A weakly non-adiabatic one-zone model of stellar pulsations: application to Mira stars

8 Pages·1.1 MB·English

by  Andreea Munteanu

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2)
A weakly non-adiabatic one-zone model of stellar
pulsations: application to Mira stars
Andreea Munteanu1, Enrique Garc´ıa–Berro1,2, & Jordi Jos´e2,3
1 Departament de F´ısica Aplicada, Universitat Polit`ecnica de Catalunya, Jordi Girona Salgado s/n, M`odul B–5, Campus Nord,
3 08034 Barcelona, Spain
0 2 Institut d’Estudis Espacials de Catalunya, Edifici Nexus, Gran Capit`a 2-4, 08034 Barcelona, Spain
0 3 Departament de F´ısica i Enginyeria Nuclear, Universitat Polit`ecnica de Catalunya, Av. V´ıctorBalaguer s/n,
2 08800 Vilanova i la Geltru´ (Barcelona), Spain

n
a
J Accepted 24January2003/Received25November2002

7
2
ABSTRACT

1 There is growing observationalevidence that the irregularchanges in the light curves
v ofcertainvariablestarsmightbeduetodeterministicchaos.Supportingtheseconclu-
8
sions,severalsimplemodelsofnon-linearoscillatorshavebeenshowntobecapableof
3
reproducingthe observedcomplex behaviour.Inthis work,we introducea non-linear,
5
non-adiabatic one-zone model intended to reveal the factors leading to irregular lu-
1
minosity variations in some pulsating stars. We have studied and characterized the
0
3 dynamical behaviour of the oscillator as the input parameters are varied. The para-
0 metric study implied values corresponding to stellar models in the family of Long
/ Period Variables and in particular of Mira-type stars. We draw the attention on cer-
h
tainsolutionsthatreproducewithreasonableaccuracytheobservedbehaviourofsome
p
peculiar Mira variables.
-
o
Key words: stars: AGB and post-AGB — stars: oscillations — stars: variables —
r
t stars: Miras
s
a
:
v
i
X
1 INTRODUCTION equationofstate,opacities,andmanymore—andrequires
r theimplementation ofahydrodynamicalcodecoupledwith
a The region in the Hertzsprung–Russell diagram which pro-
adetailedtreatmentofthetransferofradiation.Hence,this
vides the majority and, probably, the most interesting
approachdefinitelyprovidesthemostdetailedandaccurate
classes of pulsating stars is the Asymptotic Giant Branch
descriptionofpulsations.However,withinthisframeworkit
(AGB). As the stars of low and intermediate mass (from
is sometimes difficult to interpret the origin and shapes of
say ∼1 to11M⊙) evolvealong theAGBphase, they expe- the resulting light curves as the stellar parameters are var-
rience recurrent thermal instabilities and substantial mass
ied. On its hand, the second approach is complementary to
loss. Depending on the phase in the thermal pulse cycle,
theformerinthesensethatitgivesaqualitativeframework
theymayspendintermittenttimeintervalsaspulsatingstars
inwhichthegeneralfeaturesofthepulsationsareeasilyun-
(Groenewegen & Jong 1994). As a consequence, they eject
derstood and, consequently, allows to develop intuitive ex-
freshly synthesized material into the interstellar medium.
planationsintermsofafewverybasicandrelativelysimple
Thus, they also play a crucial role for our understanding of
physical processes (Buchler1993).
the chemical evolution of galaxies. Last but not least, the
corresponding stellar models are basic tools to study how The hydrodynamic simulations of pulsating stars suc-
planetarynebulaeandtheircentralobjectsform—seeHab- ceeded in reproducing the phenomenon of period-doubling
bing (1996) and Willson (2000) for reviews on the subject. (Buchler & K´ovacs 1987; K´ovacs & Buchler 1988; Aikawa
A substantial number of pulsating stars in the Galaxy 1990) and tangent bifurcations (Buchler, Goupil & K´ovacs
show irregular behaviour and have the characteristics of 1987; Aikawa 1987) found in the light curves of some spe-
AGB stars (Wood 2003). This makes them an interesting cific stars by changing the surface temperature or surface
research target for theoretical modeling. The work done so gravity.Inparticular,it hasbeenfoundthatthereisstrong
far on non-linearstellar pulsations can begrouped intotwo evidenceofunderlyinglow-dimensionalchaosfromthenon-
different categories: the full numerical hydrodynamical ap- linearanalysisofthelightcurvesoftheirregularstarsRSct,
proachandthemorestraightforwardapproachofthetheory AC Her, and SX Her (Buchler, Koll´ath & Cadmus 2001).
of dynamical systems. The former is directly based on an This paved the road to construct simple models within the
extensiveknowledgeof thephysicalprocesses inthestar— frameworkofthetheoryofdynamicalsystemswhichturned
2 A. Munteanu, E. Garc´ıa–Berro & J. Jos´e

(a) (b) (c) (d)
0.2 0.2 0.2 0.2
0.1 0.1 0.1 0.1
v v v v
0.0 0.0 0.0 0.0
-0.1 -0.1 -0.1 -0.1
-0.2 -0.2 -0.2 -0.2
3.0 3.0 3.0 3.0
1.2 1.2 1.2 1.2
1.0 1 .0 1.0 1 .0 1.0 1 .0 1.0 1 .0
0.8 0.8 0.8 0.8
p r p r p r p r
Figure 1. Period-doubling route to chaos represented in the space (r,v,p). In all panels, α = 0.1, ω = 20.1, and a = 20 have been
adopted. (a)ξ=0.08:Period-1;(b)ξ=0.09:Period-2;(c)ξ=0.108:Period:4;(d)ξ=0.12:Chaos.
out to be capable of reproducing the complex behaviour. the particular case in which the luminosity at the base of
Withinthisapproachperhapsthesimplestmodelsofstellar the mantle equals the (constant) equilibrium luminosity of
pulsations are the so-called one-zone models. In this fam- the star, L . For convenience, we use as variables the non-
⋆
ily of models the star is treated as a rigid core surrounded dimensional radius r ≡R/R , pressure p≡P/P and time
⋆ ⋆
by a homogeneous gas shell. The one-zone models are not t≡ωmτ,wherethestellarradiusandpressurewerenormal-
intendedtobeasubstituteforfullyhydrodynamicalandso- ized totheir equilibrium values, while
phisticated models. Instead, they merely seek a global and
qualitativeexplanationofwhatprocessmightbeattheroot GM
of the chaotic behaviour. In the present paper, we propose ωm ≡r R⋆3 (1)
aone-zonemodelofnon-linearnon-adiabaticstellar oscilla-
is thecharacteristic frequency of thestar.
tions. We present the parametric study of the system and
Considering that the additional perturbative accelera-
drawtheattentiononaparticularsetofnumericalsolutions
tionisproportionaltothedrivingaccelerationwithatrans-
which may have implications in the study of the variability
mission coefficient Q as in Icke et al. (1992) and following
of Long Period Variables (LPVs).
the reasoning of Saitou, Takeuti & Tanaka (1989) for the
energyequation,weobtainthefollowingequationofmotion
2 THE ONE-ZONE MODEL
dr
= v
Baker et al. (1966) were the first to introduce the one-zone dt
model as tool to study the nonlinear behaviour of stellar dv = pr2−r−2−
pulsations.Buchler&Regev(1982)andAuvergne&Baglin dt
(1985) later studied one-zone models undertheassumption −Qαω4/3sin(ωr−ωt−αω1/3sinωt) (2)
that the nonlinearity of the adiabatic coefficient Γ1 is the dp
main trigger for nonlinear pulsations, whereas Tanaka & = −3Γ1r−1vp−ξr−3(rβpδ−1),
dt
Takeuti (1988) pointed out that dynamical stability might
where
benecessary forrealistic modelsofpulsatingstars, whichis
theapproach followed here. β=a(r3p−1.2)+21.6 , δ=3.6r3p(r3p−0.2) (3)
The one-zone model approach is especially suited for
theAGBstarsasthedensitydifferencebetweenthecentral are the coefficients introduced in Saitou et al. (1989) as a
core and the outer layers is so large that these two regions saturationeffectoftheκ-mechanismwithabeingthecontrol
can be considered as effectively decoupled. Following Icke parameter.Thecharacteristicfrequencyoftheperturbation,
et al. (1992) and Munteanu et al. (2002), we consider the ω results from the use of the dimensionless time unit and
stellar pulsation tobesimulated byavariableinnerbound- fromtheassumptionthatR0 encompassesalmosttheentire
ary located well beneath the photosphere and moving with stellar mass. It is defined as
constant frequency (piston approximation). This sinusoidal
driving consists of pressure waves originating in the inte- ω≡ ωc = R0 −3/2. (4)
rior and propagating through a transition zone until they ωm (cid:16)R⋆(cid:17)
dissipateintheouterlayers.Wedenominatethedrivingos- In its simplest form, the study of stellar pulsations
cillator“theinterior”andthedrivenoscillator“themantle”. can be considered as a thermo-mechanical, coupled oscilla-
Whiletheradiusoftheformerregion,R0,canbeestimated tor problem (Gautschy & Glatzel 1990). The coupling con-
as the radius of the hydrogen burning shell, the radius of stant is given by the ratio of the dynamical to the ther-
the latter, R is determined by the dissipation of the afore- maltimescaleintheouterlayersofthestar.Wheneverthe
mentioned pressure waves. thermal time scale (τth ∼ 4πr2ρ∆rcVT/Lr) of an outer re-
Thevariationoftheinteriorradius,Rc,aroundtheequi- gion of the star of radial extension ∆r happens to become
libriumvalue,R0,isgivenbyRc =R0+αR0 sin ωcτ,where comparabletothesound-travelingtimethroughthatregion
αandωc are,respectively,thefractional amplitudeandthe (τdyn ∼∆r/cs), non-adiabatic effects are relevant. This im-
frequency of the driving. As in Stellingwerf (1972), we use plies that there is an efficient exchange of mechanical and
theequationofmotionandtheenergyequationwithouten- thermalenergiesinthatregion.Theratioofthetimescales
ergy sources and in absence of any driving force in orderto is much smaller than one throughout most of the envelope
determine the final equation of motion describing the dy- and is close to unity only in the outermost regions. Signifi-
namics of the mantle. In the present work, we will study cantnon-adiabaticeffectsarerelevantforheliumstars,very
Weakly non-adiabatic model 3
(a) (b) (c) (d)

0.3 0.3 0.3 0.3
0.1 0.1 0.1 0.1
v v v v
-0.1 -0.1 -0.1 -0.1
-0.3 -0.3 -0.3 -0.3
4 2 1.0 1.2 4 2 1.0 1.2 4 2 1.0 1.2 4 2 1.0 1.2
p 0.8 r p 0.8 r p 0.8 r p 0.8 r
7 7 7 7
5 5 5 5
L* L* L* L*
L/ 3 L/ 3 L/ 3 L/ 3
1 1 1 1
20 25 30 20 25 30 20 25 30 20 25 30
t (yr) t (yr) t (yr) t (yr)
Figure 2.Thebirthofaknot-likestructureforincreasingvaluesofQ.Toppanels:Stroboscopicsamplingoftheorbit(r,v,p)withthe
characteristic frequency ω: (a) Q=1.02, (b) Q=1.04, (c) Q=1.08and (d) Q=1.16. Bottom panels: The lightcurves corresponding
tothecasesshownintheupperpanels.
massive AGB stars, some post-AGB stars and in the ion- x˙ =F(x),with x=(x1,x2,...,xn)andF =(F1,F2,...,Fn)
ization zones of hot stars (Stellingwerf 1986; Gautschy & is critically determined by its fixed points x0 given by
Saio1995).Inthecontextofoursystem,theparameterξ in F(x0)=0. The associated eigenvalues of the Jacobian ma-
Eq.(2) is a measure of the non-adiabaticity and is given by trix (Jx ) = (∂F /∂x )x determine the nature of these
0 ij i j 0
theratioofthedynamicaltimescaletothethermaltimescale fixed points. For instance, for a fixed point to be stable,
of the shell it is required that all eigenvalues have negative real parts.
L In our case, the system has three fixed points: a trivial
ξ = ωmcV⋆mT. (5) oanned, t(wr0o,vo0t,hpe0r)fi=xed(1p,0o,in1t)s,,w(rit+h,vm+a,ipn+ly) ≈adia(0b.a6t8i,c0,o4r.i7g5in)
The system of Eq.(2) constitutes the final set of rela- and (r−,v−,p−) ≈ (8.8558,0,000020), e0ntirely due to non-
tionsfortheunknownsr,vandp.Theparametersthatmust 0 0 0
adiabatic effects (that is, they exist only for ξ 6= 0). For
be specified are Q, α, ω, Γ1, a and ξ. We consider an ideal initialconditionsclosetothetrivialfixedpoint,theperiod-
gas with an adiabatic coefficient Γ1 = 5/3. The parameter doublingroutetochaoswasobtainedbySaitouetal.(1989)
fixingthe evolutionary status of the star is ω as it provides
by decreasing the surface temperature, more precisely by
a measure of the contrast between the core and the enve-
varying the control parameter a in the range a ∈ [14,20],
lope. For AGB stars, R0/R⋆ is of the order of 15%. This whileξ waskeptconstant.Noinvestigationofanequivalent
led Icke et al. (1992) to consider a value of ω =20.1 which
effectproducedbyvaryingξ wascarriedon.Concerningthe
is thevalueadopted here.We haveperformed aparametric
dynamics near the trivial fixed point, our analysis of the
study involving the parameters of the perturbation, Q and
dependence of the Jacobian matrix on a and ξ reveals that
α. The results of the numerical integrations show that the fora<36,eithertheincreaseofξ orthedecreaseofaleads
∼
pair (Q,α) — and not the individual specific values of Q
to the same effect on the eigenvalues consisting in chaotic
and α — determines the dynamics of the system. In other
behaviourthrough theincrease of pulsational instability.In
words, given a fixed value of α, a certain range of values of
order to prove the importance of ξ in the dynamics of the
Q can be found for which the same peculiar dynamics dis-
systemandtocompletethestudyofSaitouetal.(1989),in
cussed below develops. Therefore, we present here only the
Figure 1 we present a period-doubling route to chaos with
caseofsmallinternalperturbation(α≈3−4%)andampli-
the increase of the parameter ξ as it appears in the space
fied transmission through the envelope (Q>1). It is worth
(r,v,p).Inordertoeasetheexploration ofthedynamicde-
noticing that for higher (lower) values of the parameter α,
tails, we use mainly the Poincar´e map. As we deal with a
the same behaviour is encountered if lower (higher) values
periodically driven system, the Poincar´e map reduces to a
of the coupling coefficient Q are chosen.
stroboscopic sampling of the r,v, and p valuesat multiples
of T =2π/ω.
3 NUMERICAL RESULTS
3.1 Zero perturbation 3.2 Non-zero perturbation
The system studied by Saitou et al. (1989) can be ob- We present now the properties of the non-linear oscillator
tained from thesystem given in (2) byconsidering thecase that are due to the presence of the time-dependent per-
of zero perturbation (Q = 0). In spite of the inexistence turbation. We fix a = 20 as corresponding to the regular
of internal perturbation, this system is the prototype of pulsation found by Saitou et al. (1989) and, moreover, we
non-linearself-excited oscillators duetheκ-mechanism pro- choose ξ=0.06 in orderto compare with theirwork. Addi-
vided that ξ 6= 0. The behaviour of any dynamical system tionallywefixω=20.1andα≃0.037. Ourstudyconsiders
4 A. Munteanu, E. Garc´ıa–Berro & J. Jos´e

4 0.3
0.1
L/L*2 v -0.1
-0.3
0 4 2 1.01.2
10 30 t (yr) 50 70 p 0 0.8 r

4 0.3
0.1
L/L*2 v -0.1
-0.3
0 4 2 1.01.2
10 30 t (yr) 50 70 p 0 0.8 r

0.3
6
0.1
L/L*4 v -0.1
2
-0.3
0 4 2 1.01.2
10 30 t (yr) 50 70 p 0 0.8 r

Figure3.Comparisonbetweenregularandirregulardynamics.Thelightcurvesandthecorrespondingstroboscopicmapsforthecases
α=0.037,ω=20.1and(toppanels)Q=0.8,(middlepanels)Q=1.0and(bottom panels)Q=1.048.
the coupling coefficient Q as the primary control parame- For a slightly lower value of the parameter Q (top panels),
ter for the strength of the perturbation, while ω and α are a completely irregular light curve results and the strobo-
keptconstantatthevaluespreviouslymentioned.Inthetop scopicmap clearly showsit.Finally,in thelowerpanels, we
panels of Figure 2 we present the stroboscopic map of the illustrate thedynamics of the system when Q takesa value
system for increasing values of Q. We notice the successive corresponding to the accumulation of a sequence of period-
creation of loops, finally leading to aknot-likestructurefor doublingbifurcations.
strong perturbation (Q=1.16). Forthesakeof comparison Another important feature of the dynamics is the fact
withrealastronomicaldata,wealsoplotinthebottompan- that the time interval τB between major bursts increases
els of Figure 2 the light curves for the corresponding cases. with the strength of the perturbation, as it can be seen in
The luminosity L is obtained from the condition of radia- the bottom panels of Figure 2. This is illustrated in Fig-
tivetransfer together with theperfect gas law (Stellingwerf ure 4a, where the time interval between the successive ma-
1972): jor bursts is shown as a function of the coupling coefficient
Q. As a visual guide we also plot the shape of the strobo-
L
=rβpδ, (6) scopicmapatthefixedvaluesofQwhereanadditionalinner
L
⋆
loop appears. As it can be seen in this panel and in panel
wherethenormalizingconstantL istheequilibriumstellar
⋆ 4b, τB significantly increases for Q ∈ [1.32,1.39] whereas
luminosity. The temporal scale is expressed in years and
for Q > 1.39 (Figure 4c) the separation between succes-
for thistask wehaveusedthestellar parameters associated
sive bursts increases drammatically. Also, in Figure 4b we
to a typical Mira of 1M⊙ as they result from the work of show that the above mentioned increase of τB can also be
Vassiliadis & Wood (1993).
obtained by decreasing the non-adiabaticity of the system
ThemostnoticeablefeatureofthelightcurvesofFigure
(thatis,decreasingξ).Neverthelessthemainparameterfor
2 consists in a highly energetic sporadic burst followed by
tuning the time interval between major bursts turns out to
a series of smaller peaks. The numberof small peaks which
be Q. Note as well that these major bursts favor mass loss
appear between the major ones depends on Q. Moreover,
atexceptionallyhighratesand,moreover,thetimeintervals
the creation of every new inward loop in the stroboscopic
between them are long. Hence, it is tantalizing to directly
map is equivalent to the appearance of a new small peak
connectthemwiththeperiodicitiesobservedinthecircum-
in the light curve. We also found that each change in the
stellar shellsthatcan befound surroundingsomeplanetary
number of loops is accompanied by a chaotic regime. That
nebulae(Van Horn et al. 2002).
is, the transition from a regular regime to a chaotic one
occurs through a sequence of period-doubling bifurcations
similar to Figure 1.
3.3 Mathematical interpretation of the results
For a clear exemplification of the chaotic regimes, in
Figure 3 we show the light curves and the associated stro- Inordertovalidatethenumericalresultsdiscussedabove,we
boscopic maps for a case of regular dynamics and for two briefly present the mathematical characteristics of our sys-
cases of different degrees of irregularity. The central panels tem. Given that our system is non-autonomous — that is,
correspond to the case of regular dynamics prior to the de- theHamiltonian is explicitly time-dependent— thetypical
velopment of the successive loops in the stroboscopic map. methods of analysis of the theory dynamic systems cannot
Weakly non-adiabatic model 5
((aa))
3300
...... ...... ......
(yr)(yr)BB 2200
ττ 1100
00
11..0000 11..0055 11..1100 11..1155 11..2200 11..2255 11..3300 11..3355
QQ
((((((((((((bbbbbbbbbbbb)))))))))))) ((cc))
333333333333000000000000 ξξξξξξξξξξξξ============000000000000............000000000000666666666666 11220000
11000000
880000
ττττττττττττ(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr) B B B B B B B B B B B B 121212121212121212121212000000000000000000000000 ξξξξξξξξξξξξ============000000000000............111111111111111111111111 ττττττττττττ(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr)(yr) B B B B B B B B B B B B 464600000000
220000
000000000000 00
111111111111............333333333333222222222222 QQQQQQQQQQQQ 111111111111............333333333333999999999999 11..3399 QQ 11..33990055550066

Figure 4. Time interval between bursting oscillations. In all cases, α =0.037, ω =20.1, a =20. (a) Time between major peaks as a
functionofthestrengthcoefficient Qforthecaseofξ=0.06;(b)VariationofthetimebetweenmajorpeaksasafunctionofQandfor
different values of ξ from 0.06 to 0.11; (c) The extension to higher values of Q leads to larger time intervals between major peaks for
ξ=0.06.
be used. To overcome this drawback, we used an averaging Ickeetal.(1992)concludedthatinthecaseofcompleteadi-
method(Sanders&Verhulst1985) totransform oursystem abaticity (ξ = 0), a decrease of ω leads to stronger chaotic
into an autonomous one. The high value of the character- pulsations. The values of ω used in their work were equiv-
istic frequency (ω ≫ 1) assures us that this method is ap- alent to adopting stellar models in the family of low-mass
plicable to our case. In the time-averaged framework, the stars(M ≤5−8M⊙)reachingtheAGB.Inarecentpublica-
time-dependent perturbation from Eq. (2) becomes F(r)= tion (Munteanu et al., 2002), we extended their conclusion
−Qαω4/3A sinωr,whereA=−0.04993isaconstantevalu- tointermediate-massstars(8M⊙ ≤M ≤11M⊙)alsointhe
atednumerically.Thefixedpointsoftheaveragedsystemin AGB phase, more precisely to values of ω around 3. In the
thecaseofnonzeroperturbationmustsatisfytheconditions: previoussections,wehaveshownthatinthecaseofω=20.1
apeculiarbehaviourisbornfromtheinterplaybetweennon-
p0=r0−4+Qαω4/3A r0−2 sin ωr0 (7) asydsitaebmatcicoirtryesapnodndinintegrntoalthpeerptaurrabmateitornic. Oinuterravanlal5y≤sisωof≤t2h5e
r0βpδ0−1=0 . (8) has revealed that such a behaviour is found for values of
or, equivalently, ω close to 20, that is for low-mass stars. Mathematically,
we attribute this fact entirely to the creation of new fixed
G(r)≡rβ(r−4+Qαω4/3r−2A sin ωr)δ−1=0. (9)
pointsmentionedintheprevioussectionwhichcritically al-
The roots of Eq. (9) are to be found numerically. The new ter the dynamics of the system. They exist for values of
fixed points of interest are r0 ≈ 0.65 and r0 ≈ 1.03 and ω higher than about 18. For values slightly lower than 18,
represent onlyslight displacementsof thefixedpointsmen- the dynamics resembles the one encountered for ω = 20.1
tioned before in the case of zero perturbation. Moreover, (middlepanelsofFigure5),butitdoesnotpresentthesuc-
theirassociatedeigenvaluesmaintaintheformfromthepre- cessive creation of new loops. Instead, the change of the
vious case (unstable fixed points of saddle-focus type). In- control parameter Q leads to a mixture of chaotic regimes
creasing the parameter Q, a new fixed point is created at and uncorrelated creation and dissapearance of new loops.
about Q0 ≈ 1.3855 at r ≈ 0.86. This new fixed point is For completeness, we present in Figure 5 the light curves
stable as the associated eigenvalues have all negative real and the stroboscopic maps for three values of ω. For each
parts. With thecreation of this fixedpoint,thelooping be- case, the temporal scaling factor was computed according
haviourdissappears. Forhighervalues of Q,thefixed point to the work of Vassiliadis & Wood (1993) which provides a
isreplacedbytwounstablefixedpointsagainofsaddle-focus completesetofstellarparametersforAGBstarswithinitial
type.Thedistancebetweenthemincreaseswith(Q−Q0)1/2. masses in the range 0.89 ≤ M/M⊙ ≤ 5.0. Due to the rich-
Note, however, that these very high values of Q result in nessof thedynamicsinthecase ofω=20.1, weuseit fora
transmissions through the mantle which are not very real- comparison with observations.
istic and, moreover, the time intervals between successive
burstsfor larger values of Q become very large.
4 COMPARISON WITH OBSERVATIONS
3.4 The role of ω
The values of the parameters used throughout the numer-
Before interpreting the behaviour of the system in the con- ical integrations were intended to locate the stellar models
text of real data of stellar variability, we consider worth of we are dealing with in the family of Long Period Variables
interest to explore the role of the parameter ω and to jus- (LPVs) and, specifically, in the families of semiregular and
tifytheparticularvaluewehaveusedthroughoutthiswork. Miravariables.PerhapsthemostimportantoftheMirastars
6 A. Munteanu, E. Garc´ıa–Berro & J. Jos´e

3
0.3
2 0.1
L/L * 1 v -0.1
-0.3
0 4 2 1.01.2
10 20 t (yr) 30 40 p 0 0.8 r

5
0.3
L/L * 3 v -00..11
1 -0.3
4 2 1.01.2
10 20 t (yr) 30 40 p 0 0.8 r

7
0.3
5 0.1
L/L * 3 v -0.1
1 -0.34 2 1.01.2
10 20 t (yr) 30 40 p 0 0.8 r

Figure 5.The roleof ω. Thelightcurves and the correspondingstroboscopic maps forthe cases α=0.037, Q=1.2and (top panels)
ω=10(M =5M⊙),(middlepanels)ω=15(M =3M⊙)and(bottom panels)ω=20.1(M =1M⊙).
isoCeti.Itslightcurveshowsapeculiarvariabilityconsist- leadingtochaos.Thegapsbetweenthetheoreticaldataare
inginanexceptionalpeakoccurringeverytwo,threeorfive theconsequence of thedrastic change in thecharacteristics
“cycles” (Barth´es & Mattei 1997). Our simple model natu- of our light curves when a new loop is created, and cor-
rally recovers this behaviour by tuning the strength of the respond to a very small change of the coupling coefficient.
perturbation or the coupling coefficient. Moreover, we have Hence, according to our analysis we should not find almost
shownthatwithinoursimplemodellargepeaksinthelight any star in theseregions, which is exactly what it is found.
curveare associated to large inter-pulseintervals. Another peculiarity of Mira stars is that some of them
TheMirastarsbelongingtotheLargeMagellanicCloud show alternating deep and shallow minima, giving the ap-
constitutethebestsampleofMirasconcerningbothperiod- pearance of double maxima. Some examples are R Cen,
icities and luminosities. In Figure 6a we show the observa- RNor,UCMi,RZCygandRUCyg—seeHawkins(2001)
tionaldataforsomeMirastarsintheLMC(Feast1989)and and references therein. Among these, R Cen has the most
thebestfittotheobservationaldata.Amongthem,theones persistentandstabledoublemaximainthelightcurvewhile
havingperiodslongerthanabout400daysclearlyappearto for therest of the cases the second maximum is often weak
be over-luminous with respect to the period-luminosity re- and the light curve sometimes reverts to that of a normal
lationship foundforMiras withrelatively short periods(Zi- Mira. Our simplistic model provides such light curves for
jlstraetal.1996;Beddingetal.1998).Withthisinmindwe smallvaluesoftheparameterQ.Asforthegeneral“chaotic
haveobtainedaperiod-luminosityrelationshipfromourthe- connection” for the Mira variables, the first (and unique)
oreticalmodelsusingthetimeintervalτB andaveragedval- caseofevidenceofchaoticpulsationinaMirastar(RCyg)
uesofthemajorpeaksinluminosity.Moreprecisely,wehave comes from the study by Kiss & Szatm´ary (2002). They
varied the parameter Q in the range [1,1.26] resulting in associate the long sub-segments of alternating maxima in
light curves whose major peaks have periodicities (in days) R Cygto a period-doublingevent,supportingtherefore the
withintherange2.4<∼ logτB<∼3.0.Thecorrespondingbolo- well-known scenario of period-doubling to chaos, which we
metricmagnitudeswerecomputedusingareferencevaluefor also find in our model. Buchler et al. (2001) present an
theequilibriumstellarluminosityoflog(L⋆/L⊙)=3.5(Vas- overview of observational examples of chaotic behaviour in
siliadis & Wood 1993) which is typical for Mira stars. Our some semiregular variables (SX Her, R UMi, RS Cyg, and
theoreticalperiod-luminosityrelationshipisshowninFigure V CVn). They argue that AGB stars are prone to chaotic
6b. For the sake of comparison we also show in this panel pulsations due to the fact that relative growth rates of the
all those Miras with logP > 2.4 and M < −5.0, that is lowestfrequencymodesareoftheorderofunity.Higherrel-
bol
all the stars which are found to be over-luminous in Fig- ative growth rates are a consequence of higher luminosity–
ure 6a. We have also included two other interesting Miras: mass ratios, that is more non-adiabatic stars. Hence, not
RHyaandVHya.Asitcanbeseenourtheoreticalperiod- only semiregular variables, but also Mira stars should also
luminosityrelationshipfitsverywelltheobservationaldata. becandidatesforchaoticpulsators.Inspiteofitssimplicity,
Moreover,noteaswellthattheobservationstendtocluster our toy-model presents chaotic pulsations for certain inter-
around fixed regions of the period-luminosity relationship. valsoftheparameterscharacterizing thestrengthofthein-
Theseregionsarecoincident withtheregions wherewefind ternaldrivingand,thus,couldprovidesomesupport tothe
regularoscillationswithafixednumberofloopsinthestro- conjecture that the evolution of semiregulars and of Mira
boscopic map and its sequence of period-doubling cascade stars is strongly connected.
Weakly non-adiabatic model 7
((((aaaa)))) (((bbb)))
----6666....5555 ---666...555
----6666....0000 000555111555---666666000888
000555000333---666666222000
----5555....5555
---666...000
IIIRRRAAASSS
----5555....0000 000000555555444---777333555111
bolbolbolbol bolbolbol VVV HHHyyyaaa
MMMM MMM
----4444....5555 RRR HHHyyyaaa CCC222
000555111555---666555111000
---555...555
----4444....0000
----3333....5555
IIIRRRAAASSS 000222111555222+++222888222222
RRR111000555
----3333....0000 ---555...000
2222....0000 2222....2222 2222....4444 2222....6666 2222....8888 3333....0000 222...444 222...555 222...666 222...777 222...888 222...999 333...000
lllloooogggg ((((PPPP //// ddddaaaayyyyssss)))) llloooggg (((τττ BBB /// dddaaayyysss)))
Figure 6.Period-LuminosityrelationforMirastars.(a)PLrelationforMirastarsintheLMCandthedataonwhichtherelationship
isbased(Feast1989).(b)ThePLrelationusingthedatafurnishedbyourmodel(filledcircles):theequivalentMbol ofthemajorpeaks
andthetimeinterval τB between them.Thevalues forthe parameters usedhereareα=0.037,ξ=0.06andQ∈[1,1.26].Thecrosses
representasampleofover-luminousMirastarsintheLMC.Seetextforadditional details.
5 CONCLUSIONS AND CAVEATS of non-adiabaticity turns out to be a determining factor in
the development of the period-doubling route to chaos. As
Wehaveintroducedaweaklynon-adiabaticone-zonemodel farasthetransitiontochaosisconcerned,anincreaseinthe
driven by sinusoidal pressure waves intended to reproduce degreeofnon-adiabaticity playsthesameroleasadecrease
theirregularpulsationsofMira-likevariablesforwhichother of theeffective temperaturein themodel studied by Saitou
simple models already exist (Buchler & Regev 1982; Au- etal.(1989).Moreover,duetothetime-dependentperturba-
vergne & Baglin 1985; Reid & Goldston 2002). Our model tion,aknot-likestructureiscreatedinthephasespacewhen
extends the works of Icke et al. (1992) and Saitou et al. varying the transmission coefficient (Q) while the strength
(1989).Inparticular,Ickeetal.(1992)proposedanadiabatic oftheperturbationiskeptfixed.Theresultingperiodiclight
model drivenbypressurewaves(thepiston approximation) curves are characterized by a repetitive pattern consisting
whereas Saitou et al. (1989) studied a simple non-adiabatic inamajorpeakfollowedbynminorpeaks,wherenisgiven
model without driving (the self-excited pulsation model). by the number of loops in the phase space. Our theoreti-
Our approach is justified by the large density contrast be- cal light curves are in qualitative agreement with those of
tween the interior and the outer layers. One-zone models several well-known Mira stars. In particular the prototype
havelimitationssincesomeMirastarsaresuspectedtohave of Mira stars, o Ceti, shows this pattern of alternating ma-
atleast twofrequencies(Mantegazza 1996) whichmayvary jorpeaksfollowed byseveralminorpeaks.Furthermore,for
independently, suggesting that more than one mode is in- a given choice of our input parameters the resulting light
volved. Clearly, one-zone models are unable to reproduce curves also resemble those of some peculiar Miras (R Cen,
this behaviour. Nevertheless, the majority of Miras do not R Nor, U CMi, RZ Cyg, or RU Cyg) which appear to have
show this behaviour. Another interesting approach would doublemaximaduetoalternatingdeepandshallowminima.
have been to use the modal coupling. However, although We have found as well that our dynamical system presents
this approach has been used to model the pulsation of Mi- bothchaoticregimesandpatternsofperiodicity.Thechaotic
ras(Buchler&Goupil1988),itismoreappropriateforclas- regions occur for ranges of Q between those corresponding
sical Cepheids, RR Lyrae and W Vir stars — see Buchler to the creation of a new luminosity peak between major
et al. (1993) and references therein. It is also worth not- bursts. We have also noticed that for increasing strengths
ing here that the piston approximation — first introduced of the perturbation the time interval between major bursts
by Bowen (1988) — used in this paper has been used since increases. This interval increases with Q as well. At high
then by several authors (Fleischer et al. 1995; H¨ofner et al. values of the strength of the perturbation within the range
2003)evenifithasneverbeendeeplyscrutinizedforvalidity. yielding the knot-like structure, we obtained a peculiar be-
Neverthelessthisapproximation appearstocorrectly repro- haviour: the creation of new loops stops and the resulting
ducethevelocitiesandmasslossratestypicalofAGBstars. light curvesshow atimeintervalduringwhich theluminos-
Hence, this approximation can be regarded as a reasonable ity preceeding every major peak remains constant.
first-orderapproximationofthedynamicaleffectofthepul-
We have also obtained a theoretical period-luminosity
sation on theatmosphere. Tosummarize, ourmodelshould
relationship andcompared itwiththeobservationaldataof
be considered as a toy model that qualitatively reproduces
Miras in the LMC. In particular, we have focused on those
thegeneral features of thelight curves.
peculiar Miras with long periods which are known to be
Wehavethoroughlyexploredtheinterestingparticular- over-luminous with respect to the best fit of Feast (1989).
itiesofthesystemfromboththeastrophysicalandfromthe We have found that our model provides a reasonable fit to
mathematicalpointsofview.Wehavefoundthatthedegree theperiod-luminosity relationship ofthesestarsandalsoto
8 A. Munteanu, E. Garc´ıa–Berro & J. Jos´e
the observed clustering of the over-luminous stars around
certain regions in the period-luminosity diagram. The ulti-
matereason forthisfactisclosely relatedtothecreation of
anewloopinthestroboscopic map,and,consequently,ofa
new luminosity peak in thecorresponding time series.
Finally,wewouldliketostressthatalthoughoursimple
model succeeds in furnishing reasonable comparisons with
real data, a detailed quantitative interpretation is beyond
itscapabilitiesbecauseofitscrudephysicalassumptions.In
particular, next steps towards improving the one-zone ap-
proachcouldeventuallyincludethetreatmentofconvection
whichissupposedtoplayanimportantroleintheenvelopes
of these stars. This improvement will affect also the func-
tionalform oftheluminosity whichwas rathersimplifiedin
thepresent work.
Acknowledgements. This work has been supported by the
MCYT grant AYA2000–1785, by the MCYT/DAAD grant
HA2000–0038 andbytheCIRITgrants 1995SGR-0602 and
2000ACES-00017. We would also like to acknowledge our
anonymousreferee for valuablecriticisms and suggestions.
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