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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.

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Wide class of logarithmic potentials with power-tower kink tails

We present a wide class of potentials which admit kinks and corresponding mirror kinks with either a power law or an exponential tail at the two extreme ends and a power-tower form of tails at the two neighbouring ends. We analyse kink stability equation in all these cases and show that there is no gap between the zero mode and the beginning of the continuum. Finally, we provide a recipe for obtaining logarithmic potentials with power-tower kink tails and estimate kink–kink interaction strength.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Heavy-tailed distribution of the number of papers within scientific journals

Scholarly publications represent at least two benefits for the study of the scientific community as a social group. First, they attest to some form of relation between scientists (collaborations, mentoring, heritage, …), useful to determine and analyze social subgroups. Second, most of them are recorded in large databases, easily accessible and including a lot of pertinent information, easing the quantitative and qualitative study of the scientific community. Understanding the underlying dynamics driving the creation of knowledge in general, and of scientific publication in particular, can contribute to maintaining a high level of research, by identifying good and bad practices in science. In this article, we aim to advance this understanding by a statistical analysis of publication within peer-reviewed journals. Namely, we show that the distribution of the number of papers published by an author in a given journal is heavy-tailed, but has a lighter tail than a power law. Interestingly, we demonstrate (both analytically and numerically) that such distributions match the result of a modified preferential attachment process, where, on top of a Barabási-Albert process, we take the finite career span of scientists into account.

96 KNOWLEDGE MANAGEMENT AND PRESERVATION↗
DOE OSTI2023

Measuring the giant radio galaxy length distribution with the LoTSS

Many massive galaxies launch jets from the accretion disk of their central black hole, but only ~10 3 instances are known in which the associated outflows form giant radio galaxies (GRGs, or giants): luminous structures of megaparsec extent that consist of atomic nuclei, relativistic electrons, and magnetic fields. Large samples are imperative to understanding the enigmatic growth of giants, and recent systematic searches in homogeneous surveys constitute a promising development. For the first time, it is possible to perform meaningful precision statistics with GRG lengths, but a framework to do so is missing. We measured the intrinsic GRG length distribution by combining a novel statistical framework with a LOFAR Two-metre Sky Survey (LoTSS) sample of freshly discovered giants. In turn, this allowed us to answer an array of questions on giants. For example, we can now assess how rare a 5 Mpc giant is compared with one of 1 Mpc, and how much larger – given a projected length – the corresponding intrinsic length is expected to be. Notably, we can now also infer the GRG number density in the Local Universe. We assumed the intrinsic GRG length distribution to be Paretian (i.e. of power-law form) with tail index ξ, and predicted the observed distribution by modelling projection and selection effects. To infer ξ, we also systematically searched the LoTSS for hitherto unknown giants and compiled the largest catalogue of giants to date. We show that if intrinsic GRG lengths are Pareto distributed with index ξ, then projected GRG lengths are also Pareto distributed with index ξ. Selection effects induce curvature in the observed projected GRG length distribution: angular length selection flattens it towards the lower end, while surface brightness selection steepens it towards the higher end. We explicitly derived a GRG’s posterior over intrinsic lengths given its projected length, laying bare the ξ dependence. We also discovered 2060 giants within LoTSS DR2 pipeline products; our sample more than doubles the known population. Spectacular discoveries include the largest, second-largest, and fourth-largest GRG known (l p = 5.1 Mpc, l p = 5.0 Mpc, and l p = 4.8 Mpc), the largest GRG known hosted by a spiral galaxy (l p = 2.5 Mpc), and the largest secure GRG known beyond redshift 1 (l p = 3.9 Mpc). We increase the number of known giants whose angular length exceeds that of the Moon from 10 to 23; among the discoveries is the angularly largest known radio galaxy in the Northern Sky, which is also the angularly largest known GRG ($\phi$ = 2°). Combining theory and data, we determined that intrinsic GRG lengths are well described by a Pareto distribution, and measured the index ξ = –3.5 ± 0.5. This implies that, given its projected length, a GRG’s intrinsic length is expected to be just 15% larger. Finally, we determined the comoving number density of giants in the Local Universe to be n GRG = 5 ± 2(100 Mpc) –3 . We developed a practical mathematical framework that elucidates the statistics of giant radio galaxy lengths. Through a LoTSS search, we also discovered 2060 new giants. By combining both advances, we determined that intrinsic GRG lengths are well described by a Pareto distribution with index ξ = –3.5 ± 0.5, and that giants are truly rare in a cosmological sense: most clusters and filaments of the Cosmic Web are not currently home to a giant. Thus, our work yields new observational constraints for analytical models and simulations featuring radio galaxy growth.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Footprints of the QCD Crossover on Cosmological Gravitational Waves at Pulsar Timing Arrays

Pulsar timing arrays (PTAs) have reported evidence for a stochastic gravitational wave (GW) background at nanohertz frequencies, possibly originating in the early Universe. We show that the spectral shape of the low-frequency (causality) tail of GW signals sourced at temperatures around T ≳ 1 GeV is distinctively affected by confinement of strong interactions (QCD), due to the corresponding sharp decrease in the number of relativistic species, and significantly deviates from ∼ f 3 commonly adopted in the literature. Bayesian analyses in the NANOGrav 15 years and the previous international PTA datasets reveal a significant improvement in the fit with respect to cubic power-law spectra, previously employed for the causality tail. While no conclusion on the nature of the signal can be drawn at the moment, our results show that the inclusion of standard model effects on cosmological GWs can have a decisive impact on model selection. Published by the American Physical Society 2024

Franciolini, Gabriele (ORCID:0000000268929145)↗
DOE OSTI2024

Are turbulence effects on droplet collision–coalescence a key to understanding observed rain formation in clouds?

Rain formation is a critical factor governing the lifecycle and radiative forcing of clouds and therefore it is a key element of weather and climate. Cloud microphysics–turbulence interactions occur across a wide range of scales and are challenging to represent in atmospheric models with limited resolution. Based on past experiments and idealized numerical simulations, it has been postulated that cloud turbulence accelerates rain formation by enhancing drop collision–coalescence. We provide substantial evidence for significant impacts of turbulence on the evolution of cloud droplet size distributions and rain formation by comparing high-resolution observations of cumulus congestus clouds with state-of-the-art large-eddy simulations coupled with a Lagrangian particle-based microphysics scheme. Turbulent coalescence must be included in the model to accurately represent the observed drop size distributions, especially for drizzle drop sizes at lower heights in the cloud. Turbulence causes earlier rain formation and greater rain accumulation compared to simulations with gravitational coalescence only. The observed rain size distribution tail just above cloud base follows a power law scaling that deviates from theoretical scalings considering either a purely gravitation collision kernel or a turbulent kernel neglecting droplet inertial effects, providing additional evidence for turbulent coalescence in clouds. In contrast, large aerosols acting as cloud condensation nuclei (“giant CCN”) do not significantly impact rain formation owing to their long timescale to reach equilibrium wet size relative to the lifetime of rising cumulus thermals. Overall, turbulent drop coalescence exerts a dominant influence on rain initiation in warm cumulus clouds, with limited impacts of giant CCN.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

On the physical size of the Milky Way globular cluster NGC 7089 (M2)

ABSTRACT We study the outer regions of the Milky Way globular cluster NGC 7089 based on new Dark Energy Camera observations. The resulting background-cleaned stellar density profile reveals the existence of an extended envelope. We confirm previous results that cluster stars are found out up to ∼1° from the cluster’s centre, which is nearly three times the value of the most robust tidal radii estimations. We also used results from direct N-body simulations in order to compare with the observations. We found a fairly good agreement between the observed and numerically generated stellar density profiles. Because of the existence of gaps and substructures along globular cluster tidal tails, we closely examined the structure of the outer cluster region beyond the Jacobi radius. We extended the analysis to a sample of 35 globular clusters, 20 of them with observed tidal tails. We found that if the stellar density profile follows a power law ∝ r−α, the α slope correlates with the globular cluster present mass, in the sense that, the more massive the globular cluster, the smaller the α value. This trend is not found in globular clusters without observed tidal tails. The origin of such a phenomenon could be related, among other reasons, to the proposed so-called potential escapers or to the formation of globular clusters within dark matter minihaloes.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

The Escape Velocity Profile of the Milky Way from Gaia DR3

Abstract The escape velocity profile of the Milky Way offers a crucial and independent measurement of its underlying mass distribution and dark matter (DM) properties. Using a sample of stars from the third data release of Gaia with 6D kinematics and strict quality cuts, we obtain an escape velocity profile of the Milky Way from 4 to 11 kpc in Galactocentric radius. To infer the escape velocity in radial bins, we model the tail of the stellar speed distribution with both traditional power-law models and a new functional form that we introduce. While power-law models tend to rely on extrapolation to high speeds, we find our new functional form gives the most faithful representation of the observed distribution. Using this for the escape velocity profile, we constrain the properties of the Milky Way’s DM halo modeled as a Navarro–Frenck–White profile. Combined with constraints from the circular velocity at the solar position, we obtain a concentration and mass of c 200 c DM = 13.9 − 4.3 + 6.2 and M 200 c DM = 0.55 − 0.14 + 0.15 × 10 12 M ⊙ . This corresponds to a total Milky Way mass of M 200 c = 0.64 − 0.14 + 0.15 × 10 12 M ⊙ , which is on the low end of the historic range of the galaxy’s mass, but in line with other recent estimates.

Astronomy & Astrophysics↗
DOE OSTI2022

A scale-wise analysis of intermittent momentum transport in dense canopy flows

We investigate the intermittent dynamics of momentum transport and its underlying time scales in the near-wall region of the neutrally stratified atmospheric boundary layer in the presence of a vegetation canopy. This is achieved through an empirical analysis of the persistence time scales (periods between successive zero-crossings) of momentum flux events, and their connection to the ejection–sweep cycle. Using high-frequency measurements from the GoAmazon campaign, spanning multiple heights within and above a dense canopy, the analysis suggests that, when the persistence time scales ( $t_p$ ) of momentum flux events from four different quadrants are separately normalized by $\varGamma _{w}$ (integral time scale of the vertical velocity), their distributions $P(t_p/\varGamma _{w})$ remain height-invariant. This result points to a persistent memory imposed by canopy-induced coherent structures, and to their role as an efficient momentum-transporting mechanism between the canopy airspace and the region immediately above. Moreover, $P(t_p/\varGamma _{w})$ exhibits a power-law scaling at times $t_{p}<\varGamma _{w}$ , with an exponential tail appearing for $t_{p} \geq \varGamma _{w}$ . By separating the flux events based on $t_p$ , we discover that around 80 % of the momentum is transported through the long-lived events ( $t_{p} \geq \varGamma _{w}$ ) at heights immediately above the canopy, while the short-lived ones ( $t_{p} < \varGamma _{w}$ ) only contribute marginally ( $\approx 20\,\%$ ). To explain the role of instantaneous flux amplitudes in momentum transport, we compare the measurements with newly developed surrogate data and establish that the range of time scales involved with amplitude variations in the fluxes tends to increase as one transitions from within to above the canopy.

Mechanics↗
DOE OSTI2022

Substructure at High Speed. I. Inferring the Escape Velocity in the Presence of Kinematic Substructure

Abstract The local escape velocity provides valuable inputs to the mass profile of the galaxy, and requires understanding the tail of the stellar speed distribution. Following Leonard & Tremaine, various works have since modeled the tail of the stellar speed distribution as ∝ ( v esc − v ) k , where v esc is the escape velocity, and k is the slope of the distribution. In such studies, however, these two parameters were found to be largely degenerate and often a narrow prior is imposed on k in order to constrain v esc . Furthermore, the validity of the power-law form can breakdown in the presence of multiple kinematic substructures or other mis-modeled features in the data. In this paper, we introduce a strategy that for the first time takes into account the presence of kinematic substructure. We model the tail of the velocity distribution as a sum of multiple power laws as a way of introducing a more flexible fitting framework. Using mock data and data from FIRE simulations of Milky Way-like galaxies, we show the robustness of this method in the presence of kinematic structure that is similar to the recently discovered Gaia Sausage. In a companion paper, we present the new measurement of the escape velocity and subsequently the mass of the Milky Way using Gaia eDR3 data.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

On the Statistics of Elsasser Increments in Solar Wind and Magnetohydrodynamic Turbulence

In this Letter we investigate the dependency with scale of the empirical probability distribution functions (PDF) of Elsasser increments using large sets of WIND data (collected between 1995 and 2017) near 1 au. The empirical PDF are compared to the ones obtained from high-resolution numerical simulations of steadily driven, homogeneous reduced MHD turbulence on a 2048 3 rectangular mesh. A large statistical sample of Alfvénic increments is obtained by using conditional analysis based on the solar wind average properties. The PDF tails obtained from observations and numerical simulations are found to have exponential behavior in the inertial range, with an exponential decrement that satisfies power laws of the form α l ∝ l –μ , where l is the scale size, with μ between 0.17 and 0.25 for observations and 0.43 for simulations. PDF tails were extrapolated assuming their exponential behavior extends to arbitrarily large increments in order to determine structure function scaling laws at very high orders. Our results point to potentially universal scaling laws governing the PDF of Elsasser increments and to an alternative approach to investigate high-order statistics in solar wind observations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

X-Ray Polarization of the Magnetar 1E 1841−045

We report on IXPE and NuSTAR observations beginning 40 days after the 2024 outburst onset of magnetar 1E 1841−045, marking the first IXPE observation of a magnetar in an enhanced state. Our spectropolarimetric analysis indicates that both a blackbody (BB) plus double power-law (PL) and a double blackbody plus power-law spectral model fit the phase-averaged intensity data well, with a hard PL tail (Γ = 1.19 and 1.35, respectively) dominating above ≈5 keV. For the former model, we find the soft PL (the dominant component at soft energies) exhibits a polarization degree (PD) of ≈30% while the hard PL displays a PD of ≈40%. Similarly, the cool BB of the 2BB+PL model possesses a PD of ≈15% and a hard PL PD of ≈57%. For both models, each component has a polarization angle (PA) compatible with celestial north. Model-independent polarization analysis supports these results, wherein the PD increases from ≈15% to ≈70% in the 2–3 keV and 6–8 keV ranges, respectively, while the PA remains nearly constant. We find marginal evidence for phase-dependent variability of the polarization properties, namely a higher PD at phases coinciding with the hard X-ray pulse peak. We compare the hard X-ray PL to the expectation from resonant inverse Compton scattering (RICS) and secondary pair cascade synchrotron radiation from primary high-energy RICS photons; both present reasonable spectropolarimetric agreement with the data, albeit the latter does so more naturally. We suggest that the soft PL X-ray component may originate from a Comptonized corona in the inner magnetosphere.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Seeing through noise in power laws

Despite widespread claims of power laws across the natural and social sciences, evidence in data is often equivocal. Modern data and statistical methods reject even classic power laws such as Pareto’s law of wealth and the Gutenberg–Richter law for earthquake magnitudes. We show that the maximum-likelihood estimators and Kolmogorov–Smirnov (K-S) statistics in widespread use are unexpectedly sensitive to ubiquitous errors in data such as measurement noise, quantization noise, heaping and censorship of small values. This sensitivity causes spurious rejection of power laws and biases parameter estimates even in arbitrarily large samples, which explains inconsistencies between theory and data. We show that logarithmic binning by powers of λ > 1 attenuates these errors in a manner analogous to noise averaging in normal statistics and that λ thereby tunes a trade-off between accuracy and precision in estimation. Binning also removes potentially misleading within-scale information while preserving information about the shape of a distribution over powers of λ, and we show that some amount of binning can improve sensitivity and specificity of K-S tests without any cost, while more extreme binning tunes a trade-off between sensitivity and specificity. We therefore advocate logarithmic binning as a simple essential step in power-law inference.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Emergent hydrodynamic mode on SU(2) plaquette chains and quantum simulation

We search for emergent hydrodynamic modes in real-time Hamiltonian dynamics of 2+1-dimensional SU(2) lattice gauge theory on a quasi-one-dimensional plaquette chain, by numerically computing symmetric correlation functions of energy densities on lattice sizes of about 20 with the local Hilbert space truncated at 𝑗 max = $\frac{1}{2}$. Because of the Umklapp processes, we only find a mode for energy diffusion. The symmetric correlator exhibits transport peak near zero frequency with a width approximately proportional to momentum squared at small momentum, when the system is fully quantum ergodic, as indicated by the eigenenergy level statistics. This transport peak leads to a power-law 𝑡 −$\frac{1}{2}$ decay of the symmetric correlator at late time, also known as the long-time tail, as well as diffusionlike spreading in position space. We also introduce a quantum algorithm for computing the symmetric correlator on a quantum computer and find it gives results consistent with exact diagonalization when tested on the IBM emulator. Finally we discuss the future prospect of searching for the sound modes.

Hamiltonian systems↗
DOE OSTI2023

Four Years of Type Ia Supernovae Observed by TESS: Early-time Light-curve Shapes and Constraints on Companion Interaction Models

Abstract We present 307 type Ia supernova (SN) light curves from the first 4 yr of the Transiting Exoplanet Survey Satellite mission. We use this sample to characterize the shapes of the early-time light curves, measure the rise times from first light to peak, and search for companion star interactions. Using simulations, we show that light curves must have noise <10% of the peak flux to avoid biases in the early-time light-curve shape, restricting our quantitative analysis to 74 light curves. We find that the mean power-law index t β 1 of the early-time light curves isβ 1 = 1.93 ± 0.57, and the mean rise time to peak is 15.7 ± 3.5 days. The underlying population distribution forβ 1 may instead consist of a Gaussian component with mean 2.29, width 0.34, and a long tail extending to values less than 1.0. We find that the data can rarely distinguish between models with and without companion interaction models. Nevertheless, we find three high-quality light curves that tentatively prefer the addition of a companion interaction model, but the statistical evidence for the companion interactions is not robust. We also find two SNe that disfavor the addition of a companion interaction model to a curved power-law model. Taking the 74 SNe together, we calculate 3σupper limits on the presence of companion signatures to control for orientation effects that can hide companions in individual light curves. Our results rule out common progenitor systems with companions having Roche lobe radii >31R ⊙ (separations >5.7 × 10 12 cm, 99.9% confidence level) and disfavor companions having Roche lobe radii >10R ⊙ (separations >1.9 × 10 12 cm, 95% confidence level). Lastly, we discuss the implications of our results for the intrinsic fraction of single degenerate progenitor systems.

Astronomy & Astrophysics↗
DOE OSTI2023

Seeing through noise in power laws

Despite widespread claims of power laws across the natural and social sciences, evidence in data is often equivocal. Modern data and statistical methods reject even classic power laws such as Pareto’s law of wealth and the Gutenberg–Richter law for earthquake magnitudes. We show that the maximum-likelihood estimators and Kolmogorov–Smirnov (K-S) statistics in widespread use are unexpectedly sensitive to ubiquitous errors in data such as measurement noise, quantization noise, heaping and censorship of small values. This sensitivity causes spurious rejection of power laws and biases parameter estimates even in arbitrarily large samples, which explains inconsistencies between theory and data. We show that logarithmic binning by powers of λ > 1 attenuates these errors in a manner analogous to noise averaging in normal statistics and that λ thereby tunes a trade-off between accuracy and precision in estimation. Binning also removes potentially misleading within-scale information while preserving information about the shape of a distribution over powers of λ, and we show that some amount of binning can improve sensitivity and specificity of K-S tests without any cost, while more extreme binning tunes a trade-off between sensitivity and specificity. We therefore advocate logarithmic binning as a simple essential step in power-law inference.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2021

Ion Acceleration and the Development of a Power-law Energy Spectrum in Magnetic Reconnection

How charged particles are accelerated efficiently and form a power-law energy spectrum in magnetic reconnection is a problem that is not well understood. In a previous paper, it was shown that the electron Kelvin–Helmholtz instability (EKHI) in force-free magnetic reconnection generates fast-expanding vortices that can accelerate electrons in a few tens of ion gyroperiods (less than 1 ms in the solar corona) to form a power-law energy distribution. In this paper, we present a particle-in-cell (PIC) simulation study of ion acceleration in force-free magnetic reconnection in the presence of the EKHI-induced turbulence. We find that ions are not significantly accelerated by the EKHI-induced stochastic electric field until the magnetic vortices expand to sizes comparable to the ion gyroradius. The Alfvén waves generated by the EKHI couple with the magnetic vortices, leading to resonance between the ions inside the magnetic vortices and Alfvén waves and enhanced ion heating. The induced Alfvén wave resonance results in a broken power-law energy spectrum with a breakpoint at $\sim {m}_{i}{v}_{A}^{2}$, where v A is the Alfvén velocity. Here, we show that the process that forms the nonthermal tail is a second-order Fermi mechanism and the mean spectral index is α = (1 + 4 a 2 D/R) /2, where D is the spatial scale of the inductive electric field, R is that of vortices, and a = B g /B 0 , with ratio of guide field B g and asymptotic B 0 .

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

A possible mass distribution of primordial black holes implied by LIGO-Virgo

The LIGO-Virgo Collaboration has so far detected around 90 black holes, some of which have masses larger than what were expected from the collapse of stars. The mass distribution of LIGO-Virgo black holes appears to have a peak at ~ 30 M ⊙ and two tails on the ends. By assuming that they all have a primordial origin, we analyze the GWTC-1 (O1&O2) and GWTC-2 (O3a) datasets by performing maximum likelihood estimation on a broken power law mass function f ( m ), with the result f ∝ m 1.2 for m < 35 M ⊙ and f ∝ m -4 for m > 35 M ⊙ . This appears to behave better than the popular log-normal mass function. Surprisingly, such a simple and unique distribution can be realized in our previously proposed mechanism of PBH formation, where the black holes are formed by vacuum bubbles that nucleate during inflation via quantum tunneling. Moreover, this mass distribution can also provide an explanation to supermassive black holes formed at high redshifts.

Astronomy & 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↗