Stellar Model Building

8/3/2019 Stellar Model Building

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Stellar Model Building

Anand A.Anand A.


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How to say my name?


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Summary of the Equations of

Stellar Structure TimeTime--independentindependent(static) stellar structure(static) stellar structureequationsequations

LastLast eqneqn: holds when: holds whentemp. gradient is purelytemp. gradient is purelyadiabaticadiabatic

Energycontribution dueEnergycontribution dueto gravityifnonto gravityifnon--staticstaticstarstar


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Entropy

Redefine gravitationalenergy generationRedefine gravitationalenergy generation

ratein terms ofentropyper unit massratein terms ofentropyper unit mass

Contraction and ExpansionContraction and Expansion

Contraction in relation to 2Contraction in relation to 2ndnd

Law ofLaw ofThermodynamicsThermodynamics Entropyofthe Universeis increasingEntropyofthe Universeis increasing

Star entropycarried out by neutrinos and photonsStar entropycarried out by neutrinos and photons


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Constitutive Relations

Useofequations (basic stellar structure) requiresUseofequations (basic stellar structure) requiresknowledgeon thephysicalproperties ofmatter thatknowledgeon thephysicalproperties ofmatter that

make up the starmake up the star

Need equations ofstatefor pressure,opacity, andNeed equations ofstatefor pressure,opacity, andenergy generation rateenergy generation rate

P = P(P = P(VV, T, composition), T, composition)

Most cases:Most cases: PPtt == VVkT/kT/QQmmHH + aT+ aT44/3/3

OO == OO ((

VV, T, composition), T, composition)

Presented in tablePresented in table

II ==II((VV, T, composition), T, composition)

Formulas for ppchain and CNO cycle, reaction networksFormulas for ppchain and CNO cycle, reaction networks


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Boundary Conditions

Solutions to stellar structureequations (includingSolutions to stellar structureequations (includingconstitutive relations) requires appropriate boundaryconstitutive relations) requires appropriate boundary

conditionsconditions

Help definelimits ofintegrationHelp definelimits ofintegration Central Boundary Conditions: Interior mass andCentral Boundary Conditions: Interior mass and

luminosity approach zero at thecenter ofstarluminosity approach zero at thecenter ofstar

Second set required at surface:Second set required at surface: TT,,PP, and, and VV go to zerogo to zero

Never strictlyobtained becauseNever strictlyobtained becauseTTon surfaceis not zeroon surfaceis not zero Need morecomplicated boundaryconditions for realisticNeed morecomplicated boundaryconditions for realistic

modelmodel


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The Vogt-Russell Theorem

Theorem: Themass and thecompositionTheorem: Themass and thecomposition

structure throughout a star uniquelystructure throughout a star uniquely

determineits radius,luminosity, and internaldetermineits radius,luminosity, and internal

structure, as well as its subsequentstructure, as well as its subsequent

evolutionevolution

The dependenceofstars evolution on mass andThe dependenceofstars evolution on mass and

composition is a consequenceofthechangeincomposition is a consequenceofthechangeincomposition due to nuclear burningcomposition due to nuclear burning


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umerical Modeling of the Stellar

Structure Equations Differentialequations with respectiveconstitutiveDifferentialequations with respectiveconstitutiverelations cannot be solved analyticallyrelations cannot be solved analytically

Solved by numericalintegrationSolved by numericalintegration

Numericalintegration approximates differentialNumericalintegration approximates differentialequations to differenceequationsequations to differenceequations

dP/drdP/dr ------>>((PP//((rr

Spherically symmetric shellsSpherically symmetric shells

Increment each fundamentalphysicalparameterIncrement each fundamentalphysicalparameterthrough successive applications of differenceequationthrough successive applications of differenceequation Integration carried out from a given layer to the surface and aIntegration carried out from a given layer to the surface and a

given layer to thecenter simultaneouslyin order to highlightgiven layer to thecenter simultaneouslyin order to highlightdifferences in physicalprocesses between outer an inner layersdifferences in physicalprocesses between outer an inner layers


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Polytropic Models and the Lane-

Emden Equation PolytropesPolytropes: specialcaseofapproximate: specialcaseofapproximate

solutions to the stellar structureequationssolutions to the stellar structureequations

Can be solved analyticallyCan be solved analytically J. Homer Lanefirst worked in this area andJ. Homer Lanefirst worked in this area and

Emden extended his workEmden extended his work

Based on finding simple relation betweenBased on finding simple relation between

pressure and densitypressure and density

RedefineRedefinepolytropepolytrope mathematically:mathematically: P =KP =KVVKK


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Lane-Emden Equation

Started with andStarted with andnow have a renow have a re--scaledscaleddimensionless radial term, xi,dimensionless radial term, xi,and a reand a re--scaledscaleddimensionless density termdimensionless density term

nn is theis thepolytropicpolytropic indexindexwhere thepressure andwhere thepressure anddensityofthe gas are relateddensityofthe gas are related P =KP =KVVKK

KK ==((nn +1)/n+1)/n

Pressure, temperature, andPressure, temperature, anddensityprofilecan be deriveddensityprofilecan be derivedfrom this functionfrom this function


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Limitations of L-E Eqn.

Describes only hydrostaticequilibrium andDescribes only hydrostaticequilibrium andmass conservation through highlyidealizedmass conservation through highlyidealizedpolytropicpolytropic equations ofstateequations ofstate

Only three analytical solutions,Only three analytical solutions,nn = 0, 1, and 5= 0, 1, and 5

nn = 1.5= 1.5

nn = 3= 3


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Thank you

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Stellar Model Building