Pitch Angle Distribution Function Research Articles

Probing the Low-Mass End of the Black Hole Mass Function via a Study of Faint Local Spiral Galaxies

We present an analysis of the pitch angle distribution function (PADF) for nearby galaxies and its resulting black hole mass function (BHMF) via the well-known relationship between pitch angle and black hole mass. Our sample consists of a subset of 74 spiral galaxies from the Carnegie-Irvine Galaxy Survey with absolute B-band magnitude MB>−19.12 mag and luminosity distance DL≤25.4 Mpc, which is an extension of a complementary set of 140 more luminous (MB≤−19.12 mag) late-type galaxies. We find the PADFs of the two samples are, somewhat surprisingly, not strongly dissimilar; a result that may hold important implications for spiral formation theories. Our data show a distinct bimodal population manifest in the pitch angles of the Sa–Sc types and separately the Scd–Sm types, with Sa–Sc types having tighter spiral arms on average. Importantly, we uncover a distinct bifurcation of the BHMF, such that the Sa–Sc galaxies typically host so-called “supermassive” black holes (M•≳106M⊙), whereas Scd–Sm galaxies accordingly harbor black holes that are “less-than-supermassive” (M•≲106M⊙). It is amongst this latter population of galaxies where we expect fruitful bounties of elusive intermediate-mass black holes (IMBHs), through which a better understanding will help form more precise benchmarks for future generations of gravitational wave detectors.

Spectral Power-law Formation by Sequential Particle Acceleration in Multiple Flare Magnetic Islands

Abstract We present a first-principles model of pitch-angle and energy distribution function evolution as particles are sequentially accelerated by multiple flare magnetic islands. Data from magnetohydrodynamic (MHD) simulations of an eruptive flare/coronal mass ejection provide ambient conditions for the evolving particle distributions. Magnetic islands, which are created by sporadic reconnection at the self-consistently formed flare current sheet, contract and accelerate the particles. The particle distributions are evolved using rules derived in our previous work. In this investigation, we assume that a prescribed fraction of particles sequentially “hops” to another accelerator and receives an additional boost in energy and anisotropy. This sequential process generates particle number spectra that obey an approximate power law at mid-range energies and presents low- and high-energy breaks. We analyze these spectral regions as functions of the model parameters. We also present a fully analytic method for forming and interpreting such spectra, independent of the sequential acceleration model. The method requires only a few constrained physical parameters, such as the percentage of particles transferred between accelerators, the energy gain in each accelerator, and the number of accelerators visited. Our investigation seeks to bridge the gap between MHD and kinetic regimes by combining global simulations and analytic kinetic theory. The model reproduces and explains key characteristics of observed flare hard X-ray spectra as well as the underlying properties of the accelerated particles. Our analytic model provides tools to interpret high-energy observations for missions and telescopes, such as RHESSI, FOXSI, NuSTAR, Solar Orbiter, EOVSA, and future high-energy missions.

Whistler instability driven by the sunward electron deficit in the solar wind

Context. Solar wind electrons play an important role in the energy balance of the solar wind acceleration by carrying energy into interplanetary space in the form of electron heat flux. The heat flux is stored in the complex electron velocity distribution functions (VDFs) shaped by expansion, Coulomb collisions, and field-particle interactions. Aims. We investigate how the suprathermal electron deficit in the anti-strahl direction, which was recently discovered in the near-Sun solar wind, drives a kinetic instability and creates whistler waves with wave vectors that are quasi-parallel to the direction of the background magnetic field. Methods. We combined high-cadence measurements of electron pitch-angle distribution functions and electromagnetic waves provided by Solar Orbiter during its first orbit. Our case study is based on a burst-mode data interval from the Electrostatic Analyser System (SWA-EAS) at a distance of 112 RS (0.52 au) from the Sun, during which several whistler wave packets were detected by Solar Orbiter’s Radio and Plasma Waves (RPW) instrument. Results. The sunward deficit creates kinetic conditions under which the quasi-parallel whistler wave becomes unstable. We directly test our predictions for the existence of these waves through solar wind observations. We find whistler waves that are quasi-parallel and almost circularly polarised, propagating away from the Sun, coinciding with a pronounced sunward deficit in the electron VDF. The cyclotron-resonance condition is fulfilled for electrons moving in the direction opposite to the direction of wave propagation, with energies corresponding to those associated with the sunward deficit. Conclusions. We conclude that the sunward deficit acts as a source of quasi-parallel whistler waves in the solar wind. The quasilinear diffusion of the resonant electrons tends to fill the deficit, leading to a reduction in the total electron heat flux.

Pitch-angle diffusion coefficients in test particle simulations and the estimation of the particle parallel mean free path

The transport of energetic charged particles in turbulent magnetic fields is a topic of interest in various astrophysical contexts. In order to estimate the mean free path of a particle in the direction parallel to the mean magnetic field, one can use theoretical expressions that employ pitch-angle diffusion coefficients. In this work we review some of the methods used in estimating pitch-angle diffusion coefficients from test particle computer simulations. We examine if these methods and theoretical approaches are able to provide consistent estimates of the parallel mean free path, that can also be obtained directly from computer simulations. We perform test particle simulations for synthetic turbulence models over a range of turbulence parameters and particle energies. From the trajectories of test particles, pitch-angle distribution functions and statistics of pitch-angle displacements are obtained, which are then used to estimate the pitch-angle diffusion coefficients. We find that a method using the pitch-angle flux and derivative of the pitch-angle distribution is able to provide accurate values for the parallel mean free path over the range of parameters considered. Other methods considered are accurate only for a limited range of the turbulent fluctuation strength, or must be evaluated at a specific time to provide a reasonable estimate.

Whistler wave generation by electron temperature anisotropy during magnetic reconnection at the magnetopause

Two magnetopause reconnection events of the Magnetospheric Multiscale mission with whistler wave activity are presented. The whistler mode around half of the electron cyclotron frequency is excited near the magnetospheric separatrix. In both events, there are positive correlations between the whistler wave and the lower hybrid drift instability (LHDI) activities, indicating a possible role of LHDI in the whistler wave generation. A sudden change in the electron pitch angle distribution (PAD) function of energetic electrons is observed right after intense LHDI activity. This change in the PAD leads to temperature anisotropy in energetic electrons which is responsible for the whistler wave excitation. The measured dispersion relation demonstrates that the whistler wave propagates toward the X line nearly parallel to the magnetic field line. Finally, a linear analysis with the measured distribution function verifies that the whistler mode is excited by the temperature anisotropy in energetic electrons.

Analytical forms of the first 14 moments of the cosmic ray Fokker–Planck equation

The Fokker–Planck transport equation describing the motion of energetic particles through a plasma is explored analytically. The latter equation provides a pitch-angle and position-dependent distribution function of the charged particles. In the current paper the first 14 moments of this equation are computed exactly for an arbitrary initial pitch angle. Such analytical forms are required in nonlinear treatments of perpendicular transport and other scenarios in plasma physics and astrophysics.

On the synchrotron emission in kinetic simulations of runaway electrons in magnetic confinement fusion plasmas

Developing avoidance or mitigation strategies of runaway electrons (REs) in magnetic confinement fusion (MCF) plasmas is of crucial importance for the safe operation of ITER. In order to develop these strategies, an accurate diagnostic capability that allows good estimates of the RE distribution function in these plasmas is needed. Synchrotron radiation (SR) of RE in MCF, besides of being one of the main damping mechanisms for RE in the high energy relativistic regime, is routinely used in current MCF experiments to infer the parameters of RE energy and pitch angle distribution functions. In the present paper we address the long standing question about what are the relationships between different REs distribution functions and their corresponding synchrotron emission simultaneously including: full-orbit effects, information of the spectral and angular distribution of SR of each electron, and basic geometric optics of a camera. We study the spatial distribution of the SR on the poloidal plane, and the statistical properties of the expected value of the synchrotron spectra of REs. We observe a strong dependence of the synchrotron emission measured by the camera on the pitch angle distribution of runaways, namely we find that crescent shapes of the spatial distribution of the SR as measured by the camera relate to RE distributions with small pitch angles, while ellipse shapes relate to distributions of runaways with larger the pitch angles. A weak dependence of the synchrotron emission measured by the camera with the RE energy, value of the q-profile at the edge, and the chosen range of wavelengths is observed. Furthermore, we find that oversimplifying the angular dependence of the SR changes the shape of the synchrotron spectra, and overestimates its amplitude by approximately 20 times for avalanching runaways and by approximately 60 times for mono-energetic distributions of runaways.

Martian electron foreshock from MAVEN observations

AbstractFlux enhancements of energetic electrons are always observed when the Mars Atmosphere and Volatile EvolutioN (MAVEN) spacecraft is magnetically connected to the shock. The observations indicate that the foreshock electrons consist of two populations. The most energetic (E≥237 eV) originate from a narrow region at the nearly perpendicular shock. They always appear as spikes, and their flux level reaches a maximum when the angle θBn approaches 90°. The other population emanates from the entire Martian bow shock surface, and the flux level decreases slightly from the quasi‐parallel to quasi‐perpendicular regions. A detailed examination of the pitch angle distribution shows that the enhanced fluxes are associated with electrons moving sunward. Annulus centered along the interplanetary magnetic field direction is the most stringent feature of the 3‐D angular distribution. The gyrotropic character is observed over the whole range of shock geometry. Although such signatures in the electron pitch angle distribution function strongly suggest that the reflection off the shock of a fraction of the solar wind electrons is the main mechanism for the production of Martian foreshock electrons, the decay of the flux of the second population on the other hand has yet to be understood.

SIMULATION OF ENERGETIC PARTICLE TRANSPORT AND ACCELERATION AT SHOCK WAVES IN A FOCUSED TRANSPORT MODEL: IMPLICATIONS FOR MIXED SOLAR PARTICLE EVENTS

ABSTRACT We use numerical solutions of the focused transport equation obtained by an implicit stochastic differential equation scheme to study the evolution of the pitch-angle dependent distribution function of protons in the vicinity of shock waves. For a planar stationary parallel shock, the effects of anisotropic distribution functions, pitch-angle dependent spatial diffusion, and first-order Fermi acceleration at the shock are examined, including the timescales on which the energy spectrum approaches the predictions of diffusive shock acceleration theory. We then consider the case that a flare-accelerated population of ions is released close to the Sun simultaneously with a traveling interplanetary shock for which we assume a simplified geometry. We investigate the consequences of adiabatic focusing in the diverging magnetic field on the particle transport at the shock, and of the competing effects of acceleration at the shock and adiabatic energy losses in the expanding solar wind. We analyze the resulting intensities, anisotropies, and energy spectra as a function of time and find that our simulations can naturally reproduce the morphologies of so-called mixed particle events in which sometimes the prompt and sometimes the shock component is more prominent, by assuming parameter values which are typically observed for scattering mean free paths of ions in the inner heliosphere and energy spectra of the flare particles which are injected simultaneously with the release of the shock.

THE BLACK HOLE MASS FUNCTION DERIVED FROM LOCAL SPIRAL GALAXIES

We present our determination of the nuclear supermassive black hole (SMBH) mass function for spiral galaxies in the Local Universe, established from a volume-limited sample consisting of a statistically complete collection of the brightest spiral galaxies in the Southern Hemisphere. Our SMBH mass function agrees well at the high-mass end with previous values given in the literature. At the low-mass end, inconsistencies exist in previous works that still need to be resolved, but our work is more in line with expectations based on modeling of SMBH evolution. This low-mass end of the spectrum is critical to our understanding of the mass function and evolution of SMBHs since the epoch of maximum quasar activity. A luminosity distance $\leq$ 25.4 $Mpc$ and an absolute B-band magnitude $\leq$ -19.12 define the sample. These limits define a sample of 140 spiral galaxies, with 128 measurable pitch angles to establish the pitch angle distribution for this sample. This pitch angle distribution function may be useful in the study of the morphology of late-type galaxies. We then use an established relationship between the pitch angle and the mass of the central SMBH in a host galaxy in order to estimate the mass of the 128 respective SMBHs in this sample. This result effectively gives us the distribution of mass for SMBHs residing in spiral galaxies over a lookback time $\leq$ 82.1 $h_{67.77}^{-1}$ $Myr$ and contained within a comoving volume of 3.37 $\times$ $10^4$ $h_{67.77}^{-3}$ $Mpc^3$. We estimate the density of SMBHs residing in spiral galaxies in the Local Universe is $5.54_{-2.73}^{+6.55}$ $\times$ $10^4$ $h_{67.77}^3$ $M_{\odot}$ $Mpc^{-3}$. Thus, our derived cosmological SMBH mass density for spiral galaxies is $\Omega_{BH} = 4.35_{-2.15}^{+5.14}$ $\times$ $10^{-7}$ $h_{67.77}$.

Absolute Flux Comparison of Magnetospheric Particles

Problem statement: The calculation of magnetospheric particle flux by dividing the particle count rate by the instrument geometric fac tor does not take into account the anisotropic pitch angle distribution function. A comparison of particle fluxes measured by different instruments fail to lead to the right comparison. T o circumvent the omission of anisotropy of the pitch angle distribution and to make the correct co mparison of particle fluxes, the instrument response function to different pitch angles within the sampling range of the instrument has been incorporated into the count rates over a certain re adout time. Conclusion/Recommendations: A quantity in absolute comparison of magnetospheric p article flux has been found. The newly defined quantity of different observations does lea d to the correct comparison for studying temporal variations. Investigators interested to st udy temporal features of magnetospheric particles over epochs may find the response functio ns for different instruments flown at different epochs useful for their study.

PITCH ANGLE SCATTERING IN THE OUTER HELIOSHEATH AND FORMATION OF THEINTERSTELLAR BOUNDARY EXPLORERRIBBON

The dominant and unexpected feature in the first Interstellar Boundary EXplorer (IBEX) maps is a ribbon of the enhanced energetic neutral atom (ENA) emissions. Presenting the first results from IBEX, McComas et al. identified six possible mechanisms of ribbon formation. One of the mechanisms, the so-called mechanism, was already quantitatively elaborated by Heerikhuisen et al., and they successfully reproduced the main features of the ribbon. We further study the secondary ENA mechanism by quantifying a previously omitted stage of the proton evolution between two consecutive acts of the charge-exchange in the outer heliosheath (OHS). The main findings can be summarized as follows. (1) The neutrals supplied by the supersonic near-equatorial solar wind dominate the near-equatorial source of the keV ENAs in the OHS compared to the inner heliosheath contribution. (2) The ribbon of the observed width can be produced even if only the large-scale (~102-104 AU) interstellar turbulence operates but the resulting pitch angle distribution functions (PADFs) are unstable with respect to the ion cyclotron wave generation around the locus where the line of sight from IBEX to the ribbon is perpendicular to the interstellar magnetic field beyond the heliopause. (3) A combination of the large-scale interstellar turbulence and a small-scale (~10–5to10–4 AU) turbulence generated by an unstable PADF of the energetic protons is able to make PADF marginally stable. In this case, the ribbon is still narrow because only a small part of the proton phase space distribution function can resonate with a locally generated ion cyclotron turbulence. (4) A concurrent operation of the large-scale interstellar turbulence and the locally generated small-scale turbulence might be responsible for the localized emission structures observed in the IBEX ribbon.

Energy dependence of electron pitch angle distribution widths in solar bursts

Solar activity regularly produces suprathermal electron bursts at energies below 1.4 keV. At 1 AU, these bursts have been detected at energies as low as 73 eV. The characteristics of such bursts vary considerably from event to event due in part to the physical processes involved in their propagation to Earth. In 2002, Advanced Composition Explorer/Solar Wind Electron Proton Alpha Monitor (ACE/SWEPAM) observed 101 solar electron bursts at energies below 1.4 keV, 40 of which exhibited broader pitch angle distributions than the preburst strahl. In general, the width of the pitch angle distribution from 73 to 1370 eV is a function of energy. In ∼65% of the events the pitch angle distribution width increased with energy during the burst, in ∼10% of the events it decreased with energy, and ∼25% of the cases demonstrated no clear energy dependence. The delay time between the onset of the burst and the onset of pitch angle distribution broadening also showed energy dependence. At 987 eV broadening almost always occurred after onset of the burst; however, at energies less than 519 eV broadening of the pitch angle distribution sometimes preceded the burst onset. Although energy dispersion was observed in the burst onset times, several events revealed nearly simultaneous broadening over an extended range of energies, in one case from 197 to 1370 eV. Broadening of the pitch angle distribution function during solar electron bursts almost certainly results from wave‐particle scattering. Scattering, in turn, should depend on solar wind plasma and magnetic field conditions. Nevertheless, we were not able to find any general statistical correlations between ACE plasma and magnetic field parameters and the burst pitch angle distribution width. Because ACE measurements are inherently local, whereas scattering can occur anywhere between the Sun and Earth, we also examined five bursts that we infer underwent relatively local scattering. These bursts exhibited maximum pitch angle distribution widths greater than 75°. Even these special cases failed to manifest any correlation with local plasma or magnetic field measurements.

Widths of suprathermal pitch angle distributions during solar electron bursts: ACE observations

Solar active processes frequently produce electron bursts which, as observed at 1 AU, extend down to energies less than 1.4 keV. The characteristics of these solar electron bursts vary considerably from event to event due in part to the physical processes involved in their propagation to Earth. We present a thorough survey of 71 low‐energy solar electron bursts that extended down to least 713 eV as observed by ACE/SWEPAM in 2002. This study characterizes properties of these bursts, including the preburst and maximum burst intensity, and the burst duration, as well as properties of the solar electron pitch angle distributions, including preburst strahl width and maximum burst beam width. Approximately 60% of bursts observed down to at least 713 eV have, on average, broader beams than the preceding strahl. Our analysis shows that all bursts in this survey have similar preburst strahl widths and electron intensities. Consequently, we conclude that beam broadening does not depend on preburst conditions. Furthermore, we find preferential associations between bursts that exhibit increased beam widths and several burst characteristics, namely, (1) high‐intensity bursts, (2) bursts observable below 519 eV, and (3) long‐duration bursts. Broadening of the pitch angle distribution function during solar electron bursts almost certainly results from particle scattering. Since the broadening of the pitch angle distribution function during bursts does not depend on preburst conditions and it does depend on burst characteristics, we suggest that scattering may result from self‐generated fluctuations which should increase in magnitude as the energy density of the burst beam increases. Finally, we describe two sources of wave‐particle interactions that offer plausible explanations for broadening of the electron pitch angle distribution during solar bursts.

The Compton‐Getting Effect of Energetic Particles with an Anisotropic Pitch‐Angle Distribution: An Application toVoyager 1Results at ∼85 AU

This paper provides a theoretical simulation of anisotropy measurements by the Low-Energy Charged Particle (LECP) experiment on Voyager. The model starts with an anisotropic pitch-angle distribution function in the solar wind plasma reference frame. It includes the effects of both Compton-Getting anisotropy and a perpendicular diffusion anisotropy that possibly exists in the upstream region of the termination shock. The calculation is directly applied to the measurements during the late 2002 particle event seen by Voyager 1. It is shown that the data cannot rule out either the model with zero solar wind speed or the one with a finite speed on a qualitative basis. The determination of solar wind speed using the Compton-Getting effect is complicated by the presence of a large pitch-angle distribution anisotropy and a possible diffusion anisotropy. In most high-energy channels of the LECP instrument, because the pitch-angle distribution anisotropy is so large, a small uncertainty in the magnetic field direction can produce very different solar wind speeds ranging from 0 to >400 km s-1. In fact, if the magnetic field is chosen to be in the Parker spiral direction, which is consistent with the magnetometer measurement on Voyager 1, the derived solar wind speed is still close to the supersonic value. Given the uncertainty of the magnetic field direction, only the two lowest energy channels of the LECP instrument can give a definitive result for the solar wind speed. However, these channels contain very high levels of background from their response to isotropic cosmic rays. An uncertainty of just a few percent in the background level can entirely hamper the estimate of solar wind speed.

The Compton–Getting effect of energetic particles with an anisotropic pitch-angle distribution

This paper presents a simulation of anisotropy measurements by the low-energy charged particle (LECP) experiment on Voyager 1 for cases when the particle pitch-angle distribution function in the solar wind plasma reference frame is not isotropic. The model includes both the Compton–Getting anisotropy and perpendicular diffusion anisotropy that possibly exists in the upstream region of the termination shock. The results show that the Voyager 1 data cannot rule out either the model with zero solar wind speed or the one with a finite speed on qualitative basis. The determination of solar wind speed using the Compton–Getting effect is affected by the assumption of the magnetic field direction and perpendicular diffusion anisotropy. Because the pitch-angle distribution anisotropy is so large, a small uncertainty in the magnetic field direction can produce very different solar wind speeds ranging from ∼0 to >400 km/s. In fact, if the magnetic field is chosen to be in the Parker spiral direction, which is consistent with the magnetometer measurement on Voyager 1, the derived solar wind speed is still close to the supersonic value. Only the two lowest-energy channels of the LECP instrument may give a definitive answer to the solar wind speed. However, because these channels contain a very high level of cosmic ray background, an uncertainty of just a few percent in the background can entirely hamper the estimate of solar wind speed.

Natural spectral bandwidth of electron cyclotron maser emission

We studied the spectral properties of the electron cyclotron maser emission generated by anisotropic distributions of fast electrons with power-law Magnitude distributions in solar flares. The 19 natural bandwidth of the generated spectral line depends significantly on the angle between the direction of the wave emission and the direction of the magnetic field in the Source. In addition, the line bandwidth depends on the parameters of the momentum and pitch-angle distribution functions for fast electrons. The 14 typical spectral bandwidths are shown to lie within the range 0.1-0.4%, in agreement with the minimum observable bandwidths of millisecond solar radio spikes. (C) 2004 MAIK"Nauka/lnterperiodica".

Slowing-Down Dynamics of Fast Particles in Plasmas via the Fokker-Planck Equation

A detailed discussion is given of the effects of energy diffusion and pitch-angle scattering on the slowing-down dynamics of a beam of monoenergetic particles being released with unidirectional velocity. Approximate solutions are given for characteristic averaged quantities like the pitch-angle averaged distribution function and different physically relevant velocity moments. The relation to previous exact investigations is discussed.

Return current region aurora: E∥, jz, particle energization, and broadband ELF wave activity

In this paper we consider ion heating in downward current, downward electric field, premidnight auroral regions. We show a case study, a test particle model, and a statistical study. The case study and statistical study use electron, ion, wave, and field‐aligned current data and include wave spectra and ion pitch angle distributions. The ion pitch angle distribution functions are used to calculate, through the momentum balance equation, a self‐consistent local parallel electric field. The test particle model shows ion pitch angle distributions and moments resulting from a “pressure‐cooker” arrangement of magnetic and electric fields. This simple model can reproduce the ion pitch angle distributions and the moments of these distributions as seen in the statistical database; the parallel electric field calculated from these moments is consistent with the imposed electric field profile. The statistical database uses data from 23 premidnight auroral Fast Auroral Snapshot crossings. The statistical study shows that the ion to electron energy ratio varies from 1 to 10; the broadband ELF wave power is controlled by the local value of jz · E∥; the relationship between jz and E∥ is controlled by ni; and the wave power near the cyclotron frequency is sufficient to provide the observed ion energies for oxygen in all cases and the observed ion energies for hydrogen for events near the poleward edge. The strong correlation between the wave power and jz · E∥ validates the use of the ion distributions as a probe of the local E∥. As expected from theoretical arguments, the current‐voltage relationship in these regions is seen to be controlled by the local ionospheric density. Allowing for suitable range in values of the density, the model can be applied to both strong return‐current regions within the auroral cavity and weak‐field regions at the poleward boundary.

Electron scattering loss in Earth's inner magnetosphere: 2. Sensitivity to model parameters

The sensitivity of the rate of energetic electron pitch angle scattering and precipitation loss in the Earth's magnetosphere due to Coulomb interactions with thermal plasma and resonant wave‐particle interactions with plasmaspheric hiss, lightning‐generated whistlers and VLF transmitter signals is computed for a realistic range of plasma and wave parameters. The computed scattering rates are most sensitive to the choice of mean wave frequency and the average angle of propagation; the frequency bandwidth and angular spread in propagation direction are relatively unimportant. Variations in plasma density can also strongly modulate the effectiveness of wave‐particle interactions. The average intensity of plasma waves controls electron lifetimes, but has little effect on the shape of the equilibrium pitch angle distribution function. All three classes of waves must be included together with Coulomb collisions in any realistic analysis of inner magnetospheric electron scattering. The long‐term energetic electron population beyond the inner belt is largely controlled by plasmaspheric hiss and, to a lesser extent, by lightning‐generated whistlers, while VLF transmitter signals constitute the principal loss mechanism over the range 1.3 ≤ L ≤ 2.4. The region below L = 1.3 is dominated by Coulomb collisions.