Particle acceleration by fluctuating electric fields at a magnetic field null point

# Particle acceleration by fluctuating electric fields at a magnetic field null point

Panagiota Petkaki and Alexander L. MacKinnon 1 Physical Sciences Division, British Antarctic Survey, Cambridge, CB3 0ET, UK
DACE/Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK 1221 Physical Sciences Division, British Antarctic Survey, Cambridge, CB3 0ET, UK
DACE/Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK 122
###### Key Words.:
Acceleration of particles – Waves – Sun: flares – Sun: X-rays, gamma rays
offprints: P. Petkaki
###### Abstract

Context:Particle acceleration consequences from fluctuating electric fields superposed on an X-type magnetic field in collisionless solar plasma are studied. Such a system is chosen to mimic generic features of dynamic reconnection, or the reconnective dissipation of a linear disturbance.

Aims:Explore numerically the consequences for charged particle distributions of fluctuating electric fields superposed on an X-type magnetic field.

Methods:Particle distributions are obtained by numerically integrating individual charged particle orbits when a time varying electric field is superimposed on a static X-type neutral point. This configuration represents the effects of the passage of a generic MHD disturbance through such a system. Different frequencies of the electric field are used, representing different possible types of wave. The electric field reduces with increasing distance from the X-type neutral point as in linear dynamic magnetic reconnection.

Results:The resulting particle distributions have properties that depend on the amplitude and frequency of the electric field. In many cases a bimodal form is found. Depending on the timescale for variation of the electric field, electrons and ions may be accelerated to different degrees and often have energy distributions of different forms. Protons are accelerated to -ray producing energies and electrons to and above hard X-ray producing energies in timescales of 1 second. The acceleration mechanism is possibly important for solar flares and solar noise storms but is also applicable to all collisionless plasmas.

Conclusions:

## 1 Introduction

Release of stored magnetic energy via particle acceleration is a characteristic feature of astrophysical plasmas. In the particular case of the Sun, we see this manifested in the catastrophic events of flares, as well as in quieter phenomena like radio noise storms. Similar phenomena are observed in other late-type stars, and similar physics may be involved in understanding a wide variety of astrophysical objects (see e.g. Kuijpers Kuijpers93 , Hanasz and Lesch HanaszLesch ).

The special case of solar flares involves particular challenges to theory. A large fraction (several tens of percent) of the flare energy is manifested initially in the form of fast electrons (accelerated out of the background distribution to 100 keV in about 1 second and to 100 MeV in a few seconds), which reveal their presence by producing bremsstrahlung X-rays (e.g. Miller Miller98 ; MacKinnon MacKinnon06 ). Protons are accelerated in flares to energies of several tens of MeVs in a timescale of one second (Miller Miller98 , Aschwanden Aschwanden02 ). Thus the acceleration of particles is an important part of the energy release process, rather than an energetically unimportant consequence of the flare. Moreover, radio signatures (Type I noise storms, Type III bursts away from flares) testify to particle acceleration at ”quiet” times.

Magnetic reconnection is one of the primary candidate mechanisms for releasing non-potential energy from magnetized plasmas (e.g. Priest and Forbes Priest01 ). The electric field in the current-carrying region also makes it a natural particle accelerator. Collision-dominated sheets will involve the production of some runaway particles, but almost by definition particle acceleration is not a primary consequence of such a situation (e.g. Smith Smith80 ). However, Martens (Martens88 ) gave order-of-magnitude arguments in favor of a collisionless current sheet as both the energy release mechanism and the particle accelerator in flares. Particle acceleration is energetically the primary consequence of such a situation. Collisionless reconnection thus assumes great potential importance in understanding the flare process, particle acceleration, energy conversion and release in astrophysical plasmas generally (Petkaki and MacKinnon Petkaki97 ; Heerikhuisen et al. Heerikhuisen02 ; Hamilton et al. Hamilton03 ; Turkmani et al. Turkmami06 ; Wood and Neukirch Wood05 ; Vainchtein et al. Vainchtein05 ; McClements et al. McClements06 ).

Here we present test particle calculations designed to illuminate the consequences for particle acceleration of dynamic reconnection. We have in mind particularly the picture of Craig and McClymont (Craig91 , Craig93 ), in which a linear disturbance passes through a magnetic configuration containing an X-type neutral point. The disturbance travels non-dissipatively with the local Alfvén speed until it approaches the dissipation region surrounding the neutral point, where the resistive diffusion term in the induction equation becomes important. The wave damps resistively in a few system transit times, with consequences (heating or particle acceleration) determined by the physical nature of the resistivity. Several assumptions were made in Craig and McClymont’s original discussion (linear disturbance, cold plasma limit, 2-D, Ohm’s law including only a scalar resistivity) but this essential picture still affords a qualitative guide in more complex situations (e.g. McClymont and Craig McClymont96 ; Senanayake and Craig SenCraig06 ).

In Petkaki and MacKinnon (Petkaki97 ), we examined the behavior of protons in the presence of electric and magnetic fields obtained from the Craig and McClymont (Craig91 ) analysis. Here we carry out a complementary exercise, studying test particle evolution in the presence of simple fields chosen to mimic generic features of dynamic reconnection. Our aim is to comment on particle acceleration consequences, in a parametric way that does not depend on a particular set of simplifying physical assumptions or boundary conditions. Time-dependence of the electric field is the essential ingredient reflecting the dynamic character of the reconnection. We present examples of distributions resulting from a time-independent electric field for comparison and highlight distinct features of the distributions resulting from dynamic situations.

We use the Craig and McClymont (Craig91 ) linear solution as a qualitative guide for the spatial and temporal form of the electric field. Our adopted field also resembles a linear situation in displaying a time dependence that does not change (i.e. does not develop multiple frequencies, saturate, etc.). Basing our calculations on this linear picture makes it unlikely that they will provide a complete description of what happens in a flare, although they offer useful insight. They may however be particularly relevant to non-flaring particle acceleration, e.g. in solar noise storms, or as part of the explanation of ’quiescent’ radio emission seen from RS CVn binaries (Kuijpers & van der Hulst Kuijpers85 ).

Since we aim to emulate a linear situation we may pick our test particles from an isotropic, homogeneous distribution representing the background. This is in contrast to particle studies of nonlinear reconnection, where consistency demands consideration of the motion of particles into the dissipation region. In most studies particles are injected in two opposite quadrupoles of the X-point and they subsequently are driven, by the drift due to an imposed constant electric field, to cross the nonadiabatic region or miss it depending on their initial conditions (see e.g. Burkhart et al. Burkhart90 ).

Many previous studies of test particle evolution in steady reconnection exist. Here we mention particularly the work of Martin (Martin86 ), which demonstrates that the orbits of such test particles are chaotic, and of Burkhart et al. (Burkhart90 , Burkhart91 ) who iterated from the test particle calculations to construct a self-consistent description of the diffusion region. Recent work studies regular and chaotic dynamics in 3-D reconnecting current sheets (Efthymiopoulos et al. Efthym05 ; Gontikakis et al. Gonti06 ) or studies particle orbits in the presence of 3-D magnetic nulls (Heerikhuisen et al. Heerikhuisen02 ; Dalla and Browning dalla05 ). Particularly relevant here is the exploratory, analytical study of Litvinenko (Litvinenko03 ) which looks at charged particle orbits in an oscillating electric field in a magnetic field containing a neutral line.

The next section gives details of the specific model we adopt in order to study particle acceleration in time-dependent reconnection, while Section 3 describes our results for particle distributions. Section 4 discusses some possible implications of our results.

## 2 Model for Particle Acceleration in fluctuating electric fields

We are going to study the evolution of test particles in the presence of electromagnetic fields chosen to mimic generic features of dynamic reconnection. Time-dependence of the electric field reflects the dynamic character of the reconnection.

### 2.1 Equations of motion

We solve numerically the relativistic equations of motions of test particles (particles are expected to acquire relativistic velocities) in electromagnetic fields and in the observer’s reference frame:

 d\brdt=\bpmγ (1)
 d\bpdt=q(\bE+1c(\bu×\bB)) (2)

where , .

To model the reconnection magnetic field, we adopt an idealized 2-D magnetic field containing an X-type neutral point:

 \bB=B0D(y^x+x^y). (3)

The current density vanishes for this field configuration. The field lines are the solutions of which are hyperbolae . The X-line (neutral line) lies along the z-axis. The field strength depends on position thus:

 |\bB|=B0rD (4)

where Note that this configuration has no natural scale length. Requiring the field to have a value of gauss at a typical active region distance of cm from the neutral point, fixes only gauss We are free to use other considerations to fix one of and independently, as we do below in introducing dimensionless variables. An electric field is imposed in the direction, with spatial and temporal form chosen to mimic qualitative features of dynamic reconnection (see Sec. 2.2).

We normalize distances to and times to the gyroperiod at . We denote the resulting timescales by and for the cases of electrons and protons respectively. As noted above, is as yet undetermined. It turns out to be convenient in this relativistic calculation to choose such that velocities are normalized to the speed of light. This has the consequence that takes different values and for electrons and protons respectively (Petkaki and MacKinnon, Petkaki94 , Petkaki, Petkaki96 ), such that

 Dp=(mpme)12De. (5)

Specifically, with , we find cm and cm. With our choices of and the Lorentz equations (1) and (2) become in three dimensions and in dimensionless units:

 d¯xd¯t = ¯ux=¯pxγ d¯yd¯t = ¯uy=¯pyγ d¯zd¯t = ¯uz=¯pzγ d¯pxd¯t = −ϵ¯x¯uz d¯pyd¯t = ϵ¯y¯uz d¯pzd¯t = ¯E+ϵ(¯x¯ux+¯y¯uy) (6)

where and for protons, for electrons. is the true value of the electric field and is the dimensionless electric field, with subscript taking the values for electrons or for protons. Energies are now normalised to the particle rest mass energy so that kinetic energy in dimensionless units is just . Equations (6) with appropriate initial conditions and a specific form for describe the motion of a particle.

### 2.2 Electric Field

Craig and McClymont (Craig91 ) guide us in adopting a functional form of electric field which allows us to investigate consequences of time-dependence in a parametric way. Their resistively damping, linear disturbance involves a regularly oscillating electric field whose amplitude is greatest in the region near the neutral point where the resistive term of the induction equation becomes important. Far from this region the disturbance is Alfv’enic in character and dominated by the boundary conditions, so that the electric field amplitude always maximises in the central, diffusion region (see also Petkaki and MacKinnon Petkaki97 , Section 3). Thus we adopt the following form for the electric field :

 \bE=E0sin(ωt)^zf(x,y) (7)

where describes the spatial variation of . We take

 f(x,y)=exp(−αi√(|r|)) (8)

where, , , . Consistent with our concentration on particle acceleration near the neutral point and with in situ measurements in Earth’s magnetosphere (Øieroset et al., oieroset01 ), we expect that resistivity will be primarily inertial in character (Speiser Speiser70 ). As shown in Figure 1, this form approximates the radial form of the electric field calculated from the Craig and McClymont (Craig91 ) solution, for an (inertial) resistivity estimated assuming the dominant contribution from 1 keV protons (see Petkaki and MacKinnon Petkaki97 , Section 3; Speiser, Speiser70 ). The form of the electric field is shown in Figure 1 at . The exact Craig and McClymont solution develops more complex spatial structure but the spatial form of Eq. (8) thus embodies a dissipation region on the appropriate length scale.

The frequency of oscillation of the electric field which we denote by is a free parameter. Each simulation uses one value of . We take values of such that , corresponding to a broadband wave spectrum which may propagate in such a system (see Petkaki and Mackinnon Petkaki97 ).

To compare with a simple, unvarying state, we also calculate the energy distributions that result from a constant in time imposed electric field

 \bE=E0^zf(x,y) (9)

where is defined in Equation 8. With its nonzero curl, this assumed form of cannot represent a steady state reconnection. We employ it primarily to provide a simple, unvarying state for comparison with results in the time-dependent situation.

Key to understanding particle behaviour near the neutral point is the ‘adiabaticity’ radius , the distance from the neutral point at which the Larmor radius equals the magnetic field scale length. For , particles move adiabatically. In other words, if the distance of the particle from the neutral point is of the order of its Larmor radius, then the particle is non-adiabatic. The ‘adiabaticity’ radius depends on the particle mass and velocity perpendicular to the magnetic field, , and is given by

For electrons and protons of the same energy, the electron gyroradius is smaller than the proton gyroradius.

### 2.3 Numerical Method

Due to the complexity of the orbits, their calculation cannot be done analytically. For integrating the ordinary differential equations (ODEs) describing the motion of the particles, we use the Bulirsch-Stoer method (Press et al. Press96 ). For a single particle with the same initial conditions, the orbit changes if the accuracy required of the integration routine is varied, and when the particle crosses the neutral point area (for general properties of X-type neutral point orbits see e.g. Martin Martin86 ). The statistical properties of the distribution of test particles, which are of primary interest here, are unaffected by changes in the accuracy required of the integration routine. In the absence of an electric field, the routine conserves particle energy to one part in .

We start the integration of particle orbits at t=0 and with the particles positioned randomly in a box with the following size

 −1.0≤ x0 ≤1.0 (11) −1.0≤ y0 ≤1.0 (12) z0 =0.0 (13)

in electron or proton units depending on the species. We integrate the particle orbits up to 230400 timesteps () for electrons and 5360 () for protons. With and our form of dimensionless units these times correspond to 1 second for electrons and protons. The initial velocities of the particles are picked randomly from a Maxwellian distribution of temperature ( eV), a typical coronal value. We consider only small values for , consistent with the passage of a disturbance in the linear regime (Craig and McClymont, Craig91 ). Values of 0.0001, 0.001 and 0.01 are used in the actual calculation. The value 0.001 corresponds to electric field statvolt/cm. These are moderate values for electric fields present in the solar atmosphere (see Foukal et al., Foukal86 ).

### 2.4 Particle Orbits

We are going to examine a typical proton orbit which is shown in Figure 2. The amplitude of the time-varying electric field is and the frequency is . In Fig. 2a we plot the (dimensionless) energy of the proton as a function of time for the interval 2400-5360 . In Fig. 2b is shown the projection of the same orbit on the X-Y plane and in Fig. 2c the projection of the same orbit on the X-Z plane. Our model assumes a system scale of , corresponding to a typical active region scale of cm. In Fig. 2b we zoom in close to the neutral point to observe the particle orbit in detail and we look in an area of and . In the same figure superposed in dotted line are some of the magnetic field lines showing the structure of the X-type magnetic neutral point. All field lines tend to the separatrices (shown in dashed lines) as the distance from the neutral point becomes very large.

The thick solid circle has radius for a thermal proton. Inside this region the gyroradius (Larmor radius) of most particles is not well defined since the particle is not bound to one magnetic field line and meandering motion is observed. The electric field accelerates or decelerates the proton causing further changes in the particle gyroradius and energy. This behavior resembles a stochastic-type acceleration. Stochasticity is introduced by the phase of the electric field and the phase of the particle orbit and is sustained because of the form of the magnetic field (e.g. Martin Martin86 ). Outside the magnetic neutral point area the particle is moving along a particular magnetic field line. The gyroradius and the parallel velocity decrease as the particle moves away from the neutral point. The particle mirrors and recrosses the non-adiabatic region and the process is repeated until the end of the integration time or until the particle escapes the outer boundary of the system ().

Looking back at Fig. 2a we see the variation of the particle energy as a function of time. The intervals of energy conservation correspond to the times the particle is away from the nonadiabatic region, and in regions where the electric field is decreasing. Changes in the particle energy take place during the crossing of the non-adiabatic region.

The orbits of electrons show similar properties to that of the protons. Additionally from the way we pick our initial conditions the electrons start closer to the neutral point (see Sect. 2.2). But since the form of the electric field for both species is calculated assuming proton inertial resistivity, electrons see an area much larger than their adiabaticity radius where the electric field is close to its maximum value. Consequently some of the electrons start their motion outside their adiabaticity radius and their motion is immediately adiabatic. In this case the presence of the electric field does not increase their energy except if they eventually cross the non-adiabatic region.

The amount of acceleration that particles get depends on the time they spend close to the neutral point, on the phase of the orbit and on the frequency of the electric field. We define crossing time as the time the particle needs to cross the non-adiabatic region (Sec. 2.2) and is given to order of magnitude by

where is the velocity projection in the x-y plane. So,

 tcr∼(2.828cDeBo)1/2m3/4iE1/4. (15)

It turns out that particles with the same energy satisfy

 tcrp=tcre(memp)34=280tcre (16)

where is the proton crossing time and is the electron crossing time. One would expect that in order to get particles effectively accelerated (or decelerated since the sign of the electric field is not constant) we need where is the period of fluctuation of the electric field. Thus we see potential for differences between electron and ion acceleration.

## 3 Energy Distributions of Accelerated Electrons and Protons

We calculate the kinetic energy of each particle up to maximum of a 1 second real time along with final positions and velocities. Particles in our calculation spend a relatively short time close to the neutral point but they get trapped in the magnetic configuration and re-cross the neutral point a number of times (Sect. 2.4). Particles encounter the non-adiabatic region, the process resulting in a Fermi-type acceleration. A similar phenomenon has been noted for a multiple neutral point configuration by Kliem (Kliem94 ), in the behavior of protons in the presence of an MHD disturbance by Petkaki and Mackinnon (Petkaki97 ), and was explored analytically for time-varying electric field by Litvinenko (Litvinenko03 ).

We sample the electric field frequency range for protons and for electrons. In Fig. 3 we plot histograms of the logarithm of the initial and final energy distributions of protons, for magnitude of the electric field and total number of timesteps 5360. Each distribution is generated using 50000 test protons. We also calculated the distributions resulting when the magnitude of the electric field is and . The initial Maxwellian distribution is shown in dotted lines in each panel. Panel (1) shows the distributions for electric field magnitude . We observe no change in the form of the distribution since no acceleration is taking place (see also Sect. 2.3). Panel (2) shows the energy distributions for constant electric field () to provide a comparison with steady state magnetic reconnection. The final energy distribution for has two distinct peaks, one at the initial Maxwellian distribution and a beamlike distribution close to MeV. The energy distributions for constant electric field are not power laws as found elsewhere (see e.g. Bulanov and Sasarov Bulanov76 ; Bruhwiler and Zweibel Bruhwiler92 ). In those former calculations the particles crossed the neutral point only once, whereas particles recross the neutral point numerous times in our model. The frequency of the time-varying electric field increases progressively from in panel (3) to in panel (12). For a small beamlike structure appears at MeV. The final energy distributions are bi-modal from to . For higher frequencies the proton energy distributions do not show significant energy changes.

In Fig. 4 we plot the mean of the logarithm of the initial and final proton energy distributions versus the frequency of the electric field and for three amplitudes of the electric field (dashed star line), (solid star line), and (dotted star line). The mean energy for the constant electric field is represented on this plot by . The same representation is used in Fig. 5 where we plot the standard deviation of the logarithm of the initial and final proton energy distributions versus for the same three amplitudes of the electric field as in Fig. 4. We use the mean value of the logarithm of the energy to better represent the changes in highly non-thermal distributions.

The mean energy increases monotonically with for the constant electric field case and for all frequencies of the electric field except for the highest frequency used in our model. The highest mean energy is achieved for constant electric field for all values of the electric field. For the highest energy gain for the time-varying electric field is achieved when , indicating a resonant acceleration process. For a peak in the mean energy is also present when . For protons gain most energy from the low electric field (see Fig. 4). When protons a peak in the mean energy is present when . For frequency and greater the energy distribution does not change significantly for all values of .

In Fig. 6 we plot histograms of the logarithm of the initial and final energy distributions of electrons, for magnitude of the electric field and total number of timesteps 230480. Again we generate each distribution using 50000 test electrons, and show the initial Maxwellian distribution in dotted lines in each panel. Panel (1) shows the energy distributions for non-varying electric field (). The final energy distribution (shown in solid line) includes a small beamlike component. The lower energy part of the final distribution is Maxwellian-like peaking at kinetic energy MeV, with a small beamlike component superposed at around kinetic energy MeV. The frequency of the time-varying electric field increases progressively from (panel 2) to (panel 12). The bulk of the distribution is accelerated for the frequency range to . Accelerated distribution for (panel 2) is Maxwellian-like peaking around MeV. From (panel 3) to (panel 6) the accelerated distributions have maximum close to MeV with maximum energy close to 1 MeV. From (panel 7) to (panel 9) the accelerated distributions have maximum close to MeV with maximum energy exceeding to 1 MeV. The final energy distributions are bi-modal for to containing a Maxwellian-like part at the energy range of the initial Maxwellian distribution and an accelerated part at higher energies with peak in the range to MeV. For higher frequencies the energy distributions do not show significant energy changes in the time of maximum 1 s.

The electron distribution gains energy for most of the frequencies of the electric field that we used in this model except for the highest frequencies. In Fig. 7 we plot the mean of the logarithm of the electron energy distributions versus for two amplitudes of the electric field (dashed star line), (solid star line). In the same plot we superposed as error the standard deviation in the mean for each distribution. The mean energy for the constant electric field is represented on this plot by . The highest energy gain achieved is for electric field with for amplitude of the electric field . When electrons gain most energy for the constant electric field and for .

In the Tables that follow we summarized the energy gain aspects of the acceleration mechanism. For each particle we find the energy gain using its initial and final kinetic energy, that is:

 ΔKiKiinitial=Kifinal−KiinitialKiinitial (17)

where is the initial energy of the -th particle and is the final energy of the -th particle. , where takes the values 1, 10, 100.

In Table 1 we list the accelerated percentage of the final proton distribution for each electric field frequency and for . The first column lists the frequencies of the electric field. The second, third and fourth columns list the percentage of the final proton distribution for which is greater than 1, 10 and 100 respectively (Eq. 17). The fifth column lists the percentage of the final proton distribution that has energy greater than 1 MeV and the last column list the highest energy in the final proton distribution in MeV. We see that depending on the frequency of the electric field, to of the proton distributions get accelerated to -ray producing energies in 1 s.

In Tables  2 and  3 we list the accelerated percentage of the final proton distribution for each available electric field frequency for and for . The layout of these tables is the same as for Table 1 discussed before.

In Tables 4 and  5 we list the accelerated percentage of the final electron distribution for each available electric field frequency for and for respectively. The first, second, third and fourth columns are layed out as in Table 1. The fifth column lists the percentage of the final electron distribution with energy greater than and the last column lists the highest energy in the final electron distribution in MeV. For and for most frequencies of the electric field (and for constant electric field) the bulk of the electron distributions get accelerated to X-ray producing energies in the timescale of our model. When only small percentage of the electron distribution accelerates to X-ray producing energies except for constant electric field where accelerates to X-ray producing energies and for where accelerates to X-ray producing energies.

Flare fast electrons as revealed by hard X-ray observations generally have energy distributions characterised by energy spectral indices in the range 2 - 5 (Dennis, Dennis85 ). RHESSI data allow a less crude characterisation of the energy distribution (Kontar et al. Kontar05 ), but this range nonetheless gives a reasonable starting point for comparison with our results. With their high-energy peaks, sometimes bimodal in form, many of the distributions shown in Figures 3 and  6 are clearly some way from those implied by observations. The electron distributions of Figure 6 would give very hard photon spectra, harder than usually observed in flares. When segments of the distributions appeared to decline in energy in power-law form , we fit power-laws in energy to them, finding values of between about 1 and 2.5. Although we can account for particle acceleration in this way, to very high energies, we have to appeal to some other agent to redistribute energy among the accelerated particles to be compatible with observations. Very hard energy distributions are also found in most other calculations of acceleration in either one or many dissipation regions (e.g. Turkmani et al., Turkmami06 ).

By fixing the duration of the integrations at 1 second we generate a snapshot of the distributions produced as particle acceleration proceeds. Obviously, particle energies will be less for shorter periods and greater for longer ones. As an illustration, in Figure 8 we plot the time evolution of the logarithm of the mean energy of proton distributions for , as functions of . Mean energy is plotted for 0.25 s (1340 ), 1 s, and 2 s (10720 ). We observe that the greatest changes in mean proton energy take place for the lowest frequencies . At high frequencies, on the other hand, proton mean energy apparently changes little after 1 s. A sort of steady state is approached. Since no particles escape, this indicates a decrease with proton energy of the energy increment experienced on each return to the dissipation region: protons that can be accelerated at all no longer gain much energy after this time.

## 4 Discussion and conclusions

In this work we investigate the particle acceleration consequences of time-varying electric fields superposed on a X-type magnetic field to mimic generic features of dynamic, collisionless reconnection (Craig and McClymont Craig91 ). We have shown that protons and electrons may gain relativistic energies in times 1 s; for plausible (small) electric field amplitudes and active region magnetic fields. This parametric study is meant to be complementary to Petkaki and MacKinnon (Petkaki97 ) where we attempted to tie this test-particle approach self-consistently to an MHD description of the passage of a wave. Although there are some qualitative similarities with the particle behaviour studied analytically by Litvinenko (Litvinenko03 ), his adoption of a spatially uniform electric field makes direct comparison difficult.

Before discussing some consequences of our results, we note some limitations of our calculation. First, this is a test particle approach. Particles do not interact with each other, nor do they influence the background field. In particular, the particle distribution including the accelerated component may well be unstable to growth of various sorts of waves. Obviously such wave growth would influence the motion of particles, but we neglect this possibility. We neglect also radiation losses. In the solar corona this is not a serious neglect (even for 10 MeV electrons the radiative energy loss time is s), but elsewhere in the cosmos it could become significant.

We followed Craig and McClymont (Craig91 ) in assuming a smoothly varying X-type field through the whole of our system. We took the active region lengthscale of cm to define the boundary of the system. It is possible that the field strength increases more rapidly from the neutral point, approaching a constant value at smaller distance. This would reduce both the adiabaticity radii of particles and the characteristic timescale. If we continued to scale the electric field region with the proton adiabaticity radius, nothing would change except that the integration periods correspond to smaller real times. Thus particle acceleration would proceed more rapidly; however, fewer particles would be involved. Further consideration of this question might proceed via study of more realistic configurations including a neutral point or sheet (e.g. Forbes and Priest Forbes95 ; Fletcher & Martens Fletcher98 ; Titov and Démoulin Titov99 ) or complex magnetic field structures (e.g. Malara et al. Malara00 ).

The finite width of the nonadiabatic region allows particles to gain or lose some energy randomly before returning to adiabatic motion. Together with repeated encounters with the dissipation region, the consequence of mirrorings in the extended configuration, this results in a Fermi-type, ’stochastic’ acceleration.

In our model particle acceleration takes place for geometrical reasons. The test particle calculation is numerically simpler than self-consistent approaches (e.g. Vlasov simulations, see Petkaki et al. (Petkaki03 , Petkaki06 ) and gives useful insights to the particle energization process. There is no threshold for this type of acceleration, unlike resonant interaction with low-frequency, MHD waves. The necessity for protons particularly to have threshold energies of around 25 keV is a well known difficulty when such mechanisms are invoked (e.g. Forman et al., Forman86 ). Our results indicate that low-frequency waves may themselves perform the ’first-step’ acceleration, if they propagate in a coronal structure including a neutral point. This may occur independently of, or simultaneously with, the resonant cascade scenario of Miller and Vinas (Miller93 ). Possible difficulties with the number of pre-accelerated particles may be obviated if many neutral points are present, although such a situation obviously needs separate investigation (Kliem Kliem94 ).

Most of the resulting proton distributions have a bi-modal form (see Fig. 3). Electron distributions are also bi-modal for the highest frequencies, (see Fig. 6). Whereas for the lowest frequencies of the electric field the bulk of the initial electron Maxwellian distribution is accelerated, for the highest frequencies only part of the electron distribution is accelerated (Table 4). Acceleration occurs for all frequencies when addressing the proton distributions (Table 1). The bi-modal form of the proton energy distributions might offer a way to have protons of gamma-ray producing energies ( 2 MeV) without the energetically dominant population at lower energies that is the inevitable consequence of a diffusive particle accelerator (see the Appendix of Eichler (Eichler ); and MacKinnon (MacKinnon91 )). Investigation of the velocity space stability of these distributions needs details of the angular distribution at particular points in space, and is not discussed here.

We note the effectiveness of acceleration of the two species varies according to the frequency of oscillation invoked. Electrons are accelerated for a broader spectrum of frequencies. Frequencies 0.001 to 1000 have been simulated here, corresponding to real frequencies in the range 5 Hz to 5 MHz (cf. the frequency range of waves from the base of the solar corona, probably in the range 0.01 Hz to 10KHz, e.g. Marsch et al. (Marsch82 )). Frequencies lower than 0.001 will also accelerate electrons as indicated by the net acceleration achieved for the constant electric field cases (Fig. 7), but frequencies higher than 1000 do not produce a net acceleration in the timescale of our model.

Considered as a function of , the mean energy of the accelerated electron distribution exhibits a peak in the broad range . Such a peak leads us to suspect a resonance involving two or more of the timescales in the problem. The initial gyrofrequencies of electrons lying in the adiabatic portion of the dissipation region also generally lie in this range. Inverse crossing times (, see Equation 14) comparable with might also lead to enhanced acceleration. Using Equation 14, but taking account also of the mean increase in we do indeed find upper limits in the range .

Protons are accelerated for low electric field frequencies, achieving -ray producing energies in 5360 s for frequencies and for and . A local peak in the mean energy of the accelerated proton distribution is seen at . This range of frequencies are comparable to the gyrofrequencies of protons in the adiabatic region for our set of initial conditions and to the proton inverse crossing time.

The variability of the effectiveness of acceleration of the two species according to the frequency of electric field oscillation might bear on the apparent variation of electron/proton ratios in flares (Ramaty & Murphy Ramaty87 ) and the phenomenon of ‘electron-only’ flares (Rieger Rieger89 ). As a general comment, we note that higher frequency disturbances favour electrons over ions, although more definitive statements will need a proper treatment involving a more realistic wave.

For most frequencies and for constant electric field, part of the electron distribution escapes from the system boundaries before 230400 s. Electrons on average escape in less than 0.6 seconds in the frequency range . Protons on the other hand do not escape the system boundaries on the same timescale of 5360 s for and . Electrons are accelerated more rapidly than protons to energies that do not allow them to mirror inside our system boundaries.

Here and in Petkaki & MacKinnon (Petkaki97 ) we investigate particle acceleration at a null in the presence of a linear disturbance. Such calculations may give some insight into particle acceleration in flares, although conditions then presumably depart severely from linearity, but might be most relevant to quiescent, long-lasting phenomena such as radio noise storms. Definitely involving deka-keV electrons (Raulin & Klein, Raulin94 ) and showing correlations with X-ray variations, but without chromospheric, flare-like signatures (Svestka et al Svestka82 ; Crosby et al. Crosby96 ), particle acceleration in noise storms might occur as described here, if the relevant coronal structures include null points. Electrons accelerated at a neutral point will likely encounter very large mirror ratios, trapping them in the corona (Fletcher & Martens Fletcher98 ) and accounting for the exclusively coronal phenomena accompanying noise storms.

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