Bidirectional reflectance spectroscopy: 3. Correction for macroscopic roughness

A mathematically rigorous formalism is derived by which an arbitrary photometric function for the bidirectional reflectance of a smooth surface may be corrected to include effects of general macroscopic roughness. The correction involves only one arbitrary parameter, the mean slope angle $\text{θ}$, and is applicable to surfaces of any albedo. Using physically reasonable assumptions and mathematical approximations the correction expressions are evaluated analytically to second order in $\text{θ}$. The correction is applied to the bidirectional reflectance function of B. Hapke (1981, J. Geophys. Res.86, 3039–3054). Expressions for both the differential and integral brightnesses are obtained. Photometric profiles on hypothetical smooth and rough planets of low and high albedo are shown to illustrate the effects of macroscopic roughness. The theory is applied to observations of Mercury and predicts the integral phase function, the apparent polar darkening, and the lack of limb brightness surge on the planet. The roughness-corrected bidirectional reflectance function is sufficiently simple that it can be conveniently evaluated on a programmable hand-held calculator.     

UBV photometry of Vesta

A lightcurve of Vesta, obtained on four nights between June 23 and 30 June 1978 during the coordinated campaign for the determination of the rotation period, is presented. The observations were performed at the 91-cm telescope of the Catania Observatory employing UBV filters and a photon counting photometer. The V lightcurve apparently shows a single maximum, suggesting that the 5^h20^m29^s.2 period is the correct one. Features are evident near the maximum and the minimum closely resembling those of Johnson's lightcurve of 22 December 1950 and Taylor's of January 21, 1973. The amplitude in V light is 0^m.105 and small variations are also found in the color indices. The largest color variation is for the U-V with Δm = 0.^m.05, which is slightly larger than the value 0^m.02 found by T. Gehrels [Astron. J.72, 929 (1967)]. The maximum and minimum values occur at the same phase with respect to the maximum V light as found by Gehrels, i.e., Vesta appears bluer near 0^p.25 and redder near 0^p.7. Corrections with the solar phase angle were made using the coefficients given by Gehrels for the B-V and U-V while a new value of 0.036 mag/deg was assumed for the V observations. The available amplitudes of Vesta's lightcurve were analyzed with respect to the longitude position and the solar phase angle.     

Ion momentum and energy transfer rates for charge exchange collisions

The rates of momentum and energy transfer have been obtained for charge exchange collisions between ion and neutral gases having arbitrary Maxwellian temperatures T_i and T_n and bulk transport velocities c_i and c_n. The results are directly applicable to the F-region of the ionosphere where O⁺ - O charge is the dominant mechanism affecting ion momentum and energy transfer.     

Mutual phenomena of Jupiter's satellites

Observations of an occultation of Europa by Io are fitted by a model light curve. The model has five free parameters, namely the radius of Europa, the impact parameter, the brightness ratio of the satellites, the time of midevent and the mean relative velocity. The model assumes a fixed value for the radius of Io and for the solar phase angle α, and that Europa has a uniform surface brightness. The OC residuals of the best fitting light curve are very small (～0.002 mag) and of a purely random nature; there is no evidence of albedo features. Taking α = 0 does not affect significantly the quality of the fit. Six mutual eclipses were also observed, and their times of minima agree well with the predictions of Aksnes Icarus21 (1974). For two events these predictions differ by about 20 min from those of Brinkmann and Millis Sky & Telescope45 (1973).     

Advanced method to derive the IMF direction near Mars from cycloidal proton distributions

In a previous paper, we showed a method for deriving the interplanetary magnetic field (IMF) orientation from the velocity distribution of ring-like distributed ions as measured by the Ion Mass Analyser (IMA) on board Mars Express (MEX). This method has been improved so that one can derive the IMF orientation from a very limited portion of the ring distributions, i.e., only the highest energy portion of the ring distribution. This method uses the maximum variance direction L instead of the minimum variance direction N, which are derived from manually selected ring data. Because IMA's count rate for a semi-persistent ring distribution is nearly proportional to energy squire, L is most likely aligned to the tangential direction of the ring distribution at its highest energy, and this tangential direction is parallel or anti-parallel to the electric field. A vector product of L and the solar wind direction (X) gives the IMF orientation projected to the Y-Z plane. The tilt angle of IMF toward the X direction from the Y-Z plane is the same as the angle between the X direction and the ring plane, and is obtained from two methods when the initial speed of the ring ions is estimated to be much smaller than the solar wind speed: (1) angle between the velocity of ring's maximum energy portion and the solar wind vector, and (2) energy ratio between the solar wind and the maximum energy of the ring. The present method is applied to the IMA data from 3 June 2005 (0605-0640 UT) when the Mars Global Surveyor (MGS) magnetometer data are available. Using these data, we also tried to determine the sign of the IMF direction by estimating the evolution direction of the ring ions.     

Photometry and polarimetry of Titan: Pioneer 11 observations and their implications for aerosol properties

The preliminary measurements by Pioneer 11 of the limb darkening and polarization of Titan at red and blue wavelenghts (M. G. Tomasko, 1980,J. Geophys. Res., 85, 5937–5942) are refined and the measurements of the brightness of the integrated disk at phase angles from 22 to 96° are reduced. At 28° phase, Titan's reflectivity in blue light at southern latitudes is as much as 25% greater than that at northern latitudes, comparable to the values observed by Voyager 1 (L. A. Sromovsky et al., 1981,Nature (London), 292, 698–702). In red light the reflectivity is constant to within a few percent for latitudes between 40°S and 60°N. Titan's phase coefficient between 22 and 96° phase angle averages about 0.014 magnitudes/degree in both colors—a value considerably greater than that observed at smaller phase from the Earth. Comparisons of the data with vertically homogeneous multiple-scattering models indicate that the single-scattering phase functions of the aerosols in both colors are rather flat at scattering angles between 80 and 150° with a small peak at larger scattering (i.e., small phase) angles. The models indicate that the phase integral, q, for Titan in both red and blue light is about 1.66 ± 0.1. Together with Younkin's value for the bolometric geometric albedo scaled to a radius of 2825 km, this implies an effective temperature in equilibrium with sunlight of 84 ± 2°K, in agreement with recent thermal measurements. The single-scattering polarizations produced by the particles at 90° scattering angle are quite large, >85% in blue light and >95% in red. A vertically homogeneous model in which the particles are assumed to scatter as spheres cannot simultaneously match the polarization observations in both colors for any refractive index. However, the observed polarizations are most sensitive to the particle properties near optical depth $\text{1}\text{2}$ in each color, and so models based on single scattering by spheres can be successful over a range of refractive indices if the size of the particles increases with depth and if the cross section of the particles increases sufficiently rapidly with decreasing wavelenght. For example, with n_r = 1.70, the polarization (and the photometry) are reproduced reasonably well in both colors when the area-weighted average radous of the particles, α, is given by α = (0.117 μm)(τ_red/0.5)^0.217. While this model does not reproduce the large increase in brightness from 129 to 160° phase observed by Voyager 1, the observed increase is determined by the properties of the particles in the top few hundredths of an optical depth. Thus the addition of a very thin layer of forward-scattering aerosols on top of the above model offers one way of satisfying both the Pioneer 11 and Voyager 1 observations. Of course, other models, using bimodal size distributions or scattering by nonspherical particles, may also be capable of reproducing these data.     

Exact solution of the equation of radiative transfer for semi-infinite plane-parallel isotropic scattering atmosphere with scattering albedo ω0 ≤ 1 by a method based on Laplace transform and Wiener-Hopf technique

By performing the one-sided Laplace transform on the scalar integro-differential equation for a semi-infinite plane-parallel isotropic scattering atmosphere with a scattering albedo ₀ 1, an integral equation for the emergent intensity has been derived. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity. The intensity at any optical depth for a positive scattering angle is also derived by inversion. The intensity at any optical depth for a negative scattering angle is also derived in terms of Cauchy's principal value using Plemelj's formulae.     

Photometry of Phoebe

Observatios of Phoebe (S9) in the V filter at small solar phase angles (0.2° to 1.2°) with the MIT SNAPSHOT CCD are presented. The value of Phoebe's sideral rotational period is refined to 9.282 ± 0.015hr. Assuming the Voyager-derived 110 km radius, Phoebe's observed mean opposition V magnitude of 16.176 ± 0.033 (extrapolated from small angles) corresponds to a geometric albedo of 0.084 ± 0.003. A strong opposition effect is indicated by the 0.180 ± 0.035 mag/deg solar phase coefficient observed at these small phase angles. The data are shown to be compatible with a phase function for C-type asteroids (K. Lumme and E. Bowell, 1981, Astron. J. 86, 1705–1721; K. Lumme, E. Bowell, and A. W. Harris, 1984, Bull. Amer. Astron. Soc. 16, 684), but give a poorer fit to the average asteroid phase relation of T. Gehrels and E.F. Tedesco (1979, Astron. J. 84, 1079–1087). Phoebe's rotational lightcurve in the V filter is roughly sinusoidal, with a 0.230-mag peak-to-peak amplitude and weaker higher order harmonics indicating primarily bimodal surface feature contrast. In addition to these photometric results, precise positions on 3 nights are given.     

2 Pallas: 1982 and 3 lightcurves and a new pole solution

Photoelectric lightcurves of asteroid 2 Pallas obtained in March 1982 and May 1983 display amplitudes of 0.04 and 0.10 magnitude respectively. The latter lightcurve shows that Pallas was at a V(1,0) magnitude of 4.51 ± 0.02 when it occulted 1 Vulpeculae on May 29 1983. A least-squares best fit to an amplitude-aspect relation for all available lightcurve observations of Pallas between 1951 and 1983 yields two solutions for its pole position: λ = 200, β = 40 and λ = 220, β = 15, where the uncertainty regions corresponds to an overall estimate of ± degrees. Use of phase angle bisector coordinates (A. W. Harris, J. W. Young, F. Scaltriti, and V. Zappalà (1984) Icarus57, 251–258) gave lower residuals than geocentric coordinates. The (220,15) pole position is favored since it is in very good agreement with an independent pole solution obtained by photometric astrometry (J. V. Lambert 1983 personal communication). This pole position implies that the latitude of the sub-Earth point at the time of the occultation was 22 degrees.     

Principe cosmologique,principe variationnel et theorie des groupes

The Friedmann universes are built on the cosmological principle only. The Robertson-Walker metric is common to all the theories based on a homogeneous, isotropic and irrotational universe. In the present work we examine ways of constructing a metric conformal with that of Robertson and Walker, by means of a variational principle which takes into account the cosmological principle as stated by Weinberg (1972), and based on the existence of orbits generated by a one-parameter group of diffeomorphisms of physical space. The application of the cosmological principle to variational methods allows the determination of first integrals which can characterize the physical properties of the Universe. To this end, we show that the Lagrangian of the Universe, considered as a mechanical system, can be chosen from the germs of functions, and that the form variations δq ⁱ are tangent vectors of the group orbits in a Riemannian manifold. Thus the variation of the action vanishes automatically. There appears a first integral of the Euler equations, which is δq ⁱ (?L/?q1 ⁱ ) = C ^te , and also the condition ?L/?t=0, which means the uniformity of time in a Lagrangian conservative system, and which is a direct application of the cosmological principle. These conditions allow the effective determination of a form invariant Lagrangian in the case of isometries. These conditions can be generalized to the case in which the group trajectories are a partition of physical space. Thus, it is possible to define a time from the group trajectories inV ₃: a second of the group time is a lengthm measured along any orbit θ _p of the group. Any pointp of the manifold can then be considered as the starting point of a bundle of orbits, along which the tangent vectors δq ⁱ could be calculated. From this group time, we can build a metric ds ² conformal to the initial ds ² and for which the orbits, which are geodesic, are orthogonal to the transitivity surfaces of the group in the manifold. This implies new statements of the cosmological principle:

1. At any point of space-time it is possible to construct a metric ds ² from the trajectories generated by a one-parameter group of diffeomorphisms ofV ₄.
2. Any two points of space-time can always be joined by means of trajectories of group.

The variational implications of these two principles are the appearance of spectral line shifts such as 1+z=F(p, t _p)/F(q, t_q), wherep andq are arbitrary points of the manifold, andF the transformation function which allows passage from one metric to another. The identification of group trajectories with physical trajectories depends on these two principles. The photon trajectories inV ₃ is an example of this identification. The trajectories of charged particles inV ₄ are another. Principle (b) stated an entropy condition; its application allows a new expression of action variation, this one leading to a general formulation of the shift of spectral lines by a variational method. If we choose the parabolic Friedmann universe as a realistic model, it is the expansion itself which is the generator of the diffeomorphisms allowing the establishment of a group structure in the manifold. The photons are carried away by expansion and do not resist it. The massive particles moderate this expansion locally, and their trajectories inV ₃ are the result of the reaction. In this scheme there is no theoretical difference between the treatment of particles of vanishing proper mass and massive particles. The Robertson-Walker metric fork=0 corresponds to a picture of the Universe which can be drawn by study of the movement of photons in physical space. Only the study of particles can allow the generalization of this scheme and, from this, make a real Universe which is not just a reflection of the physical properties of the photons alone.     

Photoelectric analysis of asteroid 216 Kleopatra: Implications for its shape

In some recent theoretical papers it has been suggested that gravitationally bound “rubble piles” in hydrostatic equilibrium possibly exist among the asteroids. For a higher-than-critical value of the angular momentum acquired by such a body, the instability phenomena can produce fission into a binary system. S. J. Weidenschilling [Icarus44, 807–809 (1980)] suggested that 216 Kleopatra may represent a binary asteroid, since it has a large light curve amplitude (1.3–1.4 mag). In this paper new observations of Kleopatra are presented, suggesting the equal plausibility of the single triaxial ellipsoid model. Namely, when phase and aspect effects are taken into account, the actual maximum amplitude is reduced to about 0.9 mag at 90° of aspect which is close to the value predicted by theory for the instability limit. Moreover, multiple-scattering effects [M. Poutanen, E. Bowell, and K. Lumme, Bull. Amer. Astron. Soc.13, 725 (1981)] can reduce the axial ratio a/b even more. If the single-body model is adopted, the density of Kleopatra should be on the order of 1.7 g/cm³. This low value seems reasonable for “rubble pile” models.     

Some properties of finite energy constant-α force-free magnetic fields in a half-space

Some useful properties of a finite energy, constant-α, force-free magnetic field B _α occupying a half-space D are presented. In particular:

1. Fourier and Green representations of B _α are obtained and used to derive conditions for the existence and uniqueness of a B _α having a given normal component B _z on the boundary ?D.
2. The asymptotic behaviour of B _α at infinity as well as stability results against changes in the boundary condition on ?D and in the value of α are established.
3. The energy of B _α is shown to be smaller than the energy of the open field having the same B _z on ?D, thus confirming an earlier conjecture (Aly, 1984).
4. B _α is proved to not be a Taylor-Heyvaerts-Priest state, in spite of the fact that its relative helicity H is finite and that it is the only solution of the Lagrange-Euler equation associated with the problem of minimizing the energy among all the fields having the same value of H and the same B _z on ?D.

     

Numerical simulation of cometary nuclei: III. Internal temperatures of cometary nuclei

A one-dimensional heat-diffusion model was used to calculate internal temperatures in cometary nuclei composed of either crystalline or amorphous ice, and for a range of orbits. It was found that the final central temperature, T_c, was a complex function of the comet's orbital semimajor axis, a, and eccentricity, e, as well as the functional form of the thermal conductivity. For cometary nuclei with identical thermal properties, T_c was found to decrease with eccentricity for a short-period orbit with a = 3 AU. For an intermediate-period orbit with a = 20 AU, T_c initially increased with eccentricity but then declined at large values of e for a crystalline ice nucleus, while for amorphous ice T_c increased monotonically. In addition, it was found that for conductivities of similar magnitude, crystalline ice (for which the conductivity varies inversely proportional to temperature) reached the final central temperature twice as fast as amorphouslike ice (for which the conductivity is proportional to temperature). T_c also depended on the magnitude of the conductivity. A four- to fivefold decrease in the conductivity resulted in a 3–4°K decrease in T_c at large eccentricities, while at small eccentricities T_c was only weakly dependent on the conductivity. Finally, the numerical results are compared to the analytical solutions of J. Klinger (1981, Icarus 47, 320–324) and C. P. McKay, S. W. Squyres, and R. T. Reynolds (1986, Icarus, 66, 625–629), and a numerical correction factor is derived for the McKay et al. expression for the central temperature.     

Coronal Mass Ejections as Expanding Force-Free Structures

We model solar coronal mass ejections (CMEs) as expanding force-free magnetic structures and find the self-similar dynamics of configurations with spatially constant ??, where J=?? B, in spherical and cylindrical geometries, expanding spheromaks and Lundquist fields, respectively. The field structures remain force-free, under the conventional non-relativistic assumption that the dynamical effects of the inductive electric fields can be neglected. While keeping the internal magnetic field structure of the stationary solutions, expansion leads to complicated internal velocities and rotation, caused by inductive electric fields. The structure depends only on overall radius R(t) and rate of expansion $\dot{R}(t)$ measured at a given moment, and thus is applicable to arbitrary expansion laws. In case of cylindrical Lundquist fields, magnetic flux conservation requires that both axial and radial expansion proceed with equal rates. In accordance with observations, the model predicts that the maximum magnetic field is reached before the spacecraft reaches the geometric center of a CME.     

A regularization of the three-body problem

Letr ₁,r ₂,r ₃ be arbitrary coordinates of the non-zero interacting mass-pointsm ₁,m ₂,m ₃ and define the distancesR ₁=|r ₁?r ₃|,R ₂=|r ₂?r ₃|,R=|r ₁?r ₂|. An eight-dimensional regularization of the general three-body problem is given which is based on Kustaanheimo-Stiefel regularization of a single binary and possesses the properties:

1. The equations of motion are regular for the two-body collisionsR ₁→0 orR ₂→0.
2. Provided thatR?R ₁ orR?R ₂, the equations of motion are numerically well behaved for close triple encounters.

Although the requirementR? min (R ₁,R ₂) may involve occasional transformations to physical variables in order to re-label the particles, all integrations are performed in regularized variables. Numerical comparisons with the standard Kustaanheimo-Stiefel regularization show that the new method gives improved accuracy per integration step at no extra computing time for a variety of examples. In addition, time reversal tests indicate that critical triple encounters may now be studied with confidence. The Hamiltonian formulation has been generalized to include the case of perturbed three-body motions and it is anticipated that this procedure will lead to further improvements ofN-body calculations.     

Infrared polarization of venus: Its periodic fluctuations and evidence for thin haze

Infrared polarimetry of Venus over the phase angles from 18 to 171° has been made extending previous measurements (S. Sato, K. Kawara, Y. Kobayashi, H. Okuda, K. Noguchi, T. Mukai, and S. Mukai (1980). Icarus43, 288) in both wavelength λ and phase angle θ. The results of polarization measurements at 2.25 μm ? λ ? 5.0 μm are (i) small positive and negative values at K(2.25 μm), (ii) a remarkable variation with λ in the CVF(2.2?4.2μm) filter region, (iii) a nearly smooth curve as a function of θ having a peak value of ～36% at θ ～ 90° at both 3.6 μm and L′(3.8 μm), and (iv) a decrease with increasing field of view at M(5.0 μm) due to the contamination of thermal emission from the dark crescent. Furthermore, at 3.6 μm and L′(3.8 μm), (v) higher values at the poles than at the equator and (vi) 4.5- to 5.9-day periodic fluctuations are also found. From a comparison with model calculations, the results confirm the existence of a thin haze layer consisting of submicron-size particles above the main clouds of Venus; e.g., its optical thickness is about 0.1 at λ ～ 0.94 μm. In addition, result (vi) could be explained by a variation of the optical thickness of the haze layer or that of the brightness temperature of the main clouds.     

Local Lorentz transformation and exact solution in f(T) gravity theories

A general tetrad fields, with an arbitrary function of radial coordinate, preserving spherical symmetry, is provided. Such tetrad is split into two matrices: The first matrix represents a Local Lorentz Transformation (LLT), which contains an arbitrary function. The second matrix represents a proper tetrad fields which satisfy the field equations of f(T) gravitational theory. This general tetrad is applied to the field equations of f(T). We derive a solution with one constant of integration to the resulting field equations of f(T). This solution gives a vanishing value of the scalar torsion. We calculate the energy associated with this solution to investigate what is the nature of the constant of integration.     

A model for the steady interaction of the solar wind with the Moon

The steady state interaction of the solar wind with the Moon is modeled as a uniform, magnetized, quasi-neutral, collisionless, hypersonic, and hyper-Alfvénic flow of an electronproton plasma past a perfectly ion absorbing, non-magnetized sphere. For the temperature of the electronsT much less than that of the ionsT _i , steady state equations are derived self-consistently from the Vlasov and Maxwell equations by taking advantage of the fact that the ion gyration ratio is small compared to the radius of the Moon, by employing an ordering which requires different scale lengths along the magnetic fieldB and center of mass velocity, and by expanding in a small parameter ? that measures the smallness of ?B terms compared to a dominant term retained. A partial numerical solution is presented and discussed for the limit in which ? is much less than β=(ion pressure/magnetic pressure). In addition, a simple technique is presented whereby the steady state equations can be approximately extended to cases in whichT?T _i for arbitrary ?/β.     

Phase curves of materials of Io: Interpretation in terms of Hapke's function

The photometric function developed by B. Hapke (1981,J. Geophys. Res.86, 3039–3054; 1984, Icarus59, 41–59) has been applied to near-opposition (α = 2–8°) disk-resolved phase curves for three color classes on Io, and the disk-integrated phase curve (α = 2–159°) of the satellite as a whole. Derived values of the Hapke compaction parameter h suggest that (1) a large percentage of the material on Io's surface has a porosity significantly greater significantly greater than 60%, supporting the estimate of high porosity made by D.L. Matson and D.B. Nash (1983,J. Geophys. Res.88, 4771–4783) and Nelson et al. (1984, Bull. Amer. Astron. Soc.16, 683–685; 1984,EOS65, 982–983); and (2) Average (“orange”) and Polar (“brown”) materials are significantly more porous than Bright (“white”) materials, a cottrast consistent with the Matson and Nash (1983) SO₂ cold trap model. The best-fit single particle phase function becomes more backscattering on moving from Polar to Average to Bright materials, with the surface of Io on average exhibiting significant backscattering comparable in magnitude to that of the lunar surface. For the color classes, and for Io as a whole, the degree of backscattering tends to increase toward longer wavelengths. The average macroscopic roughness of the Ionian surface, characterized by a mean slope angle of Ø ≃ 25°, is similar to that of other solid surface in the solar system. Consistency between observed limb darkening and that predicted by the Hapke model requires the presence of significant macroscopic roughness (Ø ≥ 20°) for the Average regions, but not necessarily for the Bright and Polar materials.