Search NASA⌕ Search

SEARCH · Search NASA

Results for “Particle Drag”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 253 records · Page 14
All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Creation of fully vectorized FORTRAN code for integrating the movement of dust grains in interplanetary environments

The main objective is to improve the performance of a specific FORTRAN computer code from the Planetary Sciences Division of NASA/Johnson Space Center when used on a modern vectorizing supercomputer. The code is used to calculate orbits of dust grains that separate from comets and asteroids. This code accounts for influences of the sun and 8 planets (neglecting Pluto), solar wind, and solar light pressure including Poynting-Robertson drag. Calculations allow one to study the motion of these particles as they are influenced by the Earth or one of the other planets. Some of these particles become trapped just beyond the Earth for long periods of time. These integer period resonances vary from 3 orbits of the Earth and 2 orbits of the particles to as high as 14 to 13.

Colquitt, Walter↗
DOE OSTI2021

CFD-DEM study of bubble properties in a cylindrical fluidized bed of Geldart Group D particles and comparison with prior MRI data

Numerical simulations of a freely bubbling cylindrical fluidized bed are carried out using a coupled computational fluid dynamics and discrete element method (CFD-DEM) model and compared to recent experimental data. The experiments were conducted using high-resolution and high-frequency magnetic resonance imaging providing high-fidelity data of the bubbling within a central 10 mm slice of the bed. Qualitatively, we nd more regular (i.e., less chaotic) structures in the simulated beds than observed experimentally. Quantitatively, however, the bubble diameter and number of bubbles as a function of height within the bed is predicted well by the base model. Unfortunately, the regularity in the simulations manifests as a considerable discrepancy in the speed of the (dense) emulsion phase. The simulated velocity probability distribution functions show an accumulation of low-speed regions and deficiency of high-speed regions. A simple parametric study of the base model is also carried out considering many of the most common CFD-DEM modeling parameters. It is found that the fluid grid size, geometry resolution, transfer kernel and drag law did not have a significant effect on bubble or particle dynamics.

42 ENGINEERING↗
NASA NTRS2003

Design and Preparation of a Particle Dynamics Space Flight Experiment, SHIVA

ABSTRACT This paper describes the flight experiment, supporting ground science, and the design rationale for project SHIVA (Spaceflight Holography Investigation in a Virtual Apparatus). SHIVA is a fundamental study of particle dynamics in fluids in microgravity. Gravity often dominates the equations of motion of a particle in a fluid, so microgravity provides an ideal environment to study the other forces, such as the pressure and viscous drag and especially the Basset history force. We have developed diagnostic recording methods using holography to save all of the particle field optical characteristics, essentially allowing the experiment to be transferred from space back to earth in what we call the "virtual apparatus" for on-earth microgravity experimentation. We can quantify precisely the three-dimensional motion of sets of particles, allowing us to test and apply new analytical solutions developed by members of the team as reported in the 2001 Conference (Banff, Canada). In addition to employing microgravity to augment the fundamental study of these forces, the resulting data will allow us to quantify and understand the ISS environment with great accuracy. This paper shows how we used both experiment and theory to identify and resolve critical issues and produce an optimal the study. We examined the response of particles of specific gravity from 0.1 to 20, with radii from 0.2 to 2mm. to fluid oscillation at frequencies up to 80 Hz with amplitudes up to 200 microns. To observe some of the interesting effects predicted by the new solutions requires the precise location of the position of a particle in three dimensions. To this end we have developed digital holography algorithms that enable particle position location to a small fraction of a pixel in a CCD array. The spaceflight system will record holograms both on film and electronically. The electronic holograms can be downlinked providing real time data, essentially acting like a remote window into the ISS experimental chamber. Ground experiments have provided input to a flight system design that can meet the requirements for a successful experiment on ISS. Moreover the ground experiments have provided a definitive, quantitative observation of the Basset History force over a wide range of conditions. Results of the ground experiments, the flight experiment design, preliminary flight hardware design, and data analysis procedures are reported.

Trolinger, James↗
NASA NTRS2023

Can the Streaming Instability Form the First Planetesimals in Globally Turbulent Protoplanetary Disks?

Understanding the formation of the first planetesimals remains key to deciphering the history of planet formation within our own solar system and beyond. Evidence from the vast meteorite record [e.g., 1] as well as observations [e.g., 2] strongly suggest that the first planetesimals, and perhaps giant planet core accretion occurred well within the first million years of disk evolution. Moreover, chemical and lithological mixing as well as observations of line-broadening in protoplanetary disks (PPDs) [e.g., 3-6] suggest that the solar system nebula in this epoch was weakly-to-moderately turbulent in the regions where particle growth is of the greatest interest [7-8]. It is a well-known though that global hydrodynamic turbulence complicates particle growth due to a slew of barriers that can slow or even stall particle or aggregate growth at pebble sizes (with corresponding small particle Stokes numbers St) that can lead to loss to the central star via radial drift before planetesimals can ever form [9,10], requiring that some other mechanism come into play that collects growth-frustrated pebbles into gravitationally bound multi-km bodies – objects that are “born big” [11]. The current leading candidate for such a “leap-frog” mechanism is the so-called Streaming Instability (SI), a gas-drag mediated momentum exchange resonance in which the relative velocity between the particle component and a rotating gaseous fluid can lead to high densities in the particle field [12], which has been invoked in a number of recent PPD models that include a turbulent intensity α [e.g., 13-14] as the defacto mechanism for planetesimal formation if conditions for the SI (depending on particle St and the solids-to-gas mass ratio) are satisfied. However, these previous works use conditions established from occurrence studies for the onset of SI in laminar disks [15-17] in which the only source of turbulence is that self-generated by the settling particle layer, and not externally driven global turbulence. Recent analytical theories of the SI subject to global turbulence predict much more stringent conditions for the effectiveness of the SI than the laminar case [18-19]. Thus, whether the efficient operation of the SI can be attained in realistic models of the solar nebula have yet to be established. In this work, we ask whether the conditions under which the SI can produce gravitationally bound particle overdensities can actually be met in the first million years of evolution in globally turbulent PPDs.

P R Estrada↗
NASA NTRS1994

Interplanetary dust particles collected from the stratosphere: Physical, chemical, and mineralogical properties and implications for their sources

The suggestion that significant quantities of interplanetary dust are produced by both main-belt asteroids and comets is based on the Infrared Astronomical Satellite (IRAS) detection of dust trails or bands associated with these objects. Gravitational focusing strongly biases all near-Earth collections of interplanetary dust in favor of particles with the lowest geocentric velocities, that is the dust from main-belt asteroids spiraling into the Sun under the influence of Poynting-Robertson radiation drag. The major dust bands in the main-belt appear to be associated with the catastrophic disruptions which produced the Eos, Themis and Koronis families of asteroids. If dust particles are produced in the catastrophic collision process, then Poynting-Robertson radiation drag is such an efficient transport mechanism from the main-belt to 1 AU that near-Earth collections of interplanetary dust should include, and perhaps be dominated by, this material. Interplanetary dust particles from 5 to 100 micrometers in diameter have been recovered from the stratosphere of the Earth by NASA sampling aircraft since the mid-1970s. The densities of a large fraction of these interplanetary dust particles are significantly lower than the densities of their constituent silicate mineral phases, indicating significant porosites. The majority of the particles are chemically and mineralogically similar to, but not identical to, the carbonaceous chondrite meteorites. Most stony interplanetary dust particles have carbon contents exceeding those of Allende, a carbonaceous chondrite meteorite having a low albedo. Higher albedo particles corresponding to S-type asteroids are underrepresented or absent from the stratospheric collections, and primitive carbonaceous particles seem to be overrepresented in the stratospheric collections compared to the fraction of main-belt asteroids classified as primitive. This suggests that much of the interplanetary dust may be generated by a stochastic process, probably preferentially sampling a few most recent collisional events.

Flynn, George J.↗
DOE OSTI2024

Direct numerical simulation of open-channel flow over a heterogeneous particle bed at low relative submergence

In this article, this study investigates turbulent open-channel flows over beds of irregularly arranged particles, using direct numerical simulations at a friction Reynolds number of Re T = 300⁠. Two distinct cases are examined: a polydisperse bed (P800) composed of multiple layers of randomly distributed spheres of varying sizes, and a monodisperse bed (M1015) formed by a random distribution of uniform sized spheres, with a bottommost single layer of varied-sized particles to introduce realistic randomness. Our investigation unveils a rich network of low- and high-speed streaks within the flow field, exhibiting distinctive behaviors in different bed configurations. The P800 case presents a poorly organized flow pattern induced by the varied particle sizes and arrangements, while the M1015 case shows a more regular flow pattern, marked by larger streaks. We also observe that total wall shear stress is substantially influenced by surface roughness-induced drag, extending beyond the effects documented in existing studies of open-channel flows. The present study reveals intricate secondary flow patterns over irregular particle beds. Large-scale circulations are discerned around particle crests in the P800 case and localized circulations with increased turbulence in the M1015 case. Furthermore, analysis of Reynolds stress tensor components indicates that roughness disrupts coherent turbulent eddies, consequently mitigating peak stress. We quantify correlations between drag force and local fluid velocity fluctuations. Notably, a larger deviation in drag is observed in the P800 case compared to M1015, accentuating the influence of particle size and distribution on fluid–particle interactions.

42 ENGINEERING↗
NASA NTRS1994

Detection of asteroidal dust particles from known families in near-Earth orbits

The orbital evolution of dust particles with two different sizes (diameters equal to 4 and 9 microns) originating from the Eos, Koronis, and Themis asteroidal families was studied. All the planetary perturbations, radiation pressure, Poynting-Robertson light drag, and corpuscular solar wind effects are included in the calculation. It is concluded that for particles having diameters ranging from 4 to 9 microns, Eos particles are quite different in orbital elements from Themis and Koronis particles. For Koronis and Themis particles, the best times to collect them are around April and October.

Dermott, Stanley F.↗
NASA NTRS2001

Numerical Simulation of One- and Two-Phase Flows in Propulsion Systems

In this report, we present some results of problems investigated during joint research between the Hampton University Fluid Mechanics and Acoustics Laboratory (HU/FM&AL), NASA GRC, and the LaRC Hyper-X Program. This work is supported by joint research between the NASA GRC and the Institute of Mechanics at Moscow State University (IM/MSU) in Russia under a CRDF grant. The main areas of current scientific interest of the HU/FM&AL include an investigation of the proposed and patented advanced methods for aircraft engine thrust and noise benefits. These methods are based on nontraditional 3D corrugated and composite nozzle, inlet, propeller and screw designs such as a Bluebell and Telescope nozzle, Mobius-shaped screw, etc. This is the main subject of our other projects, of which one is presented at the current conference. Here we analyze additional methods for exhaust jet noise reduction without essential thrust loss and even with thrust augmentation. Such additional approaches are: (1) to add some solid, fluid, or gas mass at discrete locations to the main supersonic gas stream to minimize the negative influence of strong shock waves formed in propulsion systems. This mass addition may be accompanied by heat addition to the main stream as a result of the fuel combustion or by cooling of this stream as a result of the liquid mass evaporation and boiling; (2) Use of porous or permeable nozzles and additional shells at the nozzle exit for preliminary cooling of the hot jet exhaust and pressure compensation for off-design conditions (so-called continuous ejector with small mass flow rate); and (3) to propose and analyze new effective methods of fuel injection into the flow stream in air-breathing engines. The research is focused on a wide regime of problems in the propulsion field as well as in experimental testing and theoretical and numerical simulation analyses for advanced aircraft and rocket engines. The FM&AL Team uses analytical methods, numerical simulations, and experimental tests at the Hampton University campus, NASA, and IM/MSU. The main outcomes during this reporting period are: (1) Publications: The AIAA Paper #01-1893 has been accepted for the AIAA/NAL-NASDA-ISAS 10th International Space Planes and Hypersonic Systems and Technologies Conference, 24-27 April 2001, Kyoto, Japan. The AIAA Paper #01-3204 has been accepted for presentation at the 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference being held on 08-11 July, in Salt Lake City, UT; (2) Grants and proposals: The HU/FM&AL was awarded the NASA grant NAG3-2495 in October 2000 and the CRDF award was granted to the NASA GRC-HU FM&AL and IM/MSU (Russia) in July 2000. A solicited proposal was submitted for the NASA NRA-01GRC-02 competition and two unsolicited proposals to NASA are in preparation; (3) Theory and numerical simulations: Analytical theory, numerical simulation, comparison of theoretical with experimental results, and modification of theoretical approaches, models, grids, etc. Such investigations have been conducted for three main problems: (a) Combustion efficiency optimization in the half-duct combustor system; (b) Drag reduction effects for blunt bodies with solid needles; and (c) Solid particle injection from the butt-end against a supersonic flow. The NASA CFL3D, HU/FM&AL, and IM/MSU GODUNOV codes were used and modified for solution of these problems. The codes are based on full Euler and Navier-Stokes solvers with and without nonequilibrium oxygen-nitrogen and air-hydrogen chemical reactions in laminar and turbulent gas flow regimes; (4) Experimental tests: In the IM/MSU supersonic wind tunnel, experimental tests were conducted with different number, location, and geometric parameters of solid needles mounted at the front of the butt-end in supersonic flow. Optimal parameters were determined that provide minimal butt-end drag in the stationary flow regime. Experimental and numerical simulation results are in good agreement; (5) Student Research Activity: Involvement of one graduate and two undergraduate students as research assistants in the current project.

Gilinsky, Mikhail↗
NASA NTRS1984

On the nature of the material surrounding Vega

Observations of Vega at 193 microns indicate that the far-infrared emission from the circumstellar material discovered by IRAS (Aumann et al. 1984) may decline more rapidly than a Planck spectrum at wavelengths greater than 100 microns. This suggests that the emitting particles may be smaller than the millimeter-sized objects proposed by Aumann et al. (1984). Small grains would be driven from the stellar system by radiation pressure, or their orbits would decay as a result of Poynting -Robertson drag. In order to maintain a state of dynamic equilibrium, a continuous supply of new particles would be required. It is hypothesized that the small grains are ejected by sublimation of volatile material from larger comet-like bodies in a partially coalesced preplanetary disk. A reservoir containing less than a few hundred earth masses could sustain the source over the lifetime of the star.

Harper, D. A.↗
NASA NTRS1983

Saturn's E, G, and F rings - Modulated by the plasma sheet?

Saturn's broad E ring, the narrow G ring, and the structured and apparently time-variable F ring(s) contain many micron and submicron-sized particles, which make up the 'visible' component. These rings (or ring systems) are in direct contact with magnetospheric plasma. Fluctuations in the plasma density and/or mean energy, due to magnetospheric and solar wind processes, may induce stochastic charge variations on the dust particles, which in turn lead to an orbit perturbation and spatial diffusion. In addition, Coulomb drag forces may be important, in particular for the E ring. The possibility that electromagnetic effects may play a role in determining the F ring structure and its possible time variations is critically examined. Sputtering of micron-sized dust particles in the E ring by magnetospheric ions yields lifetimes of 100 to 10,000 years. This effect as well as the plasma induced transport processes require an active source for the E ring, probably Enceladus.

Morfill, G. E.↗
NASA NTRS1981

Pioneer fly-by of Saturn and its rings

Results acquired by the imaging photopolarimeter on board Pioneer 11 during the spacecraft fly-by of Saturn and its rings on September 1, 1979 are reviewed. Analysis of the broadband photometry and polarimetry obtained of the Saturn atmosphere has been used to determine a cloud top height of 300 mb and a scale height of the aerosol distribution about 1/4 that of the ambient gas, and to point out differences between the forward scattering and belt and zone characteristics of the Saturn and Jupiter atmospheres. Images of Saturn's rings have been used to derive a profile of ring optical depth between 1.22 and 2.35 Saturn radii, and reveal new divisions and thin rings and azimuthal variations in the brightness of the A ring not observable from earth. Linear polarization observations of Titan in red and blue light reveal that the aerosols near the top of the atmosphere have radii less than about 0.09 micron and that the optical thickness of the small aerosol layer is about 0.6 above an effectively depolarizing surface, and indicate radii of 2845 + or - 25 km and 2880 + or - 22 km in red and blue light, respectively. Earth-based and spacecraft data are consistent with the formation of rings structures as a result of Poynting-Robertson drag and gravitational satellite resonances with the original ice and rock particles.

Gehrels, T.↗
NASA NTRS2003

Microgravity and Hypogravity Compatible Methods for the Destruction of Solid Wastes by Magnetically Assisted Gasification

This report summarizes a three-year collaborative effort between researchers at UMPQUA Research Company (URC) and the Chemical Engineering Department at Oregon State University (OSU). The Magnetically Assisted Gasification (MAG) concept was originally conceived as a microgravity and hypogravity compatible means for the decomposition of solid waste materials generated aboard spacecraft, lunar and planetary habitations, and for the recovery of potentially valuable resources. While a number of methods such as supercritical water oxidation (SCW0), fluidized bed incineration, pyrolysis , composting and related biological processes have been demonstrated for the decomposition of solid wastes, none of these methods are particularly well- suited for employment under microgravity or hypogravity conditions. For example, fluidized bed incineration relies upon a balance between drag forces which the flowing gas stream exerts upon the fluidization particles and the opposing force of gravity. In the absence of gravity, conventional fluidization cannot take place. Hypogravity operation can also be problematic for conventional fluidized bed reactors, because the various factors which govern fluidization phenomena do not all scale linearly with gravity. For this reason it may be difficult to design and test fluidized bed reactors in lg, which are intended to operate under different gravitational conditions. However, fluidization can be achieved in microgravity (and hypogravity) if a suitable replacement force to counteract the forces between fluid and particles can be found. Possible alternatives include: centripetal force, electric fields, or magnetic fields. Of these, magnetic forces created by the action of magnetic fields and magnetic field gradients upon ferromagnetic media offer the most practical approach. The goal of this URC-OSU collaborative effort was to develop magnetic hardware and methods to control the degree of fluidization (or conversely consolidation) of granular ferromagnetic media and to employ these innovations in sequential filtration and fluidized bed processes for the segregation and decomposition of solid waste materials, and for the concentration and collection of inorganic residue (ash). This required the development of numerous enabling technologies and tools.

Atwater, James E.↗
NASA NTRS2006

A Summary of the Experimental Results for a Generic Tractor-Trailer in the Ames Research Center 7- by 10-Foot and 12-Foot Wind Tunnels

Experimental measurements of a generic tractor-trailer were obtained in two wind tunnels at Ames Research Center. After a preliminary study at atmospheric conditions in the 7- by 10-Foot Wind Tunnel, additional testing was conducted at Reynolds numbers corresponding to full-scale highway speeds in the 12-Foot Pressure Wind Tunnel. To facilitate computational modeling, the 1:8-scale geometry, designated the Generic Conventional Model, included a simplified underbody and omitted many small-scale details. The measurements included overall and component forces and moments, static and dynamic surface pressures, and three-component particle image velocimetry. This summary report highlights the effects of numerous drag reduction concepts and provides details of the model installation in both wind tunnels. To provide a basis for comparison, the wind-averaged drag coefficient was tabulated for all configurations tested. Relative to the baseline configuration representative of a modern class-8 tractor-trailer, the most effective concepts were the trailer base flaps and trailer belly box providing a drag-coefficient reduction of 0.0855 and 0.0494, respectively. Trailer side skirts were less effective yielding a drag reduction of 0.0260. The database of this experimental effort is publicly available for further analysis.

Storms, Bruce L.↗
NASA NTRS2011

Benefit of "Push-pull" Locomotion for Planetary Rover Mobility

As NASAs exploration missions on planetary terrains become more aggressive, a focus on alternative modes of locomotion for rovers is necessary. In addition to climbing steep slopes, the terrain in these extreme environments is often unknown and can be extremely hard to traverse, increasing the likelihood of a vehicle or robot becoming damaged or immobilized. The conventional driving mode in which all wheels are either driven or free-rolling is very efficient on flat hard ground, but does not always provide enough traction to propel the vehicle through soft or steep terrain. This paper presents an alternative mode of travel and investigates the fundamental differences between these locomotion modes. The methods of push-pull locomotion discussed can be used with articulated wheeled vehicles and are identified as walking or inchinginch-worming. In both cases, the braked non-rolling wheels provide increased thrust. An in-depth study of how soil reacts under a rolling wheel vs. a braked wheel was performed by visually observing the motion of particles beneath the surface. This novel technique consists of driving or dragging a wheel in a soil bin against a transparent wall while high resolution, high-rate photographs are taken. Optical flow software was then used to determine shearing patterns in the soil. Different failure modes were observed for the rolling and braked wheel cases. A quantitative comparison of inching vs. conventional driving was also performed on a full-scale vehicle through a series of drawbar pull tests in the Lunar terrain strength simulant, GRC-1. The effect of tire stiffness was also compared; typically compliant tires provide better traction when driving in soft soil, however its been observed that rigid wheels may provide better thrust when non-rolling. Initial tests indicate up to a possible 40 increase in pull force capability at high slip when inching vs. rolling.

robotics↗
NASA NTRS1988

The application of single particle hydrodynamics in continuum models of multiphase flow

A review of the application of single particle hydrodynamics in models for the exchange of interphase momentum in continuum models of multiphase flow is presented. Considered are the equations of motion for a laminar, mechanical two phase flow. Inherent to this theory is a model for the interphase exchange of momentum due to drag between the dispersed particulate and continuous fluid phases. In addition, applications of two phase flow theory to de-mixing flows require the modeling of interphase momentum exchange due to lift forces. The applications of single particle analysis in deriving models for drag and lift are examined.

Decker, Rand↗
NASA NTRS2004

History of Thermally Processed Solids in the Protoplanetary Disk: Reconciling Theoretical Models and Meteoritical Evidence

In this talk we assess theoretical models of the radial, temporal, and thermal evolution of nebula solids, and their ultimate accretion into planetesimals such as we see today, using meteorite evidence as a guide. Each class of chondrites contains a characteristic suite of chondrules and CAIs that may have formed over a period of several Myr during which planetesimals were accreting in the disk. Details of the various models for transient melting of chondrules and igneous CAIs will be left to others. However, high-temperature processes of different kinds evaporation, alteration, etc did affect these constituents and their environment over this time span. Here we describe evolutionary scenarios consistent with a large time gap between CAI and chondrule formation and the presence of distinctive suites of chondrules and CAIs in each chondrite class. Particle-gas dynamical processes transport particles of all relevant sizes (microns to many meters) within the nebula and affect their evolution in a variety of important ways. Turbulent radial diffusion spreads particles radially down their concentration gradients - as one example, it can prevent CAIs from being lost into the sun on several Myr timescales [1]. Vertical diffusion spreads the dense midplane particle layer, determining its volume density, which in turn affects the particle growth rate and even the dominant growth process [2-4]. Turbulent concentration selects aerodynamically sorted particles for orders-of-magnitude density enhancement, and is applicable to porous, fluffy particles of appropriate size as well as to solid chondrules [5]. Inward radial drift under gas drag brings a surprisingly large amount of material to regions where it evaporates; these evaporation fronts cause significant chemical modification of the nebula gas over a wide range of radii [6]. Radial transport by stellar winds can be important for small particles [7].

Cuzzi, J. N.↗
NASA NTRS1994

Some considerations on velocity vector accuracy in dust trajectory analysis

The relative contributions of comets and asteroids to the reservoir of dust in the interplanetary medium is not known. There are direct observations of dust released from comets and there is evidence to associate the IRAS dust bands with possible collisions of asteroids in the main belt. A means towards sorting out the parent sources has been proposed in the establishment of a dust collector in orbit about the Earth. The purpose of such a facility would be to collect not only cosmic dust particles intact but also the state vectors, as they arrive at the detector, the idea being that one may combine analytical laboratory analysis of the physics and chemistry of the captured particles with orbital data in order to help distinguish between bodies and identify parent bodies. The theoretical study of dust particle orbits in the solar system takes on greatly more importance if we use collected trajectory data. The orbital motion of dust when radiation and forces alone are acting is well understood. When gravitational forces due to the planets are included, the motion can become quite complex. In order to characterize the orbits of particles as they crossed the Earth's orbits, a study of the long-time dust orbital evolution was undertaken. We have considered various parameters associated with these dust orbits to see if one may in a general way discriminate between particles evolved from comets and asteroids. We proceed in this study as we have done previously. That is, we considered the dust particles as ideal black bodies, of density 1 gm/cc, spherical, with radii 10-100 microns. Particles of this size are affected by radiation forces, photon pressure, and Poynting-Robertson drag. Account was also taken of solar wind drag, which amounts to about 30 percent of the Poynting-Robertson drag negligible. The gravitational forces due to the planets are included, unlike in our previous study; the planetary orbits are those of true n-body interaction so that the possibility of secular resonance is included. Our method was to calculate explicitly by a numerical procedure the orbits of dust particles after they left their parent bodies. The motion is determined numerically with the implicit Runge-Kutta integrator using Gauss-Radau spacings.

Jackson, A. A.↗