On the steady states of the spherically symmetric Einstein-Vlasov system

Using both numerical and analytical tools we study various features of static, spherically symmetric solutions of the Einstein-Vlasov system. In particular, we investigate the possible shapes of their mass-energy density and find that they can be multi-peaked, we give numerical evidence and a partial proof for the conjecture that the Buchdahl inequality $\sup_{r > 0} 2 m(r)/r < 8/9$, $m(r)$ the quasi-local mass, holds for all such steady states--both isotropic {\em and} anisotropic--, and we give numerical evidence and a partial proof for the conjecture that for any given microscopic equation of state--both isotropic {\em and} anisotropic--the resulting one-parameter family of static solutions generates a spiral in the radius-mass diagram.Comment: 34 pages, 18 figures, LaTe

The Validity of the Super-Particle Approximation during Planetesimal Formation

The formation mechanism of planetesimals in protoplanetary discs is hotly debated. Currently, the favoured model involves the accumulation of meter-sized objects within a turbulent disc, followed by a phase of gravitational instability. At best one can simulate a few million particles numerically as opposed to the several trillion meter-sized particles expected in a real protoplanetary disc. Therefore, single particles are often used as super-particles to represent a distribution of many smaller particles. It is assumed that small scale phenomena do not play a role and particle collisions are not modeled. The super-particle approximation can only be valid in a collisionless or strongly collisional system, however, in many recent numerical simulations this is not the case. In this work we present new results from numerical simulations of planetesimal formation via gravitational instability. A scaled system is studied that does not require the use of super-particles. We find that the scaled particles can be used to model the initial phases of clumping if the properties of the scaled particles are chosen such that all important timescales in the system are equivalent to what is expected in a real protoplanetary disc. Constraints are given for the number of particles needed in order to achieve numerical convergence. We compare this new method to the standard super-particle approach. We find that the super-particle approach produces unreliable results that depend on artifacts such as the gravitational softening in both the requirement for gravitational collapse and the resulting clump statistics. Our results show that short range interactions (collisions) have to be modelled properly.Comment: 10 pages, 7 figures, accepted for publication in Astronomy and Astrophysic

Existence of axially symmetric static solutions of the Einstein-Vlasov system

We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system.Comment: 32 page

Dynamical Stability of Imaged Planetary Systems in Formation: Application to HL Tau

A recent ALMA image revealed several concentric gaps in the protoplanetary disk surrounding the young star HL Tau. We consider the hypothesis that these gaps are carved by planets, and present a general framework for understanding the dynamical stability of such systems over typical disk lifetimes, providing estimates for the maximum planetary masses. We collect these easily evaluated constraints into a workflow that can help guide the design and interpretation of new observational campaigns and numerical simulations of gap opening in such systems. We argue that the locations of resonances should be significantly shifted in massive disks like HL Tau, and that theoretical uncertainties in the exact offset, together with observational errors, imply a large uncertainty in the dynamical state and stability in such disks. This presents an important barrier to using systems like HL Tau as a proxy for the initial conditions following planet formation. An important observational avenue to breaking this degeneracy is to search for eccentric gaps, which could implicate resonantly interacting planets. Unfortunately, massive disks like HL Tau should induce swift pericenter precession that would smear out any such eccentric features of planetary origin. This motivates pushing toward more typical, less massive disks. For a nominal non-resonant model of the HL Tau system with five planets, we find a maximum mass for the outer three bodies of approximately 2 Neptune masses. In a resonant configuration, these planets can reach at least the mass of Saturn. The inner two planets' masses are unconstrained by dynamical stability arguments.Comment: Accepted in ApJ. 16 pages 8 figure

Spherically symmetric steady states of galactic dynamics in scalar gravity

The kinetic motion of the stars of a galaxy is considered within the framework of a relativistic scalar theory of gravitation. This model, even though unphysical, may represent a good laboratory where to study in a rigorous, mathematical way those problems, like the influence of the gravitational radiation on the dynamics, which are still beyond our present understanding of the physical model represented by the Einstein--Vlasov system. The present paper is devoted to derive the equations of the model and to prove the existence of spherically symmetric equilibria with finite radius.Comment: 13 pages, mistypos correcte

Enterprise Social Benefit and the Economic Transition in Hungary

In Hungary as in other East Bloc countries, enterprises have given a variety of nonwage benefits to their workers, sometimes called the "social wage." We explore what has happened to non-wage compensation during the economic transition which began in 1989. During this period, the real wage has fallen, and many aspects of enterprise operations have undergone change. This paper considers three broad questions. (1) How has total compensation and its composition changed during this period of restructuring? Have changes in non-wage compensation offset or reinforced changes in wages and which elements have been increasing, which decreasing? (2) What factors can account for the change? (3) Have enterprise non-wage benefits in fact served social functions in addition to their business functions, and, if so, has the social role of benefits changed during this period

On static shells and the Buchdahl inequality for the spherically symmetric Einstein-Vlasov system

In a previous work \cite{An1} matter models such that the energy density $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leq\Omega\rho, \Omega\geq 1,$ were considered in the context of Buchdahl's inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, $[R_0,R_1], R_0>0,$ satisfies $R_1/R_0<1/4.$ Moreover, given a sequence of solutions such that $R_1/R_0\to 1,$ then the limit supremum of $2M/R_1$ was shown to be bounded by $((2\Omega+1)^2-1)/(2\Omega+1)^2.$ In this paper we show that the hypothesis that $R_1/R_0\to 1,$ can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric Einstein-Vlasov system with this property. We also prove that for this sequence not only the limit supremum of $2M/R_1$ is bounded, but that the limit is $((2\Omega+1)^2-1)/(2\Omega+1)^2=8/9,$ since $\Omega=1$ for Vlasov matter. Thus, static shells of Vlasov matter can have $2M/R_1$ arbitrary close to $8/9,$ which is interesting in view of \cite{AR2}, where numerical evidence is presented that 8/9 is an upper bound of $2M/R_1$ of any static solution of the spherically symmetric Einstein-Vlasov system.Comment: 20 pages, Late

The Einstein-Vlasov sytem/Kinetic theory

The main purpose of this article is to guide the reader to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades where the main focus has been on nonrelativistic- and special relativistic physics, e.g. to model the dynamics of neutral gases, plasmas and Newtonian self-gravitating systems. In 1990 Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models (e.g. fluid models). The first part of this paper gives an introduction to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental in order to get a good comprehension of kinetic theory in general relativity.Comment: 31 pages. This article has been submitted to Living Rev. Relativity (http://www.livingreviews.org