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We confirm that the first-, second-, and third-order derivatives of fully-normalized Legendre polynomial (LP) and associated Legendre function (ALF) of arbitrary degree and order can be correctly evaluated by means of non-singular fixed-degree formulas (Bosch in Phys Chem Earth 25:655-659, 2000) in the ordinary IEEE754 arithmetic when the values of fully-normalized LP and ALF are obtained without underflow problems, for e. g., using the extended range arithmetic we recently developed (Fukushima in J Geod 86:271-285, 2012). Also, we notice the same correctness for the popular but singular fixed-order formulas unless (1) the order of differentiation is greater than the order of harmonics and (2) the point of evaluation is close to the poles. The new formulation using the fixed-order formulas runs at a negligible extra computational time, i. e., 3-5 % increase in computational time per single ALF when compared with the standard algorithm without the exponent extension. This enables a practical computation of low-order derivatives of spherical harmonics of arbitrary degree and order. © 2012 Springer-Verlag.

Fukushima T.,Japan National Astronomical Observatory
Journal of Geodesy | Year: 2013

A recursive method is developed to compute the ratios of the oblate spheroidal harmonics of the second kind and their first-, second-, and third-order derivatives. The recurrence formulas consist of three kinds: (1) fixed-degree increasing-order, (2) mixed-degree increasing-order, and (3) fixed-order decreasing-degree. The three seed values are evaluated by rapidly convergent series. The derivatives of the ratios are recursively obtained from the values and lower-order derivatives of the same harmonic order and of the same or higher degrees. The new method precisely and quickly computes the ratios and their low-order derivatives. It provides 13 correct digits of the ratios of degree as high as 262,000 and runs 20-100 times faster than the existing methods. © 2012 Springer-Verlag Berlin Heidelberg.

Fukushima T.,Japan National Astronomical Observatory
Journal of Geodesy | Year: 2012

By extending the exponent of floating point numbers with an additional integer as the power index of a large radix, we compute fully normalized associated Legendre functions (ALF) by recursion without underflow problem. The new method enables us to evaluate ALFs of extremely high degree as 2 32 = 4,294,967,296, which corresponds to around 1 cm resolution on the Earth's surface. By limiting the application of exponent extension to a few working variables in the recursion, choosing a suitable large power of 2 as the radix, and embedding the contents of the basic arithmetic procedure of floating point numbers with the exponent extension directly in the program computing the recurrence formulas, we achieve the evaluation of ALFs in the double-precision environment at the cost of around 10% increase in computational time per single ALF. This formulation realizes meaningful execution of the spherical harmonic synthesis and/or analysis of arbitrary degree and order. © 2011 Springer-Verlag.

Martinache F.,Japan National Astronomical Observatory
Publications of the Astronomical Society of the Pacific | Year: 2013

This article introduces a novel wavefront sensing approach that relies on the Fourier analysis of a single conventional direct image. In the high Strehl ratio regime, the relation between the phase measured in the Fourier plane and the wavefront errors in the pupil can be linearized, as was shown in a previous work that introduced the notion of generalized closure-phase, or kernel-phase. The technique, to be usable as presented requires two conditions to be met: (1) the wavefront errors must be kept small (of the order of one radian or less), and (2) the pupil must include some asymmetry, which can be introduced with a mask, for the problem to become solvable. Simulations show that this asymmetric pupil Fourier wavefront sensing or APF-WFS technique can improve the Strehl ratio from 50% to over 90% in just a few iterations, with excellent photon noise sensitivity properties, suggesting that on-sky close loop APF-WFS is possible with an extreme adaptive optics system.

Niino Y.,Japan National Astronomical Observatory
Astrophysical Journal | Year: 2012

We investigate the relation between stellar mass (M*), star formation rate (SFR), and metallicity (Z) of galaxies, the so-called fundamental metallicity relation, in the galaxy sample of the Sloan Digital Sky Survey Data Release 7. We separate the galaxies into narrow redshift bins and compare the relation at different redshifts and find statistically significant (>99%) evolution. We test various observational effects that might cause seeming Z evolution and find it difficult to explain the evolution of the relation only by the observational effects. In the current sample of low-redshift galaxies, galaxies with different M* and SFR are sampled from different redshifts, and there is degeneracy between M */SFR and redshift. Hence, it is not straightforward to distinguish a relation between Z and SFR from a relation between Z and redshift. The separation of the intrinsic relation from the redshift evolution effect is a crucial issue in the understanding of the evolution of galaxies. © 2012. The American Astronomical Society. All rights reserved..

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