Effects of the position of the solar wind termination shock and the heliopause on the heliospheric modulation of cosmic rays

The effects of changing the position of the solar wind termination shock and the position of the heliopause, and therefore the extent of the heliosheath, on the modulation of cosmic ray protons are illustrated. An improved numerical model with diffusive termination shock acceleration, a heliosheath and drifts is used. The modulation is computed in the equatorial plane and at 35 heliolatitude using recently derived diffusion coefficients applicable to a number of cosmic ray species during both magnetic polarity cycles of the Sun. It was found that qualitatively the modulation results for the different heliopause positions are similar although they differ quantitatively, e.g., clearly different radial gradients are predicted for the regions beyond the termination shock compared to inside the shock. The difference between the modulation for the two solar polarity cycles are less significant at a heliolatitude of 35° than in the equatorial plane. We found that moving the termination shock from 90 to 100 AU, with the heliopause fixed at 120 AU, caused only quantitative differences so that the exact position of the TS in the outer heliosphere seems not crucially important to global modulation. Moving the heliopause outwards, to represent the modulation in the tail region of the heliosphere, causes overall decreases in the cosmic ray intensities but not linearly as a function of energy, e.g., at 1 GeV the effect is insignificant. We conclude from this modelling that the modulation of protons in the heliospheric nose and tail regions are qualitatively similar although, clear quantitative and interesting differences occur.     

The heliospheric modulation of cosmic rays: Effects of a latitude dependent solar wind termination shock

Observations made with the two Voyager spacecraft confirmed that the solar wind decelerates to form the heliospheric termination shock and that it has begun its merger with the local interstellar medium. The compression ratio of this shock affects galactic cosmic rays when they enter the heliosphere. Hydrodynamic (HD) models show that the compression ratio can have a significant latitude dependence; with the largest value in the nose direction of the heliosphere, becoming significantly less towards the polar regions. The modulation effects of such large latitude dependence are studied, using a well-established numerical drift and shock modulation model. We focus on computing the modulated spectra for galactic protons with emphasis on the radial and polar gradients in the equatorial plane and at a polar angle of θ = 55°, corresponding to the heliolatitude of Voyager 1. Two sets of solutions are computed and compared each time; with and without a latitude dependence for the compression ratio. All computations are done for the two magnetic field polarity cycles assuming solar minimum conditions. Including the termination shock in the model allows the study of the re-acceleration of galactic protons in the outer heliosphere. We find that for the A < 0 polarity cycle the intensity between ∼200 MeV and ∼1 GeV in the vicinity of the shock in the heliospheric equatorial plane may exceed the local interstellar value specified at the heliopause. Unfortunately, at θ = 55°, the effect is reduced. This seems not possible during an A > 0 cycle because significant modulation is then predicted between the heliopause and the termination shock, depending on how strong global gradient and curvature drifts are in the heliosheath. The overall effect of the shock on galactic protons in the equatorial plane is to reduce the total modulation as a function of radial distance with respect to the interstellar spectrum. Making the compression ratio latitude dependent enhances these effects at energies E < 200 MeV in the equatorial plane. At larger heliolatitudes these effects are even more significant. The differences in the modulation between the two drift cycles are compelling when the compression ratio is made latitude dependent but at Earth this effect is insignificant. A general result is that the computed radial gradient changes for galactic protons at and close to the TS and that these changes are polarity dependent. In line with previous work, large polarity dependent effects are predicted for the inner heliosphere and also close to the shock’s position in the equatorial plane. In contrast, at θ = 55°, the largest polarity effect occurs in the middle heliosphere (50 AU), enhanced by the latitude dependence of the compression ratio. At this latitude, the amount of proton modulation between the heliopause and the termination shock is much reduced. If galactic cosmic rays were to experience some diffusive shock acceleration over the 100–1000 MeV range at the shock, the radial gradient should change its sign in the vicinity of the shock, how large, depends on the compression ratio and the amount of drifts taking place in the outer heliosphere. The effective polar gradient shows a strong polarity dependence at Earth but this dissipates at θ = 55°, especially with increasing radial distance. This tendency is enhanced by making the compression ratio latitude dependent.     

The role of radial perpendicular diffusion and latitude dependent acceleration along the solar wind termination shock

A numerical model, based on Parker’s transport equation, describing the modulation of anomalous cosmic rays and containing diffusive shock acceleration is applied. The role of radial perpendicular diffusion at the solar wind termination shock, and as the dominant diffusion coefficient in the outer heliosphere, is studied, in particular the role it plays in the effectiveness of the acceleration of anomalous protons and helium when its latitude dependence is changed. It is found that the latitudinal enhancement of radial perpendicular diffusion towards the heliospheric poles and along the termination shock has a prominent effect on the acceleration of these particles. It results in a ‘break’ in the energy spectrum for anomalous protons at ∼6.0 MeV, causing the spectral index to change from E^−1.38 to E^−2.23, but for anomalous helium at ∼3.0 MeV, changing the spectral index from E^−1.38 to E^−2.30. When approaching the simulated TS, the changes in the modulated spectra as they unfold to a ‘steady’ power law shape at energies below 50 MeV are much less prominent as a function of radial distances when radial perpendicular diffusion is increased with heliolatitude.     

Modelling of galactic Carbon in an asymmetrical heliosphere: Effects of asymmetrical modulation conditions

Observations of galactic cosmic rays (GCRs) from the two Voyager spacecraft inside the heliosheath indicate significant differences between them, suggesting that in addition to a possible global asymmetry in the north–south dimensions (meridional plane) of the heliosphere, it is also possible that different modulation (turbulence) conditions could exist between the two hemispheres of the heliosphere. We focus on illustrating the effects on GCR Carbon of asymmetrical modulation conditions combined with a heliosheath thickness that has a significant dependence on heliolatitude. To reflect different modulation conditions between the two heliospheric hemispheres in our numerical model, the enhancement of both polar and radial perpendicular diffusion off the ecliptic plane is assumed to differ from heliographic pole to pole. The computed radial GCR intensities at polar angles of 55° (approximating the Voyager 1 direction) and 125° (approximating the Voyager 2 direction) are compared at different energies and for both particle drift cycles. This is done in the context of illustrating how different values of the enhancement of both polar and radial perpendicular diffusion between the two hemispheres contribute to causing differences in radial intensities during solar minimum and moderate maximum conditions. We find that in the A > 0 cycle these differences between 55° and 125° change both quantitatively and qualitatively for the assumed asymmetrical modulation condition as reflected by polar diffusion, while in the A < 0 cycle, minute quantitative differences are obtained. However, when both polar and radial perpendicular diffusion have significant latitude dependences, major differences in radial intensities between the two polar angles are obtained in both polarity cycles. Furthermore, significant differences in radial intensity gradients obtained in the heliosheath at lower energies may suggest that the solar wind turbulence at and beyond the solar wind termination shock must have a larger latitudinal dependence.     

Modeling the acceleration and modulation of anomalous cosmic ray oxygen

After the solar wind termination shock crossings of the Voyager spacecraft, the acceleration of anomalous cosmic rays has become a very contentious subject. In this paper we examine several topics pertinent to anomalous cosmic ray oxygen acceleration and transport using a numerical cosmic ray modulation model. These include the effects of drifts on a purely Fermi I accelerated spectra, the effects of introducing higher charge states of oxygen into the modulation model, examining the viability of momentum diffusion as a re-acceleration process in the heliosheath and examining energy spectra, and intensity gradients, in the inner heliosphere during consecutive drift cycles.     

Acceleration of interplanetary pick-up ions and anomalous cosmic rays

It may not be doubted anymore that anomalous cosmic rays (ACRs) are produced in the heliosphere from interplanetary pick-up ions through their acceleration at the solar wind termination shock. However, there is no general agreement in the community of heliospheric researchers concerning the mechanism of injection of the pick-up ions into the shock acceleration. We discuss here three possible ways for pick-up ions to be involved into the acceleration process at the termination shock: (1) preacceleration of pick-up ions in the whole region from the Sun up to the termination shock by solar wind turbulences and interplanetary shock waves, (2) local preacceleration of pick-up ions in a vicinity of the termination shock by shock surfing, and (3) formation of high-velocity tails in pick-up ion spectra consisting of secondary pick-up ions which are produced in the supersonic solar wind due to ionization of energetic neutral atoms entering from the inner heliosheath.     

The heliospheric modulation of 3–10 MeV electrons: Modeling of changes in the solar wind speed in relation to perpendicular polar diffusion

The discrepancy between cosmic ray model predictions representing solar minimum conditions in the heliosphere and the 3–10 MeV post-1998 electrons observations by the Kiel Electron Telescope (KET) onboard Ulysses suggests the need for consistent changes in model parameters with increasing solar activity. In order to reduce this discrepancy, an effort is made to model the KET observations realistically during periods of increased solar activity by applying an advanced three-dimensional, steady-state electron modulation model based on Parker’s transport equation including the Jovian electron source. Some elements of the diffusion tensor which were not previously emphasized are revisited. A new relation is also established between the latitudinal dependence of the solar wind speed and the perpendicular polar diffusion. Based on this relation, a transition of an average solar wind speed from solar minimum to solar maximum conditions, as observed on board the Ulysses spacecraft, is modeled on the concept of the time-evolution of large polar coronal holes. These changes are correlated to different scenarios of the enhancement of perpendicular polar diffusion. Effects of these scenarios are illustrated, as a series of steady-state solutions, on the computed 7 MeV Jovian and galactic electrons in comparison with 3–10 MeV electrons observed from the period 1998 to the end of 2003. It is shown that this approach improves compatibility with the KET observations but it also points to the need for a time-dependent electron modulation model to fully describe modulation during moderate to extreme solar maximum conditions.     

The termination shock in a striped pulsar wind

I present observational and theoretical evidence that most of the pulsar spin-down energy is transferred away as a striped pulsar wind and that this energy is released by annihilation of the alternating magnetic field at the pulsar wind termination shock. One-dimensional particle-in-cells (PIC) simulations show that the alternating fields do annihilate at the termination shock in a striped wind. The particle acceleration should be studied in multidimensional simulations. As a first step, I simulated driven annihilation of alternating fields undergoing compression by an external force. It is shown that that in the course of this process, a particle distribution function is formed, which resembles that observed in plerions.     

Particle acceleration and transport at an oblique CME-driven shock

In gradual solar energetic particle (SEP) events, protons and heavy ions are often accelerated to >100 MeV/nucleon at a CME-driven shock. In this work, we study particle acceleration at an oblique shock by extending our earlier particle acceleration and transport in heliosphere (PATH) code to include shocks with arbitrary θ_BN, where θ_BN is the angle between the upstream magnetic field and the shock normal. Instantaneous particle spectra at the shock front are obtained by solving the transport equation using the total diffusion coefficient κ, which is a function of the parallel diffusion coefficient κ_∥ and the perpendicular diffusion coefficient κ_⊥. In computing κ_∥ and κ_⊥, we use analytic expressions derived previously. The particle maximum energy at the shock front as a function of time, the time intensity profiles and particle spectra at 1 AU for five θ_BN’s are calculated for an example shock.     

Modulation of galactic cosmic rays in a north–south asymmetrical heliosphere

Observations made with the two Voyager spacecraft confirmed that the solar wind decelerates to form the heliospheric termination shock. Voyager 1 crossed this termination shock at ∼94 AU in 2004, while Voyager 2 crossed it in 2007 at a different heliolatitude, about 10 AU closer to the Sun. These different positions of the termination shock confirm the dynamic and cyclic nature of the shock’s position. Observations from the two Voyager spacecraft inside the heliosheath indicate significant differences between them, suggesting that apart from the dynamic nature caused by changing solar activity there also may exist a global asymmetry in the north–south (polar) dimensions of the heliosphere, in addition to the expected nose–tail asymmetry. This relates to the direction in which the heliosphere is moving in interstellar space and its orientation with respect to the interstellar magnetic field. In this paper we focus on illustrating the effects of this north–south asymmetry on modulation of galactic cosmic ray Carbon, between polar angles of 55° and 125°, using a numerical model which includes all four major modulation processes, the termination shock and the heliosheath. This asymmetry is incorporated in the model by assuming a significant dependence on heliolatitude of the thickness of the heliosheath. When comparing the computed spectra between the two polar angles, we find that at energies E < ∼1.0 GeV the effects of the assumed asymmetry on the modulated spectra are insignificant up to 60 AU from the Sun but become increasingly more significant with larger radial distances to reach a maximum inside the heliosheath. In contrast, with E > ∼1.0 GeV, these effects remain insignificant throughout the heliosphere even very close to the heliopause. Furthermore, we find that a higher local interstellar spectrum for Carbon enhances the effects of asymmetric modulation between the two polar angles at lower energies (E < ∼300 MeV). In conclusion, it is found that north–south asymmetrical effects on the modulation of cosmic ray Carbon depend strongly on the extent of the geometrical asymmetry of the heliosheath together with the assumed value of the local interstellar spectrum.     

Heliospheric terminal shock acceleration and modulation of the anomalous cosmic-ray component

It is shown that a non-drift, spherically-symmetric, terminal shock acceleration model gives reasonable fits to the observed anomalous oxygen spectra for 1976 and 1982 only when a compression ratio of 2 is used together with a moderate radial dependence for K_rr. Preliminary results with an axially-symmetric, terminal shock acceleration model, including drifts, show that despite a change in polarity the accelerated spectra, at 50 AU in the equatorial plane, differ negligibly. However, the calculated modulation of anomalous oxygen at 2 AU is significantly less for a qA < 0 epoch (1987) than when qA > 0 (1976).     

行星际结构与垂直无碰撞激波的相互作用

应用一维混合模拟方法数值研究了两种行星际结构──反向磁场和高密度等离子团与垂直无碰撞激波的相互作用．结果表明，随着激波上游区磁场的反向，下游区磁场将逐渐改变符号，且等离子体密度和速度分别呈现较强的湍动．激波上游和下游的物理量依然满足Rankine-Hugonoit关系．当高密度的等离子体团通过垂直无碰撞激波时，部分质子被激波反射，部分质子被加速并进入下游区域．由于质子速度分布为非Maxwell分布，在激波下游也激发出较强的湍动．     

Diffusive shock acceleration at interplanetary perpendicular shock waves: Influence of the large scale structure of turbulence on the maximum particle energy

We calculate the maximum energy that a particle can obtain at perpendicular interplanetary shock waves by the mechanism of diffusive shock acceleration. The influence of the energy range spectral index of the two-dimensional modes of the interplanetary turbulence is explored. We show that changes in this parameter lead to energies that differ in at least one order of magnitude. Therefore, the large scale structure of the turbulence is a key input if the maximum particle energy is calculated.     

Modelling of low-energy galactic electrons in the heliosheath

The modulation of cosmic ray electrons in the heliosphere plays an important role in improving our understanding and assessment of the processes applicable to low-energy galactic electrons. A full three-dimensional numerical model based on Parker’s transport equation is used to study the modulation of 10 MeV galactic electrons, in particular inside the heliosheath. The emphasis is placed on the role that perpendicular diffusion plays in causing the extraordinary large increase in the observed intensities of these electrons in the heliosheath. The modelling is compared with observations of 6–14 MeV electrons from the Voyager 1 mission. Results are shown for the radial intensity profiles of these electrons, as well as the modulation effects of varying the extent of the heliosheath by changing the location of the termination shock and the heliopause and the value of the local interstellar spectrum. We confirm that the heliosheath acts as a modulation ‘barrier’ for low-energy galactic electrons. The significance of this result depends on how wide the inner heliosheath is; on how high the very local interstellar spectrum is at these low energies (E < 100 MeV) and on how small perpendicular diffusion is inside the inner heliosheath.     

垂直激波条件下能量粒子加速机制的模拟研究

在具有湍动的磁场和垂直激波条件下对大量测试粒子的轨迹进行了数值计算,研究了激波强度和粒子初始能量对于粒子穿越激波的平均能量变化的影响,分析了漂移加速(SDA)在不同条件下对粒子加速的贡献,并给出了一个与数值结果相符合的漂移加速理论公式△E=amvivup(1-1/s).结果表明,加入磁场湍流后,垂直激波条件下粒子仍主要受到漂移加速作用,而基于粒子引导中心的耗散漂移加速理论在此条件下失效.      

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.     

Acceleration at the termination shock: Anisotropies and spectra

Low-energy termination shock particle populations observed by the Voyagers upstream of the shock exhibited strong field-aligned beaming with anisotropies of the order of unity. The Parker transport equation is valid only for nearly isotropic phase space distributions and is inapplicable to these highly beamed populations. The usual approach is to revert to the more general focused transport equation retaining pitch-angle information. We developed a complimentary technique employing a three-moment expansion of the Skilling equation using Legendre polynomials. We investigate the effects of adiabatic focusing and reflection on the diffusive acceleration process at oblique shock waves. It is shown that low-energy particle intensities are discontinuous and sharply peaked at the shock, consistent with the observations. Particle spectra are not only harder than the power laws predicted from diffusive transport theory, but also exhibit spectral gaps near the low-energy acceleration threshold due to more efficient acceleration by scattering and mirroring. Our model also predicts upstream anisotropies as high as 100% for highly oblique shocks whereas downstream distributions are nearly isotropic.     

存在初始引导场情况下的无碰撞磁场重联

采用二维三分量的全粒子模拟方法研究了不同初始引导场情况下的无碰撞磁场重联及初态为一维的Harris电流片.结果表明,Bz0＞0.5B0的强引导场不仅会显著改变粒子的运动轨迹,而且会改变重联区附近的电场和流场结构,从而影响重联率和电子加速.运用广义欧姆定律解释了不同引导场下电场的结构特征.另外,通过对扩散区附近束流电子的跟踪研究发现,在二维模型中,不论引导场强弱,位于扩散区中心垂直模拟平面的感应电场对电子加速起主要作用,而扩散区外平面电场的贡献很小.      

Heating heavy ions in the polar corona by collisionless shocks: A one-dimensional simulation

Recently a new model for explaining the observations of preferential heating of heavy ions in the polar solar corona was proposed ( and ). In that model the ion energization mechanism is the ion reflection off supercritical quasi-perpendicular collisionless shocks in the corona and the subsequent acceleration by the motional electric field E = −V × B/c. The mechanism of heavy ion reflection is based on ion gyration in the magnetic overshoot of the shock. The acceleration due to the motional electric field is perpendicular to the magnetic field, giving rise to large temperature anisotropy with T_⊥ ? T_∥, in agreement with SoHO observations. Such a model is tested here by means of a one dimensional test particle simulation where ions are launched toward electric and magnetic profiles representing the shock transition. We study the dynamics of O⁵⁺, as representative of coronal heavy ions for Alfvénic Mach numbers of 2–4, as appropriate to solar corona. It is found that O⁵⁺ ions are easily reflected and gain more than mass proportional energy with respect to protons.