# The marx Models for the Spacecraft¶

marx provides the capability to simulate the various combinations of scientific instruments onboard the Chandra satellite and includes models of the Chandra mirrors, the low- and high-energy transmission grating assemblies (LETG and HETG), and the HRC and ACIS focal plane detectors. marx also provides support for simulations of ground–based calibration tests at XRCF. The HRMA shutter assembly is modeled as well as sources at a finite distance. The purpose of this Section is to provide technical information about the implementation of the various subsystems of marx.

With very few exceptions, a marx user should not change the parameters that control the spacecraft model. Their defaults are based on the best available Chandra calibration information. Thus, only very few marx parameters are described in this Section. For reference a list of all parameters that control the spacecraft model can be found in Further marx parameters.

## The marx Coordinate System¶

The coordinate system used by marx is shown in Figure Coordinate System. The origin is in the vertical plane of symmetry of the system. The orientation is such that the X–axis is parallel to the optical axis of the telescope and increases away from the focal plane. Thus, astrophysical sources are at a distance of $$x=+\infty$$. The Z–axis is perpendicular to the horizontal plane of symmetry of the telescope passing through the X–axis and parallel to the SIM translation direction. The Y–axis completes the system and corresponds to the dispersion direction for the gratings on Chandra. The origin of the marx coordinate system is located at the center of the Rowland torus.

The MARX Coordinate System.

This coordinate system is equivalent to the Chandra coordinate system described in the Chandra Observatory Guide. More information on the marx and Chandra coordinate systems can be found in several CXC Manuals available from http://cxc.harvard.edu/ciao/manuals.html . The physical placements of the mirror, gratings, detectors and other hardware components of Chandra in marx were taken from those documents.

## HRMA model¶

marx implements two different models for the HRMA onboard Chandra. Selection between these two models is accomplished using the MirrorType parameter. The first of these models, the EA-MIRROR model, is a simpler representation of the HRMA based on effective area and point spread function tables. This model does not include any of the detailed characterization of the mirror such as misalignments, tilts, etc. present in the either the HRMA model or SAOSAC. The EA-MIRROR is limited to simulation of on-axis point sources. Use of the various spatial models listed in The spectrum and the spatial shape of the X-ray source requires the HRMA model in marx . The remainder of this discussion refers to the HRMA model which is the default model.

### HRMA Geometric Model¶

The HRMA onboard Chandra consists of four Wolter Type I mirrors. These mirrors are nested and each shell consists of a paraboloid at the front and a hyperboloid at the back. The physical geometry of the HRMA is defined externally to marx through the file EKCHDOS06.rdb. This file is produced by the CXC Calibration group and contains information about the size, shape, and placement of the hyperboloid and paraboloid mirror elements. This information includes the offsets and rotations of the various mirror shells relative to the optical axis. During the simulation, the paths of individual photons are traced through this geometric structure.

The mirror support structure is not currently modeled as part of the HRMA raytrace. The vignetting effects of these structures is instead included as a uniform reduction in the total effective area of the HRMA. The degree of vignetting can be adjusted via the HRMAVig parameter. In general, this parameter should not be modified.

### HRMA Reflectivity Model¶

When a ray encounters a mirror surface, marx calculates the probability that the photon is reflected or absorbed based on the properties of the mirror coating and the energy and incidence angle of the photon. The mirror shells are assumed to be coated with iridium and the iridium optical constants as provided by the Henke Tables are used to compute the reflectivity of the mirrors as a function of energy. Comparison of the resulting HRMA effective area with that produced by SAOSAC raytraces agree to less than 1% over the energy range 0.03–10 keV.

### HRMA Scattering Model¶

The point spread function (PSF) of the HRMA is largely determined by two components: the gross physical shape of the mirrors and scattering of photons due to small–scale surface irregularities. The physical geometry of the mirrors is implemented as discussed above. To treat the scattering properties of the HRMA, marx provides two options.

First, a simple Gaussian “blur” may be applied to photon reflections. The direction of a ray after reflection from the mirror surface is determined by the orientation of the surface normal at the point of reflection. This simple model assumes that the direction of the normal can vary from its ideal direction according to a Gaussian probability distribution whose standard deviation is given by the HRMABlur parameters. Blur parameters are specified for each of the parabolic and hyperbolic HRMA elements. This model produces a reasonable approximation to the measured core of the HRMA PSF.

In addition to a Gaussian core, the HRMA PSF exhibits energy–dependent extended tails as seen in SAOSAC simulations. By default, marx uses a scattering model based on the treatment of HRMA scattering used by the MST’s high fidelity raytrace SAOSAC. The probability that a photon’s scattered direction is displaced from a perfect reflection is determined using the results of WFOLD scattering tables which specify the probability that a photon of a given energy is scattered into a given angle. Since the previous release, the HRMA raytrace in marx has been improved to account for the individual scattering properties of the various mirror components. For reference, SAOSAC breaks the HRMA P and H optics into many different segments with an associated WFOLD scattering table for each to better treat the changes in surface properties along the mirror surfaces. The files listed in Table [tab:hrma] and used by marx represent the scattering properties for the midpoint of each hyperbolic and parabolic section of the HRMA. They do not include any alignment or gravity induced errors which are handled by the physical model of the HRMA. The normalizations of the various WFOLD scattering tables can be adjusted using the ScatFactor parameters. The WFOLD scattering model can be disabled in marx by setting the parameter HRMA_Use_WFold="no".

## Grating Modules¶

Chandra contains two distinct grating assemblies called the HETG and the LETG. The parameter GratingType selects the grating for a marx simulation. The HETG is meant to be used for high energy X–rays and the LETG is optimized for low energy X–rays. Actually, the HETG consists of two types of gratings: MEG for medium energy rays, and HEG for high energy rays. The LETG consists entirely of LEG type gratings. Each grating facet is arranged such that its geometric center lies on a Rowland Torus. The MEG torus is rotated by $$-5$$ degrees with respect to the LEG torus, and the HEG torus is rotated by $$+5$$ degrees with respect to the LEG torus.

After a ray leaves the mirror it travels towards the detector. If the gratings are being used, the ray will intersect the grating and undergo a diffraction process. Actually, a certain percentage of the rays will not strike a grating facet; instead some will be absorbed by the grating assembly. The percentage of rays that intersect with a facet is specified by the appropriate vignetting parameter, LEGVig if the LETG is being used, or HEGVig and MEGVig if the HETG is used.

marx currently knows very little about the actual location of individual grating facets. The assumption is that the HRMA and the grating assembly is aligned such that the probability of a ray striking a facet is maximized, and the percentage that miss is controlled by the vignetting factor.

The LETG includes a complex support structure consisting of a triangular “coarse” support and a mesh of “fine” wire supports. Both of these “fine” and “coarse” wire support structures result in additional diffraction patterns. The LETG grating model in marx includes the multiple diffractions due to these support structures. Roughly 10% of the detected photons will be diffracted by one or both of these support structures. The reader is referred to the http://cxc.harvard.edu/proposer/POG/html/chap9.html for more details.

### Intersection with the Rowland Torus¶

The Rowland torus is defined by the equation

(1)$(x^2 + y^2 + z^2)^2 = 4 R^2 (x^2 + z^2)$

where $$R$$ is the Rowland radius. To determine the intersection of a ray with the torus, the ray equation

(2)${\vec{x}}= {\vec{x}}_0 + {\hat{p}}t$

is substituted into the equation for the torus. This yields the fourth order equation for $$t$$

(3)$\begin{split}\begin{split} 0 = t^4 &+ 4 ({\hat{p}}\cdot{\vec{x}}_0)t^3 \\ &+ 2t^2 \big(|{\vec{x}}|^2 + 2({\hat{p}}\cdot{\vec{x}}_0)^2 - 2R^2(p_x^2 + p_z^2)\big) \\ &+ 4t \big(|{\vec{x}}_0|^2 ({\hat{p}}\cdot{\vec{x}}_0) - 2R^2(p_x x_0 + p_z z_0)\big) \\ &+ |{\vec{x}}_0|^4 - 4R^2 (x_0^2 + z_0^2) \end{split}\end{split}$

where the vector $${\vec{x}}_0$$ has components $$(x_0, y_0, z_0)$$. The four roots of this equation are a manifestation of the fact that a line can intersect the torus at four different places.

An important case is when $${\vec{x}}_0 = \vec{0}$$ where an enormous simplification occurs and the equation reduces to

$0 = t^4 - 4R^2t^2(p_x^2 + p_z^2).$

This equation has a double root at $$0$$ and non-zero roots at

(4)$t = \pm 2R\sqrt{p_x^2 + p_z^2}.$

In the coordinate employed by marx , rays travel in the negative $$x$$ direction from the HRMA to the torus. This means that the solution of interest is the most negative root of (3). Such a root corresponds to the first intersection point of the ray with the torus.

Even if $${\vec{x}}_0$$ is non-zero, one can always project the ray to the $$x = 0$$ plane to make the component $$x_0 = 0$$. One can then argue that the remaining two components $$z_0$$ and $$y_0$$ will be small (i.e., $$z_0<<R$$) since the rays from the HRMA will be converging to the focal point located at the center of the torus. The upshot is that (4) is a good zeroth order approximation to the exact solution and that one can use this value as the starting point in an iterative solution to (3). Newton’s method is used by MARX, although a closed form solution exists for the quartic equation.

Let $$t$$ be the solution to the equation $$0 = f(t)$$ and let $$t_0$$ represent an approximate root. If $$\delta t = t - t_0$$, then a taylor expansion yields

$\begin{split}\begin{split} 0 = & f(t) \\ = & f(t_0 + \delta t) \\ = & f(t_0) + \delta t f'(t_0) + \cdots \end{split}\end{split}$

or

$t = t_0 - \frac{f(t_0)}{f'(t_0)} + \cdots.$

Newton’s method follows from the last expression as an iterative solution of the form

(5)$t_{k+1} = t_k - \frac{f(t_k)}{f'(t_k)}.$

For the quartic equation

$0 = t^4 + at^3 + bt^2 + ct + d,$

Newton’s method yields the iterative scheme

(6)$t_{k+1} = \frac{(3t_k^2 + 2at_k + b)t_k^2 - d}{(4t_k + 3a)t_k^2 + 2bt_k + c}$

From (3), it follows that

$\begin{split}\begin{split} a = & 4{\hat{p}}\cdot{\vec{x}}_0 \\ b = & 2|{\vec{x}}|^2 + 4({\hat{p}}\cdot{\vec{x}}_0)^2 - 4R^2(p_x^2 + p_z^2) \\ c = & 4|{\vec{x}}_0|^2 {\hat{p}}\cdot{\vec{x}}_0 - 8R^2 p_z z_0 \\ d = & |{\vec{x}}_0|^4 - 4R^2 z_0^2 \end{split}\end{split}$

where $$x_0$$ has been set to zero in accordance with the understanding that the ray has been projected to the x = 0 plane. This means that

$t_0 = -2R\sqrt{p_x^2 + p_z^2}$

should be used to seed (6).

The previous analysis is appropriate for any torus whose symmetry axis is aligned with the marx $$y$$ axis. This is the case for the LETG; however the tori that make up the HETG differ from the LETG torus by a rotation. In particular, the MEG torus differs from the LEG torus by a rotation of $$-5$$ degrees about the $$x$$ axis. Similarly, the HEG torus is rotated by $$+5$$ degrees the other direction. In the following, we consider the more general case of a torus that is rotated by an angle $$\theta$$ about the $$x$$ axis.

Let $${\cal R}(\theta)$$ represent a rotation about the $$x$$ axis by an angle theta. It takes a vector $$\vec{v}$$ and transforms it into a new vector $$\vec{v'}$$ via

(7)$\vec{v}' = {\cal R}(\theta) \vec{v}$

where the components of $$\vec{v}'$$ satisfy

$\begin{split}\begin{split} v_x' = & v_x \\ v_y' = & v_y \cos\theta + v_z \sin\theta \\ v_z' = & -v_y \sin\theta + v_z\cos\theta. \end{split}\end{split}$

At this point (7) could be applied to points on the torus to obtain a rotated version of (1) and the preceding analysis repeated with the new, more complicated, equation. However, it is easier to work in a rotated coordinate system where the equation of the torus retains its form given in (1). So, the prescription for computing the intersection with a rotated torus looks like this:

1. After projecting $${\vec{x}}_0$$ to the $$x = 0$$ plane, rotate $${\vec{x}}_0$$ and $${\hat{p}}$$ via $${\cal R}(-\theta)$$.
2. Perform the intersection calculation outlined above using the rotated values of $${\vec{x}}_0$$ and $${\hat{p}}$$. This calculation will result in the intersection point $${\vec{x}}$$ with components expressed in the rotated frame.
3. Rotate all vectors back using $${\cal R}(\theta)$$. The result will be that the intersection point $${\vec{x}}$$ will be expressed in the unrotated frame.

To illustrate this procedure, consider the special case of $${\vec{x}}_0 = 0$$. In the unrotated case, we found (4) as the solution. For a rotation by an angle $$\theta$$, the solution in the rotated frame will be

\begin{split}\begin{aligned} {\vec{x}}' &= {\hat{p}}' t_0 \\ &= -2R {\hat{p}}' \sqrt{p_x^2 + (p_z\cos\theta + p_y\sin\theta)^2} \\ \end{aligned}\end{split}

which when rotated back to the original frame yields

${\vec{x}}= -2R {\hat{p}}\sqrt{p_x^2 + (p_z\cos\theta - p_y\sin\theta)^2}.$

### Diffraction of the Ray¶

Consider a ray with wavelength $$\lambda$$ and direction $${\hat{p}}$$ incident upon a diffraction grating of period $$d$$ located at position $${\vec{x}}$$ and normal $$\hat{n}$$. The grating lines are assumed to oriented in the direction $$\hat{l}$$. See Figure Diffraction Coordinate System. In a very general way, the diffraction equation can be written in vector form as $$\hat{p}' \times \hat{n} = \hat{p} \times \hat{n} + (m \lambda/d)\hat{l}$$. Taking the dot product of this equation with $$\hat{d}$$ and $$\hat{l}$$ respectively, it can be shown that a ray diffracting into order $$m$$ will move in a direction $${\hat{p}}'$$ determined by the conditions:

(8)\begin{split}\begin{aligned} {\hat{p}}'\cdot\hat{l} &= {\hat{p}}\cdot\hat{l} \\ {\hat{p}}'\cdot\hat{d} &= {\hat{p}}\cdot\hat{d} + \frac{m\lambda}{d} \end{aligned}\end{split}

where

$\hat{d} = \hat{n} \times \hat{l}.$

(And because these vectors form an orthonormal basis the following is also true: $$\hat{l} = \hat{d} \times \hat{n}$$ and $$\hat{n} = \hat{l} \times \hat{d}$$.)

Diffraction Coordinate System

Figure showing the orthogonal coordinate system local to an individual grating facet. The vector $$\hat{n}$$ is normal to the facet and $$\hat{l}$$ is in the direction of the grating lines. The vector $$\hat{d}$$ is in the dispersion direction. The incident ray is given by p and the diffracted ray is $$\hat{p}'$$.

Since $$\hat{n}$$, $$\hat{l}$$, and $$\hat{d}$$ form a right-handed orthonormal coordinate system, it trivially follows that

(9)${\hat{p}}' = ({\hat{p}}\cdot\hat{l})\hat{l} + ({\hat{p}}\cdot\hat{d} + \frac{m\lambda}{d})\hat{d} + \hat{n} \sqrt{1 - ({\hat{p}}\cdot\hat{l})^2 - ({\hat{p}}\cdot\hat{d} + \frac{m\lambda}{d})^2}.$

After diffraction, the ray will travel along the trajectory

${\vec{x}}(t) = {\vec{x}}+ {\hat{p}}'t.$

Note that (9) may be put into a more familiar form as follows. Since the component of the ray in the $$\hat{l}$$ direction is not changed by the grating, the effect of the diffraction is simply a rotation of $${\hat{p}}$$ about the $$\hat{l}$$ axis by some angle. Let $${\vec{p}_{\perp}}$$ denote the projection of $${\hat{p}}$$ onto the $$(\hat{d},\hat{n})$$ plane, and let $$\theta$$ be the angle between $${\vec{p}_{\perp}}$$ and $$\hat{n}$$. Define $${{\vec{p}_{\perp}}\,\!\!\!\!'}$$ and $$\theta'$$ in a similar fashion (see Figure Diffraction in a plane).

Diffraction in a plane

Diffraction in the $$(n, d)$$ plane. Here $$\theta$$ is the angle the projection of the incoming ray onto the $$\hat{d}\hat{n}$$ plane makes with respect to the normal, and $$\theta$$ is the angle between the normal and the projection of the outgoing ray.

It follows from (8) that

$p_{\perp} \sin \theta' = p_{\perp} \sin \theta - \frac{m\lambda}{d},$

where $$p_{\perp} = |{\vec{p}_{\perp}}| = |{{\vec{p}_{\perp}}\,\!\!\!\!'}|$$. In fact, the previous equation reduces to the well known diffraction equation when $${\hat{p}}$$ has no component in the $$\hat{l}$$ direction. Using these definitions, one can write (9) in the form

${\hat{p}}' = ({\hat{p}}\cdot\hat{l})\hat{l} - (p_{\perp} \sin{\theta'}) \hat{d} + (p_{\perp} \cos{\theta'}) \hat{n}.$

In general, $$\hat{n}$$ and $$\hat{l}$$ are complicated functions of the position of the grating. However, for gratings of infinitesimal size (For finite size facets, the grating normal will have to be looked up in a facet database.) positioned on the surface of the Rowland torus, $$\hat{n}$$ will be directed towards the origin, i.e.,

$\hat{n} = -\frac{{\vec{x}}}{|{\vec{x}}|}$

Similarly, $$\hat{l}$$ may be determined from the condition that the facets are arranged such that $$\hat{l}$$ has no $$y$$ component (We are working in the natural coordinate system of the torus. Thus these equations hold for the LETG and the HETG.) and that it is normal to $$\hat{n}$$. That is,

$\begin{split}\begin{split} 0 &= \hat{l}\cdot\hat{y} \\ 0 &= \hat{l}\cdot\hat{n} \\ 1 &= |\hat{l}| \end{split}\end{split}$

from which it follows that

$\begin{split}\hat{l} = \frac{1}{\sqrt{n_x^2 + n_z^2}} \begin{pmatrix} n_z\\ 0\\ -n_x \end{pmatrix}.\end{split}$

Since the LETG gratings have a support structure that also acts as a diffraction grating, we need to consider a more general orientation of the $$\hat{l}$$ axis that consists of a rotation about the $$\hat{n}$$ axis by some angle $$\theta$$. This means that the rotated vectors,

\begin{split}\begin{aligned} \hat{l}_{\theta} &= \hat{l} \cos\theta + \hat{d}\sin\theta \\ \hat{d}_{\theta} &= -\hat{l} \sin\theta + \hat{d}\cos\theta, \end{aligned}\end{split}

should be used in (9) to yield

${\hat{p}}' = ({\hat{p}}\cdot\hat{l}_{\theta})\hat{l}_{\theta} + ({\hat{p}}\cdot\hat{d}_{\theta} + \frac{m\lambda}{d})\hat{d}_{\theta} + \hat{n} \sqrt{1 - ({\hat{p}}\cdot\hat{l}_{\theta})^2 - ({\hat{p}}\cdot\hat{d}_{\theta} + \frac{m\lambda}{d})^2}.$

### Grating Efficiency¶

The grating efficiency is a function of many quantities such as the geometrical parameters that specify the bar shape, the chemical composition and thickness of the layers that make up the plating base of the grating, etc. An extensive effort has been made to quantitatively understand the relationship of these quantities to the grating efficiency (see the http://space.mit.edu/HETG/report.html).

In early versions of marx a simple, uniform rectangular bar model was used to calculate the diffraction efficiency of the HETG and LETG grating facets. Based on comparison to synchrotron measurements, the rectangular grating bar model appears to be accurate to approximately 5% over most of the HETG’s operating passband. This model does not meet the HETG calibration goal of 1%. Consequently, the current marx version uses a new grating efficiency model based on tabulated facet data from sub–assembly and XRCF data.

UseGratingEffFiles

(default: yes) Use grating efficiency tables? These efficiency tables have been provided by the HETG IPI team and include grating efficiencies for orders -11 to 11. In the case of the LETG tables, orders from -25 to 25 are included. Individual tables have been calculated for each mirror shell and include the inter-grating vignetting. Users can still access the old uniform bar facet model by setting UseGratingEffFiles=no, but this is not recommended.

Use_Unit_Efficiencies

(default: no) If yes, rays which intersect the HETG or LETG will still be diffracted but no efficiency filter will be applied. Hence all orders will have an equal probability of being populated. This mode is useful for studying the characteristics of higher order dispersed photons without having to run very large simulations in order to build up reasonable statistics.

### Facet Period Variations and Misalignments¶

The HETG onboard Chandra consists of 336 individual grating facets. During the XRCF calibration of the HETG, it was discovered that 6 MEG grating facets were mis-aligned by angles ranging from 3 to 24 arcmins. The effects of these mis-aligned facets is shown in Figure Misalignment.

An image from XRCF test D-HXH-AL-27.001 showing the MEG 3rd order Al-K defocused to 65.54 mm. The main $$K\alpha$$ line, satellite line, and $$K\beta$$/O-K lines are visible in the left panel. The enlarged view in the right panel shows the effects of the mis-aligned gratings.

marx allows to specify the mis-alignment angles and period variations of groups of facets as a function of azimuthal angle around the HETG support structure. Sector files describing the properties of the facets as a function of angle (including the mis-aligned MEG facets) have been provided by the HETG IPI team and reside in the MARX_DATA_DIR directory.

Use_HETG_Sector_Files

(default: yes) Use HETG Sector Files?

Use_LETG_Sector_Files

(default: no) Sector files are currently unavailable for the LETG, so this option is off by default when simulating LETG observations. Instead, the misalignmens is treated statistically using legdTheta parameter.

There was a long standing issue of a relative rotation between the LETG and the ACIS detector. The root of this problem was tracked down (with the help of marx) to a rotation offset between the aspect coordinate system and the focal plane detector system. This offset was masked by compensating rotations of the detectors from astrometric analysis, and manifested itself as a small rotation of the LEG dispersion arm on the ACIS detector. Changes were added to CIAO 4.3 that effectively adds an additional rotation to the LETG when used with ACIS.

LETG_ACIS_dTheta

(default: -0.0867) aspect/focalplane misaligment induced dtheta for LETG/ACIS (degrees)

## Detector Models¶

The detector models in marx are all consist of at least four components: geometry, filter transmissions, detector quantum efficiency, and spectral resolution. The specifics of these components for each of the four Chandra focal plane detectors is discussed here.

### Detector Geometry¶

The physical placement of the detectors in the Chandra focal plane is based on reference data given in the CXC coordinates documents. These data include locations and tilts in three dimensions for each CCD in the ACIS-I and ACIS-S arrays as well as all four MCPs in the HRC-I and HRC-S. The detector geometric model in marx reproduces the tilts of the ACIS-S CCD to follow the “bowl”-shaped HRMA focal surface and the arc of the six ACIS-S CCDs which follows the curved Rowland focal surface (see http://asc.harvard.edu/proposer/POG/html/index.html for a more detailed description).

Similarly, the tilts of the three MCPs in the HRC-S spectroscopic array are reproduced. Chip and plate gaps as appropriate are also included in the geometric model. marx writes the raw U and V coordinates for the HRC-S to the hrc_u.dat and hrc_v.dat files and they will appear in the events files created with marx2fits.

### Filters¶

#### ACIS¶

Both the ACIS-I and ACIS-S CCD arrays include UV/visual optical blocking filters to protect the CCDs from non-X-ray photons. marx models these filters using tabulated transmission efficiencies supplied by G. Chartas (Penn State). Separate tables are used for the filters on the ACIS-I and ACIS-S arrays. This transmission efficiency calculation can be disabled in marx using the parameter DetIdeal="yes".

#### HRC¶

The HRC-I and HRC-S detectors include a set of UV/Ion shields to block UV photons and low energy ions. In the case of the HRC-I, a single uniform UV/Ion shield covers the entire surface of the MCP. As with the ACIS optical blocking filters, marx uses an external data file containing tabulated efficiencies to model the shield’s transmission and this transmission can be disabled with DetIdeal="yes".

The UV/Ion shield configuration of the HRC-S array is slightly more complicated and includes four distinct regions each with a unique transmission efficiency. For an overview of the HRC–S shield configuration see: http://cxc.harvard.edu/proposer/POG/html/chap7.html.

marx uses four individual data files to specify the transmission of these regions. The central region of the HRC-S UV/Ion shield includes a “T” shaped region of thicker Al which can be used to preferentially reject low energy photons. This Low Energy Suppression Filter (LESF) region is included in the marx model of the HRC–S UV/Ion shield. If the LESF is to be used, the SIM should be repositioned using DetOffsetZ=-6.5 to place dispersed spectrum over the LESF. Users should consult the http://cxc.harvard.edu/proposer/POG/html/chap7.html for more information on the LESF.

As a final complication, the UV/Ion shield on the HRC-S array is physically offset from the MCP surfaces by approximately 10 mm. This separation can lead to “shadowing” near the edges of differing filter regions. This effect is included in marx and the separation is controlled with the HESFOffsetX parameter.

### Detector Quantum Efficiency¶

The detector quantum efficiency is modeled in exactly the same manner for both the ACIS and HRC detectors. External data files are used to define the quantum efficiency as a function of photon energy for each detector. Using this function, marx calculates a cumulative probability as a function of photon energy. For each photon which reaches the detector surface, a random number is then generated and compared with the cumulative probability in order to determine whether the photon was detected.

Unique quantum efficiency curves are used for the MCPs in the HRC–I and HRC-S; however, all three HRC-S MCPs are currently assumed to have the same quantum efficiency. As the Chandra calibration effort progresses, these curves will be replaced by specific curves for each MCP.

Quantum efficiency (QE) files are available for the 10 CCDs comprising the ACIS-I and ACIS-S detectors. In the previous version of marx , QE files where available only for generic front-illuminated and back-illuminated CCDs.

If the parameter DetIdeal="yes", the QE of the selected focal plane detector (including any filter transmission) will be set to unity.

### Detector Spectral Resolution¶

The detector redistribution function determines the mapping of photon energy to detected pulse height (PH). These functions determine the intrinsic spectral resolution of the different detectors. marx uses a mixture of calibration information and simple analytic forms to approximate these functions. More accurate redistribution functions can be applied to marx simulations using the marxrsp tool discussed Using Response Matrices with marx.

#### ACIS¶

The redistribution of the ACIS detector is exceedingly complex. Both the gain and spectral resolution of each of the 10 ACIS CCDs varies with position on the chip. Due to the radiation damage induced increase in charge transfer inefficiency (CTI), the spatial variations of the frontside chips are especially large. Fortunately, an extensive effort has been undertaken by the CXC Calibration group to measure these variations as a function position for each of the ACIS CCDs. Currently, the ACIS-S aimpoint CCD (chip ID 7) has calibration data specifying the gain and spectral resolution for each 32x32 pixel region on the chip. Due to reduced signal to noise, the remaining backside chip (chip ID 5) has been calibrated on 64x64 pixel regions. The remaining 8 frontside CCDs are calibrated in 256x32 pixel segments. For each of these CCD calibration regions, the CXC has determined a unique gain and functional fit to the redistribution function. The redistribution model in utilizes this calibration information when determining the observed PHA channel for a given detected event.

The CXC Calibration group currently models the ACIS redistribution using a functional form consisting of multiple Gaussian components. The internal marx redistribution function reproduces only the primary peak of the ACIS response, assuming a single Gaussian whose width is determined by the CXC CCD Calibration data mentioned above.

The gain and spectral response of the ACIS CCDs are also functions of focal plane temperature. At the time of this release, complete calibration data is available for a focal plane temperature of -110 C. A marx gain map will be released for -120 C when this data becomes available.

More information on the ACIS energy resolution can be found at http://cxc.harvard.edu/proposer/POG/html/chap6.html#tth_sEc6.7 .

#### HRC¶

The MCPs which comprise the HRC-I and HRC-S detectors have very limited spectral resolution with $$\sigma_E / E \sim 1$$. As with the ACIS CCDs, the redistribution function is assumed to be a Gaussian. The width of the MCPs distribution, however, is more complicated and is represented in marx by

$\begin{split}\sigma(E) = \left\{ \begin{array}{ll} a_0 \sqrt{ E }~~~~~~ & E < 0.5 ~\mbox{keV} \\ a_1 E^{0.1} & 0.5 < E < 2.0 ~\mbox{keV} ~~.\\ a_2 & E > 2.0 ~\mbox{keV} \end{array} \right. \label{eqn:hrc_res}\end{split}$

Here $$E$$ is the photon energy and $$a_0$$, $$a_1$$, and $$a_2$$ are constants which have been adjusted to approximately reproduce the preliminary XRCF measurements of the HRC redistribution function.

### Detector Spatial Resolution¶

#### ACIS¶

No extra blur is applied to ACIS simulations.

#### HRC¶

The physical characteristics and readout electronics of the HRC MCPs add a “blur” to the observed system point spread function in addition to the intrinsic FWHM of the HRMA. In marx, this blur is modeled as the sum of two Gaussians and a Lorentzian. Since the 2d Lorentzian diverges, it has to be cutoff at some radius, $$R_\mathrm{max}$$. Its normalized form is given by

$L(r) = \frac{L_0}{1+r^2/g^2}$

where $$1/L_0 = g^2 \pi \log(1 + R_\mathrm{max}^2/g^2)$$. Then $$1 = \int_0^{R_\mathrm{max}} (2 \pi r) L(r) \;dr$$.

The 2-d gaussian has the normalized form

$\begin{split}G(r) = \exp(-\frac{r^2}{2\sigma^2}) / (2 \pi \sigma^2)\\ 1 = \int (2 \pi r) G(r) \;dr\end{split}$

The parameters that specify the coefficients of all functions are listed in HRC-I Model Parameters and HRC-S Model Parameters.

#### HESF¶

Due to the poor intrinsic energy resolution of the HRC-S, order sorting for astrophysical spectra obtained with the LETG+HRC-S combination will be difficult. In an attempt to ameliorate this problem, the High Energy Reflection Filter (HESF) was added to the original HRC-S design. The HESF (a.k.a. Drake Flat) is a two facet filter coated with Cr and C which can be inserted into the Chandra beam by a translation of the SIM along the $$Z$$ axis. Above the Cr L and C K edges, the reflectivity of the filter is designed to be low, thus suppressing higher order photons. Figure Drake Flat shows a schematic of the HESF. More details on the HESF are available in the http://cxc.harvard.edu/proposer/POG/html/chap7.html .

Schematic of the HRC–S High Energy Suppression Filter (HESF). Figure courtesy of Dr. Jeremy Drake (SAO/CXC).

marx includes the HESF in its raytrace calculation if the parameter HRC-HESF="yes". The reflectivity of the of Cr and C surface coatings is calculated internally. If the HESF is to be used, the SIM should be repositioned using DetOffsetZ=-5.471 to place the HESF in the Chandra beam.

# Simulating ground calibration data¶

With appropriate configuration, marx can be used to simulate data taken during the calibration phase of the Chandra mission at the X–Ray Calibration facility (XRCF) in Huntsville, AL. By default, marx simulates the flight performance of the Chandra satellite. However, a number of effects contribute to differences between the flight and XRCF performance of Chandra. A brief summary of these effects are listed here.

• Additional HRMA blur: The effects of gravity on the HRMA at XRCF produce an additional “blurring” of the mirror’s point spread function (PSF) relative to the flight performance. This behavior can be adjusted with the P1Blur, P3Blur, P4Blur, and P6Blur and the H1Blur, H3Blur, H4Blur, and H6Blur parameters.
• Change of HRMA Focus position: Since the x-ray source at XRCF was at a finite distance from the HRMA, the effective location of the “focus” falls at a different location along the optical axis than the default flight configuration. Changing the DetOffsetX parameter will move the location of the focal plane relative to the HRMA.
• Finite Source Size: Due to its finite distance, the EIPS x-ray source used at XRCF was actually resolved by the HRMA resulting in a broader PSF than one would measure for a point source. A simple way to include this affect is to use the DISK source model to simulate an extended source. Alternatively, one could use the IMAGE source model in conjunction with actual FITS images of the EIPS source provided by the http://wwwastro.msfc.nasa.gov/xray/xraycal/spot .
• Finite Source Distance: By default marx, assumes that sources are sufficiently far away that photons impinging on the HRMA can be assumed to be parallel to the optical axis. At XRCF, the calibration source was not far enough away from the focal plane for this assumption to hold. For XRCF simulations, the SourceDistance parameter should be set to a value of 537.587 meters.
• Modified Rowland Diameter: The difference in the location of the focal plane at XRCF results in a different Rowland geometry for the HETG and LETG spectrometers. This geometry is controlled via the HEGRowlandDiameter, MEGRowlandDiameter, and LEGRowlandDiameter parameters.

To simulate XRCF data, these parameters should be modified in your marx.par file. The table provides a summary of the relevant parameters, their default values, and values appropriate for simulating XRCF data. An example XRCF simulation is shown in the figure below for test ID D-IXH-PI-3.003.

Parameter Default XRCF
P1Blur 0.18129215 0.362
H1Blur 0.13995037 0.280
P3Blur 0.11527828 0.230
H3Blur 0.16360829 0.327
P4Blur 0.12891340 0.258
H4Blur 0.098093014 0.196
P6Blur 0.076202759 0.152
H6Blur 0.079767401 0.160
DetOffsetX 0.0 -194.925
SourceType POINT DISK
S-DiskTheta0 0.0 0.0
S-DiskTheta1 0.0 0.0767372
SourceDistance 0.0 537.587
HEGRowlandDiameter 8632.48 8587.88
MEGRowlandDiameter 8632.48 8587.88
LEGRowlandDiameter 8637.00 8593.12

A comparison between data from XRCF test ID D-IXH-PI-3.003 and a corresponding MARX simulation. The MARX simulation was 9.7 mm out of focus like the XRCF test.