Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2S^2 and the hyperbolic plane H2H^2

The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the mathematical expressions are presented using the curvature \k as a parameter, in such a way that they reduce to the appropriate property for the system on the sphere $S^2$, or on the hyperbolic plane $H^2$, when particularized for \k>0, or \k<0, respectively; in addition, the Euclidean case arises as the particular case \k=0. In the second part we study the main properties of the Kepler problem on spaces with curvature, we solve the equations and we obtain the explicit expressions of the orbits by using two different methods: first by direct integration and second by obtaining the \k-dependent version of the Binet's equation. The final part of the article, that has a more geometric character, is devoted to the study of the theory of conics on spaces of constant curvature.Comment: 37 pages, 7 figure

Gipsy 3D: Analysis, Visualization and Vo-Tools

The scientific goals of the AMIGA project are based on the analysis of a significant amount of spectroscopic 3D data. In order to perform this work we present an initiative to develop a new VO compliant package, including present core applications and tasks offered by the Groningen Image Processing System (GIPSY), and new ones based on use cases elaborated in collaboration with ad- vanced users. One of the main goals is to provide local interoperability between GIPSY (visualization and data analysis) and other VO software. The connectivity with the Virtual Observatory environment will provide general access to 3D data VO archives and services, maximizing the potential for scientific discovery.Comment: 2 pages, 1 figure, to appear in the proceedings of the "Multi-wavelength Astronomy and Virtual Observatory" Workshop held at ESAC 1-3 Dec 200

Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and some non-compact symmetric spaces associated to SL(N+1,R) are the generic members in this family. The method encapsulates trigonometry for this whole family of spaces into a single "basic trigonometric group equation", and has 'universality' and '(self)-duality' as its distinctive traits. All previously known results on the trigonometry of CP^N and CH^N follow as particular cases of our general equations. The physical Quantum Space of States of any quantum system belongs, as the complex Hermitian space member, to this parametrised family; hence its trigonometry appears as a rather particular case of the equations we obtain.Comment: 46 pages, LaTe

A three-dimensional view of the remnant of Nova Persei 1901 (GK Per)

We present a kinematical study of the optical ejecta of GK Per. It is based on proper motions measurements of 282 knots from ~20 images spanning 25 years. Doppler-shifts are also computed for 217 knots. The combination of proper motions and radial velocities allows a unique 3-D view of the ejecta to be obtained. The main results are: (1) the outflow is a thick shell in which knots expand with a significant range of velocities, mostly between 600 and 1000 km/s; (2) kinematical ages indicate that knots have suffered only a modest deceleration since their ejection a century ago; (3) no evidence for anisotropy in the expansion rate is found; (4) velocity vectors are generally aligned along the radial direction but a symmetric pattern of non-radial velocities is also observed at specific directions; (5) the total Halpha+[NII] flux has been linearly decreasing at a rate of 2.6 % per year in the last decade. The Eastern nebular side is fading at a slower rate than the Western one. Some of the knots displayed a rapid change of brightness during the 2004-2011 period. Over a longer timescale, a progressive circularization and homogenization of the nebula is taking place; (6) a kinematic distance of 400+-30 pc is determined. These results raise some problems to the previous interpretations of the evolution of GK Per. In particular, the idea of a strong interaction of the outflow with the surrounding medium in the Southwest quadrant is not supported by our data.Comment: Accepted for publication in The Astrophysical Journal (19 pages, 17 figures). Higher resolution version of this article (2.5 MB) is available at http://www.aai.ee/~sinope/ApJ89291_liimets.pd

Lagrangian Formalism for nonlinear second-order Riccati Systems: one-dimensional Integrability and two-dimensional Superintegrability

The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are nonnatural and the forces are not derivable from a potential. The constant value $E$ of a preserved energy function can be used as an appropriate parameter for characterizing the behaviour of the solutions of these two systems. In the second part the existence of two--dimensional versions endowed with superintegrability is proved. The explicit expressions of the additional integrals are obtained in both cases. Finally it is proved that the orbits of the second system, that represents a nonlinear oscillator, can be considered as nonlinear Lissajous figuresComment: 25 pages, 7 figure

Hydrogen atom as an eigenvalue problem in 3D spaces of constant curvature and minimal length

An old result of A.F. Stevenson [Phys. Rev.} 59, 842 (1941)] concerning the Kepler-Coulomb quantum problem on the three-dimensional (3D) hypersphere is considered from the perspective of the radial Schr\"odinger equations on 3D spaces of any (either positive, zero or negative) constant curvature. Further to Stevenson, we show in detail how to get the hypergeometric wavefunction for the hydrogen atom case. Finally, we make a comparison between the ``space curvature" effects and minimal length effects for the hydrogen spectrumComment: 6 pages, v

Morpho-kinematical modelling in the molecular zoo beyond CO: the case of M 1-92

Ongoing improvements of sub-mm- and mm-range interferometers and single-dish radiotelescopes are progressively allowing the detailed study of planetary nebulae (PNe) in molecular species other than 12CO and 13CO. We are implementing a new set of tables for extending the capabilities of the morpho-kinematical modelling tool SHAPE+shapemol, so radiative transfer in molecular species beyond 12CO and 13CO, namely C17O, C18O, HCN, HNC, CS, SiO, HCO+, and N2H+, are enabled under the Large Velocity Gradient approximation with the ease of use of SHAPE. We present preliminary results on the simultaneous analysis of a plethora of IRAM-30m and HERSCHEL/HIFI spectra, and NOEMA maps of different species in the pre-PN nebula M~1-92, which show interesting features such as a previously undetected pair of polar, turbulent, high-temperature blobs, or a 17O/18O isotopic ratio of 1.7, which indicates the AGB should have turned C-rich, as opposed to the apparent nature of its O-rich nebula.Comment: To be published in the Proceedings of the IAU Symposium 384; 6 pages, 5 figures, 1 tabl

Informe acerca del museo nacional

Informe dirigido al rector de la Universidad Nacional, quien comisiona a Rafael Santander, para realizar una visita de inspección al Museo Nacional. Bogotá, octubre de 1868