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NASA NTRS1995

High-speed correction factor to the O(+)-O resonance charge exchange collision frequency

The high-speed correction factor to the O(+)-O collision frequency, resulting from drift velocities between ions and neutrals, is calculated by solving the integral expression in this factor both numerically and analytically. Although the analytic solution is valid for either small or large drift velocities between ions and neutrals, for temperatures of interest and all drift velocities considered, agreement is found between analytic and detailed numerical integration results within less than 1% error. Let T(sub r) designate the average of the ion and neutral temperatures in K, and u = nu(sub d)/alpha, where nu(sub d) is the relative drift velocity in cm/s, and alpha = 4.56 x 10(exp 3) square root of T(sub r) cm/s is the thermal velocity of the O(+)-O system. Then, as u ranges from 0 to 2, the correction factor multiplying the collision frequency increases monotonically from 1 to about 1.5. An interesting result emerging from this calculation is that the correction factor for temperatures of aeronomical interest is to a good approximation independent of the temperature, depending only on the scaled velocity u.

Omidvar, K.↗

NASA NTRS1995

The interstellar disk-halo connection in the spiral galaxy NGC 3079

We discuss the morphology and excitation of ionized gas in the nearby Sc galaxy NGC 3079. The almost edge-on orientation is ideal for studying the vertical structure of the gaseous disk, and especially the diffuse ionized medium (DIM) found between the bright H II regions. We used the Hawaii Imaging Fabry-Perot Interferometer (HIFI) to map 150,000 H-alpha + (N II) lambda lambda 6548, 6583 emission-line profiles across the entire disk, with resolution 70 km/s at subarcsecond steps, down to a flux level of approximately 10(exp -17) ergs/s/sq cm (EM approximately equal to 4 cm(exp -6) pc). The DIM contributes approximately 30% of the total disk H-alpha emission within a radius of 10 kpc. The DIM has broader emission lines and larger (N II) H-alpha flux ratios than the adjacent H II regions. Within a radius of 5 kpc, we find that the X-shaped filaments reported in previous studies emerge from the inner (R approximately equal to 1.5 kpc) disk, and rise more than 4 kpc above the disk plane. The morphology, kinematics, and excitation of the filaments suggest that they form a biconic interface between the undisturbed disk gas, and gas entrained in the wide-angle outflow. The DIM beyond 5 kpc radius is more vertically extended than the thick ionized disk detected in our Galaxy and in a few nearby edge-on systems. After correcting for dust, the vertical profile of this DIM has an exponential scale height of about 1.1 kpc, similar to that of the H I disk. The (N II) lambda 6538/H-alpha flux ratio of the DIM increases monotonically with vertical height, reaching unity for absolute value of z greater than or approximately equal to 2.5 kpc. The flux required to keep the DIM ionized at R = 8 kpc is similar to that near the solar circle of our Galaxy. Highly dilute radiation from O stars in the galactic plane probably maintains the DIM. The total mass of the DIM is of order 10(exp 8) - 10(exp 9) solar mass, representing less than 1% of the total dynamical mass of NGC 3079. Mechanical energy from intense star formation in the disk probably lifts the DIM above the disk. The several bubbles and filaments within 1 kpc of the disk plane is direct evidence for gas flow between the disk and halo.

Veilleux, Sylvain↗

NASA NTRS1995

Bolometric temperature and young stars in the Taurus and Ophiuchus complexes

We calculated bolometric temperature (T(sub bol)) and luminosity (L(sub bol)) for 128 young stellar objects (YSOs) in Taurus, 74 in the Ophiuchus 'core', and 33 in the Ophiuchus 'off-core' region. We have constructed the bolometric luminosity-temperature (BLT) diagram, the log-log plot of L(sub bol) versus T(sub bol), for the three samples. T(sub bol) is defined as the temperature of a blackbody having the same frequency as the observed continuum spectrum. It measures the redness (or coldness) of an astronomical source. The BLT diagram is analogous to the H-R diagram and allows for a direct and quantitative comparison of YSOs at a wide variety of evolutionary states, ranging from the most deeply embedded stars to T Tauri stars nearly on the main sequence. We found (1) T(sub bol) increases monotonically from embedded sources (approximately 60-500 K) to classical T Tauri stars (approximately 1000-3000 K) to weak-line T Tauri stars (approximately 2000-5000 K); (2) T(sub bol) correlates reasonably well with the age inferred from the evolutionary models of pre-main-sequence stars and protostars for embedded 'protostars' and weak-line T Tauri stars. There is no significant correlation for the classical T Tauri stars. These results can be understood in terms of dissipation of circumstellar dust envelope and disk during the early stages of stellar evolution. Sources in the three regions have different distributions in the BLT diagram. The Ophiuchus core has the highest fraction of cold sources among the three regions. These cold sources are also more luminous than the YSOs in the other regions. The Ophiuchus off-core sample is dominated by the more evolved pre-main-sequence stars. The Taurus sources have distributions intermediate in L(sub bol), T(sub bol), and age between the Ophiuchus core and off-core distributions. These may suggest differences in the star formation history, and possibly in the stellar masses and mass accretion rates in these star-forming regions.

Chen, H.↗

NASA NTRS1995

Cosmological implications of ROSAT observations of groups and clusters of galaxies

We have combined ROSAT Position Sensitive Proportional Counter (PSPC) and optical observations of a sample of groups and clusters of galaxies to determine the fundamental parameters of these systems (e.g., the dark matter distribution, gas mass fraction, baryon mass fraction, mass-to-light ratio, and the ratio of total-to-luminous mass). Imaging X-ray spectroscopy of groups and clusters show that the gas is essentially isothermal beyond the central region, indicating that the total mass density (mostly dark matter) scales as rho(sub dark) varies as 1/r squared. The density profile of the hot X-ray emitting gas is fairly flat in groups with rho(sub gas) varies as 1/r and becomes progressively steeper in hotter richer systems, with rho(sub gas) varies as 1/r squared in the richest clusters. These results show, that in general, the hot X-ray-emitting gas is the most extended mass component in groups and clusters, the galaxies are the most centrally concentrated component, and the dark matter is intermediate between the two. The flatter density rofile of the hot gas compared to the dark matter produces a gas mass fraction that increases with radius within each object. There is also a clear trend of increasing gas mass fraction (from 2% to 30%) between elliptical galaxies and rich clusters due to the greater detectable extent of the X-ray emission in richer systems. For the few systems in which the X-ray emission can be traced to the virial radius (where the overdensity delta is approximately equal 200), the gas mass fraction (essentially the baryon mass fraction) approaches a roughly constant value of 30%, suggesting that this is the true primordial value. Based on standard big bang nucleosynthesis, the large baryon mass fraction implies that Omega = 0.1 - 0.2. The antibiased gas distribution suggests that feedback from galaxy formation and hydrodynamics play important roles in the formation of structure on the scale of galaxies to rich clusters. All the groups and clusters in our sample have mass-to-light ratios of M/L(sub V) approximately 100 - 150 solar mass/solar luminosity, which strongly contrasts with the traditional view that the mass-to-light ratio of rich clusters is significantly greater than individual galaxies or groups with M/L(sub V) approximately 250 - 300 solar mass/solar luminosity. We also show that M/L(sub V is essentially constant within the virial radius of clusters (where delta is greater than or approximately 200), which is consistent with the peaks formalism of biased galaxy formation. While the mass-to-light ratios of groups and clusters are comparable (indicating a constant mass fraction of optically luminous material), the ratio of the total mass-to-luminous mass (gas plus stars) monotonically decreases between galaxies and clusters. The decrease in M(sub total)/M(sub lum) arises from two factors: (1) the composition of baryonic matter varies from a predominance of optically luminous material (stars) on the scale of galaxies (approximately 10 kpc) to a predominance of X-ray luminous material (hot gas) on the scale of rich clusters (approximately 1 Mpc), and (2) the hot gas has a more extended spatial distribution than the gravitating matter. The observed decrease M(sub total)/M(sub lum) between galaxies and clusters indicates that the universe actually becomes `brighter' on mass scales between 10(exp 12) and 10(exp 15) solar mass, in the sense that a greater fraction of the gravitating mass is observable.

David, Laurence P.↗

NASA NTRS1994

Numerical investigation of separated nozzle flows

A numerical study of axisymmetric overexpanded nozzle is presented. The flow structure of the startup and throttle-down processes are examined. During the impulsive startup process, observed flow features include the Mach disk, separation shock, Mach stem, vortex core, contact surface, slip stream, initial shock front, and shocklet. Also the movement of the Mach disk is not monotonical in the downstream direction. For a range of pressure ratios, hysteresis phenomenon occurs; different solutions were obtained depending on different processes. Three types of flow structures were observed. The location of separation point and the lower end turning point of hysteresis are closely predicted. A high peak of pressure is associated with the nozzle flow reattachment. The reversed vortical structure and affects engine performance.

Chen, C. L.↗

NASA NTRS1995

Nonlinear evolution of magnetic flux ropes. 2: Finite beta plasma

In this second paper on the evolution of magnetic flux ropes we study the effects of gas pressure. We assume that the energy transport is described by a polytropic relationship and reduce the set of ideal MHD equations to a single, second-order, nonlinear, ordinary differential equation for the evolution function. For this conservative system we obtain a first integral of motion. To analyze the possible motions, we use a mechanical analogue -- a one-dimensional, nonlinear oscillator. We find that the effective potential for such an oscillator depends on two parameters: the polytropic index gamma and a dimensionless quantity kappa the latter being a function of the plasma beta, the strength of the azimuthal magnetic field relative to the axial field of the flux rope, and gamma. Through a study of this effective potential we classify all possible modes of evolution of the system. In the main body of the paper, we focus on magnetic flux ropes whose field and gas pressure increase steadily towards the symmetry axis. In this case, for gamma greater than 1 and all values of kappa, only oscillations are possible. For gamma less than 1, however, both oscillations and expansion are allowed. For gamma less than 1 and kappa below a critical value, the energy of the nonlinear oscillator determines whether the flux rope will oscillate or expand to infinity. For gamma less than 1 and kappa above critical, however, only expansion occurs. Thus by increasing kappa while keeping gamma fixed (less than 1), a phase transition occurs at kappa = kappa(sub critical) and the oscillatory mode disappears. We illustrate the above theoretical considerations by the example of a flux rope of constant field line twist evolving self-similarly. For this example, we present the full numerical MHD solution. In an appendix to the paper we catalogue all possible evolutions when (1) either the magnetic field or (2) the gas pressure decreases monotonically toward the axis. We find that in these cases critical conditions can occur for gamma greater than 1. While in most cases the flux rope collapses, there are notable exceptions when, for certain ranges of kappa and gamma, collapse may be averted.

Osherovich, V. A.↗

NASA NTRS1995

Correlations between critical current density, j(sub c), critical temperature, T(sub c),and structural quality of Y1B2Cu3O(7-x) thin superconducting films

Correlations between critical current density (j(sub c)) critical temperature (T(sub c)) and the density of edge dislocations and nonuniform strain have been observed in YBCO thin films deposited by pulsed laser ablation on (001) LaAlO3 single crystals. Distinct maxima in j(sub c) as a function of the linewidths of the (00 l) Bragg reflections and as a function of the mosaic spread have been found in the epitaxial films. These maxima in j(sub c) indicate that the magnetic flux lines, in films of structural quality approachingthat of single crystals, are insufficiently pinned which results in a decreased critical current density. T(sub c) increased monotonically with improving crystalline quality and approached a value characteristic of a pure single crystal. A strong correlation between j(sub c) and the density of edge dislocations ND was found. At the maximum of the critical current density the density of edge dislocations was estimated to be N(sub D) approximately 1-2 x 10(exp 9)/sq cm.

Chrzanowski, J.↗

NASA NTRS1995

A satellite anemometer

This report describes the design, development, and testing of components of a satellite anemometer, an instrument for measuring neutral winds in the upper atmosphere from a satellite platform. The device, which uses four nearly identical pressure sensors, measures the angle of arrival of the bulk neutral flow in the satellite frame of reference. It could also be used in a feedback loop to control spacecraft attitude with respect to the ram velocity direction. We have now developed miniaturized ionization pressure gauges that will work well from the slip flow region near 115 km up to the base of the exosphere, which covers the entire altitude range currently being considered for Tether. Laboratory tests have demonstrated a very linear response to changes in ram angle out to +/- 20 deg. (transverse wind component of 2.7 km s(exp -1)) from the ram, and a monotonic response to out beyond 45 deg. Pitch (vertical wind) and yaw (horizontal wind) can be sampled simultaneously and meaningfully up to 10 Hz. Angular sensitivity of 30 arc seconds (approximately 1 ms(exp -1) is readily attainable, but absolute accuracy for winds will be approximately 1 deg (130 m/s) unless independent attitude knowledge is available. The critical elements of the design have all been tested in the laboratory.

Hanson, W. B.↗

NASA NTRS1995

Real-time design with peer tasks

We introduce a real-time design methodology for large scale, distributed, parallel architecture, real-time systems (LDPARTS), as an alternative to those methods using rate or dead-line monotonic analysis. In our method the fundamental units of prioritization, work items, are domain specific objects with timing requirements (deadlines) found in user's specification. A work item consists of a collection of tasks of equal priority. Current scheduling theories are applied with artifact deadlines introduced by the designer whereas our method schedules work items to meet user's specification deadlines (sometimes called end-to-end deadlines). Our method supports these scheduling properties. Work item scheduling is based on domain specific importance instead of task level urgency and still meets as many user specification deadlines as can be met by scheduling tasks with respect to urgency. Second, the minimum (closest) on-line deadline that can be guaranteed for a work item of highest importance, scheduled at run time, is approximately the inverse of the throughput, measured in work items per second. Third, throughput is not degraded during overload and instead of resorting to task shedding during overload, the designer can specify which work items to shed. We prove these properties in a mathematical model.

Goforth, Andre↗

NASA NTRS1994

Thermally induced damage in composite space structure: Predictive methodology and experimental correlation

A general analysis method is presented to predict matrix cracks in all plies of a composite laminate, and resulting degraded laminate properties, as functions of temperature or thermal cycles. A shear lag solution of the stresses in the vicinity of cracks and a fracture mechanics crack formation criteria are used to predict cracks. Damage is modeled incrementally, which allows the inclusion of the effects of temperature dependent material properties and softening of the laminate due to previous cracking. The analysis is incorporated into an easy-to-use computer program. The analysis is correlated with experimentally measured crack densities in a variety of laminates exposed to monotonically decreasing temperatures. Crack densities are measured at the edges of specimens by microscopic inspection, and throughout the specimen volumes by x-ray and sanding down of the edges. Correlation between the analytical results and the crack densities in the interiors of the specimens was quite good. Crack densities measured at specimen edges did not agree with internal crack densities (or analyses) in some cases. A free-edge stress analysis clarified the reasons for these discrepancies.

Park, Cecelia↗

NASA NTRS1995

Microcracking in composite laminates under thermal and mechanical loading

Composites used in space structures are exposed to both extremes in temperature and applied mechanical loads. Cracks in the matrix form, changing the laminate thermoelastic properties. The goal of the present investigation is to develop a predictive methodology to quantify microcracking in general composite laminates under both thermal and mechanical loading. This objective is successfully met through a combination of analytical modeling and experimental investigation. In the analysis, the stress and displacement distributions in the vicinity of a crack are determined using a shear lag model. These are incorporated into an energy based cracking criterion to determine the favorability of crack formation. A progressive damage algorithm allows the inclusion of material softening effects and temperature-dependent material properties. The analysis is implemented by a computer code which gives predicted crack density and degraded laminate properties as functions of any thermomechanical load history. Extensive experimentation provides verification of the analysis. AS4/3501-6 graphite/epoxy laminates are manufactured with three different layups to investigate ply thickness and orientation effects. Thermal specimens are cooled to progressively lower temperatures down to -184 C. After conditioning the specimens to each temperature, cracks are counted on their edges using optical microscopy and in their interiors by sanding to incremental depths. Tensile coupons are loaded monotonically to progressively higher loads until failure. Cracks are counted on the coupon edges after each loading. A data fit to all available results provides input parameters for the analysis and shows them to be material properties, independent of geometry and loading. Correlation between experiment and analysis is generally very good under both thermal and mechanical loading, showing the methodology to be a powerful, unified tool. Delayed crack initiation observed in a few cases is attributed to a lack of preexisting flaws assumed by the analysis. Some interactions between adjacent ply groups are attributed to local stress concentrations. These two effects are not captured by the analysis due to its global nature. The analysis is conservative in these cases and agrees well with data after the observed onset of cracking.

Maddocks, Jason R.↗

NASA NTRS1995

The effects of pure density evolution on the brightness distribution of cosmological gamma-ray bursts

In this work, we explore the effects of burst rate density evolution on the observed brightness distribution of cosmological gamma-ray bursts. Although the brightness distribution of gamma-ray bursts observed by the BATSE experiment has been shown to be consistent with a nonevolving source population observed to redshifts of order unity, evolution of some form is likely to be present in the gamma-ray bursts. Additionally, nonevolving models place significant constraints on the range of observed burst luminosities, which are relaxed if evolution of the burst population is present. In this paper, three analytic forms of density evolution are examined. In general, forms of evolution with densities that increase monotonically with redshift require that the BATSE data correspond to bursts at larger redshifts, or to incorporate a wider range of burst luminosities, or both. Independent estimates of the maximum observed redshift in the BATSE data and/or the range of luminosity from which a large fraction of the observed bursts are drawn therefore allow for constraints to be placed on the amount of evolution that may be present in the burst population. Specifically, if recent measurements obtained from analysis of the BATSE duration distribution of the actual limiting redshift in the BATSE data at z(sub lim) = 2 are correct, the BATSE N(P) distribution in a Lambda = 0 universe is inconsistent at a level of approximately 3 alpha with nonevolving gamma-ray bursts and some form of evolution in the population is required. The sense of this required source evolution is to provide a higher density, larger luminosities, or both with increasing redshift.

Horack, J. M.↗

NASA NTRS1995

Elevated temperature axial and torsional fatigue behavior of Haynes 188

The results are reported for high-temperature axial and torsional low-cycle fatigue experiments performed at 760 C in air on thin-walled tubular specimens of Haynes 188, a wrought cobalt-based superalloy. Data are also presented for mean coefficient of thermal expansion, elastic modulus, and shear modulus at various temperatures from room to 1000 C, and monotonic and cyclic stress-strain curves in tension and in shear at 760 C. This data set is used to evaluate several multiaxial fatigue life models (most were originally developed for room temperature multiaxial life prediction) including von Mises equivalent strain range (ASME Boiler and Pressure Code), Manson-Halford, modified multiaxiality factor (proposed in this paper), modified Smith-Watson-Topper, and Fatemi-Socie-Kurath. At von Mises equivalent strain ranges (the torsional strain range divided by the square root of 3, taking the Poisson's ratio to be 0.5), torsionally strained specimens lasted, on average, factors of 2 to 3 times longer than axially strained specimens. The modified multiaxiality factor approach shows promise as a useful method of estimating torsional fatigue life from axial fatigue data at high temperatures. Several difficulties arose with the specimen geometry and extensometry used in these experiments. Cracking at extensometer probe indentations was a problem at smaller strain ranges. Also, as the largest axial and torsional strain range fatigue tests neared completion, a small amount of specimen buckling was observed.

Bonacuse, Peter J.↗

NASA NTRS1995

Inertial acceleration as a measure of linear vection: An alternative to magnitude estimation

The present study focused on the development of a procedure to assess perceived self-motion induced by visual surround motion - vection. Using an apparatus that permitted independent control of visual and inertial stimuli, prone observers were translated along their head x-axis (fore/aft). The observers' task was to report the direction of self-motion during passive forward and backward translations of their bodies coupled with exposure to various visual surround conditions. The proportion of 'forward' responses was used to calculate each observer's point of subjective equality (PSE) for each surround condition. The results showed that the moving visual stimulus produced a significant shift in the PSE when data from the moving surround condition were compared with the stationary surround and no-vision condition. Further, the results indicated that vection increased monotonically with surround velocities between 4 and 40/s. It was concluded that linear vection can be measured in terms of changes in the amplitude of whole-body inertial acceleration required to elicit equivalent numbers of 'forward' and 'backward' self-motion reports.

Carpenter-Smith, Theodore R.↗

NASA NTRS1995

Dust in the Small Magellanic Cloud

We discuss simultaneous dust model fits to our extinction and polarization data for the Small Magellanic Cloud (SMC) using existing dust models. Dust model fits to the wavelength dependent polarization are possible for stars with small lambda(sub max). They generally imply size distributions which are narrower and have smaller average sizes compared to those in the Galaxy. The best fits for the extinction curves are obtained with a power law size distribution. The typical, monotonic SMC extinction curve can be well fit with graphite and silicate grains if a small fraction of the SMC carbon is locked up in the grains. Amorphous carbon and silicate grains also fit the data well.

Rodrigues, C. V.↗

NASA NTRS1995

Flow properties along field lines in a 3-D tilted-dipole geometry

A 3-D MHD simulation of a global, tilted-dipole solar wind flow pattern is analyzed to determine flow properties along individual magnetic field lines. In the model, flow conditions near the Sun are chosen to provide a reasonable match to the interplanetary configuration prevailing during the recent south polar passage by Ulysses, i.e., a streamer belt inclined approximately 30 deg to the solar equator and speeds ranging from 325-800 km/s. Field lines all across the stream pattern are traced from 1 to 10 AU by following the motion of marker particles embedded in the flow. It is found that those field lines threading the core of the interaction region are subject to significant latitudinal and relative longitudinal displacement over this range of heliocentric distance. Thus, observations taken at a fixed latitude in the inner solar system sample, over the course of a solar rotation, field lines which connect to a range of latitudes in the outer heliosphere. Maps of the field line displacements are presented to help visualize these connections. In addition, it is found that depending upon the location relative to the CIR structure, the radial evolution of fluid elements frozen to different field lines can deviate considerably from that of the canonical solar wind. That is, for selected subsets of field lines, large speed changes (not just at shocks) can be experienced; the density variation can be far from 1/r(exp 2), and the magnetic field intensity need not decay monotonically with distance.

Pizzo, V. J.↗

NASA NTRS1995

The 3-D effects in the long-term solar wind speed rise observed by Voyager 2 in early 1994

In early 1994, Voyager 2 at 42-43 AU near heliolatitude 10 deg S observed over a period of approximately 100 days a remarkable sequence of quasi-recurrent stream fronts, wherein the background (ambient) speed rose steadily from approximately 450 to approximately 550 km/s while the mean period of the streams decreased from the usual 25 days down to approximately 20 days. A qualitative explanation for this effect can be derived from IMP observations, which show that the amplitude of the stream structure at 1 AU increased monotonically in late 1993, concurrent with major secular evolution in the corona. The reduction in period, then, amounts to a doppler shift due to the progressive overtaking of successively faster streams in the sequence. Attempts to model this process quantitatively with 1-D dynamic simulations falter on three accounts: (1) the reduction in period is overestimated, (2) the simulation predicts many more fronts surviving to 43 AU than are observed by Voyager; (3) the density variations are much too large. It is argued that inclusion of the 3-D geometry in the simulation would resolve most all these shortcomings. Using a series of calculations executed with 1-D, 2-D, and 3-D MHD models of hypothetical tilted-dipole flows, we show that: (1) the radial propagation velocities of 3-D fronts are less than those of 1-D or 2-D fronts, owing to the tilt of (and increased shearing across) the interaction surfaces hence the overtaking rate of successive streams is reduced; (2) in a tilted-dipole geometry, the reverse fronts should largely disappear from the equatorial plane by 43 AU, effectively halving the number of fronts to be observed (see companion paper on predominance of forward fronts at Voyager); and (3) the density enhancements would be much smaller than predicted by a 1-D model.

Pizzo, V. J.↗

NASA NTRS1996

The modelling of symmetric airfoil vortex generators

An experimental study is conducted to determine the dependence of vortex generator geometry and impinging flow conditions on shed vortex circulation and crossplane peak vorticity for one type of vortex generator. The vortex generator is a symmetric airfoil having a NACA 0012 cross-sectional profile. The geometry and flow parameters varied include angle-of-attack alfa, chordlength c, span h, and Mach number M. The vortex generators are mounted either in isolation or in a symmetric counter-rotating array configuration on the inside surface of a straight pipe. The turbulent boundary layer thickness to pipe radius ratio is delta/R = 0. 17. Circulation and peak vorticity data are derived from crossplane velocity measurements conducted at or about 1 chord downstream of the vortex generator trailing edge. Shed vortex circulation is observed to be proportional to M, alfa, and h/delta. With these parameters held constant, circulation is observed to fall off in monotonic fashion with increasing airfoil aspect ratio AR. Shed vortex peak vorticity is also observed to be proportional to M, alfa, and h/delta. Unlike circulation, however, peak vorticity is observed to increase with increasing aspect ratio, reaching a peak value at AR approx. 2.0 before falling off.

Reichert, B. A.↗