arXiv subjects
Publications and source records attributed to Richard J. Mathar.
The regular 2n-gon (square, hexagon, octagon, ...) is subdivided into smaller polygons (tiles) by the subset of diagonals which run parallel to any of the 2n sides. The manuscript reports on the number of tiles up to the 78-gon.
Phase screens above a telescope pupil represent the variation of the phase of the electromagnetic field induced by atmospheric turbulence. Instances drawn from such statistics are represented by a vector of random phase amplitudes which are coefficients of a linear superposition of two-dimensional basis functions across the pupil. This work shortly reviews Fried's analysis of this modal decomposition for the case of Kolmogorov statistics of the phase covariance as a function of separation in the pupil plane. We focus on the numerical synthesis of phase screens. The statistically independent modes are transformed into the eigen-modes of a gradient matrix as time-dependence is introduced such that on short time scales the instances of the phase screens are rigidly shifted into a direction imposed by some wind velocity - known as the Taylor frozen screen approximation. This simple technique factorizes spatial and temporal variables and aims at binding the time dependence of the phase screens to the few expansion coefficients of the basis functions that obey a stochastic time-dependent differential equation.
Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the role of expansion coefficients. Founded on a known convergence acceleration for the ordinary prime zeta functions, the sums and first derivatives are tabulated with high precision for indices k=2,...,6 and integer powers s=2,...,8.
Baseline calibration of a stellar interferometer is a prerequisite to data reduction of astrometric operations. This technique of astrometry is triangulation of star positions. Since angles are deduced from the baseline and delay side of these triangles, length and pointing direction (in the celestial sphere) of the baseline vector at the time of observation are key input data. We assume that calibration follows from reverse astrometry; a set of calibrator stars with well-known positions is observed and inaccuracies in these positions are leveled by observing many of them for a common best fit. The errors in baseline length and orientation angles drop proportional to the inverse square roots of the number of independent data taken, proportional to the errors in the individual snapshots of the delay, and proportional to the errors in the apparent positions of the calibrators. Scheduling becomes important if the baseline components are reconstructed from the sinusoidal delay of a single calibrator as a function of time.
These are seven corrigenda to equations in the Lehmer article in American Mathematical Monthly 92 (1985), pp 449--457, partially reproduced in the Apelblat tables of integrals and series.
The basic geometry of a stellar interferometer with two telescopes consists of a baseline vector and a direction to a star. Two derived vectors are the delay vector, and the projected baseline vector in the plane of the wavefronts of the stellar light. The manuscript deals with the trigonometry of projecting the baseline further outwards onto the celestial sphere. The position angle of the projected baseline is defined, measured in a plane tangential to the celestial sphere, tangent point at the position of the star. This angle represents two orthogonal directions on the sky, differential star positions which are aligned with or orthogonal to the gradient of the delay recorded in the u-v plane. The North Celestial Pole is chosen as the reference direction of the projected baseline angle, adapted to the common definition of the "parallactic" angle.
The set of Gaussian Type Orbitals g(n1,n2,n3) of order (n+1)(n+2)/2, of common n=n1+n2+n3<=7, common center and exponential, is customized to define a set of 2n+1 linear combinations t(n,m) (-n<=m<=n) such that each t(n,m) depends on the azimuthal and polar angle of the spherical coordinate system like the real or imaginary part of the associated Spherical Harmonic. (Results cover both Hermite and Cartesian Gaussian Type Orbitals.) Overlap, kinetic energy and Coulomb energy matrix elements are presented for generalized basis functions of the type r^s*t(n,m) (s=0,2,4,...). In addition, normalization integrals int |g(n1,n2,n3)|d^3r are calculated up to n=7 and normalization integrals int |r^s*t(n,m)|d^3r up to n=5.
Each straight infinite line defined by two vertices of a finite square point lattice contains (covers) these two points and a - possibly empty - subset of points that happen to be collinear to these. This work documents vertex subsets of minimum order such that the sum of the infinite straight lines associated with the edges of their complete subgraph covers the entire set of vertices (nodes). This is an abstraction to the problem of sending a light signal to all stations (receivers) in a square array with a minimum number of stations also equipped with transmitters to redirect the light to other transmitters.
The radial polynomials of the 2D (circular) and 3D (spherical) Zernike functions are tabulated as powers of the radial distance. The reciprocal tabulation of powers of the radial distance in series of radial polynomials is also given, based on projections that take advantage of the orthogonality of the polynomials over the unit interval. They may play a role in the expansion of products of the polynomials into sums, which is demonstrated by some examples. Multiplication of the polynomials by the angular bases (azimuth, polar angle) defines the Zernike functions, for which we derive and tabulate transformations to and from the Cartesian coordinate system centered at the middle of the circle or sphere.
Sets of orthogonal basis functions over two-dimensional circular areas--most often representing pupils in optical applications--are known in the literature for the full circle (Zernike or Jacobi polynomials) and the annulus. This work proposes an orthogonal set if the area is two non-overlapping circular pupils of same size. The major free parameter is the ratio of the pupil radii over the distance between both circles. Increasingly higher order aberrations--as defined for a virtual larger pupil in which both pupils are embedded--are fed into a Gram-Schmidt orthogonalization to implement one unique set of basis functions. The key element is to work out the overlap integrals between a full set of primitive basis functions (products of powers of the distance from the mid-point between both pupils by azimuthal functions of the Fourier type).
The statistically independent Karhunen-Loeve modes of refractive indices with isotropic Kolmogorov spectrum of the covariance are calculated in a sphere of given radius, rendered as series of 3D Zernike functions. Many of the symmetry arguments of the associated 2D problem for the circular input pupil remain valid. The technique of efficient diagonalization of the eigenvalue problem in wavenumber space is founded on the Fourier representation of the 3D Zernike basis.
Coupling efficiencies of an electromagnetic field with a Kolmogorov phase statistics into a step-index fiber in its monomode regime of wavelengths are computed from the overlap integral between the phase screens and the far-field of the monomode at infrared wavelengths. The phase screens are composed from Karhunen-Loeve basis functions, optionally cutting off some of the eigenmodes of largest eigenvalue as if Adaptive Optics had corrected for some of the perturbations. The examples are given for telescope diameters of 1 and 1.8 m, and Fried parameters of 10 and 20 cm. The wavelength of the stellar light is in the J, H, or K band of atmospheric transmission, where the fiber core diameter is tailored to move the cutoff wavelength of the monomode regime to the edges of these bands.
The theory of summation of electromagnetic line transitions is used to tabulate the Taylor expansion of the refractive index of humid air over the basic independent parameters (temperature, pressure, humidity, wavelength) in five separate infrared regions from the H to the Q band at a fixed percentage of Carbon Dioxide. These are least-squares fits to raw, highly resolved spectra for a set of temperatures from 10 to 25 C, a set of pressures from 500 to 1023 hPa, and a set of relative humidities from 5 to 60%. These choices reflect the prospective application to characterize ambient air at mountain altitudes of astronomical telescopes.
Long-baseline interferometry detects fringes created by superposition of two beams of light collected by two telescopes pointing into a common direction. The external path difference generated by pointing away from the zenith is commonly compensated by adding a variable optical path length (delay) through air for one beam such that the optical path difference between the beams remains close to zero near the detector. The ABCD formula assigns a (wrapped) phase to the amplitudes A to D of an interference pattern shifted by multiples of 90 degrees in phase. We study the interplay between a wide band pass of the optics and the dispersion of the air in the compensating delay, which leads to small deviations between the ABCD phase and the reduced, monochromatic group-delay representation of the wave packets. In essence, this adds dispersion to the effects that have been discussed for evacuated interferometers (telescopes in space) before [J. Opt. Soc. Am. A 22 (2005) 2774].
The Feynman diagrams of the Green's function expansion of fermions interacting with a non-relativistic 2-body interaction are displayed in first, second and third order of the interaction as 2, 10 and 74 diagrams, respectively. A name convention for the diagrams is proposed and then used to tabulate the 706 diagrams of fourth order. The Hartree-Fock approximation summons up 2, 8, 40 and 224 of them, respectively.
An inverse polynomial has a Chebyshev series expansion 1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear combinations of simple functions of the polynomial roots. If the first k of the coefficients a_n are known, the others become linear combinations of these with expansion coefficients derived recursively from the b_j's. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the b_j in f(x)/sum_(j=0..k)b_j*T_j(x)=1+sum_(n=k+1..oo) a_n*T_n(x), and may be handled with a Newton method providing the Chebyshev expansion of f(x) is known.
The basic effect of the earth's atmospheric refraction on telescope operation is the reduction of the true zenith angle to the apparent zenith angle, associated with prismatic aberrations due to the dispersion in air. If one attempts coherent superposition of star images in ground-based interferometry, one is in addition interested in the optical path length associated with the refracted rays. In a model of a flat earth, the optical path difference between these is not concerned as the translational symmetry of the setup means no net effect remains. Here, I evaluate these interferometric integrals in the more realistic arrangement of two telescopes located on the surface of a common earth sphere and point to a star through an atmosphere which also possesses spherical symmetry. Some focus is put on working out series expansions in terms of the small ratio of the baseline over the earth radius, which allows to bypass some numerics which otherwise is challenged by strong cancellation effects in building the optical path difference.
Various approaches to the numerical representation of the Incomplete Gamma Function F_m(z) for complex arguments z and small integer indexes m are compared with respect to numerical fitness (accuracy and speed). We consider power series, Laurent series, Gautschi's approximation to the Faddeeva function, classical numerical methods of treating the standard integral representation, and others not yet covered by the literature. The most suitable scheme is the construction of Taylor expansions around nodes of a regular, fixed grid in the z-plane, which stores a static matrix of higher derivatives. This is the obvious extension to a procedure often in use for real-valued z.