A time-variable, phase-dependent emission line in the X-ray spectrum of the isolated neutron star RX J0822-4300
A. De Luca, D. Salvetti, A. Sartori, P. Esposito, A. Tiengo, S. Zane, R. Turolla, F. Pizzolato, R. P. Mignani, P. A. Caraveo, S. Mereghetti, G. F. Bignami
aa r X i v : . [ a s t r o - ph . H E ] D ec Mon. Not. R. Astron. Soc. , 1–5 (2011) Printed 19 September 2018 (MN LaTEX style file v2.2)
A time-variable, phase-dependent emission line in the X-rayspectrum of the isolated neutron star RX J0822–4300
A. De Luca , , ⋆ , D. Salvetti , , , A. Sartori , , , P. Esposito , A. Tiengo , ,S. Zane , R. Turolla , , F. Pizzolato , R. P. Mignani , , P. A. Caraveo , ,S. Mereghetti , and G. F. Bignami , INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica - Milano, via E. Bassini 15, I-20133 Milano, Italy Istituto Universitario di Studi Superiori, Viale Lungo Ticino Sforza 56, I-27100 Pavia, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Via Bassi 6, I-27100 Pavia, Italy Universit`a degli Studi di Pavia, Dipartimento di Fisica Nucleare e Teorica, Via Bassi 6, I-27100 Pavia, Italy INAF – Osservatorio Astronomico di Cagliari, localit`a Poggio dei Pini, strada 54, I-09012 Capoterra, Italy Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK Dipartimento di Fisica e Astronomia, Universit`a degli Studi di Padova, Via Marzolo 8, I-35131 Padova, Italy Kepler Institute of Astronomy, University of Zielona G´ora, Lubuska 2, 65-265, Zielona G´ora, Poland
Accepted . . . . Received . . . ; in original form . . .
ABSTRACT
RX J0822–4300 is the Central Compact Object associated with the Puppis A supernova rem-nant. Previous X-ray observations suggested RX J0822–4300 to be a young neutron star with aweak dipole field and a peculiar surface temperature distribution dominated by two antipodalspots with different temperatures and sizes. An emission line at 0.8 keV was also detected. Weperformed a very deep (130 ks) observation with
XMM-Newton , which allowed us to study indetail the phase-resolved properties of RX J0822–4300. Our new data confirm the existenceof a narrow spectral feature, best modelled as an emission line, only seen in the ‘Soft’ phaseinterval – when the cooler region is best aligned to the line of sight. Surprisingly, comparisonof our recent observations to the older ones yields evidence for a variation in the emissionline component, which can be modelled as a decrease in the central energy from ∼ . keVin 2001 to ∼ . keV in 2009–2010. The line could be generated via cyclotron scatteringof thermal photons in an optically thin layer of gas, or – alternatively – it could originate inlow-rate accretion by a debris disk. In any case, a variation in energy, pointing to a variationof the magnetic field in the line emitting region, cannot be easily accounted for. Key words: pulsars: general – stars: neutron – X-rays: individual: RX J0822–4300.
Central Compact Objects (CCOs) in Supernova Remnants (SNRs)are a handful of about 10 point-like, thermally-emitting X-raysources located close to the geometrical centres of non-plerionicsupernova remnants, with no counterparts at any other wavelength.CCOs are supposed to be young, isolated, radio-quiet neutron stars(see De Luca 2008, for a review).While the first discovered CCO (the one in the RCW103 SNR)turned out to be a unique object (De Luca et al. 2006), results of X-ray timing on a sub-sample of sources with fast periodicities haverecently set very useful constraints for a general picture of CCOsas a class. Analysis of multi-epoch
XMM-Newton and
Chandra observations of 1E 1207.4–5209 inside G296.5+10.0 and CXOUJ185238.6+004020 at the centre of Kes 79 ( P = 424 ms and ⋆ E-mail: [email protected] P = 105 ms, respectively) yielded unambiguous evidence for verysmall period derivatives (Halpern & Gotthelf 2010, 2011). This im-plies, under standard rotating dipole assumptions, characteristicages exceeding the age of the host supernova remnants by morethan 3 orders of magnitude, as well as very small ( . G) dipolemagnetic fields. This points to an interpretation of such sources as‘anti-magnetars’ – weakly magnetised isolated neutron stars, bornwith a spin period almost identical to the currently observed one.Such a picture has been recently strengthened by the discoveryof 112 ms pulsations from RX J0822–4300, the CCO in the PuppisA SNR (Gotthelf & Halpern 2009), in two archival
XMM-Newton datasets collected in 2001, previously used by Hui & Becker (2006)for a comprehensive spectral analysis of the neutron star and ofthe surrounding SNR. Gotthelf et al. (2010), using a 2010
Chandra observation, set a σ upper limit of . × − s s − to the periodderivative, corresponding to a dipole magnetic field B < × G, and to a characteristic age τ c > Myr, much larger than the age c (cid:13) De Luca et al. of the host SNR (3.7 kyr Winkler et al. 1988). Thus, RX J0822–4300 can be included in the anti-magnetar family.The X-ray emission properties of RX J0822–4300 are verypeculiar. Gotthelf & Halpern (2009) report on a ◦ shift in thephase of the pulse peaks occurring abruptly at ∼ . keV. Thishas been interpreted as due to the existence of two antipodal hotspots on the star surface, with two different temperatures (‘warm’and ‘hot’) – indeed seen in the emission spectrum as two differ-ent blackbody components. The star rotation bringing into view orhiding such regions makes the observed spectrum to change from asoft phase (maximum alignment of the warm spot with the line ofsight) to a hard phase (maximum alignment of the hot spot). Evenmore intriguing, Gotthelf & Halpern (2009) report on the evidencefor an emission line at ∼ . keV, possibly associated with the warmspot.Here we report on a very deep XMM-Newton observation ofRX J0822–4300, performed in 2009 and 2010, which allows us tostudy in detail the phase-resolved emission properties of the neu-tron star and to check for any time variability.
Our study is based on a very deep ( ∼ ks) observation with XMM-Newton , originally scheduled to fit into a single satellite or-bit (revolution n.1836), starting on 2009 December 17. However,launch of the
Helios 2B spacecraft required support from
XMM-Newton ground stations, which resulted in a ∼ ks data gap in themiddle of the orbit. The observation was completed on 2010 April5. We also included in our study the two archival, shorter datasets,obtained on 2001 April 15 ( ∼ ks) and 2001 November 8 ( ∼ ks), used by Gotthelf & Halpern (2009) in their previous investiga-tions. A summary of the observations is reported in Table 1.We focus on data obtained by the pn detector (Str¨uder et al.2001) of the European Photon Imaging Camera instrument (EPIC).The detector was operated in the small-window mode (time res-olution of 5.7 ms, field of view of ′ . × ′ . ). The thin opticalfilter was used in all observations. EPIC/pn Observation Data Fileswere processed with the most recent release of the XMM-Newton
Science Analysis Software (
SAS v11) using standard pipelines.
For our timing analysis, we selected the pn source events from acircular region of 30-arcsec radius, including only 1- and 2-pixelevents (
PATTERN
TBD time us-ing the
Chandra position (Gotthelf & Halpern 2009).Before the analysis presented in Gotthelf & Halpern (2009),the pulsations in RX J0822–4300 eluded detection for many years,because of a phase shift of about half a cycle between the nearlysinusoidal profiles in the soft ( < Z periodograms (Buccheri et al. 1983), we found that theenergy bands that maximise the pulsed signal are 0.46–1.17 keVand 1.25–5.12 keV. The resulting soft and hard pulse profiles ofthe individual XMM-Newton observations (Table 1) were cross-correlated, shifted and summed to create two distinct pulse pro-file templates. Owing to the higher signal-to-noise of the 1.25–5.12keV profiles, we carried out the analysis in the hard band, but wechecked that the results are fully consistent with those obtainedin the soft band. The hard pulse profiles from temporal segments C t s / s Softband0.80.91 C t s / s Hardband0.5 1 1.50.80.911.1 N o r m a li z ed HR Phase Ratio H a r den i ng H a r d pha s e S o ft en i ng S o ft pha s e Figure 1.
Background-subtracted folded light curves for RX J0822–4300 inthe soft energy range (0.46–1.17 keV, upper panel) and in the hard energyrange (1.25–5.12 keV, middle panel). The lower panel shows the hardnessratio (Hard/Soft), normalised to its average value. Phase intervals used forphase-resolved spectroscopy are also marked. of the
XMM-Newton observations were cross-correlated with thetemplate to determine times of arrival at each epoch. By means ofa phase-fitting analysis (e.g. Dall’Osso et al. 2003), we measuredthe periods given in Table 1 (we also confirm the measurements byGotthelf & Halpern 2009 for observations A and B).We attempted to obtain a full phase-connected timing solution(i.e. a solution that accounts for all the spin cycles of the pulsar)for the longest possible time. Unfortunately, it was not possibleto univocally phase-connect the solution found for the two adja-cent datasets C and D to other observations (the uncertainty on thephase – propagated along the large time span to other observations– largely exceeds 1 cycle).A linear fit of the periods in Table 1 gives for the periodderivative (9 ± × − s s − , which translates into 3 σ limits − . × − s s − < ˙ P < . × − s s − (in agreement withthe limits recently published in Gotthelf, Perna & Halpern 2010).The period derivative at the Solar system barycentre results fromthe intrinsic pulsar spin-up/down plus the contributions due toany external gravitational field and the pulsar proper motion (e.g.Phinney 1992). In the case of RX J0822–4300, for an assumed dis-tance of 2.2 kpc (Reynoso et al. 1995), the Galactic contribution isnegative and negligible ( ∼− × − s s − ) and that produced bythe proper motion ( ± mas yr − ; Winkler & Petre 2007) is . +0 . − . × − s s − . While the total (Galactic + proper motion)contribution does not impact significantly on the current limits onthe ˙ P of RX J0822–4300, we note that it is larger than the periodderivative measured for the CCO in Kes 79 ( ∼ . × − s s − ;Halpern & Gotthelf 2010).The phase of the pulse peak is energy dependent (see Fig-ure 1), with an offset of . ± . between the profile as seenat lower ( E < . keV) and higher energy ( E > . keV), thetransition occurring quite abruptly at ∼ . keV, consistent with thefindings of Gotthelf & Halpern (2009). As shown in Figure 2, thepulsed fraction decreases from ∼ in the 0.4–0.6 keV energyrange to < in the 1.1–1.3 keV range, then grows again to ∼ at E > keV, in good agreement with the model by Gotthelf et al.(2010). c (cid:13) , 1–5 ime-variable emission line in RX J0822–4300 Table 1.
Journal of the
XMM-Newton observations. a Mid-point of obser-vation. b Time between first and last event. c σ errors in the last digit arequoted in parentheses. d Gotthelf & Halpern (2009). e This observation wasbroken into two segments (see text).Dataset Obs.ID Date a Duration b Pulse period c (MJD TBD) (ks) (ms)A 0113020101 52014.471 25.1 112.799 42(5) d B 0113020301 52221.898 23.0 112.799 44(4) d C 0606280101 e
500 1000 1500 2000 P u l s ed F r a c t i on ( % ) Energy (eV)
Figure 2.
Energy dependence of the pulsed fraction (PF). The PF was eval-uated in overlapping 0.2 keV energy bins, incremented in 0.1 keV steps, asthe ratio between the number of counts above the minimum and the totalnumber of counts. Background has been subtracted. A clear trend is appar-ent, with a minimum in the 1.1–1.3 keV energy range.
Thorough phase-integrated spectroscopy of RX J0822–4300 hasbeen published by Hui & Becker (2006). We will focus here onphase-resolved spectroscopy.RX J0822–4300 lies in a very complex environment, whichmakes background subtraction a critical task. Using phase-integrated data, we evaluated an optimal selection of source eventsby maximising the signal to noise ratio in the 0.3–10 keV range as afunction of the source extraction radius. The best choice turned outto be a circle of 17.5 ′′ centred at the Chandra position. We extractedthe background spectrum from an annular region with radii of 28 ′′ and 35 ′′ centred on the source. Different source and/or backgroundregions yield consistent results within ∼ σ .We aligned the photon phases for each observation by cross-correlating the pulse profiles (see Sect. 3). We generated a com-bined event list for first-epoch observations (i.e. data collected in2001 and used by Gotthelf & Halpern 2009) and one for second-epoch observations (i.e. our new data, collected in 2009–2010), inview of the large intercurring time span. We extracted four phase-resolved spectra from both first epoch and second epoch data, basedon the intervals marked as ‘Hard phase’, ‘Soft phase’, ‘Harden-ing’ and ‘Softening’ in Fig. 1. Phase-resolved spectra were re-binned with at least 100 counts per bin and so that the energyresolution was not oversampled by more than a factor of 3. Re-sponse matrices and effective area files for each epoch were gener-ated by combining (weighted by exposure time) the correspondingfiles generated using the SAS tools
RMFGEN and
ARFGEN . Spec-tral modelling was performed with the
XSPEC − TBABS model in
XSPEC , with the . R e s i dua l s ( c t s s − ) Soft PhaseFirst epoch . R e s i dua l s ( c t s s − ) Energy (keV)
Soft PhaseSecond epoch
Figure 3.
Residuals (in counts s − ) of two-blackbody fits to the Soft phasespectra (see Fig. 1). A structure in the 0.6–0.9 keV range is seen in bothepochs, although with different shape and intensity. abundances by Wilms, Allen, & McCray (2000). We quote errorsat 90% confidence level for a single interesting parameter unlessotherwise specified.First, we repeated the exercise performed byGotthelf & Halpern (2009), fitting a double blackbody modelto our data. A simultaneous fit to the four phase-resolved spectrawas performed for each epoch. The blackbody normalisations wereallowed to vary as a function of the phase, while the temperaturesand N H were held fixed in order to constrain a single value forall phase ranges. This model yields a reduced- χ of of 1.20 for298 degrees of freedom (d.o.f.) and of 1.23 for 389 d.o.f for thefirst and second epoch data, respectively. Although modulation ofthe emitting radii accounts for the bulk of the spectral variation,structured residuals in the 0.6–0.9 keV range are apparent inthe Soft phase both in the 2001 dataset (as already reported byGotthelf & Halpern 2009) and in our deeper 2009–2010 dataset,which confirms the existence of a phase-dependent spectral feature.Very interestingly, while the phase-resolved best fit parame-ters do not change as a function of the epoch – they can be linkedin a simultaneous two-epoch fit (more details below) – the devia-tion from the continuum in the soft phase has a somewhat differentshape in 2001 with respect to 2009–2010 (see Fig. 3), suggesting apossible time variability of the spectral feature. Indeed, such vari-ation is fully apparent when plotting together the two Soft phasespectra (see Fig. 4). To quantify the significance of the spectralchange in a model-independent way, we compared the distributionsof the source events’ energies observed in the two epochs using theKolmogorov-Smirnov test. The probability of a statistical fluctua-tion producing the apparent difference in the energy range wherethe feature is seen ( ∼ × − .As a second step, in order to model the feature, we focused onthe two Soft phase spectra. Following Gotthelf & Halpern (2009),we added a Gaussian emission line to the two-blackbody model.Indeed, this yields a much better fit with no structured residuals inthe 0.6–0.9 keV range. As expected, the line component varies asa function of the time, its central energy being higher in the firstepoch ( ∼ ∼ Monte Carlo simulations, we es-timated that the significance of the line is greater than 99.99 % in the c (cid:13) , 1–5 De Luca et al. . . First epochSecond epoch no r m a li z ed c oun t s s − k e V − Soft Phase10.5 2 − Energy (keV) ∆ S χ Figure 4.
Upper panel: Soft phase spectra for the two epochs. First epoch:blue. Second epoch: green. The variation in the 0.6–0.9 keV range is appar-ent. The best-fit model, two blackbody components and a variable Gaussianemission line, is superimposed. The line component at both epochs is alsoshown. The continuum components do not change as a function of time.Lower panel: residuals to the best fit model. The model yields a very gooddescription of the two spectra ( χ ν = 1 . , 154 d.o.f.). second-epoch spectrum, and greater than 99.97 % in the first epoch.Performing a simultaneous fit to the two spectra, the two-blackbodyplus emission line model yields a reduced- χ of 1.17 (155 d.o.f.)when all parameters (both continuum and line) are linked, while byallowing the line central energy to vary between first epoch and sec-ond epoch, the fit improves to a reduced- χ of 1.02 (154 d.o.f.); seeFig. 4. The chance occurrence probability of such an improvementis ∼ × − , as evaluated by an F-test. Such a result is confirmedby Monte Carlo simulations: assuming the best fit model of secondepoch data, a line centroid as observed in first epoch could be ob-tained by chance with a probability as low as ∼ × − . The fitdoes not improve significantly by allowing further line parametersto vary. There is no evidence for any variation in the continuumparameters between the two epochs.We also tried to model the two soft phase spectra using vari-able absorption features. Adding a single Gaussian absorption line( GABS in XSPEC ) at a higher energy than the one required by theemission line model ( ∼ . keV and ∼ . keV in the first andsecond-epoch, respectively) yields a much worse fit than the emis-sion line model (reduced- χ of 1.20 for 154 d.o.f.). Adding two lines at ∼ . keV and at ∼ . keV in the first epoch and at ∼ . keV and at ∼ . keV in the second epoch still yields aremarkably worse description of the data (reduced χ = 1 . for150 dof). As a further test, we tried low- B field neutron star atmo-sphere models for the continuum ( NSA , Zavlin, Pavlov & Shibanov1996;
NSATMOS , Heinke et al. 2006). As for the blackbody case,two components with different temperatures are needed. Additionof a variable Gaussian emission line is still favoured ( χ = 1 . for 154 dof) with respect to a single variable absorption line, aswell as to two variable absorption lines ( χ = 1 . for 150 dof forthe latter model).Then, the whole analysis was repeated for the other phase in-tervals (Hard, Softening and Hardening). No significant improve- ment in the fit was obtained by adding an emission line to the two-blackbody model (the same is true using absorption features). Weassessed that, in each phase interval, the continuum did not changeas a function of the epoch and that there are no systematic varia-tions between first and second-epoch in the 0.5–1 keV energy range(such results indicate that the long-term variability of the featurecannot have an instrumental origin).As a final step, we performed a simultaneous fit to all thespectra based on the results described above. We used the two-blackbody plus emission line model. The blackbody normalizationsare phase-dependent, but not epoch-dependent; the line normalisa-tion is phase-dependent and its central energy is epoch-dependent.This yields a reduced χ = 1 . for 691 d.o.f. In such model, the N H is (5 . ± . × cm − , the warm blackbody has a tem-perature kT W = 265 ± eV and a radius ranging from 2.27 km(Soft phase) to 2.04 km (Hard phase), the hot blackbody has a tem-perature kT H = 455 ± eV and a radius ranging from 0.53 km(Soft phase) to 0.65 km (Hard phase). The line component is nar-row ( σ < eV). In 2001, the line energy is . ± . keV andthe Equivalent Width (EW) ranges from ∼
53 eV in the Soft phaseto < . +0 . − . keV and the EW rangesfrom ∼ eV (Soft phase) to < ∼ of theflux of the continuum in the same energy range. Our multi-epoch
XMM-Newton analysis shows a phase-dependent, time variable spectral feature, best modelled as an emission linewith a variable central energy, in the X-ray spectrum of the ‘anti-magnetar’ candidate RX J0822–4300.To put such a peculiar result in context, we first note that ourobservations confirm the picture of RX J0822–4300 as a weakly-magnetised neutron star. Indeed, we improved the upper limit on ˙ P , bringing the dipole component of the magnetic field down to < . × G at σ level. Based on a larger statistics, we also con-firm the geometric model by Gotthelf et al. (2010), explaining thephase-resolved thermal emission with two antipodal spots of dif-ferent temperature (compare e.g. our Fig. 2 to their Fig. 6). Lack ofany measurable time variation in the continuum properties suggeststhat the warm and hot regions are intrinsic, persistent features in thethermal map of the star. To explain such a large surface anisotropyfor RX J0822–4300 (and for CCOs in general) we may considerthat a large difference in intensity could exist between the internaland the external magnetic fields of the neutron star, as proposed byTurolla et al. (2011) to explain the properties of the low-magneticfield Soft Gamma Repeater SGR 0418+5729 (Rea et al. 2010). Inour case, an internal (toroidal + poloidal) field of a few Gwould be large enough to effectively channel the heat flux from thecore (Geppert, K¨uker, & Page 2004, 2006), but would not inducecrustal fractures with consequent magnetar-like bursting activity.The most natural interpretation of the variable emission lineis that of a cyclotron feature produced by electrons. If its cen-tral energy is associated to the fundamental e − cyclotron fre-quency, the magnetic field in the line emitting region would be 6– × (1 + z ) G (where z ∼ . is the gravitational redshift).This is quite compatible with the upper limit from the spin-downrate. The line is very narrow ( σ eV). If it is a cyclotron linethen ∆ E/E = ∆
B/B and the relative variation of B over theemitting region (conservatively) needs to be . Thus, the line c (cid:13) , 1–5 ime-variable emission line in RX J0822–4300 should be produced in a very compact region. A variation of thecentral energy of the feature would require a change either in po-sition of the emitting plasma within a non-variable magnetic field,or in the intensity of the magnetic field itself. The lack of changesin the phase-resolved continuum emission rules out simpler, purelygeometric explanations such as precession of the neutron star.To explain the generation of an emission line in the spec-trum of an isolated neutron star, the possibility of cyclotron scat-tering of surface thermal photons by a geometrically thin, opticallythin layer of plasma could be considered. However, under simpleassumptions (emission from the entire star surface; plane-parallelgeometry; pure, conservative scattering), a scattering layer wouldproduce an absorption line. One might invoke a spatially-limitedscattering medium, possibly some distance away from the star sur-face. Photons coming from the part of the surface not covered bythe layer could be scattered along the line-of-sight giving rise toan emission feature. The value of B derived from the line energyis somehow smaller than the upper limit on the surface field, sothe line could indeed form at some height in the magnetosphere. Aconfined medium seems also to be required by the results of phase-resolved spectroscopy which shows that the emission line is seenmostly when the cooler spot is into view. Still, such a picture seemsrather contrived, since the nature of the layer and the mechanismkeeping the plasma suspended and confined in a compact blob re-mains to be understood.To ease the problem, an energy source unrelated to thesurface thermal emission should be invoked to excite e − to higher Landau levels in the line emitting region. Indeed,Nelson, Salpeter, & Wasserman (1993) predicted that for neutronstars accreting at a low rate ( L accr < erg s − ), and endowedwith magnetic fields of – G, accreting ions may lose en-ergy to atmospheric electrons via magnetic Coulomb collisions.Electrons, excited to high Landau levels, radiatively decay and partof the cyclotron photons are expected to escape producing an emis-sion line. According to Nelson, Salpeter, & Wasserman (1993), at
B < G, the fraction of the accretion-powered flux escaping inthe line is expected to be very small ( ≪ ), the largest part beingreprocessed and emitted in a thermal continuum. Thus, one shouldpostulate that the bulk of the X-ray luminosity of RX J0822–4300 isaccretion-powered, at variance with observations. It would requirean accretion rate of ∼ × g s − implying – under standardrelations for propeller spin-down (Menou et al. 1999) – a ˙ P valuemore than 10 times larger than our upper limit. Although the modelby Nelson, Salpeter, & Wasserman (1993) does not fit to our case,low-level accretion of supernova fallback material (which cannotbe ruled out, based on X-ray timing, as well as on the optical upperlimits set by Mignani et al.2009) could play some role in generatingan emission line in a low- B field atmosphere. A detailed investiga-tion of such possibility is beyond the scope of this paper.We will not go into further speculations about the line emit-ting mechanism. We stress that the evidence for time evolution ofthe spectral feature is model-independent and represents to datethe first evidence for variability in an “anti-magnetar” candidate.Likely, such ‘activity’ is related to a variation of the magnetic fieldof the star. A ∼ decrease in ∼ yr seems vastly too steep tobe attributed to the large-scale dipole field. Possibly, we are wit-nessing evolution of a localised multipole component, dominatingclose to the star surface. This would hint at the presence of a largeinternal field, as proposed to explain the anisotropic thermal map ofthe star. Precise X-ray timing, assessing the ˙ P and measuring thestar dipole field, will add a crucial piece of information. Coupled tofurther sensitive phase-resolved spectroscopy to monitor spectral variability, this could help to solve the puzzles set by our resultson RX J0822–4300 which would have important implications forthe understanding of the nature of CCOs and of their relations withother families of neutron stars. ACKNOWLEDGMENTS
Based on observations obtained with
XMM-Newton , an ESA sci-ence mission with instruments and contributions directly funded byESA Member States and NASA. We thank F. Gastaldello for usefuladvice about statistics and A. Possenti for discussions. This workwas partially supported by the ASI-INAF I/009/10/0 agreement. PEacknowledges financial support from the Autonomous Region ofSardinia through a research grant under the program PO SardegnaFSE 2007–2013, L.R. 7/2007.
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