arXiv subjects

S. Mathur

Publications and source records attributed to S. Mathur.

At least 91 records · Page 5Linked to original sources

arXiv2019

Oscillations in the Sun with SONG: Setting the scale for asteroseismic investigations

Context. We present the first high-cadence multi-wavelength radial-velocity observations of the Sun-as-a-star, carried out during 57 consecutive days using the stellar échelle spectrograph at the Hertzsprung SONG Telescope operating at the Teide Observatory. Aims. The aim was to produce a high-quality data set and reference values for the global helioseismic parameters {ν_{max}}, and {Δν} of the solar p-modes using the SONG instrument. The obtained data set or the inferred values should then be used when the scaling relations are applied to other stars showing solar-like oscillations which are observed with SONG or similar instruments. Methods. We used different approaches to analyse the power spectrum of the time series to determine {ν_{max}}; simple Gaussian fitting and heavy smoothing of the power spectrum. {Δν} was determined using the method of autocorrelation of the power spectrum. The amplitude per radial mode was determined using the method described in Kjeldsen et al. (2008). Results. We found the following values for the solar oscillations using the SONG spectrograph: {ν_{max}} = 3141 {\pm} 12 μHz, {Δν} = 134.98 {\pm} 0.04 μHz and an average amplitude of the strongest radial modes of 16.6 {\pm} 0.4 cm/s. These values are consistent with previous measurements with other techniques.

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arXiv2019

HD219666b: A hot-Neptune from TESS Sector 1

We report on the confirmation and mass determination of a transiting planet orbiting the old and inactive G7 dwarf star HD219666 (Mstar = 0.92 +/- 0.03 MSun, Rstar = 1.03 +/- 0.03 RSun, tau_star = 10 +/- 2 Gyr). With a mass of Mb = 16.6 +/- 1.3 MEarth, a radius of Rb = 4.71 +/- 0.17 REarth, and an orbital period of P ~ 6 days, HD219666b is a new member of a rare class of exoplanets: the hot-Neptunes. The Transiting Exoplanet Survey Satellite (TESS) observed HD219666 (also known as TOI-118) in its Sector 1 and the light curve shows four transit-like events, equally spaced in time. We confirmed the planetary nature of the candidate by gathering precise radial-velocity measurements with HARPS@ESO3.6m. We used the co-added HARPS spectrum to derive the host star fundamental parameters (Teff = 5527 +/- 65 K, log g = 4.40 +/- 0.11 (cgs), [Fe/H]= 0.04 +/- 0.04 dex, log R'HK = -5.07 +/- 0.03), as well as the abundances of many volatile and refractory elements. The host star brightness (V = 9.9) makes it suitable for further characterisation by means of in-transit spectroscopy. The determination of the planet orbital obliquity, along with the atmospheric metal-to-hydrogen content and thermal structure could provide us with important clues on the formation mechanisms of this class of objects.

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arXiv2019

Evidence for an emerging disc wind and collimated outflow during an X-ray flare in the narrow-line Seyfert 1 galaxy Mrk 335

A triggered 140 ks XMM-Newton observation of the narrow-line Seyfert 1 (NLS1) Mrk 335 in December 2015 caught the active galaxy at its lowest X-ray flux since 2007. The NLS1 is relatively quiescent for the first ~120 ks of the observation before it flares in brightness by a factor of about five in the last 20 ks. Although only part of the flare is captured before the observation is terminated, the data reveal significant differences between the flare and quiescent phases. During the low-flux state, Mrk 335 demonstrates a reflection-dominated spectrum that results from a compact corona around a Kerr black hole. In addition to the rapid brightening, the flare is further described by spectral softening and a falling reflection fraction that are consistent with previous observations advocating at least part of the corona in Mrk 335 could be the base of an aborted jet. The spectrum during the flaring interval reveals several residuals between the 2-3 sigma level that could be attributed to absorption lines from a highly ionised plasma that is moving outward at v~0.12c. It could be that the increased luminosity during the flare enhances the radiation pressure sufficiently to launch a possible wind. If the wind is indeed responding to the change in corona luminosity then it must be located within ~80 Rg. The escape velocity at this distance is comparable to the estimated wind velocity. If confirmed, this is the first example of a radio-quiet AGN exhibiting behaviour consistent with both diffuse and collimated outflow.

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arXiv2019

The First Swift Intensive AGN Accretion Disk Reverberation Mapping Survey

Swift intensive accretion disk reverberation mapping of four AGN yielded light curves sampled $\sim$200-350 times in 0.3-10 keV X-ray and six UV/optical bands. Uniform reduction and cross-correlation analysis of these datasets yields three main results: 1) The X-ray/UV correlations are much weaker than those within the UV/optical, posing severe problems for the lamp-post reprocessing model in which variations in a central X-ray corona drive and power those in the surrounding accretion disk. 2) The UV/optical interband lags are generally consistent with $ τ\propto λ^{4/3} $ as predicted by the centrally illuminated thin accretion disk model. While the average interband lags are somewhat larger than predicted, these results alone are not inconsistent with the thin disk model given the large systematic uncertainties involved. 3) The one exception is the U band lags, which are on average a factor of $\sim$2.2 larger than predicted from the surrounding band data and fits. This excess appears due to diffuse continuum emission from the broad-line region (BLR). The precise mixing of disk and BLR components cannot be determined from these data alone. The lags in different AGN appear to scale with mass or luminosity. We also find that there are systematic differences between the uncertainties derived by javelin vs. more standard lag measurement techniques, with javelin reporting smaller uncertainties by a factor of 2.5 on average. In order to be conservative only standard techniques were used in the analyses reported herein.

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arXiv2018

Detection and characterization of an ultra-dense sub-Neptune planet orbiting the Sun-like star HD 119130

We present the discovery and characterization of a new transiting planet from Campaign 17 of the Kepler extended mission K2. HD 119130 b is a warm sub-Neptune on a 17-d orbit around a bright (V = 9.9 mag) solar-like G3 V star with a mass and radius of $M_\star = 1.00\pm0.03\,\mathrm{M_\odot}$ and $R_\star = 1.09\pm0.03\,\mathrm{R_\odot}$, respectively. We model simultaneously the K2 photometry and CARMENES spectroscopic data and derive a radius of $R_\mathrm{p} = 2.63_{-0.10}^{+0.12}\,\mathrm{R_\oplus}$ and mass of $M_\mathrm{p} = 24.5_{-4.4}^{+4.4}\,\mathrm{M_\oplus}$, yielding a mean density of $ρ_\mathrm{p} = 7.4_{-1.5}^{+1.6}\,\mathrm{g\,cm^{-3}}$, which makes it one of the densest sub-Neptune planets known to date. We also detect a linear trend in radial velocities of HD 119130 ($\dotγ_{\rm RV}= -0.40^{+0.07}_{-0.07}\,\mathrm{m\,s^{-1}\,d^{-1}}$) that suggests a long-period companion with a minimum mass on the order of $33\,\mathrm{M_\oplus}$. If confirmed, it would support a formation scenario of HD 119130 b by migration caused by Kozai-Lidov oscillations.

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arXiv2018

The interacting late-type host galaxy of the radio-loud narrow-line Seyfert 1 IRAS 20181-2244

Narrow-line Seyfert 1 galaxies (NLS1s) are a class of active galactic nuclei (AGN) which are known to be one of the few sources of $γ$ rays, which originate in a relativistic beamed jet. Becuase of their relatively large distance, a poorly investigated aspect of these jetted NLS1s is their environment, and in particular their host galaxy. In this work we present the results of a morphological analysis of the host galaxy of the jetted NLS1 IRAS 20181-2244 observed with the 6.5m Baade Telescope of the Las Campanas Observatory. The GALFIT analysis ran on the Ks image, along with additional spectroscopic observations performed with the Nordic Optical Telescope, clearly revealed the presence of an interacting system of two galaxies. The data suggest that this NLS1 is hosted by a late-type galaxy, although the result is not conclusive. This analysis, along with other results in the literature, might suggest that two populations of jetted NLS1 exist. Further morphological studies are needed to confirm or disprove this hypothesis.

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arXiv2018

FliPer: Classifying TESS pulsating stars

The recently launched NASA Transiting Exoplanet Survey Satellite (TESS) mission is going to collect lightcurves for a few hundred million of stars and we expect to increase the number of pulsating stars to analyze compared to the few thousand stars observed by the CoRoT, $\textit{Kepler}$ and K2 missions. However, most of the TESS targets have not yet been properly classified and characterized. In order to improve the analysis of the TESS data, it is crucial to determine the type of stellar pulsations in a timely manner. We propose an automatic method to classify stars attending to their pulsation properties, in particular, to identify solar-like pulsators among all TESS targets. It relies on the use of the global amount of power contained in the power spectrum (already known as the FliPer method) as a key parameter, along with the effective temperature, to feed into a machine learning classifier. Our study, based on TESS simulated datasets, shows that we are able to classify pulsators with a $98\%$ accuracy.

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arXiv2018

TESS's first planet: a super-Earth transiting the naked-eye star $π$ Mensae

We report on the confirmation and mass determination of Pi Men c, the first transiting planet discovered by NASA's TESS space mission. Pi Men is a naked-eye (V=5.65 mag), quiet G0 V star that was previously known to host a sub-stellar companion (Pi Men b) on a long-period (Porb = 2091 days), eccentric (e = 0.64) orbit. Using TESS time-series photometry, combined with Gaia data, published UCLES@AAT Doppler measurements, and archival HARPS@ESO-3.6m radial velocities, we found that Pi Men c is a close-in planet with an orbital period of Porb = 6.27 days, a mass of Mc = 4.52 +/- 0.81 MEarth, and a radius of Rc = 2.06 +/- 0.03 REarth. Based on the planet's orbital period and size, Pi Men c is a super-Earth located at, or close to, the radius gap, while its mass and bulk density suggest it may have held on to a significant atmosphere. Because of the brightness of the host star, this system is highly suitable for a wide range of further studies to characterize the planetary atmosphere and dynamical properties. We also performed an asteroseismic analysis of the TESS data and detected a hint of power excess consistent with the seismic values expected for this star, although this result depends on the photometric aperture used to extract the light curve. This marginal detection is expected from pre-launch simulations hinting at the asteroseismic potential of the TESS mission for longer, multi-sector observations and/or for more evolved bright stars.

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arXiv2018

The Changing-Look Quasar Mrk 590 is Awakening

Mrk 590 was originally classified as a Seyfert 1 galaxy, but then it underwent dramatic changes: the nuclear luminosity dropped by over two orders of magnitude and the broad emission lines all but disappeared from the optical spectrum. Here we present followup observations to the original discovery and characterization of this "changing look" active galactic nucleus (AGN). The new Chandra and HST observations from 2014 show that Mrk 590 is awakening, changing its appearance again. While the source continues to be in a low state, its soft excess has re-emerged, though not to the previous level. The UV continuum is brighter by more than a factor of two and the broad MgII emission line is present, indicating that the ionizing continuum is also brightening. These observations suggest that the soft excess is not due to reprocessed hard X-ray emission. Instead, it is connected to the UV continuum through warm Comptonization. Variability of the Fe K-alpha emission lines suggests that the reprocessing region is within about 10 light years or 3 pc of the central source. The AGN type change is neither due to obscuration, nor due to one-way evolution from type-1 to type-2, as suggested in literature, but may be related to episodic accretion events.

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arXiv2018

FliPer: A global measure of power density to estimate surface gravities of main-sequence Solar-like stars and red giants

Asteroseismology provides global stellar parameters such as masses, radii or surface gravities using the mean global seismic parameters as well as the effective temperature for thousands of low-mass stars $(0.8 M_\odot <M<3 M_\odot)$. This methodology has been successfully applied to stars in which acoustic modes excited by turbulent convection are measured. Other techniques such as the Flicker can also be used to determine stellar surface gravities, but only works for $\log{g}$ above $2.5$ dex. In this work, we present a new metric called FliPer (the acronym stands for Flicker in spectral power density, in opposition to the standard Flicker measurement which is computed in the time domain that is able to extend the range for which reliable surface gravities can be obtained ($0.1<\log{g}<4.6$ dex) without performing any seismic analysis for stars brighter than $\textit{Kp}$ $<$ 14. FliPer takes into account the average variability of a star measured in the power density spectrum in a given range of frequencies. However, FliPer values calculated on several ranges of frequency are required to better characterize a star. Using a large set of asteroseismic targets it is possible to calibrate the behavior of surface gravity with FliPer through machine learning. This calibration made with a random forest regressor covers a wide range of surface gravities from main-sequence stars to subgiants and red giants, with very small uncertainties from $0.04$ to $0.1$ dex. FliPer values can be inserted in automatic global seismic pipelines to either give an estimation of the stellar surface gravity or to assess the quality of the seismic results by detecting any outliers in the obtained $ν_{max}$ values. FliPer also constrain the surface gravities of main-sequence dwarfs using only long cadence data for which the Nyquist frequency is too low to measure the acoustic-mode properties.

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arXiv2018

The hyperluminous Compton-thick $z\sim2$ quasar nucleus of the hot DOG W1835+4355 observed by NuSTAR

We present a 155ks NuSTAR observation of the $z\sim2$ hot dust-obscured galaxy (hot DOG) W1835+4355. We extracted spectra from the two NuSTAR detectors and analyzed them jointly with the archival XMM PN and MOS spectra. We performed a spectroscopic analysis based on both phenomenological and physically motivated models employing toroidal and spherical geometry for the obscurer. In all the modelings, the source exhibits a Compton-thick column density $N_{\rm H} \gtrsim 10^{24}$ cm$^{-2}$, a 2-10 keV luminosity $L_{2-10}\approx2\times10^{45}$ erg s$^{-1}$ , and a prominent soft excess ($\sim5-10$ % of the primary radiative output), which translates into a luminosity $\sim10^{44}$ erg s$^{-1}$. We modeled the spectral energy distribution from 1.6 to 850 $μm$ using a clumpy two-phase dusty torus model plus a modified blackbody to account for emission powered by star formation in the far-infrared. We employed several geometrical configurations consistent with those applied in the X-ray analysis. In all cases we obtained a bolometric luminosity $L_{\rm bol}\approx3-5\times10^{47}$ erg s$^{-1}$, which confirms the hyperluminous nature of this active galactic nucleus. Finally, we estimate a prodigious star formation rate of $\sim$3000 $M_{\odot}\,yr^{-1}$, which is consistent with the rates inferred for $z\approx2-4$ hyperluminous type I quasars. The heavily obscured nature, together with $L_{\rm bol}$, the ratio of X-ray to mid-infrared luminosity, the rest-frame optical morphology, and the host star formation rate are indicative of its evolutionary stage. We can interpret this as a late-stage merger event in the transitional, dust-enshrouded, evolutionary phase eventually leading to an optically bright AGN.

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arXiv2018

Observations of the MIssing Baryons in the warm-hot intergalactic medium

It has been known for decades that the observed number of baryons in the local universe falls about 30-40% short of the total number of baryons predicted by Big-Bang Nucleosynthesis, as inferred from density fluctuations of the Cosmic Microwave Background and seen during the first 2-3 billion years of the universe in the so called Lyman-alpha Forest. A theoretical solution to this paradox locates the missing baryons in the hot and tenuous filamentary gas between galaxies, known as the warm-hot intergalactic medium. However, it is difficult to detect them there because the largest by far constituent of this gas - hydrogen - is mostly ionized and therefore almost invisible in far-ultraviolet spectra with typical signal-to-noise ratios. Indeed, despite the large observational efforts, only a few marginal claims of detection have been made so far. Here we report observations of two absorbers of highly ionized oxygen (OVII) in the high signal-to-noise-ratio X-ray spectrum of a quasar at redshift >0.4. These absorbers show no variability over a 2-year timescale and have no associated cold absorption, making the assumption that they originate from the quasar's intrinsic outflow or the host galaxy's interstellar medium implausible. The OVII systems lie in regions characterized by large (x4 compared to average) galaxy over-densities and their number (down to the sensitivity threshold of our data), agrees well with numerical simulation predictions for the long-sought warm-hot intergalactic medium (WHIM). We conclude that the missing baryons have been found.

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arXiv2018

The Gaia-ESO Survey: properties of newly discovered Li-rich giants

We report 20 new lithium-rich giants discovered within the Gaia-ESO Survey, including the first Li-rich giant with evolutionary stage confirmed by CoRoT data. Atmospheric parameters and abundances were derived in model atmosphere analyses using medium-resolution GIRAFFE or high-resolution UVES spectra. These results are part of the fifth internal data release of Gaia-ESO. The Li abundances were corrected for non-LTE effects. We used Gaia DR2 parallaxes to estimate distances and luminosities. The giants have A(Li) > 2.2 dex. The majority of them (14 out of 20 stars) are in the CoRoT fields. Four giants are located in the field of three open clusters but are not members. Two giants were observed in fields towards the Galactic bulge but are likely in the inner disk. One of the bulge field giants is super Li-rich with A(Li) = 4.0 dex. We identified one giant with infrared excess at 22 microns. Two other giants, with large vsin i, might be Li-rich because of planet engulfment. Another giant is found to be barium enhanced and thus could have accreted material from a former AGB companion. Otherwise, besides the Li enrichment, the evolutionary stages are the only other connection between these new Li-rich giants. The CoRoT data confirm that one Li-rich giant is at the core-He burning stage. The other giants are concentrated in close proximity to the RGB luminosity bump, the core-He burning stages, or the early-AGB. This is very clear when looking at the Gaia-based luminosities of the Li-rich giants. This is also seen when the CoRoT Li-rich giants are compared to a larger sample of 2252 giants observed in the CoRoT fields by the Gaia-ESO Survey, which are distributed all over the RGB in the Teff-logg diagram. These observations show that evolutionary stage is a major factor behind the Li enrichment in giants. Other processes, like planet accretion, contribute to a smaller scale. [abridged]

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arXiv2018

The Kepler Follow-Up Observation Program. II. Stellar Parameters from Medium- and High-Resolution Spectroscopy

We present results from spectroscopic follow-up observations of stars identified in the Kepler field and carried out by teams of the Kepler Follow-Up Observation Program. Two samples of stars were observed over six years (2009-2015): 614 standard stars (divided into "platinum" and "gold" categories) selected based on their asteroseismic detections and 2667 host stars of Kepler Objects of Interest (KOIs), most of them planet candidates. Four data analysis pipelines were used to derive stellar parameters for the observed stars. We compare the $T_{\mathrm{eff}}$, $\log$(g), and [Fe/H] values derived for the same stars by different pipelines; from the average of the standard deviations of the differences in these parameter values, we derive error floors of $\sim$ 100 K, 0.2 dex, and 0.1 dex for $T_{\mathrm{eff}}$, $\log$(g), and [Fe/H], respectively. Noticeable disagreements are seen mostly at the largest and smallest parameter values (e.g., in the giant star regime). Most of the $\log$(g) values derived from spectra for the platinum stars agree on average within 0.025 dex (but with a spread of 0.1-0.2 dex) with the asteroseismic $\log$(g) values. Compared to the Kepler Input Catalog (KIC), the spectroscopically derived stellar parameters agree within the uncertainties of the KIC, but are more precise and are thus an important contribution towards deriving more reliable planetary radii.

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arXiv2018

HD 89345: a bright oscillating star hosting a transiting warm Saturn-sized planet observed by K2

We report the discovery and characterization of HD 89345b (K2-234b; EPIC 248777106b), a Saturn-sized planet orbiting a slightly evolved star. HD 89345 is a bright star ($V = 9.3$ mag) observed by the K2 mission with one-minute time sampling. It exhibits solar-like oscillations. We conducted asteroseismology to determine the parameters of the star, finding the mass and radius to be $1.12^{+0.04}_{-0.01}~M_\odot$ and $1.657^{+0.020}_{-0.004}~R_\odot$, respectively. The star appears to have recently left the main sequence, based on the inferred age, $9.4^{+0.4}_{-1.3}~\mathrm{Gyr}$, and the non-detection of mixed modes. The star hosts a "warm Saturn" ($P = 11.8$~days, $R_p = 6.86 \pm 0.14~R_\oplus$). Radial-velocity follow-up observations performed with the FIES, HARPS, and HARPS-N spectrographs show that the planet has a mass of $35.7 \pm 3.3~M_\oplus$. The data also show that the planet's orbit is eccentric ($e\approx 0.2$). An investigation of the rotational splitting of the oscillation frequencies of the star yields no conclusive evidence on the stellar inclination angle. We further obtained Rossiter-McLaughlin observations, which result in a broad posterior of the stellar obliquity. The planet seems to conform to the same patterns that have been observed for other sub-Saturns regarding planet mass and multiplicity, orbital eccentricity, and stellar metallicity.

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arXiv2018

Probing black hole accretion in quasar pairs at high redshift

Models and observations suggest that luminous quasar activity is triggered by mergers, so it should preferentially occur in the most massive primordial dark matter haloes, where the frequency of mergers is expected to be the highest. Since the importance of galaxy mergers increases with redshift, we identify the high-redshift Universe as the ideal laboratory for studying dual AGN. Here we present the X-ray properties of two systems of dual quasars at z=3.0-3.3 selected from the SDSS-DR6 at separations of 6-8 arcsec (43-65kpc) and observed by Chandra for 65ks each. Both members of each pair are detected with good photon statistics to allow us to constrain the column density, spectral slope and intrinsic X-ray luminosity. We also include a recently discovered dual quasar at z=5 (separation of 21 arcsec, 136kpc) for which XMM-Newton archival data allow us to detect the two components separately. Using optical spectra we derived bolometric luminosities, BH masses and Eddington ratios that were compared to those of luminous SDSS quasars in the same redshift ranges. We find that the brighter component of both pairs at z=3.0-3.3 has high luminosities compared to the distribution of SDSS quasars at similar redshift, with J1622A having an order magnitude higher luminosity than the median. This source lies at the luminous end of the z~3.3 quasar luminosity function. While we cannot conclusively state that the unusually high luminosities of our sources are related to their having a close companion, for J1622A there is only a 3% probability that it is by chance.

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arXiv2017

FliPer: Checking the reliability of global seismic parameters from automatic pipelines

Our understanding of stars through asteroseismic data analysis is limited by our ability to take advantage of the huge amount of observed stars provided by space missions such as CoRoT, Kepler, K2, and soon TESS and PLATO. Global seismic pipelines provide global stellar parameters such as mass and radius using the mean seismic parameters, as well as the effective temperature. These pipelines are commonly used automatically on thousands of stars observed by K2 for 3 months (and soon TESS for at least around 1 month). However, pipelines are not immune from misidentifying noise peaks and stellar oscillations. Therefore, new validation techniques are required to assess the quality of these results. We present a new metric called FliPer (Flicker in Power), which takes into account the average variability at all measured time scales. The proper calibration of FliPer enables us to obtain good estimations of global stellar parameters such as surface gravity that are robust against the influence of noise peaks and hence are an excellent way to find faults in asteroseismic pipelines.

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arXiv2017

The WISSH Quasars Project III. X-ray properties of hyper-luminous quasars

We perform a survey of the X-ray properties of 41 objects from the WISE/SDSS selected Hyper-luminous (WISSH) quasars sample, composed by 86 broad-line quasars (QSOs) with bolometric luminosity $L_{Bol}\geq 2\times 10^{47}\,erg\, s^{-1}$, at z~2-4. All but 3 QSOs show unabsorbed 2-10 keV luminosities $L_{2-10}\geq10^{45} \,erg \,s^{-1}$. Thanks to their extreme radiative output across the Mid-IR-to-X-ray range, WISSH QSOs offer the opportunity to significantly extend and validate the existing relations involving $L_{2-10}$. We study $L_{2-10}$ as a function of (i) X-ray-to-Optical (X/O) flux ratio, (ii) mid-IR luminosity ($L_{MIR}$), (iii) $L_{Bol}$ as well as (iv) $α_{OX}$ vs. the 2500$\mathring{A}$ luminosity. We find that WISSH QSOs show very low X/O(<0.1) compared to typical AGN values; $L_{2-10}/L_{MIR}$ ratios significantly smaller than those derived for AGN with lower luminosity; large X-ray bolometric corrections $k_{\rm Bol,X}\sim$ 100-1000; and steep $-2<α_{OX}<-1.7$. These results lead to a scenario where the X-ray emission of hyper-luminous quasars is relatively weaker compared to lower-luminosity AGN. Models predict that such an X-ray weakness can be relevant for the acceleration of powerful high-ionization emission line-driven winds, commonly detected in the UV spectra of WISSH QSOs, which can in turn perturb the X-ray corona and weaken its emission. Accordingly, hyper-luminous QSOs represent the ideal laboratory to study the link between the AGN energy output and wind acceleration. Additionally, WISSH QSOs show very large BH masses ($\log[M_{\rm BH}/M_{\odot}]$>9.5). This enables a more robust modeling of the $Γ-M_{BH}$ relation by increasing the statistics at high masses. We derive a flatter $Γ$ dependence than previously found over the broad range 5 <$\log(M_{\rm BH}/M_{\odot})$ < 11.

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