Given these two methods of determining the photon interaction order in GCTs, CKD and SSD, a full event reconstruction technique incorporating other background rejection techniques can be developed. The high spectral and spatial resolution of a GCT make several powerful background rejection techniques possible. The predominance of multiple scattering events, while initially a complication, dramatically helps in the overall background rejection.

The dominant sources of background in GCTs are expected to be diffuse cosmic $\gamma$ -ray emission, induced satellite $\gamma$ -ray emission, and induced $\beta ^{+}$ , $\beta ^{-}$ radioactivities in the GeDs themselves (Jean et al. 1996; Graham et al. 1997; Gehrels 1985; Dean et al. 1991; Naya et al. 1996). Source/background photons which scatter out of the instrument before depositing all of their energy, and hence are improperly imaged, must also be included in these calculations.

$\begin{figure} \resizebox{8.8cm}{!}{\includegraphics{h2143f7.eps}} \end{figure}$

Figure 7: Peak-to-Compton ratios. Shown are the ratios of correctly ordered (imaged) photopeak events to integrated Compton continuum events for: 3+ sites before CKD rejection ( $\diamond$ ), 3+ sites after CKD rejection ( $\triangle$ ), and two-site events ( $\Box$ )

Restrictions on the acceptable events can have a dramatic effect on sensitivity of a GCT. Specifically, several factors can affect the angular resolution of the instrument as well as the background rates - such as the inclusion of backscatter events, limits on the accepted scatter angles, and the minimum acceptable lever arm - and must be included in any discussion of full reconstruction and background rejection.

6.1 Shield veto

Placing an active shield below the bottom GCT detector plane could be useful for rejecting background photons from below the instrument (induced satellite $\gamma$ -ray emission), as well as helping to reject Compton continuum and $\beta-$ decay events in the instrument. However, many of these events can be distinguished and rejected using CKD and other tests/restrictions outlined below; therefore, the usefulness of including a shield in the GCT design must be studied in detail for a given telescope configuration.

6.2 Restrictions on number of interaction sites: Pair production/ $\beta ^{+}$ decays

While it is obvious that any single-site interactions should be rejected in a Compton telescope, events with $\sim$ 8 or more interaction sites should also be rejected since these are very likely due to pair production events, as is evident from Fig. 3, or similarly $\beta ^{+}$ decays (see Sect. 6.8).

6.3 CKD $\chi ^{2}_{\rm min}$ test: Compton continuum, unresolved interactions, etc.

Using the tests outlined in Sects. 4 and 5, the most likely ordering of the interaction sites can be determined, and for 3+ site events many of the unresolved and Compton continuum events, as well as pair production and $\beta-$ decays, rejected.

Shown in Fig. 7 is the peak-to-Compton ratio for 3+ site events, here defined as the ratio of the properly imaged photopeak events to the corresponding integrated Compton continuum (photons which scatter out of the instrument before depositing all of their energy). This standard measure for $\gamma$ -ray spectroscopy instruments has an altered meaning here, since the Compton continuum events will be incorrectly imaged, and thus will appear as off-source background. The peak-to-Compton ratio is shown both before and after rejection of the continuum events with the CKD statistic. CKD rejection of the Compton continuum events increases the peak-to-Compton ratio by factors of 3-6, an important improvement for low background instruments. By rejection of events appearing to originate from below the instrument (Sect. 6.4), as well as backscattered interactions (Sect. 6.5), this ratio can be increased by further factors of 2-4.

Also shown in Fig. 7 is the photopeak-to-Compton ratio for two-site events using SSD, which is significantly lower than the corresponding ratio for 3+ site events. In fact, this ratio drops significantly below unity, which means that more background than signal is being created in the instrument for two-site events. This result questions whether two-site events should be included in actual observational analysis given the accompanying increase in unrejectable background. This conclusion is further supported by the fact that the majority of two-site events are backscatters (Sect. 6.5), which will significantly degrade the angular resolution. Even though inclusion of two-site events is unlikely to improve the overall sensitivity for GCTs, detailed background analysis for specific instrument configurations is required to determine the overall effects.

6.4 Effective TOF

Once the most likely order of interactions and the initial scatter angle are determined, it is possible to determine whether the incident photon scattered upwards or downwards in the instrument, as well as whether the initial scatter was forward or backwards. Thus, events which appear to be photons originating from below the instrument can be rejected, which include the induced satellite $\gamma$ -ray emission, many Compton continuum events which were not rejected by CKD, photons which scatter in the passive satellite material before interacting in the detectors, and many of the pair production and $\beta-$ decay events. The simulation results for the configuration in Appendix A show that $\sim 95\%$ of photons originating from below the instrument are rejected.

6.5 Backscatters

Once the most likely ordering of interaction sites has been determined, this information can also be used to accept/reject backscattered source photons. The fraction of photopeak events which backscatter during the initial interaction is not strongly energy dependent, $\sim 60-70\%$ for two site events, and $\sim 30\%$ for 3+ site events. These events can significantly increase the effective area at lower energies, where two-site events are most common, at the expense of degrading the angular resolution due to larger uncertainties in $\delta \phi_{1,E}$ for backscatters events (Eq. 4). It is unlikely that the overall sensitivity will improve by including backscattered events given the increased background rates and degraded angular resolution; however, the effects on sensitivity will depend on the exact instrument configuration and observational goals.

6.6 "Standard $\phi$ restriction''

Restrictions can also be set on the scattering angles accepted for forward-scattering photons entering the front of the instrument. These limits can be used to restrict the instrument FOV to improve imaging capabilities and background, such as the "standard $\phi$ restriction'' (Schönfelder et al. 1982). These restricitions will have to be reanalyzed in detail for specific GCT configurations.

6.7 Nonlocalized $\beta ^{-}$ decays

In a nonlocalized $\beta ^{-}$ decay, the daughter nuclide is produced in an excited state which quickly decays on timescales relative to the detector collection time, emitting a photon with energy characteristic to the daughter nuclide. Therefore, the event consists of the intial $\beta ^{-}$ decay site, plus the interaction sites of the emitted photon. Such an event can be rejected if the characteristic photon energy can be detected in any combination of the interaction site energies. The seven dominant $\beta ^{-}$ isotopes and characteristic photon energies for natural Ge are given in Table 1 (Naya et al. 1996; Gehrels 1985). In general, if the coincident $\gamma$ -ray is fully deposited, then the event will have $\chi^{2}_{\rm min} \sim 1$ , with the $\beta ^{-}$ electron interaction ordered as the initial "scatter'' site. In these cases, W₁ will have the characteristic photon energy specific to that decay. The rejection of all events with W₁ equal to one of the characteristic energies in Table 1 can dramatically decrease the $\beta ^{-}$ decay background, with only a small effect, typically $\leq 3\%$ drop in photopeak efficiencies for true photon events. More $\beta ^{-}$ decays can be rejected if every possible combination of interaction sites is tested for the decay photon energies, at the expense, however, of more rejection of photopeak events, typically $\sim 15-20\%$ . If the coincident $\gamma$ -ray is only partially deposited, the event will likely be rejected by the limits set on $\chi ^{2}_{\rm min}$ .

**Table 1:** Characteristic photon energies for the strongest nonlocalized $\beta ^{-}$ decays in natural Ge
daughter	photon energy
⁷⁵Ge	0.265 MeV
⁷³Ga	0.297 MeV
⁷²Ga	0.834 MeV
⁷¹Zn	0.512 MeV
⁷⁶As	0.559 MeV
²⁸Al	1.779 MeV
⁷⁷Ge	0.216 MeV

$\beta ^{-}$ decay background events were simulated for the instrument discussed in Appendix A. The results of these background calculations will be presented in a separate paper, but here we make preliminary use of these simulations to demonstrate the background rejection capabilities. After initial rejection of two site ( $34.4\%$ ) and 8+ site ( $2.9\%$ ) events, $62.7\%$ of the nonlocalized $\beta ^{-}$ events remain. Applying the CKD test, requiring a $5\%$ probability of $\chi ^{2}_{\rm min}$ , brings the remaining number of decays down to 17.9%. After screening the interactions for characteristic $\beta ^{-}$ decay energies, this number is reduced to 9.3% (15.1%). Finally, after rejecting events which appear to originate from below the instrument, or which appear to be backscatter events, the final number of unrejected $\beta ^{-}$ decays comes to 4.2% (6.8%), a factor of 20 (15) reduction in this background component. (First numbers give results when all combinations of interaction sites are searched for $\beta ^{-}$ decay energies, while the numbers in parenthesis are results when only W₁ is tested. Typical errors $\sim 0.2\%$ .)

6.8 Positron signatures

A further test for rejecting the pair production/ $\beta ^{+}$ background events that survive the other tests/restrictions outlined above is to search for positron annihilation signatures in the interaction energies. By analyzing all combinations of the interaction sites to see if the energies sum to $m_{\rm e}c^{2} = 0.511$ MeV, events with a positron annihilation signature can be rejected. This test typically reduces the non-pair production photopeak events by $\leq 2\%$ .

$\beta ^{+}$ background events were simulated for the instrument discussed in Appendix A. After initial rejection of two site events ( $23.0\%$ ) and 8+ site events ( $3.0\%$ ), $74.0\%$ of the events remain. After the CKD test, $16.7\%$ remain. After screening the interactions for 0.511 MeV positron annihilation signatures, the number is reduced to $5.2\%$ . Finally, after rejecting events which appear to originate from below the instrument, or which appear to be backscatter events, the final number of unrejected $\beta ^{+}$ events is $1.9\%$ , a factor of 50 reduction in this background component. Similar reductions occur when these tests are applied to pair production events. (Typical errors $\sim 0.1\%$ .)

6.9 Minimum lever arm

In general, a minimum acceptable distance between the first and second interaction sites - the lever arm - must be set. Figure 8 shows the fraction of 0.5 and 2.0 MeV photopeak events with lever arms above a given level, for the instrument configuration in Appendix A. Similar to the case of backscattered events, a smaller minimum lever arm means a higher effective area at the expense of poorer angular resolution. The exact lever arm chosen will depend on the instrument configuration and observational goals.

$\begin{figure} \resizebox{8.8cm}{!}{\includegraphics{h2143f8.eps}} \end{figure}$

Figure 8: The fraction of 0.5 and 2.0 MeV photopeak events with lever arms (separations between the first and second interaction sites) greater than the specified values for the model in Appendix A. Shown for comparison are several characteristic distances of this model

**Table 2:** Percentage of events remaining after subsequent application of rejection techniques
rejection	0.5 MeV	2.0 MeV	$\beta ^{-}$ decays	$\beta ^{+}$ decays	spacecraft
technique	photopeak	photopeak
2 site events	65.0%	82.0%	65.6%	77.0%	61.6%
8+ site events	64.9%	80.9%	62.7%	74.0%	60.8%
CKD	51.2%	61.4%	17.9%	16.7%	35.2%
$\beta$ signatures	48.4%	58.0%	15.1%	5.2%	32.2%
backscatter/TOF	35.5%	38.5%	6.8%	1.9%	12.2%
min lever arm (10 cm)	15.0%	16.7%	2.0%	1.2%	2.6%