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

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At least 325 records · Page 18
All sourcesNASA NTRSDOE OSTI
NASA NTRS2000

Bubble Formation from Wall Orifice in Liquid Cross-Flow Under Low Gravity

Two-phase flows present a wide variety of applications for spacecraft thermal control systems design. Bubble formation and detachment is an integral part of the two phase flow science. The objective of the present work is to experimentally investigate the effects of liquid cross-flow velocity, gas flow rate, and orifice diameter on bubble formation in a wall-bubble injection configuration. Data were taken mainly under reduced gravity conditions but some data were taken in normal gravity for comparison. The reduced gravity experiment was conducted aboard the NASA DC-9 Reduced Gravity Aircraft. The results show that the process of bubble formation and detachment depends on gravity, the orifice diameter, the gas flow rate, and the liquid cross-flow velocity. The data are analyzed based on a force balance, and two different detachment mechanisms are identified. When the gas momentum is large, the bubble detaches from the injection orifice as the gas momentum overcomes the attaching effects of liquid drag and inertia. The surface tension force is much reduced because a large part of the bubble pinning edge at the orifice is lost as the bubble axis is tilted by the liquid flow. When the gas momentum is small, the force balance in the liquid flow direction is important, and the bubble detaches when the bubble axis inclination exceeds a certain angle.

Nahra, Henry K.↗
NASA NTRS1980

Plasma collection by high voltage spacecraft at low earth orbit

A computer model of the three-dimensional sheath formation and plasma current collection by high voltage spacecraft has been developed. By using new space charge density and plasma collection algorithms, it is practical to perform calculations for large, complex spacecraft. The model uses NASCAP compatible objects and geometries. Results indicate that ion focusing observed in the laboratory during high voltage collection experiments is probably due to voltage gradients on the collecting surfaces.

Katz, I.↗
NASA NTRS1998

Bubble Formation and Detachment in Liquid Flow Under Normal and Reduced Gravity

Two-phase flows are present in a wide variety of applications for spacecraft thermal control systems design. Bubble formation and detachment is an integral part of the two-phase flow science. The authors objective is to experimentally investigate the effects of liquid cross velocity, gas velocity, and nozzle diameter on the bubble diameter at detachment under reduced and normal gravity and under relatively low gas flow rates. Results from ground (1 g) and reduced gravity experiments will be presented in this paper. For the 1 g experiment, a flow loop was designed and built to accommodate a range of liquid and gas flow rates. The reduced gravity experiment was conducted on the NASA DC-9 reduced gravity platform using the two-phase flow loop qualified for operation on the low-gravity platform. Flow visualization is accomplished using a high speed 500 frames/s camera. The results suggest that the existence of buoyancy force contributes to the faster detachment of bubbles. Buoyancy helps the detachment process which results in smaller bubbles being formed. In reduced gravity, although drag force is present, the virtual non-existence of buoyant force results in larger bubbles and longer times for detachment. Theoretical predictions are also presented in this paper and seem to agree with the experimental results.

Nahra, Henry K.↗
NASA NTRS2004

Formation Control of the MAXIM L2 Libration Orbit Mission

The Micro-Arcsecond Imaging Mission (MAXIM), a proposed concept for the Structure and Evolution of the Universe (SEU) Black Hole Imaging mission, is designed to make a ten million-fold improvement in X-ray image clarity of celestial objects by providing better than 0.1 microarcsecond imaging. To achieve mission requirements, MAXIM will have to improve on pointing by orders of magnitude. This pointing requirement impacts the control and design of the formation. Currently the architecture is comprised of 25 spacecraft, which will form the sparse apertures of a grazing incidence X-ray interferometer covering the 0.3-10 keV bandpass. This configuration will deploy 24 spacecraft as optics modules and one as the detector. The formation must allow for long duration continuous science observations and also for reconfiguration that permits re-pointing of the formation. In this paper, we provide analysis and trades of several control efforts that are dependent upon the pointing requirements and the configuration and dimensions of the MAXIM formation. We emphasize the utilization of natural motions in the Lagrangian regions that minimize the control efforts and we address both continuous and discrete control via LQR and feedback linearization. Results provide control cost, configuration options, and capabilities as guidelines for the development of this complex mission.

Folta, David↗
NASA NTRS2024

Starling Formation-Flying Optical Experiment: Initial Operations and Flight Results

This paper presents initial flight results for angles-only navigation of a swarm of small spacecraft, conducted during the Starling Formation-Flying Optical Experiment (StarFOX). StarFOX is one of four experiments aboard the NASA Starling mission, which consists of four CubeSats launched in July 2023. Angles-only methods apply inter-satellite bearing angles obtained by on-board cameras for navigation, increasing satellite autonomy and enabling new mission concepts. Nevertheless, prior flight demonstrations have only featured one observer and target and have relied upon a-priori target orbit knowledge for initialization, translational maneuvers to resolve target range, and external absolute orbit updates to maintain convergence. StarFOX overcomes these limitations by applying the angles-only Absolute and Relative Trajectory Measurement System (ARTMS), which integrates three novel algorithms. Image Processing detects and tracks multiple targets in images, using multi-hypothesis methods and domain-specific kinematic modeling, and computes target bearing angles. Batch Orbit Determination computes initial swarm orbit estimates from bearing angle batches, via iterative batch least squares and sampling of the weakly observable target range. Sequential Orbit Determination leverages an adaptive, efficient unscented Kalman filter with nonlinear models to refine swarm state estimates over time. Multi-observer measurements shared over an intersatellite link are seamlessly fused to enable robust absolute and relative orbit determination. Initial StarFOX experiments are the first demonstrations of autonomous angles-only navigation for a satellite swarm, including multi-target and multi-observer relative navigation; autonomous initialization of navigation for unknown targets; and simultaneous absolute and relative orbit determination. Relative navigation accuracy of 1% (single observer) and 0.2% (multi-observer) of the inter-satellite range is achieved under challenging measurement conditions. Results demonstrate promising performance with regards to ongoing StarFOX campaigns and the application of angles-only navigation to future distributed missions.

Relative Navigation↗
NASA NTRS2009

Protocol for Communication Networking for Formation Flying

An application-layer protocol and a network architecture have been proposed for data communications among multiple autonomous spacecraft that are required to fly in a precise formation in order to perform scientific observations. The protocol could also be applied to other autonomous vehicles operating in formation, including robotic aircraft, robotic land vehicles, and robotic underwater vehicles. A group of spacecraft or other vehicles to which the protocol applies could be characterized as a precision-formation- flying (PFF) network, and each vehicle could be characterized as a node in the PFF network. In order to support precise formation flying, it would be necessary to establish a corresponding communication network, through which the vehicles could exchange position and orientation data and formation-control commands. The communication network must enable communication during early phases of a mission, when little positional knowledge is available. Particularly during early mission phases, the distances among vehicles may be so large that communication could be achieved only by relaying across multiple links. The large distances and need for omnidirectional coverage would limit communication links to operation at low bandwidth during these mission phases. Once the vehicles were in formation and distances were shorter, the communication network would be required to provide high-bandwidth, low-jitter service to support tight formation-control loops. The proposed protocol and architecture, intended to satisfy the aforementioned and other requirements, are based on a standard layered-reference-model concept. The proposed application protocol would be used in conjunction with conventional network, data-link, and physical-layer protocols. The proposed protocol includes the ubiquitous Institute of Electrical and Electronics Engineers (IEEE) 802.11 medium access control (MAC) protocol to be used in the datalink layer. In addition to its widespread and proven use in diverse local-area networks, this protocol offers both (1) a random- access mode needed for the early PFF deployment phase and (2) a time-bounded-services mode needed during PFF-maintenance operations. Switching between these two modes could be controlled by upper-layer entities using standard link-management mechanisms. Because the early deployment phase of a PFF mission can be expected to involve multihop relaying to achieve network connectivity (see figure), the proposed protocol includes the open shortest path first (OSPF) network protocol that is commonly used in the Internet. Each spacecraft in a PFF network would be in one of seven distinct states as the mission evolved from initial deployment, through coarse formation, and into precise formation. Reconfiguration of the formation to perform different scientific observations would also cause state changes among the network nodes. The application protocol provides for recognition and tracking of the seven states for each node and for protocol changes under specified conditions to adapt the network and satisfy communication requirements associated with the current PFF mission phase. Except during early deployment, when peer-to-peer random access discovery methods would be used, the application protocol provides for operation in a centralized manner.

Jennings, Esther↗
NASA NTRS2017

Results of the Apogee-Raising Campaign of the Magnetospheric Multiscale Mission

This paper describes the apogee-raising campaign of the Magnetospheric Multiscale mission, where the spacecraft increased their apogee radii from 12 to 25 Earth radii in a total of 98 maneuvers. These maneuvers included an initial formation resize set to spread the spacecraft apart for safety, 32 apogee-raise delta-v maneuvers, their associated slews, four perigee-raise maneuvers and the associated slews, and finally a set of maneuvers to get back into formation. These activities were all accomplished successfully and on schedule with no anomalies, and at a fuel consumption somewhat less than predicted. As a result, MMS was set up ready to carry out in situ studies of magnetic reconnection in the magnetotail, with sufficient fuel remaining for a significant extended mission.

Mission Design↗
NASA NTRS2004

Formation acquisition sensor for the Terrestrial Planet Finder (TPF) mission

The Terrestrial Planet Finder (TPF) pre-project, an element of NASA’s Origins Program, is currently investigating multiple implementation architectures for finding earth-like planets around other stars. One of the technologies being developed is the Formation Flying Interferometer (FFI). The FFI is envisioned to consist of up to seven spacecraft, each with an infrared telescope, flying in precise formation within +/- l cm of pre-determined trajectories for synchronized observations. The spacecraft to- spacecraft separations are variable between 20 m to 100m during observations to support various interferometer configurations in the planet-finding mode. The challenges involved with TPF autonomous operations, ranging from formation acquisition and formation maneuvering, to high precision formation flying during science observations are unprecedented for deep space missions. To meet these challenges, the Formation Sensor Testbed (FST) under the TPF technology program will develop and demonstrate the key technology of the formation acquisition sensor. Key performance targets for the acquisition sensor are an instantaneous 4phi-steradian field of view and simultaneous range and bearing-angle measurements for multiple spacecraft with accuracy better than 50 cm and 1 degree, respectively. In this paper, we will describe the TPF FFI mission concept, the key formation flying challenges, the acquisition sensor design, the key design challenges, and the current plan to mitigate these design challenges.

Purcell, George H. Jr.↗
NASA NTRS2002

Integrated Orbit and Attitude Control for a Nanosatellite with Power Constraints

HokieSat is a NASA Goddard sponsored spacecraft currently being built by students at Virginia Tech. HokieSat is part of the Ionospheric Observation Nanosatellite Formation (ION-F) project. The project involves spacecraft built by three schools: Virginia Tech (VT), Utah State University (USU), and University of Washington (UW). The three spacecraft are similar in design and will perform formation flying demonstrations, and make ionospheric measurements. HokieSat uses Pulsed Plasma Thrusters (PPTs) to maintain its position in the formation. There are two pairs of PPTs on HokieSat; their position on HokieSat's hexagonal cross-section is shown. Thrusters T(sub 2) and T(sub 3) provide translation control, and Thrusters TI and T4 can provide yaw steering. Any thruster can be fired individually. However because they share a capacitor, thrusters T(sub 1) and T(sub 2) or thrusters T(sub 3) and T(sub 4) cannot be fired simultaneously. Thrusters T(sub 2) T(sub 3) can be fired simultaneously, as well as thrusters T(sub 1) and T(sub 4). Each thruster provides an impulse-bit of 56 micronN-s and fires at a rate of 1 Hz. For translation control thrusters T2 and T3 are fired together providing an impulse-bit of 112 micronN-s. All four thrusters are positioned slightly above the center of mass, and therefore exert a torque on the spacecraft. Because there are no thrusters in the zenith-nadir directions, and the communication system requires that the spacecraft remain nadir-pointing, there is no way to thrust in the radial direction. The attitude of HokieSat is controlled by 3 orthogonal magnetic torque coils. Attitude control is achieved by forcing a current through the torque coils, which interacts with the Earth's magnetic field and creates a torque. Due to magnetic field interactions between the coils and PPTs, the two actuator systems cannot be used simultaneously, and any attitude or orbit control must be performed in a piecewise fashion. Power limitations place an additional constraint on the HokieSat control subsystem. When the spacecraft is in eclipse, the power subsystem can provide only enough power to operate vital spacecraft functions.

Berry, Matthew M.↗
NASA NTRS2004

Electromagnetic Formation Flight (EMFF) for Sparse Aperture Arrays

Traditional methods of actuating spacecraft in sparse aperture arrays use propellant as a reaction mass. For formation flying systems, propellant becomes a critical consumable which can be quickly exhausted while maintaining relative orientation. Additional problems posed by propellant include optical contamination, plume impingement, thermal emission, and vibration excitation. For these missions where control of relative degrees of freedom is important, we consider using a system of electromagnets, in concert with reaction wheels, to replace the consumables. Electromagnetic Formation Flight sparse apertures, powered by solar energy, are designed differently from traditional propulsion systems, which are based on V. This paper investigates the design of sparse apertures both inside and outside the Earth's gravity field.

Kwon, Daniel W.↗
NASA NTRS2000

Autonomous Relative Navigation for Formation-Flying Satellites Using GPS

The Goddard Space Flight Center is currently developing advanced spacecraft systems to provide autonomous navigation and control of formation flyers. This paper discusses autonomous relative navigation performance for a formation of four eccentric, medium-altitude Earth-orbiting satellites using Global Positioning System (GPS) Standard Positioning Service (SPS) and "GPS-like " intersatellite measurements. The performance of several candidate relative navigation approaches is evaluated. These analyses indicate that an autonomous relative navigation position accuracy of 1meter root-mean-square can be achieved by differencing high-accuracy filtered solutions if only measurements from common GPS space vehicles are used in the independently estimated solutions.

Gramling, Cheryl↗
NASA NTRS2001

Follow That Satellite: EO-1 Maneuvers Into Close Formation With Landsat-7

As the Landsat-7 (LS-7) spacecraft continued NASA's historic program of earth imaging begun over three decades ago, NASA launched the Earth Observing-1 (EO-1) spacecraft carrying examples of the next generation of LS instruments. The validation method for these instruments was to have EO-1 fly in a close formation behind LS-7 on the same World Reference System (WRS) path. From that formation hundreds of near-coincident images would be taken by each spacecraft and compared to evaluate improvements in the EO-1 instruments. This paper will address the mission analysis required to launch and maneuver EO-1 into the formation with LS-7 where instrument validation was to occur plus a summary of completing the formation acquisition. Each EO-1 launch opportunity that occurred on a different day of a LS-7 16-day repeat cycle required a separate and distinct maneuver profile.

DeFazio, Robert L.↗
NASA NTRS1992

Near Earth asteroid rendezvous

The Spacecraft Design Course is the capstone design class for the M.S. in astronautics at the Naval Postgraduate School. The Fall 92 class designed a spacecraft for the Near Earth Asteroid Rendezvous Mission (NEAR). The NEAR mission uses a robotic spacecraft to conduct up-close reconnaissance of a near-earth asteroid. Such a mission will provide information on Solar System formation and possible space resources. The spacecraft is intended to complete a NEAR mission as a relatively low-budget program while striving to gather as much information about the target asteroid as possible. A complete mission analysis and detailed spacecraft design were completed. Mission analysis includes orbit comparison and selection, payload and telemetry requirements, spacecraft configuration, and launch vehicle selection. Spacecraft design includes all major subsystems: structure, electrical power, attitude control, propulsion, payload integration, and thermal control. The resulting spacecraft demonstrates the possibility to meet the NEAR mission requirements using existing technology, 'off-the-shelf' components, and a relatively low-cost launch vehicle.

Source record↗
NASA NTRS2002

The StarLight mission: a formation-flying stellar interferometer

The StarLight mission is designed to validate the technologies of formation flying and stellar interferometry in space. The mission consists of two spacecraft in an earth-trailing orbit that formation fly over relative ranges of 40 to 600 m to an accuracy of 10 cm.

formation-flying↗
NASA NTRS2011

Magnetospheric Multiscale (MMS) Mission Attitude Ground System Design

This paper describes the attitude ground system (AGS) design to be used for support of the Magnetospheric MultiScale (MMS) mission. The AGS exists as one component of the mission operations control center. It has responsibility for validating the onboard attitude and accelerometer bias estimates, calibrating the attitude sensors and the spacecraft inertia tensor, and generating a definitive attitude history for use by the science teams. NASA's Goddard Space Flight Center (GSFC) in Greenbelt, Maryland is responsible for developing the MMS spacecraft, for the overall management of the MMS mission, and for mission operations. MMS is scheduled for launch in 2014 for a planned two-year mission. The MMS mission consists of four identical spacecraft flying in a tetrahedral formation in an eccentric Earth orbit. The relatively tight formation, ranging from 10 to 400 km, will provide coordinated observations giving insight into small-scale magnetic field reconnection processes. By varying the size of the tetrahedron and the orbital semi-major axis and eccentricity, and making use of the changing solar phase, this geometry allows for the study of both bow shock and magnetotail plasma physics, including acceleration, reconnection, and turbulence. The mission divides into two phases for science; these phases will have orbit dimensions of l.2xl2 Earth radii in the first phase and l.2x25 Earth radii in the second in order to study the dayside magnetopause and the nightside magnetotail, respectively. The orbital periods are roughly one day and three days for the two mission phases. Each of the four MMS spacecraft will be spin stabilized at 3 revolutions per minute (rpm), with the spin axis oriented near the ecliptic north pole but tipped approximately 2.5 deg towards the Sun line. The main body of each spacecraft will be an eight-sided platform with diameter of 3.4 m and height of 1.2 m. Several booms are attached to this central core: two axial booms of 14.9 m length, two radial magnetometer booms of 5 m length, and four radial -wire booms of 60 m length. Attitude and orbit control will use a set of axial and radial thrusters. A four-head star tracker and a slit-type digital Sun sensor (DSS) provide input for attitude determination. In addition, an accelerometer will be· used for closed-loop orbit maneuver control. The primary AGS product will be a daily definitive attitude history. Due to power limitations; the star tracker and accelerometer data will not be available at all times. However, tracker data from at least 10 percent of each orbit and continuous DSS data will be provided. An extended Kalman filter (EKF) will be used to estimate the three-axis attitude (i.e., spin axis orientation and spin phase) and rotation rate for all times when the tracker data is valid. For other times, the attitude is generated by assuming a constant angular momentum vector in the inertial frame. The DSS sun pulse will provide a timing signal to maintain an accurate spin phase. There will be times when the Sun is occulted and DSS data is not available. If this occurs at the start or end of a definitive attitude product, then the spin phase will be extrapolated using the mean rate determined by the EKF.

Sedlak, Joseph E.↗
NASA NTRS2010

Magnetospheric Multiscale (MMS) Mission Attitude Ground System Design

This paper describes the attitude ground system (AGS) design to be used for support of the Magnetospheric MultiScale (MMS) mission. The AGS exists as one component of the mission operations control center. It has responsibility for validating the onboard attitude and accelerometer bias estimates, calibrating the attitude sensors and the spacecraft inertia tensor, and generating a definitive attitude history for use by the science teams. NASA's Goddard Space Flight Center (GSFC) in Greenbelt, Maryland is responsible for developing the MMS spacecraft, for the overall management of the MMS mission, and for mission operations. MMS is scheduled for launch in 2014 for a planned two-year mission. The MMS mission consists of four identical spacecraft flying in a tetrahedral formation in an eccentric Earth orbit. The relatively tight formation, ranging from 10 to 400 km, will provide coordinated observations giving insight into small-scale magnetic field reconnection processes. By varying the size of the tetrahedron and the orbital semi-major axis and eccentricity, and making use of the changing solar phase, this geometry allows for the study of both bow shock and magnetotail plasma physics, including acceleration, reconnection, and turbulence. The mission divides into two phases for science; these phases will have orbit dimensions of 1.2 x 12 Earth radii in the first phase and 1.2x25 Earth radii in the second in order to study the dayside magnetopause and the nightside magnetotail, respectively. The orbital periods are roughly one day and three days for the two mission phases. Each of the four MMS spacecraft will be spin stabilized at 3 revolutions per minute (rpm), with the spin axis oriented near the ecliptic north pole but tipped approximately 2.5 deg towards the Sun line. The main body of each spacecraft will be an eight-sided platform with diameter of 3.4 m and height of 1.2 m. Several booms are attached to this central core: two axial booms of 14.9 m length, two radial magnetometer booms of 5 m length, and four radial wire booms of 60 m length. Attitude and orbit control will use a set of axial and radial thrusters. A four-head star tracker and a slit-type digital Sun sensor (DSS) provide input for attitude determination. In addition, an accelerometer will be used for closed-loop orbit maneuver control. The primary AGS product will be a daily definitive attitude history. Due to power limitations, the star tracker and accelerometer data will not be available at all times. However, tracker data from at least 10 percent of each orbit and continuous DSS data will be provided. An extended Kalman filter (EKF) will be used to estimate the three-axis attitude (i.e., spin axis orientation and spin phase) and rotation rate for all times when the tracker data is valid. For other times, the attitude is generated by assuming a constant angular momentum vector in the inertial frame. The DSS sun pulse will provide a timing signal to maintain an accurate spin phase. There will be times when the Sun is occulted and DSS data is not available. If this occurs at the start or end of a definitive attitude product, then the spin phase will be extrapolated using the mean rate determined by the EKF.

Sedlak, Joseph E.↗
NASA NTRS2007

NASA Tech Briefs, May 2007

Topics include: Noise-Canceling Helmet Audio System; Program Analyzes Spacecraft/Ground Radio Links; Two-Way Communication Using RFID Equipment and Techniques; Six-Message Electromechanical Display System; Scanning Terahertz Heterodyne Imaging Systems; Master Clock and Time-Signal-Distribution System; Synchronous Phase-Resolving Flash Range Imaging; Integrated Radial Probe Transition From MMIC to Waveguide; Bar-Code System for a Microbiological Laboratory; MMIC Amplifier Produces Gain of 10 dB at 235 GHz; Mapping Nearby Terrain in 3D by Use of a Grid of Laser Spots; Digital Beam Deflectors Based Partly on Liquid Crystals; Narrow-Band WGM Optical Filters With Tunable FSRs; Better Finite-Element Analysis of Composite Shell Structures; Computing Spacecraft-Pointing Vectors for Limb Tracking; Enhanced Master Controller Unit Tester; Rover Graphical Simulator; Increasing Durability of Flame-Sprayed Strain Gauges; Multifunctional, High-Temperature Nanocomposites; Multilayer Impregnated Fibrous Thermal Insulation Tiles; Radiation-Shielding Polymer/Soil Composites; Film/Adhesive Processing Module for Fiber-Placement Processing of Composites; Fabrication of Submillimeter Axisymmetric Optical Components; Electrochemical Disposal of Hydrazines in Water; Statistical Model of Evaporating Multicomponent Fuel Drops; Resistively Heated SiC Nozzle for Generating Molecular Beams; Compact Packaging of Photonic Millimeter-Wave Receiver; Diffractive Combiner of Single-Mode Pump Laser-Diode Beams; Wide-Band, High-Quantum-Efficiency Photodetector; A Robustly Stabilizing Model Predictive Control Algorithm; Modeling Evaporation of Drops of Different Kerosenes; Development of Vapor-Phase Catalytic Ammonia Removal System; Several Developments in Space Tethers; Design Concept for a Nuclear Reactor-Powered Mars Rover; Formation-Initialization Algorithm for N Spacecraft; and DNSs of Multicomponent Gaseous and Drop-Laden Mixing Layers Achieving Transition to Turbulence.

Source record↗
NASA NTRS1999

Investigating Atmospheric Effects on Impact Ejecta Morphology: Possible Tool for Determining Past Climate Conditions on Mars?

The combined use of impact crater morphology and mechanics provides important information on the physical conditions of both planetary atmospheres and planetary and asteroid surfaces present during crater formation, while an understanding of the rate of crater production on the surface of asteroids provides information of their surface and spin rate evolution. The research performed with support from this project improves our understanding of (1) the mechanics of impact cratering in order to gain insights on the evolution of these physical surface conditions on planets with atmospheres and asteroids, and (2) how impact flux across an asteroid surface may vary due to anisotropic distribution of impactors in the solar system. As part of this project, we have undertaken three studies. In the first study, we investigate atmospheric effects on the morphology of ejecta excavated during a cratering event in order to determine the atmospheric and target conditions from observed crater morphologies. In the second study, we use the physical and morphological consequences of oblique impacts on an asteroid to understand how the asteroid Mathilde (recently imaged by the Near Earth Asteroid Rendezvous - NEAR- spacecraft) could have survived the formation of five giant craters. In a third study, we use a Monte Carlo method to calculate the impact flux on an asteroid given a distribution of impactors on elliptical orbits. In the following section, we present the result obtained from all three studies.

Appleby, John F.↗