Multipole Expansion Model in Gravitional Lensing

1997 ◽  
Vol 06 (04) ◽  
pp. 425-447 ◽  
Author(s):  
Takeshi Fukuyama ◽  
Yuuko Kakigi ◽  
Takashi Okamura
Keyword(s):  
Power Law ◽  
Catastrophe Theory ◽  
Multipole Expansion ◽  
Point Mass ◽  
Mass Model ◽  
Expansion Model ◽  
Best Fit

Nontransparent models of the multipole expansion model and the two point-mass model are analyzed from the catastrophe theory. Singularity behaviours of 2n-pole moments are discussed. We apply these models to the triple quasar PG1115+080 and compare with the typical transparent model, softened power law spheroids. The multipole expansion model gives the best fit among them.

scholarly journals GRAVITATIONAL LENSING AND CATASTROPHE THEORY

1995 ◽  
Vol 04 (06) ◽  
pp. 685-710 ◽  
Author(s):  
YUUKO KAKIGI ◽  
TAKASHI OKAMURA ◽  
TAKESHI FUKUYAMA
Keyword(s):  
Gravitational Lensing ◽  
Catastrophe Theory ◽  
Multipole Expansion ◽  
Numerical Calculations ◽  
Lensing Effect ◽  
Expansion Model

Singularities of caustics appearing in the gravitational lensing effect are discussed analytically. The multipole expansion model of the lensing object is mainly studied since it is tractable and universal. Our analyses are confirmed by numerical calculations and applied to the multiple quasar system of PG1115+080. Consistencies with elliptical lens models are also discussed.

A point-mass model of gibbon locomotion

1999 ◽  
Vol 202 (19) ◽  
pp. 2609-2617 ◽  
Author(s):  
J.E. Bertram ◽  
A. Ruina ◽  
C.E. Cannon ◽  
Y.H. Chang ◽  
M.J. Coleman
Keyword(s):  
Energy Exchange ◽  
Forest Canopy ◽  
Mechanical Energy ◽  
Single Point ◽  
Metabolic Cost ◽  
Point Mass ◽  
Mass Model ◽  
Natural Behavior ◽  
Continuous Contact ◽  
Mechanical Factors

In brachiation, an animal uses alternating bimanual support to move beneath an overhead support. Past brachiation models have been based on the oscillations of a simple pendulum over half of a full cycle of oscillation. These models have been unsatisfying because the natural behavior of gibbons and siamangs appears to be far less restricted than so predicted. Cursorial mammals use an inverted pendulum-like energy exchange in walking, but switch to a spring-based energy exchange in running as velocity increases. Brachiating apes do not possess the anatomical springs characteristic of the limbs of terrestrial runners and do not appear to be using a spring-based gait. How do these animals move so easily within the branches of the forest canopy? Are there fundamental mechanical factors responsible for the transition from a continuous-contact gait where at least one hand is on a hand hold at a time, to a ricochetal gait where the animal vaults between hand holds? We present a simple model of ricochetal locomotion based on a combination of parabolic free flight and simple circular pendulum motion of a single point mass on a massless arm. In this simple brachiation model, energy losses due to inelastic collisions of the animal with the support are avoided, either because the collisions occur at zero velocity (continuous-contact brachiation) or by a smooth matching of the circular and parabolic trajectories at the point of contact (ricochetal brachiation). This model predicts that brachiation is possible over a large range of speeds, handhold spacings and gait frequencies with (theoretically) no mechanical energy cost. We then add the further assumption that a brachiator minimizes either its total energy or, equivalently, its peak arm tension, or a peak tension-related measure of muscle contraction metabolic cost. However, near the optimum the model is still rather unrestrictive. We present some comparisons with gibbon brachiation showing that the simple dynamic model presented has predictive value. However, natural gibbon motion is even smoother than the smoothest motions predicted by this primitive model.

Point mass impulse-momentum model of the equine rotational fall

2019 ◽  
Vol 15 (3) ◽  
pp. 157-165
Author(s):  
M.H. Foreman ◽  
J.R. Engsberg ◽  
J.H. Foreman
Keyword(s):  
Point Mass ◽  
Future Research ◽  
Collision Time ◽  
Mass Model ◽  
Impact Forces ◽  
Force Transfer ◽  
Reference Curves ◽  
Key Variables ◽  
Cross Country ◽  
Input Variables

Rotational falls are a serious cause of injury and death to horse and rider, particularly in the cross-country phase of eventing. The forces involved when horses galloping cross-country strike an immovable fence are unknown. The objective of this study was to mathematically model those forces using existing kinematic data measured from jumping horses. Data were obtained from published research using motion capture to measure mechanics about the center of gravity of the jumping horse at take-off. A convenience method from video evidence of rotational falls was used to estimate time of collision (Δt). A point mass model using equations of impulse-momentum and incorporating key variables was systematically implemented in Matlab (r2016a). The mean collision time (Δt=0.79s) produced horizontal, vertical, and resultant impact forces of 8,580, 8,245, and 12,158 N, respectively. Reference curves of impact forces were created for ranges of relevant input variables including collision time. Proportional relationships showed that shorter impact duration led to higher magnitude of force transfer between horse and obstacle. This study presents a preliminary range of collision forces based on a simplified model and numerous assumptions related to input variables. Future research should work to build upon these estimates through more complex modelling and data collection to enhance applicability for the design of cross-country safety devices.

scholarly journals X-Ray Observations of GRS 1915+105

1997 ◽  
Vol 163 ◽  
pp. 709-710 ◽  
Author(s):  
J. Greiner ◽  
B.A. Harmon ◽  
W.S. Paciesas ◽  
E.H. Morgan ◽  
R.A. Remillard
Keyword(s):  
Power Law ◽  
Count Rate ◽  
Monitoring Data ◽  
Detector Response ◽  
X Ray ◽  
Spectral Fitting ◽  
Spectral Components ◽  
Best Fit ◽  
Intensity Evolution ◽  
Thermal Bremsstrahlung

After the discovery of GRS 1915+105 (Castro-Tirado et al. 1992) we obtained pointed ROSAT observations every six months (12 until now). The flux in the ROSAT (0.1–2.4 keV) band is strikingly different from the simultaneous BATSE (25–50 keV) flux which was obtained by integrating the best fit power law (Fig. 1). Motivated by the different intensity evolution in the soft and hard X-ray band we have selected BATSE monitoring data collected simultaneously to ROSATdata and performed joint spectral fitting with XSPEC. As a result, we never got an acceptable fit (see Fig. 2): The BATSE power law (upper dotted line) is too steep to match the ROSAT band, and even allowing for an increased absorbing column (lower dotted line) does not solve the problem. Alternatively, neither a thermal bremsstrahlung fit (solid line) nor a power law fit (lower dash-dot line) to the ROSAT data match the BATSE flux. The upper dash-dot line is a -2.5 powerlaw which would match the BATSE data while giving too much 1–2 keV emission. A similar, but less stringent result is obtained when folding the best fit BATSE power law models with the HRI detector response to compare the expected count rate with the observed one. We therefore conclude that the spectrum during all simultaneous ROSAT/BATSE observations seemingly consists of two different spectral components.

Increase the feasible step region of biped robots through active vertical flexion and extension motions

Robotica ◽  
2016 ◽  
Vol 35 (7) ◽  
pp. 1541-1561 ◽  
Author(s):  
Wei Gao ◽  
Zhenzhong Jia ◽  
Chenglong Fu
Keyword(s):  
Vertical Motion ◽  
Vertical Movement ◽  
Point Mass ◽  
Simple Point ◽  
Mass Model ◽  
Biped Robots ◽  
Inverted Pendulum Model ◽  
Pendulum Model ◽  
Flexion And Extension ◽  
New Strategies

SUMMARYThis paper investigates the active vertical motion of biped systems and its significance to the balance of biped robots, which have been commonly neglected by the use of a well-known model called the Linear Inverted Pendulum Model. The feasible step location is theoretically estimated by considering the active vertical movement on a simple point mass model. Based on the estimation, we present two new strategies, namely the flexion strategy and the extension strategy, to enable biped robots to restore balance through active upward and downward motions. The analytical results demonstrate that the robot is able to recover from much larger disturbances using our proposed methods. Simulations of the simple point mass model validate our analysis. Besides, prototype controllers that incorporate our proposed strategies have also been implemented on a simulated humanoid robot. Numerical simulations on both the simple point mass model and the realistic humanoid model prove the effectiveness of proposed strategies.

Minimizing constraint forces and moments of manipulators using teaching–learning-based optimization and octahedron point mass model

2017 ◽  
Vol 232 (19) ◽  
pp. 3500-3511 ◽  
Author(s):  
Devi Singh Kumani ◽  
Himanshu Chaudhary
Keyword(s):  
Optimization Technique ◽  
Computational Effort ◽  
Point Mass ◽  
Constraint Forces ◽  
Mass Model ◽  
Industrial Manipulator ◽  
Model Configuration ◽  
Teaching Learning Based Optimization ◽  
Teaching Learning ◽  
First Time

The octahedron seven point mass model of equimomental point masses and optimization technique “teaching–learning-based optimization” is presented to minimize constraint forces and moments at joints of an industrial manipulator. The octahedron point mass model configuration presented offers positive values for equimomental point masses to facilitate link shape formation. Equations are derived to compute point masses and their locations for the rigid links of an industrial manipulator. The flow chart of the teaching–learning-based optimization applicable in solving the manipulators problem is presented and used. The constraint moments at heavily loaded joints are reduced significantly. Moreover, the maximum values of driving torques are also reduced at joints. It is observed that teaching–learning-based optimization gives better results with less computational effort vis-à-vis genetic algorithm for the manipulator optimization problem formulated. The teaching–learning-based optimization algorithm introduced first time for manipulator balancing as optimization solver.

scholarly journals Systematic errors induced by the elliptical power-law model in galaxy-galaxy strong lens modeling

2021 ◽  
Author(s):  
Xiaoyue Cao ◽  
Ran Li ◽  
James Nightingale ◽  
Richard Massey ◽  
Andrew Robertson ◽  
...  
Keyword(s):  
Power Law ◽  
Gravitational Lensing ◽  
Time Delays ◽  
Hubble Space Telescope ◽  
Stellar Dynamics ◽  
Hubble Constant ◽  
Power Law Model ◽  
Mass Model ◽  
Mass Distributions ◽  
Model Mismatch

Abstract The elliptical power-law (EPL) mass model of the mass in a galaxy is widely used in strong gravitational lensing analyses. However, the distribution of mass in real galaxies is more complex. We quantify the biases due to this model mismatch by simulating and then analysing mock {\it Hubble Space Telescope} imaging of lenses with mass distributions inferred from SDSS-MaNGA stellar dynamics data. We find accurate recovery of source galaxy morphology, except for a slight tendency to infer sources to be more compact than their true size. The Einstein radius of the lens is also robustly recovered with 0.1\% accuracy, as is the global density slope, with 2.5\% relative systematic error, compared to the 3.4\% intrinsic dispersion. However, asymmetry in real lenses also leads to a spurious fitted `external shear' with typical strength, $\gamma_{\rm ext}=0.015$. Furthermore, time delays inferred from lens modelling without measurements of stellar dynamics are typically underestimated by $\sim$5\%. Using such measurements from a sub-sample of 37 lenses would bias measurements of the Hubble constant $H_0$ by $\sim$9\%. The next generation cosmography must use more complex lens mass models.

scholarly journals Modelling the Soft X-ray Excess in E1615+061

1994 ◽  
Vol 159 ◽  
pp. 317-317
Author(s):  
M. Bałucińska-Church ◽  
L. Piro ◽  
H. Fink ◽  
F. Fiore ◽  
M. Matsuoka ◽  
...  
Keyword(s):  
Power Law ◽  
Column Density ◽  
High Energy ◽  
X Rays ◽  
Hard Component ◽  
X Ray ◽  
Simple Power ◽  
High Energy Tail ◽  
Photon Index ◽  
Best Fit

SummaryWe report results of an international UV – X-ray campaign in 1990–1992 involving the IUE, Rosat and Ginga satellites to observe E1615+061, a Seyfert 1 galaxy with peculiar spectral and intensity behaviour over the last 20 years. The source has been found to be stable in its medium state during the observations. The Ginga (1–20 keV) spectrum of E1615+061 is adequately represented by a simple power law with a photon index α = 1.8 ± 0.1. However, α ∼ 2, as expected for the intrinsic power law component in a reflection model, cannot be ruled out statistically. The Rosat PSPC (0.1–2 keV) spectra collected during the All Sky Survey and the AO-1 phase can be well-described by a simple power law (α = 2.2 ± 0.1) with cold absorber (NH = 3.5 ± 0.3 · 10λ20 H/cmλ2). Both the photon index being significantly different than that obtained from the Ginga spectrum and the column density being smaller than the galactic column (NH ∼ 4.2 · 10λ20 H/cmλ2) give an indication of a soft excess over and above the hard component seen in the Ginga spectrum. E1615+061 has been observed with IUE in 1990 and in 1992. The source was stable and the colour excess E(B-V) derived from the data = 0.1 is in good agreement with that expected from the galactic absorption.To parameterise the soft excess we fitted the Rosat data with a two-component model consisting of a power law, and a blackbody or thermal bremsstrahlung, with a single galactic absorption term. The column density and the slope of the power law were kept constant. The blackbody temperature was 80 ± 6 eV and 63 ± 12 eV for photon index equal to 1.8 and 2.0, respectively, whereas the bremsstrahlung temperature was 220 ± 40 eV and 115 ± 30 eV for the two cases.An attempt to model the soft excess seen in the Rosat PSPC spectrum has been made assuming that the soft excess is the high energy tail of a disc spectrum which peaks in the UV part of the spectrum. Additionally it was assumed that there is a hard component contributing to the spectrum from UV to X-rays with parameters as described by the Ginga spectrum. The best fit parameters: the mass of the central source and the mass accretion rate were around 5 ± 1 · 10λ6 M⊙ and 0.2 ± 0.04 M⊙/yr, respectively.Our modelling shows that the soft X-ray excess can be described (χredλ2 < 1.2) as the high energy tail of an accretion disk spectrum if the intrinsic power law is quite steep (α = 2). The main contribution to the residuals in the Rosat PSPC range comes from 0.3–0.6 keV, with a tendency for these residuals to increase when the slope gets flatter. The accretion luminosity is ∼ 6.5 · 10λ44 erg/s for the best fit parameters, i.e. about the Eddington luminosity.

scholarly journals High-Accuracy Treecode Based on Pseudoparticle Multipole Method

2003 ◽  
Vol 208 ◽  
pp. 305-314 ◽  
Author(s):  
Atsushi Kawai ◽  
Junichiro Makino
Keyword(s):  
Arbitrary Order ◽  
Scaling Up ◽  
Multipole Expansion ◽  
High Accuracy ◽  
High Order ◽  
Point Mass ◽  
Linear Scaling ◽  
Multipole Method ◽  
Primary Advantage

We invented the pseudoparticle multipole method (P2M2), a method to express multipole expansion by a distribution of pseudoparticles. We can use this distribution of particles to calculate high order terms in both the Barnes-Hut treecode and FMM. The primary advantage of P2M2 is that it works on GRAPE. Although the treecode has been implemented on GRAPE, we could handle terms only up to dipole, since GRAPE can calculate forces from point-mass particles only. Thus the calculation cost grows quickly when high accuracy is required. With P2M2, the multipole expansion is expressed by particles, and thus GRAPE can calculate high order terms. Using P2M2, we realized arbitrary-order treecode on MDGRAPE-2. Timing result shows MDGRAPE-2 accelerates the calculation by a factor between 20 (for low accuracy) to 150 (for high accuracy). We parallelized the code so that it runs on MDGRAPE-2 cluster. The calculation speed of the code shows close-to-linear scaling up to 16 processors for N ≳ 106.

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