The Physics of Crystallization from Globular Cluster White Dwarf Stars in NGC 6397

0.2 MB·English

by  D. E. Winget

ACCEPTEDBYTHEASTROPHYSICALJOURNALLETTERS
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
THEPHYSICSOFCRYSTALLIZATIONFROMGLOBULARCLUSTERWHITEDWARFSTARSINNGC6397
D.E.WINGET1,2,S.O.KEPLER2,FABÍOLACAMPOS2,M.H.MONTGOMERY1,3,LEOGIRARDI4,P.BERGERON5,KURTISWILLIAMS1,6
(ReceivedAugust18,2008;AcceptedJanuary16,2009)
AcceptedbytheAstrophysicalJournalLetters
ABSTRACT
Weexplorethephysicsofcrystallizationinthedeepinteriorsofwhitedwarfstarsusingthecolor-magnitude
9
diagramandluminosityfunctionconstructedfrompropermotioncleanedHubbleSpaceTelescopephotometry
0
0 oftheglobularclusterNGC6397.Wedemonstratethatthedataareconsistentwiththetheoryofcrystallization
2 oftheionsintheinteriorofwhitedwarfstarsandprovidethefirstempiricalevidencethatthephasetransition
is first order: latent heat is released in the process of crystallization as predicted by vanHorn (1968). We
n outlinehowthisdatacanbeusedtoobservationallyconstrainthevalueofΓ≡E /E neartheonset
a Coulomb thermal
ofcrystallization,thecentralcarbon/oxygenabundance,andtheimportanceofphaseseparation.
J
Subjectheadings:whitedwarfs—densematter—equationofstate
9
1

1. STARFORMATIONHISTORYANDPHYSICSFROMTHEWHITE becausethediskpopulationcontainsstarsformedatdifferent
]
R DWARFSTARS timesandfromdifferentmainsequenceprogenitors. Thisis
S White dwarf stars are the inevitable progeny of nearly greatly simplified in a cluster sample and, most of all, in an
. all(≃97%)stars(e.g.,Fontaine,Brassard,&Bergeron2001, oldglobularcluster. InthispaperwefocusontheHSTpho-
h hereafterFBB).Theirdistributioncanbeusedtoextracttwo tometryofNGC6397andthedistributionofWDstarsinthe
p things:ageofthestellarpopulationandcoolingphysicsofthe color magnitudediagram. We reportevidencefor a “bump”
-
o whitedwarf(WD) stars. Thetwo are interrelated,butquali- intheLFduetothereleaseofthelatentheatofcrystallization
r tatively different. Extracting the age and history of star for- andwe show howthis can be refined to yield moreaccurate
st mationhasbecomeknownasWDcosmochronology. Anex- measuresoftheseprocesses.
a cellentreviewemphasizingthisconnectionandtheattendant
[ uncertaintiesisgivenbyFBB. 2. ANCHORINGTHEWHITEDWARFEVOLUTIONARY
SEQUENCESINTHECOLOR-MAGNITUDEPLANE
The techniques of WD cosmochronology have been suc-
1
v cessfullyappliedtothediskbyanumberofinvestigators(e.g., Wefitmainsequence,pre-WD,andWDevolutionarymod-
0 Wingetetal.1987;Wood1992;Hansen&Liebert2003,and els simultaneously. The main sequenceandpre-WD models
5 referencestherein)andarebeingcontinuouslyrefined. They we used for this work were computed with the Padova stel-
9 have also been applied to a varietyof open clustersand cal- larevolutioncode(Marigoetal.2008). Weusedavarietyof
2 ibrated against main sequence turnoff and related methods metallicitiestodeterminethebestfittothemainsequenceand
. (e.g., Kaliraietal. 2007; DeGennaroetal. 2008, and refer- WDmodels.
1
encestherein). TheHubbleSpaceTelescope(HST)photom- Our WD evolutionary models have updated constitu-
0
9 etry obtained by Richer and Hansen and their collaborators tivephysics(seee.g.,Bischoff-Kim,Montgomery,&Winget
0 (Hansenetal. 2002, 2007; Richeretal. 2008) has yielded a 2008). We place the new generation of WD evolutionary
models of DA and DB WD stars in the observed F
: new harvest of informationfor WD populations. They have 814W
v usedtheAdvancedCameraontheHSTtoreachtheterminus vs. F606W color-magnitudediagram(hereinafterCMD) using
Xi of the WD cooling sequence, giving us a qualitatively dif- P. Bergeron’s model atmosphere grids7 (for a detailed de-
ferent tool for analyzing the WD population. Hansenetal. scriptionseeBergeronetal.1995;Holberg&Bergeron2006;
r
a (2007)usedMonteCarlotechniquesinconjunctionwiththeir Holberg,Bergeron,&Gianninas2008)alongwithananalyt-
coolingmodelsto determinethe age of NGC 6397from the icalcorrectiontotheKowalski(2007)resultsfortheeffectof
WDstars,attemptingtoaccountforuncertaintiesinthebasic Ly α far red-wing absorption. This correction is small and
physicalparametersoftheWD starstodetermineanagefor willbediscussedinaforthcomingpaper.
thecluster;usinggoodnessoffitcriteria,theyarriveatanage Our best fit of the CMD in the natural ACS color sys-
forthecluster,basedonWDcooling,of11.47±0.47Gyr. temgivesametallicityofZ=0.00012±0.00001,E(606W-
Findingthe signatureofthe key physicalpropertiesof the 814W)=0.22±0.02,and(m- M)=12.49±0.05(Figure1)
WDstarsinthediskluminosityfunction(hereinafterLF)has and a main sequence turnoff age of 12-+10..05 Gyr. The age is
provenmore difficult than getting an age constraint. This is consistent with the values found by Richeretal. (2008) and
Hansenetal. (2007), even though they used the Dartmouth
1Department ofAstronomy,University ofTexasatAustin, Austin, TX, EvolutionarySequence (DES). The metallicity is a factor of
USA;[email protected] two lower than Richeretal. (2008) but is in agreementwith
2Instituto deFísica, Universidade Federal doRioGrandedoSul, Porto theindependentdirectspectroscopicdeterminationsbasedon
Alegre,RS-Brasil
VLTdata(Kornetal.2007). Thevaluesofthese parameters
3Delaware Asteroseismic Research Center, Mt. Cuba Observatory,
fixtheWDcoolingtracksinthecolor-magnitudediagramand
Greenville,DE,USA
4INAF-PadovaAstronomicalObservatory eliminatethefreedomtoslidethetracksashasbeendonein
5Département de Physique, Université de Montréal, C. P. 6128, Succ. other works (e.g., Hansenetal. 2007). This constrains the
Centre-Ville,Montréal,QuébecH3C3J7,Canada
6NSFAstronomy&AstrophysicsPostdoctoralFellow 7http://www.astro.umontreal.ca/∼bergeron/CoolingModels/
2 Wingetetal.
NGC6397 Hubble ACS FIgure 2: NGC 6397 white dwarfs with DA and DB evolutionary tracks with
5 Bergeron-Kowalski Colors and E=0.22,(m-M)=12.49
22
23
24
10
Padova 12Gyr
25
[[FFee//HH]]==--22..01 F814W 26 DA 000...454705500
0.535
15 [Fe/H]=-2.2 27 0.580
0.685
0.800
0.900
28 DB 0.450
(m-M)=12.49 00..550305
E(F606W-F814W)=0.22 29 0.580
0.685
0.800
20 0.900
30
-0.5 0 0.5 1 1.5 2
F606W-F814W
FIG.2.— NGC6397WDswithDAandDBevolutionarysequencesusing
Bergeron(2006)atmospheresadjustedwithananalyticalfittotheKowalski
25 (2007)changesforLy-αred-wingopacity.
of the reddening. We emphasize again that simultaneously
fittingthemainsequenceandwhitedwarfsequenceprovides
0 1 2 3 tightconstraintsonboththedistancemodulusandtheredden-
F606W-F814W
ing.
InFigure3weshowtheLFofboththeHansenetal.(2007)
FIG. 1.— The best fit to the proper-motion screened HST data on
andRicheretal.(2008)samples. ThepeakinbothLFsnear
NGC6397usingthePadovastellarevolutioncode(Marigoetal.2008)and
theBergeron(2006)atmospheres. Thisfitgives[Fe/H]=- 2.2, (m- M)= F814W =26.5suggeststhatevolutionslowsthroughthisre-
12.49,andE(F606W- F814W)=0.22. ThisanchorsourWDsequencesin gion. In the sample of Hansenetal. (2007) the LF contin-
theCMD.
best-fittotalWD massthroughoutthe CMD. Itisclear from
Figure2thatthetracksfitthebulkofthesamplewell.
The data (see Figure 2) are taken from the proper motion
selected sample of Richeretal. (2008). This is a more ho-
mogeneous sample than that of Hansenetal. (2007) but is
smaller because of the reduced area and magnitude limits
of the proper-motion data. Four features stand out in Fig-
ure 2: there is a gap in the distribution near F814W =24.5
thatmaybestatisticallysignificant,thereisanoticeablecon-
centration of stars near F814W = 26.5, there is a terminus
atapproximatelyF814W =27.6and a noticeableturnto the
bluebeforetheterminus. Theselasttwofeatureswerenoted
in Hansenetal. (2007). In this paper we focus on what we
canlearnfromtheconcentration,orclump,ofstarsnear26.5,
providingaphysicalexplanation.
3. PHYSICSWITHTHECMDANDLUMINOSITYFUNCTION
The CMD diagramconstrainsthe mechanicalandthermal
propertiesoftheWDstars(Richeretal.2008). Oncetheevo-
lutionary tracks have been anchored by the main sequence
andWDsequencesimultaneouslyasdescribedabovewecan
move on to exploring the physics contained in the CMD.
Hansenetal.(2007)pointoutthatthelocationoftheterminus
providesasimplelower-limittotheageoftheclusterfromthe
WDcoolingtimes. Forourmodels,thisWDcoolinglimitis
reachedatabout10.5Gyrforpurecarboncoremodels. This
agelimitisconsistentwiththevaluesquotedinHansenetal.
(2007) and Richeretal. (2008). The position of the tracks
isinsensitivetoprocessesaffectingonlytheage;toexamine
thesewemustlooktotheLF,thenumberofstarsobservedas
afunctionofmagnitude.
Fortheparametersdescribedabove,itisevidentfromFig- FIG. 3.—toppanel: TheobservedWDLF(histogram)andcompleteness
ure2thatmodelswithmassesintherangeof0.500- 0.535M⊙ rpealnaetilo:nth(deostatemdelifnoer)tohfethHeanRsiecnheerteatl.al(.2(020070)8s)asmamplpel.eFfoorrNboGthCs6a3m9p7l.elso,wweer
bestfittheregionnearthecenteroftheclump.Thisincreases notethatthecompletenesschangesfairlyslowlyovertheregionoftheob-
slightly fordecreasedvaluesof m- M andis also a function servedrapidfall-offofstarsandwhileremainingabove50%.
CrystallizationPhysicsfromWDs 3
uedtorisewellpastthispeak,soitwasnotthemaximumof
thedistribution;itcompletelydominatesthedistributionafter
the application of the proper motion selection (Richeretal.
2008).Thecompletenessestimatesofbothsampleshavebeen
carefullyconsideredbytherespectiveauthors(Figure3,dot-
tedlines)andtherelativelyslowvariationofthecompleteness
near this peak implies it is not the result of incompleteness.
We therefore seek a physicalexplanationof this peak in the
contextofaphysicalprocessthatoccursnearthispointinthe
models.
TwoprocessesoccurinthedominantDAmodelsnearthis
point:crystallizationandtheconvectivecoupling(e.g.,FBB).
Crystallization,throughthereleaseoflatentheat,slowsdown
evolutionandproducesabumpin theLF. Convection,when
itreachesdowntothedegeneracyboundary,decreasesthein-
sulation of the nondegenerate envelope and temporarily in-
creases the total temperature gradient; this serves to slow
downtheevolution,briefly,thencausesittospeedupagain.
This producesa broad feature in the WD cooling curve that
willhaveasignatureintheLF.
3.1. The ΓofCrystallization
Crystallization in the dense Coulomb plasma of WD inte-
FIG. 4.— Theobserved WDLFofNGC6397(Richeretal. 2008, his-
riors was theoretically predicted independently by Kirzhnits
togram) with LFs from theoretical evolutionary sequences of 0.5M⊙ DA
(1960), Abrikosov (1960), and Salpeter (1961) to occur modelswithpurecarboncores(lines): crystallizationwith(“Pure_C_Xtal”)
when the ratio of the Coulomb energy to the thermal en- andwithout(“Pure_C_Xtal_No_LH”)thereleaseoflatentheat,andexclud-
ergy of the ions (the ratio “Γ”) is large. For a one- ingthephysicsofcrystallization altogether(“Pure_C_No_Xtal”). Thenor-
malizationofthetheoreticalcurvesischosentominimizetheRMSresiduals
component plasma (OCP), there is universal agreement
intheneighborhoodofthepeak,betweenthemagnitudesof25.1and27.7,
amongdifferenttheoreticalapproachesthatcrystallizationoc- thefaintestvaluecalculatedforthenocrystallizationcase. Thevalueofthe
curs when Γ≃175 (e.g., Slattery,Doolen,&Dewitt 1982; averageresidualforeachcurveislistedinthelegend,e.g.,itis4.77forthe
“Pure_C_Xtal”case.
Stringfellow,Dewitt,&Slattery 1990; Potekhin&Chabrier
2000; Horowitz,Berry,&Brown 2007). There is a similar correction given explicitly in Table 4 of Richeretal. (2008)
consensus, based largely on a density-functional approach, andshowninthetoppanelofourFigure3. Finally,wenor-
thatthisvalueofΓalsoholdsforabinarycarbonandoxygen malize the resulting curves by minimizing their root-mean-
mixture. Such a mixture is likely relevant to WD interiors. squareresidualsintheneighborhoodofthepeak,betweenan
Recently, Horowitz,Berry,&Brown (2007) used a massive F814Wof25.1and27.5.
moleculardynamicscomputationtoexplorecrystallizationin In Figure 4 we show the LF of our fiducial sequence
a dense Coulomb plasma. They found Γ=175 for an OCP, (“Pure_C_Xtal”), that of a sequence with crystallization ar-
whileforaspecificmixtureofelementstheyfoundΓ≃237. tificially suppressed (“Pure_C_No_Xtal”), and that of one
Asweshow,suchadifferenceispotentiallymeasurablefrom including crystallization but artificially excluding the latent
theobservationsofWDstarsinglobularclustersorolderopen heatofcrystallization(“Pure_C_Xtal_No_LH”);allareplot-
clusters. tedoverahistogramoftheobservedLF.Thenocrystallization
andnolatentheatsequencesshowevidenceofabumpdueto
3.2. Luminosityfunctionswithandwithoutcrystallization convectivecouplingaroundF814W∼26,butdonotcontinue
Hansenetal.(2007)demonstratedintheiranalysisthatthe torise throughtheobservedmaximum;thisisclearlyincon-
entire observed sequence represents a very narrow range of sistent with the data, and no adjustment of mass or internal
WDmasses,includingmagnitudeswellbelowF814W=26.5. composition can bring them into good agreement. In terms
It is therefore reasonable for purposes of this initial explo- of χ2, for average observational errors of ∼5.5 stars/bin in
ration to adopt a fiducial mass. On the basis of the model theneighborhoodofthepeak,wehaveχ2=0.75forthecrys-
tracksin the CMD shownin Figure 2, we choose the model tallizingsequenceandχ2∼2.2forthenocrystallizationand
that passes nearest the color of the red edge of the clump no latent heat sequences, a nearly three-fold increase in χ2.
of stars corresponding to the peak in the LF; this model Thus,thesequencewithcrystallizationprovidesamuchbet-
has a mass of 0.5M⊙. For the layer masses, we assume termatchtothedata.
M /M = 10- 4 and M /M = 10- 2 for the DA sequences,
H ⋆ He ⋆
and M /M =10- 2 for the DB sequence. We adopt a car- 3.3. Constrainingcrystallization,phaseseparation,andcore
He ⋆ composition
boncoremodelincludingtheeffectsofcrystallizationforthis
sequence. In the years since vanHorn (1968) the realization that
Assumingaconstantstarformationrate,thetheoreticalLF the cores of normal mass WD stars should consist of
is proportional to the “cooling function” of an evolutionary a mixture of carbon and oxygen implied that crystal-
model sequence. This function is given by the derivative lization may also release energy resulting from phase
(dt/dm),wheremistheF814Wmagnitudeofagivenmodel separation of the carbon and oxygen (Stevenson 1977;
andt isitsage. Sincewewillbecomparingdirectlywiththe Barrat,Hansen,&Mochkovitch 1988; Segretain&Chabrier
data,wealsomultiplythetheoreticalLFbythecompleteness 1993;Segretainetal.1994;Isernetal.2000).Thisoccursbe-
4 Wingetetal.
rioroxygenabundancebecomesstronger.Takenatfacevalue,
theseresultsindicatethatthecarbon-to-oxygenratioismuch
greaterthan1,andwewillbeabletomakeamorequantitative
statementfromourfuturemorecompleteBayesianstatistical
analysis(inpreparation).
Second, as shown in Figure 4, the data are consistent and
well-fitbycarboncoremodelswithcrystallizationandthere-
leaseoflatentheat,butnotbymodelswithout.Thisconfirms
thepredictionofvanHorn(1968)thatcrystallizationisafirst
orderphase transitionand releases the latent heatof crystal-
lization. Were itnotso, crystallizationwouldleavenosharp
peakat thismagnitudein the observedLF. Thisimpactsour
understanding of solid-state physics at extremely high den-
sity: it is the first empirical confirmation of the release of
latent heat during crystallization – an important theory that
has a large impact on WD ages, as has been pointed out by
vanHorn(1968)andmanyauthorsintheinterveningyears.
Third,itispossibleapriorithatasignificantfractionofthe
observedWDs may be DBs. The mismatch of observed LF
andthe DB sequencein Figure5 essentially eliminatespure
Heatmospheresasasignificantcomponentofthesample(as
shown by Hansenetal. 2007), but not models that become
mixed (H/He) as they cool; we explore this possibility in a
FIG. 5.— The same as Figure 4 but for DA model sequences with forthcomingpaper.
uniform, 50:50 carbon/oxygen cores: crystallization only (“CO_Xtal”), Fourth, the “best-fit” fiducial sequence in Figure 4 begins
crystallization and phase separation (“CO_Xtal_PS”), and no crystalliza- crystallizingnearthevalueofΓ ≡E /E ∼170.
tion (“CO_No_Xtal”). Inaddition, weshow a pure oxygen DA sequence xtal Coulomb thermal
(“pure_O_Xtal”)andapurecarbonDBsequence(“pure_C_DB_Xtal”). All If the actual value is higher (lower) then crystallization will
modelshave0.5M⊙. occuratlower(higher)luminosities. Highervaluesallowfits
with larger amounts of oxygen in their cores. In addition,
causewhenacarbon/oxygenmixturecrystallizestheoxygen Potekhin&Chabrier(2000)showthatevenforapurecompo-
contentofthesolidshouldbeenhanced. SinceWDscrystal- sitiontheoreticaluncertaintiesinthepolarizationoftheelec-
lizefromthecenteroutwards,thisleadstoa nettransportof tron Fermi gas and quantum effects in the liquid and solid
oxygen inward and carbon outward, and because oxygen is phasecanalterthevalueofΓxtal. Infutureanalyses,anaccu-
slightlyheavierthancarbonthisdifferentiationreleasesgrav- rate determinationof the mass, distance, and reddeningwill
itationalenergy. leadtoanaccuratedeterminationofΓ andthecorecompo-
xtal
We have included this energy in our models as de- sition.
scribed in Montgomeryetal. (1999). For these compu- Finally,wenotethatthecentraldensityandtemperatureas-
tations we have assumed the carbon and oxygen abun- sociatedwithaparticularvalueofF814W throughthemodel
dances are equal throughout the core. This underesti- atmospheresissensitive onlytothe mass-radiusrelationship
mates the oxygen abundance compared to that predicted setbythedegenerateelectronpressuresupport. Thisis very
by standard stellar evolution calculations (e.g., Salarisetal. insensitive to theC/O relative abundances— these produce
1997), but the remaining uncertainty in the C(α,γ)O reac- differencesofδF814W <0.05.Thisimpliesthatthevalueof
tion rate (Metcalfe,Salaris,&Winget 2002; Assunçãoetal. Γ in the center, at the peak of the LF for example, is sensi-
2006) lead to a degree of uncertainty in the C/O ratio and tiveonlytotheinteriorcomposition. Thereforeweconclude
profile. In Figure 5 we show several LFs: crystalliza- thattheonsetofcrystallizationisdeterminedbytheparticular
tion only (“CO_Xtal”), crystallization with phase separation mixtureand the value of Γ for that mixture. Comparison of
(“CO_Xtal_PS”), no crystallization (“CO_No_Xtal”), pure thetheoreticalmodelsandthedatapromisetoprovideimpor-
oxygencorewithcrystallization(“Pure_O_Xtal”),andapure tantmeasuresoftheonsetanddevelopmentofcrystallization.
carbonDBsequence(“Pure_C_DB_Xtal”). Asisreadilyap-
parent,allofthesesequenceshaveapeakwhichistoobright 4. DISCUSSION,SUMMARY,ANDFUTURESOFEXPLORING
byatleast0.5maginF814W. WHITEDWARFPHYSICSWITHCMDS
These results have several interesting possible interpreta- Althoughwearenotfocusedonuncertainties,itisreason-
tions. First, theresultsseemtosuggestthattheoxygencon- able to examine how changing the distance modulus might
tent of these stars is relatively small or zero, since it is the affect the results. Put another way, how much does the dis-
highercrystallizationtemperatureofoxygenwhichshiftsthe tance have to change to reproduce the peak of the observed
peak in the LF to smaller magnitudes. The only way to LF with oxygen crystallization rather than carbon? The an-
accommodate more oxygen would be to have lower mass swer is containedin Figure 5. Here we see thatto makethe
models, in conflict with the distance modulus(Hansenetal. location of the peak of the LF consistent with oxygen crys-
2007). Thecolors(e.g.,Figure2)alsomakeitdifficulttoap- tallization we have to lower the distance modulusby a little
peal to lower masses with higher oxygen abundances. Ad- more than 0.5 magnitudes— this possibility is excluded by
ditionally, for plausible IFMRs the main sequence lifetime themainsequencefitting(Richeretal.2008).
for single stars becomes more problematic with lower WD We have shown that simultaneously fitting the main se-
masses even with the IFMR dependency on metallicity of quence and the WDs in a cluster gives the best possible
Meng,Chen,&Han(2008). Thusthe constraintontheinte- constraintondistance,metallicityandreddeningcorrections.
CrystallizationPhysicsfromWDs 5
Physically realistic atmospherecalculationsthen allow us to Brassard&Fontaine (2005). While certainly important, we
place evolutionarytracksin the CMD. The numberdistribu- notethatasteroseismologicalanalysesdonotprobethelatent
tion of stars contains important information on the internal heatofcrystallization;thisquantityisaccessibleonlythrough
physicsoftheWDstars. Thisallowsustoexplorethephysics the WDLF, as demonstrated in this paper. To this end, we
of crystallization. We presentevidencethat the data is most eagerly anticipate the forthcomingHST observationsof this
consistentwith a first orderphase transition, releasing latent cluster. Thesewillprovideaproper-motionscreenedsample
heatduringcrystallization, as proposedby vanHorn(1968). overalargerareaoftheclusterandtofaintermagnitudes,pro-
Thecurrentdataplacesconstraintsontheonsetofcrystalliza- vidinganexactingtestoftheideasputforthinthispaper.
tion, the central carbon/oxygen abundance, and the compo-
sition of the envelope at the degeneracy boundary. We will
improvetheseconstraintswithamorecompleteBayesiansta- The authors wish to thank A. Zabot for help computing
tisticalanalysisinthenearfuture.Thisworkalsopointstothe theisochrones,TedvonHippel,andHughvanHornforuse-
importanceofforthcomingdataonadditionalclustersaswell ful discussions, and Brad Hansen and referees for suggest-
asincreasingthesampleofstarsthroughmoreHSTfieldson ing improvements. D.E.W., S.O.K., and F.C. are fellows of
this cluster. This work also underscoresthe essential nature CNPq-Brazil. M.H.M.andK.A.W.aregratefulforthefinan-
ofmorepropermotiondatatogetthemostinformationoutof cialsupportoftheNationalScienceFoundationunderawards
thesekindsofstudies. AST-0507639 and AST-0602288, respectively, and M.H.M.
Pulsationsmayalsoallowanasteroseismologicaldetermi- acknowledgesthesupportoftheDelawareAsteroseismicRe-
nationofthecrystallizedmassfractionformassivepulsators, searchCenter.ThisworkwassupportedinpartbytheNSERC
asshownbyMetcalfe,Montgomery,&Kanaan(2004)forthe CanadaandbytheFundFQRNT (Québec). P.Bergeronisa
DAVBPM37093,althoughthisclaimhasbeenchallengedby CottrellScholarofResearchCorporation.
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