Pickup Ion-Mediated Plasma Physics of the Outer Heliosphere and Very

Zank Geosci. Lett. (2016) 3:22 DOI 10.1186/s40562-016-0055-2

REVIEW Open Access Pickup ion‑mediated plasma physics of the outer heliosphere and very local interstellar medium G. P. Zank*

Abstract Observations of plasma and turbulence in the outer heliosphere (the distant supersonic solar wind and the subsonic solar wind beyond the heliospheric termination shock) made by the Voyager Interstellar Mission and the energetic neutral atom observations made by the IBEX spacecraft have revealed that the underlying plasma in the outer heliosphere and very local interstellar medium (VLISM) comprises distinct thermal proton and electron and suprathermal pickup ion (PUI) populations. Estimates of the appropriate collisional frequencies show that the multi-component plasma is not collisionally equilibrated in either the outer heliosphere or VLISM. Furthermore, suprathermal PUIs in these regions form a thermodynamically dominant component. We review briefly a subset of the observations that led to the realization that the solar wind–VLISM interaction region is described by a non-equilibrated multi-component plasma and summarizes the derivation of suitable plasma models that describe a PUI-mediated plasma. Keywords: Solar wind, Interstellar medium, Plasma, Pickup ions, Neutral gas

Introduction interstellar boundary explorer (IBEX) of and in our very The Voyager 1 (V1) spacecraft crossed the heliopause, the local neighborhood of the LISM. boundary separating matter of solar origin from inter- It is now recognized that the interstellar medium stellar matter, and entered the local interstellar medium and heliosphere are coupled intimately through charge (LISM) during August 2012 (Stone et al. 2013; Krimi- exchange of neutral H and protons, and that the physics gis et al. 2013; Burlaga et al. 2013; Gurnett et al. 2013), of the outer heliosphere and neighboring LISM cannot be an event of enormous historical import for humankind. understood independently of each other. Voyager 1 is the first human-made object to leave the The heliosphere is the region of space filled by the confines of the heliosphere and enter interstellar space. expanding solar corona; a region extending >120 astro- With a working set of instruments, Voyager 1 begins nomical units (AU) in the direction of the Sun’s motion an epoch of extraordinary in situ discovery science in through the interstellar medium and perhaps tens of the interstellar medium. We now have the opportunity thousands of AU in the opposite or heliotail direction. to study in situ basic plasma physical processes in the Neutral interstellar hydrogen is the dominant (by mass) interstellar medium (ISM). We review briefly our under- constituent of the solar wind beyond an ionization cav- standing of the basic plasma physics model that is begin- ity of ∼6−10 AU in the upwind direction (the direction ning to emerge as a result of observations made by the antiparallel to the incident interstellar wind), and is cou- Voyager interstellar mission (Voyagers 1 and 2) and the pled weakly to the solar wind plasma via resonant charge exchange. Charge exchange produces pickup ions (PUIs) that eventually dominate the internal energy of the solar wind. *Correspondence: [email protected] If, for simplicity, we adopt initially a perspective that the Center of Space Plasma and Aeronomic Research and Department of Space Science, University of Alabama in Huntsville, Sparkman Ave, plasma can be described as a single-fluid or magnetohy- Huntsville, AL 35899, USA drodynamic (MHD) system, then the heliospheric-LISM

© 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Zank Geosci. Lett. (2016) 3:22 Page 2 of 17

−15 2 plasma environment is composed of essentially three σc = 5 × 10 cm and a total LISM number density −3 thermodynamically distinct regions: (i) the supersonic of 0.2 cm ), in the IHS (region 2) ∼2500 AU for a num- −3 solar wind, with a relatively low temperature, large radial ber density of 0.005 cm , and >200 AU in the supersonic speeds, and low densities, bounded by the heliospheric solar wind beyond 10 AU (region 3). With the excep- termination shock (HTS). The outer heliosphere is that tion of the local interstellar medium region, the charge- region of the solar wind influenced dynamically by physi- exchange mfps are so large that they exceed the expected cal processes associated with the LISM. (ii) The transition scale size of the boundary regions separating the helio- of the supersonic solar wind to a subsonic flow through sphere and LISM. The interaction of the solar wind with the HTS creates a region of heated subsonic solar wind, the LISM therefore requires the modeling of plasmas called the inner heliosheath (IHS). The IHS has much and non-equilibrated H atom gas. Despite the very large higher temperatures and densities, larger magnetic fields, charge-exchange mfps in both the supersonic solar wind and lower flow speeds than does the distant supersonic and the boundary regions, the structure of the global solar wind (SW). The IHS is bounded by a contact or tan- heliosphere is determined in large part by the non-equil- gential discontinuity called the heliopause (HP). (iii) The ibrated coupling of neutral interstellar H to supersonic HP is the boundary that separates plasma of solar ori- and subsonic solar wind plasma (Zank 1999; 2015; Zank gin from plasma of interstellar origin. The LISM in our et al. 2009; McComas et al. 2011). This makes the mod- neighborhood possesses a small plasma flow speed and eling of the solar wind interaction with the LISM very temperature, but the density is higher than in regions challenging. Nonetheless, despite these complications, (i) and (ii). A bow shock may or may not exist ahead of the basic structure illustrated in the cartoon Fig. 1 (left) the heliosphere due to the relative motion of the Sun and emerges from simulations that include the basic physics interstellar medium. The three regions are illustrated of the plasma–H charge-exchange coupling. An illustra- in Fig. 1 (left panel), where region 3 corresponds to the tive simulation of a 2D coupled model of the heliospheric supersonic solar wind, region 2 to the hot inner heli- interaction with the LISM is shown in Fig. 1 (right panel). osheath, and the interstellar region is subdivided into The top plot shows the 2D plasma temperature distribu- region 1b between the HP and a possible bow shock/ tion, clearly identifying the three distinct regions and the wave, sometimes called the outer heliosheath (OHS), and overall topology and boundaries that can exist (together region 1a beyond a bow shock or bow wave. Of course, with a further sub-division of region 1 into pre- and post- as we discuss below, the plasma system is vastly more bow shock regions 1a and 1b, respectively). The bottom complicated than that of MHD and the plasma itself pos- plot illustrates the neutral H density distribution. A more sesses multiple components coupled via charge-exchange extended summary that discusses the magnetic field and/or collisional and collisionless processes, with asso- observations in both the IHS and at the HP, together with ciated transfer of charge, momentum, and energy, and associated references and related theoretical modeling, thus thermodynamic coupling. Nonetheless, the zeroth- can be found in the review by Zank (2015). order distinction of thermodynamically distinct regions The coupling of plasma and neutral H occurs through provides a useful intuitive guide to the underlying physics the creation of PUIs via charge exchange between the of the global solar wind–LISM interaction. charged and neutral gases. Over suitably large distances, Each of the thermodynamically distinct regions is the neutral H and protons are fully equilibrated, both the source of a distinct population of hydrogen (H) possessing the same temperature and velocity. Charge atoms produced by charge exchange between the ambi- exchange in a fully equilibrated partially ionized plasma ent plasma and neutrals entering the region (Zank et al. has no essential dynamical effect, with charge exchange 1996). These three distinct neutral H populations include effectively doing no more than relabeling protons and the “splash” component produced in the fast or super- H atoms (assuming that the dominant neutral gas com- sonic solar wind, i.e., fast neutrals that acquire high ponent is H atoms—in the LISM, this is a reasonable 400 750 radially outward speeds (∼ − km/s) with a rela- assumption, although He atoms are approximately 9 % tively small thermal spread, very hot neutrals produced of the neutral gas and the remaining heavy atom neu- in the inner heliosheath with comparatively high speeds tral gas is about 1 %). However, in regions 2 and 3, the 100 (∼ km/s) and a large thermal spread (which can pro- interstellar H drift speed is different from the plasma duce ENAs with speeds even that exceed 100 km/s), flow velocity ∼( 20 km/s for H versus ∼100−750 km/s and decelerated heated atoms originating in the outer for the plasma), and H originating from regions 3 and 2 heliosheath. that splashes back into the LISM has flow speeds rang- The charge-exchange mean free path (mfp) of neutral ing from ∼100−>400 km/s, which is quite different from hydrogen atoms in the LISM (region 1) is approximately the ∼15−26 km/s speed of region 1. Thus, throughout 100 ∼ AU (assuming a charge-exchange cross-section the boundary regions and in the LISM within several 100 Zank Geosci. Lett. (2016) 3:22 Page 3 of 17

a b BS(?) T (K) 7 × 103 1.6 × 104 3.8 × 104 8.7 × 104 2.0 × 105 4.7 × 105 1.1 × 106

HP Plasma temperature 400 Region 1a

2 B Region 1b HTS ow shoc Heliopause Region Region 3 200 k VLISM VSW Termination shock

AU 0

Hy drogen wall drogen

200

400 Hydrogen density

–500 –300 –100100 300 AU n (cm–3) 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 Fig. 1 Left Schematic of the solar wind–VLISM boundary regions that correspond to distinguishable thermodynamic regions, and which act as neutral H sources whose characteristics are clearly distinct (after Zank et al. 2009). HTS heliospheric termination shock, HP is heliopause, BS is bow shock, VSW denotes the radial solar wind flow speed, andV LISM the LISM flow velocity. Right A 2D steady-state, 2-shock heliosphere showing, top plot, the logarithmic temperature distribution of the solar wind and interstellar plasma and, bottom plot, the density distribution of neutral hydrogen. The HTS, heliopause, and bow shock are labeled, and the wall of neutral hydrogen is identified. The solid lines of the top plot show the plasma stream- lines. Distances are measured in AU

AU of the HP, there is a relative drift between the back- 2002; Gary and Madland 1988, see Gary 1991; Isenberg ground plasma and some H components. Depending on 1995; Zank 1999 for extensive summaries). PUIs expe- the specific environment, the neutral gas can be ionized rience scattering and gradual isotropization by either by either solar photons (photoionization) or charged par- ambient or self-generated low-frequency electromagnetic ticles (charge exchange, electron-impact ionization) and fluctuations in the plasma. Since the newly born ions are the new ions are accelerated almost instantaneously by eventually isotropized, their bulk velocity is essentially the motional electric field of the plasma. The PUIs form that of the background plasma, i.e., they advect with the a ring-beam distribution on the time scale of the inverse plasma flow and are then said to be “picked up” by the gyrofrequency and stream along the magnetic field while flowing plasma. The isotropized PUIs form a distinct experiencing advection by the bulk plasma flow perpen- suprathermal population of energetic ions (∼1 keV ener- dicular to the mean magnetic field. Newly created PUIs gies in the supersonic SW, with a number density approx- drive a host of plasma instabilities, from fast magneto- imately 20 % of the solar wind number density in the sonic and Alfvénic waves, ion cyclotron waves, to lower vicinity of the HTS) in the plasma whose origin is either hybrid waves (e.g., Lee and Ip 1987; Cairns and Zank the interstellar medium when considering region 3 and 2 Zank Geosci. Lett. (2016) 3:22 Page 4 of 17

or the heliosphere when considering regions 2 and 1 (e.g., distribution function (Zirnstein et al. 2014), with a rela- Holzer 1972; Lee and Ip 1987; Williams and Zank 1994, tively cool thermal solar wind Maxwellian distribution see Zank 1999, 2015 for an extensive review). and two or three superimposed PUI distributions. Such Consider now the three specific regions discussed a decomposition of the IHS proton distribution func- above. PUIs are created in these regions and mediate the tion can be exploited in modeling energetic neutral atom plasma properties. Although each region is mediated by (ENA) spectra observed by the IBEX spacecraft at 1 AU PUIs, the origin of the PUI population in each is different (Desai et al. 2012; Zirnstein et al. 2014; Desai et al. 2014). in important ways. Multiple proton populations were identified in the IHS Coulomb collisions are necessary to thermally equili- and the very local interstellar medium, these being the brate a background thermal plasma, such as the solar various PUI populations described above and the ther- wind, and the PUI protons. In the case of the supersonic mal solar wind proton population (Zank et al. 2010). solar wind, (Isenberg 1986) argued that a multi-fluid Zank et al. (2014) show that in the IHS neither proton model is necessary to describe a coupled solar wind– nor electron collisions can equilibrate a PUI-thermal PUI plasma since neither proton nor electron collisions solar wind plasma in the subsonic solar wind or IHS on can equilibrate the PUI-mediated supersonic solar wind scales smaller than at least 10,000 AU, meaning that a plasma (see Zank et al. 2014). multi-component plasma description that discriminates The inner heliosheath (IHS) is complicated by the between PUIs and the subsonic solar wind plasma is microphysics of the HTS. The supersonic solar wind is necessary. decelerated on crossing the quasi-perpendicular HTS. The interstellar plasma upwind of the heliopause is The flow velocity is directed away from the radial direc- also mediated by energetic PUIs. It was noted (Zank tion and is ∼100 km/s. The interplanetary magnetic field et al. 1996) that energetic neutral H created via charge remains approximately perpendicular to the plasma flow. exchange in the IHS and fast solar wind could “splash” Voyager 2 measured the downstream solar wind tem- back into the interstellar medium where they would perature to be in the range of ~120,000–180,000 K ∼16 experience a secondary charge exchange. The secondary eV (Richardson 2008; Richardson et al. 2008), which charge exchange of hot and/or fast neutral H with cold was much less than predicted by simple MHD models. (∼7500 K—McComas et al. (2012, 2015); Schwadron Instead, the thermal energy in the IHS is dominated by et al. (2015); Bzowski et al. (2015) LISM protons leads PUIs. There are two primary sources of PUIs in the inner to the creation of a hot or suprathermal PUI population heliosheath. The first is interstellar neutrals that drift locally in region 1. The heating of the LISM in the neigh- across the HP and charge exchange with hot solar wind borhood of the Sun has been discussed in detail (Zank plasma. These newly created ions are picked up in the et al. 2013), since this results in an increased sound speed IHS plasma in the same way that ions are picked up in with a concomitant weakening or even elimination of the the supersonic solar wind. The characteristic energy for bow shock (yielding instead a bow wave) (McComas et al. 5 −5 −3 PUIs created in this manner is ∼50 eV or ∼6 × 10 K, 2012). PUIs form a tenuous (np ≃ 5 × 10 cm , (Zirn- which is about five times hotter than the IHS solar wind stein et al. 2014) suprathermal component in the plasma protons. The second primary source is PUIs created in upwind of the HP that is not collisionally equilibrated the supersonic solar wind and then convected across in the LISM on scales smaller than at least 75 AU (Zank the HTS into the IHS. The PUIs convected to the HTS et al. 2014). are either transmitted immediately across the HTS or Zank (2015) introduced an alternative definition of are reflected before transmission (Zank et al. 1996). PUI the very local interstellar medium (VLISM) than has reflection was predicted by Zank et al. (1996) to be the been used previously (Holzer 1989). Instead of defin- primary dissipation mechanism at the quasi-perpendic- ing the VLISM as the region within 0.01 pc of the Sun, ular HTS, with the thermal solar wind protons experi- i.e., within about 2000 AU (Holzer 1989), the definition encing comparatively little heating across the HTS. The that we advocate is that it is that region of the ISM sur- transmitted PUIs downstream of the HTS have tem- rounding the Sun which is modified by the deposition of 6 peratures ∼9.75 × 10 K (∼0.84 keV) and the reflected heliospheric material. A corresponding definition would 7 protons have a temperature of ∼7.7 × 10 K (∼6.6 keV) be that the VLISM is a part of the ISM surrounding the (Zank et al. 2010). PUIs, whether transmitted, reflected, Sun that is not equilibrated with either H or PUIs of or injected, dominate the thermal energy of the IHS, heliospheric origin. This is consistent with the discussion despite being only some 20 % of the thermal subsonic above, and simulations (Zank et al. 2013; Heerikhuisen solar wind number density at the HTS. The IHS pro- et al. 2014) show that the interstellar temperature is mod- ton distribution function can be approximated by a 3- ified by the creation of heliospheric PUIs to at least some (Zank et al. 2010; Burrows et al. 2010) or 4-component 700 AU of the Sun in the upwind direction. Zank Geosci. Lett. (2016) 3:22 Page 5 of 17

Review heliosheath proton distribution (Zank et al. 2010) resem- Selected observations bles a κ-distribution (Heerikhuisen et al. 2008). As a The crossing of the HTS by Voyager 2 (V2) revealed an result, a significant number of protons reside in the wings almost classical perpendicular shock structure (labeled of the distribution function, quite unlike the Maxwellian TS-3) (Burlaga et al. 2008; Richardson et al. 2008), except distribution. The close correspondence between the con- that the observed average downstream proton plasma structed distributions and the κ-distribution with index temperature was an order of magnitude smaller than pre- 1.63 is useful in allowing for simplified simulations based dicted by the MHD Rankine–Hugoniot conditions (Zank on a κ-distribution (Heerikhuisen et al. 2008; Zank et al. et al. 2009). The transmitted solar wind proton distribu- 2010, 2013; Zank 2015). Zank et al. (2010) predicted that tion is a broadened/heated Maxwellian (with a somewhat the constructed heliosheath proton distribution should flattened peak), and there is no evidence of reflected solar possess some structure that would manifest itself in ENA wind ions being transmitted downstream (Richardson spectra observed at 1 AU by IBEX, and that the micro- et al. 2008; Richardson 2008). Richardson et al. (2008); physics of the HTS plays a key role in determining the Richardson (2009) concluded that PUIs provide both form of the total downstream or heliosheath proton dis- the primary shock dissipation mechanism and the bulk tribution. Finally, we note that kinetic hybrid and PIC of the hot plasma downstream of the HTS, as predicted simulations (Lipatov and Zank 1999; Oka et al. 2011; Wu 12 years earlier by Zank et al. (1996). The basic model et al. 2009; Matsukiyo and Scholer (2011, (2014); Yang (Zank et al. 1996) for the microstructure of the HTS et al. 2015) appear to support the basic model (Zank et al. therefore appears to be supported by V2 observations. 1996, 2010) rather well. These comments are illustrated However, both the observed solar wind proton distribu- graphically in Fig. 2, where we show three panels. The left tion and a shock dissipation mechanism based on PUIs panel plots the solar wind proton distribution upstream mean that the downstream proton distribution function and downstream of the HTS observed by the Voyager is a (possibly complicated) function of the physics of 2 plasma instrument PLS (Richardson 2008). Unfortu- the HTS. Zank et al. (2010) developed a basic model of nately, the PLS instrument measures only solar wind a quasi-perpendicular HTS, mediated by PUIs, to derive energy protons and not PUIs. The observed downstream the complete downstream proton distribution function proton distribution shows clearly that the transmitted in the IHS, determine the partitioning of energy between solar wind distribution is simply a heated Maxwellian solar wind protons and PUIs, and infer the implications distribution and no reflected solar wind protons can of the constructed IHS proton distribution function for be identified. The middle panel shows the theoretically the ENA spectral flux observed by IBEX. predicted total downstream proton velocity distribu- Zank et al. (2010) introduced a three-distribution tion function Zank et al. (2010). The blue curve shows a approximation of the IHS proton distribution, compris- κ-distribution with a kappa value of 1.63 (Heerikhuisen ing core solar wind protons, transmitted (without reflec- et al. 2008). The black curves depict the distribution con- tion) PUIs, and reflected (and then transmitted) PUIs. structed from a superposition of transmitted solar wind Electrons are of course included too in the complete protons, transmitted but not reflected PUIs, and reflected plasma model. The reflected PUI population results from and then transmitted PUIs. The red curve illustrates a the reflection of some upstream PUIs at the cross-shock Maxwellian distribution for the observed downstream electrostatic potential of the quasi-perpendicular HTS. density and temperature. For this model, the heliosheath Reflected PUIs are the primary dissipation mechanism at constructed proton distribution (black curve) assumed the HTS (Zank et al. 1996; Lipatov and Zank 1999; Bur- that downstream phase mixing of PUIs caused them to rows et al. 2010). Although the post-HTS PUI distribu- evolve into an approximately Maxwellian distribution. tion is likely highly complex, as a first approximation the The assumption of a downstream PUI distribution inter- solar wind proton distribution is a Maxwellian. Since mediate to a filled shell and a Maxwellian distribution the number of PUIs reflected is comparatively small, a creates a shoulder feature in the total downstream proton simplifying assumption that the non-reflected PUI dis- distribution function (Zank et al. 2010). The right panel tribution can be approximated by either a filled-shell shows the total transmitted solar wind and PUI distri- or a Maxwellian distribution can be made (Zank et al. bution function downstream of the HTS derived from a 2010). The downstream PUI temperatures for the trans- hybrid simulation (Oka et al. 2011)—see also Yang et al. mitted and reflected PUIs can be computed (Zank et al. (2015)—assuming an upstream PUI number density of 2010), allowing the partitioning of downstream thermal 30 % of the total proton number density. The solar wind energy into transmitted solar wind protons, transmit- protons and the PUIs are denoted by the solid and dashed ted PUIs and reflected, and then transmitted PUIs to curves, respectively, and the heavy black curves are their be determined. The smoothed form of the constructed sum. Besides illustrating the correspondence of the basic Zank Geosci. Lett. (2016) 3:22 Page 6 of 17

Fig. 2 Left histograms of the solar wind thermal proton temperature distributions observed by Voyager 2 across the HTS measured in the SW and IHS: (black) solar wind distribution, (red) IHS distribution, and (blue) distribution of the solar wind temperature multiplied by 13, the ratio between the upstream solar wind and downstream IHS temperatures. No reflected solar wind protons can be identified from the distribution function (Richardson 2008). Center the IHS constructed proton distribution (black curve) assuming that the transmitted but not reflected PUIs evolve into a Maxwellian distribution. The blue curve shows a κ-distribution with a value of −1.63. The black curve is the superposition of transmitted solar wind protons, transmitted PUIs, and reflected and then transmitted PUIs. The red curve is a Maxwellian distribution assuming the observed downstream density and temperature. The particle velocity vx is normalized to the Maxwellian thermal speed vth = 2kT/mp, where k is Boltzmann’s constant, mp the proton mass, and T the total downstream temperature (Zank et al. 2010). Right one-dimensional cut of the proton velocity distribution function downstream of the HTS. The particle velocities are normalized by the upstream flow speedV 1 in the shock rest frame. The solar wind protons and the PUIs are identified by the solid and dashed curves, respectively, and the thick black curves are their sum (Oka et al. 2011)

theory introduced in Zank et al. (2010) to both Voyager to the cutoff step at v∼2U of the filled shell distribution 2 observations and simulations, the relative energies (see e.g., Gloeckler et al. 1994, 2001), and so a cutoff is of downstream solar wind ions and transmitted (both still present. However, Gloeckler et al. (1994, 2001) show directly and initially reflected) PUIs are clearly illustrated examples of the transmission of the solar wind PUI dis- in Fig. 2. tribution through a CIR reverse shock. The flat-topped To test the possibility that the microphysics of the HTS PUI distribution is transformed into a kappa-like distri- would manifest itself in IBEX ENA spectra observed at bution on transmission through the interplanetary shock, 1 AU, Desai et al. (2012) in an initial study found that i.e., a Maxwellian-like core with an extended tail that the fluxes, energy spectra, and energy dependence of emerges smoothly from the thermal distribution. These the spectral indices of ∼0.5–6 keV ENAs measured by observations, together with the theory described above, IBEX-Hi along the V1 and V2 lines of sight were consist- suggest that the observed ENAs are generated primarily ent within a factor of ~2 with the model results of Zank downstream of the HTS, after the PUI distribution has et al. (2010). The observed ENA spectra do not exhibit been processed by the HTS, rather than upstream in the sharp cutoffs at approximately twice the solar wind speed supersonic solar wind. A further interesting point con- as is typically found for shell-like PUI distributions in the cerns PUI, and hence ENA, energies higher than ∼6 keV. heliosphere. Desai et al. concluded that the ENAs meas- Since all upstream PUIs are processed by the HTS, this ured by IBEX-Hi are generated by at least two types of produces PUIs in the ∼0.5–6 keV energy range down- PUI populations whose relative contributions depend on stream which do not have a flat-topped distribution. the ENA energy: transmitted PUIs in the ~0.5 to 5 keV These energetic PUIs make up some 20 % of the pro- energy range and reflected PUIs above ~5 keV energy ton number density. Additional energization may result (Desai et al. 2012). either at the shock (via, e.g., diffusive shock acceleration The absence of sharp cutoffs in the ENA distribu- (Senanayake et al. 2015) or further downstream (Zank tion indicates that the ENA source in the ∼0.5–5 keV et al. 2015), or deep in the IHS itself (Lazarian and Opher energy range is almost certainly beyond the HTS. The 2009; Fisk and Gloeckler 2009), and this would then lead PUI distribution is modified by energization processes to a modification of the total proton spectrum in the IHS. in the supersonic solar wind, such as the formation of Although it is difficult to quantify the effect of additional the filled shell due to cooling, and it appears to develop proton energization in the IHS, if it produces a power law an extended tail beyond v–2U (U the bulk solar wind distribution from the energetic tail of the HTS-processed speed). However, the tail does not emerge smoothly from distribution, then this will simply ensure that (i) there is the flat-topped PUI distribution function but instead no cutoff at∼ 6 keV; (ii) the intensity in the energy range appears as a discontinuous, lower intensity attachment ∼0.5–6 keV will be reduced a little (bearing in mind that Zank Geosci. Lett. (2016) 3:22 Page 7 of 17

the new distribution is a power law), and (iii) this process at 1 AU. One can similarly partition the VLISM energy is likely to be of more importance to higher energy ENA into various proton populations (Zirnstein et al. 2014). observations (such as the Cassini INCA observations of The VLISM plasma consists mostly of protons, initially ENAs (Krimigis et al. 2009). ∼7500 K in the pristine LISM (McComas et al. 2015; The observed lower energy ENAs (below∼ 0.5 keV) Schwadron et al. 2015; Bzowski et al. 2015), that are par- are not well described by the theory (Zank et al. 2010), tially heated by charge exchange near the H wall and by and most existing models underestimate the ENA fluxes crossing a bow wave (McComas et al. 2012; Zank et al. between ∼0.05–0.5 keV by an order of magnitude or 2013). However, the increase in thermal energy of the more (Fuselier et al. 2012). To address the lower energies, VLISM plasma near the HP is also due to energetic PUIs, Zirnstein et al. (2014) extended the Zank et al. (2010) which are created from charge exchange between LISM model in two ways. First, they accounted for the extinc- protons and ENAs from the IHS (Zank et al. 1996). The tion of solar wind protons and transmitted and reflected majority of PUIs are in close proximity to the HP and PUIs by charge exchange with interstellar neutral H in drop off exponentially at larger distances due to the mean the composite proton distribution. The extinction pro- free path of their parent ENAs, and due to advection cess alters the distribution of energy in the IHS, com- with the LISM flow toward the HP (Zirnstein et al. 2014). pared to assuming that the relative energy densities of the As with the IHS, Zirnstein et al. (2014) determine the core SW protons and the transmitted and reflected PUIs VLISM PUI properties by partitioning the total energy remain constant. Determining an accurate partitioning from the plasma-neutral results between LISM protons of the energy is essential for understanding the role that and PUIs. Since ENAs from IHS protons may propa- PUIs play in the heliosphere and its effect on H ENA flux. gate into the VLISM and charge exchange to become The second extension introduced by Zirnstein et al. PUIs, they treat the VLISM plasma as a five-component (2014) was to include ENAs from the VLISM that were distribution, including protons from the core (and com- created by PUIs. Although ENAs are created every- pressed) VLISM plasma, and PUIs created by charge where in the solar wind–LISM interaction region, ENAs exchange from IHS ENAs. produced in the IHS easily propagate into the VLISM Figure 3a shows various sources of the H spectrum in the before charge exchange occurs, creating a population V1 and V2 direction based on an extended model (Zirn- of PUIs there. ENAs produced in the VLISM, however, stein et al. 2014) with a comparison to the corrected IBEX do not easily charge exchange in the IHS, and therefore data (Desai et al. 2014). The results illustrated in Fig. 3 are permeate the inner heliosphere and can be detected based on a single set of parameters that were introduced

a Voyager 1 b Voyager 2 104 104 2 OHS + 3 IHS 1 OHS + 3 IHS 3 IHS 103 103 Sec. ENAs OHS, ribbon 2 OHS s sr keV) s sr keV) 2 102 2 102

2 2 (ENAs/cm 10 (ENAs/cm 10 IBEX-Lo IBEX-Lo IBEX-Hi IBEX-Hi

100 100 10–1 100 101 10–1 100 101 ENA energy (keV) ENA energy (keV) Fig. 3 IBEX-Hi and IBEX-Lo ENA spectra compared with simulations based on one set of parameters considered by Zirnstein et al.—see text and Zirnstein et al. (2014) for details. Dashed green ENAs from a secondary VLISM population, forming the ribbon; dashed blue three IHS populations with Maxwellian distributions; dashed red ENAs from a hot, PUI VLISM population; solid purple superposition of all three IHS, the secondary ENAs from the ribbon, and a single completely thermalized VLISM population (not visible); solid red superposition of all three IHS, the secondary ENAs from the ribbon, and the two proton VLISM populations (Desai et al. 2014). a The left panel corresponds to observations made by IBEX in the Voyager 1 direction and b the right panel to observations made in the Voyager 2 direction Zank Geosci. Lett. (2016) 3:22 Page 8 of 17

in the model (Zirnstein et al. 2014). Specifically, Zirnstein Zank et al. 2014). As reported by Decker et al. (2008, et al. (1) considered multiple possible sources for OHS 2015), the inner heliosheath pressure contributed by PUIs whereas Desai et al. (2014) considered just one case energetic PUIs and anomalous cosmic rays far exceeds for which the source of OHS PUIs was the IHS, and (2) that of the thermal background plasma and magnetic explored different values for a heating parameterα in their field. The VLISM can also be regarded as a multi-com- simulations, whereas Desai et al. (2014) assumed a fixed ponent plasma (Desai et al. 2012, 2014; Zirnstein et al. value α = 1/4. The effect of varying these parameters was 2014). discussed in detail by Zirnstein et al. (2014), and a simi- Coulomb collisions can equilibrate a background ther- lar comparison of the theoretical model and IBEX obser- mal plasma and energetic protons. Assume that the back- vations is presented in Fig. 4 of Zirnstein et al. (2014). As ground thermal proton and electron distributions are illustrated in Fig. 3, below ∼0.5 keV, the flux is dominated Maxwellian. If we restrict our attention to PUIs, then they by ENAs from VLISM secondary PUIs, while ENAs from satisfy the ordering vts ≪ vp < vte, where vts/e denotes the HTS transmitted and reflected PUIs dominate above background proton/electron thermal speed respectively 0.5 keV. Although a small fraction of ENAs from core and vp the PUI speed. For PUIs experiencing scattering solar wind protons are visible at 1 AU, most exit the HP off thermal protons and electrons from a Maxwellian dis- and become PUIs in the VLISM, producing significant flux tribution function, the collision frequency between PUIs near ∼0.1 keV. Zirnstein et al. (2014) predict that a signif- and protons and PUIs and electrons is given by icant part of the ENA flux seen at 1 AU comes from the 1 2 n e4 ln � n e4 ln �m / VLISM. ENAs created from solar wind PUIs in the VLISM νps = s s −1, and νpe = e e s −1, ∼ s 2 2 2 3 s 2 2 3/2 2 3/2 dominate the flux below 0.2 keV, while secondary- πε0mpv ( π) ε0(kTe) mp injected, secondary-transmitted, and secondary-reflected (1) PUIs contribute a significant flux up to keV energies, comparable to the flux from the IHS. Our current detailed respectively. Here mp,e and np,e denote the proton and model (Zirnstein et al. 2014) therefore exploits the proper- electron mass and number density, respectively, e the ties of PUIs that contribute to heating the VLISM plasma, charge on an electron, Te the electron temperature, ε0 the thereby establishing that not only the low- but also the permittivity of free space, and ln the Coulomb loga- high-energy flux is a result of the coupling between the rithm. If the collisional time scale exceeds the character- IHS and VLISM plasmas through charge exchange. PUIs istic flow time of the plasma region of interest,τ f ≃ L/U , from the IHS are the source of multiple PUI species in the where L is the size of the region and U the characteris- VLISM. Simulation results (Zirnstein et al. 2014) compare tic velocity, then the PUI distribution will not equilibrate favorably with IBEX data, although perhaps somewhat low with the background thermal plasma. Expressions (1) at high energies compared to those observed by IBEX since should be used to determine whether one needs to intro- VLISM PUIs created from supersonic solar wind ENAs, or duce a plasma model that distinguishes energetic PUIs time-dependent solar wind boundary conditions were not from background or thermal plasma protons. included. Nonetheless, these results suggest strong cou- Zank et al. (2014) present detailed estimates for the pling between the IHS and VLISM plasmas through ENA equilibration times for PUIs in the supersonic solar charge exchange, and VLISM PUIs up to ∼10 keV may wind of the outer heliosphere, the subsonic solar wind dominate the globally distributed ENA flux visible at 1 AU. (the inner heliosheath), and the VLSIM using appropri- The results from the theoretical models (Zank et al. 2010; ate plasma parameters. In all three regions, the plasma Zirnstein et al. 2014) describing the interaction of the solar does not equilibrate and cannot therefore be described wind and the partially ionized LISM and the observational as a magnetized single-component plasma and at least results (Desai et al. 2012, 2014) confirm that indeed the some elements of a multi-component description are IHS and VLISM are multi-component non-equilibrated necessary. plasmas. Simplified single-fluid MHD plasma descriptions PUIs drive streaming instabilities in one form or do not capture the complexity of the plasma. The multi- another, and experience pitch-angle scattering from both component model introduced by Zank et al. (2014) is the self-excited and pre-existing Alfvénic fluctuations. The first rigorous attempt to extend basic models to incorpo- initial PUI ring-beam distribution is scattered toward rate the physics of non-thermal PUI distributions isotropy (Lee and Ip 1987; Williams and Zank 1994; Zank 1999; Cannon et al. 2014). Besides pitch-angle scatter- Modeling a pickup ion‑mediated plasma ing by Alfvénic and magnetic field fluctuations, PUIs The outer heliosphere beyond the ionization cavity (i.e., can experience diffusion in velocity space, both due to ≥∼8 AU) is dominated thermally by PUIs (e.g., Burlaga counter-propagating Alfvén waves and PUI excited lower et al. 1994; Richardson et al. 1995a; Zank 1999; 2015; hybrid waves, for example. As is typical, we assume that Zank Geosci. Lett. (2016) 3:22 Page 9 of 17

pitch-angle scattering is the fastest process associated U = U(ue, us, up, ne, ns, np, me, mp), where np and up refer with wave-particle interactions and neglect velocity dif- to PUI variables. The PUIs are governed by the Boltz- fusion terms. As we show below, pitch-angle scattering mann transport equation with a collisional term δf /δt|c, serves to introduce both a collisionless heat flux and a ∂f e δf non-isotropic pressure tensor into the transport equa- + v ·∇f + (E + v × B) ·∇vf = , (5) ∂t mp δt tions describing the PUIs. The pressure tensor modifica- c tion is expressed as a collisionless viscosity tensor. for average electric and magnetic fieldsE and B. On To describe a plasma that contains a non-equilibrated transforming the transport equation (5) into a frame PUI population, we construct an appropriate multi-com- that ensures there is no change in PUI momentum and ponent plasma description for a thermal background energy due to scattering, assuming that the cross-helicity plasma comprising electrons and protons and a non- is zero, and introducing the random velocity c = v − U , equilibrated PUI component that is subject to pitch- we obtain angle scattering by turbulence and Alfvénic fluctuations. ∂ ∂ f f e E U B e c B By making various approximations, we derive succes- + (Ui + ci) + ( + × )i + ( × )i sively simpler models. In so doing, we place on a more ∂t ∂xi mp mp (6) formal footing the derivation of the well-known two-fluid ∂Ui ∂Ui ∂f δf − − Uj + cj = . model of cosmic ray magnetohydrodynamics (Axford ∂t ∂x ∂c δt j i c et al. 1982; Webb 1983), showing, somewhat unexpect- edly and contrary to perceived wisdom, that the cosmic The velocityU is still unspecified so we chooseU such ′ ray number density is in fact included implicitly in the that E ≡ E + U × B = 0. This assumption corresponds total number density. to choosing E × B The multi‑component model U⊥ = U − U� = ≡ U, (7) B2 In deriving a multi-component plasma model that includes PUIs, we shall assume that the distribution func- since we choose U� = 0 (U is parallel to B and therefore tions for the background protons and electrons are each arbitrary). The use of the velocityU then yields Maxwellian, which ensures the absence of heat flux or ∂ ∂ f f e c B stress tensor terms for the background plasma. The exact + (Ui + ci) + ( × )i continuity, momentum, and energy equations governing ∂t ∂xi mp (8) the thermal electrons (e) and protons (s) are therefore ∂Ui ∂Ui ∂f δf − − Uj + cj = . given by ∂t ∂x ∂c δt j i c ∂ne,s +∇· ne,sue,s = 0; (2) By taking moments of (8), we can derive the evolution ∂t equations for the macroscopic PUI variables, such as 3 the number density np = fd c, momentum density u 3 ∂ e,s npup = cifd c me,pne,p + ue,s ·∇ue,s i , and energy density. Moments of the ∂t collisional term δf /δt|c are zero. The zeroth moment of = −∇Pe,s + qe,sne,sE + ue,s × B; (3) (8) yields the continuity equation for PUIs, ∂n ∂ p 0, + np Ui + upi = (9) ∂Pe,s ∂t ∂xi + u , ·∇P , + γ , P , ∇·u , = 0. (4) ∂t e s e s e s e s e s where up is the PUI bulk velocity in the guiding center Here ne,s, ue,s, and Pe,s are the macroscopic fluid variables frame. For the first moment, we multiply (8) by cj and for the electron/proton number density, velocity, and integrate over velocity space. This yields, after a little pressure, respectively, γe,s the electron/proton adiabatic algebra, the momentum equation for PUIs, E B q , index, the electric field, the magnetic field, and e s ∂ n U + u +∇·n U U + u + n u U the charge of particle. ∂t p j pj p j pj p p j The streaming instability for the unstable PUI ring- ∂ e + 3 = ε , beam distribution excites Alfvénic fluctuations. cicjfd c np jklupk Bl (10) ∂xi mp The self-generated fluctuations and in situ turbulence serve to scatter PUIs in pitch-angle. The Alfvén where εijk is the Levi-Civeta tensor. Note the presence of 3 waves and magnetic field fluctuations both propa- the term cicjfd c, which is the momentum flux or pres- gate and convect with the bulk velocity of the system sure tensor. Zank Geosci. Lett. (2016) 3:22 Page 10 of 17

To close Eq. (10), we need to evaluate the momentum from which we find the zeroth- and first-order flux, which requires that we solve (8) for the PUI distri- expressions, bution function f. In solving (8), we assume (1) that the 1 ′ ′ 3 , ′ ′ 3 0, PUI distribution is gyrotropic, and (2) that scattering of cicjf0d c = δijPp cicjµf1d c = mp PUIs is sufficiently rapid to ensure that the PUI distribu- 4π ′2 ′2 tion is nearly isotropic. We can therefore average (8) over P ≡ m c f0c dc. p p 3 (16) gyrophase, obtaining the “focused transport equation” for non-relativistic particles (Isenberg 1997). Details of the Consequently, the first-order PUI stress tensor is identi- derivation can be found in Ch. 5 of Zank (2014). To solve cally zero and the pressure is isotropic, δijPp. the gyrophase-averaged transport equation requires that The inclusion of the second-order terms yields a non- we specify the scattering or collisional operator. We make zero collisionless stress tensor. Since the PUI pressure is the simplest possible choice, which is the isotropic pitch- defined in the frame of the bulk PUI velocityu p, the dis- angle diffusion operator, tribution function over which the integral is taken needs to be evaluated in this frame. Since the expression (15) ∂ 2 ∂f νs(1 − µ ) , (11) f2 U ∂µ ∂µ for is a function of the guiding center velocity , we need to transform to the frame Up = U + up. On using where µ = cos θ is the cosine of the particle pitch-angle the solution (15) for f2, we obtain −1 θ, and νs = τs is the scattering frequency. The form of ′ 2 1 2 3 ′ ′ 2 1 2 3 ′ c (3µ − 1)f2d c = c (3µ − 1)f2d c the scattering operator (11) allows us to solve the focused x 2 y 2 transport equation using a Legendre polynomial expan- η ∂Upj 1 ∂Upi sion of the distribution function f. The second-order cor- = bibj − ; 15 ∂xi 3 ∂xi rect solution to the gyrophase-averaged form of Eq. (8) is (17) 1 2 f ≃ f0 + µf1 + (3µ − 1)f2; (12) 2 ′ 2 1 2 3 ′ 2η ∂Upj 1 ∂Upi cz (3µ − 1)f2d c =− bibj − ; 2 15 ∂xi 3 ∂xi = (x, , ); f0 f0 c t (13) (18)

cτs ∂f0 DUi τs ∂f0 ′ ′ 1 2 3 ′ f1 =− bi + bi ; (14) cicj (3µ − 1)f2d c =0, (i �=j), (19) 3 ∂xi Dt 3 ∂c 2 η cτ ∂U 1 ∂U ∂f0 where the coefficient of viscosity is defined as s j i , f2 ≃ bibj − (15) 4π ∂ 4π P τ 15 ∂xi 3 ∂xi ∂c ′4 ′ ′2 ′2 ′ p s . η ≡ ′ (c cτs)f0dc ≃ c τsf0c dc ≃ 15 ∂c 3 mp where c =|c| is the particle random speed, b ≡ B/B is a directional unit vector defined by the magnetic field, and (20) D/Dt ≡ ∂/∂t + Ui∂/∂xi is the convective derivative. The The first equality in (20) is the formal definition of the expansion terms f0, f1 and f2 are functions of position, coefficient of viscosity for the PUI gas. If we assume time, and particle random speed c, i.e., independent of (probably reasonably) that |c|≫|up|, then we obtain the µ (and of course gyrophase φ). Of particular importance second equality, which may be regarded as a PUI pressure is the retention of the large-scale acceleration, and shear moment weighted by the PUI scattering time. Finally, if terms. These terms are often neglected in the derivation we assume that τs is independent of c, we then obtain the of the transport equation describing f0 (for relativistic “classical” form (20) of the viscosity coefficient. The pres- particles, the transport equation is the familiar cosmic sure tensor may therefore be expressed as ray transport equation). In deriving a multi-fluid model, 100   η ∂U 1 ∂U retaining the various flow velocity terms is essential to = δ + 010 � pk − pm �. �Pij� Pp� ij� 15 bk bℓ 3 derive the correct multi-fluid formulation for PUIs. We  00 − 2  ∂xℓ ∂xm need to evaluate (21) 3 3 cicjfd c = (ci − upi)(cj − upj)fd c + npupiupj

′ ′ 3 If we introduce a “viscosity matrix,” ≡ cicjfd c + npupiupj η 1 Ppτsbk bℓ 1 (M ) ≡ (η ) = b b ≃ , ′ ′ 2 3 ′ kℓ kℓ k ℓ (22) ≃ c c f0 + µf1 + (3µ − 1)f2 d c + n u u , 15 15 mp i j 2 p pi pj Zank Geosci. Lett. (2016) 3:22 Page 11 of 17

and note that ηij = ηji and η/15 = η11 + η22 + η33 = ηijδij after transforming to Up. To evaluate the heat flux, we 2 (since b = 1), we can rewrite (21) in the more revealing have “classical” stress tensor form, 1 ′2 ′ 3 ′3 ′2 ′ c cif0d c = π c µbif0c dc = 0, η ∂U 1 ∂U 2 pk − pm 15 bk bℓ 3 ∂xℓ ∂xm and η ∂U ∂U ℓ 1 ∂U = kℓ pk + p − η δ pm 2 kℓ kℓ (23) mp ′2 ′ 3 ′ π ′2 ∂f0 ′2 ′ 2 ∂xℓ ∂xk 3 ∂xm µ 1 =− κ 2 c ci f d c 3 mp c ij c dc η ∂U ∂ 2 ∂ ∂xj kℓ pk Upℓ Upm . = + − δkℓ 1 ∂ 2 ∂x ∂x 3 ∂x Pp x ℓ k m =− κ¯ij = qi( , t). (28) 2 ∂xj The pressure tensor is therefore the sum of an isotropic scalar pressure Pp associated with drift and curvatur and In (28), we introduced the spatial diffusion coefficient the stress tensor, i.e., 2 c τs 10 0 κij ≡ bi bj, (29)   2 3 ηkℓ � ∂Upk ∂Upℓ ∂Upm � �Pij� = Pp�δij� + 01 0 + − δ ℓ   2 ∂x ∂x 3 k ∂x 00 − 2 ℓ k m together with PUI speed-averaged form κ¯ij ≡ Kij. The collisionless heat flux for PUIs is therefore described in ≡ PpI + �p. (24) terms of the PUi pressure gradient and consequently the The stress tensor is a generalization of the “classical” form averaged spatial diffusion introduces a PUI diffusion time in that several coefficients of viscosity are present, and of and length scale into the multi-fluid system. course the derivation here is for a collisionless charged For continuous flows, the transport equation for the gas of PUIs experiencing only pitch-angle scattering by PUI pressure Pp can be derived from (27), yielding turbulent magnetic fluctuations. Use of the pressure ten- ∂Pp ∂P 5 ∂Up 1 ∂ ∂P 2 ∂Upj sor (24) yields a “Navier-Stokes”-like modification of the + U p + P i = K p − � , pi 3 p 3 ij 3 ij PUI momentum equation, ∂t ∂xi ∂xi ∂xi ∂xj ∂xi ∂ (30) �ρpUp� +∇·�ρpUpUp + IPp� = enp�E + Up × B� ∂t illustrating that the PUI heat flux yields a spatial diffu- 10 0  η ∂U ∂U 2 ∂U sion term in the PUI equation of state together with a 01 0 kℓ � pk pℓ pm �, −∇· + − δkℓ viscous dissipation term. The PUI system of equations is  00 2  2 ∂xℓ ∂xk 3 ∂xm − properly closed and correct to the second-order. Note the = enp�E + Up × B� −∇·�p (25) typo in Zank et al. (2014) since we mistakenly omitted the viscous term of Eq. (30) in the corresponding pres- where we used the transformation Up = up + U for the sure equation. remaining velocity terms in (10) and ρp = mpnp. The full system of PUI equations can be written in the ′ If we introduce c ≡ c − up as before, we can express form the heat fluxq (x, t) through the definition ∂ρp +∇· ρ U = 0; (31) 1 2 m p p (x, ) ≡ ′ ′ 3 ′ = p 2 3 ∂t qi t mp 2c cifd c 2 c cifd c ∂ U U U I E U B 5 1 2 ρp p +∇·ρp p p + Pp + � = enp + p × ; − u P − ρ u u . ∂t 2 pi p 2 p p pi (26) (32) The equation for the total energy of the PUIs can then be ∂ 1 2 3 1 2 derived from (8), yielding ρ U + P +∇· ρ U U ∂t 2 p p 2 p 2 p p p ∂ 1 2 3 ∂ 1 2 5 5 1 ρ U + P + ρ U U + P U + � U + q 2 p p 2 p 2 p p pi 2 p pi ij pj i + P U + � · U − K ·∇P = en U · E, (33) ∂t ∂xi 2 p p p 2 p p p U B , = enpUpiEi + p × i (27) which is the form we use below. Zank Geosci. Lett. (2016) 3:22 Page 12 of 17

The full thermal electron–thermal proton–PUI multi- where ni = ns + np. Since the PUIs are not thermally fluid system is therefore given by Eqs. (2)–(4) and (31)– equilibrated with the background plasma (Ts �= Tp), we (33) or (30), together with Maxwell’s equations, need to deal separately with the Ps and Pp equations. ∂B These become = −∇ × E; (34) ∂t ∂Ps + U ·∇P + γ P ∇·U = 0; (41) ∂t i s s s i ∇×B = µ0J; (35) ∂Pp ∂Pp 5 ∂Ui 1 ∂ ∂Pp 2 ∂Uj + Ui + Pp = Kij − �ij . ∇·B = 0; (36) ∂t ∂xi 3 ∂xi 3 ∂xi ∂xj 3 ∂xi (42) J = e nsus + npUp − neue, (37) We can combine the proton Eqs. (39)–(42) with the elec- where J is the current and µ0 the permeability of free tron Eqs. (2)–(4) to obtain an MHD-like system of equa- space. The diffusion tensor is assumed to be of a simple tions. On defining the macroscopic variables, diagonal form (i.e., we do not include the off-diagonal ρ ≡ mene + mpni; q ≡−e(ne − ni); terms associated with drift and curvature–see the discus- ρU ≡ meneue + mpniUi; J ≡−e(neue − niUi), (43) sion in Zank (2014) and we specify we can express  κ⊥ 00 1 2 1 2 K = 0 κ⊥ 0 ; κ⊥ = η C0 , κ� = C0 . 3� 3� ρ − (mp/e)q ρ + ξ(mp/e)q  00κ  p p n = ≃ ρ/m ; n = ≃ ρ/m ; � e 1 p i 1 p (38) mp( − ξ) mp( + ξ) ρU − (mp/e)J mp J ρU + ξ(mp/e)J ue = ≃ U − ; ui = ≃ U, We parametrize the perpendicular component of the ρ − (mp/e)q e ρ ρ + ξ(mp/e)q heat conduction tensor by a term η<1. In estimating the (44) diffusion coefficients (38) from (29), we choose a characteristic PUI speed for the region of interest and assume where the smallness of the mass ratio ξ ≡ me/mp ≪ 1 that the scattering time can be approximated by a time has been exploited. Use of the approximations (44) allows scale greater than the corresponding gyroperiod. us to combine the continuity and momentum equations in the usual way and to rewrite the thermal electron and Single‑fluid‑like model proton pressure in terms of the single-fluid macroscopic For many problems, the complete multi-component variables. Thus, model derived above is far too complicated to solve. The ∂ρ multi-fluid system (2)–(4) and (31)–(33) or (30), together +∇·(ρU) = 0; (45) ∂t with Maxwell’s equations can be considerably reduced in complexity by making the key assumption that Up ≃ us . ∂U Up ≃ us ρ + U ·∇U = −∇(P + P + P ) + J × B −∇·�; The assumption that is quite reasonable since ∂t e s p (i) the bulk flow velocity of the plasma is dominated by the background protons since the PUI component scat- (46) ters off fluctuations moving with the background plasma ∂Ps speed and (ii) the large-scale motional electric field forces + U ·∇Ps + γsPs∇·U = 0; (47) ∂t newly created PUIs to essentially co-move with the background plasma flow perpendicular to the mean magnetic J ∂Pe mp γemp Up ≃ us = Ui + U ·∇Pe + γePe∇·U = J ·∇Pe + Pe∇· , field. Accordingly, we let be the bulk pro- ∂t eρ e ρ ton (i.e., thermal background protons and PUIs) velocity. The thermal proton and PUI continuity and momentum (48) equations are therefore trivially combined as where

∂ni  10 0  +∇·(niUi) = 0; (39) ηkℓ �∂Uk ∂Uℓ 2 ∂Um � ∂t �kℓ = 01 0 + − δkℓ .  00 − 2  2 ∂xℓ ∂xk 3 ∂xm U ∂ i mpni + Ui ·∇Ui = −∇(Ps + Pp) Since we may assume that the current density is much ∂t less than the momentum flux, i.e.,| J|≪|ρU|, we can E U B , + eni( + i × ) −∇·�p (40) simplify (48) further by neglecting the RHS. By assuming Zank Geosci. Lett. (2016) 3:22 Page 13 of 17

that γe = γs = γ, we can combine the thermal proton and the PUI pressure is required. Instead of the conserva- electron equations in a single thermal plasma pressure tion of energy Eq. (53), one could use the PUI pressure equation with P ≡ Pe + Ps, Eq. (42) for continuous flows. PUIs introduce both a col- ∂P lisionless heat conduction and viscosity into the system. + U ·∇P + γ P∇·U = 0. (49) The model Eqs. (51)–(55), despite being appropriate to ∂t non-relativistic PUIs, are identical to the so-called two- Note that at this point, no assumptions about either the fluid MHD system of equations used to describe cosmic thermal electron or proton pressures (or temperatures) ray-mediated plasmas (Webb 1983). However, the deri- have been made. vation of the two models is substantially different in that Finally, we need an equation for the electric fieldE . the cosmic ray number density is explicitly neglected in To do so, we multiply the respective momentum equa- the two-fluid cosmic ray model and a Chapman–Enskog tions by the electron or proton charge, sum, and use the derivation is not used in deriving the cosmic ray hydro- approximations (44) to obtain dynamic equations. Nonetheless, the sets of equations m 2 1 ∂J that emerge are the same indicating that the cosmic ray ξ p +∇·(JU + UJ) e ρ ∂t two-fluid equations do in fact include the cosmic ray number density explicitly. mp = ∇Pe − J × B − ξ∇(Ps + Pp) The single-fluid-like model may be extended to include, eρ e.g., anomalous cosmic rays (ACRs) as well as PUIs. In −ξ∇·�) + E + U × B. this case, the ACRs are relativistic particles. The same analysis carries over, and one has an obvious extension of The generalized Ohm’s law is therefore the model Eqs. (51)–(55) with the inclusion of the ACR mp pressure. Thus, the extension of (51)–(55) is E =−U × B − ∇Pe − J × B − ξ∇Pp, eρ (50) ∂ρ +∇·(ρU) = 0; (56) ∂t where we have retained the PUI pressure since in prin- ciple it can be a high-temperature component of the U plasma system and ξPp may be comparable to the Pe ∂ ρ + U ·∇U = −∇(P + Pp + PA) + J × B term. For typical cases of interest, however, the Pp term ∂t can be neglected in Ohm’s law (50). Neglect of the elec- −∇·�p −∇·�A; (57) tron pressure and Hall current term then yields the usual form of Ohm’s law. ∂P The reduced single-fluid model equations may there- + U ·∇P + γ P∇·U = 0; (58) ∂t fore be summarized as

∂ρ ∂Pp +∇·(ρU) = 0; (51) + U ·∇P + γ P ∇·U ∂t ∂t p p p 1 = ∇· Kp ·∇P − (γ − 1)� : (∇U); (59) ∂U 3 p p p ρ + U ·∇U = −∇(P + Pp) + J × B −∇·�; (52) ∂t ∂P A + U ·∇P + γ P ∇·U ∂t A A A ∂ 1 2 3 1 2 1 2 5 ρU + (P + Pp) + B +∇· ρU U + (P + Pp)U 1 ∂t 2 2 2µ0 2 2 = ∇·(KA ·∇P ) − (γ − 1)� : (∇U); (60) 3 A A A 1 2 1 1 + B U − U · BB + � · Up − K ·∇Pp = 0; µ0 µ0 2 (53) ∂B E =−U × B; = −∇ × E; µ0J =∇×B; ∇·B = 0, ∂P ∂t + U ·∇P + γ P∇·U = 0; (54) ∂t (61) ∂B where we have introduced the ACR pressure PA, the E =−U × B; = −∇ × E; µ0J =∇×B; ∇·B = 0. ∂t corresponding stress tensor A, the ACR diffusion tensor KA and adiabatic index γA (4/3 ≤ γA ≤ 5/3). The (55) coupled system (56)–(61) is the simplest continuum The single-fluid description (51)–(55) differs from the model to describe a non-equilibrated plasma compris- standard MHD model in that a separate description for ing a thermal proton–electron plasma with suprathermal Zank Geosci. Lett. (2016) 3:22 Page 14 of 17

particles (e.g., PUIs or even solar energetic particles) and wind region of the outer heliosphere, the anomalous relativistic energy (anomalous) cosmic rays. The system heating of the solar wind (Williams et al. 1995) has been includes both the collisionless heat flux and viscosity interpreted in terms of the dissipation of PUI-driven associated with the suprathermal and relativistic particle turbulence that leads to the heating of the solar wind distributions. plasma (Zank et al. 1996; 2012; Matthaeus et al. 1996, On reverting to Eqs. (51)–(55), we can recover the 1999; Smith et al. 2001; Adhikari et al. 2015a). In the standard form of the MHD equations if we set the heat inner heliosheath and the VLISM, the observed plasma conduction spatial diffusion tensorK = 0 and the coef- characteristics of the HTS (Zank et al. 1996; Richardson ficient of viscosity(η kl) = 0, which corresponds to 2008; Richardson et al. 2008) and the ENA observations assuming τs → 0. If the total thermodynamic pressure made by IBEX (Zank et al. 2010; Desai et al. 2012, 2014; Ptotal = P + Pp is introduced, then we recover the stand- Zirnstein et al. 2014) have been similarly interpreted in ard MHD equations (dropping the subscript “total”), i.e., terms of a multi-component plasma distribution com- ∂ρ prising various PUI populations. Estimates of the col- +∇·(ρU) = 0; (62) lisional frequency between thermal plasma components ∂t and PUIs in the outer supersonic solar wind (> ∼10 ∂U AU), IHS, and VLISM show that equilibration cannot be ρ + ρU ·∇U + (γ − 1)∇e + (∇×B) × B = 0; (63) achieved in these regions. Illustrated in Fig. 4 is a sche- ∂t matic of the solar wind–LISM interaction region with 2 colors indicating regions that have to be described in ∂ 1 2 B 1 2 ρU + e + +∇· ρU + γ eU terms of a multi-component plasma. The three colors for ∂t 2 2µ0 2 the different regions indicate that each region has a dis- 1 tinct multi-component plasma description reflecting the + B × (U × B) = 0; (64) µ0 different origins of the PUI population for each. In the

∂B =∇×(U × B); ∇·B = 0, (65) ∂t with an equation of state e = αnkBT/(γ − 1). The choice of α = 2 (or greater if incorporating the contribution of cosmic rays, etc.) corresponds to a plasma population comprising protons and electrons. In setting K = 0 and (ηkl) = 0, we have implicitly assumed that PUIs are completely coupled to the thermal plasma. With K �= 0, heat conduction reduces the effec- tive coupling of energetic particles to the thermal plasma, and their contribution to the total pressure is not as large. This will have important consequences for numeri- cal models of, e.g., the large-scale heliosphere since they incorporate PUIs into the MHD equations, without dis- tinguishing PUIs from thermal plasma and therefore neglect heat conduction. Consequently, the total pressure is over-estimated.

Conclusions Observations by Voyager 1 and 2 and the IBEX spacecraft indicate that plasma in the outer heliosphere (the super- and subsonic solar wind) and the VLISM possesses characteristics of a multi-component plasma, being Fig. 4 Schematic of the solar wind–LISM interaction showing the boundaries. The colored regions require a non-equilibrated multi- essentially a non-equilibrated distribution of background component plasma description. The different colors indicate that the thermal protons and electrons and PUIs of various ori- non-equilibrated PUI component(s) originates from different physical gins. Limitations of space prevent discussion of all the processes. The region in white surrounding the Sun corresponds to observational threads that lead to this conclusion, and we the ionization cavity where PUIs are not present in sufficient numbers list and discuss above only a few. In the supersonic solar to effectively mediate the plasma. See text and Table 1 for details Zank Geosci. Lett. (2016) 3:22 Page 15 of 17

supersonic solar wind, it is primarily PUIs created from is fundamental to the physics of PUIs—for a review, see interstellar neutral H that make up the PUI component. Zank (1999). Wave-particle scattering of PUIs introduces In the IHS, PUIs created in the supersonic solar wind and a collisionless form of PUI heat conduction and viscosity processed by the HTS are the most important PUI com- through the PUI pressure tensor. These important dissi- ponent energetically in the IHS, although there is a lower pative terms need to be included in any description of a number density PUI component due to charge exchange non-equilibrated PUI-mediated plasma. The model pre- with interstellar H as well. For the VLISM, the PUIs sented here is appropriate for use in models of the global arise from charge exchange with secondary “solar wind heliosphere. In particular, in the inner heliosheath, the or splash component neutrals” that were created in the role of PUI and ACR heat flux is to partially decouple the supersonic solar wind and IHS. Basic plasma properties full pressure contribution of the PUIs and energetic parti- are mediated by PUIs in each of the regions illustrated in cles from the overall pressure, thereby reducing the effec- Fig. 4, and some discussion of the linear wave modes in tive thermodynamic pressure balancing the interstellar these regions was presented by Zank et al. (2014). Table 1 pressure against that of the inner heliosheath. The net provides a precise breakdown of the plasma models that effect should be a thinner heliosheath than predicted by are necessary for each region, together with estimates of conventional MHD models that over-estimate the total the corresponding plasma and PUI temperature and den- pressure contribution contributed by energetic particles. sity for each species (second column). Possible simplifica- The new model equations should be used to explore the tions of the full model are listed in the third column for global structure of the solar wind–VLISM interaction the IHS case. eventually (but this is well beyond the current scope of Having motivated the need for a multi-component the paper). Finally, we note that it is not completely obvi- plasma description throughout the solar wind–VLISM ous how or whether ENA fluxes would change. At lead- interaction region, a derivation of the multi-fluid plasma ing order, the energy densities of the different ionized model was presented, based on the analysis of Zank components should be largely unchanged in the IHS. The et al. (2014) (and correcting some typos). The stand- only difference is that the heat flux associated with ener- ard approach of simply using a set of multi-fluid equa- getic particles removes energy (i.e., reduces their pres- tions under the assumption that all distributions are sure contribution) from the overall total pressure of the isotropic, as done by Zieger et al. (2015), is incorrect in inner heliosheath. Because this could lead to a thinner that it neglects the basic physics of PUI scattering by heliosheath, the ENA flux could be reduced but because pre-existing and self-excited fluctuations. Numerous the heat flux causes the energetic ionized particles to dif- observations of the flat-topped form of the PUI distribu- fuse to greater distances; the net effect may be that there tion in the solar wind show that wave-particle scattering is little change in the overall ENA fluxes.

Table 1 An explicit listing of the possible separate PUI populations for the different regions illustrated in Fig. 4 Region Plasma model Reduced model

<8 AU (within ionization Standard MHD model—PUIs essentially treated as test particles cavity) ~8 AU to HTS (i.e., beyond Multi-component model ionization cavity) Interstellar PUIs created in solar wind (T ≤ 1 keV, np ∼ 0.05−0.2ni) + thermal plasma (protons and electrons) IHS Multi-component model Multi-component model Three PUI populations: 1) Retain 1) 2) and incorporate 3) in 1) Interstellar PUIs transmitted across HTS without reflection T( ∼ 1 keV, thermal plasma+ model (i.e., neglect heat np ∼ 0.18ni) flux forT ∼ 50eV PUIs) 2) Interstellar PUIs reflected and then transmitted at HTS T( ∼ 6 –7 keV, 2) Combine models 1) and 2) and incorpo- np ∼ 0.02ni) rate 3) in thermal plasma model 3) Interstellar PUIs created in IHS (T ∼ 50 eV, np ∼ 0.015ni) thermal −3 + plasma (T ∼ 16 eV, ni ∼ 0.005 cm ) VLISM Multi-component model −5 −3 Solar wind PUIs created in VLISM (T ∼ 200 eV, np ∼ 5 × 10 cm ) −3 + thermal plasma (T ∼ 0.65 eV, ni ∼ 0.15 cm )

Characteristic temperatures and densities are given for the different PUI species. In estimatingn p for interstellar PUIs created in the IHS, we assumed a characteristic time scale of 40 AU/100 km/s ∼ 6 × 107 s Zank Geosci. Lett. (2016) 3:22 Page 16 of 17

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