Numerical Solution of Numerical Solution of Rotating StarsRotating Stars

Robert Deupree, DirectorRobert Deupree, DirectorInstitute for Computational Institute for Computational

AstrophysicsAstrophysicsSaint Mary’s UniversitySaint Mary’s University

Mea CulpaMea Culpa I have deliberately kept the talk pretty basic I have deliberately kept the talk pretty basic

because I am sufficiently removed from your because I am sufficiently removed from your discipline (which I take to be pure math) that discipline (which I take to be pure math) that I am not sure what you do and do not know I am not sure what you do and do not know about applied mathabout applied math

I do know that sometimes different I do know that sometimes different disciplines have different meanings for the disciplines have different meanings for the same words, so I have tried to define most same words, so I have tried to define most terms at the risk of seeming to patronize youterms at the risk of seeming to patronize you

So, if I have erred in making the talk too So, if I have erred in making the talk too simple or simplistic, I apologize, but I would simple or simplistic, I apologize, but I would rather do that than have the talk be so rather do that than have the talk be so incomprehensible that you could not get incomprehensible that you could not get anything out of itanything out of it

Stellar Models and Stellar Models and HistoryHistory

Spherically symmetric models of stars Spherically symmetric models of stars were one of the first (non-defense) were one of the first (non-defense) successes of finite difference equationssuccesses of finite difference equations Semi-analytic models in 1950Semi-analytic models in 1950 Detailed numerical solutions of entire life Detailed numerical solutions of entire life

history of stars by 1963history of stars by 1963 Spherically symmetric models are very Spherically symmetric models are very

good approximations for most starsgood approximations for most stars Good agreement with observationsGood agreement with observations

1D Calculations Avoid 1D Calculations Avoid ProblemsProblems

Interior mass, MInterior mass, Mrr, is an excellent , is an excellent independent variableindependent variable Runs from 0 to M, the total mass of the Runs from 0 to M, the total mass of the

starstar Increases monotonically with radiusIncreases monotonically with radius Dependent variables (P, Dependent variables (P, ρρ, T, r, , T, r,

composition) composition) are reasonable (and mostly are reasonable (and mostly monotonic) functions of interior massmonotonic) functions of interior mass

Can be subdivided as neededCan be subdivided as needed

Solution TechniqueSolution Technique

Method of solution is the finite Method of solution is the finite difference techniquedifference technique Divide star into spherical shellsDivide star into spherical shells Replace differentials by differences Replace differentials by differences

between adjacent shellsbetween adjacent shells

Not a 1D WorldNot a 1D World

While the structure of many stars While the structure of many stars may be excellently approximated by may be excellently approximated by spherical symmetry, there are a spherical symmetry, there are a number which are notnumber which are not RotationRotation Magnetic fieldsMagnetic fields Anisotropic equation of stateAnisotropic equation of state

Equation of state gives P = Equation of state gives P = P(P(ρρ,T,composition,…),T,composition,…)

Requirements for Multi-Requirements for Multi-Dimensional Significance Dimensional Significance

RotationRotation Centrifugal force must be sizeable somewhere in Centrifugal force must be sizeable somewhere in

the star when compared to the gravitational force the star when compared to the gravitational force (note that sizeable may mean only that it (note that sizeable may mean only that it modestly changes the direction of the net force modestly changes the direction of the net force vector) vector)

Expected to happen most often near the stellar Expected to happen most often near the stellar surfacesurface

Means that most deep interior (aka, core) events are Means that most deep interior (aka, core) events are relatively unchanged (and this is why 1D models are not relatively unchanged (and this is why 1D models are not a complete embarrassment even for some rotating stars)a complete embarrassment even for some rotating stars)

These interior events control the evolution of the starThese interior events control the evolution of the star What the observer sees may be strongly affectedWhat the observer sees may be strongly affected

Significant number of massive stars have Significant number of massive stars have significant rotationsignificant rotation

Requirements for Multi-Requirements for Multi-Dimensional SignificanceDimensional Significance

Magnetic FieldsMagnetic Fields Magnetic pressure (BMagnetic pressure (B22/8/8ππ) must be a reasonable ) must be a reasonable

fraction of the gas pressure, P, somewhere in fraction of the gas pressure, P, somewhere in the starthe star

Because gas pressure may be large, one may Because gas pressure may be large, one may have a large magnetic field but still have the have a large magnetic field but still have the structure essentially spherically symmetricstructure essentially spherically symmetric

Neutron stars have B ≈ 10Neutron stars have B ≈ 101212 Gauss (Earth’s B is about Gauss (Earth’s B is about 0.6 Gauss), but star can still be well approximated as 0.6 Gauss), but star can still be well approximated as spherically symmetricspherically symmetric

A relatively small number of stars have sizeable A relatively small number of stars have sizeable (in this sense) observed magnetic fields(in this sense) observed magnetic fields

Requirements for Multi-Requirements for Multi-Dimensional SignificanceDimensional Significance

Anisotropic Equation of StateAnisotropic Equation of State Off diagonal components of stress tensor Off diagonal components of stress tensor

must be a reasonable fraction of the must be a reasonable fraction of the pressure (which is related to the trace of pressure (which is related to the trace of the stress tensor)the stress tensor)

Not sure if actually happens in any star; Not sure if actually happens in any star; most likely place is neutron starsmost likely place is neutron stars

Equation of state for most stars may be treated Equation of state for most stars may be treated as that of an ideal gas plus radiation pressureas that of an ideal gas plus radiation pressure

Equation of state for very dense regions may Equation of state for very dense regions may involve the Pauli exclusion principle for either involve the Pauli exclusion principle for either electrons or free neutronselectrons or free neutrons

Where is this Talk Where is this Talk Going?Going?

Focus will be on 2D models of rotating starsFocus will be on 2D models of rotating stars 2D is believed to be a good approximation for rotating 2D is believed to be a good approximation for rotating

stars except in some extreme casesstars except in some extreme cases Massive stars are important because they become Massive stars are important because they become

supernovae and provide significant seeding to the supernovae and provide significant seeding to the composition of the interstellar medium which will form composition of the interstellar medium which will form the next starsthe next stars

Massive stars may rotate up to critical rotation (where Massive stars may rotate up to critical rotation (where centrifugal force balances gravity at the surface equatorial centrifugal force balances gravity at the surface equatorial radius)radius)

Predictions of supernovae explosions are not successful Predictions of supernovae explosions are not successful (i.e., do not agree with observations) and are sensitive (i.e., do not agree with observations) and are sensitive to details that rotation could modifyto details that rotation could modify

Focus will be on mathematical solution to the 2D Focus will be on mathematical solution to the 2D simulation of rotating starssimulation of rotating stars

The ProblemThe Problem EquationsEquations

Mass conservationMass conservation 3 components of momentum (azimuthal symmetry - 3 components of momentum (azimuthal symmetry -

2.5D calculation) conservation2.5D calculation) conservation Energy conservationEnergy conservation Composition conservation (N equations)Composition conservation (N equations)

Typically need only H and He until late in evolutionTypically need only H and He until late in evolution Poisson’s equation for gravitational potentialPoisson’s equation for gravitational potential Equation of stateEquation of state Subsidiary equations (nuclear reaction rates, thermal Subsidiary equations (nuclear reaction rates, thermal

conductivities,…)conductivities,…) VariablesVariables

ρρ, T, P, v, T, P, vrr, v, vθθ, v, vφφ,,ΦΦ, X, Y, r, , X, Y, r, θθ, t, t Boundary conditionsBoundary conditions

Some at centre, some at surface (where is it?)Some at centre, some at surface (where is it?)

DecisionsDecisions

Nearly all the important decisions Nearly all the important decisions must be made before fingers are put must be made before fingers are put to keyboardto keyboard Coordinate systemCoordinate system Independent variablesIndependent variables Determination of stellar surface Determination of stellar surface

locationlocation Determination of gravitational potential Determination of gravitational potential

boundary condition at stellar surfaceboundary condition at stellar surface

Decision 1Decision 1

Coordinate systemCoordinate system Stars may be considered as gases Stars may be considered as gases

because the temperature is so high because the temperature is so high compared to the densitycompared to the density Gravity wants the star to be sphericalGravity wants the star to be spherical Rotation wants the star to be cylindricalRotation wants the star to be cylindrical Gravity dominates throughout most of star, Gravity dominates throughout most of star,

especially at centre where evolution is especially at centre where evolution is controlled, so spherical is probably the best controlled, so spherical is probably the best choice (and the one I made)choice (and the one I made)

Decision 2Decision 2 Independent variablesIndependent variables

Spherical polar coordinate Spherical polar coordinate θθ is reasonable ( is reasonable (θθ boundaries are boundaries are polar axis and possibly equatorial axis [for equatorial polar axis and possibly equatorial axis [for equatorial symmetry])symmetry])

MMrr loses much of its value loses much of its value Radius, r, is a problem because the stellar radius, R, can grow Radius, r, is a problem because the stellar radius, R, can grow

by about a factor of 100 throughout the stellar evolutionby about a factor of 100 throughout the stellar evolution Use the fractional surface radius, x = r/RUse the fractional surface radius, x = r/R

R is here the largest surface radius (usually the equatorial radius)R is here the largest surface radius (usually the equatorial radius) Radial zoning expands and contracts like an accordion as the Radial zoning expands and contracts like an accordion as the

stellar surface expands and contractsstellar surface expands and contracts This allows me to have reasonable zoning throughout the star’s This allows me to have reasonable zoning throughout the star’s

entire evolution sequenceentire evolution sequence One consequence is that I now need another equation to One consequence is that I now need another equation to

determine R – use that the integral over the density distribution determine R – use that the integral over the density distribution gives the total mass of the stargives the total mass of the star

Consequences of Decisions Consequences of Decisions 1 and 21 and 2

Decisions 1 and 2 determine the Decisions 1 and 2 determine the actual form that the conservation actual form that the conservation equations takeequations take There is nothing to program until the There is nothing to program until the

form of the equations is determinedform of the equations is determined If you want to change, you effectively If you want to change, you effectively

have to start overhave to start over Think about the changes in the code you Think about the changes in the code you

would have to make to change a calculation would have to make to change a calculation from spherical to cylindrical coordinatesfrom spherical to cylindrical coordinates

Decision 3Decision 3

Determination of stellar surface locationDetermination of stellar surface location With MWith Mrr this is easy; it is where M this is easy; it is where Mrr = M = M R is now f(R is now f(θθ)) If rotation law is conservative (i.e., rotation If rotation law is conservative (i.e., rotation

rate rate ωω(r,(r,θθ) can be written as the gradient of ) can be written as the gradient of a potential), then the surface is an a potential), then the surface is an equipotential (of the total potential [gravity equipotential (of the total potential [gravity + rotation])+ rotation])

Assume this is true even when rotation law Assume this is true even when rotation law is not conservativeis not conservative

Decision 4Decision 4 Surface gravitational potential boundary Surface gravitational potential boundary

conditioncondition Must be continuous across the stellar surfaceMust be continuous across the stellar surface Evaluate the gravitational potential exterior to Evaluate the gravitational potential exterior to

the surface by integrating over the density the surface by integrating over the density distributiondistribution

Use method attributed to Gauss for quasi-Use method attributed to Gauss for quasi-analytic solution for a uniform circular wire analytic solution for a uniform circular wire (azimuthal symmetry)(azimuthal symmetry)

Density distribution is a very LARGE collection Density distribution is a very LARGE collection of uniform circular wires (test for spherically of uniform circular wires (test for spherically symmetric density distribution to get symmetry symmetric density distribution to get symmetry to about one part in 10to about one part in 1099))

Importance of DecisionsImportance of Decisions

All decisions (clearly there are more All decisions (clearly there are more than the four I summarized) are than the four I summarized) are important, but the first two important, but the first two (particularly the second) have (particularly the second) have several important consequences for several important consequences for the method of solutionthe method of solution

Consequences of Consequences of Decision 2Decision 2

Having the fractional surface radius as the Having the fractional surface radius as the independent variable means that the independent variable means that the independent variable is neither Eulerian nor independent variable is neither Eulerian nor LagrangianLagrangian Eulerian independent variables – fixed in spaceEulerian independent variables – fixed in space Lagrangian independent variables – move with Lagrangian independent variables – move with

fluidfluid Works great in 1D, but can get tied up in knots in >1DWorks great in 1D, but can get tied up in knots in >1D

Independent variables that are neither Eulerian Independent variables that are neither Eulerian nor Lagrangian are called Euler-Lagrangenor Lagrangian are called Euler-Lagrange

Practical Result of Practical Result of Decision 2Decision 2

AdvectionAdvection Lagrangian time derivative (D/Dt) is related to Eulerian time Lagrangian time derivative (D/Dt) is related to Eulerian time

derivative (∂/∂t) for quantity q byderivative (∂/∂t) for quantity q by Dq/dt = ∂q/∂t + v(x) ∂q/∂xDq/dt = ∂q/∂t + v(x) ∂q/∂x Spatial coordinate is xSpatial coordinate is x Velocity is vVelocity is v

Time derivative as seen by the fluid equals time derivative at a fixed Time derivative as seen by the fluid equals time derivative at a fixed location plus advectionlocation plus advection

For Euler-Lagrange independent variable, z, this becomesFor Euler-Lagrange independent variable, z, this becomes Dq/dt = ∂q/∂t + [v(z)-V(z)] ∂q/∂zDq/dt = ∂q/∂t + [v(z)-V(z)] ∂q/∂z V is the velocity of the independent variable with respect to an V is the velocity of the independent variable with respect to an

Eulerian coordinate systemEulerian coordinate system What I want to advect is the motion of the material with respect to What I want to advect is the motion of the material with respect to

my spatial independent variablemy spatial independent variable To keep x = r/R constant, one can show that To keep x = r/R constant, one can show that

V = x (DR/Dt)V = x (DR/Dt) Surface at largest R is the one place where the independent Surface at largest R is the one place where the independent

variable is actually Lagrantianvariable is actually Lagrantian

Second Practical Result of Second Practical Result of Decision 2Decision 2

Euler-Lagrange nature of independent Euler-Lagrange nature of independent variable means that I must calculate the variable means that I must calculate the velocities, even though they are quite small velocities, even though they are quite small (most of stellar evolution takes place on (most of stellar evolution takes place on time scales of millions of years [massive time scales of millions of years [massive stars] to billions of years [sun])stars] to billions of years [sun]) This leads to “evolutionary velocities” of This leads to “evolutionary velocities” of μμm/sm/s Courant condition for stability [Courant condition for stability [ΔΔt < (t < (ΔΔx/cx/css), ),

where cwhere css is the sound speed] for explicit is the sound speed] for explicit calculationscalculations

Sound speed in stars is several 100 km/s so that Sound speed in stars is several 100 km/s so that the calculations must be implicitthe calculations must be implicit

Explicit and Implicit Explicit and Implicit CalculationsCalculations

Have a differential equation ∂f/∂t = g(f,x)Have a differential equation ∂f/∂t = g(f,x) Independent variables x and tIndependent variables x and t Dependent variable fDependent variable f Function g may involve spatial derivatives of any order of Function g may involve spatial derivatives of any order of

ff Explicit solves for time step n+1 assuming Explicit solves for time step n+1 assuming

everything at time step n is known byeverything at time step n is known by (f(fn+1n+1 – f – fnn)/)/ΔΔt = g(ft = g(fnn,x),x) Solution is easy, but stability is constrained by Courant Solution is easy, but stability is constrained by Courant

conditioncondition Implicit solves byImplicit solves by

(f(fn+1n+1 – f – fnn)/)/ΔΔt = g(ft = g(fn+1/2n+1/2,x) where f,x) where fn+1/2 n+1/2 =1/2 (f=1/2 (fnn + f + fn+1n+1)) Solution is more complicatedSolution is more complicated Accuracy constraint is Accuracy constraint is ΔΔt < (t < (ΔΔx/v)x/v)

So how do I solve this?So how do I solve this?

Make a 2D mesh of my star with NMake a 2D mesh of my star with Nrr radial and Nradial and Nθθ angular zones angular zones

At each mesh point, I have NAt each mesh point, I have Nvv dependent variables and two dependent variables and two independent variables (x and independent variables (x and θθ))

Time is a global independent variableTime is a global independent variable Have (NHave (Nr r by Nby Nθθ by N by Nvv) values to solve for) values to solve for Have (NHave (Nr r by Nby Nθθ by N by Nvv) coupled finite ) coupled finite

difference expressions to solvedifference expressions to solve

What do finite difference What do finite difference expressions look like?expressions look like?

Assume differential equations have no Assume differential equations have no more than second order derivatives in more than second order derivatives in spacespace

Let qLet qi,ji,j denote a vector containing the denote a vector containing the dependent variables at mesh point i,j dependent variables at mesh point i,j

Finite difference expression at i,j need Finite difference expression at i,j need only involve qonly involve qi-1,ji-1,j , q , qi,j-1i,j-1 , q , qi,ji,j , q , qi,j+1i,j+1 , q , qi+1,ji+1,j

Solving this bevy of coupled Solving this bevy of coupled equationsequations

Assume I have a guess for the value of each of Assume I have a guess for the value of each of the qthe qi,ji,j

Assume the guess is sufficiently close that I can Assume the guess is sufficiently close that I can assume I get a solution by merely doing a assume I get a solution by merely doing a linear perturbation (linear perturbation (δδq) of all the variablesq) of all the variables

This yields equations of the formThis yields equations of the form AAi,ji,j δδqqi-1,ji-1,j + B + Bi,ji,j δδqqi,j-1i,j-1 + C + Ci,ji,j δδqqi,ji,j + D + Di,ji,j δδqqi,j+1i,j+1 + E + Ei,ji,j δδqqi+1,ji+1,j

+F+Fi,ji,j δδR + GR + Gi,ji,j = 0 = 0 A, B, C, D, and E are (NA, B, C, D, and E are (Nvv by N by Nvv) matrices) matrices F and G and the F and G and the δδq’s are column vectors of length Nq’s are column vectors of length Nvv δδR is a scalerR is a scaler

Surface gravitational boundary condition Surface gravitational boundary condition requires sum over all requires sum over all δδq’sq’s

Matrix GeneratedMatrix Generated

The collection of these expressions The collection of these expressions produces a band diagonal matrix in produces a band diagonal matrix in these (Nthese (Nvv by N by Nvv) blocks) blocks

Width of band is 2NWidth of band is 2Nθθ +1 [in these (N +1 [in these (Nvv by Nby Nvv) blocks]) blocks]

Last NLast Nθθ rows are full (corresponding rows are full (corresponding to surface gravitational potential to surface gravitational potential boundary condition at each latitude)boundary condition at each latitude)

SolutionSolution

Invert the matrix for the Invert the matrix for the δδq’sq’s Add to current values of the q’sAdd to current values of the q’s

Cut them all down so the largest Cut them all down so the largest perturbation is sufficiently small (whatever perturbation is sufficiently small (whatever that means)that means)

Because the equations are nonlinear, Because the equations are nonlinear, the A’s etc. contain the current values of the A’s etc. contain the current values of the q’s, so repeat the entire process the q’s, so repeat the entire process until all until all δδq’s are sufficiently small q’s are sufficiently small (Newton-Raphson or Henyey method)(Newton-Raphson or Henyey method)

StatusStatus

Capability is currently unique in some Capability is currently unique in some waysways

Being used toBeing used to Study effects of composition and angular Study effects of composition and angular

momentum transport through secular momentum transport through secular rotational instabilitiesrotational instabilities

Difficult, problems with stellar surfaceDifficult, problems with stellar surface Observational evidence indicates it occursObservational evidence indicates it occurs

In combination with pulsation periods to In combination with pulsation periods to determine interior angular momentum determine interior angular momentum distribution (within limits)distribution (within limits)