HYDRODYNAMICS OF ROTATING STARSPhilipp Edelmann

Heidelberg Institute for Theoretical Studies, Germany

from January: Newcastle University

credit: Solar Dynamics Observatory (NASA)

credit: Solar Dynamics Observatory (NASA)

WHY SHOULD WE CONSIDER ROTATINGSTARS?

credit: Solar Dynamics Observatory (NASA)

credit: Solar Dynamics Observatory (NASA)

credit: Solar Dynamics Observatory (NASA)

IMPORTANCE OF ROTATION IN STARSchange in mechanical structureadditional instabilities and mixinginitial rotation and magnetic field of supernova progenitors

credit: Solar Dynamics Observatory (NASA)

ROTATION PROFILES IN STARSbased on A. Maeder “Physics, Formation and Evolution of Rotating Stars” (2009)

Roche approximation: treat gravity as if mass were located at centre of star

credit: Solar Dynamics Observatory (NASA)

CYLINDRICAL ROTATION is constant on cylinders of constant .

hydrostatic equilibrium:

with

can be written as with

barotropic: equipotentials and isobars coincide

Ω ϖ = r sinϑ

∇P = −ρ∇Φ + ∇12 Ω2 ϖ2

ϖ = r sinϑ

∇P = −ρ∇Ψ Ψ = Φ − 12 Ω2ϖ2

credit: Solar Dynamics Observatory (NASA)

SHELLULAR ROTATION is constant on isobars ( ).

centrifugal force cannot be derived from potentialhydrostatic equilibrium:

with baroclinic: equipotentials and isobars do not coincidedensity and temperature vary on shell

Ω P = const

∇P = −ρ∇Ψ + Ω∇Ωϖ2

Ψ = Φ − 12 Ω2ϖ2

credit: Solar Dynamics Observatory (NASA)

WHICH ONE IS REALIZED?turbulent mixing is more efficient in horizontal direction (i.e. onisobars) Zahn (1992)magnetic fields might produce models closer to .convection zones might break shellular rotationstrong rotation can lead to Rayleigh–Taylor unstable models

Ω = const.

credit: Solar Dynamics Observatory (NASA)

ROTATION IN STELLAR MODELS

credit: Solar Dynamics Observatory (NASA)

TIMESCALESfree-fall timescale: Kelvin–Helmholtz timescale: nuclear timescale:

= 27 minτff,⊙

= 2 × yrτKH,⊙ 107

= yrτnuc,⊙ 1011

= =τnuc,⊙ 103 τKH,⊙ 1015 τff,⊙

credit: Solar Dynamics Observatory (NASA)

ONE-DIMENSIONAL MODELSpurely hydrostatic models needed to bridge vast timescalesnon-rotating hydrostatic stars are sphericalhydrodynamic phenomena (e.g. convection) are treated usingprescriptions (MLT, etc.)

rotating stars are not sphericaluse isobars as coordinate instead

credit: Solar Dynamics Observatory (NASA)

DEFORMATION OF A ROTATING STAR

credit: Solar Dynamics Observatory (NASA)

STELLAR STRUCTURE=dr

dM1

4π ρr2

= −dPdM

GM4πr4

= − +dLdM

ϵnuc ϵν ϵgrav

= − min [ , ]d ln TdM

GM4πr4 ∇ad ∇rad

=drPdMP

14πr2

P ρ̄

= −dPdMP

GMP

4πr4P

fP

= − +dLP

dMPϵnuc ϵν ϵgrav

= − min [ , ]d ln TdMP

GMP

4πr4P

fP ∇ad ∇radfTfP

credit: Solar Dynamics Observatory (NASA)

INSTABILITIESrotation can cause many new types of instabilities that causeadditional mixing e.g. dynamical shear, secular shear, GSF, ABCD,…hydrostatic 1D models cannot treat these self-consistentlyusually solved by prescribing a diffusion coefficient

credit: Solar Dynamics Observatory (NASA)

EXAMPLE FOR DYNAMICAL SHEAR

Richardson number

unstable if apply in unstable zones

Ri = N 2

(∂v/∂r)2

Ri < R =ic14

D = rΔΩΔr13

credit: Solar Dynamics Observatory (NASA)

driven by thermal imbalance onisobarsstudied since Eddington (1928)self-consistent theory (inframework of shellular rotation)by Zahn (1992)leads to transport ofcomposition and angularmomentum

credit G. Meynet

MERIDIONAL CIRCULATION

credit: Solar Dynamics Observatory (NASA)

2D STELLAR MODELSneed at least two spatial dimensions to describe structure of rotatingstarESTER code built to produce self-consistent 2D steady-state solutionsof rotating stars (Espinosa Lara & Rieutord, 2013)convection zones can be confined to certain latitudesmeridional circulationno time evolution done so farhydrodynamical instabilities still need to be prescribed

credit: Solar Dynamics Observatory (NASA)

HYDRODYNAMICShydrodynamics is a first-principle verification of stellar modelslong timescales involved make it difficult to approach

credit: Solar Dynamics Observatory (NASA)

SEVEN-LEAGUE HYDRO CODEsolves the compressible Euler equations in 1-, 2-, 3-Dexplicit and implicit time integrationflux preconditioning to ensure correct behaviour at low Mach numbersother low Mach number schemes (e.g. AUSM -up)works for low and high Mach numbers on the same gridhybrid (MPI, OpenMP) parallelization (works up to 458 752 cores)several solvers for the linear system: BiCGSTAB, GMRES, Multigrid, (direct)arbitrary curvilinear meshes using a rectangular computational meshgravity solver (monopole, Multigrid)radiation in the diffusion limitgeneral equation of stategeneral nuclear reaction network

+

credit: Solar Dynamics Observatory (NASA)

DYNAMICAL SHEAR INSTABILITIEScollaborators:

Raphael Hirschi (Keele) and Cyril Georgy (Geneva) Friedrich Röpke (Heidelberg), Leonhard Horst (Heidelberg)

credit: Solar Dynamics Observatory (NASA)

ROTATIONAL INSTABILITIES AND THEIR TREATMENT IN1D CODES

instabilities: dynamical shear, secular shear, GSF, …usually treated as diffusion in SE codes

E.G. DYNAMICAL SHEAR

Richardson number

unstable if apply in unstable zones

Ri = N 2

(∂v/∂r)2

Ri < R =ic14

D = rΔΩΔr13

credit: Solar Dynamics Observatory (NASA)

energy conservation: for instabilityRi = ( ) <N 2

ϖ ∂Ω∂r

14

credit: Solar Dynamics Observatory (NASA)

THE STELLAR MODELshould be shear unstable in stellar evolution codeshould not show other instabilities at the same timeshould be on similar time scale in stellar evolution and hydro code

Geneva stellar evolution code ZAMS star, 40% crit. rotation

post core O burning phaseC/Ne shell interfaceconvectively stableRi unstable

20M⊙

credit: Solar Dynamics Observatory (NASA)

SIMULATIONS WITH SLH2D equatorial planemore than 6 hours of physical timespecial mapping of GENEC data to keep convective stability

credit: Solar Dynamics Observatory (NASA)

0:00 / 3:42

credit: Solar Dynamics Observatory (NASA)

RICHARDSON NUMBER

credit: Solar Dynamics Observatory (NASA)

ANGULAR VELOCITY

credit: Solar Dynamics Observatory (NASA)

EFFECTIVE DIFFUSION COEFFICIENT

credit: Solar Dynamics Observatory (NASA)

SOURCE TERMS

0:00 / 0:33

credit: Solar Dynamics Observatory (NASA)

CONCLUSIONSrotation is an important effect in stellar modelsmany simplifications are needed to include it in 1D models2D models are a significant improvement but far from being ready toreplace 1D modelshydrodynamics can be used to study instabilities and improveprescriptionsdynamical shear mixing might be stronger than predicted by 1Dprescription