Convection without the Mixing Length Parameter Windermere, September 2016 Stefano Pasetto and Mark Cropper Mullard Space Science Lab, University College London + Cesare Chiosi, Emanuela Chosi, Achim Weiss, Eva Grebel Mark Cropper – 15 September 2016 (1 of 18)
Rationale for replacing Mixing Length Theory The current approach for convection is Mixing Length Theory • [Prandtl (1925), Böhm–Vitense (1958)] The universal applicability of the MLT is unproven and requires a • calibration for each star ⇒ a self-consistent theory will be a significant advance (and overdue) The correct treatment of convection is critical for stellar models • throughout the H-R diagram ⇒ affects every aspect of stellar and galactic evolution Advances in asteroseismology have allowed the internal structure of • stars to be measured directly with increasing accuracy ⇒ allows detailed confrontation with stellar models Advent of scale and accuracy of Gaia data requires stellar models of • greater fidelity to fully utilise it e.g. location of red giant tracks depends sensitively on MLT parameter Mark Cropper – 15 September 2016 (2 of 18)
Convection Theory: stability criteria Energy transfer by convection in • the classical treatment is a linear “stability study” against non- spherical perturbations Assuming that dr is small and p star +dp star = p sur +dp sur leads to the Schwarzschild/Ledoux criterion for instability p sur +dp sur p star +dp star r+dr r sur +d r sur log ρ S i.e . convection. S: stable gradient r star +d r star ad T sur +dT sur U: unstable gradient T star +dT star U ρ 1 Star Bubble ρ 2,u ρ e p sur ρ 2,s p sur is the p star r r sur pressure at the r star T sur cred cr edit: On Onno Po Pols surface of the T star bubble P P 2 1 log P Mark Cropper – 15 September 2016 (3 of 18)
Mixing Length Theory The formulation is set in terms of: • φ rad the radiative energy flux several logarithmic φ cnv the convective energy flux d log T pressure ∇ the stellar temperature gradient with respect to pressure d log P o identifi ∇ e the element temperature gradient with respect to pressure � d ln T � � ≡ � d ln P e mean veloc With the assumption that l m ≡ Λ m h P where • l m is the mean free path of a convective element h P is the distance scale of the pressure stratification Λ m is the proportionality constant (the Mixing Length Parameter) the system of equations T 4 ⎧ 4 ac ϕ rad | cnd = κ h P ρ ∇ 3 ⎪ ⎪ T 4 ⎪ 4 ac ⎪ ϕ rad | cnd + ϕ cnv = κ h P ρ ∇ rad ⎪ 3 ⎪ ⎪ ⎨ l 2 v 2 ¯ = g δ ( ∇ − ∇ e ) m 8 h P ρ c P T √ g δ l 2 ⎪ 2 h − 3 / 2 ( ∇ − ∇ e ) 3 / 2 ⎪ = ϕ cnv m √ ⎪ P ⎪ 4 ⎪ ⎪ 6 acT 3 ∇ e −∇ ad ⎪ = v , ⎩ can be solved. κρ 2 c P l m ¯ ∇−∇ e (62) Mark Cropper – 15 September 2016 (4 of 18)
A self-consistent theory: two papers Pasetto et al (2014) MNRAS, 445, 3592 • – Paper 1 formulates the problem in the reference frame of the moving convective element – This allows the identification of a self-consistent additional constraint which can be used to close the system of equations without the external imposition of a mixing-length parameter – A comparison is made of the derived parameters (e.g., sound speed) in the Sun (where the Mixing Length Theory is calibrated) Pasetto et al (2016) MNRAS, 459, 3182 • – Paper 2 presents the first stellar models using the non-MLT treatment – Evolutionary tracks are derived and compared to MLT-derived tracks – Derived internal parameters are compared between the two theories and agreement is found to be satisfactory Mark Cropper – 15 September 2016 (5 of 18)
Self-consistent Theory: stability criterion The new treatment is in the The instability criterion now • • translates to a co-moving frame of the bubble 𝒘 criterion that ≡ v � 1 ˙ ξ e 𝛐 𝑠 co-moving coordinates i.e. the new instability criterion + is a velocity criterion that the the concept of the expansion speed of the bubble is greater than the “velocity potential” speed of the bubble in the star Mark Cropper – 15 September 2016 (6 of 18)
Relation between blob size and time the unstable expansion is in terms of hyper-geometric functions • which is quadratic in time in the leading term — unstable normalised -- stable blob size normalised 3 τ 2 = time Mark Cropper – 15 September 2016 (7 of 18)
Formulation Pasetto et al (2014) derives 6 equations in 6 unknowns: • T 4 4 ac ⎧ = ϕ rad / cnd κ h P ρ ∇ 3 ⎪ ⎪ T 4 4 ac ⎪ ϕ rad / cnd + ϕ cnv = κ h P ρ ∇ rad ⎪ ⎪ 3 ⎪ ∇−∇ e − ϕ ⎪ δ ∇ µ ¯ ⎪ v 2 ¯ = ξ e g ⎪ ⎪ 3 hP v τ + ( ∇ e +2 ∇− ϕ 2 δ ∇ µ ) ⎨ 2 δ ¯ v 2 τ ρ c P T ( ∇ − ∇ e ) ¯ = ϕ cnv h P ⎪ ⎪ 4 acT 3 ⎪ ∇ e −∇ ad τ = ⎪ ⎪ ¯ κρ 2 c P ∇−∇ e ξ 2 ⎪ ⎪ e ∇−∇ e − ϕ ⎪ ¯ δ ∇ µ g ⎪ = 2 δ ∇ µ ) ¯ ξ e χ , ⎪ ⎩ 3 hP 4 v τ + ( ∇ e +2 ∇− ϕ 2 δ ¯ The two new unknowns are: • ∇−∇ ¯ the mean size of the convective element and ξ e the mean velocity ty ¯ v Mark Cropper – 15 September 2016 (8 of 18)
Solving the system of equations After substitutions and definition of new variables, the 6 equations • reduce to the following: Y 2 ( W − η ) ( η − Y ) = 1 ¯ χ 3 τ 2 where: W ≡ ∇ rad − ∇ ad > 0 , η ≡ ∇ − ∇ ad , Y ≡ ∇ − ∇ e . χ ≡ ξ e ξ 0 1 ¯ χ χ ≡ ξ e but, recall ξ 0 3 τ 2 from previous graph, so constant τ 2 = ∝ • 3 Mark Cropper – 15 September 2016 (9 of 18)
Outcome of the reduction of dimensionality The new system of equations has a • new invariant manifold on which all the solutions live The temperature gradients • at each point in any star are located on this manifold "Theorem of Unicity”: • a relation between the 4 fundamental gradients that govern the energy transfer inside a star. Mark Cropper – 15 September 2016 (10 of 18)
Another important consequence The treatment leads to a non-hydrostatic equilibrium theory, • hence non-hydrostatic equilibrium models of atmospheres (P sur – P star )/P star time Y axis is the increasing pressure difference across the blob interface compared to pressure in the same stellar layer far from the blob normalised χ ≡ ξ e blob size ξ 0 This is a fundamental advance on the MLT where equilibrium is • assumed to be reached at the end of the bubble movement Mark Cropper – 15 September 2016 (11 of 18)
Results (1): outer convective layers 10 10 Solar Model 8 9 Bertelli et al. (2008) 6 8 black: MLT ( L = 1.68) red: this work 4 7 14 12 10 8 6 14 12 10 8 6 surface surface 1 1 0.8 0.8 0.6 0.6 expanded pressure scale 0.4 0.4 0.2 0.2 0 0 7 6.5 6 5.5 5 7 6.5 6 5.5 5 Mark Cropper – 15 September 2016 (12 of 18)
Results (2): outer convective layers 10 10 8 8 2M ⊙ RGB star log L/L ⊙ =2.598, log T eff =3.593 6 Bertelli et al. (2008) 6 4 black: MLT ( L = 1.68) 4 red: this work 9 8 7 6 5 4 9 8 7 6 5 4 surface surface 2 2 1.5 1.5 1 1 0.5 0.5 0 0 7 6 5 4 7 6 5 4 Mark Cropper – 15 September 2016 (13 of 18)
Outer convective layers: comparison For Solar model: • good agreement for convective and radiative fluxes throughout – temperature gradients are in good agreement except for surface – layers Reason: treatment incomplete at the boundary For 2M ⊙ model: • good agreement for convective fluxes – divergence to lower boundary for radiative fluxes – Reason: these solutions are not constrained to match the inner solution at the transition layer temperature gradients as for Solar model – To constrain the inner solution, need to calculate full stellar models • For these full calculations, Mixing Length Theory used for the interiors • Mark Cropper – 15 September 2016 (14 of 18)
Results 3: Stellar models dots: MLT ( L = 1.68) lines: this work Mark Cropper – 15 September 2016 (15 of 18)
Results 3: Stellar models lines: MLT ( L = 1.68) dots: this work Note: away from the well-calibrated cases, care should be exercised in which approach is the considered to be the reference. Mark Cropper – 15 September 2016 (16 of 18)
Full stellar models: Overshooting The new theory does not yet include overshooting • However, it derives the acceleration acquired by convective elements • under the action of the buoyancy force in presence of the inertia of the displaced fluid and gravity. Therefore, it is also best suited to describe convective overshooting • Extension of the atmospheric modelling to include overshooting is in • preparation (Pasetto et al 2017) Mark Cropper – 15 September 2016 (17 of 18)
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