Space Sci RevDOI 10.1007/s11214-011-9819-3

Cosmic-Ray Modulation Equations

H. Moraal

Received: 6 December 2010 / Accepted: 5 August 2011© Springer Science+Business Media B.V. 2011

Abstract The temporal variation of the cosmic-ray intensity in the heliosphere is calledcosmic-ray modulation. The main periodicity is the response to the 11-year solar activitycycle. Other variations include a 27-day solar rotation variation, a diurnal variation, and ir-regular variations such as Forbush decreases. General awareness of the importance of thiscosmic-ray modulation has greatly increased in the last two decades, mainly in communitiesstudying cosmogenic nuclides, upper atmospheric physics and climate, helio-climatology,and space weather, where corrections need to be made for these modulation effects. Pa-rameterized descriptions of the modulation are even used in archeology and in planning theflight paths of commercial passenger jets.

The qualitative, physical part of the modulation is generally well-understood in thesecommunities. The mathematical formalism that is most often used to quantify it is the so-called Force-Field approach, but the origins of this approach are somewhat obscure and itis not always used correct. This is mainly because the theory was developed over more than40 years, and all its aspects are not collated in a single document.

This paper contains a formal mathematical description intended for these wider commu-nities. It consists of four parts: (1) a description of the relations between four indicators of“energy”, namely energy, speed, momentum and rigidity, (2) the various ways of how tocount particles, (3) the description of particle motion with transport equations, and (4) thesolution of such equations, and what these solutions mean. Part (4) was previously describedin Caballero-Lopez and Moraal (J. Geophys. Res, 109: A05105, doi:10.1029/2003JA0103582004). Therefore, the details are not all repeated here.

The style of this paper is not to be rigorous. It rather tries to capture the relevant tools todo modulation studies, to show how seemingly unrelated results are, in fact, related to oneanother, and to point out the historical context of some of the results. The paper adds no newknowledge. The summary contains advice on how to use the theory most effectively.

Keywords Cosmic rays · Modulation · Force field · Transport equation

H. Moraal (�)Space Research Centre, North-West University, Potchefstroom 2520, South Africae-mail: harm.moraal@nwu.ac.za

H. Moraal

1 The Four “Energy” Variables

The bulk of the cosmic-ray particles observed in the inner heliosphere have energy of theorder of 1 GeV (109 eV), which is on the borderline between non-relativistic and relativisticprotons. This means that the kinetic and rest-mass energies of the bulk of the particles are ofthe same order of magnitude, because at higher energies the intensity falls off fast, while atlower energies the particles are shielded away from the inner heliosphere by the modulationprocess.

While non-relativistic and super-relativistic particle descriptions are relatively easy, thequasi-relativistic regime, where cosmic-ray modulation is most important, is technicallymore difficult.

The description starts with the energy-momentum relationship

E2 = p2c2 + m20 c4, (1)

where E is energy, p is momentum, m0 is the rest mass of the particle, and c the speed oflight. This relationship is Lorentz-invariant. (This property is useful for theoretical develop-ment, but it need not concern the general reader.) The quantity E0 = m0c

2 is called the rest-mass energy, and the total energy consists of two parts, kinetic plus rest-mass: E = T + E0.E0 = 938 MeV (i.e. roughly 1 GeV) for protons, and 511 keV for electrons. When T � E0

the particles are relativistic and (1) reduces to E = T = pc. When T � E0 they are non-relativistic. Writing E = E0(1 + T/E0) for this case, and using the binomial expansion(1 ± ε)n = 1 ± nε + n(n − 1)ε2/2! ± . . . with T/E0 = ε � 1, (1) reduces to the standardT = p2/2m0 = 1

2 m0v2. Relativistic mass is given by m = m0/

√1 − β2 with β = v/c. If this

is substituted into (1) it leads to the well-known mass-energy relationship E = mc2.In many cosmic-ray applications it is important to refer explicitly to the number of nu-

cleons that compose a nucleus, a nuclide, or an ion. A fully-stripped light atomic nucleuswith A nucleons, of which Z are protons, has A/Z = 2, while higher up in the periodictable A/Z gradually grows larger than 2. A Hydrogen nucleus is an exception, because ithas A/Z = 1. For partially stripped atoms A/Z is also > 2. A singly-stripped oxygen atom,for example, has A/Z = 16. Such partially (mostly singly) ionized atoms are called anoma-lous cosmic rays (ACR), discovered in the 1970s. The discovery and understanding of theseACRs is described and referenced in Moraal (2001).

With this explicit reference to the number of nucleons, (1) can be rewritten as Ep =p2c2 + E2

0p , or

A2(T + E0)2 = p2c2 + A2E2

0 , (2)

where Ep and E0p are energies per particle, while E,T and E0 are energies per nucleon.The horizontal axis of almost any published cosmic-ray spectrum expresses energy as kineticenergy per nucleon, and not per particle.

Since cosmic rays are charged, they are confined by the magnetic fields pervading thecosmos. They experience the Lorentz force F = q(E + v × B), where v is the particle veloc-ity, E the electric, and B the magnetic field. In the highly conducting plasma of the cosmos,particles quickly re-arrange themselves to cancel out any electric fields so that they almostnever come into play explicitly, and the net force is purely magnetic: F = q(v × B).

Standard electrodynamics and plasma physics texts show that this force-law leads to aspiral trajectory of a charged particle about a magnetic field line as shown in Fig. 1(a), witha so-called gyro-radius or Larmor radius rg = mv sinα/(qB), where α is the pitch anglebetween the velocity vector v and the magnetic field B. When α = 90◦ the spiral becomes

Cosmic-Ray Modulation Equations

Fig. 1 Charged particle motion in a magnetic field. (a) In a uniform magnetic field the particle has a spiralorbit with a gyroradius rg = P/Bc. (b) When the field is non-uniform the particle drifts away from a fieldline due to the gradient and curvature of the field. (c) When a particle meets a kink in the field that has a scalelength � rg , all particles will progress through the kink (but they may drift to adjacent field lines while doingso). (d) Likewise, if rg � scale size of the kink, all particles will pass through it without being affected much.(e, f, g) When rg ≈ scale size of the kink, it depends on the gyrophase of the motion when the particles startsto feel the kink whether it will go through the kink (e), be reflected back (f), or effectively get stuck in thekink (g). This process is called pitch-angle scattering along the field. (h) When particles meet such a kink,there is also a scattering in phase angle, which leads to scattering across the field lines, but such that κ⊥ � κ‖

a circle with radius rg = mv/(qB) = p/(qB). This implies that the gyroradius depends ontwo particle properties, namely its momentum and charge. For this reason we introduce theconcept of rigidity, defined as P = p/q . Then rg = P/B , which says that the gyroradiusdepends on only one particle property and on the field strength.

The SI-units of rigidity are kg m s−1 C−1 or J s m−1 C−1, and this is cumbersome to use.It can be translated into the much more useful unit of Volt (V) by noting from (1) that pc

has the same units as E. Thus, if one rather defines rigidity as P = pc/q , it has dimensionsof energy per unit charge, or potential. If energy (and pc) is expressed in eV, and charge interms of the number Z of elementary charges, i.e. q = Ze where e = 1.602 × 10−19 C, thenP has units of Volt (V). Thus, the formal definition of rigidity is

P ≡ pc/(Ze),

with the gyroradius given by rg = P/Bc.Putting this into (1) gives the relationship P = (A/Ze)2

√T (T + 2E0) between the rigid-

ity of a particle and the kinetic energy per nucleon of that particle. Bearing in mind thatm = m0/

√1 − β2, one gets the universal relationship

P = pc/(Ze) = (A/Ze)√

T (T + 2E0) = (A/Ze)β(T + E0), (3)

H. Moraal

between (1) rigidity P , (2) momentum p, (3) the dimensionless speed β of a nucleuswith mass number A and atomic number Z, and (4) its kinetic energy per nucleon, T .From here one can go in any direction: For instance, the speed in terms of kinetic en-ergy per nucleon is β = √

T (T + 2E0)/(T + E0). (Notice that this is independent ofparticle species—i.e. independent of A and Z.) In terms of rigidity the particle speed is

β = P/

√P 2 + (A/Ze)2E2

0 . Other quantities such as T (β) and P (β) are also readily calcu-

lated. For example, T (P ) =√

(Ze/A)2P 2 + E20 − E0.

The anchor point of these relationships is that the rest-mass energy of a proton is E0 =938 MeV. This is near enough to 1 GeV, so that putting E0 = 1 in all these relationshipsimplies that one works in units of GeV, so that rigidity has units of GV. Then one operatesvery easily between the different quantities, such as:

1. A 1 GeV proton has a rigidity P = √3 GV.

2. A 1 GV proton has kinetic energy T = 0.4 GeV or 400 MeV.3. A 1 GV nucleus with A/Z = 2 has kinetic energy T = 120 MeV/nucleon.4. A singly-charged anomalous cosmic-ray oxygen ion with kinetic energy T = 10

MeV/nucleon has a rigidity P = 2.2 GV.

Notice that for highly relativistic protons (T � E0) T and P are numerically equal, i.e.P = T , while for non-relativistic ones (T � E0), P = √

2T . However, this last relationshipis only true if one sticks to the rule of always working in units of GeV and GV. If one workswith electrons, all the expressions yield numerically correct answers if one takes A = 1,Z = 1 and E0 = 0.000511 GeV.

Finally, particles need a rigidity of at least 17 GV to penetrate vertically through theEarth’s magnetosphere at the magnetic equator (in the vicinity of Thailand where the fieldis strongest). This is the highest so-called cut-off rigidity, Pc , in the geomagnetic field. To-wards the geomagnetic poles this cut-off rigidity reduces to zero. However, the particles,or their decay products, must also penetrate through the atmosphere to reach ground level.This requires a minimum energy of about 400 MeV for a proton, or a rigidity of 1 GV.Such ground-level measurements are typically done with neutron monitors, and hence theireffective vertical cutoff ranges from Pc ≈ 1 GV in the polar regions, to Pc ≈ 17 GV at the ge-omagnetic equator. These instruments have their maximum energy of response at P ≈ 6 GV.

2 Counting Particles

There are at least four different useful ways to count particles. Some of these, like intensity,are suited for experimental purposes, while others, like the distribution function or differ-ential density, are more useful in theory. They have to be related to one another, and onceagain, this is made more difficult because we work in the quasi-relativistic regime.

Consider an infinitesimal box with volume dxdydz, usually written as d3r . In sphericalpolar co-ordinates (r, θ,φ) this volume is d3r = r2 sin θdrdθdφ. Let there be N particlesinside the box. The density of particles, n, is defined such that N = nd3r .

In cosmic-ray physics, density is not a very useful quantity. Almost all properties ofcosmic rays are derived from cosmic-ray spectra, which count the number of particles withina given interval of kinetic energy (T ,T + dT ) (or the equivalent momentum, rigidity, orspeed interval). For this reason we define the differential density, U , such that the numberof particles in the infinitesimal box d3r , with kinetic energy per nucleon in the interval(T ,T + dT ) is dN = Ud3rdT . Evidently, the total (or integral) density in terms of the

Cosmic-Ray Modulation Equations

Fig. 2 The figure illustrates howthe intensity j that goes througha detector with area dA is relatedto the density U in the ring withthickness vdt

differential density is n = ∫U(T )dT . The use of the symbol U for differential density has

become standard; it seems that it was introduced by researchers such as L.J. Gleeson andM.A. Forman in the late 1960s.

It is important to be specific about the “energy” units used for U . We therefore callthe differential density defined above UT , because it is the differential density w.r.t. kineticenergy (per nucleon). Similarly, one can define momentum and rigidity densities, denoted asUp and UP respectively. Since the number of particles, dN , remains the same, independentof how they are counted, one must have dN = UT dT d3r = Updpd3r = UP dPd3r . From (3)and its associated expressions, it follows that dT /dp = βc/A and dT /dP = βZe/A. HenceUp = (c/Ze)UP = (βc/A)UT . This says that spectra in terms of rigidity and momentumhave the same shape (they differ only with the numerical factor c/Ze), but they look differentfrom the kinetic-energy spectrum because β is a function of T .

Experimenters do, however, not measure differential densities; they measure differentialintensities instead. This is so because detectors measure the rate at which particles gothrough detectors, instead of the number inside them. Therefore, consider a detector ele-ment with area dA, and that particles with speed v go through it, as shown in Fig. 2. Theparticles that go through the detector in the time interval dt at present, were somewherein a spherical shell with thickness vdt some time before. The dark sub-region in this shellcontains dn′ = UT d3rdT = UT r2 sin θvdtdθdφdT particles. They go in all directions, andthe fraction of them that will go through the detector is the ratio of the projected detec-tor area to the area of a sphere with radius r , i.e. dA| cos θ |/4πr2. Therefore, the num-ber of particles from the shaded part of the shell that go through the detector in time dt

is dn′′ = dn′dA| cos θ |/4πr2 = UT (dA| cos θ |/4π)vdt sin θdθdφdT . Bearing in mind thatthe θ,φ-integral over the entire shell has a value of 2π , one finally arrives at: the numberof particles passing through the detector per unit area, per unit time, per unit kinetic energyinterval is vUT /2. It is standard practice not to refer to all the particles coming form alldirections, but rather those coming from one steradian of solid angle. Hence, the differentialintensity, denoted as jT , is

jT = vUT /8π, (4)

with the dimensions of particles per unit area, per unit time, per unit solid angle, per unitkinetic energy per nucleon, or units of particles/m2/s/steradian/GeV/nucleon.

Obviously, one similarly finds that jp = vUp/8π and jP = vUP /8π .Unfortunately, the intensity spectra, j , that experimenters measure differ in shape from

the density spectra, U , that theoreticians tend to use, because of the factor v(T ). However,since UT dT = Updp = UP dP , one easily finds from (3) that

jT = (A/8π)Up = (A/8Ze)UP . (5)

H. Moraal

Fig. 3 The definition of avolume element d3r inconfiguration space, a similarelement d3p in momentumspace, and a momentum shellwith thickness dp

This is a useful result because it means that the (usually measured) differential intensityw.r.t. kinetic energy per nucleon has the same form as the (usually calculated) differentialdensities w.r.t. momentum and rigidity; they only differ with numerical factors.

Finally, it is standard practice in transport theory and plasma physics not to measure thenumber of particles in terms of the differential density U , but rather in terms of the particledistribution function.

This distribution function is defined such that the number of particles with momentum inthe interval d3p between the vectors p to p + dp, and in the volume element d3r is

dn = F(r,p, t)d3rd3p. (6)

This definition is sketched in Fig. 3. Plasma physics and transport theory texts variously usef and F , but almost all of these texts define the distribution function in terms of velocityv instead of momentum p. Since most plasmas and other flows are non-relativistic, thismakes no difference. But for the relativistic cosmic rays it turns out that using momentumas variable eliminates a large amount of unnecessary algebra.

We interrupt by noting the hierarchy of counting particles: there are F =dn/(d3rd3p) particles per unit volume per unit momentum vector between p and p + dp;there are Up = dN/(d3rdp) particles per unit volume per unit momentum magnitude fromp to p + dp; and there are n = N/d3r particles per unit volume, of all possible momenta.

The momentum interval d3p can be written, just as in the case for d3r as d3p =p2dpd = p2 sin θdθdφdp, where d is an element of solid angle in momentum space.Then, the number of particles in d3p, with momentum magnitude between p and p+dp arethose inside the shell of Fig. 3, and their number is given by dN = d3rp2dp

∫

F(r,p, t)d.But it was argued above that this number is also dN = Updpd3r . Hence Up(r,p, t) =p2

∫

F(r,p, t)d. Now define the directional average of the distribution function as

f (r,p, t) =∫

F(r,p, t)d

∫

d= 1

4π

∫

F(r,p, t)d.

Note that F is a function of vector p, while f is a function only of its magnitude, p. Fromthis it immediately follows that Up(r,p, t) = 4πp2f (r,p, t).

Finally, when one also incorporates the previously described differential intensity j , itproduces the following very useful relationship between three possible ways of countingparticles: jT = (A/8π)Up = (A/2)p2f .

Cosmic-Ray Modulation Equations

In summary, since absolute normalizations are seldom important, these relationships areusually conveniently abbreviated as

jT = Up = p2f = UP = P 2f, (7)

which means that observed intensity spectra w.r.t. kinetic energy are the same as densityspectra w.r.t. momentum and rigidity, and they differ only with a factor of p2 or P 2 fromdistribution functions w.r.t. momentum and rigidity. Often cosmic-ray spectra have powerlaw forms, i.e. f ∝ p−γ . Then it becomes second nature to immediately know that jT , Up

and UP all have power law indices 2 − γ .

3 Two Other Considerations

There is another reason why one prefers momentum above velocity as variable for the dis-tribution function. This is that the momentum distribution is Lorentz-invariant. The numberof particles with momentum in the interval d3p between the vectors p to p + dp, and inthe volume element d3r was given above by dn = F(r,p, t)d3rd3p. This same number ofparticles can also be described as the number in the interval velocity interval d3v betweenthe vectors v to v + dv, also in the volume element d3r , so that dn is also given by dn =F(r,v, t)d3rd3v. Consider now a transformation to another reference frame that moves withvelocity V relative to the first one, such as from the solar wind frame to a spacecraft frame,for instance. The velocity of a particle in this frame is then v′ = v − V. The observer in theprimed frame then measures the same number of particles, dn, but in terms of his distributionfunction it is given by dn = F ′(r,v′, t)d3rd3v′. Hence F(r,v, t)d3v = F ′(r,v′, t)d3v′. Thesame holds for momentum distribution functions: F(r,p, t)d3p = F ′(r,p′, t)d3p′. The bigdifference is however, that for transformations in which the transformation speed is muchless than the particle speed, i.e. V � v, d3p′ = d3p, but d3v′ �= d3v. Hence F ′(p′) = F(p),but F ′(v′) �= F(v). This useful property is called the Lorentz invariance of the momentumdistribution function, and was first pointed out by Forman (1970).

It is also important to distinguish between the concepts of intensity and flux in transporttheories. Intensity was defined above as the scalar quantity jα = vUα , where the subscript α

can refer to the variables kinetic energy, momentum or rigidity. On the other hand, the flux,or streaming density, is defined as the vector Sα = 〈v〉Uα . The two quantities have the samedimensions, but the intensity counts all the particles that go through the detector, irrespectiveof direction, while the flux counts the net amount of them, or the directional sum (and thusgives the direction of the net flow). In the case of a unidirectional beam of particles, all withthe same v, the flux and intensity have the same numerical value. If, in the other extreme,the intensity is isotropic, the average velocity, and hence the flux is zero.

4 The Cosmic-Ray Transport Equation

Transport equations have their origin in the continuity principle, which states that the time-rate-of-change of the number of particles in a given volume must be equal to the rate ofparticles flowing across the closed surface around that volume, plus the rate at which parti-cles are created/destroyed by physical process (such as ionization, recombination, inelasticcollisions etc.) in that volume:

dN

dt= −

∮S · da + Q, (8)

H. Moraal

where da is a surface element, and Q is a source function with dimensions of particles perunit time. The minus sign accounts for the fact that the number of particles in the volumedecreases if there is a net outflow.

According to the divergence theorem∮

S · da = ∫ ∇ · Sdτ , where dτ is the volumeelement within the closed surface

∮da. Putting this into (8), and noting that N = ∫

ndτ ,leads to the differential form of the continuity equation

∂n

∂t+ ∇ · S = q ′, (9)

where q ′ is the source/sink function per unit volume.The physics of the problem is contained in the processes that cause a given flux S. In the

solar wind, this flux consists of two parts, a diffusive flux due to scattering off the irregu-larities in the heliospheric magnetic field (HMF), and radial outward convection in the solarwind, with velocity V. The convective flux is simply Sc = nV, but the diffusive flux due toscattering in HMF irregularities has been the core of all theoretical modulation studies overthe last ∼50 years.

The panels (c), (d), (e), (f) and (g) of Fig. 1 show that when a spiraling charged particleencounters an irregularity in a magnetic field line that is of the same scale size as the gy-roradius of the particle, then the trajectory of the particle through the irregularity dependscritically on the phase of the gyromotion when the particle starts to “feel” the irregularity.Some trajectories—calculated from the force law F = q(v × B)—will go though the irregu-larity (e), others will “reflect back” along the field line (f), while still others will effectively“get stuck” in the vicinity of the irregularity (g). Basically, this means that the pitch angle ofthe particle is randomly scattered. If the density of particles along the field line is constant,this process leads to no net flux. However, if there is a density gradient along the field line,this pitch-angle scattering leads to a diffusive flux according to Fick’s law: Sd = −κ∇n.

Charged particles do not readily cross magnetic field lines, except for two effects: per-pendicular scattering and gradient/curvature drift. When particles encounter magnetic fieldirregularities, not only their pitch angle, but also their gyrophase changes. Physically thishappens because the gyroradius is suddenly squeezed when the field is suddenly stronger,or enlarged when the field is weaker. The net effect is that the particle will attach itself toa neighboring field line, as shown in Fig. 1(h). Since this is also a random process, it leadsto a diffusive flux perpendicular to the background magnetic field. Hence, the diffusive fluxconsists of two components: Sd = κ‖∇n‖ + κ⊥∇n⊥, where κ‖ and κ⊥ are the diffusion co-efficients parallel and perpendicular to the background magnetic field, and where typicallyκ⊥ � κ‖. This is called the weak-scattering limit. Perpendicular diffusion may be signif-icantly enhanced, however, by random walk of the fluctuating magnetic field lines them-selves. The upper limit for perpendicular diffusion is κ⊥ = κ‖, which is reached through asimultaneous decrease in κ‖ and an increase in κ⊥ as the amount of turbulence in the fieldincreases. This limit is reached when the fluctuations in the field become as large as its av-erage background value, so that the notion of a well-ordered background field disappears,and diffusion becomes isotropic.

The theory of scattering parallel to the background magnetic field is fairly well under-stood in terms of so-called quasi-linear theory of scattering, which holds for weak fluctua-tions, when δB2/B2 � 1. This means that one can calculate κ‖ as function of the turbulencespectrum of the fluctuations, δB2, and when these spectra are measured throughout the helio-sphere, they produce a diffusion coefficient as function of momentum (or energy or rigidity)and position. This quasi-linear theory goes back to the paper of Jokipii (1966).

Cosmic-Ray Modulation Equations

The theory of perpendicular diffusion is much more complicated, and this remains thebiggest outstanding theoretical problem in cosmic-ray modulation theory. The current theoryis non-linear in nature, and it was introduced by Matthaeus et al. (2003). Salchi (2009) givesan extensive account of the status of the field.

In comparison, transport perpendicular transport to the background field due to gradientand curvature drifts is simpler and it is elegant. In Plasma Physics texts the calculationof the two drift velocities is cumbersome, and the results are approximate. A significantcontribution from the field of cosmic rays in the heliosphere to this topic is that: for anisotropic particle distribution, or for one with at most a first-order anisotropy (i.e. weaklyanisotropic), the combined gradient and curvature drift velocity of the distribution is givenby

〈vdr〉 = βP

3∇ × B

B2. (10)

The average bracket 〈 〉 denotes that this is the velocity of the distribution, and we emphasizethat under the condition of weak anisotropy this expression is exact. Furthermore, such adrift velocity gives rise to a drift flux Sdr = βP

3B2 B × ∇n. It is important to note that theaverage drift velocity and drift flux are not the same; they need not even be in the samedirection, as some careful trajectory constructing readily reveals.

A further simplification results by formulating drift motion as an antisymmetric elementof the cosmic-ray diffusion tensor

K = κij =⎛

⎝κ‖ 0 00 κ⊥ −κT

0 κT κ⊥

⎞

⎠ , (11)

where κT = βP/(3B). Then the combined anisotropic diffusion (consisting of κ‖ and κ⊥)and drift flux can be symbolically contracted into a single term, −K · ∇n. This handling ofthe drift effect as part of the diffusion tensor is entirely equivalent to the explicit referenceto the drift velocity (10) or to the drift flux Sdr .

The elegant simplicity of this drift formalism is due to the group of J.R. Jokipii at theUniversity of Arizona, with the first paper on the topic by Jokipii et al. (1977), with a morecomprehensive version by Isenberg and Jokipii (1979). The different fluxes are schemati-cally shown in Fig. 2 of Moraal (1991). Ongoing research is conducted on how the simpleform of κT is affected by strong scattering; a summary of the relevant literature on this topicis given by Burger and Visser (2010).

The fact that the HMF and its scattering centers are convected radially outward by thesolar wind with velocity V, leads to a convective flux Vn. Hence, the total flux of cosmicrays in the heliosphere is

S = Vn − K · ∇n. (12)

When this flux is substituted into the continuity equation (9), it leads to the equation ∂n/∂t +∇ · (Vn − K · ∇n) = q ′. This is, however, an equation for the integral cosmic-ray densityn = ∫

Updp = ∫4πp2f dp. It seems that it should also hold for Up (or for f ) in the energy

interval (T ,T + dT ,) or in the momentum interval (p,p + dp,) i.e. ∂f/∂t + ∇ · (Vf − K ·∇f ) = q .

This does, however, not take into account that particles can gain or lose energy, and hencemove out of the interval (p,p + dp). This causes a flux in momentum space, similar to theflux in configuration space, which must be included in the differential equation.

H. Moraal

The form of the energy/momentum change term can readily be understood from thefollowing analogy: In spherical polar coordinates the divergence of the flux is given by∇ · S = 1

r2∂∂r

(r2〈v〉rUp)+ (two terms containing θ and φ derivatives). The velocity is v =r = rer + rθeθ + r sin θφeφ . When the particle distribution is isotropic, the average velocityreduces to 〈v〉 = 〈r〉er , because the two directional averages and the last two terms in thedivergence become zero. Thus the divergence of the flux reduces to ∇ · S = 1

r2∂∂r

(r2〈r〉Up).In an analogous fashion, when particles change the magnitude of their momentum vectorinstead of their position vector, the divergence of the flux in momentum space will be ∇ ·Sp = 1

p2∂∂p

(p2〈p〉Up). With this, the final form of the differential transport equation, i.e. forparticles in the momentum range (p,p + dp) becomes

∂f/∂t + ∇ · S + 1

p2

∂

∂p(p2〈p〉f ) = q, (13)

or

∂f/∂t + ∇ · (Vf − K · ∇f ) + 1

p2

∂

∂p(p2〈p〉f ) = q. (14)

The cosmic-ray transport equation was written down for the first time in this form as (3) inParker (1965), although in different notation. Parker then noted that in the heliosphere theonly significant energy change process is adiabatic cooling due to the fact that the particlesride with the fields in the wind, and these fields expand due to the positive divergence ofthe wind speed. This leads to a rate of change of momentum 〈p〉/p = −(1/3)∇ · V. If thiscooling rate is substituted into (14) it leads to

∂f/∂t + ∇ · (Vf − K · ∇f ) − 1

3p2(∇ · V)

∂

∂p(p3f ) = q. (15)

This is equivalent to Parker’s equation (4), and should be regarded as the original form of thecosmic-ray transport equation. Parker’s form was, however, only written down for sphericalsymmetry and for constant radial V. [We note that the second term in Parker’s form of theequation should be – in his notation – v∂U/∂r instead of v

r2∂∂r

(r2U).]

5 The Gleeson-Axford Derivation of the Transport Equation

Gleeson and Axford (1968) rederived the cosmic-ray transport equation starting with aBoltzmann equation, evaluating single-particle scatterings in the solar wind frame, then in-tegrating over direction in momentum space, and transforming back to the observer’s frameof reference. This different approach was limited to spherical symmetric geometry, but itcorrectly reproduced the spherically symmetric version of (15) above. These authors thenwent further, and in Gleeson and Axford (1968) they pointed out that the flux of parti-cles, defined as S = nV for density n in a flow field V, must be corrected for the so-calledCompton-Getting effect when measured on a differential basis, i.e. between energies T andT + dT .

The original paper, with its notation in terms of kinetic energy and differential densityis generally used, but it is not easy to read. Later, Gleeson and Urch (1973) simplified theCompton-Getting correction, and in its most elegant form it can be stated as: when a dif-ferential density w.r.t. to momentum, Up = 4πp2f , is convected with velocity V, then theflux observed is not Sp = V Up = 4πp2Vf , but rather Sp = CV Up = 4πp2CVf , where

Cosmic-Ray Modulation Equations

C = − 13

∂ lnf

∂ lnpis called the Compton-Getting coefficient. This effect is similar in nature to

the Doppler effect on photons, i.e. it is due to the fact that when a beam of particles is ob-served in the oncoming (receding) direction, the particles are observed with higher (lower)energy or momentum. The Compton-Getting coefficient is related to the logarithmic slopeof the spectrum, and for a power law of the form, f ∝ p−γ , it is simply C = γ /3.

When the Compton-Getting corrected flux

S = 4πp2(CVf − K · ∇f ) (16)

is substituted into (13) it does not produce the correct transport equation (15). But Gleesonand Webb (1978) also pointed out that the adiabatic rate of change of momentum 〈p〉/p =−(1/3)∇ · V referred to above, is actually the rate of change in the (non-inertial) solar windframe. They showed that the rate in the stationary frame is 〈p〉/p = −(1/3)V · (∇f/f ).When these two corrected expressions are inserted in (13), they produce the transport equa-tion

∂f/∂t + ∇ · (CVf − K · ∇f ) − 1

3p2

∂

∂p(p3V · ∇f ) = q. (17)

It is a simple exercise to show that this is identical to Parker’s original form (15). Thismeans that when both the flux and the rate of adiabatic cooling are correctly transformed tothe stationary frame, the two corrections cancel one another, and the transport equation isnot affected.

Finally, (15) can be readily rewritten in the simpler form

∂f/∂t + V · ∇f − ∇ · (K · ∇f ) − 1

3(∇ · V)∂f/∂ lnp = q. (18)

The three forms (15), (17) and (18) are equivalent forms of the cosmic-ray transport equa-tion, with (18) the preferred form.

The transport equation is effectively five-dimensional, in the sense that the particle dis-tribution function, or the cosmic-ray intensity depends on three spatial coordinates, plusmomentum and time. It is a second-order partial differential equation because of the doublespatial derivative in the diffusion term. It is parabolic in nature (similar to the heat flow equa-tion) and the standard numerical technique to solve it is through finite difference schemes.

One full numerical solution with this finite-difference technique in the five dimensionshas been developed by J. Kóta of the Arizona group. It was used by Kóta and Jokipii (1991)for the study of drifts due to the warped heliospheric current sheet and co-rotating interactionregions. However, due to its complexity, the solution had limited resolution. Lately, severalso-called stochastic solutions of the multi-dimensional transport have also appeared.

There are, instead, many approximate solutions of the full equation available, and thegeneral user of modulation theory will be more interested in these approximations. Thus,in the final sections of this paper we summarize the hierarchy of these approximations inincreasing order of complexity.

6 The Convection-Diffusion Solution

The lowest-order approximation is the so-called Convection-Diffusion formalism. It is sosimple that the equation that describes it precedes the transport equation itself by six orseven years. It basically says that cosmic rays diffuse inwards in a spherically symmetric

H. Moraal

heliosphere as they scatter off the irregularities in the HMF, leading to an inward radial dif-fusive flux −κ∂f/∂r , where κ is a phenomenological diffusion coefficient that is a functionof radial coordinate r and momentum p. This flux is countered by an outward convectiveflux Vf because the irregularities are frozen into the solar wind. (The 4πp2 factor is omit-ted in both terms.) Since the overall time scale of variations is long (typically several years)relative to the propagation time through the heliosphere (typically less than one year), thecosmic-ray intensity can be considered to be in quasi-equilibrium, giving

Vf − κ∂f/∂r = 0. (19)

This simple balance equation also follows form (14) or (15) when adiabatic cooling is ne-glected, when ∂/∂t = 0, when spherical symmetry is assumed, and when the full diffusiontensor (11) is contracted into a single scalar κ(r,p).

The solution of the convection-diffusion equation (19) is

f = fbe−M, where M =

∫ rb

r

V

κdr, (20)

and where fb is the so-called local interstellar spectrum (LIS) at the outer boundary rb ofthe heliosphere, typically at rb ∼ 150 AU.

Since f is the measured spectrum, the dimensionless modulation parameter M =ln(fb/f ) would be known experimentally if the LIS, fb , were known. Generally it is not;the ultimate task of modulation theory is to determine it correctly. Hence, the essence ofmodulation theory is to find M theoretically from V and κ (or more complicated versionsof its tensor form), so that fb can be derived.

Modulation changes between times t1 and t2 are, however, conveniently described with-out reference to the LIS as ln(f2/f1) = −�M = M1 − M2.

The diffusion coefficients and the drift coefficient in (11) can conveniently be written interms of a diffusion mean free path or drift length scale λ, as κi = 1/3vλi , where the indexi can stand for ‖, ⊥, or T . The most basic rigidity dependence of the three length scalesis λi ∝ P , although significant deviations are known. Hence κi ∝ βP , where β = v/c. Ittherefore follows that to first order the modulation strength is inversely proportional to βP .

Typical values are Msolar min ≈ 1.5/βP (GV) and Msolar max ≈ 4.5/βP (GV). Thus forhigh-latitude neutron monitors (cutoff rigidity ∼1 GV and median rigidity ∼15 GV) thefull change in intensity over an 11-year solar cycle is �M = �f/f = ln(f2/f1) = �M ≈(4.5 − 1.5)/15 = 0.2 (or 20%). At P = 100 GV (T = 1011 eV) modulation becomesinsignificant, while for P � 1 GV this simple approximation becomes invalid. UsingV = 400 km/s and rb ∼ 150 AU in (20) leads to typical values of the diffusion coefficient κ

in the range (2–6) × 1022βP (GV) cm2/s from solar maximum to solar minimum at 1 AUin the heliosphere.

The modulation parameter M is a convenient single parameter that describes the modu-lation as a function of (1) the solar wind speed, (2) the diffusion coefficient, and (3) the sizeof the heliosphere. It is important to note, however, that these three parameters cannot bededuced separately—only their integral effect is known, and therefore the solution containslittle physics. For example, the magnitude and radial dependence of κ are determined by theParker spiral magnetic field and its fluctuation spectrum, but these cannot be deduced froma simple measurement of the M via ln(fb/f ), or of �M = ln(f2/f1).

The next two paragraphs show that the next two levels of approximation, namely theForce-Field approach and the numerical solution of the steady-state spherically symmetricmodulation equation produce the same modulation function M .

Cosmic-Ray Modulation Equations

7 The Force-Field Solution

The Force-Field formalism is by far the most generally used approximation of modulationtheory. It produces the so-called modulation potential φ, the origin of which is not generallyappreciated. Here it is shown how the Force-Field equation relates to the full cosmic-raytransport equations (15), (17), or (18), and also that the Force-Field potential is essentiallythe same as the modulation parameter M of the previous section.

In their original paper Gleeson and Axford (1968) formulated the Force-Field solution tothe cosmic-ray transport equation as:

j (r,E)

E2 − E20

= jlis(E + �)

(E + �)2 − E20

, (21)

where j refers to the observed intensity (instead of distribution function as we have used upto now) at radial distance r , jlis is the local interstellar spectrum, E is total energy (kineticplus rest mass energy), and � is the so-called Force-Field energy loss that particles suffer asthey propagate inwards into the heliosphere. This energy loss is related to the Force-Field ormodulation potential φ by � = Zeφ. There are many variants of this formulation.

The modulation potential φ has become the most commonly used modulation parameterin the literature. However, the origin and nature of the solution are seldom quoted, and itis often used wrong because the limitations are not recognized. For instance, if the diffu-sion coefficient in the transport equation is not strictly proportional to rigidity to the powerone, � can not be interpreted as an energy loss. The identification of φ as a potential is di-mensionally correct, but it seems somewhat mysterious, and it is almost never related to theunderlying diffusion and convection parameters. To implement the solution, i.e. to calculatethe parameter φ, involves an intricate inversion process.

In view of this, Gleeson and Urch (1973) gave a much simpler and more transparentre-derivation of the Force-Field solution, which will be presented here.

The Force-Field approximation recognizes that the cosmic-ray flux needs the Compton-Getting correction as in (16). Thus, balancing the inward diffusive flux with the correctedoutward convective flux produces the equation

CVf − κ∂f/∂r = 0, (22)

instead of (19). This suggests that the solution is the same as (20), but with the modu-lation parameter modified to M = ∫ rb

r(CV/κ)dr . This is, however, not useful, because

C = −1/3(∂ lnf/∂ lnp), which represents the form of the spectrum, and this spectrumchanges with radial distance as the modulation occurs. Thus C is a function of r , and tofind this r-dependence is part of the modulation problem.

When C is introduced explicitly, (22) rather becomes

Vp

3

∂f

∂p+ κ

∂f

∂r= 0 (23)

which is a first-order partial differential equation with solution f (r,p) = constant =fb(rb,pb) along contours of the characteristic equation dp/dr = pV/3κ in (r,p) space. Thesubscript b once again designates values on the outer boundary of the modulation region.The reader should note that this statement is exactly the same as the “classical” Force-Fieldsolution (21), but in different notation.

H. Moraal

The name “Force-Field” originates from the fact that (23) can also be written in terms ofrigidity P = pc/Ze as

∂f

∂r+ V P

3κ

∂f

∂P= 0. (24)

Particle rigidity has the dimensions of electrostatic potential and therefore the coefficientV P/3κ of the second term has the dimensions of potential per unit length (SI units V/m),or units of electrostatic field—hence the name “Force-Field”.

The rigidity Pb = Pb(r,P ) is obtained by integrating the characteristic equationdP/dr = V P/3κ from the initial phase space point (r,P ) to the point (rb,Pb) at the outerboundary rb . If the diffusion coefficient is separable in the form

κ(r,P ) = βκ1(r)κ2(P ), (25)

the solution is∫ Pb(r,P )

P

β(P ′)κ2(P′)

P ′ dP ′ =∫ rb

r

V (r ′)3κ1(r ′)

dr ′ ≡ φ(r), (26)

where φ is called the Force-Field parameter. When κ2 ∝ P , which is typical, and whenβ ≈ 1, as for most ground-based observations such as with neutron monitors, the solutionreduces to the widely used form,

Pb − P = φ. (27)

This implies that the Force-Field parameter φ has the physical meaning of a rigidity loss.This, in turn, can also be transformed into an energy or a momentum loss. Since the Force-Field parameter has the dimensions of potential, it is often called the Force-Field potential.

It is, however, often forgotten that the Force-Field rigidity (or momentum) loss in theform (27) applies only to the special case of relativistic particles, β = 1, and the rigiditydependence κ ∝ P . As mentioned before, there are many cases in which this is not true. Inall such cases Pb − P is some other function of φ. Then φ alone is insufficient to describethe modulation and it does not have the dimensions of potential.

Gleeson and Urch (1973) explicitly emphasized this complication, namely that the fullForce-Field parameter is actually φ/κ2 instead of φ, but this is not generally realized. Itfollows from (20), (25) and (26) that this full Force-Field parameter is given by

φ

κ2= β

M

3, (28)

which says that the Force-Field parameter is just 1/3 of the modulation parameter M

(times β). Using the previously given values of M , typical values of φ range from 450MV at solar minimum to 1350 MV at solar maximum.

In this sense the Force-Field formalism produces no new insight over and above thesimple Convection-Diffusion mechanism. However, the description of the modulation in thetwo formalisms are entirely the opposite of one another. This is demonstrated in Fig. 4. Thisfigure shows hypothetical cosmic-ray spectra, one on the outer boundary, and the other amodulated spectrum inside the heliosphere. Specifically, it plots the distribution functionf (P ) = jT /P 2 (see (7) above) instead of the usually measured intensity spectrum jT . Thevertical line shows that the Convection-Diffusion approach describes the modulation as areduction in intensity at a given rigidity, without rigidity (or energy) loss, and the magnitudeof this reduction is given by M . The Force-Field approach, however, describes this same

Cosmic-Ray Modulation Equations

Fig. 4 Graphical representationof the description of themodulation with the Force-Fieldsolution (horizontal line withparameter �) and theConvection-Diffusion solution(vertical line with parameter M).The sloped line represents theactual modulation as acombination of intensityreduction and energy (or rigidity)loss

modulation as a rigidity (or energy or momentum) loss, � = Pb −P , which is read off froma graph such as in Fig. 4. This � is a function of φ and it is only numerically equal to φ

when κ ∝ P , β = 1, and when one deals with protons (for which A/Z = 1). In all othercases the rigidity loss that is read off from spectra as in Fig. 4 cannot be converted to amodulation potential, and it is more correct to describe the modulation in the Force-Fieldformalism with the dimensionless parameter M .

Notice the contradiction that the Force-Field formalism produces a modulation potentialthat causes energy (or rigidity or momentum) changes, while its defining (22) neglects adia-batic energy changes. This is so because the Force-Field equation (22) stems from (17) in thesteady state (∂/∂t = 0), with no sources (q = 0), and with the adiabatic momentum-changeterm also equal to zero. The energy change that results from the Force-Field equation orig-inates from the interpretation of the coefficient V P/3κ in (24) as a field, or force per unitcharge. There is no physical reason why this term, i.e. the Force-Field energy loss, is relatedin any way to the true adiabatic loss. It is fortuitous that this energy loss is a reasonable ap-proximation of the adiabatic energy loss in certain circumstances. Gleeson and Urch (1971)showed, in fact, that the Force-Field energy loss that is implicit in (26) is an upper limit ofthe true, adiabatic loss.

8 The Spherically Symmetric Steady-State Transport Equation

The above two solutions of the cosmic-ray transport equation are the only two useful ana-lytical solutions available. They were introduced in the first place because forty years agonumerical solutions of the transport equation were difficult to do and time-consuming onslow computers. This has drastically changed, however, and in this section it is shown thatthe third level of approximation is a very easy and accessible numerical solution of the

H. Moraal

one-dimensional (spherically symmetric), steady-state (∂/∂t = 0) version of (18). This ap-proximation is

V∂f

∂r− 1

r2

∂

∂r

(r2κ

∂f

∂r

)− 1

3r2

∂

∂r(r2V )

∂f

∂ lnp= 0. (29)

The difference with the previous two approximations is that it retains the effect of adiabaticcooling in its third term. In addition, it was shown that the Compton-Getting effect, i.e. theproper transformation from the solar wind to the stationary frame is also implicitly included.

This form of the equation also explicitly shows that the adiabatic cooling rate 〈p〉/p =−(1/3)∇ · V is the same as the well-known adiabatic cooling law Pτ 5/3 = constant (whereτ is a volume and P is pressure; not rigidity). In the adiabatic limit of modulation at lowenergies κ → 0 and for constant V equation (29) simplifies to ∂f/∂r = (2p/3r)∂f/∂p.Its solution is that f (r,p) = constant along the contours dp/dr = −2p/(3r) or pr2/3 =constant. For a constant solar wind speed the volume element τ in which the particles arecontained is proportional to r2 (because the element only expands in the angular directionsand not in radial thickness). This implies that pτ 1/3 = constant. Using the formulas for theinternal energy of a gas and the ideal gas law then produces Pτ 5/3 = constant.

A consequence of the fact that f (r,p) is constant in this low-energy adiabatic limit ofmodulation, is that, according to (7), kinetic energy spectra are of the form jT ∝ T , and thatthe radial gradient in the intensity is zero. Physically, that means that particles at such lowrigidities have not diffused into the heliosphere at those low rigidities, but have appearedthere due to rigidity loss.

Equation (29) is of the heat flow-type, and it is readily solved with the Crank-Nicholsonmethod, with the stepping coordinate in time, i.e. �t , replaced by −� lnp, i.e. steppingdownward in momentum because particles lose momentum. This technique was first usedby Fisk (1971). The ‘initial’ condition is that at the highest momentum there is no mod-ulation, i.e. at that momentum the LIS pervades undiminished throughout the heliosphere.The outer boundary condition at r = rb is that the LIS is maintained there at all momenta,while experience has shown that the inner boundary at r = rsun has almost no effect on thesolution—typically either f or ∂f/∂r are set equal to zero there.

Fortran and C++ versions of this numerical solution can be found at http://www.nwu.ac.za/p-csr/index.html.

Section 3 of the paper of Caballero-Lopez and Moraal (2004) contains a full solutionwith a realistic set of parameters. Figure 5 is from that paper and it shows solutions of theConvection-Diffusion, the Force-Field, and the steady-state spherically symmetric approxi-mations of the transport equation.

The basic features of the solutions are:

1. In the inner heliosphere the Force-Field equation is a much better approximation to thetransport equation (29) than the Convection-Diffusion equation, because at low energiesthe Convection-Diffusion solution for the intensity is always much lower than the numer-ical solution.

2. In the outer heliosphere the Convection-Diffusion approximation is much better than theForce-Field approximation. Here the Force-Field solution at low energies leads to a grossoverestimation of the intensity obtained from (29).

3. These above two effects can be understood because adiabatic losses are proportional to∇ · V = 1

r2∂∂r

(r2V ), and this equals 2V/r for constant V . Hence, in the inner heliospherethe adiabatic loss rate is large, which is well-represented by the Force-Field energy loss,while in the outer heliosphere it is small, which is better represented by the Convection-Diffusion approach without such a loss.

Cosmic-Ray Modulation Equations

Fig. 5 Numerical solution of thesteady-state, one-dimensionaltransport equation (full lines),together with the Force-Fieldsolution (dashed lines) and theConvection-Diffusion solution(dotted lines) for galacticcosmic-ray protons in aheliosphere with rb = 90 AU,V = 400 km/s,λ = 0.29P (GV) AU, and φ

(1 AU) = 407 MV). Intensitiesare multiplied by factors of

√10

to enhance visibility. Adaptedfrom Caballero-Lopez andMoraal (2004)

4. These effects can also be understood qualitatively by referring back to Fig. 4. Themodulation process proceeds along the diagonal line drawn from the LIS to the spec-trum inside the heliosphere: it is a combination of exclusion due to convection-diffusionalong the vertical axis, and of adiabatic loss along the horizontal axis. The vertical linemarked ‘M’ and the horizontal line marked ‘�’ therefore indicate that the Convection-Diffusion and Force-Field approaches respectively neglect the energy loss and the ex-clusion. Thus, the Force-Field energy loss is always an upper limit for the true adiabaticloss.

5. The largest deviation that occurs from the numerical solution of (29) is at 1 AU, wherethe Convection-Diffusion solution starts to diminish fast with decreasing energy atT <∼ 400 MeV (for protons). This is equivalent to a rigidity of 1 GV. Neutron monitorshave a median rigidity of response �1 GV. For applications such as the 10Be concentra-tion in ice, the yield function peaks in the range 2 to 6 GV. Both these rigidities are wellabove 1 GV, and hence for these applications the Convection-Diffusion and Force-Fieldapproximations produce essentially the same modulation parameter M as the numericalsolution.

The combination of the values rb = 90 AU, V = 400 km/s, and κ = 4.38 × 1022βP

(GV) cm2/s, which fit the observed spectra at 1 AU near solar minimum conditions, givethe modulation parameter M , defined in (20) as M = 1.22 βP (GV). This is equivalent to aForce-Field parameter φ (1 AU) = 407 MV. The question arises as to how the solutions willchange if a different solar wind, boundary distance, or κ are chosen. Equations (20) and (28)imply that the Convection-Diffusion and Force-Field solutions will remain the same for all

H. Moraal

Fig. 6 Numerical solution of thesteady-state, one-dimensionaltransport equation (29) forgalactic cosmic-ray protons usingdifferent radial dependencies ofthe diffusion coefficient, suchthat the modulation function at 1AU remains fixed at M

(1 AU) = 1.22βP (GV). Thisshows that even with adiabaticlosses included, the modulationparameter M remains anexcellent single-parameterindicator of modulation strength.Adapted from Caballero-Lopezand Moraal (2004)

combinations that give the same M or φ, i.e. that only the integrated effect of the modulationcan be detected. This is not true for the full numerical solution of (29), because the adiabaticloss term in that equation does not scale proportional to M . Therefore, in Fig. 6 we testto what extent the full numerical solution deviates from the scaling with M or φ. A setof five diffusion coefficients was chosen, being proportional to r−1, r−0.5, r0, r0.5 and r1.The magnitudes were chosen such that M (1 AU) is the same for all of these. The figureshows that the five numerical solutions at 1 AU are very similar. In contrast, the 80 AUsolutions differ dramatically, because M (80 AU) is different for each of them. At 10 MeV,for instance, the modulation from the LIS to the 1 AU intensity is a factor of 470, whilethe solutions for κ ∝ r−1 to κ ∝ r1 differ by only a factor of 2, i.e., by a relative amountof 0.4%. This shows that the modulation parameter M is a single expression that adequatelydescribes the modulation process, including its rigidity dependence, and that this is trueeven in the presence of adiabatic energy losses, which do not scale proportional to M . Theindividual values of κ, rb , and V can only be deduced from two or more spectra at differentradial distances.

9 Other Solutions of the Transport Equation

Because all the information on the modulation is captured in the modulation parameter M ,the above one-dimensional analytical and numerical solutions of the transport equation con-tain no physics that is introduced by the geometry of the heliospheric magnetic field, i.e.the so-called Parker spiral field or higher order models. However, for most “general” mod-ulation corrections, for which it is sufficient to characterize modulation strength by a singleparameter, these more involved solutions are also not needed. These solutions are brieflyreferenced.

The lowest-level model that can incorporate the physics of the Parker spiral geometry isthe two-dimensional steady-state solution of the transport equation. The two spatial variablesare radial distance r and polar angle θ . In this case the diffusion coefficients parallel and

Cosmic-Ray Modulation Equations

perpendicular to the magnetic field, κ‖ and κ⊥, can be defined separately, as well as thedrift coefficient κT , all three going into the diffusion tensor (11). It is well-established thatthe HMF in the northern hemisphere points away from the sun and that in the southernhemisphere toward it, with the two hemispheres separated by a so-called neutral sheet. Thissheet is relatively flat during solar minimum conditions when the heliomagnetic equator iswell aligned with the heliographic equator, but it becomes progressively warped towardssolar maximum, when the heliomagnetic equator tilts. This field pattern lasts for an 11-year activity cycle. In the next cycle the field reveres in direction, and this causes the driftflux to reverse direction throughout the heliosphere. These oppositely directed drifts causessharply peaked cosmic-ray maxima during solar minima at the end of uneven numberedsolar activity cycles as in 1965, 1987 and 2009, and flat-topped cosmic ray maxima at theend of even-numbered cycles solar activity cycles, such as in 1976 and 1997. The neutralsheet drift is technically more difficult to handle than through a simple off-diagonal term inthe diffusion tensor (11).

A first version that includes this geometry (but not the drifts) was written by LA Fiskin 1973 (no reference available). It was described with adaptations in Moraal and Gleeson(1975), and with further refinement of magnetic field models and the inclusion of drifts byPotgieter and Moraal (1985). A similar solution was developed by the Arizona group of JRJokipii and J Kóta, with a first detailed description by Jokipii and Korpriva (1979). Over theyears it has been shown that these two-dimensional solutions can quite effectively describethe solar rotation-averaged effects of a wavy neutral sheet in the heliospheric magnetic field,although this sheet has a three-dimensional geometry.

For the description of shorter-term and more localized effects, a three-dimensional solu-tion that includes azimuthal angle φ is needed, and such steady-state solutions are describedby Kóta and Jokipii (1983) and Burger et al. (2008).

The above solutions are all based on the finite difference method, with the spatial vari-ables on a solution grid, and with momentum the stepping variable, as explained in theprevious section. This method cannot work in the case when acceleration effects, primar-ily introduced by shocks, are introduced, because then the particles both gain and lose (theadiabatic loss) energy. The most important acceleration effect is that of the solar wind ter-mination shock at approximately 90 AU. In such cases momentum must be treated as an-other grid variable, and the solution is stepped forward in time. When one is interested inthe steady-state behavior of such a time-dependent solution, it is started from some initialcondition and run until it saturates. Typical time scales for saturation are of the order ofone year. Such two-dimensional (radial distance and polar angle) time-dependent solutionsthat include termination shock acceleration ere developed by Jokipii (1986) and Steenkamp(1995).

As mentioned previously, one full numerical solution with the finite-difference schemein three spatial dimensions plus momentum and time has been developed by J. Kóta ofthe Arizona group. It was used by Kóta and Jokipii (1991) for the study of drifts by thewarped heliospheric neutral sheet and co-rotating interaction regions. However, due to itscomplexity, the solution had to be limited in resolution. Lately, several so-called stochasticsolutions of the multi-dimensional transport equation have appeared. These solutions caneasier handle more complicated transport coefficients than finite difference solutions, butthey are generally much slower. However, the rapid advances in high-performance parallelcomputing makes them continually more attractive relative to the standard finite differencemethods. References to such stochastic solutions can be found in Pei et al. (2009).

H. Moraal

10 Summary

This paper gives the mathematical tools needed to describe the modulation of cosmic rays inthe heliosphere, starting with elementary definitions and geometrical concepts, then writingdown the cosmic-ray transport equation, and describing its solutions. The three simplest so-lutions are the Convection-Diffusion, the Force-Field, and the one-dimensional steady-statenumerical solution. They are most generally used for correction of modulation effects. Allthree of these produce a single dimensionless modulation parameter M as function of time(or phase in the solar cycle) and rigidity (or energy) that characterizes the depth of mod-ulation. For most applications this parameter is sufficient. For rigidities > 1 GV, the threesolutions produce approximately the same numerical value of M , and the simplest solutionof the three, the Convection-Diffusion solution, can be used. Alternatively, the widespreadpractice to use the Force-Field solution can be continued, but it is generally not correct toexpress its result as a modulation potential, but rather as the same dimensionless M , as waspointed out by Gleeson and Urch (1973). For rigidities < 1 GV the Convection-Diffusionand Force-Field solutions start to deviate from the numerical solution in such a way thatthe former produces a too small, and the latter a too large value of M when compared tothe numerical solution of the one-dimensional transport equation. With the rapid increaseof computing speed such a numerical solution has actually become easier to do than itstwo analytical approximations, and it is the recommended solution to use. Fortran and C++versions of this numerical solution can be found at http://www.nwu.ac.za/p-csr/index.html.

Because these three lowest-order solutions lump the effects of diffusion, convection, adi-abatic energy loss and the size of the heliosphere into a single parameter (M), these solutionsdo not produce a deeper physical insight into the separate nature of each of these processes.Neither can these solutions account for higher-order effects such as the geometry of theheliospheric magnetic field, drift motions in this field and in its wavy current sheet, andacceleration in traveling shocks and the termination shock of the heliosphere. These effectscan only be studied with more involved numerical solutions of the transport equation.

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