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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