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The current system of stellar magnitudes first introduced by Hipparchus was strictly defined by Norman Robert Pogson in 1856. He based his system on Ptolemys star catalogue `Almagest, recorded in about 137 A.D., and defined the magnitude-intensity relationship on a logarithmic scale. Stellar magnitudes observed with the naked eye recorded in seven old star catalogues were analyzed in order to examine the visual magnitude systems. Despite that psychophysists have proposed that humans sensitivities are on a power-law scale, it is shown that the degree of agreement is far better for a logarithmic magnitude than a power-law magnitude. It is also found that light ratios in each star catalogue nearly equal to 2.512, excluding the brightest (1st) and the dimmest (6th and dimmer) stars being unsuitable for the examination. It means that the visual magnitudes in old star catalogues fully agree with Pogsons logarithmic scale.
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Historical star magnitudes from catalogues by Ptolemy (137 AD), as-Sufi (964) and Tycho Brahe (1602/27) are converted to the Johnson V-mag scale and compared to modern day values from the HIPPARCOS catalogue. The deviations (or errors) are tested for dependencies on three different observational influences. The relation between historical and modern magnitudes is found to be linear in all three catalogues as it had previously been shown for the Almagest data by Hearnshaw (1999). A slight dependency on the colour index (B-V) is shown throughout the data sets and as-Sufis as well as Brahes data also give fainter values for stars of lower culmination height (indicating extinction). In all three catalogues, a stars estimated magnitude is influenced by the brightness of its immediate surroundings. After correction for the three effects, the remaining variance within the magnitude errors can be considered as approximate accuracy of the pre-telescopic magnitude estimates. The final converted and corrected magnitudes are available via the Vizier catalogue access tool (Ochsenbein, Bauer, & Marcout, 2000).
We investigate the old star clusters in the sample of cluster candidates from Froebrich, Scholz & Raftery 2007 -- the FSR list. Based on photometry from the 2-Micron All Sky Survey we generated decontaminated colour-magnitude and colour-colour diagrams to select a sample of 269 old stellar clusters. This sample contains 63 known globular clusters, 174 known open clusters and 32 so far unclassified objects. Isochrone fitting has been used to homogeneously calculate the age, distance and reddening to all clusters. The mean age of the open clusters in our sample is 1Gyr. The positions of these clusters in the Galactic Plane show that 80% of open clusters older than 1Gyr have a Galactocentric distance of more than 7kpc. The scale height for the old open clusters above the Plane is 375pc, more than three times as large as the 115pc which we obtain for the younger open clusters in our sample. We find that the mean optical extinction towards the open clusters in the disk of the Galaxy is 0.70mag/kpc. The FSR sample has a strong selection bias towards objects with an apparent core radius of 30 to 50 and there is an unexplained paucity of old open clusters in the Galactic Longitude range of 120deg < l < 180deg.
If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniquely specified by requiring the dissipators jump operators to be traceless. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the systems bare Hamiltonian.
Low rank matrix factorisation is often used in recommender systems as a way of extracting latent features. When dealing with large and sparse datasets, traditional recommendation algorithms face the problem of acquiring large, unrestrained, fluctuating values over predictions especially for users/items with very few corresponding observations. Although the problem has been somewhat solved by imposing bounding constraints over its objectives, and/or over all entries to be within a fixed range, in terms of gaining better recommendations, these approaches have two major shortcomings that we aim to mitigate in this work: one is they can only deal with one pair of fixed bounds for all entries, and the other one is they are very time-consuming when applied on large scale recommender systems. In this paper, we propose a novel algorithm named Magnitude Bounded Matrix Factorisation (MBMF), which allows different bounds for individual users/items and performs very fast on large scale datasets. The key idea of our algorithm is to construct a model by constraining the magnitudes of each individual user/item feature vector. We achieve this by converting from the Cartesian to Spherical coordinate system with radii set as the corresponding magnitudes, which allows the above constrained optimisation problem to become an unconstrained one. The Stochastic Gradient Descent (SGD) method is then applied to solve the unconstrained task efficiently. Experiments on synthetic and real datasets demonstrate that in most cases the proposed MBMF is superior over all existing algorithms in terms of accuracy and time complexity.
One of the key science goals for a diffraction limited imager on an Extremely Large Telescope (ELT) is the resolution of individual stars down to faint limits in distant galaxies. The aim of this study is to test the proposed capabilities of a multi-conjugate adaptive optics (MCAO) assisted imager working at the diffraction limit, in IJHK$_s$ filters, on a 42m diameter ELT to carry out accurate stellar photometry in crowded images in an Elliptical-like galaxy at the distance of the Virgo cluster. As the basis for realistic simulations we have used the phase A studies of the European-ELT project, including the MICADO imager (Davies & Genzel 2010) and the MAORY MCAO module (Diolaiti 2010). We convolved a complex resolved stellar population with the telescope and instrument performance expectations to create realistic images. We then tested the ability of the currently available photometric packages STARFINDER and DAOPHOT to handle the simulated images. Our results show that deep Colour-Magnitude Diagrams (photometric error, $pm$0.25 at I$ge$27.2; H$ge$25. and K$_sge$24.6) of old stellar populations in galaxies, at the distance of Virgo, are feasible at a maximum surface brightness, $mu_V sim$ 17 mag/arcsec$^2$ (down to M$_I > -4$ and M$_H sim$ M$_K > -6$), and significantly deeper (photometric error, $pm$0.25 at I$ge$29.3; H$ge$26.6 and K$_sge$26.2) for $mu_V sim$ 21 mag/arcsec$^2$ (down to M$_I ge -2$ and M$_H sim$ M$_K ge -4.5$). The photometric errors, and thus also the depth of the photometry should be improved with photometry packages specifically designed to adapt to an ELT MCAO Point Spread Function. We also make a simple comparison between these simulations and what can be expected from a Single Conjugate Adaptive Optics feed to MICADO and also the James Webb Space Telescope.
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