Cluster Mass Mixing Models.

These models are for determining distributions of gravitating mass in spherical, hydrostatic atmospheres. A hot atmosphere is approximated as a set of concentric spherical shells, each containing isothermal gas. The mixing models combine thermal spectra for the shells, with weights determined by the gravitational potential for the model, to produce projected spectra for annular regions centred on a cluster. The models have much in common with the projct mixing model, but the gas density distribution is determined by the gravitational potential and the assumption of hydrostatic equilibrium. Details of the models are discussed in Nulsen, Powell & Vikhlinin (2010) and the clmass model explained fully. For the clmass model, the gravitating matter density is assumed to be constant in each spherical shell. The monomass model is physically identical to clmass, but parametrized by the differences in mass densities for adjacent shells to ensure that the gravitating mass density is a non-increasing function of the radius. The nfwmass model treats an atmosphere as a nested set of isothermal, spherical shells, but with the Navarro, Frenk & White form for the gravitational potential.

Spectra should be extracted from concentric, circular annuli centered on a cluster (elliptical annuli are rejected). All spectra for one annulus must belong to the same data group, while spectra for distinct annuli must belong to separate data groups. Spectra must be provided for a complete set of annuli filling the range between the innermost and outermost radius. For each model, the inner radius of the innermost annulus is specified as the first model parameter.

To handle cases where the observations do not cover the whole of a cluster, X-ray emission from beyond the inner edge of the outermost annulus may be modeled with an isothermal beta model. This feature is designed to deal with background X-ray emission from parts of a cluster outside the region that has been observed. It adds a model-dependent element to the mass models. When the beta model is used, the pressure is assumed to be continuous between the two outermost shells, but the gravitational potential is ignored for the outermost shell. If the beta model is disabled, the outermost shell is treated like any other.

Using the models:

The models rely on the XFLT keywords in much the same way as projct. Each spectrum must include, at least, the XFLT0001 keyword specifying the outer radius of the corresponding annulus. If present, XFLT0002 must equal XFLT0001 (annuli must be circular). XFLT0003 is ignored. If present, the following pairs of keywords (XFLT0004/5, XFLT0006/7, etc) give ranges of angle that are summed and divided by 360 to determine the fraction of the total annulus covered by a spectrum. Note that the same effect can be achieved by specifying this fraction in the AREASCAL keyword and leaving XFLT0004/5, etc, undefined (this approach provides greater flexibility when there is more than one spectrum in each data group).

It is essential for the models to link the temperature of the thermal model for each shell to the corresponding temperature parameter of the model. All unused shell parameters must be frozen. The norms of the thermal model must be tied (equal) for all shells (which happens by default). The one free norm applies to gas occupying the intersection between the innermost spherical shell and the 3-dimensional cylinder corresponding to the innermost annulus.

Model parameters are as follows:

For clmass:

rinner - inner radius of innermost shell (same units as XFLT0001)
a - core radius for beta model (same units as XFLT0001)
beta - beta model exponent
switch - 1 enable, 0 disable beta model
kTa... - shell temperatures - must be linked to the corresponding thermal component, or frozen
dena... - gravitating matter densities for the shells - unused densities must be frozen, including the density for the outermost shell when the beta model is used

For monomass:

rinner - inner radius of innermost shell (same units as XFLT0001)
a - core radius for beta model (same units as XFLT0001)
beta - beta model exponent
switch - 1 enable, 0 disable beta model
kTa... - shell temperatures - must be linked to the corresponding thermal component, or frozen
dela... - gravitating matter density differences for the shells. In terms of clmass the parameters, dela = dena - denb, delb = denb - denc, etc. Unused parameters must be frozen, including that for the outermost shell when the beta model is used

For nfwmass:

rinner - inner radius of innermost shell (same units as XFLT0001)
a - core radius for beta model (same units as XFLT0001)
beta - beta model exponent
switch - 1 enable, 0 disable beta model
kTa... - shell temperatures - must be linked to the corresponding thermal component, or frozen
nfwa - NFW scale length (same units as XFLT0001)
nfwpot - Normalization for NFW potential

Units:

Lengths can be specified in any unit, but must be consistent. Internal units depend on the length unit. If the physical length corresponding to the unit in XFLT0001 is u, then the densities used by the model are in units of keV / (G μ mH u2), where keV is the energy of 1 keV, G is Newton's constant and μ mH is the mean mass per particle in the gas. For nfwmass, the normalization constant, nfwpot, is 4 π G ρ0 a2 μ mH in units of keV. Here, a is the NFW scale length in physical units and ρ0 is the normalizing density for the NFW potential (mass density is ρ0 /[r/a (1 + r/a)2]).

Notes:

The number of shells available in these models is determined solely by the number of entries in model.dat. If you need more shells, simply add more shell parameters to model.dat (being sure to add them in pairs for clmass and monomass) and rebuild the models.

Code for the gravitational potentials has been separated from the remainder of the code to make it relatively easy to add new potential models. See the notes with the model code.

Installing the models

Download the compressed tar file massmodels.tar.gz and untar in your chosen directory. Several sub-directories will be created. The model code is in model, useful support programs in support, tcl scripts in tcl, and test data and scripts in test. To build these models go to the model sub-directory and follow the instructions in the file USING. To build the support programs go to the support sub-directory and type make. This file was updated on 1/31/2012 with changes to avoid problems with some newer versions of g++.


Keith Arnaud, Lab. for High Energy Astrophysics, NASA/Goddard Space Flight Center

Xspec Xspec Home Page


HEASARC Home | Observatories | Archive | Calibration | Software | Tools | Students/Teachers/Public

Last modified: Tuesday, 31-Jan-2012 18:21:20 EST