Issue 
A&A
Volume 650, June 2021
Parker Solar Probe: Ushering a new frontier in space exploration



Article Number  A24  
Number of page(s)  14  
Section  The Sun and the Heliosphere  
DOI  https://doi.org/10.1051/00046361/202039352  
Published online  02 June 2021 
A new view of energetic particles from stream interaction regions observed by Parker Solar Probe
^{1}
University of New Hampshire,
Durham,
NH
03824,
USA
email: nschwadron@unh.edu
^{2}
Department of Astrophysical Sciences, Princeton University,
Princeton,
NJ
08544,
USA
^{3}
California Institute of Technology,
Pasadena,
CA
91125,
USA
^{4}
University of Texas at San Antonio,
San Antonio,
TX
78249,
USA
^{5}
University of California at Berkeley,
Berkeley,
CA
94720,
USA
^{6}
HarvardSmithsonian Center for Astrophysics,
Cambridge,
MA
02138,
USA
^{7}
Goddard Space Flight Center,
Greenbelt,
MD
20771,
USA
^{8}
University of Arizona,
Tucson,
AZ
85721,
USA
^{9}
Johns Hopkins University, Applied Physics Laboratory,
Laurel,
MD
20723,
USA
^{10}
BWX Technologies, Inc,
Lynchburg,
VA,
24504,
USA
^{11}
University of Michigan,
Ann Arbor,
MI
48109,
USA
^{12}
Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing of the National Observatory of Athens, Vas. Pavlou and I. Metaxa,
15236
Penteli,
Greece
^{13}
University of Delaware,
Newark,
DE
19716,
USA
^{14}
Jet Propulsion Laboratory, California Institute of Technology,
Pasadena,
CA
91109,
USA
Received:
7
September
2020
Accepted:
14
November
2020
Early observations from the first orbit of Parker Solar Probe (PSP) show recurrent stream interaction regions that form close to the Sun. Energetic particle enhancements were observed on the 320th–326th day of the year 2018, which corresponds to ~1–7 days after the passage of the stream interface between faster and slower solar wind. Energetic particles stream into the inner heliosphere to the PSP spacecraft near 0.33 au (71 solar radii) where they are measured by the Integrated Science Investigation of the Sun (IS⊙IS). The large 6day time interval over which energetic particles are observed after the stream passage provides a unique perspective on the development of stream interactions within the heliosphere. The long duration of energetic particle enhancements suggests that particles stream in through the inner heliosphere more directly along magnetic field lines that form a subParker spiral structure due to magnetic footpoint motion at the Sun and shearing of the magnetic field in the rarefaction region behind the stream interface. The strong buildup of energetic particle fluxes in the first 3 days after the passage of the stream interface indicates that suprathermal populations are enhanced near the interaction region through compression or other acceleration processes in addition to being diffusively accelerated. The early increases in energetic particle fluxes (in the first 3 days) in the formation of these events allows for the characterization of the acceleration associated with these suprathermal seed populations. Thus, we show that the time history of energetic particle fluxes observed by IS⊙IS provides a new view of particle acceleration at stream interaction regions throughout the inner heliosphere.
Key words: Sun: magnetic fields / solar wind / Sun: heliosphere / shock waves / acceleration of particles
© ESO 2021
1 Introduction
The NASA Parker Solar Probe (PSP) mission explores our nearest star, the Sun, and its environment (Fox et al. 2016). Comprehensive measurements of solar energetic particles (SEPs) are provided by the Integrated Science Investigation of the Sun (IS⊙IS) instrument suite (McComas et al. 2016), which is comprised of two Energetic Particle Instruments measuring Higher (EPIHi) and Lower (EPILo) energy particles (McComas et al. 2016) over the range 0.02–200 MeV/nucleon. Here, we examine energetic particles accelerated from stream interaction regions in the inner heliosphere, at radial distances beyond the PSP spacecraft (to ~5 au). The stream interfaces were identified using in situ plasma observations from the Solar Wind Electrons Alphas and Protons Investigation (SWEAP; Kasper et al. 2016) and the in situ magnetic field observations from the electromagnetic fields investigation, commonly referred to as FIELDS (Bale et al. 2016).
Particles with energies on the order of 1 MeV per nucleon are accelerated at the forward and reverse shocks and compressive waves that bound corotating interaction regions (CIRs; e.g. Barnes & Simpson 1976; McDonald et al. 1976; Fisk & Lee 1980) and stream interaction regions (SIRs). These interaction regions are formed from faster corotating solar wind streams overtaking slower solar wind streams. Typically, it is assumed that CIRs and SIRs accelerate particles beyond several au, where shocks are generally formed. However, observational studies at 1 au by Wind have shown that CIR ions are first accelerated much closer to the Sun (within 1.5 au) than previously thought (Chotoo et al. 2000; Mason 2000; Giacalone et al. 2002), and the suprathermal populations that feed into the acceleration process are largely composed of interstellar pickup ions (e.g., Schwadron et al. 1996). The conclusion that CIR acceleration begins near 1 au is supported by observations from SOHO and ACE (Yu et al. 2017).
One of the puzzles that has surrounded the acceleration of CIR particles closer to the Sun, at less than 1.5 au, is whether the acceleration process requires the presence of welldefined and relatively strong shocks (Chotoo et al. 2000). The modeling of pickup ions and a comparison with observations from STEREO shows that the acceleration process and the development of relatively hard suprathermal tails begins where CIR compression regions are first formed inside of 1 au (Chen et al. 2015).
An important feature of energetic particles in SIRs is that the energy spectrum for suprathermal particles assumes a functional form that is extremely common (Gloeckler & Fisk 2006, 2010; Fisk & Gloeckler 2006, 2008; Fisk et al. 2010) with a powerlaw index γ_{s} for the distribution function f ∝ v^{γs} as a function of particle speed v (hereafter referred to as the common particle speed distribution, or common spectrum). This form applies at relatively low energies from ~3 keV up to ~160 keV (Fisk & Gloeckler 2012). The distribution function powerlaw index assumes a value, on average, that is close to −5 corresponding to a spectral index of γ_{s}∕2 + 1 = −1.5 for the differential flux J as a function of energy E, J ∝ E^{γs∕2+1}. While this spectral form is common, Dayeh et al. (2017) found variations in the spectra for heavy ion species.
The presence of a common power law index poses a puzzle concerning the origin of suprathermal tails. Suprathermal tails observed by Ulysses typically at ~2–5 au are correlated with the levels of magnitude fluctuations in the heliospheric magnetic fields near SIRs (Schwadron et al. 1996). This correlation was attributed to the presence of magnetosonic waves that are transittime damped during the acceleration of suprathermal ions. Fisk and Gloeckler proposed a specific acceleration mechanism that creates a v^{−5} distribution (Fisk & Gloeckler 2006, 2008, 2014). The mechanism is driven by local compressive turbulence and operates much like a magnetic pump. The common form of the spectrum could also result from acceleration processes that are stochastically distributed. In this case, the observed energy spectrum is a superposition of distributions from variable acceleration processes (Schwadron et al. 2010) including diffusive acceleration and statistical acceleration. Further, thecommon spectrum is so hard that it is close to the limit of possible stationary state plasma distributions out of equilibrium (Livadiotis & McComas 2009, 2010). It is possible that rapid acceleration processes driven by local compressive turbulence push the distribution close to this limit of stationary states out of equilibrium.
Suprathermal He^{+} is produced from pickup ions (Moebius et al. 1985). These ions are predominantly photoionized from neutral He atoms that enter the heliosphere from the interstellar medium (Moebius et al. 1985; Schwadron 1998; Gloeckler & Geiss 1998, 2001). The source of interstellar pickup ions is well known, and modeled distributions based on the source reproduce observed distributions (e.g., Schwadron 1998; Schwadron et al. 1999). Despite the lack of variability in pickup ion production, the variable He^{+} suprathermal tail index and the flux (Hill et al. 2009; Möbius et al. 2019; Popecki et al. 2013; Tessein et al. 2013; Taut et al. 2018) suggest that the tails are created by stochastically distributed processes including shocks, compressions and turbulence (Schwadron et al. 2010). Particles are accelerated by discrete sources, and then transported throughout the solar wind. Bower et al. (2020) present support for this concept using superposed epoch analyses of He^{+} data from the PLASTIC instrument on STEREOA at predominantly SIR shocks at 1 au. The tail index and suprathermal flux vary systematically across the shock. The observed powerlaw index agrees with the prediction of diffusive acceleration based on the compression ratio at the shock. Statistical analysis of suprathermal PUI powerlaw slopes in shocks with widely varying compression strengths showed that values close to −5 were the most common.
Two takeaways on the formation of the energy spectrum observed in the inner heliosphere are as follows: first, despite the variability in acceleration mechanisms and in underlying distributions, the average functional dependence on speed for accelerated suprathermal tails is observed to attain a relatively common form with $${f}_{\text{s}}(v)\propto {v}^{{\gamma}_{\text{s}}}\mathrm{exp}(v/{v}_{\text{s}}),$$(1)
where γ_{s} ≈−5 and v_{s} ~ 4400 km s^{−1} is the particle speed at ~ 10× the solar wind speed or, equivalently, ~0.16 MeV. The exponential rollover at higher suprathermal energies is directly observed by the ULEIS instrument on ACE (Fisk & Gloeckler 2012). Second, there are various acceleration mechanisms associated with suprathermal tails, and strong advocates for each of them. While the underlying physical mechanism that explains these suprathermal tails is an area of active research, there is an increasing number of studies concluding that the average spectrum within SIRs attains a common form (Eq. (1)). This result is used in this paper to characterize the energetic particle spectrum near SIR compressions and shocks.
The recent observations of energetic particles from SIRs by PSP IS⊙IS (McComas et al. 2019; Desai et al. 2020; Cohen et al. 2020; Joyce et al. 2021; Allen et al. 2020a) provide some unique features. The vantage point of PSP close to the Sun allows the spacecraft to observe magnetic field lines spanning an enormous range of the inner heliosphere. Here, we discuss the profiles of energetic particles observed by IS⊙IS following the passage of a stream interaction region to discern both the magnetic structure within the rarefaction region that trails the CIRs, and the nature of particle acceleration from these structures.
This paper is a companion to the work of Joyce et al. (2021), which analyzes the SIR energetic particles observed dayofyear 320–325 in 2018 (November–21). In the inner heliosphere, typically at or inside 1 au, lower energy particles below ~1 MeV/nuc are expected to show the signatures of modulation with reduced lower energy fluxes and hardened energy spectra due to inhibited propagation of ions that are scattered as they move many au from the SIR reverse compression or shock to the point of observation. However, Joyce et al. (2021) find little or no evidence of modulation of lower energy particles. The relative lack of modulation is also found by Allen et al. (2020a) in PSP IS⊙IS and STEREO A observations of a CIR event from DOY 258–273 in 2019 (September 15–30). Our paper examines how this reduced modulation can be accounted for, and how the rapid evolution of energetic particle fluxes from SIRs can be explained. This paper offers an explanation for these fundamental questions and therefore develops a new view of energetic particles from stream interactions regions.
The paper proceeds by first discussing the observations of SIR timeprofiles in Sect. 2 and the revision to the magnetic structure needed to explain observed timeprofiles in Sect. 3. We discuss the energetic particle spectrum at SIRs in Sect. 4. We explore the implications of our findings in Sect. 5 and conclude in Sect. 6. There are five appendices in the paper that detail the subParker spiral Appendix A, the distance to the SIR compression or shock Appendix B, the injection energy into diffusive shock acceleration Appendix C, the cooling rates of SIR suprathermal populations Appendix D, and the modeling of energetic particle propagation in to the position of PSP Appendix E.
2 IS⊙IS observations of energetic particles from SIRs
Parker Solar Probe’s vantage point close to the Sun provides a unique platform to observe the progression of CIR acceleration throughout the inner heliosphere. Figure 1 shows a summary of the observations from IS⊙IS on day 318 through 325 of 2018. The passage of the stream interface occurs near the end of day 318. During passage of the stream interface, there is a small enhancement in the EPILo count rate, which possibly indicates local particle acceleration (Allen et al. 2020b), and after the passage of the stream interface, we observe the gradual buildup and subsequent decay of particle fluxes wheremagnetic field lines are connected to the SIR compression and shock that forms further out in the heliosphere.
The average integrated flux is computed from the differential flux for each bin multiplied by the width of the energy bin, then summed and divided by the total of the energy bin widths. The EPIHi data is comprised of all protons, and the EPILo fluxes are protondominated, but also include small contributions from other species. The EPIHi fluxes are from the A and B sides of LET1, which are sunward and antisunward facing, respectively (A_ and B_H_Flux from psp_isoisepihi_l2let1rates3600). The fluxes are summed over energy bins spanning 1.005.66 MeV. The data is binned hourly and then smoothed (boxcar averaged) over 5 h intervals.
The EPILo fluxes are Ion TOFonly (H_Flux_ChanT from psp_isoisepilo_l2ic). They are averaged over the three antisunwardfacing wedges (W0, W6 and W7; apertures,09 and 6079) and summed over energy bins spanning 166.47–496.27 keV. The data are binned hourly and smoothed over 11 h intervals.
The average sunward directed fluxes (red curves) and antisunward fluxes (blue curves) are shown in Fig. 1, panels 1 and 3. The observations indicate weak anisotropies, consistent with a distant source from the SIRs.
Fig. 1 Summary of energetic particles following the stream interaction region: (top) EPIHi proton fluxes (sunward directed particles in blue, antisunward particles in red) from 1–10 MeV, (panel 2) EPILo proton fluxes as a function of energy and time, (panel 3) EPILo proton fluxes as a function as a function of time from 200–500 keV, (panel 4) density measured by SWEAP,(panel 5) solar wind speed measured by SWEAP, and (panel 6) magnetic field measured by FIELDS. This figure is similar to Fig. 2 of Joyce et al. (2021); the only difference is in panel 3, which is shown from 100–500 keV H^{+} in Joyce et al. (2021). 
3 Magnetic field structure of the rarefaction region behind the SIR
The magnetic field structure within the rarefaction region behind the SIR is taken traditionally to be a Parker Spiral. However, numerous studies have shown the magnetic field structure within the solar wind conforms to a subParker spiral (Murphy et al. 2002; Schwadron 2002; Schwadron & McComas 2005; Schwadron et al. 2005). Footpoint motion at the Sun associated with differential rotation (Fisk 1996; Fisk et al. 1999) and interchange reconnection (Fisk & Schwadron 2001) create the condition where magnetic field lines are connected across the source regions of slow and fast solar wind. This results in magnetic field lines that are dragged out more quickly in the fast solar wind, creating a magnetic field structure in the inner heliosphere with a radial component significantly larger than that of a Parker Spiral magnetic field. The deviations in field direction are extremely prominent and commonly observed by Ulysses in corotating rarefaction regions (Murphy et al. 2002; Schwadron & McComas 2005; Schwadron et al. 2005). The magnetic field structure is illustrated in Fig. 2 (black field lines).
Appendix A describes the subParker magnetic field structure, and Appendix B derives the distance along magnetic field lines to the compression or shock. Figure 3 shows the distance to the SIR compressionor shock along the magnetic field (top panel), and the distance to the SIR in the radial direction (second panel). The subParker spiral (black curves) due to its larger radial component than the Parker Spiral (blue curves) provides a more direct connection through the rarefaction region to the SIR. The parameters associated with footpoint motion are taken from Schwadron (2002) and detailed in Appendices A and B.
The shorter connection to the site of acceleration for a subParker field configuration is evident in Fig. 3. Both the distance along the field to the SIR reverse compression or shock and the radial distance to this location are reduced along a subParker spiral.
The subParker magnetic structure also reduces the angle between the magnetic field and the normal of the shock or compression, and the injection energy into diffusive acceleration. Note that this field angle is reported in the corotating reference frame, a natural choice for treating particle acceleration associated with SIR Ions. Ions require a sufficient speed to be able to move upstream along the magnetic field relative to the shock or compression. This injection speed described in Appendix C is simply v_{inj} = u_{1}∕cosθ_{1} in the limit that particles are restricted to a magnetic flux tube. Here, u_{1} is the upstream speed and θ_{1} is the angle between the magnetic field and the normal to the shock or compression. This expression for the injection speed neglects crossfield diffusion and drift (associated with a small antisymmetric term in the diffusion tensor), but includes several of the key factors that control the injection speed. The inversecosine dependence indicates that ions must have higher speed to move upstream on field lines that make at large angles θ_{1} to the shock or compression. Increased plasma speed upstream also increases the injection speed. The injection speed and energy are solved for including the effects of cross field diffusion (we take κ_{⊥} ∕κ_{∥} = 0.01) and drift as detailed in Eq. (C.5) from Giacalone (2001).
The low injection speed at the SIR enables diffusive acceleration across a large range of energies from pickup ion energies (several keV) up through the energetic particle energies at > 1 MeV/nuc. The particle energy spectrum observed within SIRs attains, on average, a common function with a powerlaw for the distribution function close to v^{−5} and an exponential rollover at ~0.16 MeV, as detailed previously. In the next section, we use this common form of the spectrum and the observations from IS⊙IS to infer the rate associated with acceleration energetic particles through the inner heliosphere.
Fig. 2 Illustration of the magnetic field structure in a stream interaction region (SIR) adapted from Gosling et al. (1976). Within SIRs, Faster solar wind overtakes slower solar wind, forming a compression or shock (red dashed curves). The Parker Spiral magnetic field lines are shown as thin blue curves. With footpoint motion at the Sun, magnetic field lines (black curves) are connected between fast and slow solar wind and form the subParker spiral (Murphy et al. 2002; Schwadron & McComas 2005; Schwadron et al. 2005) in rarefaction regions where the fast solar wind stretches out magnetic field lines in the radial direction. 
Fig. 3 Distance along the subParker spiral (black curves) is shorter than along the Parker Spiral (blue curves) to Stream Interaction Regions, and a the subParker magnetic field is associated with a lower injection energy compared to a Parker field. We show (top panel) the distance along a magnetic field line to the SIR compression or shock, and (second panel) the radial distance to the SIR compression/shock. These distances are shown as a function of time from stream interfaces at r_{o} ~ 0.33 au where theobservations are made by IS⊙IS. The subParker spiral also causes a reduced angle (third panel) between the magnetic field and the normal to the SIR compression or shock. Note that the fieldtonormal angle at the SIR compression or shock is calculated in the corotating reference frame. The injection energy detailed in Appendix C is also shown (fourth panel). 
4 The energetic particle spectrum at SIRs
We use IS⊙IS observations to determine energetic particle fluxes on magnetic field lines that connect to the acceleration region in SIRs. We consider particle acceleration at SIRs involving a twostep process (Schwadron et al. 1996). In the first step suprathermal ions are accelerated, and the second step is due to diffusive shock acceleration at the compression or shock. The first step of the acceleration process occurs at lower suprathermal energies, whereas the second step is responsible for accelerating particles to higher MeV energies. As discussed further in the next session, there is no sharp energy boundary separating these two steps.
Bower et al. (2020) showed that the energetic particle spectra at SIR shocks are consistent with diffusive shock acceleration. These authors conclude that averaging over sufficiently long periods leads to distribution functions with f ∝ v^{−5.4}, close to the −5 powerlaws discussed by Fisk and Gloeckler. This supports the conclusion that the roughly fixed spectral behavior, the common particle speed distribution, may be the result of the superposition of stochastically distributed processes including shocks, compressions and turbulence (Schwadron et al. 2010).
We use the common particle speed distribution to understand the rates associated with particle acceleration in the SIR. At the SIR compression and reverse shock, we take the functional form for the distribution function specified by Eq. (1) with γ_{s} = −5.4 and the rollover begins at particle speed v_{s} = 4400 km s^{−1}. The question of how the common particle speed distribution is maintained is taken up in the next section.
In Appendix D, we show that with the common particle speed distribution and a fixed rate of particle acceleration (ȧ_{s}), the distribution function must increase exponentially with distance, as exp(ȧ_{s}r∕u). Here u is average radial solar wind speed across the SIR and the rate associated with particle acceleration is related to the average divergence of flow within the SIR, and the rate of statistical acceleration (see Eq. (D.4)).
If instead of compression and statistical acceleration within the SIR, there is cooling due to solar wind expansion, the distribution function decreases with radius, as r^{gγs∕3} where g is the solar wind expansion factor (e.g., for radial expansion, g = 2). The time profiles of energetic particles from IS⊙IS constrain first the rate associated with suprathermal acceleration during the merging of faster wind with slower wind within the SIR, and then the rate associated with cooling at distances where the SIR breaks down.
The substantial rise in both higher energy and lower energy energetic particle fluxes is observed by IS⊙IS up to τ_{peak} ~ 3 days after passage of the stream interface. For the subParker spiral, the field lines observed near τ_{peak}~ 3 days connect out to a radial distance of ~2.6 au.
This 2.6 au distance is not a coincidence, as explained in the following paragraph. The SIR forms compressions and then shocks due to the merging of the fast wind into slower wind. Typically, this process occurs with a reverse wave or shock moving backward (upstream) into the faster wind, and a forward wave or shock moving ahead (downstream) into the slower wind. The breakdown of the SIR occurs at the point where either the forward wave merges through all of the slow wind in front of it, or the reverse wave moves through the faster wind behind the SIR. PSP Observations show that prior to the passage of the interface, the slower wind persists for τ_{l} ~ 3 days. This period of slower wind constitutes a change in longitude of Δϕ_{l} = Ω_{⊙}τ_{l} = 41.5°. We observe the faster stream with radial speed V_{h} ~ 530 km s^{−1}, whereas the slower stream moves with radial speed V_{l} ~320 km s^{−1} and the intermediate speed wind within the SIR has radial speed V_{m} = 425 km s^{−1}. In the corotating frame these streams move outward along an Archimedian spiral. At a radial distance given by r_{peak}= r_{o} + τ_{l}V_{m}V_{l}∕(V_{m} − V_{l}) = 2.6 au, the slower wind spiral on the forward interface intersects with the spiral associated with intermediate speed wind observed by PSP near the stream interface. In other words, it is at the radial distance of 2.6 au where the faster wind within the SIR sweeps up the majority of slower wind in front of the SIR. This corresponds to where IS⊙IS observes the peak in energetic particle fluxes from the SIR and suggests that the particle acceleration within the SIR persists while streams merge, and accordingly suprathermal fluxes increase. After streams within the SIR have merged, the compression of the SIR gives way to expansion, causing the suprathermal distribution to cool, and accordingly energetic particle fluxes to decrease.
We analytically solve for the distribution function at the SIR reverse compression and shock (Appendix D), and numerically model the propagation of energetic particles back to the location of PSP (Appendix E). Acceleration at the rate ȧ_{s} explains the rising level of the suprathermal population out to radius r_{peak} while the faster solar wind continues to overtake slower wind within the SIR. Beyond r_{peak}, the distributions cool at a rate that is determined from the expansion of solar wind (D.6), which we parameterize according to the expansion factor g.
The propagation model solves for diffusion of energetic particles in the rarefaction region trailing the SIR. We solve for diffusion parallel and perpendicular to the magnetic field (Appendix E). The parallel diffusion coefficient is scaled with radial distance ${\lambda}_{\parallel}={\lambda}_{0}{(r/{r}_{1})}^{1.3}$ (Erdos et al. 1999) and we take κ_{⊥}∕κ_{∥} = 0.01 which is on the low end of the range generally considered by Giacalone & Jokipii (1999).
Results of the model are compared to observations in Fig. 4. The model uses a scattering mean free path of λ_{0} = 1 au, a rate associated with particle acceleration ȧ_{s} = 7.1 × 10^{−6} s^{−1} (equivalent to a timescale for acceleration of ~1.6 days), an expansion factor of g = 3.1, and a peak injection efficiency of ϵ = 1∕2. The value of the rate associated with particles acceleration is chosen to account for the rise in fluxes over the first 3 days after passage of the stream interface, and the expansion factor is chosen to account for the subsequent fall in energetic particle fluxes. These results are discussed further in the next section where we show that the deduced rate associated with particle acceleration is consistent with a combination of compression within the SIR and statistical acceleration. Further, the expansion factor is consistent with weak superradial expansion driven by the SIR overpressure that occurs after the streams have merged within the SIR.
The long scattering mean free path is needed to account for the similar time evolution of lower energy and higher energy fluxes. Witha much shorter scattering mean free path, the ratio of the lower energy to higher energy fluxes would drop quickly after passage of the stream interface. Instead, we observe both lower energy (200–500 keV fluxes) and higher energy (1–5 MeV) fluxes to persist many days after the passage of the stream interface. This indicates that both the lower and higher energy fluxes are weakly modulated by the solar wind.
Another indication of the weak modulation is found by examining the slopes in the modeled distributions (Fig. 5). Note that the powerlaw indexes are relatively stable functions of time past the stream interface. The observations early in the event (the first day) show steeper powerlaws, and afterward the powerlaws are observed to become fairly stable with time (Joyce et al. 2021), in agreement with the simulation. Strong modulation causes an increase in the powerlaw index, and a hardening of spectrum particularly at lower energies. The long scattering mean free path indicated here is also consistent with observations of pickup ions from Ulysses/SWICS that showed large anisotropies (e.g., Gloeckler et al. 1995; Isenberg 1997; Schwadron 1998).
Several important differences are observed between observations and simulations. First, the spectral index of the 1–5 MeV differential fluxes are generally in the range of −5 to −6 (corresponding to powerlaw indexes in the distribution function of −12 to −13), whereas simulations show differential flux powerlaws of approximately − 4 (corresponding to a powerlaw index for the distribution function of −10). We discuss this difference further in the next section as a possible indication of broken powerlaw distributions from SIRs, as opposed to the exponential rollover used in our empirical form for the speed distribution. Second, as already mentioned, the spectral indices during the first day in the lower energy range are observed to be approximatly − 3 (Joyce et al. 2021) (powerlaw index −8) whereas simulations show spectral indexes for the differential flux of approximately − 2.5 (powerlaw index −7) near 250 keV. This suggests that early in the SIR acceleration process, suprathermal distributions are slightly softer than later in the acceleration process when the SIR compression has formed fully into a shock.
In the next section, we further discuss the implications of IS⊙IS observationspertaining to both the magnetic field structure within the rarefaction region, and the processes responsible for suprathermal particle acceleration.
Fig. 4 Comparison between observed (blue data points) and modeled (curves) differential fluxes at higher and lower energies. Modeled energetic particles are accelerated at the SIR reverse compression or shock and propagate back to PSP along subParker field lines (black curves) and Parker field lines (blue curves). Observed differential fluxes (averaged over 1–5 MeV, top, and 200–500 keV, bottom) are compared with modeled fluxes (2 MeV, top, and 250 keV, bottom). 
Fig. 5 Spectral index s of the modeled differential fluxes as a function of time after passage of the stream interface. Powerlaw indexes γ_{s} for the distribution correspond to γ_{s} = 2(s − 1). For example, a spectral index of s = −2 corresponds to a powerlaw index of γ_{s} = −6, which is close to the −5.4 powerlaw used for the common particle speed distribution. 
5 Discussion
5.1 Magnetic field structure and footpoint motion
The magnetic structure of the rarefaction region shows much more direct connection to the SIR for the subParker spiral. This is essential in explaining the long period of time (6 days) when energetic particle enhancements are observed after passage of the stream interface. Further, the Parker Spiral field lines quickly connect out to large distances in the heliosphere, and, as a result the emission from the SIR is expected to peak in less than 2 days following passage of the stream interface.
The subParker spiral (Murphy et al. 2002; Schwadron 2002; Schwadron & McComas 2005; Schwadron et al. 2005) confirms that footpoint motion at the Sun associated provides a magnetic fieldline connection between regions of fast and slower solar wind. The resulting largescale magnetic configuration (see Fig. 2) provides access for energetic particles throughout the inner heliosphere. The IS⊙IS observationsat PSP provide further validation for this largescale magnetic field structure.
An important question at this stage is how common the subParker field structure is. The subParker spiral appears to be observed in almost all rarefaction regions studied by Ulysses, but there has not yet been a systematic study that attempts to determine how frequently the subParker magnetic structure is present. It is interesting in Fig. 4 that variations in the time evolution suggest the presence of magnetic flux tube that may have variable largescale structure. For example, in the period between day 1 and 2 past the stream interface, there appears to be a structure that conforms more closely to the Parker Spiral, at least at lower energies. Since footpoint motion is driven in part by interchange reconnection, it is likely that footpoint motion is itself nonuniform. Nonuniformity in footpoint motion should lead to variations in the magnetic fieldline connection across regions of faster and slower solar wind source regions. From source regions where footpoint motion is restricted, we would expect the largescale magnetic structure to conform more closely to a Parker Spiral, whereas regions with faster footpoint motion would conform to the subParker spiral.
This discussion reveals that IS⊙IS observations of SIRs may hold the key to understanding the variability of footpoint motion at the Sun, which remains a fundamental question linked to the source of solar wind and the global structure of the corona (Fisk 1996; Fisk et al. 1999; Schwadron et al. 1999; Fisk & Schwadron 2001; Schwadron & McComas 2003a) and energetic particle propagation through the heliosphere (Schwadron & McComas 2003b). Therefore, systematic study of IS⊙IS observations from SIRs would help us understand the largescale structure of heliospheric magnetic field, its link to the source of the solar wind and the sources of energetic particle variability within the heliosphere.
5.2 Suprathermal particle acceleration
The rapid rise in fluxes observed by IS⊙IS in the buildup to the intensity peak is associated with acceleration of the suprathermal population within the SIR. From the observations, we derive a rate associated with particle acceleration ȧ_{s} = 7.1 × 10^{−6} s^{−1}. In Appendix D, we relate this rate to compression and statistical acceleration, Eq. (D.4). Observations from Ulysses show the buildup of density (n_{s}) within SIRs in the range of n_{s}∕n_{0} = 10–15 times that (n_{0}) in the ambient surrounding solar wind (Crooker et al. 1999). Because of mass conservation, this buildup in density is used to determine the average compression within the SIR: $$\begin{array}{lll}{\dot{a}}_{\text{s}}^{\text{comp}}\hfill & =\hfill & {\gamma}_{\text{s}}{\langle \nabla \cdot u\rangle}_{\text{s}}/3\hfill \\ \hfill & =\hfill & ({\gamma}_{\text{s}}/3){u}_{\text{s}}\mathrm{ln}({n}_{\text{s}}/{n}_{0})/r.\hfill \end{array}$$(2)
Here, ${\langle \nabla \cdot u\rangle}_{\text{s}}={u}_{\text{s}}\mathrm{ln}({n}_{\text{s}}/{n}_{0})/r$ is the divergence of the bulk flow velocity within the SIR, u_{s} is the flow speed within the SIR, and r ≈ 2.6 au is the distance up to the density peak. With these quantities, we find ${\dot{a}}_{\text{s}}^{\text{comp}}$ in the range 4.5 × 10^{−6} to 5.3 × 10^{−6} s^{−1}, which is ~ 65−75% of the rate of acceleration needed to explain the buildup in the suprathermal flux.
Statistical acceleration through waveparticle interactions also contributes to the rate of suprathermal acceleration. For example, transittime damping of magnitude fluctuations in the magnetic field is a known source of rapid particle acceleration (Schwadron et al. 1996). Rates associated statistical acceleration were determined from Ulysses/SWICS observations both from observed fluctuations in the magnetic field and from the tails of suprathermal distribution functions. Rates for secondorder velocityspace diffusion were found to range from D_{pp} ∕p^{2} ~ 2 × 10^{−7} to 8 × 10^{−7} s^{−1} at ~3 times the solar wind speed. As discussed in Appendix D, this second–order diffusion contributes to the acceleration at rate ${\dot{a}}_{\text{s}}^{\text{stat}}\approx {\gamma}_{\text{s}}^{2}{D}_{pp}/{p}^{2}\approx 5.8\times {10}^{6}$ s^{−1} to 2.3 × 10^{−5} s^{−1}, which is 80–330% of the needed rate associated with particle acceleration. Both compression and statistical acceleration in SIRs contribute to the acceleration of suprathermal protons. Further, processes involving secondorder statistical acceleration appear likely to contribute the majority of the needed rate associated suprathermal acceleration.
The twostep acceleration process within SIRs requires some additional explanation to account for observations. If ions were first statistically accelerated and then injected into diffusive acceleration, there would be a clear break in the energy spectrum between the ions with energies below the injection energy and higher energy particles involved in diffusive acceleration at the SIR shock or compression. The fact that no break is observed is an indication that the twostep acceleration process does not have a clear boundary between lower energy suprathermal ions and higher energy particles that are diffusively accelerated. Simulations by Chen et al. (2015) confirm this point. Within SIRs there is a clear powerlaw from suprathermal energies up to ~ 150 keV, which is likely the result of multiple processes contributing to the acceleration of the suprathermal ions that are injected into diffusive acceleration.
Removing the first step of the acceleration process also poses problems. If we remove the first step, the diffusively accelerated powerlaws should not build up increasing fluxes with distance through the inner heliosphere. In this case, we would expect a powerlaw from the injection energy, and a rollover or a broken powerlaw (Schwadron et al. 2015a,b) to higher energies. However, IS⊙IS data show the build up of suprathermal fluxes up to the peak fluxes (Fig. 4), indicating that the first step of suprathermal acceleration must be occurring.
It is likely that both suprathermal particle acceleration via waveparticle interactions and compression build up fluxes within the SIR. STEREOA/PLASTIC observations directly show the compression and compressive turbulence that increase the pickup ion cutoff energy in SIRs (Bower et al. 2019). As the compression regions bounding the SIR form into shocks, suprathermal ions that form powerlaw tails participate in both firstorder acceleration at the shock or compression, and secondorder acceleration and compressive acceleration inside the SIR. This explains both the maintenance of the powerlaw from diffusive acceleration and the buildupof fluxes from compressive acceleration and statistical acceleration inside or near the SIR. The observed powerlaw is therefore a superposition of the contributions to suprathermal acceleration from compressions, shocks, and turbulence (Schwadron et al. 2010). This implies that the observed powerlaw is not a singular reflection of any individually observed compression ratio, and the superposition process maintains greater stability in the observed powerlaw.
Our discussion of suprathermal acceleration up to this point appears to neglect the specific acceleration process that Fisk and Gloeckler argue acts to form observed powerlaw tails (Fisk & Gloeckler 2006, 2008, 2014). However, the averaging implicit to the superposition conjecture (Schwadron et al. 2010) reproduces features associated with a variety of different acceleration mechanisms. For example, Fisk et al. (2010) consider three possible approaches for explaining the common spectral form: (1) the acceleration mechanism of Fisk and Gloeckler in which energy is redistributed from a core particle population into the suprathermal tail; (2) traditional stochastic acceleration in which particles are accelerated by damping turbulence; and (3) the statistical approach introduced by Schwadron et al. (2010) in which the −5 spectrum is formed by averaging over individual spectra. The study (Fisk et al. 2010) concludes that the statistical approach of Schwadron et al. can yield the −5 spectrum and is consistent with the results of Fisk and Gloeckler when the distribution functions for individual events and the averaging technique are compatible. The superposition conjecture can reproduce observations under a wide varietyof types of underlying variability. Bower et al. (2020) find significant levels of variability within the observations, and yet reveal the common spectrum when averaging over this variability. The emergence of the common spectrum after averaging out variability provides direct support for the superposition conjecture. Corroborating this further for the event shown in Fig. 1, we note that the event has an average spectral index of −2 (corresponding to a distribution function f ∝ v^{−6} velocity spectrum) (Joyce et al. 2021), which is similar to the index of −1.75 (corresponding to f∝ v^{−5.5} velocity spectrum) found by Bower et al. (2020).
It is notable that the early acceleration within the SIR (observed ~ 1−2 days after the stream interface passage) does not form the common particle energy spectrum. Over this period, Joyce et al. (2021) show a markedly softer spectrum in low energy fluxes. This trend can also be observed in Fig. 1 on DOY 320 where we observe an excess of 40 keV protons. This reflects the early development of the energy spectrum that hardens as the SIR compression moves beyond 1–1.5 au and begins to form into a more defined structure or a shock.
The emergence of a common spectrum from SIRs and CIRs requires that the acceleration process operate in a region where turbulence, compression and possibly shocks cause the buildup of the suprathermal particle population. The compression regions of SIRs and CIRs develop where faster solar wind streams overtake slower streams. It is here where the development of compressional acceleration, compressive waves and turbulence, and transittime damping (Schwadron et al. 1996) take place naturally. After the compression has developed fully, typically around or beyond 2.5 au, the compressions bounding SIRs and CIRs may also form into shocks. However, the development of shocks is not a necessary precursor for particle acceleration, and in many small SIRs, the faster and slower streams may merge prior to the development of a shock. We emphasize these points to avoid confusion about the development of the common spectrum: the suprathermal acceleration process occurs within or near the SIR or CIR compressed or shocked plasma, and energetic particles are transported from the sites of acceleration back to the location of PSP. As a result of the subParker spiral magnetic field, energetic particles are transported more readily from the SIR or CIR through the rarefaction region. However, it is very unlikely that significant acceleration occurs within the rarefaction region.
Starting ~3 days after passage of the stream interface, we observe a sharp decline in energetic particle fluxes (Fig. 4), which is readily explained by the expansion of the remnants of the SIR after the fast wind has overtaken the slower wind (beyond 2.6 au). The drop in fluxes is so rapid that we inferred an expansion factor of g ≈ 3, corresponding with superradial expansion. This overexpansion could be the result of the overpressure in the plasma and suprathermal ions that has built up from SIR compressions.
While this discussion has focused on H^{+}, the observed trends in energetic He ions are also consistent with ease of inward transport from the sites of acceleration into the location of PSP at ~0.33 au. In commenting on the same event reported here, but in the context of He ions, Desai et al. (2020) state: “the spectrum flattens but does not turn over as expected from energy losses due to adiabatic deceleration during transport from an SIR shock located well beyond 1 au”. In other words, the lowenergy spectrum of He also indicates the lack of modulation, and the ease of inward transport of He ions.
Desai et al. (2020) go on to describe observations of He ions in terms of “magnetic channels” invoked by McDonald & Burlaga (1985) to explain how SEPs accelerated near the Sun and Jovian electrons can propagate to Earth orbit (Mewaldt et al. 1976; Chenette 1980; Roelof et al. 1996). This magnetic channeling is likely a natural result of the more direct radial access provided by the subParker spiral magnetic field.
5.3 Higher energy particle acceleration in SIRs
The energetic particle fluxes at lower and higher energies (Fig. 4) track each other remarkably well. Notably, the excess of lower energy fluxes and the deficit at higher energies between day 1 and 2 past the interface is consistent with the acceleration process beginning at lower energies and extending to higher energies progressively as a function of distance and therefore time within or near the SIR. The powerlaw index determined from simulations (Fig. 5) is harder than observed at these higher energies (1–5 MeV), the energy spectrum appears to be a broken powerlaw (Joyce et al. 2021), and the higher energy spectral index is approximately −5.1 (Cohen et al. 2020; Joyce et al. 2021).
The broken powerlaw distribution is commonly observed in large solar energetic particle events (Mewaldt et al. 2012). One explanation is that these broken powerlaws (Schwadron et al. 2015b) are associated with CME expansion regions low in the corona that are effective accelerators over a finite spatial region. There is a rigidity regime where particles effectively diffuse away and escape from the acceleration sites that leads to the formation of broken powerlaw distributions. An extension of this idea is that SIR compressions and shocks also have a regime of high rigidity (and high energy) ions that diffuse away and escape the acceleration region to form broken powerlaws.
6 Summary and conclusions
We have examined the distributions observed by IS⊙IS when PSP was near 0.33 au following the passage of a stream interface on DOY 318–325 in 2018. Details concerning the energetic particle distributions are published in a companion paper (Joyce et al. 2021). We model distributions at the SIR compression and shock based on a common form of the particle energy distribution observed throughout the inner heliosphere (Gloeckler & Fisk 2006, 2010; Fisk & Gloeckler 2006, 2008; Fisk et al. 2010; Bower et al. 2020). We compare the results of a propagation model using a Parker field and a subParker Magnetic field configuration.
Observations extending from ~30 keV up to ~5 MeV show energetic particle flux enhancements for roughly a week past the stream interface. Five days after passage of the stream interface, a standard Parker field configuration has field lines that extend from PSP over distances of more than 12 au to the SIR. These large distances along the Parker Spiral field lines make it difficult to understand how enhancements in fluxes can be observed at PSP so long after the passage of the stream interface, particularly at low energies.
The subParker magnetic field configuration provides a much shorter connection to the SIR, and explains why IS⊙IS observes energetic particle enhancements long after the stream interface passage. The magnetic field configuration arises due to footpoint motions at the Sun driven by differential rotation and interchange reconnection. The energetic particle evolution observed by IS⊙IS provides evidence that footpoint motion at the Sun structures the magnetic field throughout the inner heliosphere.
The rapid rise in energetic particle fluxes observed before day 3 after the passage of the stream interface indicates acceleration of suprathermal particles associated both with compression within the SIR and statistical acceleration. The interplay between the acceleration of suprathermal ions in the SIR and diffusive shock acceleration to >MeV energies explains the development of the common particle energy spectrum and the rapid rise in energetic particle fluxes.
For the period following day 3 after the passage of the stream interface, energetic particle fluxes rapidly decrease, reflecting the subParker fieldline connection to the SIR as it decays. During this period, the subParker field lines connect to the SIR at radial distances beyond 2.6 au where the faster wind has fully overtaken the slower solar wind in front of it. The cooling of energetic particle distributions at distances beyond 2.6 au are consistent with overexpansion of the solar wind in this region driven by the overpressure of the plasma and suprathermal ions within the SIR.
This paper develops a methodology to analyze the observations by IS⊙IS on PSP associated with Stream Interaction Regions. The new vantage point from PSP close to the Sun enables observation of the energetic particle fluxes from the Stream Interaction Region as it develops and then decays further out in the heliosphere. Suprathermal acceleration is critical in explaining the evolution energetic particle fluxes from the Stream Interaction Region. Theevolution of energetic particles from the Stream Interaction Region provides a unique way to probe the global magnetic structure of the inner heliosphere. Thus, the new view of Stream Interaction Regions observed by IS⊙IS on Parker Solar Probe provides evidence of the deviation from the standard Parker Spiral in rarefaction regions in the inner heliosphere, and the rate associated with suprathermal particle acceleration within Stream Interaction Regions.
Acknowledgements
We are deeply indebted to everyone that helped make the Parker Solar Probe (PSP) mission possible. We thank all of the outstanding scientists, engineers, technicians, and administrative support people across all of the IS⊙IS, FIELDS and SWEAP institutions that produced and supported the IS⊙IS, FIELDS and SWEAP instrument suites, support its operations and the scientific analysis of its data. This work was supported as part of the PSP mission under contract NNN06AA01C. NAS and CMSC were also supported under LWS grant “New Insights into SEP Sources, Acceleration, and Propagation: an Integrated ObservationModeling Approach”, grant number 80NSSC19K0067. The IS⊙IS data and visualization tools are available to the community at: https://spacephysics.princeton.edu/missionsinstruments/isois; data are also available via the NASA Space Physics Data Facility (https://spdf.gsfc.nasa.gov/). Parker Solar Probe was designed, built, and is now operated by the Johns Hopkins Applied Physics Laboratory as part of NASA’s Living with a Star (LWS) program (contract NNN06AA01C). Support from the LWS management and technical team has played a critical role in the success of the Parker Solar Probe mission.
Appendix A The subParker spiral
We follow the ballistic propagation of plasma parcels from the Sun within the solar wind to determine the structure of the magnetic field in the inner heliosphere. This calculation follows Schwadron (2002), and simplifies the formalism for application to energetic particle acceleration and propagation within CIRs. Figure A.1 shows the configuration near the Sun in the corotating reference frame along a boundary surface (at radius R_{B}) where footpoint motion moves magnetic footpoints between regions of faster wind (with speed V +δV∕2) and regions of slower wind (with speed V − δV∕2). The boundary surface is at a radius where the field has roughly reached a pressure balance (balance between magnetic pressure and ram pressure). Footpoints rotate in the azimuthal direction at rate ω_{ϕ} = −ω_{ϕ} in the opposite direction of the Sun’s rigid rotation (the rigid rotation rate is Ω_{⊙}). Footpoints also move in colatitudeat rate ω_{θ} = ω_{θ}.
We follow several parcels of plasma that are magnetically connected, with footpoint motion included. The first parcel (a) of plasma is emitted at time t_{a0} = −δt∕2 from the faster wind source region with azimuthal angle ϕ_{0} + δϕ∕2 and colatitude θ_{0} − δθ∕2. The second parcel (b) of plasma is emitted attime t_{b0} = δt∕2 from the slower source region with azimuthal angle ϕ_{0} − δϕ∕2 and colatitude θ_{0} + δθ∕2. Magnetic connection with footpoint motion requires that δϕ = ω_{ϕ}δt and δθ = ω_{θ}δt.
The plasma parcels move out ballistically with the solar wind. Therefore, at time t parcel (a) has coordinates $$\begin{array}{lll}{r}_{\text{a}}\hfill & =\hfill & {R}_{\text{B}}+(V+\delta V/2)(t+\delta t/2)\hfill \\ {\theta}_{\text{a}}\hfill & =\hfill & {\theta}_{0}\delta \theta /2\hfill \\ {\varphi}_{\text{a}}\hfill & =\hfill & {\varphi}_{0}+\delta \varphi /2{\Omega}_{\odot}(t+\delta t/2),\hfill \end{array}$$
and parcel (b) has coordinates $$\begin{array}{lll}{r}_{\text{b}}\hfill & =\hfill & {R}_{\text{B}}+(V\delta V/2)(t\delta t/2)\hfill \\ {\theta}_{\text{b}}\hfill & =\hfill & {\theta}_{0}+\delta \theta /2\hfill \\ {\varphi}_{\text{b}}\hfill & =\hfill & {\varphi}_{0}\delta \varphi /2{\Omega}_{\odot}(t\delta t/2).\hfill \end{array}$$
The displacement between parcels (a) and (b) is: $$\begin{array}{lll}\Delta r\hfill & =\hfill & {r}_{\text{a}}{r}_{\text{b}}=V\delta t\left(1+\frac{\delta V}{\delta t}\frac{(r{R}_{\text{B}})}{{V}^{2}}\right)\hfill \\ \Delta \theta \hfill & =\hfill & {\theta}_{\text{a}}{\theta}_{\text{b}}\text{}=\delta t{\omega}_{\theta}\hfill \\ \Delta \varphi \hfill & =\hfill & {\varphi}_{\text{a}}{\varphi}_{\text{b}}=\delta t({\Omega}_{\odot}\delta {\omega}_{\varphi}).\hfill \end{array}$$
We solve for the change in wind speed δV as the footpoint undergoes displacement δϕ and δθ, $$\begin{array}{lll}\delta V\hfill & =\hfill & \left(\frac{\delta {V}_{\varphi}}{\delta \varphi}\right)\delta \varphi +\left(\frac{\delta {V}_{\theta}}{\delta \theta}\right)\delta \theta \hfill \\ \hfill & =\hfill & \left(\frac{\delta {V}_{\varphi}}{\delta \varphi}\right)\left{\omega}_{\varphi}\right\delta t+\left\frac{\delta {V}_{\theta}}{\delta \theta}\right{\omega}_{\theta}\delta t.\hfill \end{array}$$(A.10)
In Eq. (A.10), the terms δV_{ϕ} and δV_{θ} refer to the small changes in solar wind speed in the azimuthal and colatitude directions, respectively. By construction both of the directional contributions to δV speed changes are positive since we find the difference between parcel (a) released at time − δt∕2 on a faster stream and parcel (b) released at time δt∕2 on a slower stream. However, δV_{θ} is a negative quantity, which requires that its contribution to the net speed change is the absolute value of the speed change in the colatitude direction. As a result, we can express the rate of speed change in a more general manner, $$\begin{array}{lll}\frac{\delta V}{\delta t}\hfill & =\hfill & \left(\frac{\delta {V}_{\varphi}}{\delta \varphi}\right){\omega}_{\varphi}+\frac{\delta {V}_{\theta}}{\delta \theta}{\omega}_{\theta}\hfill \\ \hfill & =\hfill & {R}_{\text{B}}{\omega}_{\text{B}}\cdot \nabla V{}_{r={R}_{\text{B}}},\hfill \end{array}$$
where $${\omega}_{\text{B}}={\omega}_{\theta}{\text{ehat}}_{\theta}+\mathrm{sin}\theta {\omega}_{\varphi}{\text{ehat}}_{\varphi}.$$(A.13)
We notethat the R_{B} enters the equation but cancels when multiplied by the gradient operator. Further, the quantity R_{B} ω_{B} represents the transverse velocity associated with footpoint motion at the boundary radius R_{B}. Since the magnetic field must be parallel to the displacement vector, we can solve for the direction of the magnetic field: $$\begin{array}{lll}B\hfill & =\hfill & A(r)\{\left(1{R}_{\text{B}}{\omega}_{\text{B}}\cdot \nabla V{}_{r={R}_{\text{B}}}\frac{(r{R}_{\text{B}})}{{V}^{2}}\right){\text{ehat}}_{r}\hfill \\ \hfill & \hfill & \frac{{\omega}_{\text{B}}r}{V}\frac{{\Omega}_{\odot}r\mathrm{sin}\theta}{V}{\text{ehat}}_{\varphi}\}.\hfill \end{array}$$(A.14)
We take the interface between fast and slow wind to be tilted by angle Ψ with respect to the azimuthal direction. On the inner boundary surface, at radius R_{B}, the unit vector along the stream interface is defined $${\text{nnehat}}_{I}=\mathrm{sin}\Psi {\text{ehat}}_{\theta}+\mathrm{cos}\Psi {\text{ehat}}_{\varphi}.$$(A.15)
On the inner boundary the unit vector normal to the stream interface is defined, $${\text{nnehat}}_{\perp}{}_{r={R}_{\text{B}}}\approx \mathrm{cos}\Psi {\text{ehat}}_{\theta}+\mathrm{sin}\Psi {\text{ehat}}_{\varphi}.$$(A.16)
Therefore the footpoint rotation rate normal to the stream interface is $${\omega}_{\perp}={\omega}_{\theta}\mathrm{cos}\Psi +{\omega}_{\varphi}\mathrm{sin}\theta \mathrm{sin}\Psi .$$(A.17)
The footpoint rotation rate times the velocity gradient is $$\begin{array}{lll}{R}_{\text{B}}{\omega}_{\text{B}}\cdot \nabla V{}_{r={R}_{\text{B}}}\hfill & =\hfill & {\omega}_{\theta}\frac{\partial V}{\partial \theta}+{\omega}_{\varphi}\frac{\partial V}{\partial \varphi}\hfill \\ \hfill & =\hfill & {R}_{\text{B}}{\omega}_{\perp}\frac{\partial V}{\partial {s}_{\perp}},\hfill \end{array}$$
where the velocity gradient normal to the interface is $${R}_{\text{B}}\frac{\partial V}{\partial {s}_{\perp}}=\left({\omega}_{\theta}\frac{\partial V}{\partial \theta}+{\omega}_{\varphi}\frac{\partial V}{\partial \varphi}\right){({\omega}_{\theta}\mathrm{cos}\Psi +{\omega}_{\varphi}\mathrm{sin}\theta \mathrm{sin}\Psi )}^{1}.$$(A.20)
In the application to the rarefaction region considered here ω_{ϕ} is a negative quantity since longitudinal footpoint motion opposes solar rotation, and ω_{θ} is positive. This implies that ω_{⊥} < 0, $${\omega}_{\perp}=({\omega}_{\theta}\mathrm{cos}\Psi +{\omega}_{\varphi}\mathrm{sin}\theta \mathrm{sin}\Psi ).$$(A.21)
The velocity gradient in colatitude is a negative quantity, ∂V∕∂θ < 0, and the velocity gradient in azimuthal is a positive quantity, ∂V∕∂ϕ > 0. Therefore, the velocity gradient normal to the interface is a positive quantity, $${R}_{\text{B}}\frac{\partial V}{\partial {s}_{\perp}}=\left({\omega}_{\theta}\left\frac{\partial V}{\partial \theta}\right+\left{\omega}_{\varphi}\right\frac{\partial V}{\partial \varphi}\right){({\omega}_{\theta}\mathrm{cos}\Psi +{\omega}_{\varphi}\mathrm{sin}\theta \mathrm{sin}\Psi )}^{1}.$$(A.22)
Given these properties of the velocity and footpoint rotation rates, we express the magnetic field in the rarefaction region as follows $$\begin{array}{lll}B\hfill & =\hfill & A(r)\{\left(1+{R}_{\text{B}}\left{\omega}_{\perp}\right{\frac{\partial V}{\partial {s}_{\perp}}}_{r={R}_{\text{B}}}\frac{(r{R}_{\text{B}})}{{V}^{2}}\right){\text{ehat}}_{r}\hfill \\ \hfill & \hfill & \frac{{\omega}_{\text{B}}r}{V}\frac{{\Omega}_{\odot}r\mathrm{sin}\theta}{V}{\text{ehat}}_{\varphi}\}.\hfill \end{array}$$(A.23)
We solve for the magnitude A of the magnetic field in the same manner as detailed by Schwadron (2002) by requiring conservation of magnetic flux across the interface between fast wind and the rarefaction region. The unit vector along a fast wind streamline is $${\text{nnehat}}_{\text{s}}=\left({\text{ehat}}_{r}{\text{ehat}}_{\varphi}{\Omega}_{\odot}r\mathrm{sin}\theta /V\right){\left(1+{({\Omega}_{\odot}r\mathrm{sin}\theta /V)}^{2}\right)}^{1/2}.$$(A.24)
The unit vector perpendicular to the stream interface ê_{I} and perpendicular to the streamline is $$\begin{array}{lll}{\text{nnehat}}_{\perp}(r)\hfill & =\hfill & \left({\Omega}_{\odot}r\mathrm{sin}\theta \mathrm{sin}\Psi {\text{ehat}}_{r}V\mathrm{cos}\Psi {\text{ehat}}_{\theta}+V\mathrm{sin}\Psi {\text{ehat}}_{\varphi}\right)\hfill \\ \hfill & \hfill & \times {\left({\Omega}_{\odot}^{2}{r}^{2}{\mathrm{sin}}^{2}\theta {\mathrm{sin}}^{2}\Psi +{V}^{2}\right)}^{1/2}.\hfill \end{array}$$(A.25)
The magnetic field on the side of the interface with fast solar wind follows a pattern close to the Parker spiral but modified to account for footpoint motion: $${B}_{\text{f}}={B}_{\text{fB}}{\left(\frac{{R}_{\text{B}}}{r}\right)}^{2}\left({\text{ehat}}_{\text{r}}\frac{{\omega}_{\text{B}}r}{V}\frac{{\Omega}_{\odot}r\mathrm{sin}\theta}{V}{\text{ehat}}_{\varphi}\right),$$(A.26)
where B_{fB} is the magnetic field strength on the inner boundary surface. The magnetic field on the rarefaction side of the interface is consistent with Eq. (A.26): $$\begin{array}{lll}{B}_{\text{a}}\hfill & =\hfill & A(r)\{\left(1+{R}_{\text{B}}\left{\omega}_{\perp}\right{\frac{\partial V}{\partial {s}_{\perp}}}_{r={R}_{\text{B}}}\frac{(r{R}_{\text{B}})}{{V}^{2}}\right){\text{ehat}}_{\text{r}}\hfill \\ \hfill & \hfill & \frac{{\omega}_{\text{B}}r}{V}\frac{{\Omega}_{\odot}r\mathrm{sin}\theta}{V}{\text{ehat}}_{\varphi}\}.\hfill \end{array}$$(A.27)
We require conservation of magnetic flux across the interface: B_{f} ⋅ ê_{⊥} = B_{a} ⋅ê_{⊥}. Therefore, $$A(r)={B}_{\text{fB}}{\left(\frac{{R}_{\text{B}}}{r}\right)}^{2}{\left(1+{R}_{\text{B}}{\frac{\partial V}{\partial {s}_{\perp}}}_{r={R}_{\text{B}}}\frac{(r{R}_{\text{B}})}{{V}^{2}}{\Omega}_{\odot}\mathrm{sin}\theta \mathrm{sin}\Psi \right)}^{1}.$$(A.28)
In the treatment of footpoint motion used in this paper, we use the following parameters (adapted from Schwadron 2002): $$\begin{array}{lll}{\omega}_{\theta}\hfill & =\hfill & 0.1{\Omega}_{\odot}\hfill \\ {\omega}_{\varphi}\hfill & =\hfill & 0.17{\Omega}_{\odot}.\hfill \end{array}$$(A.29)
Fig. A.1 Differential motion creates the subParker spiral. Blackcurves show the streamlines associated with parcel (a) and parcel (b) in the corotating reference frame. Footpoint motion provides a magnetic connection between parcel (a) and (b), which implies that the magnetic field is directed along the displacement between these plasma parcels. 
We introduce the term α to quantify the radial speed gradient within the rarefaction region, $$\alpha ={R}_{\text{B}}\left{\omega}_{\perp}\right{\frac{\partial V}{\partial {s}_{\perp}}}_{r={R}_{\text{B}}}.$$(A.30)
Note that α represents an effective acceleration and has units of speed over time (or km s^{−2}). At the position of PSP, the stream interface rotates past the spacecraft at an angular rate of Ω_{⊙}. Given an observed reduction in solar wind speed of 55 km s^{−1} in 6 days (after the passage of the stream interface), we find a value of α = 1.8 × 10^{−5} km s^{−2}.
Appendix B Distance to the SIR compression or shock
In this appendix, we consider a spacecraft located at a specified set of heliographic coordinates (r_{o}, θ_{o}, ϕ_{o}) and derive expressions for the coordinates (r_{s}, θ_{s}, ϕ_{s}) where the solar wind magnetic field connects the observer to the SIR compression or shock. We begin by considering these expressions for a standard Parker spiral. The SIR is created by the interaction of a slower and faster streams, with speeds V_{l} and V_{h}, respectively.In the corotating coordinate system, the faster stream has the following pathline solution: $$\begin{array}{ll}r(\varphi )\hfill & ={r}_{\text{o}}\frac{(\varphi {\varphi}_{\text{o}}){V}_{\text{o}}}{{\Omega}_{\odot}}\hfill \\ \theta \hfill & ={\theta}_{\text{o}},\hfill \end{array}$$
where V_{o} is the speed at the observer’s position. We take the compression or shock to move out with intermediate speed V_{m} = (V_{l} + V_{h})∕2 and the associated pathline is given by: $$\begin{array}{ll}{r}_{\text{m}}(\varphi )\hfill & ={r}_{\text{o}}\frac{(\varphi {\varphi}_{\text{m}}){V}_{\text{m}}}{{\Omega}_{\odot}}\hfill \\ \theta \hfill & ={\theta}_{\text{o}}.\hfill \end{array}$$
where we use ϕ_{m} to represent the azimuthal angle of the initial stream interface. The intersection of these pathlines occurs at radius: $$\begin{array}{lll}{r}_{\text{sh}}\hfill & =\hfill & {r}_{\text{o}}+\frac{({\varphi}_{\text{m}}{\varphi}_{\text{o}})}{{\Omega}_{\odot}}\frac{{V}_{\text{m}}{V}_{\text{o}}}{{V}_{\text{o}}{V}_{\text{m}}}.\hfill \end{array}$$(B.5)
In the more general case of the subParker spiral in the rarefaction region, the pathline is solved for using the following differential form: $$\begin{array}{lll}\frac{\text{d}r}{\text{d}s}\hfill & =\hfill & \frac{A}{B}\left(1+{R}_{\text{B}}\left{\omega}_{\perp}\right{\frac{\partial V}{\partial {s}_{\perp}}}_{r={R}_{\text{B}}}\frac{(r{R}_{\text{B}})}{{V}^{2}}\right)\hfill \\ \frac{\text{d}\theta}{\text{d}s}\hfill & =\hfill & \frac{A}{B}\frac{{\omega}_{\theta}}{V}\hfill \\ \frac{\text{d}\varphi}{\text{d}s}\hfill & =\hfill & \frac{A}{B}\frac{{\Omega}_{\odot}\left{\omega}_{\varphi}\right}{V}.\hfill \end{array}$$
To solve the parametric pathline equations explicitly, we express the differential distance along the fieldline as, d s = (V B∕A)dτ. The pathline now takes the form $$\begin{array}{lll}\frac{\text{d}r}{\text{d}\tau}\hfill & =\hfill & V+\alpha (r{R}_{\text{B}})/V\hfill \\ \frac{\text{d}\theta}{\text{d}\tau}\hfill & =\hfill & {\omega}_{\theta}\hfill \\ \frac{\text{d}\varphi}{\text{d}\tau}\hfill & =\hfill & ({\Omega}_{\odot}{\omega}_{\varphi}).\hfill \end{array}$$
The quantity dτ represents an infinitesimal displacement in time along a fieldline history. The equation for r is solved byintroducing the displacement with respect to the inner boundary, R = r − R_{B}, and expressing (B.9) as follows, $$\mathrm{exp}\left({\displaystyle {\int}_{0}^{\tau}\frac{\alpha}{V}}\text{d}{\tau}^{\prime}\right)\frac{\text{d}}{\text{d}\tau}\left[\mathrm{exp}\left({\displaystyle {\int}_{0}^{\tau}\frac{\alpha}{V}}\text{d}{\tau}^{\prime}\right)R\right]=V.$$(B.12)
The solution is $$R(\tau )={R}_{\text{o}}\mathrm{exp}\left({\displaystyle {\int}_{0}^{\tau}\frac{\alpha}{V}}\text{d}{\tau}^{\prime}\right)+{\displaystyle {\int}_{0}^{\tau}V}\mathrm{exp}\left({\displaystyle {\int}_{{\tau}^{\prime}}^{\tau}\frac{\alpha}{V}}\text{d}{\tau}^{\prime \prime}\right)\text{d}{\tau}^{\prime}.$$(B.13)
where R_{o} = r_{o} − R_{B}. We take α = constant.
The footpoints of the fieldline on the corotating inner boundary have coordinates $$\begin{array}{lll}{\theta}_{\text{B}}(\tau )\hfill & =\hfill & {\theta}_{\text{Bo}}{\omega}_{\theta}\tau \hfill \\ {\varphi}_{\text{B}}(\tau )\hfill & =\hfill & {\varphi}_{\text{Bo}}{\omega}_{\varphi}\tau \hfill \\ \hfill & =\hfill & {\varphi}_{\text{Bo}}+\left{\omega}_{\varphi}\right\tau ,\hfill \end{array}$$
where (θ_{Bo}, ϕ_{Bo}) are the coordinates of the footpoint on the inner boundary associated with fieldline tied to the observer. The velocity along the fieldline history is $$\begin{array}{lll}V(\tau )\hfill & =\hfill & V({R}_{\text{B}},{\theta}_{\text{Bo}}{\omega}_{\theta}\tau ,{\varphi}_{\text{Bo}}{\omega}_{\varphi}\tau )\hfill \\ \hfill & \approx \hfill & {V}_{\text{o}}\tau {R}_{\text{B}}{\omega}_{\text{Bo}}\cdot \nabla V{}_{r={R}_{\text{B}}}\hfill \\ \hfill & \approx \hfill & {V}_{\text{o}}+\tau \alpha ,\hfill \end{array}$$(B.17)
where V_{o} = V (R_{B}, θ_{Bo}, ϕ_{Bo}) is the solar wind velocity at the observer location. Therefore, from Eq. (A.10) $$V(\tau )={V}_{\text{o}}+\alpha \tau .$$(B.18)
With this explicit form of the velocity, the integrals in Eq. (B.13) are directly solved, resulting in the following expression for the fieldline’s radial location R(τ): $$R(\tau )={R}_{\text{o}}\left(1+\frac{\alpha \tau}{{V}_{\text{o}}}\right)+({V}_{\text{o}}+\alpha \tau )\tau .$$(B.19)
We now solve for the point of intersection of the stream interface with the fieldline in the rarefaction region by equating the azimuthal angle of the stream interface, ϕ_{I}, with the azimuthal angle along the magnetic field line: $$\begin{array}{ll}{\varphi}_{\text{I}}\hfill & ={\varphi}_{\text{m}}\frac{{\Omega}_{\odot}(R{R}_{\text{o}})({\tau}_{\text{s}})}{{V}_{\text{m}}}\hfill \\ \varphi ({\tau}_{\text{i}})\hfill & ={\varphi}_{\text{o}}({\Omega}_{\odot}{\omega}_{\varphi}){\tau}_{\text{s}},\hfill \end{array}$$(B.20)
where ϕ_{m} is the azimuthal location of the stream interface at R = R_{o} and τ = τ_{s} is the traceback time at the intersection point. We then use Eq. (B.19) to convert Eq. (B.20) into a quadratic equation for τ: $$a{\tau}_{\text{s}}^{2}+b{\tau}_{\text{s}}\Delta \varphi =0,$$(B.21)
where Δϕ = ϕ_{m} − ϕ_{o}, a = Ω_{⊙}α∕V_{m}, and b = −(Ω_{⊙}−ω_{ϕ}) + (R_{o}α∕V_{o} + V_{o})Ω_{⊙}∕V_{m}. In the limit that α =0, Eq. (B.21) yields the solution τ_{s} = Δϕ∕[Ω_{⊙}(V_{o}∕V_{m} − 1) + ω_{ϕ}]. This particular solution yields an intersection at r_{s} given by Eq. (B.5) in the limit of small ω_{ϕ}. In the more general case that α > 0, we find that $${\tau}_{\text{s}}=\frac{b+\sqrt{{b}^{2}+4a\Delta \varphi}}{2a}.$$(B.22)
Given the solutions derived for the radial location of the connection to the compression or shock, we calculate the corresponding distance along the fieldline. In the case of the Parker field structure, the fieldline distance is calculated using $$\frac{\text{d}s}{\text{d}r}=\frac{B}{{B}_{\text{r}}}=\sqrt{1+{r}^{2}{\left(\frac{{\Omega}_{\odot}\mathrm{sin}\theta}{V}\right)}^{2}}.$$(B.23)
The integral form of Eq. (B.23) is as follows, $$\begin{array}{lll}\Delta s\hfill & =\hfill & \frac{V}{{\Omega}_{\odot}\mathrm{sin}\theta}{\displaystyle {\int}_{{x}_{\text{o}}}^{{x}_{\text{s}}}\sqrt{1+{x}^{2}}}\text{d}x\hfill \\ \hfill & =\hfill & \frac{V}{{\Omega}_{\odot}\mathrm{sin}\theta}[\frac{{x}_{\text{s}}\sqrt{1+{x}_{\text{s}}^{2}}}{2}\frac{{x}_{\text{o}}\sqrt{1+{x}_{\text{o}}^{2}}}{2}\hfill \\ \hfill & \hfill & +\frac{1}{2}\mathrm{ln}\left(\frac{{x}_{\text{s}}+\sqrt{1+{x}_{\text{s}}^{2}}}{{x}_{\text{o}}+\sqrt{1+{x}_{\text{o}}^{2}}}\right)],\hfill \end{array}$$
where x_{o} = r_{o}Ω_{⊙}sinθ∕V and x_{s} = r_{s}ω_{⊙}sinθ∕V. In the case of the magnetic field that includes footpoint motion, the differential equation along the magnetic field line is solved for using the quantity ds∕dτ: $$\begin{array}{lll}\frac{\text{d}s}{\text{d}\tau}\hfill & =\hfill & \sqrt{{\left(\frac{\text{d}r}{\text{d}\tau}\right)}^{2}+{r}^{2}{\left(\frac{\text{d}\theta}{\text{d}\tau}\right)}^{2}+{r}^{2}{\mathrm{sin}}^{2}\theta {\left(\frac{\text{d}\varphi}{\text{d}\tau}\right)}^{2}}\hfill \\ \hfill & =\hfill & \sqrt{{\left(V+\alpha R/V\right)}^{2}+{r}^{2}{\omega}_{\theta}^{2}+{r}^{2}{\mathrm{sin}}^{2}\theta {\left({\Omega}_{\odot}\left{\omega}_{\varphi}\right\right)}^{2}},\hfill \end{array}$$
where r(τ) = R(τ) + R_{B}, and V (τ) and R(τ) are specified by Eqs. (B.18) and (B.19). The resulting relation for the distance along the fieldline is, $$\begin{array}{lll}\Delta s\hfill & =\hfill & {\displaystyle {\int}_{0}^{{\tau}_{\text{s}}}\text{d}}\tau [{\left(V+\alpha R/V\right)}^{2}+{r}^{2}{\omega}_{\theta}^{2}\text{vphantom}+{r}^{2}{\mathrm{sin}}^{2}\theta {\left({\Omega}_{\odot}\left{\omega}_{\varphi}\right\right)}^{2}\hfill \\ \hfill & \hfill & {+{r}^{2}{\mathrm{sin}}^{2}\theta {\left({\Omega}_{\odot}\left{\omega}_{\varphi}\right\right)}^{2}]}^{1/2}.\hfill \end{array}$$(B.28)
A special case to consider is where α = 0 and the previous expression for the Δs can be made explicit: $$\begin{array}{lll}\Delta s\hfill & =\hfill & \frac{V}{\left\omega \right}[\frac{{x}_{\text{s}}^{\prime}\sqrt{1+{({x}_{\text{s}}^{\prime})}^{2}}}{2}\frac{{x}_{\text{o}}^{\prime}\sqrt{1+{({x}_{\text{o}}^{\prime})}^{2}}}{2}\hfill \\ \hfill & \hfill & +\frac{1}{2}\mathrm{ln}\left(\frac{{x}_{\text{s}}^{\prime}+\sqrt{1+{({x}_{\text{s}}^{\prime})}^{2}}}{{x}_{\text{o}}^{\prime}+\sqrt{1+{({x}_{\text{o}}^{\prime})}^{2}}}\right)],\hfill \end{array}$$(B.29)
where ${x}_{\text{o}}^{\prime}={r}_{\text{o}}\left\omega \right/V$, ${x}_{\text{s}}^{\prime}={r}_{\text{s}}\left\omega \right/V$ and $$\left\omega \right=\sqrt{{\omega}_{\theta}^{2}+{\mathrm{sin}}^{2}\theta {({\Omega}_{\odot}{\omega}_{\varphi})}^{2}}.$$(B.30)
In the case of the magnetic field that includes footpoint motion, the differential equation along the magnetic field line is solved
Appendix C Injectionenergy into diffusive shock acceleration
The injection energy is determined by the angle between the magnetic field and the shock normal. The shock normal is solved usingEq. (B.5) for the shock radius. This yields a shock or compression normal given by $${\widehat{e}}_{\text{sn}}=\frac{(r\mathrm{sin}\theta /{L}_{\text{s}}){\widehat{e}}_{\text{r}}+{\widehat{e}}_{\varphi}}{\sqrt{{(r\mathrm{sin}\theta /{L}_{\text{s}})}^{2}+1}},$$(C.1)
where $${L}_{\text{s}}=\frac{{V}_{\text{h}}}{{\Omega}_{\odot}}\frac{{V}_{\text{h}}+{V}_{\text{l}}}{{V}_{\text{h}}{V}_{\text{l}}}.$$(C.2)
With V_{l} = 320 km s^{−1} and V_{h} = 530 km s^{−1}, we find that L_{s} = 5.1 au. The unit vector along the magnetic field in the fast solar wind upstream of the shock or compression is $$\begin{array}{ll}{\widehat{e}}_{\text{B}}=\hfill & \left\{\left(1+\alpha \frac{(r{R}_{\text{B}})}{{V}^{2}}\right){\text{ehat}}_{\text{r}}\frac{{\omega}_{\theta}r}{V}{\text{ehat}}_{\theta}\frac{({\Omega}_{\odot}+{\omega}_{\varphi})r\mathrm{sin}\theta}{V}{\text{ehat}}_{\varphi}\right\}\hfill \\ \hfill & \times {\left\{{\left(1+\alpha \frac{(r{R}_{\text{B}})}{{V}^{2}}\right)}^{2}+{\left(\frac{{\omega}_{\theta}r}{V}\right)}^{2}+{\left(\frac{({\Omega}_{\odot}+{\omega}_{\varphi})r\mathrm{sin}\theta}{V}\right)}^{2}\right\}}^{1/2}.\hfill \end{array}$$(C.3)
The unit vector along the magnetic field for a Parker spiral is given by, $$\begin{array}{lll}{\widehat{e}}_{\text{B}}\hfill & =\hfill & \left\{{\text{ehat}}_{\text{r}}\frac{{\Omega}_{\odot}r\mathrm{sin}\theta}{V}{\text{ehat}}_{\varphi}\right\}{\left\{1+{\left(\frac{{\Omega}_{\odot}r\mathrm{sin}\theta}{V}\right)}^{2}\right\}}^{1/2}.\hfill \end{array}$$(C.4)
The injection speed v_{inj} (and associated injection energy) is found from the condition that the energetic particle anisotropy is sufficiently small (Giacalone 2001): $${v}_{\text{inj}}={u}_{\text{1}}{\left[1+\frac{{({\kappa}_{A}/{\kappa}_{\parallel})}^{2}{\mathrm{sin}}^{2}{\theta}_{1}+{(1{\kappa}_{\perp}/{\kappa}_{\parallel})}^{2}{\mathrm{sin}}^{2}{\theta}_{1}{\mathrm{cos}}^{2}{\theta}_{1}}{{\left(({\kappa}_{\perp}/{\kappa}_{\parallel}){\mathrm{sin}}^{2}{\theta}_{1}+{\mathrm{cos}}^{2}{\theta}_{1}\right)}^{2}}\right]}^{1/2},$$(C.5)
where κ_{A} = vr_{g}∕3, r_{g} is the gyroradius, u_{1} is the upstream solar wind speed, and θ_{1} is the upstream angle between the magnetic field and the shock normal. In the limit that κ_{A} → 0 and κ_{⊥} ∕κ_{∥}→ 0, the injection speed becomes, v_{inj} = u_{1}∕cosθ_{1}, which is the speed of shock (or compression) along a magnetic flux tube. Only ions with a speed larger than injection speed can move upstream relative to the shock or compression and thereby gain energy through multiple crossings of the speed gradient in the bulk plasma. Therefore, the injection speed acts as a gate for ions to begin the diffusive acceleration process.
Appendix D Rates associated with acceleration and cooling of SIR suprathermal populations
Observations have shown that there is a common form of the particle energy distribution within SIRs, as detailed in the introduction. This form of the distribution may be the result of the superposition of distribution functions from distributed sources (Schwadron et al. 2010), or from specific acceleration mechanisms. Given the form of the energy distribution (1), and the relatively low injection energy (< 1 keV, Fig. 3, panel 4), it is expected that particle acceleration at energies near the injection energy increases the magnitude of the distribution function with distance. In the corotating reference frame, the steadystate evolution of the isotropic part of the distribution function ${\tilde{f}}_{s}$ at speeds v with associated speed v > u but near injection is approximated (Fisk 1976; Schwadron et al. 1996): $$u\cdot \nabla {\tilde{f}}_{\text{s}}\frac{1}{3}\nabla \cdot up\frac{\partial {\tilde{f}}_{\text{s}}}{\partial p}\frac{1}{{p}^{2}}\frac{\partial}{\partial p}\left({p}^{2}{D}_{pp}\frac{\partial {\tilde{f}}_{\text{s}}}{\partial p}\right)=0,$$(D.1)
where u is the bulkflow velocity in the corotating reference frame, p is the particle momentum, and D_{pp} is the diffusion term in momentum space, which takes into account waveparticle interactions resulting in effects such as transittime damping (Fisk 1976; Schwadron et al. 1996). Note that this form of the Parker transport equation neglects diffusion due to the low mobility of ions for ion speeds that slightly exceed the bulk wind speed (e.g., 1 < v∕u ≤ 5 in the plasma frame, where u is the radial solar wind speed). We take the distribution function within the SIR near the injection energy to depend on particle speed and radial distance, ${\tilde{f}}_{\text{s}}(r,v)$, with the functional form for the speed detailed in the introduction (see Eq. (1)). The speeds indicated here are below the exponential rollover. Therefore, $${\tilde{f}}_{\text{s}}\approx {F}_{\text{s}}(r){v}^{{\gamma}_{\text{s}}}.$$(D.2)
Substitution of Eq. (D.2) into the transport Eq. (D.1) and averaging over the SIR longitude at a given radius yields $$u\frac{\partial {F}_{\text{s}}}{\partial r}{\dot{a}}_{\text{s}}{F}_{\text{s}}=0,$$(D.3)
where the rate associated with acceleration is defined $$\begin{array}{lll}{\dot{a}}_{\text{s}}\hfill & =\hfill & {\gamma}_{\text{s}}{\langle \nabla \cdot u\rangle}_{\text{s}}/3+{\gamma}_{\text{s}}^{2}\frac{{D}_{pp}}{{p}^{2}}+{\gamma}_{\text{s}}\frac{1}{{p}^{2}}\frac{\partial}{\partial p}\left(p{D}_{pp}\right).\hfill \end{array}$$(D.4)
The averaging within the SIR is indicated by ${\langle \dots \rangle}_{\text{s}}$. The solution to Eq. (D.3) yields the following approximation $${F}_{\text{s}}\propto \mathrm{exp}\left(\frac{{\dot{a}}_{\text{s}}r}{u}\right).$$(D.5)
The net cooling rate is scaled in terms of the solar wind expansion. We use an expansion factor g such that $$\nabla \cdot u=gu/r.$$(D.6)
For example, with a radial expansion, g = 2. The transport equation in this case is $$u\cdot \nabla {\tilde{f}}_{\text{s}}\frac{g}{3}\frac{u}{r}p\frac{\partial {\tilde{f}}_{\text{s}}}{\partial p}=0.$$(D.7)
The resulting equation for the radial dependence of the distribution is $$u\frac{\partial {F}_{\text{s}}}{\partial r}\frac{g{\gamma}_{\text{s}}}{3}\frac{u}{r}{F}_{\text{s}}=0,$$(D.8)
which has the solution, $${F}_{\text{s}}\propto {r}^{g{\gamma}_{\text{s}}/3}.$$(D.9)
We use the results in Eqs. (D.5) and (D.9) to solve for the distribution function f_{s} in SIRs throughout the inner heliosphere: $${f}_{\text{s}}={F}_{\text{s}}(r){(v/{u}_{\text{s}})}^{{\gamma}_{\text{s}}}\mathrm{exp}[(v{u}_{\text{s}})/{v}_{\text{s}}],$$(D.10)
where u is the intermediate solar wind speed in the SIR $$\begin{array}{lll}{F}_{\text{s}}(r)\hfill & =\hfill & {F}_{\text{peak}}\mathrm{exp}[(r{r}_{\text{peak}}){\dot{a}}_{\text{s}}/{u}_{\text{s}}]\text{for}r\le {r}_{\text{peak}}\hfill \\ \hfill & =\hfill & {F}_{\text{peak}}{(r/{r}_{\text{peak}})}^{g{\gamma}_{\text{s}}/3}\text{for}r{r}_{\text{peak}}.\hfill \end{array}$$(D.11)
The radial location r_{peak} of the peak distribution function associated with the radial distance where the forward or reverse shock moves through the extent of either slower wind in front of the SIR or the faster wind behind the SIR. The peak in the distribution function, F_{peak}, is related to the injection of pickup ions at r = r_{peak} (Schwadron et al. 1996): $${F}_{\text{peak}}=\frac{3{n}_{\text{H}}{\beta}_{1}}{8\pi {u}_{\text{s}}^{4}}\left(\frac{{r}_{1}^{2}}{{r}_{\text{peak}}}\right)\u03f5,$$(D.12)
where ϵ is a peak injection efficiency, r_{1} = 1 AU, β_{1} is the ionization rate referenced to 1 AU, and the neutral interstellar H density is $${n}_{\text{H}}\approx {n}_{\text{H0}}\mathrm{exp}({\lambda}_{\text{H}}/{r}_{\text{peak}}).$$(D.13)
Here, the hydrogen density in the outer heliosphere is n_{H0} = 0.1 cm^{−3}, we take λ_{H} = 4 AU and β_{1} = 6 × 10^{−7} s^{−1}. Note that the term ϵ scales the height of the suprathermal tail relative to the injected pickup ion population. In principle the peak injection efficiency ϵ should be close to 1.
Appendix E Particle transport through the rarefaction region
The Parker transport equation is written as follows, $$\frac{\partial f}{\partial t}+u\cdot \nabla f\nabla \cdot \left(\text{BarBar}\kappa \cdot \nabla f\right)\frac{1}{3}\nabla \cdot up\frac{\partial f}{\partial p}=Q.$$(E.1)
We transform the transport equation into the corotating frame where the bulk flow velocity in this frame is u′ = u + r′×Ω_{⊙}: $$\frac{\partial f}{\partial t}+{u}^{\prime}\cdot {\nabla}^{\prime}f{\nabla}^{\prime}\cdot \left(\text{BarBar}\kappa \cdot {\nabla}^{\prime}f\right)\frac{1}{3}{\nabla}^{\prime}\cdot {u}^{\prime}p\frac{\partial f}{\partial p}=Q.$$(E.2)
We invoke steady state in the corotating reference frame, and take the solar wind to be radial in the inertial frame, u = uê_{r}. Accordingly, in the corotating reference frame, the solar wind velocity is u′ = uê_{r} − Ω_{⊙}rsinθê_{ϕ} and the transport equation in this system is $${u}^{\prime}\cdot {\nabla}^{\prime}f{\nabla}^{\prime}\cdot \left(\text{BarBar}\kappa \cdot {\nabla}^{\prime}f\right)\frac{2u}{3r}p\frac{\partial f}{\partial p}=Q.$$(E.3)
We consider diffusive propagation along a fieldline with distance s, and take a fixed powerlaw in momentumspace, f ∝ p^{−α}: $${u}_{\text{s}}\frac{\partial f}{\partial s}B\frac{\partial}{\partial s}\left(\frac{{\kappa}_{\parallel}}{B}\frac{\partial f}{\partial s}\right)\frac{\partial}{\partial {s}_{\perp}}\left({\kappa}_{\perp}\frac{\partial f}{\partial s}\right)+\frac{2u\alpha}{3r}pf=0,$$(E.4)
where u_{s} = u′⋅ê_{B}. Here s_{⊥} refers to distance along a curve perpendicular to the magnetic field. We solve the problem numerically by setting up a grid of field lines out to the SIR. Each node location along the grid is specified as a function of radial distance and azimuthal angle. The diffusion equation is solved using secondorder differencing across the fieldline, and a spectral solution (equivalent to a Fourier transform solution) along the magnetic field. The powerlaw in momentum space varies across the grid in response to modulation by the solar wind.
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All Figures
Fig. 1 Summary of energetic particles following the stream interaction region: (top) EPIHi proton fluxes (sunward directed particles in blue, antisunward particles in red) from 1–10 MeV, (panel 2) EPILo proton fluxes as a function of energy and time, (panel 3) EPILo proton fluxes as a function as a function of time from 200–500 keV, (panel 4) density measured by SWEAP,(panel 5) solar wind speed measured by SWEAP, and (panel 6) magnetic field measured by FIELDS. This figure is similar to Fig. 2 of Joyce et al. (2021); the only difference is in panel 3, which is shown from 100–500 keV H^{+} in Joyce et al. (2021). 

In the text 
Fig. 2 Illustration of the magnetic field structure in a stream interaction region (SIR) adapted from Gosling et al. (1976). Within SIRs, Faster solar wind overtakes slower solar wind, forming a compression or shock (red dashed curves). The Parker Spiral magnetic field lines are shown as thin blue curves. With footpoint motion at the Sun, magnetic field lines (black curves) are connected between fast and slow solar wind and form the subParker spiral (Murphy et al. 2002; Schwadron & McComas 2005; Schwadron et al. 2005) in rarefaction regions where the fast solar wind stretches out magnetic field lines in the radial direction. 

In the text 
Fig. 3 Distance along the subParker spiral (black curves) is shorter than along the Parker Spiral (blue curves) to Stream Interaction Regions, and a the subParker magnetic field is associated with a lower injection energy compared to a Parker field. We show (top panel) the distance along a magnetic field line to the SIR compression or shock, and (second panel) the radial distance to the SIR compression/shock. These distances are shown as a function of time from stream interfaces at r_{o} ~ 0.33 au where theobservations are made by IS⊙IS. The subParker spiral also causes a reduced angle (third panel) between the magnetic field and the normal to the SIR compression or shock. Note that the fieldtonormal angle at the SIR compression or shock is calculated in the corotating reference frame. The injection energy detailed in Appendix C is also shown (fourth panel). 

In the text 
Fig. 4 Comparison between observed (blue data points) and modeled (curves) differential fluxes at higher and lower energies. Modeled energetic particles are accelerated at the SIR reverse compression or shock and propagate back to PSP along subParker field lines (black curves) and Parker field lines (blue curves). Observed differential fluxes (averaged over 1–5 MeV, top, and 200–500 keV, bottom) are compared with modeled fluxes (2 MeV, top, and 250 keV, bottom). 

In the text 
Fig. 5 Spectral index s of the modeled differential fluxes as a function of time after passage of the stream interface. Powerlaw indexes γ_{s} for the distribution correspond to γ_{s} = 2(s − 1). For example, a spectral index of s = −2 corresponds to a powerlaw index of γ_{s} = −6, which is close to the −5.4 powerlaw used for the common particle speed distribution. 

In the text 
Fig. A.1 Differential motion creates the subParker spiral. Blackcurves show the streamlines associated with parcel (a) and parcel (b) in the corotating reference frame. Footpoint motion provides a magnetic connection between parcel (a) and (b), which implies that the magnetic field is directed along the displacement between these plasma parcels. 

In the text 
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