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P. Bordas

Publications and source records attributed to P. Bordas.

At least 37 records · Page 2Linked to original sources

Sensitivity of the Cherenkov Telescope Array to spectral signatures of hadronic PeVatrons with application to Galactic Supernova Remnants

The local Cosmic Ray (CR) energy spectrum exhibits a spectral softening at energies around 3~PeV. Sources which are capable of accelerating hadrons to such energies are called hadronic PeVatrons. However, hadronic PeVatrons have not yet been firmly identified within the Galaxy. Several source classes, including Galactic Supernova Remnants (SNRs), have been proposed as PeVatron candidates. The potential to search for hadronic PeVatrons with the Cherenkov Telescope Array (CTA) is assessed. The focus is on the usage of very high energy $γ$-ray spectral signatures for the identification of PeVatrons. Assuming that SNRs can accelerate CRs up to knee energies, the number of Galactic SNRs which can be identified as PeVatrons with CTA is estimated within a model for the evolution of SNRs. Additionally, the potential of a follow-up observation strategy under moonlight conditions for PeVatron searches is investigated. Statistical methods for the identification of PeVatrons are introduced, and realistic Monte--Carlo simulations of the response of the CTA observatory to the emission spectra from hadronic PeVatrons are performed. Based on simulations of a simplified model for the evolution for SNRs, the detection of a $γ$-ray signal from in average 9 Galactic PeVatron SNRs is expected to result from the scan of the Galactic plane with CTA after 10 hours of exposure. CTA is also shown to have excellent potential to confirm these sources as PeVatrons in deep observations with $\mathcal{O}(100)$ hours of exposure per source.

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Multi-wavelength study of the galactic PeVatron candidate LHAASO J2108+5157

LHAASO J2108+5157 is one of the few known unidentified Ultra-High-Energy (UHE) gamma-ray sources with no Very-High-Energy (VHE) counterpart, recently discovered by the LHAASO collaboration. We observed LHAASO J2108+5157 in the X-ray band with XMM-Newton in 2021 for a total of 3.8 hours and at TeV energies with the Large-Sized Telescope prototype (LST-1), yielding 49 hours of good quality data. In addition, we analyzed 12 years of Fermi-LAT data, to better constrain emission of its High-Energy (HE) counterpart 4FGL J2108.0+5155. We found an excess (3.7 sigma) in the LST-1 data at energies E > 3 TeV. Further analysis in the whole LST-1 energy range assuming a point-like source, resulted in a hint (2.2 sigma) of hard emission which can be described with a single power law with photon index Gamma = 1.6 +- 0.2 between 0.3 - 100 TeV. We did not find any significant extended emission which could be related to a Supernova Remnant (SNR) or Pulsar Wind Nebula (PWN) in the XMM-Newton data, which puts strong constraints on possible synchrotron emission of relativistic electrons. The LST-1 and LHAASO observations can be explained as inverse Compton-dominated leptonic emission of relativistic electrons with a cutoff energy of $100^{+70}_{-30}$ TeV. The low magnetic field in the source imposed by the X-ray upper limits on synchrotron emission is compatible with a hypothesis of a PWN or a TeV halo. The lack of a pulsar in the neighborhood of the UHE source is a challenge to the PWN/TeV-halo scenario. The UHE gamma rays can also be explained as $π^0$ decay-dominated hadronic emission due to interaction of relativistic protons with one of the two known molecular clouds in the direction of the source. The hard spectrum in the LST-1 band is compatible with protons escaping a shock around a middle-aged SNR because of their high low-energy cut-off.

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Origin and role of relativistic cosmic particles

This white paper briefly summarizes the importance of the study of relativistic cosmic rays, both as a constituent of our Universe, and through their impact on stellar and galactic evolution. The focus is on what can be learned over the coming decade through ground-based gamma-ray observations over the 20 GeV to 300 TeV range. The majority of the material is drawn directly from "Science with the Cherenkov Telescope Array", which describes the overall science case for CTA. We request that authors wishing to cite results contained in this white paper cite the original work.

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Multi-messenger and transient astrophysics with the Cherenkov Telescope Array

The discovery of gravitational waves, high-energy neutrinos or the very-high-energy counterpart of gamma-ray bursts has revolutionized the high-energy and transient astrophysics community. The development of new instruments and analysis techniques will allow the discovery and/or follow-up of new transient sources. We describe the prospects for the Cherenkov Telescope Array (CTA), the next-generation ground-based gamma-ray observatory, for multi-messenger and transient astrophysics in the decade ahead. CTA will explore the most extreme environments via very-high-energy observations of compact objects, stellar collapse events, mergers and cosmic-ray accelerators.

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Probing extreme environments with the Cherenkov Telescope Array

The physics of the non-thermal Universe provides information on the acceleration mechanisms in extreme environments, such as black holes and relativistic jets, neutron stars, supernovae or clusters of galaxies. In the presence of magnetic fields, particles can be accelerated towards relativistic energies. As a consequence, radiation along the entire electromagnetic spectrum can be observed, and extreme environments are also the most likely sources of multi-messenger emission. The most energetic part of the electromagnetic spectrum corresponds to the very-high-energy (VHE, E>100 GeV) gamma-ray regime, which can be extensively studied with ground based Imaging Atmospheric Cherenkov Telescopes (IACTs). The results obtained by the current generation of IACTs, such as H.E.S.S., MAGIC, and VERITAS, demonstrate the crucial importance of the VHE band in understanding the non-thermal emission of extreme environments in our Universe. In some objects, the energy output in gamma rays can even outshine the rest of the broadband spectrum. The Cherenkov Telescope Array (CTA) is the next generation of IACTs, which, with cutting edge technology and a strategic configuration of ~100 telescopes distributed in two observing sites, in the northern and southern hemispheres, will reach better sensitivity, angular and energy resolution, and broader energy coverage than currently operational IACTs. With CTA we can probe the most extreme environments and considerably boost our knowledge of the non-thermal Universe.

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Sensitivity of the Cherenkov Telescope Array for probing cosmology and fundamental physics with gamma-ray propagation

The Cherenkov Telescope Array (CTA), the new-generation ground-based observatory for $γ$-ray astronomy, provides unique capabilities to address significant open questions in astrophysics, cosmology, and fundamental physics. We study some of the salient areas of $γ$-ray cosmology that can be explored as part of the Key Science Projects of CTA, through simulated observations of active galactic nuclei (AGN) and of their relativistic jets. Observations of AGN with CTA will enable a measurement of $γ$-ray absorption on the extragalactic background light with a statistical uncertainty below 15% up to a redshift $z=2$ and to constrain or detect $γ$-ray halos up to intergalactic-magnetic-field strengths of at least 0.3pG. Extragalactic observations with CTA also show promising potential to probe physics beyond the Standard Model. The best limits on Lorentz invariance violation from $γ$-ray astronomy will be improved by a factor of at least two to three. CTA will also probe the parameter space in which axion-like particles could constitute a significant fraction, if not all, of dark matter. We conclude on the synergies between CTA and other upcoming facilities that will foster the growth of $γ$-ray cosmology.

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Sensitivity of the Cherenkov Telescope Array to a dark matter signal from the Galactic centre

We provide an updated assessment of the power of the Cherenkov Telescope Array (CTA) to search for thermally produced dark matter at the TeV scale, via the associated gamma-ray signal from pair-annihilating dark matter particles in the region around the Galactic centre. We find that CTA will open a new window of discovery potential, significantly extending the range of robustly testable models given a standard cuspy profile of the dark matter density distribution. Importantly, even for a cored profile, the projected sensitivity of CTA will be sufficient to probe various well-motivated models of thermally produced dark matter at the TeV scale. This is due to CTA's unprecedented sensitivity, angular and energy resolutions, and the planned observational strategy. The survey of the inner Galaxy will cover a much larger region than corresponding previous observational campaigns with imaging atmospheric Cherenkov telescopes. CTA will map with unprecedented precision the large-scale diffuse emission in high-energy gamma rays, constituting a background for dark matter searches for which we adopt state-of-the-art models based on current data. Throughout our analysis, we use up-to-date event reconstruction Monte Carlo tools developed by the CTA consortium, and pay special attention to quantifying the level of instrumental systematic uncertainties, as well as background template systematic errors, required to probe thermally produced dark matter at these energies. "Full likelihood tables complementing our analysis are provided here [ https://doi.org/10.5281/zenodo.4057987 ]"

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An extreme particle accelerator in the Galactic plane: HESS J1826$-$130

The unidentified very-high-energy (VHE; E $>$ 0.1 TeV) $γ$-ray source, HESS J1826$-$130, was discovered with the High Energy Stereoscopic System (HESS) in the Galactic plane. The analysis of 215 h of HESS data has revealed a steady $γ$-ray flux from HESS J1826$-$130, which appears extended with a half-width of 0.21$^{\circ}$ $\pm$ 0.02$^{\circ}_{\text{stat}}$ $\pm$ 0.05$^{\circ}_{\text{sys}}$. The source spectrum is best fit with either a power-law function with a spectral index $Γ$ = 1.78 $\pm$ 0.10$_{\text{stat}}$ $\pm$ 0.20$_{\text{sys}}$ and an exponential cut-off at 15.2$^{+5.5}_{-3.2}$ TeV, or a broken power-law with $Γ_{1}$ = 1.96 $\pm$ 0.06$_{\text{stat}}$ $\pm$ 0.20$_{\text{sys}}$, $Γ_{2}$ = 3.59 $\pm$ 0.69$_{\text{stat}}$ $\pm$ 0.20$_{\text{sys}}$ for energies below and above $E_{\rm{br}}$ = 11.2 $\pm$ 2.7 TeV, respectively. The VHE flux from HESS J1826$-$130 is contaminated by the extended emission of the bright, nearby pulsar wind nebula (PWN), HESS J1825$-$137, particularly at the low end of the energy spectrum. Leptonic scenarios for the origin of HESS J1826$-$130 VHE emission related to PSR J1826$-$1256 are confronted by our spectral and morphological analysis. In a hadronic framework, taking into account the properties of dense gas regions surrounding HESS J1826$-$130, the source spectrum would imply an astrophysical object capable of accelerating the parent particle population up to $\gtrsim$200 TeV. Our results are also discussed in a multiwavelength context, accounting for both the presence of nearby supernova remnants (SNRs), molecular clouds, and counterparts detected in radio, X-rays, and TeV energies.

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Resolving the Crab pulsar wind nebula at teraelectronvolt energies

The Crab nebula is one of the most studied cosmic particle accelerators, shining brightly across the entire electromagnetic spectrum up to very high-energy gamma rays. It is known from radio to gamma-ray observations that the nebula is powered by a pulsar, which converts most of its rotational energy losses into a highly relativistic outflow. This outflow powers a pulsar wind nebula (PWN), a region of up to 10~light-years across, filled with relativistic electrons and positrons. These particles emit synchrotron photons in the ambient magnetic field and produce very high-energy gamma rays by Compton up-scattering of ambient low-energy photons. While the synchrotron morphology of the nebula is well established, it was up to now not known in which region the very high-energy gamma rays are emitted. Here we report that the Crab nebula has an angular extension at gamma-ray energies of 52 arcseconds (assuming a Gaussian source width), significantly larger than at X-ray energies. This result closes a gap in the multi-wavelength coverage of the nebula, revealing the emission region of the highest energy gamma rays. These gamma rays are a new probe of a previously inaccessible electron and positron energy range. We find that simulations of the electromagnetic emission reproduce our new measurement, providing a non-trivial test of our understanding of particle acceleration in the Crab nebula.

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H.E.S.S. observations of the flaring gravitationally lensed galaxy PKS 1830-211

PKS 1830-211 is a known macrolensed quasar located at a redshift of z=2.5. Its high-energy gamma-ray emission has been detected with the Fermi-LAT instrument and evidence for lensing was obtained by several authors from its high-energy data. Observations of PKS 1830-211 were taken with the H.E.S.S. array of Imaging Atmospheric Cherenkov Telescopes in August 2014, following a flare alert by the Fermi- LAT collaboration. The H.E.S.S observations were aimed at detecting a gamma-ray flare delayed by 20-27 days from the alert flare, as expected from observations at other wavelengths. More than twelve hours of good quality data were taken with an analysis threshold of $\sim67$ GeV. The significance of a potential signal is computed as a function of the date as well as the average significance over the whole period. Data are compared to simultaneous observations by Fermi-LAT. No photon excess or significant signal is detected. An upper limit on PKS 1830-211 flux above 67 GeV is computed and compared to the extrapolation of the Fermi-LAT flare spectrum.

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Monte Carlo studies for the optimisation of the Cherenkov Telescope Array layout

The Cherenkov Telescope Array (CTA) is the major next-generation observatory for ground-based very-high-energy gamma-ray astronomy. It will improve the sensitivity of current ground-based instruments by a factor of five to twenty, depending on the energy, greatly improving both their angular and energy resolutions over four decades in energy (from 20 GeV to 300 TeV). This achievement will be possible by using tens of imaging Cherenkov telescopes of three successive sizes. They will be arranged into two arrays, one per hemisphere, located on the La Palma island (Spain) and in Paranal (Chile). We present here the optimised and final telescope arrays for both CTA sites, as well as their foreseen performance, resulting from the analysis of three different large-scale Monte Carlo productions.

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The 2014 TeV Gamma-ray Flare of Mrk 501 Seen with H.E.S.S.: Temporal and Spectral Constraints on Lorentz Invariance Violation

The blazar Mrk 501 (z=0.034) was observed at very-high-energy (VHE, $E\gtrsim 100$~GeV) gamma-ray wavelengths during a bright flare on the night of 2014 June 23-24 (MJD 56832) with the H.E.S.S. phase-II array of Cherenkov telescopes. Data taken that night by H.E.S.S. at large zenith angle reveal an exceptional number of gamma-ray photons at multi-TeV energies, with rapid flux variability and an energy coverage extending significantly up to 20 TeV. This data set is used to constrain Lorentz invariance violation (LIV) using two independent channels: a temporal approach considers the possibility of an energy dependence in the arrival time of gamma rays, whereas a spectral approach considers the possibility of modifications to the interaction of VHE gamma rays with extragalactic background light (EBL) photons. The non-detection of energy-dependent time delays and the non-observation of deviations between the measured spectrum and that of a supposed power-law intrinsic spectrum with standard EBL attenuation are used independently to derive strong constraints on the energy scale of LIV ($E_{\rm{QG}}$) in the subluminal scenario for linear and quadratic perturbations in the dispersion relation of photons. For the case of linear perturbations, the 95% confidence level limits obtained are $E_{\rm{QG},1} > 3.6 \times 10^{17} \ \rm{GeV} $ using the temporal approach and $E_{\rm{QG},1} > 2.6 \times 10^{19} \ \rm{GeV}$ using the spectral approach. For the case of quadratic perturbations, the limits obtained are $E_{\rm{QG},2} > 8.5 \times 10^{10} \ \rm{GeV} $ using the temporal approach and $E_{\rm{QG},2} > 7.8 \times 10^{11} \rm{ GeV}$ using the spectral approach.

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Particle Transport within the Pulsar Wind Nebula HESS J1825-137

Aims: We present a detailed view of the pulsar wind nebula (PWN) HESS J1825-137. We aim to constrain the mechanisms dominating the particle transport within the nebula, accounting for its anomalously large size and spectral characteristics. Methods: The nebula is studied using a deep exposure from over 12 years of H.E.S.S. I operation, together with data from H.E.S.S. II improving the low energy sensitivity. Enhanced energy-dependent morphological and spatially-resolved spectral analyses probe the Very High Energy (VHE, E > 0.1 TeV) gamma-ray properties of the nebula. Results: The nebula emission is revealed to extend out to 1.5 degrees from the pulsar, ~1.5 times further than previously seen, making HESS J1825--137, with an intrinsic diameter of ~100 pc, potentially the largest gamma-ray PWN currently known. Characterisation of the nebula's strongly energy-dependent morphology enables the particle transport mechanisms to be constrained. A dependence of the nebula extent with energy of R $\propto$ E^αwith α= -0.29 +/- 0.04 (stat) +/- 0.05 (sys) disfavours a pure diffusion scenario for particle transport within the nebula. The total gamma-ray flux of the nebula above 1~TeV is found to be (1.12 +/- 0.03 (stat) +/- 0.25 (sys)) $\times 10^{-11}$ cm$^{-2}$ s$^{-1}$, corresponding to ~64% of the flux of the Crab Nebula. Conclusions: HESS J1825-137 is a PWN with clear energy-dependent morphology at VHE gamma-ray energies. This source is used as a laboratory to investigate particle transport within middle-aged PWNe. Deep observations of this highly spatially-extended PWN enable a spectral map of the region to be produced, providing insights into the spectral variation within the nebula.

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VHE $γ$-ray discovery and multi-wavelength study of the blazar 1ES 2322-409

A hotspot at a position compatible with the BL Lac object 1ES 2322-409 was serendipitously detected with H.E.S.S. during observations performed in 2004 and 2006 on the blazar PKS 2316-423. Additional data on 1ES 2322-409 were taken in 2011 and 2012, leading to a total live-time of 22.3h. Point-like very-high-energy (VHE; E>100GeV) $γ$-ray emission is detected from a source centred on the 1ES 2322-409 position, with an excess of 116.7 events at a significance of 6.0$σ$. The average VHE $γ$-ray spectrum is well described with a power law with a photon index $Γ=3.40\pm0.66_{\text{stat}}\pm0.20_{\text{sys}}$ and an integral flux $Φ(E>200GeV) = (3.11\pm0.71_{\rm stat}\pm0.62_{\rm sys})\times10^{-12} cm^{-2} s^{-1}$, which corresponds to 1.1$\%$ of the Crab nebula flux above 200 GeV. Multi-wavelength data obtained with Fermi LAT, Swift XRT and UVOT, RXTE PCA, ATOM, and additional data from WISE, GROND and Catalina, are also used to characterise the broad-band non-thermal emission of 1ES 2322-409. The multi-wavelength behaviour indicates day-scale variability. Swift UVOT and XRT data show strong variability at longer scales. A spectral energy distribution (SED) is built from contemporaneous observations obtained around a high state identified in Swift data. A modelling of the SED is performed with a stationary homogeneous one-zone synchrotron-self-Compton (SSC) leptonic model. The redshift of the source being unknown, two plausible values were tested for the modelling. A systematic scan of the model parameters space is performed, resulting in a well-constrained combination of values providing a good description of the broad-band behaviour of 1ES 2322-409.

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Searches for gamma-ray lines and `pure WIMP' spectra from Dark Matter annihilations in dwarf galaxies with H.E.S.S

Dwarf spheroidal galaxies are among the most promising targets for detecting signals of Dark Matter (DM) annihilations. The H.E.S.S. experiment has observed five of these systems for a total of about 130 hours. The data are re-analyzed here, and, in the absence of any detected signals, are interpreted in terms of limits on the DM annihilation cross section. Two scenarios are considered: i) DM annihilation into mono-energetic gamma-rays and ii) DM in the form of pure WIMP multiplets that, annihilating into all electroweak bosons, produce a distinctive gamma-ray spectral shape with a high-energy peak at the DM mass and a lower-energy continuum. For case i), upper limits at 95\% confidence level of about $\langle σv \rangle \lesssim 3 \times 10^{-25}$ cm$^3$ s$^{-1}$ are obtained in the mass range of 400 GeV to 1 TeV. For case ii), the full spectral shape of the models is used and several excluded regions are identified, but the thermal masses of the candidates are not robustly ruled out.

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First Ground-based Measurement of Sub-20 GeV to 100 GeV $γ$-rays from the Vela Pulsar with H.E.S.S. II

We report on the measurement and investigation of pulsed high-energy $γ$-ray emission from the Vela pulsar, PSR B0833$-$45, based on 40.3 hours of observations with the largest telescope of H.E.S.S., CT5, in monoscopic mode, and on 8 years of data obtained with the Fermi-LAT. A dedicated very-low-threshold event reconstruction and analysis pipeline was developed and, together with the CT5 telescope response model, was validated using the Fermi-LAT data as reference. A pulsed $γ$-ray signal at a significance level of more than $15σ$ is detected from the P2 peak of the Vela pulsar light curve. Of a total of 15835 events, more than 6000 lie at an energy below 20 GeV, implying a significant overlap between H.E.S.S. II-CT5 and the Fermi-LAT. While the investigation of the pulsar light curve with the LAT confirms characteristics previously known up to 20 GeV, in the tens of GeV energy range, CT5 data show a change in the pulse morphology of P2, i.e., an extreme sharpening of its trailing edge, together with the possible onset of a new component at 3.4$σ$ significance level. Assuming a power-law model for the P2 spectrum, an excellent agreement is found for the photon indices ($Γ\simeq$ 4.1) obtained with the two telescopes above 10 GeV and an upper bound of 8% is derived on the relative offset between their energy scales. Using both instruments data, it is however shown that the spectrum of P2 in the 10-100 GeV has a pronounced curvature, i.e. a confirmation of the sub-exponential cutoff form found at lower energies with the LAT. This is further supported by the weak evidence for an emission above 100 GeV obtained with CT5. In contrast, converging indications are found from both CT5 and LAT data for the emergence of a hard component above 50 GeV in the leading wing (LW2) of P2, which possibly extends beyond 100 GeV.

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The $γ$-ray spectrum of the core of Centaurus A as observed with H.E.S.S. and Fermi-LAT

Centaurus A (Cen A) is the nearest radio galaxy discovered as a very-high-energy (VHE; 100 GeV-100 TeV) $γ$-ray source by the High Energy Stereoscopic System (H.E.S.S.). It is a faint VHE $γ$-ray emitter, though its VHE flux exceeds both the extrapolation from early Fermi-LAT observations as well as expectations from a (misaligned) single-zone synchrotron-self Compton (SSC) description. The latter satisfactorily reproduces the emission from Cen A at lower energies up to a few GeV. New observations with H.E.S.S., comparable in exposure time to those previously reported, were performed and eight years of Fermi-LAT data were accumulated to clarify the spectral characteristics of the $γ$-ray emission from the core of Cen A. The results allow us for the first time to achieve the goal of constructing a representative, contemporaneous $γ$-ray core spectrum of Cen A over almost five orders of magnitude in energy. Advanced analysis methods, including the template fitting method, allow detection in the VHE range of the core with a statistical significance of 12$σ$ on the basis of 213 hours of total exposure time. The spectrum in the energy range of 250 GeV-6 TeV is compatible with a power-law function with a photon index $Γ=2.52\pm0.13_{\mathrm{stat}}\pm0.20_{\mathrm{sys}}$. An updated Fermi-LAT analysis provides evidence for spectral hardening by $ΔΓ\simeq0.4\pm0.1$ at $γ$-ray energies above $2.8^{+1.0}_{-0.6}$ GeV at a level of $4.0σ$. The fact that the spectrum hardens at GeV energies and extends into the VHE regime disfavour a single-zone SSC interpretation for the overall spectral energy distribution (SED) of the core and is suggestive of a new $γ$-ray emitting component connecting the high-energy emission above the break energy to the one observed at VHE energies.

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The starburst galaxy NGC 253 revisited by H.E.S.S. and Fermi-LAT

(Abridged) Context. NGC 253 is one of only two starburst galaxies found to emit $γ$-rays from hundreds of MeV to multi-TeV energies. Accurate measurements of the very-high-energy (VHE) (E $>$ 100 GeV) and high-energy (HE) (E $>$ 60 MeV) spectra are crucial to study the underlying particle accelerators and cosmic-ray interaction and transport. Aims. The measurement of the VHE $γ$-ray emission of NGC 253 published in 2012 by H.E.S.S. was limited by large systematic uncertainties. Here, a measurement of the $γ$-ray spectrum of NGC 253 is investigated in both HE and VHE $γ$-rays. Methods. The data of H.E.S.S. observations are reanalysed using an updated calibration and analysis chain. The $Fermi$-LAT analysis employs more than 8 years of data processed using pass 8. The cosmic-ray particle population is evaluated from the combined HE--VHE $γ$-ray spectrum using NAIMA. Results. The VHE $γ$-ray energy spectrum is best fit by a power-law with a flux normalisation of $(1.34\,\pm\,0.14^{\mathrm{stat}}\,\pm\,0.27^{\mathrm{sys}}) \times 10^{-13} \mathrm{cm^{-2} s^{-1} TeV^{-1}}$ at 1 TeV -- about 40 \% above, but compatible with the value obtained in Abramowski et al. (2012). The spectral index $Γ= 2.39 \pm 0.14^{\mathrm{stat}} \pm 0.25^{\mathrm{sys}}$ is slightly softer than but consistent with the previous measurement. At energies above $\sim$3 GeV the HE spectrum is consistent with a power-law ranging into the VHE part of the spectrum measured by H.E.S.S. Conclusions. Two scenarios for the starburst nucleus are tested, in which the gas in the starburst nucleus acts as a target for hadronic cosmic rays. In these two models, the level to which NGC\,253 acts as a calorimeter is estimated to a range of $f_{\rm cal} = 0.1$ to $1$ while accounting for the measurement uncertainties.

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