Appendix A: Example GCT configuration

Figure A1: Assumed resolutions for the GCT model. (Left) The total energy resolution (solid line) is determined by the electronic noise (dotted line) and the intrinsic resolution (dashed line) added in quadrature. (Right) The total x-, y- (solid line) and z- (dot-dashed line) spatial resolution are determined by the segmentation/signal processing limits (dotted line), and the recoil electron range (dashed line) added in quadrature. (See text for details)

While it is not the intention of this paper to fully characterize the performance of a specific GCT, it is useful to have a telescope model for which the results of the event reconstruction can be presented. The telescope configuration modeled in this study is presented in Fig. 1. The instrument consists of five planar arrays of 15 mm thick germanium, each of area 100 cm $\times$ 100 cm. In reality each array would consist of separate smaller detectors ($\sim 5$ cm $\times$ 5 cm) tiled to form the entire plane; however, the simulation performed here modeled each plane as a solid detector for simplicity. The five planar arrays are spaced 20 cm apart.

This configuration differs from historical Compton telescope configurations which generally consist of two detector planes separated by 100-150 cm. This separation distance is determined by the spatial resolution in z and the desired angular resolution. As will be discussed in a second paper, the configuration modeled here significantly improves the effective area of the telescope by letting each plane act as converter, and permitting a much wider range of scatter angles to produce good events. Allowing large-angle scatters also significantly increases the instrument FOV, and limits the effects of point spread function smearing for sources at large off-axis angles. The potential drawbacks of this configuration are increased background and degraded angular resolution.

The instrument was simulated using CERN's GEANT Monte Carlo code. The Monte Carlo simulation produces a file of interaction locations and energy depositions for each photon/$\beta-$decay event. Before performing event reconstruction on the interactions, the simulated events are modified to reflect realistic measurement uncertainties of an instrument: for each interaction, a random Gaussian-distributed uncertainty is added to the energy and position of each interaction. All interaction locations which lie within twice the instrumental spatial resolution of each other are combined into a single interaction site, to accurately reflect the resolving power of the detectors. Finally, interaction sites with energy deposits below the assumed detector threshold of 10 keV are ignored.

Two components are assumed to add in quadrature to determine the energy resolution: (i) a constant electronic noise, $W_{\rm e} = 1.0$ keV FWHM, and (ii) the intrinsic resolution W_i determined by the germanium Fano factor, F = 0.13, and average free electron-hole pair energy, $\varepsilon = 2.98$ eV, giving $W_{i} = 2.35 \sqrt{F \varepsilon E}$ FWHM. This corresponds to a resolution $\sim 1.8$ keV FWHM at 1 MeV, which is optimistic but not unrealistic. It is assumed that charge trapping and ballistic deficit do not significantly alter this energy resolution. The two components as well as the total energy resolution are shown in Fig. A1.

It is assumed that two components add in quadrature to determine the 1-D spatial resolutions, $\delta x, \delta y, \delta z$, of the detectors: (i) the range of the recoil electrons in the detector, and (ii) the positioning limits of the detector due to physical segmentation and/or signal analysis. Calculated electron ranges in germanium for different energies (Mukoyama 1976) are used here as the 1-D FWHM positional uncertainties, $\delta x_{\rm e}, \delta y_{\rm e}, \delta z_{\rm e}$. Methods to determine the event position by physically segmenting the GeD contacts into cross strips or pixels (Luke et al. 1994; Kroeger et al. 1995), as well as using advanced signal processing to interpolate to even better positions (Boggs 1998; Luke et al. 1994), are currently active fields of research - so this component of the spatial resolution remains speculative for now. Here it is assumed that signal processing will allow positional resolutions of $\sim 0.5$ mm FWHM at 100 keV, and that the discrimination capabilities go as the signal-to-noise ratio of the induced detector signal to electronic noise, i.e. as the inverse power of the interaction energy. It is also assumed that there is $\sim 1$ mm physical segmentation of the detector contacts in x, y, so that this component never exceeds this value. The z uncertainty, however, is not constrained by any such segmentation at the lowest energies. Therefore, the signal processing uncertainty is given by $\delta x_{\rm s}, \delta y_{\rm s}, \delta z_{\rm s} \sim 0.50 (E/100$ keV)^-1 mm FWHM, maximizing at 1 mm in x, y below 50 keV, and approaching, but never maximizing at 15 mm in z at low energies. The two components as well as the total spatial resolution are shown in Fig. A1.