1-D TREATMENT OF ROTATION IN STELLAR EVOLUTION CODES. Ana PALACIOS LUPM/Université Montpellier II

Transcription

1 1-D TREATMENT OF ROTATION IN STELLAR EVOLUTION CODES Ana PALACIOS LUPM/Université Montpellier II

2 OUTLINE Introduction Rotation of stars : structural effects and transport processes Effect on the total potential Effect on the radiative transfert Hydrodynamical instabilities Meridional circulation Internal gravity waves Modelling rotating evolving stars Formalism for 1D structural equations of rotating stars Formalism for transport of angular momentum by meridional circulation and HD instabilities Evolution of rotating stars The transport loop in the advective/diffusive approach Some results

3 INTRODUCTION

4 INTRODUCTION Stars rotate because it is actually quite hard for them not to A.T. Potter (PhD 01) Turbulence in collapsing molecular clouds total angular momentum 0 and rotation in a priviledged direction +? Does rotation significantly affect the structure and evolution of stars Conservation of AM (more or less How? Is the effect the same for all stars? important AM loss should occur from the dense core to the formed star phase) Rotation in stars is a normal feature of their structure. Bill Saxton, NRAO/AUI/NSF

5 ROTATION AND ANGULAR MOMENTUM PROBLEM time (yr) accretion : j disk viscosity, jets, outflows : j disk braking : j magnetized winds : j contraction : Ω expansion : Ω hydrodynamics : Ω or Maeder 010

6 IMPACT OF ROTATION ON STELLAR STRUCTURE AND EVOLUTION

7 EFFECT OF ROTATION ON STELLAR STRUCTURE ROCHE MODEL Dravins et al. 01

8 EFFECT OF ROTATION ON STELLAR STRUCTURE ROCHE MODEL Two families of approaches to evaluate the distorsion of stellar shape due to rotation : Ω/Ωcrit = 0, 0., 0.4, 0.6, 0.8, 1 - MacLaurin's spheroïds star body of ct ρ and uniform Ω - Roche equipotentials Φgrav that of star with M* = Mcent Φgrav = GM r 1 Φ rot = s Ω GM 1 GM s Ω Φ = Φ rot + Φgrav = s Ω = r s + z Surface of the star = equipotential Φ = constant. Critical angular velocity beyond which the star is unbound: Ωcrit = 8G M 3 7 R p

9 VON ZEIPEL THEOREM GRAVITY-DARKENING Relation btw radiative flux and local effective gravity in a rotating star. Case of solid-body rotating star (essentially holding for differential rotation, Maeder A&A 347, 185 (1999)) ) Total luminosity on an equipotential surface SP 1) Energy flux T (Ω, ϑ ) F (Ω, ϑ ) = χ dt =χ P (Ω, ϑ ) dp dt = ρ χ g (Ω, ϑ ) dp eff 4 act χ= 3 κρ L = S F (Ω, ϑ ). n ds P solid-body rot. + hydrostatic equilibrium P = ρχ P of Michigan) Δ Ψ = 4 π G ρming Zhao Ω(University Poisson equation 3 dt ρχ = dp dt Δ Ψ (Ω, ϑ)dv P dp V dt = ρχ ( 4π G ρ Ω ) dv P V dp L = ρχ 3) L ( dt Ψ (Ω, ϑ ). n ds P dp S P P 4π G m r 1 Ω π G ρ m L F (Ω, ϑ ) = g eff (Ω, ϑ ) 4 π GM * ) ( M*= M 1 Ω π G ρ m J. Aufdenberg and NOAO/AURA/NSF ) T eff = ( L 1/4 g eff 4π G M* )

10 VON ZEIPEL THEOREM GRAVITY-DARKENING T eff = ( L 1/4 g eff 4π G M* ) Ming Zhao (University of Michigan) geff is lower at the equator due to the centrifugal force poles hotter than equator J. Aufdenberg and NOAO/AURA/NSF Actually verified by interferometric data for MS rapidly rotating stars.

11 VON ZEIPEL'S PARADOX Von Zeipel (194) : a rotating star cannot achieve simultaneously hydrostatic equilibrium and solid-body* rotation. *also true for non-conservative rotation laws energy equation in spherical coordinates and axisymmetric case in absence of energy production 1 1 ( sin θ F ) = 0. F = ( r Fr ) + θ r sin θ θ r r radiative equilibrium radiation heat transfer equation 3 4 act T F = 3 κρ For solid-body rotation, the hydrostatic equilibrium yields P P(Ψ) with Ψ = Φ + V the total potential. Combined with the EoS, one also gets T T(Ψ) and ρ ρ(ψ) 3 4 act dt F = Ψ 3 κρ d Ψ ( ) 3 3 d 4 act dt 4 act dt. F= Ψ + Ψ=0 3 κρ d Ψ d Ψ 3 κρ d Ψ ( ) ( ) not constant on equipotentials 0 inconsistency!!

12 MERIDIONAL CIRCULATION Eddington-Vogt solution to Von Zeipel's paradox (195) The Von Zeipel paradox can be solved if meridional motions contribute to the ty i s o energy transport from the cool equators to the warmer poles. isc v f o that This meridional circulation will carry away the excess energy from warm regions e l ro y e cannot be radiated and bring energy to the otherwise cooling regions. k e h rt o f t n u o ds S = a1cc. T + T u. F+ϵ o t dt ρ d ee N gravothermal contraction meridional circulation n o i at v r e s con M A e ur s n e t o sn e o d h c oa r p ap s i Th. F ) of the meridional circulation.. Source term ( = rad Light and dark areas have positive and negative divergence. Equilibrium circulation velocity. Black arrows are streamlines; the length of grey arrows is proportional to local velocity. Close to the center of the circulation cell, the period required to perform one complete revolution is 4t KH, and it increases up to 11tKH in the outer part. Talon et al. 003, J. Of Comp. Phys., 184, 44

13 MERIDIONAL CIRCULATION Busse's 1981 and Zahn's 199 solution to the von Zeipel paradox In Eddington and Vogt's approach, the flux of AM induced by meridional circulation prevents the local AM conservation need for a compensating flux. Busse (1981) and Zahn (199) give a selfconsistent solution in which the meridional circulation is a result of AM conservation in a differentially rotating viscous fluid. See Rieutord (008) for in depth analysis. From Rieutord, 008, EAS Pub. Ser. 9, 17

14 ANGULAR MOMENTUM TRANSPORT IN STARS The mechanisms redistributing angular momentum in evolving stars can be separated in two classes according to the time scales involved. Dynamical instabilities Operate on typical time scales of the order of the free-fall time scale. Associated perturbations can be considered adiabatic. Secular instabilities Operate on typical time scales of the order of the Kelvin-Helmoltz time scale. Associated perturbations operate on long timescales allowing for energy exchange need to be explicitly considered.

15 BRÜNT-VÄISÄLÄ FREQUENCY & CONVECTION Convective instability sets in if ρbub < ρsur or d bub d sur dr 0 dr We can define an oscillation frequency for the bubble : the Brunt-Väisälä frequency N The convective instability corresponds to N ad Ledoux criterion or = NT N = g ad Hp ℜ and > 0 and sets in when rad ad Schwarzschild criterion

16 BRÜNT-VÄISÄLÄ FREQUENCY IN RADIATION ZONES In radiative zones, upwards adiabatic displacement z + δz ρbuble > of a parcel of matter with maintained pressure ρlocal equilibrium, submits the parcel to restoring force because it becomes overdense with respect to its g z surroundings. The parcel oscillates at the Brunt-Väisälä frequency ρ0 N = NT N = g Hp ad after S. Talon,003, IAUS 15, 336

17 INTERACTION OF ROTATION AND CONVECTION Centrifugal force applied to a displaced fluid element modification of N² modification of the stability criterion against convection. N = N T + N μ + N Ω sin ϑ 4 gδ ϕ 1 d (Ω s ) = ad + μ + 3 sin ϑ δ HP ds s ( ) with s = r sin ϑ the distance to the rotation Rayleigh criterion stability of an angular velocity distribution For constant density, NT + Nμ = 0 and the stability against oscillation is Ω N >0 4 1 d (Ω s ) sin ϑ > 0 3 ds s j = sω away from the rotation axis

18 HYDRODYNAMICAL INSTABILITIES IN ROTATING STARS Axisymmetric baroclinic instability : Solberg-Høiland Related to Coriolis Force. Applies to fluid in cylindrical rotation Arises when the net force applied to a parcel adiabatically displaced has components only in the direction of the displacement. s s+ξ Stability criterion against axisymmetric displacements is twofold: 1) N + N Ω 0 with N Ω = 1 d ( s Ω ) s ds ) specific angular momentum j increases from pole to equator along surfaces of constant S In the non-rotating case, cond. 1) reduces to the Schwarzschild criterion rotation acts as a stabilizing effect against convection if j/ r > 0. In a compressible fluid, the buoyancy forces allow a rotating law of the form (s,z) to be sustained. Instability applying only to baroclinic stars (Ω = Ω(s,z)). For axisymmetric and adiabatic perturbations, will occur only if the Rayleigh criterion is violated

19 HYDRODYNAMICAL INSTABILITIES IN ROTATING STARS Baroclinic instabilities Instabilities appearing in stars with non-cylindrical rotation law, when iso-surfaces of P, ρ, S and g no longer coincide. Modified Solberg-Høiland criteria obtained: GSF instability thermal diffusivity weakens the stabilizing effect of the temperature gradient ν N + N 0 Ω KT T GSF stability criterion Ω ν s < NT z KT ABCD instability molecular diffusivity weakens chemical stratification ν N + ν N + N 0 Ω KT T Kμ μ ABCD stability criterion ν N + N 0 Ω KT

20 HYDRODYNAMICAL INSTABILITIES IN ROTATING STARS Shear instability Case of differential rotation Small viscosity of stellar interiors always a scale for which horizontal shear becomes turbulent For shear to occur in a star, both Reynolds and Richardson instability criteria need to be fulfilled Richardson instability criterion N 1 Ri = Ri = crit 4 (du /dz) defines the ability of the entropy gradient to stabilize a radiative region Modified Richardson instability criterion ( Ri = Γ N +N μ Γ+1 T includes the unstabilizing effect of the thermal diffusivity Reynolds instability criterion ) (du /dz ) Ri crit = 1 4 νv vl Γ= = 6 KT 6 KT νv ν Re, c defines the ability of increase of the instability comparing turbulent and microscopic viscosities

21 INTERNAL GRAVITY WAVES Courtesy of S. talon Propagate in highly stratified media Restoring force : buoyancy exist a favored direction of propagation Frequency : 0 < ν < N Group velocity : vg k Excitation mechanism : turbulent motions (convective zones edges) / internal stresses Conserve angular momentum if not dissipated IGW generation by overshooting convective plumes in multi-d simulations. The Sun He Flash in low-mass RGB star Herwig et al., 006, ApJ 64, 1057 Rogers et al., 006, ApJ 653, 765 1

22 INTERNAL GRAVITY WAVES When gravity modes are dissipated, they become internal gravity waves. Will deposit their angular momentum from generation to dissipation region Convective zone IGW If prograde (m>0) and retrograde (m<0) waves equally excited and no differential rotation NO NET AM TRANSPORT If differential rotation : prograde and retrogade speeds dω/dz INCREASES locally Shear Layer of Oscillation filtering prograde waves momentum deposition NET EXTRACTION OF AM FROM THE INTERIOR

23 IMPACT OF ROTATION Rotation radiative flux varies on equipotential surfaces strong impact on thermal structure and radiative transfert The surface of the star is no longer spherical L, Teff and geff vary with latitude modification of the surface parameters hence the evolutionary path Thermal imbalance in the stellar radiative interiors meridional currents transport of nuclides and angular momentum modification of the chemical stratification possible modification of evolutionary path

24 APPLICATION TO (1D) STELLAR EVOLUTION MODELLING

25 1D STELLAR EVOLUTION MODELLING Reff, Teff, M center r =0 photosphere r =R* From the complex to the simple picture : spherical symmetry applied

26 EQUATIONS FOR THE STELLAR STRUCTURE The stellar structure general equations are derived from conservation equations and thermodynamic principles applied to spherical symmetric stars : 1 r = m 4π r ρ continuity equation P 1 Φ r = + m m 4πr t ( Poisson equation Φ = 4πGρ Gm g = Φ = r r motion equation / Navier Stokes ) L T P = ϵn ϵν c P +δ = ϵn ϵν + ϵg m t t ρ energy equation T GmT = 4 m 4πr P L = 4 π r (F F rad = conv +F rad d lnt d lnp ( ) d lnt =( d lnp ) = medium e 4 )= 4 16 π a c G m T rad 3κP 4 ac G T m 3 κp r elt heat transfert equation F conv = ρ c P T g δ l 4 3/ Hp 1 / ( e ) mixing length theory, Kippenhahn's formulation

27 STRUCTURAL EQUATIONS FOR A ROTATING STAR Rotation will affect the stellar structure equations in several ways : * reduction of the effective gravity away from the rotation axis by centrifugal forces impact on the hydrostatic equilibrium * deformation of equipotential surfaces (no more sphericity) impact on all equations except the equation for adiabatic convective equilibrium * Von Zeipel effect radiative flux not constant anymore on equipotential surfaces impact on the radiative equilibrium equation / possible effects on convection Kippenhahn & Thomas (1970), Maeder & Meynet (1997), Zeng (00) formalisms

28 KIPPENHAHN & THOMAS' FORMALISM CASE OF CONSERVATIVE ROTATION H Conservative rotation : centrifugal acceleration derives from a potential Roche-type potential (Ψ = Ψs + Ψc) KT70 or Kopal-type (Ψ = Ψs + Ψc + Ψd) ES76 Spherical surfaces replaced by non-spherical equipotentials on which P, T and ρ are constant if the potential Ψ is conservative. a) V Ψ = 4π 3 r 3 Ψ volume enclosed by the equipotential of surface S Ψ The form of the mass conservation equation is preserved 1 rψ = m Ψ 4 π r Ψ ρ Mean value of quantities not constant on equipotential surfaces: Local effective gravity 1 g = SΨ Ψ=cst g= dψ dn dψ ds Ψ dn f = 1 SΨ Ψ =cst not constant over equipotential surfaces g 1 1 = SΨ Ψ=cst dψ dn 1 ( ) ds Ψ f ds Ψ

29 KIPPENHAHN & THOMAS' FORMALISM CASE OF CONSERVATIVE ROTATION dv Ψ = d Ψ Ψ=cst dn 1 ds Ψ = d Ψ S Ψ g dψ ( ) dψ= dv Ψ SΨ g 1 b) d Ψ = 1 SΨ g 1 dm Ψ ρ( Ψ) Straightforward modification of structure equations using a) and b) Hydrostatic equilibrium for spherical symmetric (non-rotating) star Ψ= Gm r d Ψ = Gm dm dr dr = & 4 π r²ρ r dp = ρ( Ψ ) dψ dp Gm = 4 dm 4πr Hydrostatic equilibrium for non-spherically symmetric (rotating) star b) & dp = ρ( Ψ ) dψ dp Gm Ψ = 4 fp dm Ψ 4π r Ψ with 4 4 π rψ 1 fp = G m Ψ S Ψ g 1

30 KIPPENHAHN & THOMAS' FORMALISM CASE OF CONSERVATIVE ROTATION The new set of structure equations then becomes 4 4 π rψ 1 fp = G m Ψ S Ψ g 1 1 rψ = m Ψ 4 π r Ψ ρ 4π rψ ft = SΨ ( P Gm Ψ = 4 fp mψ 4π r Ψ ) LΨ E P =ϵ δ = ϵn + ϵg ϵν mψ t t ρ T G mψ T = Ψ 4 mψ 4 π rψp ac T T 4 ac T T 4 ac T 1 T F rad = = g = g SΨ g Ψ 3 κρ n 3κρ 3 κρ mψ 3 4 ac T 1 LΨ = g SΨ g T 3κ mψ Ψ = 3κ P LΨ f T 16 π acg T 4 mψ f P 1 g g 1

31 MAEDER & MEYNET'S FORMALISM CASE OF SHELLULAR (NON-CONSERVATIVE) ROTATION For shellular rotation, Ω = Ω(r), and the centrifucal acceleration does not derive from a potential Surface of constant Ψ 1 Ψ (r, ϑ) = Φ (r ) Ω r sin ϑ = const., colatitude r, radius Gm Φ = r gravitational potential in Roche approx. Potential gradients and effective gravity in polar coordinates Ψ = Φ Ω r sin ϑ r² sin ϑ Ω Ω r r r 1 Ψ 1 Φ 1 = Ω r sin ϑ cos ϑ r² sin ϑω Ω r ϑ r ϑ r ϑ Ψ r sin ϑω Ω g eff = g eff, r = Φ Ω r sin ϑ r g eff, ϑ = Ω r sinϑ cos ϑ P = ρ g = ρ ( Ψ + r sin ϑω Ω) eff Surfaces of constant Ψ are isobaric surfaces but are not equipotentials : the star is baroclinic

32 MAEDER & MEYNET'S FORMALISM CASE OF SHELLULAR (NON-CONSERVATIVE) ROTATION For shellular rotation, Ω = Ω(r), and the centrifugal acceleration does not derive from a potential P = ρ g = ρ ( Ψ + r sin ϑω Ω) eff For Ω constant on isobars, Ω P Ψ the surfaces of constant Ψ are isobaric surfaces but are not equipotentials : the star is baroclinic For Ω = const (solid-body rotation), isobars equipotentials : the star is barotropic For shellular rotation, stellar structure equations written on isobars instead of equipotentials The same equations as those devised by Kippenhahn & Thomas apply provided that some simplifying assumptions are made.

33 MAEDER & MEYNET'S FORMALISM CASE OF SHELLULAR (NON-CONSERVATIVE) ROTATION The new set of structure equations then becomes Gm P P = 4 fp mp 4 π rp rp 1 = m P 4 π r P ρ 4π rp 1 ft = SP g g 1 ( ) Provided that all quantities depend not on the density and temperature, but on their volume averaged means between two isobars ρ, T with : 1 ρ ( 1 r sin ϑω α ) g ρ = 1 g g 1 r sin ϑ Ω α LP = ϵn ϵν + ϵg mp G mp T T = P 4 mp 4π r P P ac T T 4 ac T 1 F rad = = g SP g T 3 κρ n 3κ mp 3 4 ac 1 T g T LP = g SP κ 3 mp 4 4 π rp 1 fp = G mp S P g 1 Ω=α Ψ, α= d Ω dψ dψ g = ( 1 r sin ϑ Ωα ) dn P = 3κ P LP f T 16π acg T 4 mp f P

34 MODELLING THE TRANSPORT OVER STELLAR EVOLUTION TIMESCALES Decressin et al. 009

35 TRANSPORT OF ANGULAR MOMENTUM CASE OF SHELLULAR ROTATION Two schools have been developed concerning the resolution of the angular momentum evolution equation in 1D stellar evolution codes. Endal & Sofia (1978), Pinsonneault (1989), Heger et al. (000) Advective property of meridional flows neglected diffusion equation Zahn (199), Maeder & Zahn (1998), Mathis & Zahn (004) Meridional circulation is an advective process advection / diffusion equation The first method is computationally much easier to implement. Both method give similar results when turbulent instabilities dominate the transport of AM.

36 TRANSPORT OF ANGULAR MOMENTUM Endal & Sofia 1976, Pinsonneault et al. 1989, Heger et al. 000 The formalism developed by Endal & Sofia implements the equation for the evolution of angular momentum as a pure diffusion equation ν is a radial viscosity accounting for all the mechanisms (instabilities) supposedly transporting AM. This viscosity (identified with a diffusivity) is alternatively of unspecified nature (Denissenkov et al. 010) or appears as the sum of different diffusivities attached to different mechanisms (Heger et al. 000): Heger et al. 000 convection semi-convection dynamical shear Solberg-Høiland secular shear Eddington-Sweet circulation instability GSF instability

37 TRANSPORT OF ANGULAR MOMENTUM Endal & Sofia 1976, Pinsonneault et al. 1989, Heger et al. 000 All the diffusion coefficients also enter the equation for nuclides transport. They are derived using a linear approach as the product of a characteristic velocity times a charactaristic action length : Case of the meridional circulation Case of the secular shear instability

38 TRANSPORT OF ANGULAR MOMENTUM CASE OF SHELLULAR ROTATION Meridian flows advect angular momentum feed differential rotation For fluid's small intrinsic viscosity, Ω(r) shaped by competition between meridional circulation and turbulent stresses. The angular momentum evolution (in radiative regions) advection then obeys an advection/diffusion equation diffusion IGW (Zahn 199, Maeder & Zahn 1998, Talon & Charbonnel 003 ) νv stands for all vertical turbulence associated with hydrodynamical instabilities. U is the vertical component of the meridional circulation velocity Φ + 1 Ω s g g eff with g eff = ρ fluctuations μ fluctuations ρ r dω Θ= = 3 g dr ρ

39 TRANSPORT OF ANGULAR MOMENTUM CASE OF SHELLULAR ROTATION Stable stratification in radiative interiors anisotropic transport ( v h, Dv Dh) with horizontal gradients smaller than vertical ones. Ω is constant over an isobar : shellular rotation Scalar fields can be projected on spherical harmonics allowed Mathis & Zahn 004 Mean Fluctuation Each vectorial field is projected on a basis of vectorial spherical harmonics Poloïdal part Toroïdal part

40 TRANSPORT OF ANGULAR MOMENTUM Zahn 199, Maeder & Zahn 1998, Maeder 010 Meridional circulation velocity vector fields expanded over spherical functions for axisymmetric fields: = U l (r )P l (cos ϑ) e r + V l (r) U l >0 l >0 Development to second order = U (r ) P (cos ϑ) e r + V (r ) U Ur dp (cos ϑ) e ϑ dϑ dp l (cos ϑ) e ϑ dϑ Uϑ P (cos ϑ) = 1 (3 cos ϑ 1) P0 constant and odd degree terms not symmetrical with respect to the equator set to 0 V obtained applying the continuity equation in the anelastic approximation:

41 TRANSPORT OF ANGULAR MOMENTUM Zahn 199, Maeder & Zahn 1998, Maeder 010 Stationary continuity equation in spherical coordinates : d P (cos ϑ) 1 (r ρu r ) 1.(ρ U) = + ρsin ϑ V =0 r sin ϑ d ϑ r r ϑ ( ) Replacing the derivative of the Legendre polynomial by its expression : ρv ρv 1 (r ρ U r ) = ( 3 9cos ϑ ) = 6 P (cos ϑ) r r r r 1 d (r ρ U (r )) 6ρV = 0 r dr gives V and Uϑ provided U during stellar evolution, expansion and contraction occur that enter the expression of U r U r = U (r )P (cos ϑ) + r

42 TRANSPORT OF ANGULAR MOMENTUM Zahn 199, Maeder & Zahn 1998, Maeder 010 Derivation of U Meridional circulation derives from thermal imbalance write the equation of energy conservation ρt ds. (χ T) + ρϵ. F = h dt thermal energy flux due to horizontal turbulence transmits heat by viscous friction S D ρc T F h = D h ρt h h p h As Dh Dv, shellular rotation structure equations developed on isobars perturbative expression of variables T (P, ϑ) = T (P)+ T (P) P (cos ϑ) d S (P ) + T (P )d S (P) P (cos ϑ) T ds (P, ϑ) = T Introducing this formalism to the energy conservation equation leads to ρt ds d S d S d S d T dp d S δ =ρ T + ρ T P (cos ϑ) = ρ ϵ + ρ T P (cos ϑ) = ρ C + ρ T P (cos ϑ) grav p ρ dt dt dt dt dt dt dt and finally : ( 6 d S + D h S = L ( E Ω* + E μ ) T dt M r [ ] * )

43 TRANSPORT OF ANGULAR MOMENTUM Zahn 199, Maeder & Zahn 1998, Maeder 010 Introduce auxiliary variables describing the horizontal fluctuations of density, temperature and mean molecular weight over an isobar baroclinic balance ρ r dω Θ= = 3 g dr ρ Λ= μ μ ψ= T T related by the general EoS at constant pressure : Θ = δ ψ + ϕ Λ Considering that the entropy variation, the horizontal fluctuations of entropy and the entropy gradients can be written in the following way for stellar interiors: ds = C P [ dt dp dμ ad +Φ T P μ ] [( ϕ S = C P ( ψ+φ Λ) = C P +Φ Λ Θ δ δ ) ] C S = P ( ad Φ μ ) r Hp Φ= Aμ U with A = μ CPT ( ) ρ,μ P ϕδ ρ, αμ

44 TRANSPORT OF ANGULAR MOMENTUM Zahn 199, Maeder & Zahn 1998, Maeder 010 We finally get an expression for the meridional circulation velocity component U C P Θ C ϕ L * 6 Θ T + U (r ) P ( ad + μ ) = (E Ω+ E μ ) + D h C P T t Hp δ δ M* r δ 4Ω R g g 3 GM 3 fϵ = ϵ ( ϵ + ϵgrav )

45 TRANSPORT OF ANGULAR MOMENTUM Zahn 199, Maeder & Zahn 1998, Maeder 010 An equation describing the evolution of the mean molecular weight fluctuations is needed to complete the formalism diffusion equation with adopted formalism for scalars writing μ μ 6 + U (r ) = D h μ t r r Λ = U (r ) 6 D Λ t HP μ r h We end up with a coupled system of 5 non-linear differential equations resulting from the splitting of the 4th order in Ω of the AM transport equation + equation for mean molecular weight fluctuations evolution Λ = U (r ) 6 D Λ t HP μ r h Independent variables : Ω, Ψ/Θ, Λ, U, A= f(θ,ω)

46 VISCOSITIES / DIFFUSIVITIES There is no non-linear instability criterion associated to the baroclinic and the Solberg-Høiland instabilities mixing and transport by hydrodynamical instabilities occurs via the turbulence they generate turbulence occurs in the non-linear regime no reliable criterion to determine the amount of mixing associated with these instabilities The instability criterion derived for secular shear is non-linear (based on energy considerations) adopted criterion to describe vertical shear instability Shear instability should be the dominant one in stellar radiative interiors. Zahn 199

47 VISCOSITIES / DIFFUSIVITIES CASE OF SHELLULAR ROTATION assumption : Dv νv Effective diffusivity transport of chemicals by meridional circulation r U (r ) D eff = 30 D h Horizontal component of turbulent shear 8 Ri crit (r d Ω/dr ) D v ν v = 5 NT Nμ + K T +D h D h Vertical component of the horizontal shear r 1 d (ρ r U) U d ln r Ω D h νh = Ch 3ρ r dr d ln r (Chaboyer & Zahn 199, Zahn 199, Talon & Zahn 1997, Maeder & Zahn 1998)

48 TRANSPORT OF NUCLIDES CASE OF SHELLULAR ROTATION The association of meridional circulation and (horizontal) shear leads to a global vertical diffusion of the chemical species. The advection/diffusion equation describing the evolution of chemical concentrations reduces to a diffusion equation: nuclear microscopic processes Turbulence + merid. circ. (Chaboyer & Zahn 199, Zahn 199, Maeder & Zahn 1998)

49 TRANSPORT OF ANGULAR MOMENTUM Boundary conditions Angular momentum the convective boundaries torque from wind No differential rotation at the convective boundaries Convective regions are supposed to rotate as solid-body ( for simplicity only)

50 TRANSPORT OF ANGULAR MOMENTUM Convective regions Young 10 Myrs from the birthline L = 0.48 L Ballot et al. 004 R = 1.1 R, RBCE = 0.45 R*, MBCE = 0.37 M* Specific AM distribution in inner part of the CE of a low-mass RGB star. Brun & Palacios 009

51 MASS-LOSS : LOW-MASS STARS Rotation may affect mass loss of low-mass, low gravity and low temperature stars. Cranmer & Saar 011 : mass loss for cool stars covering a range of parameters also encompassing the radiative PMS.

52 MASS-LOSS : LOW-MASS STARS MHD fluctuations assumed as the only source to accelerate winds from cool stars. Mass loss is associated to supersonic winds emanating from : * gas pressure in a hot corona * wave pressure in a cool extended chromosphere M M cold + M hot exp( 4 M A,TR ) f* = magnetic filling factor depends on Prot / θ = dimensionless constant btw 0 and 1 FA* = flux of energy in kink / Alfvén waves in stellar photospheres

53 MASS-LOSS : HIGH-MASS STARS Maeder & Meynet 000 In a massive rotating star where the radiative flux is important : Complete Von Zeipel theorem in a differentially rotating star (shellular rotation) deviation due to baroclinicity The radiative acceleration writes Local Eddington ratio = ratio of the actual flux to the limiting local flux where Flim is the limiting flux when gtot = 0

54 MASS-LOSS : HIGH-MASS STARS Maeder & Meynet 000 Total acceleration with an Eddington factor that is taken locally and depends on Ω ΓΩ(ϑ) max where opacity is the equator where T is lower ΓΩ-limit is where ΓΩ(ϑ) = 1, and when the total gravity gtot = 0, significant effects of radiation and rotation This is the limit that should be considered for OB-type stars, LBV, RSG and WR stars Radiative wind theory applied in the context considered here, leads the following expression for the total mass loss

55 MASS-LOSS : HIGH-MASS STARS Maeder & Meynet 000 Nathan Smith / Jon Morse / NASA LBV domain

56 WIND BRAKING : YOUNG LOW-MASS STARS The wind braking is used after the disc-braking phase. Several prescriptions available to describe the torque onto the star: Kawaler 1988 Irwin & Bouvier 009 Matt et al. 01

57 STELLAR EVOLUTION OF ROTATING STARS

58 STELLAR EVOLUTION CODES WITH ROTATION CODE References Description YREC Endal & Sofia 1976, ApJ 10, 184 Pinsonneault et al. 1989, ApJ 338, 44 Chaboyer et al. 1995, ApJ 441, 865 Yang & Bi 007, ApJ 658, L67 Endal & Sofia Formalism / Magnetic fields Mainly low-mass stars STERN Heger et al. 000, ApJ 58, 368 Brott et al. 011, A&A 530, A115-A116 Endal & Sofia formalism / Magnetic fields Mainly massive stars Padova code Krivicic, 01, PhD Thesis Endal & Sofia formalism Grid of intermediate mass stars MESA Paxton et al., 013, arxiv: Endal & Sofia formalism / Magnetic fields Should treat all masses Geneva code Eggenberger et al., 008, ApSS 316, 43 Ekström, 008, PhD Thesis Zahn's formalism / Magnetic fields Mainly massive stars STAREVOL Palacios et al., 006, A&A Decressin et al., 009, A&A Zahn's formalism / IGW Mainly low and intermediate mass stars CESTAM Marques et al. 013, A&A 549, A74 Zahn's formalism Low-mass stars Hui-Bon-Hoa, 008, ApSS 316, 55 Modified Zahn's formalism Low and intermediate mass stars Denissenkov & VandenBerg 003, ApJ 593, 509 Endal & Sofia formalism / Magnetic fields Low-mass stars TGEC Modified Paczynski Code

59 EVOLUTION OF INTERNAL ROTATION PROFILE Evolution of the transport of AM and chemical species during the MS

60 EVOLUTION OF INTERNAL ROTATION PROFILE Evolution of angular velocity profile dominated by : * mass and AM losses by stellar wind * structural changes (expansion) * wind braking * structural changes (core contraction)

61 EVOLUTION OF THE MERIDIONAL CIRCULATION Evolution of meridional circulation in the massive star: Meridional circulation extracts AM outwards (counterclockwise circulation U<0) to compensate for core contraction and envelope inflation U dominated by in the interior and by shear turbukence (Gratton-Öpik subsurface

62 EVOLUTION OF THE MERIDIONAL CIRCULATION Evolution of meridional circulation in the low-mass star: Meridional circulation driven by local transport of AM everywhere. No Gratton-Öpik term

63 EVOLUTION OF THE THERMAL RELAXATION The thermal relaxation: MC driven by local extraction of AM and structural changes advects specific entropy perturbation of the thermal equilibrium temperature fluctuations Ψ S advection by MC compensated by thermal readjustment (barotropic and thermic term) Barotropic term dominated by baroclinic corrections due to differential rotation advective non-stationary baroclinic barotropic nuclear/ gravitation

64 TRANSPORT OF CHEMICALS Transport of chemicals Shear dominates transport of nuclides in the massive star Larger efficiency of the meridional circulation in the low-mass star due to the braking

65 EVOLUTIONARYWIND PATH BRAKING OF PMS ROTATING STARS Evolutionary path of a 1 solar mass star The main effect on the PMS is that of the centrifugal force that lowers the gravity and makes the star behave as a less massive star. Pinsonneault et al Eggenberger et al. 01

66 EFFECT OF ROTATION ON STELLAR PARAMETERS M = 3 M, Z = Z Eggenberger et al., 009, A&A 509, A7

67 EFFECT OF ROTATION ON STELLAR PARAMETERS When rotation is rapid enough : Age increased Larger radii Larger luminosity 0.83 M, [Fe/H] = 1.5 Palacios et al. 006 Already on the main sequence Post-MS phases should result from INTEGRATED evolution 3 M, [Fe/H] = 0 Eggenberger et al., 010

68 EFFECT OF ROTATION ON STELLAR PARAMETERS In an intermediate mass star rotation modifies the evolutionary track as Core overshoot changing with time Lower initial stellar mass 3 M, [Fe/H] = 0 Eggenberger et al., 010

69 ROTATING MASSIVE STARS Ekström et al. 01

70 ROTATING MASSIVE STARS Stellar evolution tracks for rotating and non-rotating models at different metallicities Stars are more compact at low metallicity: the surface velocity is much higher for the same angular momentum homogeneous evolution Brott et al. 011

71 THINGS NOT W ORKING THAT WELL... Palacios et al. 006 Deheuvels et al. 01 Palacios et al. 006

72 COMPARING ROTATING MODELS No rotation rotation with v km/s

73 CONCLUSIONS

74 Stellar evolution models including rotation are becoming more common Solve the transport of AM and nuclides in a self-consistent way, taking the structural feedback into account (thanks to Kippenhahn & Thomas and to Zahn's formalism). Including rotation and rotation-induced mixing have clearly improved comparison to observations. Transport of AM and nuclides solely due to meridional circulation and hydrodynamical instabilities is not sufficient to explain all the observations other transport processes need to be introduced (IGW, thermohaline mixing, magnetic fields...) The interaction between rotation and mass-loss, and rotation and convection needs to be better taken into account.