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A. Riggio

Publications and source records attributed to A. Riggio.

At least 37 records · Page 2Linked to original sources

Fe K$α$ and Fe K$β$ line detection in the NuSTAR spectrum of the ultra-bright Z-source Scorpius X-1

Low-mass X-ray binaries hosting a low-magnetised neutron star, which accretes matter via Roche-lobe overflow, are generally grouped in two classes, named Atoll and Z sources after the path described in their X-ray colour-colour diagrams. Scorpius X-1 is the brightest persistent low-mass X-ray binary known so far, and it is the prototype of the Z sources. We analysed the first NuSTAR observation of this source to study its spectral emission exploiting the high statistics data collected by this satellite. Examining the colour-colour diagram, the source was probably observed during the lower normal and flaring branches of its Z-track. We separated the data from the two branches in order to investigate the evolution of the source along the track. We fitted the 3-60 keV NuSTAR spectra using the same models for both the branches. We adopted two description for the continuum: in the first case we used a blackbody and a thermal Comptonisation with seed photons originating in the accretion disc; in the second one, we adopted a disc-blackbody and a Comptonisation with a blackbody-shaped spectrum of the incoming seed photons. A power-law fitting the high energy emission above 20 keV was also required in both cases. The two models provide the same physical scenario for the source in both the branches: a blackbody temperature between 0.8 and 1.5 keV, a disc-blackbody with temperature between 0.4 and 0.6 keV, and an optically thick Comptonising corona with optical depth between 6 and 10 and temperature about 3 keV. Furthermore, two lines related to the K$α$ and K$β$ transitions of the He-like Fe XXV ions were detected at 6.6 keV and 7.8 keV, respectively. A hard tail modelled by a power law with a photon index between 2 and 3 was also required for both the models.

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Time Domain Astronomy with the THESEUS Satellite

THESEUS is a medium size space mission of the European Space Agency, currently under evaluation for a possible launch in 2032. Its main objectives are to investigate the early Universe through the observation of gamma-ray bursts and to study the gravitational waves electromagnetic counterparts and neutrino events. On the other hand, its instruments, which include a wide field of view X-ray (0.3-5 keV) telescope based on lobster-eye focusing optics and a gamma-ray spectrometer with imaging capabilities in the 2-150 keV range, are also ideal for carrying out unprecedented studies in time domain astrophysics. In addition, the presence onboard of a 70 cm near infrared telescope will allow simultaneous multi-wavelegth studies. Here we present the THESEUS capabilities for studying the time variability of different classes of sources in parallel to, and without affecting, the gamma-ray bursts hunt.

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GrailQuest & HERMES: Hunting for Gravitational Wave Electromagnetic Counterparts and Probing Space-Time Quantum Foam

GrailQuest (Gamma-ray Astronomy International Laboratory for Quantum Exploration of Space-Time) is an ambitious astrophysical mission concept that uses a fleet of small satellites whose main objective is to search for a dispersion law for light propagation in vacuo. Within Quantum Gravity theories, different models for space-time quantization predict relative discrepancies of the speed of photons w.r.t. the speed of light that depend on the ratio of the photon energy to the Planck energy. This ratio is as small as 1E-23 for photons in the gamma-ray band (100 keV). Therefore, to detect this effect, light must propagate over enormous distances and the experiment must have extraordinary sensitivity. Gamma-Ray Bursts, occurring at cosmological distances, could be used to detect this tiny signature of space-time granularity. This can be obtained by coherently combine a huge number of small instruments distributed in space to act as a single detector of unprecedented effective area. This is the first example of high-energy distributed astronomy: a new concept of modular observatory of huge overall collecting area consisting in a fleet of small satellites in low orbits, with sub-microsecond time resolution and wide energy band (keV-MeV). The enormous number of collected photons will allow to effectively search these energy dependent delays. Moreover, GrailQuest will allow to perform temporal triangulation of impulsive events with arc-second positional accuracies: an extraordinary sensitive X-ray/Gamma all-sky monitor crucial for hunting the elusive electromagnetic counterparts of Gravitational Waves, that will play a paramount role in the future of Multi-messenger Astronomy. A pathfinder of GrailQuest is already under development through the HERMES (High Energy Rapid Modular Ensemble of Satellites) project: a fleet of six 3U cube-sats to be launched by the end of 2022.

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Timing techniques applied to distributed modular high-energy astronomy: the HERMES project

The HERMES-TP/SP (High Energy Rapid Modular Ensemble of Satellites -- Technologic and Scientific Pathfinder) is an in-orbit demonstration of the so-called distributed astronomy concept. Conceived as a mini-constellation of six 3U nano-satellites hosting a new miniaturized detector, HERMES-TP/SP aims at the detection and accurate localisation of bright high-energy transients such as Gamma-Ray Bursts. The large energy band, the excellent temporal resolution and the wide field of view that characterize the detectors of the constellation represent the key features for the next generation high-energy all-sky monitor with good localisation capabilities that will play a pivotal role in the future of Multi-messenger Astronomy. In this work, we will describe in detail the temporal techniques that allow the localisation of bright transient events taking advantage of their almost simultaneous observation by spatially spaced detectors. Moreover, we will quantitatively discuss the all-sky monitor capabilities of the HERMES Pathfinder as well as its achievable accuracies on the localisation of the detected Gamma-Ray Bursts.

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Evidence of a non-conservative mass transfer in the ultra-compact X-ray source XB 1916-053

The dipping source XB 1916-053 is a compact binary system with an orbital period of 50 min harboring a neutron star. Using ten new {\it Chandra} observations and one {\it Swift/XRT} observation, we are able to extend the baseline of the orbital ephemeris; this allows us to exclude some models that explain the dip arrival times. The Chandra observations provide a good plasma diagnostic of the ionized absorber and allow us to determine whether it is placed at the outer rim of the accretion disk or closer to the compact object. From the available observations we are able to obtain three new dip arrival times extending the baseline of the orbital ephemeris from 37 to 40 years. From the analysis of the dip arrival times we confirm an orbital period derivative of $\dot{P}=1.46(3) \times 10^{-11}$ s s$^{-1}$. We show that the $\dot{P}$ value and the luminosity values are compatible with a mass accretion rate lower than 10\% of the mass transfer rate. We show that the mass ratio $q=m_2/m_1$ of 0.048 explains the apsidal precession period and the nodal precession period. The observed absorption lines are associated with the presence of \ion{Ne}{x}, \ion{Mg}{xii}, \ion{Si}{xiv}, \ion{S}{xvi,} and \ion{Fe}{xxvi} ions. We observe a redshift in the absorption lines between $1.1 \times 10^{-3}$ and $1.3 \times 10^{-3}$. By interpreting it as gravitational redshift, as recently discussed in the literature, we find that the ionized absorber is placed at a distance of $10^8$ cm from the neutron star with a mass of 1.4 M$_{\odot}$ and has a hydrogen atom density greater than $10^{15}$ cm$^{-3}$. (Abstract abridged)

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The INTEGRAL view of the pulsating hard X-ray sky: from accreting and transitional millisecond pulsars to rotation-powered pulsars and magnetars

In the last 25 years, a new generation of X-ray satellites imparted a significant leap forward in our knowledge of X-ray pulsars. The discovery of accreting and transitional millisecond pulsars proved that disk accretion can spin up a neutron star to a very high rotation speed. The detection of MeV-GeV pulsed emission from a few hundreds of rotation-powered pulsars probed particle acceleration in the outer magnetosphere, or even beyond. Also, a population of two dozens of magnetars has emerged. INTEGRAL played a central role to achieve these results by providing instruments with high temporal resolution up to the hard X-ray/soft gamma-ray band and a large field of view imager with good angular resolution to spot hard X-ray transients. In this article, we review the main contributions by INTEGRAL to our understanding of the pulsating hard X-ray sky, such as the discovery and characterization of several accreting and transitional millisecond pulsars, the generation of the first catalog of hard X-ray/soft gamma-ray rotation-powered pulsars, the detection of polarization in the hard X-ray emission from the Crab pulsar, and the discovery of persistent hard X-ray emission from several magnetars.

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ESA Voyage 2050 white paper -- GrailQuest: hunting for Atoms of Space and Time hidden in the wrinkle of Space-Time

GrailQuest (Gamma Ray Astronomy International Laboratory for QUantum Exploration of Space-Time) is a mission concept based on a constellation (hundreds/thousands) of nano/micro/small-satellites in low (or near) Earth orbits. Each satellite hosts a non-collimated array of scintillator crystals coupled with Silicon Drift Detectors with broad energy band coverage (keV-MeV range) and excellent temporal resolution ( below or equal 100 nanoseconds) each with effective area around 100 cm2. This simple and robust design allows for mass-production of the satellites of the fleet. This revolutionary approach implies a huge reduction of costs, flexibility in the segmented launching strategy, and an incremental long-term plan to increase the number of detectors and their performance: a living observatory for next-generation, space-based astronomical facilities. GrailQuest is conceived as an all-sky monitor for fast localisation of high signal-to-noise ratio transients in the X/gamma-ray band, e.g. the elusive electromagnetic counterparts of gravitational wave events. Robust temporal triangulation techniques will allow unprecedented localisation capabilities, in the keV-MeV band, of a few arcseconds or below, depending on the temporal structure of the transient event. The ambitious ultimate goal of this mission is to perform the first experiment, in quantum gravity, to directly probe space-time structure down to the minuscule Planck scale, by constraining or measuring a first order dispersion relation for light in vacuo. This is obtained by detecting delays between photons of different energies in the prompt emission of Gamma-ray Bursts.

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Reflection component in the Bright Atoll Source GX 9+9

GX 9+9 (4U 1728-16) is a low mass X-ray binary (LMXB) source harboring a neutron star. Although it belongs to the subclass of the bright Atoll sources together with GX 9+1, GX 3+1, and GX 13+1, its broadband spectrum is poorly studied and apparently does not show reflection features in the spectrum. To constrain the continuum well and verify whether a relativistic smeared reflection component is present, we analyze the broadband spectrum of GX 9+9 using {\it BeppoSAX} and \textit{XMM-Newton} spectra covering the 0.3-40 keV energy band. We fit the spectrum adopting a model composed of a disk-blackbody plus a Comptonized component whose seed photons have a blackbody spectrum (Eastern Model). A statistically equivalent model is composed of a Comptonized component whose seed photons have a disk-blackbody distribution plus a blackbody that mimics a saturated Comptonization likely associated with a boundary layer (Western model). Other trials did not return a good fit. The spectrum of GX 9+9 was observed in a soft state and its luminosity is $2.3 \times 10^{37}$ erg s$^{-1}$ assuming a distance to the source of 5 kpc. In the Eastern Model scenario, we find the seed-photon temperature and electron temperature of the Comptonized component to be $1.14^{+0.10}_{-0.07}$ keV and $2.80^{+0.09}_{-0.04}$ keV, respectively, while the optical depth of the Comptonizing corona is $8.9\pm0.4$. The color temperature of the inner accretion disk is $0.86^{+0.08}_{-0.02}$ keV and $0.82 \pm 0.02$ keV for the {\it BeppoSAX} and \textit{XMM-Newton} spectrum, respectively. In the Western Model scenario, instead, we find that the seed-photon temperature is $0.87 \pm 0.07$ keV and $1.01 \pm 0.08$ keV for the {\it BeppoSAX} and \textit{XMM-Newton} spectrum, respectively. (Abridged abstract)

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Timing of the accreting millisecond pulsar IGR J17591-2342: evidence of spin-down during accretion

We report on the phase-coherent timing analysis of the accreting millisecond X-ray pulsar IGR J17591-2342, using Neutron Star Interior Composition Explorer (NICER) data taken during the outburst of the source between 2018 August 15 and 2018 October 17. We obtain an updated orbital solution of the binary system. We investigate the evolution of the neutron star spin frequency during the outburst, reporting a refined estimate of the spin frequency and the first estimate of the spin frequency derivative ($\dotν \sim -7\times 10^{-14}$ Hz s$^{-1}$), confirmed independently from the modelling of the fundamental frequency and its first harmonic. We further investigate the evolution of the X-ray pulse phases adopting a physical model that accounts for the accretion material torque as well as the magnetic threading of the accretion disc in regions where the Keplerian velocity is slower than the magnetosphere velocity. From this analysis we estimate the neutron star magnetic field $B_{eq} = 2.8(3)\times10^{8}$ G. Finally, we investigate the pulse profile dependence on energy finding that the observed behaviour of the pulse fractional amplitude and lags as a function of energy are compatible with a thermal Comptonisation of the soft photons emitted from the neutron star caps.

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Broadband spectral analysis of MXB 1659-298 in its soft and hard state

The X-ray transient eclipsing source MXB 1659-298 went into outburst in 1999 and 2015. During these two outbursts the source was observed by XMM-Newton, nuSTAR, and Swift/XRT. Using these observations, we studied the broadband spectrum of the source to constrain the continuum components and to verify whether it had a reflection component, as is observed in other X-ray eclipsing transient sources. We combined the available spectra to study the soft and hard state of the source in the 0.45-55 keV energy range. We report a reflection component in the soft and hard state. The direct emission in the soft state can be modeled with a thermal component originating from the inner accretion disk plus a Comptonized component associated with an optically thick corona surrounding the neutron star. On the other hand, the direct emission in the hard state is described only by a Comptonized component with a temperature higher than 130 keV; this component is associated with an optically thin corona. We observed narrow absorption lines from highly ionized ions of oxygen, neon, and iron in the soft spectral state. We investigated where the narrow absorption lines form in the ionized absorber. The equivalent hydrogen column density associated with the absorber is close to $6 \times 10^{23}$ cm$^{-2}$ and $1.3 \times 10^{23}$ cm$^{-2}$ in the soft and hard state, respectively.

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Broadband spectral analysis of MXB 1659-298 in its soft and hard state

The X-ray transient eclipsing source MXB 1659-298 went in outburst in 1999 and 2015, respectively, during which it was observed by XMM-Newton, NuSTAR and Swift. Using these observations we studied the broadband spectrum of the source to constrain the continuum components and to verify the presence of a reflection component. We analysed the soft and hard state of the source, finding that the soft state can be modelled with a thermal component associated with the inner accretion disc plus a Comptonised component. A smeared reflection component and the presence of an ionised absorber are also requested in the best-fit model. On the other hand, the direct continuum emission in the hard state can be described by a Comptonised component with a temperature larger than 150 keV. Also in this case a reflection component and a ionised absorber are observed.

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Pulsating in unison at optical and X-ray energies: simultaneous high-time resolution observations of the transitional millisecond pulsar PSR J1023+0038

PSR J1023+0038 is the first millisecond pulsar discovered to pulsate in the visible band; such a detection took place when the pulsar was surrounded by an accretion disk and also showed X-ray pulsations. We report on the first high time resolution observational campaign of this transitional pulsar in the disk state, using simultaneous observations in the optical (TNG, NOT, TJO), X-ray (XMM-Newton, NuSTAR, NICER), infrared (GTC) and UV (Swift) bands. Optical and X-ray pulsations were detected simultaneously in the X-ray high intensity mode in which the source spends $\sim$ 70% of the time, and both disappeared in the low mode, indicating a common underlying physical mechanism. In addition, optical and X-ray pulses were emitted within a few km, had similar pulse shape and distribution of the pulsed flux density compatible with a power-law relation $F_ν \propto ν^{-0.7}$ connecting the optical and the 0.3-45 keV X-ray band. Optical pulses were detected also during flares with a pulsed flux reduced by one third with respect to the high mode; the lack of a simultaneous detection of X-ray pulses is compatible with the lower photon statistics. We show that magnetically channeled accretion of plasma onto the surface of the neutron star cannot account for the optical pulsed luminosity ($\sim 10^{31}$ erg/s). On the other hand, magnetospheric rotation-powered pulsar emission would require an extremely efficient conversion of spin-down power into pulsed optical and X-ray emission. We then propose that optical and X-ray pulses are instead produced by synchrotron emission from the intrabinary shock that forms where a striped pulsar wind meets the accretion disk, within a few light cylinder radii away, $\sim$ 100 km, from the pulsar.

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Indications of non-conservative mass-transfer in AMXPs

Context. Since the discovery of the first Accreting Millisecond X-ray Pulsar SAX J1808.4-3658 in 1998, the family of these sources kept growing on. Currently, it counts 22 members. All AMXPs are transients with usually very long quiescence periods, implying that mass accretion rate in these systems is quite low and not constant. Moreover, for at least three sources, a non-conservative evolution was also proposed. Aims. Our purpose is to study the long term averaged mass-accretion rates in all the Accreting Millisecond X-ray Pulsars discovered so far, to investigate a non-conservative mass-transfer scenario. Methods. We calculated the expected mass-transfer rate under the hypothesis of a conservative evolution based on their orbital periods and on the (minimum) mass of the secondary (as derived from the mass function), driven by gravitational radiation and/or magnetic braking. Using this theoretical mass-transfer, we determined the expected accretion luminosity of the systems. Thus, we achieved the lower limit to the distance of the sources by comparing the computed theoretical luminosity and the observed flux averaged over a time period of 20 years. Then, the lower limit to the distance of the sources has been compared to the value of the distance reported in literature to evaluate how reasonable is the hypothesis of a conservative mass-transfer. Results. Based on a sample of 18 sources, we found strong evidences of a non-conservative mass-transfer for five sources, for which the estimated distance lower limits are higher than their known distances. We also report hints for mass outflows in other six sources. The discrepancy can be fixed under the hypothesis of a non-conservative mass-transfer in which a fraction of the mass transferred onto the compact object is swept away from the system, likely due to the (rotating magnetic dipole) radiation pressure of the pulsar.

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Updated orbital ephemeris of the ADC source X 1822-371: a stable orbital expansion over 40 years

The source X 1822-371 is an eclipsing compact binary system with a period close to 5.57 hr and an orbital period derivative $\dot{P}_{\rm orb}$ of 1.51(7)$\times 10^{-10}$ s s$^{-1}$. The very large value of $\dot{P}_{\rm orb}$ is compatible with a super-Eddington mass transfer rate from the companion star, as suggested by X-ray and optical data. The XMM-Newton observation taken in 2017 allows us to update the orbital ephemeris and verify whether the orbital period derivative has been stable over the last 40 yr. We added to the X-ray eclipse arrival times from 1977 to 2008 two new values obtained from the RXTE and XMM-Newton observations performed in 2011 and 2017, respectively. We estimated the number of orbital cycles and the delays of our eclipse arrival times spanning 40 yr using as reference time the eclipse arrival time obtained from the Rossi-XTE observation taken in 1996. Fitting the delays with a quadratic model, we found an orbital period $P_{\rm orb}=5.57062957(20)$ hr and a $\dot{P}_{\rm orb}$ value of $1.475(54) \times 10^{-10}$ s s$^{-1}$. The addition of a cubic term to the model does not significantly improve the quality of the fit. We also determined a spin-period value of $P_{\rm spin}=0.5915669(4)$ s and its first derivative $\dot{P}_{\rm spin}= -2.595(11) \times 10^{-12}$ s s$^{-1}$. The obtained results confirm the scenario of a super-Eddington mass transfer rate; we also exclude a gravitational coupling between the orbit and the change in the oblateness of the companion star triggered by the nuclear luminosity of the companion star.

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Spectral analysis of the dipping LMXB system XB 1916-053

Context: XB 1916-053 is a low mass X-ray binary system (LMXB) hosting a neutron star (NS) and showing periodic dips. The spectrum of the persistent emission was modeled with a blackbody component having a temperature between 1.31 and 1.67 keV and with a Comptonization component with an electron temperature of 9.4 keV and a photon index $Γ$ between 2.5 and 2.9. The presence of absorption features associated with highly ionized elements suggested the presence of partially ionized plasma in the system. Aims: In this work we performed a study of the spectrum of XB 1916-053, which aims to shed light on the nature of the seed photons that contribute to the Comptonization component. Methods: We analyzed three Suzaku observations of XB 1916-053: the first was performed in November 2006 and the others were carried out in October 2014. We extracted the persistent spectra from each observation and combined the spectra of the most recent observations, obtaining a single spectrum with a higher statistic. We also extracted and combined the spectra of the dips observed during the same observations. Results: On the basis of the available data statistics, we infer that the scenario in which the corona Comptonizes photons emitted both by the innermost region of the accretion disk and the NS surface is not statistically relevant with respect to the case in which only photons emitted by the NS surface are Comptonized. We find that the source is in a soft spectral state in all the analyzed observations. We detect the K$α$ absorption lines of \ion{Fe}{xxv} and \ion{Fe}{xxvi}, which have already been reported in literature, and for the first time the K$β$ absorption lines of the same ions. We also detect an edge at 0.876 keV, which is consistent with a \ion{O}{viii} K absorption edge. (Abridged)

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HERMES: An ultra-wide band X and gamma-ray transient monitor on board a nano-satellite constellation

The High Energy Modular Ensemble of Satellites (HERMES) project is aimed to realize a modular X/gamma-ray monitor for transient events, to be placed on-board of a CubeSat bus. This expandable platform will achieve a significant impact on Gamma Ray Burst (GRB) science and on the detection of Gravitational Wave (GW) electromagnetic counterparts: the recent LIGO/VIRGO discoveries demonstrated that the high-energy transient sky is still a field of extreme interest. The very complex temporal variability of GRBs (up to the millisecond scale) combined with the spatial and temporal coincidence between GWs and their electromagnetic counterparts suggest that upcoming instruments require sub-ms time resolution combined with a transient localization accuracy lower than a degree. The current phase of the ongoing HERMES project is focused on the realization of a technological pathfinder with a small network (3 units) of nano-satellites to be launched in mid 2020. We will show the potential and prospects for short and medium-term development of the project, demonstrating the disrupting possibilities for scientific investigations provided by the innovative concept of a new "modular astronomy" with nano-satellites (e.g. low developing costs, very short realization time). Finally, we will illustrate the characteristics of the HERMES Technological Pathfinder project, demonstrating how the scientific goals discussed are actually already reachable with the first nano-satellites of this constellation. The detector architecture will be described in detail, showing that the new generation of scintillators (e.g. GAGG:Ce) coupled with very performing Silicon Drift Detectors (SDD) and low noise Front-End-Electronics (FEE) are able to extend down to few keV the sensitivity band of the detector. The technical solutions for FEE, Back-End-Electronics (BEE) and Data Handling will be also described.

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The broadband spectral analysis of 4U 1702-429 using XMM-Newton and BeppoSAX data

Most of the X-ray binary systems containing neutron stars classified as Atoll sources show two different spectral states, called soft and hard. Moreover, a large number of these systems show a reflection component relativistically smeared in their spectra, which gives information on the innermost region of the system. Our aim is to investigate the poorly studied broadband spectrum of the low mass X-ray binary system 4U 1702-429, which was recently analysed combining XMM-Newton and INTEGRAL data. The peculiar value of the reflection fraction brought us to analyse further broadband spectra of 4U 1702-429. We re-analysed the spectrum of the XMM-Newton/INTEGRAL observation of 4U 1702-429 in the 0.3-60 keV energy range and we extracted three 0.1-100 keV spectra of the source analysing three observations collected with the BeppoSAX satellite. We find that the XMM-Newton/INTEGRAL spectrum is well fitted using a model composed of a disc blackbody plus a Comptonised component and a smeared reflection component. We used the same spectral model for the BeppoSAX spectra, finding out that the addition of a smeared reflection component is statistically significant. The best-fit values of the parameters are compatible to each other for the BeppoSAX spectra. We find that the reflection fraction is $0.05^{+0.03}_{-0.01}$ for the XMM-Newton/INTEGRAL spectrum and between 0.15 and 0.4 for BeppoSAX ones. The relative reflection fraction and the ionisation parameter are incompatible between the XMM-Newton/INTEGRAL and the BeppoSAX observations and the characteristics of the Comptonising corona suggest that the source was in a soft state in the former observation and in a hard state in the latter.

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NuSTAR and XMM-Newton broad-band spectrum of SAX J1808.4-3658 during its latest outburst in 2015

The first discovered accreting millisecond pulsar, SAX J1808.4-3658, went into X-ray outburst in April 2015. We triggered a 100 ks XMM-Newton ToO, taken at the peak of the outburst, and a 55 ks NuSTAR ToO, performed four days apart. We report here the results of a detailed spectral analysis of both the XMM-Newton and NuSTAR spectra. While the XMM-Newton spectrum appears much softer than in previous observations, the NuSTAR spectrum confirms the results obtained with XMM-Newton during the 2008 outburst. We find clear evidence of a broad iron line that we interpret as produced by reflection from the inner accretion disk. For the first time, we use a self-consistent reflection model to fit the reflection features in the NuSTAR spectrum; in this case we find a statistically significant improvement of the fit with respect to a simple Gaussian or diskline model to fit the iron line, implying that the reflection continuum is also significantly detected. Despite the differences evident between the XMM-Newton and NuSTAR spectra, the smearing best-fit parameters found for these spectra are consistent with each other and are compatible with previous results. In particular, we find an upper limit to the inner disk radius of $\sim 12~R_g$. In all the cases, a high inclination angle ($>50^\circ$) of the system is required. This inclination angle, combined with measurements of the radial velocity of the optical companion, results in a low value for the neutron star mass ($<0.8\,M_\odot$), a result that deserves further investigation.

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