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At least 685 records · Page 38

Development of the GeoNEX Level 2G Products: Exploiting the Diurnal Variability of TOA Reflectance in Atmospheric Correction

This study develops a new atmospheric correction algorithm to generate the Level 2G products, in particular the gap-filled Surface Reflectance at 10-minute time steps, for the Geostationary-NASA Earth Exchange (GeoNEX) project. The algorithm is based on the MODIS MAIAC (Multi-Angle Implementation of Atmospheric Correction) framework but with significant modifications to exploit angular/temporal information from the diurnal variability of the GeoNEX L1G TOA (Top-of-Atmosphere) reflectance. The algorithm starts by evaluating the roughness/smoothness of the diurnal time series of the TOA reflectance. Because rapid changes in TOA reflectance are generally caused by passing clouds or shadows, rough segments of the time series are automatically filtered out while the smooth segments are further tested for brightness and temperature to identify clear-sky and snow-free observations. Next the algorithm runs the MAIAC RTM (Radiative Transfer Model) to retrieve the Ross-Thick-Li-Sparse (RTLS) BRDF model parameters and the daily-mean atmospheric optical depth (AOD) that allow the RTM to optimally simulate the observed diurnal variability of clear-sky TOA reflectance. Once the initial RTLS parameters are retrieved after the algorithm’s burn-in period, they are used as the prior information to predict the AOD level for the next days, while the subsequent clear-sky observations are used to make necessary adjustments to the RTLS parameters in an continuous fashion. This “prediction-analysis” cycle is then iterated to process the full time series of the L1G data, skipping only total-cloudy days or when surface snow is detected. We tested the algorithm over a list of selected AERONET sites. The retrieved results (the daily mean AOD and the RTLS parameters) reasonably agree with the ground-based measurements. Importantly, the results indicate that the diurnal cycles of surface reflectance are continuous functions of the illumination-view geometry. Thus we can use the retrieved RTLS model to accurately fill in data gaps on partial cloudy days. Also, our algorithm is totally independent from the traditional approaches based on the use of spectral band ratios between the shortwave infrared (e.g., 2.2µm) and the visible (e.g., 0.47µm and 0.64µm) bands. Our results thus demonstrate that the high-frequent diurnal geostationary observations contain unique information that helps us improve atmospheric correction of remote sensing data.

GeoNEX↗

Analytic theory for the selection of Saffman-Taylor fingers in the presence of thin film effects

An analytic theory is presented for the width selection of Saffman-Taylor fingers in the presence of thin film effect. In the limit of small capillary number Ca and small gap to width ratio epsilon, such that epsilon much less than Ca much less than 1, it is found that fingers with relative width lambda less than 1/2 are possible such that lambda squared (1-lambda)/(1-2 lambda) = k epsilon/Ca(sup 3/2), where the positive constant k depends on the branch of solution and equals 2.776 for the first branch. A fully nonlinear analysis is necessary in this problem even to obtain the correct scaling law. It is also shown how in principle, the selection rule for arbitrary Ca can be obtained.

Tanveer, S.↗

The effect of inlet swirl on the dynamics of long annular seals in centrifugal pumps

This paper describes additional results from a continuing research program which aims to identify the dynamics of long annular seals in centrifugal pumps. A seal test rig designed at Heriot-Watt University and commissioned at Weir Pumps Research Laboratory in Alloa permits the identification of mass, stiffness, and damping coefficients using a least-squares technique based on the singular value decomposition method. The analysis is carried out in the time domain using a multi-fiequency forcing function. The experimental method relies on the forced excitation of a flexibly supported stator by two hydraulic shakers. Running through the stator embodying two symmetrical balance drum seals is a rigid rotor supported in rolling element bearings. The only physical connection between shaft and stator is the pair of annular gaps filled with pressurized water discharged axially. The experimental coefficients obtained from the tests are compared with theoretical values.

Ismail, M.↗

Modeling and Evaluation of Canted Coil Springs as High Temperature Seal Preloading Devices

Future reusable launch vehicles will require advanced structural seals. This includes propulsion seals along edges and hinge lines in hypersonic engines, and control surface seals for movable flaps and elevons on proposed reentry vehicles. Seals must remain in sealing engagement with opposing surfaces, for multiple missions, even though the seal gap may be opening and closing due to thermal and structural loads. To meet this requirement either the seals themselves must be resilient or there must be a resilient structural element behind the seals. Case Western Reserve University is working with NASA s Glenn Research Center to develop more resilient high temperature seal components and preloading devices. Results are presented for a finite element analysis of a canted coil spring that is being considered as a high temperature seal preloading device. This type of spring is a leading candidate due to its ability to provide nearly constant force over a large deflection. The finite element analyses were verified by comparing them to experimental results of canted coil springs of three different stiffnesses, measured at Glenn Research Center. Once validated the parameterized model was combined with a scripting algorithm to assess the effects of key spring design variables (wire diameter, coils per inch, cant amplitude, eccentricity, and spring width) on spring stiffness and maximum Von Mises stress to aid in subsequent design.

Oswald, Jay J.↗

Design of the ARES Mars Airplane and Mission Architecture

Significant technology advances have enabled planetary aircraft to be considered as viable science platforms. Such systems fill a unique planetary science measurement gap, that of regional-scale, near-surface observation, while providing a fresh perspective for potential discovery. Recent efforts have produced mature mission and flight system concepts, ready for flight project implementation. This paper summarizes the development of a Mars airplane mission architecture that balances science, implementation risk and cost. Airplane mission performance, flight system design and technology maturation are described. The design, analysis and testing completed demonstrates the readiness of this science platform for use in a Mars flight project.

Braun, Robert D.↗

An Overview of the Patch Integral Method (PIM), a New Heat Transfer Analysis Tool for Hypersonic Wind Tunnel Facilities at NASA Langley

NASA Langley’s hypersonic wind tunnels are heavily leveraged for planetary missions. The data collection method in these tunnels is thermography, and surface temperature measurements of the model surface are collected and reduced to produce surface heating data, as seen in Fig 1. However, during model injection, no temperature data are collected, and thus conventional, integral heat transfer methods cannot be used to solve for surface heating. A method was developed in the 1990’s to reduce this data despite the data gap, known as the step approximation method. The method assumes that the film coefficient behaves as a step function, the model is semi-infinite, and thermal properties are constant. With these simplifying assumptions, a Laplace transform can be performed to result in an equation that takes the initial temperature of the model and a temperature at some point in time to back out the film coefficient at that time. This is the method that is used in the current thermographic data reduction software, IHEAT. While computationally light-weight, the step approximation has several issues associated with it. The time-history of temperature is not accounted for, which is vital as heat transfer is an integral process. Additionally, the required semi-infinite assumption is unnecessary and might be violated during runtime. Thermal variation of material properties can have a sizeable impact on heating results and are not modeled by the method. This method also takes multiple seconds to “collapse” to a steady state, which is undesirable from both a facility and data reduction standpoint. The method is also very sensitive to the “effective time” approximation, an approximation of when heating instantaneously starts (which is a nonphysical simplification), and a small variation in this value can result in an error in heating results.

J. S. Cheatwood↗

Unsteady Turbine Blade and Tip Heat Transfer Due to Wake Passing

The geometry and the flow conditions of the first stage turbine blade of GE s E3 engine have been used to obtain the unsteady three-dimensional blade and tip heat transfer. The isothermal wall boundary condition was used. The effect of the upstream wake of the first stage vane was of interest and was simulated by provision of a gust type boundary condition upstream of the blades. A one blade periodic domain was used. The consequence of this choice was explored in a preliminary study which showed little difference in the time mean heat transfer between 1:1 and 2:3 vane/blade domains. The full three-dimensional computations are of the blade having a clearance gap of 2 percent the span. Comparison between the time averaged unsteady and steady heat transfer is provided. It is shown that there is a significant difference between the steady and time mean of unsteady blade heat transfer in localized regions. The differences on the suction side of the blade in the near hub and near tip regions were found to be rather significant. Steady analysis underestimated the blade heat transfer by as much as 20 percent as compared to the time average obtained from the unsteady analysis. As for the blade tip, the steady analysis and the unsteady analysis gave results to within 2 percent.

Ameri, Ali A.↗

Atmospheric Correction at AERONET Locations: A New Science and Validation Data Set

This paper describes an AERONET-based Surface Reflectance Validation Network (ASRVN) and its dataset of spectral surface bidirectional reflectance and albedo based on MODIS TERRA and AQUA data. The ASRVN is an operational data collection and processing system. It receives 50x50 square kilometer subsets of MODIS L1B data from MODAPS and AERONET aerosol and water vapor information. Then it performs an accurate atmospheric correction for about 100 AERONET sites based on accurate radiative transfer theory with high quality control of the input data. The ASRVN processing software consists of L1B data gridding algorithm, a new cloud mask algorithm based on a time series analysis, and an atmospheric correction algorithm. The atmospheric correction is achieved by fitting the MODIS top of atmosphere measurements, accumulated for 16-day interval, with theoretical reflectance parameterized in terms of coefficients of the LSRT BRF model. The ASRVN takes several steps to ensure high quality of results: 1) cloud mask algorithm filters opaque clouds; 2) an aerosol filter has been developed to filter residual semi-transparent and sub-pixel clouds, as well as cases with high inhomogeneity of aerosols in the processing area; 3) imposing requirement of consistency of the new solution with previously retrieved BRF and albedo; 4) rapid adjustment of the 16-day retrieval to the surface changes using the last day of measurements; and 5) development of seasonal back-up spectral BRF database to increase data coverage. The ASRVN provides a gapless or near-gapless coverage for the processing area. The gaps, caused by clouds, are filled most naturally with the latest solution for a given pixels. The ASRVN products include three parameters of LSRT model (k(sup L), k(sup G), k(sup V)), surface albedo, NBRF (a normalized BRF computed for a standard viewing geometry, VZA=0 deg., SZA=45 deg.), and IBRF (instantaneous, or one angle, BRF value derived from the last day of MODIS measurement for specific viewing geometry) for MODIS 500m bands 1-7. The results are produced daily at resolution of 1 km in gridded format. We also provide cloud mask, quality flag and a browse bitmap image. The new dataset can be used for a wide range of applications including validation analysis and science research.

Wang, Yujie↗

The Plasma Proton Environment within Saturn’s F-G Ring Gap as Observed by the Cassini Plasma Spectrometer Ion Mass Spectrometer during Saturn Orbit Insertion

We report on the detection of protons and the potential detection of H 2 + between Saturn's F and G rings based on Cassini Plasma Spectrometer (CAPS) Ion Mass Spectrometer (IMS) time-of-flight (TOF) composition measurements acquired during Saturn Orbit Insertion (SOI) outbound pass. The range in dipole L shell is 2.3 < L < 2.8. Initial results based on TOF data were presented in E. C. Sittler et al., 2017. Here we present the latest results of our analysis. During the SOI outbound pass between the F and G rings the CAPS IMS was in a mode of reduced post-acceleration voltage at -6 kV instead of the usual -14.6 kV. This reduced voltage still allows the analysis of protons since 6 keV protons are minimally scattered by the instrument’s ultrathin carbon foils when compared to heavier ions O + and O 2 + , scattering. Background noise from penetrating radiation and ghost peaks produced by foil-scattered O + ions within the instrument were considered in our analysis. The analysis allowed determination of the proton density, temperature and flow velocity, accounting for spacecraft potential by assuming a convected Maxwellian for the proton velocity distribution function. We find average proton density n P = 3.2 ± 1.1 #/cm 3 , proton temperature T P = 1.74 ± 0.12 eV, proton rotational flow speed V P = 24 ± 1.5 km/s and spacecraft potential  SC = -0.8 ±1.5 V. These results are compared with previous theoretical estimates of H + and H 2 + ions within Saturn’s inner magnetosphere.

E C Sittler↗

Radar sensitivity and antenna scan pattern study for a satellite-based Radar Wind Sounder (RAWS)

Modeling global atmospheric circulations and forecasting the weather would improve greatly if worldwide information on winds aloft were available. Recognition of this led to the inclusion of the LAser Wind Sounder (LAWS) system to measure Doppler shifts from aerosols in the planned for Earth Observation System (EOS). However, gaps will exist in LAWS coverage where heavy clouds are present. The RAdar Wind Sensor (RAWS) is an instrument that could fill these gaps by measuring Doppler shifts from clouds and rain. Previous studies conducted at the University of Kansas show RAWS as a feasible instrument. This thesis pertains to the signal-to-noise ratio (SNR) sensitivity, transmit waveform, and limitations to the antenna scan pattern of the RAWS system. A dop-size distribution model is selected and applied to the radar range equation for the sensitivity analysis. Six frequencies are used in computing the SNR for several cloud types to determine the optimal transmit frequency. the results show the use of two frequencies, one higher (94 GHz) to obtain sensitivity for thinner cloud, and a lower frequency (24 GHz) to obtain sensitivity for thinner cloud, and a lower frequency (24 GHz) for better penetration in rain, provide ample SNR. The waveform design supports covariance estimation processing. This estimator eliminates the Doppler ambiguities compounded by the selection of such high transmit frequencies, while providing an estimate of the mean frequency. the unambiguous range and velocity computation shows them to be within acceptable limits. The design goal for the RAWS system is to limit the wind-speed error to less than 1 ms(exp -1). Due to linear dependence between vectors for a three-vector scan pattern, a reasonable wind-speed error is unattainable. Only the two-vector scan pattern falls within the wind-error limits for azimuth angles between 16 deg to 70 deg. However, this scan only allows two components of the wind to be determined. As a result, a technique is then shown, based on the Z-R-V relationships, that permit the vertical component (i.e., rain) to be computed. Thus the horizontal wind components may be obtained form the covariance estimator and the vertical component from the reflectivity factor. Finally, a new candidate system is introduced which summarizes the parameters taken from previous RAWS studies, or those modified in this thesis.

Stuart, Michael A.↗

Predissociation in N2(C'4, 1 Sigma u +) and other N2 states and its importance in the atmospheres of Titan and Triton

The objectives of this program are to further the understanding of the upper atmospheres of Titan, Triton, and the Earth in terms of the observed emissions of the 13-14 eV states of N2. These states are generated at quite high rates, yet very little emission is observed from them. The reasons are complex, involving resonance trapping and predissociation, and it is desired to quantify the effects of predissociation, particularly on the c(sub 4)' 1 Sigma(sub u),(sup +) state of N2. Earlier experiments had indicated that predissociation of the c(sub 4)' state was of little importance, yet over the last two years a growing body of evidence has shown that for levels above v = 2, predissociation is in fact a major process. It is the v = 0 level for which production by electron bombardment and photoexcitation is highest, and so it has been most important to evaluate the effects of predissociation on this particular level. The goal has been to target c(sub 4)' (v = 0) for a thorough analysis, in which both the extent of predissociation as a function of rotational level and the atomic product branching ratio, where the only possible products are N(4S) + N(4S) and N(2D) + N(4S), are determined. For the first year of funding, the intention was to demonstrate two-photon excitation of the intermediate N2(a(sup 1) Pi(sub g)) state, so that the gap to the 13 eV energy region could be bridged, and then use a second laser to reach the c(sub 4)' state itself.

Slanger, Tom G.↗

Circuits Enhance Scientific Instruments and Safety Devices

Since its founding in 1958, NASA has pioneered the use of different frequencies on the electromagnetic spectrum - including X-ray, microwave, and infrared wavelengths - to gather information about distant celestial bodies. During the 1962 Mariner 2 mission, NASA used microwave radiometers that operated in the range of 15-23 gigahertz (GHz) to assess the surface temperature of Venus and to determine the percentage of water vapor in its atmosphere. Today, there is another area on the spectrum proving uniquely useful to scientists: the terahertz (THz) range, spanning from about 100 GHz-10,000 GHz. (1 THz equals approximately 1,000 GHz.) Terahertz frequencies span the lesser-known gap on the electromagnetic spectrum between microwave radiation and infrared (and visible) light, falling within the spectral range where most simple molecules resonate. This molecular resonance makes terahertz particularly useful for chemical spectroscopy and the remote sensing of specific molecules. In the 1990s, NASA began using frequencies above 300 GHz (more than an order of magnitude higher than the instrumentation on Mariner 2) to perform spectral analysis of molecular clouds and planetary atmospheres. Instruments using these higher frequencies have included the Microwave Limb Sounder (MLS) on the Upper Atmosphere Research Satellite (UARS), deployed from 1991-2001, and the Microwave Instrument for the Rosetta Orbiter (MIRO), launched in 2004. With UARS-MLS, NASA used advanced terahertz receivers to measure the emission signatures from atmospheric molecules, providing researchers with valuable data about the changes in the Earth s protective ozone layer. MIRO, set to rendezvous with the comet 67P Churyumov-Gerasimenko in 2014, will use terahertz instrumentation to analyze the comet s dust and gases. Although NASA has been a driving force behind the development of terahertz technology, scientific equipment for terahertz research - including transmitters, receivers, and basic test and measurement equipment - is not widely available, making scientific experiments in this range between traditional electronics and quantum photonics more costly and greatly limiting commercial development in the field. Given NASA s interest in studying distant bodies in space as well as in improving life on Earth, the Agency has collaborated with private industry to develop terahertz technologies.

Source record↗

Orbit Determination Adaptations for the Cassini Grand Finale

satellite encounters. Over this period, there have been several papers describing the orbit determination process and performance up through 2016 [1-5]. In April of 2017, Cassini will enter its Grand Finale mission phase when it will traverse the gap between the D-ring and the Saturn atmosphere twenty-two times before plunging deep into the atmosphere to end the mission. The lack of targeted satellite encounters during this period necessitates updates to the nominal Cassini Orbit Determination (OD) process. This paper describes these planned adaptations for the operation of the Grand Finale. During the Equinox and Solstice Mission Phase (2008-2016), navigation analysis has been divided into segments focused on two particular targeted satellite encounters, called an “arc”. Maneuvers in an arc were usually targeted to encounter B-plane position and time, so the OD state and covariance were mapped forward to the B-plane of the encounter within the arc. Trajectory dispersions during the Grand Finale need instead to be mapped to equator crossings and targeted Cartesian positions. In addition, trajectory arcs have typically covered a few orbital revolutions (~2-8 weeks), in order to span the time between two encounters. However, the Grand Finale will encompass five months of time without an encounter which necessitates an adjusted arc strategy. A modified arc strategy was developed based on OD behavior during long multi-rev periods between encounters in the year leading up to the Grand Finale. The OD covariance study conducted for the Grand Finale mission phase will also be examined.

Criddle, Kevin↗

Data Collection and Analysis Software Development for Rotor Dynamics Testing in Spin Laboratory

Gas turbine engine components undergo high rotational loading another complex environmental conditions. Such operating environment leads these components to experience damages and cracks that can cause catastrophic failure during flights. There are traditional crack detections and health monitoring methodologies currently being used which rely on periodic routine maintenances, nondestructive inspections that often times involve engine and components dis-assemblies. These methods do not also offer adequate information about the faults, especially, if these faults at subsurface or not clearly evident. At NASA Glenn research center, the rotor dynamics laboratory is presently involved in developing newer techniques that are highly dependent on sensor technology to enable health monitoring and prediction of damage and cracks in rotor disks. These approaches are noninvasive and relatively economical. Spin tests are performed using a subscale test article mimicking turbine rotor disk undergoing rotational load. Non-contact instruments such as capacitive and microwave sensors are used to measure the blade tip gap displacement and blade vibrations characteristics in an attempt develop a physics based model to assess/predict the faults in the rotor disk. Data collection is a major component in this experimental-analytical procedure and as a result, an upgrade to an older version of the data acquisition software which is based on LabVIEW program has been implemented to support efficiently running tests and analyze the results. Outcomes obtained from the tests data and related experimental and analytical rotor dynamics modeling including key features of the updated software are presented and discussed.

Rotor Dynamics↗

Preliminary analysis of hot spot factors in an advanced reactor for space electric power systems

The maximum fuel pin temperature for nominal operation in an advanced power reactor is 1370 K. Because of possible nitrogen embrittlement of the clad, the fuel temperature was limited to 1622 K. Assuming simultaneous occurrence of the most adverse conditions a deterministic analysis gave a maximum fuel temperature of 1610 K. A statistical analysis, using a synthesized estimate of the standard deviation for the highest fuel pin temperature, showed probabilities of 0.015 of that pin exceeding the temperature limit by the distribution free Chebyshev inequality and virtually nil assuming a normal distribution. The latter assumption gives a 1463 K maximum temperature at 3 standard deviations, the usually assumed cutoff. Further, the distribution and standard deviation of the fuel-clad gap are the most significant contributions to the uncertainty in the fuel temperature.

Lustig, P. H.↗

Steady internal flow and aerodynamic loads analysis of shuttle thermal protection system

An analytical model for calculation of ascent steady state tile loading was developed and validated with wind tunnel data. The analytical model is described and results are given. Results are given for loading due to shocks and skin friction. The analysis included calculation of internal flow (porous media flow and channel flow) to obtain pressures and integration of the pressures to obtain forces and moments on an insulation tile. A heat transfer program was modified by using analogies between heat transfer and fluid flow so that it could be used for internal flow calculation. The type of insulation tile considered was undensified reusable surface insulation (RSI) without gap fillers, and the location studied was the lower surface of the orbiter. Force and moment results are reported for parameter variations on surface pressure distribution, gap sizes, insulation permeability, and tile thickness.

Petley, D. H.↗

Formal Analysis of the Remote Agent Before and After Flight

This paper describes two separate efforts that used the SPIN model checker to verify deep space autonomy flight software. The first effort occurred at the beginning of a spiral development process and found five concurrency errors early in the design cycle that the developers acknowledge would not have been found through testing. This effort required a substantial manual modeling effort involving both abstraction and translation from the prototype LISP code to the PROMELA language used by SPIN. This experience and others led to research to address the gap between formal method tools and the development cycle used by software developers. The Java PathFinder tool which directly translates from Java to PROMELA was developed as part of this research, as well as automatic abstraction tools. In 1999 the flight software flew on a space mission, and a deadlock occurred in a sibling subsystem to the one which was the focus of the first verification effort. A second quick-response "cleanroom" verification effort found the concurrency error in a short amount of time. The error was isomorphic to one of the concurrency errors found during the first verification effort. The paper demonstrates that formal methods tools can find concurrency errors that indeed lead to loss of spacecraft functions, even for the complex software required for autonomy. Second, it describes progress in automatic translation and abstraction that eventually will enable formal methods tools to be inserted directly into the aerospace software development cycle.

Havelund, Klaus↗

Non-Gimbaled Antenna Pointing: Summary of Results and Analysis

There is considerable interest at this time in developing small satellites for quick-turnaround missions to investigate near-earth phenomena from space. One problem to be solved in mission planning is the means of communication between the control infrastructure in the ground segment and the satellite in the space segment. A nominal small-satellite mission design often includes an omni-directional or similar wide-pattern antenna on the satellite and a dedicated ground station for telemetry, tracking, and command support. These terminals typically provide up to 15 minutes of coverage during an orbit that is within the visibility of the ground station; however, not all orbits will pass over the ground station so that coverage gaps will exist in the data flow. To overcome this general limitation on data transmission for low-earth orbiting satellites, the Space Network (SN), operated by the National Aeronautics and Space Administration, (NASA) has been designed to transmit data to and from user satellites through the Tracking and Data Relay Satellites (TDRS) in geostationary orbit and interfacing to the White Sands Complex (WSC) in New Mexico for the data's ground entry point. The advantage of the SN over a fixed ground station is that all low-earth-satellite orbits will be within the visibility area of at least one TDRS within the SN for a large part of the orbit and the potential exists to establish a communications link if the user satellite can point an antenna in the direction of any one of the relay satellites. Within NASA, there is considerable interest in seeing that small satellite developers are aware of the advantages to the SN and that designs to include the SN are part of the satellite design. Small satellite users have not often considered using the SN because of: (1) The 26-dB link penalty differential between direct broadcast to a ground station and transmission through a TDRS to the ground; (2) The class of satellite is too small to support high-gain antennas and associated attitude control and drive electronics; (3) The class of satellites is severely weight and power limited; (4) There are perceived problems in scheduling communications for this class of user on the SN. This report addresses the potential for SN access using non-gimbaled, i.e. fixed-pointed, antennas in the design of the small satellite using modest transmission power to achieve the necessary space-to-ground transmissions. The advantage of using the SN is in the reduction of mission costs arising from using the SN infrastructure instead of a dedicated, proprietary ground station using a similar type of communications package. From the simulations and analysis presented, we will show that a modest satellite configuration can be used with the space network to achieve the data transmission goals of a number of users and thereby rival the performance achieved with proprietary ground stations. In this study, we will concentrate on the return data link (from the user satellite through a TDRS to the ground data entry point). The forward command link (from the ground data entry point through a TDRS to the user satellite) will usually be a lower data rate service and the data volume will also be considerably lower than the return link's requirement. Therefore, we assume that if the return link requirements are satisfied, then the forward link requirements can also be satisfied.

Horan, Stephen↗