- Email: [email protected]
Heavy ion acceleration in solar flares
HEAVY ION ACCELERATION
IN SOLAR FLARES
J. A. Miller
University of Alabama in Huntsville, Department of Physics, Huntsville, AL 35899, USA ABSTRACT Stochastic acceleration by cascading MHD turbulence is able to account for a wealth of impulsive solar flare energetic particle properties. Here we show that it can also account for the interplanetary heavy ion distributions from the August 7 1999 flare, observed by the Advanced Composition Explorer.
INTRODUCTION The acceleration of particles to high energies is an ubiquitous phenomenon at sites throughout the Universe. Impulsive solar flares offer one of the most impressive examples anywhere, releasing up to 1032 ergs of energy over timescales of several tens of seconds to several tens of minutes. Much of this energy is spent on accelerating ambient electrons and ions to suprathermal and even relativistic energies, which then either remain trapped at the Sun or escape into interplanetary space. The radiations from the trapped particles (Ramaty & Murphy 1987) are important signatures of the acceleration mechanism. However, the particles that escape into space (Reames 1990) and subsequently are detected there are perhaps the most direct (assuming there is no appreciable further acceleration away from the Sun) and important diagnostic, and are refered to as solar energetic particles (SEPs). The observed SEP abundances from impulsive flares are illustrated in Table 1, which shows specifically the abundance ratios for particles >~ 1 MeV nucleon -1 (data from the "Great Debate" (1995) and Reames et al. references therein, Share & Murphy 1997, Reames 2001). (The ions labeled Kr and Xe are actually groups of ions with mean mass numbers of 85 and 128, respectively.) The abundances for the gradual events are essentially the same as those in the ambient corona, and this is the basic evidence for the shock accelerated nature of these particles. However, the abundances for the impulsive events show dramatic enhancements in several of the ratios. Namely, (1) 3He is enhanced by a factor of ~ 2000 relative to 4He; (2) 4He is enhanced by a factor of ,~ 5 relative to H (or H is suppressed by a factor of ~ 5 relative to 4He); (3) Ne, Mg, Si, Fe, and especially Kr and Xe are enhanced by progressively increasing factors relative to O; and (4) C, N, O, and 4He are not enhanced relative to one another. Also shown is the charge-to-mass ratio Q/A for the first ion (the "numerator") in the ratio, for a temperature of 3 • 106 K. At this temperature, O is fully ionized and has a Q/A of 0.5, like 4He, C, and N. Note that Q/A is also equal to that ion's cyclotron frequency in units of the H cyclotron frequency gtg. One trend is immediately apparent: below the 4He, C, N, and O cyclotron frequency of 0.5~H, the abundance enhancement of an energetic ion species is inversely proportional to its cyclotron frequency. Note that 4He is also enhanced relative to H, but the degree of this enhancement is somewhat larger than the trend of the heavier ions would suggest. Another curious point is the magnitude of the 3He enhancement, which is even more curious due to the location of its cyclotron frequency between that of H and 4He. If one assumes an acceleration mechanism that has an efficiency that increases with increasing Q/A, H should be enhanced relative to 4He, which is not the case; on the other hand, if the efficiency increases with decreasing Q/A, - 387-
J.A. Miller Ratio H/4He 3He/4He 4He/O C/O N/O Ne/O Mg/O Si/O Fe/O Kr/O Xe/O
Q/A 1 0.67 0.5 0.5 0.5 0.42 0.42 0.42 0.26 0.13 0.11
Table 1. Ion Abundance Ratios Impulsive Flares Gradual Flares (Corona) ~2 (• decrease) ~10 ~1 (• increase) ..~ 0.0005 ~46 ~55 ~0.436 ~0.471 ~0.153 ~0.128 ~0.416 (• increase) ~0.151 ~0.413 (• increase) ~0.203 ~0.405 (• increase) ~0.155 ~1.234 (• increase) ~0.155 (• 100 increase) (• 1000 increase)
then 4He should be enhanced relative to 3He, which is also clearly not the case. A second point regarding 3He is that its energy distribution is often quite different from that of the other ions in interplanetary space (e.g. Mason et al. 2002), which are in turn quite similar. It should be emphasized that these enhancements are all valuable diagnostics of the impulsive flare acceleration mechanism. Of course, the Q/A values for the ions depend upon the plasma temperature, and the above discussion is predicated upon a temperature of 3 • 106 K. However, it has been argued (Reames, Meyer, & v o n Rosenvinge 1994) that the initial plasma temperature must in fact be 3 • 106 K; otherwise, the Q/A values become either (1) interlaced in a manner which precludes any pattern at all, or (2) equal, in which case there is no known mechanism that can preferentially accelerate one ion over another. The simplest overall explanation for these observations is that 3He is subjected to an additional acceleration, from which the other ions are immune. The only known way this could occur is if 3He were stochastically accelerated by cyclotron resonance with waves that are excited in a narrow frequency range just around the 3He cyclotron frequency. This basic idea was originally proposed by Fisk (1978), and subsequent models just differ in the specific wave mode that is employed (e.g., cf. Fisk with Temerin & Roth 1992, and Miller & Vifias 1993). However, regardless of the specific model, 3He should be considered essentially as proof that stochastic acceleration at least operates in impulsive flares. THE MODEL Given that stochastic acceleration is known to at least occur in impulsive flares, we advocate it as the mechanism whereby all ions and electrons are energized from thermal to relativistic energies. In our model, we suppose that everything (including 3He) is stochastically accelerated by broad-band MHD shear Alfv~n and fast mode waves, but that 3He is also subjected to some extra acceleration via one of the models referenced above. This extra stochastic acceleration is assumed to be responsible for the spectral difference in the 3He energy distribution as well as its huge enhancement, and will not be discussed further here. Stochastic acceleration relies upon wave-particle resonance, and occurs when the condition x - w - kllv[i-
gl~21/~ = 0 is satisfied. Here, vii , "y, and 9t are the particle's parallel speed (with respect to the ambient magnetic field /30), Lorentz factor, and cyclotron frequency; while w and kll are the wave frequency and parallel wavenumber. The quantity x is the frequency mismatch parameter. If the harmonic number g ~ 0, its sign depends upon the sense of rotation of the wave electric field and the particle in the plasma frame: if both rotate in the same sense (right or left handed) relative to B0, then g > 0; if not, then g < 0. When x = 0, the frequency of rotation of the wave electric field is an integer multiple of the frequency of gyration of the particle in its guiding center frame, and the sense of rotation is the same. The particle thus sees an -
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Heavy Ion Acceleration in Solar Flares
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Fig. 1. Model fits (dotted lines) to the interplanetary 4He, O, Fe, and Ne distributions from the August 7 1999 flare (Mason et al. 2002). Data (connected dots with error bars) is obtained from ULEIS and SIS on the Advanced Composition Explorer. The model distributions are from the cascading turbulence simulation, with an injection of 5 1 e r g s c m -3 s -1 of turbulence into a region of size 108 cm; the magnetic field is 100G and the plasma density is 101~ cm -3.
electric field for a sustained length of time and will be either strongly accelerated or decelerated, depending upon the relative phase of the field and the gyromotion. This is called cyclotron or gyroresonance. If g = 0, there is matching between the parallel motion of the particle and the wave parallel electric or magnetic field. This is called L a n d a u or Cherenkov resonanae. The Alfv~n waves posses a transverse left-hand polarized electric field, which can resonate with the ions via the g = +1 (or cyclotron) resonance. We see from the resonance condition and the Alfv~n wave dispersion relation that in order for a wave to cyclotron resonate with a low-energy ion (say near the thermal speed), w must be near ft. On the other hand, as the ion gains energy, w can become much less than g/. Hence, a broad-band spectrum of Alfv~n waves extending up to fl can stochastically cyclotron accelerate ions from the tail of the thermal distribution to high energies (see also Barbosa 1979, Eichler 1979). On the other hand, Alfv~n waves are not able to accelerate electrons from the background.
- 389-
J.A. Miller The fast mode waves posses a compressive magnetic field, which can couple with either ions or electrons via the g = 0 (or Landau) resonance. We see from the resonance condition and the fast mode dispersion relation that (1) a wave of any w will only Landau resonate with a particle having a speed v greater than the Alfv6n speed VA, and (2) as the energy increases the wave propagation angle must approach 90 ~ Hence, fast mode waves having a distribution of propagation directions can resonate with particles from Alfv~nic to relativistic energies, and this leads specifically to stochastic transit-time acceleration (Miller 1997). Now, initially in a flare plasma of temperature T ~ 3 x 106 K, only electrons will be present in large numbers above va, and so these particles will be preferentially accelerated by the fast mode waves. However, should the ions become super Alfv~nic (e.g. as a result of cyclotron acceleration by Alfv~n waves), then they too will be transit-time accelerated. Lastly, even though fast mode waves of any w can resonate with super Alfv~nic particles, the acceleration rate is proportional to their wavenumber or frequency. When a particle is in resonance with a single small-amplitude wave, vii executes approximate simple harmonic motion about that parallel velocity which exactly satisfies the resonance condition (Karimabadi et al. 1992). There is no energy gain on average. The frequency Wb of oscillation is proportional to the square root of the wave amplitude, and if Ixl <_ 2Wb the particle and wave effectively are in resonance. Hence, the exact resonance condition x = 0 does not have to be satisfied in order for a strong wave-particle interaction to occur.
This brings us to resonance overlap, which is what yields large average energy gains. To understand overlap, consider two neighboring waves, i and i + 1, where i + 1 will resonate with a particle of higher energy than i will. A particle initially resonant with wave i will periodically gain and lose a small amount of vii. If the gain at some time is large enough to allow it to satisfy ]x I <_ 2Wb,i+l, where Wb,i+l is the bounce frequency for wave i + 1, then the particle will resonate with that wave next. After "jumping" from one wave to the next in this manner, the particle will have achieved a net gain in energy. If other waves are present that will resonate with even higher energy particles, the particle will continue jumping from resonance to resonance and achieve a m a x i m u m energy corresponding to the last resonance present. If the wave spectrum is discrete, then the spacing of waves is critical; however, if the spectrum is continuous (as is almost certainly the case in flares, and is the case in our model below), then resonance overlap will automatically occur. Of course, the particle can also move down the resonance ladder, but over long timescales there is a net gain in energy and stochastic acceleration is the result. An important aspect of stochastic acceleration in general is that it is not directed, as with DC electric fields (e.g. Miller et al. 1997). This allows cospatial return currents to form, which draw particles up from the denser and cooler chromosphere, ensure charge neutrality, and provide the replenishment for the acceleration region that is necessary in order to sustain the huge fluxes of hard X-rays and g a m m a rays that also accompany these events (see Miller et al. 1997). We incorporate this into our model, which consists of just a few elements:
1. (Assumption) During the primary flare energy release phase, we suppose that long wavelength, lowamplitude ( S B / B << 1) MHD Alfv~n and fast mode waves are excited. 2. (Some fact, some assumption) These waves then cascade in a Kolmogorov-like fashion to smaller wavelengths (e.g. Verma et al. 1996), forming a power-law spectral density. 3. (Fact) W h e n the mean wavenumber of the fast mode waves has increased sufficiently, the transittime acceleration rate for super Alfv~nic electrons can overcome Coulomb energy losses, and these electrons are accelerated out of the thermal distribution and to relativistic energies (Miller, LaRosa, & Moore 1996). As the Alfv~n waves cascade to higher wavenumbers, they can cyclotron resonate with progressively lower energy ions. Eventually, they will resonate with ions in the tail of the thermal distribution, which will then be accelerated to relativistic energies as well (Miller & Roberts 1995). 4. (Fact) W h e n the ions become superAlfv~nic (above ~ 1MeV nucleon-I), they too can suffer transittime acceleration by the fast mode waves and will receive an extra acceleration "kick." - 390 -
Heavy Ion Acceleration in Solar Flares
Regarding item 3, the Alfv@n waves will encounter Fe first, since it has the lowest gyrofrequency. (We do not discuss Kr and Xe here since these have not been incorporated into our simulation yet, but the same general scenario should apply to them as well.) Iron will be strongly accelerated but is not abundant enough to damp the waves. Thus, some wave energy will cascade to higher frequencies where it encounters Ne, Mg, and Si. The same way, these ions suffer strong acceleration, but the wave dissipation is not complete. Some wave energy then cascades to reach 4He, C, N, and O. Thus, iron will resonate with the most powerful waves; Ne, Mg, and Si will resonate with waves having less power; and 4He, C, N, and O will resonate with even less powerful waves. Hence, Fe should be enhanced more than Ne, Mg, and Si relative to 4He, C, N, and O. Since 4He, C, N, and O all have the same cyclotron frequency, they should not be enhanced relative to each other. In this way, cascading turbulence is qualitatively able to account for the observed abundances at, and below, the 4He cyclotron frequency ~4. We investigate this process quantitatively with a self-consistent quasilinear simulation, in which the two wave species are evolved with nonlinear diffusion equations and each particle species (electrons plus all the ion species) is described by a Fokker-Planck equation in energy space. We take into account the nonlinear cascading of the turbulence, its damping on the particles, the accompanying acceleration of the particles, their escape from the acceleration region, Coulomb losses, and replenishment by a cospatial return current. The simulation is also basically a two-parameter model, the two parameters being the turbulence injection rate and the acceleration region length. Of course, there are other parameters that must be specified, but variances in these have the same effect as varying either one (or both) of the above two quantities. We assume that the particles suffer no further acceleration once they escape from the flare site, and that interplanetary transport does not further alter the energy distributions. RESULTS AND CONCLUSIONS We cannot consider all the model results here, but do note that it is capable of accounting, simultaneously, for typical ion and electron fluxes, maximum energies, and acceleration timescales (Miller et al. 1997), as well as the energy integrated abundance ratios given in Table 1. We focus here on just one aspect of these results: namely, the agreement between the theoretical ion distributions and those observed in space. In Figure 1, we show the calculated distributions of 4He, O, Fe, and Ne obtained from our cascading turbulence model, along with the observed distributions. It is important to remember that the relative normalizations of the model spectra are fixed by the simulation, as are the spectral shapes. All that is varied in order to fit the data is the absolute normalization of one distribution. This is excellent agreement for the cascading turbulence model, which, in addition to being able to account for the bulk properties of the energetic particles, is now able to explain as well the detailed ion distributions observed in space. We conclude by discussing the one ion that has been left out thus far: namely, H. It is interesting that the most abundant ion may have the most convoluted acceleration. In the present version of our simulation, we assume that the Alfv~n wave dispersion relation was not affected by ions heavier than H. T h a t is, while these heavy ions were able to damp the cascading waves, they did not affect the dispersion properties of the plasma; in this case, the dispersion relation continued up to the H cyclotron frequency, and H could be directly accelerated out of the background by the cascading waves. This was a simplifying assumption, since the more realistic treatment (below) is quite involved. At any rate, we obtain a suppression of H acceleration due to wave absorption by 4He that is qualitatively in agreement with the observations. reality, the plasma dispersion relation is not this simple, since 4He is 10% as numerous as H and will significantly alter the dielectric tensor of the plasma. (All other heavy ions, which are much less numerous, can still be treated as test particles [i.e., neglected] as far as dispersion is concerned.) Specifically, as the wave frequency w increases from the MHD regime, the Alfv~n branch of the dispersion relation has a resonance, or "stops" at ~ 4 -- 0 . 5 ~ H . It begins again at the cutoff frequency of about 0.6~g and continues up to ~H, where there is another resonance that finally ends the Alfv~n branch altogether; the waves on this section of the Alfv@n branch are commonly called H + electromagnetic ion cyclotron (EMIC) waves.
In
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J.A. Miller Since the cascading waves on the Alfv@n branch are generated at very low frequencies, they will now cascade only up to f~4, since this is where the dispersion relation terminates. There will consequently be no wave energy cascading up to the H cyclotron frequency ~H now. Protons will thus not be stochastically accelerated directly by the cascading Alfv@n waves. However, fast mode waves, which accompany the Alfv@n waves in the turbulent cascade, have a different polarization, do not interact with the background thermal ions, and are not totally damped by the background electrons until they reach frequencies somewhat above f~H. The initial situation which should develop is thus one in which there is a near equipartition between the two wave species below ~4 but only fast mode waves above Ft4. As a result of mode-mode conversion, there will be a flow of energy from the fast mode branch to the EMIC branch, where these waves will then cyclotron resonate with and accelerate protons. This scenario qualitatively explains the enhanced 4He/H ratio: 4He is directly accelerated by the cascading waves, but H is accelerated only after an intervening nonlinear process occurs, which should decrease its overall efficiency. This qualitative picture needs further investigation, but roughly accounts for H acceleration in a realistic solar flare plasma. ACKN OWLED G EMENT S This work was supported by NASA grants NAG5-8480 and NAG5-4608. REFERENCES Barbosa, D.D., ApJ, 233, 383 (1979). Eichler, D., ApJ, 229, 413 (1979). Karimabadi, H. et al., JGR, 97, 13853 (1992). Fisk, L.A., ApJ,224, 1048 (1978). Great Debates in Sp. Phys. 1995 (articles by Miller, Hudson, and Reames), Eos Trans. AGU, 76(41), 401 (1995). Mason, G.M. et al., ApJ, in press (2002). Miller, J.A., ApJ, 491,939 (1997). Miller, J.A., and Vifias, A.F., ApJ, 412, 386 (1993). Miller, J A., and Roberts, D.A., ApJ, 452, 912 (1995). Miller, J A., LaRosa, T.N., and Moore, R.L., ApJ, 461,445 (1996). Miller, J.A. et al., JGR, 102, 14631 (1997). Ramaty, R., and Murphy, R.J. 1987, Space Sci. Rev., 45, 213 (1987). Reames, D.V., ApJS, 73, 235 (1990). Reames, D.V., ApJ Letters, 540, L l l l (2001). Reames, D.V., Meyer, J.P., and von Rosenvinge, T.T., ApJS, 90, 649 (1994). Share, G.H., and Murphy, R.J., ApJ, 485, 409 (1997). Temerin, M., and Roth, I., ApJ Letters, 391, L105 (1992). Verma, M.K., et al., JGR, 101, 21619 (1996).
- 392-
IN SOLAR FLARES
J. A. Miller
University of Alabama in Huntsville, Department of Physics, Huntsville, AL 35899, USA ABSTRACT Stochastic acceleration by cascading MHD turbulence is able to account for a wealth of impulsive solar flare energetic particle properties. Here we show that it can also account for the interplanetary heavy ion distributions from the August 7 1999 flare, observed by the Advanced Composition Explorer.
INTRODUCTION The acceleration of particles to high energies is an ubiquitous phenomenon at sites throughout the Universe. Impulsive solar flares offer one of the most impressive examples anywhere, releasing up to 1032 ergs of energy over timescales of several tens of seconds to several tens of minutes. Much of this energy is spent on accelerating ambient electrons and ions to suprathermal and even relativistic energies, which then either remain trapped at the Sun or escape into interplanetary space. The radiations from the trapped particles (Ramaty & Murphy 1987) are important signatures of the acceleration mechanism. However, the particles that escape into space (Reames 1990) and subsequently are detected there are perhaps the most direct (assuming there is no appreciable further acceleration away from the Sun) and important diagnostic, and are refered to as solar energetic particles (SEPs). The observed SEP abundances from impulsive flares are illustrated in Table 1, which shows specifically the abundance ratios for particles >~ 1 MeV nucleon -1 (data from the "Great Debate" (1995) and Reames et al. references therein, Share & Murphy 1997, Reames 2001). (The ions labeled Kr and Xe are actually groups of ions with mean mass numbers of 85 and 128, respectively.) The abundances for the gradual events are essentially the same as those in the ambient corona, and this is the basic evidence for the shock accelerated nature of these particles. However, the abundances for the impulsive events show dramatic enhancements in several of the ratios. Namely, (1) 3He is enhanced by a factor of ~ 2000 relative to 4He; (2) 4He is enhanced by a factor of ,~ 5 relative to H (or H is suppressed by a factor of ~ 5 relative to 4He); (3) Ne, Mg, Si, Fe, and especially Kr and Xe are enhanced by progressively increasing factors relative to O; and (4) C, N, O, and 4He are not enhanced relative to one another. Also shown is the charge-to-mass ratio Q/A for the first ion (the "numerator") in the ratio, for a temperature of 3 • 106 K. At this temperature, O is fully ionized and has a Q/A of 0.5, like 4He, C, and N. Note that Q/A is also equal to that ion's cyclotron frequency in units of the H cyclotron frequency gtg. One trend is immediately apparent: below the 4He, C, N, and O cyclotron frequency of 0.5~H, the abundance enhancement of an energetic ion species is inversely proportional to its cyclotron frequency. Note that 4He is also enhanced relative to H, but the degree of this enhancement is somewhat larger than the trend of the heavier ions would suggest. Another curious point is the magnitude of the 3He enhancement, which is even more curious due to the location of its cyclotron frequency between that of H and 4He. If one assumes an acceleration mechanism that has an efficiency that increases with increasing Q/A, H should be enhanced relative to 4He, which is not the case; on the other hand, if the efficiency increases with decreasing Q/A, - 387-
J.A. Miller Ratio H/4He 3He/4He 4He/O C/O N/O Ne/O Mg/O Si/O Fe/O Kr/O Xe/O
Q/A 1 0.67 0.5 0.5 0.5 0.42 0.42 0.42 0.26 0.13 0.11
Table 1. Ion Abundance Ratios Impulsive Flares Gradual Flares (Corona) ~2 (• decrease) ~10 ~1 (• increase) ..~ 0.0005 ~46 ~55 ~0.436 ~0.471 ~0.153 ~0.128 ~0.416 (• increase) ~0.151 ~0.413 (• increase) ~0.203 ~0.405 (• increase) ~0.155 ~1.234 (• increase) ~0.155 (• 100 increase) (• 1000 increase)
then 4He should be enhanced relative to 3He, which is also clearly not the case. A second point regarding 3He is that its energy distribution is often quite different from that of the other ions in interplanetary space (e.g. Mason et al. 2002), which are in turn quite similar. It should be emphasized that these enhancements are all valuable diagnostics of the impulsive flare acceleration mechanism. Of course, the Q/A values for the ions depend upon the plasma temperature, and the above discussion is predicated upon a temperature of 3 • 106 K. However, it has been argued (Reames, Meyer, & v o n Rosenvinge 1994) that the initial plasma temperature must in fact be 3 • 106 K; otherwise, the Q/A values become either (1) interlaced in a manner which precludes any pattern at all, or (2) equal, in which case there is no known mechanism that can preferentially accelerate one ion over another. The simplest overall explanation for these observations is that 3He is subjected to an additional acceleration, from which the other ions are immune. The only known way this could occur is if 3He were stochastically accelerated by cyclotron resonance with waves that are excited in a narrow frequency range just around the 3He cyclotron frequency. This basic idea was originally proposed by Fisk (1978), and subsequent models just differ in the specific wave mode that is employed (e.g., cf. Fisk with Temerin & Roth 1992, and Miller & Vifias 1993). However, regardless of the specific model, 3He should be considered essentially as proof that stochastic acceleration at least operates in impulsive flares. THE MODEL Given that stochastic acceleration is known to at least occur in impulsive flares, we advocate it as the mechanism whereby all ions and electrons are energized from thermal to relativistic energies. In our model, we suppose that everything (including 3He) is stochastically accelerated by broad-band MHD shear Alfv~n and fast mode waves, but that 3He is also subjected to some extra acceleration via one of the models referenced above. This extra stochastic acceleration is assumed to be responsible for the spectral difference in the 3He energy distribution as well as its huge enhancement, and will not be discussed further here. Stochastic acceleration relies upon wave-particle resonance, and occurs when the condition x - w - kllv[i-
gl~21/~ = 0 is satisfied. Here, vii , "y, and 9t are the particle's parallel speed (with respect to the ambient magnetic field /30), Lorentz factor, and cyclotron frequency; while w and kll are the wave frequency and parallel wavenumber. The quantity x is the frequency mismatch parameter. If the harmonic number g ~ 0, its sign depends upon the sense of rotation of the wave electric field and the particle in the plasma frame: if both rotate in the same sense (right or left handed) relative to B0, then g > 0; if not, then g < 0. When x = 0, the frequency of rotation of the wave electric field is an integer multiple of the frequency of gyration of the particle in its guiding center frame, and the sense of rotation is the same. The particle thus sees an -
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Heavy Ion Acceleration in Solar Flares
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Fig. 1. Model fits (dotted lines) to the interplanetary 4He, O, Fe, and Ne distributions from the August 7 1999 flare (Mason et al. 2002). Data (connected dots with error bars) is obtained from ULEIS and SIS on the Advanced Composition Explorer. The model distributions are from the cascading turbulence simulation, with an injection of 5 1 e r g s c m -3 s -1 of turbulence into a region of size 108 cm; the magnetic field is 100G and the plasma density is 101~ cm -3.
electric field for a sustained length of time and will be either strongly accelerated or decelerated, depending upon the relative phase of the field and the gyromotion. This is called cyclotron or gyroresonance. If g = 0, there is matching between the parallel motion of the particle and the wave parallel electric or magnetic field. This is called L a n d a u or Cherenkov resonanae. The Alfv~n waves posses a transverse left-hand polarized electric field, which can resonate with the ions via the g = +1 (or cyclotron) resonance. We see from the resonance condition and the Alfv~n wave dispersion relation that in order for a wave to cyclotron resonate with a low-energy ion (say near the thermal speed), w must be near ft. On the other hand, as the ion gains energy, w can become much less than g/. Hence, a broad-band spectrum of Alfv~n waves extending up to fl can stochastically cyclotron accelerate ions from the tail of the thermal distribution to high energies (see also Barbosa 1979, Eichler 1979). On the other hand, Alfv~n waves are not able to accelerate electrons from the background.
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J.A. Miller The fast mode waves posses a compressive magnetic field, which can couple with either ions or electrons via the g = 0 (or Landau) resonance. We see from the resonance condition and the fast mode dispersion relation that (1) a wave of any w will only Landau resonate with a particle having a speed v greater than the Alfv6n speed VA, and (2) as the energy increases the wave propagation angle must approach 90 ~ Hence, fast mode waves having a distribution of propagation directions can resonate with particles from Alfv~nic to relativistic energies, and this leads specifically to stochastic transit-time acceleration (Miller 1997). Now, initially in a flare plasma of temperature T ~ 3 x 106 K, only electrons will be present in large numbers above va, and so these particles will be preferentially accelerated by the fast mode waves. However, should the ions become super Alfv~nic (e.g. as a result of cyclotron acceleration by Alfv~n waves), then they too will be transit-time accelerated. Lastly, even though fast mode waves of any w can resonate with super Alfv~nic particles, the acceleration rate is proportional to their wavenumber or frequency. When a particle is in resonance with a single small-amplitude wave, vii executes approximate simple harmonic motion about that parallel velocity which exactly satisfies the resonance condition (Karimabadi et al. 1992). There is no energy gain on average. The frequency Wb of oscillation is proportional to the square root of the wave amplitude, and if Ixl <_ 2Wb the particle and wave effectively are in resonance. Hence, the exact resonance condition x = 0 does not have to be satisfied in order for a strong wave-particle interaction to occur.
This brings us to resonance overlap, which is what yields large average energy gains. To understand overlap, consider two neighboring waves, i and i + 1, where i + 1 will resonate with a particle of higher energy than i will. A particle initially resonant with wave i will periodically gain and lose a small amount of vii. If the gain at some time is large enough to allow it to satisfy ]x I <_ 2Wb,i+l, where Wb,i+l is the bounce frequency for wave i + 1, then the particle will resonate with that wave next. After "jumping" from one wave to the next in this manner, the particle will have achieved a net gain in energy. If other waves are present that will resonate with even higher energy particles, the particle will continue jumping from resonance to resonance and achieve a m a x i m u m energy corresponding to the last resonance present. If the wave spectrum is discrete, then the spacing of waves is critical; however, if the spectrum is continuous (as is almost certainly the case in flares, and is the case in our model below), then resonance overlap will automatically occur. Of course, the particle can also move down the resonance ladder, but over long timescales there is a net gain in energy and stochastic acceleration is the result. An important aspect of stochastic acceleration in general is that it is not directed, as with DC electric fields (e.g. Miller et al. 1997). This allows cospatial return currents to form, which draw particles up from the denser and cooler chromosphere, ensure charge neutrality, and provide the replenishment for the acceleration region that is necessary in order to sustain the huge fluxes of hard X-rays and g a m m a rays that also accompany these events (see Miller et al. 1997). We incorporate this into our model, which consists of just a few elements:
1. (Assumption) During the primary flare energy release phase, we suppose that long wavelength, lowamplitude ( S B / B << 1) MHD Alfv~n and fast mode waves are excited. 2. (Some fact, some assumption) These waves then cascade in a Kolmogorov-like fashion to smaller wavelengths (e.g. Verma et al. 1996), forming a power-law spectral density. 3. (Fact) W h e n the mean wavenumber of the fast mode waves has increased sufficiently, the transittime acceleration rate for super Alfv~nic electrons can overcome Coulomb energy losses, and these electrons are accelerated out of the thermal distribution and to relativistic energies (Miller, LaRosa, & Moore 1996). As the Alfv~n waves cascade to higher wavenumbers, they can cyclotron resonate with progressively lower energy ions. Eventually, they will resonate with ions in the tail of the thermal distribution, which will then be accelerated to relativistic energies as well (Miller & Roberts 1995). 4. (Fact) W h e n the ions become superAlfv~nic (above ~ 1MeV nucleon-I), they too can suffer transittime acceleration by the fast mode waves and will receive an extra acceleration "kick." - 390 -
Heavy Ion Acceleration in Solar Flares
Regarding item 3, the Alfv@n waves will encounter Fe first, since it has the lowest gyrofrequency. (We do not discuss Kr and Xe here since these have not been incorporated into our simulation yet, but the same general scenario should apply to them as well.) Iron will be strongly accelerated but is not abundant enough to damp the waves. Thus, some wave energy will cascade to higher frequencies where it encounters Ne, Mg, and Si. The same way, these ions suffer strong acceleration, but the wave dissipation is not complete. Some wave energy then cascades to reach 4He, C, N, and O. Thus, iron will resonate with the most powerful waves; Ne, Mg, and Si will resonate with waves having less power; and 4He, C, N, and O will resonate with even less powerful waves. Hence, Fe should be enhanced more than Ne, Mg, and Si relative to 4He, C, N, and O. Since 4He, C, N, and O all have the same cyclotron frequency, they should not be enhanced relative to each other. In this way, cascading turbulence is qualitatively able to account for the observed abundances at, and below, the 4He cyclotron frequency ~4. We investigate this process quantitatively with a self-consistent quasilinear simulation, in which the two wave species are evolved with nonlinear diffusion equations and each particle species (electrons plus all the ion species) is described by a Fokker-Planck equation in energy space. We take into account the nonlinear cascading of the turbulence, its damping on the particles, the accompanying acceleration of the particles, their escape from the acceleration region, Coulomb losses, and replenishment by a cospatial return current. The simulation is also basically a two-parameter model, the two parameters being the turbulence injection rate and the acceleration region length. Of course, there are other parameters that must be specified, but variances in these have the same effect as varying either one (or both) of the above two quantities. We assume that the particles suffer no further acceleration once they escape from the flare site, and that interplanetary transport does not further alter the energy distributions. RESULTS AND CONCLUSIONS We cannot consider all the model results here, but do note that it is capable of accounting, simultaneously, for typical ion and electron fluxes, maximum energies, and acceleration timescales (Miller et al. 1997), as well as the energy integrated abundance ratios given in Table 1. We focus here on just one aspect of these results: namely, the agreement between the theoretical ion distributions and those observed in space. In Figure 1, we show the calculated distributions of 4He, O, Fe, and Ne obtained from our cascading turbulence model, along with the observed distributions. It is important to remember that the relative normalizations of the model spectra are fixed by the simulation, as are the spectral shapes. All that is varied in order to fit the data is the absolute normalization of one distribution. This is excellent agreement for the cascading turbulence model, which, in addition to being able to account for the bulk properties of the energetic particles, is now able to explain as well the detailed ion distributions observed in space. We conclude by discussing the one ion that has been left out thus far: namely, H. It is interesting that the most abundant ion may have the most convoluted acceleration. In the present version of our simulation, we assume that the Alfv~n wave dispersion relation was not affected by ions heavier than H. T h a t is, while these heavy ions were able to damp the cascading waves, they did not affect the dispersion properties of the plasma; in this case, the dispersion relation continued up to the H cyclotron frequency, and H could be directly accelerated out of the background by the cascading waves. This was a simplifying assumption, since the more realistic treatment (below) is quite involved. At any rate, we obtain a suppression of H acceleration due to wave absorption by 4He that is qualitatively in agreement with the observations. reality, the plasma dispersion relation is not this simple, since 4He is 10% as numerous as H and will significantly alter the dielectric tensor of the plasma. (All other heavy ions, which are much less numerous, can still be treated as test particles [i.e., neglected] as far as dispersion is concerned.) Specifically, as the wave frequency w increases from the MHD regime, the Alfv~n branch of the dispersion relation has a resonance, or "stops" at ~ 4 -- 0 . 5 ~ H . It begins again at the cutoff frequency of about 0.6~g and continues up to ~H, where there is another resonance that finally ends the Alfv~n branch altogether; the waves on this section of the Alfv@n branch are commonly called H + electromagnetic ion cyclotron (EMIC) waves.
In
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J.A. Miller Since the cascading waves on the Alfv@n branch are generated at very low frequencies, they will now cascade only up to f~4, since this is where the dispersion relation terminates. There will consequently be no wave energy cascading up to the H cyclotron frequency ~H now. Protons will thus not be stochastically accelerated directly by the cascading Alfv@n waves. However, fast mode waves, which accompany the Alfv@n waves in the turbulent cascade, have a different polarization, do not interact with the background thermal ions, and are not totally damped by the background electrons until they reach frequencies somewhat above f~H. The initial situation which should develop is thus one in which there is a near equipartition between the two wave species below ~4 but only fast mode waves above Ft4. As a result of mode-mode conversion, there will be a flow of energy from the fast mode branch to the EMIC branch, where these waves will then cyclotron resonate with and accelerate protons. This scenario qualitatively explains the enhanced 4He/H ratio: 4He is directly accelerated by the cascading waves, but H is accelerated only after an intervening nonlinear process occurs, which should decrease its overall efficiency. This qualitative picture needs further investigation, but roughly accounts for H acceleration in a realistic solar flare plasma. ACKN OWLED G EMENT S This work was supported by NASA grants NAG5-8480 and NAG5-4608. REFERENCES Barbosa, D.D., ApJ, 233, 383 (1979). Eichler, D., ApJ, 229, 413 (1979). Karimabadi, H. et al., JGR, 97, 13853 (1992). Fisk, L.A., ApJ,224, 1048 (1978). Great Debates in Sp. Phys. 1995 (articles by Miller, Hudson, and Reames), Eos Trans. AGU, 76(41), 401 (1995). Mason, G.M. et al., ApJ, in press (2002). Miller, J.A., ApJ, 491,939 (1997). Miller, J.A., and Vifias, A.F., ApJ, 412, 386 (1993). Miller, J A., and Roberts, D.A., ApJ, 452, 912 (1995). Miller, J A., LaRosa, T.N., and Moore, R.L., ApJ, 461,445 (1996). Miller, J.A. et al., JGR, 102, 14631 (1997). Ramaty, R., and Murphy, R.J. 1987, Space Sci. Rev., 45, 213 (1987). Reames, D.V., ApJS, 73, 235 (1990). Reames, D.V., ApJ Letters, 540, L l l l (2001). Reames, D.V., Meyer, J.P., and von Rosenvinge, T.T., ApJS, 90, 649 (1994). Share, G.H., and Murphy, R.J., ApJ, 485, 409 (1997). Temerin, M., and Roth, I., ApJ Letters, 391, L105 (1992). Verma, M.K., et al., JGR, 101, 21619 (1996).
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