Ray-Tracing and Flux-Limited-Diffusion for simulating Stellar Radiation Feedback Rolf Kuiper 1,2 H. Klahr 1, H. Beuther 1, Th. Henning 1, C. Dullemond 3, W. Kley 2, R. Klessen 3 1 - Max Planck Institute for Astronomy Heidelberg, Germany 2 - Computational Physics, Tübingen University, Germany 3 - Institute for Theoretical Astrophysics, Heidelberg University, Germany Fire Down Below - The Impact of Feedback on Star and Galaxy Formation Kavli Institute for Theoretical Physics, Santa Barbara, CA, USA April 17, 2014
Radiative Rayleigh-Taylor Instability in ULIRGs E = 0.5 Shane Davis (on Tuesday): Flux-Limited-Diffusion vs. Variable Eddington Tensor for Thermal Radiation Feedback (T ~ 80 K). This talk: Flux-Limited-Diffusion vs. Ray-Tracing for Stellar Radiation Feedback Davis et al. (2014, submitted)
Radiation Transport Equation: Ray-Tracing (RT) 1 c @I @t ~ ~ ri ext I = c 4 ( absb scate) Radiative Flux and Radiation Energy Density: ~F = Z E = 1 c 4 Z 4 Flux-Energy Relation: ~F = R d ~ I(r,,t) d I(r,,t) 4 d ~ I(r,,t) R4 ce d I(r,,t)
Radiation Transport Equation: Flux-Limited-Diffusion (FLD) 1 c @I @t ~ ~ ri ext I = c 4 ( absb scate) Approximations: Locally Isotropic, mean angular values (integral over full solid angle): @E @t r ~ F ~ = c abs (B E) FLD approximation: ~F = D re ~ @E @t ~r(d re) ~ =c abs (B E) Gray approximation: Opacity is computed from local conditions: Conservation equation Flux-Energy Relation Diffusion equation apple(~x) =apple P/R (T rad (~x))
The Hybrid Scheme (RTFLD) Split Radiation Fields: Stellar Irradiation Thermal dust (re-)emission Different Solvers: Ray-Tracing (RT) Flux-Limited-Diffusion (FLD) not to scale Kuiper et al. (2010), A&A 511
Outline A. Radiation Transport Problem: Circumstellar Disk Temperatures Setup from Pascucci et al. (2004) benchmark test Setup from Pinte et al. (2009) benchmark test B. Radiation-Hydrodynamics: (1) Stellar Radiative Feedback (2) Radiative Rayleigh-Taylor Instability (3) The Science case: A Solution to the Radiation Pressure Problem in the Formation of the Most Massive Stars
Circumstellar Disk Temperatures Setup: Star Disk (flared) Pascucci et al. (2004) Tstar = 5800 K Rstar = 1 R Tau = 0.1...10 2 at 550 nm Pinte et al. (2009) Tstar = 4000 K Rstar = 2 R Tau =10 3...10 6 at 810 nm Methods/Codes: Monte-Carlo code RADMC as reference (scattering is neglected) MC : Hybrid : FLD : ν-dependent RT for Stellar Gray FLD for Thermal Radiation Gray FLD approximation for both - Stellar and Thermal - Radiation Kuiper & Klessen (2013), A&A 555
Results Optically thin (Tau550nm = 0.1): Hybrid accurate up to 3% FLD yields wrong Temperature slope Hybrid/RT/ MC: apple(~x) =apple apple P (T ) T @KD 3 10 2 2 10 2 1 10 2 5 10 1 3 10 1 2 10 1 o t 550 nm = 0.1 o o o o Hybrid o o = MC x = analytical: Spitzer (1978) o = analytical: gray & isotropic 1 10 1 10-2 10-1 10 0 10 1 10 2 10 3 R-R in @AUD o FLD 10 t 550 nm = 0.1 FLD: apple(~x) =apple P (T rad (~x)) -40-50 10-2 10-1 10 0 10 1 10 2 10 3 DT @%D 0-10 -20-30 R-R in @AUD Hybrid FLD Kuiper & Klessen (2013), A&A 555 Fig. 3. Temperature profiles (upper panel) in the midplane of the circumstellar disk for the case of low optical depth 550nm = 0.1. F
Results Optically thick (Tau810nm = 10 4 ): Hybrid accurate up to 46% FLD misses Shadowing effects T @KD 10 3 10 2 FLD t 810 nm = 12200 Hybrid/RT/MC: optically thin medium 10 1 = MC Hybrid 10-3 10-2 10-1 10 0 10 1 10 2 R-R in @AUD optically thick radiative barrier 300 250 200 FLD t 810 nm = 12200 FLD: optically thin medium DT @%D 150 100 50 Hybrid 0 optically thick radiative barrier -50 10-3 10-2 10-1 10 0 10 1 10 2 R-R in @AUD Kuiper & Klessen (2013), A&A 555 Fig. 6. Same as Fig. 3 for simulation runs with 810nm = 1.22 10 4. In this highly optical thick case, the results for gray RT gray FLD are identical to the frequency-dependent RT gray
Results Optically thick (Tau810nm = 10 4 ): Hybrid accurate up to 46% FLD misses Shadowing effects T @KD 10 3 10 2 FLD t 810 nm = 12200 Hybrid/RT/MC: optically thin medium ~ optically thick radiative barrier 10 1 10-3 10-2 10-1 10 0 10 1 10 2 R-R in @AUD 4 d ~ I(r,,t) 300 ~F = 250 200 directional history Hybrid = MC R R4 FLD d I(r,,t) ce t 810 nm = 12200 FLD: optically thin medium optically thick radiative barrier -50 10-3 10-2 10-1 10 0 10 1 10 2 DT @%D 150 optically thin medium = free-streaming limit 100 Hybrid 50 ~F = D ~ re ~ re! 0 re ~ ce R-R in @AUD Kuiper & Klessen (2013), A&A 555 Fig. 6. Same as Fig. 3 for simulation runs with 810nm = 1.22 10 4. In this highly optical thick case, the results for gray RT gray FLD are identical to the frequency-dependent RT gray
Conclusion 300 250 Irradiation! Long-range effect Frequency-dependence! FLD Shadowing! HDTL max @%D 200 150 100 Fig. 8. 50 gray Hybrid = ΔMC 0 10-1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 t 810 nm freq. Hybrid Kuiper & Klessen (2013), A&A 555
Radiation-Hydrodynamics: Stellar Feedback Eddington limit: L M F grav = Proto-Star G M r 2 F rad = apple L 4 r 2 c apple Interstellar medium Dust grains / Molecules / Ionized gas Radiative Force overcomes Gravity: F rad >F grav L M 4 Gc apple Scale-free! RT: apple(~x) =apple apple P (T ) Scale-free! FLD: apple(~x) =apple P (T rad (~x)) Eddington ratio decreases with distance to Star!
Radiation-Hydrodynamics: Stellar Feedback RT Kuiper et al. (2012), A&A 537
Radiation-Hydrodynamics: Stellar Feedback gray FLD Kuiper et al. (2012), A&A 537
Radiative Rayleigh-Taylor Instability E = 0.5: Fgrav ~ Frad E = 0.02: Fgrav >> Frad Davis et al. (2014, submitted) E = 2.0: Frad = 2*Fgrav In this case, the shell is accelerated efficiently and reaches the upper boundary of the domain before the RTI has time to grow appreciable. Jiang, Davis, & Stone (2013)
Analytically: Radiative Rayleigh-Taylor Instability Opacity / Radiative force is actually 1-2 orders of magnitude higher than computed within the gray FLD approximation Radiation-pressure-dominated cavities remain stable Massive Stars do not form via Radiative Rayleigh-Taylor Instability a cm s 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 R cavity 2000 AU FLD 1 FLD 2 Ray Tracing Gravity 10 15 0 10 20 30 40 50 60 M M a cm s 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 R cavity 10000 AU FLD 1 FLD 2 Ray Tracing Gravity 10 15 0 10 20 30 40 50 60 M M Kuiper et al. (2012), A&A 537
Scientific Application: Radiation Pressure Problem Kuiper et al. (2011), ApJ 732
Solving the Radiation Pressure Problem! 10-2 M core = 480 M Ṁ * @M ü yr -1 D 10-3 10-4 10-5 M core = 120 M M core = 240 M 10-6 M core = 60 M 0 20 40 60 80 100 120 M * @M ü D First simulations... Kuiper et al. (2010), ApJ 722 Including Radiation Pressure Feedback Forming stars far beyond the Radiation Pressure Barrier! mostly up to the observed upper mass limit M! 140 M
Radiation Transport Benchmark: HDTL max @%D 300 250 200 150 100 Fig. 8. Irradiation! Long-range effect 50 gray Hybrid = ΔMC 0 10-1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Radiative Stellar Feedback: Hybrid / RT Frequency Dependence! t 810 nm Shadowing! FLD freq. Hybrid gray FLD Summary Radiation Pressure Problem: Solved via Disk Accretion! Ṁ * @M ü yr -1 D 10-2 10-3 10-4 10-5 10-6 0 20 40 60 80 100 120 M * @M ü D Thanks for your attention!
Stability of radiation-pressure-dominated cavities Shell morphology: Frequency-dependent RT: pre-acceleration of layers on top of the cavity shell FLD (E ~ 1.0) RT (E ~ 20... 200) 2000 AU Kuiper et al. (2012), A&A 537
Double-Check Kuiper et al. (2012): In the RT cases, the radiation pressure exceeds gravity by 1 2 orders of magnitude. Owen, Ercolano, & Clarke (2012): FLD, Hybrid, and MC Radiation Transport [...] we find the FLD method significantly underestimates the radiation pressure by a factor of ~100. Harries, Haworth, & Acreman (2012): MC-Radiation-Hydrodynamics The development and speed of the cavities is similar to that found by Kuiper et al.