1. SPARX: Simulation Platform for Astrophysical Radiative Xfer SPARX, a new numerical program for non-LTE radiative transfer has been developed. In order to handle an arbitrary range of densities, velocities and geometries, the Accelerated Monte Carlo (AMC) method (Hogerheijde & van der Tak 2000) for calculating molecular excitation is used. The code is parallelized to take advantage of the speed provided by modern parallel computers, and other useful tasks such as uv sampling are implemented to provide users with a convenient interface for radiative transfer modeling. Preliminary code validation tests are presented and future development is discussed. ABSTRACT As astronomical instruments become more and more sophisticated and observational data become more detailed due to the increased sensitivity and resolution, radiative transfer modeling tools which can accommodate increasingly complex models are required so that accurate interpretations can be made from the data. With this in mind, SPARX was developed to meet the needs of observers: fast, multi-dimensional and easy to use. 1. INTRODUCTION Bernes (1979), A&A 73, 67-73 Goldreich & Kwan (1974), ApJ 189, 441 Hogerheijde & van der Tak (2000), A&A 362, 697 van der Tak et al. (2005), ESASP 577, 431 Bernes (1979), A&A 73, 67-73 Goldreich & Kwan (1974), ApJ 189, 441 Hogerheijde & van der Tak (2000), A&A 362, 697 van der Tak et al. (2005), ESASP 577, 431 REFERENCES Non-LTE radiative transfer is done by first calculating molecular level populations followed by ray- tracing to generate images. To accommodate an arbitrary range of densities, velocities and geometries, the Accelerated Monte Carlo (AMC) method first developed by Hogerheijde & van der Tak (2000) was chosen as the main algorithm for non-LTE radiative transfer in SPARX. In a nutshell, the AMC method solves for molecular level populations in a gridded model by calculating the detailed balance between radiative and collisional transitions Non-LTE radiative transfer is done by first calculating molecular level populations followed by ray- tracing to generate images. To accommodate an arbitrary range of densities, velocities and geometries, the Accelerated Monte Carlo (AMC) method first developed by Hogerheijde & van der Tak (2000) was chosen as the main algorithm for non-LTE radiative transfer in SPARX. In a nutshell, the AMC method solves for molecular level populations in a gridded model by calculating the detailed balance between radiative and collisional transitions 2. NON-LTE RADIATIVE TRANSFER WITH THE AMC METHOD Eric Chung ( 鍾恕 ) a, Sheng-Yuan Liu ( 呂聖元 ) a and Huei-ru Chen ( 陳惠茹 ) b a Institute of Astronomy & Astrophysics, Academia Sinica; b Institute of Astronomy, National Tsing-Hua University Eric Chung ( 鍾恕 ) a, Sheng-Yuan Liu ( 呂聖元 ) a and Huei-ru Chen ( 陳惠茹 ) b a Institute of Astronomy & Astrophysics, Academia Sinica; b Institute of Astronomy, National Tsing-Hua University Neglecting scattering effects, the mean radiation field intensity J ν for a particular cell can be approximated through the integration along random lines of sight, and subsequently solving the detailed balance self-consistently in each grid cell. The radiation field may contain contributions from the CMB, dust continuum or molecular line emission. 3 3 solve for local J ν and S ν self-consistently SνSνSνSν J local J ext 2 2 backtrack incoming photons to obtain external contribution to radiation field (J ext ) 1 1 calculate each cell in grid separately 4 4 check for convergence and loop through other cells until entire grid has converged J ext SνSνSνSν SνSνSνSν SνSνSνSν SνSνSνSν SνSνSνSν SνSνSνSν SνSνSνSν SνSνSνSν SνSνSνSν SνSνSνSν SνSνSνSν SνSνSνSν cloud with converged S ν projected sky grid integrate everything along line of sight: FFT + interpolation in uv space image cube uv data Fig. 1: The AMC method for solving molecular excitation Fig. 2: Image and visibility data generation 3. PARALLELIZATION By solving the radiation field and level populations self-consistently, convergence in optically thick cells which depend more on local conditions can be accelerated, as opposed to “traditional” Monte Carlo methods (Bernes 1979) which suffer from slow convergence in the presence of optically thick cells. node1 node2 node3 node1 node2 node3 1 1 each cell is solved separately on each node 2 2 synchronize all nodes and recalculate to account for changes Although accelerated, Monte Carlo methods are still much slower than deterministic RT methods such as LVG ro Microturbulence. To speed up the code so it may handle complex models with multi- dimensional parameter spaces, we took advantage of parallel computation which is now almost always cheaply available. Our scheme for parallelization is relatively straightforward: calculate each grid cell individually on each node of the parallel computer, then combine the results and synchronize all of the nodes so changes in the entire grid are accounted for. Although accelerated, Monte Carlo methods are still much slower than deterministic RT methods such as LVG ro Microturbulence. To speed up the code so it may handle complex models with multi- dimensional parameter spaces, we took advantage of parallel computation which is now almost always cheaply available. Our scheme for parallelization is relatively straightforward: calculate each grid cell individually on each node of the parallel computer, then combine the results and synchronize all of the nodes so changes in the entire grid are accounted for. Fig. 3: Parallelization scheme implemented in SPARX 4. VALIDATION In order to validate our implementation of the code, three problems were tested: (1) a 2-level H 2 O molecule in a static, isothermal and constant density sphere, (2) problem 1 plus a large velocity gradient and (3) test for self-consistency under LTE conditions. Problems 1 & 2 were used as benchmark problems for a comparison of available radiative transfer codes by van der Tak et al. (2005). 4.1) 2-LEVEL H 2 O IN AN ISOTHERMAL, CONSTANT DENSITY SPHERE UNDER STATIC CONDITIONS This problem consists of a spherical gas cloud (radius = 0.001 – 0.1 pc) which is static (V gas = 0), isothermal (T k = 40K) and has a constant density profile (n(H 2 ) = 10 4 cm -3. The sphere was divided into 200 radial shells, and level populations for a fictive 2-level (1 01 & 1 10 ) H 2 O molecule were calculated assuming an abundance of 10 -8 and 10 -9 relative to molecular hydrogen. The calculated populations are expressed as the excitation temperature (T ex ) for the (1 10 – 1 01 ) line transition defined according to the Boltzmann relation This problem consists of a spherical gas cloud (radius = 0.001 – 0.1 pc) which is static (V gas = 0), isothermal (T k = 40K) and has a constant density profile (n(H 2 ) = 10 4 cm -3. The sphere was divided into 200 radial shells, and level populations for a fictive 2-level (1 01 & 1 10 ) H 2 O molecule were calculated assuming an abundance of 10 -8 and 10 -9 relative to molecular hydrogen. The calculated populations are expressed as the excitation temperature (T ex ) for the (1 10 – 1 01 ) line transition defined according to the Boltzmann relation Fig. 4: Upper panels: calculated T ex of the 2-level H 2 O in the static sphere. Lower panels: results of the same calculation done with other non-LTE codes (cf. van der Tak 2005). 4.3) SELF-CONSISTENCY UNDER LTE CONDITIONS When the gas density becomes sufficiently high (> than the critical density of the molecular line), the cloud is considered to be in local thermodynamic equilibrium (LTE) and molecular line emission becomes thermalized. As a self-consistency test, we generated a model in LTE by populating the molecular levels according to the Boltzmann distribution, and did a χ 2 search by comparing multiple lines of HCO + produced from non-LTE calculations to that of the LTE model. The molecular gas number density and kinetic temperature were used as free parameters, and the HCO + J=1-0, 3-2, 5-4, 7-6, 9-8 spectra were used for calculating the reduced χ 2 according to When the gas density becomes sufficiently high (> than the critical density of the molecular line), the cloud is considered to be in local thermodynamic equilibrium (LTE) and molecular line emission becomes thermalized. As a self-consistency test, we generated a model in LTE by populating the molecular levels according to the Boltzmann distribution, and did a χ 2 search by comparing multiple lines of HCO + produced from non-LTE calculations to that of the LTE model. The molecular gas number density and kinetic temperature were used as free parameters, and the HCO + J=1-0, 3-2, 5-4, 7-6, 9-8 spectra were used for calculating the reduced χ 2 according to Fig. 6: χ 2 surface produced by comparing LTE spectra and non-LTE spectra As expected, the χ 2 search resulted in a point of minimum χ 2 which corresponded to the density and temperature values given for the LTE model, which is a good indication that the code produces consistent results. 5. EXTENDING TO MULTIPLE DIMENSIONS Since the only portion of the code that is coordinate-system dependent is the ray-tracing code responsible for approximating the local mean radiation field J ν, extending the code to different coordinate systems (e.g. 2-D cylindrical, 3-D Cartesian) is fairly easy – once ray-tracing can be achieved, the AMC “engine” can be directly used to calculate the level populations. Currently 2-D cylindrical code has already been developed for SPARX, though due to the lack of good test problems, the code has yet to be validated. Since the only portion of the code that is coordinate-system dependent is the ray-tracing code responsible for approximating the local mean radiation field J ν, extending the code to different coordinate systems (e.g. 2-D cylindrical, 3-D Cartesian) is fairly easy – once ray-tracing can be achieved, the AMC “engine” can be directly used to calculate the level populations. Currently 2-D cylindrical code has already been developed for SPARX, though due to the lack of good test problems, the code has yet to be validated. axisymmetric coordinates ray tracing code axisymmetric coordinates ray tracing code Cartesian coordinates ray tracing code (under development) spherical symmetric coordinates ray tracing code the AMC “engine” ray tracing codes SνSνSνSν SνSνSνSν J local J ext Fig. 6: extending the code to multiple dimensions 4.2) 2-LEVEL H 2 O IN AN ISOTHERMAL, CONSTANT DENSITY SPHERE WITH A LARGE VELOCITY GRADIENT For low H 2 O abundance, radiative trapping is unimportant, while as abundance increases, radiative trapping will decrease the effective critical density and drive the populations toward LTE. Fig. 4 shows the radial distribution of T ex as calculated by SPARX and other non-LTE codes. Results show that the SPARX calculations are consistent with that calculated by other codes. The second problem takes the first model, but adds a outflow velocity gradient of 100 km s -1 pc -1, resulting in gas velocities much larger than the thermal line width of ~0.2km s -1. This creates a condition in which the LVG approximation (Goldreich & Kwan 1974) is applicable, and the level populations can be solved analytically according to The second problem takes the first model, but adds a outflow velocity gradient of 100 km s -1 pc -1, resulting in gas velocities much larger than the thermal line width of ~0.2km s -1. This creates a condition in which the LVG approximation (Goldreich & Kwan 1974) is applicable, and the level populations can be solved analytically according to where N u and N l the level populations, K ul and K lu the collisional coefficients, A ul the Einstein A coefficient and β is the escape probability, n cr the critical density. For X(H 2 O) = 10 -8, the analytical solution gives T ex = 3.57K; while for X(H 2 O) = 10 -10, T ex = 3.33K. SPARX yields a solution close to the analytical solution for the lower abundance case; while for the higher abundance case, insufficient gridding of the velocity field results in underestimated optical depth towards the inner part of the sphere, which is a known artifact. This and the previous problem indicate that the accuracy of SPARX is comparable to other well known codes, while it’s parallel computation capabilities significantly decrease the amount of time required to solve such problems. For X(H 2 O) = 10 -8, the analytical solution gives T ex = 3.57K; while for X(H 2 O) = 10 -10, T ex = 3.33K. SPARX yields a solution close to the analytical solution for the lower abundance case; while for the higher abundance case, insufficient gridding of the velocity field results in underestimated optical depth towards the inner part of the sphere, which is a known artifact. This and the previous problem indicate that the accuracy of SPARX is comparable to other well known codes, while it’s parallel computation capabilities significantly decrease the amount of time required to solve such problems. Fig. 5: Radial distribution of T ex for the LVG sphere calculated by SPARX (upper panels) and other codes (lower panel, cf. van der Tak 2005)