Eccentricity Dependence Research Articles

Post-Newtonian expansion of the spin-precession invariant for eccentric-orbit nonspinning extreme-mass-ratio inspirals to 9PN and e16

We calculate the eccentricity dependence of the high-order post-Newtonian (PN) expansion of the spin-precession invariant $\psi$ for eccentric-orbit extreme-mass-ratio inspirals with a Schwarzschild primary. The series is calculated in first-order black hole perturbation theory through direct analytic expansion of solutions in the Regge-Wheeler-Zerilli formalism, using a code written in \textsc{Mathematica}. Modes with small values of $l$ are found via the Mano-Suzuki-Takasugi (MST) analytic function expansion formalism for solutions to the Regge-Wheeler equation. Large-$l$ solutions are found by applying a PN expansion ansatz to the Regge-Wheeler equation. Previous work has given $\psi$ to 9.5PN order and to order $e^2$ (i.e., the near circular orbit limit). We calculate the expansion to 9PN but to $e^{16}$ in eccentricity. It proves possible to find a few terms that have closed-form expressions, all of which are associated with logarithmic terms in the PN expansion. We also compare the numerical evaluation of our PN expansion to prior numerical calculations of $\psi$ in close orbits to assess its radius of convergence. We find that the series is not as rapidly convergent as the one for the redshift invariant at $r \simeq 10M$ but still yielding $\sim 1\%$ accuracy for eccentricities $e \lesssim 0.25$.

High-order post-Newtonian expansion of the redshift invariant for eccentric-orbit nonspinning extreme-mass-ratio inspirals

We calculate the eccentricity dependence of the high-order post-Newtonian (PN) series for the generalized redshift invariant $⟨{u}^{t}{⟩}_{\ensuremath{\tau}}$ for eccentric-orbit extreme-mass-ratio inspirals on a Schwarzschild background. These results are calculated within first-order black hole perturbation theory using Regge-Wheeler-Zerilli (RWZ) gauge. Our Mathematica code is based on a familiar procedure, using PN expansion of the Mano-Suzuki-Takasugi analytic function formalism for $l$ modes up to a certain maximum and then using a direct general-$l$ PN expansion of the RWZ equation for arbitrarily high $l$. We calculate dual expansions in PN order and in powers of eccentricity, reaching 10PN relative order and ${e}^{20}$. Detailed knowledge of the eccentricity expansion at each PN order allows us to find within the eccentricity dependence numerous closed-form expressions and multiple infinite series with known coefficients. We find leading logarithm sequences in the PN expansion of the redshift invariant that reflect a similar behavior in the PN expansion of the energy flux to infinity. A set of flux terms and special functions that appear in the energy flux, like the Peters-Mathews flux itself, are shown to reappear in the redshift PN expansion.

Eccentricity Dependence of Motion Induced Position Shifts Revealed by Continuous Motion Nulling

Eccentricity Dependence of Motion Induced Position Shifts Revealed by Continuous Motion Nulling

Eccentric-orbit extreme-mass-ratio-inspiral radiation. II. 1PN correction to leading-logarithm and subleading-logarithm flux sequences and the entire perturbative 4PN flux

In a recent paper we showed that for eccentric-orbit extreme-mass-ratio inspirals the analytic forms of the leading-logarithm energy and angular momentum post-Newtonian (PN) flux terms (radiated to infinity) can, to arbitrary PN order, be determined by sums over the Fourier spectrum of the Newtonian quadrupole moment. We further showed that an essential part of the eccentricity dependence of the related subleading-logarithm PN sequences, at lowest order in the symmetric mass ratio $\nu$, stems as well from the Newtonian quadrupole moment. Once that part is factored out, the remaining eccentricity dependence is more easily determined by black hole perturbation theory. In this paper we show how the sequences that are the 1PN corrections to the entire leading-logarithm series, namely terms that appear at PN orders $x^{3k+1} \log^k(x)$ and $x^{3k+5/2} \log^k(x)$ (for PN compactness parameter $x$ and integers $k\ge 0$), at lowest order in $\nu$, are determined by the Fourier spectra of the Newtonian mass octupole, Newtonian current quadrupole, and 1PN part of the mass quadrupole moments. We also develop a conjectured (but plausible) form for 1PN correction to the leading logs at second order in $\nu$. Further, in analogy to the first paper, we show that these same source multipole moments also yield nontrivial parts of the 1PN correction to the subleading-logarithm series, and that the remaining eccentricity dependence (at lowest order in $\nu$) can then more easily be determined using black hole perturbation theory. We use this method to determine the entire analytic eccentricity dependence of the perturbative (i.e., lowest order in $\nu$) 4PN non-log terms, $\mathcal{R}_4(e_t)$ and $\mathcal{Z}_4(e_t)$, for energy and angular momentum respectively.

Eccentric-orbit extreme-mass-ratio-inspiral radiation: Analytic forms of leading-logarithm and subleading-logarithm flux terms at high PN orders

We present new results on the analytic eccentricity dependence of several sequences of gravitational wave flux terms at high post-Newtonian (PN) order for extreme-mass-ratio inspirals. These sequences are the leading logarithms, which appear at PN orders $x^{3k} \log^k(x)$ and $x^{3k+3/2} \log^k(x)$ for integers $k\ge 0$ ($x$ a PN compactness parameter), and the subleading logarithms, which appear at orders $x^{3k} \log^{k-1}(x)$ and $x^{3k+3/2} \log^{k-1}(x)$ ($k\ge 1$), in both the energy and angular momentum radiated to infinity. For the energy flux leading logarithms, we show that to arbitrarily high PN order their eccentricity dependence is determined by particular sums over the function $g(n,e_t)$, derived from the Newtonian mass quadrupole moment, that normally gives the spectral content of the Peters-Mathews flux as a function of radial harmonic $n$. An analogous power spectrum $\tilde{g}(n,e_t)$ determines the leading logarithms of the angular momentum flux. For subleading logs, the quadrupole power spectra are again shown to play a role, providing a distinguishable part of the eccentricity dependence of these flux terms to high PN order. With the quadrupole contribution understood, the remaining analytic eccentricity dependence of the subleading logs can in principle be determined more easily using black hole perturbation theory. We show this procedure in action, deriving the complete analytic structure of the $x^6 \log(x)$ subleading-log term and an analytic expansion of the $x^{9/2}$ subleading log to high order in a power series in eccentricity. We discuss how these methods might be extended to other sequences of terms in the PN expansion involving logarithms.

Spatial scaling of illusory motion perceived in a static figure.

In a phenomenon known as the Rotating Snakes illusion (Kitaoka & Ashida, 2003), illusory motion is perceived in a static figure with a specially designed luminance profile. It is known that the strength of this illusion increases with eccentricity, suggesting that the underlying mechanism of the illusion has a spatial property that changes with eccentricity. If a change in receptive-field size of responsible neurons causes the eccentricity dependence of the illusion, its strength should be spatially scalable using a scaling factor that increases with eccentricity, because the receptive field size of neurons in visual areas with retinotopy generally obeys quantitative dependence on eccentricity. For the luminance micropatterns comprising the figure for the Rotating Snakes illusion, we varied eccentricity from 9 to 15 deg and spatial frequency from 0.25 to 1.6 cycles/deg, and measured illusion strength. Illusion strength was found to increase with decreasing spatial frequency and with increasing eccentricity. Furthermore, the profiles of illusion strength at different eccentricities were spatially scalable into a single parabola as a function of the spatially scaled visual angle. The estimated scaling factors linearly increased with eccentricity with a slope similar to the eccentricity dependence of the receptive field size of V1 neurons, suggesting the involvement of early visual areas in the generation of the illusion.

Eccentricity dependence of orientation anisotropy of surround suppression of contrast-detection threshold.

Both neurophysiological and psychophysical data provide evidence for orientation biases in nonfoveal vision-specifically, a tendency for a Cartesian horizontal and vertical bias close to fixation, changing to a radial bias with increasing retinal eccentricity. We explore whether the strength of surround suppression of contrast detection also depends on retinotopic location and relative surround configuration (horizontal, vertical, radial, tangential) in parafoveal vision. Three visual-field locations were tested (0°, 225°, and 270°, angle increasing anticlockwise from 0° horizontal axis) at viewing eccentricities of 6° and 15°. Contrast-detection threshold was estimated with and without a surrounding annulus. At 6° eccentricity, horizontally oriented parallel center-surround (C-S) configurations resulted in greater surround suppression compared to vertically oriented parallel center-surround configurations (p = 0.001). At 15° eccentricity, radially oriented parallel center-surround stimuli conferred greater suppression than tangentially oriented stimuli (p = 0.027). Parallel surrounds resulted in greater suppression than orthogonal surrounds at both eccentricities (p < 0.05). At 6° the horizontal center was more susceptible to suppression than a vertical center (p < 0.001) for both parallel and orthogonal surrounds, while at 15° a radial center was more susceptible to suppression (relative to a tangential center), but only if the surround was parallel (p = 0.005). Our data show that orientation anisotropy of surround suppression alters with eccentricity, reflecting a link between suppression strength and visual-field retinotopy.

String percolation threshold for elliptically bounded systems

It has been shown that a hot and dense deconfined nuclear matter state produced in ultra-relativistic heavy-ion collisions, can be quantitatively described by the String Percolation phenomenological model. The model address the phase transition in terms of the two-dimensional continuum percolation theory over strings, which are schematic representations of the fundamental interactions among the partons of the colliding nuclei in the initial state. In this work, we present an extension of the critical string density results including the eccentricity dependence on the initial state geometry focus on small string number with different density profile, small deviations from the different profile densities are found. The percolation threshold shows consistency with the thermodynamic limit for the uniform density profile with a large number of strings in the case of circular boundary system. A significant dependence on the eccentricity for a small number of strings compared to high occupancy systems is exhibited, the implications may become relevant in hadron–hadron or hadron–nucleus collision systems.

Mean Motion Resonances at High Eccentricities: The 2:1 and the 3:2 Interior Resonances

Abstract Mean motion resonances (MMRs) play an important role in the formation and evolution of planetary systems and have significantly influenced the orbital properties and distribution of planets and minor planets in the solar system and in exoplanetary systems. Most previous theoretical analyses have focused on the low- to moderate-eccentricity regime, but with new discoveries of high-eccentricity resonant minor planets and even exoplanets, there is increasing motivation to examine MMRs in the high-eccentricity regime. Here we report on a study of the high-eccentricity regime of MMRs in the circular planar restricted three-body problem. Numerical analyses of the 2:1 and the 3:2 interior resonances are carried out for a wide range of planet-to-star mass ratio μ, and for a wide range of eccentricity of the test particle. The surface-of-section technique is used to study the phase space structure near resonances. We find that new stable libration zones appear at higher eccentricity at libration centers that are shifted from those at low eccentricities. We provide physically intuitive explanations for these transitions in phase space, and we present novel results on the mass and eccentricity dependence of the resonance widths. Our results show that MMRs have sizable libration zones at high eccentricities, comparable to those at lower eccentricities.

Eccentricity Dependence on Iron Abundance

AbstractThe occurrence and eccentricity distribution of planets as a function of period is significantly different for iron-rich and iron-poor planet systems. We find that iron-poor stars with planets having periods between 525 and 600 days have higher eccentricity than such systems outside this range. If whole planet pollution causes the correlation of giant planet eccentricity with stellar iron abundance, then this cluster could be due to a paucity of pollution in this period range. Newly reported patterns of planet occurrence must result from planet system architectural features such as the snow line, followed by subsequent migration. Different results favor pollution or higher initial iron abundance causing the higher occurrence fraction of giant planets hosted by iron-rich stars, but the two explanations could be complementary. Relations between planet and stellar parameters are a major product of planet-finding, which promise further insights into star-planet system formation and evolution. Collaborators are sought to study these patterns. We expect a spirited debate over the relative contributions of initial abundances, disk accretion, and whole planet accretion.

Visual spatial attention enhances the amplitude of positive and negative fMRI responses to visual stimulation in an eccentricity-dependent manner

Endogenous visual spatial attention improves perception and enhances neural responses to visual stimuli at attended locations. Although many aspects of visual processing differ significantly between central and peripheral vision, little is known regarding the neural substrates of the eccentricity dependence of spatial attention effects. We measured amplitudes of positive and negative fMRI responses to visual stimuli as a function of eccentricity in a large number of topographically-organized cortical areas. Responses to each stimulus were obtained when the stimulus was attended and when spatial attention was directed to a stimulus in the opposite visual hemifield. Attending to the stimulus increased both positive and negative response amplitudes in all cortical areas we studied: V1, V2, V3, hV4, VO1, LO1, LO2, V3A/B, IPS0, TO1, and TO2. However, the eccentricity dependence of these effects differed considerably across cortical areas. In early visual, ventral, and lateral occipital cortex, attentional enhancement of positive responses was greater for central compared to peripheral eccentricities. The opposite pattern was observed in dorsal stream areas IPS0 and putative MT homolog TO1, where attentional enhancement of positive responses was greater in the periphery. Both the magnitude and the eccentricity dependence of attentional modulation of negative fMRI responses closely mirrored that of positive responses across cortical areas.

Eccentricity dependence of the curveball illusion.

When the Gaussian envelope of a Gabor patch moves straight down while its sine wave component drifts to the left, the Gabor is seen to move down to the left when viewed with peripheral vision (exhibiting the so-called curveball illusion) but straight down when viewed foveally. We used a nulling procedure to quantify the strength of the curveball illusion at a range of stimulus sizes and eccentricities to determine if the illusion is size dependent and whether size scaling can equate the strength of the illusion across the visual field. We find that the illusion is size dependent (illusion strength decreases with increasing stimulus size) but that size scaling is not sufficient to elicit an illusion at fixation that matches the strength of the illusion at 10°. These results suggest that certain changes in motion processing across the visual field differ in more than just the local scale of the mechanisms involved.

Effect of Blur Adaptation on Human Parafoveal Vision

This study was conducted to investigate whether neural compensation for induced defocus can alter visual resolution in other areas of the human retina beyond the fovea. In certain circumstances, the blur adaptation response may be influenced by refractive status. The effect of blur adaptation on the central 10° of the retina was investigated in 20 normally sighted observers (10 emmetropes and 10 myopes; median age, 21 years). Visual acuity (VA) was measured at the fovea and at five locations of the parafoveal nasal visual field (2°, 4°, 6°, 8°, and 10°) with best corrected distance vision. Myopic defocus of 1 D was introduced, and the same measurements were repeated immediately before and after a 30-minute adaptation. VA declined with increasing eccentricity in the clear, blurred, and blur-adapted viewing conditions. The rate of decline was quantified by the parameter E2, which represents the amount of eccentricity dependence of the acuity task. Foveal and parafoveal VA decreased with the introduction of optical defocus and improved significantly after a period of blur adaptation. The consistent value of E2 in each condition indicated that these changes in VA were not eccentricity dependent. Changes in VA under blurred and blur-adapted conditions were of similar magnitudes in myopic and emmetropic observers. Neural adaptation to blur improves VA under defocused conditions in the parafovea as well as the fovea, indicating that the underlying compensatory mechanism acts across a range of spatial scales and independently of retinal eccentricity. Foveal and parafoveal blur adaptation does not vary with refractive error.

Peripheral vision and pattern recognition: a review.

We summarize the various strands of research on peripheral vision and relate them to theories of form perception. After a historical overview, we describe quantifications of the cortical magnification hypothesis, including an extension of Schwartz's cortical mapping function. The merits of this concept are considered across a wide range of psychophysical tasks, followed by a discussion of its limitations and the need for non-spatial scaling. We also review the eccentricity dependence of other low-level functions including reaction time, temporal resolution, and spatial summation, as well as perimetric methods. A central topic is then the recognition of characters in peripheral vision, both at low and high levels of contrast, and the impact of surrounding contours known as crowding. We demonstrate how Bouma's law, specifying the critical distance for the onset of crowding, can be stated in terms of the retinocortical mapping. The recognition of more complex stimuli, like textures, faces, and scenes, reveals a substantial impact of mid-level vision and cognitive factors. We further consider eccentricity-dependent limitations of learning, both at the level of perceptual learning and pattern category learning. Generic limitations of extrafoveal vision are observed for the latter in categorization tasks involving multiple stimulus classes. Finally, models of peripheral form vision are discussed. We report that peripheral vision is limited with regard to pattern categorization by a distinctly lower representational complexity and processing speed. Taken together, the limitations of cognitive processing in peripheral vision appear to be as significant as those imposed on low-level functions and by way of crowding.

MSTD population responses account for the eccentricity dependence of heading discrimination thresholds

MSTD population responses account for the eccentricity dependence of heading discrimination thresholds

Holistic crowding of Mooney faces

An object or feature is generally more difficult to identify when other objects are presented nearby, an effect referred to as crowding. Here, we used Mooney faces to examine whether crowding can also occur within and between holistic face representations (C. M. Mooney, 1957). Mooney faces are ideal stimuli for this test because no cues exist to distinguish facial features in a Mooney face; to find any facial feature, such as an eye or a nose, one must first holistically perceive the image as a face. Through a series of six experiments we tested the effect of crowding on Mooney face recognition. Our results demonstrate crowding between and within Mooney faces and fulfill the diagnostic criteria for crowding, including eccentricity dependence and lack of crowding in the fovea, critical flanker spacing consistent with less than half the eccentricity of the target, and inner-outer flanker asymmetry. Further, our results show that recognition of an upright Mooney face is more strongly impaired by upright Mooney face flankers than inverted ones. Taken together, these results suggest crowding can occur selectively between high-level representations of faces and that crowding must occur at multiple levels in the visual system.

STUDY OF COMETS COMPOSITION AND STRUCTURE

The present paper focuses on the nature of the different interactions between cometary nucleus and tail with solar wind. The dynamics of the comet will impose many features that provide unique behavior of the comet when entering the solar system. These features are reviewed in this paper and few investigations are made. The calculations made in this work represent the analysis and interpretation of the different features of the comet, such as perihelion and eccentricity dependence on the gas production rate, and the dependence of the latter on the composition of the comet nucleus. The dependences of the heliocentric, bow shock, contact surface, and stand-off distances with gas production rate for many types of comets that cover linear and non-linear types are studied in this work. Important results are obtained which indicated the different physical interactions between cometary ions and solar wind. Furthermore, the important relation between mean molecular weight and gas production rate are analyzed and studied in this work and a conclusion is made that, as the gas production rate increases, the mean molecular weight will decrease exponentially. A detailed discussion for this unique relation is given.

Extrafoveal viewing reveals the nature of second-order human vision

For the majority of visual tasks, performance in extrafoveal vision can be equated with that at the fovea simply by a change in spatial scale of the stimuli (magnification). We sought to exploit this association to examine the nature of second-order vision. More specifically, we investigate the relationship between the scale of second-order vision and the scale of its first-order input. We find that sensitivity to second-order stimuli can be equated across visual space, but only for stimuli that are magnified in every respect (identical scaling of both first- and second-order characteristics). In other words, sensitivity to stimuli which poses a fixed ratio between the scale of first-order input and second-order spatial scale can be equated across the visual field using a single magnification factor. Moreover, stimuli which poses quite different ratios of first- and second-order scale can be equated across eccentricity using the same magnification factor. This argues for a strict relationship between second-order vision and the scale of its first-order input and reflects a parallel arrangement of dedicated second-order mechanisms having a common eccentricity dependence.

Pattern-onset stimulation boosts central multifocal VEP responses.

Multifocal visual evoked potentials (VEP) allow one to assess whether stimulation at specific visual field locations elicits cortical activity; it might therefore enable us to conduct objective visual field perimetry. However, due to the cortical folding, which differs markedly between subjects, a particular electroencephalogram generator may fail to project signal on some recording electrodes. This may lead to false alarms for potential scotomata. Here we compare pattern-reversal and pattern-onset stimulation in their efficacy to activate the visual cortex and recorded mfVEPs to 60 locations comprising a visual field of 44 degrees diameter. We report three main findings: (1) Pattern-onset compared to pattern-reversal enhances the amplitude by 30% for stimulation of the central visual field (<10 degrees radius), while evoking 30% less response in the periphery (>15 degrees ). (2) Although pattern-onset and pattern-reversal responses differ markedly in their eccentricity dependence, they have a similar topographical distribution. (3) By combining both stimuli, the number of false positives was reduced to less than 1.5% of the visual field locations tested. We conclude that pattern-onset and pattern-reversal activate identical visual cortical areas but target different neural mechanisms within these areas. Furthermore, pattern-onset stimulation greatly increases the sensitivity of the mfVEP to assess the cortical representation of the central 10 degrees of the visual field.

Discriminating mirror symmetry in foveal and extra-foveal vision

The ability to discriminate perfect from imperfect mirror symmetry was examined at the fovea and at eccentricities out to 10° in the nasal visual field. A 2-AFC method of constant stimuli was employed in which a bilaterally symmetric pattern was presented in one interval and a degraded version of this symmetric pattern in the other. The subject’s task was to decide which interval contained the perfectly symmetric pattern. Pattern size was varied by changing the viewing distance. Probit analysis revealed the degree of asymmetry corresponding to 75% correct performance. Given sufficient size scaling, perfectly symmetric stimuli can be discriminated from degraded symmetric stimuli in extra-foveal vision. Spatial scaling with an E 2 value similar to that for positional acuity was successful in removing the eccentricity dependence for the task.