G.Fontaine 1 ,P.Brassard 1 ,S. Charpinet 2 ,P.-O.Quirion 3 ,S.K.Randall 4 ,
andV.VanGrootel 5
1
DepartementdePhysique,UniversitedeMontreal,Montreal,Quebe ,Canada
H3C3J7
2
Laboratoired'AstrophysiquedeToulouse-Tarbes, UniversitedeToulouse,CNRS,
14av.E.Belin,31400 Toulouse, Fran e
3
InstitutforFysikogAstronomi,AarhusUniversitet,Aarhus,DenmarkDK-8000
4
EuropeanSouthernObservatory,Karl-S hwarzs hild-Str.2,85748Gar hingbei
Mun hen,Germany
5
Institutd'AstrophysiqueetdeGeophysique,UniversitedeLiege,Alleedu6ao^ut
17,B4000Liege,Belgium
1 Asteroseismology and White Dwarf Stars
Thete hniqueof asteroseismologyexploits the information ontainedin the
normal modes of vibrationthat maybe ex ited during parti ular phases in
theevolutionofastar.Su hmodesmodulatetheemergent uxofapulsating
star and manifest themselves primarily in terms of multiperiodi
luminos-ity variations. In its simplest form, observational asteroseismology onsists
in gathering light urves, i.e., monitoring the (variable) brightness of
pul-sating starsasafun tion oftime. Standardsignalpro essing methods, su h
asFourierte hniquesforexample, are thenused to extra t the periods (or,
equivalently,frequen ies),theapparentamplitudes,andtherelativephasesof
thedete tedpulsationmodes.
Thenextstep{inessen ethemostbasi omponentofthe
asteroseismo-logi al approa h as awhole { onsists in omparing the observedpulsation
periods with periods omputedfrom stellar modelswith thehopeof nding
an optimalmodel that provides agood physi al des riptionof the real
pul-satingstarunders rutiny.Toinsurethatthesear hfortheoptimalmodelin
parameterspa eis doneobje tivelyand automati allyrequires good
model-building apabilities, eÆ ient period-mat hing algorithms, and onsiderable
omputing power. Otherwise, with simpler trial-and-error sear h methods,
there always remainsadoubt about theuniqueness and validity of thebest
Theperiods of pulsation modes of stellar models angenerally be
om-putedquitea uratelyandreliablywithintheframeworkofthelineartheory
ofstellarpulsationsinitsadiabati version(seebelow).Thesameframework
(although omplementarynonadiabati al ulationsarethenne essary)may
beused to ompute rst-order orre tionstotheunperturbedemergent ux,
anapproa hthatisfundamentalintheexploitationoftheadditional
informa-tion ontainedintheobservedamplitudesandrelativephasesofthedete ted
pulsationmodes.Andindeed, onstraintsontheangulargeometryofagiven
observed pulsation mode may be obtained by omparing theoreti al
ampli-tuderatiosandphasedi eren eswiththose observedindi erentwavebands
throughmulti olorphotometryand/ortime-resolvedspe tros opy.Thesame
istruewhenexploitingtherelativeamplitudesofmodesin agivenwaveband
in onjun tionwiththoseoftheirharmoni sand ross-frequen y omponents
intheFourierdomain.Whenavailable,su h inferen esontheangular
geom-etry of pulsation modes may provide extremely valuable onstraints in the
sear hfortheoptimalmodelin parameterspa e.Intheend,ifsu essful,an
asteroseismologi alexer iseleadstothedeterminationoftheglobalstru tural
parametersofapulsatingstarandprovidesuniqueinformationonitsinternal
stru tureandevolutionarystate.Anoutstandingexampleofsu hasu essful
exer iseisthatprovidedre entlyby[11℄forthepulsatingsdB omponentof
the losee lipsingbinarysystemPG1336 018.
Thete hniqueof asteroseismology hasfound parti ularly fertile grounds
at the bottom of the HR diagram where several distin t types of pulsating
stars havebeen dis overed. These are generally referred to as the ompa t
pulsators (i.e., those with surfa e gravities log g >
5). Figure 1 illustrates
that portion of the surfa e gravity-e e tive temperature plane where these
families arefound. Thosein lude theZZCeti stars whi h areH-atmosphere
whitedwarfswithT
e
'12,000K( rstdis overedby[26℄),theV777Herstars
whi h areHe-atmosphere whitedwarfswith T
e
'25,000 K([45℄), andthe
GW Vir pulsators whi h areHe/C/O-atmosphere whitedwarfs with T
e '
120,000K([28℄).Figure1alsoindi atesthelo ationsoftwoother ategoriesof
pulsatorswhi hweredis overedmorere ently.ThesearetheV361Hyastars
whi h areshort-period pulsatinghotBsubdwarf (sdB) stars([24℄), andthe
long-periodV1093Herpulsatorswhi hare oolerandless ompa tsdBstars
([21℄).Inaddition,short-periodpulsationshavealsobeenreportedin the,so
far,uniqueobje tSDSSJ1600+0748([46℄),averyhotsdOsubdwarfwithlog
g5.9andT
e
71,000K([19℄).Thisobje tmayormaynotberelatedtoa
groupoffoursdOsubdwarfsfoundinthe luster!Cen,and lusteredaround
T
e
50,000K, for whi h pulsationalinstabilities were re entlydis overed
([35℄).Moreover,[30℄havefoundthepresen eofat leastonepulsationmode
in SDSS J1426+5752,a member ofan entirelynewand unexpe ted kindof
veryrare white dwarfs, those of the so- alled Hot DQ spe traltype, whi h
are relatively ool (T
e
' 20,000 K) stars with atmospheres dominated by
Fig.1. Regionofthelogg logT
e
planewherethe ompa tpulsatorsarefound.
Ea hof vedistin t familiesisidenti edbyitsoÆ ialIAU name,andthe yearof
the report of the dis overy of the prototype of ea h lass is also indi ated. Two
more ategories areidenti edbytheyear oftheir dis overy,2006 2011 and 2008,
respe tively.Typi alevolutionarytra ksareplotted showing 1)the tra k followed
by a0.6 M
post-AGB, H-ri hstar whi h be omes a H-atmosphere white dwarf
(dashed urve),2)thepathfollowedbya0.6M post-AGB,H-de ientstarwhi h
be omesaHe-atmospherewhitedwarf(solid urve),and3)thepathfollowedbya
0.478M post-EHBmodelwhi hleadstotheformationofalow-massH-atmosphere
whitedwarf(dotted urve).
refertoareisolatedstarsor omponentsofnon-intera tingbinaries,andtheir
luminosity variations are aused by internal partial ionization me hanisms.
For ompleteness,wefurtherpointoutthatpulsationalinstabilitieshavealso
been dis overed in several a reting white dwarfs in ata lysmi variables
([43℄;[1℄andreferen estherein).
Fig.2. Segmentsoftypi alopti allight urvesforea hofthefourknowntypesof
isolatedpulsatingwhitedwarfs.The urvesreferringtoPG0122+200,GD358,and
G207 9wereobtainedwithLAPOUNE,theMontrealportable3- hannel
photome-ter atta hedto the 3.6 mCFHT teles ope.This photometer usesphotomultiplier
tubesas dete tors. No lter was used,so these are integrated\white light" data.
In omparison,the urveforthemu hfainterstarSDSSJ2200 0741wasobtained
byBetsy Green and Patri k Dufour usingthe Mont4K CCD ameramounted on
oneea hforea hofthedi erentkinds,areshowninFigure2.Theyaretypi al
inthattheyallshowmultiperiodi variationsanda leartenden ytoexhibit
in reasinglynonlinearbehaviorwithin reasingamplitude(asexempli ed by
the ase of the large-amplitude pulsating DB white dwarf GD 358 shown
here). Pulsatingwhitedwarfsgenerallyhavelight urvesthat show
peak-to-peak amplitudes in the range from 0.4 millimag in the lowest amplitude
pulsators known to upward of 0.3 mag in the largest amplitude ones. The
luminosity variations are due to the presen e of low-degree (l=1 and l=2)
and low- to medium-ordergravity modes that are ex itedthrough apartial
ionization me hanism(see below).The observedperiods generally liein the
rangefromabout100stosome1400s,althoughtheupperlimit anextend
to severalthousandse ondsin low-gravityGWVirpulsators.
The properties of the pulsating white dwarfs have been extensively
re-viewedre entlyby[17℄andwereferthereadertothat paperfortherelevant
details.Inaddition, [44℄ haveprovided anotherreviewon thesestars. Given
theavailabilityofthosepapers,wefeltthatitwouldbemoreusefultodis uss
thetopi from anotherpoint ofviewratherthan providewhat would
ne es-sarilyamountto ashortened versionofthose manus ripts.Hen e, in a
ped-agogi alspirit,wede idedto useourallo atedspa e topresentadis ussion
oftheprin iples underlying thetheory ofpulsatingstarsasappliedto white
dwarfs. Our target audien eis therefore resear hersnot ne essarilyfamiliar
withpulsationtheoryandthosethatarebutmaynotbewell-a quaintedwith
thepe uliaritiesofwhitedwarfphysi s.
2 Normal Modes of Vibration in a Homogeneous
Continuous Elasti Medium
Itisusefulto rstre allsomeelementaryphysi s on erningnormalmodesof
vibrationinsimpleme hani alsystems.Thus, underthespe i assumption
that thedispla ementsof matterhavesmallamplitudes omparedto the
di-mensionsofthesystem(thisistherealmofthelinearapproa h),thespatial
and temporal behaviors of perturbations in a homogeneous elasti medium
aregovernedbythe lassi alwaveequation,
2 (r;t) t 2 = (k) 2 k 2 r 2 (r;t); (1)
where ( r;t) is, in general, a ontinuous fun tion of spa e and time. This
wave fun tion ould represent, for example, the (small) displa ement of an
elementof matter withrespe tto its equilibrium position.Inthis equation,
(k)istheangularfrequen yandkisthewavenumber.
Thegeneralsolutiontothisequation orrespondstoana ousti orsound
wave whi h propagates with a speed = =k, a onstant hara teristi of
thereareboundary onditionstoberespe tedatthefrontiersofthedomain,
where wavesare re e ted.Only thewaveswiththe \ orre t"wavelength or
frequen y (generallythose with a node oran extremum at the boundaries)
will resonate, theothers disappearing through destru tiveinterferen e after
several re e tions. The normal modes of vibration are the solutions of the
lassi alwaveequation inpresen eof boundary onditions(whi h transform
theproblem into aneigenvalueproblem). Only ertain solutionsareallowed
(thenormalmodesoreigenfun tions), orrespondingtoadis retespe trumof
possiblevaluesoftheangularos illationfrequen y(theeigenvalues
0 < 1 < 2
;:::). Ina normal mode, all moving parts os illate at the same frequen y
andgothroughtheirequilibrium positionsat thesametime.
Asaspe i example, onsiderthesimple aseofthetransverseos illations
ofanidealelasti re tangularmembraneofdimensionXY thatisatta hed
at the edges. In that ase, the wave fun tion is a simple s alar, (x;y;t),
and the appropriate boundary onditions are spe i ed by (x = 0;y;t) =
(x = X;y;t) = (x;y = 0;t) = (x;y = Y;t) = 0. The solution of the
lassi alwaveequationforthissimpleproblemiswellknown.Spe i ally,the
eigenfun tionsaregivenby,
(x;y;t)sin(k n x x)sin(k n y y)e i nx;ny t : (2)
Note that the amplitude remains unde ned (as in any linear problem) and
is usually normalized to some onvenient arbitrary onstant. Likewise, the
eigenvaluesaregivenby,
n x ;n y = q k 2 nx +k 2 ny ; (3)
where isthe( onstant)speedofsound,and
k n x =(n x +1) X ; n x =0;1;2;:::; and k n y =(n y +1) Y ; n y =0;1;2;:::: (4)
Hen e, in our 2D system, there are 2 \quantum" numbers, n
x and n
y , to
spe ify a mode. These numbers orrespond, respe tively, to the number of
nodal lines that ut the x and y axis. Note that if there are symmetries
(e.g.,asquaremembrane),thendi erentmodesmayhavethesameos illation
frequen y.Su hmodesare alleddegeneratemodes.Itiseasytoinferthatan
eigenmodein3Dshouldbespe i edbythreequantumnumbersrepresenting
the number of nodal surfa es that ut a ross the system. Furthermore, if
there aresymmetriessu h asthatfound inaspheri almodel ofastar,then
degenerateeigenmodesareautomati allypresent.
3 Normal Modes of Vibration in Stars
hemi al omposition.Thespeedofsoundisnotuniform(itdependsondepth)
and, furthermore, the uid elements in a star are subje ted to a variable
gravitationala elerationfrom g =0at the enter to g =g
s
atthe surfa e.
Thewavesthatpropagateinsideastararenolongerpurea ousti (orsound)
waves,buttheyareofthegravito-a ousti type.Su hwavesaresubje tedto
spe i boundary onditionsatthe enterandatthesurfa eofthestar.The
wavesarere e tedthere andoneoftenspeaksofagravito-a ousti avityto
des ribeastarin relationtoitsnormalmodesofvibration.
In its most basi form (see, in this ontext, the ex ellent textbook by
[40℄),thelineartheoryofnonradialstellaros illationsusesasastartingpoint
threewell-knownhydrodynami equationsthatgovernthebehaviorofa uid
in presen e of gravity: the equation of motion, the equation of ontinuity,
andPoisson'sequationthat relatesthegravitationalpotentialwiththe
den-sitydistribution of matter. Inthis version, theex hange of energy between
the thermal bath (the internal energy of the gas/ uid) and the os illations
(thekineti energyofthema ros opi uidmotions)isnegle ted.Thisisthe
adiabati approximation whi h only allows a des ription of the me hani al
behavior of the os illations and does not address the question of the
sta-bility of the modes (whi h involves thermal properties and the question of
energy ex hange). However, it turns out that nonadiabati e e ts generally
onlymarginallyin uen ethevalueoftheos illationfrequen yofapulsation
mode in a stellar model, so the adiabati version an be used with
on -den efor omputingfrequen ies(periods) tosuÆ ienta ura y tobeofuse
in asteroseismologi alexer ises.
Thenextstepisto onsideronlysmall perturbationsin orderto beable
tolinearize thebasi hydrodynami equations.Theunperturbedequilibrium
on guration is usually that of apurely spheri alstellar model. In keeping
with the fundamental properties of a normal vibrationmode, allperturbed
quantities ofinterestare assumedto os illatein phaseand gothrough their
equilibrium positionsat thesametime. Hen e,thedependent variablesthat
appearinthelinearizedequationsareassumedtohaveatemporaldependen e
givenbythestandardos illatorytermin linearphysi s,i.e.,e it
,where is
thefrequen yandtisthetime.Thete hniqueofseparationofvariablesisthen
used to hara terizethespatial behaviorofamodein termsofaradialpart
andanangularpart.Forunperturbedspheri almodels,theangularbehavior
ofamode omesout,notunexpe tedly, tobegivenbyaspheri alharmoni
fun tionY m
l
(; ).Oneendsupwithasystemof4lineardi erentialequations
withrealvariables(dependingnowonlyontheradial oordinate)whi h,
be- auseofboundary onditionstoberespe tedatthe enterandatthesurfa e
of thestar model, permitsonly ertainsolutions(the modes) orresponding
to spe i valuesoftheos illationfrequen y (the eigenvalues).
A stellar pulsation mode is de ned in termsof 3 dis rete numbers,k, l,
and m, the rst one giving the number of nodes in the radial dire tion of
k is termed the \radial order" and may take on all positive integer values,
i.e., k = 0, 1, 2, et . The index l is alled the \degree" of the mode and
gives the total number of nodal planes that divide the stellar sphere (l =
0, 1, 2,...). The number m is alled the \azimuthal order" and its absolute
value jmj gives the number of nodal planes that divide the stellar sphere
perpendi ulartotheequatorwhilegoingthroughthepoles.There are2l+1
possiblevaluesform( l; l+1;:::0;:::l 1;l).Non-rotating(spheri al)stars
haveeigenfrequen iesthatare(2l+1)-folddegenerateinm.Inthat ase,the
periodofapulsationmodeistriviallyrelatedtoitsangularfrequen ythrough
theexpressionP
k l
=2=
k l
,independentofm.
Itshouldbepointedoutthatthere existtwodistin tbut omplementary
behaviorsof gravito-a ousti wavesin aspheri alstar, onefundamentallyof
a ousti origin (as in a homogeneous sphere), and the other related to the
a tion of gravity. In terms of normal modes of vibration, these behaviors
are respe tively referred to as\pressure modes" (or p-modes) and \gravity
modes"(org-modes). 1
Thesedi erentbehaviorsarerelatedtotwo
fundamen-tal quantities thatappearinthelinearized pulsationequations:theso- alled
Brunt-Vaisalafrequen yN de nedby,
N 2 g 1 1 dlnP dr dln dr ; (5)
andtheLambfrequen yL
l de ned by, L 2 l l(l+1) 2 r 2 with 2 = 1 P ; (6)
where isthe lo aladiabati sound speedandthe othersymbolshavetheir
standardmeaning.
Considerin this ontext an elementof uid in equilibrium withits
envi-ronmentatsomearbitrarydepthinastar.Ifaperturbationisappliedtothis
elementsu h that it is ompressed without hanging position in the star, a
restoringfor eproportionaltothe ontrastindensity(or,equivalently,tothe
pressure gradient) betweenthe perturbed element and its environment will
establish itself. Freedof the initial onstraint, the element will rea tto the
restoringfor e byos illating in volume (density, pressure) aboutits
equilib-rium on guration.Theseos illationswillo urata hara teristi frequen y,
the Lambfrequen y, whi h is intimately related to thelo al speedof sound
at the equilibrium point in the star. These os illations are then essentially
a ousti in nature,andonespeaksofa ousti wavesor,equivalentlybe ause
the restoring for e is due to apressure gradient, pressurewaves. In normal
situations(e.g., instablestars),theos illationsareultimatelydamped.
Consider again an element of uid in equilibrium with its environment
at some arbitrary depth in a star. This time, the perturbation onsists in
1
displa ing the uid element in the verti al dire tion, say above its
equilib-rium position. The element is then heavierthan it should be (its density is
higherthanthatofitsnewsurroundings),and,freed ofitsinitial onstraint,
will return toward its equilibrium position, and will os illate up and down
aboutit. Normally, theos illationsgeneratedbythe initialperturbationare
damped over a hara teristi times ale that depends on the lo al vis osity.
These os illationswill o urat a hara teristi frequen ywhi h depends on
the lo al physi al onditions at the point of equilibrium, the Brunt-Vaisala
frequen y.In that ase,it is buoyan y that provides therestoringfor e and
sin ebuoyan yisdire tlyrelatedtothemagnitudeofthelo algravitational
a eleration,onespeaksof gravitywaves.
Itshould be learthatone annotseparate ompletelythee e tsof
pres-surefrom those ofgravity.Forinstan e, in thelast example,the ontrastin
density(pressure)betwenthe displa edelementandits surroundingsshould
also play a role. Hen e, even though it is ustomary to divide the
gravito-a ousti wavesintwoseparatebran hes(onespeaksofpressuremodesandof
gravitymodesin stellarpulsationtheory),theeigenfrequen iesofthenormal
pulsation modes ofa stardepend at thesame timeon boththe variationof
theLambfrequen ywithdepthandonthat oftheBrunt-Vaisalafrequen y.
It is possible to derive a useful dispersion relation for gravito-a ousti
wavesby makingtwosimplifying assumptions. The rst oneisthe so- alled
Cowling approximationwhi h onsists in negle tingthe perturbation ofthe
gravitationalpotential.Inthatapproximation,theadiabati pulsation
equa-tionsredu etoasystemof2lineardi erentialequations.Althoughthe
Cowl-ingapproximationisnotverygoodformodeswithlowvaluesofkandland,
moregenerally,forp-modes(itisnotusedindetailednumeri al al ulations),
itissuÆ ientinthe ontextofthepresentdis ussion.
These ond approximation, termedthe lo al approximation, fo usses on
anarbitraryshellinastellarmodelinwhi hmostofthequantitiesappearing
in the tworemaining pulsation equations (speed of sound, Lamb frequen y,
Brunt-Vaisalafrequen y, lo al gravitationala eleration) anbeseenasnot
varying verymu hin theradialdire tion overthe shellregionwhi h, atthe
sametime, oversalargenumberofradialwavelengths.This ispossibleonly
formodeswithlargevaluesoftheradialorder k.Ifthe radialvariationsare
ompletelynegle ted,thenone ndsthattheeigenfun tionshaveabehaviorof
thetypeexp(ik
r
r),wherek
r
isaradialwavenumberthatobeysthefollowing
dispersionrelation, k 2 r = 1 2 2 2 L 2 l 2 N 2 : (7)
This indi ates that theos illations propagate radially(os illatorybehavior)
if k
r
is a real quantity orare evanes ent (exponential behavior) if k
r is an
imaginaryquantity.
Spe i ally, k 2
r
is positive (and k
r
is thus real) when 2 > L 2 l , N 2 . This
orrespondstothep-modes.Likewise,k 2
r
isagainpositivewhen 2 <L 2 l ,N 2 ,
andthiso ursfortheg-modes.IntheregionsofthestarwhereL 2 l < 2 <N 2 orN 2 < 2 <L 2 l ,k 2 r is negative(k r
is imaginary),and thegravito-a ousti
wavesareevanes ent.
Itispossibletoobtainfurtherqualitativeinformationastothebehaviors
oftheeigenfrequen ies(periods)ofbothp-modesandg-modesby onsidering
two limiting ases. Spe i ally, in the limit where 2 L 2 l ;N 2 , the above
dispersionrelationleadsto,
2 = 2 p ' 2 k 2 r . (8)
Thisisthelimitwhereessentiallypurea ousti orpressuremodesarefound
sin eonlythespeedofsoundisinvolvedintherighthandsidetermofequation
(8).One anseethatthefrequen y(period)ofap-modein reases(de reases)
whenk
r
(i.e, thenumberofnodesintheradialdire tion)in reases.
Intheopposite limitwhere 2 L 2 l ;N 2
,equation(7)redu es to,
2 = 2 g ' l(l+1)N 2 k 2 r r 2 . (9)
Thisisthelimitofpuregravitymodessin eonlytheBrunt-Vaisalafrequen y
(and noa ousti term)is involvedin theexpression. One aninfer thatthe
frequen y (period) of a g-mode de reases (in reases) when the radial order
in reases. In addition, the frequen y (period) of a g-mode of given radial
order in reases (de reases) when the degreeindex l in reases.Furthermore,
g-modes with adegreeindex l=0{ these would beradial modes {do not
exist astheyhaveafrequen yofzero.
Inbrief,gravito-a ousti modesbehaveasalmostpurepressuremodesat
veryhighfrequen ies,whiletheybehaveasalmostpuregravitymodesatvery
lowfrequen ies.Inboth ases,theselimits orrespondtoverylargevaluesof
theradialorder,i.e.,k 1,sometimesreferredtoasthe\asymptoti limit".
Asindi atedabove,itisimportantto realizethatforgravito-a ousti modes
of low radial order (su h as those generally observed in white dwarf stars
forinstan e), theeigenfrequen iesdepend atthesametimeonbotha ousti
and gravitye e ts. Despite this,the usualnomen lature in stellarpulsation
theory is to keepthe terms p-modesand g-modes for all modes with radial
orderk1.
Figure3summarizesthese onsiderationsonthebasisofanexa t
al ula-tionofthelow-order, low-degreeperiodspe trumof atypi almodelofaZZ
Cetipulsator.Thismodelis hara terizedbyatotalmassM=0:6M
andan
e e tivetemperatureT
e
=11,800K.Ithasapure arbon oresurroundedby
aheliummantle ontaining10 3
ofthetotalmassofthestar,itselfsurrounded
byahydrogenouterlayer ontaining10 6
ofthetotalmassofthestar. The
un-Fig. 3. Low-orderandlow-degreeperiodspe trumforarepresentativemodelofa
ZZCeti star. Resultsare shownfor valuesof thedegree indexl =0,1, 2,and 3.
Thevaluesof the radial orderk are alsoindi ated up to k =5.Thetwofamilies
of gravito-a ousti modes, the p-modes and the g-modes, are learly illustrated.
Sometimes,dependingonthe onventionadopted,theradialindi esofg-modesare
assignednegativevalues,sothat,onafrequen ys ale,thevaluesofkin reasewith
in reasingfrequen yoverthefullspe trum.
are fullyrepresentativeofa ZZCeti star. This equilibriumstru ture will be
usedagainbelowasareferen emodel.
We omputedallpulsationmodeswithvaluesofl=0,1,2,and3inthe
periodwindow1 1000sforourreferen emodel.Figure3neatlyillustratesthe
p-modeandg-modebran hesofthegravito-a ousti periodspe trum,aswell
astheso- alled\fundamentalmode"formodeswithl2.Thefundamental
modehasnonodeintheradialdire tion(k =0)andfallsbetweenthep-and
g-bran h.Inwhitedwarfs,thatmodeismoreakintoap-modethanag-mode,
although,on eagain,it doesdepend onbotha ousti andgravitye e ts. It
is noteworthy that an isolated star annot havea fundamental mode ifthe
degreevalue is l= 1sin ethis would implya motionof the enter ofmass
with l =0 areknown asradialmodesbe ause allmotions are restri tedto
theradialdire tion asthere isnoangulardependen e (Y 0
0
(; )= onstant).
They are simply a parti ular ase of nonradial p-modes. Their eigenperiod
spe trumdoes ontainafundamental mode.
It should be pointed out that a stellar atmosphere loses its apa ity to
re e t ba k towardthe interioroutgoing gravito-a ousti wavesin the limit
of very high radial order k. Indeed, in this limit, pulsation eigenmodes are
stronglydampedbyoutwardlypropagatingwavesintheatmosphereandare
not expe ted to belong to the gravito-a ousti avity mentioned above. In
otherwords,highradialordermodes\leakthrough"theatmosphereandtheir
energyislosttotheoutside.Amethodforestimatingthese uto {or riti al
{periodsinawhitedwarf ontexthasbeendevelopedby[23℄.Applyingthis
approa hto therepresentativeZZ Ceti starmodelused inFigure 3,we nd
that p-modeswithperiods lessthan0.1sand g-modeswithperiods larger
than6000/ p
l(l+1) sarenotexpe tedto beofinterestinthatmodel.
4 Properties of Pulsation Modes in White Dwarfs
In the adiabati approximation, there are four distin t eigenfun tions that
omeoutofthenumeri aleigenvalueproblemandwhi hspe ifyagiven
pul-sationmode.Itis possibletoexpress twoofthose 2 dire tly intermsof r (r) and h
(r), the radialand horizontal omponent of theLagrangian
displa e-mentve tor,respe tively.Thelatterisgivenby,
= r (r); h (r) ; h (r) 1 sin Y m l (; )e it ; (10)
and hasanobviousphysi al interpretation.In thefollowing,weuse thetwo
distin t(radial)eigenfun tions,
r
(r)and
h
(r),toillustratesomeinteresting
propertiesof pulsationmodesin whitedwarfs. Notethat, foragiven mode,
both
r
(r)and
h
(r)have,of ourse,thesamenumberofnodesintheradial
dire tion(thatistheradialorderkofthemode),butthosenodesdonot
over-lapinspa e.Forg-modes,thenodesofthe
r
eigenfun tionaresystemati ally
higherin thestarthanthe orrespondingnodesofthe
h
eigenfun tion. The
oppositeistrueforp-modes.
4.1 Propagationdiagram
Figure4isknownasa\propagationdiagram"andillustratesthebehaviorof
themodesinrelationtothepro lesoftheBrunt-Vaisalafrequen yandofthe
Lamb frequen y asfun tions ofdepth. Theplot refersto ourrepresentative
2
Inpra ti e,atthenumeri allevel,thea tualeigenfun tionsare onveniently
\Dziem-Fig.4. Propagationdiagramforquadrupole(l=2)modes omputedusinga
rep-resentative modelof a ZZ Ceti pulsator. Thesolid urve shows the pro leof the
logarithm ofthe squareof theBrunt-Vaisalafrequen y asa fun tionoffra tional
mass depth.Thevalue oflogq =0 orrespondstothe enterofthestellarmodel.
The lo ations of two atmospheri layers, thosewith R = 10 2
and R = 1, are
alsoindi atedbytheverti al dottedlinesonthats ale.Likewise,thedashed urve
givesthe logarithmi pro leofthesquareoftheLambfrequen yfor modeswithl
= 2.Thelabelledhorizontal dotted linesshowthe low-order frequen y spe trum,
again ona s ale involving the logarithm ofthe square ofthe frequen y.The dots
givethe lo ations ofthe nodesof the(radial) eigenfun tion
r
(r) for thedi erent
modesillustrated.
ZZ Ceti starmodel andfeatures anabs issade ned bythe logarithmofthe
fra tional mass depth, a s ale hosen to emphasize the outer layers where
most of the \a tion" goes on in terms of modal behavior in white dwarfs.
Thiss aleisusedrepeatedlybelowinother gures.Thesolid urveshowsthe
distribution of the logarithm of the square of the Brunt-Vaisala frequen y.
Thewell inthepro leisdueto thepresen eofasuper ial onve tionzone
ausedbyHpartialionizationinthemodel.Inaddition,therearetwobumps
inthedistributionoftheBrunt-Vaisalafrequen y,one enteredaroundlogq
' 6and asso iated with the H/He ompositional transition zone, andthe
ontrastin mean mole ular weightbetween Hand Hethan betweenHeand
C.
The dashed urve in Figure 4 shows the pro le of the logarithm of the
squareoftheLambfrequen yforquadrupole(l =2)modes,theones hosen
forthisillustrativeexample.Bumpsdueagaintothepresen eof ompositional
transitionzonesarepresent,buttheoneasso iatedwiththeHe/Ctransition
region is quite weak and is hardly visible in the plot. These ompositional
featuresaresigni antasthey\pin h"theeigenfun tionsandprodu emode
trapping and mode on nement, ultimately leadingto a nonuniform period
distribution. Thee e t is parti ularlyimportant in g-modes as an be seen
in thelow-ordermodesof thekindillustratedin Figure3above.
Thehorizontal dottedlinesgivethevaluesofthelogarithmofthesquare
oftheeigenfrequen iesforquadrupolemodesinarange overingfromthek=
6g-mode(termed g
6
)atthelow-frequen yendto thek=6p-mode(termed
p
6
) at the high-frequen yend of the retained interval. Note, in passing,the
nonuniformdistributionofeigenfrequen iesthatisparti ularlyevidentforthe
low-order g-modes onsidered here,a hara teristi that wasmentionedjust
aboveinthepreviousparagraph.Thedotsin the guregivethelo ationsof
thenodesintheradialdire tionforea hofthemodesillustrated.Hen e,the
f-modeshowsnonode, onsistentwithitsradialorderk =0.Thelo ations
of the nodes indi ate where, in astellar model, a given pulsationmode has
an os illatory behavior in the radial dire tion, a region where the mode is
saidtopropagate.Figure4then learlyrevealsthatp-modespropagatewhen
2 >L 2 l ,N 2
,whileg-modesdowhen 2 <L 2 l ,N 2
. Thisresult, omingfrom
detailednumeri al omputations,isentirely onsistentwiththe onsiderations
presentedin thepreviousse tion.Inaddition,Figure4showsthatlow-order
p-modes in whitedwarfspropagate in mu h deeper layersthan low-order
g-modes, whi h insteadpropagateinthe outermostlayers.Quiteinterestingly,
thisistheexa toppositeofthebehavioren ounteredinmainsequen estars
andin allnondegeneratestarsin general.
Wenote, in this ontext, that the sear h for g-modesin theSun is
on-sidered of fundamental interest in helioseismology, as asu essful dete tion
wouldallowtheextensionofasteroseismologi alprobingof thesolarinterior
tomu hdeeperregionsthanis urrentlypossibleonthebasisoftheobserved
p-modes.Inwhitedwarfs,onlylow-to medium-orderg-modeshavebeen
ob-servedso far,and itis thelow-orderp-modes thatwouldallowto probethe
orebest.Although thepresen e ofex ited p-modesis expe ted from
nona-diabati pulsationtheory,su h modeshaveyet tobefound in whitedwarfs.
Thelatest(unsu essful)attempttodete tp-modesinwhitedwarfsusingthe
VLThasbeenreportedby[39℄.
4.2 Angularmodaldependen e
of whi h anbemodeledin terms of temperature, radius,and surfa e
grav-ityperturbations. Inpulsating white dwarfs, it was rst shownby [36℄ that
the luminosity variations are ompletely dominated by temperature
pertur-bations,aresultthathasbeenformally on rmedby[34℄,althoughthelatter
authorspointedoutthatthisisnotne essarilytrueinthe oresofabsorption
lines.Hen e,toagoodapproximation,theluminosityvariationsofapulsating
whitedwarfmaybevisualizedsolely intermsof temperaturewavesa rossa
diskthat otherwisedoesnot hangeinshapeorsurfa earea.
Figure 5 illustrates typi al angular geometries for nonradial pulsation
modes expe ted in whitedwarfs, giventhat the modesfor whi h the degree
indexhasbeenidenti edsofarhavevaluesofl=1orl=2.Onere ognizes,
of ourse,thegeometriesofspheri alharmoni fun tions.Ea h olumnrefers
to amodewith xed angular indi es land m, and showsthe instantaneous
temperaturedistributionon the visible disk over half apulsation y le
ov-eredby vedistin tphases.Forea hofthe vedi erentmodesdepi ted(the
ve olumns), it is assumedthat the stellarsphere is in linedsu h that the
anglebetweentheline-of-sightandthesymmetryaxisofthepulsationmode
isequalto1radian(toberegardedasarepresentativevalue).
The rsttwo olumnsrefertodipole(l =1)modes,i.e., toageometryin
whi hthere isonenodalplane that dividesthestellar sphereintotwoequal
hemispheres. The ase m = 0 ( rst olumn) orresponds to a nodal plane
that isthesameastheequatorialplane(i.e.,the utisperpendi ulartothe
symmetryaxis),whilethe asejmj=1(se ond olumn) orrespondstoanodal
planethatisakintoagreatmeridianplane(i.e.,the utisalongthesymmetry
axis). It should beunderstood that, at any giventime, all eigenfun tionsof
interest havezero amplitude onan angular nodal plane, from the enter to
thesurfa eofthemodel.
Thelastthree olumnsrefertoquadrupole(l=2)modes,i.e.,theyinvolve
twonodal planes.The ase m=0(third olumn) orrespondsto twonodal
planes parallel tothe equatorialplane, the ase jmj =1(fourth olumn) to
onenodalplanefusedintotheequatorialplaneandonenodalplanealongthe
symmetryaxis,andthe asejmj=2( fth olumn)totwonodalplanesgoing
throughthepulsationaxisandperpendi ulartoea hother.Itisinterestingto
pointoutthat modeswithm=0havenodalplanesthatdonot hangewith
time, whereas modes with jmj 6= 0have jmj planes that \rotate"aboutthe
symmetryaxis.Fromthatpointof view,theangular omponentofam =0
modemaybe onsideredasastandingwave,whilethatofajmj6=0modemay
be onsideredasarunningwave.Notethatthetwomodeswiththesamevalue
ofjmjbutwithdi erentsignshavesimilartemperaturedistributions(orother
eigenfun tionsofinterest)that\rotate"atthesameabsolutespeedaboutthe
symmetry axis,but one lo kwise and theother ounter lo kwise.It should
benoted that the \rotationspeed" is dire tly related to theeigenfrequen y
.
Fig.5.Instantaneoustemperaturedistributionsonthevisiblediskofawhitedwarf
modelfor di erent modal angular geometries.Ea h olumnrefers to agivenpair
(l ,m), and overshalf apulsation y lein vephases. The pulsationfrequen y is
arbitraryand remainsunde nedinthisplot.Inea h ase, theangleofin lination
betweentheline-of-sight andtheaxis ofsymmetryofthepulsationhasbeen xed
to 1 radian. The olor ode is su hthat the deepest blue may orrespond to the
highestlo altemperature(abovetheaverageone),thedeepestredmay orrespond
disk twi e during a pulsation y le. This is well illustrated in the rst and
third olumn for thephase orrespondingto 1/4ofafull y le. In ontrast,
runningangular waves(jmj 6=0)nevershow aphasewhen thetemperature
distribution is uniform a ross the visible disk. Finally, for ompleteness, it
should be remembered that pulsation modes with a value of the azimuthal
order m =0 are sometimes referredto as\zonal modes", while those with
l=jmjare alled \se torialmodes".
4.3 Radialmodaldependen e
Figure6illustratesthetypi alradialdependen e ofthe
r and
h
eigenfun -tions orresponding, respe tively, to theradialand horizontal omponentof
thedispla ementve torinapulsatingwhitedwarf.Weagainrefertoour
rep-resentativeZZ Cetistarmodel,andwe onsiderthelowestorderquadrupole
modesfrom g
5 to p
5
,in ludingthef-mode.
The upper panel of the gure depi ts the pro le of
r
as a fun tion of
depthforthef-mode,threelow-orderp-modes(p
1 ,p
3 ,andp
5
),andthethree
orrespondinglow-orderg-modes(g
1 ,g
3 ,andg
5
).Themodeswithradialorder
k=2andk=4werenotplottedinordertonot lutterthediagramtoomu h.
InkeepingwithourremarkaboveinSe tion2relatedtothelinearapproa h,
theamplitudesoftheeigenfun tionsinthelineartheoryofstellarpulsations
havetobearbitrarilynormalizedatsome onvenientvalueandlo ation.Here,
theamplitudeofthe
r
eigenfun tionhasbeennormalizedtothevalueofone
at the surfa eof the model for ea h mode onsidered, i.e.,
r
(r = R ) = 1.
Hen e,a omparisonofthe
r
pro lesforvariousmodesbe omesmeaningful
ifoneunderstandsthattheamplitudes ofallthemodesarenormalizedtobe
thesameat thesurfa eofthestar.
One ansee that the amplitudes of the low-order p-modes are generally
mu h larger than those of the low-order g-modes in the deeper layers. The
latter exhibit nodes mu h higher in the star than the former (these nodes
are, of ourse, thesameas those shownin Figure 4above).It is interesting
to observethat thef-modekeepsthelargest amplitudeoverthefull stellar
model.Also,theg
3
modebehavesdi erentlyfromtheg
1 andg
5
modesinthat
itsamplitudeinthestellar oreremains omparativelylarge,whereasthetwo
othermodesshowverysmallamplitudevaluesatthesedepths.Thisisbe ause
the g
3
mode is partially on ned below the H/He transition layer due to a
resonan e ondition:these ondnodein
r
( ountingfromthesurfa einward)
falls just below the H/He ompositionaltransition zone, while, at the same
time,the rstnodein
h
fallsjustabove.The orollary onditionfortrapping
ofag-modeabovea ompositionaltransitionzoneisthatthen th
nodeofthe
r
eigenfun tionfallsjustabovethetransitionregion,whilethen th
nodeofthe
h
eigenfun tionfallsjustbelow.Thisnotionofmode on nementandmode
Fig. 6. Upper panel: Pro le of theradial eigenfun tion
r
as a fun tionof depth
for thelowestorderquadrupolemodesinourreferen eZZCetistar model.Lower
panel:Absoluteratioofthehorizontaltotheradial omponentofthedispla ement
ve torforthesamemodes.
ofresonan e.Thereaderisreferredtothatpaperformoredetails.Webrie y
revisitthe on eptofmode on nement/trappingbelow.
ThelowerpanelofFigure 6showstheabsoluteratioof thehorizontalto
radial omponentsof thedispla ementve torasafun tion of depth forthe
same modes. This is a parti ularly interesting plot as the results are
inde-pendent of the a tual normalization adopted. That is, whatever the a tual
amplitudeofamode, lineartheorypredi tsagivenamplituderatiobetween
thenodesin the
r
eigenfun tion,whiletheminima orrespondtothenodes
in the
h
eigenfun tion.Asindi atedabove,thenumberofnodesforagiven
mode is thesamefor theradial andhorizontal displa ementeigenfun tions,
but theydonotfallatthesamelo ationsin thestar.
It is quite instru tive to fo us on the amplitude ratio at the surfa e of
the model. Thus, in the observable atmospheri layersof apulsating white
dwarf,matterisdispla edmu h morehorizontallythanverti allyduringthe
pulsation y leofag-mode.Indeed,a ordingtothelowerpanelofthe gure,
the ratio of the horizontal to radial omponents of the displa ement ve tor
alreadyex eeds100forak=3mode,anditin reasesrapidlywithin reasing
radial order. Conversely, the ratio of the radial to horizontal displa ement
in reases rapidly with in reasing radial order for p-modes, rea hing nearly
100forthep
5
modeillustratedin Figure6.The ontrastbetweenhorizontal
and radial displa ements is even larger for l = 1 modes, being some three
timeslargerforthek=3g-modethantheg
3
quadrupolemodeshowninthe
gure. It is therefore ertain that the g-modes observed in pulsating white
dwarfs orrespondto material motionsthat are essentiallyhorizontal in the
super iallayers.Theverylargesurfa egravity hara teristi ofwhitedwarfs
(logg 8) isattheoriginofthisphenomenon.
4.4 Kineti energy
Aninterestingglobalpropertyofapulsationmodeisitskineti energyde ned
bythegeneralrelation,
E kin 1 2 Z V v 2 dV , (11)
wheretheintegrationis arriedoutoverthetotalvolumeo upiedbythestar.
If one negle ts the rotation of the star and any other possible ma ros opi
velo ity eld (su h as onve tion ormeridional ir ulation),one anexpress
thekineti energyintermsofaradialintegralinvolvingthetwoeigenfun tions
r and h , E kin 2 2 R Z 0 r (r) 2 +l(l+1) h (r) 2 r 2 dr . (12)
Given thearbitrary normalization ofthe eigenfun tions in lineartheory,
thekineti energyofamodeisknownonlytowithinamultipli ativefa tor.In
pra ti e,thismeansthat onlyrelative omparisonsbetweenthekineti
ener-giesofdi erentmodeshaveaphysi alsense.Hen e,toex iteamodewiththe
sameobservableamplitudeatthesurfa eofastarasthatofareferen emode
requires (moreor less)energy,and thatin rementisgivenbythedi eren e,
E
kin
,betweenthekineti energiesofthetwomodes.
Fig. 7. Kineti energyasafun tionofperiodfor thefamilyof quadrupolemodes
with periods inthe rangefrom 1s to 1000 s as omputed usingourreferen e ZZ
Cetimodel.
1s and 1000 s as omputed onthe basis ofour representativeZZ Ceti star
model. Given the behavior of its displa ement eigenfun tions { as an be
observedinFigure 6{in onjun tionwithequation (12),itisnotsurprising
thatthef-mode omesoutwiththelargestkineti energyofthelot.Indeed,
theamplitudeofthefundamentalmode(bothin
r and
h
)isbasi allylarger
than anyother mode overthe whole stellar model. This means that the f
-mode requires the most energy to be ex ited to a given surfa e amplitude.
Sometimes,su hamodeisreferredtoastheonewiththemost\inertia".
Thefa t that thelow-orderp-modeshavesigni antlylargeramplitudes
than their g-mode ounterparts in thedeeperregions of a white dwarf (see
againFigure6), ombinedwiththelargevaluesofthedensityatthesedepths
ompared toits envelopevalues,readilyexplainswhythekineti energiesof
thep-modesaremu hlargerthanthoseoftheg-modes.Thisisanimportant
hara teristi ofwhitedwarfstars.Inaddition,theeigenfun tionsdepi tedin
Figure6forbothp-modesandg-modesexhibitanoutward\migration"with
the signi ant drop of the kineti energy with in reasing radialorder along
thetwobran hesofnonradialmodes.
The ase of g-modes merits further dis ussion as these are the modes
observedinrealpulsatingwhitedwarfs.Forinstan e,amodelikeg
3
inFigure
7 shows a lo al maximum in kineti energy along the g-bran h. As brie y
alluded to above, su h a mode an be seen as partially on ned below the
H/He transition zone. It exhibits the larger amplitudes below that region
as ompared to other g-modesand, therefore, it hasalargerkineti energy.
Conversely,theg
6
modeshowsalo alminimumin kineti energyduetothe
fa tthatittendstobepartiallytrappedabovetheH/Hetransitionzone,with
the onsequen ethat ithasloweramplitudes belowthat regionas ompared
toitsimmediateadja entmodes.
4.5 Weight fun tion
Ithasbeenshownby[27℄(andseealso[7℄)thattheeigenvaluesoftheadiabati
pulsationequations anbeestimatedfromavariationalapproa h.Forapurely
spheri almodel,thesquareofaneigenfrequen yisgivenby,
2 = D A , (13)
where D and A are two integralexpressions involving the eigenfun tionsof
that eigenfrequen y. This expression an be used, after the solution of the
eigenvalueproblemhasbeenobtained,to deriveavariationalestimateofthe
eigenfrequen y.Whilethisisgenerallylessa uratethantheresultprovided
bytheeigenvalueitself, inpartdue tothefa tthat theboundary onditions
used in the variational approa h are notexa tly the same asthose used in
the eigenvalueproblem, theapproa h isoften used asameasure of internal
onsisten y.Ofgreaterinterest,however,isthefa tthatequation(13)involves
integralexpressionsoverthestellarmodel.WhiletheintegralAappearingin
the denominatoris simply proportionalto the kineti energy de ned above,
theintegrandoftheintegralDappearinginthenumeratorprovidesameasure
of the ontribution of ea h shell in the stellar model to the overall integral.
As su h, theintegrandofthe integralD foragivenpulsationmode isoften
referredtoasthe\weightfun tion" ofthemode.This on eptisveryuseful
forinferringwhi hregionsofastellarmodel ontributemosttotheformation
ofamode.TheintegralD maybewritten,
D= R Z 0 2 r N 2 + (P 0 ) 2 1 P + 0 P 0 1 P + r N 2 g r 2 dr , (14)
where twootheradiabati eigenfun tionsexpli itlyappear: 0
(the Eulerian
perturbationofthegravitationalpotential),andP 0
Fig. 8. Upper panel: Normalizedweight fun tionas afun tionofdepthfor afew
low-orderquadrupolemodesinourreferen eZZCetistarmodel.Thenormalization
insuresthattheareaunderea h urveisthesame.Lowerpanel:Normalizedweight
fun tionfora on nedmode(g3)andatrappedmode(g6).
Figure8illustratestheweightfun tionsforafewofthelow-orderquadrupole
pulsationmodesthat havebeendis ussed aboveforourrepresentativewhite
dwarfmodel.The weightfun tion of ea h mode isnormalizedsu h thatthe
areaunderea h urveisthesame.Inkeepingwithourpreviousdis ussion,it
isnotsurprisingtoobserveintheupperpanelthatap-modeofagivenradial
thephysi al onditionsintheinteriorthang-modesdo.Thelatteraremostly
envelopemodesin oolpulsatingwhitedwarfssu hasZZ Cetistars.
ThelowerpanelofFigure8showstheweightfun tionsofthequadrupole
modes g
3 and g
6
that have been dis ussed above. The former is partially
on ned belowthe H/Hetransition zone entered onlog q ' 6, whilethe
latterispartiallytrappedabovethat ompositionaltransitionregion.Clearly,
their weight fun tions re e t this state of a air. It should be evident that
this on ept of mode on nement/trapping must be seenas the result of a
(relativelyweak)partialwavere e tioninthetransition layers,allowingthe
on nedmodetoextendabovethetransitionregion,andthetrappedmodeto
extendbelowthat region.Themodesremainglobal(asalways),buttheyare
moresensitivetothe onditionsfoundaboveorbelow,dependingonwhether
ornottheyaretrappedor on ned.
It is very instru tive to investigate the evolution of the weight fun tion
ofapulsationmodealongtheevolutionarytra kfollowedbya oolingwhite
dwarf.Inthisway,one anappre iatethe hangeofregimefromtheGWVir
hotphaseofevolutiontothe oolZZCetiphase(whi hhasbeenemphasized
throughtheuseofourreferen emodelin theillustrativeexamplespresented
sofar).Alongthe oolingtra k(see,e.g.,the urvesshownin Figure1),the
overalldegenera yofawhitedwarfmodelin reasesandthispushestheregion
of g-mode formationoutwards.This implies that mode sensitivity to model
parameters hanges from theGWVir phasetothe V777Herphaseandthe
ZZCeti regime.
This notion of hangingmode sensitivity along the white dwarf ooling
tra k is well illustrated in the toppanel of Figure 9.In order to avoidthe
ompli ations ausedby hemi allayering,i.e.,mode on nementand
trap-pingwhi hwould ausetheweightfun tionnottobehavemonotoni allyand
onfusetheplot,theevolutionofapureCmodelwas onsidered.The gure
showstheweightfun tion ofthelowest-order(k =1)dipoleg-modein terms
of depth, andin terms ofdi erent phasesof oolingasquanti edby the
ef-fe tivetemperature.Theweightfun tionisagainnormalizedsothatthearea
under ea h urveisthesame.Figure9 learlyrevealstheoutwardmigration
of theregionofg-mode formationwith ooling.Thisimpliesthat the
pulsa-tionmodesofawhitedwarf(theg-modesasobserved)progressivelylosetheir
abilitytoprobethedeepinterior(GWVirregime)andbe omemoresensitive
to the details of the outermostlayers(ZZ Ceti regime) as oolingpro eeds.
It is parti ularlyobviousherethat the k =1, l=1g-modein the10,117 K
model doesnotprobethe oreverywell.
Thepanelat thebottomofFigure9illustrates,in ontrast,thefa t that
there is littlemigration of theregion of mode formationfor ap-modein an
evolving white dwarf. There is some outward migration, parti ularly below
30,000K,butthee e t remainsmild.Hen e,evenatT
e
=10,117K,thep
1
modestillprobesthedeep oreasitdidatmu hhighere e tivetemperatures.
Fig. 9. Upper panel:Evolution ofthe normalizedweight fun tionfor the
lowest-orderdipoleg-modeinaseries of models ulled fromanevolutionarysequen eof
pureCwhitedwarfs.Lower panel:Similar,butforthelowest-orderdipolep-mode.
reason forthis is to befound in the di erent behavior ofthe Brunt-Vaisala
frequen y omparedtothat oftheLamb frequen y.
4.6 Rotational splitting
Thevariationalapproa hputforwardby[27℄isagainofgreatusefor
estimat-ingthee e tsofslowrotationontheeigenfrequen yspe trumofapulsating
tationasaperturbationandleadsto orre tionstotheeigenfrequen iesthat
are given in terms of integral expressions involving the unperturbed
eigen-fun tions,i.e.,those omingoutofthesolutionoftheeigenvalueproblemfor
purelyspheri allysymmetri models(su has
r and
h
en ountered above).
Themain e e t of slow rotationis to destroythespheri al symmetryof
the star and, asa onsequen e, the (2l+1)-fold degenera y that exists for
theeigenfrequen iesofmodeswithdi erentvaluesofmbutbelongingtothe
samepair(k,l)inanonrotatingmodelislifted.Undertheassumptionthatthe
angularrotationfrequen yisasimplefun tion ofdepth,(r),thefrequen y
of a mode, now de ned by the three indi es k, l, and m, is given through
rst-orderperturbationtheoryby,
k lm ' k l m Z R 0 (r)K k l (r)dr , (15) where k l
is the frequen y of the degenerate modes (k,l) in the absen e of
rotation,andthese ondtermontherightsideisthe rst-order orre tionto
that frequen y,withmtaking onthevalues l; l+1;:::;l 1;l.The
quan-tity K
k l
(r) appearing in the orre tion termis referredto as the rst-order
rotation kernel. It obviously plays the role of a weight fun tion,verymu h
similar to the weight fun tion dis ussed above for the eigenfrequen y, but,
thistime, referringtotheregions ontributingtothefrequen ysplittingdue
torotation.Itisgivenbythefollowingexpressioninvolvingtheunperturbed
eigenfun tions, K k l (r)= f 2 r +[l(l+1) 1℄ 2 h 2 r h gr 2 R R 0 f 2 r +l(l+1) 2 h gr 2 dr . (16)
Ifthestarisfurtherassumedto rotateasasolidbody(6=(r)),equation
(15)redu esto, k lm ' k l m(1 C k l ) , (17)
whereisthe(uniform)angularrotationfrequen y,andC
k l
isa
dimension-lessquantitynamedthe rst-ordersolidbodyrotation oeÆ ient(orLedoux
oeÆ ient).Itisgivenby,
C k l = R R 0 f2 r h + 2 h gr 2 dr R R 0 f 2 r +l(l+1) 2 h gr 2 dr . (18)
Giventhesolutionoftheeigenvalueproblemforamode,equation(18)allows
theevaluationoftheC
k l
oeÆ ientforthatmode.Notethat rst-ordersolid
body rotation leads to a set of equally-spa ed frequen ies with a splitting
betweenadja entfrequen y omponentsgivenby,
Figure10showsthevaluesofthe rst-ordersolidbodyrotation oeÆ ient
for a series of quadrupole modes (the same as those illustrated in Fig. 7)
overing the period interval 1 1000 s, and referring to our representative
whitedwarfmodel. Asis the asefor alltypesofstars,thevaluesofC
k l for
p-modes tend to be ome verysmall ompared to 1 in the asymptoti limit
of highradialorder.In ontrast,thevaluesofC
k l
forg-modestend towards
anonzero value,1=(l(l+1)), in the samelimit,as anbeseenin the gure
forl=2.Notethatthelowest-orderg-modes{ofmostinterestforpulsating
white dwarfs { show signi antly di erent values of C
k l
from one mode to
another,agood thingbe ausethismayhelpin onstrainingtheradialorder
ofamodeversusanotheronein presen eof observedrotationalsplitting.
Fig. 10. First-order solidbodyrotation oeÆ ient asa fun tionof periodfor the
familyofquadrupolemodeswithperiodsintherangefrom1sto1000sas omputed
onthe basisof ourreferen e ZZCeti starmodel. Theformat is similartothat of
Fig.7above,andthemodes onsideredarethesame.
Figure11showsexamplesoffrequen yspe tra,withandwithoutrotation,
obtainedfromourreferen ewhitedwarfmodel.Arelativelysmallvalueof3
hwasassumedfortherotationperiodofthemodelinorderto learlyseethe
rotational splitting has beendete ted and used to inferthe rotation period
of some 14 pulsatingwhite dwarfsso far (see Table 4of [17℄). The inferred
rotationperiodsvaryfrom 5hto55h, whi h isagainquiteslow.
Fig. 11. Rotationalsplitting inour representative ZZ Ceti star model. The
low-orderg-modefrequen y spe trafor bothdipole andquadrupole modes,with and
withoutrotationturnedon, areillustrated.Itisassumedthatthestarrotatesasa
solidbodyandwitha(relatively)shortperiodof3h.Degeneratedipolemodessplit
into triplets,whilequadrupolemodessplitintoquintuplets. Thespa ings between
adja ent omponentswithin a given multipletare the samein frequen yspa e as
shownhere.
5 Period Evolution
Anotherobservableofhighpotentialinterestforwhitedwarfasteroseismology
istherateofperiod hangeofamodeinagivenpulsatingstar.Iftheperiod
hangeisdueprimarilytothese ularevolutionofthestar,andnottoexternal
planet-propertiesor on rm/testthe propertiesderivedfrom usingtheperioddata
in astandardasteroseismologi alexer ise.
Fig. 12. Representative examples of period evolution for dipole g-modes during
the GWVir, V777Her,and ZZCetiphases. Notetheo setinagebetweenthese
di erentphasesofevolution.Theradialorderk isindi atedforsomeofthemodes
depi tedhere.
Figure12ismeanttoillustratehowpulsationperiodstypi allyevolvefor
thethreemain ategories ofpulsatingwhitedwarfsthatareknown:theGW
Vir,V777Her,andZZCetistars.Inea h ase,theevolvingmodelisspe i ed
byatotalmassof0.6M
andauniform ore ompositionmadeof arbonand
oxygen in the sameproportions by mass fra tion. Theenvelope in the GW
Vir model is made of a representative PG1159 omposition (X(He)=0.38,
X(C)=0.40,andX(O)=0.20)and ontains10 2
ofthetotalmassofthestar.
TheenvelopeoftheV777Hermodel alsohasanenveloperepresenting10 2
of the total mass of the star, but it is made of pure helium. In the ase of
theZZCeti model, thereisapureheliummantle ontainingamassfra tion
of10 2
surroundedbyapurehydrogenenvelope ontaining10 4
ofthetotal
mass.Giventhedi erenttimes alesinvolved{theevolutionofawhitedwarf
respe ttosomearbitraryzeropoint)astheabs issaandweintrodu edashift
of 0.80( 1.45)dexfortheV777Her(ZZCeti)model.
Forea htypeof pulsatorofinterest, thelowerpartof thedipoleg-mode
spe trum (starting with k = 1) was omputed for several equilibrium
on- gurations mapping the instability strip. Inthe ase of the GW Vir phase,
theequilibriummodels onsidered overthe\turningofthebend"intheHR
diagram(see,e.g.,Fig.1) orrespondingtothe nal ontra tionphaseduring
whi h thestar gets hotter andmore ompa tand the pulsation period of a
modede reases,followedbythebeginningofthe oolingphaseduring whi h
thestargets oolerand(slightly)more ompa tandthepulsationperiod
in- reases.Theperiodofag-modeinaGWVirmodelgoesthroughaminimum
pra ti allywhenthemodelrea hesitsmaximume e tivetemperatureinits
ex ursion in theHR diagram.Things are simplerfor the V777Her and ZZ
Ceti families whi h orrespondtopurely oolingphases and,therefore, su h
starsexhibitpulsationperiodsthatin reasemonotoni allywithpassingtime,
althoughnotatthesameratefromonemodetoanotheras anbeappre iated
in Figure 12. Thein rease of the pulsation period of ag-modein a ooling
whitedwarfisintimatelyrelatedtotheoutwardlygrowingdegenerateregion
whi hhasthee e t ofloweringthevalueoftheBrunt-Vaisalafrequen y. 3
It should be noted that the phases overed in Figure 12 en ompass the
empiri alinstabilitystripsandaremeantto beprimarilyillustrative.For
in-stan e,the oolestmodel onsideredintheGWVirphaseissomewhat ooler
at T
e
' 77,000 K than the a tual observed red edge. Also, the V777 Her
phasemapsarangeofe e tivetemperaturesfromabout28,000Kto 21,000
K,whiletheZZ Cetiphase oversanintervalfrom about13,000Kto about
10,300K,somewhatlargerthanthewidthsoftheobservedinstabilitystrips.
It shouldfurther benotedthat the gurerevealssigni antwavystru tures
in the omputed period distributions,most obvious for the GW Vir phase.
These stru tures, sometimes referredto as\mode bumpings" and \avoided
rossings",areduetothephenomenonofmodetrapping/ on nementat
om-positionaltransitionlayersinourwhitedwarfmodels.Purelyradiativemodels
withauniform hemi al ompositionwouldnotshowthese wavyfeatures.
Inthe aseoftheGWVirmodel,inparti ular,thenodesofagivenradial
overtone rstmigrateinwardsandthen outwardsasthemodelturns around
the bend in the spe tros opi HR diagram. These nodes su essively pass
through the omposition transition region at the interfa e of the C/O ore
andtheenvelope,where onditionsforpartialmode on nementortrapping
aremet.Hen e,agivenmodeisalternativelypartially trappedandpartially
on ned as afun tion oftime, and this produ es theperiod variationsthat
anbeseeninFigure12.Forhigherordermodes,thenodesaremore
numer-ousand losertogether, sothere aremore\waves"in theirtemporal period
distributionas analsobeobservedin the gure.
3
Brunt-Fig.13.Upperpanel:Evolutionoftheratesofperiod hangeforthreerepresentative
dipolemodes(k=1,k=20,andk=40)a rosstheGWVirinstabilitystrip.The
evolutionarypathsareshownbythesolid urvesandstartwithadot.Thedotted
urvesshowthevaluesofdP=dtforthe40overtonesatthreedistin tepo hsduring
theevolution.Middlepanel:Similar,butforthek=1,k=14,andk=27overtones
a rosstheV777Herinstabilitystrip.Also,onlytheinitialand naldistributionsfor
the 27modesare shownbythe dotted urves. Lower panel:Similar tothemiddle
panel,butforthek=1,k=13,andk=25modesintheZZCetistrip.
Figure13showssomeresultsforvaluesoftheratesofperiod hange
om-putedfromtheseevolutionarysequen es.Asexpe tedfromthewavystru ture
ompli ated. Forinstan e, in theupperpanelof Figure 13,theevolutionary
pathsfollowedbydP=dtfordipoleg-modeswithk=1,k=20,andk=40are
illustratedby thesolid urves.The startingpointfor ea hpath isindi ated
byasmalldot,andthis orrespondstoanearlyepo hwhenthemodelenters
theGWVir regionatlowgravityand (relatively) lowe e tivetemperature,
still ontra ting and getting hotter. The dotted urve onne ting the three
\starting"pointsshowsthespe trumof dP=dt valuesforthe rst40 dipole
overtoneswhenthemodelentersthestrip.Thequasi-periodi behavioralong
the spe trum is again due to mode trapping/ on nemente e ts. The
spe -trum is equivalent to measuring the slope for ea h of the 40 modes onthe
left side of the 40 urves shown in Figure 12 and referring to the GW Vir
regime.Inthisearlyphase,allslopesarenegativeandthedP=dtvaluesfora
ontra tingmodelofapre-whitedwarfareallnegative.
In ontrast, by thetime the model exits the GWVir domain asa
high-gravity ooling white dwarf with T
e
' 80,000 K, all values of dP=dt are
positive(topdotted urveintheupperpanelofthe gure).Inbetween,there
exist epo hs when, in agiven model, one an nd bothmodeswith positive
valuesoftherateofperiod hangeandmodeswithnegativevalues,depending
on theirradial order.This is illustratedby the middle dotted urvein that
samepanel, whi h orrespondsto anevolutionary phasenearthe turningof
thebendin theHR diagram.Thisparti ular ir umstan e,ifobserved,may
be of high value for pinning down the pre ise evolutionary status of a GW
Virpulsator.
Theevolutionarypaths shownin Figure 13 are ompli ated in that they
againre e tthee e tsofmodetrapping/ on nement.Forinstan e,thepath
forthek=40mode rossesthezerovalueseveraltimes,meaningthattherate
of period hange for that parti ularmodeis initiallynegative,then hanges
signafewtimesaroundtheturningofthebend,and nallytakesonapositive
valuebythetime thestarleavestheGW Virregion.Thispathisequivalent
to measuringtheslopealongtheGWVirk =40 urveinFigure 12.
For their part, dP=dt values for g-modes due to se ular evolution are
alwayspositivefor V777Her and ZZCeti pulsatorsas an be seenin both
Figures 12and 13. This isdue to the fa t that whitedwarfsin these
evolu-tionaryphasesarepurely oolingbodiesand thattheoverallde reaseofthe
Brunt-Vaisalafrequen yintheirinternalregionsduetothegrowing
degener-a ypushestheperiodsofg-modestohighervalues.Ofnotableinterest,Figure
13alsorevealsthat thevaluesofdP=dtformodesof omparableradialorder
de rease substantiallyalongthe whitedwarf oolingsequen e.Forinstan e,
the typi alorder of magnitude for the rate of period hange for a GW Vir
pulsator is dP=dt10 11
s/s,foraV777 Herstar itis dP=dt10 13
s/s,
and for aZZ Ceti pulsator it drops to the low valueof dP=dt 10 14
s/s.
These numberssimplyre e ttheverydi erentevolutionarytimes alesthat
hara terizethesethreephases.Thenumbersshowthatitismu hmore
ItshouldberealizedthatthemeasurementofdP=dtforamodeina
pul-satingwhitedwarf,assumingthatsu hmeasurementisavailableand redible,
is noteasyto interpretwithouthaving rstderivedareliableseismi model
forthepulsator under s rutiny. Su h amodel, obtainedfromthe analysisof
theperiod data(andadditionalinputsu hasmulti olorphotometryasmay
bethe ase),providesestimates ofthestru tural parametersof thepulsator
andamodeidenti ation.Theseareessentialingredientsrequiredfor
inter-pretingthemeasurementoftherateofperiod hange.Andindeed,it anbe
shown that agiven value of dP=dt may be shared by modes with di erent
radial order or degree index, or by models with di erent masses, envelope
layerings, ore ompositions, and e e tive temperatures. Hen e, it is
abso-lutelyne essaryto knowtheseparameterstoagoodlevelofa ura ybefore
attemptingto exploitthedP=dtdata.
Figure14summarizestheresultsofnumeri alexperimentsfeaturing
rep-resentativemodesofinterest(dipoleg-modeswithk=1,8,and15)formodels
of ZZ Ceti stars. It shows how the values of dP=dt for these modes hange
asthemodels oolthroughtheZZ Cetiinstabilitystrip.Intheupperpanel,
twosimilar models areused: they both haveatotalmass of0.6 M
,a
uni-form ore ompositionmadeof arbonandoxygeninthesameproportionsby
massfra tion,andapureheliummantle ontainingamassfra tion of10 2
,
but theydi erin thatone(dashed urves) hasapureH outermostenvelope
ontainingamassfra tionof10 4
(letus allitthe\referen emodel"inthis
Se tion),whiletheother(solid urves)hasamu hthinnerhydrogenlayerof
massfra tion 10 10
.Inthesamespirit, themiddlepanelof the gurerefers
to thereferen eevolutionarymodel(dashed urves)and asimilar one(solid
urves) di eringonly in its total mass, now equalto 0.9 M
. Likewise, the
lowerpanelreferstoadditionalmodelssimilartothereferen emodel(dashed
urves), but one having a pure C ore omposition (solid urves), and the
otherapureO ore omposition(dotted urves).
Therearesomedi eren esbetweenthevaluesofdP=dtforthelowest-order
(k = 1) mode onsidered in Figure 14 depending on themodel parameters,
butthesedi eren esremainsmallonthes alesusedhere,andtheyaremu h
smallerthanthosefoundforthehigherordermodes.Generallyspeaking,the
rates of period hange are highly dependent on modal and model
parame-ters.It shouldthereforebe learfromthisthat theinterpretation ofadP=dt
measurementmustrestontheavailabilityofareliableseismi model.
6 The Nonadiabati Approa h
6.1 Basi onsiderations
Initsfull version,thelineartheory ofnonradialstellarpulsationstakesinto
Fig. 14. E e ts of varying the envelope layering, the total mass, and the ore
ompositionontheratesofperiod hangeforrepresentativedipoleg-modes(k=1,
8,and15)inevolvingZZCetistarmodels.
energy onservationequationandtheenergytransferequation,arethus alled
uponattheoutset.Afterlinearization,oneendsupwithasystemof6linear
di erentialequationswith6dependent omplexeigenfun tions. 4
As ompared
to the adiabati approximation (whi h leads to a set of 4 linear equations
4
Thisistrueformodelsinwhi honlyradiative(and ondu tive)transportistaken
into a ount, although theone extreme treatment of onve tion that has been
usedmostoftenforwhitedwarfs,theso- alledfrozen onve tionapproximation,
with real variables), the full approa h be omes a mu h more ompli ated
problemfromanumeri alpointofview.Thisisthepri etopay,however,to
verifyif apulsationmode isex ited ornotin a stellarmodel. Likewise,the
full nonadiabati approa h is ne essaryto understand why apulsating star
pulsates,andtomapinstabilitystrips,amongotherthings.
Inthefullnonadiabati version,theeigenfrequen y ofamodeisa
om-plexnumberanditstemporaldependen etakesontheform,
e it = e i(R+iI)t = e iRt e It , (20) where R
is the (angular) os illation frequen y of the mode, and
I (also
expressedin rads 1
)isdire tlyrelatedtotheso- allede-foldingtimeofthe
mode de ned by
e
= 1=
I
. (In somereferen es, the term \growth rate"
is also de ned by the expression =
I = R ). When I is positive, the
amplitudeof theinitialperturbationde aysa ordingto equation(20),and
the mode issaid to bestable, damped, ornot ex ited, andthe mode is not
expe tedto beobservable.Conversely,when
I
isnegative,theamplitudeof
theinitialperturbationblowsupexponentially(thisislineartheory),andthe
modeissaidtobeunstable,driven,orex ited.Su hamodemaygrowto an
observableamplitudein a realpulsator.Note that, quitegenerallyin stellar
modelsandespe iallyforlow-ordermodes,j
R j j
I
j,whi hpartlyexplains
whytheadiabati approximationisusuallyjusti edfor omputingos illation
frequen ies(periods)atasuÆ ientlevelofa ura y.
Innonadiabati pulsationtheory,oneveryuseful on eptisthatofa\work
integral"whi hmaybeevaluatedfromthe(nonadiabati )eigenfun tionsafter
the solution of the eigenvalue problem has been obtained. In a way similar
to the weight fun tion dis ussed in Subse tion 4.5 above, the integrand of
theworkintegralindi ateswhi hregionsofastellarmodel ontributetothe
driving of apulsation mode, and whi h regions ontributeto damping. The
integrandoftheworkintegralmaybewritten intheform,
dW dM r = R Re ÆT T Æ N Æ 1 r(F R +F C , (21)
where the terms have their standard meaning. The nu lear term, Æ
N , has
yettoproveitsrelevan eforpulsatingwhitedwarfs,althoughit oulddrive,
throughtheappropriatelynamed-me hanism,short-periodg-modesinsome
modelsofGWVirstarswithresidualHeshellburninga ordingto[37℄(and
seealso[12℄).Shortperiodpulsationsofthetypehaveyettobedis overedin
GWVirstars,however,anditisnowgenerallya knowledgedthatthemodes
observedinthepulsatorsofthekindaredrivenbytheso- alled-me hanism
involvingthe modulation of the radiative ux F
R
around an opa ity bump
in the envelope (see, e.g., [31℄, and referen es therein). The -me hanism,
thistime basedonHshellburning, ouldalsodriveshort-period (40 125s)
thedotted urveshowninFigure1indi ates thepredi tedinstabilityregion.
Nolow-massDAOwhitedwarfhasbeenfoundto pulsateyet(see,in
parti -ular,[22℄),buttheseobje tsarequiterareandthejuryisstill outaboutthe
possibility.Thisisparti ularlytrueinthelightofthere entstudyof[20℄who
haveshownthatthepresumedpost-EHBDAOwhitedwarfsstudiedfor
vari-abilityby[22℄ are,in fa t,post-AGBwhitedwarfs.Inthe asesof theV777
HerandZZCetistars,residualshellburningis ompletelynegligibleandthe
-me hanism annotoperate.Instead, themodulations ofboththeradiative
and onve tive uxmustbetakenintoa ountinthedrivingpro essasthese
oolerwhitedwarfshavedevelopedextensivesuper ial onve tionzonesthat
intera t with the pulsations.Forla kof a better approa h, stabilitystudies
of white dwarfsof thekind have largelybeen basedon the so- alled frozen
onve tion approximation,whi h onsists in negle ting the perturbations of
the onve tive ux.Fortunately,progresshasbeenmadere entlyonthisfront
byimplementing,for the rsttime, anapproa hbasedonatime-dependent
treatmentof onve tioninawhitedwarf ontext([15℄,[33℄, [42℄)
The sign of the integrand dW=dM
r
indi ates if, lo ally, a region of the
star hasastabilizingordestabilizinge e t onamode.If dW=dM
r
<0,the
region will be short of energy after a pulsation y le and that energy will
be taken from the kineti energy of the mode. The amplitude of the mode
tends thento de reaselo ally,and theregion ontributesto dampingofthe
mode.If,ontheotherhand,dW=dM
r
>0,theregionendsupwithapositive
energy in rement after a y le, whi h is transfered as extra kineti energy
to the mode. The amplitude has then a tenden y to grow lo ally, and the
region ontributestodriving.Theglobalstabilityofamodeisdeterminedby
summingoverthe ontributionsofallregionsinamodel,andthisisthework
integralgivenby,
W = M Z 0 dW dM r dM r . (22)
If W <0,damping dominatesoverdriving and themodeis globally stable.
Conversely,ifW >0,themodeisgloballyex ited.
6.2 Ex itation ofpulsation modes in white dwarfs
Thephenomenonultimatelyresponsiblefordrivingpulsationmodesin white
dwarfs is the partial ionization of the main envelope onstituents. Indeed,
theK-shellele tronsof arbonandoxygenionizeandthenre ombineduring
the ex ursion around the bend in the GW Vir phase of the evolution of a
H-de ient,post-AGBstar,heliumre ombinesinthestri ly oolingphaseof
aHe-atmospherewhitedwarf orrespondingtotheV777Herregime,andso
doeshydrogenintheeven oolerphaseofaH-atmospherestar orresponding
on-ea h ase,partial ionizationleads to averyimportant in reaseof the
enve-lopeopa ity,and thistendsto hoketheoutgoingenergy ux.Inthe aseof
V777 Her models, and even moreso in Hot DQ models and ZZ Ceti stars,
theopa itybump be omesso largethat asuper ial onve tionzone
devel-opsasaresultofthebuildupofasuperadiabati temperaturegradient,and
this signi antly a e ts the me hani sof the a tual pro ess responsible for
theex itationofpulsationmodes.Conve tiveenergytransportmustthenbe
taken into a ount in addition to the usual radiative hannel. Figure 15
il-lustrates some opa ity pro les and onve tive ux pro les in the envelopes
of typi al models of pulsating white dwarfs. The monotoni in reaseof the
opa itymaximumwithde reasinge e tivetemperature(GWVir,V777Her,
Hot DQ,ZZCeti)isnoteworthy.
Fig. 15. Opa itypro les(heavy urves)intheenvelopesofrepresentativemodels
ofthefourtypesofpulsatingwhitedwarfs.Theratioofthe onve tivetototal ux
is also plotted (thin urves), ex eptfor the GW Vir modelin whi h there is no
onve tion.
