Infrared Polarimetry and the Torus

Polarimetry at near-infrared wavelengths is shown to be an important tool in understanding the properties of the obscuring torus in NGC1068. We present evidence for the existence of aligned grains in the torus, which produce polarized flux through dichroism, and show how the torus itself can be observed in silhouette in polarized flux images. This revised version was published online in July 2006 with corrections to the Cover Date.     

Intrinsic stability of periodic orbits

Families of orbits of a conservative, two degree-of-freedom system are represented by an unsteady velocity field with componentsu(x, y, t) andv(x, y, t). Intrinsic stability properties depend on velocity field divergence and curl, whose dynamical evolution is determined by a matrix Riccati equation. Near equilibrium, divergence-free or irrotational fields are dynamically compatible with the conservative force field. It is shown that a necessary condition for stable periodic orbits is satisfied when the orbitaveraged divergence is zero, which results in bounded normal variations. A sufficient condition for stability is derived from the requirement that tangential variations do not exhibit secular growth.In a steady, divergence-free field, velocity component functionsu(x, y) andv(x, y) may be continuedanalytically from any initial condition, except when velocity is parallel to U or at equilibria. In an unsteady field, the orbit-averaged divergence is zero when the vorticity function is periodic. When such a field exists, initial conditions for stable periodic orbits (i.e., characteristic loci) may be determinedanalytically.     

Isovortical orbits of autonomous,conservative, two degree-of-freedom dynamical systems

For an autonomous, conservative, two degree-of-freedom dynamical system, vorticity (the curl of velocity) is constant along the orbit if the velocity field is divergence-free such that: $$u\left( {x, v} \right) - \psi _y , v\left( {x, y} \right) = - \psi _x .$$ Isovortical orbits in configuration space are level curves of a scalar autonomous function Ψ (x, v) satisfying a second-order, non-linear partial differential equation of the Monge-Ampere type: $$2\left( {\psi _{xx} \psi _{yy} - \psi _{xy}^2 } \right) + U_{xx} + U_{yy} = 0,$$ where U(x. y) is the autonomous potential function. The solution Soc the time variable is reduced to a quadrature following determinatio of Ψ. Self-similar solutions of the Monge-Ampere equation under Birkhoff's one-parameter transformation group are derived for homogeneous (power-law) potential functions. It is shown that Keplerian orbits belong to the class of planar isovortical flows.     

Orbits near critical inclination,including lunisolar perturbations

An improved theory is presented of long period perigee motion for orbits near the critical inclinations 63.4° and 116.6°. Inclusion of lunisolar perturbations andall measured zonal harmonic coefficients from a recent Earth model are significant improvements over existing theories. Phase portraits are used to depict the interaction between eccentricity magnitude and argument of perigee. The Hamiltonian constant can be chosen as the parameter to display a family of phase plane trajectories consisting of libration, circulation, and asymptotic motion along separatrices near equilibrium points. A two parameter family of phase portraits is defined by the other two integrals, the average semimajor axis and component of angular momentum resolved along the Earth's polar axis. There are regions of the parameter space where the stability and total number of equilibria can change, or two separatrices can coalesce. These phenomena signal large qualitative changes in phase portrait topology. Numerical studies show that lunisolar perturbations control stability of equilibria for orbits with semimajor axes exceeding 1.4 Earth radii. Moreover, a theory which includes lunisolar perturbations predicts larger maximum fluctuations in eccentricity and faster oscillations near stable equilibria compared to a theory which models only the zonal harmonics.     

A new integration theory for conservative two degree-of-freedom systems

A two degree-of-freedom, conservative system is reduced to a single degree-of-freedom, kinematic system with Hamiltonian integral under the change of independent variable: $$dt = \zeta dt (\zeta = \upsilon _x - \upsilon _y )$$ where ζ is the curl (or vorticity) of the velocity field with cartesian inertial componentsu(x, y, t) andv(x, y, t). In the autonomous case whenu _t=v _t=0, orbits are globally represented by the level curves of an autonomous Hamiltonian functionH(x,y) satisfying a second-order quasilinear partial differential equation (Szebehely's Equation): $$2(H + U)\left( {H_{xx} H_y^2 - 2H_{xy} H_x H_y + H_{yy} H_x^2 } \right) + (H_x U_x + H_y U_y )\left( {H_x^2 + H_y^2 } \right) = 0$$ whereU(x, y) is the autonomous potential function. An inversion of dependent and independent variables reduces this equation to a second-order, ordinary differential equation for a function specifying the orbital curve. The true time variable is recovered by evaluating a quadrature. Fundamental differences exist between this approach and Hamilton-Jacobi theory.