A Modified Gooding Method to Calculate Inclination Functions

In this paper, the expression of V k_lm(I) in the Gooding method is rewritten to be the form convenient for calculation, and a standard recursive lm procedure is used to calculate Ak_lm(I). We have rewritten the Gooding's program under the assumption that l and k have the same odd-even parity, this makes the program be shorten for one half, the computational effciency and readability of the program be raised, the computing time be shortened for 41%, and the computational accuracy and stability are also slightly improved.     

Two Methods for Deriving the Recursion Formula of Inclination Function

Two kinds of recursion relations of hypergeometric series were derived, and hereby the Gooding's recursion formula of inclination function. In addition, it was demonstrated that this recursion formula can be derived also on the basis of the recursion relation of Jacobi polynomials. Comparing with the hypergeometric series, the recursion process based on the Jacobi polynomials is much simpler, indicating that the Gooding's method is the recursion of Jacobi polynomials in essence.     

Fourier analysis of the light curves of eclipsing variables. XIII

New expressions for the fractional loss of light _l ⁰ have been derived in the simple forms of rapidly converging expansions to the series of Chebyshev polynomials, Jacobi polynomials, and Kopal'sJ-integrals. In these expansions, which are a supplement to those given by Kopal (1977b), variablesk andh occur in different products that simplify the numerical computation. The treatment follows the new definition of _l ⁰ which has been recently developed by Kopal (1977a).     

倾角函数递推公式的误差传递分析

在倾角函数递推关系这个题目上,一些研究者做过工作,如Challe,A(1969),R.H.Googing(1971),J.A.Campbell(1972),G.E.O.Giacaglia(1976)和童付(1979)。最好的结果是由童付获得的,递推关系仅由二个或三个相邻的函数来表达。然而,当倾角较小时,童付的公式存在严重的计算误差传递。本文导出了一个分析公式来估计此种计算误差传递,并在分析计算误差传递机制的基础上,指出可避免计算误差传递的途径,导出的分析公式见文中第(9)式。     

Moments of inertia and the Chandlerian period for two-and three-layer models of the Galilean satellite Io

We consider two-layer (Fe-FeS core+silicate mantle) and three-layer (Fe-FeS core+silicate mantle+crust) models of the Galilean satellite Io. Two parameters are known from observations for the equilibrium figure of the satellite, the mean density ρ₀ and the Love number k₂. Previously, the Radau-Darwin formula was used to determine the mean moment of inertia. Using formulas of the Figure Theory, we calculated the principal moments of inertia A, B, and C and the mean moment of inertia I for the two-and three-layer models of Io using ρ₀ and k₂ as the boundary conditions. We concluded that when modeling the internal structure of Io, it is better to use the observed value of k₂ than the moment of inertia I derived from k₂ using the Radau-Darwin formula. For the models under consideration, we calculated the Chandlerian wobble periods of Io. For the three-layer model, this period is approximately 460 days.     

Near-opposition limb darkening of solids of planetary interest

This paper presents a laboratory study of the limb darkening near opposition, of particulate materials of planetary interest and concentrates on the wavelength dependence of this limb darkening. We find that near zero phase the scattering properties of most particulate materials can be described adequately by Minnaert's law. However, there are materials for which such a representation is totally inadequate. Examples are bronzite and graphite, materials that tend to fracture into flakes having mirrorlike surfaces. In addition, there are materials, such as olivine, whose scattering properties within deep absorption bands show definite departures from Minnaert's law at large angles of incidence or emission. Our Minnaert parameters, k and B₀, were measured at a phase angle of α = 4°. For samples of comparable surface texture and roughness, k and B₀ are approximately linearly related, k increasing as B₀ increases. Very dark materials tend to have k ～ 0.5 to 0.6, while very bright materials tend to have k ～ 1. The linear relationship between k and B₀ can be explained in terms of the varying importance of multiple scattering in the surface layer. Thus for materials for which B₀ is strongly wavelength dependent, so is k. For example, for olivine, k varies from 0.73 to 0.87 between 0.4 and 1.2 μm. These variations are closely correlated with those in B₀: the value of k is relatively high outside of absorption bands and relatively low within them. For bright materials, k is very sensitive to surface roughness. For example, for BaSO₄ powder, k can be changed from ～1.0 to ～0.8 by this effect alone, a fact which has relevance to the photometry of frost-covered satellites. For dark materials, the effects of surface roughness on k are smaller and more subtle.     

Can Cassini detect a subsurface ocean in Titan from gravity measurements?

Recent models of Titan's interior predict that the satellite contains an ocean of water and ammonia under an icy layer. Direct evidence for the presence of an ocean can be provided on the Cassini mission only by radio science determination of Titan Love number k₂. Simulations that use the five flybys T11, T22 T33, T45, and T68 (the latter two belonging to the extended mission) lead to the result that in the elastic case, where the Love number is real, k₂ will be determined with a one-sigma accuracy of 0.1. In the viscoelastic case, where k₂ is complex, the real and imaginary parts of k₂ will be determined with one sigma accuracies of 0.138 and 0.115, respectively. Ocean and oceanless models that include a viscoelastic rheology are built. In the viscoelastic case, there is a 93% probability to correctly predict the presence or absence of an ocean; this probability improves to 97% in the elastic case.     

The photometric properties of natural snow and of snow-covered planets

Using the white light measurements of Knowles Middleton and Mungall (1952), the Minnaert constants k and B₀ are derived for six types of snow surfaces for phase angles up to 80°. The conclusion is that snow is in general a quasi-Lambert scatterer (k = 1.04?1.35). Even in an extreme case of specular reflection (a “glazed rain crust”), k is less than 2 at these phase angles. The range in k and B₀ suggested by these data are then used to estimate some fundamental photometric parameters of smooth snow-covered planets: geometric albedos, phase integrals, Bond albedos, and phase coefficients.     

Out-of-plane motion about libration points: Nonlinearity and eccentricity effects

Out-of-plane motion about libration points is studied within the framework of the elliptic restricted three-body problem. Nonlinear motion in the circular restricted problem is given to third order in the out-of-plane amplitudeA _z by Jacobi elliptic functions. Linear motion in the elliptic problem is studied using Mathieu's and Hill's equations. Additional terms needed for a complete third-order theory are found using Lindsted's method. This theory is constructed for the case of collinear libration points; for the case of triangular points, a third-order nonlinear solution is given separately in terms of Jacobi elliptic functions.     

On the determination of the long period tidal perturbations in the elements of artificial earth satellites

In the present article we develop the theory of the long period tidal effects in the motion of artificial satellites assuming the variability of elastic parameters of the Earth (Love numbers) across the parallels. The dependence of Love numbers on the longitude produces perturbations of the period of one day or less and hence is neglected in the present theory. In this respect we follow in the footsteps of Kaula (1969). If the deviations ofk ₂ andk ₃ from pure constants are not taken into consideration, then the perturbations caused by the variability ofk ₂ andk ₃ across the parallels will be misinterpreted as the perturbations caused byk ₄...-terms, and the spurious values ofk ₄... will be deduced. It is extremely doubtful, however, that the real effects caused byk ₄,k ₅,..., are significant enough to be detected. The short period effects with the period of the revolution of the satellite, or less, were removed from the differential equations for the variation of elements of the satellite by the averaging over the orbit of the satellite. These differential equations are in the form convenient for numerical integration over a long interval of time and also suitable for developing the tidal effects into trigonometric series with the arguments ω, Ω of the satellite andl, l′, F, D, Γ of the Moon. The numerical integration can be performed using some simple quadrature formula, without resorting to a predictor-corrector system.     

Fourier analysis of the light curves of eclipsing variables,XXII

The theoretical values of the momentsA _2m for any type of eclipses, expressed in terms of the elementsL ₁,a andc ₀, have been derived in the simple forms of rapidly convergent expansions to the series of Chebyshev polynomials, Jacobi polynomials and KopalJ-integrals (Kopal, 1977c) and hold good for any real (not necessarily integral) value ofm0.The aim of the present paper has been to establish explicit expressions for the Jacobian and its fast enough computation in the light changes of close eclipsing systems, arising from the partial derivative of different pairs ofg-functions (Kopal and Demircan, 1978, Paper XIV) with respect toa andc ₀ ² , for any type of eclipses (be these occultations or transit, partial, total or annular) and for any arbitrary degreel of the adopted law of limb-darkening. The functional behaviour of this Jacobian would determine the reasonable light curve in connection with geometrical determinacy of the parametersa andc ₀. In the expansion of Jacobian, the terms consist of two polynomials which satisfy certain three-term recursion relations having the eclipse parametersa andc ₀, as their arguments.Closed form expressions forf-functions, as well as of the Jacobian (e.g.,m=1, 2, 3), obtaining in the case of total eclipses, are given for a comparative discussion with the theoretical values of Jacobian derived from partial derivative of different pairs ofg-functions.The numerical magnitude of Jacobian would determine the best combination of the momentsA _2m in the different pairs ofg-functions and definite results would follow in the subsequent paper of this series (Edalati, 1978c, Paper XXIV).     

Mid-infrared reflectance spectra and optical constants of six iron oxide/oxyhydroxide phases

We have determined the real and imaginary indices of refraction (n and k) for six iron oxide/oxyhydroxide phases—magnetite, maghemite, goethite, lepidocrocite, akaganéite, and ferrihydrite. A single crystal of magnetite was used to derive bulk n and k values from 100-2000 cm⁻¹ (5-100 μm). Synthetic nanocrystalline samples of maghemite, goethite, lepidocrocite, akaganéite, and ferrihydrite were pressed into compact pellets used to determine bulk n and k values from 100-1200 cm⁻¹ (8.33-100 μm). All values of n and k (the optical constants) were determined from specular reflectance spectra acquired at 2 cm⁻¹ spectral sampling using classical Lorentz-Lorenz dispersion theory. In this paper, we present the optical constants of all six minerals and the oscillator parameters with which they were modeled. Use of these optical constants could aid in radiative transfer models of terrestrial dust as well as Mars, the Moon, and airless bodies in the Solar System.     

Limb darkening of meteorites and asteroids

This paper presents the results of a laboratory study of the limb darkening, near opposition, of the carbonaceous chondrites Orgueil (C1), Murchison (C2), and Allende (C3), the ordinary chondrite Bruderheim (L6), and a stainless-steel powder. These materials represent possible analogs for the surface materials of C, S, and M asteroids respectively. At low phase angles, the limb-darkening behavior of all materials studied is well represented by Minnaert's law. For carbonaceous chondrites, the Minnaert limb-darkening parameter k is nearly independent of wavelength for wavelengths between 0.4 and 0.9 μm, with a typical value of k = 0.55. The reflectance parameter, B₀, varies from 0.045 to 0.065 over the same range of wavelengths. Both k and B₀ are larger for the stainless-steel powder and the ordinary chondrite, due to the increased importance of multiple scattering in the surface layer. If no limb darkening were present, k would equal $\text{1}\text{2}$ and the geometric albedo (p) of an asteroid would equal the normal reflectance $\text{(r}{}_{\mathrm{<mtext>n</mtext>}}\text{? B}{}_0\text{)}$ of its surface material. For bodies whose surface material is appreciably limb darkened, the geometric albedo measured at the telescope will be lower than the true normal reflectance of surface material; we estimate that for S and M objects r_n ? 1.05 p. In the case of nonspherical asteroids, because the distribution of incidence and emission angles varies as the asteroid rotates, the geometric albedo must change with aspect. If limb darkening is not considered when interpreting asteroid light curves, the values of b/a derived will be too extreme. This effect is probably too small to be observed for C asteroids, because of their intrinsically low reflectances, but could be appreciable for S and M objects.     

Exploring the 7:4 mean motion resonance—II: Scattering evolutionary paths and resonance sticking

In our preliminary study, we have investigated basic properties and dynamical evolution of classical TNOs around the 7:4 mean motion resonance with Neptune (a∼43.7 AU), motivated by observational evidences that apparently present irregular features near this resonance (see [Lykawka and Mukai, 2005a. Exploring the 7:4 mean motion resonance—I. Dynamical evolution of classical trans-Neptunian objects (TNOs). Space Planet. Sci. 53, 1175-1187]; hereafter “Paper I”). In this paper, we aim to explore the dynamical long-term evolution in the scattered disk (but not its early formation) based on the computer simulations performed in Paper I together with extra computations. Specifically, we integrated the orbital motion of test particles (totalizing a bit more than 10,000) placed around the 7:4 mean motion resonance under the effect of the four giant planets for the age of the Solar System. In order to investigate chaotic diffusion, we also conducted a special simulation with on-line computation of proper elements following tracks in phase space over 4-5 Gyr. We found that: (1) A few percent (1-2%) of the test particles survived in the scattered disk with direct influence of other Neptunian mean motion resonances, indicating that resonance sticking is an extremely common phenomenon and that it helps to enhance scattered objects longevity. (2) In the same region, the so-called extended scattered TNOs are able to form via very long resonance trapping under certain conditions. Namely, if the body spends more than about 80% of its dynamical lifetime trapped in mean motion resonance(s) and there is the action of a k+1 or (k+2)/2 mean motion resonance (e.g., external mean motion resonances with Neptune described as (j+k)/j with j=1 and 2, respectively). According to this hypothetical mechanism, 5-15% of current scattered TNOs would possess thus probably constituting a significant part of the extended scattered disk. (3) Moreover, considering hot orbital initial conditions, it is likely that the trans-Neptunian belt (or Edgeworth-Kuiper belt) has been providing members to the scattered disk, so that scattered TNOs observed today would consist of primordial scattered bodies mixed with TNOs that came from unstable regions of the trans-Neptunian belt in the past.Considering the three points together, our results demonstrated that the scattered disk has been evolving continuously since early times until present.     

Expansions of elliptic motion based on elliptic function theory

New expansions of elliptic motion based on considering the eccentricitye as the modulusk of elliptic functions and introducing the new anomalyw (a sort of elliptic anomaly) defined byw=u/2K–/2,g=amu–/2 (g being the eccentric anomaly) are compared with the classic (e, M), (e, v) and (e, g) expansions in multiples of mean, true and eccentric anomalies, respectively. These (q,w) expansions turn out to be in general more compact than the classical ones. The coefficients of the (e,v) and (e,g) expansions are expressed as the hypergeometric series, which may be reduced to the hypergeometric polynomials. The coefficients of the (q,w) expansions may be presented in closed (rational function) form with respect toq, k, k=(1–k ²)^1/2,K andE, q being the Jacobi nome relatedk whileK andE are the complete elliptic integrals of the first and second kind respectively. Recurrence relations to compute these coefficients have been derived.on leave from Institute of Applied Astronomy, St.-Petersburg 197042, Russia     

Substorm energy

It is shown that the area A_k(× 10⁶km²) covered by brightest auroras and the area A_q bounded by the auroral oval have a simple relation given by

$\text{A}{}_{\mathrm{k}}\text{= 0.05(A}{}_{\mathrm{q}}\text{? A}{}_0\text{)}{}^2$

, where A₀ denotes the area of the minimum size oval and the quantity (A_q ? A₀)² is proportional to the energy ε_q which is stored in the magnetotail and is available for substorms. Following the definition of the intensity of solar flares, A_k may be chosen as a measure of the intensity of substorms. It is also found that the joule heat energy produced by the auroral electrojet is also proportional to (A_q ? A₀)². Thus, it may be concluded that the intensity of substorms is proportional to the energy ε_q stored in the magnetotail.     

Disk-resolved photometry of Io: I. Near-opposition limb darkening

We have determined the Minnaert limb-darkening parameters (B₀, k) for three color classes of regions on Io near opposition (solar phase angle = 4–5°). Bright (“white”) areas show k ≈ 0.6 independent of wavelength. k for Average (“orange”) regions starts slightly below 0.5 in the Voyager ultraviolet filter (≈0.35 μm), rises monotonically toward longer wavelengths, and reaches ≈0.6 in the orange (≈0.59 μm), a trend consistent with the globally averaged results of McEwen and Soderblom (NASA Technical Memorandum 86246, pp. 261–262 (1984)). For Polar (“brown”) materials, k lies in the range 0.6–0.7 in all filters longward of ultraviolet. The Bright and Average regions show significantly smaller limb darkening, and smaller wavelength dependence of k, than laboratory samples of sulfur, a discrepancy that might represent the influence of macroscopic surface roughness. From a study of the relative merits of two different techniques for determining limb darkening parameters, we find that the single-image method, involving a single spacecraft image and the choice of a group of “similar” regions, has serious consistency problems when applied to the multicolored surface of Io. In contrast, an adaptation of the two-image solution of McEwen and Soderblom, involving the use of two or more images and computation of the limb darkening for individual surface regions, gives consistent results for spectrally similar regions. We conclude that the two-image technique is the only way to accurately determine the limb darkening of Ionian surface materials.     

Fission origin of the Moon: Cause and timing

A new scenario is offered for the origin of the Moon. It is assumed that the Earth formed initially with about the maximum amount of angular momentum consistent with dynamical stability. This state is approximated by the secularly unstable Maclaurin spheroids (highly flattened, hamburger-shaped bodies). It is shown that the Earth cannot depart from this state at a resonable rate as long as its viscosity is in the range of liquid rock. Since core formation supplies about 1600 kJ kg^?1 the Earth will not leave this state until core formation is complete. When cooling produces a rise in viscosity, the Earth will necessarily evolve along a path which is approximated by the Riemann ellipsoids (which have rapid internal motion). The evolution is toward a Jacobi ellipsoid, but it is intercepted by the development of a third-harmonic (pearshaped) instability, which is catastrophic and leads to fission. The process of fission itself may be fundamentally analogous to the breaking of a wave in water. We cannot exclude the possibility that some other planets evolved similarly.     

Uniform density equilibrium model of self-gravitating stellar systems

A class of equilibrium solutions of the Vlasov equation for self-gravitating systems is discussed. The density and the potential are derived in form of Jacobi polynomials, which in a special case give rise to a model with uniform density.     

Om diagnostic for viscous Cardassian universe

Om diagnostic is a useful geometric method to differentiate between different cosmological models without the accurate current value of matter density. We investigate the Om diagnostic for viscous Cardassian universe and find that the model can be easily distinguished from LCDM. We also investigate the influence of the bulk viscosity coefficient τ on the evolutive behavior of Om with respect to redshift z. According to the value of Om(z=0) for viscous Cardassian models, we obtain the current value of equation of state w _k0.