f(J) models code

Code for generating action-based distribution function models for spheroids, as described in Posti et al. (2015).

Features

The code generates an axisymmetric self-consistent model by specifying a Distribution Function (DF) which is a double-power law function of the action integrals. The model's density (and higher moments) are computed integrating the DF, then the self-consistent potential is found by iterative procedure (convergence criteria are currently in the form of distance from the potential of the previous iteration and on the tensor Virial theorem). The action integrals are computed in the Staeckel Fudge (Binney 2012) approximation.

• the models have a double-power law density profile: the central and outer slopes in the density profile are determined mainly by the DF's power law slope in the small- and large-action regimes. The code has currently hardwired such DF's parameters such that the final profile follows that of classical models, such as Hernquist, Isochrone and NFW.
• the code stores the coefficients of the multipole expansion of e.g., density, potential and velocity dispersion distributions in output files at every iterations. Such output files can be managed easily with the Python tool pyfJmod.
• the models can currently be
  • self-consistent single-component
  • single-component with external potential (to be fully implemented...)
  • self-consistent two-components
• it uses a 22-points tanh-sinh quadrature (see Takahasi & Mori 1973) three-dimensional integrator to optimize a (typically) highly peaked DF at the centre. Different choices for the integration rule (e.g., Gauss-Legendre) and for the number of points used are implemented and can be switched on.
• the code benefits from a multi-threaded OpenMP implementation, vector-register instructions (SSE/AVX) and compiler's optimization flags.

Installation

Prerequisites

• Gnu Scientific Library - GSL. If you use a Debian-like Linux distribution (e.g., Ubuntu) you can download from the official repositories the packages libgsl0ldbl libgsl0-dev.

Get the code

Use one of the two following methods (the first one is recommended).

Use git (recommended)

Clone the repository in your preferred directory:

git clone https://github.com/lposti/fJmodels

Download release

Download the .tar or compressed .tar.gz package from the release tab at the beginning of the code homepage, then unpack the package.

Compile

cd to the cloned/unpacked directory and

./configure && make

At this step you have to make sure that either GSL are installed in standard paths (e.g., /usr/lib, /usr/local/lib, etc.) or that you pass the prefix of the local GSL installation to the configure script. The latter can be done configuring with the option

./configure --with-gslprefix=/absolute/path/to/prefix

If GSL are still not found an error will occur: please check that you have a functional GSL installation in the standard paths (e.g., by installing from official repositories) or use the correct prefix of the local install (please remember not to append /lib or /include to the prefix, which should be a directory containing both of them!)

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After running make, if the installation was successful an executable named fJmodels was created in the current directory.

Usage

The code can be launched as follows:

./fJmodels <input_filename>

where the argument is the filename of the parameter file and it is optional: if not specified the code assumes that the input parameter file is param.txt. The outputs will be written in the directory named models, which must exist.

Parameter file

The code can currently compute both 1-component and 2-component models, for which all the parameters are specified in the parameter file. An example is given as the file param.txt.

All the DFs are of the following general form:

and A, B are slopes defined by the model type desired (see Posti et al. 2015). The above DF is an even function in the azimuthal velocity (since it depends only on the absolute value of the vertical angular momentum ![alt text] (doc/imgs/Jphi.png "vertical angular momentum")), so the models do not roatate. We parametrize rotation adding an odd component in ![alt text] (doc/imgs/Jphi.png "vertical angular momentum"): ![alt text] (doc/imgs/rot.png)

The hyberbolic tangent is an odd function and the parameter \chi controls the steepness of the model's rotation curve. If \chi is greater than zero, it is assumed k=0.5 (for a maximally rotating configuration). If \chi=0, k=0 and the model does not rotate.

Follows a schematic descriptions of the implemented parameters that can be set in the file to specify the model.

Parameter Name	Description: mandatory for the first component, optional for the second	Default Value
model / 2:model	Model type: currently Hernquist, Isochrone, NFW	Hernquist
A, B / 2:A, 2:B	Slopes of the DF for small and large actions. They are ignored if model is specified	None
h(J):dphi / 2:h(J):dphi	DF parameter \delta_\phi for h(J)	0.55
h(J):dz / 2:h(J):dz	DF parameter \delta_z for h(J)	0.55
g(J):dphi / 2:g(J):dphi	DF parameter \delta_\phi for g(J)	1.0
g(J):dz / 2:g(J):dz	DF parameter \delta_z for g(J)	1.0
chi / 2:chi	Controls the steepness of the rotation curve. If greater than 0 an odd part is added to the DF	0.0
mass / 2:mass	Component's mass (parameter M_0 in Posti et al. 2015)	1.0
r0 / 2:r0	Component's scale radius (defines parameter J_0=sqrt(GM_0*r_0) as in Posti et al. 2015)	1.0
q / 2:q	Flattening of the initial guess potential	1.0
itermax	Defines the number of iterations computed	5

The default configuration generates a nearly-isotropic, spherical, non-rotating, Hernquist-like model.

Output

Use pyfJmod package for analysis and plotting f(J) models data.

References

Main paper: Posti et al. (2015)

Staeckel Fudge: Binney (2012)