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At least 1,693 records · Page 94
All sourcesNASA NTRSDOE OSTI
NASA NTRS2011

Numerical Treatment of Degenerate Diffusion Equations via Feller's Boundary Classification, and Applications

A numerical method is devised to solve a class of linear boundary-value problems for one-dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite-difference scheme, grid, and treatment of the boundary data. Second-order accuracy, unconditional stability, and unconditional convergence of solutions of the finite-difference scheme to a constant as the time-step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration.

Cacio, Emanuela↗
NASA NTRS2012

End-to-end Coronagraphic Modeling Including a Low-order Wavefront Sensor

To evaluate space-based coronagraphic techniques, end-to-end modeling is necessary to simulate realistic fields containing speckles caused by wavefront errors. Real systems will suffer from pointing errors and thermal and motioninduced mechanical stresses that introduce time-variable wavefront aberrations that can reduce the field contrast. A loworder wavefront sensor (LOWFS) is needed to measure these changes at a sufficiently high rate to maintain the contrast level during observations. We implement here a LOWFS and corresponding low-order wavefront control subsystem (LOWFCS) in end-to-end models of a space-based coronagraph. Our goal is to be able to accurately duplicate the effect of the LOWFS+LOWFCS without explicitly evaluating the end-to-end model at numerous time steps.

coronagraphs↗
NASA NTRS2012

Atmospheric Dust Modeling from Meso to Global Scales with the Online NMMB/BSC-Dust Model Part 2: Experimental Campaigns in Northern Africa

The new NMMB/BSC-Dust model is intended to provide short to medium-range weather and dust forecasts from regional to global scales. It is an online model in which the dust aerosol dynamics and physics are solved at each model time step. The companion paper (Perez et al., 2011) develops the dust model parameterizations and provides daily to annual evaluations of the model for its global and regional configurations. Modeled aerosol optical depth (AOD) was evaluated against AERONET Sun photometers over Northern Africa, Middle East and Europe with correlations around 0.6-0.7 on average without dust data assimilation. In this paper we analyze in detail the behavior of the model using data from the Saharan Mineral dUst experiment (SAMUM-1) in 2006 and the Bodele Dust Experiment (BoDEx) in 2005. AOD from satellites and Sun photometers, vertically resolved extinction coefficients from lidars and particle size distributions at the ground and in the troposphere are used, complemented by wind profile data and surface meteorological measurements. All simulations were performed at the regional scale for the Northern African domain at the expected operational horizontal resolution of 25 km. Model results for SAMUM-1 generally show good agreement with satellite data over the most active Saharan dust sources. The model reproduces the AOD from Sun photometers close to sources and after long-range transport, and the dust size spectra at different height levels. At this resolution, the model is not able to reproduce a large haboob that occurred during the campaign. Some deficiencies are found concerning the vertical dust distribution related to the representation of the mixing height in the atmospheric part of the model. For the BoDEx episode, we found the diurnal temperature cycle to be strongly dependant on the soil moisture, which is underestimated in the NCEP analysis used for model initialization. The low level jet (LLJ) and the dust AOD over the Bodélé are well reproduced. The remaining negative AOD bias (due to underestimated surface wind speeds) can be substantially reduced by decreasing the threshold friction velocity in the model.

photometers↗
NASA NTRS2014

A Review of Current and Prospective Factors for Classification of Civil Unmanned Aircraft Systems

While progress is being made on integrating unmanned aircraft systems (UAS) into our national airspace on a broad scale, much work remains to establish appropriate certification standards and operational procedures, particularly with respect to routine commercial operations. This paper summarizes research to examine the extent to which today's civil aircraft taxonomy applies to UAS, and, if needed, how that taxonomy could be amended to better cover different UAS designs and operations. Factors that shape the current taxonomy, as defined in the Federal Aviation Regulations, were assessed for applicability to UAS, potential incompatibilities were identified, and additional factors were proposed that might be useful for an updated aircraft taxonomy intended to cover UAS. The results suggest the possibility of constructing new groups in the taxonomy for UAS under a restricted category that share common airworthiness standards. Establishing distinct groups for UAS and associated standards that enable low risk operations for compensation or hire could be a timely step toward full integration. Such a step would allow the civil aviation industry and regulators to gain valuable experience with UAS while carefully controlling access and potential harm to the aviation system as a whole.

Hayhurst, Kelly J.↗
NASA NTRS2014

Thermal Analysis System

A thermal fluid system modeler including a plurality of individual components. A solution vector is configured and ordered as a function of one or more inlet dependencies of the plurality of individual components. A fluid flow simulator simulates thermal energy being communicated with the flowing fluid and between first and second components of the plurality of individual components. The simulation extends from an initial time to a later time step and bounds heat transfer to be substantially between the flowing fluid, walls of tubes formed in each of the individual components of the plurality, and between adjacent tubes. Component parameters of the solution vector are updated with simulation results for each of the plurality of individual components of the simulation.

DiStefano, III, Frank James↗
NASA NTRS2014

An Examination of Parameters Affecting Large Eddy Simulations of Flow Past a Square Cylinder

Separated flow over a bluff body is analyzed via large eddy simulations. The turbulent flow around a square cylinder features a variety of complex flow phenomena such as highly unsteady vortical structures, reverse flow in the near wall region, and wake turbulence. The formation of spanwise vortices is often times artificially suppressed in computations by either insufficient depth or a coarse spanwise resolution. As the resolution is refined and the domain extended, the artificial turbulent energy exchange between spanwise and streamwise turbulence is eliminated within the wake region. A parametric study is performed highlighting the effects of spanwise vortices where the spanwise computational domain's resolution and depth are varied. For Re=22,000, the mean and turbulent statistics computed from the numerical large eddy simulations (NLES) are in good agreement with experimental data. Von-Karman shedding is observed in the wake of the cylinder. Mesh independence is illustrated by comparing a mesh resolution of 2 million to 16 million. Sensitivities to time stepping were minimized and sampling frequency sensitivities were nonpresent. While increasing the spanwise depth and resolution can be costly, this practice was found to be necessary to eliminating the artificial turbulent energy exchange.

Mankbadi, M. R.↗
NASA NTRS2009

Neutralizer Hollow Cathode Simulations and Comparisons with Ground Test Data

The fidelity of electric propulsion physics-based models depends largely on the validity of their predictions over a range of operating conditions and geometries. In general, increased complexity of the physics requires more extensive comparisons with laboratory data to identify the region(s) that lie outside the validity of the model assumptions and to quantify the uncertainties within its range of application. This paper presents numerical simulations of neutralizer hollow cathodes at various operating conditions and orifice sizes. The simulations were performed using a two-dimensional axisymmetric model that solves numerically a relatively extensive system of conservation laws for the partially ionized gas in these devices. A summary of the comparisons between simulation results and Langmuir probe measurements is provided. The model has also been employed to provide insight into recent ground test observations of the neutralizer cathode in NEXT. It is found that a likely cause of the observed keeper voltage drop is cathode orifice erosion. However, due to the small magnitude of this change, is approx. 0.5 V (less than 5% of the beginning-of-life value) over 10 khrs, and in light of the large uncertainties of the cathode material sputtering yield at low ion energies, other causes cannot be excluded. Preliminary simulations to understand transition to plume mode suggest that in the range of 3-5 sccm the existing 2-D model reproduces fairly well the rise of the keeper voltage in the NEXT neutralizer as observed in the laboratory. At lower flow rates the simulation produces oscillations in the keeper current and voltage that require prohibitively small time-steps to resolve with the existing algorithms.

electric propulsion↗
NASA NTRS2012

Joint Chance-Constrained Dynamic Programming

This paper presents a novel dynamic programming algorithm with a joint chance constraint, which explicitly bounds the risk of failure in order to maintain the state within a specified feasible region. A joint chance constraint cannot be handled by existing constrained dynamic programming approaches since their application is limited to constraints in the same form as the cost function, that is, an expectation over a sum of one-stage costs. We overcome this challenge by reformulating the joint chance constraint into a constraint on an expectation over a sum of indicator functions, which can be incorporated into the cost function by dualizing the optimization problem. As a result, the primal variables can be optimized by a standard dynamic programming, while the dual variable is optimized by a root-finding algorithm that converges exponentially. Error bounds on the primal and dual objective values are rigorously derived. We demonstrate the algorithm on a path planning problem, as well as an optimal control problem for Mars entry, descent and landing. The simulations are conducted using a real terrain data of Mars, with four million discrete states at each time step.

Ono, Masahiro↗
NASA NTRS2014

Evaluating an Automated Approach for Monitoring Forest Disturbances in the Pacific Northwest from Logging, Fire and Insect Outbreaks with Landsat Time Series Data

Forests are the largest aboveground sink for atmospheric carbon (C), and understanding how they change through time is critical to reduce our C-cycle uncertainties. We investigated a strong decline in Normalized Difference Vegetation Index (NDVI) from 1982 to 1991 in Pacific Northwest forests, observed with the National Ocean and Atmospheric Administration's (NOAA) series of Advanced Very High Resolution Radiometers (AVHRRs). To understand the causal factors of this decline, we evaluated an automated classification method developed for Landsat time series stacks (LTSS) to map forest change. This method included: (1) multiple disturbance index thresholds; and (2) a spectral trajectory-based image analysis with multiple confidence thresholds. We produced 48 maps and verified their accuracy with air photos, monitoring trends in burn severity data and insect aerial detection survey data. Area-based accuracy estimates for change in forest cover resulted in producer's and user's accuracies of 0.21 +/- 0.06 to 0.38 +/- 0.05 for insect disturbance, 0.23 +/- 0.07 to 1 +/- 0 for burned area and 0.74 +/- 0.03 to 0.76 +/- 0.03 for logging. We believe that accuracy was low for insect disturbance because air photo reference data were temporally sparse, hence missing some outbreaks, and the annual anniversary time step is not dense enough to track defoliation and progressive stand mortality. Producer's and user's accuracy for burned area was low due to the temporally abrupt nature of fire and harvest with a similar response of spectral indices between the disturbance index and normalized burn ratio. We conclude that the spectral trajectory approach also captures multi-year stress that could be caused by climate, acid deposition, pathogens, partial harvest, thinning, etc. Our study focused on understanding the transferability of previously successful methods to new ecosystems and found that this automated method does not perform with the same accuracy in Pacific Northwest forests. Using a robust accuracy assessment, we demonstrate the difficulty of transferring change attribution methods to other ecosystems, which has implications for the development of automated detection/attribution approaches. Widespread disturbance was found within AVHRR-negative anomalies, but identifying causal factors in LTSS with adequate mapping accuracy for fire and insects proved to be elusive. Our results provide a background framework for future studies to improve methods for the accuracy assessment of automated LTSS classifications.

Landsat↗
NASA NTRS2016

A Nonlinear Modal Aeroelastic Solver for FUN3D

A nonlinear structural solver has been implemented internally within the NASA FUN3D computational fluid dynamics code, allowing for some new aeroelastic capabilities. Using a modal representation of the structure, a set of differential or differential-algebraic equations are derived for general thin structures with geometric nonlinearities. ODEPACK and LAPACK routines are linked with FUN3D, and the nonlinear equations are solved at each CFD time step. The existing predictor-corrector method is retained, whereby the structural solution is updated after mesh deformation. The nonlinear solver is validated using a test case for a flexible aeroshell at transonic, supersonic, and hypersonic flow conditions. Agreement with linear theory is seen for the static aeroelastic solutions at relatively low dynamic pressures, but structural nonlinearities limit deformation amplitudes at high dynamic pressures. No flutter was found at any of the tested trajectory points, though LCO may be possible in the transonic regime.

Goldman, Benjamin D.↗
NASA NTRS2016

Computational Analysis of Multi-Rotor Flows

Interactional aerodynamics of multi-rotor flows has been studied for a quadcopter representing a generic quad tilt-rotor aircraft in hover. The objective of the present study is to investigate the effects of the separation distances between rotors, and also fuselage and wings on the performance and efficiency of multirotor systems. Three-dimensional unsteady Navier-Stokes equations are solved using a spatially 5th order accurate scheme, dual-time stepping, and the Detached Eddy Simulation turbulence model. The results show that the separation distances as well as the wings have significant effects on the vertical forces of quadroror systems in hover. Understanding interactions in multi-rotor flows would help improve the design of next generation multi-rotor drones.

Multi-Rotor Flows↗
NASA NTRS2016

A First Look at the Upcoming SISO Space Reference FOM

Simulation is increasingly used in the space domain for several purposes. One example is analysis and engineering, from the mission level down to individual systems and subsystems. Another example is training of space crew and flight controllers. Several distributed simulations have been developed for example for docking vehicles with the ISS and for mission training, in many cases with participants from several nations. Space based scenarios are also used in the "Simulation Exploration Experience", SISO's university outreach program. We have thus realized that there is a need for a distributed simulation interoperability standard for data exchange within the space domain. Based on these experiences, SISO is developing a Space Reference FOM. Members of the product development group come from several countries and contribute experiences from projects within NASA, ESA and other organizations. Participants represent government, academia and industry. The first version will focus on handling of time and space. The Space Reference FOM will provide the following: (i) a flexible positioning system using reference frames for arbitrary bodies in space, (ii) a naming conventions for well known reference frames, (iii) definitions of common time scales, (iv) federation agreements for common types of time management with focus on time stepped simulation, and (v) support for physical entities, such as space vehicles and astronauts. The Space Reference FOM is expected to make collaboration politically, contractually and technically easier. It is also expected to make collaboration easier to manage and extend.

Crues, Edwin↗
NASA NTRS2015

Time Accurate Unsteady Pressure Loads Simulated for the Space Launch System at a Wind Tunnel Condition

Using the Fully Unstructured Three-Dimensional (FUN3D) computational fluid dynamics code, an unsteady, time-accurate flow field about a Space Launch System configuration was simulated at a transonic wind tunnel condition (Mach = 0.9). Delayed detached eddy simulation combined with Reynolds Averaged Naiver-Stokes and a Spallart-Almaras turbulence model were employed for the simulation. Second order accurate time evolution scheme was used to simulate the flow field, with a minimum of 0.2 seconds of simulated time to as much as 1.4 seconds. Data was collected at 480 pressure taps at locations, 139 of which matched a 3% wind tunnel model, tested in the Transonic Dynamic Tunnel (TDT) facility at NASA Langley Research Center. Comparisons between computation and experiment showed agreement within 5% in terms of location for peak RMS levels, and 20% for frequency and magnitude of power spectral densities. Grid resolution and time step sensitivity studies were performed to identify methods for improved accuracy comparisons to wind tunnel data. With limited computational resources, accurate trends for reduced vibratory loads on the vehicle were observed. Exploratory methods such as determining minimized computed errors based on CFL number and sub-iterations, as well as evaluating frequency content of the unsteady pressures and evaluation of oscillatory shock structures were used in this study to enhance computational efficiency and solution accuracy. These techniques enabled development of a set of best practices, for the evaluation of future flight vehicle designs in terms of vibratory loads.

Alter, Stephen J.↗
NASA NTRS2016

The Impact of New Trends in Satellite Launches on Orbital Debris Environment

The main goal of this study is to examine the impact of new trends in satellite launch activities on the orbital debris environment and collision risk. Starting from the launch of the first artificial satellite in 1957, space borne technology has become an indispensable part of our lives. More than 6,000 satellites have been launched into Earth orbit. Though the annual number of satellites launched stayed flat for many decades, the trend has recently changed. The satellite market has been undergoing a major evolution with new space companies replacing the traditional approach of deploying a few large, complex and costly satellites with an approach to use a multitude of smaller, less complex and cheaper satellites. This new approach creates a sharp increase in the number of satellites and so the historic trends are no longer representative. As a foundation for this study, a scenario for satellite deployments based on the publicly announced future satellite missions has been developed. These constellation-deploying companies include, but are not limited to, Blacksky, CICERO, EROS, Landmapper, Leosat, Northstar, O3b, OmniEarth, OneWeb, Orbcomm, OuterNet, PlanetIQ, Planet Labs, Radarsat, RapidEye Next Generation, Sentinel, Skybox, SpaceX, and Spire. Information such as the annual number of launches, the number of orbital planes to be used by the constellation, as well as apogee, perigee, inclination, spacecraft mass and area were included or approximated. Besides the production of satellites, a widespread ongoing effort to enhance orbital injection capabilities will allow delivery of more spacecraft more accurately into Earth orbits. A long list of companies such as Microcosm, Rocket Lab, Firefly Space Systems, Sierra Nevada Corporation and Arca Space Corporation are developing new launch vehicles dedicated for small satellites. There are other projects which intend to develop interstages with propulsive capabilities which will allow the deployment of satellites into their desired orbits beyond the restrictions of the launch vehicle used. These near future orbital injection technologies are also covered in the developed scenario. Using the above-mentioned background information, this study aims to examine how the orbital debris environment will be affected from the new dynamics of the emerging space markets. We developed a simulation tool that is capable of propagating the objects in a given deployment scenario with variable-sized time-steps as small as one second. Over the course of the run, the software also detects collisions; additional debris objects are then created according to the NASA breakup model and are fed back into the simulation framework. Examining the simulation results, the total number of particles to accumulate in different orbits can be monitored and the number of conjunctions can be tracked to assess the collision risks. The simulation makes it possible to follow the short- and long-term effects of a particular satellite or constellation on the space environment. Likewise, the effects of changes in the debris environment on a particular satellite or constellation can be evaluated. It is authors hope that the results of this paper and further utilization of the developed simulation tool will assist in the investigation of more accurate deorbiting metrics to replace the generic 25-year disposal guidelines, as well as to guide future launches toward more sustainable and safe orbits.

New Space↗
NASA NTRS2014

Implementation of Active Thermal Control (ATC) for the Soil Moisture Active and Passive (SMAP) Radiometer

NASA's Earth Observing Soil Moisture Active and Passive (SMAP) Mission is scheduled to launch in November 2014 into a 685 kilometer near-polar, sun-synchronous orbit. SMAP will provide comprehensive global mapping measurements of soil moisture and freeze/thaw state in order to enhance understanding of the processes that link the water, energy, and carbon cycles. The primary objectives of SMAP are to improve worldwide weather and flood forecasting, enhance climate prediction, and refine drought and agriculture monitoring during its three year mission. The SMAP instrument architecture incorporates an L-band radar and an L-band radiometer which share a common feed horn and parabolic mesh reflector. The instrument rotates about the nadir axis at approximately 15 revolutions per minute, thereby providing a conically scanning wide swath antenna beam that is capable of achieving global coverage within three days. In order to make the necessary precise surface emission measurements from space, the electronics and hardware associated with the radiometer must meet tight short-term (instantaneous and orbital) and long-term (monthly and mission) thermal stabilities. Maintaining these tight thermal stabilities is quite challenging because the sensitive electronics are located on a fast spinning platform that can either be in full sunlight or total eclipse, thus exposing them to a highly transient environment. A passive design approach was first adopted early in the design cycle as a low-cost solution. With careful thermal design efforts to cocoon and protect all sensitive components, all stability requirements were met passively. Active thermal control (ATC) was later added after the instrument Preliminary Design Review (PDR) to mitigate the threat of undetected gain glitches, not for thermal-stability reasons. Gain glitches are common problems with radiometers during missions, and one simple way to avoid gain glitches is to use the in-flight set point programmability that ATC affords to operate the radiometer component away from the problematic temperature zone. A simple ThermXL model (10 nodes) was developed to exercise quick trade studies among various proposed control algorithms: Modified P control vs. PI control. The ThermXL results were then compared with the detailed Thermal Desktop (TD) model for corroboration. Once done, the simple ThermXL model was used to evaluate parameter effects such as temperature digitization, heater size and gain margin, time step, and voltage variation of power supply on the ATC performance. A Modified P control algorithm was implemented into the instrument flight electronics based on the ThermXL results. The thermal short-term stability margin decreased by 10 percent with ATC and a wide temperature error band (plus or minus 0.1 degrees Centigrade) compared to the original passive thermal design. However, a tighter temperature error band (plus or minus 0.1 degrees Centigrade) increased the thermal short-term stability margin by a factor of three over the passive thermal design. The current ATC design provides robust thermal control, tighter stability, and greater in-flight flexibility even though its implementation was prompted by non-thermal performance concerns.

Mikhaylov, Rebecca↗
NASA NTRS2014

Water Balance in the Amazon Basin from a Land Surface Model Ensemble

Despite recent advances in land surfacemodeling and remote sensing, estimates of the global water budget are still fairly uncertain. This study aims to evaluate the water budget of the Amazon basin based on several state-ofthe- art land surface model (LSM) outputs. Water budget variables (terrestrial water storage TWS, evapotranspiration ET, surface runoff R, and base flow B) are evaluated at the basin scale using both remote sensing and in situ data. Meteorological forcings at a 3-hourly time step and 18 spatial resolution were used to run 14 LSMs. Precipitation datasets that have been rescaled to matchmonthly Global Precipitation Climatology Project (GPCP) andGlobal Precipitation Climatology Centre (GPCC) datasets and the daily Hydrologie du Bassin de l'Amazone (HYBAM) dataset were used to perform three experiments. The Hydrological Modeling and Analysis Platform (HyMAP) river routing scheme was forced with R and B and simulated discharges are compared against observations at 165 gauges. Simulated ET and TWS are compared against FLUXNET and MOD16A2 evapotranspiration datasets andGravity Recovery and ClimateExperiment (GRACE)TWSestimates in two subcatchments of main tributaries (Madeira and Negro Rivers).At the basin scale, simulated ET ranges from 2.39 to 3.26 mm day(exp -1) and a low spatial correlation between ET and precipitation indicates that evapotranspiration does not depend on water availability over most of the basin. Results also show that other simulated water budget components vary significantly as a function of both the LSM and precipitation dataset, but simulated TWS generally agrees with GRACE estimates at the basin scale. The best water budget simulations resulted from experiments using HYBAM, mostly explained by a denser rainfall gauge network and the rescaling at a finer temporal scale.

GPCP↗
NASA NTRS2016

A First Look at the Upcoming SISO Space Reference FOM

Spaceflight is difficult, dangerous and expensive; human spaceflight even more so. In order to mitigate some of the danger and expense, professionals in the space domain have relied, and continue to rely, on computer simulation. Simulation is used at every level including concept, design, analysis, construction, testing, training and ultimately flight. As space systems have grown more complex, new simulation technologies have been developed, adopted and applied. Distributed simulation is one those technologies. Distributed simulation provides a base technology for segmenting these complex space systems into smaller, and usually simpler, component systems or subsystems. This segmentation also supports the separation of responsibilities between participating organizations. This segmentation is particularly useful for complex space systems like the International Space Station (ISS), which is composed of many elements from many nations along with visiting vehicles from many nations. This is likely to be the case for future human space exploration activities. Over the years, a number of distributed simulations have been built within the space domain. While many use the High Level Architecture (HLA) to provide the infrastructure for interoperability, HLA without a Federation Object Model (FOM) is insufficient by itself to insure interoperability. As a result, the Simulation Interoperability Standards Organization (SISO) is developing a Space Reference FOM. The Space Reference FOM Product Development Group is composed of members from several countries. They contribute experiences from projects within NASA, ESA and other organizations and represent government, academia and industry. The initial version of the Space Reference FOM is focusing on time and space and will provide the following: (i) a flexible positioning system using reference frames for arbitrary bodies in space, (ii) a naming conventions for well-known reference frames, (iii) definitions of common time scales, (iv) federation agreements for common types of time management with focus on time stepped simulation, and (v) support for physical entities, such as space vehicles and astronauts. The Space Reference FOM is expected to make collaboration politically, contractually and technically easier. It is also expected to make collaboration easier to manage and extend.

Mueller, Bjorn↗
NASA NTRS2016

The Impact of New Trends in Satellite Launches on the Orbital Debris Environment

The main goal of this study is to examine the impact of new trends in satellite launch activities on the orbital debris environment and collision risk. As a foundation for the study, we developed a deployment scenario for satellites and associated rocket bodies based on publicly announced future missions. The upcoming orbital injection technologies, such as the new launch vehicles dedicated for small spacecraft and propulsive interstages, are also considered in this scenario. We then used a simulation tool developed in-house to propagate the objects within this scenario using variable-sized time-steps as small as one second to detect conjunctions between objects. The simulation makes it possible to follow the short- and long-term effects of a particular satellite or constellation in the space environment. Likewise, the effects of changes in the debris environment on a particular satellite or constellation can be evaluated. It is our hope that the results of this paper and further utilization of the developed simulation tool will assist in the investigation of more accurate deorbiting metrics to replace the generic 25-year disposal guidelines, as well as to guide future launches toward more sustainable and safe orbits.

conjunction analysis↗