Grazing Orbit Research Articles

A Blind Search for Transit Depth Variability with TESS

The phenomenon of transit depth variability offers a pathway through which processes such as exoplanet atmospheric activity and orbital dynamics can be studied. In this work we conduct a blind search for transit depth variations among 330 known planets observed by the Transiting Exoplanet Survey Satellite within its first four years of operation. Through an automated periodogram analysis, we identify four targets (KELT-8b, HAT-P-7b, HIP 65 Ab, and TrES-3b) that appear to show significant transit depth variability. We find that KELT-8b’s transit depth variability likely comes from contaminating flux from a nearby star, while the apparent variabilities of HIP 65 Ab and TrES-3b are probable artifacts due to their grazing orbits. HAT-P-7b indicates signs of variability that possibly originate from the planet or its host star. A population-level analysis does not reveal any significant correlation between transit depth variability and the effective temperature and mass of the host star; such correlation could arise if stellar activity was the cause of depth variations via the transit light source effect. Extrapolating our ∼1% detection rate to the upcoming Roman mission, predicted to yield of order 100,000 transiting planets, we expect that ∼1000 of these targets will be found to exhibit significant transit depth variability.

Higher order discontinuity mapping for double grazing bifurcations in a slender rigid block confined between side-walls

We investigate the near grazing dynamics in a slender rigid block confined between two side-walls. We obtain conditions of the existence of double grazing orbits of the system, namely, periodic orbits that have two grazing points. By extending the method of discontinuity mapping for grazing orbit with a single grazing point, we compute the lower and the higher order approximations of the Poincaré map respectively near the double grazing orbit. The results of computing Monte Carlo bifurcation diagrams obtained by the lower and the higher order maps respectively are compared with those from direct simulations of the original system. We find that there are large disagreements between the lower order map and the original system. Thus the lower order map is not accurate enough to study double grazing bifurcations. On the other hand, the higher order map can effectively reduce such disagreements and we expect that it plays an important role in the study of the near grazing dynamics of the system.

First Return Time to the contact hyperplane for $ N $-degree-of-freedom vibro-impact systems

<p style='text-indent:20px;'>The paper deals with the dynamics of conservative <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula>-degree-of-freedom vibro-impact systems involving one unilateral contact condition and a linear free flow. Among all possible trajectories, grazing orbits exhibit a contact occurrence with vanishing incoming velocity which generates mathematical difficulties. Such problems are commonly tackled through the definition of a Poincaré section and the attendant First Return Map. It is known that the First Return Time to the Poincaré section features a square-root singularity near grazing. In this work, a non-orthodox yet natural and intrinsic Poincaré section is chosen to revisit the square-root singularity. It is based on the unilateral condition but is not transverse to the grazing orbits. A detailed investigation of the proposed Poincaré section is provided. Higher-order singularities in the First Return Time are exhibited. Also, activation coefficients of the square-root singularity for the First Return Map are defined. For the linear and periodic grazing orbits from which bifurcate nonlinear modes, one of these coefficients is necessarily non-vanishing. The present work is a step towards the stability analysis of grazing orbits, which still stands as an open problem.</p>

15 Climate Change and the Evolving Fulani Pastoralists Cattle Production System in Nigeria: A Need to Retool Extension and Innovation Services

Abstract The Fulani pastoralists of Nigeria, as in other parts of West Africa, are traditionally semi- sedentary or nomadic herders, and are often described as the largest nomadic group in the world. The combined effects of climate change and instability due to terrorist activities have led to massive displacement of many Nigerian Fulani households to Southwestern Nigeria in search of refuge, pasture and water for their cattle. The objective of this project was to examine the current Fulani cattle production system. In addition to obtaining data via focus groups and direct observation, a survey was administered to members of 600 Fulani pastoralist households to determine cattle production techniques and herding strategies adopted. The observation period occurred over a 24 month period in three Southwestern Nigeria states (Ogun, Kwara and Oyo). Herding activities were mapped by using hand-held Global Positioning System (GPS) to obtain the coordinates of grazing routes and settlements. The results showed that the Fulani operate a system of male social contract and kinship networks in labor utilization and movement of herds. Among semi-sedentary and sedentary groups, the grazing orbit was usually within 5km to 10km radius of the settlement, depending on the season. Of the households observed, very few had access to tap or treated water (0.9% in Oyo, 1.4% in Ogun and 0.2% in Kwara). In Ogun, rivers and streams provided water for the largest number of households, while in Kwara, hand-dug wells were the largest providers. In Oyo, streams and rivers provided water for a large percentage of the Fulani households. In addition to a lack of basic water infrastructure, the results further revealed low productivity of cattle due to poor management and the utilization of cattle breeds with low genetic potential. It is recommended that micro-ranches, enhanced by extension services that provide training on cattle management, be implemented for this population.

Dust Observations by the Radio and Plasma Wave Science Instrument During Cassini's Grand Finale

AbstractDust particles in the Saturn system can be detected by the Radio and Plasma Wave Science (RPWS) instrument on board Cassini via antenna voltage signals induced by dust impacts. These impact signals have been simulated in the laboratory by accelerating dust particles onto a Cassini model with electric field antennas. RPWS dust measurements have been shown to be consistent with the Cosmic Dust Analyzer. During the Grand Finale orbits, Cassini flew through the gap between the D ring and Saturn's atmosphere 22 times. In situ measurements by RPWS helped quantify the hazards posed to the spacecraft and instruments on board, which showed a micron‐sized dust density orders of magnitude lower than that observed during the Ring Grazing orbits. Close inspection of the waveforms indicated a possible dependence of the impact signal decay time on ambient plasma density.

Mapping spiral waves and other radial features in Saturn’s rings

We have analyzed the highest-quality images to be obtained by Cassini of Saturn’s main rings after the Saturn Orbit Insertion (SOI) and before the Ring Grazing Orbits (RGO) and Grand Finale (GF). These images are comparable to those of SOI in fidelity, though not in nominal resolution, due to their high signal-to-noise. We have systematically searched for radial structure in these images by reducing them to a single dimension (distance from Saturn’s center) and using the continuous wavelet transform technique. We discuss the resonant theory of spiral waves and discuss the proper method for deriving the local surface mass density from the wavelet signature of a spiral wave. We present 1) individual features of interest found in our data, including several classes of waves that have not previously been reported; 2) a radial profile of surface mass density in Saturn’s rings, which is more definitive for the A ring than any previously presented and which corrects some errors in previous profiles; and 3) an atlas of resonant features that indicates whether each feature is or is not expressed in the rings and that is organized graphically by resonance strength.

Drift-resonant, relativistic electron acceleration at the outer planets: Insights from the response of Saturn’s radiation belts to magnetospheric storms

The short, 7.2-day orbital period of Cassini’s Ring Grazing Orbits (RGO) provided an opportunity to monitor how fast the effects of an intense magnetospheric storm-time period (days 336–343/2016) propagated into Saturn’s electron radiation belts. Following the storms, Cassini’s MIMI/LEMMS instrument detected a transient extension of the electron radiation belts that in subsequent orbits moved towards the inner belts, intensifying them in the process. This intensification was followed by an equally fast decay, possibly due to the rapid absorption of MeV electrons by the planet’s main rings. Surprisingly, all this cycle was completed within four RGOs, effectively in less than a month. That is considerably faster than the year-long time scales of Saturn’s proton radiation belt evolution. In order to explain this difference, we propose that electron radial transport is partly controlled by the variability of global scale electric fields which have a fixed local time pointing. Such electric fields may distort significantly the orbits of a particular class of energetic electrons that cancel out magnetospheric corotation due to their westward gradient and curvature drifts (termed “corotation-resonant” or “local-time stationary” electrons) and transport them radially between the ring current and the radiation belts within several days and few weeks. The significance of the proposed process is highlighted by the fact that corotation resonance at Saturn occurs for electrons of few hundred keV to several MeV. These are the characteristic energies of seed electrons from the ring current that sustain the radiation belts of the planet. Our model’s feasibility is demonstrated through the use of a simple test-particle simulation, where we estimate that uniform but variable electric fields with magnitudes lower that 1.0 mV/m can lead to a very efficient transport of corotation resonant electrons. Such electric fields have been consistently measured in the magnetosphere, and here we provide additional evidence showing that they may be constantly present all the way down to the outer edge of Saturn’s main rings, further supporting our model. The implications of our findings are not limited to Saturn. Corotation resonance at Jupiter occurs for electrons with energies above about 10 MeV throughout the quasi-dipolar, energetic particle-trapping region of the magnetosphere. The proposed process could in principle then lead to rapid transport and adiabatic acceleration electrons into ultra-relativistic energies. The observation by Galileo’s EPD/LEMMS instrument of an intense Jovian acceleration event at the orbital distance of Ganymede during the mission’s C22 orbit, when > 11 MeV electron fluxes were preferentially enhanced, provides additional support to our transport model and insights on the origin of that orbit’s extreme energetic electron environment. Finally, if the mode of radial transport that we describe here is a dominant one, radial diffusion coefficients (DLL) would be subject to strong energy, pitch angle and species dependencies.

Toward Detection of Exoplanetary Rings via Transit Photometry: Methodology and a Possible Candidate

Abstract The detection of a planetary ring of exoplanets remains one of the most attractive, but challenging, goals in the field of exoplanetary science. We present a methodology that implements a systematic search for exoplanetary rings via transit photometry of long-period planets. This methodology relies on a precise integration scheme that we develop to compute a transit light curve of a ringed planet. We apply the methodology to 89 long-period planet candidates from the Kepler data so as to estimate, and/or set upper limits on, the parameters of possible rings. While the majority of our samples do not have sufficient signal-to-noise ratios (S/Ns) to place meaningful constraints on ring parameters, we find that six systems with higher S/Ns are inconsistent with the presence of a ring larger than 1.5 times the planetary radius, assuming a grazing orbit and a tilted ring. Furthermore, we identify five preliminary candidate systems whose light curves exhibit ring-like features. After removing four false positives due to the contamination from nearby stars, we identify KIC 10403228 as a reasonable candidate for a ringed planet. A systematic parameter fit of its light curve with a ringed planet model indicates two possible solutions corresponding to a Saturn-like planet with a tilted ring. There also remain two other possible scenarios accounting for the data; a circumstellar disk and a hierarchical triple. Due to large uncertain factors, we cannot choose one specific model among the three.

HATS-22b, HATS-23b and HATS-24b: three new transiting super-Jupiters from the HATSouth project

We report the discovery of three moderately high-mass transiting hot Jupiters from the HATSouth survey: HATS-22b, HATS-23b and HATS-24b. These planets add to the numbers of known planets in the ~2MJ regime. HATS-22b is a 2.74+/-0.11 MJ mass and 0.953+0.048/-0.029 RJ radius planet orbiting a V = 13.455 +/- 0.040 sub-silar mass (M_star = 0.759+/-0.019 M_sun; R_star = 0.759+/-0.019 R_sun) K-dwarf host star on an eccentric (e = 0.079 +/- 0.026) orbit. This planet's high planet-to-stellar mass ratio is further evidence that migration mechanisms for hot Jupiters may rely on exciting orbital eccentricities that bring planets closer to their parent stars followed by tidal circularisation. HATS-23b is a 1.478 +/- 0.080 MJ mass and 1.69 +/- 0.24 RJ radius planet on a grazing orbit around a V = 13.901 +/- 0.010 G-dwarf with properties very similar to those of the Sun (M_star = 1.115 +/- 0.054 M_sun; R_star = 1.145 +/- 0.070 R_sun). HATS-24b orbits a moderately bright V = 12.830 +/- 0.010 F-dwarf star (M_star = 1.218 +/- 0.036 M_sun; R_star = 1.194+0.066/-0.041 R_sun). This planet has a mass of 2.39 +0.21/-0.12 MJ and an inflated radius of 1.516 +0.085/-0.065 RJ.

Nonsmooth modal analysis of a N-degree-of-freedom system undergoing a purely elastic impact law

The dynamics of a N-degree-of-freedom autonomous oscillator undergoing an energy-preserving impact law on one of its masses is investigated in the light of nonlinear modal analysis. The impacted rigid foundation provides a natural Poincaré section of the investigated system from which is formulated a smooth First Return Map well-defined away from the grazing trajectory. In order to focus on the impact-induced nonlinearity, the oscillator is assumed linear. Continuous one-parameter families of T-periodic orbits featuring one impact per period and lying on two-dimensional invariant manifolds in the state-space are shown to exist. The geometry of these piecewise-smooth manifolds is such that a linear “flat” portion (on which contact is not activated) is continuously attached to a purely nonlinear portion (on which contact is activated once per period) exhibiting a velocity discontinuity through a grazing orbit. These features explain the newly introduced terminology “Nonsmooth modal analysis”. The stability of the periodic orbits lying on the invariant manifolds is also explored by calculating the eigenvalues of the linearized First Return Map. Internal resonances and multiple impacts per period are not addressed in this work. However, the pre-stressed case is succinctly described and extensions to multiple oscillators as well as self-contact are discussed.

Investigation of the near-grazing behavior in hard-impact oscillators using model-based TS fuzzy approach

An impact oscillator is a non-smooth dy- namical system with discontinuous state jumps whose dynamical behavior illustrates a variety of non-linear phenomena including a grazing bifurcation. This spe- cific phenomenon is difficult to analyze because it co- incides with an infinite stretching of the phase space in the neighborhood of the grazing orbit, resulting in the well-known problem of the square-root singular- ity of the Jacobian of the discrete-time map. A novel Takagi-Sugeno fuzzy model-based approach is pre- sented in this paper to model a hard impacting system as a non-smooth dynamical system including discon- tinuous jumps. Employing non-smooth Lyapunov the- ory, the structural stability of the system is analyzed to predict the onset of the destabilizing chaotic be- havior. The proposed stability results, formulated as a Linear Matrix Inequality (LMI) problem, demonstrate how the new method can detect the loss of stability just before the grazing bifurcation.

Singularities in soft-impacting systems

In this paper, the character of the normal form map in the neighbourhood of a grazing orbit is investigated for four possible configurations of soft impacting systems. It is shown that, if the spring in the impacting surface is relaxed, the impacting side of the map has a power of 3 / 2 , but if the spring is pre-stressed the map has a square root singularity. The singularity appears only in the trace of the Jacobian matrix and not in the determinant. Under all conditions, the determinant of the Jacobian matrix varies continuously across the grazing condition. However, if the impacting surface has a damper, the determinant decreases exponentially with increasing penetration. It is found that the system behaviour is greatly dependent upon a parameter m , given by 2 ω 0 / ω forcing , and that the singularity disappears for integer values of m . Thus, if the parameters are chosen to obtain an integer value of m , one can expect no abrupt change in behaviour as the system passes through the grazing condition from a non-impacting mode to an impacting mode with increasing excitation amplitude. The above result has been tested on an experimental rig, which showed a persistence of a period-1 orbit across the grazing condition for integer values of m , but an abrupt transition to a chaotic orbit or a high-period orbit for non-integer values of m . Finally, through simulation, it is shown that the condition for vanishing singularity is not a discrete point in the parameter space. This property is valid over a neighbourhood in the parameter space, which shrinks for larger values of the stiffness ratio k 2 / k 1 .

Inelastic impact dynamics of ships with one-sided barriers. Part II: experimental validation

This paper is the second part of a two-part study of impact interaction of a ship roll motion with one-sided ice barrier. The first part was devoted to analytical and numerical simulations for the case of inelastic impact. The analytical approach was based on Zhuravlev and Ivanov non-smooth coordinate transformations. Extensive numerical simulations were carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequency of external wave-induced roll moment. The basins of attraction of safe operation revealed the coexistence of different response regimes such as non-impact periodic oscillations, modulated impact motion, period added impact oscillations, chaotic impact motion and roll-over dynamics. This part presents an experimental investigation conducted on a small ship model in a tow tank. In particular, the experimental tests reveal complex dynamic response characteristics such as multi-frequency wave motion caused by the wave reflection from the tank end wall. Measured results show a good agreement with the predicted results by for small angles of the barrier relative to the ship unbiased position. However, deviation becomes significant as the angle increases. This deviation is mainly attributed to the uncertainty of the coefficient of restitution, which is found to depend on the velocity of impact in addition to the geometry and material properties of the model and barrier.

Inelastic impact dynamics of ships with one-sided barriers. Part I: analytical and numerical investigations

This two-part paper deals with impact interaction of ships with one-sided ice barrier during roll dynamics. The first part presents analytical and numerical studies for the case of inelastic impact. An analytical model of a ship roll motion interacting with ice is developed based on Zhuravlev and Ivanov non-smooth coordinate transformations. These transformations have the advantage of converting the vibro-impact oscillator into an oscillator without barriers such that the corresponding equation of motion does not contain any impact term. Such approaches, however, account for the energy loss at impact times in different ways. The present work, in particular, demonstrates that the impact dynamics may have qualitatively different response characteristics to different dissipation models. The difference between localized and distributed equivalent damping approaches is discussed. Extensive numerical simulations are carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequency of external wave roll moment. The basins of attraction of safe operation are obtained and reveal the coexistence of different response regimes such as non-impact periodic oscillations, modulation impact motion, period added impact oscillations, chaotic impact motion and rotational motion. The second part will consider experimental validations of predicted results.

Vanishing singularity in hard impacting systems

It is known that the Jacobian of the discrete-time map of an impactoscillator in the neighborhood of a grazing orbit depends on thesquare-root of the distance the mass would have gone beyond theposition of the wall if the wall were not there. This results in aninfinite stretching of the phase space, known as the square-rootsingularity. In this paper we look closer into the Jacobian matrix andfind out the behavior of its two parameters---the trace and thedeterminant, across the grazing event. We show that the determinantof the matrix remains invariant in the neighborhood of a grazingorbit, and that the singularity appears only in the trace of thematrix. Investigating the character of the trace, we show that thesingularity disappears if the damped frequency of the oscillator is anintegral multiple of half of the forcing frequency.

Modelling and analysis of ship roll oscillations interacting with stationary icebergs

Impact dynamic interaction of ships with solid ice or stationary rigid structures is a serious problem that affects the safe operation and navigation in arctic regions. The purpose of this study is to present two analytical models of impact interaction between ship roll dynamics and one-side rigid barrier. These models are the phenomenological modelling represented by a power law in stiffness and damping forces, and Zhuravlev non-smooth coordinate transformation. Extensive numerical simulations are carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequencies of external wave roll moment. The basins of attraction of safe operation are obtained and reveal the coexistence of different response regimes such as non-impact periodic oscillations, modulation impact motion, period-added impact oscillations, chaotic impact motion, and unbounded rotational motion. The results are summarized in the bifurcation diagram in terms of response-excitation amplitudes plane. The stability fraction index is obtained for different values of excitation frequency based on the ratio of the area of bounded roll oscillations to the total area of the grazing orbit.

The nature of the normal form map for soft impacting systems

Soft impacting mechanical systems—where the impacting surface is cushioned with a spring–damper support—are common in engineering. Mathematically such systems come under the description of switching dynamical systems, where the dynamics toggle between two (or more) sets of differential equations, determined by switching conditions. It has been shown that the Poincaré map of such a system would have a power of 1/2 (the so-called square-root singularity) if the vector fields at the two sides of the switching manifold differ, and a power of 3/2 if they are the same. These results were obtained by concentrating on the leading order terms in a Taylor expansion of the zero-time discontinuity map, and are true in the immediate neighbourhood of a grazing orbit. In this paper we investigate how the character of the two-dimensional map changes over a large parameter range as the system is driven from a non-impacting orbit to an impacting orbit. This study leads to vital conclusions regarding the character of the normal form of the map not only in the immediate vicinity of the grazing orbit, but also away from it, as dependent on the system parameters. We obtain these characteristics by experiment and by simulation.

Character of the map for switched dynamical systems for observations on the switching manifold

It has been proposed to obtain the discrete-time models of switching dynamical systems by observing the states at the switching instants. Apart from the lowering of dimension, such switching maps or impact maps offer advantage in modeling systems that exhibit chattering. In this Letter we derive the nature of the switching map for the special case of grazing orbits. We show that the map is discontinuous in the neighborhood of a grazing orbit, and that it has a square root slope singularity on one side of the discontinuity. We illustrate the above by obtaining the switching maps for two example systems: the Colpitt's oscillator in the electrical domain and the soft impact oscillator in the mechanical domain.

A land-use system and the challenge of sustainable ago- pastoral production in southwestern Nigeria

SUMMARY The study is focused on land use in agro-pastoral production in Ogun State. Using a combination of data from ethno-social surveys, remote sensing and geographical information systems (GIS), the study has characterized the overall agro-pastoral production system, with emphasis on population, land use and herd productivity. It was found that yields of arable crops on agro pastoralists' fields were generally low due, perhaps, to the low genetic potentials of seed varieties used, poor soil conditions and poor land management practices. Livestock productivity was similar on all parameters to levels in similar contexts elsewhere across the West African subregion but generally below possible potentials and research station findings. Much of the land cover had been considerably altered around the areas of pastoral household settlements compared with the situation 20 years ago. Agro-pastoral households currently use between 221 km2 and 523 km2 of land in Ogun State. Pastoral grazing orbits extend beyond the boundaries of the State in the northeastern areas and beyond the boundaries of the country into the Republic of Benin in the northwest. These areas represent potential conflict areas between crop farmers and agro-pastoralists. There was a direct and positive relationship between population density in pastoral household settlements and the area of land disturbed. It was recommended that there was a need for deliberate land-use planning intervention by government to ensure long-term peaceful co-existence of farmers and agro-pastoral producers. It was concluded that further studies should focus on establishing a clear link between the land-use practices of agro-pastoralists and their specific effects on soil conditions, ecological succession and the overall farming systems.

Grazing Orbits and Related Local Bifurcations of an Oscillator with Continuous and Piecewise-Linear Restoring Force

One critical case for the motion of a periodically excited oscillator with continuous and piecewise-linear restoring force is that the motion happens to graze a switching plane between two linear regions of the restoring force. This article presents a numerical scheme for locating the periodic grazing orbit first. Then, through a brief analysis, the article shows that the grazing phenomenon turns the stability trend of the periodic orbit so abruptly that it may be impossible to predict an incident local bifurcation with the variation of a control parameter from the concept of smooth dynamic systems. The numerical simulation in the article well supports the scheme and the analysis, and shows an abundance of grazing phenomena in an engineering range of the excitation frequency.