New radiation transport methods for Radiation-Hydrodynamicskuiper/Rolf... · 2018. 3. 26. · Rolf Kuiper Dust Radiative Transfer 2013 October 10, 2013 not to scale Basic idea of

(Dis)advantages of approximate radiation transport methodsfor Radiation-Hydrodynamics

Rolf Kuiper1,2

R. Klessen3, C. Dullemond3, H. Klahr1, Th. Henning1, W. Kley2

1 - Max Planck Institute for Astronomy Heidelberg, Germany2 - Computational Physics, Tübingen University, Germany3 - Institute for Theoretical Astrophysics, Heidelberg University, Germany

Dust Radiative Transfer 2013Grenoble, FranceOctober 10, 2013

Rolf Kuiper October 10, 2013 Dust Radiative Transfer 2013

Modeling Massive Star Formation

Kuiper et al. (2011), ApJ 732

Appetizer:

• 3D

• freq.-dep. RT

• 1AU resolution

✓3D freq.-dep. RT as fast as Hydrodynamics!


not to scale

Basic idea of hybrid schemesSplit Radiation Fields and Solvers (not domains!):

• Gray or Frequency-dependent Ray-Tracing (RT) for stellar irradiation

• Gray Flux-Limited Diffusion (FLD) for thermal dust (re-)emission

Kuiper et al. (2010), A&A 511


Setup:

• From Pascucci et al. (2004) and Pinte et al. (2009)

• Solar type star R = 1 R⊙◉☉ & T = 5800 K R = 2 R⊙◉☉ & T = 4000 K

• Flared circumstellar disk

• Various optical depths from 10-1 up to 10+2 (550 nm) and from 10+3 up to 10+6 (810 nm)

Methods:

•Monte-Carlo (MC) RADMC as reference (scattering is neglected!)

• Flux-Limited Diffusion approximation (FLD)

• Gray Ray-Tracing (RT) + Gray FLD

• Frequency-dependent RT + Gray FLD

ConfigurationR. Kuiper and R. S. Klessen: On the reliability of approximate radiation transport methods for irradiated disk studies 3

Fig. 1. Visualization of the disk configuration and final temper-ature for the case ⌧550nm = 100. Solid lines denote iso-densitycontours. Colors denote the final temperature distribution.

T⇤ = 5800 K. In the Pinte et al. (2009) radiation benchmarktest (⌧810nm � 1000), the star has a radius of R⇤ = 2 R� and atemperature of T⇤ = 4000 K. A visualization of the disk setupand the final temperature distribution is given in Fig. 1.

The opacity table used is the same as in the original bench-mark test of Pascucci et al. (2004) taken from Draine & Lee(1984). These opacities are shown in Fig. 2. The dust to gas massratio is set constant to 1%. Scattering is neglected.

ÊÊÊÊÊÊÊÊÊÊÊÊ

ÊÊÊ

ÊÊ

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ

Ê

ÊÊÊÊÊÊÊ

ÊÊ‡‡

‡‡‡‡‡

‡‡

‡‡

‡‡

‡

‡

‡

‡

‡

µ n2.0508

1012 1013 1014 101510-1

100

101

102

103

104

105

n @HzD

k@cm

2g-

1 D

Fig. 2. Frequency-dependent opacities from Draine & Lee(1984). The solid line denotes an exponential fit to the longwavelength regime.

3.2. Numerical configuration

In the Pascucci et al. (2004) radiation benchmark test (⌧550nm 100), the computational domain goes from Rmin = 1 AU up toRmax = 1000 AU. In the Pinte et al. (2009) radiation bench-mark test (⌧810nm > 1000), the computational domain goes from0.1 AU up to 400 AU. In both setups, the polar angle covers thefull domain from 0 to 180 degree.

The boundary conditions in the polar direction are chosento not allow any radiative flux over the polar axis. The bound-ary condition at the outer radial direction is chosen to allow fora radiative flux out of the computational domain computed inthe optically thin limit. The boundary condition for the thermalradiation field at the inner radial direction is in the RT simula-tions given by a low value of the thermal radiation field (the totalradiation field at the inner disk rim is dominated by the irradia-tion). The boundary condition for the thermal radiation field atthe inner radial direction is in the simulations using only FLDdetermined from the surface temperature of the central star as-suming an optically thin limit between the stellar surface and theinner disk rim (as assumed in the original benchmark tests).

The grids in spherical coordinates are chosen to resolve thelocal optical depths of the circumstellar disks in the polar direc-tion appropriately to guarantee the computation of correct cool-ing properties. For all cases up to ⌧810nm = 1.22 ⇥ 10+3 the gridconsists of n

r

⇥n✓ = 128⇥360 cells. For ⌧810nm = 1.22⇥10+4 thegrid has n

r

⇥ n✓ = 64⇥ 2880 grid cells. For ⌧810nm = 1.22⇥ 10+6

the grid has n

r

⇥ n✓ = 16 ⇥ 105 grid cells.We note that results of the same accuracy are obtained by

using 128 ⇥ 360 grid cells for the higher optical depths runsas well, but make use of a numerical ’trick’ similar to the onedescribed in Pinte et al. (2009) for the ProDiMo code. For theProDiMo code, the density distribution was altered to limit theoptical depth in Pinte et al. (2009). Here, we promote a modifiedalgorithm of the same idea: The resolutions of a 128 ⇥ 360 gridis su�cient for the high optical depth runs as well, if an upperlimit of the optical depth of ⌧max 1 is applied to each individ-ual grid cell during the radiation transport. Although we focusin the following discussion on the results of the high resolutiongrids only, a solution of a simulation run using this numericalgimmick is shown for comparison for the most optically thickcase of ⌧810nm = 1.22 ⇥ 106 in Fig. 7.

The numerical configuration of the Monte-Carlo codeRADMC is given in Pascucci et al. (2004) and Pinte et al. (2009)with the di↵erence that we do not consider scattering here.

3.3. Runs performed

We are computing the equilibrium temperature distribution inthe disk and envelope using di↵erent radiation transport meth-ods. We study the gray FLD approximation, the gray irradia-tion + gray FLD approximation, and the frequency-dependentirradiation + gray FLD approximation. As a reference solution,we use results obtained with the Monte-Carlo code RADMC,respectively. We are performing simulations for various opti-cal depths of the disk’s midplane from ⌧550nm = 0.1 up to⌧810nm = 1.22⇥ 10+6. An overview of runs performed is given inTable 1.

Kuiper & Klessen (2013), A&A 555


ResultsR. Kuiper and R. S. Klessen: On the reliability of approximate radiation transport methods for irradiated disk studies 5

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Fig. 3. Temperature profiles (upper panel) in the midplane of thecircumstellar disk for the case of low optical depth ⌧550nm = 0.1.In the lower panel the deviations of the three di↵erent radia-tion transport methods to the comparison code result are dis-played. Solid lines denote results for the frequency-dependentRT + gray FLD method. Dotted lines denote results for the grayRT + gray FLD method. Dashed lines denote results for the grayFLD method. Pluses “+” denote results for the comparison codeRADMC. In this optical thin case, the results for gray RT + grayFLD are almost identical to the frequency-dependent RT + grayFLD. Here, analytical estimates for the optically thin approxi-mation are given: Crosses “x” mark the analytical estimate bySpitzer (1978) for irradiated regions far away from the star, cir-cles “o” mark the analytical estimate in the gray and isotropicapproximation (T / r

�1/2).

ton propagation in the optically thick regions alone is most eas-ily described in the di↵usion limit, the (flux-limited) di↵usionapproximation fails to account for shadowing e↵ects in multi-dimensional problems due to the loss of angular information inits derivation.

The simulation results for the most optically thick case areshown in Fig. 7. The RT + gray FLD runs are in overall goodagreement with the comparison results from RADMC. The max-imum deviation of 43% corresponds to the maximum di↵erence

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Fig. 4. Same as Fig. 3 for simulation runs with ⌧550nm = 102.

between the various radiation transport codes of the originalbenchmark test (Pinte et al. 2009). As expected in this limitingcase of very high optical depth, the frequency-dependent andgray irradiation routines give identical results.

The gray FLD approximation fails to reproduce the shadowbehind the optically thick inner disk rim. In this method, the stel-lar radiative flux di↵uses around this obstacle and heats the diskthrough the optically thin atmosphere at larger radii. The result-ing midplane temperature is up to a factor of 2.8 higher than inthe comparison run.

5.1.3. The regime of moderate optical depth

The frequency-dependent and gray irradiation routines di↵eronly in the case of moderate optical depth, shown in Fig. 4. Inthis intermediate regime, the disk’s midplane is optically thickfor the higher-frequency part of the stellar irradiation spectrum,but optically thin in the long wavelength regime. Therefore, onlythe frequency-dependent irradiation routine is able to resemblethe heating of the irradiated midplane at larger radii correctly. Inthe case of gray irradiation, the long wavelength flux is absorbedalready at the inner disk rim and the temperature at larger radiifollows the slope of the gray FLD method.

Hybrid

FLD

FLD

Hybrid

+ = MCo = isotropic assumptionx = analytical, Spitzer (1978)

Tau = 0.1:

• Hybrid accurate up to 3%

• Gray approximation is valid

• FLD yields wrong temperature slope due to local isotropic assumption




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FLD

gray Hybrid

freq. HybridTau = 100:


• Gray approximation yields too cool disk

• FLD approximation yields too hot disk



Stellar spectrum

R. Kuiper et al.: Frequency-dependent radiation transport for hydrodynamics simulations in massive star formation

Fig. 2. Regarding the frequency dependenceof stellar irradiation feedback. Frequency de-pendent opacities ! ("), Planck mean opaci-ties !P (T!), and stellar black body spectrumB" (",T!) as function of the frequency ".

corresponding part of the stellar black body Planck functionB" (", T!) in the frequency dependent interval:

|F! (R!, "i) | =c4

("i+"i+1)/2!

("i"1+"i)/2

B" (", T!) d". (20)

2.4. The generalized minimal residual method

The FLD Eqs. (8) or (15) are solved in our newly implementedapproximate radiation transport module for hydrodynamics in animplicit fashion via a generalized minimal residual (GMRES)solver. The GMRES solver is a Krylov subspace (KSP) methodfor solving a system of linear equations Ax = b via an ap-proximate inversion of the large but sparse matrix A and is anadvancement of the minimal residual (MinRes) method. TheGMRES method was developed in 1986 and is described in Saad& Schultz (1986). This method is much better than the conju-gate gradient (CG) or successive over-relaxation (SOR) method,and at least as good as the well-known improved stabilized Bi-Conjugate Gradient (BiCGstab) method used, e.g., in Yorke &Sonnhalter (2002).

The general idea of minimal residual solvers based onthe KSP method is the following: The ith KSP is defined asKi = span{b, Ab, A2b, ..., Ai"1b}. In each solver iteration i theGMRES method increments the subspace Ki used with an ad-ditional basis vector Ai"1b and approximates the solution of thesystem of linear equations by the vector xi which minimizes thenorm of the residual r = |Axi" b|. This method converges mono-tonically and theoretically reaches the exact solution after per-forming as many iterations as the column number of the matrix A(which equals the number of grid cells). Of course, in practicethe iteration is already stopped after reaching a specified relativeor absolute tolerance of the residual, which normally takes onlya small number of iterations. The computation of each iterationgrows like O

"i2#. In the current implementation, we use the so-

called “GMRES restarted” by default. GMRES restarted neverperforms all the iterations to reach the exact solution. After apriorily fixed number of internal iterations n, the solver starts asecond time in the first subspace K1 but with the last approximatesolution xn. Due to the growing computational e!ort with O

"i2#

this approach generally results in a speedup of the computation.

The radiation transport module is parallelized for distributedmemory machines, using the message passing interface (MPI)language. The results of a detailed parallel performance test ofthe whole radiation transport module, including this GMRESsolver is presented in Sect. 4.

3. Frequency dependent test of the approximateradiation transport

The approximate radiation transport introduced in the previoussection can now be tested for realistic dust opacities in a stan-dard benchmark test for irradiated circumstellar disk models.The setup of the following comparison is adopted from Pascucciet al. (2004) and includes a central solar-type star, an irradiatedcircumstellar flared disk and an envelope. We have to choosea low-mass central star, because no benchmark for high-massstars was performed so far. However, the tests should not dependon the actual size and luminosity of the central star. For com-parison, we choose a standard full frequency dependent Monte-Carlo based radiation transport code. The comparison is donefor two di!erent (low and high) optical depths taken from theoriginal radiation benchmark test. To test each of the compo-nents (gray and frequency dependent irradiation as well as FluxLimited Di!usion) of the proposed approximate radiative trans-fer separately, we perform several test runs with and without thedi!erent physical processes (absorption and re-emission) withboth the Monte-Carlo based (see Table 1) and the approximate(see Table 2) radiative transfer code.

3.1. Setup

3.1.1. Physical setup of the star, the disk, and the envelope

The stellar parameters are solar-like: The e!ective temperatureof the star is 5800 K (T! = 5800 K) and the stellar radius is fixedto 1 solar radius (R! = 1 R#). The disk ranges from rmin = 1 AUup to rmax = 1000 AU.

Although the numerical setup of the gas density is done inspherical coordinates, the analytic setup of the gas density, asdescribed in the original benchmark test, is given in cylindricalcoordinates:

# (r, z) = #0 f1 (r) f2 (r, z) (21)

Page 5 of 16

Frequency-dependent absorption

Gray absorption

UV-endIR-tail

Results




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FLD

gray Hybrid



• Gray approximation yields too cool disk due to missing IR flux

• FLD approximation yields too hot disk



Ray-Tracing:

Flux-Limited Diffusion:

a single emitting source

radiative barrier

optically thin medium

Shadows? ... Shadows!


~Frad = �D~rErad



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• FLD approximation yields too hot disk due to shadowing effects



PerformanceR. Kuiper and R. S. Klessen: On the reliability of approximate radiation transport methods for irradiated disk studies 5

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FLD

FLD

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orders of magnitude difference in cpu-time!

gray Hybrid




• FLD approximation yields too hot disk due to shadowing effects

•MC codes of Pascucci et al. (2004) differ about 5% in the inner and up to 20% in the outer disk (test cases without scattering!)

+ = MC



Hybrid & optically thick:

•Accurate up to 10%, if frequency dependence is considered

•For comparison: Different Monte-Carlo codes (Pascucci et al. 2004) differ about

•5% in the inner disk region (< 200 AU)

•15% towards the outer boundary

•Correct transition from optically thick disk to optically thin envelope

A&A 511, A81 (2010)

Fig. 4. Radial cut through the temperature profile in the midplane inthe most optically thick case !550 nm = 100 without di!usion. Upperpanel: radial temperature slope of gray irradiation (dashed line), fre-quency dependent irradiation (solid line), and the Monte-Carlo routinein the “one-photon-limit” (dots). Lower panel: deviations of the gray(dashed line) and frequency dependent method (solid line) from theMonte-Carlo code in percent.

Monte-Carlo simulation in the most optically thick case!550 nm = 100.

Also in this case, we found that the frequency dependent ir-radiation is necessary to achieve the accuracy needed for realis-tic radiative feedback in hydrodynamics studies (e.g. in massivestar formation or in irradiated accretion disks). Gray irradiationcombined with the FLD approximation leads to deviations up to38.4% in the resulting temperature slope (see Fig. 5). Includingfrequency dependent irradiation and FLD the radial temperatureprofiles agree within 0.7% at the inner boundary up to a maxi-mum deviation of 11.1% at roughly r ! 200 AU. The compari-son between the complete radiation scheme and the correspond-ing Monte-Carlo run is shown in radial and polar temperatureprofiles in Figs. 5 and 6 respectively. The turnover point (at apolar angle from the midplane of " ! 19") from the opticallythin envelope to the optically thick disk region is reproducedvery well.

Further remarks and discussion of these results are given inthe last Sect. 3.4 at the end of the comparison section.

3.3.4. Radiative force

Stellar radiative feedback on the dynamics of the environmentplays a crucial role in the formation of massive stars. The heat-ing will probably prevent further fragmentation of the cloud

Fig. 5. Radial cut through the temperature profile in the midplane in themost optically thick case !550 nm = 100 including irradiation and FluxLimited Di!usion. Upper panel: radial temperature slope of gray irra-diation plus Flux Limited Di!usion (dashed line), frequency dependentirradiation plus Flux Limited Di!usion (solid line), and the correspond-ing Monte-Carlo routine (dots). Lower panel: deviations of the gray(dashed line) and frequency dependent run (solid line) from the Monte-Carlo code in percent.

Fig. 6. Polar cut through the temperature profile at r = 2 AU of the fre-quency dependent irradiation plus Flux Limited Di!usion run for themost optically thick case !550 nm = 100. The profile reproduces theturnover point at a polar angle of " ! 19" above the midplane fromthe optically thin envelope to the optically thick disk region. Solid line:Frequency dependent irradiation plus Flux Limited Di!usion. Dots:Data from the corresponding Monte-Carlo comparison run. The ver-tical axis covers only a small temperature range from 200 to 260 K tobetter visualize the small deviations.

Page 8 of 16

r = 2 AU


Results

Hybrid


6 R. Kuiper and R. S. Klessen: On the reliability of approximate radiation transport methods for irradiated disk studies

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Fig. 5. Same as Fig. 3 for simulation runs with ⌧810nm = 1.22 ⇥103. In this fairly optical thick case, the results for gray RT +gray FLD are almost identical to the frequency-dependent RT +gray FLD, small di↵erences are visible at the innermost rim ofthe disk only.

5.2. Validity of approximations

5.2.1. The gray flux-limited diffusion approximation for the

thermal dust emission

In all our approximate radiation transport simulations, the ther-mal (re-)emission of dust grains is computed in the gray FLDapproximation. This approximation reduces the cpu time signif-icantly (up to several orders of magnitude for a high-resolution3D scheme) to allow the usage of the radiation transport methodwithin magneto-hydrodynamics simulations. If the stellar irra-diation is accounted for within a frequency-dependent RT step,the results of the approximate radiation transport simulations arein accordance with the deviations given by the di↵erences fromthe various high-level radiation transport methods from the orig-inal radiation benchmark tests (Pascucci et al. 2004; Pinte et al.2009). Hence, we conclude that the gray FLD approximationis an e�cient and accurate method for solving the thermal dust

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Fig. 6. Same as Fig. 3 for simulation runs with ⌧810nm = 1.22 ⇥104. In this highly optical thick case, the results for gray RT +gray FLD are identical to the frequency-dependent RT + grayFLD.

(re-)emission of circumstellar disks. In general, the FLD methodsu↵ers from the issues explained in Sect. 5.2.3.

5.2.2. The gray approximation for the stellar irradiation

The frequency-dependent treatment of the stellar irradiationspectrum improves the results for configurations of moderateoptical depth (for the particular test case of a Sun-like star and⌧550nm ⇠ 100), in which the disk’s midplane is optically thick forthe high-frequency part of the stellar spectrum, while it is opti-cally thin for a substantial part of the long wavelength regime.In the limiting cases of very low and very high optical depth thegray irradiation is in agreement with the frequency-dependentruns.

Results

Kuiper & Klessen, accepted at A&A

FLD

+ = MC

freq. Hybrid

FLD

freq. Hybrid

gray Hybrid

Tau = 10+3:



• FLD approximation up to 282% off due to increasing shadowing effects


6 R. Kuiper and R. S. Klessen: On the reliability of approximate radiation transport methods for irradiated disk studies

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Fig. 5. Same as Fig. 3 for simulation runs with ⌧810nm = 1.22 ⇥103. In this fairly optical thick case, the results for gray RT +gray FLD are almost identical to the frequency-dependent RT +gray FLD, small di↵erences are visible at the innermost rim ofthe disk only.

5.2. Validity of approximations

5.2.1. The gray flux-limited diffusion approximation for the

thermal dust emission

In all our approximate radiation transport simulations, the ther-mal (re-)emission of dust grains is computed in the gray FLDapproximation. This approximation reduces the cpu time signif-icantly (up to several orders of magnitude for a high-resolution3D scheme) to allow the usage of the radiation transport methodwithin magneto-hydrodynamics simulations. If the stellar irra-diation is accounted for within a frequency-dependent RT step,the results of the approximate radiation transport simulations arein accordance with the deviations given by the di↵erences fromthe various high-level radiation transport methods from the orig-inal radiation benchmark tests (Pascucci et al. 2004; Pinte et al.2009). Hence, we conclude that the gray FLD approximationis an e�cient and accurate method for solving the thermal dust

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Fig. 6. Same as Fig. 3 for simulation runs with ⌧810nm = 1.22 ⇥104. In this highly optical thick case, the results for gray RT +gray FLD are identical to the frequency-dependent RT + grayFLD.

(re-)emission of circumstellar disks. In general, the FLD methodsu↵ers from the issues explained in Sect. 5.2.3.

5.2.2. The gray approximation for the stellar irradiation

The frequency-dependent treatment of the stellar irradiationspectrum improves the results for configurations of moderateoptical depth (for the particular test case of a Sun-like star and⌧550nm ⇠ 100), in which the disk’s midplane is optically thick forthe high-frequency part of the stellar spectrum, while it is opti-cally thin for a substantial part of the long wavelength regime.In the limiting cases of very low and very high optical depth thegray irradiation is in agreement with the frequency-dependentruns.

Results

FLD

+ = MC

FLD

Hybrid

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Tau = 10+4:






Results R. Kuiper and R. S. Klessen: On the reliability of approximate radiation transport methods for irradiated disk studies 7

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Fig. 7. Same as Fig. 3 for simulation runs with ⌧810nm = 1.22 ⇥106. In this highly optical thick case, the results for gray RT+ gray FLD are identical to the frequency-dependent RT +gray FLD. Here, the dotted lines represent the results for thefrequency-dependent RT + gray FLD method in a low resolu-tion simulation if an upper limit of the optical depth per gridcell of ⌧max 1 is used, see Sect. 3.2 for a description of thisnumerical gimmick.

5.2.3. The flux-limited diffusion approximation for the stellar

source term

In general, treating the stellar source term within the FLD ap-proximation involves two major problems. First, the FLD ap-proximation assumes a locally isotropic radiation field. This as-sumption yields the wrong radial slope of the temperature evenin highly optically thin regions as depicted in Fig. 3.

Second, in the derivation of the FLD approximation the mo-mentum expansion of the radiation transport equation is closedby relating the radiative flux to the gradient of the radiation en-ergy. This assumption prevents the FLD approximation to ac-count for shadows behind radiative obstacles such as the opti-cally thick inner rim of circumstellar disks. The radiative flux inFLD simulations tends to di↵use from the stellar surface around

the optically thick inner disk rim and heats the disk midplane atlarger radii to unrealistic high values.

Hence, we conclude that the FLD approximation results inlarge errors if used for stellar irradiation sources. In (magneto-)hydrodynamic studies of accretion disks, the strong increase ofthe midplane temperature would e.g. result in an overestimate ofthe disk stability based on the Toomre criterion (Toomre 1964).

5.2.4. The hybrid approximate radiation transport method

In general, the hybrid approximate radiation transport methodusing a frequency-dependent RT step for the stellar irradiationand a gray FLD solver for the thermal dust emission results inhigh accuracy for the whole range of optical depths. By usingonly a small fraction of the cpu-time needed for the more so-phisticated comparison code, this method fully agrees for casesof optical depths ⌧550nm 100 as well as for the high opticaldepth cases. Only in the intermediate regime of a total opticaldepth of about ⌧810nm ⇠ 1000, the approximate method over-estimates the temperature in the disk midplane behind the mostoptically thick inner rim by about 48% compared to only 20%deviation of the various original benchmark codes in Pinte et al.(2009).

6. Summary

We checked the reliability of approximate radiation transportmethods using flared disk configurations of standard radiationbenchmark tests. We compare the resulting temperature distri-butions for a wide range of optical depths from a fairly opti-cally thin case (⌧550nm = 0.1) up to a highly optically thick case(⌧810nm = 1.22⇥10+6). The outcome for the various methods anddi↵erent regimes of optical depth are summarized in Fig. 8.

Approximating the stellar irradiation spectrum by a grayPlanck opacity is in accordance with a frequency-dependenttreatment in the optically thin and thick limit. For intermedi-ate optical depths, a gray approximation of the stellar irradiationspectrum yields a slightly hotter inner rim and a slightly coolermidplane of the disk at larger radii.

The gray FLD approximation fails to compute an appropri-ate temperature profile in all regimes of optical depth, the maxi-mum deviations to the comparison runs are 50% in the opticallythin and up to 280% in the optically thick limit. For low opticaldepth, the isotropic assumption within the FLD method yields atoo steep decrease of the radial temperature slope. For higher op-tical depths, the FLD approximation does not render the shadowbehind the optically thick inner rim of the circumstellar disk,yielding artificial heating at larger disk radii. This issue of ne-glected shadows is expected to be even more problematic forincreasing stellar luminosity.

The frequency-dependent RT + gray FLD approximation de-notes a highly accurate treatment of the radiation field in the op-tically thin and thick regime. Only in the intermediate regime ofmoderate optical depth ⌧810nm ⇠ 1000 the deviations of the radi-ation transport codes from the original benchmark tests by Pinteet al. (2009) are lower than for the hybrid approximate radiationtransport method. The high accuracy of the hybrid approximateradiation transport is attended by very low computational costscompared with the very expensive Monte-Carlo calculations.

Hence, we close this comparison study by pointing out thatthe frequency-dependent RT + gray FLD method is ideallysuited for follow-up (magneto-)hydrodynamical studies of cir-cumstellar accretion disks.

FLD

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Conclusions8 R. Kuiper and R. S. Klessen: On the reliability of approximate radiation transport methods for irradiated disk studies

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Fig. 8. Maximum temperature deviation in the disk’s midplaneas function of the optical depth of the circumstellar disk. Thesolid line denotes the results for the frequency-dependent RT +gray FLD method. The dotted line denotes the results for thegray RT + gray FLD method. The dashed line denotes the resultsfor the gray FLD method. The pluses ”+” denote the deviationsof the results of the various high-level radiation transport codesused in the original benchmark tests of Pascucci et al. (2004)and Pinte et al. (2009). These values are read o↵ Figs. 4 and 5 inPascucci et al. (2004) and Fig. 10 in Pinte et al. (2009).

Acknowledgements. We thank Cornelis Dullemond for providing the RADMCreference simulation data and for fruitful discussions of approximate ra-diation transport solvers in the context of passively irradiated circumstel-lar disks. Author R. K. acknowledges financial support from the Deutsche

Forschungsgemeinschaft (DFG) via the collaborative research project SFB 881”The Milky Way System” in sub-project B2. Both authors thank the DFG forsupport via the priority program SPP 1573 ”Physics of the ISM” as well as theBaden-Wurttemberg Foundation for support via contract research (grant P- LS-SPII/18) via their program Internationale Spitzenforschung II.

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arXiv, 1211, 6345Bjorkman, J. E. & Wood, K. 2001, ApJ, 554, 615Boley, A. C., Durisen, R. H., Nordlund, Å., & Lord, J. 2007, ApJ, 665, 1254Boss, A. P. 1984, MNRAS, 209, 543Chiang, E. I. & Goldreich, P. 1997, ApJ, 490, 368Draine, B. T. & Lee, H. M. 1984, ApJ, 285, 89Dullemond, C. P. & Dominik, C. 2004a, A&A, 421, 1075Dullemond, C. P. & Dominik, C. 2004b, Extrasolar Planets: Today and

Tomorrow, 321, 361Dullemond, C. P. & Turolla, R. 2000, A&A, 360, 1187Dullemond, C. P., van Zadelho↵, G. J., & Natta, A. 2002, A&A, 389, 464Flaig, M., Ruo↵, P., Kley, W., & Kissmann, R. 2012, MNRAS, 420, 2419Fouchet, L., Alibert, Y., Mordasini, C., & Benz, W. 2012, A&A, 540, 107Gammie, C. F. 2001, ApJ, 553, 174Godon, P. & Livio, M. 2000, ApJ, 537, 396Harries, T. J. 2011, MNRAS, 416, 1500Harries, T. J., Haworth, T. J., & Acreman, D. 2012, eprint arXiv, 1209, 1512Hawley, J. F. & Balbus, S. A. 1991, ApJ, 376, 223Hirose, S. & Turner, N. J. 2011, ApJL, 732, L30Johnson, B. M. & Gammie, C. F. 2005, ApJ, 635, 149Kenyon, S. J. & Hartmann, L. 1987, ApJ, 323, 714Klahr, H. & Bodenheimer, P. 2003, ApJ, 582, 869Kley, W. & Lin, D. N. C. 1992, ApJ, 397, 600Kley, W., Papaloizou, J. C. B., & Lin, D. N. C. 1993a, ApJ, 416, 679Kley, W., Papaloizou, J. C. B., & Lin, D. N. C. 1993b, ApJ, 409, 739Kratter, K. M. & Matzner, C. D. 2006, MNRAS, 373, 1563Kratter, K. M., Matzner, C. D., & Krumholz, M. R. 2008, ApJ, 681, 375Kuiper, R., Klahr, H., Beuther, H., & Henning, T. 2010a, ApJ, 722, 1556Kuiper, R., Klahr, H., Beuther, H., & Henning, T. 2011, ApJ, 732, 20Kuiper, R., Klahr, H., Beuther, H., & Henning, T. 2012, A&A, 537, 122Kuiper, R., Klahr, H., Dullemond, C. P., Kley, W., & Henning, T. 2010b, A&A,

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AAA032.107.029Mordasini, C., Alibert, Y., & Benz, W. 2009a, A&A, 501, 1139Mordasini, C., Alibert, Y., Benz, W., Klahr, H., & Henning, T. 2012, A&A, 541,

97Mordasini, C., Alibert, Y., Benz, W., & Naef, D. 2009b, A&A, 501, 1161Owen, J. E., Ercolano, B., & Clarke, C. J. 2012, eprint arXiv, 1208, 6243Pascucci, I., Wolf, S., Steinacker, J., et al. 2004, A&A, 417, 793Pinte, C., Harries, T. J., Min, M., et al. 2009, A&A, 498, 967Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, Icarus, 124, 62Spitzer, L. 1978, New York Wiley-Interscience, 333Toomre, A. 1964, ApJ, 139, 1217Umurhan, O. M. & Regev, O. 2004, A&A, 427, 855Vaidya, B., Fendt, C., & Beuther, H. 2009, ApJ, 702, 567

freq. Hybrid

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Shadowing!Non-isotropy! Frequency-dependence!




gray Flux-Limited Diffusion (FLD)

Application: Massive Star Formation


not the same color coding!


- gray FLD run -

Krumholz et al. (2009), Science

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Radiative Rayleigh-Taylor instability?


R. Kuiper et al.: On the stability of radiation-pressure-dominated cavities 11

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Fig. 8. Radiative and gravitational acceleration at a position Rcavity = 2000 AU (left panel) and Rcavity = 10000 AU (right panel) above the massiveproto-star as a function of the stellar mass M! for three di!erent radiation transport approaches: The label “FLD 1” denotes the FLD approximationwith the opacity description of Krumholz et al. (2007). The label “FLD 2” refers to the FLD approximation with the opacities used in Kuiper et al.(2010a, 2011) and in this paper. The label “Ray-Tracing” refers to the RT step as in the hybrid radiation transport scheme.

the infrared regime) is computed in the FLD solver step. Thecorresponding length scale is given by l! = (!!(") #cavity)"1 andtherefore depends on the frequency " of the photons as well ason the actual shell density #cavity. This absorption length scale l!is shown as a function of frequency " for di!erent densities # ofthe cavity shell in Fig. 2.

8. Comparisons8.1. Comparison with a 3D gray FLD simulation

Krumholz et al. (2009) presented a self-gravity radiation hydro-dynamics simulation of the collapse of a 100 M# pre-stellar core.The outer core radius was chosen to be 0.1 pc and the den-sity profile declines in proportion to r"1.5. The initial isother-mal temperature of the core is 20 K. The pre-stellar core is ini-tially in solid-body rotation without any turbulent motion. Themodel describes a so-called monolithic collapse scenario. Theapplied radiation transport method is the gray flux-limited fif-fusion approximation. These properties of this configuration arethe same as in our simulation run “C-FLD”. The equations aresolved on a 3D adaptive mesh refinement grid in cartesian coor-dinates. Densities above the Jeans density on the finest grid levelare represented by sink particles.

During the simulation, bipolar “radiation-filled bubbles” areblown into the environment of the forming massive star. To clar-ify our terminology, these “radiation-filled bubbles” correspondto the “radiation-pressure-dominated cavities” and the “bubblewall” is called a “cavity shell” in this paper. At an extent ofroughly from 1200 to 2000 AU (from Krumholz et al. (2009),Fig. 1 therein) the cavity shell is subject to a “radiative Rayleigh-Taylor instability”, meaning that the radiation pressure expandsthe optically thin region only at specific solid angles, while atother solid angles the concentrated mass load on top of the cav-ity shell is able to penetrate into the low-density region.

This mechanism of shell instability, the focus of radiationpressure onto solid angles with lower optical depth, and the col-lapse of condensed material from the infalling envelope at othersolid angles, precisely matches the outcome of our simulations,if the FLD approximation is used for the stellar radiation feed-back. After the epoch of instability – the caving-in of material– the continuing evolution of their simulation di!ers in terms ofmorphology from our simulations because of the evolving non-

axisymmetric structures. We comment further on this di!erenceafter the following paragraph on the RT + FLD simulations.

In our simulations using the frequency-dependent RT stepfor the stellar irradiation, the mass load on top of the cavityshell causes a dependence of the optical depth on the solid an-gle (owing to the decreasing centrifugal forces towards the pole).However, in these simulations, the caving-in of material at solidangles of high optical depth is prohibited; this is most likelycaused by the treatment of the direct stellar irradiation by meansof a RT approach preserving the natural isotropy of the irradi-ation up to the first absorption layer. Furthermore, the RT ap-proach accounts for the respective frequency-dependent absorp-tion coe"cients as demonstrated in the previous sections.

The cavity is a!ected by Rayleigh-Taylor-like instabilities,when the FLD approximation is used for the radiation transport.This implication is verified by the axial symmetric simulationsperformed herein as well as by the 3D simulation in Krumholzet al. (2009). Improving the radiation transport scheme by in-corporating a ray-tracing of the direct stellar irradiation correctsfor this behavior and leads to a stable cavity growth. This impli-cation is verified by the axial symmetric simulations performedherein as well as by the 3D simulation in Kuiper et al. (2011).The largest di!erences between the numerical treatment in thesimulation of Krumholz et al. (2009) and the simulations withsimilar initial conditions presented herein are twofold: the radi-ation transport approaches and our assumed axial symmetry. Asmentioned above, the simplification to axial symmetry increasesthe di"culty of a comparison of epochs after the occurrence ofthe instability, but the overall simulation results indicate that theoutcome, regardless of whether the instability occurs, does notdepend on the axial symmetric approach:

– The 2D simulations in the FLD approximation match theoutcome of the 3D simulation in the FLD approximation,namely the instability of the cavity.

– In the 2D simulations with the improved radiation transportscheme (ray-tracing + FLD), the cavity remains stable.

– In our 3D simulation (Kuiper et al. 2011), using a frequency-dependent RT step to account for stellar irradiation, the out-flow cavity contains strong non-axisymmetric features, butno instability is detected.


Analytically:

•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 does not form via Radiative Rayleigh-Instability




Kuiper et al. (2012), A&A 5372000 AU

RT + FLDFLD

Cavity morphology:

• frequency-dependent RT: pre-acceleration of layers on top of the cavity shell


Generalized splitting scheme:

• Split total radiation field in s subcomponents

•MC up to n photon interactions (absorption or scattering events)

• Remnant radiation energy + component(s) in approximate RT

• gray / frequency-dependent FLD

• gray / frequency-dependent M1

Outlook: Boosting RT codes for RHD?!

•MC in its full glory

• previous steps give very good initial conditions/guess (e.g. already converged in the most optically thick regions)

‣stop here

• adjusting n gives either a more accurate or faster solver

• convergence of the splitting scheme is given for n to infinity


Text

ARTIST (LIME + DustPol + LinePol)

Adaptable Radiation Transport Innovations for Submillimeter Telescopes (PI: J. Jorgensen)Check: http://youngstars.nbi.dk/artist/

A&A 523, A25 (2010)

Fig. 2. The leftmost panel shows a random point distribution. In the middle panel, the points have been Delaunay triangulated. The rightmost panelshows the corresponding Voronoi tessellation.

shape of its cells, Voronoi grids do not su!er from the aliasinge!ects which are inherent to regular Cartesian grids.

When the grid points have been distributed throughout themodel domain, it is inevitable, due to the stochastic nature of thesampling method, that some points end up much closer than thelocal separation expectation value and some will be much fur-ther apart. This results in a Delaunay triangulation that is veryirregular with some triangles being very long and narrow. Thisirregularity can be remedied by applied what is known as Lloydsalgorithm (Lloyd 1982; Springel 2010), which iteratively movesa grid point slightly toward its Voronoi cells center of mass. Inour implementation, each grid point is moved slightly away fromits nearest neighbor for a preset number of iterations. The e!ectis illustrated in Fig. 3 for a random 2D sampling of a Gaussiandensity profile. The top panels shows the initial unsmoothed dis-tribution while the lower panels showed the triangulated pointdistribution after the smoothing algorithm has been applied. Theplots in the right column show the neighbor distances as a func-tion of radius. In the smoothed grid, the distances are much lessscattered and follows the Gaussian profile (shown as the lightcolored full curve) much more accurately than in the top panel.The smoothing strategy should not be exaggerated, i.e., movingthe points too slowly and iterating trough too many steps, be-cause the algorithm will then act as an annealing process and itwill result in a perfectly regular grid where all variations in thepoint distribution due to the underlying density field is smearedout. By doing it right, however, a smooth grid can be obtainedwhile the underlying density structure is still preserved in thegrid. We have found empirically that by using 25 iterations andmoving the closest neighbors about 10% of the distance awayfrom each other results in a su"ciently smooth grid that pre-serves the underlying physical structure well.

During gridding of our source model, we also distributea number of points randomly on the surface of a sphere sur-rounding our model. These points are also Delaunay triangu-lated and connected to the model grid points, but they do notrepresent anything except the surface of our computational do-main. Whenever a photon reaches one of these sink points, it isconsidered to have escaped the model.

3.2. Photon propagation

The photon transport itself goes along Delaunay lines only,from one point to another, which makes integration of Eq. (1)

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particularly simple and very fast. In the three-dimensionalDelaunay triangulation, the expectation value for the number oflines attached to a grid point is approximately 16 (Ritzerveld &Icke 2006) and the spatial sampling of J! is thus limited to thisnumber of directions. However, we still need to trace a num-ber of photons along each Delaunay line, not only in order tosample the frequency band properly, but also because we can-not conserve momentum stringently with a single photon on thisgrid. In principle, a photon passing a grid point from a certaindirection should continue to travel in the exact same direction.This is in general not possible due to the random orientation ofthe Delaunay lines, so instead we choose one of the two outgo-ing Delaunay lines ("1 and "2) that make the smallest angle with

Page 4 of 12

Example of Delaunay Tesselation and Voronoi Grid

Capabilities:

•Molecular Line Radiation Transport

•Dust Polarization

• Line Polarization (coming soon)

User-friendly:

•Open Source, free to download

• Graphical User Interface (GUI)

•Model Library

• FITS output

Setup:

• arbitrary density distribution

• arbitrary temperature distribution

• arbitrary molecule(s) and transition(s)

• arbitrary magnetic field morphology

• ...


Summary

Hybrid Scheme:

• can be as accurate as MC for certain setups while being tremendously faster (as fast as Magneto-Hydrodynamics)

• can accelerate MC Radiation Transport for Radiation-Hydrodynamic Simulations

Massive Star Formation:

• Long-range radiative forces keep the outflow cavities stable

ARTIST

•Molecular Line Radiation Transport

•Dust Polarization

• Line Polarization

Documents

New radiation transport methods for Radiation-Hydrodynamicskuiper/Rolf... · 2018. 3. 26. · Rolf Kuiper Dust Radiative Transfer 2013 October 10, 2013 not to scale Basic idea of