SPARX: Simulation Platform for Astrophysical Radiative Xfer

1. backtrack incoming photons to obtain external contribution to radiation field (Jext) calculate each cell in grid separately synchronize all nodes and recalculate to account for changes each cell is solved separately on each node ray tracing codes 2 1 1 2 the AMC “engine” Sν Sν Sν Sν spherical symmetric coordinates ray tracing code Sν node3 node1 node2 solve for local Jν and Sν self-consistently Sν 3 check for convergence and loop through other cells until entire grid has converged 4 axisymmetric coordinates ray tracing code Jext Jlocal Jext Jlocal Sν Sν Cartesian coordinates ray tracing code (under development) node1 node2 node3 Jext cloud with converged Sν projected sky grid uv data FFT + interpolation in uv space image cube integrate everything along line of sight: SPARX: Simulation Platform for Astrophysical Radiative Xfer Eric Chung (鍾恕)a, Sheng-Yuan Liu (呂聖元)a and Huei-ru Chen (陳惠茹)b aInstitute of Astronomy & Astrophysics, Academia Sinica; bInstitute of Astronomy, National Tsing-Hua University ABSTRACT 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. 4.2) 2-LEVEL H2O IN AN ISOTHERMAL, CONSTANT DENSITY SPHERE WITH A LARGE VELOCITY GRADIENT 1. INTRODUCTION 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. 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 2. NON-LTE RADIATIVE TRANSFER WITH THE AMC METHOD 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 where Nu and Nl the level populations, Kul and Klu the collisional coefficients, Aul the Einstein A coefficient and β is the escape probability, ncr the critical density. Fig. 5: Radial distribution of Tex for the LVG sphere calculated by SPARX (upper panels) and other codes (lower panel, cf. van der Tak 2005) Neglecting scattering effects, the mean radiation field intensity Jν 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. For X(H2O) = 10-8, the analytical solution gives Tex = 3.57K; while for X(H2O) = 10-10, Tex = 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. 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 χ2 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 χ2 according to Fig. 1: The AMC method for solving molecular excitation Fig. 2: Image and visibility data generation 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. 3. PARALLELIZATION 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. 6: χ2 surface produced by comparing LTE spectra and non-LTE spectra Fig. 3: Parallelization scheme implemented in SPARX As expected, the χ2 search resulted in a point of minimum χ2 which corresponded to the density and temperature values given for the LTE model, which is a good indication that the code produces consistent results. 4. VALIDATION In order to validate our implementation of the code, three problems were tested: (1) a 2-level H2O 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). 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 Jν, 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. 4.1) 2-LEVEL H2O 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 (Vgas = 0), isothermal (Tk = 40K) and has a constant density profile (n(H2) = 104 cm-3. The sphere was divided into 200 radial shells, and level populations for a fictive 2-level (101 & 110) H2O 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 (Tex) for the (110 – 101) line transition defined according to the Boltzmann relation Fig. 6: extending the code to multiple dimensions For low H2O abundance, radiative trapping is unimportant, while as abundance increases, radiative trapping will decrease the effective critical density and drive the populations toward LTE. REFERENCES Fig. 4: Upper panels: calculated Tex of the 2-level H2O in the static sphere. Lower panels: results of the same calculation done with other non-LTE codes (cf. van der Tak 2005). 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 Fig. 4 shows the radial distribution of Tex as calculated by SPARX and other non-LTE codes. Results show that the SPARX calculations are consistent with that calculated by other codes.