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We compute the semiclassical formulas for the partition functions obtained using two different Lagrangians: the Chern-Simons functional and the symplectic action functional.
Using a new 2D radiative transfer algorithm, we have calculated the thermally reradiated infrared spectra of the compact dust tori which are thought to surround many AGN. These tori radiate anisotropically. Face-on tori may be from one-half to a few orders of magnitude brighter than edge-on tori throughout the infrared. Their spectra at nearly all viewing angles are basically 'bumps' which are about 50 percent wider than blackbodies and peak in the mid-infrared at a wavelength determined mainly by the flux of nuclear radiation on the inner edge of the torus. The infrared color temperatures are hotter for face-on tori than edge-on tori by about 100 K. The 10 micron silicate feature often associated with dust can appear in absorption, emission, or not at all. There is a rough tendency for absorption features to be more prominent for edge-on tori than for face-on tori.
We present new mid-infrared imaging data for three Type-1 Seyfert galaxies obtained with T-ReCS on the Gemini-South Telescope at subarcsecond resolution. Our aim is to enlarge the sample studied in a previous work to compare the properties of Type-1 and Type-2 Seyfert tori using clumpy torus models and a Bayesian approach to fit the infrared (IR) nuclear spectral energy distributions. Thus, the sample considered here comprises 7 Type-1, 11 Type-2, and 3 intermediate-type Seyferts. The unresolved IR emission of the Seyfert 1 galaxies can be reproduced by a combination of dust heated by the central engine and direct active galactic nucleus (AGN) emission, while for the Seyfert 2 nuclei only dust emission is considered. These dusty tori have physical sizes smaller than 6 pc radius, as derived from our fits. Unification schemes of AGN account for a variety of observational differences in terms of viewing geometry. However, we find evidence that strong unification may not hold and that the immediate dusty surroundings of Type-1 and Type-2 Seyfert nuclei are intrinsically different. The Type-2 tori studied here are broader, have more clumps, and these clumps have lower optical depths than those of Type-1 tori. The larger the covering factor of the torus, the smaller the probability of having a direct view of the AGN, and vice versa. In our sample, Seyfert 2 tori have larger covering factors (C{sub T} = 0.95 {+-} 0.02) and smaller escape probabilities (P{sub esc} = 0.05% {+-} {sup 0.08}{sub 0.03}%) than those of Seyfert 1 (C{sub T} = 0.5 {+-} 0.1; P{sub esc} = 18% {+-} 3%). All the previous differences are significant according to the Kullback-Leibler divergence. Thus, on the basis of the results presented here, the classification of a Seyfert galaxy as a Type-1 or Type-2 depends more on the intrinsic properties of the torus rather than on its mere inclination toward us, in contradiction with the simplest unification model.
A Newtonian model of non-conductive, charged, perfect fluid tori orbiting in combined spherical gravitational and dipolar magnetic fields is presented and stationary, axisymmetric toroidal structures are analyzed. Matter in such tori exhibits a purely circulatory motion and the resulting convection carries charges into permanent rotation around the symmetry axis. As a main result, we demonstrate the possible existence of off-equatorial charged tori and equatorial tori with cusps that also enable outflows of matter from the torus in the Newtonian regime. These phenomena qualitatively represent a new consequence of the interplay between gravity and electromagnetism. From an astrophysical point of view, our investigation can provide insight into processes that determine the vertical structure of dusty tori surrounding accretion disks.
We consider multiplicative functions and Prym differentials on variable tori. We prove counterparts of the theorem on the total sum of residues for Prym differentials of any integer order on tori. As a consequence, reciprocity laws are proved. We construct elementary Prym differentials of all kinds with any integer order which holomorphically depend on the moduli of tori and characters. We derive an analogue of the Appell decomposition formula for functions with characters. We also study vector bundles of Prym differentials of any integer order over the product of Teichmueller spaces for a torus and a group of characters.
This letter restudies the Nosé-Hoover oscillator. Some new averagely conservative regions are found, each of which is filled with different sequences of nested tori with various knot types. Especially, the dynamical behaviors near the border of “chaotic region” and conservative regions are studied showing that there exist more complicated and thinner invariant tori around the boundaries of conservative regions bounded by tori. Our results suggest an infinite number of island chains in a “chaotic sea” for the Nosé-Hoover oscillator.
According to recent models, the accretion disk and black hole in active galactic nuclei (AGNs) are surrounded by a clumpy torus. We investigate the NIR flux variation of the torus in response to a UV flash for various geometries. Anisotropic illumination by the disk and the torus self-occultation contrast our study with earlier works. Both the waning effect of each clump and the torus self-occultation selectively reduce the emission from the region with a short delay. Therefore, the NIR delay depends on the viewing angle (where a more inclined angle leads to a longer delay), and the time response shows an asymmetric profile with negative skewness, opposing the results for optically thin tori. The range of the computed delay coincides with the observed one, suggesting that the viewing angle is primarily responsible for the scatter of the observed delay. We also propose that the red NIR-to-optical color of type 1.8/1.9 objects is caused not only by the dust extinction but also the intrinsically red color. Compared with the modest torus thickness, both a thick and a thin tori display weaker NIR emission. A selection bias is thus expected such that NIR-selected AGNs tend to possess moderately thick tori. A thicker torus shows a narrower and more heavily skewed time profile, while a thin torus produces a rapid response. A super-Eddington accretion rate leads to much weaker NIR emission due to the disk self-occultation and the disk truncation by self-gravity. A long delay is expected from an optically thin and/or a largely misaligned torus. Very weak NIR emission, such as in hot-dust-poor active nuclei, can arise from a geometrically thin torus, a super-Eddington accretion rate, or a slightly misaligned torus.
We present an analytical approach for the equilibrium of a self-gravitating charged fluid embedded in a spherical gravitational and dipolar magnetic fields produced by a central mass. Our scheme is proposed, as a toy model, in the context of gaseous/dusty tori surrounding supermassive black holes in galactic nuclei. While the central black hole dominates the gravitational field and remains electrically neutral, the surrounding material has a non-negligible self-gravitational effect on the torus structure. By charging mechanisms it also acquires non-zero electric charge density, so the two influences need to be taken into account to achieve a self-consistent picture. Using our approach we discuss the impact of self-gravity, represented by the term d{sub t} (ratio of the torus total mass to the mass of the central body), on the conditions for existence of the equilibrium and the morphology and typology of the tori. By comparison with a previous work without self-gravity, we show that the conditions can be different. Although the main aim of the present paper is to discuss a framework for the classification of electrically charged, magnetized, self-gravitating tori, we also mention potential astrophysical applications to vertically stratified fluid configurations.
Most unstable periodic orbits of the planar circular restricted 3-body problem (PCRTBP) persist as invariant tori when a periodic forcing is added to the equations of motion. Among the persisting orbits are unstable resonant periodic or- bits. In this study, we first compute invariant tori in the planar concentric cir- cular restricted 4-body problem corresponding to exterior Jupiter-Europa and in- terior Jupiter-Ganymede PCRTBP resonant periodic orbits. Motivated by the 2:1 Laplace resonance between the orbits of Europa and Ganymede around Jupiter, we then use the computed tori to characterize relations between m:n Jupiter-Ganymede and m:2n Jupiter-Europa PCRTBP resonant orbits.
We prove that curved noncommutative tori are Leibniz quantum compact metric spaces and that they form a continuous family over the group of invertible matrices with entries in the image of the quantum tori for the conjugation by modular conjugation operator in the regular representation, when this group is endowed with a natural length function.
In this paper, we construct Liouville Quantum Field Theory (LQFT) on the toroidal topology in the spirit of the 1981 seminal work by Polyakov [Phys. Lett. B 103, 207 (1981)]. Our approach follows the construction carried out by the authors together with Kupiainen in the case of the Riemann sphere [“Liouville quantum gravity on the Riemann sphere,” e-print arXiv:1410.7318]. The difference is here that the moduli space for complex tori is non-trivial. Modular properties of LQFT are thus investigated. This allows us to integrate the LQFT on complex tori over the moduli space, to compute the law of the random Liouville modulus, therefore recovering (and extending) formulae obtained by physicists, and make conjectures about the relationship with random planar maps of genus one, eventually weighted by a conformal field theory and conformally embedded onto the torus.
This paper is devoted to the classical Knotting Problem: for a given manifold N and number m describe the set of isotopy classes of embeddings N{yields}S{sup m}. We study the specific case of knotted tori, that is, the embeddings S{sup p} Multiplication-Sign S{sup q}{yields}S{sup m}. The classification of knotted tori up to isotopy in the metastable dimension range m {>=} p + 3/2q + 2, p{<=}q, was given by Haefliger, Zeeman and A. Skopenkov. We consider the dimensions below the metastable range and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension. Bibliography: 35 titles.
The paper looks at quite a wide class of nonlinear evolution equations in a Banach space, including the typical boundary value problems for the main wave equations in mathematical physics (the telegraph equation, the equation of a vibrating beam, various equations from the elastic stability and so on). For this class of equations a unified approach to the bifurcation of invariant tori of arbitrary finite dimension is put forward. Namely, the problem of the birth of such tori from the zero equilibrium is investigated under the assumption that in the stability problem for this equilibrium the situation arises close to an infinite-dimensional degeneracy. Bibliography: 28 titles.
The possible occurrence of equilibrium off-equatorial tori in the gravitational and electromagnetic fields of astrophysical compact objects has been recently proved based on non-ideal magnetohydrodynamic theory. These stationary structures can represent plausible candidates for the modeling of coronal plasmas expected to arise in association with accretion disks. However, accretion disk coronae are formed by a highly diluted environment, and so the fluid description may be inappropriate. The question is posed of whether similar off-equatorial solutions can also be determined in the case of collisionless plasmas for which treatment based on kinetic theory, rather than a fluid one, is demanded. In this paper the issue is addressed in the framework of the Vlasov-Maxwell description for non-relativistic, multi-species axisymmetric plasmas subject to an external dominant spherical gravitational and dipolar magnetic field. Equilibrium configurations are investigated and explicit solutions for the species kinetic distribution function are constructed, which are expressed in terms of generalized Maxwellian functions characterized by isotropic temperature and non-uniform fluid fields. The conditions for the existence of off-equatorial tori are investigated. It is proved that these levitating systems are admitted under general conditions when both gravitational and magnetic fields contribute to shaping the spatial profiles of equilibrium plasma fluid fields. Then, specifically, kinetic effects carried by the equilibrium solution are explicitly provided and identified here with diamagnetic energy-correction and electrostatic contributions. It is shown that these kinetic terms characterize the plasma equation of state by introducing non-vanishing deviations from the assumption of thermal pressure.
Astrophysical fluids may acquire nonzero electrical charge because of strong irradiation or charge separation in a magnetic field. In this case, electromagnetic and gravitational forces may act together and produce new equilibrium configurations, which are different from the uncharged ones. Following our previous studies of charged test particles and uncharged perfect fluid tori encircling compact objects, we introduce here a simple test model of a charged perfect fluid torus in strong gravitational and electromagnetic fields. In contrast to ideal magnetohydrodynamic models, we consider here the opposite limit of negligible conductivity, where the charges are tied completely to the moving matter. This is an extreme limiting case which can provide a useful reference against which to compare subsequent more complicated astrophysically motivated calculations. To clearly demonstrate the features of our model, we construct three-dimensional axisymmetric charged toroidal configurations around Reissner-Nordstroem black holes and compare them with equivalent configurations of electrically neutral tori.
Substantial short time regularity, even in the chaotic regions of phase space, is found for what is seen as a large class of systems. This regularity manifests itself through the behavior of approximate constants of motion calculated by Pade summation of the Birkhoff-Gustavson normal form expansion; it is attributed to remnants of destroyed invariant tori in phase space. The remnant torus-like manifold structures are used to justify Einstein-Brillouin-Keller semiclassical quantization procedures for obtaining quantum energy levels, even in the absence of complete tori. They also provide a theoretical basis for the calculation of rate constants for intramolecular mode-mode energy transfer. These results are illustrated by means of a thorough analysis of the Henon-Heiles oscillator problem. Possible generality of the analysis is demonstrated by brief consideration of classical dynamics for the Barbanis Hamiltonian, Zeeman effect in hydrogen and recent results of Wolf and Hase (1980) for the H-C-C fragment.
The inner satellites of Saturn are icy bodies imbedded in a plasma environment in which they are continuously bombarded by energetic ions, corotating plasma, and solar radiation. Laboratory sputtering experiments indicate that this should result in the injection of substantial amounts of neutral H, H2, OH, H2O, and O2 into the magnetosphere. The atomic processes affecting these neutrals and the neutrals and ions formed from them are modeled, and the steady state neutral and ion densities expected in the plasma tori of Enceladus, Dione-Tethys, and Rhea are calculated. Comparison with observations shows that recombination can limit the Enceladus and Dione-Tethys tori to the observed densities, but that transport rates of at least 4 x 10 to the -8th Saturn radii squared/s are required to limit torus densities at Rhea to the observed values.
While satellite tori are morphologically similar, they divide into dynamically distinct groups whose global balances of ions and neutrals are presently modeled in order to determine how a torus behaves as a consequence of whether it generates itself and whether it disperses or recombines in place. Six possible types of behavior result; attention is given to the equilibrium and stability of each type. The behavior of a perturbation is characterized by a trajectory in the two-dimensional space formed by the ion density and neutral density perturbations. The equilibrium is stable if a perturbation follows its trajectory toward the origin, but is unstable if the motion is away from it. An attempt is made to classify the Jupiter and Saturn tori in this light.