The spectrum and the spatial shape of the X-ray source

marx provides support for both extended and point sources, which can be either on or off–axis. The sources can have a flat energy spectrum or they can use an energy spectrum supplied by the user and in the next section we describe those options. Then, we explain the different models for the spatial distribution that these source can have on the sky. The following sources are currently available:

  • POINT: Point Source
  • GAUSS: Radially symmetric Gaussian
  • BETA: Cluster Beta model
  • DISK: Disk or annulus (e.g. SN remnant)
  • LINE: Straight line
  • IMAGE: Input from a FITS image file
  • SAOSAC: Input from an SAOSAC FITS rayfile
  • RAYFILE: Input from a marx rayfile
  • SIMPUT: Input from a fits file following the SIMPUT standard.
  • USER: Dynamically linked user–supplied model

First, it is important to have an understanding of what a source means in the context of marx. It is particularly important to understand how marx interacts with a source model for the proper implementation of a user–defined source. As far as marx is concerned, a source is characterized by the distribution of rays in energy, time, and direction that it produces at the entrance aperture of the telescope. Implicit in this characterization is the assumption that the source is sufficiently far away that the flux does not vary with position across the telescope.

To be more precise, let

\[F(t,E,{\hat{p}},{\vec{x}}) ({\hat{p}}\cdot{\mbox{d}\vec{A}\;}) {\mbox{d}\Omega\;} {\mbox{d}E\;} {\mbox{d}t\;}\]

denote the number of rays with energy between \(E\) and \(E + dE\), and whose directions lie in a solid angle \(d\Omega\) about the direction \({\hat{p}}\), crossing an area element \({\mbox{d}\vec{A}\;}\) at \({\vec{x}}\) during the time \(dt\). That is, \(F(t,E,{\hat{p}},{\vec{x}})\) is a space-time dependent flux density. If the source is very far away, as is the case for objects of astrophysical interest, the spatial dependence of the flux density may be ignored when \({\vec{x}}\) refers to points near the telescope. For this reason, \(F\) will be assumed to be independent of \({\vec{x}}\). Similarly, since the angular acceptance of the telescope is very small (on the order of arc-minutes), to first order we need only consider area elements \({\mbox{d}\vec{A}\;}\) whose normals are in the direction of the source. Hence, \({\hat{p}}\cdot{\mbox{d}\vec{A}\;}\) will simply be written as \({\mbox{d}A\;}\).

We can consider various classes of source models according to whether or not \(F\) factors. For example, all marx sources (with the exception of USER sources) produce fluxes that are time-independent with an energy spectrum that is uncorrelated with the direction of the flux (see USER Source for more information about the USER source). For the rest of this section, we will only consider the class of models for which \(F\) can be written

\[F(t,E,{\hat{p}}) = f(t) \cdot f_E(E) \cdot f_p({\hat{p}}).\]

When such a factorization is possible, it will be assumed that \(f_E(E)\) and \(f_p({\hat{p}})\) are normalized, i.e.,

\[1 = \int_{\Omega} {\mbox{d}\Omega\;} f_p({\hat{p}}),\]

and

\[1 = \int{\mbox{d}E\;} f_E(E) .\]

With this normalization,

\[f(t) = \int {\mbox{d}E\;} \int_{\Omega} {\mbox{d}\Omega\;} F(t, E, {\hat{p}})\]

is simply the total flux, which can depend upon time. In the following, we shall call \(f_E\) the energy spectrum, and \(f_p\) the angular distribution of the source.

Spectrum of the simulated X-ray source

In the marx parameter file, marx.par, the parameter SpectrumType is used to specify the function \(f_E\) for the following source types: "POINT", "LINE", "GAUSS", "BETA", "DISK", and "IMAGE" (the other source have a source specific way to input the energy, e.g. for a "SAOSAC" source the energy of each ray is already included in the ray file).

Currently, \(f_E\) can only be a FLAT spectrum or a FILE spectrum. Similarly, when \(f(t)\) is time-independent, as it for all marx sources in this class, then its value is specified by the SourceFlux parameter. For simulations with SpectrumType="FLAT", the function \(f_E\) is assumed to be constant over the specified energy range with a normalization given by SourceFlux. Alternatively, the FILE type will use a tabulated spectral energy distribution read from an external ASCII file. The following parameters describe the source spectrum:

SpectrumType

(default: FLAT) Select spectrum type. Can be FLAT or FILE.

SourceFlux

(default: 0.003) Incoming ray flux (photons/sec/cm2). For SpectrumType="FLAT" this number must be positive. If SpectrumType="FILE" this number can be positive to renormalize the spectrum file to the given source flux. If it is negative, then the normalization from the SpectrumFile will be used.

SpectrumFile

(default: flux.dat) Input spectrum filename (only used if SpectrumType="FILE"). The file has to consist of two columns of data with no header line. The first column contains the energy of the upper bin edge in keV, the second the flux density in photons/s/cm^2/keV in that bin (the flux in the first row is ignored, because there is no row before which would define the lower energy edge of the bin). Various tools exist to help in generating this file:

MinEnergy

(default: 1.775) MIN ray energy in keV (only used if SpectrumType="FLAT")

MaxEnergy

(default: 1.775) MAX ray energy in keV (only used if SpectrumType="FLAT")

Spatial distribution of the simulated source

The distribution function \(f_p({\hat{p}})\) characterizes the angular distribution of the flux and, hence, the angular distribution of the source. The nominal aimpoint of the observation (given by RA_Nom and Dec_Nom) can differ from the source position (given by SourceRA and SourceDEC) to simulate off-axis sources.

By convention, \(f_p({\hat{p}})\) is assumed to be normalized to unity, i.e.,

\[1 = \int_{0}^{\pi} \sin\theta {\mbox{d}\theta\;} \int_0^{2\pi} d{\phi} f_p(\theta, \phi) ,\]

where \({\hat{p}}\) has been expressed in spherical coordinates. For an azimuthally symmetric source, \(f_p\) is independent of \(\phi\) and the normalization condition reduces to

\[1 = 2\pi \int_{0}^{\pi} {\mbox{d}\theta\;} \sin\theta f_p(\theta) .\]

In marx the following parameter selects model for the spatial distribution of the source:

SourceType

(default: POINT) The following values are allowed: "POINT", "LINE", "GAUSS", "BETA", "DISK", "IMAGE", "SAOSAC", "RAYFILE", "SIMPUT", and "USER". Depending on the source model chosen, further parameters (such as the radius of the disk) may be required.

Each availble model is now described in more detail.

POINT Source

The POINT source corresponds to an angular distribution function given by

\[f_p(\theta, \phi) = \frac{1}{2\pi} \delta (\phi) \delta(1 - \cos \theta)\]

A POINT source requires no further parameter to specify the spatial distribution.

LINE Source

The LINE source corresponds to an angular distribution function given by

\[f_p(\theta, \phi) = \frac{1}{\theta_0\theta}\cdot \frac{1}{2} \big[\delta(\phi - \phi_0) + \delta(\phi - \phi_0 - \pi) \big]\]

for \(\theta < \theta_0\) and zero otherwise.

S-LinePhi

(default: 0) Line source orientation angle \(\phi_0\) (degrees)

S-LineTheta

(default: 1800) Line source length \(\theta_0\) (arcsec)

GAUSS Source

The GAUSS source corresponds to an angular distribution function given by

\[f_p(\theta, \phi) = \frac{1}{\pi} e^{-\theta^2/\theta_0^2}\]

where \(\theta_0\) determines the width of the Gaussian distribution:

S-GaussSigma

(default: 60) gauss source sigma (arcsec)

BETA Source

The BETA source corresponds to an angular distribution function given by

\[f_p(\theta, \phi) = \frac{1}{2\pi} \cdot \frac{6}{\theta_c}(\beta - \frac{1}{2}) \big[ 1 + (\frac{\theta}{\theta_c})^2 \big]^{-3\beta + \frac{1}{2}}.\]

This distribution is used to model galaxy clusters.

S-BetaCoreRadius

(default: 10) Core radius \(\theta_c\) (arcsec)

S-BetaBeta

(default: 0.7) \(\beta\) value

DISK Source

The DISK source corresponds to an angular distribution function given by

\[f_p(\theta, \phi) = \frac{1}{2\pi} \cdot \frac{2}{\theta_1^2 - \theta_0^2}\]

for \(\theta_0 <= \theta < \theta_1\). Outside this region, it is zero. This source actually generates a ring structure and is useful for modeling a supernova remnant.

S-DiskTheta0

(default: 0) Min disk \(\theta_0\) (arcsec)

S-DiskTheta1

(default: 60) Max disk \(\theta_1\) (arcsec)

IMAGE Source

This option creates photons distributed on the sky according to an input image. The probability that a ray starts at a given position is proportional to the pixel value at this point. Within a pixel, the position is randomized. marx inspects the header of the file for a WCS specification and extracts the pixel scale. However, it does not extract the position or orientation on the sky. marx will just assume that the image is centered on the optical axis and that the axes directions are aligned with the detector axes.

S-ImageFile

(default: image.fits) fits filename for IMAGE source

SAOSAC Source

The SAOSAC source allows SAOSAC raytrace files to be used as input for marx. SAOSAC is a high-fidelity raytracer for the Chandra mirrors, with a much higher level of detail than the module supplied with marx. Only in very rare cases is this needed for the end-user. More details can be found in Using marx with SAOSAC.

SAOSACFile

(default: saosac.fits) marx input source/output ray filename

RAYFILE Source

The RAYFILE source can be used to dublicate the source properties of a previous marx simulation. Using this as a source keeps the photon properties energy and position as specified in the ray file. Thus, the source properties are identical to those used to generate the original ray file, but the Chandra response to them might be different, e.g. if a different detector or dither is chosen. Rayfiles are produced by setting DumpToRayFile=yes.

RayFile

(default: marx.output) marx input source/output ray filename

SIMPUT Source

marx supports the SIMPUT standard, which is a fits based description of sources, that allows a large number of sources with different spectra, light curves, and shapes on the sky. This file format is supported by a number of other simulators (e.g. for ATHENA), so integrating it in marx allows users to use the same source specification for different X-ray missions. The support in marx is through the SIMPUT code which needs to be installed separately and is linked dynamically at runtime if SourceType="SIMPUT".

S-SIMPUT-Source

(default: CDFS_cat_galaxies.fits) Filename of SIMPUT Catalog

S-SIMPUT-Library

(default: /melkor/d1/guenther/soft/simput/lib/libsimput.so) Path to dynamically linked file libsimput.so

USER Source

The USER source is the most versatile of the marx sources. With a user–defined source, each ray may be given an independent energy, time, and direction. This flexibility means that one does not need to require that the flux density factorize as was assumed for the other marx sources. Using a USER source model, sources whose spectrum changes with time, complex extended objects, etc. can be simulated.

UserSourceFile

(default: ../doc/examples/user-source/pnts.so) dynamically linked source filename

UserSourceArgs

(default: pnts.dat) user source parameter

A user-defined source model must be created by the user using a language such as C and then compiled as a shared object. During run-time, marx will dynamically link to this shared object and use it to generate rays. To use this source, first and foremost, the underlying operating system must support dynamic linking. Operating systems such as Linux and Solaris support dynamic linking while others such as NeXT do not. It is important to understand that creating a user-defined source does not mean that marx must be recompiled. If that were the case, then there would be no value to a user-defined source.

Creating a such a source is relatively simple and is best accomplished using the C programming language. The C source file must define three functions that marx will call during run-time:

user_open_source
user_close_source
user_create_ray

The user_open_source function will be called by marx before any rays are generated. The purpose of this function is to initialize any data structures required by the user_create_ray function. The user_create_ray function will be called one time for each ray generated. The purpose of this routine is to assign an energy, time, and direction to a ray. Finally, the user_close_source function will be called when marx has finished processing rays. Each of these functions are described in more detail below.

user_open_source

The user_open_source function has the prototype:

int user_open_source (char **argv, int argc,
                      double area,
                      double cosx,
                      double cosy,
                      double cosz);

The value of the marx.par parameter UserSourceArgs will be broken into an array of whitespace separated strings and passed to user_open_source via the argv parameter. The parameter argc indicates the number of such strings. The actual meaning of these strings will depend upon the details of the user-defined source. For example, if the user-defined source needs to read an external data file, the parameter can represent the name of the data file.

The area parameter specifies the area in cm\(^2\) of the entrance aperture of the mirror. Knowledge of this value is necessary to compute the time interval between rays since the incoming flux must be multiplied by this value to generate the total incoming photon rate.

The other three parameters cosx, cosy, and cosz are the direction cosines of a ray from a reference point on the source to the origin of the marx coordinate system. These numbers are derived from the marx parameter file SourceRA and SourceDEC parameters. For an on axis source, cosy and cosz will be set to zero, but cosx will be set to -1. If the reference point of the user defined source is always on axis, these parameters may be ignored and the actual parameter values for SourceRA and SourceDEC will have no affect on the rays generated by source. However, if one would like to position the source off-axis via the SourceRA and SourceDEC parameters, the values of the direction cosines will need to be taken into account. An example of this is presented below.

Upon success, user_open_source must return 0. If for any reason it fails, e.g, unable to open a file, it must return -1.

The simplest example of user_open_source is one which does nothing:

int user_open_source (char **argv, int argc,
                      double cosx,
                      double cosy,
                      double cosz)
{
   return 0;   /* Success */
}

user_close_source

The user_close_source function has the prototype:

void user_close_source (void);

Its purpose is to free up any resources acquired by the source. For example, if the source dynamically allocated memory, user_close_source should deallocate it.

user_create_ray

The user_create_ray function is the function that actually defines the source by endowing each ray with a direction, energy, and time. It has the following prototype:

int user_create_ray (double *delta_t, double *energy,
                     double *cosx, double *cosy, double *cosz);

Since the purpose of this routine is to assign a ray an energy, time, and direction, the parameters are actually pointer types and the requested information is passed back to the calling routine via the parameter list. It is important to note that the ray is completely undefined prior to calling this function.

The delta_t parameter is used to give the ray a time-stamp. Actually it does not refer directly to the absolute time of the ray; rather, its value should refer to the time since the last ray was generated. For example, if a ray is generated every second,

*delta_t = 1.0;

should be used. If *delta_t is set to -1.0, then marx will generate the time based on the SourceFlux parameter. Otherwise, the value should be set in a manner consistent with the flux and the geometric area of the mirror.

The meaning of the other parameters that specify the energy and direction cosines should be rather clear. If energy is set to -1.0, then marx will use the setting of the SpectrumType parameter to assign an energy to the ray.

Compiling a User-Defined Source

The procedure for compiling a user-defined source as a shared object will depend upon the operating system. For details, consult you compiler and linker manual. For the purposes of this section, it is assumed that the file containing the code for the user-defined source is called mysource.c. This may be compiled as a shared object under Linux using gcc via the command:

gcc -shared mysource.c -o mysource.so

If mysource.c requires other libraries, they should also be included on the command line. The syntax is slightly different under Solaris:

cc -G mysource.c -o mysource.so

To actually use the source in marx , set the marx.par parameter SourceType to "USER" and also set the parameter UserSourceFile to point to the full absolute filename for mysource.so. It is usually necessary to use an absolute filename because of the way the dynamic linker searches for shared objects. Finally, set the parameter UserSourceArgs to a value that is appropriate to your source.

If running marx using your dynamically linked source causes it to crash, do not assume that the bug is in marx . Rather, it is most likely a bug in your code. Make sure that the interface routines are properly prototyped and that the routines return the proper values to marx . If you use dynamic memory allocation, check the return status of routines such as malloc. Finally, look at the examples provided with the marx distribution and try to run those.

Examples of User-Defined Sources

The simplest source is that of a point source. Although marx already provides built-in support for this source, it is instructive to write it as a user-defined source. Here is the complete C code for such a source:

#include <stdio.h>

static double Source_CosX;
static double Source_CosY;
static double Source_CosZ;

int user_open_source (char **argv, int argc, double area,
                      double cosx, double cosy, double cosz)
{
   Source_CosX = cosx;
   Source_CosY = cosy;
   Source_CosZ = cosz;
   return 0;
}

void user_close_source (void)
{
}

int user_create_ray (double *delta_t, double *energy,
                     double *cosx, double *cosy, double *cosz)
{
   *cosx = Source_CosX;
   *cosy = Source_CosY;
   *cosz = Source_CosZ;

   *delta_t = -1.0;
   *energy = -1.0;

   return 0;
}

First of all, note that energy and delta_t have been set equal to -1.0 in user_create_ray. This indicates to marx that it should compute the time and energy of the ray via the SpectrumType and SourceFlux parameters. For this reason, the area parameter was not used by user_open_source. Since the direction cosines passed to user_open_source refers to the vector from the position of the source to the origin where the telescope is located, those values were saved and used in user_create_ray.

For more complex examples, look at the files under marx/doc/examples in the marx distribution.