Abstract:
We present a purely algebraic formulation (i.e. polynomial equations only) of the minimum-cost multi-impulse orbit transfer problem without time constraints, while keeping all the variables with a precise physical meaning. We apply general algebraic techniques to solve these equations (resultants, Gr\"obner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, we provide a proof of the optimality of the Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit transfer problem, and we provide a general formula for the optimal 2-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which we conjecture that can be removed).

Abstract:
We study a more complex case of Hohmann orbital transfer of a satellite by considering non-coplanar and elliptical orbits, instead of planar and circular orbits. We use as parameter the angle between the initial and transference planes that minimizes the energy, and therefore the fuel of a satellite, through the application of two non-tangential impulses for all possible cases. We found an analytical expression that minimizes the energy for each configuration. Some reasonable physical constraints are used: we apply impulses at perigee or apogee of the orbit, we consider the duration of the impulse to be short compared to the duration of the trip, we take the nodal line of three orbits to be coincident and the three semimajor axes to lie in the same plane. We study the only four possible cases but assuming non-coplanar elliptic orbits. In addition, we validate our method through a numerical solution obtained by using some of the actual orbital elements of Sputnik I and Vanguard I satellites. For these orbits, we found that the most fuel-efficient transfer is obtained by applying the initial impulse at apocenter and keeping the transfer orbit aligned with the initial orbit.

Abstract:
Basic resources for communication satellites are communication radio-frequencies and satellite orbits. An orbit is the trajectory followed by the satellite. The communication between the satellite and a ground station is established only when the satellite is consolidated in its own orbit and it is visible from the ground station. Different types of orbits are possible, each suitable for a specific application or mission. Most used orbits are circular, categorized as low, medium and geosynchronous (geostationary) orbits based on the attitude above the Earths surface. The launching process heading the satellite in geostationary orbit, by the first step places the satellite in a transfer orbit. The transfer orbit is elliptical in shape with low attitude at perigee, and the apogee of the geostationary orbit attitude. The apogee of the parking orbit depends on the injection velocity applied at perigee. Simulation approach of injection velocity at perigee to attain different apogees, considering an incremental step is presented in this paper.

Abstract:
The regular or chaotic character of orbits of stars moving in the meridional plane (R,z) of an axially symmetric elliptical galaxy with a dense, massive spherical nucleus and a dark matter halo component is under investigation. In particular, we explore how the flattening of an elliptical galaxy influences the overall orbital structure of the system, by computing in each case the percentage of chaotic orbits, as well as the percentages of orbits composing the main regular families. In an attempt to discriminate safely and with certainty between regular and chaotic motion, we use the Smaller ALingment Index (SALI) method to extensive samples of orbits obtained by integrating numerically the basic equations of motion as well as the variational equations. In addition, a technique which is based mainly on the field of spectral dynamics that utilizes the Fourier transform of the time series of each coordinate is used for classifying the regular orbits into different families and also to recognize the secondary resonances that usually bifurcate from them. Three cases are considered in our work: (i) the case where the elliptical galaxy is prolate (ii) the case where a spherically symmetric elliptical galaxy is present and (iii) the case where the elliptical galaxy has an oblate shape. Comparison between the current results and early related work is also made.

Abstract:
As is well known, when an SU(2) operation acts on a two-level system, its Bloch vector rotates without change of magnitude. Considering a system composed of two two-level systems, it is proven that for a class of nonlocal interactions of the two subsystems including \sigma_i\otimes\sigma_j (with i,j \in {x,y,z}) and the Heisenberg interaction, the geometric description of the motion is particularly simple: each of the two Bloch vectors follows an elliptical orbit within the Bloch sphere. The utility of this result is demonstrated in two applications, the first of which bears on quantum control via quantum interfaces. By employing nonunitary control operations, we extend the idea of controllability to a set of points which are not necessarily connected by unitary transformations. The second application shows how the orbit of the coherence vector can be used to assess the entangling power of Heisenberg exchange interaction.

Abstract:
It is shown that the MOdified Newtonian Dynamics (MOND) explains the tilt of the fundamental plane of elliptical galaxies without the need of non-baryonic dark matter. Results found for elliptical galaxies extends to globular clusters and galaxy clusters, showing that MOND agrees with observations over 7 order of magnitude in acceleration.

Abstract:
The Sersic r^(1/n) index n of an elliptical galaxy (or bulge) has recently been shown to correlate strongly (r=0.8) with a galaxy's central velocity dispersion. This index could therefore prove extremely useful and cost-effective (in terms of both telescope time and data reduction) for many fields of extragalactic research. It is a purely photometric quantity which apparently not only traces the mass of a bulge but has additionally been shown to reflect the degree of bulge concentration. This paper explores the affect of replacing the central velocity dispersion term in the Fundamental Plane with the Sersic index n. Using a sample of early-type galaxies from the Virgo and Fornax clusters, various (B-band) `Photometric Planes' were constructed and found to have a scatter of 0.14-0.17 dex in log(r_e), or a distance error of 38-48% per galaxy (the higher values arising from the inclusion of the S0 galaxies). The corresponding Fundamental Plane yielded a 33-37% error in distance for the same galaxy (sub-)samples (i.e. ~15-30% less scatter). The gains in using a hyperplane (i.e. adding the Sersic index to the Fundamental Plane as a fourth parameter) were small, giving a 27-33% error in distance, depending on the galaxy sample used. The Photometric Plane has been used here to estimate the Virgo-Fornax distance modulus; giving a value of \Delta\mu=0.62(+/-0.30) mag (cf. 0.51(+/-0.21), HST Key Project on the Extragalactic distance Scale). The prospects for using the Photometric Plane at higher redshift appears promising. Using published data on the intermediate redshift cluster Cl 1358+62 (z=0.33) gave a Photometric Plane distance error of 35-41% per galaxy.

Abstract:
Evolution and disruption of galaxies orbiting in the gravitational field of a larger cluster galaxy are driven by three coupled mechanisms: 1) the heating due to its time dependent motion in the primary; 2) mass loss due to the tidal strain field; and 3) orbital decay. Previous work demonstrated that tidal heating is effective well inside the impulse approximation limit. Not only does the overall energy increase over previous predictions, but the work is done deep inside the secondary galaxy, e.g. at or inside the half mass radius in most cases. Here, these ideas applied to cannibalization of elliptical galaxies with fundamental-plane parameters. In summary, satellites which can fall to the center of a cluster giant by dynamical friction are evaporated by internal heating by the time they reach the center. This suggests that true merger-produced multiple nuclei giants should be rare. Specifically, secondaries with mass ratios as small as 1\% on any initial orbit evaporate and those on eccentric orbits with mass ratios as small as 0.1\% evolve significantly and nearly evaporate in a galaxian age. Captured satellites with mass ratios smaller than roughly 1\% have insufficient time to decay to the center. After many accretion events, the model predicts that the merged system has a profile similar to that of the original primary with a weak increase in concentration.

Abstract:
The group PGL(3) of linear transformations of the projective plane acts naturally on the projective space parametrizing curves of a given degree. In this note we begin the study of the orbits of smooth curves under this action: we construct a resolution of the closure of the orbit of a given curve, and we use it to compute its degree. This turns out to depend only on the degree of the curve, the order of its automorphism group, and on the number and type of its flexes. This paper will appear on the Journal of Algebraic Geometry.

Abstract:
We present a study focusing on the nature of compact groups through the study of their elliptical galaxies. We determine velocity dispersions ($\sigma$) for 18 18 bright elliptical galaxies located in the core of Hickson compact groups and a control sample of 12 bright bona fide ellipticals located in the field or very loose groups. Several tests are carried out to avoid sources of systematic effects in $\sigma$ measurements. We use these velocity dispersions to compare the position of 11 compact group galaxies in the Fundamental Plane to that of a large and homogeneous sample of elliptical galaxies (Burstein et al. 1987). We find that little or no significant difference exists, as far as the Fundamental Plane is concerned, between ellipticals in compact groups and their counterparts in other environments.