An estimator of the underlying size distribution of overlapping impact-craters

The stochastic model of lunar type impact-crater formation which assumes (a) random impacts, (b) circular craters, each obliterating any portions of earlier craters lying within, and (c) a probability P_i(t) that a newly formed crater (primary or secondary) has an area a_i is analyzed to develop a method of estimating P_i from the final overlapping pattern. It is found that if each crater is weighted by the fraction of the rim which is visible and which lies in an observation area A, then the expected value of the weighted sum $\text{Ω}{}_{\mathrm{i}}$ of craters of area a_i is simply proportional to P_i for any degree of coverage under several conditions, including (a) constant P_i for all i, and (b) P_i stepping from a constant early value to zero (for some i's) with otherwise arbitrary bombardment. Furthermore, in the general case, the expected value of the contribution $\text{ΔΩ}{}_{\mathrm{i}}\text{(t}{}_0\text{)}\text{to}\text{Ω}{}_{\mathrm{i}}$ produced during t₀ ± Δt/2 is found to be proportional to P_i(t₀). Thus measurement of $\text{Ω}{}_{\mathrm{i}}$ in the first two cases, or of $\text{ΔΩ}{}_{\mathrm{i}}$ if crater age data is available in the last case, provides an estimate of the desired P_i. Therefore the $\text{Ω}{}_{\mathrm{i}}$ introduce the correct weighting factors that just compensate for the effect of overlap.Expressions for the variances of $\text{Ω}{}_{\mathrm{i}}\text{and}\text{Ω = Σ}{}_{\mathrm{i}}\text{Ω}{}_{\mathrm{i}}$ are derived from which it is shown that under the above conditions, $\text{Ω}{}_{\mathrm{i}}\text{/Ω}\text{or}\text{ΔΩ}{}_{\mathrm{i}}\text{/ΔΩ}$ are consistent estimators of P_i. Formal evaluation of the variances is carried out in the special case of constant P_i and no secondary cratering. A criterion for the degree of coverage is given; in particular it is shown that the expectation of $\text{σ = Σ}{}_{\mathrm{i}}\text{a}{}_{\mathrm{i}}\text{Ω}{}_{\mathrm{i}}$ at saturation is just A.     

Analysis of HI content in galaxies

We have collected data on 241 galaxies from 13 sources and made a statistical analysis after reduction to a uniform system. We found that the Hubble sequence is one of increasing $\text{〈}\text{M}{}_{\mathrm{<mtext>H</mtext>}}\text{M}{}_{\mathrm{<mtext>T</mtext>}}\text{〉}$ and $\text{〈}\text{M}{}_{\mathrm{<mtext>H</mtext>}}\text{L}{}_{\mathrm{<mtext>B</mtext>}}\text{〉}$, these mean values increasing monotonically from .0016 and .024 at E to .084 and .83 at Im, but the dispersion is large.The HI content in barred spiral is greater than that in ordinary spirals, and this is consistent with their statistics of angular momentum and colour.The HI content is related to colour; it is greater in bluer systems. The large dispersion suggests that it also depends on some other factors, but these are smoothed out when averaged over each type, resulting in a linear relation between $\text{〈log(}\text{M}{}_{\mathrm{<mtext>M</mtext>}}\text{M}{}_{\mathrm{<mtext>T</mtext>}}\text{〉}$ and 〈(B ? V^O_T)〉. Unlike the colour-colour diagram, the large dispersion on the $\text{log}\text{(}\text{M}{}_{\mathrm{<mtext>H</mtext>}}\text{L}{}_{\mathrm{<mtext>B</mtext>}}\text{) ? (B ? V}{}^0{}_{\mathrm{<mtext>T</mtext>}}\text{)}$ is not related to peculiar galaxies.     

New high-speed photometry of EH Lib and a binary interpretation of its period change

Six times of maxima of the ultrashort-period cepheid variable EH Librae were measured in 1980 May to June and in 1981 January, with a three-channel photocounting high-speed photoelectric photometer. These, together with all the photoelectric times of maxima over the past 30 years, are used to re-examine the nature of the change of the period. We found that we can fix the times of maxima by the following formula

$\text{T}{}_{\max }\text{= T}{}_0\text{+P}{}_0\text{E+}\text{1}\text{2}\text{βE}{}^2\text{+A}\text{sin}\text{2π}\text{EP}{}_0\text{E}{}_0$

where T₀ = HJD 2433438.6088 and P₀ = 0.0884132445 d represent the initial maximum epoch and the pulsation period, β = ?2.8 × 10^?8/yr; A = 0.0015 d, P₀ = 6251 d = 17.1 yr are the semi-amplitude and the period of the sine curve, and E is the number of periods elapsed since T₀, and (E₀ = 70700).If we interpret this 17.1 year periodicity as a modulation of the phase of maximum by binary motion, then the semi-amplitude of the orbital radial velocity variation is K = 2πasini/E₀ = 0.45 km/s and the mass function is

$\text{f(m)=}\text{m}{}^3{}_2\text{sin}{}^3\text{i}\text{(m}{}_1\text{m}{}_2\text{)}{}^2\text{=}\text{(a}\text{sin}\text{i)}{}^3\text{E}{}^2{}_0\text{=6 x 10}{}^{\mathrm{?5}}\text{M}{}_{\mathrm{\odot }}$

     

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.     

Residual mass from atmospheric ablation of small meteoroids

Numerical solutions of the equations of meteor ablation in the Earth's atmosphere have been obtained using a variable step size Runge-Kutta technique in order to determine the size of the residual mass resulting from atmospheric flight. The equations used include effects of meteoroid heat capacity and thermal radiation, and a realistic atmospheric density profile. Results were obtained for initial masses in the range 10^?7–10^?2 g, and for initial velocities less than 24 km s^?1 (results indicated no appreciable residual mass for meteors with velocities above 24 km s^?1 in this mass range). The following function has been obtained to provide the logarithm of the ratio of the residual mass following atmospheric ablation to the original preatmospheric mass

$\text{log r = 4.7 ?0.33v}{}_{\mathrm{\infty }}\text{?0.013v}{}_{\mathrm{\infty }}{}^2\text{+ 1.2 log m}{}_{\mathrm{\infty }}\text{+ 0.08 log}{}^2\text{m}{}_{\mathrm{\infty }}\text{?0.083v}{}_{\mathrm{\infty }}\text{log m}{}_{\mathrm{\infty }}\text{M}$

The pre-atmospheric mass and velocity are represented by m_∞ and v_∞.When the results are expressed in terms of the size of the residual mass following atmospheric ablation as a function of the initial mass and velocity, it is found that the final residual mass is almost independent of the original mass of the meteoroid, but very strongly dependent on the original velocity. For example, the residual mass is very nearly 10^?7 g for a meteoroid with velocity 18 kms^?1 for initial masses from 10^?7 to 10^?3 g. On the other hand, a slight change in the initial velocity to 20 km s^?1 will shift the residual mass to approx. 10^?8 g. This strong velocity dependence coupled with the weak dependence on the original mass has important consequences for the sampling of ablation product micrometeorites.     

The Moon's zonal tidal effect in the variation of the earth's spin rate

Using the time observations obtained by 8 instruments in the Chinese Joint System during the years 1966–1980, we analyse the Moon's zonal tidal effect. The results show that the effects of the M_f and M_m waves are obvious. From this, the parameters $\text{K}\text{C}$ of the zonal tide are estimated and the weighted averages of the 8 instruments are $\text{(}\text{K}\text{C}\text{)}\text{M}{}_{\mathrm{f}}\text{= 0.909 ± 0.114}\text{and}\text{(}\text{K}\text{C}\text{)}\text{M}{}_{\mathrm{m}}\text{= 0.905 ± 0.083}$ respectively.     

Electric fields and potential patterns in the high-latitude ionosphere for different situations in interplanetary space

The potential ? of the electric field at high latitudes has been obtained by solving numerically the second order differential equation in spherical coordinates:

$\text{?}\text{1}\text{2}\text{(rσ}{}^{\mathrm{H}}\text{?}{}_{\mathrm{\theta }}\text{)}{}_{\mathrm{\theta }}\text{+}\text{1}\text{r}\text{(σ}{}^{\mathrm{H}}\text{?λ)}{}_{\mathrm{\lambda }}\text{+}\text{1}\text{r}\text{(σ}{}^{\mathrm{P}}\text{?}{}_{\mathrm{\lambda }}\text{)}{}_{\mathrm{\theta }}\text{?(σ}{}^{\mathrm{P}}\text{?}{}_{\mathrm{\theta }}\text{)}{}_{\mathrm{\lambda }}\text{=}\text{1}\text{r}\text{(rψ}{}_{\mathrm{\theta }}\text{)}{}_{\mathrm{\theta }}\text{+}\text{1}\text{r}{}^2\text{ψ}{}_{\mathrm{\lambda \lambda }}$

, where θ is colatitude, λ is longitude, σ^H and σ^P are the height-integrated Hall and Perdersen ionospheric conductivities, r = sinθ, and ψ is the current function. The boundary condition is ? = 0 on the geomagnetic parallel θ = 34°. Values of ψ are determined from geomagnetic field variations at the Earth's surface from geomagnetic field variations at the Earth's surface for various conditions in interplanetary space. σ^P and σ^H are taken to vary with season, local time, tilt of the geomagnetic dipole axis (UT), and intensity of corpuscular precipitation (the model proposed by Wallis and Budzinski, 1981). The model distributions of ?^M and E^M = -▽?^m so obtained are compared with observational results. The feasibility has been demonstrated of interpreting the statistical results and individual measurement data in terms of a unified dynamic model of ionospheric electric fields. The model makes allowance for the changes of electromagnetic “weather” in interplanetary space.     

The semi-detached binary system RZ draconis

The 1978 photoeletric observations of the late type close binary RZ Dra were reanalyzed with the Wilson and Devinney approach. Photometric parameters were determined (in Tab. 1.). The system is found to be semi-detached where the less massive component fills its Roche surface, whereas the other component almost does so. The configuration of the system is shown in Fig. 1. The absolute dimensions of the system are found to be M₁ = 0.61M⊙, $\text{M}{}_2\text{= 0.41}\text{M}\text{⊙}\text{R}{}_1\text{= 1.15}\text{R}\text{,}\text{?}\text{and}\text{R}{}_2\text{= 0.96}\text{R}\text{⊙}$. Both components appear to be overluminous and oversized for their masses and spectral types. Its evolutionary stage is also discussed. The variability in the brightness of the primary mlnimum(Fig. 4) indicates mass loss from the vicinity of L₂, which would be mainly responsible for the long-term decrease in its period.     

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.     

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.     

Habitable zones about main sequence stars

Calculations show that a main sequence star which is less massive than the Sun has a continuously habitable zone about it which is not only closer in than the corresponding zone about the Sun, but is also relatively narrower. Let L(t) represent the luminosity after t billion years of a main sequence star of mass M, and let r_inner and r_outer represent the boundaries of the continuously habitable zone about such a star—that is, the zone in which an Earthlike planet will undergo neither a runaway greenhouse effect in the early stages of its history nor runaway glaciation after it develops an oxidizing atmosphere. Then our computer results indicate that $\text{r}{}_{\mathrm{<mtext>outer</mtext>}}\text{r}{}_{\mathrm{<mtext>inner</mtext>}}$ is roughly proportional to $\text{[}\text{L(3.5)}\text{L(1.0)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. This ratio is smaller for stars less massive than the Sun (because they evolve more slowly), and the width of the continuously habitable zone about a main sequence star is therefore a strong function of the initial stellar mass. Our calculations show that r_inner = r_outer for $\text{M～0.83M?}$ (i.e., K1 stars), and it therefore appears that there is no continuously habitable zone about most K stars, nor any about M stars.     

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.     

On fourier-analysis of geomagnetic pulsations pc-1, “two-fold” and “three-fold” pulsations pc-1 and fundamental frequencies of the magnetospheric cavity—I

The paper gives the results of detailed studies of the frequency spectra $\text{S}{}_{\mathrm{s}}\text{(?)}$ of the chain of the wave packets F_s(t) of geomagnetic pulsations PC-1 recorded at the Novolazarevskaya station. The bulk of the energy of F_s(t) is concentrated in the vicinity of the central frequencies $\text{?}{}_{\mathrm{s0}}$ of spectra—the carrier frequencies of the signals. The velocity $\text{V}{}_0\text{≌ 6.10}{}^3\text{km s}{}^{\mathrm{?1}}$ of the flux of protons generating these signals correspond to them. The spectra of the signals have oscillations—“satellites” irregularly distributed in frequency. These satellites, as the authors believe, testify to the presence of the individual groups of protons of low concentration whose velocities vary within 10³–10⁴ km s^?1.Their energy is only of the order of 10^?2–10^?3 of the energy of the main proton flux. Clearly pronounced maxima on double and triple frequencies $\text{? = 2?}{}_{\mathrm{s0}}\text{and}\text{3?}{}_{\mathrm{s0}}$ are detected. They show that the generation of pulsations PC-1 is accompanied by the generation on the overtones of wave packets called in this paper “two-fold” and “three-fold” pulsations PC-1. Intensive symmetrical satellites of a modulation character have been discovered on frequencies $\text{?}{}^{\mathrm{\pm }}{}_{\mathrm{sK}}$. Frequency differences $\text{Δ?}{}_{\mathrm{sK}}{}^{\mathrm{\pm }}\text{=}\text{¦?}{}_{\mathrm{s0}}\text{? ?}{}_{\mathrm{sK}}{}^{\mathrm{\pm }}\text{¦}\text{= (0.011,0.022}\text{and}\text{0.035)}\text{Hz}$ correspond to them. The authors believe that the values of Δ?^±_sK are resonance frequencies of the magnetospheric cavity in which geomagnetic pulsations PC-1 are generated. It is established that the values of Δ?^±_sK coincide closely with the carrier frequencies of geomagnetic pulsations PC-3 and PC-4 generated in the magnetosphere. This leads to the conclusion that the resonance oscillations of the magnetospheric cavity are their source. Thus, the generation of geomagnetic pulsations of different types and resonance oscillations in the magnetosphere are integrated into a unified process. The importance of the results obtained and the necessity to check further their trustworthiness and universality, using experimental data gathered in different conditions, is stressed.     

Ring current simulation in connection with interplanetary space conditions

A ring current model has been obtained which permits calculations ofD_st variations on the Earth's surface during magnetic storms. The changes in D_st are described by the equation

$\text{d}\text{d}\text{t}\text{D}{}_{\mathrm{sto}}\text{= F(EM)?}\text{D}{}_{\mathrm{sto}}\text{tau;}$

where $\text{D}{}_{\mathrm{sto}}\text{= D}{}_{\mathrm{st}}\text{-bp}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+~tc}$; p = mnv² is solar wind pressure; F(EM) is the function, controlled by the electromagnetic parameters of interplanetary medium, of injection into ring current ; τ is the constant of ring current decay. $\text{C}\text{= C}\text{u}\text{τ?=18}\text{nT}$, where C is the level of the D_st-variation field measurements; ? is the injection function characterizing the quasisteady-state injection of energy into the ring-current region. The constant Ç is determined from the condition that the change of the ring current energy from magnetic storm commencement to end should equal the difference between the injected and dissipated energy throughout the storm. The values of the factors b and τ were found by the method of minimizing the sum of the quadratic deviations of the calculated D_st from the values observed throughout the storm : b = 0.23 nT/(eV cm^?3)^{$\text{1}\text{2}$}, τ = 8.2 h at D_st? ? 55 nT and τ = 5.8 h at -120 ? D_st ? — 55 nT. The injection function F(EM) is of the form F(EM) = d(E_y? A) at the values of the azimuthal component of the solar wind electric field E_y ? A, and F(EM) =0 at A?E_y.d = ? 1.2 × 10^?3 Ts^?1 (mV/m)^?1 and A = ? 0.9 mV m^?1 . Thus, the injection to ring current is possible at the northward B_z component of the IMF.     

A third-order solution of Vinti's problem with explicit expressions for the poisson brackets

We have derived an explicit third-order solution of Vinti's problem including J₃. Poisson brackets for the elements a, e, s, M_s, ψ_s and $\text{Ω}{}_{\mathrm{s}}$ are given. These may be used in the construction of a third-order theory of artificial satellites.     

Small-scale turbulence in the atmosphere of Venus

Previous studies based on radio scintillation measurements of the atmosphere of Venus have identified two regions of small-scale temperature fluctuations located in the vicinity of 45 and 60 km. A global study of the fluctuations near 60 km, which are consistent with wind-shear-generated turbulence, was conducted using the Pioneer Venus measurements. The structure constants of refractive index fluctuations c_n² and temperature fluctuations c_T² increase poleward, peak near 70° latitude, and decrease over the pole; c_n² varies from 2 × 10^?15 to 1.5 × 10^?14m^{$\text{2}\text{3}$} and c_T² from 4 × 10^?3 to 7 × 10^?2°K²m^{$\text{?2}\text{3}$}. These results indicate greater turbulent activity at the higher latitudes. In the region near 45 km the refractive index fluctuations and the corresponding temperature fluctuations are substantially lower. Based on the analysis of one representative occultation measurement, $\text{c}{}_{\mathrm{<mtext>n</mtext>}}{}^2\text{= 2 × 10}{}^{\mathrm{?16}}\text{m}{}^{\mathrm{<mtext>?2</mtext><mtext>3</mtext>}}\text{and}\text{c}{}_{\mathrm{<mtext>T</mtext>}}{}^2\text{= 7.3 × 10}{}^{\mathrm{?4}}\text{°}\text{K}{}^2\text{m}{}^{\mathrm{<mtext>?2</mtext><mtext>3</mtext>}}$ in the 45-km region. The fluctuations in this region also appear to be consistent with wind-shear-generated turbulence. The turbulence level is considerably weaker than that at 60 km; the energy dissipation rate ε is 4.9 × 10^?5m²sec^?3 and the small-scale eddy diffusion coefficient K is 2 × 10³ cm² sec^?1.     

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.     

Models of the formation of the solar nebula

Models of the collapse of a protostellar cloud and the formation of the solar nebula reveal that the size of the nebula produced will be the larger of $\text{R}{}_{\mathrm{<mtext>CF</mtext>}}\text{≡ J}{}^2\text{/k}{}^2\text{GM}{}^3\text{and}\text{R}{}_{\mathrm{<mtext>V</mtext>}}\text{≡ (GMv/2c}{}_{\mathrm{<mtext>c</mtext>}}{}^3\text{)}\text{1}\text{2}$ (where J, M, and c_s are the total angular momentum, total mass, and sound speed of the protosetellar material; G is the gravitational constant; k is a number of order unity; and v is the effective viscosity in the nebula). From this result it can be deduced that low-mass nebulas are produced if P ≡ (R_V/R_CF)² ? 1; “massive” nebulas result if P ? 1. Gravitational instabilities are expected to be important for the evolution of P ? 1 nebulas. The value of J distinguishes most current models of the solar nebula, since P ∝ J^?4. Analytic expressions for the surface density, nebular mass flux, and photospheric temperature distributions during the formation stage are presented for some simple models that illustrate the general properties of growing protostellar disks. It does not yet seem possible to rule out either P ? 1 or P < 1 for the solar nebula, but observed or possible heterogeneities in composition and angular-momentum orientation favor P < 1 models.     

Interactions of plasmas with magnetic field boundaries

The magnetopause, the boundary layer, or current sheath, which separates the magnetosphere from the solar wind, is the particular interaction considered in this paper.The collision free electron skin depth, $\text{ξ}{}_{\mathrm{e}}\text{=}\text{c}\text{ω}{}_{\mathrm{pe}}$, where c is the velocity of light and ω_pe, is the plasma frequency, gives a classical measure of the penetration depth of a collisionless plasma by an electromagnetic field. This penetration depth is small compared with the dimensions of the magnetosphere and hence the boundary layer may be conveniently considered in one dimension.In General all one dimensional solutions lie within an order of magnitude of the value of ξ_e, the only exception being the important one, in which the electric field perpendicular to the current sheath plane is not present, either due to a particular trapped particle distribution or due to a short circuiting end effect. For this exception the thickness is increased by the factor ($\text{m}{}_{\mathrm{i}}\text{i/m}{}_{\mathrm{e}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.The current sheath solutions discussed are equilibrium solutions but not necessarily stable equilibrium solutions.The extension of the models to three dimensions has a larger effect than might at first be expected. The effect may be intuitively understood as a consequence of flux conservation in the sheath. The one dimensional solutions then correspond to the current sheath profiles at the thinnest point of the three dimensional sheath.