On the molecular scattering in the terrestrial atmosphere : An empirical formula for its calculation in the homosphere

A recent determination by D. R. Bates of the Rayleigh scattering cross section (σ_RS) for air from 0.2 to 1 μm leads to a simple empirical formula (λ in μm) $\text{σ}{}_{\mathrm{RS}}\text{=}\text{4.02 × 10}{}^{\mathrm{?28}}\text{λ}{}^{\mathrm{4+x}}\text{cm}{}^2$ where $\text{x =}\text{0.389λ + 0.09426}\text{λ ? 0.3228}$ for the spectral region 0.2 μm < λ < 0.55 μm ; the accuracy is within ±0.5%. From the visible at 0.55 μm to the infrared (i.r.) at 1 μm, the same accuracy can be obtained using a constant value, x = 0.04. The formula accounts for the degree of depolarization which varies with the wavelength according to the latest determination by Bates.     

Deuteron whistler and trans-equatorial propagation of the ion cyclotron whistler

New ion cyclotron whistlers which have the asymptotic frequency of one half the local proton gyrofrequency, $\text{G}{}_{\mathrm{p}}\text{2}$, and the minimum (or equatorial) proton gyrofrequency, G_pm, along the geomagnetic field line passing through the satellite have been found in the low-latitude topside ionosphere from the spectrum analysis of ISIS VLF electric field data received at Kashima, Japan. Ion cyclotron whistlers with asymptotic frequency of G_pm or $\text{G}{}_{\mathrm{pm}}\text{2}$ are observed only in the region of $\text{B}{}_{\mathrm{m}}\text{>}\text{B}\text{2}$ or rarely $\text{B}{}_{\mathrm{m}}\text{>}\text{B}\text{4}$, where B is the local magnetic field and B_m is the mini magnetic field along the geomagnetic field line passing through the satellite.The particles with one half the proton gyrofrequency may be the deuteron or alpha particle. Theoretical spectrograms of the electron whistlers (R-mode) and the ion cyclotron whistlers (L-mode) propagating along the geomagnetic field lines are computed for the appropriate distributions of the electron density and the ionic composition, and compared with the observed spectrograms.The result shows that the ion cyclotron whistler with the asymptotic frequency of $\text{G}{}_{\mathrm{p}}\text{2}$ is the deuteron whistler, and that the ion cyclotron whistlers with the asymptotic frequency of G_pm or $\text{G}{}_{\mathrm{pm}}\text{2}$ are caused by the trans-equatorial propagation of the proton or deuteron whistler from the other hemisphere.     

Low-frequency upstream wave as a probable source of low-latitude Pc 3–4 magnetic pulsations

Daytime Pc 3–4 pulsation activities observed at globally coordinated low-latitude stations [SGC (L = 1.8,λ = 118.0°W), EWA(1.15,158.1°W), ONW(1.3,141.5°E)] are evidently controlled by the cone angle θ_XB of the IMF observed at ISEE 3. Moreover, the Pc 3–4 frequencies ($\text{?}$) at the low latitudes and high latitude (COL; L = 5.6 and λ = 147.9°W) on the ground and that of compressional waves at geosynchronous orbit (GOES 2; L = 6.67 and λ = 106.7°W) are also correlated with the IMFmagnitude(B_IMF).The correlation of $\text{?}$ of the compressional Pc 3–4 waves at GOES 2 against B_IMF is higher than those of the Pc 3–4 pulsations at the globally coordinated ground stations, i.e., γ = 0.70 at GOES 2, and (0.36,0.60,0.66,0.54) at (COL, SGC, EWA, ONW), respectively. The standard deviation ($\text{σ}{}_{\mathrm{n}}\text{= ± Δ?}\text{mHz}$) of the observed frequencies from the form $\text{?}$ (mHz) = 6.0 × B_IMF (nT) is larger at the ground stations than at GOES 2, i.e., $\text{Δ? = ± 6.6}\text{mHz at}$GOES 2, and ±(13.9, 9.1, 10.7, 12.1) mHz at (COL, SGC, EWA, ONW), respectively. The correlations between the IMF magnitude B_IMF and Pc 3–4 frequencies at the low latitudes are higher than that at the high latitude on the ground, which can be interpreted by a “filtering action” of the magnetosphere for daytime Pc 3–4 magnetic pulsations. The scatter plots of pulsation frequency $\text{?}$ against the IMF magnitude B_IMF for the compressional Pc 3–4 waves at GOES 2 are restricted within the forms $\text{? = 4.5 × B}{}_{\mathrm{<mtext>IMF</mtext>}}\text{and}\text{? = 7.5 × B}{}_{\mathrm{<mtext>IMF</mtext>}}$. The frequency distribution is in excellent agreement with the speculation ($\text{<ω}{}_{\mathrm{sc}}\text{Ω}{}_{\mathrm{i}}\text{= 0.3 ～ 0.5}$) of the spacecraft frame frequency of the magnetosonic right-hand waves excited by the anomalous ion cyclotron resonance with reflected ion beams with V₆ = 650 ～ 1150 km s^?1 in the solar wind frame observed by the ISEE satellite in the Earth's foreshock. These observational results suggest that the magnetosonic right-handed waves excited by the reflected ion beams in the Earth's foreshock are convected through the magnetosheath to the magnetopause, transmitted into the magnetosphere without significant changes in spectra, and then couple with various HM waves in the Pc 3–4 frequency range at various locations in the magnetosphere.     

Investigation of shock waves and tangential discontinuities of the cosmic plasma by the radio interference technique

The possibility of investigation of the cosmic plasma dynamics by the radio interference technique based on a finite time of radio wave propagation between the sounding and responding stations is shown. By locating the sounding station on a spacecraft the greatest contribution to the phase difference ΔΦ(t)or the phase difference growth rate $\text{Δ?}$ between the sounding and response signals are caused by disturbances in close proximity to the spacecraft. This method permits interplanetary shock waves and tangential discontinuities to be registered and the velocities and plasma density changes on their fronts to be determined. By using experimental data of ΔΦ(t) or $\text{Δ?(t)}$ one can also obtain information about plasma concentration jump, location and motion of bow shock wave and magnetopause and plasmapause. Available experimental data about different disturbances of cosmic plasma were analysed and the requirements on frequency stability of spacecraft-borne and groundbased radio equipment were estimated to register those disturbances. In most cases relative stability 10^?11–10^?13 provided by present atomic frequency standards is sufficient.     

An ionospheric origin for Pi 1 micropulsations

Cosmic noise absorption pulsation events observed with fast response riometers at Macquarie Island in the southern auroral zone have almost always been accompanied by Pi 1 micropulsations. A cross-spectral analysis of fast response riometer data and vertical component induction magnetometer data for one such event showed that, after the low frequency components are removed, the absorption A(t) is better correlated with the absolute value of the field Z(t) than with the recorded quantity $\text{d}\text{Z}\text{dt}$. The peaks in Z(t) lag those in A(t) by one second while A(t) lags $\text{d}\text{Z}\text{d}\text{t}$ by abou second. Furthermore, many of the pulsations in Z(t) show a similar time asymmetry to that commonly observed in c.n.a. pulsations, viz. a more rapid onset time than decay time.These results are taken as evidence that the Pi 1 micropulsations observed from the ground during the recovery phase of an auroral substorm are brought about by fluctuations in the ionospheric currents which give rise to the magnetic bay, these fluctuations being due to conductivity changes resulting from particle precipitation. The lag between A(t) and Z > (t) is attributed to the self-inductance of the electrojet.     

Convex profiles from asteroid lightcurves

Integral geometry is used to solve a two-dimensional simplification of the three-dimensional lightcurve inversion problem, and a method is introduced for obtaining a convex profile, $\text{P}$, from asteroid lightcurve data. Whenever four ideal conditions are satisfied, $\text{P}$ is an estimator for the asteroid's “mean cross section,” $\text{C}$, a convex set defined as the average of all cross sections $\text{C}$ cut by planes a distance z above the asteroid's equatorial plane. $\text{C}$ is therefore a two-dimensional average of the asteroid's three-dimensional shape. The ideal conditions are that (A) each curve $\text{C}\text{(z)}$ is convex. (B) the asteroid's scattering law is uniform and geometric, (C) the astrocentric declinations of the Sun and Earth are zero, and (D) the solar phase angle θ ≠ 0. If Condition C is known to hold, the extend to which the lightcurve can be accounted for by a geometrically scattering convex object can be quantified in terms of an appropriate “goodness-of-fit” static. If the solar phase angle is zero, as for radar “lightcurve,” then (i) method yields a profile $\text{P}{}_{\mathrm{<mtext>s</mtext>}}$ the symmetrization $\text{C}{}_{\mathrm{<mtext>s</mtext>}}$ of $\text{C}$; (ii) Condition A need not hold and if it does not, then the inversion yields the symmetrization of the asteroid's mean convex hull; and (iii) Fourier analysis of the lightcurve can reveal violation of Condition B. Doppler-frequency resolution of radar echoes at several rotational phases adds information by constraining the convex hull $\text{H}{}_{\mathrm{<mtext>p</mtext>}}$ of the asteroid's (not necessarily convex) polar silhouette. Estimation of a convex profile from a photoelectric or radar lightcurve is a problem in weighted-least-squares optimization subject to inequality constraints. The solution uses a recursive quadratic programming algorithm to derive a Fourier parameterization for $\text{P}$ from the coefficients in the lightcurve's Fourier expansion. The method has been tested by inverting analytically generated lightcurves for geometrically scattering ellipsoids with semiaxes a ? b ? c, and the inversion yields $\text{P}\text{=}\text{P}{}_{\mathrm{<mtext>s</mtext>}}\text{?}\text{C}\text{=}\text{C}{}_{\mathrm{<mtext>s</mtext>}}\text{=}\text{H}{}_{\mathrm{<mtext>p</mtext>}}$ when the viewing geometry (Condition C) is close to ideal. For situations when the asteroid's pole direction is unknown, a test is offered of the hyphothesis that a given lightcurve can be due to a geometrically scattering ellipsoid with $\text{a}\text{c}\text{? ?}$, where ? is an priori upper bound on the maximum axis ratio. Convex profiles are presented for 15 Eunomia, 118 Peitho, 246 Asporina, 281 Lucretia. 790 Pretoria, 1685 Toro, and 1978 CA.     

Magnetospheric chorus: Occurrence patterns and normalized frequency

New characteristics of VLF chorus in the outer magnetosphere are reported. The study is based on more than 400 hours of broadband (0.3–12.5 kHz) data collected by the Stanford University/Stanford Research Institute VLF experiment on OGO 3 during 1966–1967. Bandlimited emissions constitute the dominant form of whistler-mode radiation in the region $\text{4? L? 10}$. Magnetospheric chorus occurs mainly from 0300 to 1500 LT, at higher L at noon than at dawn, and moves to lower L during geomagnetic disturbance, in accord with ground observations of VLF chorus. Occurrence is moderate near the equator, lower near 15°, and maximum at high latitudes (far down the field lines). The centre frequency ? of the chorus band varies as L^?3> and at low latitudes is closely related to the electron gyrofrequency on the dipole field line through the satellite. Based on the measured local gyrofrequency $\text{?}{}_{\mathrm{H}}$, the normalized frequency distribution of chorus observed within 10° of the dipole equator shows two peaks, at $\text{?}\text{?}{}_{\mathrm{H}}\text{? 0.53}$ and $\text{?}\text{?}{}_{\mathrm{H}}\text{? 0.34}$. This bimodal distribution is a persistent statistical feature of near equatorial chorus, independent of L, LT and K_p. However there is considerable variability in individual events, with chorus often observed above, below, and between these statistical peaks; in particular, it is not unusual for single emissions to cross $\text{?}\text{?}{}_{\mathrm{H}}\text{= 0.50}$. When two bands are simultaneously present individual emission elements only rarely show one-to-one correlation between bands. For low K_p the median bandwidth of the upper band, gap and lower band are all ～16% of their centre frequencies, independent of L; for higher K_p the bandwidth of the lower band increases. Bandwidth also increases with latitude beyond ~10°. Starting frequencies of narrowband emissions range throughout the band. The majority of the emissions rise in frequency at a rate between 0.2 and 2.0 kHz/sec; this rate increases with K_p and decreases with L. Falling tones are rarely observed at dipole latitudes <2.5°. The observations are interpreted in terms of whistler-mode propagation theory and a gyroresonant feedback interaction model. An exact expression is derived for the critical frequency, $\text{?}\text{?}{}_{\mathrm{H}}\text{? 0.5}$, at which the curvature of the refractive index surface vanishes at zero wave normal angle. Near this frequency rays with initial wave normal angles between 0° and ?20° are focused along the initial field line for thousands of km, enhancing the phase-bunching of incoming gyroresonant electrons. The upper peak in the bimodal normalized frequency distribution is attributed to this enhancement near the critical frequency, at latitudes of ~5°. Slightly below the critical frequency interference between modes with different ray velocities may contribute to the dip in the bimodal distribution. The lower peak may reflect a corresponding peak in the resonant electron distribution, or guiding in field-aligned density irregularities. The observations are consistent with gyroresonant generation of emissions near the equator, followed by spreading of the radiation over a range of L shells farther down the field lines.     

Planetary spectra for anisotropic scattering

We examine here some of the effects on planetary spectra that would be produced by departures from isotropic scattering. The phase function $\text{}\text{?}\text{gw(1 + a}\text{cos}\text{θ)}$ is the simplest departure to handle analytically and the only phase function, other than the isotropic one, that can be incorporated into a Chandrasekhar first approximation. This approach has the advantage of illustrating effects resulting from anisotropies while retaining the simplicity that yields analytic solutions.The curve of growth is the sine qua non of planetary spectroscopy. Our discussion emphasizes the difficulties and importance of ascertaining curves of growth as functions of observing geometry. A plea is made to observers to analyze their empirical curves of growth, whenever it seems feasible, in terms of coefficients of $\text{(1?}\text{}\text{?}\text{gw)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and $\text{(1?}\text{}\text{?}\text{gw)}$, which are the leading terms in radiative-transfer analysis.An algebraic solution to the two sets of anisotropic H functions is developed in the appendix. It is readily adaptable to programmable desk calculators and gives emergent intensities accurate to 0.3%, which is sufficient even for spectroscopic analysis.     

On the photodissociation of water vapour in the mesosphere

The photodissociation of water vapour in the mesosphere depends on the absorption of solar radiation in the region (175–200 nm) of the O₂ Schumann-Runge band system and also at H-Lyman alpha. The photodissociation products are OH + H, OH + H, O + 2H and H₂ + O at Lyman alpha; the percentages for these four channels are 70, 8, 12 and 10%, respectively, but OH + H is the only channel between 175 and 200 nm. Such proportions lead to a production of H atoms corresponding to practically the total photodissociation of H₂O, while the production of H₂ molecules is only 10% of the H₂O photodissociation by Lyman alpha.The photodissociation frequency (s^?1) at Lyman alpha can be expressed by a simple formula

$\text{J}{}_{\mathrm{<mtext>Ly</mtext><mtext>\alpha </mtext>}}\text{H}{}_2\text{O=4.5 ×10}{}^{\mathrm{?6}}\text{1+0.2}\text{F}{}_{10.7}\text{?65}\text{100}\text{exp}\text{[?4.4 ×10}{}^{\mathrm{?19}}\text{N}{}^{0.917}\text{]}$

where F_{10.7 cm} is the solar radioflux at 10.7 cm and N the total number of O₂ molecules (cm^?2), and when the following conventional value is accepted for the Lyman alpha solar irradiance at the top of the Earth's atmosphere ($\text{Δλ = 3.5}\text{A}\text{?}$) q_Lyα,∞ = 3 × 10¹¹ photons cm^?2 s^1?.The photodissociation frequency for the Schumann-Runge band region is also given for mesospheric conditions by a simple formula

$\text{J}{}_{\mathrm{<mtext>SRB</mtext>}}\text{(}\text{H}{}_2\text{O}\text{) = J}{}_{\mathrm{<mtext>SRB</mtext><mtext>,\infty </mtext>}}\text{(}\text{H}{}_2\text{O}\text{) exp [?10}{}^{\mathrm{?7}}\text{N}{}^{0.35}\text{]}$

where J_SRB,∞(H₂O) = 1.2 × 10^?6 and 1.4 × 10^?6 s^?1 for quiet and active sun conditions, respectively.The precision of both formulae is good, with an uncertainty less than 10%, but their accuracy depends on the accuracy of observational and experimental parameters such as the absolute solar irradiances, the variable transmittance of O₂ and the H₂O effective absorption cross sections. The various uncertainties are discussed. As an example, the absolute values deduced from the above formulae could be decreased by about 25-20% if the possible minimum values of the solar irradiances were used.     

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.     

Changes in the ionospheric profile and the faraday factor M with Kp

Using incoherent scatter data from Millstone Hill, we investigated the variations in the shape of the daytime, mid-latitude ionospheric electron density profile associated with changes in geomagnetic activity. The analysis performed was to deduce the dependence upon the 3-hr geomagnetic index Kp of h(N_m), h(0·7 N_m) above and below N_m, the plasma scale height H_T in the range 500–1000 km, and the ratio $\text{N(1000)}\text{N(h}{}_{\mathrm{m}}\text{)}$. The electron density data used spanned the solar maximum years 1968–1971. Daytime data from the period 1000 to 1600 LT were averaged separately for summer, winter and spring-fall. It is shown that the mean value $\text{M}$ of the factor M = B cos θ sec χ used by Titheridge (1972) to relate the Faraday rotation $\text{Ω}$ from a geostationary satellite to the total electron content N_N up to 2000 km is practically the same (to within 1–2 per cent) as the $\text{M}$ value used to relate the N_T and $\text{Ω}$ values both computed up to 1000 km. Taking advantage of this identity, we have used the linear relationship obtained between the ionospheric parameters and Kp to deduce the height at which $\text{M}$ should be evaluated as a function of Kp.     

Angular momentum drain: A mechanism for despinning asteroids

It is proposed that a new mechanism—angular momentum drain—helps account for the relatively slow rotation rates of intermediate-sized asteroids. Impact ejecta on a spinning body preferentially escape in the direction of rotation. This material systematically drains away spin angular momentum, leading to the counterintuitive result that collisions can reduce the spin of midsized objects. For an asteroid of mass M spinning at frequency ω, a mass loss δM correspond to an average decrease in rotation rate $\text{δω ≈}\text{ωδM}\text{M}$. A. W. Harris' (1979), Icarus40, 145–153) theory for the collisional evolution of asteroidal spins is significantly altered by inlusion of this effect. While the modified theory is still somewhat artificial, comparison of its predictions with the data of S. F. Dermott, A. W. Harris, and C. D. Murray (1984, Icarus57, 14–34) suggests that angular momentum drain is essential for understanding the statistics of asteroidal rotations.     

Modification of the phase methods for investigation of non-stationary processes in the ionosphere,magnetosphere and interplanetary space

The method of elimination of the nonrelativistic Doppler effect, suggested in 1960 by Badessa et al. allows the possibility of eliminating the influence of the first derivative of electron content $\text{?N}\text{?t}$ along the signal path. This gives in principal the possibility of measuring the second derivative of N and therefore of studying rapid small amplitude processes in the medium. Other possible reasons for the influence of the medium on the phase and frequency of signal stabilized with high accuracy ($\text{Δω}\text{ω}\text{?10}{}^{\mathrm{?12}}$) are investigated. For example, the frequency shift appears due to the motion of the ionized medium. That allows under favourable conditions the study of large scale unbalanced electric fields in the ionosphere, magnetosphere and interplanetary space.     

Phase function and polarization curve of interplanetary scatterers from zodiacal light photopolarimetry

Wide-angle ecliptic measurements of zodiacal light brightness (Z) and polarization (P) lead to fundamental results about optical properties of interplanetary scatterers, under a few reasonable assumptions (that they depend upon heliocentric distance by a r^?n law, and suffer no significant distortion of their scattering indicatrix between 0.5 and 2 a.u.): 1. The phase function σ(θ) is expressed (Equation 6) as a function of n and of (Z) data. 2. At the elongation ? = 90°, the derivative $\text{dZ}\text{d?}$ yields an absolute determination of the intensity $\text{T}$ scattered at right angles from the Sun by a single unit-volume of interplanetary medium (Equation 7). 3. The polarization degree $\text{P}$(θ) of the sunlight scattered by a single volume is derived (Equation 12) from n and from (Z + P) data. For two special values of the scattering angle θ, n vanishes in Equation (12), so that a fair knowledge of the polarization curve (Fig. 2) is reached prior to any assumption, or any forthcoming Jupiter-probe measure, about the value of n.Should n be provided by the Pioneers, then a thorough treatment of the whole problem of phase function and polarization curve can be performed by means of Equations (6) and (12) supplied with available zodiacal light photopolarimetric observations.     

The locating of hydromagnetic whistler sources and determination of their generating proton spectra

Results are given of the calculations of the group delay time propagating τ(ω, φ₀) of hydromagnetic whistlers, using outer ionospheric models closely resembling actual conditions. The τ(ω, φ₀) dependencies were compared with the experimental data of τ_exp(ω, φ₀) obtained from sonagrams. The sonagrams were recorded in the frequency range $\text{? ? (0.5?2.5) Hz}$ at observation points located at geomagnetic latitudes φ₀ = (53?66)° and in the vicinity of the geomagnetic poles. This investigation has led us to new and important conclusions.The wave packets (W.P.) forming hydromagnetic whistlers (H.W.) are mainly generated in the plasma regions at L = 3.5?4.0. This is not consistent with ideas already expressed in the literature that their generation region is $\text{L ? 3?10}$. The overwhelming majority of the τ_exp values differ considerably from the times at which wave packets would, in theory, propagate along the magnetic field lines corresponding to those of the geomagnetic latitudes φ₀ of the observation points. The second important fact is that the W.P. frequency ω is less than $\text{Ω}{}_{\mathrm{H}}$ everywhere along its propagation trajectory, including the apogee of the magnetic force line ($\text{Ω}{}_{\mathrm{H}}$ is the proton gyrofrequency). Proton flux spectra $\text{E ? (30?120)}$ keV, responsible for H.W. generation, were determined. Comparison of the Explorer-45 and OGO-3 measurements published in the literature, with our data, showed that the proton flux density energy responsible for the H.W. excitation $\text{N}{}_{\mathrm{p}}\text{(}\text{MV}{}_6{}^2\text{2}\text{) ? (5 × 10}{}^{\mathrm{?3}}\text{?10}{}^{\mathrm{?1}}\text{)}\text{H}{}_{\mathrm{a}}{}^2\text{8π}$ where H_a is the magnetic field force in the generation region of these W.P. The electron concentration is $\text{N}{}_{\mathrm{a}}\text{? (10}{}^2\text{?10}{}^3\text{) cm}{}^{\mathrm{?3}}$. The values given in the literature are $\text{N}{}_{\mathrm{a}}\text{? (10}{}^{\mathrm{?10}}\text{?10}{}^3\text{) cm}{}^{\mathrm{?3}}$. The e data considered also leads to the conclusion that the generating mechanism of the W.P. studied probably always co-exists with the mechanism of their amplification.     

Use of equivalency in estimating the historical size distribution of impact craters

A previous analysis of a stochastic model of lunar-type impact cratering is extended to utilize geological age data by defining a more general statistic $\text{Ω}{}_{\mathrm{i}}\text{(t)}$ to be the number of equivalent whole craters of original diameter d_i and age ≤t in an observational area A; each crater is taken to be equivalent to the fraction of its rim (or area) that is in A and not occluded by later craters. By integration of a new gain-loss differential equation, a generalization of the previous basic equation is obtained that relates the expected value $\text{ω}{}_{\mathrm{i}}\text{(t) = E[Ω}{}_{\mathrm{i}}\text{(t)]}$ to the process functions specifying the size distribution and flux of craters (primary or secondary) as they form. The results are specialized to the plausible case in which the cratered body can be subdivided into geological provinces of increasing ages t₁, t₂, …, t_i … and the size probability distribution can be approximated as constant within each of the periods t_i+1 - t_i. It is shown that use of the $\text{Ω}{}_{\mathrm{i}}$ permits, in principle, a reconstruction of the historical values of the process functions and correctly compensates for the effect of overlap by removing the false bias favoring large craters that results from the usual method of crater counting. Possible generalizations of the gain-loss equation are indicated.     

Mars: Subsurface properties from observed longitudinal variation of the 3.5-mm brightness temperature

Measurements at 3.5 mm of the disk-average brightness temperature of Mars during the 1978 opposition can be represented by

$\text{T}{}_{\mathrm{B}}\text{(Mars, 3 5 mm, Jan/Feb 1978)}\text{=}\textcolor{red}{}$

(The errors cited are from the internal scatter; the estimated absolute calibration uncertainty is 3%.) This longitudinal variation must be taken into account if Mars is to be used as a calibration source at millimeter wavelengths. The total range of the 3.5-mm variation is three to four times larger than both the 2.8-cm and 20-μm variations. This unexpected result can possibly be explained by subsurface scattering from rocks ?1.5-cm radius.     

Electric conductivities,electric fields and auroral particle energy injection rate in the auroral ionosphere and their empirical relations to the horizontal magnetic disturbances

Recent progress in modeling ionospheric current systems requires global conductivity models which can reflect substorm conditions on an instantaneous basis. For this purpose, empirical relations of the North-South component (ΔH) of the magnetic disturbance field observed at College with the Pedersen (Σ_p) and Hall (Σ_H) conductivities deduced from the Chatanika radar data and their ratio ($\text{Σ}{}_{\mathrm{H}}\text{Σ}{}_{\mathrm{p}}$) are examined. These empirical formulas allow us to construct approximate distribution patterns of Σ_p and Σ_>H over the entire polar region on the basis of the distribution of ΔH at given instants by devising an appropriate weighting function for both the polar cap and the subauroral region. The global conductivity distributions thus obtained are compared with those employed by Kamide et al. (1981) and Spiro et al. (1982). The comparisons show that the gross features are similar among them. In addition, we also examine the relationship of ΔH with the North-South component of the electric field with the particle energy injection rate (u_A) estimated from the Chatanika radar data. Based on the empirical relation between Σ_H and u_A the global distribution of the latter over the entire polar region at particular instants can also be obtained.     

Zodiacal light gathered along the line of sight: The vicinity of the terrestrial orbit studied with photopolarimetry and with Doppler spectrometry

The expression for the zodiacal brightness integral is especially simple if the integrand contains the ‘directional scattering coefficient’, $\text{D}$, (a.u.^?1), or equivalently the scattering cross-section per unit-volume. The two intersections of the terrestrial orbit with a line of sight lying in the ecliptic offer the possibility of isolating the contribution of the chord, with a conservative assumption of steadiness, but without the controversial assumption of a homogeneous zodiacal cloud. The zodiacal brightnesses between 60 and 120° elongation can be used to derive $\text{D}$₀ and $\text{D}$, the value of $\text{D}$ and its heliocentric radial derivative, both at 1 a.u. and at a scattering angle of 90°. A polarimetric treatment leads to the local polarization degree, $\text{P}$₀, and to its heliocentric derivative, $\text{P}$. Applied to all three available observational sources, this method invalidates the assumption of homogeneity, leading to a rather high relative gradient $\text{P}\text{P}{}_0$ near 1 a.u. (? 12, ? 16 or ? 24%, according to the source, as the Sun's distance decreases from 1.0 to 0.9 a.u.).The method is extended to Doppler spectrometry, taking advantage of the two equal projections on the line of sight of the Earth's velocity vector. The brightness Z₀ and the Dopplershift Δλ₀ observed at 90° elongation, together with the derivatives w.r.t. elongation ε, of the brightness, $\text{Z}\text{?}$ and of the Dopplershift, Δλ, can be used to retrieve the mean orbital velocity, v, of the interplanetary scatterers in the region of the terrestrial orbit. The two most reliable observational sources lead, with fair agreement, to a relative excess $\text{(}\text{v ? V)}\text{V}$, over the terrestrial velocity, of the order of + 25%.     

Particle size distributions in Saturn's rings from voyager 1 radio occultation

Observations of microwave opacity τ[λ] and near forward scatter from Saturn's rings at wavelengths λ of 3.6 and 13 cm from the Voyager 1 ring occultation experiment contain information regarding ring particle sizes in the range of about a = 0.01 to 15 m radius. The opacity measurements τ[3.6] and τ[13] are sufficient to constrain the scale factor n(a₀) and index q of a power law incremental size distribution n(a) = n(a₀)[a₀/a]^q, assuming known minimum and maximum sizes and a many-particle-thick model. The families of such distributions are highly convergent in the centimeter-size range. Forward scatter at 3.6 cm can be used to solve for a general distribution over the radius range $\text{1 ? a ? 15}\text{m}$ by integral inversion and inverse scattering methods, again assuming a many-particle-thick slab-type radiative transfer model. Distributions n(a) valid over $\text{0.01 ? a ? 15}\text{m}$ are obtained by combining the results from the two types of measurements above. Mass distributions may be computed directly from n(a). Such distributions, partly measured and partly synthesized, have been obtained for four features in the ring system centered at 1.35, 1.51, 2.01, and 2.12 Saturn radii (R_s). The size and mass distributions both cut off sharply at a ? 4–5 m; the mass distribution peaks over the narrow size range $\text{3 ? a ? 4}\text{m}$ for all four locations. No single power law distribution is consistent with the data over the entire interval $\text{0.01 ? a ? 5}\text{m}$, although a power law-type model is consistent with the data over a limited size range of $\text{0.01 ? a ? 1}\text{m}$, where the indices q = 3.4 and 3.3 are obtained from the slab model for the features located at 1.51 and 2.01 R_s. The fractional contribution of the suprameter particles to the microwave opacity in each feature appears to be about $\text{1}\text{3}\text{,}\text{1}\text{3}\text{,}\text{2}\text{3}\text{,}\text{and}\text{1}$, respectively, with the fraction at 2.12 R_s being the least certain. The cumulative surface mass per unit area obtained for the classical slab model is approximately 11, 16, 41, and 132 g/cm² for the four features, respectively, if the particles are solid H₂O ice. Both the fractional opacity and the mass density estimates represent upper bounds implied by the assumption of a uniformly mixed set of particles in a many-particle-thick vertical profile; lower estimates would result if the rings were assumed to be nearly a monolayer or if the vertical distribution of particles were size dependent.