OBSERVATIONS OF SOLAR ENERGETIC PARTICLES FROM ³He-RICH EVENTS OVER A WIDE RANGE OF HELIOGRAPHIC LONGITUDE

Solar energetic particles (SEPs) are commonly thought to arise from two distinct acceleration mechanisms: in "gradual" SEP events, shocks driven by coronal mass ejections (CMEs) are responsible for energizing the particles while in "impulsive" SEP events, the acceleration is driven by magnetic reconnection. Various tabulations of the characteristics distinguishing the two types of events have been attempted (e.g., Reames 1999; Mewaldt et al. 2007a). While abundance ratios in gradual events typically resemble those in coronal or solar wind material to within factors ≲2 (Mewaldt et al. 2007b), impulsive events are characterized by enhancements of ³He/⁴He up to at least 10³, leading to the designation of "³He-rich" for these events. In addition, impulsive events often have the ratio Fe/O enhanced by factors ∼10 and are frequently accompanied by near-relativistic electron events with electron-to-proton ratios significantly greater than typically observed in gradual SEP events.

Individual gradual events are observed over a wide range of heliographic longitudes, which is attributed to the injection of accelerated particles on a wide range of magnetic field lines as the shock intersects these field lines near the Sun. In contrast, in impulsive events, the spatially compact acceleration region should give energetic particles access only to a small cluster of open field lines involved in the reconnection (Shimojo & Shibata 2000; Reames 2002). The relatively narrow range of heliographic longitudes occupied by this group of field lines at 1 AU is thought to be responsible for the reported narrow longitudinal extent of ³He-rich events.

The primary observational evidence for the narrow longitudinal spread of impulsive SEPs has come from studies in which ³He-rich events were detected at a spacecraft located near 1 AU and the injection of the particles at the Sun could be temporally associated with an X-ray flare (Reames et al. 1998; Reames 1993, 1999). The distribution of heliographic longitude differences between the spacecraft and the flare was found to have a mean value corresponding to that expected for a Parker-spiral magnetic field for a velocity of ∼400 km s⁻¹, which is typical of the slow solar wind. Reames (1999) reported an approximately Gaussian distribution of longitudinal offsets with a standard deviation in the range 15°–20° and noted that much of this could be attributed to event-to-event differences in the solar wind speed, which determines how tightly the Parker spiral is wound.

There have been a few studies of the longitudinal breadth of individual impulsive SEP events observed at multiple spacecraft. Data from the two Helios spacecraft were used in conjunction with observations from near-Earth spacecraft to look for correlated detections of energetic electrons (Wibberenz & Cane 2006) or ³He (Reames & Stone 1986; Reames et al. 1991). In the Wibberenz & Cane (2006) study, it was found that individual events could often be observed at two spacecraft located on field lines with Parker spiral connection points at the Sun that were widely separated, sometimes by more than 80° in heliographic longitude. These authors suggested that some diffusive propagation process in the low corona may be required to explain such events. In contrast, the ³He studies found that correlated event detections were largely confined to times when the longitudinal separation between the spacecraft were relatively narrow, consistent with the longitudinal spreads inferred from the single-spacecraft studies. One instance was reported (Reames et al. 1991) of a ³He-rich event which may have been observed by three spacecraft spread over ∼60° in heliographic longitude, but the data were not sufficient to identify ³He on one of the spacecraft, allowing for the possibility that they were not all observing the same event.

Using data from the two Solar Terrestrial Relations Observatory (STEREO) spacecraft and the Advanced Composition Explorer (ACE), Wiedenbeck et al. (2010) reported an impulsive event that occurred on 2008 November 3–4 when the STEREOs had longitudinal separations of ±41° from ACE. In this event, ³He was detected at STEREO-B and ACE while electrons were detected at all three spacecraft.

In this paper, we summarize the results of a search for additional multispacecraft ³He-rich SEP events and discuss in detail the event of 2010 February 7, which was detected at both STEREOs and ACE when the spacecraft spanned 136° in heliographic longitude. We also consider possible causes and consequences of this wide spread.

After having been placed into ∼1 AU heliocentric orbits around the beginning of 2007, the two STEREO spacecraft (launched on 2006 October 25) have been providing SEP measurements that can be used in conjunction with data from ACE and other near-Earth spacecraft to obtain multipoint observations of individual SEP events. The STEREO orbits are designed such that each of these spacecraft has a separation in heliographic longitude from Earth that increases at a rate of ∼225 yr⁻¹, with STEREO-A leading and STEREO-B trailing the Earth.

For the present study, we have used data from the STEREO Low Energy Telescope (LET; Mewaldt et al. 2007a), the Suprathermal Ion Telescope (SIT; Mason et al. 2007), and the Solar Electron and Proton Telescope (SEPT; Müller-Mellin et al. 2008), together with ACE data from the Ultra-Low-Energy Isotope Spectrometer (ULEIS; Mason et al. 1998), the Solar Isotope Spectrometer (SIS; Stone et al. 1998), and the Electron, Proton, and Alpha Monitor (EPAM; Gold et al. 1998). We performed a systematic search of the STEREO/LET data sets for ³He-rich SEP events occurring between 2007 January when the STEREOs and ACE were all located near Earth, and 2011 January when the STEREOs were leading and trailing the Earth by almost 90°. Since SEP intensities typically decrease with increasing energy, our search for ³He-rich events relied primarily on ³He with energies in the range 2.3–3.3 MeV nucleon⁻¹, which is just above the STEREO/LET energy threshold for this isotope.

Figure 1 shows event-integrated mass histograms from each of the instruments for the 2010 February 7 event, which is discussed in detail below. The identification of ³He in STEREO/LET instruments relies on observing the peak for this isotope in the He mass histogram for particles detected within a given range of total energies because the energy per nucleon cannot be calculated until the mass has been determined. In a few cases, the ³He peak in the histogram for 6.9–9.9 MeV He, which is used to select the 2.3–3.3 MeV nucleon^{−1 3}He, was obscured by spillover from the ⁴He peak. In these cases, the 12.9–24.0 MeV total-energy interval (4.3–8.0 MeV nucleon⁻¹ for ³He) was used to establish the presence of the ³He-rich event.

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Table 1 lists the ³He-rich SEP events that were detected by one or both of the STEREO/LET instruments during the 49 months covered by this study. For each event, the table shows the locations of the two STEREOs and indicates which instruments detected the event. Some of the detections are uncertain, as explained in the footnotes. The table shows that detections of the same event by one STEREO/LET and by ACE are not uncommon, even when the spacing between the two spacecraft is >60°. In a few cases, the longitudinal spread of an event is large enough to allow detection at both STEREOs and ACE, as in the case of two events in 2010 February when the three spacecraft spanned 136°. There were no instances in which an event was detected at both STEREO spacecraft but not at ACE. This is not surprising because during the time interval studied, the longitudinal separation between the two STEREO spacecraft on the Earth-facing side of the Sun (≡ ϕ_AB) was less than the separation on the opposite side of the Sun (360° − ϕ_AB) and because the ACE/ULEIS instrument, which measures ³He near Earth down to ∼0.15 MeV nucleon⁻¹, is significantly more sensitive to small ³He-rich events than the STEREO/LETs. A number of additional ³He-rich events that were detected only by ACE/ULEIS and not by either STEREO/LET were not included in the present study.

In several cases, the ³He intensity detected at one or both of the STEREO/LETs was only slightly above the background ³He level. To confirm that the observed ³He actually was associated with an ³He-rich SEP event, we used data from other instruments to check for additional signatures of an impulsive event, including enhancements of Fe/O, which are observable with the STEREO/SIT and ACE/ULEIS, and intensity increases of near-relativistic electrons, which are detected with STEREO/SEPT and ACE/EPAM.

The first two years of the survey period occurred during the extremely quiet portion of the solar minimum between cycles 23 and 24. Thus, the only events detected with the STEREO/LETs were those discussed by Wiedenbeck et al. (2010): a 2007 January event when the three spacecraft were all near Earth and a 2008 November event when the STEREOs were 41° from ACE. The rate of detection of ³He-rich events gradually increased through 2009 and 2010, but was still low enough that the occurrence of chance coincidences between detections of ³He-rich events from widely spaced active regions was improbable. The study ended in 2011 February when solar activity had started to increase significantly and a combination of higher SEP backgrounds and increased probability of chance associations of events began to complicate the analysis.

In the remainder of this paper, we examine in detail the best-measured three-spacecraft event that was detected. A preliminary discussion of this event has been presented by Wiedenbeck et al. (2011). Figure 2(a) shows ³He intensity versus time plots for this 2010 February 7 event as observed at ACE, STEREO-A (leading ACE by 65°), and STEREO-B (trailing ACE by 71°). Figure 2(b) shows the events observed during this period in ∼0.07–0.1 MeV electrons, which require ∼20 minutes to traverse the nominal 1.2 AU length of a Parker spiral field line between the Sun and 1 AU. From the event onset at STEREO-B (indicated approximately by the leftmost arrow on the ³He plot), it appears that the ³He was not directly associated with either of the two most prominent electron events, but possibly corresponds to a small inflection seen late on 2010 February 7 in the electron observations from STEREO-B and ACE. Radio wave observations⁷ from both STEREOs and from the Wind spacecraft, which was located at the L1 Lagrange point near ACE, indicate the occurrence of numerous type III radio bursts during the last ∼1/3 of 2010 February 7, further indicating the escape of electrons from the Earth-facing side of the Sun.

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Differences between the ³He onset times (or peak times) at STEREO-A and STEREO-B exceed by a factor ≳8 the ∼2 hr required for ∼3 MeV nucleon⁻¹ ions with 0° pitch angle to traverse the ∼1.2 AU length of the nominal Parker spiral field line from the Sun to 1 AU. Furthermore, the peak ³He intensities also differ significantly between the two STEREO measurements. The ACE/SIS data that are shown correspond to a higher ³He energy than those from the STEREO/LETs, so quantitative comparison of the peak intensity at ACE with those observed at the STEREOs requires additional analysis.

From Figure 2(c), which shows the times and peak intensities of X-ray flares of GOES class C1.0 or greater compiled by the Space Weather Prediction Center⁸ (SWPC), it is evident that we are not able to associate the ³He injection with a specific flare among the large number that were detected. Nevertheless, the location of the injection is reasonably well known because all of the flares during this time period came from the same active region, AR11045, which was located at N23W01 at 0000 UT on 2010 February 8. Inspection of coronal images from SOHO/EIT and STEREO/EUVI shows that the three C-class flares that occurred after 2100 UT on February 7 were located between E02 and W01, so we determine the injection point to be located at the central meridian to within ±2°.

Daily active region reports from the SWPC were examined to check for possible sources for an SEP event from the backside of the Sun that could have been the source of the ³He observed at STEREO-A. AR11043, which was at W62 on 2010 February 7, and AR11041, which was last recorded at W58 on 2010 January 31, both had simple (alpha) magnetic configurations with no sunspots. AR11042, which was still producing minor X-ray flares when it rotated past the west limb on 2010 January 27, would have been at ∼E110 at the time of the 2010 February 7 ³He-rich event, with a connection point at 1 AU lying approximately 125° west of STEREO-A. Moreover, STEREO/EUVI images of these regions do not show evidence of jets, which have been associated with impulsive SEP events (Wang et al. 2006; Nitta et al. 2008). Thus, we find no plausible alternative to AR11045 as the source for the STEREO-A event.

The geometrical arrangement of the three spacecraft, the flare, and selected Parker spiral field lines are illustrated in Figure 3(a). The field lines drawn through the spacecraft locations were calculated using the mean solar wind speed measured during the second half of 2010 February 7 at the respective spacecraft (STEREO-B: 480 km s⁻¹; ACE: 355 km s⁻¹; STEREO-A: 410 km s⁻¹). The field line from the active region (dashed line), which is drawn assuming a speed of 400 km s⁻¹, connects to 1 AU at east 60°.

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Figure 3(b) shows ³He fluence spectra for the 2010 February 7 event at each of the spacecraft. The vertical dashed lines indicate the boundaries of the 2.3–3.3 MeV nucleon⁻¹ energy interval over which we compare fluences. Although data points from ACE/ULEIS and ACE/SIS fall below and above this interval, together they provide reasonably good constraints on fluence at these intermediate energies and can be used to obtain the 2.3–3.3 MeV nucleon⁻¹ fluence by interpolation, as indicated by the dotted line. Data from the 2007 January 24 ³He-rich event were previously used to verify the intensity calibrations between the two LETs and the combination of ULEIS and SIS (Wiedenbeck et al. 2010). In Figure 3(c), the resulting 2.3–3.3 MeV nucleon⁻¹ averaged fluences are shown as a function of spacecraft heliographic longitude measured from the longitude to which the active region is connected at 1 AU.

The smooth curve, which is the result of a two-parameter Gaussian fit to the three data points with the center of the distribution constrained to lie at the connection longitude, does a reasonable job of representing the measurements. The standard deviation of the Gaussian obtained from the fit is 48°. Varying the assumed center location by 2° results in a change of ∼1° in the derived width. For comparison, a 13 km s⁻¹ change in solar wind speed would also cause a shift of the connection point at 1 AU by ∼2°. The fact that the shape of the observed longitudinal distribution is approximately Gaussian is suggestive since diffusion from a point source leads to a Gaussian spatial distribution.

3.1. Origin of the Longitudinal Spread

The observed broad spread of ³He in heliographic longitude is not expected from the simple model commonly used to describe the release of energetic particles from impulsive SEP events. In this model (Shimojo & Shibata 2000; Reames 2002), closed field lines in an active region undergo reconnection with adjacent open field lines so that some of the particles accelerated in the reconnection event have access to the heliosphere along these open field lines. Because of the small size of the reconnection region, one expects that the open field will span a relatively small range of angles as it expands into the heliosphere.

The rate of particle diffusion perpendicular to the mean magnetic field, which could result from cross-field scattering, field-line random walk, or a combination of both effects, is not tightly constrained but under normal conditions it is expected to be much smaller than parallel diffusion (e.g., Dröge et al. 2010). Interruptions in the intensity of impulsive SEP events lasting several hours are sometimes observed at 1 AU. These "dropouts" (Mazur et al. 2000) are attributed to convection past the observer of successive magnetic flux tubes with different connection points at the Sun such that some have been populated with particles from an impulsive event and others have not. The existence of these relatively sharp boundaries between different particle populations suggests that cross-field diffusion is not responsible for transport over a large range of heliographic longitudes in these events. Dröge et al. (2010) found that the ratio between perpendicular and parallel diffusion coefficients required to account for such dropouts is κ_⊥/κ_|| ≲ 10⁻⁴.

Using observations included in our preliminary report of the 2010 February 7 event (ACE News #137 ⁹), Giacalone & Jokipii (2012) attempted to account for the observed characteristics using an interplanetary transport model and found that reasonable agreement could be obtained by assuming parallel scattering mean free paths of a few tenths of an AU and κ_⊥/κ_|| ≃ 10⁻². These authors point out that large longitudinal spreads and the presence of dropouts do not need to be mutually exclusive since it is possible that the longitudinal spreading could result from the field line random walk associated with footpoint motions at the photosphere rather than from cross-field scattering of particles in the interplanetary medium. Furthermore, since dropouts were not observed in the February 7 event, significant cross-field scattering is not ruled out by the data. However, lacking an independent measure of the interplanetary scattering conditions, we are unable to test whether the conditions assumed by Giacalone & Jokipii (2012) actually existed. Thus, we also considered other possible causes for the longitudinal spread.

It is well established that field lines from an active region can undergo significant spreading, and possibly displacement, between the solar photosphere and the "source surface" at which the field becomes radial and opens out into the heliosphere (e.g., Liewer et al. 2004). The fact that the 2010 February 7 event, which originated from an active region at N23, was observed in the ecliptic plane suggests that at least this amount of angular (latitudinal) displacement occurred between the photosphere and the source surface, assuming that perpendicular diffusion effects were not responsible for the latitudinal transport. Potential field source surface (PFSS) models (e.g., Schrijver & DeRosa 2003) have had some success in describing coronal fields (e.g., Liewer et al. 2004). However, attempts to trace flows of energetic electrons (Rust et al. 2008) or ions (Nitta et al. 2006) from a solar flare site to 1 AU using the PFSS model field in the corona and a Parker spiral field in the heliosphere have had ≲50% success. Klein et al. (2008) investigated several electron events in which the longitudinal transport was consistent with as much as 50° field-line displacement between an active region and the source surface, as predicted by calculations. We have examined PFSS estimates of the coronal field at the time of the 2010 February 7 ³He-rich event using information available from SolarSoft¹⁰ and from the GONG Web site.¹¹ Although there are significant differences between maps intended to represent the same field, a consistent feature is that in each map field lines originating near AR11045 and reaching the source surface near the solar equatorial plane encompass a wide range of heliographic longitudes, but that range is nevertheless ≲2/3 the longitudinal separation between the two STEREOs.

To further check on the possibility that magnetic field expansion between the photosphere and the source surface could, at times, produce a longitudinal spread as large as the 136° we observe, we have examined daily PFSS maps covering a 14 year time interval (1998 through 2011). For each map, field lines reaching the source surface along the heliographic equator were traced back to their points of origin at the photosphere and instances in which a large number of these field lines were rooted within a specified angular radius, R, of one another were identified. Among the various field line "clusters" that were found for a given day, the one with the largest longitudinal spread at the source surface was selected. Figure 4 shows integral distributions of these largest cluster longitudinal spreads for several different values of R. Each curve is based on ∼5100 PFSS daily maps covering more than a full solar cycle. As seen from these distributions, spreading of field lines originating within a radius ⩽10° at the photosphere to cover as much as 136° (the separation between the two STEREOs at the time of the 2010 February 7 event) along the heliographic equator must be regarded as an extremely rare occurrence. Thus, we find it unlikely that spreading of the magnetic field in the corona could have been the main contributor to the observed longitudinal width of this event.

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It is known that impulsive SEP events are sometimes accompanied by CMEs (Kahler at al. 2001; Nitta et al. 2006), although often these CMEs are too slow to drive a shock that would accelerate particles. Indeed, in the case of the 2010 February 7 event, there was no fast CME observed and also no evidence for a shock, either in the form of a type II radio burst or an in situ detection of shock signatures.

However, a flaring active region can produce a sequence of SEP events and CMEs and thus it is not uncommon to have SEPs from a ³He-rich event released into an environment that has experienced the launch of one or more CMEs within the preceding several days. Such CMEs can distort the magnetic field and possibly lead to reconnection that further alters the field geometry and the paths of subsequently released particles. Figure 2(d) indicates the speeds and detection times of identified CMEs.¹² Of particular note is the halo CME shown around 0400 UT on 2010 February 7, which has been identified¹³ as the source of an interplanetary CME (ICME) detected at ACE starting around 0800 UT on 2010 February 11. The leading edge of this ∼421 km s⁻¹ CME, which was directed generally toward Earth, would have been at ∼0.15–0.2 AU at the release time of the 2010 February 7 ³He-rich event and thus could have been in a position to significantly influence the large-scale distribution of particles released behind it. Determining whether such preceding CMEs play a crucial role in causing the large longitudinal spreads observed in some ³He-rich SEP events will require a future statistical study including additional ³He-rich events.

There have been a number of reports of "sympathetic flares" (see Schrijver & Title 2011 and references therein) in which two solar flares are observed at well separated locations on the Sun with small enough differences in their times of occurrence that a causal connection appears probable. In one example of this type, Schrijver & Title (2011) report evidence for magnetic separatrices connecting distant sites of activity and conclude that the evolution of the field at one location can be "essential to the evolution, and possibly even to the initiation, of the flares and eruptions over an area that spans at least 180° in longitude." Thus, it is possible that sympathetic flares could lead to nearly simultaneous ³He-rich events that can be observed at widely separated longitudes in the heliosphere. However, in the 2010 February 7 event, there was no good candidate active region with a complex magnetic configuration near the nominal connection point of STEREO-A that would provide a likely source for a sympathetic flare that could be triggered by the activity at AR11045. In addition, the STEREO/EUVI data did not show evidence for flares or jets in this region of the corona. It should be noted, however, that Mason et al. (2009) reported a few instances in which ³He-rich SEP events were observed when a single, magnetically simple western-hemisphere active region was the only candidate source, but no detectable X-ray flare and only minor brightenings in the EUV were observed.

Another mechanism that could lead to a causal connection between flares at widely separated locations involves wave propagation from a coronal disturbance caused by one flare to a remote site where it can destabilize the field and trigger a subsequent event (Krucker et al. 1999). STEREO/EUVI data do show some large-scale disturbances ("EIT waves") in connection with several flares that occurred during February 7–8, but they did not occur during the time range of interest, late on February 7.

Recent work on the origin of the slow solar wind (Antiochos et al. 2011; Edmondson 2011 and references therein) has considered a puzzle similar to that posed by the wide longitudinal spread we are finding in impulsive SEP events. The slow wind, with its composition similar to that of closed-field regions in the corona, is thought to be released by means of reconnection at the boundary between open and closed field regions, yet the slow wind extends over a much wider range in heliographic latitude than does this boundary region. Calculations discussed by Antiochos et al. (2011) indicate that distinct open field regions are connected by narrow open-field corridors that map into a web of separatrices and quasi-separatrix layers in the heliosphere that extends far from the heliospheric current sheet and provides widely distributed source regions that can contribute to the slow solar wind. There is a possibility that an extensive network of open-field regions of the sort envisioned by these authors could, at times, provide a mechanism for energetic particle transport over a wide range of heliographic longitudes. Although further model calculations would be needed to explore the plausibility of this suggestion, it is conceivable that energetic particle transport could provide a useful means for probing these complex magnetic structures.

3.2. Additional Constraints

3.3. Comparison with Single-spacecraft Study Results

3.4. Relationship to Other ³He Observations

The distribution in heliographic longitude of energetic ³He from impulsive SEP events has been studied using data from energetic particle instruments on the two STEREO spacecraft and on ACE as the range of longitudes spanned by the three spacecraft increased from 0° to nearly 180° during the solar minimum period between 2007 January and 2011 January. Multispacecraft detections of ³He from individual events were not uncommon, even when the ACE-to-STEREO spacings were >60°. In the 2010 February 7 event, which is the best example observed to date of an event in which ³He was detectable at three widely spaced spacecraft (136° longitudinal separation between the two STEREOs), the ³He fluence was found to have a strong dependence on separation from the heliographic longitude connected to the source region, differing by a factor >30 between the two STEREOs. In this event, fluence as a function of heliographic longitude was found to be well fitted by a Gaussian with standard deviation ∼48° centered at the best-connected longitude.

Previous results indicating that X-ray flares associated with ³He-rich SEP events have a relatively narrow distribution in heliographic longitude (Reames 1993, 1999) may be reconcilable with the present work by taking into account the effects of instrument sensitivity thresholds for ³He detection in conjunction with the strong dependence of fluence on longitudinal separation from the field line connected to the source region. The wide longitudinal spread of ³He-rich SEP events may also be relevant to issues such as the frequency with which suprathermal ³He is present as seed material for acceleration by coronal and interplanetary shocks.

The physical mechanism responsible for the broad spread in heliographic longitude is, at present, not clearly established. Several possible mechanisms have been considered and found to merit examination in a future statistical study of ³He-rich events using multispacecraft observations. It has been found that interplanetary diffusion perpendicular to the mean magnetic field can do a reasonable job of reproducing the ³He observations if a sufficiently large value of κ_⊥/κ_|| is assumed (Giacalone & Jokipii 2012), but at present we do not have an independent test of whether κ_⊥ had a large enough value at the time of the February 7 event. If interplanetary diffusion turns out to be a major contributor to the longitudinal spreading of energetic particles in impulsive SEP events, then multispacecraft observations of the particle distributions may provide a useful probe of scattering conditions in the interplanetary medium.

The longitudinal spreading of magnetic field lines between their photospheric footpoints near an active region and the equatorial plane at the source surface must be considered. Although we find that this mechanism rarely produces longitudinal spreads as broad as the 136° separation of the STEREOs on 2010 February 7, it frequently leads to spreads sufficient to explain many of the ³He-rich SEP events that we have observed at two spacecraft (Table 1).

Finally, the effects of distortion or reconnection in the large-scale coronal and/or heliospheric magnetic fields deserves additional study. Particle propagation in the 2010 February 7 event might have been influenced by a slow, halo CME that occurred ∼1 day earlier and resulted in an ICME that was detected at Earth several days later. Finally, it must be acknowledged that the longitudinal spreading of energetic particles in ³He-rich events may be due to a combination of these effects and that the dominant cause could differ from one event to the next. Future statistical studies should help clarify the relative importance of the various physical mechanisms.

We thank Justin Edmondson for his comments on the possible relationship between our results and recent work on the origin of the slow solar wind. Information on CMEs was obtained from the catalog generated and maintained at the CDAW Data Center by NASA and The Catholic University of America in cooperation with the Naval Research Laboratory. This work was supported by NASA at Caltech (under grants NNX08AI11G, NNX10AQ68G, and NNX10AQ68G, and through UC Berkeley under contract NAS5-03131), JPL, APL (under grant NNX10AT75G), and LMSAL (under grants NNX07AN13G and NNX08AB23G). The STEREO/SEPT work was supported under grant 50 OC 0902 by the German Bundesministerium für Wirtschaft through the Deutsches Zentrum für Luft- und Raumfahrt (DLR).

As mentioned in Section 3.4, the strong dependence of ³He fluence on the observer's longitudinal separation from the field line that is best connected to the solar source can lead to a bias in the derived longitudinal distribution of a solar indicator of the source location such as X-ray flares. Using S₀ to denote the ³He fluence at the best-connected longitude, we write the distribution of this quantity as f(S₀), which we assume to be independent of heliographic longitude when averaged over a long period of time. Using g(ϕ) to denote the dependence of fluence on longitudinal separation, ϕ, from this best-connected longitude, an observer at this separation would observe a fluence S = S₀ g(ϕ). If the ³He fluence measurement has a finite detection threshold, S_thresh, then the number of ³He-rich events that would be detected with a longitudinal separation of between ϕ and ϕ + dϕ will be N(ϕ) dϕ, with

$$\begin{equation} N(\phi) = \int _{S_\mathrm{thresh}/g(\phi)}^{\infty } f(S_0)\,dS_0. \vspace*{6pt} \end{equation} \tag{ A1 }$$

Taking g(ϕ) to be a Gaussian with standard deviation σ as observed in the 2010 February 7 event (i.e., g(ϕ)∝exp [ − ϕ²/2σ²]) and the event size distribution to be a power law with index −α (i.e., f(S₀)∝S^−α₀ with α > 1), Equation (A1) gives

$$\begin{eqnarray} N(\phi) \; &\propto & \; \int _{S_\mathrm{thresh}/g(\phi)}^{\infty } S_0^{-\alpha }\,dS_0 \;\propto \; g(\phi)^{\alpha -1} \nonumber\\ &\propto &\; \exp [-\phi ^2/2 (\sigma /\sqrt{\alpha -1})^2]. \vspace*{4pt} \end{eqnarray} \tag{ A2 }$$

That is, the distribution of detected source longitudes is a Gaussian with standard deviation $\sigma /\sqrt{\alpha -1}$, which for α > 2 is narrower than the Gaussian characterizing the dependence of fluence on longitude. Although the assumed threshold, S_thresh, affects the total number of detected ³He-rich events, in the special case where the event size distribution is a power law, it does not affect the shape of the derived source location distribution.

Comparison of the width of the two measured Gaussian distributions (fluence versus longitude and source location versus longitude) could, in principle, provide information about the exponent of the power-law fluence distribution (or of a power-law approximation evaluated near the fluence sensitivity threshold in the case of a more general event-size distribution), this exercise is hindered by several facts. First, the derived flare longitude histogram is, in all likelihood, broadened by variations in the solar wind speed and, second, the longitudinal distribution of the fluence may not be well characterized by the measurements in the 2010 February 7 event, which is one of the few events that have been detected at all three spacecraft.