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NASA NTRS1994

Technical report series on global modeling and data assimilation. Volume 3: An efficient thermal infrared radiation parameterization for use in general circulation models

A detailed description of a parameterization for thermal infrared radiative transfer designed specifically for use in global climate models is presented. The parameterization includes the effects of the main absorbers of terrestrial radiation: water vapor, carbon dioxide, and ozone. While being computationally efficient, the schemes compute very accurately the clear-sky fluxes and cooling rates from the Earth's surface to 0.01 mb. This combination of accuracy and speed makes the parameterization suitable for both tropospheric and middle atmospheric modeling applications. Since no transmittances are precomputed the atmospheric layers and the vertical distribution of the absorbers may be freely specified. The scheme can also account for any vertical distribution of fractional cloudiness with arbitrary optical thickness. These features make the parameterization very flexible and extremely well suited for use in climate modeling studies. In addition, the numerics and the FORTRAN implementation have been carefully designed to conserve both memory and computer time. This code should be particularly attractive to those contemplating long-term climate simulations, wishing to model the middle atmosphere, or planning to use a large number of levels in the vertical.

Suarex, Max J.↗

NASA NTRS2017

Mapping Application for Penguin Populations and Projected Dynamics (MAPPPD): Data and Tools for Dynamic Management and Decision Support

The Mapping Application for Penguin Populations and Projected Dynamics (MAPPPD) is a web-based, open access, decision-support tool designed to assist scientists, non-governmental organizations and policy-makers working to meet the management objectives as set forth by the Commission for the Conservation of Antarctic Marine Living Resources (CCAMLR) and other components of the Antarctic Treaty System (ATS) (that is, Consultative Meetings and the ATS Committee on Environmental Protection). MAPPPD was designed specifically to complement existing efforts such as the CCAMLR Ecosystem Monitoring Program (CEMP) and the ATS site guidelines for visitors. The database underlying MAPPPD includes all publicly available (published and unpublished) count data on emperor, gentoo, Adelie) and chinstrap penguins in Antarctica. Penguin population models are used to assimilate available data into estimates of abundance for each site and year.Results are easily aggregated across multiple sites to obtain abundance estimates over any user-defined area of interest. A front end web interface located at www.penguinmap.com provides free and ready access to the most recent count and modelled data, and can act as a facilitator for data transfer between scientists and Antarctic stakeholders to help inform management decisions for the continent.

Humphries, G. R. W.↗

NASA NTRS1989

Conservation equations and physical models for hypersonic air flows over the aeroassist flight experiment vehicle

The code development and application program for the Langley Aerothermodynamic Upwind Relaxation Algorithm (LAURA), with emphasis directed toward support of the Aeroassist Flight Experiment (AFE) in the near term and Aeroassisted Space Transfer Vehicle (ASTV) design in the long term is reviewed. LAURA is an upwind-biased, point-implicit relaxation algorithm for obtaining the numerical solution to the governing equations for 3-D, viscous, hypersonic flows in chemical and thermal nonequilibrium. The algorithm is derived using a finite volume formulation in which the inviscid components of flux across cell walls are described with Roe's averaging and Harten's entropy fix with second-order corrections based on Yee's Symmetric Total Variation Diminishing scheme. Because of the point-implicit relaxation strategy, the algorithm remains stable at large Courant numbers without the necessity of solving large, block tri-diagonal systems. A single relaxation step depends only on information from nearest neighbors. Predictions for pressure distributions, surface heating, and aerodynamic coefficients compare well with experimental data for Mach 10 flow over an AFE wind tunnel model. Predictions for the hypersonic flow of air in chemical and thermal nonequilibrium over the full scale AFE configuration obtained on a multi-domain grid are discussed.

Gnoffo, Peter A.↗

NASA NTRS2017

Dawn Spacecraft Performance at Ceres: Results of Hybrid Control for Ceres Mapping

Dawn is a low-thrust interplanetary spacecraft currently orbiting the dwarf planet Ceres, to better understand the early creation of the solar system. Launched in September 2007, Dawn arrived at Vesta in July 2011. After a 16-month successful science campaign at Vesta, Dawn departed for Ceres, arriving in early 2015. The Dawn spacecraft uses both reaction wheel assemblies (RWA) and a reaction control system (RCS) to provide 3-axis attitude control for the spacecraft. Reaction wheels were designed to be the primary system for attitude control, however two wheels have shown high friction anomalies and have been removed from service. The project has implemented a hybrid control algorithm using two reaction wheels and RCS thrusters. This hybrid control capability enabled Dawn to achieve very high science return, while significantly conserving remaining hydrazine propellant. With only two remaining healthy RWA’s, hybrid control became part of the baseline plan for Ceres science operations. The Dawn team developed specific operational approaches in which sequences were developed with careful consideration of science versus resource trades. Commanding and sequence planning also incorporated contingency planning, in the event that another reaction wheel may fail. Despite the differences in operational approach between Vesta and Ceres, both campaigns achieved very rich scientific data return. This paper highlights Dawn’s recent flight experience with hybrid attitude control during Ceres orbit operations. The discussion includes the approaches utilized by the Dawn team to address unique operational challenges presented by the hybrid approach, and reviews spacecraft performance under hybrid control in low orbit at Ceres. Additionally, methods used to optimize hydrazine use and thereby extend the science campaign will be presented. Finally, a preliminary assessment of an orbit transfer with two reaction wheels, during extended mission operations, is discussed.

Smith, Brett A.↗

NASA NTRS1994

Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres

Although neglecting polarization and replacing the rigorous vector radiative transfer equation by its approximate scalar counterpart has no physical background, it is a widely used simplification when the incident light is unpolarized and only the intensity of the reflected light is to be computed. We employ accurate vector and scalar multiple-scattering calculations to perform a systematic study of the errors induced by the neglect of polarization in radiance calculations for a homogeneous, plane-parallel Rayleigh-scattering atmosphere (with and without depolarization) above a Lambertian surface. Specifically, we calculate percent errors in the reflected intensity for various directions of light incidence and reflection, optical thicknesses of the atmosphere, single-scattering albedos, depolarization factors, and surface albedos. The numerical data displayed can be used to decide whether or not the scalar approximation may be employed depending on the parameters of the problem. We show that the errors decrease with increasing depolarization factor and/or increasing surface albedo. For conservative or nearly conservative scattering and small surface albedos, the errors are maximum at optical thicknesses of about 1. The calculated errors may be too large for some practical applications, and, therefore, rigorous vector calculations should be employed whenever possible. However, if approximate scalar calculations are used, we recommend to avoid geometries involving phase angles equal or close to 0 deg and 90 deg, where the errors are especially significant. We propose a theoretical explanation of the large vector/scalar differences in the case of Rayleigh scattering. According to this explanation, the differences are caused by the particular structure of the Rayleigh scattering matrix and come from lower-order (except first-order) light scattering paths involving right scattering angles and right-angle rotations of the scattering plane.

Mishchenko, M. I.↗

NASA NTRSDate not supplied

Feasibility Study to Interactive Workshop: Building End-user Capacity to Integrate Earth Observation Data into Federally Endangered Atlantic Salmon (Salmo salar) Habitat Monitoring in Main

Changes in temperature and precipitation patterns, along with alterations in land cover threaten ongoing conservation efforts for Federally Endangered Atlantic salmon (Salmo salar) in Maine. Earth observation data offers a unique perspective for habitat monitoring that can complement habitat restoration and conservation activity on the ground. As a dual capacity building program, the NASA DEVELOP National Program strives to build the capacity of program participants by leveraging Earth observation data to address environmental concerns across the globe, while also building capacity in partner organizations to integrate Earth observation data into their decision making practices. Between September 2021 and August 2022, three NASA DEVELOP teams demonstrated the feasibility of utilizing NASA Earth observations including Aqua Moderate Resolution Imaging Spectroradiometer (MODIS), Terra MODIS, Landsat 5 Thematic Mapper (TM), Landsat 8 Operation Land Imager (OLI), and Shuttle Radar Topography Mission (SRTM) in conjunction with Sentinel-2 MultiSpectral Instrument (MSI) to assess temperature, precipitation, and land use land cover (LULC) over time throughout salmon habitat in Maine. While the first two teams completed projects that were categorized as NASA DEVELOP’s traditional feasibility projects, the third and final project team generated resources and planned an interactive workshop to transfer project methods to end-user organizations. Ultimately, the goal of this work was to not only inform the partner’s ongoing salmon population recovery and habitat restoration initiatives but provide tools that allow partner organizations to continue integrating Earth observation data into their work beyond their partnership with the program. This project serves as a case study within the NASA DEVELOP Program and provides lessons learned for moving beyond traditional feasibility studies to more interactive partner engagement and knowledge transfer practices.

Nicole Ramberg-Pihl↗

NASA NTRS1994

Diffusion model for lightning radiative transfer

A one-speed Boltzmann transport theory, with diffusion approximations, is applied to study the radiative transfer properties of lightning in optically thick thunderclouds. Near-infrared (lambda = 0.7774 micrometers) photons associated with a prominent oxygen emission triplet in the lightning spectrum are considered. Transient and spatially complex lightning radiation sources are placed inside a rectangular parallelepiped thundercloud geometry and the effects of multiple scattering are studied. The cloud is assumed to be composed of a homogeneous collection of identical spherical water droplets, each droplet a nearly conservative, anisotropic scatterer. Conceptually, we treat the thundercloud like a nuclear reactor, with photons replaced by neutrons, and utilize standard one-speed neutron diffusion techniques common in nuclear reactor analyses. Valid analytic results for the intensity distribution (expanded in spherical harmonics) are obtained for regions sufficiently far from sources. Model estimates of the arrival-time delay and pulse width broadening of lightning signals radiated from within the cloud are determined and the results are in good agreement with both experimental data and previous Monte Carlo estimates. Additional model studies of this kind will be used to study the general information content of cloud top lightning radiation signatures.

Koshak, William J.↗

NASA NTRS2021

Long-term Coherent Timing of the Accreting Millisecond Pulsar IGR J17062–6143

We report on a coherent timing analysis of the 163 Hz accreting millisecond X-ray pulsar IGR J17062–6143. Using data collected with the Neutron Star Interior Composition Explorer and XMM-Newton, we investigated the pulsar evolution over a time span of four years. We obtained a unique phase-coherent timing solution for the stellar spin, finding the source to be spinning up at a rate of (3.77 ± 0.09) × 10−15 Hz s−1. We further find that the 0.4–6 keV pulse fraction varies gradually between 0.5% and 2.5% following a sinusoidal oscillation with a 1210 ± 40 day period. Finally, we supplemented this analysis with an archival Rossi X-ray Timing Explorer observation and obtained a phase-coherent model for the binary orbit spanning 12 yr, yielding an orbital period-derivative measurement of (8.4 ± 2.0) × 10−12 s s−1. This large orbital period derivative is inconsistent with a binary evolution that is dominated by gravitational wave emission and is suggestive of highly non-conservative mass transfer in the binary system.

stars: neutron↗

NASA NTRS1998

Self-Similar Compressible Free Vortices

Lie group methods are used to find both exact and numerical similarity solutions for compressible perturbations to all incompressible, two-dimensional, axisymmetric vortex reference flow. The reference flow vorticity satisfies an eigenvalue problem for which the solutions are a set of two-dimensional, self-similar, incompressible vortices. These solutions are augmented by deriving a conserved quantity for each eigenvalue, and identifying a Lie group which leaves the reference flow equations invariant. The partial differential equations governing the compressible perturbations to these reference flows are also invariant under the action of the same group. The similarity variables found with this group are used to determine the decay rates of the velocities and thermodynamic variables in the self-similar flows, and to reduce the governing partial differential equations to a set of ordinary differential equations. The ODE's are solved analytically and numerically for a Taylor vortex reference flow, and numerically for an Oseen vortex reference flow. The solutions are used to examine the dependencies of the temperature, density, entropy, dissipation and radial velocity on the Prandtl number. Also, experimental data on compressible free vortex flow are compared to the analytical results, the evolution of vortices from initial states which are not self-similar is discussed, and the energy transfer in a slightly-compressible vortex is considered.

vonEllenrieder, Karl↗

NASA NTRS2017

Hydrazine Conservation for the Dawn Spacecraft Operations at the Dwarf Planet Ceres

Dawn is a low-thrust interplanetary spacecraft currently orbiting the dwarf planet Ceres. After successfully completing its Vesta and Ceres prime missions, Dawn is now in its extended mission, continuously exploring Ceres. After losing the second reaction wheel assembly at the end of the Vesta mission, the feasibility of the Ceres mission was at risk due to the potentially significant increase in hydrazine consumption from using only the reaction control system thrusters. This paper summarizes the intense, collaborative efforts undertaken by the project to conserve hydrazine prior to Vesta departure, during cruise to Ceres, and throughout the Ceres mission. The project’s efforts in minimizing the number of turns are discussed. A special emphasis is given to describing various efforts taken by the Attitude Control Subsystem team in an attempt to reduce hydrazine consumption. These efforts include: changing default slew rates, control gain tuning, changing nadir pointing strategy with Ahead Cross Nadir pointing, hybrid control implementation using hydrazine based thrusters and two remaining reaction wheels, optimal science turn location analysis, and reaction wheel angular momentum management strategies for reducing the number of momentum unloadings. In addition to the Attitude Control Subsystem team’s efforts, the Mission Design and Navigation team’s efforts taken in designing hydrazine friendly orbit transfers are discussed. The trade study performed collaboratively by the Attitude Control Subsystem team and the Science Operations Support Team in choosing hydrazine friendly off-nadir targets is also discussed. Various simulation results are presented and compared against actual flight data obtained. Lastly, remaining hydrazine status at the end of the prime Ceres mission and the hydrazine management planning for the extended mission are discussed.

Salami, Mana↗

NASA NTRS2007

Composite Overwrap Pressure Vessels: Mechanics and Stress Rupture Lifing Philosophy

The NASA Engineering and Safety Center (NESC) has been conducting an independent technical assessment to address safety concerns related to the known stress rupture failure mode of filament wound pressure vessels in use on Shuttle and the International Space Station. The Shuttle's Kevlar-49 fiber overwrapped tanks are of particular concern due to their long usage and the poorly understood stress rupture process in Kevlar-49 filaments. Existing long term data show that the rupture process is a function of stress, temperature and time. However due to the presence of load sharing liners and the complex manufacturing procedures, the state of actual fiber stress in flight hardware and test articles is not clearly known. Indeed non-conservative life predictions have been made where stress rupture data and lifing procedures have ignored the contribution of the liner in favor of applied pressure as the controlling load parameter. With the aid of analytical and finite element results, this paper examines the fundamental mechanical response of composite overwrapped pressure vessels including the influence of elastic-plastic liners and degraded/creeping overwrap properties. Graphical methods are presented describing the non-linear relationship of applied pressure to Kevlar-49 fiber stress/strain during manufacturing, operations and burst loadings. These are applied to experimental measurements made on a variety of vessel systems to demonstrate the correct calibration of fiber stress as a function of pressure. Applying this analysis to the actual qualification burst data for Shuttle flight hardware revealed that the nominal fiber stress at burst was in some cases 23% lower than what had previously been used to predict stress rupture life. These results motivate a detailed discussion of the appropriate stress rupture lifing philosophy for COPVs including the correct transference of stress rupture life data between dissimilar vessels and test articles.

Thesken, John C.↗

NASA NTRS1995

Evolution of Vapor Bubbles Nucleation Sites in Low Gravity

When liquid is expelled by a vapor bubble growing at a nucleation site on a superheated surface, a thin microlayer underneath the bubble is left behind. It is evaporated from the free microlayer surface that provides for bubble growth. The average thickness of the microlayer determining the evaporation rate increases with time if the latter does not exceed a threshold value associated with the burn-out crisis. The bubble is described as a spherical segment with its flattened part adjoining the microlayer. This introduces two independent variables - the radius of the spherical part of the bubble surface and the polar angle that defines the relative area of the flattened part. They are to be found out from a set of two strongly nonlinear equations resulting from mass and momentum conservation laws. The first one depends on both microlayer thickness and nonmonotonously changing bubble base area. The second involves two major factors favoring bubble detachment - the buoyancy and a force due to the initial momentum of vapor input into the bubble. The former force depends on gravity whereas the latter one does not. It is why the limiting regimes of bubble evolution that correspond to normal or moderately reduced gravity and to microgravity feature drastically different properties. In the first case, the buoyancy dominates and the bubble evolves in such a manner as to become a full sphere at a moment that can be viewed as that of detachment. The detachment volume grows as gravity decreases. In the second case, the buoyancy is negligible and the bubble stays near the surface, while its volume continues to increase for a sufficiently long time. The findings are discussed in connection with experimental data obtained under different gravity conditions, some unpublished experiments being included. They help to understand why the pool boiling heat transfer coefficient frequently increases as gravity falls down and eventually vanishes.

Buyevich, Yu A.↗

NASA NTRS1986

Transfer function analysis of thermospheric perturbations

Applying perturbation theory, a spectral model in terms of vectors spherical harmonics (Legendre polynomials) is used to describe the short term thermospheric perturbations originating in the auroral regions. The source may be Joule heating, particle precipitation or ExB ion drift-momentum coupling. A multiconstituent atmosphere is considered, allowing for the collisional momentum exchange between species including Ar, O2, N2, O, He and H. The coupled equations of energy, mass and momentum conservation are solved simultaneously for the major species N2 and O. Applying homogeneous boundary conditions, the integration is carred out from the Earth's surface up to 700 km. In the analysis, the spherical harmonics are treated as eigenfunctions, assuming that the Earth's rotation (and prevailing circulation) do not significantly affect perturbations with periods which are typically much less than one day. Under these simplifying assumptions, and given a particular source distribution in the vertical, a two dimensional transfer function is constructed to describe the three dimensional response of the atmosphere. In the order of increasing horizontal wave numbers (order of polynomials), this transfer function reveals five components. To compile the transfer function, the numerical computations are very time consuming (about 100 hours on a VAX for one particular vertical source distribution). However, given the transfer function, the atmospheric response in space and time (using Fourier integral representation) can be constructed with a few seconds of a central processing unit. This model is applied in a case study of wind and temperature measurements on the Dynamics Explorer B, which show features characteristic of a ringlike excitation source in the auroral oval. The data can be interpreted as gravity waves which are focused (and amplified) in the polar region and then are reflected to propagate toward lower latitudes.

Mayr, H. G.↗

NASA NTRS2015

BioSentinel: Mission Development of a Radiation Biosensor to Gauge DNA Damage and Repair Beyond Low Earth Orbit on a 6U Nanosatellite

We are designing and developing a 6U (10 x 22 x 34 cm; 14 kg) nanosatellite as a secondary payload to fly aboard NASAs Space Launch System (SLS) Exploration Mission (EM) 1, scheduled for launch in late 2017. For the first time in over forty years, direct experimental data from biological studies beyond low Earth orbit (LEO) will be obtained during BioSentinels 12- to 18-month mission. BioSentinel will measure the damage and repair of DNA in a biological organism and allow us to compare that to information from onboard physical radiation sensors. In order to understand the relative contributions of the space environments two dominant biological perturbations, reduced gravity and ionizing radiation, results from deep space will be directly compared to data obtained in LEO (on ISS) and on Earth. These data points will be available for validation of existing biological radiation damage and repair models, and for extrapolation to humans, to assist in mitigating risks during future long-term exploration missions beyond LEO. The BioSentinel Payload occupies 4U of the spacecraft and will utilize the monocellular eukaryotic organism Saccharomyces cerevisiae (yeast) to report DNA double-strand-break (DSB) events that result from ambient space radiation. DSB repair exhibits striking conservation of repair proteins from yeast to humans. Yeast was selected because of 1) its similarity to cells in higher organisms, 2) the well-established history of strains engineered to measure DSB repair, 3) its spaceflight heritage, and 4) the wealth of available ground and flight reference data. The S. cerevisiae flight strain will include engineered genetic defects to prevent growth and division until a radiation-induced DSB activates the yeasts DNA repair mechanisms. The triggered culture growth and metabolic activity directly indicate a DSB and its successful repair. The yeast will be carried in the dry state within the 1-atm PL container in 18 separate fluidics cards with each card having 16 independent culture microwells, with integral microchannels and filters to supply nutrients and reagents, confine the yeast to the wells, and enable optical measurement. The measurement subsystem will monitor each subgroup of culture wells continuously for several weeks, optically tracking DSB-triggered cell growth and metabolism. BioSentinel will also include physical radiation sensors based on the TimePix sensor, as implemented by JSCs RadWorks group, which record individual radiation events including estimates of their linear-energy-transfer (LET) values. Radiation-dose and LET data will be compared directly to the rate of DSB-and-repair events measured by the S. cerevisiae biosentinels.

Radiation Biosensor↗

NASA NTRS2014

Icing Analysis of a Swept NACA 0012 Wing Using LEWICE3D Version 3.48

Icing calculations were performed for a NACA 0012 swept wing tip using LEWICE3D Version 3.48 coupled with the ANSYS CFX flow solver. The calculated ice shapes were compared to experimental data generated in the NASA Glenn Icing Research Tunnel (IRT). The IRT tests were designed to test the performance of the LEWICE3D ice void density model which was developed to improve the prediction of swept wing ice shapes. Icing tests were performed for a range of temperatures at two different droplet inertia parameters and two different sweep angles. The predicted mass agreed well with the experiment with an average difference of 12%. The LEWICE3D ice void density model under-predicted void density by an average of 30% for the large inertia parameter cases and by 63% for the small inertia parameter cases. This under-prediction in void density resulted in an over-prediction of ice area by an average of 115%. The LEWICE3D ice void density model produced a larger average area difference with experiment than the standard LEWICE density model, which doesn't account for the voids in the swept wing ice shape, (115% and 75% respectively) but it produced ice shapes which were deemed more appropriate because they were conservative (larger than experiment). Major contributors to the overly conservative ice shape predictions were deficiencies in the leading edge heat transfer and the sensitivity of the void ice density model to the particle inertia parameter. The scallop features present on the ice shapes were thought to generate interstitial flow and horse shoe vortices which enhance the leading edge heat transfer. A set of changes to improve the leading edge heat transfer and the void density model were tested. The changes improved the ice shape predictions considerably. More work needs to be done to evaluate the performance of these modifications for a wider range of geometries and icing conditions

Ice Accretion Software↗

NASA NTRS2014

Icing Analysis of a Swept NACA 0012 Wing Using LEWICE3D Version 3.48

Icing CFD↗

NASA NTRS2021

Fitting strategies of accretion column models and application to the broadband spectrum of Cen X-3

Due to the complexity of modeling the radiative transfer inside the accretion columns of neutron star binaries, their X-ray spectra are still commonly described with phenomenological models, e.g., a cut off power law. While the behavior of these models is well understood and they allow for a comparison of different sources and studying source behavior, the extent to which the underlying physics can be derived from the model parameters is very limited. During recent years, several physically motivated spectral models have been developed to overcome these limitations. Their application, however, is generally computationally much more expensive and they require a high number of parameters which are difficult to constrain. In particular, Becker & Wolff(2007) presented an analytical solution to the radiative transfer equation inside the accretion column assuming a velocity profile that is linear in the optical depth. An implementation of this model that is both fast and accurate enough to be fitted to observed spectra is available in XSPEC. The main difficulty of this implementation is that some solutions violate energy conservation and therefore have to be rejected by the user. We propose a novel fitting strategy that ensures energy conservation during theχ2-minimization which simplifies the application of the model considerably. We demonstrate this approach as well as a study of possible parameter degeneracies with a comprehensive Markov-Chain Monte Carlo analysis of the complete parameter space for a combined NuSTAR and Swift/XRT dataset of Cen X-3.The derived accretion-flow structure features a small column radius of∼63 m and a spectrum dominated by bulk-Comptonization of bremsstrahlung seed photons, in agreement with previous studies.

X-rays: binaries↗

NASA NTRS1994

Can a variable alpha induce limit cycle behavior and exponential luminosity decay in transient soft x ray sources?

There has been, recently, a revival of the stability problem of accretion disks. Much of this renewed interest is due to recent observational data on transient soft X-ray novae, which are low-mass X-ray binaries. It is widely believed that nonsteady mass transfer from the secondary onto the compact primary, through an accretion disk, is the reason for the observed spectacular events in the form of often repetitive outbursts, with recurrence times ranging from 1 to 60 yr and duration time on the scale of months. Though not having reached yet a consensus about the nature of the mechanism that regulates the mass transfer, the disk thermal instability model seems to be favored by the fact that the rise in the hard X-ray luminosity is prior to the rise in the soft X-ray luminosity, while the mass transfer instability model seems to be hindered by the fact that the luminosity during quiescence is unable to trigger the thermal instability. However, it should be stressed that, remarkably, the X-ray light curves of these X-ray novae all show overall exponential decays, a feature quite difficult to reproduce in the framework of the viscous disk model, which yields powerlike luminosity decay. Taking into account this observational constraint, we have studied the temporal evolution of perturbations in the accretion rate, under the assumption that alpha is radial and parameter dependent. The chosen dependence is such that the model can reproduce limit cycle behavior (the system is locally unstable but globally stable). However, the kind of dependence we are looking for in alpha does not allow us to use the usual Shakura and Sunyaev procedure in the sense that we no longer can obtain a linearized continuity equation without explicit dependence on the accretion rate. This is so because now we cannot eliminate the accretion rate by using the angular momentum conservation equation.

Meirellesfilho, C.↗