Scaling of the asymptotic entropy jump in the superadiabatic layers of stellar atmospheres

0.36 MB·

by  Zazralt Magic

Astronomy&Astrophysicsmanuscriptno.asymp (cid:13)cESO2016
January11,2016
Scaling of the asymptotic entropy jump in the superadiabatic
layers of stellar atmospheres
Z.Magic1,2
1 NielsBohrInstitute,UniversityofCopenhagen,JulianeMariesVej30,DK–2100Copenhagen,Denmark
2 CentreforStarandPlanetFormation,NaturalHistoryMuseumofDenmark,ØsterVoldgade5-7,DK–1350Copenhagen,Denmark
e-mail:[email protected]
Received...;Accepted...
6
1
0 ABSTRACT
2
Context.Stellarstructurecalculationsareabletopredictpreciselythepropertiesofstarsduringtheirevolution.However,convection
n
isstillmodelled by themixing length theory; therefore, theupper boundary conditions near theoptical surface donot agree with
a
asteroseismicobservations.
J
Aims.Wewanttoimprovehowtheouterboundaryconditionsaredeterminedinstellarstructurecalculations.
8 Methods.Westudyrealistic3Dstellaratmospheremodelstofindalternativeproperties.
Results.Wefindthattheasymptoticentropyrunofthesuperadiabaticconvectivesurfacelayersexhibitadistinctuniversalstratifica-
] tionwhennormalizedbytheentropyminimumandjump.
R
Conclusions. Thenormalized entropy can berepresented bya 5thorder polynomial very accurately, and a3rdorder polynomial
S alsoyieldsaccuratecoefficients. Thisgenericentropystratificationorthesolarstratification,whenscaledbytheentropyjumpand
. minimum,canbeusedtoimprovethemodellingofsuperadiabaticsurfacelayersinstellarstructurecalculations. Furthermore,this
h
findingindicatesthatsurfaceconvectionoperatesinthesamewayforallcoolstars,butrequiresfurtherscrutinyinordertoimprove
p
ourunderstandingofstellaratmospheres.
-
o
Key words. convection – hydrodynamics – radiative transfer – stars: abundances – stars: atmospheres – stars: fundamental
r
t parameters–stars:general–stars:late-type–stars:solar-type
s
a
[

1. Introduction tive hydrodynamic(RHD) simulationsto improvestellar struc-
1
turecalculations,whichis expectedto enhancethe accuracyof
v
Instellarevolutioncalculations,convectioninthesuperadiabatic effective temperature, T , and radius predictions (Magicetal.
5 eff
5region(SAR)iscommonlytreatedwiththemixinglengththeory 2015; Trampedachetal. 2014a,b; Salaris&Cassisi 2015). To
8(MLT;seeBöhm-Vitense1958).However,asaresultofthenon- improve MLT models, can be specified the entropy minimum
1local and non-linearnature of surface convection, MLT cannot and the entropy jump, which can be done by considering cali-
0correctly model the complex SAR. The bulk of the convection brations of T(τ) relations and a variable mixing length, α ,
MLT
.zoneisnearlyadiabatic;i.e. comparedtotheadiabaticentropy from 3D simulations. Nonetheless, even with this approach,
1
0value, sad, entropy fluctuations, δs, are small. At the optical MLTwouldstillnotproperlyaccountforthetruestructureofthe
6surface (τRoss =1) the optical mean free path for a photon be- SAR.Rosenthaletal.(1999)hasalreadyshownthat,byappend-
1comesverylarge,leadingtoradiativelossesandconsequentlyto ingdepthdependenth3Distratificationofasolarmodeldirectly
:large-amplitudefluctuations,essentiallydrivingconvection(see onto a 1D structure, p-mode oscillation frequency calculations
v
Stein&Nordlund 1998; Nordlundetal. 2009). The radiative canbeimprovedconsiderably. Therefore,itwouldbedesirable
i
Xcooling at the surface leads to an entropy minimum, smin, and to do the same for the computation of stellar structures, even
rdeterminestheupperboundaryofthephotospherictransitionre- if implementing and interpolating h3Di stratifications onto 1D
agion. Theresultingentropy-deficientplasmaisbuoyantlyaccel- modelsishighlynon-trivial.
erateddownwardsandsubsequentmixingofthedowndraftwith
thestablebackgroundwillrapidlydiminishtheentropyfluctua-
tionswithinafewpressurescaleheights.Ultimately,thiscreates
anasymptoticentropystratificationintheSAR.
Precisestellarevolutionarycalculationsareimportantforde- In the present work, we report on our findings regarding a
termining the age of stellar clusters, extragalactic population universal asymptotic stratification of the entropy jump in nor-
synthesis, and the characterization of exoplanet hosts. How- malized h3Di entropy stratifications from the Stagger-grid, a
ever,theSARisamajorsourceofuncertaintyinstellarstructure grid of 3D RHD stellar atmosphere models (see Magicetal.
theory. This is reflected in asteroseismology, where so-called 2013a). These generic depth dependent entropy stratifications
near-surfaceeffectshavetobecorrectedfor,whenstellarstruc- areeasilyscaled,andcanthereforepotentiallyimprove1Dstel-
turemodelsarecomparedwithobservedp-modeoscillationsfre- larstructurecalculations. Inaddition,suchscalingrelationsare
quencies(seeKjeldsenetal.2008;Ball&Gizon2014). Efforts alsoparamountforthetheoreticalunderstandingofsurfacecon-
are currently being made to implement results from 3D radia- vection.
Articlenumber,page1of6
(a) (b)
Fig.1: Entropyvs. densityshownfordifferentstellarparameterswithoutandwithnormalization(figure(a)and(b),respectively).
Each panel shows models with the same surface gravity, logg, and metallicity, [Fe/H], but different effective temperature, T
eff
(orange/brownlines). Furthermore,thelocationoftheopticalsurface,τ =1,andmaximumintheentropygradient,ds| ,are
Ross max
indicated(circleandtriangle, respectively). Inthe leftfigure,the adiabaticentropy, s , valuesarealso shown(horizontaldotted
ad
lines). Wenotethedifferencesintheaxesbetweenthetopandbottompanelsinfigure(a).
2. Asymptoticentropystratification Next,normalizingthedensitybyitsvalueatthepeakoftheen-
tropygradient,ds| ,andthentakingthesquareroot:
Theaverage(specific)entropy1plottedagainstthedensityshows max
a very steep drop towards the optical surface from the interior ρ
with decreasing density (Fig. 1a). Each entropy stratification ρ∗ = . (2)
is characterizedby the entropy minimum, s , and an asymp- rρ(ds|max)
min
totic increase towards the adiabatic entropy of the deeper con-
TheresultsofthenormalizationareshowninFig. 1b. Theen-
vection zone s with increasing density. For higher effective
ad tropy stratifications are now almost indistinguishable from one
temperature,T ;lowersurfacegravity,logg;andlowermetal-
eff another and can basically be scaled from one to another. The
licity, [Fe/H], the h3Di stratifications2 tend towards lower en-
simplicityoftheoutcomeisremarkable,particularlywhencon-
tropy and density. Concomitantly, the entropyjump decreases,
sideringthecomplexityofthe3DRHDsimulationsfromwhere
but the asymptotic structure looks similar despite the different
the normalized entropy jump was derived. At the optical sur-
depth scales. In the metal-poor case ([Fe/H]=−2), it can be face, i.e. where hτ i=1, the opacity and density decreases
Ross
seen in Fig. 1a that, at high T , the entropy jump and mini-
eff significantly,whichleadstoradiativecoolinginthethinphoto-
mumaresimilartothesolarmetallicitycase,whileatlowerTeff, sphere3,andgeneratestheentropyfluctuations.Thefluctuations
both ∆s and s , are much smaller relative to the solar metal-
min arepartlyadvectedbythehorizontaldeflectionofvelocityfield,
licitycase. Thisisduetoalackofelectronsatlow[Fe/H]and
but primarilyaccelerated downwardsby buoyancy. The down-
T that are required for the dominant H−-opacity. At higher
eff flowsshearandmix(Kelvin-Helmholtzinstability)withthelay-
T , ionization of hydrogen results in more free electrons (see
eff ersbelow,whichrapidlyreducestheentropyfluctuations. This
Magicetal.2013a).
causestheasymptoticconvergenceoftheentropystratification.
We consider the followingnormalizations: Shifting the en-
Furthermore, the process furthermore appears to be universal.
tropy by its minimum and normalizing it by the entropy jump,
The surface gravity sets the geometrical depth scale in the hy-
∆s=sad−smin,i.e. drostatic equilibrium, while the effective temperature, which is
s = s−smin. (1) givenbytheradiativeflux,setstheheightoftheentropyjump.
∗
∆s Wehavealsomarkedthelocationsoftheopticalsurfaceand
1 We computed the specific (thermodynamic) entropy by integrating the peak in entropy gradient in Fig. 1. One can see that the
entropyminimum,whichmarkstheupperboundaryofthecon-
the first law of thermodynamics ds=(dε−p /ρdlnρ)/T. Then we
th
determined the spatial and temporal averages at constant geometrical vectionzone,lieswellabovetheopticalsurfacetowardshigher
height,whicharetheonlyaveragesthatpreservethehydrostaticequi- effectivetemperature. Thisisduetothevigorousvelocitiesand
librium(seeMagicetal.2013b,formoredetails).
2 Inthefollowing,welabelspatiallyandtemporallyaveraged3Dmod- 3 IntheSunthephotospheremeasures500km,whiletheconvection
elsoverlayersofconstantgeometricaldepthwithh3Di. zoneis200Mmdeep.
Articlenumber,page2of6
Z.Magic:Scalingoftheasymptoticentropyjumpinthesuperadiabaticlayersofstellaratmospheres
Fig. 2: Normalized radiative flux F∗ vs. normalized density Fig.3:Fillingfactoroftheup-anddownflows(solidanddashed
rad
ρ for different stellar parameters. The location of the entropy lines)vs. normalizeddensityρ fordifferentstellarparameters.
∗ ∗
minimumisindicated(square).
temperature,while the LHS of Eq. 4 containsthe velocity, the
overshooting,whichrenderstheuseofT(τ)relations,oftenfixed entropyandgeometricaldepthscale. Thisillustratesthesystem-
toτ=2/3instellarstructurecomputations,questionable. Also, aticvariationsofthesevalueswithstellarparameterswithinthe
the T(τ) relations are valid only for specific conditions. The Stagger-grid,whichwereportedinMagicetal.(2013a).
metal-poor cool dwarf (right bottom panel in Fig. 1b), shows In Fig. 3, we show the filling factorsof the up- and down-
larger deviations from the universal entropy stratification most flows.Thesearealsoverysimilarbetweenthedifferent3Dmod-
likelybecausethismodelexhibitsamoreadiabaticstratification, els. Theupflowshaveafillingfactorof2/3andthedownflows
whichshouldbe consideredas a limitingcase forthe universal 1/3, which was already reported by Stein&Nordlund (1998)
entropyscaling. for the solar case. This means that on average the downflows
are compressedby a universalratio of 2:1. This and the above
findingsarealsoconnectedtothescalingofthegranulationpat-
3. Radiativecoolingandconvection
tern, which we foundto scale with the pressurescale heightin
Sincetheradiativecoolingterm,q ,isresponsibleforcreating allStagger-gridmodels(seeMagic&Asplund2014),andsug-
rad
theentropyjump,itisworthwhiletostudytheradiativefluxfor gests that surface convection operates in the same way for all
anyadditionalscalingproperties.Whencomparingtheradiative cool stars. Furthermore, we note that the asymptotic character
fluxnormalizedtothetotalflux, F∗ =F /F , withρ , then oftheentropyjumpisalsoimprintedinthetemperatureanden-
rad rad tot ∗
differentstellarparametersexhibitthesamebehaviour(Fig. 2). thalpy,andbecomesapparentwhendecomposingthemintothe
Wefindthepeakoftheentropygradientcoincideswiththepeak non-adiabaticandadiabaticparts.
oftheradiativecoolinggradient.
Inordertomaketherelationbetweentheentropyandthera-
4. Fittingthenormalizedentropyjump
diativefluxclearer,weconsidertheentropyconservationequa-
tion Wefindthatthenormalizedentropyjumpcanbereasonablyfit
1 withanasymptoticfunction,ζn,ofthefunctionalform
∂ s+u·∇s = − ∇·F +q , (3)
t rad visc
T s = ζ (x)∆s+s , (5)
n min
which states thatthe radiativelosses are a resultofthe entropy
with x=logρ . Aformulationfortheζ thatmatches s wellis
advection. The time derivative of the entropy becomes zero, ∗ n ∗
agenericpolynomial:
while the viscous dissipations are negligible. Then, the spatial
andtemporalaverage,denotedbyh...i,gives n
ζ = 1+ a 1−xk/2 . (6)
n k
2hux∂xsi+huz∂zsi = hqradi/hTi, (4) Xk=1 (cid:16) (cid:17)
whichrelatestheradiativecoolinginthephotospheretothetotal Initially, we applied an asymptotic function; however, the re-
entropyadvection.Thehorizontalentropyadvectionissymmet- sulting fits were insufficient, so that we found the polynomial
ric,i.e.u ∂ s=u ∂ s;therefore,thetwofoldofx-directiongives to give a better fit. In Fig. 4 we show the results of the fitting
x x y y
thehorizontalentropyadvection.TheRHSofEq.4isthederiva- forfourdifferentstellarmodelswithtwodifferentmetallicitiesin
tiveoftheverticalradiativeflux,whichdeterminestheeffective comparisonwiththeh3Dientropystratification.Thedifferences,
Articlenumber,page3of6
Fig.4:Fittingsofthenormalizedh3Dientropyjump(dashedline)withthefunctionsζ andζ (redandbluelines,respectively)for
3 5
differentmodels:Sun(T /logg=5777/4.44),turnoffstar(6500/4.0),redgiant(4000/2.0),anddwarf(4500/5.0)forsolarmetallicity
eff
([Fe/H]=0) and metal-poor ([Fe/H]=−2.0) models (upperand lower panels). We also depict the difference between the h3Di
modelandthefittedfunctionsζ andζ .
3 5
ζ∗ = 1−2 1−x1/2 +2 1−x2/2 −3/4 1−x3/2 (7)
3
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
fitsthenormalizedentropyjumponaverage. Wefindthesmall-
estresidualsforn=5,asshowninFig. 4. However,wecannot
findanycorrelationsbetweenthefittingcoefficientsandthestel-
larparameters. Abettersuitedfunctionalbasiscouldbefound;
inparticular,at the bottomthe functionζ doesnotexactlyex-
n
hibitanasymptoticbehaviourbydefinition.Sinceanasymptotic
functional basis resulted in worse fits, we preferred to use the
presentedgenericpolynomial.
5. Depthdependentboundaryforstellarstructures
Instellarstructurecalculations,theentropycanbeconveniently
obtainedbyintegrating
ds c
= − P (∇−∇ ), (8)
ad
dz H
P
wherec is the specific heatat constantpressure, H the pres-
P P
sure scale height, and ∇ =∇−∇ the superadiabatic gradi-
sad ad
ent.Sincetheasymptoticentropyjumpisuniversal,wecancon-
Fig.5:Thedifferentcoefficientsforζ fromfitstodifferentmod-
3 structthethermalstratificationinthesuperadiabaticlayersusing
els. Thedifferentmetallicitiesareindicated.
thegenericentropystratificationscaledbytheentropyjumpand
minimum. In Fig. 6, we show a constructedentropystratifica-
tion for the Sun using the generic function ζ∗ given in Eq. 7.
δ=s∗−ζn,areverysmallasshown.Infact,therootmeansquare Comparedwiththeh3Dimodel,thedifference3sbetweenthetwo
and the maximum of the differences averaged over all models areremarkablysmall.
are 7.32×10−3 and 9.45×10−3 for the 3rd order function and To construct such a stratification, we first consider the
2.33×10−3and3.61×10−3forthe5thorderpolynomial. generic relation between s and ρ from Eq. 7. Then, for a
∗ ∗
Forthelowerorder,n=3,wefindthatthecoefficientsvary givenchoiceofstellarparametersT ,logg,and[Fe/H],wede-
eff
arounda =−2,+2,−0.75,ascanclearlybeseenin Fig. 5, i.e. termine∆sands ,whichareprovidedinMagicetal.(2013a)
k min
thefunction intheformoffunctionalfits. Wenotethattheentropyminimum
Articlenumber,page4of6
Z.Magic:Scalingoftheasymptoticentropyjumpinthesuperadiabaticlayersofstellaratmospheres
Fig.6:Reconstructed(redline)stratificationsoftheentropy,temperatureandpressure(top,middleandbottompanels,respectively)
fromthegenericfunctionζ∗ (Eq. 7)forthesolar,redgiantandturnoffmodels(fromlefttoright). Weshowalsotheh3Dimodel
3
(dashedline)incomparison.
sets theouterboundary,while theentropyjumpdeterminesthe the mixing length is encoded in ∆s and the outer boundary in
temperaturegradient(∇=∇ +∇ ). WithEq. 5,wecannow s . Such a simplified prescription of the superadiabatic lay-
sad ad min
scale the entropystratification. The density can be determined erscouldpossiblyimprovethe p-modefrequencydisagreement
from Eq. 2 and the entropy peak, which is approximately lo- (Christensen-Dalsgaardetal. 1988; Rosenthaletal. 1999). We
cated at the optical surface, i.e. ds| ≈τ =2/3. There- notethatouraiminthepresentedstudyistoillustratethepoten-
max Ross
fore, the adiabatic stratification with s , where the entropyis tialapplicationofourfindingsforstellarstructures.Themethod
min
constant, can be helpful for an initial estimate of the value of summarized above clearly needs further development for im-
thedensity,atwhichρ(ds| ). Fromtheresultingρand s, we provedresults.
max
compute the thermal pressure, p , from the equation of state.
th
Then, the geometrical depth can be retrieved from the hydro-
6. Conclusions
staticequilibrium,dp /dz=ρg. Wenotethatthesurfacegrav-
th
ity is responsiblefor the scaling of the geometricaldepth. The Thepresentedscalingrelationfortheentropyjumpindicatesthat
turbulentpressure(andvelocity)hastobeincludedinthehydro- radiativelossesincoolstarstakesplaceinafairlysimilarfash-
static equilibrium because it will elevate atmospheric structure ion. Wefoundthatthenormalizedentropy,asafunctionofthe
(i.e. ptot = pth+ρu2z). However, for simplicity we neglect the normalized density, can be fitted very well with a polynomial
latter. Thelocationoftheopticalsurfacecanberetrievedfrom function. A robust and generic, yet simple description of the
theopticaldepth, dτ/dz=ρκ. Finally, withtheconstructeden- normalized entropy stratification can be achieved with a three-
tropy and density stratification one can compute other thermo- parameter function. This could be helpful for stellar structure
dynamicquantities,suchastemperatureorinternalenergy,from computationsbyprovidingasimpledescriptionoftheotherwise
the equation of state. We note that in the differentconstructed non-trivialsuperadiabaticregion in cool stars. In particular, as
modelsshowninFig. 6,weappendedanisothermalatmosphere correctionsforthenear-surfaceeffectsinasteroseismologythis
abovethesurface. would constitute a large step forward in stellar structure mod-
This procedure can be used to construct depth dependent elling. We have shown that from the generic normalized en-
thermal stratifications to be used as outer boundary conditions tropy,wecanconstructstratificationsbyscalingthenormalized
for stellar structure computations. The physical complexity of entropywiththeentropyjumpandminimum,asoutlinedabove.
theradiativetransferequationisthenhiddeninthedependence However, the method can benefit from some refinement in the
of ∆s and s on stellar parameters. This approach would future,forexamplebyincludingtheturbulentpressureinthehy-
min
reduce the deficiencies of MLT and the T(τ) relations since drostaticequilibriumequation.Superadiabaticconvectionseems
Articlenumber,page5of6
totakeplaceinasimilarmanner,despitelargedifferencesinthe
physical conditions. The reason behind this has yet to be ex-
plored.
Acknowledgements. Thisworkwassupportedbyaresearchgrant(VKR023406)
fromVILLUMFONDEN.
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