Magnetic Induction in Hot Jupiter Atmospheres University of Toronto May 12 th, 2010
Outline An Overview of Extra solar Giant Planets Hot Jupiter General Circulation in the Atmosphere Preliminaries: Ionization, Dipole Field, Zonal Flow Derivation of the Induced Current s: Ohmic Dissipation & Magnetic Drag Magnetic Drag on Atmospheric Winds Ohmic Dissipation and the Size Problem
Extra solar Giant Planets Hot Jupiter General Circulation Part I Hot Jupiter Overview General Circulation of the Atmosphere
Extra solar Giant Planets Hot Jupiter General Circulation Extra solar Giant Planets [BS10] Many Exo Planets were discovered in the last 15 yr Most of them very close to their star and most of them Gas Giants (not all though) Methods of detection include doppler spectroscopy and transits. are Gas Giant in very close orbits. We will focus on HD 209458b
Extra solar Giant Planets Hot Jupiter General Circulation Like Cold Jupiters mostly composed of H and He Orbit Radius > 0.03 AU Orbital Period > 30 hours Skin Temperature < 2000 K Radius < 1.2 Jupiter radii Mass 0.2-2 Jupiter masses Density > 0.3 kg/m 3 The low density/large radius is not fully explained Tidal locking, extreme solar irradiance, and slow rotation lead to a unique regime of atmospheric circulation!
Extra solar Giant Planets Hot Jupiter General Circulation Horizontal Wind and Temperature Pattern Two regimes of wind pattern: Upper atmospheric flow away from sub stellar point; supersonic winds 4 km/s Strong superrotating zonal jet in the lower atmosphere, wind subsonic 1 km/s Difference to Jupiter is the slow rotation [RM10]
Extra solar Giant Planets Hot Jupiter General Circulation Vertical Temperature Profile and Zonal mean Wind Regimes of radiative forcing Extreme solar irradiance on day side Cooling on night side Radiatively forced layer above 10 Bar Inert layer below 10 Bar Deep zonal Jet extends through entire radiative layer [RM10]
Preliminaries Part II Magnetic Induction in the Atmosphere Preliminaries: Ionization, Dipole Field, Zonal Flow Derivation of the Induced Current Some Scaling Relations
Preliminaries Ionization: The Saha Equation The Saha Equation describes thethermal ionization of gas n + j n e n j n + j = ( me k B T 2π 2 ) 3 ( ) 2 Ij exp k B T. We are interested in conductivity of a gas σ given by σ = n e n e 2 πme m e A 8k B T. Ionized species are primarily alkali metals [BS10]
Preliminaries Magnetic Reynolds Number of the Atmosphere [PMR10] Conductivity scales as σ 4 T e 1 T Magnetic Reynolds Number Re m = U L η c s H σ c s T, H T Re m << 1 for most of the atmosphere, hence no dynamo generation
Preliminaries An Idealized Model Some assumptions: Dipolar magnetic field Only zonal flow Only kinematics Expected behaviour: Zonal flow (toroidal) induces equatorward current (poloidal) Current will close in lower atmosphere [BS10]
Preliminaries The Magnetic Induction Equation Magnetic Induction as we know it B t = ( v B ) + λ 2 B Decompose the magnetic field into dipole and induced field B = B dip + B ind, assuming B dip = 0 And we also need Ampere s Law B = µ 0 J
Preliminaries The Magnetic Induction Equation Magnetic Induction one step back B t = ( v B ) ( ) λ B Decompose the magnetic field into dipole and induced field B = B dip + B ind, assuming B dip = 0 And we also need Ampere s Law B = µ 0 J
Preliminaries Assuming a steady state ( B t = 0), and using Ampere s law ( v B ) ind = ( ) λ Bdip Solving for the induced current (uncurl) Jind = σ ( v B dip Φ ) Using continuity, J = 0, derive PDE for electric potential Φ σ Φ = σ ( v B ) dip There are also some simplified analytical solutions [PMR10, BS10]
Preliminaries Assuming a steady state ( B t = 0), and using Ampere s law ( v B ) dip = ( ) Jind /σ Solving for the induced current (uncurl) Jind = σ ( v B dip Φ ) Using continuity, J = 0, derive PDE for electric potential Φ σ Φ = σ ( v B ) dip There are also some simplified analytical solutions [PMR10, BS10]
Preliminaries Assuming a steady state ( B t = 0), and using Ampere s law ( v B ) dip = ) λ (µ 0 Jind Solving for the induced current (uncurl) Jind = σ ( v B dip Φ ) Using continuity, J = 0, derive PDE for electric potential Φ σ Φ = σ ( v B ) dip There are also some simplified analytical solutions [PMR10, BS10]
Preliminaries Assuming a steady state ( B t = 0), and using Ampere s law ( v B ) dip = ) λ (µ 0 Jind Solving for the induced current (uncurl) Jind = σ ( v B dip Φ ) Using continuity, J = 0, derive PDE for electric potential Φ σ Φ = σ ( v B ) dip There are also some simplified analytical solutions [PMR10, BS10]
Preliminaries Assuming a steady state ( B t = 0), and using Ampere s law ( v B ) dip = ) λ (µ 0 Jind Solving for the induced current (uncurl) Jind = σ ( v B dip Φ ) Using continuity, J = 0, derive PDE for electric potential Φ σ Φ = σ ( v B ) dip There are also some simplified analytical solutions [PMR10, BS10]
Preliminaries Assuming a steady state ( B t = 0), and using Ampere s law ( v B ) dip = ) λ (µ 0 Jind Solving for the induced current (uncurl) Jind = σ ( v B dip Φ ) Using continuity, J = 0, derive PDE for electric potential Φ σ Φ = σ ( v B ) dip There are also some simplified analytical solutions [PMR10, BS10]
Preliminaries Some Remarks Induced current is poloidal, and hence can only induce a toroidal magnetic field, which does not interact with the zonal flow Some Scaling Relations: Current scales linearly with v, B, and conductivity σ J v B J σ 4 T e 1 T
Magnetic Drag Ohmic Dissipation Part III The Effects of the Induced Currents Magnetic Drag on Zonal Winds Ohmic Dissipation: Inflating Hot Jupiter s
Magnetic Drag Ohmic Dissipation Magnetic Drag: the Lorentz Force Ion drag / bulk Lorentz Force ρ v t Drag Time scale J B τ drag ρ v J ind B dip ρ σ B dip 2 High drag on day side in upper layer; low drag at equator in lower layer [PMR10]
Magnetic Drag Ohmic Dissipation A Simple Rayleigh drag Experiment Several computed drag magnitudes were imposed on atmospehric circulation B = 3G has little effect on jet speed, wind 3 km/s B = 30G slows the jet down to 1 km/s Drag highest in upper layer Zonal momentum compensates at higher level [PMR10]
Magnetic Drag Ohmic Dissipation Ohmic Dissipation Ohmic Dissipation Power P = V J 2 σ(r) dr 3 σ v 2 B 2 Ohmic dissipation rivals insolation in the upper atmosphere and deposits energy in the inert layer [RM10]
Magnetic Drag Ohmic Dissipation The Hot Jupiter Size Problem Many EGP are too large for their mass Small amounts of energy input in the inert layer can explain the size [BS10] looked at three of them and estimated the effect of Ohmic dissipation HD 189733b is of normal size and Ohmic dissipation is negligible below 10 Bar HD 209458b is too large and Ohmic dissipation can account for it s size For Tres-4b Ohmic dissipation yoields too much heating Tres-4b is hotter and magnetic drag may reduce the zonal wind (and dissipation)
Magnetic Drag Ohmic Dissipation Summary & Conclusion High temperatures lead to ionization in the atmosphere Strong zonal winds and the magnetic field induce meridional currents Magnetic drag on induced currents affects atmospheric currents Ohmic dissipation affects temperature profile and internal structure
Magnetic Drag Ohmic Dissipation Bibliography I [BS10] Konstantin Batygin and David J. Stevenson. Inflating hot jupiters with ohmic dissipation. The Astrophysical Journal Letters, 714:L28, 2010. [PMR10] Rosalba Perna, Kristen Menou, and Emily Rauscher. Magnetic drag on hot jupiter atmospheric winds. submitted to Astrophysical Journal, 2010. [RM10] Emily Rauscher and Kristen Menou. Three dimensional modeling of hot jupiter atmospheric flows, 2010. [SCFM08] Adam P. Showman, Curtis S. Cooper, Jonathan J. Fortney, and Mark S. Marley. Atmospheric circulation of hot jupiters: Three-dimensional circulation models of hd 209458b and hd 189733b with simplified forcing. submitted to Astrophysical Journal, 2008. [SFL + 09] Adam P. Showman, Jonathan J. Fortney, Yuan Lian, Mark S. Marley, Richard S. Freedman, Heather A. Knutson, and David Charbonneau. Atmospheric circulation of hot jupiters: Coupled radiative-dynamical general circulation model simulations of hd 189733b and hd 209458b. Astrophysical Journal, 699:564 584, 2009.