Key Points: atmosphere of exoplanets and its magnetic field

 

Ohmic heating: The variations of the time varying magnetic field can induce currents in the upper atmosphere of some exoplanets, which dissipate and locally heat it up; it is called ohmic heating.

Equation for Induction and heating within a planet (which is described from ohm’s law, Ampere law, Faraday’s law) is:

where

 

Ap(x) is the steady magnetic field, the conductivity tensor σα depends on the magnetic field, and time.

Parallel conductivity is for the direction parallel to the magnetic field line.  

Pedersen conductivity is for the direction vertical to the magnetic field and parallel to the electric field. It is denoted as σp.

Hall conductivity is for the direction vertical to both the magnetic and electric fields. It is denoted as σH. In the ionosphere this conductivity is due to the drift motion of the electron (ExB drift).

Heating Rate,

Skin depth δ is defined as the depth where the current density is 1/e (about 37%) of the value at the surface; It depends on the oscillation frequency Ω  of the time-varying field and on the Pedersen conductivity in the atmosphere σp.

For constant σP, we see the maximum heating rate decreases with increasing orbital period.

Here In study, Trappist 1b and πMenc exoplanets are observed.

The differences in the electron number density lead to significant differences in the parallel, Hall and Pedersen conductivities of the upper atmospheres of the Earth, Trappist-1 b and π Men c.

Electron density of planets: Earth <Trappist 1b< πMenc.

The Pedersen conductivity is determined by the ions and electrons mixture in the atmosphere and planetary Magnetic Field.

The Pedersen conductivity of Trappist-1 b is dominated by the contribution from H2O+ and O+2 , whereas in π men c it is dominated by H+ , C+ and O+.

Due to the different XUV flux the planets receive and the π Men c atmosphere presents more electrons than Trappist-1 b, π Men c  have much stronger upper atmosphere conductivities than Trappist-1 b. therefore more efficient Ohmic heating.

Ohmic heating is expected to be important for the atmosphere only on a range of intermediate magnetic Bp values. For Trappist-1 b it matters if Bp ∈ [0.08, 0.2]G, and in the case of π Men c if BP ∈ [0.03, 0.1]G.

Source: https://arxiv.org/abs/2412.14072#:~:text=Indeed%2C%20close%2Din%20exoplanets%20are,it%20up%20through%20Ohmic%20heating.