On spurious steady-state solutions of explicit Runge-Kutta schemes

The bifurcation diagram associated with the logistic equation v sup n+1 = av sup n (1-v sup n) is by now well known, as is its equivalence to solving the ordinary differential equation u prime = alpha u (1-u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. Runge-Kutta schemes applied to both the equation u prime = alpha u (1-u) and the cubic equation u prime = alpha u (1-u)(b-u) were studied computationally and analytically and their behavior was contrasted with the explicit Euler scheme. Their spurious fixed points and periodic orbits were noted. In particular, it was observed that these may appear below the linearized stability limits of the scheme and, consequently, computation may lead to erroneous results

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit

Ionospheric refraction effects on orbit determination using the orbit determination error analysis system

The influence of ionospheric refraction on orbit determination was studied through the use of the Orbit Determination Error Analysis System (ODEAS). The results of a study of the orbital state estimate errors due to the ionospheric refraction corrections, particularly for measurements involving spacecraft-to-spacecraft tracking links, are presented. In current operational practice at the Goddard Space Flight Center (GSFC) Flight Dynamics Facility (FDF), the ionospheric refraction effects on the tracking measurements are modeled in the Goddard Trajectory Determination System (GTDS) using the Bent ionospheric model. While GTDS has the capability of incorporating the ionospheric refraction effects for measurements involving ground-to-spacecraft tracking links, such as those generated by the Ground Spaceflight Tracking and Data Network (GSTDN), it does not have the capability to incorporate the refraction effects for spacecraft-to-spacecraft tracking links for measurements generated by the Tracking and Data Relay Satellite System (TDRSS). The lack of this particular capability in GTDS raised some concern about the achievable accuracy of the estimated orbit for certain classes of spacecraft missions that require high-precision orbits. Using an enhanced research version of GTDS, some efforts have already been made to assess the importance of the spacecraft-to-spacecraft ionospheric refraction corrections in an orbit determination process. While these studies were performed using simulated data or real tracking data in definitive orbit determination modes, the study results presented here were obtained by means of covariance analysis simulating the weighted least-squares method used in orbit determination

First-passage theory of exciton population loss in single-walled carbon nanotubes reveals micron-scale intrinsic diffusion lengths

One-dimensional crystals have long range translational invariance which manifests as long exciton diffusion lengths, but such intrinsic properties are often obscured by environmental perturbations. We use a first-passage approach to model single-walled carbon nanotube (SWCNT) exciton dynamics (including exciton-exciton annihilation and end effects) and compare it to results from both continuous-wave and multi-pulse ultrafast excitation experiments to extract intrinsic SWCNT properties. Excitons in suspended SWCNTs experience macroscopic diffusion lengths, on the order of the SWCNT length, (1.3-4.7 um) in sharp contrast to encapsulated samples. For these pristine samples, our model reveals intrinsic lifetimes (350-750 ps), diffusion constants (130-350 cm^2/s), and absorption cross-sections (2.1-3.6 X 10^-17 cm^2/atom) among the highest previously reported.and diffusion lengths for SWCNTs.Comment: 6 pages, 3 figure

Pair Analysis of Field Galaxies from the Red-Sequence Cluster Survey

We study the evolution of the number of close companions of similar luminosities per galaxy (Nc) by choosing a volume-limited subset of the photometric redshift catalog from the Red-Sequence Cluster Survey (RCS-1). The sample contains over 157,000 objects with a moderate redshift range of 0.25 < z < 0.8 and absolute magnitude in Rc (M_Rc) < -20. This is the largest sample used for pair evolution analysis, providing data over 9 redshift bins with about 17,500 galaxies in each. After applying incompleteness and projection corrections, Nc shows a clear evolution with redshift. The Nc value for the whole sample grows with redshift as (1+z)^m, where m = 2.83 +/- 0.33 in good agreement with N-body simulations in a LCDM cosmology. We also separate the sample into two different absolute magnitude bins: -25 < M_Rc < -21 and -21 < M_Rc < -20, and find that the brighter the absolute magnitude, the smaller the m value. Furthermore, we study the evolution of the pair fraction for different projected separation bins and different luminosities. We find that the m value becomes smaller for larger separation, and the pair fraction for the fainter luminosity bin has stronger evolution. We derive the major merger remnant fraction f_rem = 0.06, which implies that about 6% of galaxies with -25 < M_Rc < -20 have undergone major mergers since z = 0.8.Comment: ApJ, in pres

Chemical aging of m-xylene secondary organic aerosol: laboratory chamber study

Secondary organic aerosol (SOA) can reside in the atmosphere for a week or more. While its initial formation from the gas-phase oxidation of volatile organic compounds tends to take place in the first few hours after emission, SOA can continue to evolve chemically over its atmospheric lifetime. Simulating this chemical aging over an extended time in the laboratory has proven to be challenging. We present here a procedure for studying SOA aging in laboratory chambers that is applied to achieve 36 h of oxidation. The formation and evolution of SOA from the photooxidation of m-xylene under low-NO_x conditions and in the presence of either neutral or acidic seed particles is studied. In SOA aging, increasing molecular functionalization leads to less volatile products and an increase in SOA mass, whereas gas- or particle-phase fragmentation chemistry results in more volatile products and a loss of SOA. The challenge is to discern from measured chamber variables the extent to which these processes are important for a given SOA system. In the experiments conducted, m-xylene SOA mass, calculated under the assumption of size-invariant particle composition, increased over the initial 12–13 h of photooxidation and decreased beyond that time, suggesting the existence of fragmentation chemistry. The oxidation of the SOA, as manifested in the O:C elemental ratio and fraction of organic ion detected at m/z 44 measured by the Aerodyne aerosol mass spectrometer, increased continuously starting after 5 h of irradiation until the 36 h termination. This behavior is consistent with an initial period in which, as the mass of SOA increases, products of higher volatility partition to the aerosol phase, followed by an aging period in which gas- and particle-phase reaction products become increasingly more oxidized. When irradiation is stopped 12.4 h into one experiment, and OH generation ceases, minimal loss of SOA is observed, indicating that the loss of SOA is either light- or OH-induced. Chemical ionization mass spectrometry measurements of low-volatility m-xylene oxidation products exhibit behavior indicative of continuous photooxidation chemistry. A condensed chemical mechanism of m-xylene oxidation under low-NO_x conditions is capable of reproducing the general behavior of gas-phase evolution observed here. Moreover, order of magnitude analysis of the mechanism suggests that gas-phase OH reaction of low volatility SOA precursors is the dominant pathway of aging in the m-xylene system although OH reaction with particle surfaces cannot be ruled out. Finally, the effect of size-dependent particle composition and size-dependent particle wall loss rates on different particle wall loss correction methods is discussed

MOA-2011-BLG-293Lb: First Microlensing Planet possibly in the Habitable Zone

We used Keck adaptive optics observations to identify the first planet discovered by microlensing to lie in or near the habitable zone, i.e., at projected separation $r_\perp=1.1\pm 0.1\,$AU from its $M_{L}=0.86\pm 0.06\,M_\odot$ host, being the highest microlensing mass definitely identified. The planet has a mass $m_p = 4.8\pm 0.3\,M_{\rm Jup}$, and could in principle have habitable moons. This is also the first planet to be identified as being in the Galactic bulge with good confidence: $D_L=7.72\pm 0.44$ kpc. The planet/host masses and distance were previously not known, but only estimated using Bayesian priors based on a Galactic model (Yee et al. 2012). These estimates had suggested that the planet might be a super-Jupiter orbiting an M dwarf, a very rare class of planets. We obtained high-resolution $JHK$ images using Keck adaptive optics to detect the lens and so test this hypothesis. We clearly detect light from a G dwarf at the position of the event, and exclude all interpretations other than that this is the lens with high confidence (95%), using a new astrometric technique. The calibrated magnitude of the planet host star is $H_{L}=19.16\pm 0.13$. We infer the following probabilities for the three possible orbital configurations of the gas giant planet: 53% to be in the habitable zone, 35% to be near the habitable zone, and 12% to be beyond the snow line, depending on the atmospherical conditions and the uncertainties on the semimajor axis.Comment: Accepted by ApJ, 21 pages, 4 figure

α-pinene photooxidation under controlled chemical conditions – Part 2: SOA yield and composition in low- and high-NO_x environments

The gas-phase oxidation of α-pinene produces a large amount of secondary organic aerosol (SOA) in the atmosphere. A number of carboxylic acids, organosulfates and nitrooxy organosulfates associated with α-pinene have been found in field samples and some are used as tracers of α-pinene oxidation. α-pinene reacts readily with OH and O_3 in the atmosphere followed by reactions with both HO_2 and NO. Due to the large number of potential reaction pathways, it can be difficult to determine what conditions lead to SOA. To better understand the SOA yield and chemical composition from low- and high-NO_x OH oxidation of α-pinene, studies were conducted in the Caltech atmospheric chamber under controlled chemical conditions. Experiments used low O_3 concentrations to ensure that OH was the main oxidant and low α-pinene concentrations such that the peroxy radical (RO_2) reacted primarily with either HO_2 under low-NO_x conditions or NO under high-NO_x conditions. SOA yield was suppressed under conditions of high-NO_x. SOA yield under high-NO_x conditions was greater when ammonium sulfate/sulfuric acid seed particles (highly acidic) were present prior to the onset of growth than when ammonium sulfate seed particles (mildly acidic) were present; this dependence was not observed under low-NO_x conditions. When aerosol seed particles were introduced after OH oxidation, allowing for later generation species to be exposed to fresh inorganic seed particles, a number of low-NO_x products partitioned to the highly acidic aerosol. This indicates that the effect of seed acidity and SOA yield might be under-estimated in traditional experiments where aerosol seed particles are introduced prior to oxidation. We also identify the presence of a number of carboxylic acids that are used as tracer compounds of α-pinene oxidation in the field as well as the formation of organosulfates and nitrooxy organosulfates. A number of the carboxylic acids were observed under all conditions, however, pinic and pinonic acid were only observed under low-NO_x conditions. Evidence is provided for particle-phase sulfate esterification of multi-functional alcohols

On spurious asymptotic numerical solutions of explicit Runge Kutta methods

The bifurcation diagram associated with the logistic equation vn+1 = avn(1 - vn) is by now well known, as is its equivalence to solving the ordinary differential equation (ODE) u\u27 = αu(1 - u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. We investigate, both analytically and computationally, Runge-Kutta schemes applied to the equation u\u27 = f(u), for f(u) = αu(1 - u) and f(u) = αu (1 - u)(b - u), contrasting their behavior with the explicit Euler scheme. We determine and provide a local analysis of bifurcations to spurious fixed points and periodic orbits. In particular we show that these may appear below the linearised stability limit of the scheme, and may consequently lead to erroneous computational results