indicated on the figure. The solid
displays the real part of eigenfrequencies of p-modes as a line correspondstothe Riccatimethodandthe dashedline
function of the degree l and the lower panel the imaginary to the Henyey method.
part,inunitsofσ .Eachcurveislabelled,ontherighthand
0
side,accordingtothemodalclassification.Notethatforthe
fundamental mode, the curves corresponding to the values are ambiguousthere. In fact, both for radialand nonradial
of σ obtained with both methods are indistinguishable. modeswefoundasensitivedependenceonboundarycondi-
i
tionsevenwithinthephysicallyadmissibleones.Theresults
showninthepaper,however,arebasedonthemostconser-
Both for low and high degree modes we have tested vativeconditionwithrespecttostability.I.e.,wehavecho-
the dependence on the outer boundary conditions of the sen the conditionwhich providesthe least unstable modes.
results of the stability analysis. This is necessary, since It corresponds to the requirement of vanishing Lagrangian
the outer boundary of the stellar model (the photosphere) pressureandtemperatureperturbation.Allotherboundary
does not correspond to the physical boundary of the star. conditions, in particular at the inner boundary, are unam-
Thereforethethermalandmechanicalboundaryconditions biguous.
6 Mulet-Marquis et al.: Nonradial oscillations in classical Cepheids
Incontrasttothethermalandmechanicalouterbound-

aryconditionstheinfluenceofmetallicityonthestabilityof
12
all the models investigated is rather weak. Relative differ-
ences between the results for Z=0.01 and Z=0.02 amount
11
to 10 per cent at maximum for σ . 5H
i
10
4.2. Results for low-degree p-modes 4H
9
Modes of degree below l ≈6 are extremely difficult to find
with matrix methods. Indeed, only high overtones can be 8
3H
found (e.g. 5H for l = 2). Some results obtained in this
regime are based on the Riccati method and shown in Fig. σr 7
5 . Both the frequencies of p- and g-modes decrease with
6 2H
the harmonic degree. However, while p-modes reach finite
frequency values for l→0,the frequencies ofg-modes van-
5
ishinthis limit. The stellarmodels consideredhereexhibit 1H
a propagation region for gravity waves close to the center
4
of the star which for high values of the harmonic degree
is well detached and shielded from the acoustic propaga- 3 F
tion region by an efficient evanescent barrier. It allows for
gravitymodes with frequencies in the range of p-modes up 2
to quite high order. Since the decrease of frequencies with 1 2 3 4 5 6 7 8
l is stronger for the g-modes, this leads to multiple cross- l
ings and resonances between p- and g-modes. Due to the
efficient evanescent barrier for high values of l the interac- 0.16
tion of p- and g-modes is weak at resonances above l ≈ 6
andleavesthem unaffected.For l <6,however,the barrier 0.14 5H
becomes weaker and the resonances at the crossing of g-
and p-modes imply significant interaction and unfold into
0.12
avoidedcrossings.Bumpsbothinthereal(lesspronounced)
4H
and imaginary parts of the eigenfrequencies as a function
0.1
ofl foundinFig.5arepartsoftheseavoidedcrossings.For
illustration a part of the run of a g-mode with l is shown
0.08
there as a dashed line and l is taken as a real parame-
ter. The latter allows one to follow modes continuously (of σi
0.06
course,onlyintegervaluesarephysicallymeaningful).Asa
3H
consequenceoftheresonances,modeswithl <6,inpartic-
ulartheso-calledp-modes,shouldnotbeclassifiedaseither 0.04
p- or g-modes. Physically they rather exhibit properties of
both types of modes. 0.02
F
0
5. Discussion and conclusion 2H
1H
The stellar model whose parameters are thought to match -0.02
1 2 3 4 5 6 7 8
the properties of the Cepheid observed by Moskalik et al.
l
(2004) is found unstable both with respect to radial and
nonradial perturbations. Following modes up to l = 500, Fig.5.Variationofσ andσ ,inunitsofσ ,forthemodes
r i 0
we find the following unstable modes: the fundamental p- indicatedasafunctionoftheharmonicdegreeforlowvalues
mode for l =0 and between 9≤l ≤368, the first overtone of l and the Cepheid model having m=5M⊙, logL/L⊙ =
for l = 0 and between 6 ≤ l ≤ 13 and the second overtone 3.1,T =5930KandZ=0.01.Allresultsarebasedonthe
eff
forl =0,5 and6.No nonradialinstabilitywasfound below Riccati method. The bumps for low values of l are due to
l=5:SpecificstudiesbasedontheRicattimethodforl=1 aninteractionofp-andg-modesthroughavoidedcrossings.
and2havenotrevealedanyinstabilityinthep-moderange. The dashed line indicates the run of a selected g-mode.
Moreover,fromextrapolatingthecurvesinFig.5wedonot
expectinstabilityforl =3and4.Wecanthusconcludethat
if low degree p-modes with l ≤2 were necessary to explain same is true for nonradial modes with l ≤ 10. However,
the observations of Moskalik et al. (2004), they cannot be fundamental p-modes with l > 10 may still be unstable at
attributed to a linear instability of these modes. Rather temperaturesabove6200K,wherenoradialmodeisfound
other effects, such as, e.g., nonlinear mode coupling, have to be unstable. For example, the fundamental p-mode for
to be invoked for an explanation of their excitation. l = 15 is unstable up to 6290 K, the corresponding mode
With respect to the dependence on effective tempera- for l =60 is unstable up to 6640 K. Although observation-
tureoftheinstabilityofradialandnonradialmodeswefind ally difficult to detect, we thus conclude that it might be
the unstable radial modes (F, 1H, 2H) to stabilize above worthwhile to searchfor nonradialpulsations with l≥5 in
6200K, for our test case luminosity log L/L⊙ = 3.1. The Cepheids.Particularlypromisingappeartobeobservations
Mulet-Marquis et al.: Nonradial oscillations in classical Cepheids 7
of objects with T > 6200K, where no radial pulsations
eff
are expected and – according to theory – only nonradial
pulsations should prevail. Observations of nonradial pul-
sations in Cepheids could then provide an interesting and
unique tool to sound their inner structure.
Acknowledgements. I.B. thanks warmly the Academy of Sciences of
Go¨ttingenfortheGaussprofessorshipandforsupportingthisproject
and the Institute of Astronomy in Go¨ttingen for hospitality during
completion of this work. The calculations were performed using the
computer facilitiesofthe Centre deCalcul Recherche et Technologie
(CCRT)ofCEA.
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