Scaling laws to quantify tidal dissipation in star-planet systems P - PowerPoint PPT Presentation

Scaling laws to quantify tidal dissipation in star-planet systems P . Auclair-Desrotour, S. Mathis, C. Le Poncin-Lafitte OHP 2015 – Twenty years of giant exoplanets

General context A revolution in Astrophysics: the discovery of new planetary systems and the characterisation of their host stars Kepler – K2 CFHT ; SPIRou CoRoT CHEOPS & TESS PLATO Stellar and planetary rotation history Orbital architecture Kepler 11 Mercury orbit Lissauer et al. (2011) Albrecht et al. (2012); Gizon et al. (2013) Bolmont et al. (2014) à Need to understand angular momentum exchanges within star-planet systems à TIDES à 2 OHP 2015 – 09/10/2015

State of the art In studies of star-planet systems, bodies are treated as point-mass objects or solids with prescriptions for tides calibrated on observations or on formation scenarii. However their complex internal structure, rotation, and magnetism impact tidal dissipation. Host star (M in M ¤ ¤ ) Planets à Need of an ab-initio physical modeling à 3 OHP 2015 – 09/10/2015

Tidal waves in stars and fluid planetary layers Excitation by each Fourier component of the tidal potential Brünt-Vaïsälä frequency 0 ω A 2 Ω N f L σ o Alfvén waves Acoustic waves Inertial waves Inertia frequency Internal gravity waves Mixed waves: Ω s and B ϕ are Ω B Magneto-Gravito-Inertial perturbations ( Ω s and B ϕ can not be B ( Ω Mathis & Remus (2013) treated as perturbations) 4 OHP 2015 – 09/10/2015

A resonant erratic tidal dissipation spectrum Dissipation spectrum by turbulent friction Forced (gravito-) inertial waves ( ) ∝ D − 1 ω E=10 -7 ( ) Q = f ω à resonant response à ν T , 2 Ω K , N Q=10 5 F M Inertial waves E.T. E.T. 2(n- Ω )/ Ω Ogilvie & Lin (2004) : the case of Jupiter Dintrans & Rieutord (2000) Ogilvie & Lin (2007) Rieutord & Valdetarro (2010) Baruteau & Rieutord (2013) 5 OHP 2015 – 09/10/2015 Guenel et al. (2015)

A reduced local model to understand tidal dissipation in fluids - Cartesian geometry - Rotating and inclined - Possible stable stratification - Viscous and thermal dissipation Control parameters: ✓ N ◆ 2 Stratification A = , Coriolis 2 Ω Ogilvie & Lin (2004) E = 2 π 2 ν Viscous force Ω L 2 , Auclair-Desrotour, Le Poncin-Lafitte, Mathis (2015) Coriolis K = 2 π 2 κ Thermal diffusivity Coriolis Ω L 2 6 OHP 2015 – 09/10/2015

Tidal hydrodynamics in the reduced local model Archimedean force Viscous friction Coriolis 1 0 � N Ek r 2 u � b = f , Dynamics ∂ T u + e z ^ u + 2 Ω L ρ r p Mass conservation Perturbation r · u = 0 . ∂ T b + A w = N di ff r 2 b , Heat transport Thermal diffusion Stratification • ✓ κ Z 8 ◆ N 2 , 0 D therm = N 2 B r 2 B > ρ d V if Z Z > > ⇣ ⌘ D visc = E forcing = ν u · r 2 u < ρ d V , ρ ( u · F ) d V , V > D therm = 0 N 2 = 0 > if V > V : Viscous friction Forcing Thermal diffusion 7 OHP 2015 – 09/10/2015

Tidal dissipation in the reduced local model X Expansion of the solution in Fourier series: u mn e i 2 π ( mX + nZ ) , u = v Influence of the perturbation Viscous diffusivity ⇣ m 2 + n 2 ⌘ ω ( n f mn � mh mn ) � n cos θ g mn i ˜ 8 ω = ω + iE ˜ u mn = n , > > > � m 2 + n 2 � ˜ ω 2 � n 2 cos 2 θ � Am 2 ˜ ω > ⇣ m 2 + n 2 ⌘ > > ω = ω + iK ˆ . > > > ω ˆ > > > > > > Inertial response Thermal diffusivity X ⇣ m 2 + n 2 ⌘ ⇣� ⌘ ζ visc = 2 π E � � � � � � u 2 � v 2 � w 2 , Viscous friction � + � + � � � � � � mn mn mn � ( m , n ) 2 Z ⇤ 2 X | b mn | 2 , ⇣ m 2 + n 2 ⌘ ζ therm = 2 π KA � 2 Thermal diffusion ( m , n ) 2 Z ⇤ 2 8 OHP 2015 – 09/10/2015

An evolving behaviour Deacrising viscosity / increasing rotation E=10 -3 E=10 -5 E=10 -4 Increasing stratification E=10 -4 , A=25 9 OHP 2015 – 09/10/2015

The four main regimes Dissipation controlled by viscosity Pr = E K a b Inertial Gravito- waves inertial waves CZ Stable Zone c d ✓ N ◆ 2 A = , 2 Ω Dissipation controlled by thermal diffusivity 10 Auclair-Desrotour, Mathis, Le Poncin-Lafitte (2015) OHP 2015 – 09/10/2015

The four main regimes Dissipation controlled by viscosity Pr = E K Viscous friction a b Inertial Gravito- waves inertial waves CZ Stable Zone c d Thermal diffusion ✓ N ◆ 2 A = , 2 Ω Dissipation controlled by thermal diffusivity 11 Auclair-Desrotour, Mathis, Le Poncin-Lafitte (2015) OHP 2015 – 09/10/2015

� � � � − − � − � � − � − � � � − − − − − − − − − − − − − � � � � − � � − � � � � − � � � � − � � − � � � − − − − The complex erratic tidal dissipation spectrum − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − E = 10 -4 , A = 0, K = 0, θ = 0 E = 10 � 4 4 2 n 2 cos 2 θ + Am 2 ⌘ ⇣ m 2 + n 2 ⌘ Am 2 K + E ⇣ l mn = . n 2 cos 2 θ + Am 2 � � log 10 � [J.kg � 1 ] � ⌘ 3 2 ⇣ ⌘ ⇣ A + cos 2 θ 2 cos 2 + A − � Ξ = 1 i . � ⇥ AK + � 2 cos 2 + A � E ⇤ 2 h − 2 in cos 2 θ + C 1 C 1 grav A − 0 − 2 n 2 cos 2 θ + Am 2 ⌘ ⇣ n 2 cos 2 θ + Am 2 ⌘ ⇣ 8 π F 2 E − H mn = ⇥ Am 2 K + � 2 n 2 cos 2 θ + Am 2 � E ⇤ 2 , − 2 m 2 n 2 � m 2 + n 2 � 2 − − − − − − − − − (49) � � � � − 4 � in cos 2 θ C 1 grav A + C 1 H bg = 4 π F 2 E � A + cos 2 θ � 2 1 ⌘ 3 8 ⇣ 2 cos 2 θ + A ⌘ ⇣ A + cos 2 θ 9 0 0.5 1 1.5 2 4 1 > > > > > > N kc ⇠ < = . ⇥ AK + � 2 cos 2 + A � E ⇤ 2 h in cos 2 θ + C 1 i > 2 > C 1 grav A > > > > : ; � à Complete characterization ! − − − − − − − − à Viscous friction 12 12 OHP 2015 – 09/10/2015 � � � � − − − − − − − − � � � � � � � � − − − − − − − − − − − − − − − −

Asymptotic scaling laws D omain A ⌧ A 11 A � A 11 a b n m p l mn / E m 2 + n 2 cos θ l mn / E A ω mn / ω mn / p p m 2 + n 2 P r � P reg r ;11 H mn / E � 1 N kc / E � 1 / 2 H mn / E � 1 N kc / A 1 / 4 E � 1 / 2 Ξ / E � 2 H bg / A � 1 E Ξ / AE � 2 H bg / E f c p n m l mn / EP � 1 l mn / E ω mn / m 2 + n 2 cos θ ω mn / A p p r m 2 + n 2 P r � P diss P r � P r ;11 N kc / A 1 / 4 E � 1 / 2 P 1 / 2 H mn / E � 1 P � 1 N kc / E � 1 / 2 H mn / E � 1 P 2 r ;11 r r r H bg / EP � 1 Ξ / E � 2 H bg / A � 1 E Ξ / AE � 2 P 2 P r ⌧ P reg r r r ;11 d e n m p l mn / AEP � 1 l mn / EP � 1 m 2 + n 2 cos θ A ω mn / ω mn / p p r r m 2 + n 2 P r ⌧ P diss P r ⌧ P r ;11 N kc / A � 1 / 2 E � 1 / 2 P 1 / 2 N kc / A 1 / 4 E � 1 / 2 P 1 / 2 H mn / A � 2 E � 1 P r H mn / A � 1 E � 1 P r r ;11 r r H bg / EP � 1 Ξ / A � 2 E � 2 P 2 H bg / A � 2 EP � 1 Ξ / AE � 2 P 2 r r r r Table 14. Scaling laws for the properties of the energy dissipated in the di erent asymptotic regimes. P diss indicates the transition zone between 13 OHP 2015 – 09/10/2015

Asymptotic scaling laws D omain A ⌧ A 11 A � A 11 a b n m p l mn / E m 2 + n 2 cos θ l mn / E A ω mn / ω mn / p p m 2 + n 2 P r � P reg r ;11 H mn / E � 1 N kc / E � 1 / 2 H mn / E � 1 N kc / A 1 / 4 E � 1 / 2 − 1 Ξ / E � 2 H bg / A � 1 E Ξ / AE � 2 H bg / E − 2 f c p n m A = 10 � 4 l mn / EP � 1 l mn / E ω mn / m 2 + n 2 cos θ ω mn / A p p − 3 r m 2 + n 2 A = 10 � 3 log 10 l 11 A = 10 � 2 P r � P diss P r � P r ;11 N kc / A 1 / 4 E � 1 / 2 P 1 / 2 − 4 H mn / E � 1 P � 1 N kc / E � 1 / 2 H mn / E � 1 P 2 A = 10 � 1 r ;11 r r r A = 10 0 H bg / EP � 1 Ξ / E � 2 H bg / A � 1 E Ξ / AE � 2 P 2 − 5 A = 10 1 P r ⌧ P reg r r r ;11 A = 10 2 d e − 6 A = 10 3 n m p l mn / AEP � 1 l mn / EP � 1 m 2 + n 2 cos θ A ω mn / ω mn / p p r r m 2 + n 2 − 7 P r ⌧ P diss P r ⌧ P r ;11 N kc / A � 1 / 2 E � 1 / 2 P 1 / 2 N kc / A 1 / 4 E � 1 / 2 P 1 / 2 H mn / A � 2 E � 1 P r H mn / A � 1 E � 1 P r Width r ;11 r r − 8 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 H bg / EP � 1 Ξ / A � 2 E � 2 P 2 H bg / A � 2 EP � 1 Ξ / AE � 2 P 2 r r r r log 10 E Table 14. Scaling laws for the properties of the energy dissipated in the di erent asymptotic regimes. P diss indicates the transition zone between 14 OHP 2015 – 09/10/2015