Nonlinear stability analysis of the Emden-Fowler equation

In this paper we qualitatively study radial solutions of the semilinear elliptic equation $\Delta u + u^n = 0$ with $u(0)=1$ and $u'(0)=0$ on the positive real line, called the Emden-Fowler or Lane-Emden equation. This equation is of great importance in Newtonian astrophysics and the constant $n$ is called the polytropic index. By introducing a set of new variables, the Emden-Fowler equation can be written as an autonomous system of two ordinary differential equations which can be analyzed using linear and nonlinear stability analysis. We perform the study of stability by using linear stability analysis, the Jacobi stability analysis (Kosambi-Cartan-Chern theory) and the Lyapunov function method. Depending on the values of $n$ these different methods yield different results. We identify a parameter range for $n$ where all three methods imply stability.Comment: 12 pages; new reference added; 3 new references added; fully revised versio

(In)finiteness of Spherically Symmetric Static Perfect Fluids

This work is concerned with the finiteness problem for static, spherically symmetric perfect fluids in both Newtonian Gravity and General Relativity. We derive criteria on the barotropic equation of state guaranteeing that the corresponding perfect fluid solutions possess finite/infinite extent. In the Newtonian case, for the large class of monotonic equations of state, and in General Relativity we improve earlier results

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

Resolved Images of Large Cavities in Protoplanetary Transition Disks

Circumstellar disks are thought to experience a rapid "transition" phase in their evolution that can have a considerable impact on the formation and early development of planetary systems. We present new and archival high angular resolution (0.3" = 40-75 AU) Submillimeter Array (SMA) observations of the 880 micron dust continuum emission from 12 such transition disks in nearby star-forming regions. In each case, we directly resolve a dust-depleted disk cavity around the central star. Using radiative transfer calculations, we interpret these dust disk structures in a homogeneous, parametric model framework by reproducing their SMA visibilities and SEDs. The cavities in these disks are large (R_cav = 15-73 AU) and substantially depleted of small (~um-sized) dust grains, although their mass contents are still uncertain. The structures of the remnant material at larger radii are comparable to normal disks. We demonstrate that these large cavities are common among the millimeter-bright disk population, comprising at least 20% of the disks in the bright half of the millimeter luminosity (disk mass) distribution. Utilizing these results, we assess some of the physical mechanisms proposed to account for transition disk structures. As has been shown before, photoevaporation models do not produce the large cavity sizes, accretion rates, and disk masses representative of this sample. It would be difficult to achieve a sufficient decrease of the dust optical depths in these cavities by particle growth alone: substantial growth (to meter sizes or beyond) must occur in large (tens of AU) regions of low turbulence without also producing an abundance of small particles. Given those challenges, we suggest instead that the observations are most commensurate with dynamical clearing due to tidal interactions with low-mass companions --young brown dwarfs or giant planets on long-period orbits.Comment: ApJ, in pres

Săpăturile de la Otomani (r. Marghita) / Les fouilles d’Otomani

Horedt K., Rusu Mircea, Ordentlich Ivan. Săpăturile de la Otomani (r. Marghita) / Les fouilles d’Otomani. In: Materiale şi cercetări arheologice, N°8 1962. pp. 317-324

Modeling and simulations of plasma and sheath edges in warm-ion collision-free discharges

It has been shown recently by Kos et al. [Phys. Plasmas 25, 043509 (2018)] that the common plasma-sheath boundary is characterized by three well defined characteristic points, namely the plasma edge (PE), the sheath edge (SE) and the sonic point. Moreover, it has been shown that the sheath profiles, when properly normalized at the SE, as well as the potential drop in the plasma–sheath transition region (PST), (region between between PE and SE) in collision-free (CF) discharges are rather independent of discharge parameters, such as the plasma source profile, ion temperature and plasma density, providing that the sheath thickness is kept well bellow the plasma length. While these findings were obtained by theoretical means under idealized discharge conditions, the question arises whether and to which extent they are relevant under more complex physical scenarios. As a first step toward answering this question the CF discharge with warm ions is examined in this work via kinetic simulation method in which some of the model assumptions, such as independence of time and the Boltzmann distribution of electrons can hardly be ensured. Special attention is payed to effects of ion creation inside the sheath. It is found that only with considerably increased sheath thickness the sonic point always shifts from SE towards the wall. Whether the absolute value of ion directional velocity at the sonic point will increase or decrease depends on the ion temperature and the source strength inside the sheath. In addition preliminary comparison of results obtained under CF assumption with the representative ones obtained with strongly enhanced Coulomb collisions (CC), indicate the relevancy of hypothesis that the VDF of B&J can be considered as a universal one in future reliable kinetic modeling and solving the plasma boundary and sheath problem in both collisional and collision-free plasmas