The Use of Spherical Harmonic Functions in Mathematical Physics

1930 ◽  
Vol 15 (209) ◽  
pp. 200
Author(s):  
S. Chapman
Keyword(s):  
Spherical Harmonic ◽  
Harmonic Functions ◽  
Mathematical Physics ◽  
Spherical Harmonic Functions

scholarly journals Non-radially pulsating stars as microlensing sources

2020 ◽  
Vol 498 (1) ◽  
pp. 223-234
Author(s):  
Sedighe Sajadian ◽  
Richard Ignace
Keyword(s):  
Light Curve ◽  
Spherical Harmonic ◽  
Harmonic Functions ◽  
Source Size ◽  
Light Curves ◽  
Time Dependent ◽  
High Magnification ◽  
Stellar Radius ◽  
Pulsating Stars ◽  
Spherical Harmonic Functions

ABSTRACT We study the microlensing of non-radially pulsating (NRP) stars. Pulsations are formulated for stellar radius and temperature using spherical harmonic functions with different values of l, m. The characteristics of the microlensing light curves from NRP stars are investigated in relation to different pulsation modes. For the microlensing of NRP stars, the light curve is not a simple multiplication of the magnification curve and the intrinsic luminosity curve of the source star, unless the effect of finite source size can be ignored. Three main conclusions can be drawn from the simulated light curves. First, for modes with m ≠ 0 and when the viewing inclination is more nearly pole-on, the stellar luminosity towards the observer changes little with pulsation phase. In this case, high-magnification microlensing events are chromatic and can reveal the variability of these source stars. Secondly, some combinations of pulsation modes produce nearly degenerate luminosity curves (e.g. (l, m) = (3, 0), (5, 0)). The resulting microlensing light curves are also degenerate, unless the lens crosses the projected source. Finally, for modes involving m = 1, the stellar brightness centre does not coincide with the coordinate centre, and the projected source brightness centre moves in the sky with pulsation phase. As a result of this time-dependent displacement in the brightness centroid, the time of the magnification peak coincides with the closest approach of the lens to the brightness centre as opposed to the source coordinate centre. Binary microlensing of NRP stars and in caustic-crossing features are chromatic.

scholarly journals Characteristics of a Spectral Inverse of the Laplacian Using Spherical Harmonic Functions on a Cubed-Sphere Grid for Background Error Covariance Modeling

2017 ◽  
Vol 145 (1) ◽  
pp. 307-322 ◽  
Author(s):  
Hyo-Jong Song ◽  
In-Hyuk Kwon ◽  
Junghan Kim
Keyword(s):  
Spherical Harmonic ◽  
Harmonic Functions ◽  
Inverse Method ◽  
Background Error ◽  
Computational Environment ◽  
Eigenvalues Of The Laplacian ◽  
The One ◽  
Cubed Sphere ◽  
Spectral Finite Element ◽  
Spherical Harmonic Functions

Abstract In this study, a spectral inverse method using spherical harmonic functions (SHFs) represented on a cubed-sphere grid (SHF inverse) is proposed. The purpose of the spectral inverse method studied is to help with data assimilation. The grid studied is the one that results from a spectral finite element decomposition of the six faces of the cubed sphere on Gauss–Legendre–Lobatto (GLL) points with equiangular gnomonic projection. For a given discretization of the cube in this form, as the total wavenumber of the test functions increases, there comes a point at which the cube’s eigenstructure fails to be able to replicate the spherical harmonic functions. The authors call this point a limit wavenumber in using the SHF inverse. In common with the authors’ previous research, the allowable total wavenumber of the SHF inverse increases more effectively with an enhanced polynomial order. The use of the eigenvectors and eigenvalues of the Laplacian, discretized on the grid spacing used in this study, to the Poisson equation is compared with the benchmark set by using the spherical harmonics solution to the problem. In terms of accuracy, the SHF inverse is superior to a direct inverse of the Laplacian using eigendecomposition. The feasibility of SHF inverse in operational implementation is examined under a massive computational environment.

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