Pulsating low-mass white dwarfs in the frame of new evolutionary sequences: IV. The secular rate of period change

2.3 MB·

by  Leila M. Calcaferro

Astronomy&Astrophysicsmanuscriptno.paper (cid:13)c ESO2017
February1,2017
Pulsating low-mass white dwarfs in the frame of
new evolutionary sequences
IV. The secular rate of period change
LeilaM.Calcaferro1,2,AlejandroH.Co´rsico1,2,andLeandroG.Althaus1,2
7
1
0 1 Grupo deEvolucio´n EstelaryPulsaciones, FacultaddeCiencias Astrono´micasy Geof´ısicas,
2
UniversidadNacionaldeLaPlata,PaseodelBosques/n,1900,LaPlata,Argentina

n
2 Instituto de Astrof´ısica La Plata, CONICET-UNLP, Paseo del Bosque s/n, 1900, La Plata,
a
J Argentina

1
e-mail:lcalcaferro,acorsico,[email protected]
3

Received;accepted
]
R
S ABSTRACT
.
h
p Context.Anincreasingnumberoflow-mass(M⋆/M⊙ . 0.45)andextremelylow-mass(ELM,
-
o M⋆/M⊙.0.18−0.20)white-dwarfstarsarecurrentlydiscoveredinthefieldoftheMilkyWay.
r Someofthesestarsexhibitlong-periodg-modepulsations,andarecalledELMVvariablestars.
t
s
a Also,somelow-masspre-whitedwarfstarsshowshort-period p-mode(andlikelyradial-mode)
[
photometricvariations, and aredesignated aspre-ELMVvariable stars.Theexistence ofthese

1 newclassesofpulsatingwhitedwarfsandpre-whitedwarfsopenstheprospectofexploringthe
v
binaryformationchannelsoftheselow-masswhitedwarfsthroughasteroseismology.
0
8 Aims.Wepresentatheoreticalassessmentoftheexpectedtemporalratesofchangeofperiods(Π˙)
8
forsuchstars,basedonfullyevolutionary low-massHe-corewhitedwarfandpre-whitedwarf
8
0 models.
.
1 Methods.Ouranalysisisbasedonalargesetofadiabaticperiodsofradialandnonradialpulsa-
0
tionmodescomputedonasuiteoflow-massHe-corewhitedwarfandpre-whitedwarfmodels
7
1 with masses ranging from 0.1554 to 0.4352M⊙, which were derived by computing the non-
:
v conservativeevolutionofabinarysystemconsistingofaninitially1M⊙ZAMSstaranda1.4M⊙
i
X neutronstarcompanion.
r Results.Wecomputethesecularratesofperiodchangeofradial(ℓ=0)andnonradial(ℓ=1,2)
a
gand pmodesforstellarmodelsrepresentative ofELMVandpre-ELMVstars,aswellasfor
stellarobjectsthat areevolving justbeforetheoccurrence of CNOflashesattheearlycooling
branches.WefoundthatthetheoreticallyexpectedmagnitudeofΠ˙ ofgmodesforpre-ELMVs
arebyfarlargerthanforELMVs.Inturn,Π˙ ofgmodesformodelsevolvingbeforetheoccur-
renceofCNOflashesarelargerthanthemaximumvaluesoftheratesofperiodchangepredicted
forpre-ELMVstars.Regarding pandradialmodes,wefoundthatthelargerabsolutevaluesof
Π˙ correspondtopre-ELMVmodels.
Conclusions.Weconcludethatanyeventualmeasurementofarateofperiodchangeforagiven
pulsatinglow-masspre-whitedwarforwhitedwarfstarcouldshedlightabout itsevolutionary
status.Also,inviewofthesystematicdifficultiesinthespectroscopicclassificationofstarsofthe
1
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ELMSurvey,aneventualmeasurementofΠ˙ couldhelptoconfirmthatagivenpulsatingstaris
anauthenticlow-masswhitedwarfandnotastarfromanotherstellarpopulation.
Keywords.asteroseismology,stars:oscillations,whitedwarfs,stars:evolution,stars:interiors,
stars:variables:general
1. Introduction
Low-mass (M /M . 0.45) white dwarfs (WD) are probably produced by strong mass-loss
⋆ ⊙
episodesattheredgiantbranchphaseoflow-massstarsinbinarysystemsbeforetheHe-flashonset
(Althausetal.2010).SinceHeburningisavoided,theyareexpectedtoharborHecores,incontrast
to average-mass(M ∼ 0.6M ) C/O-coreWDs. Inparticular,binaryevolutionisthe mostlikely
⋆ ⊙
originfortheso-calledextremelylow-mass(ELM)WDs,whichhavemassesM ∼0.18−0.20M .
⋆ ⊙
Theevolutionoflow-massWDsisstronglydependentontheirstellarmassandtheoccurrenceof
elementdiffusionprocesses(Althausetal.2001).Althausetal.(2001,2013);Istrateetal.(2016b)
havefoundthatelementdiffusionleadstoadichotomyregardingthethicknessoftheHenvelope,
whichtranslatesintoadichotomyintheageoflow-massHe-coreWDs.Specifically,forstarswith
M &0.18−0.20M ,theWDprogenitorexperiencesmultiplediffusion-inducedCNOthermonu-
⋆ ⊙
clearflashesthatconsumemostoftheHcontentoftheenvelope,andasaresult,theremnantenters
its finalcooling track with a verythin H envelope.The resulting objectis unableto sustain sub-
stantialnuclearburningwhileitcools,anditsevolutionarytimescaleisrathershort(∼ 107−108
yr). On the contrary,if M . 0.18−0.20M , the WD progenitordoesnot experienceH flashes
⋆ ⊙
at all, and the remnant enters its terminal cooling branch with a thick H envelope. In this case,
residual H nuclear burning via pp-chain becomes the main energy source, that ultimately slows
downtheevolutionofthestar,inwhichcasethecoolingtimescaleisoftheorderof∼109yrs.The
agedichotomyhasbeenalsosuggestedbyobservationsofthoselow-massHe-coreWDsthatare
companionstomillisecondpulsars(Bassaetal.2003).
In the past few years, numerous low-mass WDs, including ELM WDs, have been detected
through the ELM survey and the SPY and WASP surveys (see Koesteretal. 2009; Brownetal.
2010,2012;Maxtedetal.2011;Kilicetal.2011,2012;Brownetal.2013;Gianninasetal.2014;
Kilicetal.2015;Gianninasetal.2015;Brownetal.2016a,b).Theinterestinlow-massWDshas
beengreatlypromotedbythediscoveryofpulsationsinsomeofthem(Hermesetal.2012,2013c,a;
Kilicetal. 2015;Belletal. 2015,2016)1.Thediscoveryofpulsatinglow-massWDs (hereinafter
ELMVs2)constitutesauniquechanceforprobingtheinteriorofthesestarsandeventuallytotest
their formation scenarios by employing the methods of asteroseismology. Theoretical adiabatic
pulsationalanalyzesof these stars (Steinfadtetal. 2010;Co´rsicoetal. 2012c; Co´rsico&Althaus
2014a) show that g modes in ELM WDs are restricted mainly to the core regions, providing
the chance to constrain the core chemical structure. Also, nonadiabatic stability computations
(Co´rsicoetal.2012c;VanGrooteletal.2013;Co´rsico&Althaus2016)showthatmanyunstable
g and p modesare excited by a combinationof the κ−γ mechanism(Unnoetal. 1989) and the
1 ThestarSDSSJ135512+195645discoveredbyBelletal.(2016)islikelyahigh-amplitudeδScutipul-
satorwithanoverestimatedsurfacegravity,aspointedbytheseauthors.
2 Forsimplicity,hereandthroughoutthepaperwerefertothepulsatinglow-massWDsasELMVs,evenif
M⋆&0.18−0.20M⊙.
2
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“convectivedriving”mechanism(Brickhill1991),bothofthemactingattheH-ionizationzone.In
addition,theεmechanismduetostableHburningcouldcontributetodestabilizesomeshort-period
gmodesinELMWDs(Co´rsico&Althaus2014b).
Apart from ELMVs, pulsations in several objects that are likely the precursors of low-mass
WD stars have been detected in the last years (Maxtedetal. 2013, 2014; Zhangetal. 2016;
Gianninasetal. 2016; Cortietal. 2016)3. Nonadiabatic stability computations for radial modes
(Jeffery&Saio 2013) and nonradial p and g modes (Co´rsicoetal. 2016a; Gianninasetal. 2016;
Istrateetal. 2016a) have revealed that the excitation of pulsationsin these pre-WDs is the κ−γ
mechanismactingmainlyatthezoneofthesecondpartialionizationofHe,withaweakercontri-
butionfromtheregionofthefirstpartialionizationofHeandthepartialionizationofH.So,the
abundanceofHeintheenvelopesofthisnewclassofpulsatingstars(hereinafterpre-ELMVs4)isa
crucialingredientfordestabilizingthepulsationmodes(Co´rsicoetal.2016a;Istrateetal.2016a).
Theg-modepulsationperiods(Π)ofWD starsexperienceaseculardriftastheycool,giving
placetoadetectablerateofperiodchange,Π˙ ≡dΠ/dt.Specifically,asthetemperatureinthecore
ofaWDdecreases,theplasmaincreasesitsdegreeofdegeneracysothattheBrunt-Va¨isa¨la¨(buoy-
ancy)frequency—thecriticalfrequencyofg-modepulsations(Unnoetal.1989)—decreases,and
the pulsationalspectrum of the star is shifted to longer periods. On the other hand, gravitational
contraction(ifpresent)actsintheoppositedirection,favoringtheshorteningofthepulsationpe-
riods.Thecompetitionbetweentheincreasingdegeneracyandgravitationalcontractiongivesrise
to adetectableΠ˙. Inparticular,ithasbeenshownbyWingetetal.(1983) thatthe rateof change
oftheg-modepulsationperiodsisrelatedtotherateofchangeofthetemperatureattheregionof
theperiodformation,T˙,andtherateofchangeofthestellarradius,R˙ ,accordingtothefollowing
⋆
order-of-magnitudeexpression:
Π˙ T˙ R˙
≈−a +b ⋆, (1)
Π T R
⋆
where a and b are constants whose values depend on the details of the WD modeling (however
a,b ≈ 1). The first term in Eq. (1) corresponds to the rate of change in period induced by the
coolingoftheWD, andsinceT˙ < 0,itisapositivecontribution.Thesecondtermrepresentsthe
rateofchangeduetogravitationalcontraction(R˙ <0),anditisanegativecontribution.
⋆
In the cases in which robust measurements of secular period drifts of pulsating WDs can be
achieved,anumberofimportantapplicationscanbe—inprinciple—carriedout(Mukadametal.
2003). In particular, the derived values of Π˙ could help in calibrating the WD cooling curves,
thus reducing the theoretical uncertainties of WD cosmochronology to constrain the age of the
Galacticdisk(e.g.,Harrisetal.2006),halo(e.g.,Isernetal.1998),andGalacticglobularclusters
(e.g.,Hansenetal.2013)andopenclusters(Garc´ıa-Berroetal.2010).ThemeasurementofΠ˙ also
couldallowustoinferthechemicalcompositionofthecoreofaWD(Kepleretal.2005).Thisis
becausetherateofcoolingofWDs, andso,therateofperiodchangeofagivenpulsationmode,
dependprimarilyonthecorecompositionandthestellarmass.Atfixedmass,Π˙ islargerforhigher
meanatomicweightofthecore.Thisallowstoplaceconstraintsonthecorechemicalcomposition.
3 ThenatureofthevariablestarsreportedbyCortietal.(2016)isunclear,astheycouldbeprecursorsof
low-massWDstarsor,alternatively,δScuti/SXPhe-likestars,aspointedbytheseauthors.
4 Hereandthroughoutthepaperwerefertothepulsatinglow-masspre-WDsaspre-ELMVs,evenifM &
⋆
0.18−0.20M⊙.
3
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AnotherpossibleapplicationofthemeasurementofΠ˙ isthedetectionofplanets.Theorbitalmotion
of a pulsating WD around the center of mass of the system due to the possible presence of a
planet modify the light travel time of the pulses. As a result, the observed arrival time on Earth
changes,thusprovidingan alternativemethodto detectthe planet(Mullallyetal. 2008). Finally,
the rates of periodchange in WDs allow, in principle,to place constraintson axions(Isernetal.
1992; Co´rsicoetal. 2001; Bischoff-Kimetal. 2008; Co´rsicoetal. 2012a,b, 2016b; Battichetal.
2016),neutrinos(Wingetetal.2004;Co´rsicoetal.2014),andthepossiblesecularrateofvariation
ofthegravitationalconstant(Co´rsicoetal.2013).Notethat,however,inordertoestablishrobust
constraints based on the rate of change of periods, it is necessary to know with some degree of
accuracythetotalmass,theeffectivetemperature,thecorecomposition,andtheenvelopelayering
ofthetargetstar(Fontaine&Brassard2008).
The rate of change of the periods can be measured, in principle, by monitoring a pulsat-
ing WD over a long time interval when one or more very stable pulsation periods are present
in their power spectrum. In the case of pulsating DA (H-rich atmosphere) and DB (He-rich at-
mosphere) WDs, also called DAV and DBV stars, respectively, cooling dominates over gravita-
tional contraction, in such a way that the second term in Eq. (1) is usually negligible, and only
positive values of the observed rate of change of period are expected (Winget&Kepler 2008;
Fontaine&Brassard2008;Althausetal.2010).ForC/O-coreDAVs,theexpectedratesofperiod
changearein therange10−15−10−16 s/s(Bradleyetal. 1992;Bradley1996),inexcellentagree-
ment with the measured values for G117−B15A (Π˙ = 4.19 ± 0.73 × 10−15 s/s, Kepler 2012),
R548 (Π˙ = 3.3±1.1×10−15 s/s, Mukadametal. 2013) and L19−2 (Π˙ = 3.0±0.6×10−15 s/s,
Sullivan&Chote 2015), although in strong conflict with the value derived for WD 0111+0018
(Π˙ > 10−12 s/s, Hermesetal. 2013b). For DBVs, an estimate of the rate of period change has
beenobtainedforPG1351+489(Π˙ = 2.0±0.9×10−13 s/s,Redaellietal.2011),inlinewiththe
theoreticalexpectations(Π˙ ∼ 10−13−10−14 s/s; Wingetetal. 2004;Co´rsico&Althaus2004). In
the case of pulsating hot WD and pre-WD stars, also called GW Vir or pulsating PG1159 stars
(He-, C-, and O-rich atmosphere),theoreticalmodelspredictrates of period changein the range
10−11−10−12s/s(Kawaler&Bradley1994;Co´rsico&Althaus2006;Co´rsicoetal.2008).Forthe
higheffectivetemperaturescharacterizingtheGWVirinstabilitystrip,gravitationalcontractionis
stillsignificant,tosuchadegreethatitsinfluenceonΠ˙ canovercometheeffectsofcooling.Inthis
case thesecondtermin Eq.(1) isnotnegligibleand,therefore,eitherpositiveornegativevalues
of Π˙ are possible. Π˙ for severalg modes has been measured in the case of the prototypicalGW
Virstar,PG1159−035(Costa&Kepler2008).Thestarexhibitsamixtureofpositiveandnegative
Π˙ values of large magnitude, up to ∼ 4 × 10−10 s/s. In particular, the rate of period change of
the modewith periodΠ = 517.1s is Π˙ = 1.52±0.05×10−10 s/s, an order of magnitudelarger
thanthetheoreticalpredictions(Kawaler&Bradley1994;Co´rsico&Althaus2006;Co´rsicoetal.
2008). Althausetal. (2008) have foundthatthis discrepancycouldbe alleviated if PG1159−035
is characterizedby a thin He-rich envelope,leading to remarkablylarge magnitudesof the rates
of period change. A measurement of Π˙ in another GW Vir star, PG0112+200, has been carried
outby Vauclairetal. (2011). The derivedratesof periodchangeare muchlarger than those pre-
dictedbytheoreticalmodels(Co´rsicoetal.2007),callingforthepresenceofothermechanism(s)
apartfromneutrinocoolingtoexplainthedisagreement.Inparticular,amechanismthatcouldbe
playingadominantroleisresonantmodecouplinginducedbytherotation(Vauclairetal.2011).
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Acautionarynoteregardingtheinterpretationofthemeasuredratesofperiodchangeinpulsating
WDsisneededhere.ThestudiesbyHermesetal.(2013b)fortheDAVstarWD0111+0018,and
Vauclairetal. (2011) for the GW Vir star PG0112+200(among others), indicate that our under-
standingoftheratesofperiodchangeinpulsatingWDsisfarfromcomplete,andthisshouldbe
keptinmindwhenusingΠ˙ intheapplicationsmentionedbefore.
Inthispaper,thefourthoneofaseriesdevotedtolow-massWDandpre-WDstars,wepresent
forthefirsttimeadetailedassessmentofthetheoreticaltemporalratesofperiodchangeofELMV
and pre-ELMV stars. According to the theoretically estimated rates of cooling of these stars
(Althausetal. 2013), low-mass WDs cool slower than low-mass pre-WDs. On these grounds, it
isexpectedthatELMVswillhavesmallerratesofperiodchangethanpre-ELMVs.Therefore,the
eventualmeasurementoftherateofperiodchangefora givenpulsatingstarcouldbepotentially
usefultodistinguishinwhichevolutionarystagethestar is. Also,aneventualmeasurementofΠ˙
couldhelp,inprinciple,todistinguishgenuineELMWDs(M .0.18−0.20M )thathavethickH
⋆ ⊙
envelopesandlongcoolingtimescales,fromlow-massWDs(M &0.18−0.20M ),characterized
⋆ ⊙
by thinner H envelopesand shorter coolingtimescales. However,we must keep in mind that the
coolingratesofthiskindofstarscouldbesoslow,thatanysecularperiodchangewouldbevery
difficulttodetect.Notwithstanding,someofthesestarsmaynotbeontheirterminalcoolingtracks
butrathermaybeonthepre-WDstage,orevengoingtoaCNOflash,andthushavemuchhigher
(andmuchmoreeasilymeasurable)Π˙ values.Inotherwords,thedetectionofanysignificantsec-
ularperiodchangewouldbestrongevidencefortheobjecttobenotonitsfinalcoolingtrackasan
ELMVstar.AlthoughthemeasurementofΠ˙ foranyofthesestarsisnotexpectedshortly,itcould
beachievedinthenextyearsbymeansofcontinuousphotometricmonitoringoftheseobjects.
The paper is organizedas follows. In Sect. 2 we brieflydescribe ournumericaltools andthe
mainingredientsoftheevolutionarysequencesweemploytoassesstheratesofperiodchangeof
low-massHe-coreWDsandpre-WDs.InSect.3wepresentindetailourresultsofΠ˙ forELMV
and pre-ELMV models. In particular, we study the dependence of the rates of period change of
nonradialdipole (ℓ = 1) g and p modes, and radial (ℓ = 0) modes with the stellar mass and the
effectivetemperature.WeexpandtheanalysisbyincludingtheassessmentofΠ˙ forstellarmodels
evolving at stages previous of the developmentof thermonuclear CNO flashes during the early-
coolingphase.Finally,inSect.4wesummarizethemainfindingsofthepaper.
2. Modeling
2.1.Evolutionarycode
Thefullyevolutionarymodelsoflow-massHe-coreWDandpre-WDstarsonwhichthisworkis
basedweregeneratedwiththeLPCODEstellarevolutioncode.LPCODEcomputesindetailthecom-
pleteevolutionarystagesleadingtoWDformation,allowingonetostudytheWDandpre-WDevo-
lutioninaconsistentwaywiththeexpectationsoftheevolutionaryhistoryofprogenitors.Details
of LPCODEcan be foundin Althausetal. (2005, 2009, 2013, 2015, 2016) andreferencestherein.
Here,wementiononlythosephysicalingredientswhicharerelevantforouranalysisoflow-mass,
He-coreWDandpre-WDstars(seeAlthausetal.2013,fordetails).ThestandardMixingLength
Theory(MLT)forconvectionintheversionML2isused(Tassouletal.1990).Themetallicityof
the progenitorstars has been assumed to be Z = 0.01.Radiative opacities for arbitrarymetallic-
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5,5
0.1650 0.1612 0.1554 blue edge (l= 1) SDSS J1735
0.1706 SDSS J2228
6 SDSS J2139
SDSS J1112
0.1762 SDSS J1355
PSR J1738 SDSS J1614
SDSS J1840
0.1805
6,5
0.1863 SDSS J1618
g g 0.1917
o
l 0.2019
SDSS J1518
0.2389
7
0.2707
0.3205
0.3624
7,5
ELMV stars
0.4352
Template models
11000 10000 9000 8000 7000
T [K]
eff
Fig.1. T − logg plane showing the low-mass He-core WD evolutionary tracks (final cooling
eff
branches)ofAlthausetal.(2013).Numberscorrespondtothestellarmassofeachsequence.The
locationofthetenknownELMVs(Hermesetal.2012,2013c,a;Kilicetal.2015;Belletal.2015,
2016) are marked with large red circles (T and logg computed with 3D model atmospheres
eff
corrections). Stars observed not to vary (Steinfadtetal. 2012; Hermesetal. 2012, 2013c,a) are
depictedwithsmallblackcircles.TheGraysquaresandtrianglesontheevolutionarytracksindicate
thelocationofthetemplatemodelsanalyzedinthe text.Thedashedbluelinecorrespondstothe
blueedgeoftheinstabilitydomainofℓ = 1gmodesaccordingtothenonadiabaticcomputations
ofCo´rsico&Althaus(2016)usingML2(α=1.0)versionoftheMLTtheoryofconvection.
ity in the range from 0 to 0.1 are from the OPAL project (Iglesias&Rogers 1996). Conductive
opacities are those of Cassisietal. (2007). The equation of state during the main sequence evo-
lutionis thatof OPAL forH- and He-richcompositions.Neutrinoemission ratesfor pair,photo,
and bremsstrahlung processes have been taken from Itohetal. (1996), and for plasma processes
weincludedthetreatmentofHaftetal.(1994).FortheWDregimewehaveemployedanupdated
versionoftheMagni&Mazzitelli(1979)equationofstate.Thenuclearnetworktakesintoaccount
16elementsand34thermonuclearreactionratesforpp-chains,CNObi-cycle,Heburning,andC
ignition.Time-dependentelementdiffusionduetogravitationalsettlingandchemicalandthermal
diffusionof nuclearspecieshasbeentakenintoaccountfollowingthe multicomponentgastreat-
mentofBurgers(1969).Abundancechangeshavebeencomputedaccordingtoelementdiffusion,
nuclearreactions,andconvectivemixing.Thisdetailedtreatmentofabundancechangesbydiffer-
entprocessesduringtheWDregimeconstitutesakeyaspectintheevaluationoftheimportanceof
residualnuclearburningforthecoolingoflow-massWDs.
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2.2.Pulsationcode
The rates of periodchangeof radialmodesand nonradial p and g modescomputedin this work
were derived from the large set of adiabatic pulsation periods presented in Co´rsico&Althaus
(2014a).Theseperiodswerecomputedemployingtheadiabaticradialandnonradialversionsofthe
LP-PULpulsationcodedescribedindetailinCo´rsico&Althaus(2006,2014a),whichiscoupledto
theLPCODEevolutionarycode.TheLP-PULpulsationcodeisbasedonageneralNewton-Raphson
techniquethatsolvesthefourth-order(second-order)setofrealequationsandboundaryconditions
governinglinear,adiabatic,nonradial(radial)stellarpulsationsfollowingthedimensionlessformu-
lationofDziembowski(1971)(seealsoUnnoetal.1989).Theprescriptionwefollowtoassessthe
runoftheBrunt-Va¨isa¨la¨frequency(N)istheso-called“LedouxModified”treatment(Tassouletal.
1990;Brassardetal.1991).
2.3.Evolutionarysequences
Realisticconfigurationsforlow-massHe-coreWDandpre-WDstarswerederivedbyAlthausetal.
(2013)bymimickingthebinaryevolutionofprogenitorstars.Binaryevolutionwasassumedtobe
fully nonconservative, and the loss of angular momentum due to mass loss, gravitational wave
radiation,and magnetic brakingwas considered.All of the He-core pre-WD initial modelswere
derivedfromevolutionarycalculationsforbinarysystemsconsistingofanevolvingMainSequence
low-mass component (donor star) of initially 1M⊙ and a 1.4M⊙ neutron star companion as the
othercomponent.Atotalof14initialHe-corepre-WDmodelswithstellarmassesbetween0.1554
and0.4352M⊙werecomputedforinitialorbitalperiodsatthebeginningoftheRochelobephase
intherange0.9to300d.Inthispaper,we focusontheassessmentoftheratesofperiodchange
values corresponding to the complete evolutionary stages of these models down to the range of
luminositiesofcoolWDs,includingsomestagesprevioustothethermonuclearCNOflashesduring
thebeginningofthecoolingbranch.
3. Thetheoreticalratesofperiodchange
Inthiswork,theratesofperiodchangeareassessed assimpledifferencingoftheperiodsofsuc-
cessivemodelsineachevolutionarysequence.Specifically,therateofchangeoftheperiodΠ at
k
thetimeτ isestimatedas:
i
Π (τ)−Π (τ )
Π˙ (τ)= k i k i−1 , (2)
k i ∆τ
i
where∆τ =τ −τ istheevolutionarytimestep,andΠ (τ)andΠ (τ )arethepulsationperiods
i i i−1 k i k i−1
ofthemodewithradialorderkevaluatedatthetimesτ andτ ,respectively.Inourcomputations,
i i−1
thetimestep∆τ issmallenoughastoensurethatthissimplenumericalrecipeyieldsveryprecise
i
resultsforΠ˙.Wecomputetherateofperiodchangefornonradialℓ=1,2gand pmodes,andalso
radial(ℓ=0)modes.Thesetofpulsationmodesconsideredinthisworkcoversaverywiderange
ofperiods(upto∼7000s),embracingalltheperiodicitiesdetectedinELMVandpre-ELMVstars
uptonow.
Low-massWDshaverealpossibilitiesofbeingobservedatthreestages(Althausetal.2013):
thefinalcoolingbranch(WDphase),thestagesatconstantluminosityfollowingtheendofRoche
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−15/dt [10 s/s] 2345600000 00000000000000..............11111112223341667788903726351506061180025520625379975424 −15/dt [10 s/s] −−−−0000.... 86420 00000000000000..............11111111222334566778890372635150606118002542062537997542
Pd Pd −1
10
−1.2
0 −1.4
−10 0 1000 2000 3000 4000 5000 6000 7000 −1.6 0 20 40 60 80 100 120 140 160
P[s] P[s]
0
0.1554
0.1612
−0.2 00..11675006
0.1762
−0.4 00..11880653
0.1917
−0.6 00..22031899
−15/dt [10 s/s] −0−.18 0000....2334726300257542
Pd
−1.2
−1.4
−1.6
−1.8
0 20 40 60 80 100 120 140 160
P[s]
Fig.2.Left:Theratesofperiodchangeofℓ=1gmodesversusthepulsationperiods,correspond-
ingtoWDmodelscharacterizedbyaneffectivetemperatureofT ∼ 9000Kanddifferentstellar
eff
masses(0.1554≤ M⋆/M⊙ ≤0.4352).Middle:Sameasinleftpanel,butforℓ=1 pmodes.Right:
Sameasinleftpanel,butforradial(ℓ=0)modes.
lobe overflow (pre-WD phase), and for M⋆ & 0.18M⊙, the evolutionary stages prior to the oc-
currenceofCNOflashesontheearlycoolingbranches(pre-flashstages)5.Below,weexplorethe
magnitude and sign of the rates of period changes of low-mass WD stars at these evolutionary
stages.Inalltheregimesconsideredinthiswork,therateofperiodchangevaluesforℓ = 2areof
thesameorderofmagnitudethanforℓ=1.Thus,wewillconcentrateonshowingresultsonlyfor
thecaseℓ = 1,althoughwemustkeepinmindthatalsomodeswithℓ = 2canbeobservedinthis
typeofpulsatingstars.
3.1.WDphase:ELMVs
Next,weshallexaminetheeffectofchangingthestellarmassandtheeffectivetemperatureonthe
rateofperiodchangeofELMVmodels,thatis,low-massWDmodelsalreadyevolvingintheirfinal
coolingbranches.Theadiabaticandnonadiabaticpulsationpropertiesofpulsatinglow-massWD
stars,orELMVs,havebeenexploredindetailinCo´rsicoetal.(2012c);VanGrooteletal.(2013);
Co´rsico&Althaus(2014a,2016).InFig.1wepresentaT −loggdiagramshowingthelow-mass
eff
He-core WD evolutionary tracks of Althausetal. (2013), with the stellar masses indicated with
small numbers. For illustrative purposes, we also include the location of the ten known ELMVs
(Hermesetal. 2012, 2013c,a; Kilicetal. 2015; Belletal. 2015, 2016) with red circles, and stars
observed not to vary (Steinfadtetal. 2012; Hermesetal. 2012, 2013c,a), displayed with small
5 During the rapid incursions of the starsinthe logT −logg diagram whilethey arelooping between
eff
theCNOflashes, theevolutionissofast thattheprobability offindingastarinthosestagesisalmost null
(Althausetal.2013).Forthisreason,wewillnotconsiderthoseevolutionarystagesinthiswork.
8
Pleasegiveashorterversionwith:\authorrunning and/or \titilerunning priorto \maketitle
56 10987111400019932KKKK −0. 10 10987222311282172KKKK
−0.2
4
−0.3
−15/dt [10 s/s] 23 −15/dt [10 s/s] −−00..54
Pd Pd
−0.6
1
−0.7
0 −0.8
−1 0 1000 2000 3000 P 4[0s0]0 5000 6000 7000 −0.9 0 10 20 30 40 P 5[0s] 60 70 80 90
0
−0.2
−0.4
−15/dt [10 s/s] −0.6
Pd
−0.8
−1
10318K
9337K
8338K
7382K
−1.2
0 10 20 30 40 50 60 70 80 90
P[s]
Fig.3. Left: The rate of period change of ℓ = 1 g modes versus the pulsation periods, corre-
spondingto WD modelscharacterizedbya stellar mass M⋆ = 0.1762M⊙ anddifferenteffective
temperatures.Middle:Similartoleftpanel,butforℓ=1 pmodes.Right:Similartoleftpanel,but
forradial(ℓ=0)modes.
black circles. The gray squares (triangles) on the evolutionarytracks indicate the location of the
templatemodelstobeanalyzedinFig.2(Figs.3and4)below.
InassessingthedependenceofΠ˙ forgmodeswith M andT inWDstars,itisusualtocon-
⋆ eff
siderthepredictionsofthesimplecoolingmodelofMestel(1952)forcomparisonwithnumerical
results. Within the framework of the Mestel’s cooling law, Kawaleretal. (1986) have derived a
relationbetweenΠ˙ and M andT (theirEq.3),whichpredictsthatΠ˙ islargerforlowerstellar
⋆ eff
masses and higher effective temperatures. The mass dependence can be understood by realizing
thatthelowerthe mass, the largerthe radiatingsurfaceandthe lowerthetotalheatcapacity.For
afixedT value,lessmassivemodelshavehigherluminositiesandthuscoolfasterwithalarger
eff
Π˙. Concerningthe dependenceof Π˙ with the effective temperature,for a fixed M , modelswith
⋆
higher T cool faster, with the consequencethat Π˙ is larger. This simple picture becomesmore
eff
complicatedwhenthereexistanotherenergysourcelikenuclearburning,apartfromtheheatreser-
voirstoredintheionsduringthepreviousevolutionaryphases.Thisisthecaseoflow-massWDs
with M⋆ . 0.18−0.20M⊙(ELMWDs),whicharecharacterizedbyintenseHburning.Thus,the
simplepredictionsbasedontheMesteltheorycannotbeappliedtoWDsinthismassrange.
WedepictinFig.2theratesofperiodchangeintermsofperiodsforℓ=1gmodes(leftpanel),
ℓ = 1 p modes (middle panel), and ℓ = 0 radial modes(right panel), for low-mass WD models
withTeff ∼9000Kandseveralstellarmasses(0.1554≤ M⋆/M⊙ ≤0.4352).Thelocationofthese
stellarmodelsintheT −loggdiagramismarkedwithgraysquaresinFig.1.Forgmodes(left
eff
panel of Fig. 2), Π˙ linearly increases with the radial order k and thus with period, which is the
reflectionoftheincreaseofΠwithk.TwowelldistinguishablebranchesoftheΠ˙ vsΠrelationship
9
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5600 19870333366656438KKKK −0.0 50 10987333355668843KKKK
−0.1
40
−0.15
−15/dt [10 s/s] 2300 −15/dt [10 s/s] −−00.2.25
Pd Pd
−0.3
10
−0.35
0 −0.4
−10 0 1000 2000 3000P[s] 4000 5000 6000 −0.45 0 5 10 15 20 P 2[5s] 30 35 40 45
0
10358K
−0.05 98335684KK
7363K
−0.1
−0.15
−15/dt [10 s/s] −−−000.2..325
Pd −0.35
−0.4
−0.45
−0.5
−0.55
0 10 20 30 40 50 60
P[s]
Fig.4. Left: The rate of period change of ℓ = 1 g modes versus the pulsation periods, corre-
spondingto WD modelscharacterizedbya stellar mass M⋆ = 0.1863M⊙ anddifferenteffective
temperatures.Middle: Similar to the leftpanel, butfor ℓ = 1 p modes.Right: Similar to the left
panel,butforradial(ℓ=0)modes.
arevisibleinthefigure,oneofthemcorrespondingtomodelswith M⋆ & 0.18M⊙,andtheother
one associated to models with M⋆ . 0.18M⊙. In the first group of models, nuclear burning is
not relevant,and so, the rate of period change is generallylarger for lower stellar mass (at fixed
T ∼9000K)andalltheΠ˙ valuesarepositive.Notethatforthissetofmodels,gmodesarevery
eff
sensitivetotheHe/Hcompositiongradient(seeFig.8ofCo´rsico&Althaus2014a).Thefactthat
Π˙ > 0 implies that the rates of period change in models with M⋆ & 0.18M⊙ are dominated by
cooling(firstterminEq.1).Theratesofperiodchangeforthissetofmodelsrangefrom∼ 10−15
s/sfortheshortestg-modeperiods6upto∼7×10−14s/sforΠ∼6000s.
TheΠ˙ valuesforthegroupofmodelswith M⋆ . 0.18M⊙, ontheotherhand,arelowerthan
∼ 10−14 s/s, and are indeed substantially smaller than for the first group of models. This is due
to the fact that, for models with stellar masses lower than the threshold mass of ∼ 0.18M⊙, the
evolution is dominated by nuclear burning.As a result, the WD cooling is markedlydelayed, in
such a way that the rates of period change are smaller in magnitude as compared with the case
in which nuclear burning is negligible (M⋆ & 0.18M⊙). Note that g modes in this mass range
mainlyprobethecoreregions(see Fig.7ofCo´rsico&Althaus2014a).TheΠ˙ valuesformodels
M⋆ . 0.18M⊙ are smaller for lower stellar mass, as it can be seen in left panel of Fig. 2. This
trendisoppositetothatpredictedbythesimpleformulaofKawaleretal.(1986)(seeabove).The
fact that in this mass range (0.15 . M⋆/M⊙ . 0.18) the lowest-mass models are characterized
bysmallerΠ˙ valuesisduetothatthesemodelshavemoreintensenuclearburning,whichimplies
smaller cooling timescales. In this context, it is expected that the first term in Eq. (1) (cooling)
6 NotethattheΠ˙ valueforsomelow-ordermodesisveryclosetozero,orevennegative.
10