Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Power Law”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 721 records · Page 40
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Herschel Photometric Observations of Little Things Dwarf Galaxies

We present here far-infrared photometry of galaxies in a sample that is relatively unexplored at these wavelengths: low-metallicity dwarf galaxies with moderate star formation rates. Four dwarf irregular galaxies from the Little Things survey are considered, with deep Herschel PACS and SPIRE observations at 100, 160, 250, 350, and 500 μm. Results from modified blackbody fits indicate that these galaxies have low dust masses and cooler dust temperatures than more actively star-forming dwarfs, occupying the lowest L {sub TIR} and M {sub dust} regimes seen among these samples. Dust-to-gas mass ratios of ∼10{sup −5} are lower, overall, than in more massive and active galaxies but are roughly consistent with the broken power-law relation between the dust-to-gas ratio and metallicity found for other low-metallicity systems. Chemical evolution modeling suggests that these dwarf galaxies are likely forming very little dust via stars or grain growth and have very high dust destruction rates.

47 OTHER INSTRUMENTATION↗
DOE OSTI2021

OGLE-2017-BLG-1434Lb: Confirmation of a Cold Super-Earth Using Keck Adaptive Optics

The microlensing event OGLE-2017-BLG-1434 features a cold super-Earth planet that is 1 of 11 microlensing planets with a planet–host-star mass ratio of q < 1 × 10{sup -4}. We provide an additional mass–distance constraint on the lens host using near-infrared adaptive optics photometry from Keck/NIRC2. We are able to determine a flux excess of K {sub L} = 16.96 ± 0.11, which most likely comes entirely from the lens star. Combining this with constraints from the large Einstein ring radius, θ {sub E} = 1.40 ± 0.09 mas, and OGLE parallax we confirm this event as a super-Earth with a mass of m {sub p} = 4.43 ± 0.25M {sub ⊕}. This system lies at a distance of D {sub L} = 0.86 ± 0.05 kpc from Earth and the lens star has a mass of M {sub L} = 0.234 ± 0.012M {sub ⊙}. We confirm that with a star–planet mass ratio of q = 0.57 × 10{sup -4}, OGLE-2017-BLG-1434 lies near the inflexion point of the planet–host mass-ratio power law.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Lens Modeling of STRIDES Strongly Lensed Quasars Using Neural Posterior Estimation

Strongly lensed quasars can be used to constrain cosmological parameters through time-delay cosmography. Models of the lens masses are a necessary component of this analysis. To enable time-delay cosmography from a sample of $\mathcal{O}(10^3)$ lenses, which will soon become available from surveys like the Rubin Observatory’s Legacy Survey of Space and Time and the Euclid Wide Survey, we require fast and standardizable modeling techniques. To address this need, we apply neural posterior estimation (NPE) for modeling galaxy-scale strongly lensed quasars from the Strong Lensing Insights into the Dark Energy Survey (STRIDES) sample. NPE brings two advantages: speed and the ability to implicitly marginalize over nuisance parameters. We extend this method by employing sequential NPE to increase precision of mass model posteriors. We then fold individual lens models into a hierarchical Bayesian inference to recover the population distribution of lens mass parameters, accounting for out-of-distribution shift. After verifying our method using simulated analogs of the STRIDES lens sample, we apply our method to 14 Hubble Space Telescope single-filter observations. We find the population mean of the power-law elliptical mass distribution slope, γ lens , to be $\mathcal{M}_γ$ lens = 2.13 ± 0.06. Our result represents the first population-level constraint for these systems. This population-level inference from fully automated modeling is an important stepping stone toward cosmological inference with large samples of strongly lensed quasars.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Search for PeV Gamma-Ray Emission from the Southern Hemisphere with 5 Yr of Data from the IceCube Observatory

The measurement of diffuse PeV gamma-ray emission from the Galactic plane would provide information about the energy spectrum and propagation of Galactic cosmic rays, and the detection of a pointlike source of PeV gamma-rays would be strong evidence for a Galactic source capable of accelerating cosmic rays up to at least a few PeV. This paper presents several unbinned maximum-likelihood searches for PeV gamma-rays in the Southern Hemisphere using 5 yr of data from the IceTop air shower surface detector and the in-ice array of the IceCube Observatory. The combination of both detectors takes advantage of the low muon content and deep shower maximum of gamma-ray air showers and provides excellent sensitivity to gamma-rays between ~0.6 and 100 PeV. Our measurements of pointlike and diffuse Galactic emission of PeV gamma-rays are consistent with the background, so we constrain the angle-integrated diffuse gamma-ray flux from the Galactic plane at 2 PeV to 2.61 × 10 -19 cm -2 s -1 TeV -1 at 90% confidence, assuming an E -3 spectrum, and we estimate 90% upper limits on pointlike emission at 2 PeV between 10 -21 and 10 -20 cm -2 s -1 TeV -1 for an E -2 spectrum, depending on decl. Furthermore, we exclude unbroken power-law emission up to 2 PeV for several TeV gamma-ray sources observed by the High Energy Spectroscopic System and calculate upper limits on the energy cutoffs of these sources at 90% confidence. We also find no PeV gamma-rays correlated with neutrinos from IceCube's high-energy starting event sample. Lastly, these are currently the strongest constraints on PeV gamma-ray emission.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Measurements of Effective Optical Depth in the Lyα Forest from the BOSS DR12 Quasar Sample

We measure the effective optical depth in the Lyα forest using 40,035 quasar spectra from the Twelfth Data Release (DR12) of the Baryon Oscillation Spectroscopic Survey of Sloan Digital Sky Survey IV. A rigorous selection based on spectral index and the equivalent width of the C iv emission line is applied to choose seven uniform samples with minimal intrinsic variations across redshifts. Modeling the redshift evolution of the effective optical depth with a power law, ${\tau }_{\mathrm{eff}}={\tau }_{0}{(1+z)}^{\gamma },$ produces ${\tau }_{0}=(5.54\pm 0.64)\times {10}^{-3}$ and $\gamma =3.182\pm 0.074$. The 2.2% precision estimate on γ is dominated by systematic errors, likely arising from the bias and uncertainties in spectral index estimates. Even after incorporating the systematic errors, this work provides the most precise estimates of optical depth parameters to date. Finally, using the reconstructed Lyα forest continuum to directly measure the transmitted flux ratio as a function of redshift, we find deviations of less than 2.5% from the predictions from the global model and no convincing evidence for signal associated with He ii reionization.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

A Search for Neutrino Point-source Populations in 7 yr of IceCube Data with Neutrino-count Statistics

The presence of a population of point sources in a data set modifies the underlying neutrino-count statistics from the Poisson distribution. This deviation can be exactly quantified using the non-Poissonian template fitting technique, and in this work we present the first application of this approach to the IceCube high-energy neutrino data set. Using this method, we search in 7 yr of IceCube data for point-source populations correlated with the disk of the Milky Way, the Fermi bubbles, the Schlegel, Finkbeiner, and Davis dust map, or with the isotropic extragalactic sky. No evidence for such a population is found in the data using this technique, and in the absence of a signal, we establish constraints on population models with source-count distribution functions that can be described by a power law with a single break. Here, the derived limits can be interpreted in the context of many possible source classes. In order to enhance the flexibility of the results, we publish the full posterior from our analysis, which can be used to establish limits on specific population models that would contribute to the observed IceCube neutrino flux.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

X-Ray Afterglows from the Gamma-Ray Burst “Large-angle” Emission

In this work, we derive basic analytical results for the timing and decay of the gamma-ray burst (GRB) counterpart and delayed afterglow light curves for a brief emission episode from a relativistic surface endowed with angular structure, consisting of a uniform core of size ${\theta }_{c}$ (Lorentz factor ${{\rm{\Gamma }}}_{c}$ and surface emissivity ${i}_{\nu ^{\prime} }^{{\prime} }$ are angle independent) and an axially symmetric power-law envelope (${\rm{\Gamma }}\sim {\theta }^{-g}$). In this "large-angle emission" model, radiation produced during the prompt emission phase (GRB) at angles $\theta \gt {\theta }_{c}$ arrives at the observer well after the burst (delayed emission). The dynamical time range of the very fast decaying GRB "tail" and of the flat afterglow "plateau" and the morphology of the GRB counterpart/afterglow are all determined by two parameters: the core's parameter ${{\rm{\Gamma }}}_{c}{\theta }_{c}$ and the envelope's Lorentz factor index g, leading to three types of light curves that display three post-GRB phases (type 1: tail, plateau/slow decay, post-plateau/normal decay), two post-GRB phases (type 2: tail and fast decay), or just one (type 3: normal decay). We show how X-ray light-curve features can be used to determine core and envelope dynamical and spectral parameters. Testing of the large-angle emission model is done using the Swift/XRT X-ray emission of two afterglows of type 1 (GRB 060607A, GRB 061121), one of type 2 (GRB 061110A), and one of type 3 (GRB 061007). We find that the X-ray afterglows with plateaus require an envelope Lorentz factor ${\rm{\Gamma }}\sim {\theta }^{-2}$ and a comoving-frame emissivity ${i}_{\nu ^{\prime} }^{{\prime} }\sim {\theta }^{2}$; thus, for a typical afterglow spectrum ${F}_{\nu }\sim {\nu }^{-1}$, the lab-frame energy release is uniform over the emitting surface.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

The Rise and Fall of ASASSN-18pg: Following a TDE from Early to Late Times

Here, we present nearly 500 days of observations of the tidal disruption event (TDE) ASASSN-18pg, spanning from 54 days before peak light to 441 days after peak light. Our data set includes X-ray, UV, and optical photometry, optical spectroscopy, radio observations, and the first published spectropolarimetric observations of a TDE. ASASSN-18pg was discovered on 2018 July 11 by the All-Sky Automated Survey for Supernovae (ASAS-SN) at a distance of $\textit{d}$ = 78.6 Mpc; with a peak UV magnitude of $\textit{m}$ ≃ 14, it is both one of the nearest and brightest TDEs discovered to-date. The photometric data allow us to track both the rise to peak and the long-term evolution of the TDE. ASASSN-18pg peaked at a luminosity of $\textit{L}$ ≃ 2.4 × 10 44 erg s -1 , and its late-time evolution is shallower than a flux ∝$\textit{t}^{–5/3}$ power-law model, similar to what has been seen in other TDEs. ASASSN-18pg exhibited Balmer lines and spectroscopic features consistent with Bowen fluorescence prior to peak, which remained detectable for roughly 225 days after peak. Analysis of the two-component Hα profile indicates that, if they are the result of reprocessing of emission from the accretion disk, the different spectroscopic lines may be coming from regions between ~10 and ~60 lt-days from the black hole. No X-ray emission is detected from the TDE, and there is no evidence of a jet or strong outflow detected in the radio. Our spectropolarimetric observations indicate that the projected emission region is likely not significantly aspherical, with the projected emission region having an axis ratio of ≳0.65.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Solar Wind Turbulence from 1 to 45 au. I. Evidence for Dissipation of Magnetic Fluctuations Using Voyager and ACE Observations

As part of a published effort to study low-frequency magnetic waves excited by newborn interstellar pickup ions seen by the Voyager spacecraft, we developed a set of control intervals that represent the background turbulence when the observations are not dominated by wave excitation. This paper begins an effort to better understand solar wind turbulence from 1 to 45 au while spanning greater than one solar cycle. Here, we first focus on the diagnostics marking the onset of dissipation. This includes an expected break in the power spectrum at frequencies greater than the proton cyclotron frequency and a resultant steepening of the spectrum at higher frequencies. Contrary to what is established at 1 au, we only see the spectral break in rare instances. The expected scaling of the spectral index with the turbulence rate is seen, but it is not as clearly established as it was at 1 au. We also find that both Voyager data from 1 to 45 au and Advanced Composition Explorer data from 1 au show significant bias of the magnetic helicity at dissipation scales when the dissipation-range power-law spectral index steepens. We conclude that dissipation dynamics are similar throughout the heliosphere in so far as we have examined to date.

Pine, Zackary B.↗
DOE OSTI2020

Inverse Compton Cooling in the Coronae of Simulated Black Hole Accretion Flows

Here we present a formulation for a local cooling function to be employed in the diffuse, hot corona region of 3D GRMHD simulations of accreting black holes. This new cooling function calculates the cooling rate due to inverse Compton scattering by considering the relevant microphysics in each cell in the corona and approximating the radiation energy density and Compton temperature thereby integrating over the thermal seed photon flux from the disk surface. The method either assumes the ion and electron temperatures are equal (1T) or calculates them separately (2T) using an instantaneous equilibrium approach predicated on the actual relevant rate equations (Coulomb and Compton). The method is shown to be consistent with a more detailed ray-tracing calculation where the bulk of the cooling occurs, but is substantially less costly to perform. As an example, we apply these methods to a harm3d simulation of a 10M⊙, non-spinning black hole, accreting at nominally 1% the Eddington value. This new approach leads to radiative efficiency values sime65% above Novikov–Thorne, with a larger fraction of total cooling in the corona as compared to simulations performed using the original target-temperature cooling function. Time-averaged post-processing reveals that the continuum spectral observations predicted from these simulations are qualitatively similar to actual X-ray binary data, especially so for the 1T approach, which yields a harder power-law component (Γ = 2.25) compared to the 2T version (Γ = 2.53).

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Large-scale Gravitational Lens Modeling with Bayesian Neural Networks for Accurate and Precise Inference of the Hubble Constant

We investigate the use of approximate Bayesian neural networks (BNNs) in modeling hundreds of time delay gravitational lenses for Hubble constant (H 0 ) determination. Here, our BNN was trained on synthetic Hubble Space Telescope quality images of strongly lensed active galactic nuclei with lens galaxy light included. The BNN can accurately characterize the posterior probability density functions (PDFs) of model parameters governing the elliptical power-law mass profile in an external shear field. We then propagate the BNN-inferred posterior PDFs into an ensemble H 0 inference, using simulated time delay measurements from a plausible dedicated monitoring campaign. Assuming well-measured time delays and a reasonable set of priors on the environment of the lens, we achieve a median precision of 9.3% per lens in the inferred H 0 . A simple combination of a set of 200 test lenses results in a precision of 0.5 km s –1 Mpc –1 (0.7%), with no detectable bias in this H 0 recovery test. The computation time for the entire pipeline—including the generation of the training set, BNN training and H0 inference—translates to 9 minutes per lens on average for 200 lenses and converges to 6 minutes per lens as the sample size is increased. Being fully automated and efficient, our pipeline is a promising tool for exploring ensemble-level systematics in lens modeling for H 0 inference.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Identifying RR Lyrae Variable Stars in Six Years of the Dark Energy Survey

We present a search for RR Lyrae stars using the full six-year data set from the Dark Energy Survey covering ∼5000 deg2 of the southern sky. Using a multistage multivariate classification and light-curve template-fitting scheme, we identify RR Lyrae candidates with a median of 35 observations per candidate. We detect 6971 RR Lyrae candidates out to ∼335 kpc, and we estimate that our sample is >70% complete at ∼150 kpc. We find excellent agreement with other wide-area RR Lyrae catalogs and RR Lyrae studies targeting the Magellanic Clouds and other Milky Way satellite galaxies. We fit the smooth stellar halo density profile using a broken-power-law model with fixed halo flattening (q = 0.7), and we find strong evidence for a break at with an inner slope of and an outer slope of . We use our catalog to perform a search for Milky Way satellite galaxies with large sizes and low luminosities. Using a set of simulated satellite galaxies, we find that our RR Lyrae-based search is more sensitive than those using resolved stellar populations in the regime of large (r h ≳ 500 pc), low-surface-brightness dwarf galaxies. A blind search for large, diffuse satellites yields three candidate substructures. The first can be confidently associated with the dwarf galaxy Eridanus II. The second has a distance and proper motion similar to the ultrafaint dwarf galaxy Tucana II but is separated by ∼5 deg. The third is close in projection to the globular cluster NGC 1851 but is ∼10 kpc more distant and appears to differ in proper motion.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Spatial and Kinematic Clustering of Stars in the Galactic Disk

The Galactic disk is expected to be spatially and kinematically clustered on many scales due to both star formation and the Galactic potential. In this work we calculate the spatial and kinematic two-point correlation functions (TPCF) using a sample of 1.7 × 10 6 stars with radial velocities from Gaia DR2. Clustering is detected on spatial scales of 1–300 pc and a velocity scale of 15 km s –1 . After removing bound structures, the data have a power-law index of γ ≈ –1 for 1 pc < Δr < 100 pc and γ ≲ –1.5 for Δr > 100 pc. We interpret these results with the aid of a star-by-star simulation of the Galaxy, in which stars are born in clusters orbiting in a realistic potential that includes spiral arms, a bar, and giant molecular clouds. We find that the simulation largely agrees with the observations at most spatial and kinematic scales. In detail, the TPCF in the simulation is shallower than the data at ≲20 pc scales, and steeper than the data at ≳30 pc. We also find a persistent clustering signal in the kinematic TPCF for the data at large Δv (>5 km s –1 ) that is not present in the simulations. We speculate that this mismatch between observations and simulations may be due to two processes: hierarchical star formation and transient spiral arms. Furthermore, we also predict that the addition of ages and metallicities measured with a precision of 50% and 0.05 dex, respectively, will enhance the clustering signal beyond current measurements.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Collapsar R-process Yields Can Reproduce [Eu/Fe] Abundance Scatter in Metal-poor Stars

It is unclear if neutron star mergers can explain the observed r-process abundances of metal-poor stars. Collapsars, defined here as rotating massive stars whose collapse results in a rapidly accreting disk around a black hole that can launch jets, are a promising alternative. We find that we can produce a self-consistent model in which a population of collapsars with stochastic europium yields synthesizes all of the r-process material in metal-poor ([Fe/H] < - 2.5) stars. Our model reproduces the observed scatter and evolution of scatter of [Eu/Fe] abundances. We find that if collapsars are the dominant r-process site for metal-poor stars, r-process synthesis may be linked to supernovae that produce long γ-ray bursts. Our results also allow for the possibility that core-collapse supernovae beyond those that launch γ-ray bursts also produce r-process material (e.g., potentially a subset of Type Ic-BL supernovae). Furthermore, we identify collapsar jet properties (isotropic energy, engine luminosity, or engine time) that may trace r-process yield and verify that the amount of r-process yield produced per collapsar in our model ( ~ 0.07M⊙) is consistent with other independent estimates. In the future, achieving 0.05 dex precision on distribution scatter or a reliable selection function would further constrain our probe of r-process production. Our model would also hold for another prompt r-process site with a power-law yield, and work is needed to determine if, for example, fast-merging neutron stars can also explain abundance scatter.

Astronomy & Astrophysics↗
DOE OSTI2021

A Possible Tidal Disruption Event Candidate in the Black Hole Binary System of OJ 287

The BL Lacertae OJ 287 is a supermassive black hole binary system with complex physics of its irregular flares. During 2016 October–2017 April, a surprising outburst in the X-ray, UV, and optical bands was detected, while no variability was seen in the γ-ray light curve. During the outburst, the X-ray light curves were dominated by the soft X-rays, whose peak in luminosity was ~10 46 erg s -1 —more than 10 times higher than the mean level before the outburst—and a "softer-when-brighter" phenomenon was exhibited. These above phenomena have been reported by some previous works. The hardness ratio showed negligible evolution with flare time and soft X-ray luminosity. Critically, the luminosity of the soft X-rays decayed following a power law of $t$ -5/3 , which occurs in most tidal disruption events (TDEs), and a similar trend can be seen in the UV and optical bands during the soft X-ray declining period. Helium and oxygen narrow emission lines are strengthened prominently in the optical spectra of post outburst epochs, that could be attributed to the surrounding gas caused by TDE. In conclusion, we discuss three possible origins of the event, including the jet's precession, the aftereffects of the black hole–disk impaction, and the TDE. Our results show that the TDE is the more likely scenario to explain the outburst.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Spin and Accretion Rate Dependence of Black Hole X-Ray Spectra

We present a survey of how the spectral features of black hole X-ray binary systems depend on spin, accretion rate, viewing angle, and Fe abundance when predicted on the basis of first-principles physical calculations. The power-law component hardens with increasing spin. The thermal component strengthens with increasing accretion rate. The Compton bump is enhanced by higher accretion rate and lower spin. The Fe Kα equivalent width grows sublinearly with Fe abundance. Strikingly, the Kα profile is more sensitive to accretion rate than to spin because its radial surface brightness profile is relatively flat, and higher accretion rate extends the production region to smaller radii. The overall radiative efficiency is at least 30%–100% greater than as predicted by the Novikov–Thorne model.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Quasar UV Luminosity Function at 3.5 < z < 5.0 from SDSS Deep Imaging Data

We present a well-designed sample of more than 1000 type 1 quasars at 3.5 < z < 5 and derive UV quasar luminosity functions (QLFs) in this redshift range. These quasars were selected using the Sloan Digital Sky Survey (SDSS) imaging data in the Stripe 82 and overlap regions with repeat imaging observations that are about 1 mag fainter than the SDSS single-epoch data. The follow-up spectroscopic observations were conducted by the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) as one of the BOSS ancillary programs. Reaching i ~ 21.5 mag, our sample bridges previous samples from brighter and deeper surveys. We use a 1/V a method to derive binned QLFs at 3.6 < z < 4.0, 4.0 < z < 4.5, and 4.5 < z < 4.9 and then use a double power-law model to parameterize the QLFs. We also combine our data with literature QLFs to better constrain the QLFs across a much wider luminosity baseline. The faint- and bright-end slopes of the QLFs in this redshift range are around –1.7 and –3.7, respectively, with uncertainties from 0.2 to 0.3 to >0.5. The evolution of the QLFs from z ~ 5 to 3.5 can be described by a pure density evolution model (∝10 kz ) with a parameter k similar to that at 5 < z < 7, suggesting a nearly uniform evolution of the quasar density at z = 3.5–7.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

The Curious Case of ASASSN-20hx: A Slowly Evolving, UV- and X-Ray-Luminous, Ambiguous Nuclear Transient

Abstract We present observations of ASASSN-20hx, a nearby ambiguous nuclear transient (ANT) discovered in NGC 6297 by the All-Sky Automated Survey for Supernovae (ASAS-SN). We observed ASASSN-20hx from −30 to 275 days relative to the peak UV/optical emission using high-cadence, multiwavelength spectroscopy and photometry. From Transiting Exoplanet Survey Satellite data, we determine that the ANT began to brighten on 2020 June 22.8 with a linear rise in flux for at least the first week. ASASSN-20hx peaked in the UV/optical 30 days later on 2020 July 22.8 (MJD = 59052.8) at a bolometric luminosity of L = (3.15 ± 0.04) × 10 43 erg s −1 . The subsequent decline is slower than any TDE observed to date and consistent with many other ANTs. Compared to an archival X-ray detection, the X-ray luminosity of ASASSN-20hx increased by an order of magnitude to L x ∼ 1.5 × 10 42 erg s −1 and then slowly declined over time. The X-ray emission is well fit by a power law with a photon index of Γ ∼ 2.3–2.6. Both the optical and near-infrared spectra of ASASSN-20hx lack emission lines, unusual for any known class of nuclear transient. While ASASSN-20hx has some characteristics seen in both tidal disruption events and active galactic nuclei, it cannot be definitively classified with current data.

79 ASTRONOMY AND ASTROPHYSICS↗