Self-similar space-times

An exact solution of the transport equation in radiative transfer for an axially symmetric Rayleigh scattering problem in semi-infinite planetary atmosphere both for emergent intensity and intensity at any optical depth has been derived with the help of the Laplace transform and the Wiener-Hopf technique, and by use of the constancy of net flux. Chandrasekhar's results for emergent intensity have been verified. New expressions for theH _l andH _r functions have been obtained.     

Long term modulation of cosmic rays and inferable electromagnetic state in solar modulating region

It is demonstrated that the long term variation in cosmic ray intensity I(t) can be described by an integral equation,

$\text{I(t)=I}{}_{\mathrm{\infty }}\text{?}\text{∫}\text{0}\text{∞}\text{f(τ)S(t?τ)}\text{d}\text{τ}$

, which is derived from a generalization of Simpson's coasting solar wind model. A source function S(t?τ) is given by some appropriate solar activity index at a time t?τ(τ ? 0) and the characteristic functionf(τ)(?0 forτ ? 0) expresses the time dependence of the efficiency of the intensity depression due to solar disturbances represented by S(t ?τ) when the disturbances generated at the solar surface propagate through the modulating region with the solar wind. It is demonstrated further that the equation can be derived from the general diffusion-convection theory on some assumptions, and as a result, the source and characteristic functions can be related to diffusion coefficient and its transition in space. Assuming the sunspot number R (or two activity indices including R) as the source function, the characteristic function f(τ) [or f(τ)'s] is obtained with data of the cosmic ray intensity extended over several decades. Based on the theory, one can obtain from f(τ) the following physical quantities in space, such as the transition and life time of solar disturbances, the boundary of the modulating region, and the radial and time dependences of the diffusion coefficient, radial density gradient and modulated intensity of cosmic rays. Results deduced from the present analysis are consistent with those obtained directly or indirectly by space observations.     

Correlations, spectra, and instability of phase-space density fluctuations in open-cluster models

The dynamical evolution of six open star cluster models is analyzed using the correlation and spectral analysis of phase-space density fluctuations. The two-time and mutual correlation functions are computed for the fluctuations of the phase-space density of cluster models. The data for two-time and two-particle correlations are used to determine the correlation time for phase-space density fluctuations ((0.1–1) τ _v.r., where τ _v.r. is the violent relaxation time of the model) and the average phase velocities of the propagation of such fluctuations in cluster models. These velocities are 2–20 times smaller than the root mean square velocities of the stars in the cluster core. The power spectra and dispersion curves of phase-space density fluctuations are computed using the Fourier transform of mutual correlation functions. The results confirm the presence of known unstable phase-space density fluctuations due to homologous fluctuations of the cluster cores. The models are found to exhibit a number of new unstable phase-space density fluctuations (up to 32–41 pairs of fluctuations with different complex conjugate frequencies in each model; the e-folding time of the amplitude growth of such fluctuations is (0.4–10) τ _v.r. and their phases are distributed rather uniformly). Astrophysical applications of the obtained results (irregular structure of open star clusters, formation and decay of quasi-stationary states in such clusters) are discussed.     

The wilson effect and the transparency of sunspot models

Hydrostatic models of sunspot penumbra and umbra are evaluated using Bode's tables of monochromatic absorption coefficients andT-τ-relations given by Makita and Morimoto (1960) and by Zwaan (1965). These models are placed side by side to simulate a complete sunspot corresponding to an area of 480×10^?6 of a hemisphere. Intensity profiles are evaluated for aspect angles up to 85° and compared to observations. The primary aim was to study the influence of spot transparency, which is closely related to the gas-pressure, on the Wilson-effect and on other changes in the intensity profile that appear close to the solar limb. The gas-pressures at the zero-level in the geometrical depth (z=0) corresponding to optical depth, τ=10^?3, both in the umbra,P ₀ ^u , and in the penumbra,P ₀ ^p , appear as adjustable parameters. When curvature is taken into account, the Wilson-effect cannot be reproduced without depressing the zero-point in the geometrical scale in the umbra relative to the same layer in the photosphere. A depression of 400 km will give a reasonably good fit for the Wilson-effect providedP ₀ ^u <P ₀ ^P <P ₀ ^Ph . The model we found to give the best fit is based on Makita and Morimoto'sT-τ-relations withP ₀ ^P =3200 andP ₀ ^u =800. We have here chosen an umbra pressure that gives a small limb-side intensity peak at the penumbra border, assuming that the bright points described by Bray and Loughhead (1964) may be interpreted in terms of a transparency effect. Other parameters measured by Wilson and Cannon (1968) are evaluated, and for some a good agreement was obtained, while for others only a qualitative effect in the same direction could be found. Surfaces along which the optical path is constant (isodiaphanous surfaces) are functions of aspect angle and well suited for visualizing the transparency in spots. It is shown how for a wide range of models the isodiaphanous surfaces get asymmetric close to the limb. This has consequences for the interpretation of the Evershed-effect. In fact, under certain conditions, a ‘masking’ effect may take place because the greater transparency in the penumbra will allow observations of a deep laying flow, which will not be visible through the more opaque photosphere. Due to the asymmetry this effect is different on the limb side and the center side. We also found the spot to show an apparent displacement away from the limb, which at a heliocentric distance of 85° amounts to about one second of arc. Intensity profiles in the near infrared at 8206 Å and at 16482 Å are evaluated, and the importance of observations in these spectral regions is emphasized.     

Formation of terrestrial planets in a dissipating gas disk with Jupiter and Saturn

We have performed N-body simulations on final accretion stage of terrestrial planets, including the eccentricity and inclination damping effect due to tidal interaction with a gas disk. We investigated the dependence on a depletion time scale of the disk, and the effect of secular perturbations by Jupiter and Saturn. In the final stage, terrestrial planets are formed through coagulation of protoplanets of about the size of Mars. They would collide and grow in a decaying gas disk. Kominami and Ida [Icarus 157 (2002) 43-56] showed that it is plausible that Earth-sized, low-eccentricity planets are formed in a mostly depleted gas disk. In this paper, we investigate the formation of planets in a decaying gas disk with various depletion time scales, assuming disk surface density of gas component decays exponentially with time scale of τ_gas. Fifteen protoplanets with are initially distributed in the terrestrial planet regions. We found that Earth-sized planets with low eccentricities are formed, independent of initial gas surface density, when the condition (τ_cross+τ_growth)/2?τ_gas?τ_cross is satisfied, where τ_cross is the time scale for initial protoplanets to start orbit crossing in a gas-free case and τ_growth is the time scale for Earth-sized planets to accrete during the orbit crossing stage. In the cases satisfying the above condition, the final masses and eccentricities of the largest planets are consistent with those of Earth and Venus. However, four or five protoplanets with the initial mass remain. In the final stage of terrestrial planetary formation, it is likely that Jupiter and Saturn have already been formed. When Jupiter and Saturn are included, their secular perturbations pump up eccentricities of protoplanets and tend to reduce the number of final planets in the terrestrial planet regions. However, we found that the reduction is not significant. The perturbations also shorten τ_cross. If the eccentricities of Jupiter and Saturn are comparable to or larger than present values (∼0.05), τ_cross become too short to satisfy the above condition. As a result, eccentricities of the planets cannot be damped to the observed value of Earth and Venus. Hence, for the formation of terrestrial planets, it is preferable that the secular perturbations from Jupiter and Saturn do not have significant effect upon the evolution. Such situation may be reproduced by Jupiter and Saturn not being fully grown, or their eccentricities being smaller than the present values during the terrestrial planets' formation. However, in such cases, we need some other mechanism to eliminate the problem that numerous Mars-sized planets remain uncollided.     

Smoothing of force functions and the fluctuation spectra of an open star cluster model

We perform the correlation and spectral analysis of phase-space density and potential fluctuations in a model of an open star cluster for various values of the smoothing parameter ? of the force functions in the equations of motion of cluster stars, and compute the mutual correlation functions for the fluctuations of potential U and phase-space density f of the cluster model at different clustercentric distances. We use the Fourier transform of the mutual correlation functions to compute the power spectra and dispersion curves of the potential and phase-space density fluctuations. The spectrum of potential fluctuations proves to be less complex than that of phase-space density fluctuations. The most powerful potential fluctuations are associated with phase-space density fluctuations, and their spectrum lies in the domain of low frequencies ν < 3/τ _v.r.; at intermediate and high frequencies (ν > 3/τ _v.r.), the contribution of potential fluctuations to those of the phase-space density is small or equal to zero (here τ _v.r. is the violent relaxation time scale of the cluster). We find a number of unstable potential fluctuations in the core of the cluster model (up to 30 pairs of fluctuations with different complex conjugate frequencies). We also find and analyze the dependences of the spectra and dispersion curves of phase-space density and potential fluctuations on ?. We find a “repeatability” (significant correlation) of the spectra at some values of parameter ?. The form of the dispersion curve is unstable against small variations of ?. We discuss the astrophysical applications of our results: the break-up in the cluster core of the phase-space density wave running from the cluster periphery toward its center into several waves with frequencies commensurable to that of the external (tidal) influence; emission and reflection of phase-space and potential waves near the cluster core boundary; possible wavelength and phase discretization of the phase-space and potential waves in the cluster model.     

The total photospheric motion field

Brief consideration is given to the conception of the total photospheric motion field. A synthesis of the most thorough investigations is made and the radial (V ^rad) and tangential (V ^tg) components of the velocity amplitude of the total photospheric motion field are deduced. At depth logτ₅ = ?3.0 V ^rad and V ^tg have average values of 1.2 and 1.7 km s^?1 respectively. They increase smoothly with depth and reach their maximum values of V ^rad=3.0, V ^tg=3.4 km s^?1 at depths logτ₅= ?0.2 and logτ₅ = +0.4 respectively. In the deep photospheric layers both components seem to decrease with depth.     

A semi-analytic approach to angular momentum transport in stellar radiative interiors

A novel semi-analytic spectral method, based on eigenfunction expansions, is applied to model the angular momentum transport in stellar radiative interiors. The advantages of our approach are shown by applying it to a spin-down model for a 1M _⊙ main-sequence star. The evolution of the coupling between core and envelope is investigated for different values of the viscosity and different geometries and intensities of the poloidal field. We suggest that a quadrupolar poloidal field may explain the short coupling timescale (τ _c～10 Myr) needed to reproduce the observed rotational evolution of fast rotators on the zero age main sequence, while a dipolar geometry is indicated in the case of slow rotators (τ _c～100 Myr). Our method provides a rigorous analytic treatment of a classic MHD problem and allows us to explore the influence of various parameters on the rotational history of radiative interiors.     

Probable causes of long-period variations in the Uranian geometrical albedo

We use the technique developed in our previous studies to determine the ratios of optical depth components τ _a /τ _R and τ_κ/τ _S on the basis of the observational data on the Uranian geometrical albedo for the years 1981, 1993, and 1995. Here τ _a and τ _R are the aerosol and gas scattering components, τ _S = τ _a + τ _R , and τ_κ is the absorption component of the effective optical depth at which the intensity of the diffusely reflected radiation is formed. The ratios turn out to be different for different years. This phenomenon is caused by the horizontal inhomogeneity of the aerosol component distribution over the Uranian disk.     

Evolution of temperature in granule and intergranular space

The temporal evolution of temperature in a dissolving granule and in an adjacent intergranular space is presented. The semi‐empirical evolutionary models have been calculated using an inversion method applied to 4‐min time series of Stokes I spectral line profiles. The models are presented in the form of the functional dependence of temperature T(log τ₅, t) on optical depth τ₅ at 500 nm and time t. The observed disappearance of the granule is accompanied with overall cooling of the granular photosphere. Temperature changes greater than 100 K have been found in deeper (log τ₅ ≥ 0) and upper layers (log τ₅ ≤ –2) whereas the intermediate layers are thermally stable. The intergranular space, which is 2 arcsec off the granule, keeps the temperature structure of the layers from log τ₅ = 0.5 to log τ₅ = –2 without global evolutionary changes except short‐term and spatially confined heating. Finally, the significant temperature changes in the upper layers (log τ₅ ≤ 2.5) observed during the time interval of 4 min are found to be typical for the granular and intergranular photosphere.     

Exact solution of the transport equation for radiative transfer with scattering albedo ωO < 1 using the Laplace transform and the Wiener-Hopf technique and an expression ofH-function

We have considered the transport equation for radiative transfer to a problem in semi-infinite non-conservative atmosphere with no incident radiation and scattering albedo ₀ < 1. Usint the Laplace transform and the Wiener-Hopf technique, we have determined the emergent intensity and the intensity at any optical depth. We have obtained theH-function of Dasgupta (1977) by equating the emergent intensity with the intensity at zero optical depth.     

A simple expression for vertical convective fluxes in planetary atmospheres

We explore the vertical convective flux F_c in a radiative-convective grey atmosphere. An expression of the form F_c=F_sτ_o/(C+Dτ_o) appears useful, where F_s is the shortwave flux absorbed at the base of an atmosphere with longwave optical depth τ_o and C and D are constants. We find excellent agreement with an idealized grey radiative-convective model with no shortwave absorption for D=1 and C=1∼2 depending on the surface-atmosphere temperature contrast and on the imposed critical lapse rate. Where shortwave absorption is correlated with longwave opacity, as in the atmospheres of Earth and Titan, C=2, D=2 provides an excellent fit, validated against the present terrestrial situation and the results of a nongrey model of Titan's strongly antigreenhouse atmosphere under a wide range of conditions. The expression may be useful for studying the energetics of planetary climates through time where there is insufficient data to constrain more elaborate models.     

Atmospheric waves (≈ 10 min–30 days) in the mesosphere and thermosphere at saskatoon (52°N, 107°W), October 1978–September 1979

Partial reflection radiowave wind observations have been made continuously at Saskatoon for 1 year (September 1978–August 1979). The winds have been obtained in real-time from a twin micro-processor system, that produces profiles (32 heights from 49 to 142km) at 5-min intervals: daytime/night-time profiles normally involve values from $\text{60}\text{70}$ to 120 km.These data have been analyzed to give the daily mean wind and the amplitudes and phases of planetary, tidal (24, 12 h) and internal gravity (I.G.) waves (0.20 ? τ ? 6h). There are oscillations in the mean daily wind, with periods of 2–30 days: the disturbance from 60 to 100 km caused by the stratwarm of February and March is documented. The characteristics of the semi-diurnal tide undergo significant changes in vertical wavelength and time of maxima between summer months (May–August) and winter months (November–February): during summer the S²₂ mode (λ_z ? 150km) seems to dominate, and in other months a mixture of modes (λ_z ? 50km) exist. The diurnal tide is less regular, but during winter and spring λ_z ? 30–60 km, and for other months is very large, suggesting the S_?1¹ mode. Energy densities and scale heights (h₀) are given, and the relative magnitudes of these for the various waves are discussed with respect to height and season: there is a general trend for the growth of the wave amplitude with height (shown by h₀) to increase in the order planetary, tidal and I.G. waves (h₀, ～3–15 km): largest values are in winter and spring. Wave energies are largest in winter, decrease during spring and increase throughout late summer and fall.     

Scattering in the Atmosphere of Venus. I. Line profiles and curves of growth for isotropic scattering

We have computed line profiles and curves of growth for both reflected and transmitted radiation for typical lines in CO₂ bands (in the photographic infrared) which occur in the spectrum of Venus. In our model the pressure variation with altitude was considered and the base of the cloud deck was set at the 2 bar level. The temperature was held constant at 250K and a Voigt profile was used for the lineshape. We also assumed that the scale height of the cloud particles was equal to the scale height of the gas. The calculations were made for four values of the scattering optical thickness (τ_c = 0.1, 1.0, 10, and 100) using a continuum single scattering albedo $\text{ω}{}_{\mathrm{<mtext>c</mtext>}}\text{= 0.9975}$ (which gives a Bond albedo of 0.896 for τ_c = 100, the value observed for Venus at these wavelengths). Curves of growth are also presented for reflected radiation which has been averaged over the visible disk for three values of the Venus phase angle (0, 86, and 166°).     

Growth rate and decay of magnetospheric ring current

The rate of energy input to the ring current is studied as a function of solar wind parameters. The ring current dissipation rate is also examined. The decay constant τ in the main phase of a storm has been shown to be independent of its intensity and to equal (4 ± 2) h. In the recovery phase τ rises with increasing storm intensity.     

The structure and dynamics of young star clusters: King 16, NGC 1931, NGC 637 and NGC 189

In this paper, using 2MASS photometry, we study the structural and dynamical properties of four young star clusters viz. King 16, NGC 1931, NGC 637 and NGC 189. For the clusters King 16, NGC 1931, NGC 637 and NGC 189, we obtain the limiting radii of 7′, 12′, 6′ and 5′ which correspond to linear radii of 3.6 pc, 8.85 pc, 3.96 pc and 2.8 pc respectively. The reddening values E(B?V) obtained for the clusters are 0.85, 0.65–0.85, 0.6 and 0.53 and their true distances are 1786 pc, 3062 pc, 2270 pc and 912 pc respectively. Ages of the clusters are 6 Myr, 4 Myr, 4 Myr and 10 Myr respectively. We compare their structures, luminosity functions and mass functions (φ(M)=dN/dM ∝ M ^?(1+χ)) to the parameter τ=t _age /t _relax to study the star formation process and the dynamical evolution of these clusters. We find that, for our sample, mass seggregation is observed in clusters or their cores only when the ages of the clusters are comparable to their relaxation times (τ≥1). These results suggest mass seggregation due to dynamical effects. The values of χ, which characterize the overall mass functions for the clusters are 0.96±0.11, 1.16±0.18, 0.55±0.14 and 0.66±0.31 respectively. The change in χ as a function of radius is a good indicator of the dynamical state of clusters.     

Saturn's rings. I. Optical thickness of rings A,B, D and structure of ring B

A photometric study of high-resolution (～0″.3) plates of Saturn taken at the Lowell Observatory in 1943 and 1945 is presented. N-S scans were taken over both the planet and rings. The excess brightness due to the planet seen through the rings is found by taking the difference between the central meridian (CM) scans and scans displaced by 5″.7. Adopting a value for the albedo of the planet, it is possible to obtain the optical thickness, τ_CM(r). In particular, for the regions of maximum brightness in rings A and B, we find τ_CM(I_{A max}) = 0.38 ± 0.11 and τ_CM(I_{B max}) = 0.61 ± 0.11. Observations by Barnard made in 1890 show evidence of ring D, recently discovered by Guerin (1969). The value for the optical thickness of this ring is τ_D(I_{D max}) = 0.03 ± 0.01. Ring B exhibits a pronounced (7–10%) decrease in brightness from the extremity of the major axis to the CM. After considering several possible explanations, we conclude that the ring particles are nonspherical and are in synchronous rotation around the planet with their long axis toward it. The mean value for the ratio of major to minor axis for the particles at 15″ is $\text{(a/b)}\text{? 1.08}$. Because of the shape and orientation of the particles, the optical thickness at the extremity of the major axis and at the CM are different for any saturnicentric latitude B ≠ 90°. Under these circumstances, only a minimum value for τ at the extremity can be derived.     

Pitch angle dependence of the charge-exchange lifetime of ring current ions

Previous work has parameterized the pitch angle dependence of the charge-exchange lifetime τ of ring current ions in terms of γ, the power of the cosine of the mirror latitude λ_m of the particles, such that $\text{τ(λ}{}_{\mathrm{m}}\text{)}\text{τ(0) ≌}\text{cos}{}^{\mathrm{\gamma }}\text{λ}{}_{\mathrm{m}}$ at given L. Using the atomic hydrogen density model of Johnson and Fish, previous authors have suggested values of γ = 5 or 6. We here evaluate γ as a function of λ_m and L using the more recent Chamberlain density models, and show that γ = 3?4 is more appropriate over most of the pitch angle and L range. Consequently, ion distributions in the ring current decay phase are expected to become rather less anisotropic in pitch angle due to chargeexchange than previously believed. We have also investigated the use of several other simple approximate analytic forms for $\text{τ(λ}{}_{\mathrm{m}}\text{)}\text{τ(0)}$, one of which gives far better agreement with the numerical results than the cos^γ λ_m, variation, and should hence be used in future studies.     

Self-gravity wake parameters in Saturn’s A and B rings

An increasing body of evidence shows that, at the sub-km level, Saturn’s main A and B rings are dominated by an ever-changing pattern of elongated, canted structures known as self-gravity wakes. Best known for causing azimuthal variations in the rings’ reflectivity, these structures also have a profound influence on how the transmission of the rings varies with both longitude and opening angle, B (Colwell et al. [2006] Geophys. Res. Lett. 33, 7201; Colwell et al. [2007] Icarus 190, 127-144; Hedman et al. [2007] Astron. J. 133, 2624-2629). We use data from three stellar occultations observed by Cassini’s Visual and Infrared Mapping Spectrometer (VIMS) to measure the transmission of the rings as a function of B, when viewed parallel to the wakes. These data are used to constrain properties of the self-gravity wakes as a function of radius across the A and B rings: specifically the fractional width of the gaps between the wakes, G/λ, and the average normal optical depth in the gaps, τ_G. We find that the overall normal optical depth of the rings, τ_n is primarily controlled by G/λ, which varies between <0.05 and ∼0.70 in the A and B rings. The gaps, however, are not completely empty, being filled by material — possibly cm-sized ring particles — with an average normal optical depth which varies from 0.12 to ∼0.4. In addition to regional variations, local variations in τ_G are seen in the regular structure which dominates the inner B ring, and in the environs of strong density waves in the A ring. The same model applied to the lower optical depth Cassini Division reveals very little evidence of self-gravity wakes, except where τ_n exceeds ∼0.25.     

Effects of temperature and velocity gradients on doppler widths

We have investigated how the gradients of temperature and expansion velocities will change the emergent profiles from an extended medium in spherical symmetry. Variation of the source function and expansion velocities are assumed. The following variations of temperature are employed:

1. T(r) ; T₀ (isothermal case)
2. T(r) ; T₀(r/r₀)^1/2
3. T(r) ; T₀(r/r₀)^-1
4. T(r) ; T₀(r/r₀)^-2
5. T(r) ; T₀(r/r₀)^-3

The profiles calculated present an interesting feature of broadening.