A Diffusion Model of Radial Distributions of Energetic Protons in Saturn's Magnetosphere

A simple diffusion model of Saturn's radiation belts with a constant internal source and particle losses at the boundaries is considered. The radial diffusion coefficient is supposed to be D _LL L ⁴. The analytical solution to the stationary diffusion equation describes well the observed radial profiles of the distribution function of energetic protons in Saturn's magnetosphere.     

Radial profile of pressure in a storm ring current as a function of D st

Using satellite data obtained near the equatorial plane during 12 magnetic storms with amplitudes from ?61 down to ?422 nT, the dependences of maximum in L-profile of pressure (L _m) of the ring current (RC) on the current value of D _st are constructed, and their analytical approximations are derived. It is established that function L _m(D _st ) is steeper on the phase of recovery than during the storm’s main phase. The form of the outer edge of experimental radial profiles of RC pressure is studied, and it is demonstrated to correspond to exponential growth of the total energy of RC particles on a given L shell with decreasing L. It is shown that during the storms’ main phase the ratio of plasma and magnetic field pressures at the RC maximum does not practically depend on the storm strength and L _m value. This fact reflects resistance of the Earth’s magnetic field to RC expansion, and testifies that during storms the possibilities of injection to small L are limited for RC particles. During the storms’ recovery phase this ratio quickly increases with increasing L _m, which reflects an increased fraction of plasma in the total pressure balance. It is demonstrated that function L _m(D _st ) is derived for the main phase of storms from the equations of drift motion of RC ions in electrical and magnetic fields, reflecting the dipole character of magnetic field and scale invariance of the pattern of particle convection near the RC maximum. For the recovery phase it is obtained from the Dessler-Parker-Sckopke relationship. The obtained regularities allow one to judge about the radial profile of RC pressure from ground-based magnetic measurements (data on the D _st variation).     

Size quantization in high-temperature superconducting cuprates and a link to Einstein's diffusion law

Optimum doping of high-temperature superconductors (HTSC) defines a superconducting unit volume for each HTSC. For a single-mode HTSC, e.g., a cuprate with one CuO₂ plane, the volume is given by V_sc=cx², where c is the unit cell height and x the doping distance. The experimental resistivity at T_c is connected to the structure by ρ(exp)≈c×h/(2e²). Combining this result with the classical definition of resistivity leads to an equation similar to Einstein's diffusion law x²/(2τ)=h/(2M_eff)=D, where τ is the relaxation time, M_eff=2m_e and D the diffusion constant. It has also been shown that the mean free path d=x. The Einstein–Smoluchowski diffusion relation D=μk_BT_c provides a connection to T_c.     

Modeling the time behavior of the D st index during the main phase of magnetic storms generated by various types of solar wind

Results of modeling the time behavior of the D _st index at the main phase of 93 geomagnetic storms (?250 < D _st ≤ ?50 nT) caused by different types of solar wind (SW) streams: magnetic clouds (MC, 10 storms), corotating interaction regions (CIR, 31 storms), the compression region before interplanetary coronal ejections (Sheath before ICME, 21 storms), and “pistons” (Ejecta, 31 storms) are presented. The “Catalog of Large-Scale Solar Wind Phenomena during 1976–2000” (ftp://ftp.iki.rssi.ru/pub/omni/) created on the basis of the OMNI database was the initial data for the analysis. The main phase of magnetic storms is approximated by a linear dependence on the main parameters of the solar wind: integral electric field sumEy, dynamic pressure P _d , and fluctuation level sB in IMF. For all types of SW, the main phase of magnetic storms is better modeled by individual values of the approximation coefficients: the correlation coefficient is high and the standard deviation between the modeled and measured values of D _st is low. The accuracy of the model in question is higher for storms from MC and is lower by a factor of ～2 for the storms from other types of SW. The version of the model with the approximation coefficients averaged over SW type describes worse variations of the measured D _st index: the correlation coefficient is the lowest for the storms caused by MC and the highest for the Sheath- and CIR-induced storms. The model accuracy is the highest for the storms caused by Ejecta and, for the storms caused by Sheath, is a factor of ～1.42 lower. Addition of corrections for the prehistory of the development of the beginning of the main phase of the magnetic storm improves modeling parameters for all types of interplanetary sources of storms: the correlation coefficient varies within the range from r = 0.81 for the storms caused by Ejecta to r = 0.85 for the storms caused by Sheath. The highest accuracy is for the storms caused by MC. It is, by a factor of ～1.5, lower for the Sheath-induced storms.     

Ionization losses of the Earth’s radiation belt protons according to the radial diffusion theory

Using modern models of the plasmasphere and exosphere, radial profiles of the rates of ionization losses of protons with μ = 0.3–10 keV/nT (μ is the first adiabatic invariant) of the Earth’s radiation belts (ERBs) have been constructed. To calculate Coulomb losses of protons, we used the ISEE-1 satellite data at L = 3–9 and CRRES satellite data at L ≤ 3 (L is the McIlwain parameter). The relation of contributions of Coulomb losses and charge exchange in the rate of ionization losses of protons has been considered. We have discovered the effect of subtracting Coulomb losses from charge exchange of ERB protons for small μ and L, which can imitate a local particle source. It has been demonstrated that, with decreasing L, the rate of ionization losses of ERB protons decreases as a whole. The radial dependence of this rate only has a negative gradient in the narrow range (ΔL ~ 0.5) in the region of the plasmapause and only for protons with μ > 1.2 keV/nT.     

Gravitation effect on a flux of sporadic micrometeoroids in the vicinity of near-Earth orbits

The new approach to gravitation effect determination in calculating the flux of sporadic micrometeoroids in the near-Earth space is proposed. The technique is based on integration of the equations of motion of sporadic micrometeoroids with accounting for bending their trajectories when particles are approaching the Earth. The technique and results of calculation of the gravitational focusing factor k_g for various conditions are presented. The feature of the proposed technique for calculating coefficient k_g consists in the fact that this coefficient does not explicitly depend on the values of particles velocity at the last point. The results of investigation of coefficient k_g have shown that, for the given initial velocity of micrometeoroids, the values of this coefficient depend on deflection of its direction from the direction to the Earth center. It is shown that for low-altitude orbits the flux density can increase up to 60%. The distribution of probabilities of various directions of particles flying to spacecraft structural elements is found to be non-uniform.     

Energy dependence of the fraction of protons of ionospheric origin in the ring current

The energy dependence of a fraction of ring current protons of ionospheric origin is calculated using the AMPTE/CCE data for a typical strong magnetic storm (max|D _st | ≈ 120 nT). It is shown that this fraction monotonically decreases from ～ 83 to 25–30% with an increase in proton energy from 5 to 315 keV at L = 6–7 (L is the McIlwain parameter) and is 30–40% at energy 40–50 keV corresponding to the maximum of proton energy density at L = 6–7. It is demonstrated that the core of the ring current (L = 3.7–4.7) was enriched by solar protons with E ≈ 10–200 keV during the active phase of the storm (the maximum effect is reached at E ≈ 20–50 keV).     

Study of notches in the Earth’s plasmasphere based on data of the <Emphasis Type="Italic">MAGION-5</Emphasis> satellite

Depleted narrow (localized in longitude) regions (field tubes) in the plasmasphere, recently discovered in He⁺ radiation measurements on the IMAGE spacecraft, were first directly observed by the Magion-5 satellite. The low-density regions (notches) occupy <～ 10–30° in longitude and extend from L ～ 2–3 to the plasmasphere boundary in neighboring plasmasphere regions with larger densities. The Magion-5 data give evidence that in the low-density regions temperature is enhanced as compared to the neighboring denser plasmasphere regions. Formation of notches in the plasmasphere is, apparently, associated with AE intensification during weak magnetic storms, while strong magnetic storms usually result in the overall reduction of plasmasphere dimensions. However, even a strong magnetic storm on April 6–7, 2000 (max K _p = 9-and min D _st ～ ?290 nT), but accompanied by an isolated AE impulse, resulted in a density decrease only in the longitudinally limited post-midnight sector of the plasmasphere.     

Variations of the magnetopause position versus the level of geomagnetic activity (according to data of the INTERBALL-1 Satellite for 1995–1997)

The variations in the deviation of the observed position of the magnetosphere boundary from its mean position predicted by the Shue at al., 1997 (Sh97) model [7] are studied as a function of the substorm activity level (the AE-index value) and magnetic storm intensity (the value of the corrected D _st ^* index). The results obtained make it possible to state that the amplitude of motion of the magnetospheric boundary on the dayside and in the low-latitude tail is small. It is likely that the position of the boundary is either independent of the AE and D _st ^* indices or this dependence is weak. At the same time, the boundary of the high-latitude tail shifts inward on the average by 1.5R _E with an increase of the AE-index in the case of absence of magnetic storms (contraction of the magnetospheric tail). On the contrary, in the presence of magnetic storms, this boundary shifts outward by up to 3R _E with an increase of the AE-index (inflation of the magnetospheric tail). It is also shown that the boundary of the high-latitude tail moves outward with an increase of the D _st ^* index, both at low substorm activity and in periods of high substorm activity. The amplitude of the outward motion of the high-latitude tail of the magnetosphere is by a factor of two higher for moderate magnetic storms with strong substorms than for moderate magnetic storms with weak substorms.     

Numerical investigation on cavity structure of solid-fuel scramjet combustor

The influences of miscellaneous combustor structures for solid fuel scramjet combustion on the performance are investigated, including a detailed interaction analysis between shocks/waves and combustion. Hydroxyl-terminated polybutadiene is chosen as the solid fuel with the non-premixed equilibrium probability density function combustion model. The results show combustion enhancement when structure of combustor is modified. The radical emphasis is to examine the sensitivity of the properties due to variations on the length-to-depth ratio of cavity, aft wall angle, and offset ratio. It is noted that there is an appropriate structure of cavity (L/D=4, θ=45°, and D_d/D_u=1.25–1.5) regarding the combustion efficiency, total pressure loss and specific impulse. The observation of function for combustor components provides instructional insight into the design considerations for a combustor of a solid-fuel scramjet.     

On the belt of energetic electrons at <Emphasis Type="Italic">L</Emphasis> = 2.75 in the Earth’s magnetosphere

In 1964, during flights of the ELECTRON satellites the narrow belts of energetic electrons (E _e ≈ 6MeV) have been discovered in the Earth’s magnetosphere at L ≈ 2.75. The same structures approximately at the same magnetic shells were found in 2004 by the CORONAS-F and SERVIS-1 satellites. A comparison of the results of these experiments is presented. It is shown that the additional narrow belts of energetic electrons occur after intense magnetic storms (D _st > 100 nT), in our cases, having a double-triple structure. The lifetime of these belts is a few months and their disappearance had a gradual character. The obtained results separated in time by 40 years suggest the constancy of the sources of particles of the Earth’s radiation belts and processes occurring in the magnetosphere, which ensures not only existence of the radiation belts, but also the recurrence of various exotic phenomena in the belts similar to the belt of energetic electrons at the inner magnetic shells.     

On Saturn’s rotation relative to a center of mass under the action of the gravitational moments of the Sun and Jupiter

Saturn’s rotation relative to a center of mass is considered within an elliptic restricted three-body problem. It is assumed that Saturn is a solid under the action of gravity of the Sun and Jupiter. The motions of Saturn and Jupiter are considered elliptic with small eccentricities e_S and e_J, respectively; the mean motion of Jupiter n_J is also small. We obtain the averaged Hamiltonian function for a small parameter of ε = n_J and integrals of evolution equations. The main effects of the influence of Jupiter on Saturn’s rotation are described: (α) the evolution of the constant parameters of regular precession for the angular momentum vector I₂; (β) the occurrence of new libration zones of oscillations I₂ near the plane of the celestial equator parallel to the plane of the Jupiter’s orbit; (γ) the occurrence of additional unstable equilibria of vector I₂ at the points of the north and south poles of the celestial sphere and, as a result, the existence of homoclinic trajectories; and (δ) the existence of periodic trajectories with arbitrarily large periods near the homoclinic trajectory. It is shown that the effects of (β), (γ), and (δ) are caused by the eccentricity e of the Jupiter’s orbit and are practically independent of Jupiter’s mass (within satellite approximation).     

On the Precession of Saturn

The precession of Saturn under the effect of the gravity of the Sun, Jupiter and planet’s satellites has been investigated. Saturn is considered to be an axisymmetric (A = B) solid body close to the dynamically spherical one. The orbits of Saturn and Jupiter are considered to be Keplerian ellipses in the inertial coordinate system. It has been shown that the entire set of small parameters of the problem can be reduced to two independent parameters. The averaged Hamiltonian function of the problem and the integrals of evolutionary equations are obtained disregarding the effect of satellites. Using the small parameter method, the expressions for the precession frequency and the nutation angle of the planet’s axis of rotation caused by the gravity of the Sun and Jupiter are obtained. Considering the planet with satellites as a whole preceding around the normal to the unmovable plane of Saturn’s orbit, the satellites effect on the Saturn rotation is taken into account via the corrections in the formula for the undisturbed precession frequency. The satellites are shown to have no effect on the nutation angle (in the framework of the accepted model), and the disturbances from Jupiter to make the main contribution to the nutation angle evolution. The effect of Jupiter on the nutation angle and the precession period is described with regard to the attraction of satellites.     

Simultaneous Response of the Mid-Latitude Ionosphere to a Sudden Commencement of Strong Magnetic Storms

Using the new technology of global GPS detection of ionospheric disturbances (GLOBDET), it is found that a sharp increase of the time derivative of the magnetic field strength during magnetic storms is accompanied by a simultaneous decrease of the mid-latitude total electron content (TEC) over the entire dayside of the globe. The corresponding negative correlation coefficient is no less than 0.8, and the delay relative to the sudden commencement of magnetic storm is about 3–10 min. The effect is especially clearly pronounced for magnetic storms with sudden commencements (SSC). The analysis is carried out for a set of 90 to 300 GPS stations for 10 days (January 6 and April 23, 1998; April 6, June 8, July 13, 14, and 15, 2000; March 31, April 4 and 11, 2001) with various levels of geomagnetic activity (D _st and K _p varied from –6 to –295 nT and from 0 to 9, respectively). The amplitude of the response in the total electron content for the events considered was 0.1–0.4 × 10¹⁶ m^–2 (which is a deviation of 0.2–2.6% from the TEC background value). The velocity of the disturbance motion from the dayside to the nightside was about 10–20 km/s. The results obtained agree with the data of ionospheric parameter measurements conducted earlier by methods with high temporal resolution.     

Statistical study of interplanetary condition effect on geomagnetic storms

Based on the archive OMNI data for the period 1976–2000 an analysis has been made of 798 geomagnetic storms with D _st < −50 nT and their interplanetary sources-large-scale types of the solar wind: CIR (145 magnetic storms), Sheath (96), magnetic clouds MC (62), and Ejecta (161). The remaining 334 magnetic storms have no well-defined sources. For the analysis, we applied the double method of superposed epoch analysis in which the instants of the magnetic storm beginning and minimum of D _st index are taken as reference times. The well-known fact that, independent of the interplanetary source type, the magnetic storm begins in 1–2 h after a southward turn of the IMF (B _z < 0) and both the end of the main phase of a storm and the beginning of its recovery phase are observed in 1–2 h after disappearance of the southward component of the IMF is confirmed. Also confirmed is the result obtained previously that the most efficient generation of magnetic storms is observed for Sheath before MC. On the average parameters B _z and E _y slightly vary between the beginning and end of the main phase of storms (minimum of D _st and D _st ^* indices), while D _st and D _st ^* indices decrease monotonically proportionally to integral of B _z and E _y over time. Such a behavior of the indices indicates that the used double method of superposed epoch analysis can be successfully applied in order to study dynamics of the parameters on the main phase of magnetic storms having different duration.     

Luni-solar perturbations in the extended phase space representation of the Vinti problem

Perturbation theory is applied to the Vinti problem—motion about an oblate spheroid—to include the gravitational effects of the sun and moon. The problem is formulated using the extended phase space method which introduces a new independent variable similar to the true anomaly. The disturbing Hamiltonian H₁ for third bodies is of order J₂² (second order) and the final goal is a theory including second order short and long period terms and third order secular terms. The current paper however carries the development only to the second order in the secular terms and the first order in the periodic terms. Problems of including the higher orders are discussed. Therefore, in the development of H₁ all terms of order 10^?9 or larger are retained. The lunar emphemeris retains terms to e′² in the lunar eccentricity. The perturbation analysis is carried out by means of Lie series and is developed through the first order only which is consistent with the final accuracy desired. The generating function W₁ is obtained and separated into the long period, short period and secular terms. From W₁ the coordinates are defined from the Lie series by means of a transformation equation. These coordinates are non-singular for small eccentricity and inclination. Because of the complexity of the equations all algebraic computations were accomplished by means of a computerized Poisson series manipulator developed at the Naval Research Laboratory.     

On the limiting mass of a dense stellar matter

In the present work, we have derived an expression $\text{M}{}_{\mathrm{L}}\text{? 4.738 M}{}_{\mathrm{\odot }}$ (M_⊙ = 1.985 × 10³³ g = mass of the Sun) giving the “limiting” value of the mass of a dense stellar matter, by introducing the concept of nuclear size correction in the theory of relativistic Thomas Fermi model for a compressed atom. We find that $\text{M}{}_{\mathrm{L}}\text{? 5.1571 M}{}_{\mathrm{<mtext>Ch</mtext>}}\text{and}\text{=3.2750(M}{}_{\mathrm{<mtext>O</mtext>}}\text{)}{}_{\mathrm{<mtext>Prev</mtext>}}$ [M_Ch and (M_O)_Prev denote respectively the Chandrasekhar and author's “limiting” masses]. By making a comparative study with those of previous results it has been shown that our present treatment would provide satisfactory results for the density ranges from ? ? 10⁸ up to $\text{? ? 10}{}^{11}\text{g}\text{/}\text{cm}{}^3$. Other results of cognate interest in the non-relativistic regime 10³ < ? ? 10⁵ (without the nuclear size effect) are presented. The astrophysical implications of the results are mentioned.     

Protons with energy of E P ≥ MeV under the earth’s radiation belts

Variations in fluxes of quasi-trapped energetic protons were studied on the basis of the data of the CORONAS-I satellite. These variations are characterized by an increase in the proton fluxes with E _P ≥ 1 MeV both in the vicinity of the geomagnetic equator and in the high-latitude region of the magnetosphere. The analysis of structural features of the proton distributions in the regions at L ～ 1–1.1; 3 < L < 4; and L > 4, was performed and made it possible to detect reliably the type of the proton flux increase in this region. The mechanisms of particle scattering leading to the precipitation of energetic protons under conditions of various types of geomagnetic disturbances are considered.     

Halo orbits in the vicinity of the L 2 libration point in the Sun-Earth system

Methods are proposed for constructing the orbits of spacecraft remaining for long periods of time in the vicinity of the L ₂ libration point in the Sun-Earth system (so-called halo orbits), and the trajectories of uncontrolled flights from low near-Earth orbits to halo orbits. Halo orbits and flight trajectories are constructed in two stages: A suitable solution to a circular restricted three-body problem is first constructed and then transformed into the solution for a restricted four-body problem in view of the real motions of the Sun, Earth, and Moon. For a halo orbit, its prototype in the first stage is a combination of a periodic Lyapunov solution in the vicinity of the L ₂ point and lying in the plane of large-body motion, with the solution for the linear second-order system describing small deviations of the spacecraft from this plane along the periodic solution. The desired orbit is found as the solution to the three-body problem best approximating the prototype in the mean square. The constructed orbit serves as a similar prototype in the second stage. In both stages, the approximating solution is constructed by continuation along a parameter that is the length of the approximation interval. Flight trajectories are constructed in a similar manner. The prototype orbit in the first stage is a combination of a solution lying in the plane of large-body motion and a solution for a linear second-order system describing small deviations of the spacecraft from this plane. The planar solution begins near the Earth and over time tends toward the Lyapunov solution existing in the vicinity of the L ₂ point. The initial conditions of both prototypes and the approximating solutions correspond to the spacecraft’s departure from a low near-Earth orbit at a given distance, perigee, and inclination.