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ASTRONGTESTOFELECTRO-WEAKTHEORYUSINGPULSATINGDBWHITEDWARFSTARS
ASPLASMONNEUTRINODETECTORS
D.E.WINGET1,D.J.SULLIVAN2,T.S.METCALFE3,S.D.KAWALER4,M.H.MONTGOMERY5
AcceptedforpublicationinApJLetters
ABSTRACT
We demonstrate that plasmon neutrinos are the dominant form of energy loss in model white dwarf stars
down to T ∼25,000 K, dependingon the stellar mass. The lower end of this range overlapsthe observed
eff
temperaturesfortheV777Herstar(DBV)instabilitystrip.Theevolutionofwhitedwarfsatthesetemperatures
isdrivenpredominantlybycooling,sothisdirectlyaffectsthestellarevolutionarytimescaleinproportiontothe
4 ratiooftheneutrinoenergylosstothephotonenergyloss.Thisevolutionarytimescaleisobservablethroughthe
0 timerateofchangeofthepulsationperiods.Althoughtheunifiedelectro-weaktheoryofleptoninteractionsthat
0 iscrucialforunderstandingneutrinoproductionhasbeenwelltestedinthehighenergyregime,theapproach
2
presentedhereshouldresultinaninterestinglow-energytestofthetheory.Wediscussobservationalstrategies
n toachievethisgoal.
a Subjectheadings:neutrinos—stars:interiors—stars:oscillations—whitedwarfs
J
9
1. INTRODUCTION probe of plasmon neutrino emissions in just the tempera-
3 ture range where the neutrino luminosity was near its max-
Not long after Fermi’s theory of the weak interaction was
v imum. We later reported the observational detection of an
first generalized to include, among other things, the possi-
3 evolutionary rate of period change for one DOV, PG 1159-
bility of a direct interaction between the electron and the
0 035 (Wingetetal. 1985; Costa,Kepler,&Winget1999), but
relevant neutrino (e.g., Feynman&Gell-Mann 1958), a va-
3
thetheoreticalinterpretationofthisresultremainselusive(see
2 riety of theoretical papers pointed out that this interaction,
O’Brien&Kawaler2000, fora recentsummary). Theoreti-
1 thoughextremelyfeeble,couldhaveamajorimpactinthehot
cal models for PG 1159-035have been unable to reproduce
3 dense plasmas to be found in the astrophysical domain (see
the observed period structure and the rate of period change
0 Fowler&Hoyle1964, foranearlyreview). However,given
simultaneously. Since the mechanical and thermal structure
/ theweaknessofneutrinointeractions,directmeasurementsof
h ismorecloselycoupledthanincoolerwhitedwarfstars,itis
theseratesinastrophysicalobjectshaveproveda greatchal-
p moredifficulttoidentifythesourceofthisproblem.
lenge. Thesemeasurementsrelyonenormousdetectingvol-
-
Nosuchdifficultiesshouldarisefortheheliumatmosphere
o umesinordertoobtainasignificantnumberofevents.Infact,
r only in the case of the Sun and SN 1987A have direct links DBV white dwarfs, since their cooler temperatures lead to
t greater degeneracy and hence a decoupling of their thermal
s beenmadebetweendetectionsofneutrinosandparticularas-
a tronomicalobjects. andmechanicalstructure(see section 3.1 below). As a con-
: sequence,weshouldbeabletomeasuretheplasmonneutrino
v Itispossibletomeasureneutrinorates,albeitindirectly,in
flux from these objects, testing the calculationsof Itohetal.
i athirdclassofobjects,thewhitedwarfstars. Forboththehot
X (1996)andmeasuringtheneteffectofplasmonneutrinos.
white dwarf and a pre-white dwarf stars the theoretical neu-
r trino energy losses exceed the photon energy losses, and so 2. PLASMONNEUTRINOS
a
essentiallycontroltheevolutionarytimescale. Theastrophys-
Of the many possible neutrino emission processes, it is
icalsignificanceofneutrinoemissioninhotpre-whitedwarf
theplasmonneutrinoprocessthatdominatesneutrinoproduc-
stars has been investigated extensively by many authors,
tioninhotwhitedwarfinteriors—thebremsstrahlungprocess
building on the foundations of work undertaken at the Uni-
comesa distantsecond. Theexistenceof plasmonneutrinos
versityofRochester(e.g.Vila1966;Kutter&Savedoff1969;
is an important test of our understanding of the universality
Savedoff,VanHorn&Vila 1969). The principal observable
oftheweakinteraction.Thenatureoftheprocessthatcreates
signature of these neutrinos was thought to be in the white
themiswelldescribedbyClayton(1968),andwesummarize
dwarfluminosityfunction(e.g.,seeLamb&VanHorn1975).
ithere.
In the mid-eighties we (e.g., Kawaler,Winget,&Hansen
A free photon cannot decay into a neutrino and an anti-
1985) demonstrated that evolutionary frequency drifts in
neutrino because the constraint of coupling to a spin-1 par-
the non-radial gravity modes present in pulsating hot pre-
ticle requires the neutrinos to be emitted in opposite direc-
white dwarf (DOV) stars might provide a very sensitive
tions; as a result, energy and momentum cannot simultane-
ously be conserved and the process is forbidden. A photon
1DepartmentofAstronomyandMcDonaldObservatory,TheUniversity
propagatinginaplasmadoesnothavethisproblem—itcon-
ofTexas,Austin,TX78712
2 School of Chemical and Physical Sciences, Victoria University of servesbothenergyandmomentumbycouplingtotheplasma;
Wellington,Wellington,NZ suchacoupledphotonisreferredtoasa“plasmon”. Aplas-
3Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cam- monwith sufficientenergycan decay into a neutrinoandan
bridge,MA02138
anti-neutrino;theneutrinoscreatedinthisprocessaretermed
4DepartmentofPhysicsandAstronomy,IowaStateUniversity,Ames,IA
50011 plasmonneutrinos:
5InstituteofAstronomy,UniversityofCambridge,MadingleyRoad,Cam- γ∗→ν+ν¯.
bridgeCB30HA,UK
Undernormalstellar conditions, includingwhite dwarf inte-
2 WINGETETAL.
riors,weexpecttheneutrinoscreatedinthiswaytoleavethe
starwithoutfurtherinteraction,resultinginnetenergyloss.
Foraphysicalpictureofwhataplasmonactuallyis,wecan
consider the photon to be a classical electromagnetic wave
movingthroughadielectricmedium. Ifωisthefrequencyof
thephoton,thenwehave
ω2=k2c2+ω2,
0
whereω istheplasmafrequency. SinceE =¯hω and p=¯hk
0
foraphoton,thiscanbere-writtenas
E2=p2c2+m2c4,
pl
where
¯hω
m = 0
pl c2
actsastheeffectivemassoftheplasmon. Thus,theplasmon
behaves like a particle with a non-zero rest mass, and it is
thiseffectivemassthatenablesittodecayintoaneutrinoand
anti-neutrinopair.
The size of this “rest mass” energy, ¯hω , is important for
0
tworeasons.First,inageneralstellarenvironmenttheseplas-
monstateswillbeinthermalequilibriumwiththeirsurround-
ings,sosignificantnumbersofplasmonswillbeexcitedonly FIG. 1.—(a)theratioofneutrinotophotonluminosityforwhitedwarf
ifthetypicalthermalenergy,k T,islargerthantheplasmon modelsofvariousstellarmass. Intheregionofinterest(nearthetempera-
B
energy,¯hω . Second, the amountof energyliberated by the turesofthepulsatorsGD358andEC20058),neutrinolossesforthehigh-
0 massmodelsarenegligible,whiletheyareverysignificantforthelessmas-
decayofaplasmonisrelatedtoitsmass,withhighmassplas- sive models. (b) Thephoton (dashed line) and neutrino (solid line) lumi-
monsliberatingmoreenergythanlowmassones.Inlowden- nositiesfora0.6M⊙ whitedwarf,alongwiththefractionalcontributionof
sitynon-degenerategases,theplasmafrequencyisgivenby bremsstrahlung neutrinos (dotted line). The neutrino luminosity is nearly
twicethephotonluminosityatthetemperatureofEC20058,andthetwoare
4πn e2 approximatelyequalatthetemperatureofGD358. Theneutrinoluminosity
ω2= e , inthisrangeisalmostentirelyduetotheplasmonreaction.
0 m
e
and we typically find that ¯hω ≪k T. Thus, even though
0 B
there are many plasmons, their rest mass energy is so small 3. EVOLUTIONARYCALCULATIONS
thattheirdecayreleaseslittleenergy.Inthedenseplasmasof For the present work, we have computed several white
degenerate stellar interiors the plasma frequency, now given dwarf evolutionarysequences in order to explore the effects
approximatelyby of plasmon neutrinos in the DBV temperature range. We
have calculated sequences using polytropic starting models
- 1
ω2= 4πnee2 1+ ¯h 2 3π2n 32 2, aswellasfullevolutionarysequencescalculatedfromvarious
0 m m c e main-sequence progenitor models, including different para-
e " (cid:18) e (cid:19) # metrictreatmentsofmassloss. Ashasbeenwellestablished
(cid:0) (cid:1)
by numerousauthors (e.g. Kawaler,Winget,Iben,&Hansen
is muchlarger,andplasmonneutrinoproductioncan leadto
1986), the thermal history is erased by the time these mod-
significant energy loss. However, if the density is too large
els cool to 35,000 K and the model sequences become in-
then only the highest energy fluctuations out on the thermal
distinguishable for our purposes. We have calculated se-
tailwillbelargeenoughtoexciteplasmons,leadingtoasup-
quences with and without plasmon neutrino emission in-
pressionofenergylossesthroughneutrinoemission.
cluded, using the rates of Itohetal. (1996) in our models
This has important observational consequences. Theo-
(see Figure 1). At the energies relevant to our calcula-
retical evolutionary calculations show that for temperatures
greaterthanT ∼35,000K(seeFigure1a),theratioofneu- tions,thedifferencebetweenthesenewerratesandtheolder
eff Beaudet,Petrosian,&Salpeter(1967)ratesare∼10percent.
trinotophotonluminosityishighestformassivewhitedwarf
Formorecompletedetailsaboutthecomputationofthemod-
models.Higherplasmafrequenciesresultfromhighercentral
els,seeMetcalfe,Nather,&Winget(2000).
densities in more massive pre-white dwarf models, yielding
higherplasmonenergiesandcorrespondinglyhigherneutrino
3.1. PeriodStructure
energyfluxes.Eventually,asthemodelscool,thethermalen-
ergiesbecometoosmallforphotonstoexciteplasmons.This Theproblemofthecouplingofthemechanicalandthermal
happensfirstforthemoremassivemodel,andaccountsforthe structure,prominentintheDOVs,isnearlynonexistentbythe
factthatthe neutrinorates ofthe less massivemodelsdomi- time the models reach the temperature domain of the DBV
natefortemperatureslessthan35,000K.Forthetemperature stars. Forexample,wecalculatedtheperiodsofmodelswith
range in which we are most interested, the DBV instability T ∼ 28,000 K from two sequences: a sequence including
eff
strip, neutrino losses still dominate for models with masses plasmonneutrinoenergylosses,andasequencewiththeneu-
nearthetypicalmassofthewhitedwarfstars,whiletheyare trinoproductionratessettozero.Thedifferencesbetweenthe
completelynegligibleforthemoremassivemodels. correspondingpulsationperiodsofthesemodelswerealways
ASTRONGTESTOFELECTRO-WEAKTHEORY 3
FIG. 2.—Therateofchangeoftheeffectivetemperatureforwhitedwarf FIG. 3.—Therateofchangeforpulsationperiodsinmodelsonthehot
modelsthatinclude(solidpoints)andignore(openpoints)neutrinoenergy endoftheDBVinstabilitystrip,including(solidpoints)andignoring(open
losses. Theeffectivetemperaturesoftwopulsatingwhitedwarfsareshown points)theeffectsofneutrinos.Anobservationalmeasurementoftherateof
as dashed lines. The models differ bya factor of4 at the temperature of periodchangeforhotterDBVstarsshouldallowustodetectthesignatureof
EC20058andafactorof2atthetemperatureofGD358. neutrinoemission.
.1second,withatypicaldifferenceof0.24seconds.Wenote causethedominantpulsationperiodsinawhitedwarfshould
thatcurrentuncertaintiesintheinteriormechanicalstructure, be tracers of the thermal timescale at the base of the par-
suchasthedetailedC/Oabundanceprofilesandthe“double- tial ionization/convection zone. Since the observed periods
layered”envelopesdiscussedbyFontaine&Brassard(2002) are much shorter in EC 20058 than in GD 358, EC 20058
andMetcalfe,Montgomery&Kawaler(2003),onlyaffectthe shouldhaveashallowerconvectionzone,andhenceahigher
periodsatthissamelevelof1secondorless. effectivetemperature(e.g.,seeWinget1998, andreferences
Theenergylosstoplasmonneutrinosprimarilyresultsina therein).
reduced central core temperature compared to models with- Atthese effectivetemperaturesthe evolutionisdominated
outneutrinos(forthecaseconsideredabove,thisreductionin bycooling,withgravitationalcontractionplayingadiminish-
coretemperaturewas∼20percent).Thepulsationperiodsare ing role. The cooling depletes the thermal energy stored in
onlyslightlyaffectedbecausetheBrunt-Väilsäläfrequencyis the ions. In models that include neutrinos, the thermal en-
small in the core, where the pressure support is dominated ergy deep in the core is further depleted by the emission of
by the strongly degenerateelectrons. As a consequence, the plasmon neutrinos, and the evolutionarytimescale is shorter
g-modepulsationsarelargelyinsensitivetothethermalstruc- thanin modelsthatignorethe effectsof neutrinos. Figure2
tureinthecore,whichistheonlythingthatchangesdramati- makesitclearthattheeffectoftheplasmonneutrinosonthe
callyasaresultofplasmonneutrinoproduction. evolutionisnotsubtle. Nearthehightemperatureendofthe
observedDBVinstabilitystriptheevolutionarytimescalesof
3.2. RatesofPeriodChange
themodelsdifferbyafactorofaboutfour,andbythemiddle
Although the change to the thermal structure due to plas- oftheinstabilitystriptheystilldifferbyafactoroftwo.
monneutrinosdoesnotsignificantlyaltertheperiodstructure, Thissuggeststhatmeasurementoftheevolutionaryperiod
it does have a significanteffect on DBV models; it shortens changeinhotDBVwhitedwarfstarswouldmakeanexcellent
theevolutionarytimescaleinthetemperaturerange30,000K probeoftheplasmonneutrinoproductionrates.Asaproofof
& T & 25,000 K. This is easily seen in Figure 2, where principle,weshowinFigure3theratesofperiodchangefor
eff
we show the effect of plasmon neutrinoson the time rate of theoreticalevolutionarymodelswith temperaturesin the hot
changeoftheeffectivetemperature.Thedashedlinesindicate halfoftheDBVinstabilitystrip.
thetemperatureestimatesofBeauchampetal.(1999)fortwo Wehavealreadyestablishedthatmodeluncertaintiesatthe
DBVstars: thehotteroneisEC20058- 5234,andthecooler currentleveldonotaltertheindividualperiodsinasignificant
one is GD 358. This should correspond roughly to the hot way for these measurements, and Figure 3 shows that exact
halfoftheDBVpulsationinstabilitystrip(Bradley&Winget radialorderidentificationisalsounnecessary.Ifweknowthe
1994). observedperiodandthesphericalharmonicdegreeofamode,
Although the temperature of EC 20058 deduced by thenwecandeterminetheradialorderwithin±1,whichcor-
Beauchampetal.(1999)isrelativelyuncertainduetoapoor respondstoaperiodrangeof60–80sformodelswithmasses
qualityspectrum,acomparisonoftheobservedperiodsinthe near0.6M .Foratypicalrangeofperiods,thisgivesussuffi-
⊙
twostarsprovidesindependentasteroseismologicalevidence cientinformationtomeasuretheplasmonneutrinoproduction
thatEC20058issignificantlyhotterthanGD358.Thisisbe- rateswithaprecisionof∼10percent. Wenotethatthethese
4 WINGETETAL.
values are only sensitive to spherical harmonicdegree (ℓ) at seismologically, spectroscopically, or both, the better limits
.10 percent: the differencebetweenℓ=1 and ℓ=2 for the wecanplaceondP/dt. Themodelratesofperiodchangealso
sameperiodis∼10percent,andthedifferencebetweenℓ=2 dependon the pulsation modeℓ value, so knowledgeof this
andℓ=3is.1percent. indexwillminimizetheuncertainties.
Third, the overallprogram will be strengthened by access
4. DISCUSSION to a range of DBVs with differentmasses. Figure 1a shows
Ourworkdemonstratesthepotentialofdirectlymeasuring that massive white dwarf stars (∼0.9 M⊙) are not produc-
the effect of plasmon neutrino energy loss rates in the hot ing significant plasmon neutrinos in this temperature range,
DBV white dwarf stars. Currently only one known DBV, andsocanserveasacontrol. Theratesofperiodchangefor
EC 20058- 5234, has both the high temperature and period thesestarsshouldbeinsensitivetoplasmonneutrinoproduc-
stabilitynecessarytomeasureanevolutionaryperiodchange. tionrates. Inthiswaytheywillallowustocalibrateoutany
AfteritsdiscoverybyKoenetal.(1995),thisobjectwasob- possible systematic effects. Comparative plasmon neutrino
servedina1997multi-siteWholeEarthTelescopecampaign fluxeswill also allow us to calibrate the fluxes as a function
(XCOV15, Sullivanetal. 2003), with regular single-site ob- ofdensity,andthereforeplasmonenergy.
servationssince then (Sullivan 2003). Work on this particu- Fourth,we needto undertakelong-termphotometriccam-
lar object is in progress, but to fully exploit the potential of paigns (in addition to the on-going EC 20058 observations)
theDBV starsas plasmonneutrinodetectorswe needtoun- on suitable DBV candidate stars. We estimate that an ob-
dertake a comprehensive research program that includes the servationaltimebaseofbetweenthreeandsixyearsduration,
followingcomponents. depending on individual timing accuracies, will be required
First, it will be important to expand the sample of known toachievethegoalofobservingthepredictedratesofperiod
DBV stars well beyond the 9 currently known. This should change.Two-meterclasstelescopesandefficientframetrans-
provideaccesstoaselectionofhotDBV pulsatorsthathave fer time-series photometerswill probablybe required to un-
the required period stability for measuring evolutionary pe- dertakethesecampaigns,sincethenewlydetectedDBVswill
riod changes. Fortunately, this goal appears to be feasible undoubtedlybe,onaverage,significantlyfainterthanthosein
using candidate white dwarfs identified in the Sloan Digital theexistingsample.
Sky Survey, as has been adequately demonstratedby the re- Insummary,ourworksuggeststhatnotonlycanwedemon-
centlargeincreasein thenumberofknownDAV starsusing strate the reality of the plasmon neutrino process, but we
thissource(Mukadametal.2003). can also quantitatively constrain the production rates in the
Second,wemustobtainbettereffectivetemperaturesforthe temperature-densitydomainrelevanttowhitedwarfinteriors.
hotDBV starsofinterest. Althoughnotasimportant,itwill The authors thank S.J. Kleinman, A. Nitta, R.E. Nather,
also be useful to identify the ℓ values of relevant pulsation F. Mullally, and A. Mukadam for many useful discussions.
modes. Appropriate HST observations will be an important Thework was supportedin partby grantsNAG5-9321from
factorhere.BothFigure2andFigure3indicatethatthemore NASA’s Applied Information Systems Research Program,
accurately we can determine the temperature, either astero- andARP-0543fromtheTexasAdvancedResearchProgram.
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