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

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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

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SLIDE 1

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

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SLIDE 2

A revolution in Astrophysics: the discovery of new planetary systems and the characterisation of their host stars

Lissauer et al. (2011) Bolmont et al. (2014)

Stellar and planetary rotation history Orbital architecture

Kepler 11 Mercury

  • rbit

CoRoT Kepler – K2 Albrecht et al. (2012); Gizon et al. (2013) CHEOPS & TESS PLATO

General context

à Need to understand angular momentum exchanges within star-planet systems à à TIDES

CFHT ; SPIRou OHP 2015 – 09/10/2015

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SLIDE 3

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.

State of the art

Host star (M in M¤

¤)

Planets à à Need of an ab-initio physical modeling

3

OHP 2015 – 09/10/2015

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SLIDE 4

N 2Ω fL Alfvén waves Internal gravity waves Acoustic waves Mixed waves: Magneto-Gravito-Inertial (Ωs and Bϕ can not be treated as perturbations) Ωs and Bϕ are perturbations

σo

ωA Inertial waves Excitation by each Fourier component

  • f the tidal potential

Brünt-Vaïsälä frequency

Inertia frequency

Mathis & Remus (2013)

Tidal waves in stars and fluid planetary layers

B B Ω (Ω

4

OHP 2015 – 09/10/2015

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SLIDE 5

Forced (gravito-) inertial waves à à resonant response

νT, K 2Ω , N FM Q=105 E=10-7 Inertial waves

E.T. E.T.

2(n-Ω)/Ω Dissipation spectrum by turbulent friction Ogilvie & Lin (2004): the case of Jupiter Dintrans & Rieutord (2000) Ogilvie & Lin (2007) Rieutord & Valdetarro (2010) Baruteau & Rieutord (2013) Guenel et al. (2015)

A resonant erratic tidal dissipation spectrum

Q = f ω

( )∝ D−1 ω ( ) 5

OHP 2015 – 09/10/2015

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SLIDE 6
  • Cartesian geometry
  • Rotating and inclined
  • Possible stable stratification
  • Viscous and thermal dissipation

6

A = ✓ N 2Ω ◆2 ,

E = 2π2ν ΩL2 , K = 2π2κ ΩL2

Control parameters:

A reduced local model to understand tidal dissipation in fluids

Stratification Coriolis Viscous force Coriolis Thermal diffusivity Coriolis

Ogilvie & Lin (2004) Auclair-Desrotour, Le Poncin-Lafitte, Mathis (2015)

OHP 2015 – 09/10/2015

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SLIDE 7

∂Tb + Aw = Ndiffr2b,

∂Tu + ez ^ u + 1 2ΩLρrp

0 NEkr2u b = f,

Tidal hydrodynamics in the reduced local model

r · u = 0.

Dynamics Mass conservation Heat transport Coriolis Viscous friction Thermal diffusion Stratification Perturbation Archimedean force

  • Eforcing =

Z

V

ρ (u · F) dV,

Dvisc = Z

V

ρ ⇣ ν u · r2u ⌘ dV,

8 > > > < > > > : Dtherm = Z

V

ρ ✓ κ N2 B r2B ◆ dV if N2 , 0 Dtherm = 0 if N2 = 0

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Viscous friction Thermal diffusion Forcing

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SLIDE 8

ζvisc = 2πE X

(m,n)2Z⇤2

⇣ m2 + n2⌘ ⇣

  • u2

mn

  • +
  • v2

mn

  • +
  • w2

mn

  • ⌘

, ζtherm = 2πKA2 X

(m,n)2Z⇤2

⇣ m2 + n2⌘ |bmn|2 ,

˜ ω = ω + iE ⇣ m2 + n2⌘ ˆ ω = ω + iK ⇣ m2 + n2⌘ .

Viscous diffusivity Thermal diffusivity Influence of the perturbation Inertial response 8 > > > > > > > > > > > > > > > umn = n i ˜ ω (n fmn mhmn) n cos θgmn m2 + n2 ˜ ω2 n2 cos2 θ Am2 ˜ ω ˆ ω ,

Tidal dissipation in the reduced local model

u = X umnei2π(mX+nZ), v

Expansion of the solution in Fourier series:

Viscous friction Thermal diffusion

OHP 2015 – 09/10/2015

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An evolving behaviour

Deacrising viscosity / increasing rotation Increasing stratification E=10-4 E=10-5 E=10-3 E=10-4, A=25

OHP 2015 – 09/10/2015

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SLIDE 10

A = ✓ N 2Ω ◆2 ,

Pr = E K

The four main regimes

Inertial waves CZ Dissipation controlled by viscosity Dissipation controlled by thermal diffusivity Gravito- inertial waves Stable Zone

Auclair-Desrotour, Mathis, Le Poncin-Lafitte (2015)

a b c d

OHP 2015 – 09/10/2015

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SLIDE 11

A = ✓ N 2Ω ◆2 ,

Pr = E K

The four main regimes

Inertial waves CZ Gravito- inertial waves Stable Zone

a b c d

OHP 2015 – 09/10/2015

11

Viscous friction Thermal diffusion Dissipation controlled by viscosity Dissipation controlled by thermal diffusivity

Auclair-Desrotour, Mathis, Le Poncin-Lafitte (2015)

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SLIDE 12

−4 −2 2 4 0.5 1 1.5 2

log10 [J.kg1]

  • E = 104

− − − − − − − −

  • Ξ = 1

2 ⇣ 2 cos2 +A ⌘ ⇣ A + cos2 θ ⌘3 ⇥AK + 2 cos2 +A E⇤2 h C1

in cos2 θ + C1 gravA

i.

− − − − − − − −

  • −

− − − − − − −

  • −

− − − − − − −

  • −

− − − − − − −

  • Nkc ⇠

8 > > > < > > > : 1 2 ⇣ 2 cos2 θ + A ⌘ ⇣ A + cos2 θ ⌘3 ⇥AK + 2 cos2 +A E⇤2 h C1

in cos2 θ + C1 gravA

i 9 > > > = > > > ; 1 4 .

− − − − − − − −

  • Hbg = 4πF2E

C1

gravA + C1 in cos2 θ

A + cos2 θ2

− − − − − − − − − − − − − −

  • −

− − − − − − − − − − − − −

  • −

− − − − − − − − − − − − − − −

  • −

− − − − − − − − − − − − − − −

  • Hmn =

8πF2E m2n2 m2 + n22 ⇣ 2n2 cos2 θ + Am2⌘ ⇣ n2 cos2 θ + Am2⌘ ⇥Am2K + 2n2 cos2 θ + Am2 E⇤2 , (49)

  • lmn =

⇣ m2 + n2⌘ Am2K + ⇣ 2n2 cos2 θ + Am2⌘ E n2 cos2 θ + Am2 .

12

à à Complete characterization !

E = 10-4 , A = 0, K = 0, θ = 0

The complex erratic tidal dissipation spectrum

OHP 2015 – 09/10/2015

12

Viscous friction

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SLIDE 13

Domain A ⌧ A11 A A11 Pr Preg

r;11

lmn / E ωmn / n p m2 + n2 cos θ lmn / E ωmn / m p m2 + n2 p A Hmn / E1 Nkc / E1/2 Hmn / E1 Nkc / A1/4E1/2 Hbg / E Ξ / E2 Hbg / A1E Ξ / AE2 Pr ⌧ Preg

r;11

Pr Pr;11 lmn / E ωmn / n p m2 + n2 cos θ Pr Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / E1P1

r

Nkc / E1/2 Hmn / E1P2

r

Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / E2 Hbg / A1E Ξ / AE2P2

r

Pr ⌧ Pr;11 lmn / AEP1

r

ωmn / n p m2 + n2 cos θ Pr ⌧ Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / A2E1Pr Nkc / A1/2E1/2P1/2

r

Hmn / A1E1Pr Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / A2E2P2

r

Hbg / A2EP1

r

Ξ / AE2P2

r

Table 14. Scaling laws for the properties of the energy dissipated in the di erent asymptotic regimes. Pdiss indicates the transition zone between

a b c d e f

Asymptotic scaling laws

OHP 2015 – 09/10/2015

13

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SLIDE 14

Domain A ⌧ A11 A A11 Pr Preg

r;11

lmn / E ωmn / n p m2 + n2 cos θ lmn / E ωmn / m p m2 + n2 p A Hmn / E1 Nkc / E1/2 Hmn / E1 Nkc / A1/4E1/2 Hbg / E Ξ / E2 Hbg / A1E Ξ / AE2 Pr ⌧ Preg

r;11

Pr Pr;11 lmn / E ωmn / n p m2 + n2 cos θ Pr Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / E1P1

r

Nkc / E1/2 Hmn / E1P2

r

Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / E2 Hbg / A1E Ξ / AE2P2

r

Pr ⌧ Pr;11 lmn / AEP1

r

ωmn / n p m2 + n2 cos θ Pr ⌧ Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / A2E1Pr Nkc / A1/2E1/2P1/2

r

Hmn / A1E1Pr Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / A2E2P2

r

Hbg / A2EP1

r

Ξ / AE2P2

r

Table 14. Scaling laws for the properties of the energy dissipated in the di erent asymptotic regimes. Pdiss indicates the transition zone between

a b c d e f

Asymptotic scaling laws

−8 −7 −6 −5 −4 −3 −2 −1 −9 −8 −7 −6 −5 −4 −3 −2

log10 l11 log10 E

A = 104 A = 103 A = 102 A = 101 A = 100 A = 101 A = 102 A = 103

Width

OHP 2015 – 09/10/2015

14

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SLIDE 15

Domain A ⌧ A11 A A11 Pr Preg

r;11

lmn / E ωmn / n p m2 + n2 cos θ lmn / E ωmn / m p m2 + n2 p A Hmn / E1 Nkc / E1/2 Hmn / E1 Nkc / A1/4E1/2 Hbg / E Ξ / E2 Hbg / A1E Ξ / AE2 Pr ⌧ Preg

r;11

Pr Pr;11 lmn / E ωmn / n p m2 + n2 cos θ Pr Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / E1P1

r

Nkc / E1/2 Hmn / E1P2

r

Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / E2 Hbg / A1E Ξ / AE2P2

r

Pr ⌧ Pr;11 lmn / AEP1

r

ωmn / n p m2 + n2 cos θ Pr ⌧ Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / A2E1Pr Nkc / A1/2E1/2P1/2

r

Hmn / A1E1Pr Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / A2E2P2

r

Hbg / A2EP1

r

Ξ / AE2P2

r

Table 14. Scaling laws for the properties of the energy dissipated in the di erent asymptotic regimes. Pdiss indicates the transition zone between

a b c d e f

Asymptotic scaling laws

−8 −6 −4 −2 2 4 0.5 1 1.5 2

log10 [J.kg1]

  • vis

the

OHP 2015 – 09/10/2015

15

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SLIDE 16

Domain A ⌧ A11 A A11 Pr Preg

r;11

lmn / E ωmn / n p m2 + n2 cos θ lmn / E ωmn / m p m2 + n2 p A Hmn / E1 Nkc / E1/2 Hmn / E1 Nkc / A1/4E1/2 Hbg / E Ξ / E2 Hbg / A1E Ξ / AE2 Pr ⌧ Preg

r;11

Pr Pr;11 lmn / E ωmn / n p m2 + n2 cos θ Pr Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / E1P1

r

Nkc / E1/2 Hmn / E1P2

r

Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / E2 Hbg / A1E Ξ / AE2P2

r

Pr ⌧ Pr;11 lmn / AEP1

r

ωmn / n p m2 + n2 cos θ Pr ⌧ Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / A2E1Pr Nkc / A1/2E1/2P1/2

r

Hmn / A1E1Pr Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / A2E2P2

r

Hbg / A2EP1

r

Ξ / AE2P2

r

Table 14. Scaling laws for the properties of the energy dissipated in the di erent asymptotic regimes. Pdiss indicates the transition zone between

a b c d e f

Asymptotic scaling laws

−8 −6 −4 −2 2 4 5 10 15 20

log10 [J.kg1]

  • vis

the

OHP 2015 – 09/10/2015

16

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SLIDE 17

Domain A ⌧ A11 A A11 Pr Preg

r;11

lmn / E ωmn / n p m2 + n2 cos θ lmn / E ωmn / m p m2 + n2 p A Hmn / E1 Nkc / E1/2 Hmn / E1 Nkc / A1/4E1/2 Hbg / E Ξ / E2 Hbg / A1E Ξ / AE2 Pr ⌧ Preg

r;11

Pr Pr;11 lmn / E ωmn / n p m2 + n2 cos θ Pr Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / E1P1

r

Nkc / E1/2 Hmn / E1P2

r

Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / E2 Hbg / A1E Ξ / AE2P2

r

Pr ⌧ Pr;11 lmn / AEP1

r

ωmn / n p m2 + n2 cos θ Pr ⌧ Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / A2E1Pr Nkc / A1/2E1/2P1/2

r

Hmn / A1E1Pr Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / A2E2P2

r

Hbg / A2EP1

r

Ξ / AE2P2

r

Table 14. Scaling laws for the properties of the energy dissipated in the di erent asymptotic regimes. Pdiss indicates the transition zone between

a b c d e f

Asymptotic scaling laws

−8 −6 −4 −2 2 4 0.5 1 1.5 2

log10 [J.kg1]

  • vis

the

OHP 2015 – 09/10/2015

17

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SLIDE 18

Domain A ⌧ A11 A A11 Pr Preg

r;11

lmn / E ωmn / n p m2 + n2 cos θ lmn / E ωmn / m p m2 + n2 p A Hmn / E1 Nkc / E1/2 Hmn / E1 Nkc / A1/4E1/2 Hbg / E Ξ / E2 Hbg / A1E Ξ / AE2 Pr ⌧ Preg

r;11

Pr Pr;11 lmn / E ωmn / n p m2 + n2 cos θ Pr Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / E1P1

r

Nkc / E1/2 Hmn / E1P2

r

Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / E2 Hbg / A1E Ξ / AE2P2

r

Pr ⌧ Pr;11 lmn / AEP1

r

ωmn / n p m2 + n2 cos θ Pr ⌧ Pdiss

r;11

lmn / EP1

r

ωmn / m p m2 + n2 p A Hmn / A2E1Pr Nkc / A1/2E1/2P1/2

r

Hmn / A1E1Pr Nkc / A1/4E1/2P1/2

r

Hbg / EP1

r

Ξ / A2E2P2

r

Hbg / A2EP1

r

Ξ / AE2P2

r

Table 14. Scaling laws for the properties of the energy dissipated in the di erent asymptotic regimes. Pdiss indicates the transition zone between

a b c d e f

Asymptotic scaling laws

−8 −6 −4 −2 2 4 5 10 15 20

log10 [J.kg1]

  • vis

the

OHP 2015 – 09/10/2015

18

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SLIDE 19

— Dependence of the spin/orbital dynamics on the resonant tidal fluid

dissipation:

à height, width, non-resonant background level

— Dependence of the characteristics of these resonances on the

physical parameters of the fluid:

à viscosity, thermal diffusivity, stratification, etc.

— Generalization to magnetic stars and planets:

à Alfvén waves, new asymptotic behaviors (in development)

— Local model = general method and qualitative results

à Need of a global model (in development)

Conclusions and prospects

For more details, see:

  • Auclair-Desrotour & al. 2014,
  • Auclair-Desrotour & al. 2015

Tidal dissipation Internal structure Spin/orbit

OHP 2015 – 09/10/2015

19

Ready to interpret

  • bservational data