Nonlinear stability of the libration points in the photogravitational restricted three body problem

The stability of the triangular libration points in the case when the first and the second order resonances appear was investigated. It was proved that the first order resonances do not cause instability. The second order resonances may lead to instability. Domains of the instability in the two-dimensional parameter space were determined.     

About stability of libration points in the restricted photogravitational three body problem

Nonlinear stability of the triangular libration point in the photogravitational restricted three body problem was investigated in the whole range of the parameters. Some results obtained earlier are corrected. The method for proper determination of cases when stability cannot be determined by four order terms of the hamiltonian was proposed.     

Periodic rotations of a rigid body located at the libration point

Periodic rotations of a rigid body close to the flat motions were found. Their orbital stability was investigated. Analysis was done up to second order of the small parameter. It was proved that solutions found are orbitally stable except of the third order resonance case. This resonance do not appear if terms up to the first order of small parameter are considered only.     

Periodic Orbits and Accretion Disks

The effect of the radiation pressure on the periodic motion of a small particle around one ‘primary’ body of the restricted photogravitational three-body problem is examined. Simple periodic orbits are used to determine the maximum size of the accretion disk in close binary systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Identifying mechanisms of shore platform erosion using Structure-from-Motion (SfM) photogrammetry

Shore platforms control wave energy transformation which, in turn, controls energy delivery to the cliff toe and nearshore sediment transport. Insight into shore platform erosion rates has conventionally been constrained at millimetre-scales using micro-erosion metres, and at metre-scales using cartographic data. On apparently slowly eroding coasts, such approaches are fundamentally reliant upon long-term observation to capture emergent erosion patterns. Where in practise timescales are short, and where change is either below the resolution or saturates the mode of measurement, the collection of data that enables the identification of the actual mechanisms of erosion is hindered. We developed a method to monitor shore platform erosion at millimetre resolution within metre-scale monitoring plots using Structure-from-Motion photogrammetry. We conducted monthly surveys at 15 0.25 m² sites distributed across the Hartle Loup platform in North Yorkshire, UK, over one year. We derived topographic data at 0.001 m resolution, retaining a vertical precision of change detection of 0.001 m. We captured a mean erosion rate of 0.528 mm yr^-1, but this varied considerably both across the platform and through the year. We characterized the volume and shape of eroded material. The detachment volume–frequency and shape distributions suggest that erosion happens primarily via removal of shale platelets. We identify that the at-a-point erosion rate can be predicted by the distance from the cliff and the tidal level, whereby erosion rates are higher closer to the cliff and at locations of higher tidal duration. The size of individual detachments is controlled by local micro-topography and rock structure, whereby larger detachments are observed on more rough sections of the platform. Faster erosion rates and larger detachments occur in summer months, rather than in more energetic winter conditions. These results have the potential to form the basis of improved models of how platforms erode over both short- and long-timescales. © 2019 John Wiley & Sons, Ltd.     

Geomorphometric analysis of a rocky coastline: an example from Hornsund,Svalbard

We describe a method to extract a number of quantitative morphometric characteristics of a rocky coastline. The algorithm, which is implemented in the free open-source software GNU Octave, takes as its input an ordered set of coordinate pairs defining the coastline. It finds vertices of specific angularity at two different levels, first approximating convex features (headlands) and then looking for local maxima (embayment edges) in their vicinity. It extracts numerical parameters for each embayment delimited by the former points, including basic geometric parameters (e.g. area, depth and width) and coefficients expressing shapes of embayments (e.g. form factor, coefficient of asymmetry and compactness). These we adapted from morphometric studies of other geomorphic features. Complex headlands and random sectors of the coastline which are delimited by embayment edges are automatically removed. The method was developed and tested on Hornsund in Svalbard, where 20 headlands, 28 embayment edges and 20 embayments were identified, but could be adapted to a wide range of coastal situations. We consider a range of possible applications of the methods, including to the classification of coastlines, change detection and identifying controls on their morphologies, and we discuss the choice of optimum parameters for the algorithm.