Radiation Transfer and Radiation Hydrodynamics

Transcription

1 2018/02/02 Winter SHAO Radiation Transfer and Radiation Hydrodynamics Kengo TOMIDA ( 富田賢吾, Osaka University) Some Slides are provided by Yan-Fei Jiang (UCSB)

2 Textbooks Rybicki & Lightman, Radiative Process in Astrophysics Mihalas & Mihalas, Foundations of Radiation Hydrodynamics Shu, The Physics of Astrophysics, Volume I. Radiation Castor, Radiation Hydrodynamics

3 Heat Transport Mechanisms There are three different modes in heat transport Conduction - diffusion of thermal energy not always efficient, sometimes anisotropic relatively easy to implement Convection advection (+mixing) of thermal energy automatically included in the hydrodynamics although high resolution may be required for accuracy Radiation transport by photons (neutrinos) works in many different applications in astrophysics can be diffusion like (optically thick) or streaming (thin) can transport not only energy but also momentum can have much faster time scale than dynamics frequency dependent complicated behaviors

4 Radiation Transfer PROBLEM 1: Photons are collisionless. PROBLEM 2: Photons travel (always) at light speed. Collision less = photons properties are not determined locally. (c.f. hydrodynamics = gas properties are fully set by ρ & T) We have to solve photons properties non-locally, including their distribution not only in space but also in directions. Radiation Transfer equation (Boltzmann equation for photons) Intensity I(r, n, ν, t) [erg s -1 cm -2 Hz -1 sr -1 ] : power carried by photons per unit area, unit solid angle, unit frequency, at position r, time t, frequency ν, and traveling to the direction of n 6 (3 positions, 2 directions, 1 frequency) +1 (time) dimensions!

5 Radiation Transfer Equation Radiation transfer equation: evolution of I along each ray The simplest case free streaming: I ν t + cn I ν = 0 or: I ν + c I ν = 0 (s: distance along the ray) t s Interaction with material source and sink terms: 1 I ν c t + n I ν = j ν k ν I ν j: emissivity, k: absorption (note this term is proportional to I) Note: the source term can depend not only on the gas properties, but also radiation field (I in different directions) e.g. scattering 1 I ν c t + n I ν = j ν k ν + σ ν I ν + σ ν J ν (J ν : mean intensity)

6 Formal Solutions If the radiation time scale is much shorter than the dynamical one, we can omit the time derivative term. Then, along the ray, di ν ds = (k ν+σ ν )I ν + j ν + σ ν J ν Using α ν = k ν + σ ν, dτ ν = α ν ds, S ν = j ν+σ ν J ν α ν di ν dτ ν = I ν + S ν Formal solution: I ν (τ ν ) = I ν (0)e τ ν + 0 τ ν dτ ν S ν (τ ν)e (τ ν τ ν ) Constant source function: I ν τ ν = I ν 0 e τ ν + S ν 1 e τ ν τ ν : optical depth, mean free path: l ν = 1 Τα ν, τ ν ~l ν α ν ~1 Optically thick (τ ν 1): I ν τ ν Optically thin (τ ν 1): I ν τ ν S ν = I ν 0 (1 τ ν ) + S ν τ ν

7 Radiation Moments It is convenient to calculate some quantities integrated (averaged) over directions (angle) J ν = 1 4π න I νdω H ν = 1 4π න I νndω K ν = 1 4π න I νnndω Note: n is the normal unit vector in which the ray is traveling. Note: you can define higher order moments.

8 Radiation Moment Equations It is more convenient to modify them into physically relevant units E ν = 1 c I ν dω Radiation energy density F ν = I ν ndω P ν = 1 c I ν nndω Radiation energy flux Radiation pressure tensor Using these quantities, you can rewrite the transfer equations into moment equations (assuming sources and sinks are isotropic): E ν t + F ν= c(s ν σ ν E ν ) F ν t + c2 P ν = cσ ν F ν Note: the moment equations continue forever

9 Gray Approximation When the radiation field of your interest has a broad spectrum (continuum radiation), it is appropriate to take frequency average. This is often used when we consider thermal radiation (I ν = B ν ). 1 I t + n I = σ a B T I c E t + F = c(σ aat 4 σ E E) F t + c2 P = cσ F F Note: at 4 Stefan-Boltzmann, a= x erg cm 3 K 4 Note: opacities σ must be averaged with an adequate weighting The moment equations are similar to hydrodynamic equations (only) if we can truncate it at a small order of moments

10 Comoving frame Define the radiation quantities on the comoving frame with fluid Relatively simple, but non-conservative terms appear. Laboratory frame Define the radiation quantities on the laboratory (static) frame Transport part remains simple but the RHS becomes a mess Mixed frame Define radiation on the laboratory frame and interaction on the commoving frame, and connect accordingly getting popular Coupling to Moving Fluid When we want to couple radiation with moving fluids, we need to deal with frame transformation, since interaction between fluid and radiation is usually defined on the fluid frame. As radiation is intrinsically relativistic, we need Lorentz transform.

11 Comoving Frame The Transport equation becomes (Lowrie et al. 2000, JQSRT): And the (frequency integrated) moment equations: E t + (ue) + F + P: u = c(σ aat 4 σ E E) F t + uf + c2 P + F u = cσ F F The additional terms appear due to the motion of the frame (conservative) and velocity gradient (non-conservative). Note: The comoving frame equations are not exact, but the approximation is acceptable in non-relativistic regimes.

12 Source Terms in Hydro Equations The gas interacts with the radiation field in two ways: Absorption/emission and radiation force Mass conservation: ρ t + (ρu) = 0 Eq. of motion: ρu t + ρuu + pi = σ F c F Gas energy: e t + e + p u = c σ aat 4 σ E E + σ F c F u Radiation energy: E t + (ue) + F + P: u = c(σ aat 4 σ E E) Radiation flux: F t + uf + c2 P + F u = cσ F F Note: if frequency-dependency is included, we need to incorporate the Doppler shift effect in the frequency space.

13 e.g. Jiang et al. 2012, 2014 Mixed Frame Eq. of motion: ρu + ρuu + pi = S t m e Gas energy: + e + p u = cs t e Radiation energy: E + F = cs t e Radiation flux: 1 c 2 F t + P += S m Mom. coupling: S m = σ a+σ s c Ene. coupling: S e =σ a at 4 E + (σ a σ s ) u c [F (ue + u P)] + u c σ a at 4 E 2 [F (ue + u P)]

14 Different Levels of RHD Level 0: only local source terms of radiation heating/cooling Level 0.5: local source terms with simple optical-depth estimates --- radiation hydro in a broad sense --- Level 1: solve radiation fields but in simplified situations e.g. plane-parallel or direct radiation from a star, ionization only --- radiation hydro with some approximation --- Level 2: solve radiation fields but in an approximated way e.g. flux limited diffusion and other moment methods --- high-level radiation hydro --- Level 3a: solve radiation fields using ray-tracing e.g. variable Eddington factor, direct method (e.g. Jiang+ 2014) Level 3b: solve frequency-dependent radiation fields Level 3c: solve radiation and chemical reactions consistently Level 3d: solve atomic/molecular line transfers

15 Moment Methods The transport equation is a 6 (gray: 5) + 1(time) dimensional problem computationally expensive 1 I c t + n I = σ a B T I The moment equations are 3D (if gray) +1 (time). E t + F = c(σ aat 4 σ E E) F t + c2 P = cσ F F In general the moment equations continue forever. However, IF we can find a physically reasonable closure relation to truncate the moment equations, it can greatly reduce the costs.

16 Simplest Closure: Diffusion Approx. Closure: a relation between higher-moment and lower-moment. The simplest closure: in the optically-thick limit where radiation fields become almost isotropic. The radiation tensor becomes: P = 1 3 EI (I: unit matrix) And you can ignore the time dependency in the flux equation the radiation flux becomes a simple diffusion flux: F = c 3σ E Finally you obtain the radiation diffusion equation: E t c 3σ E = c σ aat 4 σ E E The system is reduced into a single 3D equation!

17 Levermore & Pomraning 1981 Levermore 1984 Flux Limited Diffusion The diffusion approximation is invalid in the optically thin region: it can violate the causality. The flux must be limited by light speed: F = cen, n = E E Flux limited diffusion forces this limit using a flux limiter function F = cλ σ E 2 + R E λ R =, R = 6 + 2R + R2 σe The radiation pressure tensor is modified to connect both limits: P = DE, D = 1 χ 3χ 1 I + nn, χ = λ + λ 2 R Note: although FLD satisfies causality, it is merely a diffusion approximation anyway and not accurate in optically thin regions. We will see this later. Note2: Do not use it with relativity because it still violates causality.

18 Dubroca & Feugeas 1999 Chan et al M1 Closure Scheme Is there any better option? Obviously, it should be better to solve higher order moment, while FLD solves only the zeroth moment. For this, a closure relation for higher-order moment is needed: Eddington Tensor: P=DI D = 1 χ 3χ 1 I + nn, n = F F 2 2 How to calculate χ? Assume either (note: they are equivalent) Radiation field must be isotropic on a certain frame, and is represented by its Lorentz boosting Entropy of radiation should be maximized χ = f 2 F, f = f 2 ce Note: this is Lorentz covariant and is compatible with relativity

19 1D Test FLD M1 A very simple test: radiation flux propagation FLD requires a gradient to produce energy flow. Also FLD requires short timestep because of the non-linearity in the limiter

20 Gonzalez et al D Shadow Test FLD M1 An optically thick obstacle placed in optically thick gas. Radiation comes from left, and should produce a shadow. Obviously FLD fails this test, because it cannot produce a shadow because the scheme is intrinsically isotropic. M1 closure, on the other hand, produces a reasonably good shadow, although there is non-zero diffusion.

21 2D Crossing Beam Test However, M1 closure fails when there are two or more directions in the radiation field, because it assumes the radiation field has only one directionality. We need to calculate the closure directly, or go to the direct method that solves the intensity equations directly.

22 Ray Tracing Long characteristic: cast rays between all the cells Short Characteristic: trace rays cell by cell ART: cast long rays across the grid Monte-Carlo: samples photon tracks randomly Long and Monte-Carlo: O(N 6 ), Short and ART: O(N 5 ) Note: Monte-Carlo require many photons to converge, but it is relatively easy to implement and parallelize, including complicated physics like lines and scattering

23 Variable Eddington Tensor Usually, the time scale of radiation is much faster than dynamics. So we can omit the time-dependency in the transport equations. Then we calculate the Eddington tensor D using ray tracing. Ray tracing itself is relatively cheap as it is just 1D transport equations along rays. (note: scattering makes this complicated.) Once D is given, we update the radiation quantities using the moment equation. Wait we already calculate the radiation intensities. Why do we need to use the moment equations? Can t we directly calculate the radiation moments from the intensities?

24 Advantages of Moment Methods Moment equations are 3D (with a good closure), while the transfer equations are 5D (with the gray approximation). This greatly reduces the computational costs. This is especially beneficial if we use implicit time integrators that requires inversion of a huge matrix to get a consistent solution between all the variables. Also, the moment equations can be treated similarly to the hydrodynamics, and conservation laws can be explicitly applied. On the other hand, if the hydrodynamics is fast enough (e.g. black hole accretion disks), or the reduced speed of light approximation is applicable, the direct method works.

25 (Based on Yan-Fei s slide) Shadow Test with Two Beams Sadowski et al McKinney t al Jiang et al VET can solve difficult situations with more than two directions where M1 closure fails.

26 (Based on Yan-Fei s slide) Disk Test Radiation fields from optically thick disks. Here, M1 closure produces solutions too-strongly directed, while FLD cannot reproduces the shadow. Which scheme do you think is better?

27 (Based on Yan-Fei s slide) Binary Test Radiation fields from two points sources. Again, M1 closure produces structures like a shock in radiation, which is not realistic but an artifact due to treatment like fluid. Interestingly, higher order moment is not necessarily better! Choose your scheme wisely based on their behaviors.

28 Direct Method Instead of solving radiation intensity along rays, it is possible to consider the radiation intensities as volume-averaged quantities within a cell, keeping the time derivative time. This is analogy to the Boltzmann equation. (Jiang et al. 2014) (Ohsuga & Takahashi 2016) Note: this method is recently developed, and is good especially for high-energy system because they both use explicit time integrators. If an implicit integrator is needed, it must be much more expensive.

29 Time Integrators Usually we utilize an explicit time integrator for hydrodynamic simulations and most of additional physics, because it is much simpler to implement, and yet more accurate. U n+1 U n = f(u t n ) However, we have to use small time steps for this method t < x x2 hyperbolic, t < v 2D parabolic or t < τ (where τ is typical time scale) An implicit method is unconditionally stable: U n+1 U n = f(u t n+1 ) But we need to solve the matrix to calculate the unknow U n+1, and often iterations are required if the system is non-linear.

30 Matrix Inversion (FLD case) To solve the non-linear system of radiation transfer (with FLD approx.) we use Newton-Raphson iterations. (I m showing them just to tell you how depressing they are.) The matrix is often ill behaving (not diagonally dominated) A robust sparse matrix solver is needed, but it is difficult (robustness vs scalability).

31 Caution on Implicit Methods Stable does not mean accurate It will approach the steady state in the limit of Δt But the time scale is different (exponential vs polynominal) You often need to use simple, crude, less accurate schemes Large time step does not mean fast or scalable Global matrix inversion is memory and communication intensive Some times the matrix is ill conditioned and difficult to converge It is not trivial to get good performance and scalability Use it only if The accuracy is tolerable for your problem You can get reasonable speed up with it You can develop it within a reasonable time frame

32 Some Useful Techniques Splitting the transport/diffusion term and the source term You can split them and use an implicit integrator for the source. If the stiffness comes from the source term this will work well. Applying the reduced speed of light (signal) approximation When the transport/diffusion speed is very fast, the field should reach a quasi steady state quickly. In this case, it is OK to reduce the transport/diffusion speed as long as the relation between different terms is unchanged. (e.g. Skinner & Ostriker 2013) (Caution: this does not work when there are regions with different time scales) Super Time Stepping (Alexiades et al. 1996) Some schemes special integrators allow larger time steps while keeping the explicit formulation. There are certain conditions (e.g. not work well with hyperbolic systems) but easy to implement.

33 S. JAMSTEC Note on Operator Splitting We cannot split the terms if they are both faster than dynamics!

34 Star Formation Applications Note: these are not exhaustive at all in any sense. 1D radiation hydrodynamics (thermal radiation) Larson 1969, MNRAS, 145, 271 The first simulation of star formation was already radiation hydrodynamics in 1D, implicit, with diffusion approximation Masunaga et al. 1998, Masunaga & Intusuka D RHD (VEF) simulations on low-mass star formation Vaytet et al. 2013, 2017 Systematic survey of 1D star formation models with M1 closure

35 Star Formation Applications Multi-dimensional R(M)HD simulations (thermal, diffuse) Krumholz et al. 2009, Myers et al D R(M)HD (FLD) simulations of massive star formation Whitehouse & Bate 2006, Bate 2010, D SPH RHD (FLD) simulations on low-mass star formation Commercon et al. 2010, Tomida et al. 2010, D RMHD (FLD) simulations of low-mass star formation Multi-dimensional RHD simulations (thermal, direct + diffuse) Yorke & Sonnhalter 2002, Kuiper et al. 2011, D RHD (direct+fld) of massive star formation with feedback. Rosen et al D RHD (direct+fld) of massive star formation with feedback. Harries et al D RHD (Monte-Carlo) of massive star formation with feedback.

36 Radiation Effects in Star Formation Tomida et al The temperature starts to rise when the radiation cooling becomes inefficient The outer region is heated up by radiation from the central hot core.

37 Disk Structure No Radiation With Radiation Radiation heats up the disk and stabilizes against gravity

38 Rosen et al Massive Star Formation Radiation feedback reduces the star formation efficiency 150Ms in the initial cloud ~50Ms after one free-fall time

39 Summary Radiation hydrodynamics is still a developing field. As radiation is the fundamental physics in astrophysics, it will get more and more important (and common). Although the current version only supports source functions, in the near future Athena++ will offer : Direct method for high energy systems Flux Limited Diffusion with an implicit integrator Variable Eddington Factor Method Stay tuned, and have a happy simulation life!