Dynamics of Radiation Belt Particles
Abstract
This paper reviews basic concepts of particle dynamics underlying theoretical aspect of radiation belt modeling and data analysis. We outline the theory of adiabatic invariants of quasiperiodic Hamiltonian systems and derive the invariants of particle motion trapped in the radiation belts. We discuss how the nonlinearity of resonant interaction of particles with small-amplitude plasma waves, ubiquitous across the inner magnetosphere, can make particle motion stochastic. Long-term evolution of a stochastic system can be described by the Fokker-Plank (diffusion) equation. We derive the kinetic equation of particle diffusion in the invariant space and discuss its limitations and associated challenges which need to be addressed in forthcoming radiation belt models and data analysis.
Keywords
RBSP mission Radiation belts Quasi-linear diffusion Chaos Particle dynamics1 Introduction
The stability of charged particles trapped in Earth’s magnetic field was well established by 1960 (e.g., Northrop and Teller 1960) providing a theoretical basis for the existence of radiation belts discovered by pioneering space missions (Van Allen 1959; Vernov et al. 1959). It was shown that in the approximately dipole field of the inner magnetosphere charged particles undergo three quasiperiodic motions each associated with an adiabatic invariant. A set of three invariants defines a stable drift shell encircling Earth. Subsequent experiments revealed that particle intensities across the belts can vary significantly with time (see review by Roederer 1968), which requires violation of one or more of the adiabatic invariants. Theoretical interpretation of the variability of radiation belt intensities was largely inspired by experiments in particle acceleration by random-phased electrostatic waves in synchrocyclotron devices (e.g., Burshtein et al. 1955; Keller and Schmitter 1958) and the development of quasi-linear theory of weak plasma turbulence (e.g., Drummond and Pines 1961; Romanov and Filippov 1961; Vedenov et al. 1961). It was concluded that in the absence of large-amplitude perturbations in the electric and magnetic fields the adiabatic invariants of trapped particles can be violated by waves, which can resonantly interact with the quasiperiodic particle motions. Since both the density and energy density of radiation belt particles is negligible compared to other plasma populations, their motion does not affect the fields that govern it. Thus, in accordance with the quasi-linear theory it was suggested that the evolution of radiation belt intensities can be described as a diffusion in the adiabatic invariants under the action of prescribed wave fields, with the diffusion coefficients determined by resonant wave-particle interactions (see reviews Dungey 1965; Trakhtengerts 1966; Tverskoy 1969). While the diffusion framework of radiation belt particle acceleration and loss was well developed within the first decade after the discovery of the belts (e.g., Falthammar 1965; Kennel and Engelmann 1966), the micro-physical origins of particle diffusion and the limitations of the diffusion framework were not fully realized until the development of nonlinear dynamics in 1980–90s.
The goal of this paper is to review basic physical concepts of particle dynamics underlying theoretical apparatus of radiation belt modeling and data analysis. The review is intended primarily for graduate students and non-experts in radiation bet physics who wish to have a brief yet systematic introduction into the field. The material for this review is based on classical monographs on radiation belt particle dynamics such as Roederer (1970) and Schulz and Lanzerotti (1974), several monographs on nonlinear dynamics including (Lichtenberg and Lieberman 1983; Sagdeev et al. 1988; Zaslavsky 2005), as well as a number of original research papers referenced in the text.
We start with outlining the theory of adiabatic invariants of quasiperiodic Hamiltonian systems, then we discuss the motion of charged particles trapped in a quasi-dipole magnetic field of the inner magnetosphere and derive the adiabatic invariants for each of the three quasiperiodic motions of trapped particles. In Sect. 3 we discuss resonant interaction of particles with small-amplitude regular wave fields. We show that particles at resonance with a given harmonic of the spectrum can be trapped in the wave potential where they undergo nonlinear oscillations and phase mixing. The overlap of particle populations at resonance with adjacent harmonics of the spectrum results in stochasticity of particle motion. In the space of adiabatic invariants particle dynamics then resembles random motion of Brownian particles due to collisions with gas molecules. In Sect. 4 we derive the equation of quasi-linear diffusion in the invariant space, often used in radiation belt models, and discuss its relation to the Fokker-Plank kinetic equation of long-term evolution of stochastic systems governed by Markov processes. In Sect. 5 we focus on some limitations underlying the diffusion approximation and associated challenges which need to be addressed in forthcoming radiation belt models and data analysis. In summary we provide a reference table of the most commonly used formulas discussed in this review.
2 Quasiperiodic Motion and Adiabatic Invariants
Adiabatic invariants are approximate constants of motion of a slowly changing system. The change of an adiabatic invariant approaches zero asymptotically as some physical parameter approaches zero. Adiabatic invariants are of great importance for the analysis of stability of the quasiperiodic particle motion in radiation bets in the presence of small perturbation forces, such as various plasma waves or slow variation of the ambient magnetic field due to changing solar wind and geomagnetic conditions.
2.1 General Considerations
It has to be noted that the change in the invariant is no longer exponentially small if the oscillation frequency goes to zero, \(\bar {\omega}\sim\varepsilon\). This corresponds to the case when the system crosses a phase space separatrix, which can result in non-negligible violation of the invariant (e.g., Cary et al. 1986; Neishtadt 1986). Implications of invariant violation at separatrix crossings to dynamics of the outer belt particles is discussed in some detail in Sect. 5.3, until then we assume that the estimate (17) holds.
2.2 Adiabatic Invariants of Radiation Belt Particles
Even in the guiding-center approximation the first invariant is a non-local quantity, which can significantly complicate its derivation from in situ particle measurements. While the perpendicular momentum in expression (26) is defined at the location of particle measurements, the magnetic field intensity has to be estimated at the gyrocenter, not sampled by the measurement.
The guiding-center approximation holds well for particles inside the electron and the inner proton belts. The description of particle motion, in this case, can be significantly simplified. Equations for the guiding-center motion were originally derived by expanding the Lorentz equation of motion about the guiding center and then removing fast oscillations by the gyrophase averaging (e.g., Landau and Lifshitz 1959; Sivukhin 1965; Northrop 1963). This procedure, however, has two fundamental shortcomings. First, the equations obtained by the gyrophase averaging do not have the Hamiltonian structure of the original Lorentz equation (e.g., Balescu 1988). As a result they do not conserve the phase space density and therefore are in violation of the Liouville’s theorem. Consequently these equations cannot be used for the description of collective phenomena in plasmas. Second, the obtained equations do not conserve energy in time-independent fields. Nonconservation appears in second order terms in the Larmor radius expansion (e.g., Cary and Brizard 2009) and can present difficulties in modeling long-term effects in particle dynamics.
3 Particle Dynamics in Wave Fields
In this section we discuss resonant interaction of particles with small amplitude waves. We show that even if wave fields are regular and no external randomness is introduced into the system, the nonlinearity of resonant wave-particle interaction combined with the overlap of particle populations in resonance with adjacent harmonics of the wave spectrum result in a stochastic particle motion. In the space of adiabatic invariants particles exhibit random walk motion similar to the Brownian motion of heavy particles due to collisions with light molecules in gasses. In our consideration we use a specific example of resonant interaction between the drift motion of the outer belt electrons and the ULF waves, resulting in a stochastic radial motion of electrons across the drift shells. However, the discussed properties of the stochastic motion are general and are equally applicable to resonant interaction of waves with the bounce and the gyromotion of trapped particles.
3.1 Nonlinear Resonance
As a result of nonlinear dependence of the oscillation period on the resonance proximity, trapped particles undergo phase mixing. Particles who originally had the same phase but slightly different values of δI oscillate about the center of the resonant island at different frequencies. Consequently their phases gradually separate and the motion becomes eventually uncorrelated. Phase mixing is illustrated in Fig. 5. Panels (b)–(g) of the figure show evolution of the phase distribution function computed for an ensemble of 10^{5} particles trapped at resonance with a single wave. Initially all particles have the same phase (Ψ=0) but were evenly distributed over δI from −2Ω_{NL} to 2Ω_{NL}. After first several oscillation periods, T_{0}, particles spread over almost entire phase range from −π to π, but still exhibit strong phase bunching indicated by multiple pronounced peaks in the distribution function. Eventually the peaks break down and spread over the phase interval. After a large number of drift periods the process produces a smooth distribution function accept for a singular peak around Ψ=0 corresponding to the particles at exact resonance \(\varPsi=\dot{\varPsi}=0\).
3.2 Resonance Overlap
Wave perturbation in the Hamiltonian (49) consists of multiple wave harmonics. From our previous considerations it follows that there is a layer of trapped particles of the width ΔI centered at each of the spectral harmonics. The larger is the wave amplitude K the wider are these layers (see Eq. (53)). If the amplitude increases to the point when the resonant width ΔI exceeds the spacing Δω between spectral harmonics m_{0} and m_{0}±1, the resonant populations trapped by adjacent harmonics overlap, and the particle motion is no longer bounded to a single resonance. From this qualitative consideration it follows that resonances overlap, if K≳π/2 (Chirikov 1960). Particle population initially at resonance with one wave harmonic can then spread over the entire system (maximum spread restricted only by the width of the spectrum: ΔI=M). Phase mixing in this case results in exponential divergence of particle trajectories with similar initial condition, which is an attribute of chaotic dynamics. Generally speaking, chaotic systems are the systems described by regular dynamical equations (the Lorentz equations (21) in this case) with no stochastic coefficients, but at the same time with solutions that are similar to some stochastic processes.
4 Transition to Kinetic Description
The motion of individual charged particles is described by the Lorentz equations of motion (21). Particle distribution function or the phase space density evolves in accordance with the Liouville’s theorem. If particle motion becomes stochastic, which in a collision-free plasma can be caused by interactions with waves, then correlations among dynamics of individual particles decay. Consequently, the description of long-term evolution of particle distribution function can be reduced to a Fokker-Planck equation, similar to the description of diffusion in gas, which we discuss in this section.
4.1 Phase Space Density
4.2 Diffusion in Action Space
The above system evolves at two characteristic time scales. It exhibits rapid oscillations in the angle variable θ and slow change in the action I due to resonant wave-particle interactions. An ensemble of particles with initially same values of I but distributed in θ will initially rotate coherently, since \(\dot{\theta}\simeq \omega(I)\). However, if the system is stochastic, then I of different particles will undergo different small variations due to their interactions with the wave field (see previous section). After some time the ensemble will spread in I and will rotate at different frequencies. Consequently the ensemble will exhibit phase mixing, i.e. correlations between particle θ(t) and its initial values will decay, and eventually particle distribution in I will become independent of the initial distribution in θ. On timescales longer than the phase correlation decay time (τ_{c}), it then become possible to derive a reduced description of long-term evolution of the system by averaging the Liouville’s equation over the fast angular variable θ.
The existence of random phase shits ψ_{m}∈[0,2π) between different harmonics of the wave spectrum in Eq. (49), considerably changes the dynamic properties of the system. If resonances overlap (K>1), the islands of regular motion are completely destroyed by random shifts, and the system becomes stochastic everywhere across the phase space. The analysis of particle transport at different values of K (Cary et al. 1990; Helander and Kjellberg 1994) show that the diffusion coefficient in this case can still exhibit large deviations from the quasi-linear value, D_{QL}: while it never gets smaller than D_{QL}, at K≃18 it reaches the maximum of 2.3D_{QL} (see Fig. 8b).
In reality, additional stochasticity may be introduced into the system due to random nature of the wave fields. Phase shifts ψ_{m} at different harmonics of the spectrum in Eq. (49) may no longer be stationary in this case. Their values can change in some characteristic time intervals T corresponding to the spatial or temporal coherence of the problem. For instance, variations of the solar wind dynamic pressure is one of the dominant drivers of the ULF waves in the inner magnetosphere (e.g., Takahashi and Ukhorskiy 2007). ULF waves can violate the third adiabatic invariant of trapped electrons in the process of resonant interaction with their drift-bounce motion discussed in Sect. 3. Oscillations in dynamic pressure are attributed to the Alfvén turbulence in the solar wind. The phase shifts between different harmonics of the ULF wave spectrum therefore change on the time scales of the autocorrelation time of the solar wind turbulence, ∼3 hr (e.g., Jokipii and Coleman 1968).
Electromagnetic ion cyclotron (EMIC) waves are considered to be one of the dominant local mechanism of electron losses from the outer radiation belt (e.g., Thorne and Kennel 1971; Horne and Thorne 1998; Summers et al. 1998; Ukhorskiy et al. 2010). Resonant interaction of EMIC waves with electron gyromotion breaks the first adiabatic invariant and can cause electron scattering into the atmospheric loss cone and their subsequent loss via precipitation. Free energy for the EMIC wave growth is supplied by the positive temperature anisotropy (T_{⊥}>T_{∥}) of energetic (∼10–100 keV) ions (e.g., Cornwall 1965; Kennel and Petscheck 1966). EMIC waves grow to observable amplitudes at frequencies of maximum growth rate out of small-amplitude electromagnetic noise propagating along the field lines through the regions of positive anisotropy (e.g., Gomberoff and Neira 1983; Horne and Thorne 1994). EMIC wave activity can extend over >10° about the magnetic equator (e.g., Erlandson and Ukhorskiy 2001) and last for tens of minutes producing pitch-angle scattering of radiation belt electrons over many bounce periods. Detailed spectral analysis (Anderson et al. 1996; Denton et al. 1996) revealed that wave events consist of many short (∼30 sec) wave packets. Consequently phase shits among the harmonics of EMIC spectra vary at time scales comparable to the duration of individual wave packets.
Numerical simulations (Ukhorskiy and Sitnov 2008) showed that if additional extrinsic stochasticity is introduced into the system by varying phase shifts ψ_{m} among spectral harmonics of the wave perturbation (49) at time intervals comparable to the time T between effective collisions (90), then particle motion becomes stochastic even if resonances do not overlap. The diffusion coefficient in this case agrees well with its quasi-linear estimate (91). At the time scales longer than the correlation decay time τ_{c} the system can then be described by diffusion equation (88) with quasi-linear diffusion coefficients. The correlation decay time τ_{c} in this case depends on both the collision time and wave amplitude similar to expression (90).
5 Limitations and Challenges
During over five decades since the discovery of radiation belts the concept of diffusion in the invariant space has been successfully applied for the analysis of transport, acceleration, and loss of radiation belt particles. Radial diffusion due to drift-resonant interaction with solar-wind driven ULF fields was the first mechanism proposed to explain acceleration of electrons and protons in radiation belts (Kellog 1959; Tverskoy 1964; Dungey 1965; Falthammar 1965). Subsequent analysis showed that radial diffusion causes not only acceleration but loss of particles from the outer belt (e.g., Bortnik et al. 2006; Shprits et al. 2006) and can be driven by variety of plasma waves including waves excited internally by instabilities in ring current plasma such as stormtime Pc5 waves (Lanzerotti et al. 1969; Ukhorskiy et al. 2009). Local resonant interaction of electron gyromotion with whistler waves was initially considered to be primarily responsible for electron losses from the belts (Dungey 1963; Cornwall 1964; Kennel and Petscheck 1966). Local wave-particle interactions are now recognized as both loss and acceleration mechanisms. As was mentioned in the previous section EMIC waves are considered to be one of the primary mechanism of local losses outside of the plasmasphere. Whistler chorus (e.g., Horne and Thorne 1998; Summers et al. 1998) and magnetosonic (e.g., Horne et al. 2007) waves were identified as mechanisms of local acceleration of radiation belt electron, more efficient than energization due to radial diffusion (e.g., Horne 2007). A number of recent review papers (Hudson et al. 2008; Shprits et al. 2008a, 2008b; Thorne 2010) provide detailed discussions and reference lists on diffusion theory of the radiation belt processes. In this section we discuss to what extent particle motion in the belt can be described in terms of three adiabatic invariants, some limitations of the diffusion approximation and associated challenges which need to be addressed in forthcoming radiation belt models and data analysis.
Diffusion approximation applies to the situations when in zero order radiation belt particles are stably trapped in quasiperiodic motion associated with three adiabatic invariants. This implies that the magnetic field has a slow-varying quasi-dipole configuration, such that the time scales of the three periodic motions are well separated, and the electric field is small, such that the E×B drift is negligible compared to the gradient-curvature drift. In this case particle invariants can be violated only in the process of resonant wave-particle interaction. Reducing the description from the full Vlasov equation to a Fokker-Plank equation in the invariant space also requires that waves have small enough amplitudes, such that nonlinear phase-dependent effects can be neglected, and the characteristic time scales of the described processes are longer than the phase correlation decay time. Variability of radiation belt intensities do not always satisfy these conditions.
5.1 Large Perturbations
Early observations of depletions of the outer belt intensities during storm main phase (Dessler and Karplus 1960; McIlwain 1966) were attributed to an adiabatic response of relativistic electrons to a slow (compared to electron drift period) increase in ring current intensity, which is referred to as the D_{st}effect. An increase in ring current intensity decreases the magnetic flux Φ enclosed by electron drift-bounce orbits. To conserve Φ electrons move outward to regions of lower magnetic field intensity. Since \(\mu=p_{\bot}^{2}/2mB\) is also conserved, the outward motion decreases electron energies. Thus, measurements of electrons within a fixed energy at a fixed radial location after increase in ring current register electrons previously located at lower radial distances where their energy was higher and their phase space density lower so that a lower intensity is measured. In recent years with the development of more quantitative empirical models of storm-time magnetic field (e.g., Tsyganenko and Sitnov 2005, 2007; Sitnov et al. 2008), it became apparent that the ring current has much more profound effect on the outer radiation belt. Test particle simulations (Ukhorskiy et al. 2006b) show that storm-time intensification of highly asymmetric partial ring current produces fast outward expansion of electron gradient-curvature drift orbits leading to their loss through the magnetopause. Depending on the storm magnitude, particles from a broad L-range of outer belt can be permanently lost. These theoretical predictions were recently confirmed by the observational analysis of multi-spacecraft data (e.g., Millan et al. 2010; Turner et al. 2012). Due to its rapid nature and dependence on the magnetic local time (azimuthal angle) this effect can be described only with full Liouville’s equation (69).
Typically pitch-angle and energy diffusion coefficients in radiation belt models are computed based on statistical properties of waves derived from time-averaged spectral intensity data. For whistler chorus waves characteristic time-averaged wave amplitude is ∼0.5 mV/m (Meredith et al. 2001). Recent analysis of instantaneous wave data (Cattell et al. 2008; Kellog et al. 2011; Wilson et al. 2011) showed that whistler chorus waves can have very large amplitudes >200 mV/m (Fig. 9b). Such large-amplitude whistler waves can accelerate electrons by more than an MeV in less than a second (Cattell et al. 2008), trap electrons (Kellog et al. 2010), and cause their prompt scattering into the loss cone and consequent precipitation into the atmosphere (Kersten et al. 2011). While it was suggested that some aspects of particle response to large-amplitude coherent waves can be described with a Fokker-Planck equation (Albert 2010), bounce and gyrophase dependent aspects of wave particle interactions require fully kinetic treatment.
5.2 Non-diffusive Transport
5.3 Drift Orbit Bifurcations
Many observational techniques rely on computing electron phase space density as a function of three adiabatic invariants. In particular, radial (L^{∗}) profiles of electron phase space density computed at constant values of the first and second invariants are used as a diagnostic of relative roles of local and global acceleration mechanisms across the outer electron belt (e.g., Green and Kivelson 2004; Chen et al. 2007). If the phase space density has a local peak at some L^{∗} value, much exceeding the phase space density value at the outer boundary of the belt, it is considered to be an indication of additional electron acceleration process operating locally at this L^{∗} value. Recent studies (e.g., Öztürk and Wolf 2007; Wan et al. 2010; Ukhorskiy et al. 2011) suggest that this argument should be used with great caution.
At drift orbit bifurcations the particle phase space trajectory crosses a separatrix (Figs. 11(B)g and 11(B)h), which divides the (p_{∥},s) phase plane into three distinct regions. The region outside the separatrix corresponds to the bounce motion across the equator, while two lobes connected at a saddle point correspond to trajectories trapped below and above the equator. As the particle approaches the separatrix, its instantaneous bounce period increases logarithmically (as discussed in Sect. 2) and in some small vicinity of the separatrix becomes comparable to the drift period. In this vicinity the quasiperiodic character of the bounce motion is broken, since the effective potential of the motion there is changing at the time scales of the instantaneous bounce period and can no longer be considered slowly varying (ε in Eq. (8) is no longer small). Close to the separatrix the second invariant is therefore not conserved. At two consecutive separatrix crossings corresponding to bifurcations off the equator and back, the invariant exhibits jumps. As a result by the time the particle resumes its motion across the equator it accumulates a nonzero change in the second invariant. Each bifurcation also leads to radial and pitch angle jumps. Consequently when the particle drifts back to its initial location on the nightside, the drift orbit does not close on itself as in the case of stably trapped particles (Fig. 11(A)).
The range of the second invariant (or equatorial pitch-angle) values affected by bifurcations at given radial locations depends on the degree of the day-night asymmetry in the geomagnetic field, which is mostly controlled by the solar wind dynamic pressure (P_{dyn}). To quantify the extent of the phase space region affected by bifurcations, we calculated the Earthward boundary of the bifurcating orbits at three different values of equatorial pitch angle as function of P_{dyn} using guiding-center simulations in the Tsyganenko 96 magnetic field model (Tsyganenko 1996) at moderate values of the dynamic pressure (P_{dyn}<10 nPa). The radial location of the boundary was quantified by L at midnight, L_{M}. The results are shown in Fig. 11(C). As can be seen from the figure, a broad range of the outer belt trajectories is affected by bifurcations. At geosynchronous orbit, for instance, at P_{dyn}>6 nT all orbits with the equatorial pitch angles α_{eq}>50° (which constitutes most of the pitch-angle distribution) exhibit bifurcations.
In the bifurcating region particle drift motion around Earth is no longer quasiperiodic (i.e. does not have three independent integrals of motion): there is no slow varying control parameter λ in the Hamiltonian function (see Eqs. (6) and (7)), which can be adjusted to turn the bifurcations off. For the drift motion, bifurcations are a property of the unperturbed Hamiltonian. The third adiabatic invariant therefore is undefined for bifurcating orbits and particle phase space density cannot be transformed into the invariant space. An alternative methodology is required for the analysis of relative roles of various acceleration mechanisms extending into the bifurcation region of the outer belt phase space.
6 Summary
Reference table
Definition (cgs) | Comments | Eqn. | |
---|---|---|---|
Relativistic factor | γ=[1−(v/c)^{2}]^{−1/2} | v is the velocity magnitude, c is the speed of light | |
Energy and (mechanical) momentum | \(\mathcal{E}+mc^{2}=m\gamma c^{2}\); p=mγv | \(\mathcal{E}\) is the kinetic energy, m is the rest mass, v is the velocity | |
Electromagnetic field | \(\mathbf{E}=-\nabla\varphi-\frac {1}{c}\frac{\partial\mathbf{A}}{\partial t}\); B=∇×A | φ is the electrostatic potential, A is the magnetic field vector potential | (20) |
Canonical momentum | \(\mathbf{P}=\mathbf{p}+\frac{e}{c}\mathbf{A}\) | e is the electric charge | (19) |
Lorentz equation | \(\frac{d\mathbf{p}}{dt}=e\mathbf{E}+\frac {e}{c}\mathbf{v}\times \mathbf{B}\) | Hamiltonian equation of charged particle motion in electromagnetic field | (21) |
Adiabatic invariant (action) | \(I=\frac{1}{2\pi}\oint\mathbf{p}(\mathbf{q},H)\cdot d\mathbf{q}\) | Integration is carried out along an unperturbed periodic orbits, such that the Hamiltonian H=const | (2) |
First invariant | \(\mu=\frac{p_{\bot}^{2}}{2mB}\) | p_{⊥} is the relativistic momentum component perpendicular to magnetic field | (22) |
Magnetic moment | M=μ/γ | Magnetic moment is an adiabatic invariant of motion only in non-relativistic limit (γ≃1) | (23) |
The guiding-center motion in static magnetic fields | \(\left\{ \begin{array}{l} \dot{p}_{\|}=-\frac{\mu}{\gamma}\hat{\mathbf{b}} \cdot\nabla B \\ [8pt] \dot{\mathbf{R}}=\frac{p_{\|}}{m\gamma}\hat{\mathbf{b}}+\mathbf{U}_{D} \\ [5pt] \mathbf{U}_{D}=\frac{c}{\gamma e} (\frac{p^{2}_{\|}}{mB}+\mu )\frac {\hat{\mathbf{b}}\times\nabla B}{B} \end{array} \right.\) | This approximation assumes that the magnetic field is curl-free, a more general case is treated by (29)–(32); p_{∥} is the momentum component and \(\hat{\mathbf{b}}\) is the unit vector parallel to magnetic field, U_{D} is the guiding-center velocity perpendicular to magnetic field | (32) |
Second invariant | I_{2}=2pJ; \(J=\int_{m}^{m^{\prime}}{\sqrt{1-\frac{B(s)}{B_{m}}}ds}\) | The integration is carried out along a fixed magnetic line between conjugate bounce points | (38) |
Third invariant | Φ=∮_{C}A⋅dl | C is any contour on the drift-bounce surface closed around Earth | (40) |
Dipole L | r=LR_{E}sin^{2}ϑ | Is constant along a dipole field line, it measures the distance from the dipole center to the field line at the equator in Earth radii (R_{E}≃6380 km); r is the radial distance and ϑ is the co-latitude | |
Generalized L value | \(L^{\ast}=-\frac{2\pi B_{0}R_{E}^{2}}{\varPhi }\) | L^{∗} is the radial distance (in R_{E}) to the equatorial points of the drift-bounce shell on which the particle would be, if all nondipolar contributions to the magnetic field would be adiabatically turned off; B_{0}≃31000 nT is the magnetic field intensity on Earth’s surface at the equator | (41) |
Liouville’s equation | \(\frac{\partial f}{\partial t}+\dot {\mathbf{q}}\cdot\frac{\partial f}{\partial\mathbf{q}}+\dot{\mathbf{p}}\cdot\frac {\partial f}{\partial\mathbf{p}}=0\) | f=f(p,q) is the density of particles in the phase space (PSD), (p,q), conserved along particle trajectories | (63) |
Intensity | \(j_{\alpha}(\mathcal{E},\mathbf{r})=p^{2}f(\mathbf{p},\mathbf{r})\) | Is the number of particles with kinetic energy \(\mathcal{E}\) coming from a given direction α which impinge per unit time, unit solid angle and unit energy, on a surface of unit area oriented perpendicular to their direction of incidence | (68) |
Course-grained phase space density | \(\begin{array}{l} F(I,t)=\langle f(I,\theta,t)\rangle\\ \phantom{F(I,t)}=\frac{1}{2\pi}\int_{0}^{2\pi}f(I,\theta,t)d\theta \end{array}\) | Variable reduction valid on the time scales longer than the phase correlation decay time τ_{c} | (75) |
Diffusion equation | \(\frac{\partial F}{\partial t}=\sum_{ij}\frac{\partial}{\partial I_{i}}D_{ij}\frac{\partial F}{\partial I_{j}}\) | Averaging over the fast angular variable reduces the Liouville’s equation for the exact PSD to a Fokker-Planck equation for course-grained PSD | (88) |
Diffusion coefficient | \(D_{ij}=\frac{\langle\Delta I_{i}\Delta I_{j}\rangle}{2T}\) | T>τ_{c} is the characteristic time of diffusion, ΔI_{i} is the adiabatic invariant change over T | (81) |
Radial diffusion | \(\frac{\partial f}{\partial t}=L^{\ast 2}\frac{\partial}{\partial L^{\ast}}\frac{1}{L^{\ast2}}D_{L^{\ast}L^{\ast}}\frac{\partial f}{\partial L^{\ast}}\) | Implies conservation of the fist and the second adiabatic invariants | (89) |
Footnotes
Notes
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