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The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic Region Bo-Chih Huang Department of Mathematics, National Central University, Taiwan January 22, 2016 2016 Annual Workshop on Differential Equations Joint work with John M. Hong, Shih-Wei Chou(NCU) and Chien-Chang Yen(Fu Jen Catholic University). Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region Outline 1. The hydrodynamic escape phenomenon in planetary atmospheres 2. The hydrodynamic escape model without conduction 3. Global transonic solution for initial-boundary value problem for HEP 4. The hydrodynamic escape region Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region Thermal escape process (Neutral processes) Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region Layer of the atmoesphere Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region The hydrodynamic escape model The three-dimensional inviscid hydrodynamic equations for a single-constituent atmosphere is govern by the following system of equations: ✩ ❇ t ρ � ∇ ☎ ♣ ρ u q ✏ 0 , ✬ ✬ ✬ ✫ ❇ t ♣ ρ u q � ∇ ☎ ♣♣ ρ u q ❜ u q � ∇ P ✏ ✁ GM ρ ⑤ r ⑤ 3 r , (2.1) ✬ ✬ ❇ t E � ∇ ☎ ♣♣ E � P q u q ✏ ✁ GM ρ ✬ ✪ ⑤ r ⑤ 3 ♣ u ☎ r q � q � ∇ ☎ ♣ κ ∇ T q . Here the total energy E and the state equation are given respectively by E ✏ 1 2 ρ ⑤ u ⑤ 2 � ρ e ✏ 1 P 2 ρ ⑤ u ⑤ 2 � γ ✁ 1 , P ✏ ρ RT . (2.2) ρ : gas density u : gas flow velocity P : the pressure G : the universal gravitational constant M : the mass of planet E : the total energy T : the temperature γ : the adiabatic constant with 1 ➔ γ ➔ 5 ④ 3 r : the position vector form center of the planet to the particle of gas q ✏ q ♣ r q : the heat profile κ ✏ κ ♣ T q : thermal conductivity Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region The hydrodynamic escape model We are concerned with in spherical symmetric space-time model of system (2.1) without conduction, that is, set κ ✏ 0 ✩ � ρ x 2 ✟ � ρ ux 2 ✟ ✬ ❇ t � ❇ x ✏ 0 , ✬ ✫ � ρ ux 2 ✟ � ρ u 2 x 2 � Px 2 ✟ ❇ t � ❇ x ✏ ✁ GM ρ � 2 Px , (2.3) ✬ � Ex 2 ✟ � ♣ E � P q ux 2 ✟ ✬ ✪ ✏ ✁ GM ρ u � qx 2 . ❇ t � ❇ x Let m : ✏ ρ u , U : ✏ ♣ ρ, m , E q T , using relation (2.2), system (2.3) can be written in compact form U t � f ♣ U q x ✏ h ♣ x q g ♣ x , U q , (2.4) where h ♣ x q ✏ ✁ 2 x and ✁ ✁ ✠✠ T m 2 m 2 m , 3 ✁ γ ρ � ♣ γ ✁ 1 q E , m γ E ✁ γ ✁ 1 f ♣ U q ✏ , 2 ρ 2 ρ ✁ ✁ ✠ ✠ T m , m 2 m 2 ρ � GM 2 x ρ, m γ E ✁ γ ✁ 1 � GM 2 x m ✁ xq g ♣ x , U q ✏ . ρ 2 ρ 2 Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region Motivation and Previous Works Motivation: [1] Feng Tian , Thermal Escape From Super Earth Atmospheres in the Habitable Zones of M Stars , The Astrophysical Journal 703 (2009) , pp. 905–909. [2] H. D. Sterk, S. Rostrup, and F. Tian , A fast and accurate algorithm for computing radial transonic flows , J. Comput. Appl. Math., 223 (2009) , pp. 916–928. [3] X. Zhu, D. F. Strobel, and J. T. Erwin , The density and thermal structure of Plutos atmosphere and associated escape processes and rates , Icurus, 228 (2014) , pp. 301–314. Previsous Works: [4] J. M. Hong, C.-C. Yen, B.-C. Huang , Characterization of the transonic stationary solutions of the hydrodynamic escape problem , SIAM J. Appl. Math., 74 (2014), pp. 1709–1741. Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region The hydrodynamic escape model The complete model of hydrodynamic escape problem (HEP) model is given by the following free boundary value problem ✩ U t � f ♣ U q x ✏ h ♣ x q g ♣ x , U q , ♣ x , t q P Σ ✑ r x B , x T s ✂ r 0 , ✽q , ✬ ✬ ✬ ✬ ✬ U ♣ x , 0 q ✏ U 0 ♣ x q P Ω , x P r x B , x T s , ✫ (HEP) (2.5) ρ ♣ x B , t q ✏ ρ B ♣ t q , m ♣ x B , t q ✏ m B ♣ t q , t → 0 , ✬ ✬ ✬ ✞ ✞ ✞ ✬ Σ , m ✬ ✞ ✞ ✞ ✪ Σ → 0 , Kn ♣ x , U q Σ ↕ 1 , ρ ρ where the exobase of the atmosphere x ✏ x T (as well as Σ), is need to be decided, Kn ♣ x , U q stands for the Knudsen number. Difficult to solve Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region The hydrodynamic escape model The complete model of hydrodynamic escape problem (HEP) model is given by the following free boundary value problem ✩ U t � f ♣ U q x ✏ h ♣ x q g ♣ x , U q , ♣ x , t q P Σ ✑ r x B , x T s ✂ r 0 , ✽q , ✬ ✬ ✬ ✬ ✬ U ♣ x , 0 q ✏ U 0 ♣ x q P Ω , x P r x B , x T s , ✫ (HEP) (2.5) ρ ♣ x B , t q ✏ ρ B ♣ t q , m ♣ x B , t q ✏ m B ♣ t q , t → 0 , ✬ ✬ ✬ ✞ ✞ ✞ ✬ Σ , m ✬ ✞ ✞ ✞ ✪ Σ → 0 , Kn ♣ x , U q Σ ↕ 1 , ρ ρ where the exobase of the atmosphere x ✏ x T (as well as Σ), is need to be decided, Kn ♣ x , U q stands for the Knudsen number. Difficult to solve Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region The Knudsen number The Knudsen number is useful for determining whether statistical mechanics or the continuum mechanics formulation of fluid dynamics should be used. It is defined as Kn : ✏ ℓ ④ H , the ratio of the mean free path of the molecules (the average distance traveled by a moving molecules between successive collisions) 1 ℓ ✏ ❄ , 2 τ n to the density scale height (the increase in altitude for which the atmospheric density decreases by a factor of e ) k B T H ✏ GM m ④ x 2 , of the atmosphere. Here τ ✓ 10 14 π cm 2 is the collision cross section, n is the number density, m is the mass of a molecule, T is the temperature of the gas and k B is the Boltzmann constant. Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic

The hydrodynamic escape phenomenon in planetary atmospheres The hydrodynamic escape model without conduction Global transonic solution for initial-boundary value problem for HEP The hydrodynamic escape region The Knudsen number (Cont.) Assuming the gas is ideal, that is, P ✏ ρ RT ✏ n k B T . The Knudsen number can be computed as follows: Kn ♣ x , U q ✏ ℓ GM p m GM p m γ GM p m ❄ ❄ ❄ H ✏ ✏ ✏ 2 τ x 2 ρ c 2 . 2 τ n x 2 k B T 2 τ x 2 P The hydrodynamic equations are applied well where Kn ➔ 1 so that many collisions occur over relevant length scales keeping the gas in thermal equilibrium. If Kn ➙ 1, the continuum assumption of fluid mechanics maybe no longer a good approximation since there are few collisions in this level to inhibit a molecule from escaping. Bo-Chih Huang Global Transonic Solutions of Planetary Atmospheric Escape Model in Hydrodynamic