The evolution of white dwarfs with a varying gravitational constant

0.1 MB·English

by  L. G. Althaus

Astronomy&Astrophysicsmanuscriptno.gnew (cid:13)c ESO2011
January6,2011
The evolution of white dwarfs with a varying gravitational constant
L.G.Althaus1,2,3,A.H.Co´rsico1,2,3,S.Torres4,5,P.Lore´n–Aguilar4,5,J.Isern6,5,andE.Garc´ıa–Berro4,5
1 Facultad de Ciencias Astrono´micas y Geof´ısicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata,
Argentina
2 InstitutodeAstrof´ısicaLaPlata,IALP,CONICET-UNLP,Argentina
3 MemberoftheCarreradelInvestigadorCient´ıficoyTecnolo´gico,CONICET,Argentina
4 DepartamentdeF´ısicaAplicada,UniversitatPolite`cnicadeCatalunya,c/EsteveTerrades,5,08860Castelldefels,Spain
5 InstituteforSpaceStudiesofCatalonia,c/GranCapita`2–4,Edif.Nexus104,08034Barcelona,Spain
6 InstitutdeCie`nciesdel’Espai(CSIC),CampusUAB,08193Bellaterra,Spain
1 e-mail:althaus,[email protected]; santi,loren,[email protected]; [email protected]
1
January6,2011
0
2
ABSTRACT

n
a Context.Withinthetheoreticalframeworkofsomemodernunificationtheoriestheconstantsofnaturearefunctionsofcosmological
J time.WhitedwarfsofferthepossibilityoftestingapossiblevariationofGand,thus,toplaceconstraintstothesetheories.

5 Aims.WepresentfullwhitedwarfevolutionarycalculationsinthecasethatGdecreaseswithtime.
Methods.Whitedwarfevolutioniscomputedinaself-consistentway,includingthemostup-to-datephysicalinputs,non-graymodel

] atmospheresandadetailedcorechemicalcompositionthatresultsfromthecalculationofthefullevolutionofprogenitorstars.
R Results.Wefindthatthemechanical structureandtheenergybalanceof whitedwarfsarestrongly modifiedbythepresence ofa
S varyingG.Inparticular,forarateofchangeofGlargerthanG˙/G=−1×10−12 yr−1,theevolutionofcoolwhitedwarfsismarkedly
. affected.TheimpactofavaryingGismorenotoriousinthecaseofmoremassivewhitedwarfs.
h
Conclusions.In view of the recent results reporting that a very accurate white dwarf cooling age can be derived for the old and
p
metal-richopenclusterNGC6791,ourstudysuggeststhatthisclustercouldbeapotentialtargettoconstrainordetectahypothetical
- secularvariationofG
o
r Keywords.stars:interiors—stars:evolution—stars:whitedwarfs
t
s
a
[
1. Introduction (Gaztan˜agaetal.2002).Finally,BigBangNucleosynthesisalso

1 provides limits on possible variations in Newton’s Constant,
v One of the principles of General Relativity — the equivalence −3×10−13yr−1 <∼(G˙/G)<∼4×10−13yr−1(Copietal.2004).
6 principle — requires that the fundamental constants should be
White dwarfs provide an independent way of constraining
8 independent of location. However, in several modern grand-
any hypothetical variation of G. There are several reasons for
9
unification theories, the constants of nature are supposed to be
this. First, white dwarfs are extremely long-lived stars. Thus,
0
. functions of low-mass dynamical scalar fields — see, for in- theeffectsofavaryingG canbecomeprominent,evenforvery
1 stance, Lore´n–Aguilar et al. (2003) and references therein. If
smallratesofchange.Second,whitedwarfsaretheend-pointof
0 these theories are correct, we expect them to experience slow
stellar evolution for the vast majority of stars. Hence, they are
1 changesovercosmologicaltimescales, and thusto also depend
numerous. Third, white dwarfs are compact objects, and their
1
onlocation.Inrecentyears,severalconstraintshavebeenplaced
structureisverysensitivetotheprecisevalueofG.Finally,the
:
v onthevariationofthefine structureconstant—seethe review evolutionofwhitedwarfsisrelativelywellunderstood,andcan
Xi papers of Uzan (2003) and Garc´ıa–Berro et al. (2007) for re- bewelldescribedasasimplegravothermalprocess.Hence,for
centrevisionsofthe moststringentupperlimits totheir rateof sufficientlylowtemperaturestheirluminosityisderivedentirely
r change. In sharp contrast with the vivid debate about whether
a froma close balance betweenthe thermalandthe gravitational
(or not) there is evidence for a varying fine structure constant, energies.Consequently,asecularlyvaryingGlargelyaffectsthe
relativelyfew workshavebeendevotedto studya hypothetical
gravothermalbalanceofwhitedwarfsand,thus,theirluminosi-
variation of G. The reason probably lies on the intrinsic diffi-
ties.
cultyofmeasuringthevalueofthisconstant(Mohretal.2008).
Oneofthewaysinwhichwhitedwarfscanbeusedtocon-
Actually,G is the fundamentalconstantfor which we have the
strain a variation ofG considersthe dependenceof the secular
lessaccuratedetermination,andtheseveralmeasuresofGdiffer
rateofchangeoftheperiodofpulsationofvariablewhitedwarfs
considerably.Thetightestconstraintsontherateofvariationof
on its cooling rate (Benvenuto et al. 2004; Biesiada & Malec
GcomefromLunarLaserRanging,G˙/G<∼(2±7)×10−13yr−1 2004).Ithasbeenshownthattheirsecularrateofchangeofthe
(Mu¨ller & Biskupek 2007), but these are purely local. At in-
periodnotonlydependsonthecoolingratebutalsoontherate
termediate cosmological ages the Hubble diagram of Type Ia
ofchangeofG. Benvenutoetal.(2004)appliedthismethodto
supernovae can also be used to constrain the rate of variation
thewellstudiedvariablewhitedwarfG117-B15Aandobtained
of the gravitational constant, G˙/G <∼ 10−11 yr−1 at z ∼ 0.5 a rather loose bound, −2.5×10−10 yr−1 ≤ G˙/G ≤ 0. Another
methodtoconstrainapossiblevariationofGistousethewhite
Sendoffprintrequeststo:E.Garc´ıa–Berro dwarfluminosityfunction.Thenumbercountsofwhitedwarfs
2 L.G.Althausetal.:Whitedwarfevolutionwithavaryinggravitationalconstant
Table1.Whitedwarfevolutionarysequencescomputedinthiswork.We listthewhitedwarfstellarmass(insolarunits)and,for
eachvalueoftherateofchangeofG,G˙/G(inunitsofyr−1),theinitialvalueofGatthebeginningofthewhitedwarfcoolingphase,
Gi/G0.Thenumbersinbracketsgivethestellarluminosity,log(L/L⊙),atwhichthepresentvalueofthegravitationalconstant,G0,
occurs.
MWD/M⊙ Gi/G0
G˙/G=−5×10−11 G˙/G=−1×10−11 G˙/G=−1×10−12
0.525 1.40(−4.40) 1.10(−4.37) 1.010(−4.33)
0.525 1.30(−4.20) 1.05(−4.05) 1.005(−4.00)
0.525 1.20(−4.05) 1.02(−3.66)
0.525 1.10(−3.77)
0.609 1.50(−4.83) 1.20(<−5) 1.100(<−5)
0.609 1.40(−4.40) 1.10(−4.30) 1.050(<−5)
0.609 1.30(−4.19) 1.05(−4.02) 1.020(<−5)
0.609 1.20(−4.02) 1.02(−3.60)
0.609 1.10(−3.68)
1.000 1.24(−4.64) 1.10(−4.55) 1.020(<−5)
1.000 1.20(−4.12) 1.05(−3.68) 1.010(−4.30)
1.000 1.10(−3.23) 1.02(−3.10) 1.005(−3.56)
depend sensitively on the characteristic cooling time of white isnotcorrect.Insummary,althoughtheevolutionaryproperties
dwarfsinthecorrespondingluminosityinterval,sodoesthepo- of white dwarfs can be potentially used to set upperboundsto
sitionofthecut-offofthewhitedwarfluminosityfunctionatlow the rate of variation of G, very few studies have been done up
luminosities.Garc´ıa–Berroetal.(1995)usedasimplifiedtreat- to now,and thosealreadydonearenotcompleteor are flawed.
ment to check which could be the effects of a slowly varying Moreover,noneoftheseworkstookintoaccountthatifGvaries
G in the white dwarf luminosity function. They assumed that itsvalueinthepastdiffersfromitspresentvalue,andallthecal-
G˙/G was small enough to ensure that white dwarfs have time culationsweredoneusingthepresentvalueofG,thusneglecting
toadjustitsmechanicalstructuretothe presentvalueofG ina apotentiallyimportanteffect.Thispaperaimsatfillingthisgap.
timescalemuchshorterthanthatofthecoolingtimescale.Under
this assumption,energyconservationleads to (Garc´ıa–Berroet
al.1995): 2. Inputphysics
dB G˙ Inthisworkwefollowinaself-consistentwaytheevolutionof
L+Lν =− + Ω. (1) whitedwarfsinthecaseofavaryingG.Intheinterestofsimplic-
dt G
ity we haveassumed thatG˙/G remainsconstantwith time. All
where Bisthe bindingenergyofthe whitedwarf, B = U +Ω, the calculations have been done using the LPCODE stellar evo-
U is the total internal energy, Ω is the total gravitational en- lutionary code — see Althaus et al. (2010) and Renedo et al.
ergy and L is the neutrino luminosity. The evolution of the (2010) for recent applications of this code. The main physical
ν
luminosity was then obtained from a series of static models, inputs considered in LPCODE comprise the following. We have
not from fully evolutionary calculations, assuming a relation- included22Nediffusionanditsassociatedenergyrelease—see
ship betweenthe luminosityand the coretemperatureobtained Garc´ıa–Berro et al. (2008; 2010) and Althaus et al. (2010) for
fromfitstoevolutionarycalculationswithconstantG.Despiteof details. The energy sources arising from crystallization of the
thissimplifiedtreatmenttheywereabletoobtainanupperlimit white dwarf core — namely, the release of latent heat and of
G˙/G≤−(1.0±1.0)×10−11yr−1.Later,Biesiada&Malec(2004) gravitationalenergyassociatedwithcarbon-oxygenphasesepa-
casteddoubtsonthistreatment.Inparticular,theyclaimedthat ration(Garc´ıa–Berroetal.1988;Isernetal.1997;2000)—are
thelastterminEq.(1),involvingG˙/G,cancelswhentheexpan- fullytakenintoaccountfollowingthetreatmentofAlthausetal.
sionworkisfurtherexpanded.However,theircriticismwasnot (2010).Weusedthecarbon-oxygenphasediagramofSegretain
correctbecausealltherelevanttermsintheenergyconservation &Chabrier(1993).Otherphasediagramsforthecarbon-oxygen
equationwerecarefullytakenintoaccountinGarc´ıa–Berroetal. mixturehavebeenrecentlycomputed(Horowitzetal.2010),but
(1995). aslongasweareconcernedwithrelativedifferencesinthewhite
What neither Garc´ıa–Berro et al. (1995) nor Biesiada & dwarfcoolingtimeswhenarunningGisadopted,theimpactof
Malec (2002) took into account is the fact that the relation- a differentchoice of the phase diagram on the boundsonG˙/G
ship between the core temperatureand the luminosityof white shouldbeminor.Forthe12C(α,γ)16Oreactionrateweadoptthe
dwarfsalsodependsonG.Forinstance,adoptingthesimplified prescription of Angulo et al. (1999). Again, we stress that al-
Mestelcoolinglaw(Mestel1952)itturnsoutthat L ∝ G. This though the precise value of this poorly known reaction rate —
means that a full stellar evolutionary code is needed if an ac- see,forinstanceAssunc¸a˜oetal.(2006)—isimportanttoobtain
curatetreatementofwhitedwarfcoolingwhenG changeswith the chemical stratification of white dwarf progenitorsand thus
time is to be done. To our knowledge,the only calculations of to compute absolute white dwarf cooling ages, the relative ef-
cooling white dwarfs with a time-varyingG employing an up- fect of a varyingG in the modelsdoesnot appreciablydepend
to-date stellar evolutionary code are those of Benvenuto et al. on it. We use non-graymodelatmospheresto provideaccurate
(1999),buttheiranalysisturnedouttobeflawed.Specifically,in outerboundaryconditionsforourmodels.Ouratmospheresin-
thiswork the energyconservationequationfora runningG in- clude non-idealeffects in the gas equation of state and chemi-
cludedthedifferentialversionofthelasttermofEq.(1),which calequilibriumbasedonthe occupationprobabilityformalism.
L.G.Althausetal.:Whitedwarfevolutionwithavaryinggravitationalconstant 3
-2 -2
G= 1.5 G G= 1.1 G
i o i o
GGi== 11..43 GGo Gi= 1.2 Go
i o G= 1.24 G
G= 1.2 G i o
i o G= G
-3 G= 1.1 G -3 o
i o
G=G
o
n n
u u
s s
L L
L/ L/
g g
o o
L L

-4 -4
M = 0.609 M M = 1.0 M
WD sun . WD sun
. -11 -1
-11 -1 G /G= -5x10 yr
G /G = -5x10 yr
-5 -5
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Age (Gyr) Age (Gyr)
Fig.1. Surface luminosity versus age for a 0.609M⊙ white Fig.2.Surfaceluminosityversusagefora1.0M⊙ whitedwarf
dwarf assuming G˙/G = −5 × 10−11 yr−1 and different initial assumingG˙/G = −5×10−11 yr−1 anddifferentinitialvaluesof
values of G at the start of the cooling track. We also show us- G at the start of the cooling track. We also show using square
ingsquaresymbolstheluminosityandageatwhichthepresent symbolstheluminosityandageatwhichthepresentvalueofG
valueofGisreachedinthecoolingtrack. isreachedinthecoolingtrack.
They consider also collision-inducedabsorption due to H2-H2, the rate of change of G, namely G˙/G = −5 × 10−11 yr−1,
H2-He, and H-He pairs, and the Lyα quasi-molecular opacity G˙/G = −1 × 10−11 yr−1, and G˙/G = −1 × 10−12 yr−1. As it
thatresultfromperturbationsofhydrogenatomsbyinteractions willbecomeclearlater,theevolutioninthecaseofavaryingG
with other particles, mainly H and H2 (Rohrmannet al. 2010). is stronglydependenton theinitialvalueofG atthe beginning
Our sequences also incorporate element diffusion in the outer of the cooling phase. Accordingly, for each value of G˙/G we
layers(Althausetal.2003). havecomputedseveralsequenceswithdifferentvaluesofG/G ,
i 0
The radiative opacities used are those of the OPAL project whereG standsforthevalueofG atthebeginningofthecool-
i
(Iglesias&Rogers1996),andincludecarbon-andoxygen-rich ingtrackathigheffectivetemperature,andG correspondstothe
0
compositions.Theseopacitiesaresupplementedatlowtemper- presentvalueofG.Toobtainstartingwhitedwarfconfigurations
atures with the molecular opacities for pure hydrogen compo- withtheappropriatevalueofG,wesimplychangedG byG at
0 i
sitionofMarigo&Aringer(2009).Theconductiveopacitiesof thebeginningofthecoolingphaseandwecalculatedtheresult-
Cassisietal.(2007)havealsobeenused.Neutrinoemissionrates ingstructure.Afterabrieftransitorystage,wegettheinitialcon-
forpair,photo,andbremsstrahlungprocessesarethoseofItohet figurationsforoursequences.Thisartificialproceduretoobtain
al.(1996),whileforplasmaprocessesweincludethetreatment the initial white dwarf configurationsis justified since the sub-
of Haft et al. (1994).We use the equationof state of Segretain sequentevolutiondoesnotdependon the way the initial white
et al. (1994) for the high-density regime. For the low-density dwarf structuresare obtained.All in all, we have computed30
regime, we used an updatedversion of the equation of state of whitedwarfevolutionarysequences,whicharelistedinTable1.
Magni&Mazzitelli(1979). In addition, in this table we also list the surface luminosity (in
Giventhatouraimistocomputeself-consistentlythecool- solarunits)forwhichG =G .Notethattheelectionoftheval-
0
ingofwhitedwarfsinthepresenceofavaryingG,wewritethe uesofG hasbeenmadeinsuchawaythatthepresentvalueof
i
localluminosityequationas Goccursatadvancedstagesofwhitedwarfevolution,whenthe
surfaceluminosityrangesfromlog(L/L⊙)=−3to−5,thetypi-
∂L ∂u P ∂̺
r =−ǫ − + . (2) calluminositiesoffieldwhitedwarfs.Allthesesequenceshave
∂m ν ∂t ̺2 ∂t beencomputeddownto log(L/L⊙) = −5,a luminositysmaller
thanthatofthedimmestfieldwhitedwarfs.
and we allow G to vary. This is a fair approach and has been
Despiteof thesmallratesofchangeofG adoptedhere,the
adopted in previous studies of this kind (degl’Innocenti et al.
evolution of our white dwarf sequences is strongly modified.
1996).Consequently,thedensityprofileofthewhitedwarfnow SincewehaveadoptedG˙/G<0,thecentraldensityofthecool-
variesnotonlybecausethewhitedwarfcools,butalsobecause
ingsequencesdecreaseswithtime.Thisisatoddswiththeevo-
Gvaries.Naturally,thisinfluencesthewhitedwarfcoolingages.
lutionofstandardmodels,inwhichwhitedwarfscoolatapprox-
imatelyconstantdensity.Sincetheenergyofcoolwhitedwarfs
is essentially of gravothermal origin, this has important impli-
3. Evolutionaryresults
cationsfor the evolutionof old white dwarfs, andeven a small
We have computed the evolution of white dwarf sequences of change in G alters the energy balance of the star and thus its
masses 0.525, 0.609 and 1.0M⊙ considering three values for luminosity. In addition, because of the larger value ofG at the
4 L.G.Althausetal.:Whitedwarfevolutionwithavaryinggravitationalconstant
-2 -2
. .
G /G = -1x10-12 yr-1 G /G = -1x10-12 yr-1
. .
G /G = -1x10-11 yr-1 G /G = -1x10-11 yr-1
. .
G /G = -5x10-11 yr-1 G /G = -5x10-11 yr-1
. .
G /G = 0 G /G = 0
-3 -3
n n
u u
Ls Ls
L/ L/
g g
o o
L L

-4 -4
M = 0.609 M
WD sun M = 1.0 M
WD sun
-5 -5
0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12
Age (Gyr) Age (Gyr)
Fig.3.Surfaceluminosityversusageforseveral0.609M⊙white Fig.4. Surface luminosity versus age for several 1.0M⊙ white
dwarfsequences,adoptingdifferentvaluesofG˙/G.Seetextfor dwarfsequences,adoptingdifferentvaluesofG˙/G.Seetextfor
details. details.
beginningofthecoolingtrack,22Nesedimentationintheliquid isalmostindistinguishablefromthatofthestandardcaseofcon-
stantG,particularlyforthelow-masswhitedwarf.Thisvalueof
phaseandcrystallizationofthewhitedwarfcoreoccuratlarger
G˙/G constitutesalowerlimitfortherateofchangeofG above
luminosities. Hence, the gravitational energy released by these
whichwecanexpectthattheevolutionofwhite dwarfswillbe
processestakesplacemuchearlier,thusstronglymodifyingthe
influencedbyavaryingG.
cooling rate when compared with that obtained in the case in
whichGisconstant.
The white dwarf cooling times are sensitive to the value of
4. Discussionandconclusions
G at the beginningof the cooling track. This can be better un-
derstoodbyexaminingFigs.1and2whichshowtherelationbe- Recently,Garc´ıa–Berroetal.(2010)havedemonstratedthatthe
tweenthesurfaceluminosityandtheageforthe0.609M⊙ and slowdownofthewhitedwarfcoolingrateowingtotherelease
1.0M⊙whitedwarfsequencesassumingG˙/G =−5×10−11yr−1 of gravitational energy from 22Ne sedimentation and carbon-
anddifferentvaluesofG/G .Inthesefigureswealsoshowus- oxygen phase separation upon crystallization is of fundamen-
i 0
ingsquaresymbolstheluminosityandageatwhichthepresent talimportancetoreconciletheagediscrepancyinthe veryold,
value ofG is reached in the cooling track. Note that the larger metal-rich open cluster NGC 6791. This raises the possibility
thevalueofG/G , thelowertheluminosityatwhichG = G of using this cluster to constrain a hypothetical variation of G
i 0 i 0
occurs. Obviously, this means that if the correct value of G is with time. In a step forward to do this, in this work we have
to be obtained at present time an initially more compact white computed a new set of white dwarf evolutionary sequences in
dwarfisneeded.Also,itisworthnotingthatthecoolingagesin which we allow G to vary. We have assumed that G˙/G re-
thecaseofavaryingGaremuchsmalleratlowluminositiesthan mains constant with time and we have followed the evolution
the coolingages obtainedwhenG is constant.Finally, because of white dwarf model sequences of masses 0.525, 0.609 and
of the stronger gravity of massive white dwarfs, the impact of 1.0M⊙ considering three values for the rate of change of G,
adecreasingGonthecoolingtimeincreasesmarkedlywiththe namelyG˙/G = −5×10−11 yr−1,G˙/G = −1×10−11 yr−1, and
stellarmass. G˙/G = −1×10−12yr−1.Tothebestofourknowledgetheseare
Toillustratethedependenceofcoolingontherateofchange theonlyself-consistentevolutionarysequencesofcoolingwhite
of G, we show in Figs. 3 and 4 the temporal evolution of dwarfsincorporatingtheeffectsofavaryingG.
the luminosity for the 0.609M⊙ and 1.0M⊙ white dwarf se- We have found that the mechanical structure and the en-
quencesassumingG˙/G =−1×10−12yr−1,−1×10−11yr−1,and ergy balance of cool white dwarfs are strongly modified when
−5×10−11 yr−1. The initial values of G have been selected in a slowly varyingG is adopted,and that the white dwarf evolu-
such a way that for each sequence the present value of G oc- tionissensitivetothe initialvalueofG atthebeginningofthe
curs at log(L/L⊙) ≈ −4.0. To further quantify the effects of a coolingphase.Becauseofthecompactnatureofmassivewhite
varyingGwealsoshowintable2thewhitedwarfcoolingages dwarfs, the impact of a decreasing G on the cooling time in-
at log(L/L⊙) = −4.0 of these evolutionary sequences and the creases markedlywith the stellar mass. We have foundas well
correspondingages for the models in whichG˙/G = 0. As can thatforarateofG˙/G =−5×10−11yr−1,thecoolingagesofdim
beseeninthesefiguresandintable2,thereisamarkeddepen- whitedwarfsresultmuchsmallerthanthecoolingagesgivenby
dence of the white dwarf ages on the assumed rate of change thestandardevolutionatconstantG.Thecalculationspresented
ofG.Thisdependenceismorenotoriousinthecaseofmassive hereshowthatthewhitedwarfevolutionisverysensitivetothe
whitedwarfs.However,forG˙/G=−1×10−12yr−1theevolution exactvalueofG˙/G andthatadetailedfittothewhitedwarflu-
L.G.Althausetal.:Whitedwarfevolutionwithavaryinggravitationalconstant 5
Table2.Coolingtimes,inGyr,atlog(L/⊙)=−4.0fordifferent Rohrmann,R.D.,Althaus,L.G.,&Kepler,S.O.2010,A&A,inpress
valuesofG˙/G and two representativewhite dwarf masses. For Segretain,L.,&Chabrier,G.,1993,A&A,271,L13
these cooling sequences the present value of G occurs at this Segretain, L.,Chabrier, G.,Hernanz, M.,Garc´ıa-Berro, E.,&Isern, J.,1994,
ApJ,434,641
valueofthestellarluminosity.
Uzan,J.-P.,2003,Rev.Mod.Phys.,75,403
G˙/G MWD/M⊙
0.609 1.0
0 5.200 7.798
−1×10−12 5.148 7.502
−1×10−11 4.677 6.718
−5×10−11 3.435 3.311
minosityfunctionofNGC6791mayplacetightconstraintsona
possiblerateofchangeofG.Lastbutnotleast,wementionthat
calculations to study the evolution of the progenitors of white
dwarfs, which may also affect the cooling times, are currently
underway.
Acknowledgements. This research was supported by AGAUR, by MCINN
grants AYA2008–04211–C02–01 and AYA08-1839/ESP, by the ESF
EUROCORES Program EuroGENESIS (MICINN grant EUI2009-04170), by
theEuropeanUnionFEDERfunds,byAGENCIA:ProgramadeModernizacio´n
Tecnolo´gicaBID1728/OC-AR,andbyPIP2008-00940fromCONICET.LGA
alsoacknowledgesaPIVgrantoftheAGAURoftheGeneralitatdeCatalunya.
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