Particle acceleration at a dynamic termination shock



[1] We investigate the effects of a large-scale interplanetary disturbance, such as a merged interaction region (MIR), on the properties of the heliospheric termination shock and particle population accelerated at the shock. The interaction process between transient structures and the shock is reflected in energetic particle spectra through variability in shock location, its compression ratio, and particle mean free paths that control their acceleration rate. It is shown that particle distributions observed at a specific moment of time exhibit features, such as spectral breaks, from earlier acceleration stages with lowest energy particles produced most recently and the more energetic anomalous cosmic rays (ACRs) accelerated at earlier times. We also show that a MIR collision can produce an extended period of anomalously low radial velocity in the heliosheath if density and velocity variations are correlated within a MIR and a following rarefaction.

1. Introduction

[2] The solar wind (SW) is a highly unsteady plasma flow with both density and velocity varying over multiple time scales. During periods around solar maxima the SW plasma flow in the outer heliosphere is dominated by large scale propagating structures (MIRs) involving dynamic pressure variations of ∼5–10 over a few months. Density and speed within MIRs are positively correlated [Richardson et al., 2003]. The two Voyager spacecraft were exposed to high levels of solar activity since the year 2000 and most of the observations pertaining to the TS crossing were performed during the period dominated by MIRs.

[3] It is generally recognized that the heliospheric termination shock (TS) is moving toward or away from the Sun in response to changes in the SW dynamic pressure. Transient structures are expected to result in rapid TS motion with speeds reaching 100 km/s [Barnes, 1993; Naidu and Barnes, 1994; Story and Zank, 1995]. The latter is particularly interesting because the TS was apparently moving toward the Sun when Voyager 1 crossed it at the end of 2004 and the subsequent low velocity flow inferred for the heliosheath [Burlaga et al., 2005; Krimigis et al., 2005] implied the shock was moving inward with speeds ∼100 km/s. In addition to position, TS compression ratio and magnetic field (both the mean and the turbulent components) up- and downstream of the shock also vary due to MIRs. As we show below, this has important consequences for acceleration and propagation of energetic particles produced at the shock.

[4] In this Letter we introduce a simple 1-dimensional model to study charged particle acceleration at a TS that is subjected to a collision with an MIR. We focus our attention on intermediate and high energies of the ion spectra (1 MeV and higher for protons) that are presumably diffusively accelerated at the shock. Lower energy particles are probably energized by a different mechanism, e.g., shock surfing [Zank et al., 1996; Lee et al., 1996] or stochastic acceleration [Fisk and Gloeckler, 2006]. Using a model that contains most of the essential physics of TS acceleration and SW modulation, we show that variations in spectral slopes are to be expected if the shock is moving and evolving. Some of the spectral features obtained here, such as multiple power laws, appear to be similar to the Voyager observations [Stone et al., 2005], although caution should be exercised in comparing them at the lowest energies.

[5] It should be noted that Jokipii and Giacalone [2003] suggested an alternative concept featuring a spatially-limited shock incursion, rather than global shock motions on fast time scales discussed below. In both models the same basic physical picture emerges with low-energy particles spectrally separated from the ACRs owing to the former's higher sensitivity to spatial or temporal irregularities in the shock structure. However, we are able to study the lower-energy part of the spectra in more detail here given the more quantitative nature of our model.

2. Model Description

[6] Medium and high energy protons are likely to be accelerated at the TS by the diffusive shock acceleration (DSA) mechanism [e.g., Jokipii, 1986; Florinski et al., 2004]. At energies below 2 MeV, a different acceleration mechanism may be operating that is responsible for producing a w−5 spectra (w being the particle velocity) observed downstream of the TS [Fisk and Gloeckler, 2006]. Here we assume that a population of pre-accelerated pickup ions exists near the shock and that a certain constant fraction of these ions are injected into the DSA process. Low-energy intensities are relatively steady in the heliosheath [Krimigis et al., 2005] therefore our assumption about steady injection is plausible.

[7] As shown in Florinski and Jokipii [2003], DSA at a spherical shock may be studied with a 1D model by placing a perfectly reflective wall upstream of the TS. This was found to produce similar modulation and spectral features obtained from more sophisticated models using spherical geometry because the wall both reflects and cools particles that experience only overtaking collisions in the plasma frame. The background plasma flow structure is obtained by solving a system of gas-dynamic conservation laws inside the simulation domain (60–120) AU with the TS placed initially at rs = 90 AU with a compression ratio s = 2.9. The location of the reflective left boundary (60 AU) was chosen such as to produce a high-energy rollover in the accelerated ion spectrum at approximately the energy where it is observed (∼100 MeV). The plasma component of the model is essentially identical to Story and Zank [1995] and the reader is referred to that paper for details. The following initial upstream flow conditions are used: velocity u0 = 330 km/s, number density n0 = 10−3 cm−3, and temperature T0 = p0/(2n0k) = 5 × 105 K, where p is thermal pressure and k is Boltzmann's constant. A high upstream temperature is required to account for the pickup ion contribution to the thermal energy of the plasma. Initial downstream conditions were calculated from Rankine-Hugoniot relations. The MIR is initiated by increasing either density or velocity at the left (inflow) boundary for a period of 3 months in such a way that the dynamic pressure increased to 2.5 times the initial value (ρ0u02/2) with a subsequent decrease in density (velocity) to 0.4 times the initial dynamic pressure in order to simulate the rarefaction region following an MIR. The remaining two upstream flow parameters (u and p or ρ and p, respectively) were kept fixed at their undisturbed levels. The first scenario produces a pair of contact discontinuities (CDs), while the second produces a forward shock (FS)/reverse shock (RS) pair, followed by a forward rarefaction (FR)/reverse rarefaction (RR) pair. Accordingly, we label the models 2CD and 2S2R, respectively.

[8] It is well known from gas dynamics that a collision with a RS, FR, and a CD with a density increase makes the TS stronger, while a FS, RR, and a CD with a density drop make it weaker. All six types of interaction inject two additional waves into the heliosheath: a CD and a FS or a centered rarefaction wave for shock-shock and shock-CD collisions [Barnes, 1993; Naidu and Barnes, 1994; Story and Zank, 1995], and an entropy wave (EW) and a rarefaction or compression wave for rarefaction-shock collisions. Both our 2CD and 2S2R models involve a total change in dynamic pressure by a factor of 6.25 at the inflow boundary. Figure 1 shows the plasma dynamic pressure that a stationary spacecraft located upstream of the TS at 70 AU heliocentric distance would measure. The compression propagates unchanged in the 2CD model, but evolves in the 2S2R model with the plasma regions confined between FS, RS and FR, RR pairs expanding with time. Superimposed on the plot is the 26-day average of the SW dynamic pressure measured by Voyager 2 (data from the MIT plasma instrument, PI Richardson), time-shifted to allow comparison with the model. The change in dynamic pressure across the modeled structure is similar to the observed MIRs and our two models thus represent two extreme cases with dominant density or velocity variations.

Figure 1.

Dynamic pressure variations at 70 AU in the 2S2R and 2CD models compared with the Voyager 2 plasma instrument data (where t = 0 corresponds to year 2004.1).

[9] Particle acceleration and propagation is described using the 1D Parker transport equation for the phase space density f(x, p),

equation image

where κxx is the diffusion coefficient, and Sinj is the injection term. Interplanetary diffusion is described with a generic relationship of the form

equation image

where we used λ0 = 1.5 AU and α = 1.5. Equation (2) differentiates between low energy particles (T ≪ 2 MeV) that resonantly scatter in the inertial turbulent range and those with high energies that resonate in the energy range [see Zank et al., 2004, and references therein]. In addition, we ran the 2S2R model with a flatter dependence of the mean free path on energy with λ0 = 0.25 AU and α = 0.5. The diffusive mean free path depends on the number density n because of the Faraday's law By = nBy0/n0, where By is the dominant (azimuthal) magnetic field component perpendicular to the flow, and the specific dependence of κxx on the particle's gyroradius [Zank et al., 2004]. Here we use β = 2 to account for a reduction in κxx in compressed regions due to an increase in both the mean magnetic field and the turbulent magnetic variance. Injection occurs monoenergetically at 1 MeV at a fixed rate at the TS only. A steady-state energetic particle distribution (i.e., a power law at the shock) was used as initial condition in our simulations.

3. TS Evolution and Particle Response

[10] Temporal evolution of TS properties is shown in Figures 2a and 2b for the 2CD and 2S2R models, respectively. In both scenarios the TS is first driven outward and then inward in response to prescribed dynamic pressure variations. The amplitude of the shock motion is ±5 AU in both cases. However, in the 2CD case the shock strength increases first during the collision of the leading edge of the MIR and subsequently decreases as the trailing edge passes the TS. In the 2S2R scenario the compression ratio decreases at the onset of the MIR collision, but increases during the FR passage and remains relatively high until the end of the simulation. The plasma speed downstream of the TS (u2) increases first during the MIR collision and drops during the collision with the following rarefaction [see also Jokipii, 2005]. This effect is much more pronounced in the 2S2R model where the plasma actually flows toward the Sun for 50 days following the passage of the FR wave. By comparison, u2 variations are rather modest in the 2CD scenario.

Figure 2.

TS compression ratio s (solid lines), location in AU rs (dashed lines) and downstream velocity u2 (dash-dotted lines) as a function of time for the (a) 2CD and (b) 2S2R models.

[11] Figures 3 and 4 show snapshots of plasma density and particle spectra at the TS at four different moments of time in the 2S2R and 2CD models, respectively. The first plot, early in the simulation, shows the MIR approaching the TS. Particle spectra at this time are still unchanged from the initial conditions and consist of a power law followed by the cooling-induced exponential cutoff. As the MIR hits the shock, two waves (a FS and a CD) are injected into the heliosheath (second plot). Because the density is high at this time, mean free paths are short and acceleration times accordingly small. A consequence of this is that the uniform power paw can be quickly established up to ACR energies and the distribution is devoid of spectral breaks. However, at this and later times, a portion of lower-energy (and, consequently, less mobile) particles becomes trapped between the high-density propagating structures in the heliosheath and the population becomes depleted at the TS. This leads to a separation of the more pervasive high-energy ACR population, which is less affected by this because of the larger diffusive mean free path, from the lower energy population.

Figure 3.

Temporal evolution of the (left) plasma density and (right) particle spectra at the current location of the TS in the 2S2R model. Solid lines refer to α = 1.5 and dashed lines to α = 0.5. The initial α = 1.5 spectrum is shown with dotted lines for comparison.

Figure 4.

Same as Figure 3 but for the 2CD scenario.

[12] After the rarefaction collides with the shock, the density decreases and acceleration rates drop. The third and fourth plots in Figures 3 and 4 show a number of CDs and rarefaction waves propagating through the heliosheath region, interacting with each other and generating more waves in the process. Most importantly, the new power law spectrum is now forming at low energies according to the current shock compression ratio. Because acceleration is slow, intermediate energy intensities are still depressed and the spectra has a characteristic rollover at several MeV. This feature is more prominent in the 2S2R model because the shock is stronger during this time (see Figure 2) and the low-energy spectrum is harder. The rollover gradually shifts to higher energy, but the acceleration time is longer than the duration of a rarefaction (3–4 months) and the spectrum does not reach a steady state. Once the following MIR hits the shock, lower energy particles will be swept away again and the distribution will not form a continuous spectrum all the way to ACR energies until the SW returns to the quiet (solar minimum) state.

[13] The α = 0.5 spectra exhibit similar properties (cf. solid and dashed lines in Figure 3), although the distinction between the populations is somewhat less obvious. This is a consequence of a flatter dependence of κxx on particle energy with less difference between DSA timescales at low and high energies in the model. Finally, some of the spectra (third plots in Figures 3 and 4) are similar to the conceptual drawing of Jokipii and Giacalone [2003] of a two-component spectra at a TS excursion.

4. Discussion and Conclusions

[14] The results of the previous section show that large scale interplanetary disturbances, such as MIRs, could have a profound effect on the TS and accelerated particles, a trend confirmed by observations [Richardson et al., 2005]. We can identify two principal physical effects that influence the particle spectra: (1) changes in the TS compression ratio and (2) variations in the mean free paths (and consequently, acceleration rates) caused by MIRs. Trapping of lower energy particles by propagating structures generated in the heliosheath is another important effect that may be responsible for a clear separation between the high-energy modulated ACR component and the lower-energy power laws. Particles accelerated at the TS during periods dominated by MIRs simply do not have enough time to be accelerated to ACR energies because they are being periodically swept farther into the heliosheath by the oncoming structures.

[15] The model presented is intended to illustrate some aspects of diffusive acceleration at an evolving shock and should not be used directly to interpret Voyager spectra. A 1D model, for example, excludes magnetic field topology and is incapable of accurately predicting low-energy intensities seen by an upstream observer because they strongly depend on field-line connection to the shock. For the same reason we could not include multiple acceleration sites located at different latitudes (e.g., in the ecliptic [see Florinski et al., 2004]) or at different longitudes [McComas and Schwadron, 2006]. Injection rate may also vary in response to changes in properties of the TS which would affect particle intensities as low energies. We do not expect significant changes to the overall picture, however, because injection rate scales as u1 u2 [Zank et al., 1993], which varies by no more than a factor of 2 in our models.

[16] It is plausible, therefore, that several of the spectral features observed by Voyagers, such as spectral breaks and multiple power laws [Stone et al., 2005], are associated with MIR collisions. If this is the case, particle distributions observed by Voyager 2 in the immediate vicinity of the TS during times near solar minimum (i.e., in the absence of MIRs) should differ substantially from Voyager 1 observations and resemble a continuous, rather than a multiple component, spectrum. Another interesting aspect of the problem is TS and particle responses to multiple MIRs, which are expected to increase the separation between the ACR and low-energy components.

[17] The MIR model that included large velocity fluctuations (2S2R model) produced more dramatic variations in TS properties and spectral shapes. Additionally, this model predicted negative plasma velocity in the heliosheath, albeit for a time period shorter than 4 months observed by Voyager 1 [Krimigis et al., 2005]. The downstream velocity was never small in the 2CD model, which can be easily understood from the condition for the TS speed required to attain u2 = 0:

equation image

This condition is more easily met by the 2S2R model during the rarefaction passage because u1 is small and s is large (Figure 2), whereas s becomes small in the 2CD model during this time period (upstream velocity does not vary in that case). Because in real MIRs both density and velocity are varied, the effect is expected to fall in between these two extreme cases.

[18] In conclusion, our model demonstrates the sensitivity of the spectra of energetic particles accelerated at the SW TS to space weather conditions in the outer heliosphere. Evidently, steady state models are inadequate to study particle acceleration at a realistic TS and time-dependent, multidimensional models should be used to investigate dynamic shock acceleration in its full complexity.


[19] This research was supported, in part, by NSF grant ATM-0428880 and NASA grant NNG-06GD43G.