Slowing down
4.6 Spectral Analysis
For each component of the component-wise celestial description with 2D Gaussians, a spectrum of sky emission is determined from the ts of intensity coecients for each energy bin in the 40 keV wide range around the 511 keV line. These spectra are characterised in more detail by deriving the 511 keV line intensity (IL), the width, characterised as kinematic broadening (F W HMSKY), the centroid shift, interpreted as Doppler-shift from bulk motion (∆E0 = Epeak−Elab), the o-Ps intensity (IO), and the Ps fraction (fP s). The expected spectral components are represented by a
Gaussian 511 keV line, an o-Ps continuum (Ore & Powell 1949, see also Eq. (2.76)), and a power-law representing the diuse Galactic gamma-ray continuum - each convolved with the SPI spectral response function plus the kinematic broadening, Eq. (3.19). Monte Carlo sampling is used to determine the uncertainties of the tted spectral characteristics, parametrised through the 511 keV line centroid, width, and amplitude, the o-Ps amplitude at the measured line centroid, and the continuum ux-density at 511 keV. The power-law index for the diuse Galactic continuum is xed a priori to −1.7 (Kinzer et al. 1999, 2001; Strong et al. 2005; Jean et al.
2006; Churazov et al. 2011; Bouchet et al. 2011). This is done because it is poorly determined in the used spectral band, and in any case has rather small impact on the annihilation component values (.3%). Likewise, the power-law indices for the Crab and Cyg X-1 continua are set to the literature value for the Crab of −2.23 (Jourdain & Roques 2009).
4.6.1 The Bulge Component
490 500 510 520 530
0.0 0.1 0.2 0.3 0.4
490 500 510 520 530
Energy [keV]
0.0 0.1 0.2 0.3 0.4
Flux [10−3 ph cm−2 s−1 (0.5 keV)−1 ]
Total Bulge spectrum (best fit parameters):
γ−Continuum: AC = 0.27±0.20
Narrow line: I = 0.96±0.07, FWHMSKY = 2.59±0.17 keV, ∆E0 = 0.09±0.08 keV Ortho−P’s: I = 6.14±0.76, fPs = 1.080±0.029
Figure 4.18: Spectrum of annihilation gamma-rays from the bulge (black crosses). The best t spectrum is shown (continuous black line), as decomposed into a single 511 keV positron annihilation line (dashed red), the continuum from annihilation through ortho-positronium (dashed olive), and the diuse gamma-ray continuum emission (dashed blue). Fitted and derived parameters are given in the legends. See text for details.
"The bulge" is dened as the superposition of the NB and BB component, because the central region of the Milky Way was shown to require two independent 2D Gaussian components whose intensities, however, strongly anti-correlate (−.836).
The bulge shows a 511 keV line intensity of (0.96±0.07)×10−3 ph cm−2 s−1 and is detected with an overall signicance of56σ. The bulge annihilation emission can be characterised by a 511 keV line of astrophysical width (2.59±0.17) keV, and a Ps fraction of (1.080±0.029), consistent with other studies (e.g. Jean et al. 2006;
Churazov et al. 2011). The line peak appears at (511.09±0.08) keV. The diuse Galactic gamma-ray continuum is a minor contribution in the bulge; its intensity is (0.27±0.20)×10−5 ph cm−2 s−1 keV−1. The spectral t quality is found adequate from a χ2 value of 66.47 with 74 degrees of freedom. Here, given the signicant correlation between NB and BB, the propagation of uncertainties for each energy bin has to take the covariance between the two components into account when merging the bulge spectra:
σBulge,k = q
θ0,k2 σ20,k+θ1,k2 σ21,k+ 2θ0,kθ1,kσ01,k (4.7) In Eq. (4.7), σBulge,k is the propagated uncertainty in energy bin k, as shown in Fig. 4.18, σi,k are the uncertainties for components i = 0 (NB), and i = 1 (BB), in energy bin k from the maximum-likelihood t, Eqs. (3.12) and (3.14), and σ01,k is the covariance between the NB and the BB for each energy bin. Neglecting the covariance term would overestimate the statistical uncertainties by about 250%.
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Bulge 511 keV line flux
1σ
Line flux [10−3 ph cm−2 s−1]
(a) Bulge 511 keV intensity as a function of disk size.
20 40 60 80 100 120 140
(b) Uncertainty of the bulge 511 keV intensity as estim-ated from1σlikelihood slices.
Figure 4.19: Dependence of the 511 keV line intensity in the bulge as a function of the choice of the disk extent (1-σ Gaussian width value). Line intensities are shown as shading, see scale on right-hand axis. Overlaid are the uncertainty contours for the disk size, as derived from the maximum likelihood ts in the grid-scan (left panel). On the right panel, the uncertainty estimation procedure using likelihood slices in the disk size plane is shown. See text for details.
The size of the disk might have an inuence on the intensities derived for the other sources. Its impact can be estimated by calculating the intersection of specic likelihood contours with the tangents of equal ux7. In Fig. 4.19, the dependence on the 511 keV line intensity in the bulge as a function of the disk extent is shown in the left panel, together with the procedure of how to estimate the uncertainty from that in the right panel. The star symbol is marking the point with the largest likelihood at 0.961×10−3 ph cm−2 s−1. Lines of equal ux, touching the (2∆C = 1)-, and (2∆C = 4)-contours, respectively, are shown and marked by blue dots.
These tangents correspond to the 1 and 2σ uncertainties of the line ux in the bulge with respect to the disk size (longitude and latitude extent). The resulting 1 and 2σ-uncertainties are (0.961±0.009)×10−3 ph cm−2 s−1, and (0.961+0.033−0.031)× 10−3 ph cm−2 s−1, respectively. The uncertainty from the disk size inuence can be compared to the "raw" statistical uncertainty from the measured counts. One realisation of the spectrum is shown in Fig. 4.18. It can be seen that the extent of the disk has a 1% eect on the estimate of the 511 keV line intensity of the bulge.
Other parameters of the bulge spectrum may also be inuenced by the size of the disk. These are shown in Fig. 4.20. All other spectral parameters are hardly inu-enced by the chosen size of the disk, even within≈5σ uncertainty on the disk, with the exception of the gamma-ray continuum. In general, the spectral parameters
7This is not to be interpreted as "an error on an error" but rather the dependence on a particular - tted or derived - value in another dimension, over which has to be marginalised to infer the correct uncertainty in the interesting parameter.
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5
Bulge 511 keV line shift
1σ
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Bulge 511 keV line width
1σ
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Ortho−positronium flux [10−3 ph cm−2 s−1]
(c) Ortho-positronium intensity,IO.
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Continuum intensity at 511 keV [10−5 ph cm−2 s−1 keV−1]
(d) Gamma-ray continuum ux density,CO. Figure 4.20: Dependence of spectral parameters of the bulge as a function of disk size. The notation is similar to
Fig. 4.20. See text for details.
show a smooth dependence on the disk size. On the left-hand side of the panels in Fig. 4.20, i.e. when the disk is modelled with small longitude extents, σdiskl . 35◦, the maximum likelihood method cannot distinguish between the model components.
Their overlap is large, and the increased degeneracy produces confusion also in the spectral domain. In the case of the galactic gamma-ray continuum, the overlap between bulge and disk increases the covariance in the continuum bins. For this reason, the continuum component in the bulge could be about twice as high. Espe-cially the line shape parameters, i.e. centroid and width, are insensitive with respect to the disk extent.
4.6.2 The Disk Component
The spectrum of the entire disk is shown in Fig. 4.21 for the best tting disk model, with a longitude extent around 60◦ and a latitude extent around 10.5◦. The 511 keV line intensity is (1.66± 0.35) ×10−3 ph cm−2 s−1. The line width in the disk is (2.47± 0.51) keV (FWHM). This is in concordance with the bulge valu. There might be systematic discrepancies between the spectral parameters of the single components, as discussed in Sec. 4.6.6. The 511 keV line shift,(0.16±0.18) keV, is consistent with zero for the disk as an entity. The o-Ps continuum has an intensity of (5.21±3.25)×10−3 ph cm−2 s−1, for which the Ps fraction obtains a value of (0.90±0.19). The galactic gamma-ray continuum ux density amounts to (5.85±
490 500 510 520 530
490 500 510 520 530
Energy [keV]
Total Disk spectrum (best fit parameters):
γ−Continuum: AC = 5.58±1.03
Narrow line: I = 1.66±0.35, FWHMSKY = 2.47±0.51 keV, ∆E0 = 0.16±0.18 keV Ortho−P’s: I = 5.21±3.25, fPs = 0.902±0.192
Figure 4.21: Spectrum of annihilation gamma-rays from the best tting disk model (black crosses). The best t spectrum is shown (continuous black line), as decomposed into a single 511 keV positron annihilation line (dashed red), the continuum from annihilation through ortho-positronium (dashed olive), and the diuse gamma-ray continuum emission (dashed blue). Fitted and derived parameters are given in the legends. See text for details.
1.05)×10−5 ph cm−2 s−1 keV−1 at 511 keV. This value corresponds to(5.99±1.07)× 10−6 ph cm−2 s−1 sr−1 keV−1 integrated across the full sky8, consistent with results by Strong et al. (2005) and Bouchet et al. (2011), for example9. The χ2 value for the disk spectrum is71.98with 74 degrees of freedom, indicating a fully acceptable model to describing the data.
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Disk 511 keV line flux
1σ
Line flux [10−3 ph cm−2 s−1]
(a) Line ux,I0.
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Disk 511 keV line shift
1σ
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Disk 511 keV line width
1σ
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Continuum intensity at 511 keV [10−5 ph cm−2 s−1 keV−1]
(d) Gamma-ray continuum uxdensity,CO.
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Ortho−positronium flux [10−3 ph cm−2 s−1]
(e) Ortho-positronium intensity,IO.
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
(f) Positronium fraction,fP s.
Figure 4.22: Dependence of spectral parameters of the disk as a function of disk size. The notation is similar to Fig. 4.20. See text for details.
The uncertainties of the derived spectral parameters are shown in Fig. 4.22, similar
8Here, the emission is truncated at 1% of the maximal surface brightness; along the galactic plane the intensity always is above that threshold and is therefore taken into account as2π; towards higher latitudes, 99% of the emission are enclosed within≈70◦. Hence, the disk emission encloses a solid angle of3.11πsr. Note, that this value is therefore model and threshold dependent.
9Strong et al. (2005) and Bouchet et al. (2011) focused on a broader energy range and on the central part of the Milky Way.
to the bulge case. As expected from the nature of the t, the spectral-shape para-meters of the disk are the most varying ones. This results from the chosen shape to model the disk. At increasing disk size in longitude and latitude, the detection of low surface-brightness regions increases, and more line and continuum ux can be found. However, in the 80 bin spectral analysis, the longitude and latitude sizes are constrained very well, albeit biased by other parameters (e.g. bulge shape and num-ber of components). This is evident by the 1σ contours in the panels of Fig. 4.22, varying by 20% at most. The line shift is hardly inuenced by the chosen disk size, whereas the line width could well be smaller, being formally consistent with the bulge value (see also further discussion of the line width below). The diuse galactic gamma-ray and o-Ps continuum are dicult to disentangle in this energy band for the disk. Their general relative uncertainties, depending on the disk size, are of similar magnitude and shape. Likewise, the Ps fraction follows surprisingly the contours of the disk extent, being a function of the line intensity and o-Ps in-tensity only. Although this might be a coincidence, and over-interpretation of the shape of probability functions should be avoided, the value of the Ps fraction in the disk is always smaller than in the bulge, though formally consistent.
490 500 510 520 530
0.0
490 500 510 520 530
Energy [keV]
Total Disk l>0 spectrum (best fit parameters):
γ−Continuum: AC = 2.66±0.49
Narrow line: I = 0.87±0.14, FWHMSKY = 3.07±0.34 keV, ∆E0 = 0.30±0.14 keV Ortho−P’s: I = 2.68±1.37, fPs = 0.897±0.157
(a) Diskl >0◦.
490 500 510 520 530
0.0
490 500 510 520 530
Energy [keV]
Total Disk l<0 spectrum (best fit parameters):
γ−Continuum: AC = 3.08±0.44
Narrow line: I = 0.80±0.12, FWHMSKY = 1.59±0.19 keV, ∆E0 = 0.07±0.17 keV Ortho−P’s: I = 2.10±1.18, fPs = 0.848±0.180
(b) Diskl <0◦.
Figure 4.23: Spectrum of annihilation gamma rays from the eastern (a) and western (b) hemisphere of the Galaxy's disk. The tted parameters are given in the legends, colours are the same as in Fig. 4.21. No ux asymmetry is found. See text for details.
The statistics from eleven years of data allow to derive spectral parameters separ-ately for the eastern (l > 0◦) and the western (l < 0◦) hemisphere, see Fig. 4.23.
Here, the Gaussian-shaped disk component is masked on alternate sides (l > 0◦, and l < 0◦), which results in tting now seven individual sky components. The χ2 t values are 83.79 and 68.42 for the l > 0◦ and l < 0◦ regions, respectively, with 74 degrees of freedom each. Although the value of83.79might indicate a bad t for 74 degrees of freedom, it is still within the 1σ tolerance band of the χ2 statistics10. The 511 keV line intensities are (0.87±0.14)×10−3 ph cm−2 s−1 for the (l > 0◦) region, and(0.80±0.12)×10−3 ph cm−2 s−1 for the (l <0◦) region. Thus, there is no disk asymmetry in the line uxes, and the east-west ratio is 1.09±0.24. This is in contrast to an earlier report by Weidenspointner et al. (2008a). The asymmetry is reduced, if not completely removed, as the narrow-bulge component is shifted away from the centre by about −1.25◦ in longitude and −0.25◦ in latitude (see also Skinner et al. 2014). The east/west ratio for the o-Ps continuum is (1.28±0.97),
10The expectation value of the χ2 statistic equals the number of degrees of freedom, hereν = 74. Since the probability distribution has a certain width (standard deviation; second moment) of√
2ν≈12.17the1σtolerance band is74.00±12.17so that the value of83.79still represents an adequate t quality.
and for the diuse galactic gamma-ray continuum it is (0.86±0.20). The western hemisphere of the disk shows a smaller line width (FWHM of (1.59±0.19) keV) than the eastern hemisphere ((3.07±0.34) keV). The two hemispheres have dierent spectra overall, with a statistical signicance of 2.8σ. Each of the two halves dier therefore from the combined disk spectrum by1.4σ.
In fact, the statistics in the central regions of the Galaxy allow to test for even more discriminations of emission regions. This leads to a measurement of the kinematical signatures in positron annihilation gamma-rays along the inner part of the Milky Way, and will be discussed in Sec. 4.7.3.
4.6.3 The Galactic Centre Source
490 500 510 520 530
−0.01 0.00 0.01 0.02 0.03 0.04
490 500 510 520 530
Energy [keV]
−0.01 0.00 0.01 0.02 0.03 0.04
Flux [10−3 ph cm−2 s−1 (0.5 keV)−1]
Total GCS spectrum (best fit parameters):
γ−Continuum: AC = 0.06±0.05
Narrow line: I = 0.80±0.19, FWHMSKY = 3.46±0.64 keV, ∆E0 = −0.27±0.31 keV Ortho−P’s: I = 2.83±1.77, fPs = 0.935±0.187
Figure 4.24: Spectrum of annihilation gamma rays from the point-like source (GCS) superimposed onto the extended bulge model in the Galaxy's centre. The t and its components are indicated as above in Fig. 4.21.
The immediate vicinity in the direction of the centre of the Milky Way could be dis-criminated in Skinner et al. (2014) as a separate source. A point source at the posi-tion of Sgr A* (the GCS) is detected with a signicance of 5σ. A rst spectrum from the annihilation emission of this region is provided in Fig. 4.24. The 511 keV line in-tensity is(0.80±0.19)×10−4ph cm−2 s−1. Its annihilation emission is characterised by a broadened line with a width of(3.46±0.64) keV (FWHM above instrumental resolution), and a Ps fraction of (0.94±0.19). This is still consistent within2σ uncer-tainties, compared to bulge and disk (total). There is a hint of an underlying broad continuum with a ux density estimate of (0.06±0.05)×10−5 ph cm−2 s−1 keV−1. There is no evident time variability down to scales of months. The annihilation line is centred at(510.73±0.31) keVif the spectrum is described by the above mentioned spectral model, indicating a red-shift. The o-Ps continuum is consistent with zero (<2σ), and assuming there is none leads to a smaller value for the line centroid of (510.59±0.35) keV. The t quality as measured by χ2 is 64.94for 74 dof.
The inuence of the disk extent on this component is hard to trace, as slight vari-ations in the disk size may lead to strong eects in particular energy bins. As an example11, the GCS line ux is shown as a function of the disk size, similar to
11Additional gures concerning the other spectral parameters of the GCS with respect to the disk size can be found in Appendix A.
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5
GCS 511 keV line flux
1σ
Line flux [10−4 ph cm−2 s−1]
Figure 4.25: Dependence of GCS 511 keV line ux as a function of disk size. The notation is similar to Fig. 4.20.
See text for details.
Fig. 4.25. Figure 4.25 shows the general trend of the GCS 511 keV line intensity, increasing from small disk axis ratios, σσlb, (top left) towards higher disk axis ratios (bottom right). This is true for also the other parameters. However, the absolute magnitude of this eect is only of the order of 4% (<3σ).
4.6.4 Continuum Sources
The Crab pulsar (Jourdain & Roques 2009) and Cygnus X-1 (Jourdain et al. 2012) are the only known Galactic sources strong enough to signicantly inuence the maximum likelihood analysis. They are thus included as constant point sources, i.e. not varying in time. The Crab is detected in the 40 keV energy band at 31σ signicance. The ux density found is (2.20±0.07)×10−5 ph cm−2 s−1 keV−1 at 511 keV, sing a power-law with a xed photon index of−2.23 (Jourdain & Roques 2009). The ux in this energy band is consistent with the analysis across the full energy range of SPI (Jourdain & Roques 2009), though on the high side. This point source contribution is equivalent to about 40% of the total diuse Galactic gamma-ray continuum emission.
Cygnus X-1 is also clearly detected at 11σ signicance. Its spectrum is described well by a single power-law spectrum with a (xed) power-law index of −2.23, and a ux density of (0.65±0.06)×10−5 ph cm−2 s−1 keV−1. Cyg X-1 is known to be time variable, with a hard, a soft, and possibly an intermediate state (McConnell et al. 2002; Rodriguez et al. 2015b, see also Sec. 5.2). At 511 keV, the measured ux dierence between the hard and the soft state is about1.5×10−4ph cm−2 s−1 keV−1 (McConnell et al. 2002). The measured average of the dierent possible spectral states of Cyg X-1 are in good agreement with recent measurements of Rodriguez et al. (2015b). Cyg X-1 is only a weak source in the energy range around 511 keV, and included as a signicant point source to optimise the maximum likelihood t.
No time variability has been tested.
The spectra of the Crab and Cyg X-1 are shown in Fig. 4.27. The chosen energy range is not enough to constrain the power-law index. The tted values for the indices are (−1.41±1.52) for the Crab, and (−2.9±4.5) for Cyg X-1. These are
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5
Continuum intensity at 511 keV [10−5 ph cm−2 s−1 keV−1]
(a) Crab.
7.5 22.5 37.5 52.5 67.5 82.5 97.5 112.5 127.5 142.5 157.5 Disk longitude extent [deg]
Continuum intensity at 511 keV [10−5 ph cm−2 s−1 keV−1]
(b) Cyg X-1.
Figure 4.26: Dependence of the point source continuum uxes as a function of disk size. Shown are the derived ux densities for the Crab (a) and Cyg X-1 (b) normalised to an energy of 511 keV. The notation is similar to Fig. 4.20. See text for details.
consistent with the literature values. The resulting ux values change by less than 0.3%, if the photon index is considered a free parameter. The correlations between the continuum sources and the other sky model components are negligible, except for the disk. The point source ux depends on the size of the disk emission model in either case, and more so for Cyg X-1. If the disk is chosen to be (unrealistically) short, the ux density of Cyg X-1 captures (erroneously) a part of this disk emission.
The continuum ux densities of the two sources as a function of chosen disk size is shown in Fig. 4.26. The Crab ux does not show any strong correlations with the extent of the disk (<2%).
490 500 510 520 530
0.00
490 500 510 520 530
Energy [keV]
Total Crab spectrum (best fit parameters):
γ−Continuum: AC = 2.20±0.07
(a) Crab.
490 500 510 520 530
0.00
490 500 510 520 530
Energy [keV]
Total CygX1 spectrum (best fit parameters):
γ−Continuum: AC = 0.65±0.06
(b) Cyg X-1.
Figure 4.27: Spectrum of the Crab (a) and Cygnus X-1 (b) between 490 and 530 keV. The tted parameters are given in the legends. There are no annihilation signals detected (<2σ). See text for details.
The Crab pulsar was historically identied to be an annihilation source but the meas-ured signal was questionable (Agrinier et al. 1990; Massaro et al. 1991; Ulmer et al.
1994). Likewise, there is a high-energy "bump" measured for Cyg X-1 (McConnell et al. 2002) which either is due to positron annihilation or Comptonisation of the corona surrounding the black hole (see also Sec. 5.2 for more details on microquas-ars). There are no (<2σ) annihilation signals detected in these two point sources.
Upper limits are given for both sources in Tab. 4.8, assuming either a "disk-like", i.e. narrow 511 keV line (1.6 keV FWHM), or a "GCS-like", i.e. broad 511 keV line (3.5keV FWHM), respectively. Underlying broad erannihilation features (>50keV FWHM) may still be compatible with the data.
Narrow line Broad line
Crab 6.62 7.44
Cyg X-1 1.27 1.89
Table 4.8: Upper limits (2σ, in units10−5 ph cm−2s−1) for 511 keV gamma-ray line emission originating from the Crab or Cyg X-1. Limits for a narrow (1.6keV FWHM), and a broad line (3.5keV FWHM) are given.
4.6.5 Additional Sources
A search for possible point-like annihilation emission in the satellite galaxies of the Milky Way was performed. This was done due to the hypothesis that light dark matter particles may also be a candidate source of positrons (Boehm et al. 2004;
Gunion et al. 2006; Hooper et al. 2004; Picciotto & Pospelov 2005; Pospelov et al.
2008), and dwarf galaxies are believed to be dark matter dominated (e.g. Simon &
Geha 2007; Strigari et al. 2008b). There was one3σ (six2σ) sources among the 39
Geha 2007; Strigari et al. 2008b). There was one3σ (six2σ) sources among the 39