Cosmic-Ray Modulation Equations

Transcription

1 Space Sci Rev DOI /s Cosmic-Ray Modulation Equations H. Moraal Received: 6 December 2010 / Accepted: 5 August 2011 Springer Science+Business Media B.V Abstract The temporal variation of the cosmic-ray intensity in the heliosphere is called cosmic-ray modulation. The main periodicity is the response to the 11-year solar activity cycle. Other variations include a 27-day solar rotation variation, a diurnal variation, and irregular variations such as Forbush decreases. General awareness of the importance of this cosmic-ray modulation has greatly increased in the last two decades, mainly in communities studying cosmogenic nuclides, upper atmospheric physics and climate, helio-climatology, and space weather, where corrections need to be made for these modulation effects. Parameterized descriptions of the modulation are even used in archeology and in planning the flight paths of commercial passenger jets. The qualitative, physical part of the modulation is generally well-understood in these communities. The mathematical formalism that is most often used to quantify it is the socalled Force-Field approach, but the origins of this approach are somewhat obscure and it is not always used correct. This is mainly because the theory was developed over more than 40 years, and all its aspects are not collated in a single document. This paper contains a formal mathematical description intended for these wider communities. 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 to count particles, (3) the description of particle motion with transport equations, and (4) the solution of such equations, and what these solutions mean. Part (4) was previously described in Caballero-Lopez and Moraal (J. Geophys. Res, 109: A05105, doi: /2003ja ). 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 to do modulation studies, to show how seemingly unrelated results are, in fact, related to one another, and to point out the historical context of some of the results. The paper adds no new knowledge. 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 Africa harm.moraal@nwu.ac.za

2 H. Moraal 1 The Four Energy Variables The bulk of the cosmic-ray particles observed in the inner heliosphere have energy of the order of 1 GeV (10 9 ev), which is on the borderline between non-relativistic and relativistic protons. This means that the kinetic and rest-mass energies of the bulk of the particles are of the same order of magnitude, because at higher energies the intensity falls off fast, while at lower energies the particles are shielded away from the inner heliosphere by the modulation process. While non-relativistic and super-relativistic particle descriptions are relatively easy, the quasi-relativistic regime, where cosmic-ray modulation is most important, is technically more difficult. The description starts with the energy-momentum relationship E 2 = p 2 c 2 + m 2 0 c4, (1) where E is energy, p is momentum, m 0 is the rest mass of the particle, and c the speed of light. This relationship is Lorentz-invariant. (This property is useful for theoretical development, but it need not concern the general reader.) The quantity E 0 = m 0 c 2 is called the restmass energy, and the total energy consists of two parts, kinetic plus rest-mass: E = T + E 0. E 0 = 938 MeV (i.e. roughly 1 GeV) for protons, and 511 kev for electrons. When T E 0 the particles are relativistic and (1) reduces to E = T = pc. WhenT E 0 they are nonrelativistic. Writing E = E 0 (1 + T/E 0 ) for this case, and using the binomial expansion (1 ± ɛ) n = 1 ± nɛ + n(n 1)ɛ 2 /2!±... with T/E 0 = ɛ 1, (1) reduces to the standard T = p 2 /2m 0 = 1 2 m 0v 2. Relativistic mass is given by m = m 0 / 1 β 2 with β = v/c.ifthis is substituted into (1) it leads to the well-known mass-energy relationship E = mc 2. In many cosmic-ray applications it is important to refer explicitly to the number of nucleons that compose a nucleus, a nuclide, or an ion. A fully-stripped light atomic nucleus with A nucleons, of which Z are protons, has A/Z = 2, while higher up in the periodic table A/Z gradually grows larger than 2. A Hydrogen nucleus is an exception, because it has 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 anomalous cosmic rays (ACR), discovered in the 1970s. The discovery and understanding of these ACRs is described and referenced in Moraal (2001). With this explicit reference to the number of nucleons, (1) can be rewritten as E p = p 2 c 2 + E 2 0p,or A 2 (T + E 0 ) 2 = p 2 c 2 + A 2 E 2 0, (2) where E p and E 0p are energies per particle, while E,T and E 0 are energies per nucleon. The horizontal axis of almost any published cosmic-ray spectrum expresses energy as kinetic energy per nucleon, and not per particle. Since cosmic rays are charged, they are confined by the magnetic fields pervading the cosmos. They experience the Lorentz force F = q(e + v B),wherev is the particle velocity, 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 almost never 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 a spiral trajectory of a charged particle about a magnetic field line as shown in Fig. 1(a), with a so-called gyro-radius or Larmor radius r g = mv sin α/(qb), whereα is the pitch angle between the velocity vector v and the magnetic field B. Whenα = 90 the spiral becomes

3 Cosmic-Ray Modulation Equations Fig. 1 Charged particle motion in a magnetic field. (a) In a uniform magnetic field the particle has a spiral orbit with a gyroradius r g = P/Bc.(b) When the field is non-uniform the particle drifts away from a field line due to the gradient and curvature of the field. (c) When a particle meets a kink in the field that has a scale length r g, all particles will progress through the kink (but they may drift to adjacent field lines while doing so). (d) Likewise, if r g scale size of the kink, all particles will pass through it without being affected much. (e, f, g)whenr g scale size of the kink, it depends on the gyrophase of the motion when the particles starts to feel the kink whether it will go through the kink (e), be reflected back (f), or effectively get stuck in the kink (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 r g = mv/(qb) = p/(qb). This implies that the gyroradius depends on two particle properties, namely its momentum and charge. For this reason we introduce the concept of rigidity, defined as P = p/q. Thenr g = P/B, which says that the gyroradius depends 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 dimensions of energy per unit charge, or potential. If energy (and pc) is expressed in ev, and charge in terms of the number Z of elementary charges, i.e. q = Ze where e = C, then P has units of Volt (V). Thus, the formal definition of rigidity is P pc/(ze), with the gyroradius given by r g = P/Bc. Putting this into (1) gives the relationship P = (A/Ze) 2 T(T + 2E 0 ) between the rigidity of a particle and the kinetic energy per nucleon of that particle. Bearing in mind that m = m 0 / 1 β 2, one gets the universal relationship P = pc/(ze) = (A/Ze) T(T + 2E 0 ) = (A/Ze)β(T + E 0 ), (3)

4 H. Moraal between (1) rigidity P,(2) momentump, (3) the dimensionless speed β of a nucleus with 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 energy per nucleon is β = T(T + 2E 0 )/(T + E 0 ). (Notice that this is independent of particle species i.e. independent of A and Z.) In terms of rigidity the particle speed is β = P/ P 2 + (A/Ze) 2 E0 2. Other quantities such as T(β)and P(β)are also readily calculated. For example, T(P)= (Ze/A) 2 P 2 + E0 2 E 0. The anchor point of these relationships is that the rest-mass energy of a proton is E 0 = 938 MeV. This is near enough to 1 GeV, so that putting E 0 = 1 in all these relationships implies that one works in units of GeV, so that rigidity has units of GV. Then one operates very easily between the different quantities, such as: 1. A 1 GeV proton has a rigidity P = 3GV. 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 E 0 )T and P are numerically equal, i.e. P = T, while for non-relativistic ones (T E 0 ), P = 2T. However, this last relationship is only true if one sticks to the rule of always working in units of GeV and GV. If one works with electrons, all the expressions yield numerically correct answers if one takes A = 1, Z = 1andE 0 = GeV. Finally, particles need a rigidity of at least 17 GV to penetrate vertically through the Earth s magnetosphere at the magnetic equator (in the vicinity of Thailand where the field is strongest). This is the highest so-called cut-off rigidity, P c, in the geomagnetic field. Towards 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 their effective vertical cutoff ranges from P c 1 GV in the polar regions, to P c 17 GV at the geomagnetic 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 differential density, are more useful in theory. They have to be related to one another, and once again, this is made more difficult because we work in the quasi-relativistic regime. Consider an infinitesimal box with volume dxdydz, usually written as d 3 r. In spherical polar co-ordinates (r,θ,φ) this volume is d 3 r = r 2 sin θdrdθdφ. Let there be N particles inside the box. The density of particles, n, is defined such that N = nd 3 r. In cosmic-ray physics, density is not a very useful quantity. Almost all properties of cosmic rays are derived from cosmic-ray spectra, which count the number of particles within a given interval of kinetic energy (T, T + dt) (or the equivalent momentum, rigidity, or speed interval). For this reason we define the differential density, U, such that the number of particles in the infinitesimal box d 3 r, with kinetic energy per nucleon in the interval (T, T + dt) is dn = Ud 3 rdt. Evidently, the total (or integral) density in terms of the

5 Cosmic-Ray Modulation Equations Fig. 2 The figure illustrates how the intensity j that goes through a detector with area da is related to the density U in the ring with thickness 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 and M.A. Forman in the late 1960s. It is important to be specific about the energy units used for U. We therefore call the differential density defined above U T, because it is the differential density w.r.t. kinetic energy (per nucleon). Similarly, one can define momentum and rigidity densities, denoted as U p and U P respectively. Since the number of particles, dn, remains the same, independent of how they are counted, one must have dn = U T dtd 3 r = U p dpd 3 r = U P dpd 3 r.from(3) and its associated expressions, it follows that dt/dp = βc/a and dt/dp = βze/a. Hence U p = (c/ze)u P = (βc/a)u T. This says that spectra in terms of rigidity and momentum have the same shape (they differ only with the numerical factor c/ze), but they look different from the kinetic-energy spectrum because β is a function of T. Experimenters do, however, not measure differential densities; they measure differential intensities instead. This is so because detectors measure the rate at which particles go through detectors, instead of the number inside them. Therefore, consider a detector element with area da, and that particles with speed v go through it, as shown in Fig. 2. The particles that go through the detector in the time interval dt at present, were somewhere in a spherical shell with thickness vdt some time before. The dark sub-region in this shell contains dn = U T d 3 rdt = U T r 2 sin θvdtdθdφdt particles. They go in all directions, and the fraction of them that will go through the detector is the ratio of the projected detector area to the area of a sphere with radius r, i.e. da cos θ /4πr 2. Therefore, the number of particles from the shaded part of the shell that go through the detector in time dt is dn = dn da cos θ /4πr 2 = U T (da cos θ /4π)vdt sin θdθdφdt. Bearing in mind that the θ,φ-integral over the entire shell has a value of 2π, one finally arrives at: the number of particles passing through the detector per unit area, per unit time, per unit kinetic energy interval is vu T /2. It is standard practice not to refer to all the particles coming form all directions, but rather those coming from one steradian of solid angle. Hence, the differential intensity, denoted as j T,is j T = vu T /8π, (4) with the dimensions of particles per unit area, per unit time, per unit solid angle, per unit kinetic energy per nucleon, or units of particles/m 2 /s/steradian/gev/nucleon. Obviously, one similarly finds that j p = vu p /8π and j P = vu P /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 U T dt = U p dp = U P dp, one easily finds from (3) that j T = (A/8π)U p = (A/8Ze)U P. (5)

6 H. Moraal Fig. 3 The definition of a volume element d 3 r in configuration space, a similar element d 3 p in momentum space, and a momentum shell with thickness dp This is a useful result because it means that the (usually measured) differential intensity w.r.t. kinetic energy per nucleon has the same form as the (usually calculated) differential densities 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 the number of particles in terms of the differential density U, but rather in terms of the particle distribution function. This distribution function is defined such that the number of particles with momentum in the interval d 3 p between the vectors p to p + dp, and in the volume element d 3 r is dn = F(r, p,t)d 3 rd 3 p. (6) This definition is sketched in Fig. 3. Plasma physics and transport theory texts variously use f and F, but almost all of these texts define the distribution function in terms of velocity v instead of momentum p. Since most plasmas and other flows are non-relativistic, this makes no difference. But for the relativistic cosmic rays it turns out that using momentum as variable eliminates a large amount of unnecessary algebra. We interrupt by noting the hierarchy of counting particles: there are F = dn/(d 3 rd 3 p) particles per unit volume per unit momentum vector between p and p + dp; there are U p = dn/(d 3 rdp) particles per unit volume per unit momentum magnitude from p to p + dp; andtherearen = N/d 3 r particles per unit volume, of all possible momenta. The momentum interval d 3 p can be written, just as in the case for d 3 r as d 3 p = p 2 dpd = p 2 sin θdθdφdp,whered is an element of solid angle in momentum space. Then, the number of particles in d 3 p, with momentum magnitude between p and p + dp are those inside the shell of Fig. 3, and their number is given by dn = d 3 rp 2 dp F(r, p,t)d. But it was argued above that this number is also dn = U p dpd 3 r. Hence U p (r,p,t) = p 2 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 F(r, p,t)d. 4π Note that F is a function of vector p, while f is a function only of its magnitude, p. From this it immediately follows that U p (r,p,t)= 4πp 2 f(r,p,t). Finally, when one also incorporates the previously described differential intensity j, it produces the following very useful relationship between three possible ways of counting particles: j T = (A/8π)U p = (A/2)p 2 f.

7 Cosmic-Ray Modulation Equations In summary, since absolute normalizations are seldom important, these relationships are usually conveniently abbreviated as j T = U p = p 2 f = U P = P 2 f, (7) which means that observed intensity spectra w.r.t. kinetic energy are the same as density spectra w.r.t. momentum and rigidity, and they differ only with a factor of p 2 or P 2 from distribution functions w.r.t. momentum and rigidity. Often cosmic-ray spectra have power law forms, i.e. f p γ. Then it becomes second nature to immediately know that j T, U p and U P 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 distribution function. This is that the momentum distribution is Lorentz-invariant. The number of particles with momentum in the interval d 3 p between the vectors p to p + dp, andin the volume element d 3 r was given above by dn = F(r, p,t)d 3 rd 3 p. This same number of particles can also be described as the number in the interval velocity interval d 3 v between the vectors v to v + dv, also in the volume element d 3 r,sothatdn is also given by dn = F(r, v,t)d 3 rd 3 v. Consider now a transformation to another reference frame that moves with velocity 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 the primed frame then measures the same number of particles, dn, but in terms of his distribution function it is given by dn = F (r, v,t)d 3 rd 3 v. Hence F(r, v,t)d 3 v = F (r, v,t)d 3 v.the same holds for momentum distribution functions: F(r, p,t)d 3 p = F (r, p,t)d 3 p. The big difference is however, that for transformations in which the transformation speed is much less than the particle speed, i.e. V v,d 3 p = d 3 p,butd 3 v d 3 v. Hence F (p ) = F(p), but F (v ) F(v). This useful property is called the Lorentz invariance of the momentum distribution function, and was first pointed out by Forman (1970). It is also important to distinguish between the concepts of intensity and flux in transport theories. Intensity was defined above as the scalar quantity j α = vu α, 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 S α = v U α. The two quantities have the same dimensions, but the intensity counts all the particles that go through the detector, irrespective of direction, while the flux counts the net amount of them, or the directional sum (and thus gives the direction of the net flow). In the case of a unidirectional beam of particles, all with the 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 timerate-of-change of the number of particles in a given volume must be equal to the rate of particles flowing across the closed surface around that volume, plus the rate at which particles are created/destroyed by physical process (such as ionization, recombination, inelastic collisions etc.) in that volume: dn dt = S da + Q, (8)

8 H. Moraal where da is a surface element, and Q is a source function with dimensions of particles per unit time. The minus sign accounts for the fact that the number of particles in the volume decreases if there is a net outflow. According to the divergence theorem S da = Sdτ, wheredτ is the volume element 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.Inthe solar wind, this flux consists of two parts, a diffusive flux due to scattering off the irregularities in the heliospheric magnetic field (HMF), and radial outward convection in the solar wind, with velocity V. The convective flux is simply S c = nv, but the diffusive flux due to scattering in HMF irregularities has been the core of all theoretical modulation studies over the last 50 years. The panels (c), (d), (e), (f) and (g) of Fig. 1 show that when a spiraling charged particle encounters an irregularity in a magnetic field line that is of the same scale size as the gyroradius of the particle, then the trajectory of the particle through the irregularity depends critically 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 irregularity (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 of the 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: S d = κ n. Charged particles do not readily cross magnetic field lines, except for two effects: perpendicular scattering and gradient/curvature drift. When particles encounter magnetic field irregularities, not only their pitch angle, but also their gyrophase changes. Physically this happens 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 to a neighboring field line, as shown in Fig. 1(h). Since this is also a random process, it leads to a diffusive flux perpendicular to the background magnetic field. Hence, the diffusive flux consists of two components: S d = κ n + κ n,whereκ and κ are the diffusion coefficients parallel and perpendicular to the background magnetic field, and where typically κ κ. This is called the weak-scattering limit. Perpendicular diffusion may be significantly enhanced, however, by random walk of the fluctuating magnetic field lines themselves. The upper limit for perpendicular diffusion is κ = κ, which is reached through a simultaneous decrease in κ and an increase in κ as the amount of turbulence in the field increases. This limit is reached when the fluctuations in the field become as large as its average 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 understood in terms of so-called quasi-linear theory of scattering, which holds for weak fluctuations, when δb 2 /B 2 1. This means that one can calculate κ as function of the turbulence spectrum of the fluctuations, δb 2, and when these spectra are measured throughout the heliosphere, 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).

9 Cosmic-Ray Modulation Equations The theory of perpendicular diffusion is much more complicated, and this remains the biggest outstanding theoretical problem in cosmic-ray modulation theory. The current theory is non-linear in nature, and it was introduced by Matthaeus et al. (2003). Salchi (2009)gives an extensive account of the status of the field. In comparison, transport perpendicular transport to the background field due to gradient and curvature drifts is simpler and it is elegant. In Plasma Physics texts the calculation of the two drift velocities is cumbersome, and the results are approximate. A significant contribution from the field of cosmic rays in the heliosphere to this topic is that: for an isotropic particle distribution, or for one with at most a first-order anisotropy (i.e. weakly anisotropic), the combined gradient and curvature drift velocity of the distribution is given by v dr = βp 3 B B 2. (10) The average bracket denotes that this is the velocity of the distribution, and we emphasize that under the condition of weak anisotropy this expression is exact. Furthermore, such a drift velocity gives rise to a drift flux S dr = βp B n. It is important to note that the 3B 2 average drift velocity and drift flux are not the same; they need not even be in the same direction, as some careful trajectory constructing readily reveals. A further simplification results by formulating drift motion as an antisymmetric element of the cosmic-ray diffusion tensor κ 0 0 K = κ ij = 0 κ κ T, (11) 0 κ T κ where κ T = 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 of the drift effect as part of the diffusion tensor is entirely equivalent to the explicit reference to the drift velocity (10) or to the drift flux S dr. The elegant simplicity of this drift formalism is due to the group of J.R. Jokipii at the University of Arizona, with the first paper on the topic by Jokipii et al. (1977), with a more comprehensive version by Isenberg and Jokipii (1979). The different fluxes are schematically shown in Fig. 2 of Moraal (1991). Ongoing research is conducted on how the simple form of κ T is affected by strong scattering; a summary of the relevant literature on this topic is given by Burger and Visser (2010). The fact that the HMF and its scattering centers are convected radially outward by the solar wind with velocity V, leads to a convective flux Vn. Hence, the total flux of cosmic rays 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 density n = U p dp = 4πp 2 fdp. It seems that it should also hold for U p (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 hence move out of the interval (p, p + dp). This causes a flux in momentum space, similar to the flux in configuration space, which must be included in the differential equation.

10 H. Moraal The form of the energy/momentum change term can readily be understood from the following analogy: In spherical polar coordinates the divergence of the flux is given by S = 1 r 2 r (r2 v r U p )+ (two terms containing θ and φ derivatives). The velocity is v = ṙ =ṙe r + r θe θ + r sin θ φe φ. When the particle distribution is isotropic, the average velocity reduces to v = ṙ e r, because the two directional averages and the last two terms in the divergence become zero. Thus the divergence of the flux reduces to S = 1 r 2 r (r2 ṙ U p ). In an analogous fashion, when particles change the magnitude of their momentum vector instead of their position vector, the divergence of the flux in momentum space will be S p = 1 p 2 p (p2 ṗ U p ). With this, the final form of the differential transport equation, i.e. for particles in the momentum range (p, p + dp) becomes f/ t + S + 1 p 2 p (p2 ṗ f)= q, (13) or f/ t + (Vf K f)+ 1 p 2 p (p2 ṗ f)= q. (14) The cosmic-ray transport equation was written down for the first time in this form as (3) in Parker (1965), although in different notation. Parker then noted that in the heliosphere the only significant energy change process is adiabatic cooling due to the fact that the particles ride with the fields in the wind, and these fields expand due to the positive divergence of the wind speed. This leads to a rate of change of momentum ṗ /p = (1/3) V. Ifthis cooling rate is substituted into (14) it leads to f/ t + (Vf K f) 1 ( V) 3p2 p (p3 f)= q. (15) This is equivalent to Parker s equation (4), and should be regarded as the original form of the cosmic-ray transport equation. Parker s form was, however, only written down for spherical symmetry and for constant radial V. [We note that the second term in Parker s form of the equation should be in his notation v U/ r instead of v r 2 r (r2 U).] 5 The Gleeson-Axford Derivation of the Transport Equation Gleeson and Axford (1968) rederived the cosmic-ray transport equation starting with a Boltzmann equation, evaluating single-particle scatterings in the solar wind frame, then integrating over direction in momentum space, and transforming back to the observer s frame of reference. This different approach was limited to spherical symmetric geometry, but it correctly reproduced the spherically symmetric version of (15) above. These authors then went further, and in Gleeson and Axford (1968) they pointed out that the flux of particles, defined as S = nv for density n in a flow field V, must be corrected for the so-called Compton-Getting effect when measured on a differential basis, i.e. between energies T and T + dt. The original paper, with its notation in terms of kinetic energy and differential density is generally used, but it is not easy to read. Later, Gleeson and Urch (1973) simplified the Compton-Getting correction, and in its most elegant form it can be stated as: when a differential density w.r.t. to momentum, U p = 4πp 2 f, is convected with velocity V, then the flux observed is not S p = VU p = 4πp 2 Vf, butrathers p = CVU p = 4πp 2 CVf, where

11 Cosmic-Ray Modulation Equations C = 1 ln f is called the Compton-Getting coefficient. This effect is similar in nature to 3 ln p the Doppler effect on photons, i.e. it is due to the fact that when a beam of particles is observed in the oncoming (receding) direction, the particles are observed with higher (lower) energy or momentum. The Compton-Getting coefficient is related to the logarithmic slope of the spectrum, and for a power law of the form, f p γ,itissimplyc = γ/3. When the Compton-Getting corrected flux S = 4πp 2 (CVf K f) (16) is substituted into (13) it does not produce the correct transport equation (15). But Gleeson and Webb (1978) also pointed out that the adiabatic rate of change of momentum ṗ /p = (1/3) V referred to above, is actually the rate of change in the (non-inertial) solar wind frame. They showed that the rate in the stationary frame is ṗ /p = (1/3)V ( f/f). When these two corrected expressions are inserted in (13), they produce the transport equation f/ t + (CVf K f) 1 3p 2 p (p3 V f)= q. (17) It is a simple exercise to show that this is identical to Parker s original form (15). This means that when both the flux and the rate of adiabatic cooling are correctly transformed to the stationary frame, the two corrections cancel one another, and the transport equation is not affected. Finally, (15) can be readily rewritten in the simpler form f/ t + V f (K f) 1 ( V) f/ ln p = q. (18) 3 The three forms (15), (17) and(18) are equivalent forms of the cosmic-ray transport equation, with (18) the preferred form. The transport equation is effectively five-dimensional, in the sense that the particle distribution function, or the cosmic-ray intensity depends on three spatial coordinates, plus momentum and time. It is a second-order partial differential equation because of the double spatial derivative in the diffusion term. It is parabolic in nature (similar to the heat flow equation) 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 dimensions has 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 interaction regions. However, due to its complexity, the solution had limited resolution. Lately, several so-called stochastic solutions of the multi-dimensional transport have also appeared. There are, instead, many approximate solutions of the full equation available, and the general 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 in increasing order of complexity. 6 The Convection-Diffusion Solution The lowest-order approximation is the so-called Convection-Diffusion formalism. It is so simple that the equation that describes it precedes the transport equation itself by six or seven years. It basically says that cosmic rays diffuse inwards in a spherically symmetric

12 H. Moraal heliosphere as they scatter off the irregularities in the HMF, leading to an inward radial diffusive flux κ f/ r,whereκ is a phenomenological diffusion coefficient that is a function of radial coordinate r and momentum p. This flux is countered by an outward convective flux Vf because the irregularities are frozen into the solar wind. (The 4πp 2 factor is omitted 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), the cosmic-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 neglected, when / t = 0, when spherical symmetry is assumed, and when the full diffusion tensor (11) is contracted into a single scalar κ(r,p). The solution of the convection-diffusion equation (19)is f = f b e M, where M = rb r V dr, (20) κ and where f b is the so-called local interstellar spectrum (LIS) at the outer boundary r b of the heliosphere, typically at r b 150 AU. Since f is the measured spectrum, the dimensionless modulation parameter M = ln(f b /f ) would be known experimentally if the LIS, f b, were known. Generally it is not; the ultimate task of modulation theory is to determine it correctly. Hence, the essence of modulation theory is to find M theoretically from V and κ (or more complicated versions of its tensor form), so that f b can be derived. Modulation changes between times t 1 and t 2 are, however, conveniently described without reference to the LIS as ln(f 2 /f 1 ) = M = M 1 M 2. The diffusion coefficients and the drift coefficient in (11) can conveniently be written in terms of a diffusion mean free path or drift length scale λ, asκ i = 1/3vλ i, where the index i can stand for,, ort. The most basic rigidity dependence of the three length scales is λ i P, although significant deviations are known. Hence κ i βp,whereβ = v/c. It therefore follows that to first order the modulation strength is inversely proportional to βp. Typical values are M solar min 1.5/βP (GV) and M solar max 4.5/βP (GV). Thus for high-latitude neutron monitors (cutoff rigidity 1 GV and median rigidity 15 GV) the full change in intensity over an 11-year solar cycle is M = f/f = ln(f 2 /f 1 ) = M ( )/15 = 0.2 (or 20%). At P = 100 GV (T = ev) modulation becomes insignificant, while for P 1 GV this simple approximation becomes invalid. Using V = 400 km/s and r b 150 AU in (20) leads to typical values of the diffusion coefficient κ in the range (2 6) βp (GV) cm 2 /s from solar maximum to solar minimum at 1 AU in the heliosphere. The modulation parameter M is a convenient single parameter that describes the modulation as a function of (1)thesolarwindspeed,(2) the diffusion coefficient, and (3)thesize of the heliosphere. It is important to note, however, that these three parameters cannot be deduced separately only their integral effect is known, and therefore the solution contains little physics. For example, the magnitude and radial dependence of κ are determined by the Parker spiral magnetic field and its fluctuation spectrum, but these cannot be deduced from a simple measurement of the M via ln(f b /f ),orof M = ln(f 2 /f 1 ). The next two paragraphs show that the next two levels of approximation, namely the Force-Field approach and the numerical solution of the steady-state spherically symmetric modulation equation produce the same modulation function M.

13 Cosmic-Ray Modulation Equations 7 The Force-Field Solution The Force-Field formalism is by far the most generally used approximation of modulation theory. It produces the so-called modulation potential φ, the origin of which is not generally appreciated. Here it is shown how the Force-Field equation relates to the full cosmic-ray transport equations (15), (17), or (18), and also that the Force-Field potential is essentially the same as the modulation parameter M of the previous section. In their original paper Gleeson and Axford (1968) formulated the Force-Field solution to the cosmic-ray transport equation as: j(r,e) E 2 E0 2 = j lis(e + ), (21) (E + ) 2 E0 2 where j refers to the observed intensity (instead of distribution function as we have used up to now) at radial distance r, j lis is the local interstellar spectrum, E is total energy (kinetic plus rest mass energy), and is the so-called Force-Field energy loss that particles suffer as they propagate inwards into the heliosphere. This energy loss is related to the Force-Field or modulation potential φ by = Zeφ. There are many variants of this formulation. The modulation potential φ has become the most commonly used modulation parameter in the literature. However, the origin and nature of the solution are seldom quoted, and it is often used wrong because the limitations are not recognized. For instance, if the diffusion coefficient in the transport equation is not strictly proportional to rigidity to the power one, can not be interpreted as an energy loss. The identification of φ as a potential is dimensionally correct, but it seems somewhat mysterious, and it is almost never related to the underlying diffusion and convection parameters. To implement the solution, i.e. to calculate the parameter φ, involves an intricate inversion process. In view of this, Gleeson and Urch (1973) gave a much simpler and more transparent re-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 corrected outward 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 modulation parameter modified to M = r b r (CV /κ)dr. This is, however, not useful, because C = 1/3( ln f/ ln p), which represents the form of the spectrum, and this spectrum changes with radial distance as the modulation occurs. Thus C is a function of r, andto find 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 = f b (r b,p b ) along contours of the characteristic equation dp/dr = pv/3κ in (r, p) space. The subscript 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-Field solution (21), but in different notation.

14 H. Moraal The name Force-Field originates from the fact that (23) can also be written in terms of rigidity P = pc/ze as f r + VP f = 0. (24) 3κ P Particle rigidity has the dimensions of electrostatic potential and therefore the coefficient VP/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 P b = P b (r, P ) is obtained by integrating the characteristic equation dp/dr = VP/3κ from the initial phase space point (r, P ) to the point (r b,p b ) at the outer boundary r b. If the diffusion coefficient is separable in the form κ(r,p) = βκ 1 (r)κ 2 (P ), (25) the solution is Pb (r,p ) β(p )κ 2 (P ) rb dp V(r ) = P P r 3κ 1 (r ) dr φ(r), (26) where φ is called the Force-Field parameter. When κ 2 P, which is typical, and when β 1, as for most ground-based observations such as with neutron monitors, the solution reduces to the widely used form, P b 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 the form (27) applies only to the special case of relativistic particles, β = 1, and the rigidity dependence κ P. As mentioned before, there are many cases in which this is not true. In all such cases P b P is some other function of φ. Thenφ alone is insufficient to describe the modulation and it does not have the dimensions of potential. Gleeson and Urch (1973) explicitly emphasized this complication, namely that the full Force-Field parameter is actually φ/κ 2 instead of φ, but this is not generally realized. It follows from (20), (25) and(26) that this full Force-Field parameter is given by φ κ 2 = 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 450 MV at solar minimum to 1350 MV at solar maximum. In this sense the Force-Field formalism produces no new insight over and above the simple Convection-Diffusion mechanism. However, the description of the modulation in the two formalisms are entirely the opposite of one another. This is demonstrated in Fig. 4. This figure shows hypothetical cosmic-ray spectra, one on the outer boundary, and the other a modulated spectrum inside the heliosphere. Specifically, it plots the distribution function f(p)= j T /P 2 (see (7) above) instead of the usually measured intensity spectrum j T.The vertical line shows that the Convection-Diffusion approach describes the modulation as a reduction in intensity at a given rigidity, without rigidity (or energy) loss, and the magnitude of this reduction is given by M. The Force-Field approach, however, describes this same

15 Cosmic-Ray Modulation Equations Fig. 4 Graphical representation of the description of the modulation with the Force-Field solution (horizontal line with parameter )andthe Convection-Diffusion solution (vertical line with parameter M). The sloped line represents the actual modulation as a combination of intensity reduction and energy (or rigidity) loss modulation as a rigidity (or energy or momentum) loss, = P b P, which is read off from a 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 other cases the rigidity loss that is read off from spectra as in Fig. 4 cannot be converted to a modulation potential, and it is more correct to describe the modulation in the Force-Field formalism with the dimensionless parameter M. Notice the contradiction that the Force-Field formalism produces a modulation potential that causes energy (or rigidity or momentum) changes, while its defining (22) neglects adiabatic energy changes. This is so because the Force-Field equation (22) stems from (17) in the steady state ( / t = 0), with no sources (q = 0), and with the adiabatic momentum-change term also equal to zero. The energy change that results from the Force-Field equation originates from the interpretation of the coefficient VP/3κ in (24) as a field, or force per unit charge. There is no physical reason why this term, i.e. the Force-Field energy loss, is related in any way to the true adiabatic loss. It is fortuitous that this energy loss is a reasonable approximation 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 of the 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 analytical solutions available. They were introduced in the first place because forty years ago numerical solutions of the transport equation were difficult to do and time-consuming on slow computers. This has drastically changed, however, and in this section it is shown that the third level of approximation is a very easy and accessible numerical solution of the

16 H. Moraal one-dimensional (spherically symmetric), steady-state ( / t = 0) version of (18). This approximation is V f r 1 ( r 2 κ f ) 1 r 2 r r 3r 2 r (r2 V) f = 0. (29) ln p The difference with the previous two approximations is that it retains the effect of adiabatic cooling in its third term. In addition, it was shown that the Compton-Getting effect, i.e. the proper 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 ṗ /p = (1/3) V is the same as the well-known adiabatic cooling law Pτ 5/3 = constant (where τ is a volume and P is pressure; not rigidity). In the adiabatic limit of modulation at low energies 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 pr 2/3 = constant. For a constant solar wind speed the volume element τ in which the particles are contained is proportional to r 2 (because the element only expands in the angular directions and not in radial thickness). This implies that pτ 1/3 = constant. Using the formulas for the internal 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 of modulation, is that, according to (7), kinetic energy spectra are of the form j T T,andthat the radial gradient in the intensity is zero. Physically, that means that particles at such low rigidities have not diffused into the heliosphere at those low rigidities, but have appeared there due to rigidity loss. Equation (29) is of the heat flow-type, and it is readily solved with the Crank-Nicholson method, with the stepping coordinate in time, i.e. t, replaced by ln p, i.e. stepping downward in momentum because particles lose momentum. This technique was first used by Fisk (1971). The initial condition is that at the highest momentum there is no modulation, i.e. at that momentum the LIS pervades undiminished throughout the heliosphere. The outer boundary condition at r = r b is that the LIS is maintained there at all momenta, while experience has shown that the inner boundary at r = r sun has almost no effect on the solution typically either f or f/ r are set equal to zero there. Fortran and C++ versions of this numerical solution can be found at za/p-csr/index.html. Section 3 of the paper of Caballero-Lopez and Moraal (2004) contains a full solution with a realistic set of parameters. Figure 5 is from that paper and it shows solutions of the Convection-Diffusion, the Force-Field, and the steady-state spherically symmetric approximations 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 the transport equation (29) than the Convection-Diffusion equation, because at low energies the Convection-Diffusion solution for the intensity is always much lower than the numerical solution. 2. In the outer heliosphere the Convection-Diffusion approximation is much better than the Force-Field approximation. Here the Force-Field solution at low energies leads to a gross overestimation of the intensity obtained from (29). 3. These above two effects can be understood because adiabatic losses are proportional to V = 1 r (r2 V), and this equals 2V/r for constant V. Hence, in the inner heliosphere r 2 the 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.

17 Cosmic-Ray Modulation Equations Fig. 5 Numerical solution of the steady-state, one-dimensional transport equation (full lines), together with the Force-Field solution (dashed lines) andthe Convection-Diffusion solution (dotted lines) for galactic cosmic-ray protons in a heliosphere with r b = 90 AU, V = 400 km/s, λ = 0.29P (GV) AU, and φ (1 AU) = 407 MV). Intensities are multiplied by factors of 10 to enhance visibility. Adapted from Caballero-Lopez and Moraal (2004) 4. These effects can also be understood qualitatively by referring back to Fig. 4. The modulation process proceeds along the diagonal line drawn from the LIS to the spectrum inside the heliosphere: it is a combination of exclusion due to convection-diffusion along the vertical axis, and of adiabatic loss along the horizontal axis. The vertical line marked M and the horizontal line marked therefore indicate that the Convection- Diffusion and Force-Field approaches respectively neglect the energy loss and the exclusion. Thus, the Force-Field energy loss is always an upper limit for the true adiabatic loss. 5. The largest deviation that occurs from the numerical solution of (29) is at 1 AU,where the Convection-Diffusion solution starts to diminish fast with decreasing energy at T< 400 MeV (for protons). This is equivalent to a rigidity of 1 GV. Neutron monitors have a median rigidity of response 1 GV. For applications such as the 10 Be concentration in ice, the yield function peaks in the range 2 to 6 GV. Both these rigidities are well above 1 GV, and hence for these applications the Convection-Diffusion and Force-Field approximations produce essentially the same modulation parameter M as the numerical solution. The combination of the values r b = 90 AU, V = 400 km/s, and κ = βp (GV) cm 2 /s, which fit the observed spectra at 1 AU near solar minimum conditions, give the modulation parameter M,definedin(20)asM = 1.22 βp (GV). This is equivalent to a Force-Field parameter φ(1 AU) = 407 MV. The question arises as to how the solutions will change 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