CAT

Chandrasekhar Anisotropic Theory

Aim of the code

Computation of the non-resonant (NR) diffusion coefficients used in the orbit-averaged Fokker-Planck equation for an anisotropic Plummer model.

Installation

Install Julia by following the instruction at https://julialang.org/downloads/platform/.

To invoke Julia in the Terminal, you need to make sure that the julia command-line program is in your PATH.

On MacOS, we must create a link in /usr/local/bin (here for Julia 1.5):

$ sudo ln -s /Applications/Julia-1.5.app/Contents/Resources/julia/bin/julia /usr/local/bin/julia

Julia packages

Open the terminal in the folder packages and type

$ julia Install-pkg.jl

to install the following packages:

• HDF5
• ArgParse
• HypergeometricFunctions
• SpecialFunctions
• StaticArrays
• Interpolations
• PolynomialRoots

!! WARNING !!

DO NOT INTERRUPT THE DOWNLOADING OF THE PACKAGES !!!!

The root finding algorithm employed in this library is described in

• J. Skowron & A. Gould, 2012, "General Complex Polynomial Root Solver and Its Further Optimization for Binary Microlenses", arXiv:1203.1034

This algorithm aims to be fast and precise, more than the well known zroots procedure described in Numerical Recipes books, whose implementations in C and Fortran are not available as free software, according to the definition of the Free Software Foundation.

Compute the local diffusion coefficients in energy-angular momentum space

To compute the local NR diffusion coefficients in energy-momentum, one needs to open code/compute/ComputeCoeffsEnergy.jl.

Then, one needs to write for which radius r, energy E, angular momentum L and field star mass m one wants to compute the diffusion coefficients. Once this is done, save the file and run the command

$ julia ComputeCoeffsEnergy.jl --q 0.0

where --q 0.0 sets the cluster's anisotropy to q=0.0. This parameter needs to be a Float64.

Compute the diffusion coefficients in action space

To compute the NR diffusion coefficients in action space (Jr,L), one needs to open code/compute/ComputeCoeffsAction.jl.

Then, one needs to write for which radial action Jr, angular momentum L and field star mass m one wants to compute the diffusion coefficients. Once this is done, save the file and run the command

$ julia ComputeCoeffsAction.jl --q 0.0

where --q 0.0 sets the cluster's anisotropy to q=0.0. This parameter needs to be a Float64.

Compute a diffusion coefficients map in action space

To compute an action space map of the NR diffusion coefficients for a cluster with 10^5 single-mass stars, one needs to access the code/compute folder and run the following command in the terminal:

$ julia ActionOrbitalMap.jl --parallel no --q 0.0

where --q 0.0 sets the cluster's anisotropy to q=0.0 and -- parallel no runs the code without parallelisation.

If one wants to run this with parallelisation, one needs to run the following commands (supposing one is using bash)

$ export JULIA_NUM_THREADS=12
$ export JULIA_CPU_THREADS=12
$ julia -t 12 ActionOrbitalMap.jl --parallel yes --q 0.0

where 12 is the number of parallelised threads. One can check the number of threads by opening the Julia terminal and by running the command

julia> Threads.nthreads()

The resulting file will be created in the folder code/data under the name Dump_Diffusion_Coefficients_Action_Orbital_Map_q_0.0.hf5, where the suffix _q_0.0. depends on the anisotropy parameter given in argument --q.

This .hf5 files contains the following tables:

• q : value of the anisotropy parameter of the cluster.
• nbJrMeasure: number of sampling loci in radial action.
• nbLMeasure: number of sampling loci in angular momentum.
• tabL : 1D-table of angular momentum values used for the evaluation of the coefficients.
• tabJr : 1D-table of radial action values used for the evaluation of the coefficients.
• tabLJr: 1D-table of the action space loci (L,Jr) used for the evaluation of the coefficients.
• tabDNRJr: 1D-table of the coefficients DNR_Jr corresponding to the action space loci (L,Jr) in table tabLJr.
• tabDNRJr: 1D-table of the coefficients DNR_L corresponding to the action space loci (L,Jr) in table tabLJr.
• tabDNRJrJr: 1D-table of the coefficients DNR_JrJr corresponding to the action space loci (L,Jr) in table tabLJr.
• tabDNRJrL: 1D-table of the coefficients DNR_JrL corresponding to the action space loci (L,Jr) in table tabLJr.
• tabDNRLL: 1D-table of the coefficients DNR_LL corresponding to the action space loci (L,Jr) in table tabLJr.

Note that the sampling in action space is done in log-log. One can modify the region of sampling by differently modifying the section Action space parameter in the file ActionOrbitalMap.jl.

Those files can be recovered using, for example, Mathematica, through the command

Import[NotebookDirectory[] <> "data/Dump_Diffusion_Coefficients_Action_Orbital_Map_q_0.0.hf5", {"Datasets", "tabDNRJr"}]

here for the (already computed )table tabDNRJr for a cluster with anisotropy q=0.0.

Compute the diffusion coefficients in action space

To compute the NR diffusion coefficients in action space (Jr,L), one needs to open code/compute/ComputeCoeffsAction.jl.

Then, one needs to write for which radial action Jr, angular momentum L and field star mass m one wants to compute the diffusion coefficients. Once this is done, save the file and run the command

$ julia ComputeCoeffsAction.jl --q 0.0

where --q 0.0 sets the cluster's anisotropy to q=0.0. This parameter needs to be a Float64.

Compute dF/dt in action space

To compute an action space map of dF/dt for a cluster with 10^5 single-mass stars, one needs to access the code/compute folder and run the following command in the terminal:

$ julia MappingdFdt.jl --parallel no --q 0.0

where --q 0.0 sets the cluster's anisotropy to q=0.0 and -- parallel no runs the code without parallelisation.

If one wants to run this with parallelisation, one needs to run the following commands (supposing one is using bash)

$ export JULIA_NUM_THREADS=12
$ export JULIA_CPU_THREADS=12
$ julia -t 12 MappingdFdt.jl --parallel yes --q 0.0

where 12 is the number of parallelised threads. One can check the number of threads by opening the Julia terminal and by running the command

julia> Threads.nthreads()

The resulting file will be created in the folder code/data under the name Dump_dFdt_Map_q_0.0.hf5, where the suffix _q_0.0. depends on the anisotropy parameter given in argument --q.

This .hf5 files contains the following tables:

• q : value of the anisotropy parameter of the cluster.
• nbJrMeasure: number of sampling loci in radial action.
• nbLMeasure: number of sampling loci in angular momentum.
• Mtot: the total mass of the cluster.
• Npart: the number of stars in the cluster.
• tabL : 1D-table of angular momentum values used for the evaluation of the coefficients.
• tabJr : 1D-table of radial action values used for the evaluation of the coefficients.
• tabLJr: 1D-table of the action space loci (L,Jr) used for the evaluation of the coefficients.
• tabdFdt: 1D-table of the values of dF/dt corresponding to the action space loci (L,Jr) in table tabLJr.

Note that the sampling in action space is done linearly. One can modify the region of sampling by differently modifying the section Action space parameter in the file ActionOrbitalMap.jl.

Those files can be recovered using, for example, Mathematica, through the command

Import[NotebookDirectory[] <> "data/Dump_dFdt_Map_q_0.0.hf5", {"Datasets", "tabdFdt"}]

here for the (already computed )table tabdFdt for a cluster with anisotropy q=0.0.