Reply to comment on “Origins of energetic ions in the cusp” by R. Sheldon, J. Chen, and T. A. Fritz

Abstract

[1] In their comment, Sheldon et al. [2003, hereinafter referred to as SETAL] express their concern about the analysis of Cusp Energetic Particle (CEP) events by Trattner et al. [2001, hereinafter referred to as TETAL]. This study concluded that energetic ions in the cusp are more likely shock accelerated ions, transported into the cusp together with magnetosheath plasma along reconnected field lines. Ions locally accelerated in the cusp by a still undefined and unmodeled process, as suggested by Chen et al. [1998], appear to be less likely to explain the observations. Apart from a typo on the altitude of the CEP observations (the CEP events were indeed observed at higher altitudes, as indicated by SETAL), the concerns in the comment are based on misunderstandings of the methods used by TETAL as well as a lack of understanding of shock acceleration theory. These issues will be clarified in this reply. However, since the comment also addresses several other papers by other authors who also criticize the local acceleration source but are not part of the paper in question, we will focus our reply only on concerns related to TETAL. These concerns can be summarized as follows: (1) reducing the original number of CEP events, (2) the fit of CEP spectra and subsequent calculated moments based on this fit, and (3) correlation with solar wind parameters. We believe that our explanations in this reply will mitigate the concerns voiced by SETAL. [2] SETAL surmise that TETAL “cherry picked” only 53 of their original published 75 Cusp events [Chen et al., 1998] in an attempt to support the bow shock source over the local acceleration source. We have indeed selected only 53 cusp events because our analysis of all CEP events showed that the missing 22 events are not or are only partially in the cusp. Figure 1 shows Polar/TIMAS observations of the CEP event on 27 September 1996, from the list published by Chen et al. [1998]. Plotted are H+ omni-directional flux measurements (1/(cm2 s sr keV/e)) from 2200 UT to 2330 UT. This CEP time interval covers three distinctly different plasma regions in the magnetosphere where Polar was actually on lobe field lines (2200 UT to 2207 UT), a rather large interval on closed field lines (2305 UT to 2330 UT), and the actual cusp crossing (2207 UT to 2305 UT). [3] The cusp is defined by the presence of downward precipitating ions from the magnetosheath [e.g., Reiff et al., 1977] on field lines opened by reconnection at the magnetopause. Definitive evidence for this scenario comes from the fact that the incoming magnetosheath distribution is truncated as it crosses the magnetopause. This truncated distribution with a characteristic “D” shaped distribution has been predicted by Cowley [1982] and observed by, for example, Fuselier et al. [1991a, 2000]. To conclude truthfully on the characteristics of simultaneously observed energetic ions in the cusp [see also Fuselier et al., 2002b] and the origin of these “Cusp” Energetic Particle events the observing satellite should be at least in the cusp. This was not the case for 22 of the events published by Chen et al. [1998]. Those events were therefore excluded from the study. [4] There is a general disagreement about spectral breaks in CEP spectra. TETAL identified characteristic spectral breaks at about 2, 10, and 150 keV/e (see Figure 2 in TETAL) and used 4 maxwellians to describe the spectrum and subsequently calculate density and temperature. As noted in TETAL these characteristic breaks are meant as a generalization with rounded numbers. These three breaks do vary slightly in position, most likely depending on the actual solar wind and bow shock conditions. [5] SETAL suggests that the use of two power laws with a break at about 50 keV is a better method to describe CEP events. It should be noted that the use of maxwellians to fit CEP spectra is not unique as SETAL assumes. For a CEP event on 27 August 1996, Chen and Fritz [2001] (coauthors on the SETAL comment) used “For comparison, a 2-keV/e maxwellian distribution curve…” to fit the low energy end of the CEP spectra. They also identified spectral breaks at 20 keV/e (for He++) and at about 10, 30, and 100 keV for protons. It is difficult to understand why Chen and Fritz, who introduced CEP events, are sometimes identifying the same spectral breaks as TETAL while also describing CEP events sometimes with power laws and a single spectral break at about 50 keV. [6] It should be noted that in Figures 2 and 3 of TETAL only 4 × 2 = 8 parameters are being used, not 12 as implied by SETAL. A single maxwellian that is not drifting has three parameters, density, temperature, and particle mass, but the mass is known in each case, either directly (TIMAS, CAMMICE) or indirectly (HYDRA). [7] The TETAL fitting procedure, exemplified in their Figures 2, 3, and 4, is also not nearly as arbitrary as inferred by SETAL. First of all, as indicated above, the particle mass m is known in each case. Second, with the fitting function being a nondrifting maxwellian, the position and width of a given differential flux peak are uniquely coupled via its temperature. That leaves two “free” parameters per peak, density n and temperature T, in principle. However, by specifying over what energy interval a single maxwellian fit is to be made, its n and T become mutually coupled as well, leaving no free parameter. [9] In each case this procedure did yield numbers for the standard deviations of a and b in the ordinary least-squares fashion and hence for n and T, but these numbers were not collected, being that they were usually small compared with the substantial differences between different sets of data. This may seem careless in retrospect, but we do not think the propagated standard deviations would have added significantly to the fairly large error margins resulting from variations between samplings. [10] One may view the density of a maxwellian component as the product of amplitude and width of its differential flux peak, but that is not how it was derived by TETAL. As outlined above, a much simpler method was used, namely the least-squares fitting of a straight line to a section of the phase space density distribution ln[f(E)]. This yields n and T as two independent numbers characterizing position and slope of that line (again two parameters, not three), and it does so with no assumption but the implicit one that the samplings of ln(f) have a gaussian random distribution about a straight line a + b · E across that limited energy interval. They do in fact characterize distributions and not integral moments of the fitting functions. These distributions are also not “highly dissimilar in shape.” On the contrary, the different distributions f are all continuous and monotonically declining functions of energy. The three or four fits made to the entire distribution f are also mutually independent in a statistical sense, since they are done for different energy intervals. [11] In a physical sense there is a small compound error among the different maxwellians in that each component is derived without subtracting the overlap in f from its neighbors. However, this only affects a few data points at the break between the different slopes, and we have independently determined the corresponding errors are at the 1% level or lower in the respective n and T. In general we believe the absolute errors in n and T are much less than 100%. [12] To address again the different definitions of spectral breaks, it should be noted that the spectral break at around 20 keV is not only separating the shocked solar wind regime from the energetic ions in TETAL, but can be also clearly identified in Figure 3 in SETAL. The SETAL definition of spectral breaks differ from that of TETAL, as do their functional fits, and this may be a cause for some of the differences in physical interpretation. In the TETAL definition, as energy E increases, a “break” occurs where the ion flux J stops falling off according to one maxwellian fit and begins to follow the shape of the next more energetic maxwellian (e.g., Figure 2 in TETAL). This break has in fact been identified by the convex bend between two near-straight line segments in a plot of ln(f) versus E, where the phase space density f is fairly well approximated by a sum of maxwellian components fi = (Ji/E) (m2/2) ∝ exp(−E/kTi). When plotted as ln(Ji) versus ln(E), these same nearly linear segments appear as concave “bumps.” If instead ln{J[ln(E)]} is fitted by a series of straight lines, that is by power-law spectra, as done by SETAL, the breaks are concave and will fall on these “bumps,” that is between the TETAL breaks, one of them at say 40 to 100 keV/e, as in Figure 3 of SETAL. [13] Figures 5 and 6 in TETAL merely report the actual occurrence of ratios of numbers that we believe have much less than 100% error each, but we have not performed a formal statistical test to prove this assertion. [14] For shock accelerated ions at the quasi-parallel bow shock, in principal, an increase in solar wind density causes an increase in the density of shock accelerated ions [see Trattner et al., 1994] while an increase in solar wind velocity causes a increase in the exponential spectral slope of shock accelerated ions as outlined in shock acceleration models [e.g., Ellison, 1981; Lee, 1982]. Figures 7 and 8 in TETAL illustrate the dependency of E0 on solar wind velocity for bow shock accelerated ions and CEPs, respectively, and is therefore a link to the acceleration method. Linear regression is easily performed, and we think the resulting straight-line segments help to illustrate the considerable similarity between the two data sets within the 340 to 620 km s−1 range for which there are data. [15] Figure 3 in SETAL shows spectra for two CEP events with considerably different flux levels for ions >30 keV, which includes the well-known energy range for shock accelerated ions from about 10 keV to 150 keV [e.g., Fuselier, 1994; Trattner et al., 1994]. Since the flux maximum for energetic ions accelerated at the quasi-parallel bow shock is located between 20 keV/e and 40 keV/e, depending on the initial solar wind conditions, we would expect to see a change in the energetic ion spectra at about this energy if conditions at the shock change. For both intervals the solar wind velocity was very similar. This fact allowed SETAL to conclude that shock accelerated particles cannot be the source of these ions. An analysis using all relevant parameters for shock acceleration at the quasi-parallel shock [see also Trattner et al., 1994] shows, however, that this example is a very good case for shock acceleration source. [16] Figures 2 and 3 show the solar wind conditions for the CEP events on 22 September 1996, 0040 to 0051 UT and 0104 UT to 0115 UT, respectively. Plotted are solar wind density N (cm−3), solar wind velocity V (km s−1), and the magnetic field components Bx (black line), By (green line), and Bz (filled area). The data in both plots have been propagated by about 17 min to account for the travel time from the Wind spacecraft to the magnetopause. As reported by SETAL, the solar wind velocity is indeed almost unchanged at about 545 km s−1 for both events. If these energetic particles in the CEP spectra are coming from the quasi-parallel region of the bow shock, the expected spectral slope should be very similar (see Trattner et al. [1994] and Figures 7 and 8 in TETAL). This is indeed the case, as shown in Figure 3 of SETAL. [17] The second CEP event (0104 UT to 0115 UT) shows a considerably higher flux level for ions >30 keV. The solar wind conditions for the two CEP events reveal that there is an almost 30% increase in density for the second event compared to the first. In agreement with shock accelerated ions there should be an increase in flux, since more ions can be injected into the acceleration process. However, the reported increase by a factor of 18 cannot be explained solely by the density increase in the solar wind. [18] The magnetic field conditions for the two CEP events show that both events occurred during strong southward Bz. For the second CEP event all interplanetary magnetic field (IMF) components have about the same magnitude, while for the first CEP event Bz was the dominant component and Bx was the weakest. This change in IMF direction has a considerably influence on the location of the quasi-parallel shock. [19] Figures 4 and 5 show the Farris et al. [1991] bow shock as seen from the Sun. Color-coded is θBn, the angle between the shock normal and the IMF direction, using the IMF conditions for the CEP events on 22 September 1996, 0040 UT to 0052 UT and 0104 to 0115 UT, respectively. The circle represents the shock size at the X = 0 plane, separating sunward and tailward regions of the shock. Dotted lines in the green and blue parts of the plot mark the boundaries to the quasi-parallel shock region at θBn = 45° and 25° regions. [20] To show which regions of the bow shock are connected to the magnetopause (where reconnection will occur) we have used the model by Kobel and Flückiger [1994] and mark the shock location of the IMF field lines that envelope the magnetopause. The Kobel and Flückiger [1994] model considers the influence of the bow shock on the IMF field lines as they pass the shock and subsequently drapes them over the magnetopause. We have traced the IMF field lines that envelope the magnetopause and which will subsequently reconnect with geomagnetic field lines, allowing magnetosheath plasma to enter the cusp. The locations where these field lines intersect and penetrate the shock are shown by the Xs on the figure. For IMF conditions with almost no Bx component, the shape of these symbols will form a U at the high latitude shock while an IMF with only a Bx component will form a point at the subsolar point. [21] Since both CEP events occurred during a southward IMF we can safely assume that the entire dayside magnetopause will be open [see Fuselier et al., 2002a]. All the IMF field lines enveloping the dayside of the magnetopause will reconnect at the magnetopause and these field lines will connect the reconnection region with the bow shock, allowing shock accelerated ions on these field lines to access the cusp. Figure 4 shows that the dayside field lines connect to a quasi-parallel bow shock with θBn = 45°, while Figure 5 shows that the reconnection region is connected to a quasi-parallel bow shock with θBn < 25°. This connection difference along with the solar wind density increase explains the observed flux difference. With increasing θBn the injection efficiency of solar wind ions into the acceleration process will dramatically decrease [see, e.g., Ellison et al., 1995]. This reduced efficiency will primary cause a significantly reduced energetic ion flux despite the fact that the shock acceleration conditions (depending ideally only on the compression ratio at the shock), and therefore the spectral slope are basically the same for both CEP events. To our knowledge, there is still no detailed qualitative observational analysis of this effect, although the principal dependency has been discussed and documented in theory and computer simulations [e.g., Ellison et al., 1995]. [22] As outlined by Trattner et al. [1994] and TETAL, there are many details, dependencies, and overlapping effects with shock accelerated particles which still need to be investigated. However, as described by TETAL, CEP events and shock-accelerated ions exhibit many similarities and similar trends. As far as the “probability” of the TETAL process is concerned, the following needs to be taken into account: (1) It is well established that ion acceleration does occur at the quasi-parallel bow shock, producing ions with energies in excess of 100 keV/e. (2) These accelerated ions are transported through the magnetosheath [e.g., Fuselier et al., 1991b]. (3) Magnetosheath plasma does enter the cusps through reconnected field lines. That plasma is likely from the subsolar bow shock, in analogy with the high-speed streaming of a gas around a blunt object. (4) Energetic ions from the bow shock move with the IMF, and if the reconnection site is connected to the quasi-parallel bow shock, these ions are able to enter geomagnetic field lines together with the magnetosheath ion. (5) Once on reconnected field lines, the shock-origin energetic ions can move along geomagnetic field lines into the cusp [Chang et al., 1998; Fuselier et al., 2002a]. [23] We believe that the distant origin source is a more likely scenario, since the local acceleration model would need to explain why only solar wind magnetosheath ions are able to enter the cusp while the shock accelerated ions are stopped at the magnetopause. In the local acceleration model, part of these magnetosheath ions in the cusp now need to be accelerated in a confined space. These accelerated ions would have spectral characteristics (spectral breaks, dependency on solar wind conditions, composition) very similar to shock accelerated ions that are presumably prevented from entering the magnetosphere. [24] The TETAL multistep process thus seems quite probable at least from a qualitative perspective. The remaining issue is whether it can account for all CEP events in a quantitative sense, and our view is that it can. Whether >100 keV ions can be colocated with ∼1 keV ions in a narrow diamagnetic cavity is not only an issue with the TETAL process but also an issue with a local acceleration process, unless the acceleration to hundreds of keV is faster than the ion gyration. [25] KJT thanks D.C. Ellison, H. Kucharek, M. Lee and O.W. Lennartsson for helpful discussions. We acknowledge the use of ISTP KP database. Solar wind observations were provided by K. Ogilvie at NASA/GSFC (Wind/SWE); magnetic field observations were provided by R. Lepping at NASA/GSFC (Wind/MFI). The work at Lockheed was supported by NASA contracts NAS5-30302 and NAG5-8072. [26] Lou-Chuang Lee thanks David G. Sibeck for the assistance in evaluating this paper.