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Acceleration and transport of energetic particles observed in the inner heliosphere
ARTICLE IN PRESS
Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 442–449 www.elsevier.com/locate/jastp
Acceleration and transport of energetic particles observed in the inner heliosphere J.R. Jokipii University of Arizona, Tucson, AZ 85721, USA Accepted 27 August 2007 Available online 29 September 2007
Abstract A number of distinct species of energetic particles—ranging from those accelerated at the Sun and heliosphere to galactic cosmic rays, are observed in the inner heliosphere. In turn, a number of different acceleration mechanisms—including stochastic fluctuations, shocks and more-gradual compressions—have been proposed to explain the various observations and each mechanism has its adherents. In this tutorial review, various popular mechanisms are introduced, and the differences between statistical acceleration and acceleration at shocks at quasi-perpendicular and quasi-parallel shocks and compressions are discussed. Spatial transport is an important part of the acceleration process, and the accelerated particles must propagate through the turbulent solar wind to the point of observation. Both the acceleration and the subsequent transport of the particles depend fundamentally on the nature of the ambient, generally turbulent, plasma and electromagnetic fields. New constraints imposed by recent in situ and remote observations of the energetic particles in the inner heliosphere will be discussed. r 2007 Published by Elsevier Ltd. Keywords: Energetic particles; Solar wind; Heliosphere; Sun
1. Introduction This tutorial talk gives a brief introduction to the basic acceleration and transport of energetic charged particles in turbulent, collisionless plasmas such as the interplanetary medium. In addition, a brief discussion is given of some current observational issues and possible theoretical solutions. Energetic charged particles or cosmic rays are found in nature wherever the ambient density is low enough that collisional losses are negligible. The Sun accelerates energetic particles (SEPs), in events associated with energy releases, such as coronal E-mail address: [email protected] 1364-6826/$ - see front matter r 2007 Published by Elsevier Ltd. doi:10.1016/j.jastp.2007.08.026
mass ejections or flares. The resulting SEP events are often power laws in energy with a similar powerlaw index, but with a cutoff at high energies. Occasionally evidence for multiple power laws is fount. The first solar energetic particle events were observed in the mid 1940s by Forbush (1954). The time history of a typical SEP event and the energy spectra of various energetic particle species during a typical solar particle event are shown in Figs. 1 and 2. Charged particles are also accelerated in the solar wind both by propagating transient shocks and corotating shocks, and at the solar-wind termination. In addition, a continual flux of energetic particles are observed in the inner heliosphere at quiet times (undisturbed by solar activity). These particles
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Fig. 1. Energy spectra during a typical SEP event (Reames, 1999).
extend over a very wide range of kinetic energies from just above thermal energies to more than 1020 eV. The anisotropy (relative to the local fluid frame) is small—being less than or of order 102 at 1018 eV and smaller at lower energies. These particles come from the galaxy (and perhaps beyond at the very highest energies) and are termed galactic cosmic rays. Fig. 3 shows the observed spectrum of galactic cosmic rays at Earth over the energy interval 105 21020 eV, compiled from a number of sources. Apparent at low energies, below ’ 1 GeV are a turnover and various other features caused by the interaction of cosmic rays with the sun and the solar wind. The energetic particle spectrum, at the very lowest energies, which can be observed only in situ, is observed to merge smoothly into the background thermal plasma distribution. The energy spectrum of galactic cosmic rays is a remarkably-smooth power law above some 109 eV, with only a minor change in slope occurring between 1015 and 1016 eV (the ‘‘knee’’), and possibly a small flattening at 1019 eV (the ‘‘ankle’’). The anomalous cosmic-ray oxygen, also seen in Fig. 3, is a consequence of the interaction of the heliosphere with the interstellar medium.
Fig. 2. An observed solar event reported by Bieber et al. (2004), showing excellent theoretical fits to intensity and anisotropies.
Photons emitted from distant galaxies, between galaxies and other regions of our galaxy as a consequence of the interaction of cosmic rays with ambient matter or electromagnetic fields, show that cosmic rays exist in many places. Moreover, the inferred energy spectra of the cosmic rays in these remote sources are also power laws with slopes not much different from that observed at Earth (Kronberg, 1996). The slope of the power law is similar to that of the solar particles discussed above. This fact, together with the smoothness of the observed spectrum at Earth over a large range in energy, suggests a common acceleration mechanism for the bulk of the cosmic rays. 2. Acceleration and transport of energetic charged particles We may neglect particle–particle collisions in nearly all astrophysical plasmas of interest. The energy change and transport are determined to a very good approximation by the electric and
ARTICLE IN PRESS J.R. Jokipii / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 442–449
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where c is the speed of light. The physical problem of acceleration, then, consists of solving Eqs. (1) and (2) for a given situation. Note that since the electric field E is determined by U and B, it need not appear explicitly in the transport equation. Electric fields occurring in (small) regions where magnetohydrodynamics breaks down, such as in shock fronts or in reconnection regions, often are not given by Eq. (2). The reader is referred to Dmitruk et al. (2003), for a discussion. 2.1. The transport equation
Fig. 3. Observed energy spectrum of cosmic rays during quiet times. Shown are the all-particle spectrum and superposed, for illustration, the Oxygen spectrum, including the anomalous component at 107 2108 eV. The turnup in intensity below 106 2107 eV is thought to be solar in origin. Compiled from a variety of sources.
magnetic fields in the ambient plasma. In this case, the change in energy DT of a particle having energy T and velocity w, moving an electric field Eðr; tÞ in the time interval Dt may be written in general as: Z tþDt DT ¼ q w Eðr; tÞ dt, (1) t
where the integrand must be evaluated along the actual particle trajectory in the ambient electric and magnetic fields. From this we see that in order to evaluate the energy change, we must know the particle trajectory in the electromagnetic field. This leads to the general requirements that acceleration and spatial transport be intimately coupled and both Eðr; tÞ and Bðr; tÞ must be considered together. Here we consider energetic particles, for which the relevant spatial plasma scales are the energeticparticle gyro-radii. At these scales the ambient plasmas are generally hydromagnetic, and the electric field is determined from the fluid velocity U and magnetic field by the relation E¼
UB , c
(2)
A robust and widely applicable transport theory, applicable to particles whose speed w is significantly larger than U has been developed and applied extensively over the past four decades. It may be considered as an expansion in powers of the small ratio U=w. Unfortunately, a general theory for lower-energy particles for which U=w is not small is not available—for such particles one must use numerical simulations. The available theory utilizes the fact that the plasmas in space are turbulent, with broadband fluctuations over scales including those comparable with the particle gyro-radii. The particle motions are described statistically—the turbulent magnetic fluctuations ‘‘scatter’’ the particles in angle, driving them to near-isotropy in the fluid frame (or, more generally, the frame of the scatterer) with a time scale tscat . This scattering occurs much more rapidly than energy change. To zeroth order in U=w the resulting motion may be approximated as diffusive, and the phase space distribution function f ðp; r; tÞ of the particles having a momentum magnitude p at position r and time t satisfies a diffusion equation qf q qf ¼ kij þ Q, (3) qt qxi qxj where kij is the diffusion tensor, determined by the spectrum of the magnetic fluctuations, and Q represents any source. Magnetic fluctuations at scales of the order of the energetic-particle gyroradius are the most important in determining kij , although the field-line random walk or braiding, which affects the cross-field transport depends on very long length scales. The flow velocity U does not appear in this order, and because E ¼ 0 in this limit, neither convection nor energy change appear. Eq. (3) gives a crude, but sometimes acceptable description of the behavior of very high energy
ARTICLE IN PRESS J.R. Jokipii / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 442–449
particles occasionally emitted by the Sun. However, it is too simple for most purposes. To the next (first) order in U=w we obtain a much more useful equation, which can be written qf q qf qf 1 qU i qf ¼ þ Q. (4) kij þ Ui qt qxi qxj qxi 3 qxi q‘nðpÞ In this equation we have, in addition to the diffusion, convection and acceleration/deceleration caused by the electric field. Note that the electric field does not appear explicitly—it is nonetheless contained in the terms containing the flow velocity U. Eq. (4) was first written down by Parker (1965). It is the basis of most current work on cosmic-ray transport and acceleration, and is often called the Parker equation. The equation is a good approximation for energetic particles (U=w51) if the scattering by magnetic irregularities is faster than the macroscopic time scales, so the distribution is nearly isotropic. It applies at shocks (discontinuities in U) and the entire theory of diffusive shock acceleration may be obtained from the equation by simply putting a step-function flow velocity U into it. From this we see that it applies equally to parallel and perpendicular shocks (Jokipii, 1982, 1987). If the divergence of the flow velocity U is zero, there is no energy change to this order in U. If we carry out the expansion to second order in U=w, we obtain new effects involving the viscosity of the cosmic rays and the acceleration of the flow (Earl et al., 1988). These give additional terms representing acceleration of the cosmic rays caused by the fluid velocity shear, other gradients in velocity, and fluid acceleration. They can give acceleration in a divergence-free flow, which is absent in the first-order equation. The above approximations neglect the possible random motions of the scattering magnetic irregularities relative to the flow velocity. If these are included (e.g., forward and backward moving Alfve´n waves), further acceleration is introduced. This is the well-known 2nd-order Fermi acceleration, introduced by Fermi (1949), long before the present transport equation was developed. This mechanism, slightly generalized, is also called statistical acceleration in the modern literature. The effect of statistical or 2nd-order Fermi acceleration on the distribution function f may be written qf 1 q qf p2 Dpp ¼ 2 (5) qt 2ndorder Fermi p qp qp
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and represents diffusion in momentum, with momentum diffusion coefficient, Dpp . This term, which is generally much slower than shock acceleration, may simply be added to the right side of Eq. (4) and is often invoked where shocks are believed not to be present. Since the Alfve´n speed and fluid speed are often comparable, the viscous acceleration and 2ndorder Fermi acceleration are formally of the same nature, although the Fermi term is the one mostoften discussed. In many cases the viscous terms can be neglected because of the effects of the magnetic field in suppressing transport normal to the field. 3. Acceleration of fast charged particles The basic transport equation discussed above has been used in a variety of contexts to discuss the acceleration of energetic particles. The most successful and widely applicable is that of standard diffusive shock acceleration, which results from the application of the first-order equation (4) to a planar shock wave. As we will see, this has the singular property of producing a power law momentum spectrum which is not sensitive to the parameters, and which is close to that observed. For a general review, see Jones and Ellison (1991). 3.1. Diffusive shock and compression acceleration Consider first a steady, plane shock propagating in a uniform medium. Define the x-direction as the direction of propagation and let particles be introduced uniformly and steadily at the shock, at an injection momentum p0 . Work in the shocknormal coordinate system, with the shock at the fixed position x ¼ xsh . The shock ratio r is defined as the ratio of upstream to downstream flow speed U 1 =U 2 . It is readily found that the steady solution to the Parker equation in this case is given by f ðpÞ ¼ Ap3r=ðr1Þ Hðp p0 ÞF ðx; pÞ,
(6)
where HðpÞ is the Heaviside step function and the spatial dependence F ðx; pÞ is independent of x and p at the shock, is independent of x behind the shock, and decreases exponentially upstream as exp½U 1 ðxsh xÞ=kxx ðpÞ. Note that in the limit of a strong shock, where r ! 4 the momentum dependence becomes f ðpÞ / p4 , which corresponds to an energy spectrum dj=dT ¼ p2 f / p2 which is not far from the observed spectrum at relativistic energies (e.g., Fig. 2). This energy dependence is independent of shock speed, diffusion coefficients and other
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parameters. Since shocks in astrophysics tend to be strong, this may be the desired ‘‘universal’’ spectrum. This precise spectrum is not seen in nature, because of the idealizations involved, but these cause relatively small corrections to the basic power law, and also tend to be similar in many places. In particular, there will be a high-energy cutoff. In addition, in some cases, the accelerated particles will modify the shock (for a discussion, see Jones and Ellison, 1991). The acceleration by the shock is not instantaneous, of course. Solving the time-dependent version of Eq. (3), with the injection at momentum p0 turned on at a time t0 , reveals that the spectrum above p0 is still the universal power law given in Eq. (4), but with a high-momentum cutoff, pc which increases at a rate dpc =dt 4U 21 =kxx .
(7)
Hence, since for quasi-perpendicular shocks kxx ¼ k? which is generally significantly smaller than kk , quasi-perpendicular shocks will in general accelerate particles significantly faster than will quasi-parallel shocks. In fact, the physics of acceleration at quasiperpendicular shocks is quite different than that at quasi-parallel shocks (Jokipii, 1982). Nonetheless, the mathematics is the same, and the conclusions concerning the energy spectrum remain unchanged. The acceleration rate in the lower solar atmosphere, where the magnetic field can be 100 G or larger, is high enough to accelerate 1 GeV protons in a time of the order of a few seconds or less. Jokipii et al. (2003) pointed out a new acceleration mechanism which is basically similar to diffusive shock acceleration but which does not require a shock transition. A compressive disturbance will accelerate particles very efficiently, just a shocks do, if the thickness of the compression Lc is less than or of the order of the diffusive skin depth kxx =U1. This can occur at a single compression, in which case it is similar to diffusive shock acceleration, or it can occur in turbulent compressive fluctuations. Jokipii et al. termed this acceleration diffusive compression acceleration, in analogy with diffusive shock acceleration. 3.2. Statistical acceleration 2nd-Order Fermi or statistical acceleration, contained in Eq. (5), has an extensive history. It is quite often the mechanism of choice for possible diffuse re-acceleration of cosmic rays in the interstellar
medium and in solar flares. Statistical acceleration has at least one considerable disadvantage relative to diffusive shock acceleration. Statistical acceleration does not produce a power law spectrum with the desired value in a robust way. The shape of the spectrum depends sensitively on the transport parameters in the acceleration region. Nonetheless, it remains popular. The effect of 2nd-order Fermi acceleration on the distribution function f may be written qf 1 q qf p2 Dpp ¼ 2 (8) qt 2ndorder Fermi p qp qp and represents diffusion in momentum, with momentum diffusion coefficient, Dpp . This term may simply be added to the right side of Eq. (3) and is often invoked where shocks are believed not to be present. Since the Alfve´n speed and fluid speed are often comparable, the viscous acceleration and 2ndorder Fermi acceleration are formally of the same general order of magnitude, although the Fermi term is the one most-often discussed. Consider an example which illustrates the nature of the energy spectrum expected in statistical acceleration. A commonly-used expression for the momentum diffusion may be written in terms of the momentum p, the Alfve`n speed V a and the time for scattering by magnetic irregularities, tscatt as: 1 ðDpÞ2 Dpp ¼ ¼ p2 =tacc ¼ ðV 2a =c2 Þðp2 =tscatt Þ, 2 Dt (9) where the acceleration time scale tacc ¼ tscatt c2 =V 2a . In the standard leaky box model with loss time tloss , and if tcoll and tloss are constant, we obtain a power law. f ðpÞ ¼ Apa , where A is a constant and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 9 tacc þ a¼ þ . 2 4 tloss
(10)
(11)
The highly uncertain and variable parameters, V a ; tacc and tloss , must vary considerably from location to location and event to event, producing quite variable spectral slopes. As emphasized by Syrovatsky (1961), who presented a quite similar analysis, this argues against this form of statistical acceleration. Similar sensitivity to the parameters is expected in other models using statistical acceleration. Hence the similar power laws in many sources
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would be difficult to understand if statistical acceleration were the major accelerator. However, it is also important to remember that the only plausible mechanism for producing the extremely large enhancements of 3 He relative to 4 He which has been worked out is statistical acceleration. For this reason, if for no other, we must consider statistical acceleration as a possibility. Another possibility is that a statistical, resonant, process accelerates the particles to a superthermal energy, enhancing 3 He, and these are then further accelerated to energetic-particle energies by a shock, thus producing the desired energy spectrum. 4. Issues of current importance The preceding introduction has introduced the basic concepts involved in current discussions of charged-particle transport and acceleration in space plasmas. Recent observations have raised basic issues involved in the transport and acceleration of low energy solar energetic particles.
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numerical simulation orbits in the turbulent magnetic field, B ¼ B0 þ dB. In the limit dB2 =B20 51, the quasilinear approximation gives a decent approximation to the Giacalone and Jokipii’s results. In this case the scattering may be expressed in terms of the power spectrum of dB at wavenumbers resonant with the helical gyro-motion in B0 . The results for a more general magnetic field are complicated, and the results are quite different from the simple scattering model discussed expressed in Eq. (12). Giacalone and Jokipii (1999) found that for a Kolmogorov turbulence spectrum similar to that observed, the ratio k? =kk is of the order of a few times 102 , as shown in Fig. 4. This is much larger than the value expected from standard scattering theory, but not as large as that reported by Dwyer et al. (1997). In any event, it appears that perpendicular transport plays a larger role in
4.1. Spatial transport A current question of interest concerns the rate of energetic-particle transport across the average magnetic field. The energetic-particle diffusion tensor is anisotropic, and reflects the structure of both the magnetic fluctuations and the average magnetic field. An often-used construct is: kk ¼
w2 tc , 3
k? ¼ kk
1 , 1 þ tc o2g
(12)
which corresponds to the classical billiard-ball analogy. For typical interplanetary scattering mean free paths, this ratio is of the order of 104 or smaller. The transport of energetic particles in the direction normal to the average magnetic field is found observationally to be much larger than expected from the simple relaxation time picture. A morecareful analysis finds that field-line meandering, or random walk plays a major role, and k? =kk may approach unity (Dwyer et al., 1997). Giacalone and Jokipii (1999) published a detailed study of transport across a turbulent magnetic field. They found it necessary to carry out a careful
Fig. 4. The perpendicular and parallel diffusion coefficients obtained by Giacalone and Jokipii (1999).
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at the radius where the shock formed and subsequently propagated in to the point of observation. Giacalone et al. (2005) pointed out that the particles may have been accelerated by the compression, in a manner very similar to diffusive shock acceleration (see above). Detailed numerical simulations showed that the observations were well accounted for naturally by this process, with no free parameters.
Fig. 5. Observation of dropouts observed by Mazur et al. (2000).
interplanetary transport of energetic particles than commonly assumed. An interesting aspect of cross-field transport was revealed by the ‘‘dropouts’’ observed in impulsive SEP events and reported by Mazur et al. (2000), and which are related to earlier observations of Fig. 5 illustrates the effect. In a companion paper, Giacalone et al. (2000) pointed out that the dropouts could be the transient initial phase of a particle event where the particles have not yet scattered appreciably and are running along meandering and braided magnetic flux tubes. The dropouts are then just empty flux tubes, not connected to the particle source, being convected past the point of observation. This is supported by orbit simulations. Chuychai et al. (2005) have presented an alternative point of view involving properties of the random magnetic field and not related to the connection to the particle source. 4.2. Acceleration mechanisms Particles are observed in the inner heliosphere which are clearly accelerated there. The mechanism of diffusive shock acceleration, as discussed above, has been invoked to explain most of these particles. However, particles are observed which are not clearly associated with shocks. Two distinct mechanisms, related to the general mechanisms discussed above, have been proposed to to accelerated the observed particles. 4.2.1. Diffusive compression acceleration Mason (2000) reported observations of particles associated with co-rotating interaction regions near 1 AU well-inside of the radius where the regions form shocks. It was possible to rule out the possibility that the particles had been accelerated
4.2.2. Suprathermal tails Gloeckler et al. (2000) emphasized the apparent ubiquity of the observed very low-energy particles or superthermal tails which he finds to have a spectrum which is always very nearly a power law with an index f ðvÞ / v5 . The problem of the mechanism for the acceleration of the particles forming the observed superthermal tails is very intriguing. If the tails are the result of diffusive shock acceleration, the particles must propagate large distances and this seems unlikely. Diffusive compression acceleration, discussed above is a possibility. However, Giacalone and Jokipii (1999) concluded that the rate of acceleration was too slow to compete with the ongoing adiabatic cooling in the expanding solar wind. Gloeckler and Fisk (2006) have more recently suggested a generalized statistical mechanism, associated with feedback between the turbulent thermal plasma and the superthermal tail. This problem remains without a consensus solution. 5. Summary The acceleration and transport of energetic charged particles observed in the inner heliosphere is determined by the ambient magnetic and electric fields. The transport equation first written down by Parker (1965) provides an initial point for understanding the various processes involved. Acceleration by propagating shock waves and more-gradual compressions appears capable of acceleration most of these particles and has the virtue of robustly producing naturally a power-law spectrum with a spectral index which varies over a relatively small range for a wide variety of conditions. This provides an attractive interpretation of observations. Statistical acceleration is also widely used, and appears to be able to account for the 3 He-rich SEP events. It has the considerable disadvantage of not robustly producing a power law. Recent observations of transport across the magnetic field and accelerated particles not associated
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with shocks currently present significant challenges to theory. Acknowledgments This work was supported, in part, by NASA under Grants NAG5-6620, NAG5-7793, NAG512919, and by the NSF under Grant ATM9616547 and ATM0327773. I acknowledge helpful discussions regarding many of these matters with my colleagues J. Ko´ta and J. Giacalone. References Bieber, J.W., et al., 2004. Astrophysical Journal 601, L103. Chuychai, P., et al., 2005. Astrophysical Journal 633, L49. Dmitruk, W.H., et al., 2003. Astrophysical Journal 597, L81. Dwyer, J.R., et al., 1997. Astrophysical Journal 490, L115. Earl, J.A., Jokipii, J.R., Morfill, G., 1988. Astrophysical Journal 331, L91–L94. Fermi, E., 1949. Physical Review 75, 1169–1174.
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Forbush, S.E., 1954. Physical Review 70, 771–772. Giacalone, J., Jokipii, J.R., 1999. Astrophysical Journal 520, 204. Giacalone, J., Jokipii, J.R., Mazur, J.E., 2000. Astrophysical Journal 532, L8. Giacalone, J., Jokipii, J.R., Ko`ta, J., 2005. Geophysical Monograph 156, 41. Gloeckler, G., Fisk, L.A., 2006. Astrophysical Journal 640, L79. Gloeckler, G., et al., 2000. American Institute of Physics Conference Proceedings 528, 221. Jokipii, J.R., 1982. Astrophysical Journal 255, 716–720. Jokipii, J.R., 1987. Astrophysical Journal 313, 842. Jokipii, J.R., Giacalone, J., Ko`ta, J., 2003. In: Proceedings of the 28th International Cosmic Ray Conference, July 31–August 7, 2003. Tsukuba, Japan, p. 3685. Jones, F.C., Ellison, D.C., 1991. Space Science Reviews 58, 259–346. Kronberg, P.P., 1996. Space Science Reviews 75, 387–399. Mason, G.L., 2000. American Institute of Physics Conference Proceedings 528, 234. Mazur, J.E., et al., 2000. Astrophysical Journal 532, L79. Parker, E.N., 1965. Planetary Space Science 13, 9. Reames, D.V., 1999. Space Science Reviews 90, 413–491. Syrovatsky, S.A., 1961. Journal of Experimental and Theoretical Physics (USSR) 40, 1788–1793.
Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 442–449 www.elsevier.com/locate/jastp
Acceleration and transport of energetic particles observed in the inner heliosphere J.R. Jokipii University of Arizona, Tucson, AZ 85721, USA Accepted 27 August 2007 Available online 29 September 2007
Abstract A number of distinct species of energetic particles—ranging from those accelerated at the Sun and heliosphere to galactic cosmic rays, are observed in the inner heliosphere. In turn, a number of different acceleration mechanisms—including stochastic fluctuations, shocks and more-gradual compressions—have been proposed to explain the various observations and each mechanism has its adherents. In this tutorial review, various popular mechanisms are introduced, and the differences between statistical acceleration and acceleration at shocks at quasi-perpendicular and quasi-parallel shocks and compressions are discussed. Spatial transport is an important part of the acceleration process, and the accelerated particles must propagate through the turbulent solar wind to the point of observation. Both the acceleration and the subsequent transport of the particles depend fundamentally on the nature of the ambient, generally turbulent, plasma and electromagnetic fields. New constraints imposed by recent in situ and remote observations of the energetic particles in the inner heliosphere will be discussed. r 2007 Published by Elsevier Ltd. Keywords: Energetic particles; Solar wind; Heliosphere; Sun
1. Introduction This tutorial talk gives a brief introduction to the basic acceleration and transport of energetic charged particles in turbulent, collisionless plasmas such as the interplanetary medium. In addition, a brief discussion is given of some current observational issues and possible theoretical solutions. Energetic charged particles or cosmic rays are found in nature wherever the ambient density is low enough that collisional losses are negligible. The Sun accelerates energetic particles (SEPs), in events associated with energy releases, such as coronal E-mail address: [email protected] 1364-6826/$ - see front matter r 2007 Published by Elsevier Ltd. doi:10.1016/j.jastp.2007.08.026
mass ejections or flares. The resulting SEP events are often power laws in energy with a similar powerlaw index, but with a cutoff at high energies. Occasionally evidence for multiple power laws is fount. The first solar energetic particle events were observed in the mid 1940s by Forbush (1954). The time history of a typical SEP event and the energy spectra of various energetic particle species during a typical solar particle event are shown in Figs. 1 and 2. Charged particles are also accelerated in the solar wind both by propagating transient shocks and corotating shocks, and at the solar-wind termination. In addition, a continual flux of energetic particles are observed in the inner heliosphere at quiet times (undisturbed by solar activity). These particles
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Fig. 1. Energy spectra during a typical SEP event (Reames, 1999).
extend over a very wide range of kinetic energies from just above thermal energies to more than 1020 eV. The anisotropy (relative to the local fluid frame) is small—being less than or of order 102 at 1018 eV and smaller at lower energies. These particles come from the galaxy (and perhaps beyond at the very highest energies) and are termed galactic cosmic rays. Fig. 3 shows the observed spectrum of galactic cosmic rays at Earth over the energy interval 105 21020 eV, compiled from a number of sources. Apparent at low energies, below ’ 1 GeV are a turnover and various other features caused by the interaction of cosmic rays with the sun and the solar wind. The energetic particle spectrum, at the very lowest energies, which can be observed only in situ, is observed to merge smoothly into the background thermal plasma distribution. The energy spectrum of galactic cosmic rays is a remarkably-smooth power law above some 109 eV, with only a minor change in slope occurring between 1015 and 1016 eV (the ‘‘knee’’), and possibly a small flattening at 1019 eV (the ‘‘ankle’’). The anomalous cosmic-ray oxygen, also seen in Fig. 3, is a consequence of the interaction of the heliosphere with the interstellar medium.
Fig. 2. An observed solar event reported by Bieber et al. (2004), showing excellent theoretical fits to intensity and anisotropies.
Photons emitted from distant galaxies, between galaxies and other regions of our galaxy as a consequence of the interaction of cosmic rays with ambient matter or electromagnetic fields, show that cosmic rays exist in many places. Moreover, the inferred energy spectra of the cosmic rays in these remote sources are also power laws with slopes not much different from that observed at Earth (Kronberg, 1996). The slope of the power law is similar to that of the solar particles discussed above. This fact, together with the smoothness of the observed spectrum at Earth over a large range in energy, suggests a common acceleration mechanism for the bulk of the cosmic rays. 2. Acceleration and transport of energetic charged particles We may neglect particle–particle collisions in nearly all astrophysical plasmas of interest. The energy change and transport are determined to a very good approximation by the electric and
ARTICLE IN PRESS J.R. Jokipii / Journal of Atmospheric and Solar-Terrestrial Physics 70 (2008) 442–449
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where c is the speed of light. The physical problem of acceleration, then, consists of solving Eqs. (1) and (2) for a given situation. Note that since the electric field E is determined by U and B, it need not appear explicitly in the transport equation. Electric fields occurring in (small) regions where magnetohydrodynamics breaks down, such as in shock fronts or in reconnection regions, often are not given by Eq. (2). The reader is referred to Dmitruk et al. (2003), for a discussion. 2.1. The transport equation
Fig. 3. Observed energy spectrum of cosmic rays during quiet times. Shown are the all-particle spectrum and superposed, for illustration, the Oxygen spectrum, including the anomalous component at 107 2108 eV. The turnup in intensity below 106 2107 eV is thought to be solar in origin. Compiled from a variety of sources.
magnetic fields in the ambient plasma. In this case, the change in energy DT of a particle having energy T and velocity w, moving an electric field Eðr; tÞ in the time interval Dt may be written in general as: Z tþDt DT ¼ q w Eðr; tÞ dt, (1) t
where the integrand must be evaluated along the actual particle trajectory in the ambient electric and magnetic fields. From this we see that in order to evaluate the energy change, we must know the particle trajectory in the electromagnetic field. This leads to the general requirements that acceleration and spatial transport be intimately coupled and both Eðr; tÞ and Bðr; tÞ must be considered together. Here we consider energetic particles, for which the relevant spatial plasma scales are the energeticparticle gyro-radii. At these scales the ambient plasmas are generally hydromagnetic, and the electric field is determined from the fluid velocity U and magnetic field by the relation E¼
UB , c
(2)
A robust and widely applicable transport theory, applicable to particles whose speed w is significantly larger than U has been developed and applied extensively over the past four decades. It may be considered as an expansion in powers of the small ratio U=w. Unfortunately, a general theory for lower-energy particles for which U=w is not small is not available—for such particles one must use numerical simulations. The available theory utilizes the fact that the plasmas in space are turbulent, with broadband fluctuations over scales including those comparable with the particle gyro-radii. The particle motions are described statistically—the turbulent magnetic fluctuations ‘‘scatter’’ the particles in angle, driving them to near-isotropy in the fluid frame (or, more generally, the frame of the scatterer) with a time scale tscat . This scattering occurs much more rapidly than energy change. To zeroth order in U=w the resulting motion may be approximated as diffusive, and the phase space distribution function f ðp; r; tÞ of the particles having a momentum magnitude p at position r and time t satisfies a diffusion equation qf q qf ¼ kij þ Q, (3) qt qxi qxj where kij is the diffusion tensor, determined by the spectrum of the magnetic fluctuations, and Q represents any source. Magnetic fluctuations at scales of the order of the energetic-particle gyroradius are the most important in determining kij , although the field-line random walk or braiding, which affects the cross-field transport depends on very long length scales. The flow velocity U does not appear in this order, and because E ¼ 0 in this limit, neither convection nor energy change appear. Eq. (3) gives a crude, but sometimes acceptable description of the behavior of very high energy
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particles occasionally emitted by the Sun. However, it is too simple for most purposes. To the next (first) order in U=w we obtain a much more useful equation, which can be written qf q qf qf 1 qU i qf ¼ þ Q. (4) kij þ Ui qt qxi qxj qxi 3 qxi q‘nðpÞ In this equation we have, in addition to the diffusion, convection and acceleration/deceleration caused by the electric field. Note that the electric field does not appear explicitly—it is nonetheless contained in the terms containing the flow velocity U. Eq. (4) was first written down by Parker (1965). It is the basis of most current work on cosmic-ray transport and acceleration, and is often called the Parker equation. The equation is a good approximation for energetic particles (U=w51) if the scattering by magnetic irregularities is faster than the macroscopic time scales, so the distribution is nearly isotropic. It applies at shocks (discontinuities in U) and the entire theory of diffusive shock acceleration may be obtained from the equation by simply putting a step-function flow velocity U into it. From this we see that it applies equally to parallel and perpendicular shocks (Jokipii, 1982, 1987). If the divergence of the flow velocity U is zero, there is no energy change to this order in U. If we carry out the expansion to second order in U=w, we obtain new effects involving the viscosity of the cosmic rays and the acceleration of the flow (Earl et al., 1988). These give additional terms representing acceleration of the cosmic rays caused by the fluid velocity shear, other gradients in velocity, and fluid acceleration. They can give acceleration in a divergence-free flow, which is absent in the first-order equation. The above approximations neglect the possible random motions of the scattering magnetic irregularities relative to the flow velocity. If these are included (e.g., forward and backward moving Alfve´n waves), further acceleration is introduced. This is the well-known 2nd-order Fermi acceleration, introduced by Fermi (1949), long before the present transport equation was developed. This mechanism, slightly generalized, is also called statistical acceleration in the modern literature. The effect of statistical or 2nd-order Fermi acceleration on the distribution function f may be written qf 1 q qf p2 Dpp ¼ 2 (5) qt 2ndorder Fermi p qp qp
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and represents diffusion in momentum, with momentum diffusion coefficient, Dpp . This term, which is generally much slower than shock acceleration, may simply be added to the right side of Eq. (4) and is often invoked where shocks are believed not to be present. Since the Alfve´n speed and fluid speed are often comparable, the viscous acceleration and 2ndorder Fermi acceleration are formally of the same nature, although the Fermi term is the one mostoften discussed. In many cases the viscous terms can be neglected because of the effects of the magnetic field in suppressing transport normal to the field. 3. Acceleration of fast charged particles The basic transport equation discussed above has been used in a variety of contexts to discuss the acceleration of energetic particles. The most successful and widely applicable is that of standard diffusive shock acceleration, which results from the application of the first-order equation (4) to a planar shock wave. As we will see, this has the singular property of producing a power law momentum spectrum which is not sensitive to the parameters, and which is close to that observed. For a general review, see Jones and Ellison (1991). 3.1. Diffusive shock and compression acceleration Consider first a steady, plane shock propagating in a uniform medium. Define the x-direction as the direction of propagation and let particles be introduced uniformly and steadily at the shock, at an injection momentum p0 . Work in the shocknormal coordinate system, with the shock at the fixed position x ¼ xsh . The shock ratio r is defined as the ratio of upstream to downstream flow speed U 1 =U 2 . It is readily found that the steady solution to the Parker equation in this case is given by f ðpÞ ¼ Ap3r=ðr1Þ Hðp p0 ÞF ðx; pÞ,
(6)
where HðpÞ is the Heaviside step function and the spatial dependence F ðx; pÞ is independent of x and p at the shock, is independent of x behind the shock, and decreases exponentially upstream as exp½U 1 ðxsh xÞ=kxx ðpÞ. Note that in the limit of a strong shock, where r ! 4 the momentum dependence becomes f ðpÞ / p4 , which corresponds to an energy spectrum dj=dT ¼ p2 f / p2 which is not far from the observed spectrum at relativistic energies (e.g., Fig. 2). This energy dependence is independent of shock speed, diffusion coefficients and other
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parameters. Since shocks in astrophysics tend to be strong, this may be the desired ‘‘universal’’ spectrum. This precise spectrum is not seen in nature, because of the idealizations involved, but these cause relatively small corrections to the basic power law, and also tend to be similar in many places. In particular, there will be a high-energy cutoff. In addition, in some cases, the accelerated particles will modify the shock (for a discussion, see Jones and Ellison, 1991). The acceleration by the shock is not instantaneous, of course. Solving the time-dependent version of Eq. (3), with the injection at momentum p0 turned on at a time t0 , reveals that the spectrum above p0 is still the universal power law given in Eq. (4), but with a high-momentum cutoff, pc which increases at a rate dpc =dt 4U 21 =kxx .
(7)
Hence, since for quasi-perpendicular shocks kxx ¼ k? which is generally significantly smaller than kk , quasi-perpendicular shocks will in general accelerate particles significantly faster than will quasi-parallel shocks. In fact, the physics of acceleration at quasiperpendicular shocks is quite different than that at quasi-parallel shocks (Jokipii, 1982). Nonetheless, the mathematics is the same, and the conclusions concerning the energy spectrum remain unchanged. The acceleration rate in the lower solar atmosphere, where the magnetic field can be 100 G or larger, is high enough to accelerate 1 GeV protons in a time of the order of a few seconds or less. Jokipii et al. (2003) pointed out a new acceleration mechanism which is basically similar to diffusive shock acceleration but which does not require a shock transition. A compressive disturbance will accelerate particles very efficiently, just a shocks do, if the thickness of the compression Lc is less than or of the order of the diffusive skin depth kxx =U1. This can occur at a single compression, in which case it is similar to diffusive shock acceleration, or it can occur in turbulent compressive fluctuations. Jokipii et al. termed this acceleration diffusive compression acceleration, in analogy with diffusive shock acceleration. 3.2. Statistical acceleration 2nd-Order Fermi or statistical acceleration, contained in Eq. (5), has an extensive history. It is quite often the mechanism of choice for possible diffuse re-acceleration of cosmic rays in the interstellar
medium and in solar flares. Statistical acceleration has at least one considerable disadvantage relative to diffusive shock acceleration. Statistical acceleration does not produce a power law spectrum with the desired value in a robust way. The shape of the spectrum depends sensitively on the transport parameters in the acceleration region. Nonetheless, it remains popular. The effect of 2nd-order Fermi acceleration on the distribution function f may be written qf 1 q qf p2 Dpp ¼ 2 (8) qt 2ndorder Fermi p qp qp and represents diffusion in momentum, with momentum diffusion coefficient, Dpp . This term may simply be added to the right side of Eq. (3) and is often invoked where shocks are believed not to be present. Since the Alfve´n speed and fluid speed are often comparable, the viscous acceleration and 2ndorder Fermi acceleration are formally of the same general order of magnitude, although the Fermi term is the one most-often discussed. Consider an example which illustrates the nature of the energy spectrum expected in statistical acceleration. A commonly-used expression for the momentum diffusion may be written in terms of the momentum p, the Alfve`n speed V a and the time for scattering by magnetic irregularities, tscatt as: 1 ðDpÞ2 Dpp ¼ ¼ p2 =tacc ¼ ðV 2a =c2 Þðp2 =tscatt Þ, 2 Dt (9) where the acceleration time scale tacc ¼ tscatt c2 =V 2a . In the standard leaky box model with loss time tloss , and if tcoll and tloss are constant, we obtain a power law. f ðpÞ ¼ Apa , where A is a constant and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 9 tacc þ a¼ þ . 2 4 tloss
(10)
(11)
The highly uncertain and variable parameters, V a ; tacc and tloss , must vary considerably from location to location and event to event, producing quite variable spectral slopes. As emphasized by Syrovatsky (1961), who presented a quite similar analysis, this argues against this form of statistical acceleration. Similar sensitivity to the parameters is expected in other models using statistical acceleration. Hence the similar power laws in many sources
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would be difficult to understand if statistical acceleration were the major accelerator. However, it is also important to remember that the only plausible mechanism for producing the extremely large enhancements of 3 He relative to 4 He which has been worked out is statistical acceleration. For this reason, if for no other, we must consider statistical acceleration as a possibility. Another possibility is that a statistical, resonant, process accelerates the particles to a superthermal energy, enhancing 3 He, and these are then further accelerated to energetic-particle energies by a shock, thus producing the desired energy spectrum. 4. Issues of current importance The preceding introduction has introduced the basic concepts involved in current discussions of charged-particle transport and acceleration in space plasmas. Recent observations have raised basic issues involved in the transport and acceleration of low energy solar energetic particles.
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numerical simulation orbits in the turbulent magnetic field, B ¼ B0 þ dB. In the limit dB2 =B20 51, the quasilinear approximation gives a decent approximation to the Giacalone and Jokipii’s results. In this case the scattering may be expressed in terms of the power spectrum of dB at wavenumbers resonant with the helical gyro-motion in B0 . The results for a more general magnetic field are complicated, and the results are quite different from the simple scattering model discussed expressed in Eq. (12). Giacalone and Jokipii (1999) found that for a Kolmogorov turbulence spectrum similar to that observed, the ratio k? =kk is of the order of a few times 102 , as shown in Fig. 4. This is much larger than the value expected from standard scattering theory, but not as large as that reported by Dwyer et al. (1997). In any event, it appears that perpendicular transport plays a larger role in
4.1. Spatial transport A current question of interest concerns the rate of energetic-particle transport across the average magnetic field. The energetic-particle diffusion tensor is anisotropic, and reflects the structure of both the magnetic fluctuations and the average magnetic field. An often-used construct is: kk ¼
w2 tc , 3
k? ¼ kk
1 , 1 þ tc o2g
(12)
which corresponds to the classical billiard-ball analogy. For typical interplanetary scattering mean free paths, this ratio is of the order of 104 or smaller. The transport of energetic particles in the direction normal to the average magnetic field is found observationally to be much larger than expected from the simple relaxation time picture. A morecareful analysis finds that field-line meandering, or random walk plays a major role, and k? =kk may approach unity (Dwyer et al., 1997). Giacalone and Jokipii (1999) published a detailed study of transport across a turbulent magnetic field. They found it necessary to carry out a careful
Fig. 4. The perpendicular and parallel diffusion coefficients obtained by Giacalone and Jokipii (1999).
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at the radius where the shock formed and subsequently propagated in to the point of observation. Giacalone et al. (2005) pointed out that the particles may have been accelerated by the compression, in a manner very similar to diffusive shock acceleration (see above). Detailed numerical simulations showed that the observations were well accounted for naturally by this process, with no free parameters.
Fig. 5. Observation of dropouts observed by Mazur et al. (2000).
interplanetary transport of energetic particles than commonly assumed. An interesting aspect of cross-field transport was revealed by the ‘‘dropouts’’ observed in impulsive SEP events and reported by Mazur et al. (2000), and which are related to earlier observations of Fig. 5 illustrates the effect. In a companion paper, Giacalone et al. (2000) pointed out that the dropouts could be the transient initial phase of a particle event where the particles have not yet scattered appreciably and are running along meandering and braided magnetic flux tubes. The dropouts are then just empty flux tubes, not connected to the particle source, being convected past the point of observation. This is supported by orbit simulations. Chuychai et al. (2005) have presented an alternative point of view involving properties of the random magnetic field and not related to the connection to the particle source. 4.2. Acceleration mechanisms Particles are observed in the inner heliosphere which are clearly accelerated there. The mechanism of diffusive shock acceleration, as discussed above, has been invoked to explain most of these particles. However, particles are observed which are not clearly associated with shocks. Two distinct mechanisms, related to the general mechanisms discussed above, have been proposed to to accelerated the observed particles. 4.2.1. Diffusive compression acceleration Mason (2000) reported observations of particles associated with co-rotating interaction regions near 1 AU well-inside of the radius where the regions form shocks. It was possible to rule out the possibility that the particles had been accelerated
4.2.2. Suprathermal tails Gloeckler et al. (2000) emphasized the apparent ubiquity of the observed very low-energy particles or superthermal tails which he finds to have a spectrum which is always very nearly a power law with an index f ðvÞ / v5 . The problem of the mechanism for the acceleration of the particles forming the observed superthermal tails is very intriguing. If the tails are the result of diffusive shock acceleration, the particles must propagate large distances and this seems unlikely. Diffusive compression acceleration, discussed above is a possibility. However, Giacalone and Jokipii (1999) concluded that the rate of acceleration was too slow to compete with the ongoing adiabatic cooling in the expanding solar wind. Gloeckler and Fisk (2006) have more recently suggested a generalized statistical mechanism, associated with feedback between the turbulent thermal plasma and the superthermal tail. This problem remains without a consensus solution. 5. Summary The acceleration and transport of energetic charged particles observed in the inner heliosphere is determined by the ambient magnetic and electric fields. The transport equation first written down by Parker (1965) provides an initial point for understanding the various processes involved. Acceleration by propagating shock waves and more-gradual compressions appears capable of acceleration most of these particles and has the virtue of robustly producing naturally a power-law spectrum with a spectral index which varies over a relatively small range for a wide variety of conditions. This provides an attractive interpretation of observations. Statistical acceleration is also widely used, and appears to be able to account for the 3 He-rich SEP events. It has the considerable disadvantage of not robustly producing a power law. Recent observations of transport across the magnetic field and accelerated particles not associated
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with shocks currently present significant challenges to theory. Acknowledgments This work was supported, in part, by NASA under Grants NAG5-6620, NAG5-7793, NAG512919, and by the NSF under Grant ATM9616547 and ATM0327773. I acknowledge helpful discussions regarding many of these matters with my colleagues J. Ko´ta and J. Giacalone. References Bieber, J.W., et al., 2004. Astrophysical Journal 601, L103. Chuychai, P., et al., 2005. Astrophysical Journal 633, L49. Dmitruk, W.H., et al., 2003. Astrophysical Journal 597, L81. Dwyer, J.R., et al., 1997. Astrophysical Journal 490, L115. Earl, J.A., Jokipii, J.R., Morfill, G., 1988. Astrophysical Journal 331, L91–L94. Fermi, E., 1949. Physical Review 75, 1169–1174.
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Forbush, S.E., 1954. Physical Review 70, 771–772. Giacalone, J., Jokipii, J.R., 1999. Astrophysical Journal 520, 204. Giacalone, J., Jokipii, J.R., Mazur, J.E., 2000. Astrophysical Journal 532, L8. Giacalone, J., Jokipii, J.R., Ko`ta, J., 2005. Geophysical Monograph 156, 41. Gloeckler, G., Fisk, L.A., 2006. Astrophysical Journal 640, L79. Gloeckler, G., et al., 2000. American Institute of Physics Conference Proceedings 528, 221. Jokipii, J.R., 1982. Astrophysical Journal 255, 716–720. Jokipii, J.R., 1987. Astrophysical Journal 313, 842. Jokipii, J.R., Giacalone, J., Ko`ta, J., 2003. In: Proceedings of the 28th International Cosmic Ray Conference, July 31–August 7, 2003. Tsukuba, Japan, p. 3685. Jones, F.C., Ellison, D.C., 1991. Space Science Reviews 58, 259–346. Kronberg, P.P., 1996. Space Science Reviews 75, 387–399. Mason, G.L., 2000. American Institute of Physics Conference Proceedings 528, 234. Mazur, J.E., et al., 2000. Astrophysical Journal 532, L79. Parker, E.N., 1965. Planetary Space Science 13, 9. Reames, D.V., 1999. Space Science Reviews 90, 413–491. Syrovatsky, S.A., 1961. Journal of Experimental and Theoretical Physics (USSR) 40, 1788–1793.