GRAnada COde for the resolution of the adiabatic and non-adiabatic stellar oscillations. General scheme

Transcription

1 GRAnada COd for th rsolution of th adiabatic and non-adiabatic stllar oscillations A. Moa Instituto d Astrofísica d Andalucía, CSIC, Granada, Spain Gnral schm Equilibrium modls Standard intrior Connctin lar Equilibrium modls Modls: CESAM, Granada cod (A. Clart) or J. MCDonald cod for dwarfs and sub-dwarfs. τ Atmosphr mods: Kurucz, Eddinton, tc. Stllar oscillations computation Adiabatic and non-adiabatic quations Unno t al. (989) τ -pulsation intraction nonadiabatic quations M.A. Duprt t al. (00) Obtain adiabatic quantitis: Frquncis and infunctions Obtain non-adiabatic quantitis: φ T, η (Growth rat), δt, T δ W us th Hn rlaxation mthod dscribd in Unno t al.

2 Prturbativ quations in th intrior and th d ln Adiabatic rsolution Th complt star solvd with th sam quations (Unno t al.) ( ) ( ) V 3 V x C ω ( C A ) ( A U ) A 3 ω 3 U UA UV ( ) V [ ( ) UV ] 3 U4 3 Prturbativ quations in th intrior and th Adiabatic rsolution With th boundar conditions Cω 0 0 and 3 4 in r0 b b 3 0 α α 3 b b 0 and ( ) in rr ξr P' σ ξh ; Φ' ; 3 Φ'; 4 r r ρ r dφ' dr

3 Prturbativ quations in th intrior and th Adiabatic rsolution For radial rsolution w can us LAWE or l0 in ths quations Prturbativ quations in th intrior and th Non-adiabatic rsolution W can divid th star in two zons : ) Intrior: Main part of th star. Hr w follow th adiabatic and non-adiabatic quations showd in Unno t al (89). Assumptions:. No rotation. No mantic filds 3. Diffusion approximation for th radiativ flux 4. Frozn convction

4 d ln ( ) ( ) V 3 V x C ω V 3 υ t 5 d ln ( C A ) ( A U ) A 3 ω 3 U UA UV ( ) υ t [ ( ) UV ] 3 U4 5 υ t ( ( ) ( ) ) ( ) ad U Cω 4 ad C V ( ad ) C VC 3 V a4 V ( s ) 5 V 6 5 V 4 x C ω κ 6 ad C 3 dlnl ( ) C3 V ad C3 V ( ) ad R εad ad ( ) C 3 adv 3 C3 s ic4 5 6 dlnx ε C ε ε ω ω V dlnr 5 ( ) Prturbativ quations in th intrior and th With th boundar conditions Cω ( ) 0 3 4, 3 4 and 0 5 or ( ) ( ) in rr in r0 4 ω ω ω 3 V V ( 4 ) 4 V ( ) 4 0 V ad ad ξ P' σ ξ r h ; Φ' ; 3 Φ'; 4 ; 5 ; 6 r r ρ r dr C p dφ' δs δl L R R

5 Prturbativ quations in th intrior and th ) Atmosphr: Onl in non-adiabatic calculations. Th xtrnal par of th star, whr th photosphr is. W follow Duprt t al (0). Assumptions:. No rotation. No mantic filds 3. Plan-paralll. 4. Frozn convction 5. Radiativ quilibrium Prturbativ quations in th intrior and th ) Stllar : Assumptions: Radiativ quilibrium in th local δ T lnt δ T lnt δ lnt δτ Tmpratur distribution: T lnt T ln lnτ τ Thrmal rlaxation tim << Pulsational priod δ F ln F δ T ln F δ Monocromatic flux: F lnt T ln ( s ) ( h 5 d ) δ h ln h δ T ln h δ ln h δµ Limb darknin: h lnt T ln ln µ µ No rotation-pulsation intraction Frozn convction Sphrical smmtr at quilibrium Sphrical harmonics

6 Prturbativ quations in th intrior and th Conctin lar: Is th transition lar btwn both dscriptions. Extrnal boundar conditions with : Impos continuit at a ivn optical dpth δ P P 0 r dφ l φ 0 dr r lim τ 0 δτ τ δτ τ Non-adiabatic obsrvabls W obtain δ T T δ ϕ T ϕ δ T T ξ r ϕ R η M 0 M 0 WdM W dm And th rowth rat r r W Whr δt δε T ( F ) N ρ

7 Rsults - Non-adiabatic obsrvabls l0,,,3 l0,,,3 Rsults - Non-adiabatic obsrvabls

8 δ m - Multicolor Photomtr.5 ln 0 ( l )( l ) cos( σ t) m ε P (cosi) b l l Surfac Equilibrium distortion atmosphric modls (Kurucz 993) Rsults lnf lnt lnf ln Influnc of th variations Non-adiabatic of th local ctiv tmpratur computation δ lnbl lnt lnbl ln δ T T δ δ ( Φ/ r) cos( σ t ψt ) cos( σ t) σ Influnci of th variation of th local ctiv ravit δ r Rsults - Multicolor photomtr

9 Summar. Prsnt a cod solvin first ordr diffrntial quations of th adiabatic and non-adiabatic stllar pulsations, includin or not th -pulsation intraction: Granada with non-adiabatic CESAM McDonald Pulsational cod without adiabatic non-adiabatic Plus first ordr prturbativ rotation Summar For this Task is important to rmark:. Th adiabatic infunctions ar: a) Radial displacmnt ξ r b) Horizontal displacmnt ξ h, rlatd with th ulrian prturbation of th prssur and th ravitational potntial c) Eulrian prturbation of th ravitaitonal potntial Φ d) Drivativ of Φ

10 Summar. Th constants ar thos prscribd in Task. 3. Th cod do not r-msh th rid 4. Th stllar radius is rardd as th on ivn for th photosphr b th quilibrium modl Conclusions Fotomtría multicolor Idntificación modal Astrosismoloía no adiabática Mu útil n futuras misions spacials Mjorar los modlos d atmósfra Prspctivas futuras Intracción rotación - pulsación Intracción convcción - pulsación

11 Prturbativ quations in th intrior Adiabatic quations Prturbativ quations in th intrior and th With th sphrical smmtr approximation for th star ) Stllar intrior: Innr boundar Imposs: conditions (r 0):. Mass consrvation. Kintic momnt consrvation 3. Poisson quation dφ lφ dr r m f ( Volvr δrs, θ, ϕ0, t) fr ( r) Yl ( θ, ϕ) 4. Enr consrvation 5. Diffusion approximation for th radiativ flux iσt