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Solar Modulation of Cosmic Rays

Marius Potgieter

arXiv:1306.4421v1physics.space-phastro-ph.SR

TL;DR

This overview addresses how cosmic rays are modulated globally in the heliosphere, including the observed 11-year, 22-year, and charge-sign-dependent variations. It synthesizes transport theory, numerical modeling, observational milestones, progress, controversies, and future challenges.

  • Problem

    Understanding the global features and causes of cosmic-ray modulation, including its cyclic and charge-sign-dependent behavior, requires a focused framework within this broad field.

  • Method

    The review explains basic transport paradigms and uses numerical modeling alongside selected theoretical and observational milestones to examine solar modulation.

  • Results

    The present understanding of global galactic cosmic-ray modulation is considered essentially correct, while heliosphere-scale knowledge has advanced through space missions and modeling.

  • Takeaways & Limitations

    Global solar modulation can be studied within an established theoretical framework, but improved diffusion-coefficient knowledge remains a central challenge.

  • Takeaways & Limitations

    Low-energy galactic spectra remain poorly known because solar modulation disguises their spectral shapes, while heliospheric geometry and HCS evolution remain uncertain.

Abstract

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This is an overview of the solar modulation of cosmic rays in the heliosphere. It is a broad topic with numerous intriguing aspects so that a research framework has to be chosen to concentrate on. The review focuses on the basic paradigms and departure points without presenting advanced theoretical or observational details for which there exists a large number of comprehensive reviews. Instead, emphasis is placed on numerical modeling which has played an increasingly significant role as computational resources have become more abundant. A main theme is the progress that has been made over the years. The emphasis is on the global features of CR modulation and on the causes of the observed 11-year and 22-year cycles and charge-sign dependent modulation. Illustrative examples of some of the theoretical and observational milestones are presented, without attempting to review all details or every contribution made in this field of research. Controversial aspects are discussed where appro- priate, with accompanying challenges and future prospects. The year 2012 was the centennial celebration of the discovery of cosmic rays so that several general reviews were dedicated to historical aspects so that such developments are brie y presented only in a few cases.

1 Introduction

Solar modulation describes how galactic cosmic rays entering the heliosphere undergo position- and time-dependent changes in intensity and energy. This review concentrates on global modulation, solar-cycle effects, charge-sign dependence, and numerical modeling.

  • Scope: Cosmic rays encounter a turbulent solar wind and embedded heliospheric magnetic field, producing global and temporal variations in intensity and energy.The review considers cosmic rays above 1 MeV/nuc, mainly originating outside the heliosphere, with anomalous cosmic rays as an exception.
  • Scope: The review focuses on modulation below ∼30 GeV/nuc, emphasizing global features, 11-year and 22-year cycles, and charge-sign dependent effects.Solar energetic particles, compositional abundances, isotopes, galactic acceleration, and variations shorter than one solar rotation are excluded.
  • Approach: It explains how the heliosphere responds to solar activity and places transport theory, recurrent cosmic-ray behavior, numerical modeling, and observational highlights in that context.The overview is intended to be informative and didactic rather than comprehensive in advanced theoretical or observational detail.

2 The Global Heliosphere and its Main Features

The global heliosphere is a dynamic, asymmetric environment whose boundaries, magnetic geometry, solar-wind structure, and heliosheath conditions shape cosmic-ray modulation. Hybrid HD/MHD modeling and spacecraft observations have progressively constrained this environment while leaving important transport uncertainties.

  • Physical boundaries: The heliosphere contains a termination shock, heliopause, and bow wave, with inner and outer heliosheaths between these interfaces.Voyager exploration has targeted the termination shock and heliopause as principal heliospheric boundaries.
  • Physical boundaries: Voyager 1 might have crossed the heliopause at the end of August 2012, while Voyager observations revealed differing and variable inner-heliosheath conditions.Voyager 2 measured plasma flow remaining above 100 km s–1 in 2012, unlike the near-zero flow inferred near Voyager 1.
  • Global geometry: MHD models generally predict nose-tail asymmetry, with an upwind-to-downwind termination-shock distance ratio of ∼1:2 and solar-cycle-dependent shock motion.The exact dependence remains unknown, and large HMF transients may also shift or locally oscillate the termination shock.
  • Global geometry: Improved HD and MHD backgrounds enabled hybrid cosmic-ray transport models, while interstellar magnetic fields and ENA observations support a tilted, asymmetric heliosphere.IBEX observations sustain the modeled north-south asymmetry despite continuing MHD-related controversy.
  • Heliosheath modulation: At 100 MeV, modeled outer-heliosheath effects produce a radial intensity gradient of 0.2% to 0.4% per AU when the bow wave is assumed at 250 AU.Whether the outer heliosheath affects cosmic rays entering from the interstellar medium remains to be determined.
  • Global magnetic field geometry: The HMF follows a spiral geometry, with ψ ≈45° at Earth and increasing to ∼90° beyond 10 AU in the equatorial plane.The field has an average value of 5 nT at Earth, and ψ denotes the angle between the radial and average HMF directions.
  • Global magnetic field geometry: The HCS divides opposite magnetic polarities, varies over an ∼11-year cycle, and uses tilt angle α as a solar-activity proxy in modulation modeling.Typical tilt angles are α = 75° during high activity and α = 3°–10° during minimum activity.
  • Global magnetic field geometry: The HCS may require additional transport physics in the heliosheath because compression could trigger magnetic reconnection or alter transport when layer spacing approaches cosmic-ray gyro-radii.Its behavior in the nose and tail directions and its role in cosmic-ray transport remain under study.

3 Cosmic Rays in the Heliosphere

The overview distinguishes galactic cosmic rays, anomalous cosmic rays, and termination-shock particles while examining their spectra, heliospheric modulation, and observational evolution. Voyager measurements and modeling expose unresolved questions about ACR acceleration, local interstellar spectra, and long-term modulation cycles.

  • ACRs typically occupy 10–100 MeV/nuc, have softer spectra than galactic cosmic rays, and are primarily singly ionized hydrogen, helium, nitrogen, oxygen, neon, and argon.
  • Voyager observations found that higher-energy ACRs were largely unaffected at the termination shock but gained energy and intensity inside the inner heliosheath.
  • Including multiply charged ACRs in modulation models could explain anomalous oxygen’s unusual 10–70 MeV per nucleon spectrum.
  • Cosmic rays in this overview are fully ionized nuclei, antiprotons, electrons, and positrons with kinetic energies generally above 1 MeV, excluding heliospheric production.
  • 3.2.1 Local interstellar spectra: Local interstellar spectra must be specified as model inputs, but their relationship to galactic, interstellar, and heliopause spectra remains uncertain.
  • 3.2.2 Main cosmic ray modulation cycles: The 11-year cycle dominates solar-activity-related cosmic-ray variation, while the 22-year cycle follows heliospheric magnetic-field polarity reversals; shorter variations are usually below 1%.

4 Solar Modulation Theory

Solar-modulation theory has established basic transport processes, but relating observed galactic and anomalous cosmic-ray modulation to physical causes across global and microphysical scales remains difficult.

  • The basic processes of cosmic-ray transport in the heliosphere are considered known after roughly 50 years of theoretical development.
  • A continuing challenge is connecting cosmic-ray modulation to its true causes over a solar cycle or longer, from global behavior to microphysics.
  • Parker’s transport theory and later numerical models enabled progress in understanding solar-activity-related cosmic-ray phenomena.

4.1 Basic transport equation and theory

The basic transport framework describes cosmic-ray evolution through diffusion, drifts, convection, and adiabatic energy changes, while numerical solutions support spectral comparison with observations. Its steady-state form omits short-term effects and requires modification for large anisotropies.

  • The Parker transport equation provides the foundational framework, with the force-field solution later surpassed by numerical models.
  • A steady-state solution sets ∂f/∂t = 0, neglecting modulation periods shorter than one solar rotation; this is insignificant for galactic cosmic rays but crucial for ACRs.
  • Large particle anisotropies near the Sun, Jupiter, or termination shock can require replacing or modifying the standard transport equation with Fokker–Planck-based treatments.
  • The transport equation represents diffusion through tensor components, particle drifts, radial solar-wind convection, adiabatic energy loss, and optional source functions.
  • Drift effects decrease implicitly when diffusion reduces cosmic-ray intensity gradients, differing from explicitly changing the drift coefficient.
  • Differential intensity and related transformations connect the distribution function with rigidity, momentum, kinetic energy, particle density, and speed for comparison with observed spectra.

4.2 Basic diffusion coefficients

Numerical modulation models use diffusion tensors whose coefficients encode rigidity, magnetic-field, spatial, and directional dependence. Perpendicular diffusion is treated as anisotropic, with enhanced polar transport helping reproduce observed latitudinal gradients.

  • The spherical-coordinate diffusion coefficients combine parallel and radial perpendicular diffusion, while polar and azimuthal terms depend on the chosen diffusion parameters and magnetic-field geometry.
  • Consensus holds that polar perpendicular diffusion exceeds radial perpendicular diffusion away from equatorial regions.
  • The parallel diffusion coefficient is modeled as two smoothly joined rigidity power laws, with parameters controlling normalization, magnetic-field scaling, slopes, and the break rigidity.
  • Perpendicular diffusion is commonly parameterized as a fraction of parallel diffusion, although the ratio is expected to depend on energy.
  • Numerical modeling uses enhanced latitudinal diffusion toward the poles to reproduce the small proton latitudinal gradients observed by Ulysses during the 1994 solar minimum.

4.3 The drift coefficient

The drift coefficient describes guiding-center drifts under a weak-scattering assumption, with polarity determining global drift patterns and reduced drifts required at low rigidity. These reductions help account for the small latitudinal gradients observed by Ulysses.

  • The pitch-angle averaged drift velocity is expressed through the curl of the drift coefficient tensor divided by the background magnetic-field direction.The background field is generally modeled with Parkerian geometry, modified near the poles to avoid unrealistically large drifts.
  • Under weak scattering, the drift coefficient is parameterized so that (Kd)0 = 1.0 represents 100% drifts.This corresponds to full weak-scattering gradient and curvature drifts.
  • The polarity parameter A = ±1 distinguishes A > 0 and A < 0 cycles, with positively charged CRs entering mainly through equatorial regions during A < 0 cycles.Those particles therefore have a high probability of encountering the wavy heliospheric current sheet.
  • The formal description of global curvature, gradient, and current-sheet drifts remains unsettled because the spatial and rigidity dependence of Kd is based on weak scattering.A deviation from this form progressively reduces drifts below Pd0, in GV.
  • Below Pd0, particle drifts are progressively reduced relative to the weak-scattering case to explain the small low-rigidity latitudinal gradients observed by Ulysses.Theoretical and numerical studies also require reduced drifts, particularly with increasing solar activity.

4.4 Gradient, curvature, and current sheet drifts

Gradient, curvature, and current-sheet drifts produce charge-sign dependence and a robust 22-year modulation cycle through polarity-dependent access paths and interactions with the wavy heliospheric current sheet. Numerical and observational results show that these effects vary with energy, location, solar cycle, and heliospheric region.

  • Gradient and curvature drifts were supported by numerical models that explained charge-sign dependent modulation and a 22-year cycle as the solar magnetic field reverses polarity.Opposite charges reach Earth from different heliospheric directions.
  • During A < 0 cycles, inward-drifting protons mainly follow equatorial regions and are progressively reduced as increasing solar activity makes the wavy HCS more pronounced.The HCS tilt angle is therefore a useful modulation parameter.
  • Computed ACR oxygen intensity distributions differ between polarity cycles in the inner heliosphere and beyond the termination shock, with intensity decreasing toward Earth.The illustrative acceleration region is near the equatorial plane close to the heliopause at 140 AU.
  • A 22-year variation appears in the direction of the galactic cosmic-ray daily anisotropy vector and in neutron-monitor differential response functions between polarity cycles.These observations provided additional evidence for gradient and curvature drifts.
  • The wavy HCS plays a significant role in establishing the 22-year cycle, while opposite charges experience different modulation because they sample different heliospheric regions.The heliosheath may exhibit drift effects different from those upstream of the termination shock.
  • Cosmic-ray radial and latitudinal gradients differ significantly between the two HMF polarity cycles, producing a repeated 22-year cycle under ideal modulation conditions.Figure 9 illustrates energy-dependent radial-gradient changes at multiple heliospheric positions.
  • Proton spectra are softer during A > 0 cycles, with A > 0 solar-minimum spectra higher below 500 MeV than corresponding A < 0 spectra.The spectra for consecutive solar minima cross at a few GeV because adiabatic energy losses differ between polarity cycles.
  • Drift effects vary across 11-year cycles, and their significance in the heliosheath remains uncertain even though they influence re-acceleration at the termination shock.The recent solar minimum differed from other A < 0 cycles.

4.5 Aspects of diffusion and turbulence theory relevant to solar modulation

Diffusion and turbulence theory are central to cosmic-ray transport, especially for cross-field motion, but a complete ab initio description remains unavailable for global modulation models. Consequently, phenomenological approaches remain useful alongside developing turbulence theories.

  • Recent turbulence research extends beyond standard quasi-linear theory by incorporating dynamical fluctuations and three-dimensional magnetic-field geometry.A composite turbulence model historically used approximately 20% slab and 80% 2D fluctuations.
  • At low rigidities, dynamical turbulence theory predicts more efficient ion scattering and shorter parallel mean free paths, while low-energy electron mean free paths can be much larger than standard theory predicts.Standard QLT underestimated low-energy electron mean free paths by almost two orders of magnitude in one study.
  • Cross-field diffusion is essential for cosmic rays to reach Earth without requiring unrealistically large parallel mean free paths and anisotropies.Its theoretical description remains puzzling and depends strongly on the dimensionality of heliospheric turbulence.
  • Nonlinear guiding-center theory is promising for perpendicular transport, but integrating turbulence quantities into diffusion coefficients throughout the heliosphere remains a work in progress.Current ab initio approaches do not yet reproduce all observations consistently in global modulation models, so phenomenological approaches remain useful.

4.6 Development of numerical modulation models

Numerical modulation models evolved from one-dimensional steady-state calculations to multidimensional, time-dependent, shock-acceleration, and stochastic approaches. SDE-based modeling provides detailed illustrations of drift, propagation-time, and energy-loss effects in global heliospheric transport.

  • Fast computers and limited in-situ observations have made numerical modeling increasingly important for broadening understanding of solar modulation.Comprehensive models require transport theory, reliable numerical schemes, boundary conditions, and local interstellar spectra.
  • Model development progressed from Fisk’s 1D steady-state TPE solution to 2D axisymmetric models and then to time-dependent models including drifts and propagating GMIRs.These extensions enabled studies of long-term cosmic-ray modulation effects.
  • Termination-shock modeling incorporated time dependence, diffusion shock acceleration, steady-state streaming boundaries, multidimensional geometry, and applications to anomalous cosmic rays.Later models also allowed nonspherical heliospheric boundaries and discontinuous or continuous solar-wind transitions.
  • Stochastic differential equations became popular because high-dimensional transport equations can cause numerical instability in conventional approaches.SDE modeling offers additional ways to represent cosmic-ray modulation.
  • 4.6.1 Illustrations of SDE based modeling: For 100 MeV protons in an A < 0 cycle, transport follows HCS-covered latitudes and is strongly affected by HCS waviness mainly at small tilt angles.Diffusion can nearly erase idealized drift patterns, showing that modulation is fundamentally a convection-diffusion process.
  • 4.6.1 Illustrations of SDE based modeling: 100 MeV electron propagation times are approximately 240 days for A > 0, 110 days for A < 0, and 400 days at the peak of the no-drift distribution.Drifts create preferred transport directions, with shorter A < 0 times attributed to easier escape through heliospheric poles.
  • 4.6.1 Illustrations of SDE based modeling: For 100 MeV protons in A < 0 cycles, propagation time increases significantly above HCS tilt angle α = 40°, while energy loss levels off above α = 40°.The protons propagate from the heliopause to Earth in the illustrated calculation.

4.7 Charge-sign dependent modulation

Charge-sign dependent modulation is a central feature of cosmic-ray transport, especially during opposite magnetic-polarity cycles and solar minimum. Drift models reproduce polarity-dependent differences among electrons, protons, positrons, and antiprotons, although early predictions overstated some effects.

  • Drift modulation: Particle drifts became a competitive modulation process alongside convection, diffusion, and adiabatic energy losses, but their full-cycle importance remains debated.Drifts were introduced seriously around 1976–1978 and took about a decade to gain broad acceptance.
  • Polarity dependence: Electrons behaved differently from protons between the 1965 A < 0 and 1977 A > 0 solar minima, revealing 22-year modulation and charge-sign dependence.The comparison used electron and proton differential-intensity ratios as functions of energy.
  • Energy dependence: 50 MeV–5 GeV is the principal electron drift-sensitive range, with the largest modeled effect around 200–500 MeV and dependence on distance from the Sun.Below 50 MeV, Earth-based electron observations can be contaminated by Jovian electrons.
  • Temporal signatures: Electron-to-proton ratios are predicted to form an inverted V during A > 0 cycles and an upright V during A < 0 cycles around minimum modulation.The associated electron profile is sharper than the proton profile; a V-shape is also reported for the e−/He ratio.
  • Antiparticle modulation: A factor-of-10 change in the p̄/p ratio at 200 MeV was predicted between A > 0 and A < 0 solar minima, while newer modeling found the maximum effect below ∼100 MeV and negligible differences above 10 GeV.The observed charge-sign effect appears significant but smaller than first-generation predictions.
  • Observational tests: PAMELA enabled simultaneous proton–antiproton and electron–positron measurements down to ∼100 MeV, supporting more precise tests of drift modulation.In 2009, e−/e+ increased from 6 at 200 MeV to 15 at 8 GeV, motivating comprehensive drift modeling.

4.8 Main causes of the complete 11-year and 22-year solar modulation cycles

The complete 11-year and 22-year modulation cycles arise from interacting time-dependent transport processes, including drifts, changing heliospheric structures, diffusion, and heliosheath effects. Compound numerical models reproduce many observed long-term features but require refinement and better outer-heliosphere physics.

  • Cycle mechanisms: A complete 22-year modulation cycle can be simulated to first order by combining particle drifts, time-dependent HCS tilt angles, and GMIRs.This combination also addresses the step-like modulation superposed on the solar cycles.
  • Diffusion changes: Global HMF changes can explain 11-year modulation at neutron-monitor energies, but lower rigidities require diffusion coefficients that depend on both solar activity and rigidity.With n = 1 and an observed factor-of-2 change in B(t), diffusion coefficients vary by only a factor of 2.
  • Compound modeling: HCS tilt angles alone match solar-minimum observations, whereas a compound approach reproduces solar-maximum modulation and many 22-year observations.The compound model produced the correct modulation amplitude and most steps for 1.2 GV electrons and helium, though some steps had incorrect magnitude or phase.
  • Latitude dependence: Enhanced polar perpendicular diffusion is required to reproduce proton latitude dependence and the weaker electron latitude dependence observed along Ulysses.The modification reduces the large latitudinal gradients produced by unmodified drifts.
  • Heliosheath effects: Heliosheath modulation is strongly energy-dependent, exceeding 80% below ∼0.02 GeV under some conditions and becoming zero at condition-dependent energies.Negative percentages indicate reacceleration at the termination shock under the assumed conditions.
  • Open problems: Outer-heliosphere modeling remains limited by uncertain structure merging beyond 20 AU, diffusion tensors beyond the termination shock, and the reduction of drifts with solar activity.The treatment of the heliosheath current sheet and non-radial solar-wind components also requires investigation.

5 A Few Observational Highlights

Observations from PAMELA, Ulysses, and Voyager revealed unusual modulation signatures across the heliosphere, including record intensities, latitude-dependent behavior, and abrupt outer-heliospheric changes. These measurements exposed both the reach of solar modulation and unresolved heliospheric physics.

  • Unusual solar minimum: During the prolonged 2007–2009 minimum, the HMF was unusually weak, high-rigidity cosmic rays reached record intensities, and the delay to maximum intensity was at least three times longer than in previous even-numbered cycles.The minimum persisted through the end of 2009 while the HCS tilt angle eventually reached its minimum.
  • PAMELA observations: PAMELA observed the highest space-age CR proton spectrum in December 2009, contrary to the usual drift-model ordering of proton spectra below a few GeV across polarity cycles.PAMELA also measured proton fluxes from July 2006 through the end of 2009 with high statistics for each Carrington rotation.
  • Inner-heliosphere gradients: Ulysses found a north–south asymmetry in galactic CR modulation, while small minimum-time latitude gradients showed that the LIS cannot be observed in the inner polar heliosphere.Proton latitude gradients peaked around 2 GV and were much smaller below those rigidities than early drift models predicted.
  • Solar-cycle dependence: Latitudinal gradients were essentially absent at solar maximum, indicating that drifts diminished strongly as solar activity increased.Electron-to-proton ratios likewise indicated large drifts near solar minimum and reduced drifts toward solar maximum.
  • Outer heliosphere: Voyager observations showed anomalous particle behavior in the outer heliosphere, including increasing ACR intensity beyond the termination shock and rapid low-energy electron increases in the heliosheath.By about 2008.5, 6–14 MeV galactic electron intensity was five times its post-shock value.
  • Voyager 1 boundary event: In August 2012, Voyager 1 recorded over 90% decreases in dominant low-energy ACR and termination-shock-particle intensities alongside approximately factor-of-2 electron increases and 30–50% increases in higher-energy nuclei.The event was interpreted as a possible heliopause or heliocliff crossing, but magnetic-field confirmation was still absent.

6 Constraints, Challenges, and Future Endeavours

Future progress depends on improving heliospheric inputs and transport models, especially beyond the termination shock. Key open issues include poorly constrained interstellar spectra, heliospheric geometry, magnetic-field structure, diffusion, drifts, and cosmic-ray species coverage.

  • Low-energy galactic cosmic-ray spectra remain uncertain because local interstellar spectra are poorly known and solar modulation disguises ion and antiproton shapes below ∼10 GeV.
  • The heliosphere’s asymmetry, tail, modulation volume, termination-shock motion, and boundary geometry remain unresolved influences on cosmic-ray modulation.
  • The wavy heliospheric current sheet is established in global models, but its outward evolution and heliosheath drift effects remain unclear.
  • Improved knowledge of the solar-wind profile and heliospheric magnetic field remains crucial for effective modulation modeling.
  • Fisk-type magnetic fields remain difficult for standard numerical codes, while more realistic fields require increasingly complex diffusion and drift descriptions.
  • Three-dimensional Jovian-electron modeling separates galactic and Jovian contributions, finding Jovian electrons dominate inner equatorial regions up to ∼10–20 AU.
  • Time-dependent models lack well-constrained diffusion-coefficient dependencies and require an extended diffusion tensor for the region beyond the termination shock.
  • Future work should model additional cosmic-ray species and isotopes, clarify the modulation onset rigidity, and sustain spacecraft and ground-based measurements.

7 Summary

Space missions and modeling have substantially advanced knowledge of cosmic-ray modulation and heliospheric structure. The global modulation framework is considered essentially correct, while diffusion-coefficient dependencies and other details remain active challenges.

  • Space missions measuring cosmic rays across wide energy ranges stimulated major theoretical and numerical progress.
  • Voyager crossings of the termination shock and approach to the heliopause have made heliospheric structures and mechanisms less freely adjustable in models.
  • The global solar-modulation mechanisms for galactic cosmic rays are considered essentially correct, supporting the basic framework of Parker’s theory.
  • The main remaining obstacles are insufficiently known spatial, rigidity, and especially temporal dependencies of diffusion coefficients.
  • Many aspects of cosmic-ray modulation remain a work in progress.
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