Detailed simulation of a Lobster-eye telescope

The concept of an x-ray telescope based on the optics of the eye of certain types of crustacea has been in currency for nearly thirty years. However, it is only in the last decade that the technology to make the telescope and the opportunity to mount it on a suitable space platform have combined to allow the idea to become a reality. Accordingly, we have undertaken a detailed simulation study, updating previous simplified models, to properly characterise the performance of the instrument in orbit. The study reveals details of how the particular characteristics of the lobster-eye optics affect the sensitivity of the instrument and allow us to implement new ideas in data extraction methods. © 2009 Optical Society of America OCIS codes: (340.7440) X-ray imaging; (340.0340) X-ray optics. References and links 1. W. C. Priedhorsky, A. G. Peele, and K. A. Nugent, “An X-ray all-sky monitor with extraordinary sensitivity,” Mon. Not. R. Astron. Soc. 279(3), 733–750 (1996). 2. A. G. 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Introduction
In common with many other space-borne instruments, the lobster-eye telescope [1,2,3,4] has undergone a range of incarnations and possible platforms during its developmental stages [5].We have undertaken a detailed simulation study to characterise the performance of the instrument in orbit for a particular configuration -Lobster-ISS [5].The study answers a broad range of questions regarding the performance of a lobster-eye telescope that are generic to the optics and the reality of an orbiting platform.Additionally, the study is flexible enough that it can readily be adapted to reflect changes in configuration of the optics and in the particular details of the platform and orbital parameters.
In previous lobster-eye work the focal plane distribution for a single static point source was obtained by ray tracing through an ideal array that had been modified by a statistical distribution of defects -tilted, rotated, non-square and surface roughened channels -designed to mimic experimental data [6].The resulting cruciform point spread function (PSF) was then convolved with source distributions to investigate the sensitivity of the telescope [1,5].In such work the effects of orbital parameters on the telescope's duty cycle were dealt with on an averaged basis only, as were the impact of enforced detector shutdowns due to Sun or Moon avoidance, and the passage of the low Earth orbit platform through the South Atlantic Anomaly (SAA).In more recent work exposure maps taking these effects into account have been generated [7].It has also been pointed out [8] that a simulation of much improved verisimilitude could be obtained by incorporating orbital motion of the satellite into the ray trace, using a model of the telescope that held an actual rather than a statistical representation of the optics and using a realistic map of the x-ray background.
Here we implement that approach to produce realistic event maps of detected photons from the x-ray sky for a lobster-eye telescope on a moving platform.Removing the streaking of a source across the detector due to the motion of the telescope can be achieved in the usual way; by correcting for the attitude of the telescope.However, the lobster-eye telescope suffers from additional effects.Firstly, the cruciform structure appears rotated with every orbital pass due to the varying orientation of the telescope.Secondly, the effective area of the telescope is position dependant due to the nature of the optics and their modular deployment.While, both of these effects are present in most other satellite-borne telescopes, they are particularly pernicious here because of the extended PSF, and a strategy to counter them has not previously been demonstrated.We show that we can correct for these effects and produce maps of detected sources and measure their strengths.Earlier work [1,5,9] reviews the types of astronomical objects that will be observable using the lobster-eye telescope, the range and depth of which provides the rationale for this study.Here we also determine the ability of a lobster-eye telescope to study time varying astrophysical sources such as active galactic nuclei (AGN) and X-ray novae.

Description of the Simulation
The LOBSTER-ISS simulations implemented here aim to model the telescope using the international space station (ISS) as an orbital platform.Telemetry data for the position of the ISS was calculated using the Simplified General Perturbations Satellite Orbit Model 4 algorithm [10].The parameters of the telescope are based on those of [8] and are summarised in Fig. 1.For the gas detector [11] we assume a detector efficiency limited only by the entrance window of 1µm of Mylar with a total of 200 Å aluminum coating.These detectors have single photon detection and contribute essentially no detector background to the estimate of detector sensitivity.Specific improvements to earlier versions of the simulation include: accurate normalisation of source and background strengths, spectra and distribution based on ROSAT catalogues [12,13,14], implementing Sun, Moon and SAA avoidance, parallelising the code, and allowing simulation of dynamic sources such as AGN and X-ray novae.Ray-tracing improvements include absolute positioning of all channels and defects (instead of random distribution), which for a real array can be ascertained from microscope images of the front and back of the array [15], along with optimisations of ray path calculations.In addition we demonstrate a suite of analysis software that operates on the raw data allowing one to build time and position dependant exposure and effective area maps, correct for telescope motion and rotation, and perform source intensity calculations.Automated code to find bright sources and subtract their flux from data sets has also been written, this enhances the detection of fainter x-ray sources under the skewed crucifix arms of a bright source.
Parallelisation of the LOBSTER-ISS code allowed it to be executed on a supercomputing cluster.Orbital ray-tracing simulations run at approximately 1  2 real time on a single Pentium 4 node.Utilising a cluster of 32 dual nodes a 180 day simulation is completed in approximately 2 to 3 days.The output from the simulation is a raw event map with orbital telemetry, akin to that which would be produced by the actual LOBSTER telescope.It is predicted the telescope will produce approximately 2 gigabytes of data per day, collecting from all 6 modules.

Results and Analysis
A typical raw output of the simulation is shown in Fig. 2(a), with a corresponding movie (Media 1) demonstrating the collection of data over a single 90 minute ISS orbit, highlighting Sun, Moon and SAA avoidance in the simulations.The first step in the analysis of the data is to put photon events back to their apparent position on the sky by correcting for the orbital path of the telescope.This is shown in Fig. 2(b), resulting in the characteristic crucifix shaped point spread function.For different orbital epochs the telescope will have some rotation angle around the pointing direction, which produces an effect we term crucifix skew due to the fact that the focal arms parallel the channel wall orientation.This is readily corrected for by storing the telescope telemetry with each photon event, then correcting for the attitude and position by applying Euler matrix rotations in a spherical coordinate system.Attitude and position should be corrected for in reality by using the telemetry from co-mounted [5] star-trackers.
Another distortion effect arises from the choice of geometry in tiling the module.This choice defines the orientation of the cruciform arms from different regions of the array -again because they parallel the channel walls.It also defines how well a module approximates spherical symmetry.Simulations of sources at various positions on a module for different tiling geometries demonstrate the advantage of a double equatorial tile geometry.This is shown in Figs.3(c) and 3(e), here the mis-alignment of crucifix arms under double meridional geometry is obvious.This effect can only be well corrected for if the source is stationary with respect to the telescope and the center of the PSF maintains a fixed position on its surface, or if the point of reflection of each photon is known.Neither is possible in practice.Placing tiles on a module surface with a double equatorial geometry minimises this uncorrectable distortion in the PSF.This result indicates the need for a redesign of the module geometry from that used in previous work [5,6].

Effective Area Calculations
For the purpose of calculating the intensity of sources detected we must understand the effective area of the telescope.This response function allows us to convert between the flux detected by the telescope, F det (E) (photons s −1 ), and the incident intensity of photons at the telescope, I src (E) (ergs s −1 cm −2 ), given that we can calculate the duty cycle of the telescope (exposure time).Using LOBSTER-ISS simulations we calculated the effective area, A e f f (E), of the telescope based on a source modeled at the center of a module.The effective area was calculated for both a "standard" LOBSTER module with channels made from Schott 8092 glass [16] with a density of 3.01 g/cm 3 , and for Nickel coated channels which theoretically provide an increase in reflectivity.The effective area curves are shown in Fig. 4, highlighting the advantages of Nickel coated channel walls.At the time of this work glass was the chosen construction material [5] due to the difficulties in demonstrating a high quality Ni coating, hence it was used in The telescope is also not perfectly spherically symmetric.Boundaries between tiles and modules, and the tiling geometry mean that the effective area will vary for different source positions.Accordingly, we calculate the position dependant effective area, A e f f (x, y, E) (see Fig. 4(b)), across the surface of a module.Knowing A e f f (x, y, E) allows one to convert between the flux recorded from a source at a particular x, y position on the detector to an actual source intensity via the function: where x, y and E define the source position and specific photon energy, F det (x, y, E) is the recorded flux, and I src (E) is the actual intensity of the source.
The position dependant effective area maps were created by taking 0.5 • steps across a quadrant of a LOBSTER module.For each of these positions a flat spectrum source was modeled with a count rate of 100.0 cnts s −1 for 5000 seconds to provide sufficient photon counting statistics.For comparison, the Crab x-ray source provides a count rate of 2.5 cnts s −1 in the 0.1 to 2.1 keV band.The effective area was then calculated for this source within a collection region 4 arc minutes in diameter.We can measure the flux in the 4 arc minute region and thus determine the source intensity.The noise level in Fig. 4(a) gives an indication of the exposure time necessary to get good statistics for weaker sources.A sample of the resulting effective area maps for three separate photon energies is shown in Fig. 4(b).Here the tile boundaries can be clearly seen and the necessity of position dependant effective area calculations is highlighted.
It should be noted that these effective area maps are based on the photons in the central maxima of the point spread function, positioned at the point x, y on the module surface.In reality there is no way to determine if a detected photon is part of the central maxima, part of a crucifix arm, or if it has undergone higher order or zero reflections from the channel walls.This fact places an inherent error in the calculation of source intensities in LOBSTER telescopes.
The result of using A e f f (x, y, E) to calculate the flux from a source modeled on the Crab nebula is discussed in section 6.

Dynamic Source Subtraction
A moving LOBSTER telescope looking at an astrophysical source will form a skewed crucifix pattern.Therefore, we must remove the photons that make up this crucifix in order to find the sources that lie below the arms, and to calculate the intensity of those sources.This leads to the requirement of an iterative approach for source finding.
In many astrophysical applications it is necessary to group objects into both energetically and/or spatially bound groups.This is particularly common in N-Body simulations and group finding in galaxy redshift surveys.One of the most common methods to do this uses the "Friends of Friends" group finding method [17].This method links pairs of particles with a given separation less than some linking length, b FOF , which is often defined as a fraction of the mean inter-particle separation within the volume or area of space, B. Once all particles separated by distances ≤ b FOF are partitioned into groups, all groups with members less than some minimum population, M, are discarded.
Studies have shown [18,19] that an effective linking length to use in the friends-of-friends algorithm is 0.15 times the mean inter-particle separation.However, in our case a trial and error approach applied to fields of view from 1 • to 5 • results in b FOF and M defined as: (2) where N is the number of particles within the area or volume of space considered.
Once the maxima of the point spread function for the brightest source in the field of view is found using the values of b FOF and M described in equation 2 and 3, we can rotate event data around that central maxima to correct for crucifix skew.Group finding can then be performed again to find both the central maxima and the now aligned crucifix arms.The photons in these arms and central maxima are subtracted from the event list in that region, allowing us to calculate the total flux from that particular source and to continue the iteration to find the next brightest source.
The results of this dynamic source subtraction algorithm are shown in Fig. 5.As seen, the resultant field is free from the majority of photons belonging to the bright source (Fig. 5(d)).Ultimately, the sensitivity to faint sources underlying brighter sources will depend on the statistical residual from the subtraction of the bright source and the faint source.This will depend on factors such as the relative brightness of the sources, their position, and the trajectory of the telescope, which determines the relative exposure for each source.

Sensitivity Predictions
In order to measure the intensity of a source we require not only an accurate measure of the position dependant effective area, but also accurate recording of the exposure time.The duty cycle of the telescope is affected by factors such as telescope position and the satellite's orbital state (for example, it might be within the SAA).Our simulations predict duty cycles of approximately 5% depending on the epoch for LOBSTER-ISS, whereas previous analytical estimates [1] assumed the lobster duty cycle to be as high as 16%.Clearly, critical parameters can have a significant effect on the net flux collected by the instrument.
We define a detection event to be when a flux collection area with a diameter of 4.0 arcmin has a count rate that is 3σ above the local (2 • diameter) background, after subtracting previously detected sources.
For the source discussed in section 4, the source flux is accurately recovered to within a error range over the input energy spectrum, defined by the nominal ROSAT bands of 0.1 keV to 2.1 keV [14] for both the broader ROSAT energy bins, and for energy bins of 10 eV (see Fig.  calculated x-ray background flux) was used as an input to the simulation to attain sensitivity measurements for the nominal ROSAT band of 0.1 keV to 2.1 keV.After a single day of simulated observations (for which there was an average duty cycle of approximately 3%) 25% of these sources are detected.After 30 days more than 70% of the sources are detected.If we only consider sources that were detected within a 20% error margin of the catalogue count rate, the faintest sources above the detection limit after 1 day and 30 days of simulation were 1.5 × 10 −12 ergs s −1 cm −2 and 2.5 × 10 −13 ergs s −1 cm −2 respectively.These limits are similar to those found in previous work [1], but for different reasons: here the duty cycle degrades the detection limit, while more accurate modeling of the effective area and sky backgrounds with a more efficient source finding method appears to have improved it.

Detection of Novae and AGN
To determine the ability of LOBSTER-ISS to detect highly time variable sources and re-create their light curves.We simulated an X-Ray Nova from the measured light curve of X-Ray Nova XTE J1550-564 [20,21].This event was measured in September 1998, reaching a detected flux in excess of 1.6 × 10 −7 ergs cm −2 s −1 (6.8 crab) in the 2 − 10 keV energy band [20].
The light curve (see Fig. 6(b)) and count rate for XTE J1550-564 were extrapolated from the data provided by [20] and placed into the simulations, which ran for 180 days of observing time.To simulate the entire sky for this period using the ROSAT bright and faint source catalogues as well as the x-ray background with the XTE J1550-564 source as an additional input equated to approximately 3 days computing on 64 dual core supercomputing nodes, though selected regions of sky can be simulated orders of magnitude faster.
The count rates were calculated every day during the 180 day period.Periods with high flux could be measured at smaller intervals, providing higher time resolution.The reconstructed curve matches the theoretical light curve, both shown in Fig. 6 N deviation.Errors are larger in the region from 0 to 20 days as the count rate for the detected source was calculated more frequently.The agreement between the recovered and actual count rates is very good.Some small discrepancies, particularly in regions of low count rate, can still be seen.These could be further improved by more sophisticated use of the position dependant effective area.

Conclusion
We have presented work on a flexible ray tracing simulation of the LOBSTER telescope that can use as inputs satellite telemetry, Sun, Moon and SAA avoidance, realistic x-ray source and background information based on catalogue data, and losses in the instrument from a realistic model of the arrays that combine to create the LOBSTER telescope.
Our simulations have resulted in exposure maps that agree with previous work [7], while including more detail relating to the telescope modules.The necessity for two dimensional mapping of the effective area has been highlighted by creating position dependant maps for various energies and observing the variability in the LOBSTER effective area.
Dynamic source finding and subtraction algorithms were described and their effectiveness in removing a source from the field of view was displayed.These algorithms were combined with the position dependant effective area maps to find and catalogue sources after 1 day and 30 day simulations, resulting, for the nominal ROSAT band of 0.1 to 2.1 keV, in a 1 day sensitivity of 10 −12 ergs s −1 cm −2 and a 30 day sensitivity limit of 2.5 × 10 −13 ergs s −1 cm −2 .
The reconstruction of light curves from a simulated x-ray nova demonstrates the ability of LOBSTER-ISS to discover and track highly variable intense x-ray sources.This provides an important predictor of our ability to detect future events such as supernovae, and highlights the strengths of all-sky monitors at monitoring these events.

Fig. 2 .
Fig. 2. (a) Raw event map from a simulation showing measured photon positions on the detector for two sources.(b) Event map for two sources after telemetry correction has been applied (crucifix skew is still present).Note that the upper arm of the right most source is faint due to the source being close to the edge of the module during the exposure.(c)Telemetry corrected event map for a single 90 minute orbit of the ISS using all 6 LOBSTER modules (Media 1).Note the "stripes" without coverage due to insufficient overlap in the field of view between modules for the particular layout simulated (this is due to a drop off in the position dependant effective area at the edges of a module as discussed in section 4).

Fig. 3 .
Fig. 3. (a) Double meridional and (c) Double equatorial geometry used in stacking tiles.Four sources simulated in different quadrants on the detector using double meridional geometry (b) and double equatorial geometry (d).After combining exposures (c) and (e), the rotation due to source position on the detector cannot be corrected for during analysis if double meridional geometry is used.

Fig. 4 .
Fig. 4. (a) Effective area in the center of a module for different channel coatings and (b) Position dependant effective Area for the 0.2 keV, 1.0 keV and 1.5 keV.

Fig. 5 .
Fig. 5. (a) Plot of all events in a 5 • × 5 • field after correcting for telescope motion, as in Fig. 2(b).(b) Particles rotated around the brightest maxima, note the faint arcs indicate the presence of the other sources, as circled in (a).(c) The particles which belong to the source and (d) the particles remaining after one iteration of the source subtraction algorithm.The sources left in the field of view have been circled.
6(a)).Accordingly, it should be noted that measurement of sources with more complex spectra is possible, depending on the strength of the source and the energy resolution of the detector.An arbitrary 10 • × 10 • region of sky from the ROSAT catalogue (centered at galactic coordinates L, B = −30 • , 0 • ) containing 293 bright and faint sources (along with the ROSAT

Fig. 6 .
Fig. 6.(a) Reconstruction of a Crab type source.The green line indicates the original source flux for each of the 3 ROSAT bands.The blue points show the reconstructed intensity within the 3 ROSAT bands, where the spectrum has been broken into smaller energy bins of 10 eV to get a feel for the performance at higher spectral resolution.The red dot at the center of each bin represents the recovered counts averaged over the width of the particular ROSAT energy band.The error for the recovered count is smaller than the dot.(b) Reconstructed light curve for XTE J1550-564 using simulated measurements across the entire ROSAT band.The green line indicates the original source flux.