Maurizio Salaris Astrophysics Research Institute – Liverpool John Moores University EaHS12 - Oxford The determination of mass, radius, Teff and spectral energy distribution of planet host stars are necessary to characterize the properties of the planets. Stellar ages are also critical to provide evolutionary scenarios for these planets (and ‘biological’ scenarios in case signs of life will one day be detected) Theoretical stellar evolution models are the essential tool to determine the properties of exoplanet host stars Equation of stellar evolution, inputs Boundary conditions Convection Mass loss Atomic diffusion + radiative levitation Additional mixings Equations of ‘classical’ stellar evolution models Mechanical | evolution | | Chemical evolution For each element s we have chemical reactions of the type w production reactions nh h+nk knp s l destruction reactions nd s + nj j nz z Timescale for mechanical evolution τKH≡|Ω|/L Timescale for chemical evolution τn ≡En/L In general, but for the most advanced evolutionary phases of massive stars, τKH« τn The inequality between τKH and τn means that the equations of chemical evolution can be decoupled from the other 4 equations. One can solve the mechanical part of the star at a given instant t with a given set of chemical abundances, then apply a timestep Δt, solve the equations for the chemical evolution to determine the new abundances using the values of the physical variables determined at the previous instant t. The equations of the mechanical structure are then integrated again at time t+Δt using these chemical abundances, and so on. The independent variables are the mass coordinate m and time t Unknowns are r, P, T, L, (Xs s=1, …I) At each stellar layer we have 4+I equations for 4+I unknowns Energy generation (and neutrino energy loss) coefficients, equation of state, opacities and nuclear cross sections have to be known The total stellar mass M and initial chemical composition (assumed to be uniform at the start) must be specified Additional element transport mechanisms Diffusion equation diffusion velocity ws A stellar astronomer once said: “We provide evolutionary tracks, yields and physics from the pre-main sequence through to the thermally pulsing AGB. It’s certainly wrong, but it’s freely available.” Isochrones Ingredients • Surface boundary conditions • Equation of state • Opacity • Nuclear cross sections • Convection • Mass loss • Atomic diffusion • Radiative levitation How do input physics/macro- and microscopic processes affect the theoretical stellar models of fixed initial mass and chemical composition ? • Equation of State luminosity – effective temperature - age • Opacity luminosity – effective temperature - age • Nuclear reaction rates luminosity – age – surface abundances (in some cases) • Efficiency/extension of the convective energy transport luminosity - effective temperature – surface abundances • Microscopic diffusion/radiative levitation luminosity – effective temperature – surface abundances • Mass loss luminosity - effective temperature (not always) – surface abundances (not always) • Boundary conditions effective temperature Treatment of surface boundary conditions The surface boundary of stellar model calculations is located at a layer where the diffusion approximation for the radiative energy transport starts to be applicable. This corresponds to a layer where the optical depth τ~10 Calculations of model atmospheres provide the pressure at the photosphere Ps as a function of the effective temperature Teff and surface gravity g (that depends on M and R). The surface boundary condition is ‘critical’ for stellar models with convective envelopes •The RGB based on model atmospheres shows a slightly different slope, crossing over the RGB of the models computed using the Krishna-Swamy solar T() •Differences are within ~50K The treatment of convection in stellar modelling has to be able to determine: i) when a layer becomes convectively unstable Chemical composition gradient Uniform chemical composition ii) whether convection can extend beyond its formal border; iii) the timescale of mixing; iv) the temperature gradient in the convective region; Desirable (and/or necessary) features of the chosen treatment are: Coverage of as many evolutionary phases as possible Economy of computer time + simplicity The problem with convection Turbulence is a flow regime characterized by chaotic and stochastic property changes. Richard Feynman described turbulence as "the most important unsolved problem of classical physics." Superadiabatic convection: The mixing length theory (Böhm-Vitense 1958) Almost universally used in stellar evolution codes. Simple, local, time independent model, that assumes convective elements with mean size l, of the order of their mean free path. l=α Hp mixing length Superadiabatic convection: The mixing length theory l=α Hp mixing length a b c BV58 ⅛ ½ 24 ML1 ML2 ⅛ ½ 1 2 24 16 ML3 1 2 16 α calibrated 1.0 0.6 - 1.0 2.0 The value of α affects strongly the effective temperature of stars with convective envelopes The’canonical’ calibration is based on reproducing the solar radius with a theoretical solar models (Gough & Weiss 1976) We should always keep in mind that there is a priori no reason why α should stay constant within a stellar envelope, and when considering stars of different masses and/or at different evolutionary stages BV58 formalism Run of the superadiabaticity and the ratio of the convective to the total energy flux as a function of the radial location in the outer layers of the solar convection zone. Solid lines represent the ML2 model, dashed lines the BV58 model. The dashed dotted line displays the ratio between the local pressure scale height and the geometrical distance from the top of the convective region. BV58 α=2.01 ML2 α=0.63 Salaris & Cassisi (2008) surface Sound speed relative differences: ML2 vs BV58 Bottom envelope convection surface Dashed – BV58 α=2.01 Solid – ML2 α=0.63 Superadiabatic convection: The Full Spectrum of Turbulence model The Full Spectrum of Turbulence (FST) theory (Canuto& Mazzitelli 1991, Canuto Goldman & Mazzitelli 1996) is a local theory based on an expression for the convective flux derived from a model of turbulence. Turbulent pressure (Pturb ≈ ρ vc2) is also accounted for Superadiabatic convection: The Full Spectrum of Turbulence model The turbulent scale length Λ is assumed to be equal to the harmonic mean of the distances from the top and bottom of the convective boundaries CM models and solar calibrated MLT models are not equivalent Interplay between boundary condition and mixing length calibration Boundary conditions play an important role. Solar calibrated models with different boundary conditions predict different RGB temperatures From Salaris et al. (2002) From Montalban et al. (2004) 2D hydro-simulations Hydro-calibration Previous attempts by Deupree & Varner (1980) Lydon et al (1992, 1993) Extended grid of 2D hydromodels by Ludwig, Steffen & Freytag Static envelope models based on the mixing length theory calibrate α by reproducing the entropy of the adiabatic layers below the superadiabatic region from the hydro-models. A relationship α=f(Teff,g) is produced, to be employed in stellar evolution modelling (Ludwig et al.1999) From Freytag & Salaris (1999) How do models from different authors compare ? ~200K From Salaris et al. (2002) Models from different authors, all with solar calibrated mixing length, predict different RGB temperatures Calibration on binary systems Contradicting results so far Yildiz et al (2006) find the following dependence by modelling Hyades binaries: Other authors obtain opposite results The α Centauri system is a good example for showing the uncertainties in this type of calibrations: Miglio & Montalban (2005) found a mixing length 5-10 % smaller for the more massive component, taking into account both seismic and non-seismic constraints. Yildiz (2007) finds a larger mixing length for the more massive component if the seismic constraint is neglected, whereas the opposite trend is found when the seismic constraint is taken into account RADII OF LOW MASS STARS Boyajian et al (2010) Analyses of low-mass eclipsing binary stars have unveiled a disagreement between the observations and predictions of stellar structure models. Results show that theoretical models (with solar calibrated mixing length) underestimate the radii (by up to 10-20%) and overestimate the effective temperatures of low-mass stars but yield luminosities that accord with observations. Chabrier et al. (2007) suggest that reduced convective efficiency, due to fast rotation and large magnetic field strengths, and/or to magnetic spot coverage of the radiating surface significantly affect their evolution, leading to a reduced heat flux and thus larger radii and cooler effective temperatures than for regular objects. But problems also for single stars (?) Mechanical overshooting a<0 v>0 a<0 v=0 a>0 v>0 RCC + a Hp Convective Core overshooting The H central burning lifetime is longer; The size of the He core at the end of the central H-burning phase is larger; The mean luminosity during the central He-burning phase is larger. The central He-burning lifetime is shorter; How to decrease the overshooting efficiency as the star mass decreases? Since the pressure scale height steadily increases when moving deeper and deeper inside a star (HP when r0), OV has to decrease to zero for stars with small convective cores (M1.5M, Roxburgh 92, Woo & Demarque 01) Different prescription for the “ramping” are adopted The trend of OV with mass introduces an additional degree of freedom in stellar evolution model calculations Calibration of the overshooting extension with: CMD morphology Eclipsing binaries NG2420 2 Gyr (canonical – solid) 3 Gyr oversh dashed solid 5.37 Gyr oversh (0.08 Hp) dashed [Fe/H]=-0.44 From Pietrinferni et al. (2004) Calibration of the overshooting extension with star counts An example: The case of NGC1978 in the LMC TO mass ~1.5Mo Age 2-3 Gyr Preferred value Λ~0.10 From Mucciarelli et al. (2007) Calibrations with eclipsing binaries: an example Schroeder et al. (1997) Pols et al. (1997) Ribas et al. (2000) = Prad/Pgas = 0.12 From Schroeder et al. (1997) Approximate trend of λov with mass (from eclipsing binaries) M~1.2-1.3 Mo M~1.5 Mo M ~2.0-2.5 Mo M ~3.0-4.0 Mo M ~6.0 Mo M ~10.0 Mo λov~0.0 Hp λov~0.10-0.15 Hp λov~0.20 Hp λov~0.25 Hp λov~0.30 Hp λov~0.40-0.50 Hp Miglio et al. (2007) found, for 12Boötis, M=1.42 λov~0.37 Hp M=1.37 λov~0.15 Hp The mixing scheme during core He-burning: connection between nuclear burning & convection rad d ln T κLr d ln P rad He (low k) -> C/O (high k) convective rad ad radiative He C/O induced overshooting (Castellani, Giannone, Renzini 1971) II phase: Development of a ‘partial mixing zone’ When Yc decreases below ~0.7, a ‘partial mixing’ (semiconvective) zone develops beyond the boundary of the convective core. Semiconvective region Semiconvection and HRD evolution Semiconvection increases central He-burning lifetime by a factor ~1.5 -2 Breathing Pulses (Start when Yc~0.10) Numerical artifact ?? Parameter R2=Nagb/Nhb Observations R2=0.14±0.02 Semiconv +BPs R2~0.08 Semiconv no BPs R2~0.12 Mimicking semiconvection with overshooting Caloi & Mazzitelli (1990) Sweigart (1990) Extension of mixing (by ~0.1Hp) in regions beyond the boundary of all convective regions forming within the He-rich core The edge of the convective core is let propagate with velocity High overshooting ~ 1 Hp Seismic data of pulsating White Dwarfs (Metcalfe et al. 01 – 02, GD358) can provide constraints on the internal chemical stratification: central O abundance q Model M core M tot XO 0.840.03 0.490.01 q The C/O profile O C No semiconvection No overshooting 0.56 Semiconvection + no Breathing Pulses 0.79 Semiconvection + no Breathing Pulses (by Dorman & Rood) 0.58 “Low” Overshooting 0.60 0.43 – 0.47 “High” Overshooting 0.56 0.49 – 0.53 O 0.48 - 0.52 C 041 – 0.45 O C Time dependent mixing: An example See also Eggleton (1972), Pols & Tout (2001), Langer et al. (1983), Ventura & Castellani (2005) Herwig et al. (1997) Radiative region Convective region f=0.02 vc , vo from MLT Overshooting Following Freytag et al. (1996) hydro-simulations Time dependent mixing in convective cores Ventura & Castellani (2005 – but see also, e.g., Eggleton 1972, Deng et al. 1996, Salasnich et al. 1999, Woosley et al. 2002) Approach similar to Herwig et al. (1997) Mass loss on the RGB The luminosity and effective temperature of HB stars depends on the amount of mass lost along the RGB Mass loss on the RGB Different parametrizations different results dM RGB L 4 1013 R dt gR high mass loss efficiency Origlia et al. 2007 Uncertain dependence the metallicity on intermediate mass loss efficiency Reimers low mass loss efficiency Atomic diffusion + radiative levitation Radiative levitation alters the surface abundances when <grad> is larger than the local acceleration of gravity g below the base of the convective envelope g grad 0.8M - Z=0.00017 Richard et al. 2002 [Fe/H]=-2 Predicted surface abundances at the TO Are atomic diffusion + radiative levitation really efficient ? M4 Mucciarelli, Salaris et al. (2011) In the Sun YES (helioseismology) In Pop II stars …. maybe not ….. But why ? diffusion ‘ Ad hoc’ solution proposed by Richard et al. (2005) T0 is a reference temperature, and is a free parameter This additional term adds to the counterbalancing effect of chemical element gradients Same type of turbulent transport as rotational induced mixing, when the combined effect of meridional circulation and horizontal turbulence leads to vertical diffusion (Chaboyer & Zahn 1992) 0.8M - Z=0.00017 Richard et al. 2002 ADDITIONAL TRANSPORT MECHANISMS “The H-burning front moves outward into the stable region, but preceding the Hburning region proper is a narrow region, usually thought unimportant, in which 3He burns. The main reaction is 3He (3He, 2p)4He: two nuclei become three nuclei, and the mean mass per nucleus decreases from 3 to 2. Because the molecular weight (µ) is the mean mass per nucleus, but including also the much larger abundances of H and 4He that are already there and not taking part in this reaction, this leads to a small inversion in the µ gradient. “ Eggleton et al. (2006) 1Mo solar composition Gratton et al. (2000) Free parameter Thermoaline mixing Lagarde & Charbonnel (2010) Superadiabatic convection MLT (3 + 1 free parameters) - FST (2 + 1 free parameters) Non-local Time-independent ›››› Affects depth of convective envelopes, Teff , P/rho stratification in the convective envelopes, surface abundances/nucleosynthesis, asteroseismic properties. i) Calibration of the MLT parameters (and the FST free parameter) depends on input physics, MLT formalism and boundary conditions chosen. ii) Uncertainties associated to the calibrators. iii) Different solar calibrations may provide different results in earlier or later evolutionary phases (uncertainties in boundary conditions play a role here !) “Semiconvection” + “Overshooting” Free parameters (or assumptions) related to the extension and efficiency of mixing. Instantaneous or time-dependent mixing ››››››› Affect evolutionary times (star counts), luminosities, Teff , loops in the Colour-Magnitude-Diagrams, predicted populations of variable stars in stellar populations, chemical profiles, asteroseismic properties. Empirical calibrations are needed. The calibrated ‘extended’ size of the mixed regions might be due not just to convective overshooting but also to other neglected physical processes and/or inadequacies of the input physics. How do we move forward ? Need to keep the number of equations and free parameters to a minimum (ideally no free parameters) i) Introduce additional ingredients in the MLT (or FST) formalisms (this usually increases the number of free parameters ) ? (e.g. Gough 1977, Balmforth 1992) ii) Calibrate the MLT parameter(s) (and overshooting) on 2D or 3D hydro simulations ? (e.g. Ludwig et al. 1999, Freytag & Salaris 1999) iii) Employ Reynolds stress models ? (e.g. Xiong 1978, Canuto 1992, 1993, Yang & Li 1997) … again free parameters/assumptions (Yang & Li model has 6 free parameters Efficiency of atomic diffusion + radiative levitation Affects effective temperatures luminosities, evolutionary timescales Not clear why it is not fully efficient in Pop II stars. Current modelling of a moderating ‘turbulent mixing’ are completely ‘ad hoc’ and calibrated on observations Empirical calibrations are (again) necessary Additional mixing on the RGB Affects surface chemical abundances of important elements like CNO Thermoaline mixing is the current most accepted candidate. Free parameter to be calibrated Hydro-simulations suggest that thermoaline mixing is insufficient to explain the observations RGB mass loss Affects mainly the mass and CMD location during the He-burning phase. No theory about RGB mass loss (but see recent paper by Cranmer & Saar 2011) Empirical calibrations are needed WILL WE EVER BE ABLE TO MAKE REALISTIC FULL HYDRO-SIMULATIONS OF A WHOLE STAR ? WILL FULL HYDRO-SIMULATIONS OF THE ‘IMPORTANT’ STELLAR REGIONS BE SUFFICIENT TO SOLVE THESE PROBLEMS ? WILL ASTEROSEISMOLOGY BE ABLE TO LEAD US TOWARDS A SOLUTION FOR THESE (OR AT LEAST SOME) PROBLEMS ?