XIM—A VIRTUAL X-RAY OBSERVATORY: INVESTIGATING THE X-RAY APPEARANCE AND LINE PROFILE FUNCTION OF VORTEX RINGS IN GALAXY CLUSTERS

One of the key Chandra legacies is the discovery of X-ray cavities in the centers of virtually all cool core clusters (e.g., McNamara et al. 2000; Blanton et al. 2001; Heinz et al. 2002; Rafferty et al. 2006, and references therein). The diffuse radio synchrotron emission inside the cavities indicates their origin: they are inflated by jets from the supermassive black holes of the central cluster elliptical galaxy.

Energy accounting indicates that the cavities could be responsible for the heat required to keep cool core clusters at their current central temperatures, though the exact details of how energy is transferred from the bubbles to the intracluster medium (ICM) is still unclear.

Similarly intriguing has been the discovery of sound and weak to moderate strength shock waves in the same group of clusters, traveling outward from the active galactic nucleus (AGN). These are the wakes from the expanding cavities, and they, too, have been suggested as possible energy sources (e.g., via viscous dissipation or diffusion).

Heinz & Churazov (2005) first pointed out that the waves created by a recent outburst must encounter and interact with cavities from previous episodes of jet activity. A bubble that is overrun by a pressure discontinuity (as expected in a shock or nonlinear sound wave) is turned into rotating vortex ring by a process called the Richtmyer–Meshkov instability (RMI). As such, the bubble extracts energy from the wave and deposits it in the vortex.

In their investigation, Heinz & Churazov (2005) discussed the energetics and general application of this process to the ICM, limited to two-dimensional simulations by computing power at the time. In the following demonstration of the capabilities of XIM, we take their simulations to three dimensions and investigate the observable characteristics of this process, both from an imaging and a spectroscopic perspective.

Understanding the dynamics and observational appearance of vortex rings is important for cluster physics for a number of reasons.

• 1.  
  As demonstrated by Heinz & Churazov (2005), vortex rings can extract energy from even weak shock waves, as they are found copiously in clusters. While weak discontinuities dissipate energy at low rate and propagate out to large cluster radii faster than the sound speed, the vortices they create stay behind and can store the energy, which can then be dissipated locally on a viscous timescale due to the differential rotation of the vortex or through a turbulent cascade.
• 2.  
  As discussed in Enßlin & Brüggen (2002), it is likely that radio relics in the outskirts of clusters are vortices generated by the RMI, as a relic is generally believed to be produced by a shock passing over an aged radio lobe that has risen buoyantly to the outer regions of the cluster.
• 3.  
  A low-density bubble in a hydrostatic atmosphere will rise buoyantly and turn itself into a vortex in the process. The detailed dynamics, the role of this process in the production of X-ray cavities, and the role of magnetic fields have been discussed at length in the literature (Churazov et al. 2001; Brüggen et al. 2002) and continue to be a topic of ongoing work.

A more detailed discussion of the general hydrodynamics, energetics, and underlying physics of the RMI in clusters will be discussed in a forthcoming companion paper (S. Friedman et al. 2011, in preparation). In this paper, we present a detailed analysis of the appearance of these vortices in clusters, based on hydrodynamic simulations of a vortex ring.

While hydro simulations can address many of the key questions in the study of clusters, it is important that the output data from such simulations can be compared with X-ray observations in the most direct and faithful way possible. Generally, galaxy clusters are simple to simulate compared to, for example, galaxies, where complex interstellar medium physics and star formation seriously complicate modeling. For that reason, it is much easier to produce realistic simulations of clusters that can be faithfully compared with observations.

This comparison requires computer codes that can turn the output from a hydro simulation into a virtual observation with a given X-ray telescope—a virtual X-ray observatory.

In this paper, we present such a virtual telescope, the IDL code XIM, which is now publicly available at the Web site http://www.astro.wisc.edu/~heinzs/XIM.

XIM is best suited to process output from Eulerian (grid-based) hydro codes and its direct application is the virtual observation of thermal emission from cosmic gas. While it was written to directly simulate Chandra, and the next generation X-ray observatory IXO (International X-ray Observatory), it is easy to adapt it to simulate any arbitrary X-ray telescope for a set of user-supplied response matrix files (a set of responses for XMM-Newton is delivered with XIM as well).

Since X-ray telescopes are imaging spectrographs (each photon is time- and energy-tagged, thus allowing a time- and energy-resolved image to be constructed), the output from a virtual X-ray observation is a spectral imaging cube (such as would be obtained from an integral field spectrograph in optical spectroscopy, for example). Figure 1 demonstrates the operation of XIM in calculating such a data volume.

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Because photon energies are tracked to arbitrarily high precision, such a code is also an ideal tool to derive the emission-line profiles from hydrodynamic processes in cases where an analytic description of the flow profile is not possible. For example, we know that galaxy clusters are permeated with bubbles of relativistic gas released by the jets of AGNs. The dynamical evolution of these bubbles, both in the gravitational potential of the cluster and the presence of sound and shock waves moving through the gas (themselves excited by the presence of AGN jets, see, e.g., Kaiser & Alexander 1997; Heinz et al. 1998; Reynolds et al. 2001; Forman et al. 2005; Graham et al. 2008; Morsony et al. 2010), leads to the development of vortices. A numerical simulation of such a vortex, as presented in this paper, allows us, for the first time, to derive the line profile function for this structure, as would be expected in the vicinity of such a vortex.

As the astronomical community prepares for the launch of Astro-H and considers the possible launch of IXO, both ultrahigh-resolution spectrographs, it is critical that we construct detailed models of what we can expect in terms of spectral line signatures from complex systems such as galaxy clusters and supernova remnants. Hydrodynamic simulations and subsequent virtual observation present one of the best ways to explore the incredible wealth of information such data will offer.

At the same time, long imaging X-ray observations of diffuse objects with Chandra and XMM-Newton can benefit greatly from prior numerical simulation and it is generally recommended that observers perform ray-tracing simulations when proposing long observations. XIM can perform such simulations and presents a powerful tool in constructing mock observations of possible targets.

This paper is organized as follows: in Section 2, we present a brief overview of the main capabilities of XIM and some examples of its application; Section 3 presents a numerical simulation of vortex generation in galaxy clusters by the RMI and derives the line profile for such a vortex. Section 4 presents a brief summary. The Appendix presents a more detailed list of the implementation and capabilities of XIM.

XIM takes as input data from any rectilinear, regularly gridded mesh. Input required to calculate the thermal emissivity of a cell of plasma are the density, temperature, and velocity (the latter required to calculate the Doppler shifted emission), as well as the metallicity of the gas and the filling factor of the thermal gas.

Ionization states and recombination rates can then be calculated self-consistently. In the case of XIM, this is done using the publicly available APEC code, assuming thermal equilibrium (an assumption implicit in most hydro schemes).

In order to calculate the intrinsic photoelectric absorption, the density of neutral hydrogen inside each cell must be specified, in which case radiative transfer for the case of pure absorption is straightforward (including Doppler shift of the cross section to the comoving frame of the fluid). Calculating foreground absorption requires specification of a Galactic neutral hydrogen column toward a simulated sources.

A realistic virtual observation must include estimated background events as well, from the cosmic X-ray background as well as energetic particles.

Given these requirements, one can then formulate several useful observables that a virtual observatory should calculate from a simulated data cube (e.g., of a galaxy cluster).

• 1.  
  Projected specific surface brightness maps (along a specified line of sight (LOS)).
• 2.  
  Projected photon flux maps.
• 3.  
  Projected count rate maps for a specific instrument on a specific X-ray satellite.
• 4.  
  Integrated spectra of the entire data cube or masked sub-sections of the image, along with the associated response files, to be analyzed by a spectral analysis package like XSPEC.

XIM is capable of generating these four output types. The formats of the events and pulse height amplitude (PHA) files follow the OGIP/92-009 standard format for Flexible Image Transport System (FITS) files and are compatible with standard data reduction tools such as DS9 and XSPEC.

The detailed steps taken by XIM to derive these data products, as well as some of the options and code limitations, are presented in the Appendix of this paper.

Instrument-independent quantities (like surface brightness) can be derived with arbitrary precision by such a method. However, virtual observations with a specific instrument require accurate representations of an instrument's spectral response, its effective area, and its point-spread function (PSF), as a function of energy and position on the detector.

Such detailed information is rarely available, especially for future telescopes such as Astro-H and IXO. To the degree of accuracy with which such information can be predicted, it should usually be sufficient to use uniform response and effective area files, and a uniform PSF.

However, in the case of the Chandra X-ray Observatory, it is possible to construct a detailed virtual observation by passing the output from XIM to the ray-tracing software MARX (Wise et al. 2003). Together with blank-sky background count rates derived from actual observations, it is thus possible to generate high-fidelity virtual Chandra observations with XIM that are indistinguishable from real data from the perspective of a data reduction thread.

A virtual observatory, such as XIM, has obvious applications beyond the calculation of thermal emission from a simulation data cube. For example, it is possible to calculate line profile functions from simulated velocity profiles of a simulation by using a delta-function input spectrum, an example of which is presented in Section 3.

Before presenting a detailed example application of XIM in Section 3, we list several generic categories of possible applications of XIM.

• 1.  
  Direct morphological comparison of simulated X-ray surface brightness images with features in real observations, such as sound waves, ripples, cavities in galaxy clusters with actively driven AGN jets. Examples for simulations that have been analyzed using XIM can be found in Brüggen et al. (2009), Heinz et al. (2010), and Morsony et al. (2010). An example output image from a virtual Chandra observation of a simulated radio galaxy in cluster, modeled to reproduce Cygnus A as closely as possible in power and environment, is shown in Figure 2. More details can be found in Heinz et al. (2006) and Morsony et al. (2010).
• 2.  
  Virtual spectral cubes of X-ray emission by dynamical objects to show emission-line structures across the image. This can be used to disentangle the velocity structure of cosmic objects such as clusters. For example, spectro-imaging observations of radio galaxies in clusters with IXO will allow us to determine the expansion velocity and thus total power generated by AGN jets to high accuracy, as was shown in Heinz et al. (2010). An example of an emission-line map that shows the Doppler shifts from the expanding cavity as a split emission line is shown in Figure 3. The simulation is the same used to generate the image in Figure 2.
• 3.  
  Simulated color maps of multi-temperature gas distributions, as seen in cool core clusters and across shocks with discernible temperature jumps. For example, X-ray astronomy makes use of object colors for selection effects. For example, Forman et al. (2007) used a color cut on a long Chandra observation of the Virgo Cluster to detect a weak shock driven into the cluster by the radio galaxy M87 at the center.
• 4.  
  Design of search filters in redshift surveys from simulated color and spectral maps for cosmological simulations of galaxies and clusters.
• 5.  
  An integrated spectrum of the velocity field in a galaxy group or cluster, as could be observed with Astro-H. Comparing clusters in different dynamic states, such virtual observations could be used to predict whether Astro-H will be capable of detecting, for example, radial gradients in velocity dispersion due to the presence of AGNs (such enhancements have been predicted, for example, in Heinz et al. 2010).

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Multi-color imaging is particularly easy and fast using XIM, given that only a small number of spectral channels have to be calculated (APEC calculates the integrated photon luminosity for a given energy band, so no interpolation is involved when calculating broad band images).

As a visualization tool, XIM is most suitable for generating virtual X-ray observations of simulated data on grids. As such, it is significantly more specialized than generic visualization tools available in the public domain, such as VisIt and YT, which are powerful, publicly available, and non-commercial tools to render a variety of data in many different ways. These tools, however, do not include the functionalities needed to produce faithful, realistic virtual X-ray observations, namely, calculating Doppler shifted thermal emission, convolving the surface brightness with an instrument-specific response and a telescope PSF, adding backgrounds and noise, and interfacing with MARX (in the case of Chandra observations).

XIM is similar in scope and capabilities to the visualization tool X-MAS (Gardini et al. 2004; Rasia et al. 2008). X-MAS is a visualization tool for smoothed particle hydrodynamic (SPH; N-body) simulations. Like XIM, X-MAS uses a plasma thermal emission code and is specifically geared toward simulating Chandra and XMM-Newton data. As such, XIM is a good complement to X-MAS in that it takes grid-based data from codes such as FLASH or ENZO. XIM's capabilities to simulate IXO data are unique and have been used in several publications. As future mission profiles become available (such as the concept for a Wide-Field X-ray Telescope suggested in the 2010 decadal survey), XIM can easily be updated with mission-specific parameters.

Heinz & Churazov (2005) analyzed two-dimensional simulations of vortex creation by the RMI. We extend the simulations presented in Heinz & Churazov (2005) to three dimensions.

We use the publicly available FLASH3.2 code (Fryxell et al. 2000) which is a modular block-structured adaptive mesh refinement (AMR) code. It solves the Riemann problem on a Cartesian grid using the piecewise-parabolic method.

We simulate a spherical bubble overrun by a semi-infinite Mach 2 shock. The initial density contrast of the bubble is set to 100, with initial pressure equilibrium between bubble and surroundings. This is certainly a simplification compared with the scenario of X-ray cavities in galaxy clusters, but has the advantage of being a clean system to analyze and giving scale free results that can be applied to a wide array of physics systems where RMI operates.

We chose outflow boundary conditions on all but the upstream face (which we set to impose inflow boundary conditions satisfying the Rankine–Hugoniot jump conditions for a Mach 2 shock).

Our numerical box has a resolution of 1024 × 512 × 512. While FLASH is capable of AMR, we chose a uniform grid for this particular simulation to avoid issues of non-uniform numerical dissipation and to capture the velocity field at large distances from the vortex accurately.

Rendered frames of the simulations are shown in Figure 4 to show the creation of the vortex ring by the shock passage. Shown are an isodensity surface (red) and an isopressure surface (blue), indicating the location of the bubble/vortex and the shock, respectively.

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The generation of vortex rings by the RMI has been discussed at length elsewhere and we refer the reader to the literature on the mechanisms underlying this process. We are interested in investigating the observational signatures of such a vortex ring for parameters appropriate to the ICM.

In Figure 5 (top row), we present simulated Chandra X-ray images for a vortex with parameters appropriate for the Virgo "radio ear" (size, distance, temperature, and density are set to correspond to the parameters in Virgo at the position of the vortex) for different viewing angles. The vortex is clearly visible as a ring when viewed from close to the vortex axis, while it appears like a regular X-ray cavity when viewed more side-on. The vortex in this case has been placed at the redshift of Virgo. The virtual observation is a 250 ks observation using the ACIS-S3 detector on Chandra.

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Simulated synchrotron radio images of the vortex are shown below in Figure 5. By-eye morphological comparison suggests that the "radio ear" in Virgo could be a vortex ring (created either by RMI or buoyancy, as had first been suggested by Churazov et al. 2000) seen at roughly 30° to the LOS relative to the vortex axis (corresponding to the third panel from the left in Figure 5).

For comparison, Figure 6 shows a 20 cm image of the vortex in Virgo (Owen et al. 2000).

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Bernoulli's theorem indicates that the pressure in the parts of the vortex moving fastest should be lower than the static pressure at large distances (the vortex is partially rotationally supported). This implies a lower temperature in the thermal gas around the vortex. Looking at the projected X-ray images, this effect is visible, as indicated in Figure 5, which shows projected color maps for a virtual observation without any counting noise added. However, this effect is so subtle (at the few percent level) that it will not be visible in actual observations (in simulated observations with Poisson noise, no color variation is visible). Nonetheless, this effect might be measurable in other astrophysical scenarios or in lab experiments, where signal to noise is not as constrained as in X-ray observations. It should be noted that the hot regions seen in the interior of the vortex arise in the figure because of numerical mixing between the bubble fluid and the thermal exterior fluid.

Apart from realistic simulations of thermal emission, XIM allows easy calculation of line profiles for arbitrary velocity distributions by using a delta function as the spectral model.

We use this capability to calculate the line profile for our vortex simulation for different LOS angles relative to the vortex axis. The resulting plots are shown in Figure 7 for different times after shock passage. In this instance, we assume that the line emissivity is proportional to n².

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While it is clear that parts of the line are time variable for viewing angles close to the LOS, it is also apparent from the plots that the line core is (1) similar for different viewing angles and (2) stays constant throughout the simulation. That is, the RMI produces a characteristic line profile within the core of the line, while the line profile at frequency shifts close to the maximum (corresponding to velocities close to the vortex surface) vary and depend strongly on viewing angle. This is best seen in the logarithmically space plot in Figure 7.

From the log–log plot, it is clear that the line profile in the line core follows a power law in δν/ν₀ with an index of −2. That is, the line core follows the classic line profile for Lorentzian wings, with a cutoff at frequencies corresponding to the characteristic vortex velocity, which is the velocity differential across the shock.

The asymmetry in the vortex velocity field manifests itself in the velocity profile as a bump in the line profile in the direction of the bulk flow, which is the material streaming through the center of the vortex, where the density is higher due to compression (also visible as a surface brightness enhancement in Figure 5).

In the previous section we showed that a vortex ring exhibits a line profile that follows a quadratic dependence on δν in the line core and sharp cutoffs at frequencies close to the maximum velocity of the vortex.

We can understand this behavior qualitatively. As was already discussed in Heinz & Churazov (2005), the characteristic velocity of the vortex is the relative velocity between the upstream and downstream frames. Velocities much larger than this cannot be exhibited by any significant volume of fluid for the simple reason of energy conservation.

The asymmetry in the cutoff seen for viewing angles at which the LOS is close to the vortex axis is also easy to understand: the material rotating through the center of the vortex must be moving faster than matter on the outside of the vortex from the continuity equation.

This implies that one can measure the orientation of the vortex: if the line skews blueward, the spine of the vortex is moving toward the observer, and the shock having created the vortex must have moved toward the observer. Similarly, if the vortex is created through buoyancy (as suggested for the radio ear in Virgo by Churazov et al. 2001), it implies that the bubble is rising toward the observer. In both cases, it implies that the bubble is located closer to the observer than the cluster center.

Finally, we can understand the quadratic behavior of the line core, for small δν/ν₀ (where ν₀ is the rest frequency of the line). As laid out in Bachelor (1970) and S. Friedman & S. Heinz (2011, in preparation), one can show that, for asymptotically large distances from the vortex center, the vortex velocity field should fall off as v_rot ∝ r⁻³, distributed over a sphere following a roughly dipole-like pattern. That is, the velocity is self-similar. Examination of the velocity field of the Mach 2 vortex simulation we present in this paper does indeed follow this distribution.

Suppose, then, that at large enough radii away from the vortex the LOS velocity along a particular radius away from the vortex is proportional to v_LOS ∝ r⁻³ (with constant density and temperature and thus total line emissivity n²Λ(T) assumed along the radius).

The frequency shift of photons emitted along the radius are thus

$$\begin{equation} \delta \nu _{r} \propto r^{-3} \end{equation} \tag{ 1 }$$

or

$$\begin{equation} \frac{d\delta \nu }{dr}=-3\frac{\delta \nu _{r}}{r}. \end{equation} \tag{ 2 }$$

The emission from the entire sphere of radius r will follow some profile f that only depends on frequency through f(δν/δν_r), where δν_r is the maximum Doppler shift measured on that sphere. f(x) will depend on viewing angle, but is independent of r.

Since δν_r is the maximum measurable Doppler shift, f will be zero at frequencies above/below than δν_r. That is, emission at some δν will only be emitted from spheres within a certain radius r_δν such that δν_r ⩾ δν.

The total specific luminosity of the vortex at frequency ν is then given by integrating the emissivity over all spheres within r_δν:

$$\begin{eqnarray} L_{\nu }=\left|\frac{dE}{d\nu }(\delta \nu)\right| & = & \frac{d}{d\nu }\int _{0}^{r_{\delta \nu }}4\pi r^2dr\,n^2\Lambda (T) f\left(\frac{\delta \nu }{\delta \nu _{r}}\right) \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} & = & \left[\frac{d}{dr_{\delta \nu }}\int _{0}^{r_{\delta \nu }}4\pi r^2 dr\,n^2\Lambda (T) f\left(\frac{\delta \nu }{\delta \nu _{r}}\right)\right] \left|\frac{dr_{\delta \nu }}{d\nu }\right| \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} & = & 4\pi r_{\delta \nu }^2\,n^2\Lambda (T) f(1)\frac{r_{\delta \nu }}{3\delta \nu } \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&\hspace{-4pc} \propto \frac{r_{\delta \nu }^3}{\nu } \propto \nu ^{-2}. \end{eqnarray} \tag{ 6 }$$

Thus, emission from large radii dominates the line core, and the self-similar nature of the velocity profile, which scales only with r⁻³, sets the line profile function of the vortex line to follow ν⁻².

In realistic atmospheres, the intrinsic line profile will be thermally broadened. For light ions, this would wash out the expected line signature, since the thermal width would be comparable to the line width expected for the vortex. However, for heavy ions and other heavy tracers (e.g., dust particles), the thermal velocity would be lower than the characteristic vortex velocity by a factor of $\sqrt{1/A}$, where A is the atomic weight of the species. The heaviest abundant species is iron, and the thermal width of the line will be reduced by a factor of 0.13, in which case we can reasonably expect to see an observable expression of the vortex broadening.

The bottom panel of Figure 7 shows overlays of a thermally broadened Gaussian line (dark gray) and an actually thermally broadened vortex line (light gray) for an atomic weight corresponding to iron, given the post-shock temperature in the simulation (the latter simulated using a thermally broadened Gaussian instead of a delta function as the line Kernel). The vortex wings of the line are visible above the Gaussian beyond about 2.5 FWHM away from line center.

While the simulated line is integrated across the entire data cube, careful selection of the extraction region on sky could help in further boosting the vortex signature in practice: because the strongest line shifts occur around the vortex itself, the contribution from the cluster background could be minimized in an iterative procedure by extracting multiple concentric shells of spectra around the vortex and picking the set that shows the strongest spectral deviations.

The line profile derived above is, of course, not limited to astrophysical applications but should be observable whenever vortex creation through buoyancy or the RMI is operating. Lab experiments with shock tubes could be used to verify the line signature.

XIM is directly suited for grid-based hydro simulations on rectilinear grids. AMR is only possible in the sense of staggered meshes, not yet in the sense of completely independent cell sizes.

The main input variables for XIM are the particle density, the gas temperature, and the gas velocity in the form of three-dimensional arrays. Additionally, XIM can accept as input the local metallicity (relative to solar, tied to the relative solar radios for lithium and heavier ions) and a local volume filling factor of the emitting gas.

In addition, physical coordinate vectors (and optional telescope pointing, roll angle, and offset parameters) are used to calculate the emitting volume and angular size of each volume element of the input data, given an object redshift and cosmology. By default, XIM uses concordance cosmological parameters of $H_{0}=70\,\rm km\, s^{-1}$, Ω = 1.0, Ω_Λ = 0.7, and Ω_matter = 0.3.

The simulation output is calculated on a user specified energy grid, which can be set to correspond one to one to the energy channel mapping of the instrument. This allows the user to easily create full resolution spectra as well as multi-color images, or low-resolution spectra for reduced computational expense (high-resolution spectra at, e.g., IXO resolution can be very costly computationally).

A.2.1. Thermal Spectral Calculation

A.2.2. Other Spectral Models

A.2.3. Foreground Absorption

The spectral projection along the LOS is performed along an arbitrary viewing angle. For each cell of the input data, a unique spectrum is calculated for the cell temperature, cell density, and its proper Doppler shift, given the LOS velocity of that cell.

To project along an arbitrary LOS, the input data arrays are rotated (using tri-linear interpolation) onto a new grid that orients the x-axis along the LOS vector. The input velocity arrays are also projected onto the LOS vector to calculate an LOS velocity for every cell. LOS integration then proceeds along a coordinate axis of the new array.

It is possible to specify fill conditions for parts of the array that are rotated into view that are not specified in the original data (e.g., it is possible to clip the array to only show data cubes that were fully specified, or assume periodic boundary conditions when rotating the cube).

A.3.1. Doppler Shift and Energy Gridding

The Doppler frequency correction from the emitting to the observed frame is

$$\begin{equation} \nu _{\rm obs}=\delta \nu _{\rm emit}={\nu _{\rm emit}}\sqrt{\frac{1 - v_{\rm LOS}/c}{1 + v_{\rm LOS}/c}}. \end{equation} \tag{ A1 }$$

On a logarithmic energy grid, Doppler shift is a simple linear shift along the grid, which is computationally trivial and is the approach adopted by XIM for computational speed.

Thus, given an output energy grid (either user-defined or fixed to the instrument channel map), XIM creates an oversampled logarithmically spaced energy grid that is padded on both sides by the maximum Doppler shift encountered in the input data and takes into account the energy resolution of the instrument, additionally extending the grid to enclose a minimum of 99% of the energy at the grid edge, given the response matrix function of the instrument.

Doppler shift is performed by linear interpolation between the nearest two integer shifts along the energy axis that straddle the actual Doppler shift value of each cell. This reproduces the mean line centroid energy and leads to a small error in line width that is well below the energy bin width and well below the instrument resolution.

A.3.2. Line-of-sight Integration

The output from the LOS projection encapsulates the object's cosmological redshift, but it is still instrument independent (though often the energy grid will be chosen to be suitable to a particular instrument to keep computational expense and memory requirements at a reasonable level; e.g., for virtual Chandra observations, emphasis should be placed on spatial resolution, while for virtual IXO observations, the energy grid should be at the highest resolution possible).

Starting from the output file of the LOS integration, virtual observations with different target telescopes and instruments are performed in a number of steps described below.

A.4.1. PSF Simulation

A.4.2. Convolution with Instrument Responses

A.4.3. Virtual Detection

A.4.4. Backgrounds

A.4.5. Virtual Chandra Observations with MARX

Finally, we will briefly discuss some of the current limitations on the software that are not likely to be remedied or implemented in the near future.

A.5.1. Grid Geometries

A.5.2. Spectral Models

A.5.3. Imaging Restrictions: Vignetting and Off-axis PSF