Non-radial stellar oscillations: a perspective from potential scattering (I)

This article is the first in a series designed to gain insight into the stellar oscillation problem from a somewhat novel point of view: that of potential scattering, well-known in the quantum mechanical literature. In this paper the known theoretical foundations are developed and applied in the context of the astrophysical problem, wherein the star itself (rather than any portion of it) is the potential which scatters waves and traps them. The basis for the identification of a precisely defined scattering problem is the existence of a linear Schrödinger equation associated both globally (Section 2) and locally (Section 8) with the nonlinear eigenvalue equation for nonradial stellar pulsations. The paper is also designed to be a fairly complete account of the relevant mathematical topics that are germane to a study of this kind. This paper is dedicated to the memory of Professor Zdenèk Kopal, who was a great source of professional encouragement to me during the last fifteen years of his life.     

Stellar irradiance variations due to surface temperature inhomogeneities

Observations of rotational modulation of continuum brightness and photospheric and chromospheric spectral-line profiles of late-type stars indicate the presence of very inhomogeneous surface temperature distributions. We present three stellar examples (VY Ari, HR 7275, HU Vir) where time-series photometry is used to trace the evolution of spotted regions. Simultaneous spectroscopy and Doppler imaging for one of the three stars (HU Virgo, Fig. 1) makes it possible to compute the temperature distribution of the photosphere and the relative intensity distribution of parts of the chromosphere (from CaII K and H line profiles). The combination of time-series spot modeling and Doppler imaging enabled us to determine thesign and amount of differential surface rotation on HU Vir. We found a big, cool polar spot (see figure below) and a differential (surface) rotation law where higher-latitude regions rotate faster than lower-latitude regions (opposite to what we see on the Sun). Currently, this ensemble of techniques - time-series photometry and photospheric and chromospheric Doppler imaging - is only applicable to stars overactive by approximately a factor of 100 as compared to the active Sun, e.g. the evolved components in RS CVn-type binaries and some rapidly-rotating, single, pre-main sequence stars or giant stars. Stellar rotation is a fundamental parameter for (magnetic) activity. Starspots, or any other surface inhomogeneities, allow one to derive very precise stellar rotation rates and, if coupled with seismological observations of solar-type stars, could provide information on the internal angular momentum distribution in overactive late-type stars.To be published in Astronomy & Astrophysics.     

Effect of perturbed potentials on the non-linear stability of libration pointL 4 in the restricted problem

The non-linear stability of the libration pointL ₄ in the restricted problem has been studied when there are perturbations in the potentials between the bodies. It is seen that the pointL ₄ is stable for all mass ratios in the range of linear stability except for three mass ratios depending upon the perturbing functions. The theory is applied to the following four cases:

Particle acceleration and the decay of soft X-ray non-thermal line broadening in solar flares

The relationship between the production of -ray emitting particles and non-thermal soft X-ray line broadening is investigated. A model of particle acceleration via the stochastic interaction with MHD turbulence is assumed and the time development of the wave energy density derived under the condition of energy conservation between waves and particles. The inferred numbers and energy distribution of accelerated protons for four -ray flares are used to define the wave energy density and its temporal development. The presence of Alfvén wave turbulence is considered as the source of the non-thermal motions in the ambient plasma. These motions are observed as excess widths in the soft X-ray line emission from these events. The decay of the waves via the particle acceleration process is compared with the observed decays of this non-thermal line broadening. Our results show that both the -ray emission and excess soft X-ray line widths in these flares can be explained by the single physical phenomenon of Alfvén wave turbulence.     

An integrable case of a rotational motion analogous to that of Lagrange and Poisson for a gyrostat in a Newtonian force field

The scope of the present paper is to provide analytic solutions to the problem of the attitude evolution of a symmetric gyrostat about a fixed point in a central Newtonian force field when the potential function isV ⁽²⁾.We assume that the center of mass and the gyrostatic moment are on the axis of symmetry and that the initial conditions are the following: (t ₀)=₀, (t ₀)=₀, (t ₀)=(t ₀)=₀, ₁(t ₀)=0, ₂(t ₀)=0 and ₃(t ₀)= ₃ ⁰ .The problem is integrated when the third component of the total angular momentum is different from zero (B ₁ 0). There now appear equilibrium solutions that did not exist in the caseB ₁=0, which can be determined in function of the value ofl ₃ ^r (the third component of the gyrostatic momentum).The possible types of solutions (elliptic, trigonometric, stationary) depend upon the nature of the roots of the functiong(u). The solutions for Euler angles are given in terms of functions of the timet. If we cancel the third component of the gyrostatic momentum (l ₃ ^r =0), the obtained solutions are valid for rigid bodies.     

The arrangement in mean elements space of the periodic orbits close to that of the Moon

In a simplified model of the Earth-Moon-Sun system based on the restricted circular 3-dimensional 3-body problem, it is possible to find numerically a set of 8 periodic orbits whose time evolutions closely resemble that of the Moon's orbit. These orbits have a period of 223 synodic months (i.e. the period of the Saros cycle known for more than two millennia as a means of predicting eclipses), and are characterized by a secular rotation of the argument of perigee . Periodic orbits of longer durations exhibiting this last feature are very abundant in Earth-Moon-Sun dynamical models. Their arrangement in the space of the mean orbital elements- for various values of the lunar mean motion is presented.     

A stability study of asteroid families near the 3:1 and 5:2 resonance with Jupiter

By using theD-criterion Lindblad (1992) has identified 14 asteroid families from a sample of 4100 numbered asteroids with proper elements from Milani and Kneevi (1990). Taxonomic types and other physical properties for a significant number of objects in five of the families show strong homogeneity within each family, further strengthening their internal relationship.To test the hypothesis of a common origin in, e.g., a catastrophic collision event, we have set out to integrate the orbits of the members of the Maria, Dora and Oppavia-Gefion families over some 10⁶ years. The mean distance for the Maria family is close to the 3:1 mean-motion resonance with Jupiter, while the other two families lie close to the 5:2 resonance.We used a simplified solar system model which included the perturbations by Jupiter and Saturn only and implemented Everhart's variable stepsize integrator RA15. All close encounters between the family members (within 0.1 AU) were recorded as well. Preliminary results from integrations over 4×10⁵ years are presented here.The statistics of close encounters show pronounced peaks for several members within each family, while for others no significant levels above the background of random encounters or even very low frequencies were found. This indicates a subclustering within the families. Quite a lot of very close (<0.005 AU) mutual encounters are found, which suggest that, at least for the larger members in a family, the mutual gravitational interactions could be of some importance for the real orbital evolutions.The encounter statistics between the Dora and Oppavia family members suggest a possible interrelationship between this two groups.     

H2 in interstellar and extragalactic ices: infrared characteristics, ultraviolet production, and implications

H2 is the most abundant molecule in the universe. We demonstrate that this molecule may be an important component of interstellar and possibly intergalactic ices, both because it can be formed in situ, within the ices, and because gas phase H2 can freeze out onto dust grains in some astrophysical environments. The condensation-sublimation and infrared spectral properties of ices containing H2 are presented. We show that solid H2 in H20-rich ices can be detected by an infrared absorption band at 4137 cm-1 (2.417 micrometers). The surface binding energy of H2 to H2O ice was measured to the delta Hs/k = 555 +/- 35 K. Surface binding energies can be used to calculate the residence times of H2 on grain surfaces as a function of temperature. Some of the implications of these results are considered.     

Propagation and absorption of cyclotron maser radiation in solar microwave spike bursts

It has been argued that the loss-cone-driven electron cyclotron maser instability can account for the properties of millisecond microwave spike bursts observed during some solar flares. However, as it propagates outward from the corona, maser radiation undergoes gyroresonance absorption when its frequency is a harmonic of the local electron-cyclotron frequency. Existing analytical models using slab geometries predict that this absorption should be sufficiently strong to prevent the radiation from being seen at the observed levels, except under highly restrictive conditions or for unrealistic plasma parameters. A more comprehensive analysis is presented here to determine if and when maser radiation can escape to produce microwave spike bursts. This analysis employs numerical raytracing and incorporates propagation and absorption of fundamental maser emission in a realistic model of a coronal flux loop. It is found that ranges of physical conditions do exist under which maser radiation can escape to an observer and that these conditions are much more limiting for fundamental emission in the extraordinary ()-mode than in the ordinary (o)-mode. Detailed investigation implies that escaping radiation in the -mode is highly directional and chiefly observable toward the center of the solar disk, while escapingo-mode radiation is found to emerge from the corona over a much wider range of directions, with some cases corresponding to radiation observable near the solar limb.     

The effect of non-thermal excitation and ionization on the hydrogen emission in impulsive solar flares

This is a quantitative investigation of the electron beam effect on the hydrogen line profiles and continuum intensity distribution during the impulsive phase of flares. The flaring atmosphere is suggested to be a hydrogenic one and its physical condition corresponds to the gas dynamics problem solution. The radiative transfer, steady-state and particle conservation equations are solved for the three-level hydrogen model atoms with continua. Return-current losses were neglected. Hydrogen line profiles are found to be slightly sensitive to nonthermal impacts with beam electrons in the cores and more sensitive in the wings. With the initial energy flux,F ₀, rising and energy spectral index, , decreasing, the wing intensities begin to increase, and the H lines are shown to have rather extended wings as is often observed. The hydrogen continua are shown to be strongly affected by nonthermal impacts. The bigger the value ofF ₀ and the smaller the value of , the greater absolute intensities of the hydrogen continua heads. This effect is more noticeable for the Balmer and Paschen continua. The head intensity slopes of them can be used for determination of these electron beam parameters on depths of the hydrogen emission origin and their following comparison with the same parameters for the coronal heights from the X-ray observations.