Proton Distribution Function Research Articles

Stability analysis of double‐peaked proton distribution functions in the solar wind

Twelve proton velocity distribution functions as measured for different solar wind speeds by the plasma analyzer at the Helios spacecraft are modelled analytically by a superposition of two bi‐Maxwellians. The stability of these distributions is investigated by solving the linear dispersion equation for parallel propagation of both left‐ and right‐hand polarized modes. Due to the temperature anisotropy of the main proton component the presence of the resonant left‐hand ion cyclotron instability is found to be typical for high‐speed streams and is independent of an ion beam. The results indicate that the ratio of the beam density to total particle density and the beam drift velocity are generally close to the threshold values necessary to drive a highly unstable nonresonant right‐hand mode. This instability is found for one proton distribution and is a consequence of the beam particles. It is concluded that both the ion cyclotron instability and the ion beam instability act as a regulating mechanism on the shape of solar wind double humped proton distribution functions.

Solutions to bi-Maxwellian transport equations for SAR-arc conditions

We have compared solutions obtained from the bi-Maxwellian based 16-moment transport equations with those obtained from the Maxwellian based 13-moment transport equations for conditions leading to the steady state, subsonic flow of a fully-ionized electron-proton plasma along geomagnetic field lines in the vicinity of the plasmapause. The bi-Maxwellian based equations can account for large temperature anisotropies and the flow of both parallel and perpendicular thermal energy, while the Maxwellian based equations account for small temperature anisotropies and only the total heat flow. Our comparison indicates that for Stable Auroral Red arc (SAR-arc) conditions leading to strong field-aligned heat flows (temperatures of 8000 K and temperature gradients of4K. km −1 at 1500 km), the bi-Maxwellian based equations predict a different thermal structure in the topside ionosphere than the less rigorous Maxwellian based equations. In particular, the bi-Maxwellian based equations predict proton and electron temperature anisotropies with T ∥ > T ⊥, while the Maxwellian based equations predict the opposite behavior for the same boundary conditions. This difference is related to the way in which the temperature anisotropies and heat flows are treated in the two formulations. For the bi-Maxwellian based equations, the inclusion of separate heat flows for parallel and perpendicular thermal energy allows for the development of a pronounced tail in both the electron and proton distribution functions, which leads to temperature anisotropies with T ∥ > T ⊥. For the Maxwellian based equations, on the other hand, the tail development is restricted because only the total heat flow is considered. Consequently, as the heat flows down, the presence of an increasing magnetic field acts to produce an anisotropy with T ⊥ > T ∥, and this process dominates tail formation for the Maxwellian based equations.

Charged particle distributions and electric field measurements from S3‐3

Comparisons of electron and proton distribution functions are made with dc electric field measurements as the S3‐3 satellite crossed a series of high‐latitude acceleration regions near local dusk on July 29, 1976. Potential drops, from parallel electric fields, are inferred from charged particle distributions using adiabatic theory and compared with potentials calculated from measured perpendicular electric fields. There exists a one to one spatial relationship between strong dc electric field enhancements in the form of electric field reversals and acceleration signatures of auroral particles. From the charged particle and electric field analysis, we conclude that electric fields, parallel to B, can extend above and below the satellite altitude near 7500 km when strong electrostatic field reversals are encountered. Within the uncertainties of our analysis, the potential drop below the satellite is comparable to the potential calculated from the individual electrostatic structures crossed by the satellite. There are, in addition, particle signatures that imply a source of strong acceleration was located above and relatively close to the satellite. These accelerated electrons are coincident with auroral hiss and ion conics. In addition to regions of acceleration that extended only fractions of a degree in latitude, there was a broad ‘inverted V structure’ near the polar cap boundary where strong perpendicular electric fields were embedded, giving rise to large variations in the inferred potential below the satellite.

The distribution function of upstream energetic particle populations

In constructing the proton distribution function at low energies, one usually applies Compton‐Getting corrections to the observed anisotropies. This implies, among other things, that the spectrum as far as its functional form is concerned has to be defined with respect to a moving frame (solar wind). The justification for applying such corrections, in particular in case of the upstream events, will be discussed. Tools to investigate this ‘frame problem’ will be described, and some appropriate examples, taken from the many burst events as observed with the low energy proton spectrometer aboard ISEE 3, will be presented. It is found that none of the directional characteristics can be properly accounted for by Compton‐Getting effects.

Interplanetary acceleration of low energy protons in association with shock waves: The 11 February 1979 event

The problem of interplanetary acceleration of low energy protons in association with shock waves is examined in the context of the specific event observed on 11 February 1979 on board the ISEE-3 spacecraft. This event has been selected for special study as it apparently was not associated with a solar flare event. The low energy proton telescope system on ISEE-3 measures the proton distribution function with good spectral, directional and temporal resolution from E p = 35 keV. The evolution of the anisotropies and of the energy spectrum during the event are consistent with particle acceleration taking place in the vicinity of the shock wave.

A New Estimate of the H+ Ion Concentration in Pure Water. Part 1: A Critical Analysis of Current Treatments

AbstractIt is shown that attempts to determine the H+ concentration in pure water involve a search for a species which is not analytically well‐defined; thus appears the problem of what is the nature of the physical statements now available from pertinent measurements? Furthermore, it is demonstrated that both the conductance and the EMF determination of Kw erroneously involve an extrapolation from an aqueous solution to the pure solvent, water. Kw is not a strictly measurable quantity. Configurational definitions of the H+ and OH− ions are given which, using an intramolecular proton distribution function, essentially converts a statement concerning the distortion of a water molecule to species concentrations. The temperature dependence of the conventional Kw is also incorporated in the characterization of non‐analytical H+ and OH− ions.

Meson Structure Functions from High-Transverse-Momentum Hadron Interactions

Data on large--transverse-momentum ..pi../sup 0/ production by proton, pion, and kaon beams is used to extract the large-x behavior of meson structure functions. Good agreement is found with the proton and pion quark distribution functions obtained from deep inelastic and dimuon production experiments. This study provides the first look at the kaon structure function. A significantly faster falloff is found as x approaches unity than for that of pions.

Velocity distribution function and cyclotron wave growth in a modified bi‐Maxwellian two‐ion component solar plasma

Observed double‐humped proton velocity distributions of the solar plasma are represented analytically by a superposition of modified bi‐Maxwellians. Stability and growth rates of the cyclotron mode are calculated in linear Vlasov theory for a warm solar wind at 1 AU. Proton distribution function and cyclotron wave growth are discussed as functions of the beam to plasma particle density and the temperature anisotropy. The presence of a second ion component causes increased growth rates and a second resonant wave particle interaction.

Ion acoustic waves and related plasma observations in the solar wind

This paper presents an investigation of solar wind ion acoustic waves and their relationship to the macroscopic and microscopic characteristics of the solar wind plasma. Comparisons with the overall solar wind corotational structure show that the most intense ion acoustic waves usually occur in the low‐velocity regions ahead of high‐speed solar wind streams. Of the detailed plasma parameters investigated, the ion acoustic wave intensities are found to be most closely correlated with the electron to proton temperature ratio Te/Tp and with the electron heat flux. Investigations of the detailed electron and proton distribution functions also show that the ion acoustic waves usually occur in regions with highly non‐Maxwellian distributions characteristic of double‐proton streams. The distribution functions for the double‐proton streams are usually not resolved into two clearly defined peaks, but rather they appear as a broad shoulder on the main proton distribution. Two main mechanisms, an electron heat flux instability and a double‐ion beam instability, are considered for generating the ion‐acoustic‐like waves observed in the solar wind. Both mechanisms have favorable and unfavorable features. The electron heat flux mechanism can account for the observed waves at moderate to large ratios of Te/Tp but has problems when Te/Tp is small, as sometimes occurs. The ion beam instability appears to provide more flexibility in the Te/Tp ratio; however, detailed comparisons using observed distribution functions indicate that the ion beam mode is usually stable. Possible resolutions of these difficulties are discussed.

Ion-cyclotron noise excitation in the topside ionosphere according to observations at the Intercosmos-14 satellite

A band-limited ELF emission having very distinct upper as well as lower cutoff frequencies, most probably determined by local parameters of the plasma, were discovered by the VLF experiment on board the Intercosmos-14 satellite (apogee height 1707 km, perigee 345 km; inclination 74°). This type of noise was observed in the day-time, at middle and low latitudes, at altitudes 600 km< h < 1400 km. The very clear case observed on March 5, 1976 is discussed in detail and arguments are listed that this ELF emission is locally generated ion-cyclotron noise. Calculations of the growth of the proton cyclotron resonance instability show that the growth may be positive if the loss cone angle of the energetic protons is sufficiently great. This condition is fulfilled at low altitudes near the mirror points. Depending on the width of the distribution function of energetic protons the growth of the Instability may be positive in the whole frequency band of the proton branch of ion-cyclotron waves or only in certain parts of that frequency band.

Some characteristics of rapidly flowing magnetotail plasmas: Further comments on ‘On hot tenuous plasmas, fireballs, and boundary layers in the Earth's magnetotail’ by L. A. Frank, K. L. Ackerson, and R. P. Lepping

Evaluations of proton and electron temperatures and proton bulk flows are reported for several time intervals on April 18, 1974. It was found that proton thermal energies exceeded their flow energies and greatly exceeded the electron thermal energies. Also, a disagreement between the cold proton distribution function and that reported previously was rated. 5 references. (JFP)

Ion cyclotron instability in the solar wind

An ion cyclotron instability, arising because of the relative drift between the beam and the main components of the proton distribution function in the solar wind at 1 AU, is studied. The instability is excited in a bounded range of wave numbers provided the relative drift exceeds a certain minimum value called instability threshold. For \(\beta M = ({{8\pi NMT_\parallel M} \mathord{\left/ {\vphantom {{8\pi NMT_\parallel M} {B_0^2 )}}} \right. \kern-\nulldelimiterspace} {B_0^2 )}}\) ≳ 1, the instability threshold is smaller than or equal to the threshold of magnetosonic and Alfven instabilities. The growth rates are enhanced by increasing relative drift and ratio of beam to main proton number density and by decreasing the wave numbers.

Evidence for extended solar wind heating by fast hydromagnetic waves

In order to reduce the discrepancy between solar wind measurements at 1 AU and the corresponding predictions of the basic two-fluid model a nonthermal heating of solar wind protons in the outer corona up to distances of about 0.1 AU from the sun was postulated. In the absence of thermal energy transfer to an anisotropic plasma the quantities c∥ = T∥B²/n² and c⊥ = T⊥/B (where T∥ and T⊥ denote proton temperature components parallel and perpendicular to the magnetic field with intensity B and n denotes the proton number density) constitute adiabatic invariants. Accepting the hypothesis that damping of fast-mode hydromagnetic waves near the sun strongly dominates the heating and acceleration of the solar wind, we argue that c∥ is increased thereby. Depending on the coupling between parallel and perpendicular degrees of freedom in the turbulent solar envelope, c⊥ will be raised to some extent too. If we may neglect interplanetary irreversible processes, these values are transported to the earth and may give some direct information on the turbulent outer corona. We therefore invoke c⊥ as a new natural ordering parameter instead of the customarily employed bulk velocity in a statistical presentation of solar wind data measured on the earth-orbiting satellite Heos 2. This choice reveals completely novel features that have been missed in the conventional presentation. We interpret these features as a signature of an extended heating process which survives the transit of the solar wind plasma to 1 AU. We note that the purely wave-heated two-fluid model of Barnes and co-workers was unable to reach flow velocities exceeding 450 km/s. Our data consistently indicate that for larger flow velocities additional heating and acceleration processes (e.g., nonspherically symmetric outflow from coronal holes) come into play. We suggest that a suitable combination of a corotating model and the wave-heated two-fluid model should be able to predict both the observed macroscale relation between proton temperature and bulk velocity and the dynamic effects in the moderately active solar wind. Further, we propose that proton double streams and related deformations of the proton distribution functions might be a relic of the interaction with a fast-mode hydromagnetic wave field which propagates from the inner corona.

Nonlinear Alfvén waves in high-speed solar wind streams

A nonlinear proton distribution function that is an exact stationary solution of the nonlinear Vlasov equation and Maxwell's equations and which supports a single nonlinear transverse Alfven (ion cyclotron) wave that is circularly polarized and nondispersive is proposed for most of the observations during high-speed solar wind streams. This nonlinear distribution removes the strong Alfven wave instability, inconsistent with the persistence of the observed proton distribution functions in high-speed streams, found by the linear stability analysis. Model temperature anisotropies and drift velocities of the two spatially inhomogeneous bi-Maxwellian components are consistent with typical proton velocity distributions measured in high-speed streams at 1 AU. Two derived relations for each of the wave number and the phase velocity of the wave are obeyed within experimental uncertainties by two typical proton measurements. Our model also predicts that the alpha particle bulk flow velocity exceeds the proton particle bulk flow velocity, as is observed.

Electromagnetic ion-cyclotron wave growth rates and their variation with velocity distribution function shape

The sensitivity of electromagnetic ion-cyclotron wave growth rates to the details of the shape of proton velocity distribution functions is explored. For this purpose two different forms of bi-Lorentzian for the proton distribution functions were adopted. The growth rates for the two types of bi-Lorentzians and the biMaxwellians for the beam (hot) protons are compared. Although the growth rates for the three shapes depend on the velocity moments of the different velocity distributions in a similar way, their magnitudes were found to vary considerably.

On the azimuthal drift of substorm-injected protons

The longitude and time dependence of the pitch angle distribution function of protons injected during substorms on a given L shell is studied. The results are compared with the experimental data of Konradi et al. (1973). The theoretical importance of the anisotropy of the initial distribution function and of the longitudinal width of the injection region is stressed. It is suggested that measurements be made at low pitch angles (< or =20degree). (AIP)

Rocket-borne double focusing analyzer for particle spectroscopy in the lower magnetosphere

We describe a rocket-borne instrument for measurement of the low energy electron and proton distribution function in the magnetosphere. It consists of an optimized programmable electrostatic toroidal analyzer as an energy selector, coupled with an electron multiplier as a particle detector, and of the associated electronics. Much attention has been paid to the experimental calibration of the instrument which gives us some understanding of the real behavior of the detector and the way it influences the analyzer properties.

Effects of the secular magnetic variation on the distribution function of inner-zone protons

The equation governing the distribution function of high-energy protons in the earth's radiation belt has been modified to include the effects of the secular change of the earth's magnetic dipole moment in addition to the effects of albedo neutron decay, collisional energy loss, and radial diffusion, which have been studied in the past. Because the time scale of the magnetic-field change (∼2000 years) is much longer than the time scale of proton trapping (∼200 years) an approximate solution of the equation can be obtained. This solution indicates that the shrinking dipole increases the distribution function at the geomagnetic equator for 1.15 < L < 1.7 and 15 < E < 160 Mev. The greatest increase occurs at L = 1.2, where it is as large as a factor of 5. The addition of this effect improves the agreement between the theoretically computed distribution function and the experimental data.

Higher moment equations and the distribution function of the solar-wind plasma

Study of the higher-moment equations for a collisionless fully ionized plasma. For a collisionless, heat-conducting plasma, the distribution function f is cylindrically symmetric about the direction of the magnetic field. It is shown that under a certain assumption the fourth moments of f can be expressed as simple functions of lower moments. Thus no higher-moment terms appear in the third-moment equations. The two third-moment equations, which are obtained in a simple form, join other lower-moment equations to form a closed set of moment equations. The new equations can be used to study the thermal anisotropy and the heat flux of the solar-wind proton. A special case of the cylindrically symmetric distribution function f is found to resemble the proton distribution function reconstructed from solar-wind data, and this resemblance justifies the assumption needed for decoupling the moment equations.

Possible mechanisms for formation of the ring current belt

A numerical model for the magnetosphere is used to investigate possible mechanisms for the injection of low-energy protons into the magnetosphere. The proton distribution function is assumed to consist of two monoenergetic components. The effects of day-night asymmetries in the geomagnetic field, the earth's rotation, and a finitely conducting ionosphere are included in the model. The model was used to determine if one group of particles could result in the injection of another group of particles. The calculations indicated that coupling between particles of different energies would be ineffective in injecting particles. Another numerical experiment sought to determine the effect of the internal polarization fields of the injected particles on particle injection by an externally imposed convection electric field. It was found that the external field tended to be canceled out by the internally generated field. The calculations indicate that an external convection field is unlikely to result in the injection of significant particles deep within the magetosphere unless the effective ionospheric conductivity is very high, as might exist in an ionosphere bombarded by auroral ionization. Another experiment was used to investigate the effect of the earth's rotation and the asymmetric geomagnetic field in causing injection. It was found that significant injection of 80-ev/γ particles occurred on a time scale of a day without any assist from external electric fields. It is pointed out that much faster and deeper injection may occur if a self-consistent model of the geomagnetic field were used in the calculations. A mechanism of this type would also require the existence of a highly conductive ionosphere. A final numerical experiment suggests that certain features of the auroral substorm could result from rapid changes in the effective ionospheric conductivity.