Stellar Over-densities in the Outer Halo of the Milky Way

This study presents a tomographic survey of a subset of the outer halo (10-40 kpc) drawn from the Sloan Digital Sky Survey Data Release 6. Halo substructure on spatial scales of $>3$ degrees is revealed as an excess in the local density of sub-giant stars. With an appropriate assumption of a model stellar isochrone it is possible for us to then derive distances to the sub-giant population. We describe three new candidate halo substructures; the 160- and 180-degree over-densities (at distances of 17 and 19 kpc respectively and radii of 1.3 and 1.5 kpc respectively) and an extended feature at 28 kpc that covers at least 162 square degrees, the Virgo Equatorial Stream. In addition, we recover the Sagittarius dwarf galaxy (Sgr) leading arm material and the Virgo Over-density. The derived distances, together with the number of sub-giant stars associated with each substructure, enables us to derive the integrated luminosity for the features. The tenuous, low surface brightness of the features strongly suggests an origin from the tidal disruption of an accreted galaxy or galaxies. Given the dominance of the tidal debris of Sgr in this region of the sky we investigate if our observations can be accommodated by tidal disruption models for Sgr. The clear discordance between observations and model predictions for known Sgr features means it is difficult to tell unambiguously if the new substructures are related to Sgr or not. Radial velocities in the stellar over-densities will be critical in establishing their origins.Comment: 14 pages, 7 figures, PASA accepte

Integrable Abel equations and Vein's Abel equation

We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein's Abel equation whose solutions are expressed in terms of the third order hyperbolic functions and a phase space analysis of the corresponding nonlinear oscillator is also providedComment: 12 pages, 4 figures, 17 references, online at Math. Meth. Appl. Sci. since 7/28/2015, published 4/201

Ermakov-Lewis Invariants and Reid Systems

Reid's m'th-order generalized Ermakov systems of nonlinear coupling constant alpha are equivalent to an integrable Emden-Fowler equation. The standard Ermakov-Lewis invariant is discussed from this perspective, and a closed formula for the invariant is obtained for the higher-order Reid systems (m\geq 3). We also discuss the parametric solutions of these systems of equations through the integration of the Emden-Fowler equation and present an example of a dynamical system for which the invariant is equivalent to the total energyComment: 8 pages, published versio

Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in its first kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers-Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second-order nonlinear equationsComment: 6 pages, 7 figures, published versio

Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping

We introduce a special type of dissipative Ermakov-Pinney equations of the form v_{\zeta \zeta}+g(v)v_{\zeta}+h(v)=0, where h(v)=h_0(v)+cv^{-3} and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel equation. When h_0(v) is a linear function, h_0(v)=\lambda^2v, general solutions are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type. In addition, the same kinds of general solutions are constructed for the cases of higher-order Reid nonlinearities. The Chiellini dissipative function is actually a dissipation-gain function because it can be negative on some intervals. We also examine the nonlinear case h_0(v)=\Omega_0^2(v-v^2) and show that it leads to an integrable hyperelliptic caseComment: 15 pages, 5 figures, 1 appendix, 21 references, published versio