Appendix B Line fitting procedures in analyses of CDG and ²⁶Al

Many astrophysical analyses, such as that of the cosmic diffuse gamma-ray background (CDG), of the galactic 1.8 MeV line emission from ²⁶Al, or of $\gamma$-ray line emission from supernovae, can be optimized by detailed modelling of the instrumental line background. Below, the (similar) procedures used to determine the background event rates due to individual isotopes employed by Weidenspointner et al. (2001); Oberlack (1997) and Plüschke et al. (2000) in investigations of the CDG and of the ²⁶Al emission, respectively, are described.

  
B.1 CDG analysis

Figure B.1: A schematic representation of the simulated E1-E2 distributions of the eight isotopes identified in the instrumental line background for CDG event selections. The "vertical'' and "diagonal'' bands represent multiple photon and single photon events, respectively (compare to the detailed individual diagrams in Sect. 4)

Figure B.2: An illustration of the E1-E2 ranges of the three E2 spectra and the $E_{\rm tot}$ spectrum used to determine the event rates due to the background isotopes. Also plotted are lines of constant $\bar{\varphi}$ (solid) and $E_{\rm tot}$ (dashed)

Accounting for the event rates due to the instrumental line background, in particular its long-lived components, is important for measuring the CDG at MeV energies (e.g. Kappadath et al. 2000; Weidenspointner et al. 2001). As explained above, the background contributions of long-lived isotopes have to be subtracted before prompt and short-lived backgrounds can be eliminated by veto rate extrapolation. The event rates due to the eight identified background isotopes producing the major instrumental lines are determined, as a function of veto rate, in an iterative procedure by fitting a set of three E₂ spectra and one $E_{\rm tot}$ spectrum for each veto rate interval.

The rationale of the iterative fitting procedure is illustrated in Figs. B.1 and B.2. The E₁-E₂ distributions of the eight isotopes as obtained from Monte Carlo simulations are schematically depicted for CDG event selections in Fig. B.1. In general, there is considerable overlap in the E₁-E₂distributions of individual isotopes (in particular around 1.3 MeV in E₂), which precludes an independent determination of the isotopes' background contributions. Therefore, an iterative procedure was introduced, which starts at the highest energies in E₁ and E₂, where ambiguities are minimal, and then proceeds down to the increasingly complex structures at lower E₁ and E₂energies. The E₁-E₂ ranges of the three E₂ spectra and the $E_{\rm tot}$ spectrum, chosen such as to enhance or suppress individual lines or spectral features, are indicated in Fig. B.2. The E₁-E₂ ranges covered by the second and third E₂ fit and by the $E_{\rm tot}$ fit overlap, hence the results of these fits are not statistically independent. The overlap is caused by the $E_{\rm tot}$ fit, which was introduced to properly separate the background events from ²D and ²⁸Al. Inclusion or omission of the $E_{\rm tot}$ fit therefore is a trade-off between systematic and statistical uncertainty. By iteratively fitting the second and third E₂ spectrum and the $E_{\rm tot}$ spectrum, both the systematic and the statistical uncertainty in the ²D and ²⁸Al event rates are minimized (other isotopes are hardly affected by the overlap, see below). In additon, the iterative approach ensures the self-consistency of the determined isotope background contributions. Also included in Fig. B.2 are lines of constant $\bar{\varphi}$ and $E_{\rm tot}$. Comparison of Figs. B.1 and B.2 provides a first indication of the many options for fitting individual lines, which can be enhanced or suppressed through the choice of the fit regions, and in addition through selections on $E_{\rm tot}$ and/or $\bar{\varphi}$. In particular, event selections may be used to suppress unidentified, long-lived spectral features, which cannot be eliminated by veto rate extrapolation (see Kappadath et al. 2000; Weidenspointner et al. 2001).

The E₁-E₂ ranges of the three E₂spectra and the $E_{\rm tot}$ spectrum indicated in Fig. B.2 were chosen for the iterative fitting procedure for the following reasons. The first E₂ spectrum, covering the E₁-E₂region of 950-1250 keV in E₁ and 2000-3500 keV in E₂(see Figs. B.1 and B.2), allows us to determine the event rate from ²⁴Na. The signal from this isotope is optimized by selecting E₁ energies around the Compton edge of the 1.37 MeV photon interacting in D1 and around the photopeak of the 2.75 MeV photon interacting in D2 (see Fig. 5). The ²D event rate is determined from fitting the $E_{\rm tot}$spectrum ( $E_{\rm tot}$ 1810-2800 keV, E₁ 70-950 keV, and E₂ 730-2800 keV, see Figs. B.1 and B.2). The second E₂ spectrum (E₁ 500-950 keV, E₂1500-3500 keV, see Figs. B.1 and B.2) is used to determine the event rate from ²⁸Al. Finally, the third E₂spectrum is intended for determining the background contributions from the $\beta^+$-decays of ²²Na, ⁵²Mn, and ⁵⁷Ni, with the 270-350 keV range in E₁ being optimized for the Compton edge of 511 keV photons, and the E₂ range covering the energies 1100-3500 keV (see Figs. B.1 and B.2). To optimize the signal of the instrumental lines, which originate in the D1 detector material, a ToF range of 2.5-7.5 ns was selected for the spectra (compare Fig. 3).

The three E₂ spectra and the $E_{\rm tot}$ spectrum are analyzed in an iterative procedure consisting of eight fits, listed below. The contributions from the primordial radio-nuclides ⁴⁰K and ²⁰⁸Tl are not determined from the fits, but calculated from their known (⁴⁰K, see Sect. 4.3) or estimated (²⁰⁸Tl, see Sect. 4.7) activities based on Monte Carlo simulations. Similarly, once the background contribution of an isotope has been determined from any spectrum, the isotope's contribution to any other spectrum can be predicted based on Monte Carlo simulations. The eight identified isotopes can account for the major instrumental lines, however, some weak lines or spectral features remain unidentified at this time. In the fits, some of these unidentified lines were described by Gaussians to minimize systematic errors in the determination of the event rates of the identified components. The unidentified components are genuinely different from those identified since their variation with cosmic-ray intensity, as well as their dependence on event parameter selections, are different (see Kappadath et al. 2000; Weidenspointner et al. 2001). The identified isotopes are represented by templates obtained from Monte Carlo simulation. These templates have not been smoothed as smoothing inevitably increases the systematic uncertainty due to distortions of the template shape. The small "spikes'' in the templates in Figs. B.3-B.6 are an artifact of the plotting software. The eight fit steps are:

Figure B.3: An example of a fit of the first E2 spectrum, which is used to determine the event rate due to 24Na. In addition to the total fit, the 24Na template (solid line), the two unidentified features (dashed and dash-dotted lines), and the exponential continuum are indicated

Figure B.4: An example of a fit of the $E_{\rm tot}$ spectrum for determining the count rate in the 2D 2.22 MeV line. In additon to the total fit, the 2D template (solid line), an unidentifed component at about 2.3 MeV (dashed line), the continuum contributions from 24Na and 208Tl (solid and dash-dotted lines), and the exponential continuum (dashed line) are indicated

1.

E₂ Fit 1: The event rate due to the isotope ²⁴Na is determined (see Fig. B.3);

2.

$E_{\rm tot}$ Fit: The ²D event rate is estimated; the contributions from ²⁴Na and ²⁰⁸Tl are fixed. All other isotopes are neglected;

3.

E₂ Fit 2: The ²D and ²⁸Al event rates are estimated; the contributions from ²⁴Na and ²⁰⁸Tl are fixed;

4.

E₂ Fit 3: The background contributions due to the $\beta^+$-decays of ²²Na, ⁵²Mn, and ⁵⁷Ni are estimated; the contributions from ²D, ²⁴Na, ²⁸Al, ⁴⁰K, and ²⁰⁸Tl are fixed;

5.

$E_{\rm tot}$ Fit: The ²D event rate is determined; the contributions from all other isotopes (²²Na, ²⁴Na, ²⁸Al, ⁴⁰K, ⁵²Mn, ⁵⁷Ni, and ²⁰⁸Tl) are fixed (see Fig. B.4);

6.

E₂ Fit 2: The event rate due to the isotope ²⁸Al is determined; the contributions from ²D, ²⁴Na, and ²⁰⁸Tl are fixed (see Fig. B.5);

7.

E₂ Fit 3: The event rates due to the $\beta^+$-decays of ²²Na, ⁵²Mn, and ⁵⁷Ni are determined; all other isotopes are fixed (see Fig. B.6);

8.

$E_{\rm tot}$ Fit: The ²D event rate is re-determined as in Fit 5.

In this procedure, Fits 5-7 re-iterate Fits 2-4 to ensure convergence, which is tested by comparing the results of Fits 5 and 8. In general, the two fits yield nearly identical results. This self-consistent determination of the event rates of long-lived isotopes inevitably requires the determination of all isotope contributions, whether long-lived, short-lived, or prompt. In the CDG analysis, subtraction of the contributions from ²D, ²⁴Na, and ²⁸Al are based on Fits 8, 1, and 6, respectively (see Weidenspointer et al. 2001). The event rates due to the $\beta^+$-decays of ²²Na, ⁵²Mn, and ⁵⁷Ni are derived from the results of Fit 7. As pointed out before, the contributions from the primordial radio-nuclides ⁴⁰K and ²⁰⁸Tl need not be determined from the line fits, but can be computed based on the known or estimated activities.

Figure B.5: An example for a fit of the second E2 spectrum, which is used to determine the event rate from 28Al. In addition to the total fit, the templates for 2D (the line feature at $\sim$1.6 MeV, fixed), 28Al (the 1.78 MeV line), 24Na (the strong line at 2.75 MeV, fixed) and 208Tl (the weaker line at 2.61 MeV, fixed), the three unidentified features (dashed, dashed-dotted, and dashed lines), and the exponential continuum are indicated

Figure B.6: An example for a fit of the third E2 spectrum, which is used to determine the event rates from 22Na (dotted line), 52Mn (thick solid line), and 57Ni (thin solid line). Also depicted are the fixed components (2D: solid line at 1.9 MeV, 24Na: dash-dotted line, 28Al: weak solid component below 1.9 MeV, 40K: dashed line, 208Tl: solid line at 2.6 MeV), as well as the total fit, the exponential continuum, and the unidentified 2.93 MeV feature

  
B.2 ²⁶Al analysis

The analysis of the galactic 1.8 MeV line emission from ²⁶Al is, similar to that of the CDG, affected by the instrumental line background and its temporal variation. The galactic 1.8 MeV line emission is determined in the energy band 1.7-1.9 MeV, using adjacent energy bands for constructing a model for the background in this so-called line interval (comprehensive descriptions of this approach can be found in, e.g., Knödlseder 1997 & Oberlack 1997). In particular, the scatter angle $(\bar{\varphi})$ distribution of the 1.7-1.9 MeV background model is derived from an interpolation of the $\bar{\varphi}$ distributions in narrow adjacent energy intervals (1.6-1.7 MeV and 1.9-2.0 MeV). Due to the long-term variation of the instrumental line background (see e.g. Fig. 8), the ratio of the number of counts in the line interval and in the adjacent energy intervals is decreasing with time (see Fig. B.7), mostly due to the build-up of ²²Na, a major component in the 1.6-1.7 MeV band in $E_{\rm tot}$ (Oberlack 1997). To eliminate the time dependent contamination of the background reference from adjacent energies, the number of counts due to each component of the instrumental line background has to be determined for each individual observation period, employing a procedure outlined below. After subtraction of the instrumental line background, the ratio of counts in the line interval and the adjacent energy intervals to a good approximation is constant in time (see Fig. B.7), and data from individual observation periods can be summed to analyze the galactic 1.8 MeV line emission.

Figure B.7: Count ratios (number of counts in the line interval divided by the number of counts in the adjacent energy intervals) as determined for observations at galactic latitude $\vert b\vert > 40^{\circ }$. The ratios, and a linear model of their time variation, are shown before (grey) and after (black) subtraction of the instrumental line background

The procedure used in the ²⁶Al analysis to determine the background contributions of the eight identified background isotopes is a modified version of the CDG procedure described in Appendix B.1. The modifications are motivated by differences in the event selections applied in the two analyses, particularly differences in the elimination of atmospheric background (see Appendix A.1 and A.2). One of the consequences is that in the ²⁶Al imaging analysis events with larger $\bar{\varphi}$values are accepted than in the CDG analysis. For some isotopes, notably ²⁴Na, this results in significant changes of the E₁-E₂ distribution as compared to that for CDG event selections. Consequently, the optimal fit regions in E₁-E₂ space are somewhat different. Also, in contrast to the CDG analysis, in the ²⁶Al analysis the $\bar{\varphi}$ distribution of the accepted events is different for each observation period due to the orbit dependent rejection of atmospheric background. The simulated energy distributions for each background isotope therefore have to be corrected for the specific $\bar{\varphi}$ distribution of each individual observation period (Oberlack 1997). The correction for selections to reject atmospheric background has been calculated assuming a homogeneous illumination of the D1 detector. Since there is a certain edge enhancement in the illumination of the D1 modules for background produced in the D1 structure, slight differences in these corrections result in small distortions of the templates, and thus add to the observed scatter in the determined isotope event rates (see Fig. 8). As each observation period is independently analyzed for its instrumental line background contamination, the background modelling in the ²⁶Al analysis yields as a welcome by-product the long-term variation of the event rates due to the eight identified background isotopes depicted in Fig. 8.

Figure B.8: An example for an $E_{\rm tot}$ fit in the 26Al analysis. The upper panel depicts the $E_{\rm tot}$ spectrum before (line histogram) and after (data points) subtracting the contributions from the identified background isotopes (indicated by smooth curves with different line types). The middle panel shows the $E_{\rm tot}$ spectrum after isotope subtraction, together with the continuum background fit. The bottom panel gives the residuum of the fit

The four steps of the iterative line fitting procedure in the ²⁶Al analysis are similar to those in the CDG analysis, hence we content ourselves with describing only the differences. In the first step, two E₁-E₂ regions are fitted simultaneously to obtain the ²⁴Na event rate (Oberlack 1997): the E₁-E₂ region depicted in Fig. B.2, and an additional E₁-E₂ region extending from 2000 to 2700 keV in E₁ and from 1100 to 2000 keV in E₂. Analogous to the first E₁-E₂region, the additional E₁-E₂ region optimizes the ²⁴Na signal from simultaneous interactions of the 1.37 MeV and 2.75 MeV photons in D2 and D1, respectively, which are no longer suppressed due to the larger accepted $\bar{\varphi}$ values for imaging selections. The E₁-E₂ region for the second step extends down to 700 keV in E₁, and is used for determining the ²⁸Al rate and for obtaining start parameters to determine the event rates of the remaining isotopes, particularly ²²Na, in Fit 3. However, unlike in the CDG analysis, the contributions of the primordial isotopes ⁴⁰K and ²⁰⁸Tl are not fixed in the ²⁶Al analysis, but determined in Fit 3 and 1, respectively. As far as determining the contributions of each individual isotope is concerned, the procedure ends here, since it has been demonstrated in the CDG analysis that one fit cycle is sufficient to obtain self-consistent isotope rates. This advanced background treatment provided the basis for the latest COMPTEL ²⁶Al all-sky maps (e.g. Oberlack 1997; Plüschke et al. 2000).

Figure B.9: The distribution of the $\chi ^2_{\nu }$ values for each of the four fit steps in determining the instrumental line contamination in the 26Al analysis (Plüschke et al. 2000)

To investigate the extent to which the identified isotopes can account for spectral features in $E_{\rm tot}$, an additional fit is performed (see Fig. B.8). In this $E_{\rm tot}$fit ( E₁ > 70 keV, E₂ > 650 keV), the contributions from the eight background isotopes are fixed at the values obtained from the three previous E₂ fits; only the continuum background, modelled by a power law with an exponential turn-over at low energies, is varied. As can be seen, the major spectral features are accounted for by the eight identified isotopes, however, some minor features remain unidentified at this time.

The distribution of the $\chi ^2_{\nu }$ values for each of the four fit steps, as obtained in the analysis of Plüschke et al. (2000), is shown in Fig. B.9. The quality of Fits 1 and 3, in which the event rates of most of the background isotopes are determined, is acceptable. The $\chi ^2_{\nu }$distribtion for the $E_{\rm tot}$ fit, however, indicates that further improvement of the instrumental line background modelling is still possible.