Shock acceleration of solar cosmic rays

We investigated the acceleration of solar cosmic rays (SCRs) by the shock waves produced by coronal mass ejections. We performed detailed numerical calculations of the SCR spectra produced during the shock propagation in the solar corona in terms of a model based on the diffusive transport equation using a realistic set of physical parameters for the corona. The resulting SCR energy spectrum N(ε) ∝ ε^?γ exp [? (ε/ε_max)^α] is shown to include a power-law portion with an index γ?2 that ends with an exponential tail with α ? 2.5 ? β, where β is the spectral index of the background Alfvén turbulence. The maximum SCR energy lies within the range ε_max = 1–300 MeV, depending on the shock velocity. Because of the steep spectrum of the SCRs, their backreaction on the shock structure is negligible. The decrease in the Alfvén Mach number of the shock due to the increase in the Alfvén velocity with heliocentric distance r causes the efficient SCR acceleration to terminate when the shock reaches a distance of r = 2–3R_⊙. Since the diffusive SCR propagation in this case is faster than the shock expansion, SCR particles intensively escape from the shock vicinity. A comparison of the calculated SCR fluxes expected near the Earth’s orbit with available experimental data indicates that the theory satisfactorily explains all of the main observed features.     

A solar wind model with the power spectrum of Alfvénic fluctuations

A new solar wind model has been developed by including in the model the Alfvénic fluctuation power spectrum equation proposed by Tu et al. (1984). The basic assumptions of the model are as follows: (1) for heliocentric distances r > 10 R _⊙, the radial variation of the power spectrum of Alfvénic fluctuations is controlled by the spectrum equation (1), (2) for heliocentric distances r < 10 R _⊙, the radial variation of the fluctuation amplitude is determined by the Alfvén wave WKB solution, (3) no energy cascades from the low-frequency boundary of the Alfvénic fluctuation power spectrum into the fluctuation frequency range, and the energy which cascades from the high-energy boundary of the spectrum into the higher frequency range is transported to heat of the solar wind flow. Some solutions of this model which, on one hand, describe the major properties of the Alfvénic fluctuations and the high-speed flow observed by Helios in the space range between 0.3–1 AU and, on the other hand, are consistent with the observational constraints at the coronal base have been obtained under the following conditions: (1) the spectrum index of the fluctuations is near to -1 for almost the whole frequency range at 10 R _⊙, (2) the particle flux density at 1 AU is not greater than 3 × 10⁸ cm^?2 s^?1, (3) the solution is for spherically-symmetric flow geometry or the solution passes through the outermost of the three critical points of the rapidly diverging flow geometry with f _max = 7. Some solutions passing through the innermost critical point of the rapidly diverging flow geometry with f _max = 7 have been found, however, with too low pressure at the coronal base to compare with the observational constraints. Heat addition or other kind of momentum addition for r < 10 R _⊙ is required to modify this model to yield better agreement with observations. A cascade energy flux function which leads to Kolmogorov power law in the high-frequency range of Alfvénic fluctuations is presented in Appendix A. More detailed discussions about the characteristics, the boundary conditions and the solution of the spectrum equation (1) are given in Appendix B.     

Solution of one-fluid model equations with short range retarding magnetic forces for the quiet solar wind

The discrepancy of the low predicted versus the observed coronal particle densities is investigated by considering radial magnetic forces acting at the base of the corona in the one fluid model equations with anomalous thermal conductivity for the quiet solar wind. If the short range retarding magnetic force is taken to fall asr ^?5,r being the heliocentric distance, then in order to obtain satisfactory agreement between the predicted and observed (about 3×10⁸ cm^?3 at 1R _⊙) coronal densities, the strength of the retarding magnetic force at 1R _⊙ should be 1.2 times that of the gravitational force.     

Spectrographic study of the eclipsing binary system SZ Camelopardalis

Thirteen high-dispersion spectrographs of the eclipsing binary star SZ Cam have been studied with a view of determining more accurate information on: (i) the spectral type and luminosity classifications, (ii) absolute parameters for the component stars, (iii) the stellar environment of SZ Cam. The main results in these categories are as follows: (i) O9.5 Vnk, (ii)m _g=19±2M _⊙,m _s=6.5±1M _⊙;r _g=9.7±3.6R _⊙,r _s=4.8±1.7R _⊙;T _e～30000 K,T _e～23000 K; (iii) there is a local concentration of absorbing material which may reach a density of 2M _⊙ pc^?3, and the distance of the star is found to be 600±150 pc. The determined overluminosity of the secondary star and the local concentration of absorbing material are two topics which provide the basis for a discussion section.     

Large-Scale Variation of Solar Wind Electron Properties from Quasi-Thermal Noise Spectroscopy: Ulysses Measurements

The transport of energy in space plasmas, especially in the solar wind, is far from being understood. Measuring the temperature of the electrons and their non-thermal properties is essential to understand the transport properties in collisionless plasmas. Quasi-thermal noise spectroscopy is a reliable tool for measuring the electron temperature accurately since it is less sensitive to the spacecraft perturbations than particle detectors. We apply this method to Ulysses radio data obtained during the first pole-to-pole fast latitude scan in the high-speed solar wind, using a kappa function to describe the electron velocity distribution. We deduce the variations with heliocentric distance between 1.5 and 2.3 AU in the fast solar wind at high latitude in terms of three fitting parameters: the electron density varies as n _e??R ^?1.96±0.08, the electron temperature as T _e??R ^?0.53±0.15, and the kappa index of the distribution remains constant at ??=2.0±0.2. These observations agree with the predictions of the exospheric theory.     

Centrifugal instability of the jovian magnetosphere and its interaction with the solar wind

The outer regions (r > 2.3 R_j; R_j = radius of Jupiter) of the magnetosphere of Jupiter will systematically accumulate plasma. If sufficient plasma accumulates, the field lines must open to allow the plasma to escape. Available energy sources appear able to supply plasma at a high enough rate to keep the field lines constantly open beyond about 60 R_J. We suggest that the solar wind interaction with Jupiter may be essentially different from that with the Earth, with the Jovian magnetosphere opening up to form a planetary wind.     

The solar modulation of electrons in the heliosphere

The propagation and modulation of electrons in the heliosphere play an important part in improving our understanding and assessment of the modulation processes. A full three-dimensional numerical model is used to study the modulation of galactic electrons, from Earth into the inner heliosheath, over an energy range from 10 MeV to 30 GeV. The modeling is compared with observations of 6–14 MeV electrons from Voyager 1 and observations at Earth from the PAMELA mission. Computed spectra are shown at different spatial positions. Based on comparison with Voyager 1 observations, a new local interstellar electron spectrum is calculated. We find that it consists of two power-laws: In terms of kinetic energy E, the results give E ^?1.5 below ～500 MeV and E ^?3.15 at higher energies. Radial intensity profiles are computed also for 12 MeV electrons, including a Jovian source, and compared to the 6–14 MeV observations from Voyager 1. Since the Jovian and galactic electrons can be separated in the model, we calculate the intensity of galactic electrons below 100 MeV at Earth. The highest possible differential flux of galactic electrons at Earth with E=12 MeV is found to have a value of 2.5×10^?1 electrons m^?2?s^?1?sr^?1?MeV^?1 which is significantly lower (a factor of 3) than the Jovian electron flux at Earth. The model can also reproduce the extraordinary increase of electrons by a factor of 60 at 12 MeV in the inner heliosheath. A lower limit for the local interstellar spectrum at 12 MeV is estimated to have a value of (90±10) electrons m^?2?s^?1?sr^?1?MeV^?1.     

The effect of interplanetary acceleration on the propagation of energetic solar particles in prompt events

Crank-Nicholson solutions are obtained to the time-dependent Fokker-Planck equation for propagation in the interplanetary medium following a point in time injection of energetic solar particles and including the acceleration terms $$\frac{\partial }{{\partial T}}\left( {D_{TT} \frac{{\partial U}}{{\partial T}}} \right) - \frac{\partial }{{\partial T}}\left( {\frac{{D_{TT} U}}{{2T}}} \right)$$ . The diffusion coefficient in kinetic energyD _TT is allowed to be either independent of radial distance,R(AU), or follow the lawD _TT=D₀T²R ₀ ² /(A²+R²) in either case with the 1 AU value ofD _TT at 10 MeV ranging between 10^?4 (MeV)² s^?1 and zero. The spatial diffusion mean free path at the Earth's orbit is fixed at λ_‖ AU at 10 MeV according to numerical estimates made by Moussas and Quenby. However, a variety ofR dependences are allowed. Reasonable agreement with experimental data out to 4 AU is obtained with the above values ofD _TT and the spatial diffusion coefficientK _r=K₀R^?2 forR«1 andK _r=K₀R^0.4 forR»1 AU. It is only in the decay phases of prompt events as seen at 2–4 AU that significant differences in the temporal behaviour of the events can be distinguished, depending on the value ofD _TT chosen within the above range. Experimental determination of the decay constant is difficult.     

On the exponential decay of western hemisphere solar particle events

Numerical solutions of the Fokker-Planck equation governing the transport of solar protons are obtained in three dimensions (time t, radial distance r, energy T) with the diffusion coefficient represented by κ = κ₀r^bT^a. The October 4, 1968, solar flare particle event is re-examined, and the rise and decay of the proton flux profiles for > 10, ;30 and > 60 MeV particles can be reasonably well reproduced with an instantaneous injection and a distant (10 AU) free escape boundary. The best fit is achieved with a diffusion coefficient $\text{κ = 1.4 × 10}{}^{20}\text{r}{}^{0.5}\text{T}{}^{0.75}\text{cm}{}^2\text{sec}$ where r is in AU and T in MeV.     

Shock acceleration of solar cosmic rays

The solar cosmic ray (SCR) acceleration by the shocks driven by coronal mass ejections is studied by taking into account the generation of Alfvén waves by accelerated particles. Detailed numerical calculations of the SCR spectra produced during the shock propagation through the solar corona have been performed within a quasi-linear approach with a realistic set of coronal parameters. The resultant SCR energy spectrum is shown to include a power-law part N ∝ ?^-γ with an index γ = 1.7–3.5 that ends with an exponential tail. The maximum SCR energy lies within the range ? _max = 0.01–10 GeV, depending on the shock velocity V _S = 750–2500 km s^?1. The decrease of the shock Alfvénic Mach number due to the increase Alfvén velocity with heliocentric distance r leads to the end of the efficient SCR acceleration when the shock size reaches R _S ≈ 4R _⊙. In this case, the diffusive SCR propagation begins to exceed the shock velocity; as a result, SCRs escape intensively from the shock vicinity. The self-consistent generation of Alfvén waves by accelerated particles is accompanied by a steepening of the particle spectrum and an increase of their maximum energy. Comparison of the calculated SCR fluxes expected near the Earth’s orbit with the available experimental data shows that the theory explains the main observed features.     

Relation Between Type II Bursts and CMEs Inferred from STEREO Observations

The inner coronagraph (COR1) of the Solar Terrestrial Relations Observatory (STEREO) mission has made it possible to observe CMEs in the spatial domain overlapping with that of the metric type II radio bursts. The type II bursts were associated with generally weak flares (mostly B and C class soft X-ray flares), but the CMEs were quite energetic. Using CME data for a set of type II bursts during the declining phase of solar cycle 23, we determine the CME height when the type II bursts start, thus giving an estimate of the heliocentric distance at which CME-driven shocks form. This distance has been determined to be ～1.5R _s (solar radii), which coincides with the distance at which the Alfvén speed profile has a minimum value. We also use type II radio observations from STEREO/WAVES and Wind/WAVES observations to show that CMEs with moderate speed drive either weak shocks or no shock at all when they attain a height where the Alfvén speed peaks (～3R _s?–?4R _s). Thus the shocks seem to be most efficient in accelerating electrons in the heliocentric distance range of 1.5R _s to 4R _s. By combining the radial variation of the CME speed in the inner corona (CME speed increase) and interplanetary medium (speed decrease) we were able to correctly account for the deviations from the universal drift-rate spectrum of type II bursts, thus confirming the close physical connection between type II bursts and CMEs. The average height (～1.5R _s) of STEREO CMEs at the time of type II bursts is smaller than that (2.2R _s) obtained for SOHO (Solar and Heliospheric Observatory) CMEs. We suggest that this may indicate, at least partly, the density reduction in the corona between the maximum and declining phases, so a given plasma level occurs closer to the Sun in the latter phase. In two cases, there was a diffuse shock-like feature ahead of the main body of the CME, indicating a standoff distance of 1R _s?–?2R _s by the time the CME left the LASCO field of view.     

Energy losses and modulation of galactic cosmic rays

Numerical solutions of the cosmic-ray transport equation for the interplanetary region and including the effects of diffusion, convection, and energy loss, have been obtained for a galactic differential number-density spectrum of gaussian form, central kinetic energyT ₀, and a width at half height maximum of 0.1T ₀. These solutions have been used to show the redistribution in energy, and the reduction in number density, within the solar cavity. The mean energy loss is shown to be usefully approximated by the force-field potential when /T ₀1/2. The principal finding is that galactic cosmic-ray protons and helium nuclei, with kinetic energies less than about 80 MeV/nucleon, virtually, are completely excluded from near Earth by convective effects. As a result of this exclusion, it is found that it is not possible to demodulate the proton and helium-nuclei spectra, observed in 1964–65, to obtain information about low-energy galactic cosmic rays.     

The heating of the earth during its formation

The thermal state of the Earth accumulating from solid bodies is investigated. The conductivity equation is deduced for a growing spherically symmetrical planet which takes into account heating by impacts of bodies, by radioactivity, and by compression of its material. The cooling is produced mainly by impact mixing, which is approximated by extrapolating the parameters from known impact craters to larger sizes. The solution of a more simple conductivity equation for a uniformly heated plane parallel layer with moving boundaries is found. It can be considered as an approximate quasi-stationary solution for the temperature distribution in the outer parts of the growing Earth. The result depends substantially on the sizes of impacting bodies but almost not at all on the time scale of the accumulation. The latter only weakly affects the surface temperature and does not affect the temperature distribution in the layer. For bodies of small radii, r′ < r₁, where the size of the crater is not affected appreciably by gravitation (for the present mass of the Earth r₁ ≈ 1 km), the heating is small. For bodies with r′ > r₁, the heating of the layer is roughly proportional to the ratio $\text{r′}\text{r}{}_1$. Toward the end of the Earth's accumulation the melting point can be reached in the outer layer at r′_M ? 60 km, where r′_M is the radius of the largest body in the power law size spectrum of falling bodies. The estimates of the initial temperature of the Earth can vary within wide limits depending on the mass distribution of large protoplanetary bodies, which at the present time is not known correctly. The initial melting of an upper layer of the Earth a few hundred kilometers thick seems to be possible.     

The role of solar-wind velocity-waves in comet outburst activity

Comet outburst activity and the structure of solar wind streams were compared on the basis of Pioneer 10, 11, Vela 3 and IMP 7, 8 measurements at the heliocentric distance r ≈ 1–6 AU. It is shown that the solar wind velocity waves which are evolving into corotating shock waves beyond the Earth orbit may be responsible for comet outburst activity. The correlation between variations of comet outburst activity with heliocentric distance and the behavior of the solar wind velocity waves is established. The closeness of the characteristic times for the velocity waves and comet outburst activity (7–8 days at r = 1 AU) as well as the simultaneous growth of both the characteristic times with r are noted. The observed distribution of the comet outburst activity parameters during the 11-year cycle is also in good agreement with the phase distributions during the 11-year cycle of variations of the coronal hole areas and the rate of change of the sunspot area δS _p.     

Cosmic-ray streaming and anisotropies

The principal result of this paper is the demonstration that in interplanetary space the electric-field drifts and convective flow parallel to the magnetic field of cosmic-ray particles combine as a simple convective flow with the solar wind. In addition there are diffusive currents and transverse gradient drift currents. With this interpretation direct reference to the interplanetary electric-field drifts is eliminated and the study of steady-state and transient cosmic-ray anisotropies is both more systematic and simpler. Following a discussion of our present knowledge of the diffusion coefficient in the interplanetary medium, the theory is applied to steady-state anisotropies near Earth in the kinetic energy (T) range 7.5 MeV<T<20 GeV. First the theory of the diurnal variation atT>-2 GeV is examined and it is suggested that the azimuthal streaming associated with the observations be regarded simply as proof that there is no significant net radial flow of cosmic rays at these energies. Second, it is predicted that, near Earth, the radial anisotropy will have a (+?+) variation with energy and this prediction is very insensitive to the precise values of the parameters used: intensity spectrum, solar wind speed, radial density gradient, and diffusion coefficient. Then, third, the small and radial steady-state anisotropies reported by Raoet al. (1967) in the intervals 7.5<T<45 MeV and 45<T<90 MeV are re-examined and it is found that the gradients and diffusion coefficients required to produce the reported anisotropies in 7.5<T<45 MeV are inconsistent with those expected from other data.     

The entry of solar protons over the polar caps

Observations of solar protons at energies from 1 MeV to 360 MeV are examined in relation to the information that these particles give about the magnetosphere, magnetotail and magnetopause. Trajectory integrations in a realistic model of the geomagnetic field out to 25R_E and a tail field model fitted to observations from 15R_E to 80R_E are used to obtain a better understanding of the particle motion. The mean free path of protons in the tail is found to be 700R_E and 200R_E for 100 MeV and 1 MeV protons respectively, which indicates that trajectory calculations in a static field model are valid.     

A regularization of the three-body problem

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

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

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

The thermal structure of the dayside upper atmosphere of venus above 125 km

A one-dimensional model of the Venus thermosphere has been constructed which includes computation of the heating efficiency of solar ultraviolet radiation, heat loss by radiation to space of infrared-active species, thermal transport by molecular and eddy conduction, and viscous dissipation. By comparing model predictions with results obtained from the Pioneer Venus Orbiter space-craft, the results indicate that energy transport parameterized by eddy heat conduction plays a dominant role in determining thermospheric temperature T. It is suggested that there exists a feedback mechanism linking heating and thermospheric circulation such that eddy cooling maintains an asymptotic temperature T_∞～300°K for both solar-maximum and solar-minimum conditions. We also study the variation in thermospheric temperature with solar zenith angle, atomic oxygen-mixing ratio, rate of vibrational excitation of CO₂ by ground-state O atoms, and the assumed transfer of O(¹D) electronic energy to CO₂ vibrational energy.     

Evolution of the Solar Flare Energetic Electrons in the Inhomogeneous Inner Heliosphere

Solar flare accelerated electrons escaping into the interplanetary space and seen as type III solar radio bursts are often detected near the Earth. Using numerical simulations we consider the evolution of energetic electron spectrum in the inner heliosphere and near the Earth. The role of Langmuir wave generation, heliospheric plasma density fluctuations, and expansion of magnetic field lines on the electron peak flux and fluence spectra is studied to predict the electron properties as could be observed by Solar Orbiter and Solar Probe Plus. Considering various energy loss mechanisms we show that the substantial part of the initial energetic electron energy is lost via wave–plasma processes due to plasma inhomogeneity. For the parameters adopted, the results show that the electron spectrum changes mostly at the distances before ～?20 R _⊙. Further into the heliosphere, the electron flux spectrum of electrons forms a broken power law relatively similar to what is observed at 1 AU.     

Analysis of EUV limb-brightening observations from ATM

Limb-brightening curves for EUV resonance lines of O vi and Mg x have been constructed from spectroheliograms (5″ resolution) of quiet limb regions observed with the Harvard experiment on Skylab. The observations are interpreted with a simple model for the transition layer and the corona. A comparison of theoretical and observed limb-brightening curves indicates that the lower boundary of the corona, where T _e= 10⁶K, is at a height of about 8000 km in typical quiet areas. For 1.01 R _⊙?r1.25 R _⊙, the corona can be represented by a homogeneous model in hydrostatic equilibrium with a temperature of 10₆K for 1.01 R _⊙?r<1.1 R _⊙ and 1.1 × 10₆K for r?1.1 R _⊙. The model for the transition layer is inhomogeneous, with the temperature gradient a factor of 3 shallower in the network than in the intranetwork regions. It appears that spicules should be included in the model in order to account for the penetration into the corona of cool (T _e<10⁶K) EUV-emitting material to heights up to 20000 km above the limb.