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All sourcesNASA NTRSDOE OSTI
NASA NTRS2013

Comparisons of a Constrained Least Squares Model versus Human-in-the-Loop for Spectral Unmixing to Determine Material Type of GEO Debris

Spectral reflectance data through the visible regime was collected at Las Campanas Observatory in Chile using an imaging spectrograph on one of the twin 6.5-m Magellan telescopes. The data were obtained on 1-2 May 2012 on the 'Landon Clay' telescope with the LDSS3 (Low Dispersion Survey Spectrograph 3). Five pieces of Geosynchronous Orbit (GEO) or near-GEO debris were identified and observed with an exposure time of 30 seconds on average. In addition, laboratory spectral reflectance data was collected using an Analytical Spectral Device (ASD) field spectrometer at California Polytechnic State University (Cal Poly) in San Luis Obispo on several typical common spacecraft materials including solar cells, circuit boards, various Kapton materials used for multi-layer insulation, and various paints. The remotely collected data and the laboratory-acquired data were then incorporated in a newly developed model that uses a constrained least squares method to unmix the spectrum in specific material components. The results of this model are compared to the previous method of a human-in-the-loop (considered here the traditional method) that identifies possible material components by varying the materials and percentages until a spectral match is obtained. The traditional model was found to match the remotely collected spectral data after it had been divided by the continuum to remove the space weathering effects, or a reddening of the materials. The constrained least-squares model also used the de-reddened spectra as inputs and the results were consistent with those obtained through the traditional method. For comparison, a first-order examination of including reddening effects into the constrained least-squares model will be explored and comparisons to the remotely collected data will be examined. The identification of each object s suspected material component will be discussed herein.

Abercromby, Kira J.↗
NASA NTRS2014

Bulk Composition of Vesta as Constrained by the Dawn Mission and the HED Meteorites

Of the objects in the main asteroid belt, Vesta is of particular interest as it is large enough to have experienced internal differentiation (520 km diameter), and it is known to have a basaltic surface dominated by FeO-bearing pyroxenes. Furthermore, visible-IR spectra of Vesta and associated Vestoids are remarkably similar to laboratory spectra of Howardite-Eucrite-Diogenite (HED) meteorites, leading to the paradigm that the HEDs ultimately came from Vesta. Geochemical and petrological studies of the HEDs confirm the differentiated nature of the near-surface region of their parent body, and imply that crust extraction occurred well within the first 10 Ma of solar system history Vesta is therefore a prime target for studies that aim to constrain the earliest stages of planet building, and it is within this context that the NASA Dawn spacecraft orbited Vesta from July 2011 to September 2012. The results of the Dawn mission so far have significantly reinforced the HED-Vesta connection, confirming a significant degree of internal differentiation, a surface mineralogy compatible with that of the HEDs, and near-surface ratios of Fe/O and Fe/Si consistent with HED lithologies. The combination of data from the HED meteorites and the Dawn mission thus presents an unprecedented opportunity to use Vesta as a natural laboratory of early differentiation processes in the early solar system. However, the bulk composition of Vesta remains a significant unknown parameter, but one that plays a key role on the physical and chemical properties of the internal and surface reservoirs (core, mantle, crust). Several attempts have been made to constrain the bulk composition of the eucrite parent body, early endeavours relying on petrological or cosmochemical constraints. More recently, individual chondrite class compositions, or mixtures thereof, have been considered, constrained by considerations such as O-isotopes, trace-element ratios and siderophile element concentrations of the eucrites. The work presented here builds upon these latter studies, with the primary aims of: i) illustrating the potential diversity of the geochemical and geophysical properties of a fully differentiated Vesta-sized parent body, and ii) assessing which, if any, of the known chondritic bulk compositions are plausible analogues for proto-Vesta.

Toplis, M. J.↗
NASA NTRS2014

Constraining the Structure of the Transition Disk HD 135344B (SAO 206462) by Simultaneous Modeling of Multiwavelength Gas and Dust Observations

Context: Constraining the gas and dust disk structure of transition disks, particularly in the inner dust cavity, is a crucial step toward understanding the link between them and planet formation. HD 135344B is an accreting (pre-)transition disk that displays the CO 4.7 micrometer emission extending tens of AU inside its 30 AU dust cavity. Aims: We constrain HD 135344B's disk structure from multi-instrument gas and dust observations. Methods: We used the dust radiative transfer code MCFOST and the thermochemical code ProDiMo to derive the disk structure from the simultaneous modeling of the spectral energy distribution (SED), VLT/CRIRES CO P(10) 4.75 Micrometers, Herschel/PACS [O(sub I)] 63 Micrometers, Spitzer/IRS, and JCMT CO-12 J = 3−2 spectra, VLTI/PIONIER H-band visibilities, and constraints from (sub-)mm continuum interferometry and near-IR imaging. Results: We found a disk model able to describe the current gas and dust observations simultaneously. This disk has the following structure. (1) To simultaneously reproduce the SED, the near-IR interferometry data, and the CO ro-vibrational emission, refractory grains (we suggest carbon) are present inside the silicate sublimation radius (0.08 is less than R less than 0.2 AU). (2) The dust cavity (R is less than 30 AU) is filled with gas, the surface density of the gas inside the cavity must increase with radius to fit the CO ro-vibrational line profile, a small gap of a few AU in the gas distribution is compatible with current data, and a large gap of tens of AU in the gas does not appear likely. (4) The gas-to-dust ratio inside the cavity is >100 to account for the 870 Micrometers continuum upper limit and the CO P(10) line flux. (5) The gas-to-dust ratio in the outer disk (30 is less than R less than 200 AU) is less than 10 to simultaneously describe the [O(sub I)] 63 Micrometers line flux and the CO P(10) line profile. (6) In the outer disk, most of the gas and dust mass should be located in the midplane, and a significant fraction of the dust should be in large grains. Conclusions: Simultaneous modeling of the gas and dust is required to break the model degeneracies and constrain the disk structure. An increasing gas surface density with radius in the inner cavity echoes the effect of a migrating Jovian planet in the disk structure. The low gas mass (a few Jupiter masses) throughout the HD 135344B disk supports the idea that it is an evolved disk that has already lost a large portion of its mass.

inner dust cavity↗
NASA NTRS2016

An Observationally Constrained Evaluation of the Oxidative Capacity in the Tropical Western Pacific Troposphere

Hydroxyl radical (OH) is the main daytime oxidant in the troposphere and determines the atmospheric lifetimes of many compounds. We use aircraft measurements of O3, H2O, NO, and other species from the Convective Transport of Active Species in the Tropics (CONTRAST) field campaign, which occurred in the tropical western Pacific (TWP) during January-February 2014, to constrain a photochemical box model and estimate concentrations of OH throughout the troposphere. We find that tropospheric column OH (OHCOL) inferred from CONTRAST observations is 12 to 40% higher than found in chemical transport models (CTMs), including CAM-chem-SD run with 2014 meteorology as well as eight models that participated in POLMIP (2008 meteorology). Part of this discrepancy is due to a clear-sky sampling bias that affects CONTRAST observations; accounting for this bias and also for a small difference in chemical mechanism results in our empirically based value of OHCOL being 0 to 20% larger than found within global models. While these global models simulate observed O3 reasonably well, they underestimate NOx (NO +NO2) by a factor of 2, resulting in OHCOL approx.30% lower than box model simulations constrained by observed NO. Underestimations by CTMs of observed CH3CHO throughout the troposphere and of HCHO in the upper troposphere further contribute to differences between our constrained estimates of OH and those calculated by CTMs. Finally, our calculations do not support the prior suggestion of the existence of a tropospheric OH minimum in the TWP, because during January-February 2014 observed levels of O3 and NO were considerably larger than previously reported values in the TWP.

Hydroxyl radical↗
NASA NTRS2017

Constraining the Depth of Polar Ice Deposits and Evolution of Cold Traps on Mercury with Small Craters in Permanently Shadowed Regions

Earth-based radar observations revealed highly reflective deposits at the poles of Mercury [e.g., 1], which collocate with permanently shadowed regions (PSRs) detected from both imagery and altimetry by the MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) spacecraft [e.g., 2]. MESSENGER also measured higher hydrogen concentrations at the north polar region, consistent with models for these deposits to be composed primarily of water ice [3]. Enigmatic to the characterization of ice deposits on Mercury is the thickness of these radar-bright features. A current minimum bound of several meters exists from the radar measurements, which show no drop in the radar cross section between 13- and 70-cm wavelength observations [4, 5]. A maximum thickness of 300 m is based on the lack of any statistically significant difference between the height of craters that host radar-bright deposits and those that do not [6]. More recently, this upper limit on the depth of a typical ice deposit has been lowered to approximately 150 m, in a study that found a mean excess thickness of 50 +/- 35 m of radar-bright deposits for 6 craters [7]. Refining such a constraint permits the derivation of a volumetric estimate of the total polar ice on Mercury, thus providing insight into possible sources of water ice on the planet. Here, we take a different approach to constrain the thickness of water-ice deposits. Permanently shadowed surfaces have been resolved in images acquired with the broadband filter on MESSENGER's wide-angle camera (WAC) using low levels of light scattered by crater walls and other topography [8]. These surfaces are not featureless and often host small craters (less than a few km in diameter). Here we utilize the presence of these small simple craters to constrain the thickness of the radar-bright ice deposits on Mercury. Specifically, we compare estimated depths made from depth-to-diameter ratios and depths from individual Mercury Laser Altimeter (MLA) tracks to constrain the fill of material of small craters that lie within the permanently shadowed, radar bright deposits of 7 north polar craters.

Deutsch, Ariel N.↗
NASA NTRS2018

Constraining the Models' Response of Tropical Clouds to SST Forcings Using CALIPSO Observations

Here we present preliminary results from the analysis of the low cloud cover (LCC) and cloud radiative effect (CRE) interannual changes in response to sea surface temperature (SST) forcings in two GISS climate models, and 12 other climate models. We further classify them as a function of their ability to reproduce the vertical structure of the cloud response to SST change against 10 years of CALIPSO observations: “the constrained models, which match the observation constraint, and the unconstrained models”. The constrained models replicate the observed interannual LCC change particularly well (ΔLCC(sub con)=-3.49 ±1.01 %/K vs. ΔLCC(sub obs)=-3.59 ±0.28 %/K) as opposed to the unconstrained models, which largely underestimate it (ΔLCC(sub unc) = -1.32 ± 1.28 %/K). As a result, the amount of short-wave warming simulated by the constrained models (ΔCRE(sub con)=2.60 ±1.13 W/m2/K) is in better agreement with the observations (ΔCRE(sub obs)=3.05 ±0.28 W/m2/K) than the unconstrained models (ΔCRE(sub con)=0.87 ±2.63 W/m2/K). Depending on the type of low cloud, the observed relationship between cloud/radiation and surface temperature varies. Over the stratocumulus regions, increasing SSTs generate higher cloud top height along with a large decrease of the cloud fraction below as opposed to a slight decrease of the cloud fraction at each level over the trade cumulus regions. Our results suggest that the models must generate sustainable stratocumulus decks and moist processes in the planetary boundary layer to reproduce these observed features. Future work will focus on defining a method to objectively discriminate these cloud types that can be applied consistently in both the observations and the models.

Cesana, Gregory↗
NASA NTRS2019

The Ability of GeoCarb to Constrain the Interannual Variability of Carbon Gases over the Amazon

We perform a number of idealized assimilation experiments with the GEOS constituent data assimilation system to test the ability of GeoCarb retrievals of CO, CO2, and CH4 to constrain the interannual variability of these gases over the Amazon. Retrievals for instruments on other satellites which observe in similar channels (e.g. MOPITT, GOSAT, and OCO-2) are limited due to persistent cloud coverage. Given its ability to sample the same location multiple times in one day, the expectation is that GeoCarb retrievals will return more soundings than those from previous missions. The goal of the assimilation experiments is to understand which scanning strategies lead to the best sounding densities and thus have the best chance of constraining interannual variability in the carbon species. The experiments each begin by picking a given year at random from a nature run (i.e., a model simulation meant to represent the truth). The model fields are sampled according to a given strategy and then screened to account for cloud coverage. Next, we pick another year at random and assimilate the synthetic GeoCarb samples into the GEOS model for that year. The output of the assimilation, 6-hourly, 3D fields of each constituent, is then directly comparable to the nature run. This comparison allows us to evaluate the ability of GeoCarb measurements to constrain the interannual variability of each gas.

Weir, B.↗
NASA NTRSDate not supplied

Creating Seasonal Climatologies of Aerosol Lidar Ratios Over Ocean Using MODIS AOD Constrained Retrievals and GOCART Model Simulations

The current CALIPSO algorithms assign one lidar ratio (i.e., extinction-to-backscatter ratio; LR) value globally for each of the seven tropospheric aerosol types. In a future data products release, the CALIPSO project aims to improve these algorithms through the development of regional and seasonal LR climatologies. In this work, aerosol LRs are inferred through CALIOP backscatter profiles constrained by collocated aerosol optical depth (AOD) from Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) data. This analysis is subsampled for those profiles that are cloud-free and contain only one CALIOP-classified aerosol type (e.g., marine). The CALIOP profiles are then collocated with aerosol volume fractions obtained through Goddard Chemistry Aerosol Radiation and Transport (GOCART) model simulations. In this poster, we show twelve-year (June 2006-August 2018) mean spatial distributions of inferred aerosol LRs for CALIOP-classified marine aerosols and how they correspond inversely with patterns of GOCART sea salt volume fraction (SSVF). Near land masses, smaller SSVFs and larger LRs are found (due to the influence of over-land aerosols). In the remote ocean regions (likely less impacted by over-land aerosols), larger SSVFs and smaller LRs are found. The developed relationship between the GOCART model SSVFs and MODIS AOD constrained LRs is used to create model-assisted seasonal LR maps. Additionally, we show maps of inferred LRs from constrained retrievals using the CALIPSO Ocean Derived Column Optical Depth (ODCOD) product and comparisons with those from the MODIS analyses. The technique demonstrated here benefits CALIPSO in the near-term, but similar methods can also be applied to the next generation space-based elastic backscatter lidars with collocated passive sensors, such as those of the upcoming NASA Atmosphere Observing System (AOS).

Travis Toth↗
NASA NTRSDate not supplied

Mapping Aerosol Lidar Ratios Over Ocean Using Constrained Retrievals and A Global Aerosol Model

The current NASA Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) algorithms assign one lidar ratio (i.e., extinction-to-backscatter ratio; LR) value globally for each of the seven tropospheric aerosol types. In a future data products release, the CALIPSO project aims to improve these algorithms through the development of regional and seasonal LR climatologies. In this work, aerosol LRs are inferred through Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) backscatter profiles constrained by collocated aerosol optical depth (AOD) from Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) data. This analysis is subsampled for those profiles that are cloud-free and contain only one CALIOP-classified aerosol type (e.g., marine). The CALIOP profiles are then collocated with aerosol volume fractions obtained through Goddard Chemistry Aerosol Radiation and Transport (GOCART) model simulations. In this presentation, we show that the twelve-year (June 2006-August 2018) mean spatial distributions of inferred aerosol LRs for CALIOP-classified marine aerosols correspond inversely with patterns of GOCART sea salt volume fraction (SSVF). For example, smaller SSVFs (< 65%) and larger LRs (> 55 sr), are found near land masses. This is indicative of the influence of over-land aerosols (e.g., pollution and biomass burning smoke). In the remote oceans (i.e., regions likely less impacted by non-sea salt aerosols), larger SSVFs (> 95%) and smaller LRs (< 25 sr) are found. The developed relationship between the GOCART SSVFs and MODIS AOD constrained LRs (polynomial fit intersect values of ~56 sr for SSVF of 0% and ~21 sr for SSVF of 100%) is used to produce model-assisted climatological LR maps on seasonal scales. Additionally, we show maps of inferred LRs from constrained retrievals using the CALIPSO Ocean Derived Column Optical Depth (ODCOD) product and comparisons with those from the MODIS analyses. The technique demonstrated in this study not only benefits CALIPSO in the near-term, but similar methods can be applied to future spaceborne elastic backscatter lidars with collocated passive sensors, such as those associated with the upcoming NASA Atmosphere Observing System (AOS).

Travis Toth↗
NASA NTRS2024

Constraining the Thermal and Textural Evolution of Evolved Achondrite DOM 10100: Implications for Volatile Loss

Identification of evolved achrondritic clasts suggests that some early planetesimals experienced extensive differentiation and formed chemically evolved crusts. Many of these samples contain volatile-bearing phases (i.e., apatite, merrillite, nominally anhydrous minerals), which can be used to constrain volatile budgets and abundancies via in-situ measurements. These ancient crustal rocks are subject to high heat, moderate pressures, potentially hot mineral-rich fluids, and degassing that has likely modified many of these volatiles, and their abundancies do not represent the composition of the primary magma from which these achondrites formed. In this work we attempt to deconvolve the thermal evolution of these samples to provide petrologic information that can be used for making inferences regarding their volatile history, and ultimately constraining volatile budgets on planetary bodies in the early Solar System. Here, we combine in-situ major element geochemistry, thermodynamic modeling, and detailed textural and microstructural observations to reconstruct the thermal and textural evolution of howardite DOM 10100, which contains a dacite within its matrix. Previous analyses of several sections of DOM 10100 suggest that the dacitic clast is evidence for partial melting of Vesta’s differentiated basaltic crust, making it an ideal candidate for evolved crustal studies. While its igneous petrogenesis has been well constrained, there is still opportunity to better characterize its post-crystallization thermal history, including possible thermal metamorphism and cooling.

J S Gorce↗
NASA NTRSDate not supplied

Mapping Aerosol Lidar Ratios Over Ocean using MODIS AOD Constrained Retrievals and GOCART Model Simulations

After 17 years, the NASA Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) mission ceased science operations in August 2023. For the final CALIPSO data products release (Version 5), the CALIPSO project seeks to improve the accuracy of its aerosol extinction by advancing knowledge of aerosol lidar ratios (i.e., extinction-to-backscatter ratios; LRs) for various aerosol types. The current algorithm assigns one LR value globally for each of the seven tropospheric aerosol types. The CALIPSO team aims to improve the retrieval algorithm through the development of regional and seasonal LR climatologies for the same aerosol types. In this study, aerosol LRs are inferred through Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) backscatter profiles constrained by collocated aerosol optical depth (AOD) from Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) data over oceans during daytime. This analysis is subsampled for those profiles that are cloud-free and contain only one CALIOP-classified aerosol type. The CALIOP profiles are then collocated with aerosol volume fractions obtained through Goddard Chemistry Aerosol Radiation and Transport (GOCART) model simulations. This presentation will reveal findings that the 12-year (June 2006-August 2018) mean spatial distributions of inferred aerosol LRs for CALIOP-classified marine and dusty marine aerosols correspond inversely with patterns of GOCART sea salt volume fraction (SSVF). For example, smaller SSVFs (< 65%) and larger LRs (> 55 sr), are found near land masses (Fig. 1). This indicates the influence of advected anthropogenic aerosols (e.g., pollution and biomass burning smoke). In the remote oceans (i.e., regions likely less impacted by non-sea salt aerosols), the SSVFs are larger (> 95%) and the LRs are smaller (< 25 sr) (Fig. 1). A polynomial fit of the MODIS AOD constrained LRs to the corresponding GOCART SSVFs (intersect values of ~58 sr for SSVF of 0% and ~21 sr for SSVF of 100%) is further used to produce model-assisted climatological LR maps on seasonal scales. Additionally, we will show results of a LR validation analysis for which we compare the revised CALIPSO AODs obtained by applying the seasonal/regional constrained LRs against CALIPSO Version 4.51 Ocean Derived Column Optical Depth (ODCOD). While the majority of the presentation will focus on LRs for CALIOP-classified marine and dusty marine aerosols, an overview of LR results will show preliminary results for other aerosol types over ocean, such as dust and elevated smoke. The technique demon-strated in this study highlights the benefits not only to the final planned CALIPSO data release in 2025, but similar methods can be applied to future spaceborne elastic backscatter lidars with collocated passive sensors (e.g., such as those associated with NASA’s proposed Atmosphere Observing System).

Travis D Toth↗
NASA NTRS2024

A Multiwavelength Approach to Constraining the Merger Properties of ACT-CL J0034.4+0225

ACT-CL J0034.4+0225 is a previously unrecognized merging galaxy cluster at z = 0.38588 ± 0.00068. Our primary evidence is provided by a 21 ks Chandra image that shows two surface brightness peaks separated by ∼49″ (259 kpc) surrounded by an extended cluster gas distribution. Each gas peak contains a brightest cluster galaxy, offset from the gas peak. We collect new South African Large Telescope optical spectra that, when augmented by archival data, yield redshifts for the two BGCs and 58 other cluster members. Archival Giant Metrewave Radio Telescope and MeerKAT data reveal a radio halo that encompasses the X-ray peaks. We provide and compare three X-ray-based mass estimates (5.0 × 1014Me, 6.4 × 1014Me, and 8.6 × 1014Me). The Planck and ACT Sunyaev–Zel’dovich masses are ≈5.8 × 1014 Me. We constrain the merger state and properties by comparing them to an existing suite of N-body/hydrodynamical models using the measured gas peak separation (259 kpc, projected) and radial velocity difference (0–1000 km s−1 ). This constrains the epoch of the merger to be within ∼100 Myr of first pericenter passage. A strong lensing analysis constrains the mass ratio to be in the range 1:1–1:20, while the cluster morphology prefers values near the equal-mass range.

Peter Doze↗
NASA NTRS2004

On the Constrained Attitude Control Problem

In this paper, we consider various classes of constrained attitude control (CAC) problem in single and multiple spacecraft settings. After categorizing attitude constraints into four distinct types, we provide an overview of the existing approaches to this problem. We then proceed to further expand on a recent algorithmic approach to the CAC problem. The paper concludes with an example demonstrating the viability of the proposed algorithm for a multiple spacecraft constrained attitude reconfiguration scenario.

semidefinite programming↗
NASA NTRS1973

CONMIN: A FORTRAN program for constrained function minimization: User's manual

CONMIN is a FORTRAN program, in subroutine form, for the solution of linear or nonlinear constrained optimization problems. The basic optimization algorithm is the Method of Feasible Directions. The user must provide a main calling program and an external routine to evaluate the objective and constraint functions and to provide gradient information. If analytic gradients of the objective or constraint functions are not available, this information is calculated by finite difference. While the program is intended primarily for efficient solution of constrained problems, unconstrained function minimization problems may also be solved, and the conjugate direction method of Fletcher and Reeves is used for this purpose. This manual describes the use of CONMIN and defines all necessary parameters. Sufficient information is provided so that the program can be used without special knowledge of optimization techniques. Sample problems are included to help the user become familiar with CONMIN and to make the program operational.

Vanderplaats, G. N.↗
NASA NTRS1973

An evaluation of a constrained test method for obtaining free body responses

A method for obtaining free body responses from dynamic tests on a constrained structure is investigated for practical feasibility. The method is based on the principle that a constrained structure can be considered to be a free body acted upon by multiple forces which include the forces of constraint. By measuring these forces and by exciting the structure so as to develop linearly independent sets of forces, the response of the free body to one force at a time can be computed. Techniques for producing these independent forces are discussed. The development of the theory, computer simulations of tests of representative aerospace vehicles (including experimental error), and a description and listing of the computer programs developed are included. The procedure appears to be a practical method for obtaining in-flight characteristics of such vehicles.

Berman, A.↗
NASA NTRS1978

Characteristics of constrained optimum trajectories with specified range

Necessary conditions of optimality are derived for trajectories whose structure is limited to climb, steady cruise, and descent segments. The performance function consists of the sum of fuel and time costs, referred to as direct operating cost (DOC). The state variable is range to go and the independent variable is energy. In this formulation a cruise segment always occurs at the optimum cruise energy for sufficiently large range. At short ranges (500 n. mi. and less) a cruise segment may also occur below the optimum cruise energy. The existence of such a cruise segment depends primarily on the fuel flow vs thrust characteristics and on thrust constraints. If thrust is a free control variable along with airspeed, it is shown that such cruise segments will not generally occur. If thrust is constrained to some maximum value in climb and to some minimum in descent, such cruise segments generally will occur. Computer calculations of typical short-range trajectories obtained about a 1% cost penalty for constraining the thrust.

Erzberger, H.↗
NASA NTRS1982

Constrained and unconstrained modes - Some modeling aspects of flexible spacecraft

Spacecraft that are partially rigid and partially flexible may be dynamically modeled in terms of either 'constrained' modes of vibration, for which the rigid part is held motionless, or the 'unconstrained' modes of the whole vehicle. A model 'completeness index' is defined for each expansion and used as a truncation criterion. Using as an example a large flexible platform with a rigid body at the center, it is shown that as the rigid portion of the vehicle becomes very small, a great many constrained modes are needed to achieve satisfactory model completeness. Otherwise, one finds an erroneous distribution of angular momentum among the modes. It is also shown that the importance of unconstrained modes is not necessarily ordered by frequency.

Hablani, H. B.↗
NASA NTRS1982

Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier

The influence of the choice of the Lagrange multiplier on constrained linear inversions is explored, with reference made to applications in inferring the columnar aerosol size distributions from spectral aerosol optical depth measurements. A range of the Lagrange multiplier is examined to find all positive solutions for the solution vector, which represents modifying factors to the assumed form of the size distribution. An iterative method is devised to constrain the calculations to consideration of only positive quantities and a requirement that the regression fit to data be consistent with measurement errors. The determination of the variances and covariances is formulated and applied to existing data sets for optical depth. Variances in the solution are found to be large for particle radii when the information content of the data is small.

King, M. D.↗