Basics of radiation hydrodynamics Let there be light Jiˇr´ı Krtiˇcka Masaryk University Desription of radiation Definition of specific intensity The specific intensity of radiation can be defined using an ideal apparatues (a pinhole camera). The energy collected by the detector during time ∆t in a bandwidth ∆ν is ∆E = Iν A1A2 r2 ∆t∆ν. r A1 A2 1 Intricate dependences r A1 A2 The specific intensity I(Ö, Ò, ν, t) depends on • Ö: location of the pinhole, • t: time, • Ò: orientation of the screen, • ν: frequency.    tangent spaces (Ò, ν) attached to fourdimensional spacetime manifold (Ö, t). 2 Radial dependency r A1 A2 ∆E = Iν A1A2 r2 ∆t∆ν. Here ∆Ω = A2/r2 is the solid angle subtended by A2 at the aperture. This means that the intensity does not depend on the location of the detector. Intensity does not change as the bundle of radiation moves. 3 Radiation transport equation Intensity does not change as the bundle of radiation moves over time τ: ∆Iν = Iν (Ö + Òcτ, Ò, t + τ) − Iν(Ö, Ò, t) = 0. Taylor-expanding the left-hand side and discarding terms of order τ2 and higher we obtain radiation transport equation in an empty space 1 c ∂Iν ∂t + Ò · ∇Iν = 0. Change of the intensity due to absorption over time τ: ∆Iν = −kνcτIν . Change of the intensity due to emission over time τ: ∆Iν = jνcτ. Summing all the contributions we arive at the (nonrelativistic) radiation transport equation in the form of 1 c ∂Iν ∂t + Ò · ∇Iν = jν − kνIν. 4 Radiation transport equation: the quantities 1 c ∂Iν ∂t + Ò · ∇Iν = jν − kνIν • Iν is the specific intensity, which is connected with the phase space density of photons f (Ö, Ô, t) = c2 h4ν3 Iν, where f (Ö, Ô, t) appears in the Boltzmann equation. (Note: radiation transport equation can be derived from the Boltzmann equation.) • kν is the absorption (extinction) coefficient • jν is the emission coefficient (emissivity) 5 Coupling with hydrodynamics First moment of the radiation transport equation Taking advantage of the constancy of Ò: 1 c ∂Iν ∂t + ∇ · (ÒIν) = jν − kνIν . Integrating over the angles: 1 c ∂ ∂t Iν dΩ + ∇ · ÒIν dΩ = (jν − kνIν ) dΩ. Denoting the radiation energy density Eν and the vector flux ν Eν = 1 c Iν dΩ ν = ÒIν dΩ we rewrite the first moment of the radiation transport equation as ∂Eν ∂t + ∇ · ν = (jν − kνIν ) dΩ. 6 First moment of the radiation transport equation Integrating the momentum equation ∂Eν ∂t + ∇ · ν = (jν − kνIν ) dΩ. over frequencies we obtain ∂E ∂t + ∇ · = dν (jν − kνIν) dΩ, where frequency-integrated radiation energy density is E = dν Eν and the vector flux is = dν ν. Derived equation represents the equation of energy conservation. The terms on the left-hand side represent the conservation law with energy density and energy flux. The terms on the right-hand side describe the rates of gain (due to emission) and loss (due to absorption) of radiation energy per unit of volume. 7 Second moment of the radiation transport equation Multiplying the radiation transport equation 1 c ∂Iν ∂t + ∇ · (ÒIν) = jν − kνIν by Ò and integrating over the angles: 1 c ∂ ∂t ÒIν dΩ + ∇ · ÒÒIν dΩ = (Òjν − kνÒIν ) dΩ. Denoting the vector flux ν and the pressure tensor Pν ν = ÒIν dΩ Pν = 1 c ÒÒIν dΩ and assuming isotropy of jν and kν we derive the second momentum equation 1 c ∂ ν ∂t + c∇ · Pν = −kν ν. 8 Second moment of the radiation transport equation Integrating the second momentum equation 1 c ∂ ν ∂t + c∇ · Pν = −kν ν. over frequencies wind dividing by c we obtain 1 c2 ∂ ∂t + ∇ · P = − dν kνc ν c2 , where frequency-integrated vector flux is = dν ν and the pressure tensor is P = dνPν . The radiation momentum density is /c2 and the momentum flux is P. Therefore the terms on the left-hand side represent the conservation law with momentum density and momentum flux. The right-hand side represents the momentum lost per unit time. 9 Coupling with Euler’s equaions The continuity equation remains to be the same: ∂ρ ∂t + ∇ · (ρÚ) = 0. The loss of photon momentum is the gain of momentum of matter. Therefore, the negative of the photom momentum loss rate is the radiative force that shall be included in the momentum equation ρ ∂Ú ∂t + ρÚ · ∇Ú = −∇p + ρ + 1 c dν kν ν. Similarly, the loss of radiation energy is the gain of energy of matter. As a result, the negative of the energy loss rate shall be included in the equation for energy ∂ ∂t ρǫ + ρv2 2 +∇· ρÚ ǫ + v2 2 + pÚ = ρÚ − dν (jν − kνIν ) dΩ. 10 Field criterion of thermal instability Equilibrium between heating and cooling In absence of macroscopic motion, there should be equilibrium between heating and cooling: dν (jν − kνIν ) dΩ = 0. For optically thin gas in LTE assuming external source of heat this simplifies to (Field 1965, Lepp et al. 1985) L(ρ, T) = 0. log log T ρ 2 6 4 11 Equilibrium between heating and cooling L(ρ, T) = 0 log log T ρ 2 6 4 • excitation of rotational levels of molecules and fine structure levels of atoms, strong dependence of L on T 12 Equilibrium between heating and cooling L(ρ, T) = 0 log log T ρ 2 6 4 • rotational levels of molecules and fine structure levels of atoms exctited, corresponding Boltzmann factors e−ǫ/kT ∼ 1, weak dependence of L on T 12 Equilibrium between heating and cooling L(ρ, T) = 0 log log T ρ 2 6 4 • excitation of levels of atoms and ions, strong dependence of L on T 12 Equilibrium between heating and cooling L(ρ, T) = 0 log log T ρ 2 6 4 • levels of atoms and ions excited, corresponding Boltzmann factors e−ǫ/kT ∼ 1, weak dependence of L on T 12 Equilibrium between heating and cooling L(ρ, T) = 0 log log T ρ 2 6 4 • matter is strongly ionized, excitation of inner shells of atoms, strong dependence of L on T 12 Cooling L(ρ, T) < 0 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 log log T ρ 2 6 4 13 Heating L(ρ, T) > 0 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 log log T ρ 2 6 4 14 Bubble with perturbed temperature • small perturbation of temperature and density in a bubble • bubble in a mechanical equilibrium with external environment: p ∼ ρT = const. 15 Bubble with perturbed temperature • small perturbation of temperature and density in a bubble • bubble in a mechanical equilibrium with external environment: p ∼ ρT = const. 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 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0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 log log T ρ 2 6 4 ρT = const. • increase of temperature and decrease of density ⇒ more cooling ⇒ stability 15 Bubble with perturbed temperature • small perturbation of temperature and density in a bubble • bubble in a mechanical equilibrium with external environment: p ∼ ρT = const. 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 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1111111111111 log log T ρ 2 6 4 ρT = const. • decrease of temperature and increase of density ⇒ more heating ⇒ stability 15 Bubble with perturbed temperature • small perturbation of temperature and density in a bubble • bubble in a mechanical equilibrium with external environment: p ∼ ρT = const. 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 log log T ρ 2 6 4 ρT = const. • region of stability 15 Bubble with perturbed temperature • small perturbation of temperature and density in a bubble • bubble in a mechanical equilibrium with external environment: p ∼ ρT = const. 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 log log T ρ 2 6 4 ρT = const. • increase of temperature and decrease of density ⇒ more heating ⇒ instability 15 Bubble with perturbed temperature • small perturbation of temperature and density in a bubble • bubble in a mechanical equilibrium with external environment: p ∼ ρT = const. 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 log log T ρ 2 6 4 ρT = const. • decrease of temperature and increase of density ⇒ more cooling ⇒ instability 15 Bubble with perturbed temperature • small perturbation of temperature and density in a bubble • bubble in a mechanical equilibrium with external environment: p ∼ ρT = const. 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 log log T ρ 2 6 4 ρT = const. • region of instability 15 Bubble with perturbed temperature • small perturbation of temperature and density in a bubble • bubble in a mechanical equilibrium with external environment: p ∼ ρT = const. 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 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2 6 4 ρT = const. Field criterion of thermal stability: ∂L ∂T p < 0. 15 Three phases of interstellar medium 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 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1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 log log T ρ 2 6 4 cool gas 17 Three phases of interstellar medium 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 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Three phases of interstellar medium 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 0000000000000000000000 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 1111111111111111111111 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 log log T ρ 2 6 4 very hot gas 19 Existence of solar transition region 103 104 105 106 -500 0 500 1000 1500 2000 2500 3000 3500 T[K] h [km] Solar interior solar wind fotosphere chromospherechromosphere corona transition region Tmin r = 1R⊙ 20 Radiative vs. adiabatic shocks Post-shock temperature distribution In the post-shock region, the gas can be typically cooled-down either adiabatically or radiatively. The importance of these processes can be determined from the energy equation ∂ (ρǫ) ∂t + ∇ · (ρǫÚ) = −ρ2 Λ(T) − p∇ · Ú, where Λ(T) is the cooling function. Assuming stationary spherically symmetric post-shock flow with constant flow velocity 3 2 k µ v r2 d r2 ρT dr = −ρ2 Λ(T) − ρkT µ 2v r . From continuity equation at constant speed r2 ρ =const. the temperature gradient is dT dr = − 4T 3r − 2 3 µ k ρ v Λ(T). r 21 Radiative vs. adiabatic shocks The first right-hand side term in energy equation dT dr = − 4T 3r − 2 3 µ k ρ v Λ(T) describes adiabatic cooling, while the second right-hand side term stands for radiative cooling. When adiabatic cooling dominates, from the energy equation follows that the post-shock region is large, comparable with r. On the other hand, the post-shock region is significantly thinner when radiative cooling dominates. 22 Suggested reading J. Castor: Radiation hydrodynamics D. Mihalas & B. W. Mihalas: Foundations of Radiation Hydrodynamics F. H. Shu: The physics of astrophysics: II. Hydrodynamics Y. B. Zeldovich, Y. P. Raizer: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena 23