Direct solution of laplace's equation for coronal magnetic fields using line-of-sight boundary conditions

A new fast mathematical method is described for computing potential magnetic field solutions in the solar atmosphere from the observed line-of-sight component of the photospheric magnetic field. As in a standard Neumann boundary problem the orthogonality relation of the spherical harmonics is used to determine the coefficients of the harmonic expansion. This leads to a very simple set of recursion formulae that determines the harmonic coefficients successively.     

Rotational dynamics of a solar system body under solar radiation torques

The rotational dynamics of a small solar system body subject to solar radiation torques is investigated. A set of averaged evolutionary equations are derived as an analytic function of a set of spherical harmonic coefficients that describe the torque acting on the body due to solar radiation. The analysis also includes the effect of thermal inertia. The resulting equations are studied and a set of possible dynamical outcomes for the rotation rate and obliquity of a small body are found and characterized.     

Ellipsoidal Harmonic expansions of the gravitational potential: Theory and application

Small bodies of the solar system are now the targets of space exploration. Many of these bodies have elongated, non-spherical shapes, and the usual spherical harmonic expansions of their gravity fields are not well suited for the modelling of spacecraft orbits around these bodies. An elegant remedy is to use ellipsoidal harmonic expansions instead of the usual spherical ones. In this paper, we present their mathematical theory as well as a real application: the simulation of a landing on the surface of a kilometer-sized comet. We show that with an ellipsoidal harmonic expansion up to degree 5, the error on the landing position is at the meter level, while the corresponding error for the spherical harmonic expansion can reach tens of meters.This revised version was published online in October 2005 with corrections to the Cover Date.     

The exact transformation from spherical harmonic to ellipsoidal harmonic coefficients for gravitational field modeling

The spherical and ellipsoidal harmonic series of the external gravitational potential for a given mass distribution are equivalent in their mutual region of uniform convergence. In an instructive case, the equality of the two series on the common coordinate surface of an infinitely large sphere reveals the exact correspondence between the spherical and ellipsoidal harmonic coefficients. The transformation between the two sets of coefficients can be accomplished via the numerical methods by Walter (Celest Mech 2:389–397, 1970) and Dechambre and Scheeres (Astron Astrophys 387:1114–1122, 2002), respectively. On the other hand, the harmonic coefficients are defined by the integrals of mass density moments in terms of the respective solid harmonics. This paper presents general algebraic formulas for expressing the solid ellipsoidal harmonics as a linear combination of the corresponding solid spherical harmonics. An exact transformation from spherical to ellipsoidal harmonic coefficients is found by incorporating these connecting expressions into the density integral. A computational procedure is proposed for the transformation. Numerical results based on the nearly ellipsoidal Martian moon, Phobos, are presented for validation of the method.     

Lunar gravity derived from long-period satellite motion — A proposed method

A new method has been devised to determine the spherical harmonic coefficients of the lunar gravity field. This method consists of a two-step data reduction and estimation process. In the first step, a weighted least-squares empirical orbit determination scheme is applied to Doppler tracking data from lunar orbits to estimate longpperiod Kepler elements and rates. Each of the Kepler elements is represented by an independent function of time. The long-period perturbing effects of the Earth, Sun, and solar radiation are explicitly modeled in this scheme. Kepler element variations estimated by this empirical processor are then ascribed to the non-central lunar gravitation features. Doppler data are reduced in this manner for as many orbits as are available. In the second step, the Kepler element rates are used as input to a second least-squares processor that estimates lunar gravity coefficients using the long-period Lagrange perturbation equations.Pseudo Doppler data have been generated simulating two different lunar orbits. This analysis included the perturbing effects of the L1 lunar gravity field, the Earth, the Sun, and solar radiation pressure. Orbit determinations were performed on these data and long-period orbital elements obtained. The Kepler element rates from these solutions were used to recover L1 lunar gravity coefficients. Overall results of this controlled experiment show that lunar gravity coefficients can be accurately determined and that the method is dynamically consistent with long-period perturbation theory.     

Small body surface gravity fields via spherical harmonic expansions

Conventional gravity field expressions are derived from Laplace’s equation, the result being the spherical harmonic gravity field. This gravity field is said to be the exterior spherical harmonic gravity field, as its convergence region is outside the Brillouin (i.e., circumscribing) sphere of the body. In contrast, there exists its counterpart called the interior spherical harmonic gravity field for which the convergence region lies within the interior Brillouin sphere that is not the same as the exterior Brillouin sphere. Thus, the exterior spherical harmonic gravity field cannot model the gravitation within the exterior Brillouin sphere except in some special cases, and the interior spherical harmonic gravity field cannot model the gravitation outside the interior Brillouin sphere. In this paper, we will discuss two types of other spherical harmonic gravity fields that bridge the null space of the exterior/interior gravity field expressions by solving Poisson’s equation. These two gravity fields are obtained by assuming the form of Helmholtz’s equation to Poisson’s equation. This method renders the gravitational potentials as functions of spherical Bessel functions and spherical harmonic coefficients. We refer to these gravity fields as the interior/exterior spherical Bessel gravity fields and study their characteristics. The interior spherical Bessel gravity field is investigated in detail for proximity operation purposes around small primitive bodies. Particularly, we apply the theory to asteroids Bennu (formerly 1999 RQ36) and Castalia to quantify its performance around both nearly spheroidal and contact-binary asteroids, respectively. Furthermore, comparisons between the exterior gravity field, interior gravity field, interior spherical Bessel gravity field, and polyhedral gravity field are made and recommendations are given in order to aid planning of proximity operations for future small body missions.     

The Photospheric Convection Spectrum

Spectra of the cellular photospheric flows are determined from observations acquired by the MDI instrument on the SOHO spacecraft. Spherical harmonic spectra are obtained from the full-disk observations. Fourier spectra are obtained from the high-resolution observations. The p-mode oscillation signal and instrumental artifacts are reduced by temporal filtering of the Doppler data. The resulting spectra give power (kinetic energy) per wave number for effective spherical harmonic degrees from 1 to over 3000. Significant power is found at all wavenumbers, including the small wavenumbers representative of giant cells. The time evolution of the spectral coefficients indicates that these small wavenumber components rotate at the solar rotation rate and thus represent a component of the photospheric cellular flows. The spectra show distinct peaks representing granules and supergranules but no distinct features at wavenumbers representative of mesogranules or giant cells. The observed cellular patterns and spectra are well represented by a model that includes two distinct modes – granules and supergranules.     

The Magnetic Field at the Inner Boundary of the Heliosphere Around Solar Minimum

STEREO A and B observations of the radial magnetic field between 1 January 2007 and 31 October 2008 show significant evidence that in the heliosphere, the ambient radial magnetic field component with any dynamic effects removed is uniformly distributed. Based on this monopolar nature of the ambient heliospheric field we find that the surface beyond which the magnetic fields are in the monopolar configuration must be spherical, and this spherical surface can be defined as the inner boundary of the heliosphere that separates the monopole-dominated heliospheric magnetic field from the multipole-dominated coronal magnetic field. By using the radial variation of the coronal helmet streamers belts and the horizontal current – current sheet – source surface model we find that the spherical inner boundary of the heliosphere should be located around 14 solar radii near solar minimum phase.     

Kinematic analysis of stellar radial velocities by the spherical harmonics

A method for a kinematic analysis of stellar radial velocities using spherical harmonics is proposed. This approach does not depend on the specific kinematic model and allows both low-frequency and high-frequency kinematic radial velocity components to be analyzed. The possible systematic variations of distances with coordinates on the celestial sphere that, in turn, are modeled by a linear combination of spherical harmonics are taken into account. Theoretical relations showing how the coefficients of the decomposition of distances affect the coefficients of the decomposition of the radial velocities themselves have been derived. It is shown that the larger the mean distance to the sample of stars being analyzed, the greater the shift in the solar apex coordinates, while the shifts in the Oort parameter A are determined mainly by the ratio of the second zonal harmonic coefficient to the mean distance to the stars, i.e., by the degree of flattening of the spatial distribution of stars toward the Galactic plane. The distances to the stars for which radial velocity estimates are available in the CRVAD-2 catalog have been decomposed into spherical harmonics, and the existing variations of distances with coordinates are shown to exert no noticeable influence on both the solar motion components and the estimates of the Oort parameter A, because the stars from this catalog are comparatively close to the Sun (no farther than 500 pc). In addition, a kinematic component that has no explanation in terms of the three-dimensional Ogorodnikov-Milne model is shown to be detected in the stellar radial velocities, as in the case of stellar proper motions.     

3-D non-linear evolution of a magnetic flux tube in a spherical shell: The isentropic case

We present recent 3-D MHD numerical simulations of the non-linear dynamical evolution of magnetic flux tubes in an adiabatically stratified convection zone in spherical geometry, using the anelastic spherical harmonic (ASH) code.We seek to understand the mechanism of emergence of strong toroidal fields from the base of the solar convection zone to the solar surface as active regions. We confirm the results obtained in cartesian geometry that flux tubes that are not twisted split into two counter vortices before reaching the top of the convection zone. Moreover, we find that twisted tubes undergo the poleward-slip instability due to an unbalanced magnetic curvature force which gives the tube a poleward motion both in the non-rotating and in the rotating case. This poleward drift is found to be more pronounced on tubes originally located at high latitudes. Finally, rotation is found to decrease the rise velocity of the flux tubes through the convection zone, especially when the tube is introduced at low latitudes. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The troposphere-stratosphere radiation field at twilight: A spherical model

Time-dependent calculations of trace constituent distributions require as input the dissociating radiation field as a function of altitude and solar zenith angle. An isotropic, spherical, multiple scattering model of the radiation field has been developed to determine the radiation field at twilight. Comparison of the spherical model with a plane parallel model at twilight shows that: (1) for solar zenith angles less than 92°, plane parallel solutions for the source function are suitable if the initial deposition of solar energy is calculated for a spherical atmosphere; (2) for solar zenith angles greater than 92°, the plane parallel radiation field can be several orders of magnitude smaller than that calculated with the spherical model; (3) at altitudes above 40 km and at all solar zenith angles, the spherical model predicts 10–20% less radiation than the radiation field calculated with the plane parallel model. Calculations of the rate of photodissociation of NO₂ in the troposphere and stratosphere show that the spherical model yields significantly higher values at solar zenith angles greater than 92°.     

Green's function methods for potential magnetic fields

The Green's function method to calculate potential magnetic field on the Sun, which was first established by Schmidt (1964) in the case that the field component normal to a flat boundary plane is specified, is extended to the following three cases: (a) The field component along the line of sight, which is not generally normal to the flat boundary plane, is specified; (b) the line of sight component on a spherical boundary surface is specified; (c) the normal component on a spherical surface is specified, together with the condition that the field becomes approximately radial on an outer spherical surface (the so-called source surface). Properties of these Green's functions are examined, and the applicability of these methods to solar magnetic data is discussed.On leave of absence from Department of Astronomy, University of Tokyo.     

Tabulation of the harmonic coefficients of the solar magnetic fields

Tables of spherical harmonic coefficients for the global photospheric magnetic field between 1959 and 1974 are now available on microfilm. (These are the same coefficients which were used to construct the maps of the coronal magnetic atlas.)The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Cautions for the estimation of equivalent current systems from geomagnetic data

Spherical harmonic geomagnetic data analysis is a convenient technique to estimate equivalent current systems and to separate the external from the internal fields. However, there are certain cautions which need to be taken in the procedure, as demonstrated by some examples of the analyses of geomagnetic daily variations during solar active and quiet periods. Questions concerning the proper level of series truncation, problems with sharp gradients of the northward amplitude in narrow latitudinal zones, and dependence of results on which horizontal component is selected (northward, eastward or both) are discussed for the northern hemispheric equivalent current systems.     

Extrapolation of the solar magnetic field within the potential-field approximation from full-disk magnetograms

This paper is concerned with the Laplace boundary-value problem with the directional derivative, corresponding to the specific nature of measurements of the longitudinal component of the photospheric magnetic field. The boundary conditions are specified by a distribution on the sphere of the projection of the magnetic field vector into a given direction, i.e., they exactly correspond to the data of daily magnetograms distributed across the full solar disk. It is shown that the solution of this problem exists in the form of a spherical harmonic expansion, and uniqueness of this solution is proved. A conceptual sketch of numerical determination of the harmonic series coefficients is given. The field of application of the method is analyzed with regard to the peculiarities of actual data. Results derived from calculating magnetic fields from real magnetograms are presented. Finally, we present differences in results derived from extrapolating the magnetic field from a synoptic map and a full-disk magnetogram.     

An iterative method for obtaining a nonlinear solution for the temperature distribution of a spinning spherical body irradiated by a central star

We developed an iterative method for determining the three-dimensional temperature distribution in a spherical spinning body that is irradiated by a central star. The seasonal temperature change due to the orbital motion is ignored. It is assumed that material parameters such as the thermal conductivity and the thermometric conductivity are constant throughout the spherical body. A general solution for the temperature distribution inside a body is obtained using spherical harmonics and spherical Bessel functions. The surface boundary condition contains a term obtained using the Stefan–Boltzmann law and is nonlinear with respect to temperature because it is dependent on the fourth power of temperature. The coefficients of the general solution are fitted to satisfy the surface boundary condition by using the iterative method. We obtained solutions that satisfy the nonlinear boundary condition within 0.1% accuracy. We calculated the rate of change in the semimajor axis due to the diurnal Yarkovsky effect using the linear and nonlinear solutions. The maximum difference between the rates calculated using the two sets of solutions is 13%. Therefore current understanding of the diurnal Yarkovsky effect based on linear solutions is fairly good.     

A simple method to compute spherical harmonic coefficients for the potential model of the coronal magnetic field

It is impossible to make a direct measurement of the coronal magnetic field from the ground. The coronal magnetic field is, then, usually inferred by extrapolation of the observed photospheric magnetic field. The so-called potential model has been used for this extrapolation. We have to solve the Laplacian equation of the magnetic scalar potential. This magnetic scalar potential can be expanded into a spherical harmonic series. In this paper, new simple recursion formulae are proposed to solve the Laplacian equation; that is, to determine the spherical harmonic coefficients.     

Non-radial motion of eruptive filaments

We develop a simple model to explain the non-radial motion of eruptive solar filaments under solar minimum conditions. The global magnetic field is derived from the first and third components of the spherical harmonic expansion of a magnetic scalar potential. The filament is modeled as a toroidal current located above the mid-latitude polarity inversion line. We investigate the stability of the filament against changes in the filament current and attempt to explain the non-radial motion and acceleration of the eruptive filament. We also discuss the limitations of this model.