Calculation of Diffusion Coefficient

Diffusive foreshock populations have been measured extensively in the quasi-parallel regime of the terrestrial bow shock. A critical symptom of the diffusive acceleration process thought to accelerate these populations is the spatial decay in the population density away from the bow shock. Comparisons of the spatial profile of the phase space density of these diffuse populations can be used to estimate the diffusion coefficient parallel to the magnetic field for this diffusive process.

The essential components used in this analysis for measurements in the terrestrial foreshock are outlined in Kis et al. (2004) and Kronberg et al. (2009). A spatially resolved phase space density distribution must be obtained at variable distances from the bow shock in the quasi-parallel region of the foreshock. With the measurement of these PSD distributions assembled, one must look at how the PSD at a given energy varies with distance from the bow shock. For a given energy, the PSD will fall off exponentially with increasing distance from the bow shock, i.e.,

𝑓𝑓(𝐸𝐸)~ exp �−_{𝜅𝜅(𝐸𝐸)}𝑥𝑥 𝑉𝑉𝑆𝑆𝑆𝑆

�

(5.1)

, where 𝑓𝑓 is the PSD, x is the distance from the bow shock along the IMF, 𝜅𝜅(𝐸𝐸) is the diffusion coefficient which is a function of energy, and 𝑉𝑉_{𝑆𝑆𝑆𝑆} is the solar wind bulk velocity. Thus if one simply plots the measured phase space density at a given energy against the distance from the bow shock, the e-folding value of this plot will be directly related to the diffusion coefficient.

For our analysis this diffusion coefficient will be estimated in two different ways, with the difference depending on how the PSD profile vs. distance from the bow shock is estimated. In the first method we will look at three diffuse events that have relatively long time duration, all on the order of 1-2 hours. The three events chosen for this

analysis are 1) 2011-11-18T23:00:00 to 2011-11-19T00:00:00, 2) 2012-02-27T10:42:00 to 2012-02-27T12:24:00, and 3) 2012-05-26T16:00:00 to 2012-05-26T17:24:00 (all times in UTC). During these time periods the spacecraft was traveling at a speed of

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roughly 3 km/s, and at distances ranging from 1800 km to 13,000 km from the bow shock (along the IMF direction). These events were then broken into 6 minute time intervals and the proton PSD and mean distance from the bow shock is computed for each of these intervals. The variation of the measured PSD with distance from the bow shock was analyzed for each E/q bin corresponding to the diffuse population. An example of this analysis is shown for two E/q bins in Figure 5.14. In this case we show the E/q steps corresponding to proton velocities of 710 km/s and 1050 km/s. The three different long diffuse event periods are shown in three different colors with errorbars corresponding to uncertainty in the distance from the bow shock and counting uncertainty in the recovered PSD values.

Figure 5.14: The observed phase space density in a single velocity interval as a function of distance from the bow shock. This phase space density is measured for three different diffuse events, as indicated by the different colored data points. In a) the measured phase space densities for the velocity bin centered on 710 km/s are shown. In b) the measure phase space densities for the velocity bin centered on 1050 km/s are shown. In both cases the linear fits to the decay of measured phase space density with increasing distance from the shock are shown.

When presented in a format with a logarithmic y-axis, the e-folding distance at these E/q steps can be read off as the inverse of the slope of a linear fit to the measurements for each event. In Figure 5.14, this best fit line is overplotted on the data points. With the average solar wind velocity calculated for each long event, using the methods of

Gershman et al. (2012) described earlier, the diffusion coefficient can be estimated with (5.1). The results of this calculation for a subrange of the E/q bins analyzed is shown in

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Figure 5.15. The uncertainties shown in this figure are propagated uncertainties in PSD and bow shock distance as shown in Figure 5.14. We have limited the velocity range of our diffusion coefficient recovery to E/q bins that have several 10s of counts and where significant decay in the PSD was observed. For the higher energies steps, as shown in Figure 5.14b, the scatter in the PSD vs bow shock distance due to variability in the magnetic field and counting statistics is large relative to the decay trend we hope to recover. Furthermore at the higher energies the e-folding distance of the density decay can become comparable to the maximum distance that MESSENGER travels away from the bow shock. Observing less than one e-fold in the density of the diffuse population complicates the recovery of the e-folding value.

Figure 5.15: Estimate of the diffusion coefficient based on the rate of decay of the PSD in several velocity bins. The diffusion coefficient is calculated for each of three diffuse events, but only for velocity bins where a clear decrease of the PSD with increasing distance from the bow shock was observed.

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The recovered diffusion coefficients in the range of velocities we sampled are around 1- 2E6 km2/s in the velocity range of 700-800 km/s. This is as high as can be reliably recovered before uncertainties dominate the slope recovery.

As a way to confirm the diffusion coefficients recovered in this first method we also present the recovery of the diffusion coefficient based on about 130 diffuse events. The majority of these events are shorter in duration than the three diffuse events discussed previously. As such we use the entire time period of each event to calculate the average PSD and average distance from the bow shock for each event. Although this method has less temporal resolution and a greater variability in the ambient solar wind conditions between events, it allows us to average over a larger number of observations, lessening the effect of statistical outliers and counting statistics.

Just as for the three diffuse event scenario, we construct a PSD vs. bow shock distance plot for each E/q step of FIPS that corresponds to a diffuse population (see Figure 5.7 for typical diffuse population distribution). Each data point on these plots now corresponds to a single diffuse event observed, instead of multiple time subsets of the same diffuse event. The slopes of these plots are again used to compute the diffusion coefficient at each energy. We have to assume an average solar wind velocity for the 130 diffuse events to estimate the diffusion coefficient, and this is estimated at about 350 km/s from the diffuse events that have valid solar wind measurements. The recovered diffusion coefficients are shown in Figure 5.16.

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Figure 5.16: Calculated diffusion coefficients as a function of observed velocity bin in diffuse events observed upstream of the bow shock. This is a compilation of about 120 diffuse events. Details of the diffusive coefficient calculation can be found in the text.

We see that the diffusion coefficient estimated from this method is about 2E6 km2/s, which is very similar to the value derived from the three diffuse event method. It is curious that the recovered diffusion coefficient is constant over the velocity range we have analyzed, but again this has to do with the sources of variability in the measurement dominating over the actual decay in density of the plasma. To get a feel for how our diffusion coefficients and their energy dependence compare with those made in the terrestrial foreshock we examine the work by Kronberg et al. (2009). The key figure from that work is shown in Figure 5.17, where the diffusion coefficient from protons (and helium) has been estimated over a range of energies.

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Figure 5.17: Adapted from Kronberg et al. 2009. The diffusion coefficient versus energy per charge; black points are for protons and red for helium. The first four points at low energy ranges are obtained using the CIS instrument on Cluster, for comparison. The straight line is a linear fit for the CIS and RAPID data points.

The energy range corresponding to our FIPS measurements is at the lowest energy end of the measurements displayed in Figure 5.17. However, we see that our value of about 2E6 km2/s is quite similar to the values recovered around 10 keV/e in the terrestrial foreshock. As far as the energy dependence of the diffusion coefficient in the terrestrial bow shock, we estimate the slope of the linear relation between the diffusion coefficient and particle energy is about 7E5 (km2/s)/ (keV/e). Thus over the range of energies that we have evaluated diffusion coefficients over (about 3-6 keV/e), if the diffusion coefficient had the same E/q dependence as at Earth, we would expect the diffusion coefficient to vary

by about �7𝐸𝐸5 𝑘𝑘𝑚𝑚2 𝑠𝑠 𝑘𝑘𝑘𝑘𝑘𝑘/𝑘𝑘 ∗ 3 𝑘𝑘𝑘𝑘𝑘𝑘 𝑘𝑘 � ≈ 2𝐸𝐸6 𝑘𝑘𝑚𝑚2

𝑠𝑠 over our measurement range. However, if the

diffusion coefficient varied by this much the e-folding length for the diffuse populations at the higher energy range of our measurements would be about 𝐿𝐿(𝐸𝐸) = 𝜅𝜅(𝐸𝐸)

𝑘𝑘𝑆𝑆𝑆𝑆 ≈

10,000 𝑘𝑘𝑘𝑘, which is comparable to MESSENGER’s max distance from the bow shock (see Figure 5.14). We would expect to have difficulty recovering the true e-folding values of the higher energy portion of the diffuse population if they are much larger than the distance that MESSENGER traverses away from the bow shock. The main reason for this is that variations in the ambient solar wind conditions and errors in the tracing to the bow shock are much larger relative to the density decay we are trying to measure for these higher energy cases. Thus we must confine ourselves to report a single diffusion

coefficient for the E/q range of about 3-6 keV/e of 𝜅𝜅 = 2𝐸𝐸6𝑘𝑘𝑚𝑚2

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with measurements at Earth. Future missions to Mercury by the BepiColumbo spacecraft should carry instruments such as the HEP-ion instrument on the Mercury Magnetospheric Orbiter (Saito et al. 2010) that will measure plasma ions from 30 keV to 1.5 MeV and will be ideally suited to further explore the energy dependence of the diffusion coefficient in the Hermean foreshock.