Astrophysical implications of the proton-proton cross section updates

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Astrophysical

implications

of

the

proton–proton

cross

section

updates

E. Tognelli

a

,

c

,

S. Degl’Innocenti

b

,

c

,

∗

,

L.E. Marcucci

b

,

c

,

P.G. Prada Moroni

b

,

c a_{DepartmentofPhysics,UniversityofRomaTorVergata,ViadellaRicercaScientifica1,00133,Roma,Italy}

b_{DepartmentofPhysics‘E.Fermi’,UniversityofPisa,LargoBrunoPontecorvo3,56127,Pisa,Italy} c_{INFN,SectionofPisa,LargoBrunoPontecorvo3,56127,Pisa,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received17November2014

Receivedinrevisedform16January2015 Accepted22January2015

Availableonline26January2015 Editor:W.Haxton

Keywords:

Stellarhydrogenburning Stellarevolution,ages Solarneutrinos

Proton–protonweakcapture

Thep(p, e+

ν

e)2H reactionrateisanessentialingredientfortheoreticalcomputationsofstellarmodels.

Inthepastseveralvaluesofthecorresponding

S-factor

havebeenmadeavailablebydifferentauthors. Promptedbyarecent evaluationof S(E),weanalysed theeffectofthe adoptionofdifferent proton– protonreactionratesonstellarmodels,focusing,inparticular,ontheageofmidandoldstellarclusters (1–12 Gyr)andonstandardsolarmodelpredictions.Bycomparingdifferentwidelyadoptedp(p, e+

ν

e)2H

reactionrates,wefoundamaximum differenceinthetemperatureregimes typicalofmain sequence hydrogen-burning stars (5×106–3×107K) ofabout 3%.Suchavariation translates intoachangeof cluster age determination lower than 1%. A slightly larger effect is observed in the predicted solar neutrinofluxeswithamaximumdifference,intheworstcase,ofabout 8%.Finallywealsonoticethatthe uncertaintyevaluationofthepresentproton–protonrateisattheleveloffewh,thusthep(p, e+

ν

e)2H

reactionratedoesnotconstituteanymoreasignificantuncertaintysourceinstellarmodels.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Atthe energies of interest forstellar nucleosynthesis the rate oftheproton–proton(p–p)weakcaptureistoolowtobedirectly measuredin laboratoryandit can be determinedonly by means of nuclear physics calculations. The p(p,e+

ν

e

)

2H reaction drives theefficiency ofthe p–p chain,which is fundamental for hydro-gen burning in stars. Thus, a variation of the p–p reaction rate adopted in the stellar models potentially influences the charac-teristics and the evolutionary times (at least) during the central hydrogen burning phase (Main Sequence,

MS

) of low-mass stars andthustheagedeterminationofoldstellarclusters(seee.g. dis-cussions in Refs. [1–4], andreferences therein). Moreover, asthe Sunburnshydrogenmainly bymeansofthe p–pchain,a change inthep–pcrosssectionadoptedinthemodelscanaffectthe pre-dictedsolarstructure and, consequently,both theneutrino fluxes andthehelioseismologicalobservables.

Thep–p reaction rateis expressed interms ofthe astrophys-ical S-factor S

(

E

)

[5,6],where theenergy, E, is measured inthe

*

Correspondingauthor.

E-mailaddresses:tognelli@df.unipi.it(E. Tognelli),scilla@df.unipi.it (S. Degl’Innocenti).

two-proton centre-of-mass frame. S

(

E

)

is oftenexpressed as the firstthreetermsofaMaclaurinseriesin

E:

S

(

E

)

=

S

(

0

)

+

S

(

0

)

×

E

+

1 2S



₍

₀

₎

_×

_E2

_{+ · · ·}

₍₁₎

where

S



(0)

and

S



(0)

arethefirstandsecondderivativesof S

(

E

)

evaluated atzeroenergy.The recentreactionratecompilationby Adelberger et al. [6] reviewed in a detailed way different eval-uations for S

(0)

and S

(0)

(see also the discussion in Ref. [7]), recommending the values: S

(0)

= (

4.01

±

0.04)

×

10−25MeV b and S

(0)/

S

(0)

= (

11.2

±

0.1)MeV−1. The

NACRE99

[8] com-pilation, still widely adopted in the literature, suggested at the time: S

(0)

=

3.94

×

10−25_{MeV b, S}

_(0)/

_S

₍₀₎

₌

_{11.7 MeV}−1 _and S

(0)/

S

(0)

= (

150

±

20)MeV−2.

Recently,theastrophysical

S-factor

hasbeencalculatedby Mar-cucci et al. [7] (hereafter

MSV13

) applying the so-called chiral effective field theory framework, which allows for a better de-termination of the theoretical uncertainty. Taking into account

two-photon and vacuum polarisation contributions beyond

sim-ple Coulomb interaction,and, moreover, the contributions of the S- and P-partial wavesinthe initialp–p state, thecorresponding valuefor

S

(0)

hasbeenfoundtobe

(4.030

±

0.006)

×

10−25MeV b, withtheuncertaintyreducedbyaboutafactorsevenwithrespect to previous evaluations. Notethat theP-partial waveshave been

http://dx.doi.org/10.1016/j.physletb.2015.01.033

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

foundtogiveacontributionofabout0.02

×

10−25_{MeV b,}

explain-ing thedifference withthe S

(0)

value ofRef.[6].Marcucci et al.

[7] provided values for S

(0),

S

(0)

and even higher derivatives to be used in Eq. (1), by fitting S

(

E

)

in the 0–100 keV energy range. However, we preferred to directly obtain the p–p rate in therequiredenergyrangeandwiththechosenenergyresolution byusingaroutine,basedonthe S

(

E

)

evaluatedinRef.[7],which ismadeavailable atthelink http://astro.df.unipi.it/stellar-models/ pprate/.

Inthiswork,relevantstellarevolutionaryquantitiescalculated by adopting the Marcucci et al. [7] p–p reaction rate with the inclusion ofthe S- andP-partial waves (hereafter

MSV13(S+P)

) are compared with those obtained by using different p–p rates widely adopted in the literature, and with the

MSV13

rate cal-culated without the inclusion of the P-partial wave contribution (hereafter

MSV13(S)

).We concentrateon theagedetermination ofstellarclustersandonthesolarmodelcharacteristics,including thesolarneutrinofluxes.

Section 2 describes the calculation of the p(p,e+

ν

e

)

2H reac-tionrate, withthe relateduncertainty;theobtainedreactionrate isthencomparedwithpreviousevaluations.InSection3the char-acteristics of the stellar models and ofthe adopted evolutionary code are briefly described, while in Section 4 and Section 5 the effects of the p–p rate update on age estimation of stellar clus-tersandonstandardsolarmodelcharacteristicsarediscussed.We concludewithasummaryinSection6.

2. Thep(p,e+

ν

e

)

2H reactionrate

The rate R for the p–p reaction is commonly expressed in cm3mol−1s−1,suchthat(seee.g.Ref.[9])

R

≡

NA



σ

v



=

3

.

73



×

1010

ˆ

μ

T₉3 ∞



0 S

(

E

)

exp



−

2

π η

−

11

.

605E T9



dE (2)

where NA is the Avogadro number,



σ

v



is the

Maxwellian-averagerate,

μ

ˆ

=

0.504 isthe p–p reducedmassinatomicmass units (1 amu

=

931.494 MeV/c2_{), T}

9 is the temperature in units

of 109K, and

η

is the Sommerfeld parameter expressed as

η

=

0.1575(μ_Eˆ

)

1/2.Theintegrationoverthecentre-of-massenergyE in

Eq. (2) can be performed numerically with standard techniques. The crucialinput inthiscalculation is S

(

E

).

InRef. [7], S

(

E

)

has been calculated in the range E

=

0–100 keV, with the inclusion of S- and P-partial waves in the initial p–p state. A realistic es-timation of the theoretical uncertainty can be performed, since

S

(

E

)

iscalculatedwithintheso-calledchiraleffectivefieldtheory framework[7],whichallowstoreduce thetheoretical uncertainty to the order of few

h

, by constraining systematically both the nuclear potential and the nuclear current operator witha strin-gentcontemporaryfitofthetrinucleonbindingenergyandtritium

β-decay

lifetime.ThisisbetterthanwhathasbeendoneinRef.[6], wherethestudyinthe“old-fashion”potential modelapproachof Ref.[10]wasprimaryused,andtheuncertaintiesmostofall aris-ing fromtwo- and three-body potentialsandcurrents leadto an accuracy for S

(0)

not better than 1%. Using the uncertainty on

S

(

E

)

ofRef.[7],weobtainanupperandlowervaluefor

S

(

E

)

ata givenvalueof

E.

Consequentlyitispossibletoderiveanupperand lowervalue forthe rateR, or,alternatively,toprovidea theoreti-cal uncertainty



R. Twomoreoptions arepresentinthe on-line releaseofthecomputer programwhichcalculates R:(i) the con-tributionsfromtheP-partialwavesoftheinitialp–pstate canbe excluded;(ii) ratherthanusingthecalculatedvaluesof

S

(

E

),

tabu-latedon101gridpointswithstepsof1 keVstartingfrom

E

=

0,it

Table 1

ValuesforS(0)anditsfirst,secondandthirdderivativesasobtainedbyMarcucci et al. [7],including(MSV13(S+P))andnotincluding(MSV13(S))theP-partial wave contribution. The corresponding values for the reaction rate compilations quotedinSection2arealsolisted(ifavailableintheliterature).

S(0) [×10−23_{MeV fm}2_] S _(0)/S(0) [MeV−1_] S _(0)/S(0) [MeV−2_] S _(0)/S(0) [MeV−3_] MSV13(S+P) 4.030(6) 11.94(1) 248.8(2) −1183(8) MSV13(S) 4.008(5) 11.42(1) 239.6(5) −1464(5) NACRE99 3.94 11.7 150(20) AD11 4.01(4) 11.2(1)

Fig. 1. Proton–protonreactionrateratiosbetweendifferentcompilationspresentin theliteratureandourreferenceone,MSV13(S+P),basedontheMarcucciet al.[7] S(E)calculation.MSV13(S)indicatestheratecalculatedbyadoptingtheMarcucci et al. S(E)evaluationwithouttheinclusionofP-partialwavecontributions.The verticaldashedlinemarksthesolarcentraltemperature.

ispossibletousetheMaclaurinseriesin

E (see

Eq.(1)),with

S

(0),

S

(0),

S

(0)

and S

(0)

givenby Marcucciet al.[7],andfittedto reproduce S

(

E

)

in the 0–100 keV range(see Table 1). Then, the energy range and energy resolution can be chosen aspreferred. No appreciable difference is seen in the calculation of R by us-ing the101gridpointof S

(

E

)

orthefittedvaluesofS

(0)

andits derivatives.Forabetter comparisonofthedifferencesamongthe selected evaluationsofthep–p rate, andforfuturereference, we reportinTable 1thevaluesof

S

(0),

S



(0),

S

(0)

and

S



(0)

as ob-tainedbyMarcucciet al.[7],comparedwiththoseavailableinthe literature,namelyinthe

NACRE99

[8]and

AD11

[6]compilations. Itistobenoticedthat,asalreadymentionedinRef.[7],a goodfit of S

(

E

)

withtheMaclaurinseriesin E in therange0–100 keVis obtainedonlyincludinguptothethirdderivative S

(0).

Fig. 1showstheratiobetweenthepresentp(p,e+

ν

e

)

2H reac-tionratecalculatedbyadoptingthe

S

(

E

)

evaluationofRef.[7]and those reported by the

NACRE99

[8],

AD11

[6], and

JINA

[11], widely used in the literature. The ratio between our reference choice (

MSV13(S+P)

)andthep–preactionratecalculated with-out the inclusionof theP-partial wave contribution(

MSV13(S)

) isalsoshown.Theresultsareshownonlyinthetemperaturerange ofastrophysicalinterestforcentralhydrogen-burningstars.

Inthe temperaturerangeofinterest,the largestrelative varia-tionofthereactionrate(about3–4%)withrespecttothereference oneisfoundforthe

NACRE99

compilation,whichadoptsan S

(0)

value differentby about2% fromthe

MSV13(S+P)

.The temper-aturebehaviour ofthe

JINA

and

AD11

rateisdifferentfromthe presentone,buttherelativechangesremain within

≈

2% (

AD11

) or

≈

2–3% (

JINA

)forthewhole selectedtemperaturerange.The effect on the

MSV13

rates ofthe inclusion ofthe P-partial wave

contributionisof theorderof 1%. It isalso worthto discussthe differences among the various rates in the light of the quoted uncertainties.Noticethat theerrorofthepresent

MSV13

p–p re-actionrateisofthe orderoffew

h

.Thedifference betweenthe

NACRE99

andthepresentreactionrateinthetemperaturerange of interest is smaller than the

NACRE99

estimated uncertainty, whichrangesbetween

−

3% and

+

7% (seeTable 2inRef.[8]). How-ever,ithastobeemphasised thatthevaluesof

S

(0)

andits deriva-tivesadoptedby

NACRE99

appearto be outdatedinthe lightof morerecentcalculations;thus,theadoptionofthe

NACRE99

p–p rateshouldbeavoided.Thedifferencesbetween

AD11

and

MSV13

reactionrates are larger than the uncertaintyon S

(0)

quoted by

AD11

(about1%).However, theglobaluncertainty on

AD11

S

(

E

)

also dependson the error on S

(0)

and on the lack of the sec-ondderivative S

(0).

Numericalsimulationsshow thatthelackof

S

(0)

affects the resulting reaction rate by less than 1%, in the temperaturerange of interest (see also Ref. [12]). Finally we are notabletodiscussthedifferencesbetweenthe

JINA

and

MSV13

ratesbecauseno informationabouttheuncertaintyon the

JINA

rateareavailableonthe

JINA

webpage. 3. Stellarmodels

Stellarmodelshavebeencalculatedbymeansofthe

PROSECCO

stellar evolutionary code [13,14]. A detailed discussion of the adoptedinputphysicscanbefoundinRefs.[15–17];herewejust summarise themostrecentupdates.

Themain noveltyconcerns the adoptednuclear reactionrates relevantforthepresentwork.Inparticulartherecent

AD11

rates havebeenadoptedinplaceofthe

NACRE99

onesforthe follow-ing reactions: 3_He(3_He,2p)4_He, 3_He(4_He,

_γ

₎

7_{Be, p(}7_Be,4_He)4_He,

p(12_C,

_γ

₎

13_N,andp(16_O,

_γ

₎

17_{F.Forthebottleneckreactionofthe}

CNOcyclep(14N,

γ

)

15O,weadoptedthe

LUNA

results[18]. Bare nuclei reactions have been corrected to account for the plasmaelectronscreeningforweak[19],weak-intermediate-strong

[20,21],andstrong[22,23]screening.Onlyinthecaseofstandard solar model calculations (see Section 5), following the choice of mostauthors,the Salpeterformulaforweak-screening isadopted forallthe p–p chainandCNO-cyclereactions (see e.g. Ref.[24]). Atomicdiffusionhasbeenincluded,takingintoaccounttheeffects of gravitational settling and thermal diffusion, with coefficients givenbyThoulet al.[25].

Regardingthe chemical composition,models havebeen calcu-latedfortwo initial [Fe/H]1 _{values,namely}

_[

_Fe/H

_]

_{= +}

_0.0

(corre-spondingtothesolarcomposition)and

[

Fe/H

]

= −

1.0,totakeinto accounttherangeofvaluesforthisquantity observedinGalactic stars.[Fe/H]hasbeenconvertedintoinitialhelium(Y )andmetal ( Z )massfractionsbyadoptingEqs. (1)and (2)inRef.[26],which is valid for solar-scaled metal distribution. Present models have

been computed by adopting the recent solar metals abundances

givenbyAsplundet al.(2009)[27].Theinitial

Y and Z for

thetwo setsof[Fe/H]valuesare:

(

Y

,

Z

)

= (

0.274,0.013)for

[

Fe/H

]

= +

0.0 and

(

Y

,

Z

)

= (

0.250,0.001)for

[

Fe/H

]

= −

1.0.

It is worth to emphasise that all the analysis presented in this paper have been performed in a differential way, i.e., the results obtained with the reference p(p,e+

ν

e

)

2H reaction rate (

MSV13(S+P)

)havebeencomparedwiththoseobtainedwiththe otherevaluationsofthep–pratediscussedabove,keepingallthe other physical parameters and the stellar chemical composition fixed. Thus, the resultsare expected to be weakly dependent on thechemicalcompositionandontheinputphysicsadoptedinthe models(seeRefs.[3,4]).

1 _[Fe/H]def_=log(NFe/NH)

(N_Fe/N_H) .

4. Proton–protoncrosssectionandagedeterminationinstellar clusters

Stellarclustersprovideaseverebenchmarkforstellarevolution models,sincetheyconsistofstarssharingthesameage,chemical composition and distancefrom the Earth. The agedetermination ofstellarclustersintheMilkyWayandintheothergalaxiesisof paramountimportance,asitprovidesinformationaboutthe evolu-tionaryhistoryofthehostgalaxies.Theageofthestellarclustersis determined throughtheluminosity ofthepoint correspondingto thecentral H-exhaustion(Turn-Off,

TO

,forold clustersand Over-allContraction,

OC

,fortheyoungerones).Theolderisthecluster andtheloweristhemassatthe

TO

/

OC

point,and,thus,thelower is its luminosity. Therefore, the efficiency of the p–p reaction is relevant forstellarcluster dating. Forstars inclustersolderthan about10 Gyr, the p–pchain isthe mainH-burning channel dur-ingthe

MS

phase,whileforstarsclosetothe

TO

phaseinyounger clusterstheCNO-cycle dominates,althoughthep–p chain contri-bution remains not negligible at least forages older than about 1 Gyr.Thisroughlycorrespondstomassesofabout1.5M (for so-larchemicalcomposition).Wewill notdiscusspost-

MS

phasesas thecontributionofthep–pchainbecomesnegligiblewithrespect totheCNO-cycleone.

We computed stellar evolutionary tracks2 _{in the mass range}

[

0.4,1.5

]

M (in steps of 0.1M ) from the pre-main sequence phaseuptothesub-giantbranchandisochrones3 foragesfrom1 to12 Gyr,tocoverallthepossibleclusteragesofinterestforthis analysis.Forcompleteness,thedifferentialanalysisoftheeffectsof thep–pratevariationhasbeenperformedfortwodifferent chem-icalcompositions(

[

Fe/H

]

= +

0.0 and

[

Fe/H

]

= −

1.0 as discussed inSection3).

Low-mass stellar tracks(or isochronesofold clusters) present at the central H-exhaustion the

TO

feature, which is theoreti-callyidentifiedasthetrack/isochrone(MainSequence) pointwith the highest effectivetemperature. However the

TO

-region in the

HR

/

CM

diagrams is almost vertical at the

TO

-point (i.e., there is alarge luminosityvariationatessentiallythesameeffective tem-perature) and thismakes the precise identificationof the

TO

lu-minosity quite difficult. Thus, to reduce the intrinsic luminosity variationofthecanonical

TO

,followingatechniquesimilartothe oneadoptedinotherworks[28,1,3,4],wedecidedtomeasurethe luminosity of a pointbrighter andwithan effective temperature of 100 K lower than the

TO

. Stars with intermediate and high mass (or isochroneswith intermediate/lowages)show atthe H-exhaustionthe

OC

feature.Wedefinedthe

OC

similarlytothe

TO

point.

Fig. 2 showstheratioofthe H-exhaustion timecalculated for theratesanalysedinSection2totheonecalculatedwiththe ref-erencep–prate,asafunctionofthestellarmass,forthelabelled chemical compositions.As expected, the

NACRE99

rate, which is themostdifferentwithrespect tothereferenceone,leadsto the maximum differencein the H-exhaustiontime; however,even in thiscasetherelativechangeislowerthanabout5–6

h

.Neglecting thecontributionoftheP-partialwave intheratecalculationgives aneffectlowerthanabout2

h

.

The same quantities have been calculated for

[

Fe/H

]

= −

1.0 (Y

=

0.250 and Z

=

0.001, bottom panel of Fig. 2). In thiscase, the sensitivity to the p–p rateis slightly reduced, in agreement

2 _{Theevolutionarytrackisthetemporalevolutionofastellarmodelofagiven} mass.

3 _{Theisochroneisthetheoreticalcounterpartofanobservedstellarclusterin} theHertzsprung-Russell(HR)/Colour-Magnitude(CM)diagram,i.e.thelocusof evo-lutionarymodelswithdifferentmassbutwiththesameageandchemical compo-sition;itiscomputedfromtheevolutionarytracks.

Fig. 2. RatiooftheH-exhaustiontime,asafunctionofthestellarmass,calculated forthelabelledp–prates totheonecalculatedwiththereferencereactionrate,

MSV13(S+P).Upperpanel:Y=0.274 andZ=0.013 (solarvalues);bottompanel: thesamebutforY=0.250 andZ=0.001.

withtheresultsobtainedbyValleet al.[4],whichshowedthatthe evolutionaryeffectsofavariationofthep–p rateare verymildly affectedby reasonable changes inhelium andmetal abundances. Differences inthe track

TO

/

OC

luminosity are always lower than 1

h

andthuscompletelynegligible.

Fig. 3 shows the luminosity differences at the

TO

/

OC

forthe

isochrones as a function of the age, between models computed

withthequotedp–preactionratesandthereferenceone.Forthe solar chemical composition (upper panel), the differences in the

TO

/

OC

luminosityarelowerthanabout5

h

forthemostofcluster ages,reachingavalueof1–1.5%onlyforagesintherange2–4 Gyr. Themajor sensitivitytoap–p ratechangeforclustersinthe age range2–4 Gyr isdueto thefact that theH-exhaustion region of the

HR

diagramoftheseclusters ispopulatedbystarsofmassesin therange

[

1.1,1.5

]

M .These arejustthe transitionmasses(the precisevaluedepending alsoonthestellarchemical composition) among stars which burn hydrogen in the central regions mainly throughthep–pchain(lowermainsequencestars)orthroughthe CNO-cycle(uppermainsequencestars).SuchadifferentH-burning producesalsoa differenttrack/isochrone morphologyclosetothe

TO

/

OC

region.Thus,it isnotsurprisingthattheeffectofthep–p ratechangeinthe

TO

/

OC

luminosityislargerinthisagerange.In anycase,we wanttoemphasise thatsuch aneffectisvery small andit has a minor relevance incluster agedetermination when compared to the other sources of uncertainty (see e.g. Refs. [1, 29–31,3,4]).

ThebottompanelofFig. 3showsthesamequantitiesasthetop one butforY

=

0.250 and Z

=

0.001.In agreementwiththe re-sultsfoundforthestellartracks,thesensitivitytoachangeofthe

Fig. 3. LuminositydifferenceattheTO/OCpoint,asafunctionoftheclusterage,for isochronescalculatedwiththelabelledp–pratesandtheonescalculatedwiththe

MSV13(S+P)one.Upperpanel:Y=0.274 andZ=0.013 (solarvalues);bottom panel:thesamebutforY=0.250 andZ=0.001.

p–pratesisslightlyreducedandthefeatureintheregion2–4 Gyr isnomorevisible.

Giventheabove resultsandthevery smalluncertaintyonthe

MSV13

S

(

E

),

wecanconcludethatthep(p,e+

ν

e

)

2H reactionrate isnow knownwithsuch ahighprecisionthat itdoesnot consti-tute anymorea significant uncertaintysource intheage evalua-tionofstellarclusters.

5. TheSunandthep(p,e+

ν

e

)

2H crosssection

The Sun is unique among stars because several observational quantities areknown withavery highprecision. Thus,it isclear that thecomparison betweenatheoretical solarmodel(Standard SolarModel,

SSM

)andtheactualSunisastrongtestofthevalidity oftheoreticalstellarcomputations.

An

SSM

is definedasa 1M model whichreproduces, atthe age of the Sun(t ), within a givennumerical tolerance,the ob-served propertiesof the Sun, by adopting a set of input physics (seee.g.Refs.[32,33]andreferencestherein).

The present

SSM

has beencomputed adopting M

=

1.989

×

1033g, L

=

3.8418

×

1033erg s−1, and R

=

6.9598

×

1010cm

[34].RegardingtheageoftheSun,wehaveadoptedt

= (

4.566

±

0.005)Gyr, as estimated from age determination for meteorites combinedwithmodelsofthesolarsystemformation[35].Wehave used the recent spectroscopically determined ratio of metals-to-hydrogen inthe solarphotosphere by Asplund et al. (2009) [27], which results in

(

Z

/

X

)

ph,

=

0.0181. We emphasise that the

present observed photospheric composition is different from the initial one due to microscopic diffusion, whose efficiency must

betheoreticallyestimated.Moreover,presentphotospherichelium abundanceisnotstronglyconstrainedbydirectobservations,since helium lines are not observed in photosphere, and the external convection efficiency cannot be theoretically obtained in a firm way. Thus, one has the freedom of adjusting the initial helium,

Y , metallicity,

Z ,

andexternalconvectionefficiencyrequiredinthe calculation.

Wehaveevolvedaninitiallyhomogeneoussolarmassfromthe pre-main sequence phase up to the solar age.To obtain L , R and

(

Z

/

X

)

ph, att ,wehavetuned,bymeansofaniterative

pro-cedure,threeparameters:theinitialheliumabundance Y (mainly

influencingtheluminosity),theinitialmetalabundance Z (mainly

affecting the present

(

Z

/

X

)

ph, surface value) and the

α

ML

pa-rameterofthe mixinglength theory [36], relatedtothe external convection efficiency(mainly influencing radius/effective temper-ature). The precision with which luminosity, radius and present

(

Z

/

X

)

ph, surfacevalue arereproducedinour

SSM

isbetterthan, respectively,10−5,10−4,and4

×

10−4.

The observed solar luminosity fixes the efficiency of the p–p chain,whose energy productionmust counterbalance theenergy lossesfromthesurface (witha contributionofabout1% fromthe CNO-cycle). Thus, the temperature (and the density) in the so-lar interior, neededto produce the required energy, depends on the p(p,e+

ν

e

)

2H cross section. An increase of the p(p,e+

ν

e

)

2H cross section would lead to a decrease of the stellar tempera-ture,thus directlyaffectingtheneutrinofluxes(seee.g. Refs.[37, 32,38–44]).Thep–p neutrinos(togetherwith

pep and hep

neutri-nos)aredirectlyconnectedto thesolarluminosityandthus they areexpectedto bevery weaklydependent onp(p,e+

ν

e

)

2H cross section change. On the other hand, all the other neutrinofluxes are much more sensitive to temperature variations and, conse-quently,are more affectedby the adoptedp–p reaction rate(see e.g.Ref.[32]).

Wecalculated

SSM

s byadoptingasp(p,e+

ν

e

)

2H reaction rate thedifferentevaluations discussedin Section 2,keepingfixed all

the other input physics. As reference model we have adopted

the one calculated with the

MSV13(S+P)

p–p rate. Among all

the models, the differencesin the original helium and metallic-ityabundances andinthemixinglengthvalue requiredtoobtain astandardsolarmodelarenegligible.

Asexpected, being thesolar luminosityfixed by observations, thesolarcentraltemperaturedecreaseswhenthep(p,e+

ν

e

)

2H re-actionefficiencyincreasestocounterbalancetheincreasednuclear energyproduction(seee.g.Ref.[37]).Thelargestdifferenceinthe centraltemperature(forthe

SSM

withp–pratefromthe

NACRE99

compilation)is oftheorder of 3

h

.Dueto thevery small differ-encesinthechemicalcompositionandcentraltemperature,allthe calculatedmodelsarenotexpectedtoshowvariationsinthe pre-dictedhelioseismicquantities;we checkedthatthisisthecaseat the level of few

h

. However, due to the high temperature de-pendence ofneutrinofluxesbutthe p–p one, theeffect onsolar neutrinosisnottotallynegligible,evenifsmall.

Table 2showstherelativedifferencesforthesolarcentral tem-perature andthe solar neutrino fluxes. The maximum difference forthe solar neutrino fluxes corresponds to the adoption of the

NACRE99

p–p reaction rate, reaching a maximum of about 8%. Thisis not a negligible difference, however, asdiscussed in Sec-tion2,inouropiniontheadoptionoftheNACRE99p–prateshould bediscouraged.The

∼

1% effectoftheP-partialwavestothep–p reactionrateturnsoutintoamaximumof3% effectonthe8B,15O and17F neutrinofluxes.

Asforthestellarclusterage,theverysmallerroronthepresent p(p,e+

ν

e

)

2H rate determination (see Section 2) has a negligible effecton

SSM

calculations.

Table 2

Thefirstcolumnliststheresultsforcentraltemperature(K)andneutrinofluxes (s−1_cm−2_{)ofourreferencemodel(i.e.calculatedwiththeMSV13(S+P)p–prate).} Theotherscolumnslisttherelativedifferencebetweentheresultsobtainedforthe referenceSSMandforSSMscomputedwiththelabelledp–prates.Boldfont: rela-tivedifferencesabove1%.

MSV13(S+P)

reference

MSV13(S) NACRE99 AD11 JINA

relative differences Tc[107K] 1.54794 −1h −3h −2h −1h Φν pp[1010] 6.020 1h 2h 2h 1h Φpepν [108] 1.446 −2h −6h −2h −1h Φhepν [10 3_{] 8.584 −1h −3h} <1h 2h ΦBeν-7[109] 4.503 −1% −3% −1% −9h Φν B-8[106] 3.694 −3% −7% −4% −2% ΦNν-13[10 8_{] 2.417 −2% −6% −3% −1%} ΦOν-15[10 8_{] 1.811 −3% −8% −4% −2%} ΦFν-17[10 6_{] 3.373 −3% −8% −4% −2%} 6. Summary

An updated p(p,e+

ν

e

)

2H reactionratehas beencalculatedby adoptingthe evaluationoftheastrophysical S-factor byMarcucci et al.[7],whichtakesintoaccounttwo-photonandvacuum polar-isation contributions and,for thefirst time, itincludes all the P-partialwavesintheincomingp–pchannel.Theuncertaintyonthe rateevaluationisnowestimatedattheleveloffew

h

.A releaseis available(atthelink:http://astro.df.unipi.it/stellar-models/pprate/) forthecalculationofthepresentp–prate(

MSV13

)withthe pos-sibilitytoselecttheenergyrangeandresolution.

The comparison with other widely adopted p–p rates shows

maximum differences ofabout 3–4%. The effects on stellar clus-ter age determination of the adoption of different p(p,e+

ν

e

)

2H reactionrates havebeendiscussed.We found thatthe maximum variation is obtained for the still widely used p–p rate by the

NACRE99

compilation;however,takingintoaccounttheother un-certainty sources,thisdifference isof minorrelevanceforcluster agedetermination.

Theinfluenceoftheadoptionofdifferentp–prateprescriptions onthestandardsolarmodelscalculationshasbeenalsoanalysed. The change of the p–p rateevaluation isnegligible for the solar structure characteristics.However,the relatedtiny changes inthe central temperaturereflectinsmall, butnot negligible,variations oftheneutrinofluxes,butthep–pone,duetotheirhigh sensitiv-itytotemperaturevariations.

Finally,wehavealsoshownthatthep(p,e+

ν

e

)

2H reactionrate isnowobtainedwithsuchahighaccuracythatitdoesnot consti-tuteanymoreasignificantuncertaintysourceinstellarmodels. Acknowledgements

We warmly thank the anonymous referee for a careful

read-ingofthemanuscriptandusefulsuggestionswhichimprovedthe

paper. This work has been supported by PRIN-MIUR 2010–2011

(Chemical

and dynamical evolution of the Milky Way and Local Group

galaxies, PIF.Matteucci),PRIN-INAF2011(Tracing

the formation and

evolution of the Galactic Halo with VST, PIM.Marconi),andPRIN-INAF 2012(The

M4 Core Project with Hubble Space Telescope,

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