PhasesPhases ofof StellarStellar EvolutionEvolution PhasesPhases ofof StellarStellar EvolutionEvolution

PrePre--MainMain SequenceSequence MainMain SequenceSequence PostPost--MainMain SequenceSequence TheThe PrimaryPrimary definitiondefinition isis thusthus whatwhat isis thethe MainMain SequenceSequence LocusLocus ofof ““corecore”” HH burningburning BurningBurning ProcessProcess cancan bebe eithereither pppp oror CNOCNO ZAMS:ZAMS: ZeroZero AgeAge MainMain SequenceSequence -- locuslocus ofof initiationinitiation ofof HH burningburning

Phases of Stellar Evolution 2 WhatWhat isis Happening?Happening?

PrePre--MainMain Sequence:Sequence: GravitationalGravitational CollapseCollapse toto ignitionignition ofof HH BurningBurning PostPost--MainMain Sequence:Sequence: CollapseCollapse ofof HH exhaustedexhausted corecore toto finalfinal endend

THETHE DETERMININGDETERMINING AGENTAGENT ININ AA STARSTAR’’SS LIFELIFE ISIS GRAVITYGRAVITY

Phases of Stellar Evolution 3 VirialVirial TheoremTheorem

EE == UU ++ ΩΩ ------NonNon--RelativisticRelativistic TotalTotal EnergyEnergy 2U2U ++ ΩΩ == 00 UU == InternalInternal EnergyEnergy ΩΩ == GravitionalGravitional BindingBinding EnergyEnergy DifferentialDifferential Form:Form: ∆∆UU == --1/21/2 ∆∆ΩΩ PrePre--MainMain SequenceSequence isis dominateddominated byby thethe VirialVirial TheoremTheorem

Phases of Stellar Evolution 4 GeneralGeneral PrinciplesPrinciples ofof StellarStellar EvolutionEvolution TheThe initialinitial effecteffect ofof nuclearnuclear burningburning isis toto increaseincrease thethe meanmean molecularmolecular weight.weight. Hydrogen Burning: 41H → 4He means μ → 8/3μ When μ increases the pressure is lowered and the core contracts, then T and P increase and thus hydrostatic equilibrium may be restored. WhenWhen TT increases,increases, thethe temperaturetemperature gradientgradient increases.increases. ThisThis causescauses anan increaseincrease inin luminosityluminosity (energy(energy flowflow increases)increases) In order to balance energy generation and luminosity, the star must increase its luminosity (which does happen on the MS). As the temperature is rising the energy generation goes up and thus so does the luminosity.

Phases of Stellar Evolution 5 MainMain SequenceSequence ConfigurationConfiguration

Thermonuclear core (pp/CNO) The core is either convective (CNO burners) or radiative (pp burners)

Envelope: Convective or Radiative (Opposite of core).

Phases of Stellar Evolution 6 LaterLater StageStage ConfigurationConfiguration

In a more massive star the rising core temperature will (might) cause ignition of higher mass nuclei. Outside is a region of processed material and outside that could be a region where the previous stage is still occurring. This is shell burning.

Phases of Stellar Evolution 7 EvolutionEvolution ofof AA ProtoStarProtoStar We consider only the immediate Pre-Main Sequence

PrinciplePrinciple Constituents:Constituents: H,H, HH2,, He,He, (dust)(dust) γγ (c(cp//ccv)) isis belowbelow 4/3:4/3: inducedinduced byby thethe ionizationionization ofof H,H, He,He, andand thethe dissociationdissociation ofof HH2 WhenWhen HH andand HeHe areare fullyfully ionizedionized γγ →→ 5/35/3 andand thethe collapsecollapse becomesbecomes quasiquasi--staticstatic VirialVirial TheoremTheorem sayssays ½½ ofof thethe energyenergy ofof collapsecollapse goesgoes intointo heatingheating andand ½½ intointo radiation.radiation. Bolometric Magnitude of a 1 pc cloud of radius 1 pc L = 4(206265*1.496(1011))2T4 = 1.2(1034) 5.67(10-8) 1004 34 L = 6.8(10 ) J/s = 34000 L

Phases of Stellar Evolution 8 WhereWhere DoesDoes thethe EnergyEnergy Go?Go?

FirstFirst CompletelyCompletely IonizeIonize thethe HeHe EE == (#He/(#He/gm)gm)(Mass(Mass Fraction)Fraction)(Ionization(Ionization Energy)Energy)

EE == (N(N0/4)/4) YY EEHe EEHe == EEHe I ++ EEHe II == 78.9878.98 eVeV SimilarlySimilarly forfor HH andand H2H2

EEI == NN0XEXEH ++ ½½ NN0XEXED ++ 1/41/4 NN0YEYEHe == 1.9(101.9(1013)[1)[1--0.2X]0.2X]

EED == EnergyEnergy ofof dissociationdissociation forfor HH2

Phases of Stellar Evolution 9 InternalInternal EnergyEnergy

FromFrom thethe VirialVirial TheoremTheorem 2 MEMEI ☺☺ ½½ αα GMGM /R/R (M(M isis thethe MassMass collapsing)collapsing) α depends on the order of the polytrope n = 1.5 (γ = 5/3) α = 6/7 α is always of order unity NowNow solvesolve forfor thethe radius:radius:

R/R = 43.2(M/M) / [1 - 0.2X] This is the maximum radius for a stable star at the beginning of its evolution If larger then ionization and disassociation will not be complete.

Once 43.2(M/M) / [1 - 0.2X] is achieved quasistatic evolution is possible.

Phases of Stellar Evolution 10 WhatWhat isis thethe CentralCentral Temperature?Temperature?

5 TTc == 3(103(10 ))μμ(1(1--0.2X)0.2X) ThisThis isis alwaysalways lessless thanthan ignitionignition temperaturetemperature (10(107K)K) soso thethe energyenergy sourcesource isis gravitationalgravitational collapse.collapse.

EET == TotalTotal EnergyEnergy ofof thethe StarStar

LuminosityLuminosity == EnergyEnergy FlowFlow // TimeTime == dEdET // dtdt

2 dET 1 d⎛⎞ GM L ==α ⎜⎟ dt2 dt⎝⎠ R GM2 R =− α 2 RR

Phases of Stellar Evolution 11 TimeTime ScaleScale α GM2 R L =− 2 RR

LL isis aa positivepositive numbernumber soso R(dotR(dot)) mustmust bebe negative;negative; ieie,, thethe starstar isis contractingcontracting TimeTime ScaleScale ∝∝ ∆∆EE // LL ∆∆tt == ∆∆EE // LL ~~ --1/21/2 ∆∆ΩΩ /L/L ~~ GMGM2/(2RL)/(2RL) 7 2 ∆∆tt == 1.6(101.6(10 )) (M/M(M/M)) (R(R/R)(L/R)(L/L)/L) yearsyears SoSo forfor thethe SunSun ∆∆tt ~~ 2(102(107)) yearsyears 5 AtAt 1010 MM::LL ~~ 1010 LL andand RR ~~ 500500 RR soso ∆∆tt ~~ 3232 years!years!

Phases of Stellar Evolution 12 HayashiHayashi TrackTrack II

CentralCentral TemperaturesTemperatures areare lowlow OpacitiesOpacities areare largelarge duedue toto contributionscontributions fromfrom ionizationionization processesprocesses (bf(bf transitions!)transitions!) RadiativeRadiative processesprocesses cannotcannot movemove thethe energyenergy soso convectionconvection dominatesdominates ForFor anan idealideal gasgas TT3//ρρ variesvaries slowlyslowly (except(except inin thethe photosphere)photosphere) soso thethe objectobject isis fullyfully convective.convective. TheThe HayashiHayashi TrackTrack isis thethe pathpath aa fullyfully convectiveconvective contractingcontracting starstar takestakes inin thethe HRHR diagram.diagram.

Phases of Stellar Evolution 13 HayashiHayashi TrackTrack IIII

OneOne cancan approximateapproximate thethe HayashiHayashi TrackTrack asas

LogLog LL == 1010 log(Mlog(M)) -- 7.247.24 loglog TTeff ++ constconst ThisThis isis aa veryvery steeplysteeply descendingdescending functionfunction WhyWhy doesdoes LL decrease?decrease?

Star is contracting and Teff is increasing L ∝ R2T4 but R is decreasing very quickly and the radius decrease is dominating the luminosity ForFor ““lowlow”” massmass starsstars aa betterbetter approximationapproximation isis loglog LL == --38.738.7 loglog TTeff -- 6.746.74 LogLog MM ++ constconst whichwhich meansmeans thethe luminosityluminosity decreasesdecreases veryvery rapidly.rapidly.

Phases of Stellar Evolution 14 LaterLater PhasePhase LowLow MassMass ProtostarsProtostars

AsAs thethe interiorinterior temperaturetemperature increasesincreases thethe opacitiesopacities changechange (bf(bf givesgives wayway toto electronelectron scatteringscattering whichwhich meansmeans lessless opacity)opacity) AA radiativeradiative corecore developsdevelops OnceOnce thethe radiativeradiative corecore developsdevelops thethe corecore isis notnot sensitivesensitive toto thethe envelope.envelope. ThisThis yieldsyields thethe ““constantconstant”” luminosityluminosity phasephase asas thethe starstar movesmoves toto thethe leftleft towardstowards thethe MS.MS.

ForFor thisthis phasephase loglog LL == 0.80.8 loglog TTeff ++ 4.44.4 loglog MM ++ CC

Phases of Stellar Evolution 15 PrePre--MainMain SequenceSequence TracksTracks

Phases of Stellar Evolution 16 TracksTracks byby IckoIcko IbenIben

Phases of Stellar Evolution 17 AA 11 SolarSolar MassMass TimeTime HistoryHistory

Phases of Stellar Evolution 18 NGCNGC 22642264

Phases of Stellar Evolution 19 EvolutionaryEvolutionary TracksTracks

TheThe evolutionevolution ofof aa starstar isis givengiven byby evolutionaryevolutionary tracks.tracks. TheyThey givegive thethe positionposition ofof thethe starstar inin HRHR diagramdiagram asas itit movesmoves inin temperaturetemperature andand luminosity.luminosity. TheThe amountamount ofof timetime betweenbetween successivesuccessive (T,L)(T,L) pointspoints isis variablevariable andand dependsdepends onon thethe mass.mass. AnAn isochroneisochrone isis (T,L)(T,L) forfor aa sequencesequence ofof massesmasses atat aa fixedfixed timetime plottedplotted inin thethe HRHR diagram.diagram.

Phases of Stellar Evolution 20 AA 55 MM TrackTrack byby IckoIcko IbenIben,, Jr.Jr.

Phases of Stellar Evolution 21 AA MovieMovie

RantRant aboutabout QuickTimeQuickTime Here.Here.

ThenThen ShowShow thethe Movie.Movie.

Phases of Stellar Evolution 22 EvolutionaryEvolutionary SequencesSequences

StellarStellar evolutionevolution isis moremore difficultdifficult thanthan stellarstellar structure.structure. Structure is static but by its nature evolution is dynamic. Some of the changes take place on free-fall timescales. The structure and its rate of change depend on the previous structure. The problem becomes one of choosing time steps that are sufficiently small with respect to the rate of change, yet practical from the point of view of computer time.

Phases of Stellar Evolution 23 StellarStellar StructureStructure EquationsEquations

Time Dependent Versions

HydrodynamicHydrodynamic ∂ρPGmdr2 EquilibriumEquilibrium =−22 − ∂rr dt m ∂ = 4 r2 ρ MassMass ContinuityContinuity r ∂ πρ TLr3() =−∂κρ EnergyEnergy FlowFlow racrT16 22 ∂π LdS2 ⎡ ⎤ EnergyEnergy GenerationGeneration =−4()()∂ rr rT rdt⎣⎢ ⎦⎥ Phases of Stellar Evolution ∂ πρ ε 24 RememberRemember

TheThe followingfollowing areare functionsfunctions ofof rr P,P, T,T, ρρ,, m,m, κκ,, L,L, εε NoteNote thatthat equationequation 33 (energy(energy flow)flow) containscontains allall ofof atomicatomic physicsphysics inin κκ!! EquationEquation 44 hashas allall ofof nuclearnuclear physicsphysics inin εε!! ThermodynamicsThermodynamics isis inin 11 && 4:4: TT ((dS/dtdS/dt)) isis thethe energyenergy ofof collapsecollapse expressedexpressed inin termsterms ofof thethe entropyentropy change.change.

Phases of Stellar Evolution 25 TheThe MainMain SequenceSequence ZAMS: Zero Age Main Sequence

NoteNote thatthat thethe observationalobservational MSMS hashas aa finitefinite widthwidth duedue toto thethe admixtureadmixture ofof ages.ages. AsAs aa starstar evolvesevolves onon thethe MSMS itit evolvesevolves upup inin luminosityluminosity andand downdown inin TT TheThe dividingdividing pointpoint forfor thethe energyenergy generationgeneration cyclescycles occursoccurs atat aboutabout 2(102(107)) KK < 2(107) K uses pp with radiative core and convective envelope > 2(107) K uses CNO with convective core and radiative envelope.

Phases of Stellar Evolution 26 ConsiderConsider thethe CNOCNO CycleCycle

ForFor 1.2(101.2(107)) KK << TT << 5(105(107)) KK thethe energyenergy generationgeneration n goesgoes asas 2020 2 nn 2 1313 inin εε == εε0ρρTT ThisThis meansmeans thatthat thethe starstar willwill developdevelop aa veryvery largelarge temperaturetemperature gradientgradient duedue toto thethe sensitivitysensitivity ofof thethe energyenergy generationgeneration toto TT To see this: ∂L/∂r ∝ ε and ∂T/∂r ∝ L ThisThis alsoalso meansmeans thesethese starsstars havehave aa highlyhighly centralizedcentralized corecore inin termsterms ofof energyenergy generation:generation: aa 2%2% decreasedecrease inin TT yieldsyields aa 33%33% decreasedecrease inin energyenergy generationgeneration (n=20)(n=20) LargeLarge temperaturetemperature gradientsgradients meansmeans convectionconvection whichwhich dominatesdominates CNOCNO corescores

Phases of Stellar Evolution 27 ForFor thethe pppp CycleCycle

ForFor thethe pppp cyclecycle atat 4(104(106)) << TT << 2.4(102.4(107)) KK n 66 2 nn 2 3.53.5 inin εε== εε0ρρTT ThisThis meansmeans thatthat thethe temperaturetemperature gradientgradient isis muchmuch smaller.smaller. ThereThere isis lessless centralizationcentralization inin thethe energyenergy generationgeneration andand littlelittle tendencytendency forfor convectionconvection inin thethe core.core. pppp corescores areare radiativeradiative

Phases of Stellar Evolution 28 EnvelopeEnvelope StructuresStructures

ReverseReverse ofof thethe corecore structurestructure ““LowLow”” massmass starsstars havehave convectiveconvective envelopesenvelopes instigatedinstigated byby ““largelarge”” HH andand HeHe ionizationionization zones.zones. NoteNote thatthat μμ changeschanges dramaticallydramatically inin anan ionizationionization zonezone andand theythey areare intrinsicallyintrinsically unstable.unstable. ““HighHigh”” massmass starsstars havehave ““shallowshallow”” ionizationionization zoneszones whichwhich dodo notnot perturbperturb thethe structurestructure asas much.much.

Phases of Stellar Evolution 29 TheThe ZAMSZAMS

AtAt thethe timetime ofof entryentry ontoonto thethe MSMS thethe corecore temperaturetemperature isis sufficientsufficient toto initiateinitiate burning:burning: pp starts 12C → 14N by 2 proton captures This happens competitively with pp at the initial core temperature Leads to an equilibrium configuration of CN cycle 14N We at once convert all 12C to 14N but to continue the CNO process we must do 14N(p,γ) 15O. This is very slow and stops the CN processing if T < Tcrit for the CNO cycle. WhyWhy isis thisthis important?important? 12 14 19 Because of the T sensitivity. For C → N ε = ε0ρT The core becomes convective!

Phases of Stellar Evolution 30 WhyWhy doesdoes LL Drop?Drop?

TheThe corecore previousprevious toto 12CC →→ 14NN waswas radiative.radiative. ItIt becomesbecomes convectiveconvective whichwhich isis centrallycentrally condensed,condensed, haltshalts contraction,contraction, andand doesdoes workwork againstagainst gravity.gravity. Energy goes into work not luminosity - 80% in fact. AfterAfter 12CC burnsburns therethere areare twotwo possibilities:possibilities: In a low mass star a radiative core is reestablished due to pp domination In a high mass star CNO dominates and a convective core remains. Fresh 12C from convection or 3α

Phases of Stellar Evolution 31 EvolutionEvolution onon thethe MainMain SequenceSequence Really Slow!

2 Timescale:Timescale: ttn ~~ mcmc /L/L 2 33 20 33 For the Sun tn ~ mc /L ~ 2(10 ) 9(10 ) / 4(10 ) s This is about 1.4(1013) years for complete conversion so the process does not need to be efficient! OneOne cancan assumeassume thatthat staticstatic structurestructure equationsequations willwill hold.hold. AsAs timetime passespasses therethere willwill bebe chemicalchemical evolutionevolution inin thethe corecore throughthrough nuclearnuclear burning.burning. AugmentAugment thethe staticstatic structurestructure byby timetime dependentdependent burning.burning.

Phases of Stellar Evolution 32 BurningBurning HydrogenHydrogen

AssumeAssume XX (protons)(protons) andand YY (alpha(alpha particles)particles) areare thethe onlyonly species.species. X,Y specify a static model at any time The time rate of change of (X,Y) are then related to energy generation rates and the energy release per gram of matter. ForFor X:X: ReductionReduction isis byby pp,pp, CN,CN, oror CNOCNO Let us find dX/dt For the pp chain Q = 26.73 MeV for each 4 H converted.

Epp = Qpp / 4mH = energy / gram

Phases of Stellar Evolution 33 WhatWhat isis dX/dtdX/dt??

Epp = Qpp / 4mH = energy / gram

ButBut whatwhat wewe wantwant isis dX/dtdX/dt whichwhich hashas unitsunits ofof gramgram // ss == (energy(energy // s)s) (gram(gram // energy)energy)

εεpp hashas unitsunits ofof energyenergy // ss EEpp hashas unitsunits ofof energyenergy // gramgram

ForFor pppp onlyonly dX/dtdX/dt == --εεpp/E/Epp

pppp ++ CN:CN: dX/dtdX/dt == -- εεpp/E/Epp -- εεCN/E/ECN LowLow Mass:Mass: TT << 2(102(107)K)K pppp dominatesdominates HighHigh Mass:Mass: TT >> 2(102(107)K)K CNCN dominatesdominates

Epp and ECN are constants εpp and εCN depend on the structure (T and ρ)

Phases of Stellar Evolution 34 dY/dtdY/dt

TheThe sinksink isis 33αα,, thethe sourcesource isis dX/dtdX/dt NoteNote thatthat dX/dtdX/dt isis intrinsicallyintrinsically negativenegative

dY/dtdY/dt == --33εεα/E3/E3α -- (1/4)dX/dt(1/4)dX/dt 4H4H →→ 11 HeHe ThisThis isis forfor aa staticstatic zone;zone; thatthat is,is, aa radiativeradiative lowlow massmass core.core.

Phases of Stellar Evolution 35 CoreCore ConvectionConvection

TimeTime ScaleScale forfor convectionconvection isis ofof orderorder monthsmonths Instantaneous with respect to the time scale of the reactions ThisThis meansmeans thatthat forfor aa convectiveconvective zonezone therethere isis anan ““averageaverage”” composition:composition: XXc,, YYc,, ((ZZc)) For a correct treatment we must consider the intrusion of the convection into the radiative zone but neglect this for now.

TheThe raterate ofof changechange ofof XXc isis εε/E/E (per(per process)process) averagedaveraged throughthrough thethe zone.zone.

For pp: dXc/dt = -εpp /Epp dM/∆m integrate between m1 and m2 and ∆m = m2 - m1. Note that the mass of the convective zone ≠ mass of the core (necessarily)

Phases of Stellar Evolution 36 ProcessProcess toto CalculateCalculate aa SequenceSequence

AssumeAssume X,Y:X,Y: calculatecalculate structurestructure EstimateEstimate ∆∆X,X, ∆∆Y:Y: dX/dtdX/dt ∆∆tt wherewhere dX/dtdX/dt isis thethe instantaneousinstantaneous raterate andand ∆∆tt isis thethe timetime step.step.

TheThe compositioncomposition atat tt0 ++ ∆∆tt isis thenthen XX == XX ++ ∆∆XX andand YY == YY ++ ∆∆Y.Y.

Phases of Stellar Evolution 37 TheThe LowerLower MainMain SequenceSequence

To Reiterate

EnergyEnergy generationgeneration byby pppp chainchain 7 TTc << 2(102(10 )) KK MM 1 22 SolarSolar MassesMasses RadiativeRadiative corescores andand convectiveconvective envelopesenvelopes CoreCore sizesize decreasesdecreases withwith totaltotal massmass CoreCore isis initiallyinitially homogeneoushomogeneous 2 3.5 to 6 EnergyEnergy generationgeneration rate:rate: εε == εε0ρρXX TT

Phases of Stellar Evolution 38 EvolutionEvolution onon thethe MSMS 2 3.5 to 6 ε = ε0ρX T NoteNote thethe XX2 dependencedependence inin εε.. AsAs XX decreasesdecreases soso doesdoes εε unlessunless TT oror ρρ increase.increase. IfIf εε decreasesdecreases thenthen soso doesdoes PP Contraction follows: Virial Theorem allocates ½ the resulting energy to radiation and ½ to heating. This means ρ increases (contraction) and T increases (Virial Theorem) ε increases Slight increase in core radius and envelope L will also increase

Teff will not increase much due to increase in R

Phases of Stellar Evolution 39 AtAt thethe SolarSolar AgeAge T = 4.3 Gigayears

NoteNote thatthat 90%90% ofof thethe massmass r/Rr/R == 0.5.0.5. XX normalizednormalized toto 1:1: depletiondepletion limitedlimited toto r/Rr/R << 0.30.3 (0.5(0.5 inin m/M)m/M)

LL == LL atat rr == 0.3R.0.3R.

Phases of Stellar Evolution 40 JustJust BeforeBefore LeavingLeaving

r < 0.03R is isothermal He. H burning: 0.03 < r < 0.3 R ε = dL/dm is just the slope of L. ε is now large with respect to previous values. The development of an inhomogeneous structure marks the end of the MS in this mass range.

Phases of Stellar Evolution 41 UpperUpper MainMain SequenceSequence

7 Tc > 2(10 ) K

HH burningburning byby thethe CNOCNO cyclescycles ConvectiveConvective corescores Homogeneous evolution in the core even though H burning is more rapid in the center than outer edges. Opacity: Kramer’s 2-3 Solar Masses Electron Scattering > 3 solar masses n εε == εε0XZXZCNOTT Since ε goes as X the generation rate is not so sensitive to composition changes This means the core contraction brought on by H depletion is not as severe as on the lower main sequence

Phases of Stellar Evolution 42 HighHigh MassMass EvolutionEvolution

AsAs thethe massmass increasesincreases

R, L, Teff, and Tc all increase but ρc does not. If the opacity is electron scattering then there is a smaller dependence of luminosity and Teff on mass. NoteNote thatthat thethe mainmain differencedifference betweenbetween highhigh massmass andand lowlow massmass evolutionevolution isis thatthat highhigh massmass starsstars dodo notnot formform thickthick burningburning shellsshells aboutabout thethe HeHe corecore asas thethe starstar agesages onon thethe MS.MS. InIn fact,fact, HeHe corescores dodo notnot formform untiluntil ““allall”” HH burningburning ceases.ceases. This is due to convection homogenizing the core. So burning merely continues until X reaches about 0.05 when ε falls below the amount needed to support the core.

Phases of Stellar Evolution 43 TheThe IsothermalIsothermal CoreCore Low Mass Stars Only L(0)L(0) == 00 andand dL/drdL/dr ≈≈ 44ππrr2ρερε(r)(r) IfIf εε == 00 throughoutthroughout aa regionregion thenthen LL == 00 asas well.well. ButBut dT/drdT/dr ~~ L(rL(r)) soso ifif LL == 00 thenthen TT == ConstantConstant SoSo whatwhat supportssupports thethe corecore (it(it isis NOTNOT degenerate)?degenerate)? AA steepsteep densitydensity gradient.gradient. ThereThere isis aa limitinglimiting massmass forfor thisthis casecase -- itit isis calledcalled thethe SchSchöönbergnberg--ChandrasekharChandrasekhar limitlimit andand isis approximatelyapproximately 0.120.12 solarsolar masses.masses.

A core with Mc < MSC is stable but a burning shell above it will continually add mass. The result is that the limit will be exceeded. The core will start to collapse

Phases of Stellar Evolution 44 TerminationTermination ofof thethe MSMS

Here is where we start ShellShell source:source: 0.10.1 << m(rm(r)/M)/M << 0.30.3 LL increasesincreases StarStar expands.expands. TT shouldshould gogo upup butbut RR increasesincreases toto suchsuch anan extentextent thatthat TT actuallyactually falls.falls. ThisThis takestakes aboutabout 12%12% ofof thethe MSMS lifetime.lifetime. EventuallyEventually thethe corecore massmass exceedsexceeds thethe SchSchöönbergnberg-- ChandrasekharChandrasekhar limitlimit (in(in allall butbut thethe lowestlowest masses)masses) andand thethe corecore isis forcedforced toto contract.contract. TheThe MSMS isis over.over.

Phases of Stellar Evolution 45 HighHigh Mass:Mass: MM >> 1.251.25 SolarSolar MassesMasses

No isothermal cores are formed X decreases to about 5% of its original value and core shuts down. Contraction starts L increases

Teff increases initially The contraction induces a shell to start. The MS is over. The shell provides L The core contracts The envelope expands: T must decrease.

Phases of Stellar Evolution 46 PostPost--MainMain SequenceSequence EvolutionEvolution A Dramatic Series of Events Events are less certain as the possibilities become wider in scope Dynamical Effects. Observational data more limited but what is known agrees rather well with the theory especially in single stars. One cannot make certain events happen numerically ab initio Pulsation in Variables Ignition of 3α in a “controlled” fashion , esp low mass SN dynamics Deflagrations are a problem Binary evolution

Phases of Stellar Evolution 47 Tracks:Tracks: SingleSingle StarsStars withwith NoNo MassMass LossLoss

Phases of Stellar Evolution 48 WhatWhat DoDo WeWe havehave toto Add?Add?

TheThe mostmost importantimportant newnew featurefeature toto allowallow forfor isis chemicalchemical inhomogeneitiesinhomogeneities andand associatedassociated shellshell sources.sources. ActiveActive Shells:Shells: CurrentlyCurrently burningburning InactiveInactive Shell:Shell: ChemicalChemical inhomogeneityinhomogeneity TheThe behaviorbehavior ofof expansionsexpansions andand contractionscontractions changechange overover activeactive shells.shells. VolumeVolume changeschanges reversereverse overover activeactive shells.shells.

Phases of Stellar Evolution 49 WhatWhat DoDo InhomogeneitiesInhomogeneities Do?Do?

InhomogeneousInhomogeneous compositioncomposition leadsleads toto greatergreater centralcentral condensationcondensation TheThe positionposition ofof aa burningburning shellshell remainsremains fairlyfairly constantconstant inin radius.radius. VolumeVolume changeschanges (contraction,(contraction, expansion)expansion) reversereverse atat eacheach nuclearnuclear burningburning shell,shell, butbut remainremain unaffectedunaffected byby thethe presencepresence ofof inactiveinactive shells.shells.

Phases of Stellar Evolution 50 DesideraDesidera

NuclearNuclear ashash hashas (usually)(usually) μμ >> μμfuel LargerLarger μμ ==>==> largerlarger centralcentral corecore valuesvalues forfor ρρ NoteNote thatthat thethe corecore γγ == 4/34/3 (n(n == 3).3). StationaryStationary shells:shells: If it tries to burn inwards then T increases and so does ε which means that P will increase forcing the shell back out. Also trying to burn inwards means that one is moving to zones depleted in the current fuel. Cannot go out (in radius) because T will be too low to support the burning.

Phases of Stellar Evolution 51 VolumeVolume ReversalsReversals

VIP

IfIf oneone hashas aa contractingcontracting corecore withwith anan activeactive shellshell ==>==> envelopeenvelope expansion.expansion. IfIf oneone hashas aa contractingcontracting corecore withwith twotwo activeactive shellsshells ==>==> envelopeenvelope contraction.contraction. Why?Why? ConsiderConsider thethe followingfollowing (long)(long) argument.argument.

Phases of Stellar Evolution 52 CentralCentral CondensationCondensation

AA measuremeasure ofof thethe centralcentral condensationcondensation forfor aa particularparticular volumevolume isis thethe ratioratio ρρ(r(r)) (the(the locallocal valuevalue ofof thethe density)density) toto <<ρρ(r(r)>)> thethe meanmean densitydensity ofof thethe materialmaterial interiorinterior toto r.r. WeWe definedefine UU ≡≡ 33ρρ(r)/<(r)/<ρρ(r(r)>)> wherewhere <<ρρ(r(r)>)> == 3m(r)3m(r) // 44ππrr3 OneOne cancan show:show: UU == d(ln(m(r))/d(ln(rd(ln(m(r))/d(ln(r)))) == dd ln(qln(q)) // dd ln(rln(r)) wherewhere qq ≡≡ m(rm(r)/M)/M WhatWhat areare thethe limitslimits onon U?U? At r = R ρ(r) = 0 U = 0 at the upper boundary

At r = 0 ρ(r) = ρc and <ρ(r)> = ρc U = 3

Phases of Stellar Evolution 53 μ ,ρ μ ,ρ ConsiderConsider thethe InterfaceInterface c c e e

CoreCore EnvelopeEnvelope interfaceinterface withwith nono burningburning IfIf thethe starstar isis toto bebe stablestable TT andand PP mustmust bebe continuouscontinuous acrossacross thethe interface.interface.

AssumeAssume thethe idealideal galgal law:law: PP == nkTnkT == ρρkT/mkT/mH whichwhich meansmeans PVPV // TT == PP′′VV′′ // TT′′ NowNow rearrangerearrange usingusing equalequal volumesvolumes oror VV == VV′′ soso P/TP/T == PP′′ /T/T′′ oror ρρ//μμ == ρρ′′ //μμ′′ oror inin termsterms ofof ourour interfaceinterface ρρc//μμc == ρρe//μμe.. NowNow μμc >> μμe soso ρρc >> ρρe IfIf thethe boundaryboundary isis sharp:sharp: <<ρρ(r(r)>)> == constantconstant

Phases of Stellar Evolution 54 InterfacesInterfaces

NoteNote thatthat ifif therethere isis aa ρρ(r(r)) discontinuitydiscontinuity thenthen therethere isis aa U(rU(r)) discontinuitydiscontinuity SinceSince UU ≡≡ 33ρρ(r)/<(r)/<ρρ(r(r)>)> andand <<ρρ(r(r)>)> == constantconstant forfor aa sharpsharp interfaceinterface UU goesgoes asas ρρ.. ThisThis meansmeans sincesince ρρc//μμc == ρρe//μμe thatthat UUc//μμc == UUe//μμe.. ThereThere areare twotwo additionaladditional characterizations:characterizations: V ≡ -d lnP/ d lnr (Pressure scale height) N+1 ≡ d lnP/ d lnr = -V These can be evaluated adiabatically and if done that way N is related to 2 An adiabatic process is one in there is no heat flow A free expansion is an adiabatic process No Heat Flow No Work

Phases of Stellar Evolution 55 CharacterisiticsCharacterisitics ofof ShellShell SourcesSources

GasGas isis ideal:ideal: ρρ == KTKTn dP/drdP/dr == dP/dTdP/dT dT/drdT/dr

== dT/drdT/dr ((d/dTd/dT ((ρρkT/kT/μμmmH))))

== kkρρ(n+1)/(n+1)/μμmmH dT/drdT/dr dP/drdP/dr cancan bebe specifiedspecified byby thethe rs hydrostatichydrostatic equationequation

mmc isis concentratedconcentrated nearnear rrs

Phases of Stellar Evolution 56 ShellShell SourcesSources IIII

m(rm(r)) ☺☺ mmc forfor rr >>>> rrs

TTs == ((μμmmHGmGmc/(k(n+1)))/(k(n+1))) (1/r(1/rs)) ++ constconst

NoteNote thatthat TsTs ~~ 1/r1/rs NoNo MotionMotion ofof thethe shellshell

ForFor Example,Example, inin aa 11 MM starstar thethe shellshell locationlocation isis atat RR ~~ 0.03R0.03R

Phases of Stellar Evolution 57 VolumeVolume ChangesChanges andand ShellShell StructureStructure

1 dqln ln(Rr )=+ ln(s ) ∫qs U

RR == StellarStellar RadiusRadius

qqs == mmc/m/m (mass(mass fractionfraction inin core)core)

rrs == constantconstant (so(so thethe stellarstellar radiusradius dependsdepends onon thethe integral)integral) UU == 33ρρ(r)/<(r)/<ρρ(r(r)>)>

Phases of Stellar Evolution 58 WhatWhat Happens?Happens?

ConsiderConsider ρρc increasingincreasing duedue toto corecore condensationcondensation

rrs remainsremains fixed.fixed. ρρ atat thethe edgeedge ofof thethe shellshell (inside)(inside) willwill decreasedecrease <<ρρ(rs(rs)>)> isis constantconstant (to(to firstfirst orderorder :: ∆∆MM == 0)0)

ThereforeTherefore wewe havehave aa decreasedecrease inin UUc butbut ifif UUc decreasesdecreases soso mustmust UUe Therefore:Therefore: reducereduce U(rU(r)) inin thethe lowestlowest levelslevels ofof thethe envelopeenvelope wherewhere 1/q1/q isis largestlargest ==>==> decreasedecrease thethe density.density.

Phases of Stellar Evolution 59 AA PracticalPractical StatementStatement

AsAs thethe centralcentral condensationcondensation growsgrows thethe densitydensity nearnear thethe bottombottom ofof thethe shellshell decreasesdecreases andand toto maintainmaintain continuitycontinuity thethe envelopeenvelope respondsresponds byby expandingexpanding (above(above thethe burningburning shell).shell). NoteNote thatthat inin thethe casecase ofof twotwo burningburning shellsshells oneone getsgets envelopeenvelope expansion!expansion!

Phases of Stellar Evolution 60