Time dependent Convection vs. frozen convection approximations. Plan
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1 Time epenent onvection vs. fozen convection appoximations A. Gigahcène, M-A. Dupet an. Gaio oto Nov 006 lan Intouction Fozen onvection Appoximations Time Depenent onvection onclusion
2 Intouction etubation of enegy consevation equation: [ F ξ(. F ( ξ F] iωtδs δε. In the stella inteio, the aiative enegy tansfe can be escibe by the iffusion appoximation: 4ac F T T κ a aiative pessue constant c κ The spee of light The opacity of the matte efeences Bake & Kippenhahn, 96, 965 Ano & Osaki, 975 Saio & ox, 980 ox et al., 987 Li, 000 Löffle ffle, 00 Gigahcène ne, h thesis, Univ. Ganaa, 004.
3 Fozen convection appoximations Ignoe the euleian vaiation of the convective luminosiy L' c 0 Ignoe the lagangian vaiation of the convective luminosiy δl Ignoe the lagangian vaiation of the convective flux δf c 0 Ignoe the lagangian vaiation of the convective flux δ ( 4 π, 0 c F c δ (. 0 c F Ignoing the euleian vaiation of convective luminosity: etube Equation: Equation Wok integal (aial case: δf c, h 0 ( π λλ 4 ( δfc, λλ iσtδs δε λλ ( λλ ( δt ξ L 4π T λλ ( Lc Lc ξ W δε 0 M ( Γ Lc ξ iσtδs L * δ σ δε ε λλ ε ( δt ξ L Lc T h δ ε V. pth ' 0 L c iσtδs δε λλ λλ ( δt ξ L 4π T Lc ξ W 0 M ( Γ Lc ξ F' c, h 0 ( L * δ σ Lc δε ξ
4 Ignoing the lagangian vaiation of convective luminosity: etube Equation: Equation (esnell esnell, 990 δf c, h 0 iσ T δs λ 4 iσtδs δε λλ λλ ( δt ξ L 4π T λλ ( L c 4π λ ( λ π L ( λ δ ( L F δt T c, h δ ε δε ε ε ξ L V. p δl c 0 λ ξ ( λ iσtδs δε λλ λλ ( δt ξ L 4π T c th ξ h h F' c, h 0 ( L Wok integal (aial case: W δ δε σ ( Γ * M 0 W δ δε σ ( Γ * M 0 Ignoing the lagangian vaiation of convective flux: δf c 0 In this case etube Equation: δl L c c ξ iσtδs δε λ( λ δt L 4π T λ( λ ξ h Lc 4π λ ( λ ξ ξ Lc ξ h L Wok integal (aial case: W δ σ * M ( Γ δε 0 ξ Lc
5 Ignoing the lagangian vaiation of convective flux: δ (. 0 c F In this case etube Equation: δl L c c ξ iσtδs δε λλ ( δt L 4π T Lc ( ξ ( λλ ξ L Wok integal (aial case: W ( Γ Lc ( * δ δε ξ σ M 0 Effects on peios (LI, 99
6 Effects on instability coefficients (Li, 99 Moe effects Wane et al., 00 esnell oscillations coe
7 t X G p p G p X Time epenent convection p β Hyoynamic equations ( v 0 t (v ( vv - Φ t U ( U v εn- F v t ( X F aiative Flux v pv t : essue tenso ; p : its iagonal component. v t Gigahcène et al., 005 onvective Fluctuation Splitting the vaiables y y y v u V Mean Equations ( u 0 t u Φ t s T t V ε V t ( pg p pt ( βg β βt ( F F εn ε V ( pg p ( pg p F VV β F p T T Tenso of eynols Flux of the kinetic enegy V V of tubulence p Tubulence pessue T V
8 ε ε T s VT s V ( F F ( T ω s M. Gabiel s Theoy Fluctuation Equations ( V 0 t V 8V p p V u t τ ( T s s Γ V s s T t t τ Dissipation ate of kinetic enegy of tubulence into heat pe unit volume. Appoximations of Gabiel s Theoy 8V ( βg β βt ( βg β βt τ ( β V ε G β ( F F ( T s τ ω τ Γ ω τ The invese of the chaacteistic time of aiative enegy lost by tubulent eies. Life time of the convective elements. onvective efficiency. In the static case, assuming constant coefficients (Hp>>l!, we have solutions which ae plane waves ientical to the ML solutions. etubation of the mean equations Linea pulsation equations Equation of mass consevation δ ( δ l( l δφ δp δ σ δ g σ δp δφ aial component of the equation of momentum consevation A A pt δ δ 0 j ( jβt Tansvesal component of the equation of momentum consevation σ δ H δp δφ ViscH A pt δ δ A H
9 Equation of Enegy consevation iσtδs δε l N ( l l 4π δ L ( δ ( T / δ ( l ( p p δt L FH δε δ V δ L L L G δ H FH Amplitue of the hoizontal component of the convective flux ML etubation The main souce of uncetainty in any ML theoy of convection-pulsation inteaction is in the way to petub the mixing-length. In the esults pesente below, we use : δl l δh H ( στ δh H Time-epenent teatment Time-epenent teatment σ τ when στ << ( στ 0 when στ >> Angula pulsation fequency H Life time of the convective elements essue scale
10 aial Moes.8 M 0, α.5 Fozen onvection Time-epenent convection δl l ( στ H δh p 8 p 7 p 6 p 5 p 4 p p p p 8 p 7 p 6 p 5 p 4 p p p Figue Figue 4 λ moes.8 M 0, α.5 Fozen onvection Time-epenent convection δl l ( στ H δh p 7 p 6 p 5 p 4 p p p f g g g 4 5 g 7 g 8 g 6 p 7 p 6 p 5 p 4 p p p f g g g g 4 g 5 g 6 g 7 g 8 Figue 6 Figue 7
11 δl l ( στ H δh δ Scuti Instability Stips M.4-. M 0, α.5, λ0 p 7 p. M 0 p M 0.8 M 0.6 M 0.4 M 0 Figue 9 δl l ( στ H δh δ Scuti Instability Stips M.4-. M 0, α.5, λ f B p 6 f g 7. M 0 M 0.8 M 0.6 M 0.4 M 0 Figue
12 δl l ( στ H δh γ Do Instability moes M.5 M 0, α, λ Figue δl l ( στ H δh ompaison between γ Do Instability Stips (λ fo α an α.5 α.5.8 M 0 α.6 M 0.4 M 0 Figue 6
13 δl l ompaison between δ Scuti Instability Stip (λ, δh an γ Do Instability Stip (λ fo α.5 ( στ H. M 0 M 0.8 M 0.6 M 0.4 M 0 Figue 7 Metallicity Log(L/L o α.8, α ov Log(T eff Z0.0Z Z0.05 Z In the intenal stuctue the metallicity has the following effects: Geat sensibility of ZAMS points to Z. Shift of all the tack to high effective tempeatue with lage Z The with of the main sequence euces a little with lage Z.
14 Effects of iffeent values of metallicity on g Do Instability stip Log(L/L α.8, Z0.0 y Z0.0, l.4 Mo.9.85 Log(T ef.4 Mo.8 The γ Doaus instability stip is not influence by metallicity. We have nealy the same Instability Stip fo moels with iffeent matallicities. ompaison: Effects on fequencies Dupet et. Al, 005
15 شكرا Obigao Thank you
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