Interaction of Inner Heliosheath Ions with Electromagnetic Ion Cyclotron Waves

Inner heliosheath (IHS) ions are expected to be exposed to various waves, shocks, and turbulence, which can affect ion distributions and thus their charge-exchange rates with interstellar neutral atoms. This work addresses the potential significance of electromagnetic ion cyclotron (EMIC) waves under expected IHS conditions. From a kinetic dispersion relation, we find the possibility of frequent triggering of EMIC instability in the IHS. The threshold anisotropy of proton temperatures required for the instability is small, (T ⊥–T ∣∣)/T ∣∣ ≈ 0.1 or less, mainly due to high plasma β (ratio of the plasma to magnetic pressures). Numerical calculations on the scattering of ions (protons, He+, and He2+ with energy of 0.01–50 keV) based on two models for EMIC waves with a moderate intensity indicate significant scattering in the pitch angle (mostly a few tens of degrees) and energy (mostly a few tens of percent) although details depend on the energy and pitch angle of each species and adopted EMIC wave models. This occurs on a short timescale (<100 times the gyro-period of each ion species). Resonant scattering in a few to a few tens of keV (corresponding to the expected pickup ion energy in the IHS) is easily expected unless the wavenumber is too large. The scattering effect is distinguished among different species such that for lower gyrofrequency ions (He+ versus He2+and He2+ versus protons), the main scattering effect moves toward a lower energy domain. All these results imply continuous disturbance of ion distributions by possibly prevailing EMIC waves in the IHS.


Introduction
The inner heliosheath (IHS) is the region in which the generation of energetic neutral atoms (ENAs) is highly expected via charge exchange between ions and neutral atoms of mostly interstellar origin.Observations by the Cassini spacecraft revealed a "belt" of ENAs at ∼5-55 keV that is likely produced by energetic protons in the IHS (Krimigis et al. 2009;Dialynas et al. 2013).Images obtained from the Interstellar Boundary Explorer (IBEX) have shown the existence of a background ENA flux (the globally distributed flux) at 0.2-6 keV, the origin of which is attributed to termination-shock-heated protons in the IHS (Funsten et al. 2009;McComas et al. 2009;Schwadron et al. 2011).
The observed ENA fluxes may depend sensitively on the ion distributions in the IHS.In addition to protons, the ions of main interest here include He + and He 2+ , originating from the outflowing solar wind, and pickup ions (PUIs), which are ionized interstellar particles in the heliosphere (Vasyliunas & Siscoe 1976;Möbius et al. 1985;Geiss et al. 1994;Gloeckler et al. 1997;Randol et al. 2013;McComas et al. 2017McComas et al. , 2021)).Both theoretically and observationally, the PUI distributions are characterized by a sharp cutoff at twice the solar wind speed in the solar inertial frame.For example, from the New Horizons observations at ∼46 au (McComas et al. 2021), the cutoff energies were found to be ∼2.8 keV for H + and ∼10.4 keV for He + , which are approximately 4 times the peak energies of the distributions of the corresponding solar wind protons and He + (the solar wind He + ions arepresumed to be produced from charge exchange between interstellar neutrals and the solar wind He 2+ ; Swaczyna et al. 2019).
Both observations and theories have suggested that as the outflowing ions cross the termination shock, the heliosheath ion distribution must be a mixture of transmitted solar wind ions and PUIs that are energized by the termination shock.Voyager observations indicate that the solar wind proton temperature changed from ∼2 × 10 4 K upstream to ∼1.8 × 10 5 K downstream (Richardson et al. 2008).According to the estimates by Zank et al. (2010), the downstream temperatures of solar wind protons, transmitted PUIs, and reflected PUIs that are transmitted downstream are ∼1.25 × 10 5 K, ∼9.75 × 10 6 K, and ∼7.7 × 10 7 K, respectively.
It is expected that the heliosheath ions are exposed to various plasma waves, shocks, and turbulence during convection and before they charge exchange with neutrals.Unfortunately, observational information on waves, shocks, and turbulence in the IHS is limited.Nevertheless, related reports and suggestions are available in various aspects.Burlaga et al. (2006Burlaga et al. ( , 2009) ) and Burlaga & Ness (2009) report on the existence of downstream magnetic turbulence observed by Voyagers 1 and 2. Zank et al. (2018) described the transmission of solar wind turbulence into the IHS through the termination shock based on a theoretical model that incorporates lowfrequency turbulence into PUIs and solar wind thermal plasma.Turbulence has been considered to demonstrate the acceleration and evolution of PUIs near the termination shock and within the IHS (Chalov et al. 2003;Ye et al. 2016;Giacalone et al. 2021;Zirnstein et al. 2021).Additionally, previous works imply the occurrence of plasma instabilities of solar wind ions and PUIs.For example, Lee & Ip (1987) suggested that the ring distribution of PUIs is unstable to the generation of hydromagnetic waves with growth timescales shorter than the continual pickup process and solar wind convection.Fahr & Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Siewert (2007) expected an anisotropic distribution function of ions after passage over the termination shock, which may cause instability to the mirror mode and then drive low-frequency magnetoacoustic or magnetosonic waves.Simulations of the termination shock and the IHS (Zieger et al. 2020) indicate the growth of fast magnetosonic waves that steepen into shocks.
The aim of the present work is to illustrate first (in Section 2), the potential occurrence of electromagnetic ion cyclotron (EMIC) waves under expected conditions in the IHS and then (in Section 3) the extent to which the heliosheath ions are scattered by unstable EMIC waves.The EMIC waves are expected to be unstable when the perpendicular temperature T ⊥ of energetic protons (or ions) is greater than their parallel temperature T || (e.g., Cornwall 1965;Kennel & Petschek 1966).To the authors' knowledge, this is the first study in which EMIC waves are applied to IHS conditions in a comprehensive way, whereas many reports on the applications of EMIC waves exist in other fields, such as the Earth's magnetosphere and solar wind.In this work, we do not attempt to quantitatively assess the extent to which EMIC wavescattered ions affect the production of ENAs but instead prefer to illustrate the basic physics of ion scattering by EMIC waves.Nevertheless, we infer some implications for ENA generation from the results of ion scattering (in Section 3.3).The calculations here are applied to protons, He + , and He 2+ with comparisons among them.Implications from the results are discussed in Section 4.

Threshold Proton Anisotropy for Ion Cyclotron Instability
In this section, we aim to estimate the possibility of ion cyclotron instability by solving a kinetic dispersion relation.Specifically, we solve the kinetic dispersion relation for parallel propagating L-mode waves in a plasma consisting of four species (electrons, protons, He 2+ , and He + ), each of which is further distinguished by cold (T = 0) and hot (nonzero T) populations (e.g., Lee et al. 2017).Moreover, we employ bi-Lorentzian (κ) distributions to effectively include suprathermal populations (the hot populations), which have been considered in previous works (e.g., Heerikhuisen et al. 2008;Prested et al. 2008Prested et al. , 2010)).The dispersion equation is as follows (Summers & Thorne 1991;Xue et al. 1996;Xiao et al. 2007): Here, ω pe , ω pi , and ω p hot s refer to the plasma frequencies of each population; the subscript "hot s" refers to the hot population of species "s"; Ω E and Ω i refer to the electron and ion gyrofrequencies; and A s is the temperature anisotropy of the hot population of species "s," as defined below: is the modified plasma dispersion function derived by Summers & Thorne (1991), where the argument is defined by v and .
As the representative case in this work, we take T = 10 8 K for all of the hot populations based on previous reports (e.g., Zank et al. 2010).Additionally, we take κ = 2 to represent the existence of energetic populations.This choice of κ value is motivated by those used in previous reports, such as κ = 1.5-2.5 in Livadiotis et al. (2011), κ = 1.63 in Heerikhuisen et al. (2008), and κ = 2 in Mostafavi et al. (2019).
We remark that the cold populations are intended to represent the core populations of the solar wind.While the Results for different parametric choices while the other parameters remain the same as for the blue dashed curve in (a): the black curve is the case with reduced hot proton density by 50% (and increased cold protons accordingly), the magenta curve is the case with reduced temperatures by 1 order for all of the hot populations, and the olive and orange curves are the cases with κ = 4 and 5, respectively.f is defined by the ratio between the electron plasma and gyrofrequencies, ω pe /Ω e .actual temperature of the core solar wind population is ∼10 4 K, which is far lower than that of the hot populations (T = 10 8 K), we take the simple assumption of T = 0 for the cold populations in the dispersion equation.This corresponds to the limitation of in the argument of the plasma dispersion function.This means that we deal with the cold population particles of nonresonant character (Gary et al. 1994).Furthermore, the cold populations contain no free energy for the instability.It is the hot resonant populations with an anisotropic distribution that can drive ion cyclotron instability.Naturally, we focus on the parametric dependence of the instability on the anisotropic hot populations.The isotropic cold populations take part in the quasi-neutrality condition.
For this work, we consider two different combinations of the background plasma density and magnetic field intensity, (n e [cm −3 ], B o [nT]) = (0.00225, 0.13) and (0.001, 0.3).These values lie in the ranges found from the observations of Voyager (Richardson 2011;Burlaga et al. 2018).These two sets of n e and B o provide different ratios between the electron plasma and gyrofrequencies, that is, f ≡ ω pe /Ω E ∼ 117 and 34, respectively.This is approximately proportional to the plasma β (to be discussed further below).Each of these two combinations is further divided by setting the anisotropy of one of the helium ions to zero (that is, either A 0 He = + or A 0 He 2 = + ), while the other is varied as a free parameter.The helium ion anisotropy is unknown and thus is treated as a free parameter.In total, we test four different combinations, for which we take the same set of density ratios, 72.65% (cold protons), 26.4% (hot protons), 0.2% (cold He + ), 0.7% (hot He + ), 5% (cold He 2+ ), 0.025% (hot He 2+ ), and 10% (hot electrons), all relative to the cold electron density n e .Here, the hot protons are regarded as representing PUIs, and their chosen density is constrained by the observations by New Horizons in the supersonic solar wind (McComas et al. 2017(McComas et al. , 2021)).However, this may be different (possibly increased) in the IHS, which is unknown observationally.The chosen density ratio of the cold He 2+ is intended to mostly represent the typical solar wind alpha particles.With the chosen density ratio of hot electrons, we allow their existence based on previous reports (Chalov & Fahr 2013;Chashei & Fahr 2013;Fahr & Siewert 2015;Fahr et al. 2015;Zieger et al. 2015).The others (cold protons, cold and hot He + , hot He 2+ ) are chosen based on the numbers available in Swaczyna et al. (2017) and by requiring the quasi-neutrality condition.
Therefore, the four combinations among n e , B o , A He + , and A He 2+ are the representative cases we choose, all being given by the common choice of hot populations' T = 10 8 K, κ = 2, and the ion density models described in the previous paragraph.Althoughwe regard them to best represent the heliosheath conditions, we additionally test the following cases to check the sensitivity of the results to different parametric choices: (i) the hot proton density is reduced by 50%, and the cold proton density is increased accordingly; (ii) the temperature of the hot populations is reduced by 1 order (that is, T = 10 7 K); and (iii) the larger κ values, κ = 4 and 5, are used, while in each of these additional cases, all the other parameters remain fixed to the same values.All these additional cases should provide a higher threshold anisotropy of hot protons.
With the parametric choices described above, we attempt to find the threshold anisotropy of hot protons, A p , required for ion cyclotron instability.This can be obtained by numerically finding the conditions in which the growth rate becomes zero in the kinetic dispersion Equation (1) above.Practically, the growth rate is set to be 10 −4 times the real frequency.The results are shown in Figure 1.The key feature in Figure 1(a) is that the threshold proton anisotropy for the four representative  cases is quite small, less than 0.1 over the broad parameter range of the helium anisotropies.Figure 1(b) indicates that the additional cases with different parametric choices increase the threshold anisotropy values only modestly up to approximately 0.1 or slightly above.This result implies that the heliosheath is susceptible to ion cyclotron instability if the chosen parameter conditions are indeed typical.
This important conclusion can be explained based on the inverse relation between the threshold anisotropy and plasma β.Specifically, the threshold anisotropy can be expressed as an inverse power law of hot plasma β hp as follows: "C" can include a complicated dependence on the densities and temperatures of the plasma populations and background magnetic conditions.Such an inverse relation for EMIC waves is well known in the Earth's magnetosheath and magnetosphere conditions (e.g., Gary & Lee 1994;Noh et al. 2018Noh et al. , 2021)), as in the cases of the well-known fire-hose and mirror instability criteria.Here, it is important to note that for the IHS, the hot plasma β hp is large, for example, ∼5 for 1 keV protons with a density ratio of 10% relative to cold electrons (n e = 0.00225 cm −3 ) under a background magnetic field of 0.13 nT in the IHS.For comparison, the hot plasma β hp is ∼0.04 for the same protons but under a much higher background electron density (n e = 1 cm −3 ) and magnetic field of 100 nT, such as in a region of the inner magnetosphere of the Earth.This implies that the prevailing high β hp condition in the IHS is responsible for the low-threshold anisotropy, providing a favorable condition for easy triggering of ion cyclotron instability.

Ion Scattering in the Pitch Angle and Kinetic Energy by EMIC Waves
Clearly, details of the particle scattering effect by EMIC waves are dependent on a specific form of the wave spectrum (e.g., Lee 2021).From this viewpoint, we choose two specific solutions of unstable ion cyclotron waves from the kinetic dispersion equation described in Section 2. The two solutions are in contrast by different wavenumber ranges (and correspondingly different wave frequency ranges).Differences in the ion scattering effect are highlighted between the two wavenumber models.

Small Wavenumber Model
This model is achieved by taking the parameter values as follows.First, we choose (n e [cm −3 ], B o [nT]) = (0.00225, 0.13), which is a combination considered in Figure 1.Then, to obtain an unstable solution, we choose the following set of anisotropies: . Although this choice is rather arbitrary, it lies in the unstable domain in Figure 1.For this condition, Figure 2 shows three branches of the solution, that is, three pairs of the real frequencies (upper panel) and corresponding growth rates (lower panel) as distinguished by the line types.The unstable solution (growth rate γ > 0) exists only for the lowest frequency branch below the He + gyrofrequency (thick lines) in a rather limited (small) wavenumber domain (shaded region).
Figure 3(a) is an expanded view of the unstable domain shown in Figure 2.For the specific calculations of ion scattering, we consider the EMIC waves defined in the unstable wavenumber domain (∼ 0.04-0.12)kc/ωpp and the corresponding frequency range ( ∼ 0.07-0.11)Ωp (the shaded region in Figure 2).Resonant scattering (meeting the condition, ω − k || v || = Ω i ) is expected to be dominant.Figure 3(b) shows the minimum kinetic energy of ions required for the resonant interaction with the EMIC waves in the wavenumber space.In the unstable wavenumber domain, it ranges from ∼1 to ∼10 keV for protons, ∼0.9-9 keV for He 2+ , and ∼0.1-1.5 keV for He + .Below, we will see that they indeed determine the threshold energies above which resonant ion scattering occurs most significantly.
We adopt a model in which the EMIC waves are provided by a simple supposition of numerous circularly polarized parallel propagating L-mode waves with different wavenumbers k ||i and frequencies ω i .These values are obtained from the unstable wave solution shown in Figure 3. Specifically, they are expressed as follows: where the summation runs up to the chosen number of wave modes (350 here), and the wave electric and magnetic fields are linked by Maxwell equations.The wave intensity of each wave mode is assumed to be a Gaussian function of the wavenumber, that is, The ions are exposed to head-on collisions with EMIC waves having an average amplitude of 5% of the background magnetic field.By solving the equation of motion numerically, their trajectories are followed until 100 times the gyro-period of each ion after launching with an initial pitch angle α o , kinetic energy KE o , and gyro-phases is reached.We present the main effects of the waves on the ion dynamics in terms of "diffusive" and "advective" changes in the pitch angle and kinetic energy, each being referred to as , and KE adv ( ) D (Bortnik et al. 2010;Lee et al. 2018Lee et al. , 2020)).Specifically, regarding the pitch angle, they are defined as follows: .
Regarding the kinetic energy, they are defined in the same way.
Here, α is the local pitch angle of individual protons, and a á ñ indicates the pitch angle averaged over the gyro-phases of the same group of ions that started with the same initial pitch angle but with different initial gyro-phases.This definition of the diffusive pitch-angle change in Equation (9) conveniently represents the average extent to which the individual pitch angles of a group of ions deviate from the average pitch angle at a later time.The advective pitch-angle change defined in Equation (10) conveniently represents the extent to which the phase-averaged pitch angle, a á ñ, drifts over time from the initial pitch angle, α 0 , in either a positive or negative direction.This provides a simple measure of the nondiffusive pitch-angle change and is a convenient way to represent the nonlinear effects that are not addressed in quasi-linear resonance diffusion theory.However, it does not explicitly reflect phase bunching and trapping effects.More descriptions of nonlinearity can be found in the literature (Inan et al. 1978;Albert & Bortnik 2009;Liu et al. 2012;Lee et al. 2018Lee et al. , 2020)).
Figure 4 demonstrates the above definitions with a specific example for 48 protons launched with the same initial kinetic energy and pitch angle (KE o = 5 keV, α o = 135°) but different initial gyro-phases.Figure 4(a) shows the local pitch angles in time for selected protons (only six are selected for visual clarity) with the average pitch angle a á ñ estimated from all 48 protons (thick line).In Figure 5 and the subsequent figures in the same format (figures with four panels), we show the diffusive and advective changes in the pitch angle and kinetic energy ) averaged over 100 times the gyro-period of each ion species over a broad range of initial kinetic energies (KE o = 0.01-50 keV, which encompasses the most relevant energy range for the ENA production in the IHS) and initial pitch angles (between α o = 90°and 180°, it should be noted that the wave propagates in the background B o direction, while the ions are launched in the opposite direction for head-on collision).For each pair of the same initial kinetic energy and pitch angle, 48 ions of different gyro-phases were traced.
Figure 5 shows the results for protons.Specifically, Figures 5(a The energy domain for the main scatterings in Figure 5 is approximately consistent with the minimum resonant energy described above in Figure 3.In Figure 6, we show the positions (open black circles) of a specific proton in velocity space at numerous selected times during an interval of 100 gyro-periods after leaving the start point with initially KE o = 5 keV and α o = 135°(solid blue circle).The proton trajectory remains roughly along the single wave characteristics (red curve), implying dominantly resonant interaction (Kennel & Engelmann 1966).The proton suffers from mostly large pitch-angle changes and relatively smaller energy changes.
In the same format as Figure 5, Figure 7 shows the results for He 2+ .Overall, the results are similar to the case of the protons in Figure 5.Nevertheless, some differences exist.The energy domain of the main scattering in the pitch angle now begins from an energy slightly less than 1 keV (the ellipsoidal contours in Figures 7(a) and (b)), which is slightly lower than that for the proton case in Figure 5.Not surprisingly, this is consistent with the lower-minimum resonant energy for He 2+ having a lower gyrofrequency.In the same pitch angle-energy domain, the diffusive energy change is mostly approximately 10% (the ellipsoidal contour in Figure 7(c)), and the largest advective energy change is above 10% in the initial pitch-angle range α o > ∼ 135°(red region within the ellipsoidal contour in Figure 7(d)), which is slightly more significant than in the proton case in Figure 5.
Additionally, we identify lower energy domains (KE o < ∼ 0.1 keV, corresponding to the nonresonant regime) in Figure 7, where both the pitch-angle and energy changes are non-negligible in contrast to the proton case in Figure 5.Most notably, we identify the following domains: Regarding He + ions with an even lower gyrofrequency and accordingly lower resonant energies, Figures 8(a) and (b) show that the energy domain of the main resonant pitch-angle scattering (the ellipsoidal contours) moves down to KE o ∼ 0.1 keV although with strong dependence on the initial pitch angle.Specifically, diffusive and advective changes of a few to several tens of degrees prevail mostly at energies between KE o ∼ 0.1 keV and a few keV, and advective changes of a few tens of degrees occur even at tens of keV domain, as marked by the thick arrow in Figure 8(b).In the same pitch angle-energy regime as marked by the ellipsoidal contours, Figures 8(c) and (d) indicate that the maximum diffusive energy change is above 10% and the maximum advective energy change is above 30%.These energy changes are overall more significant than those for protons (Figures 5(c) and (d)) and He 2+ (Figures 7(c) and (d)).We recall from Figures 2 and 3 that the chosen wave spectrum covers the wave frequency range below the He + gyrofrequency.Clearly, this frequency domain is effective in increasing the kinetic energy more for He + ions than for protons and He 2+ although it is dependent on the initial pitch angle α o .
Additionally, as in the case of He 2+ in Figure 7, there are lower energy (nonresonant scattering) domains (KE o < ∼0.05 keV) for He + in Figure 8, where both the pitch-angle and energy changes (with initial pitch-angle dependence) are notable.They are indicated by dotted lines as in Figure 7 to help visual identification.Both the pitch-angle and energy changes are overall more significant than for the case of He 2+ in Figure 7.

Large Wavenumber Model
Now, we choose another unstable wave solution from the kinetic dispersion relation, which is shown in Figure 9.This is obtained using the same parameter values as in the solution in Figure 3 except for the increased proton anisotropy, A p = 0.45.The unstable solution now exists in the wavenumber domain that is nearly ∼5 times larger.The corresponding frequency lies above the He + gyrofrequency but below the He 2+ gyrofrequency (blue in Figure 9(a)).Therefore, since we consider head-on collisions only, no resonant interaction is expected for He + ions, and the minimum resonance energy is shown only for protons and He 2+ in Figure 9(b).Mainly due to the larger wavenumber domain, the minimum resonant energy is now lower than for the short wavenumber model in Section 3.1: specifically, ∼0.05-0.1 keV for protons and ∼0.02-0.05keV for He 2+ .Any scattering of He + ions should be nonresonant.
The simulation results using this larger wavenumber model are shown in Figures 10, 11, and 12 for each of the three ions.First, Figure 10 shows the results for protons.Figures 10(a) and (b) indicate that the overall features in these plots are similar to those for protons with the smaller wavenumber model in Figure 5 with some differences.The major difference is that the main scattering effect in the pitch angle is now most significant in the much lower energy range between KE o > ∼0.05 keV and ∼1 keV, which is primarily resonant scattering.In the same energy domain, the diffusive and advective energy changes (Figures 10(c) and (d)) are mostly ∼10% or larger locally (thick arrow in (d)), which are slightly more enhanced than the smaller wavenumber model case (Figures 5(c) and (d)).Additionally, we identify advective pitch-angle changes of a few tens of degrees at energy KE o < ∼0.05 keV and in the pitch-angle regime α o > ∼160°(corresponding to nonresonant regime; thick arrow in Figure 10(b)).
The results for He 2+ are shown in Figure 11 (compare this with Figure 7 for the smaller wavenumber model).Again, we have identified the main resonant scattering domain marked by the ellipsoidal contours and other notable scattering regimes by arrows.Notably, the ellipsoidal contours now lie in the lower energy domain than those for the smaller wavenumber model in Figure 7. Figures 11(a  Figures 11(c) and (d)), which are more significant than for He 2+ with the smaller wavenumber model in Figure 7. Additionally, the pitch-angle changes at higher energies (KE o > ∼0.5 keV) are non-negligible although with limitations to specific pitch-angle domains (the thin arrows in Figures 11(a The results for He + are shown in Figure 12, which clearly reveals the main pitch angle-energy domains for the largest scatterings.They are dramatically different from those for He + with the smaller wavenumber model in Figure 8.The scattering effect here is nonresonant.First, Figure 12 (   Figure 12(c)), although limited to the pitch-angle regime of α o < 160°.The most notable advective energy changes (a few tens of percentage or larger, up to ∼100% at a maximum) are seen below KE o = ∼0.5 keV but become limited to more perpendicular pitch-angle domains with increasing energy (Figure 12(d)).Therefore, while the pitch angle-energy domains for the main scatterings in the pitch angle and energy here are quite different from the case of the smaller wavenumber model (Figure 8), the degree of nonresonant scattering with the large wavenumber model is overall quite significant.

Implications for ENA Generation
Because of the complicated dependence of the scattering details on the initial pitch angle, kinetic energy, and wave models as described in Sections 3.1 and 3.2, it is not straightforward to quantitatively estimate the extent to which EMIC wave-scattering of ions affects the ENA generation rate.Nevertheless, we infer some implications from the ion scattering results in the previous sections.The observed ENA flux at energy band E in a certain direction can be approximately represented by where P surv is the survival probability, σ ex is the chargeexchange cross section between ions and neutrals, n ISN is the interstellar neutral atom density, j ion is the ion flux, and the integration is done along the line of sight (LOS) to an observing spacecraft (e.g., IBEX or Cassini) within the IHS region.To determine j ENA , one needs j ion , which reflects scattered ion distributions at all intersection points of plasma streamlines with the LOS.The ions with a velocity vector toward an observing satellite at the time when they undergo charge exchange with neutrals contribute to the observed j ENA with the survival probability P surv .The pitch angle of such ions (heading toward the observing spacecraft) relative to the background magnetic field B o can differ depending on the specific direction of B o relative to the LOS under consideration.Figure 13 depicts two possible cases of LOS where the pitch angle of ion velocity α o is 90°in one case (LOS 0 representing the noseward direction on the equatorial plane) and greater than 90°in the other case (LOS 1 ).According to Figure 5 (the small wavenumber model results in Section 3.1), protons of 5 keV, for example, are hardly scattered for α o = 90°or so, whereas the same energy protons of α o > ∼110°suffer from substantial changes in pitch angle (tens of degrees) and energy (∼± 10%).Consequently, in this example, we expect that the ENA flux at 5 keV along LOS 0 is little affected by the EMIC waves, whereas that along LOS 1 may be affected substantially due to scattering off of the protons away from the LOS 1 direction that would otherwise contribute entirely to the ENA flux.However, note the energy dependence of ion scattering effects in Figure 5 (and in Figure 10 for the large wavenumber model), indicating that the pitch-angle scattering effects at higher energies move toward a more perpendicular pitch angle (closer to α o = 90°) domain.Therefore, the EMIC wave effects on the resulting ENA flux along a given LOS are energy dependent.Additionally, j ENA requires the LOS integration of j ion defined at different intersection points of streamlines with the LOS, and there is no reason that ions at different intersection points are always exposed to the same EMIC wave structure (marked "waves" in Figure 13).Recall that the EMIC wave model with the large wavenumber model results in a scattering effect in energy regimes (Figures 10-12) different from those with the small wavenumber model.Therefore, the resulting j ion at different intersection points should differ in general.Given no observational information of the actual EMIC wave distributions in the IHS, the task of evaluating this integration effect realistically is difficult to perform.Nevertheless, it is natural to expect a nonuniform generation of ENAs at different intersection points where the disturbances in j ion can differ.

Summary and Discussion
The main results of the present work can be summarized in two aspects.First, the kinetic dispersion relation implies a possibly frequent triggering of ion cyclotron instability in the IHS provided that the physical conditions adopted in this work are realistic and prevail to a large extent to represent typical heliosheath conditions.The main reason for this conclusion can be understood via the inverse relation between the threshold proton anisotropy and the plasma β of hot protons: that is, the plasma β in the IHS is sufficiently high to provide relatively small threshold anisotropy (A p < ∼ 0.1).
Second, EMIC waves of a moderate intensity, as modeled based on the growing wave solutions from the kinetic dispersion, can significantly affect ion distributions with energies of 0.01-50 keV in the IHS.Often, scattering in the pitch angle can be up to several tens of degrees, and the energy changes are significant (mostly by a few tens of percent but sometimes much larger for low-energy heavy ions if scattered by large wavenumber waves) although details of the scattering depend on the initial pitch angle and energy.Both resonant and nonresonant scattering occur and can be significant but at different energy domains.The scattering effect is sensitive to the specific wave spectrum such that the main resonant scattering effect moves toward a lower energy domain for a larger wavenumber model.Additionally, the scattering effect is distinguished among different species such that for lower gyrofrequency ions (He + versus He 2+ and He 2+ versus protons), the main scattering effect moves toward a lower energy domain.
A possible source of ion anisotropy is shock acceleration.Previous works (Fahr & Siewert 2007;Liu et al. 2007) suggested that ion distributions become anisotropic downstream of shocks.Here, shocks include the termination shock and shocks within the IHS, which are generated by collision of interplanetary disturbance in the solar wind with the termination shock (Mostafavi et al. 2019).The resulting downstream ion distributions are dependent on upstream solar wind conditions, and the degree of ion anisotropy should differ accordingly.
The energy regime considered here encompasses that of PUIs (a few to a few tens of keV), which are considered responsible for ENAs originating from the heliosheath due to charge exchange with cold neutrals of interstellar origin.The two EMIC wave models taken in this work imply that the resonant scattering of ions in this energy regime is easily expected unless the wavenumber is too large.That is, the energy range of the main resonant scattering effect can often coincide with the expected PUI energy ranges in the IHS provided that EMIC waves with proper wavenumber structures occur.In addition, such scattering occurs on a timescale <100 times the gyro-period of each ion species, which is far shorter than the typical timescales (years) of charge exchange and convection.Therefore, the convecting ions may suffer from fast scattering by EMIC waves continuously or intermittently depending on the persistence of the EMIC waves.
Such a fast scattering process should change the PUI distributions over time, which should in turn affect the hot proton anisotropy that drives ion cyclotron instability.Because of the rather sensitive dependence of the scattering effect (particularly in pitch angle) on energy and specific wave structure (Figure 5 for the small wavenumber model versus Figure 10 for the large wavenumber model), it is nontrivial to quantitatively determine proton anisotropy changes due to the PUI scattering effect.A precise determination of this process requires a self-consistent simulation.For example, hybrid simulations of nonlinear EMIC wave evolution have been performed for a typical condition of the Earth's magnetosphere by Omidi et al. (2010).The simulation results indicate that after initial growth of the waves for approximately 1000 ion gyroperiods, the wave amplitudes decrease by approximately 50% and remain nearly constant thereafter, and the temperature anisotropy of the hot protons decreases.A similar simulation is desired for inner heliospheric conditions to quantitatively determine the extent to which PUI scattering affects the time evolution of hot proton temperature anisotropy.
There are other factors and issues that must be considered to improve the present work in the future.The details of scattering should depend on the specific wave structure.Taking values of plasma populations other than those taken here will lead to wave solutions of different spectra and thus quantitatively different scattering results to some extent.The significance of scattering should also be proportional to the wave amplitude, the observational information of which is unknown in the IHS.The present work is based on parallel propagating waves only, and future work should include more realistic situations with obliquely propagating wave components (e.g., Lee et al. 2017).A more rigorous way is to perform a simulation of the selfconsistent excitation of EMIC waves and the resulting particle scattering (e.g., Bortnik et al. 2010;Liu et al. 2010)

Figure 1 .
Figure 1.Threshold temperature anisotropy of the hot protons (A p ) required for ion cyclotron instability shown as a function of the anisotropy of one of two helium ions (A He+ , A He2+ ) with the other set to 0. (a) Results for four different sets of background conditions (plasma density n e in per cubic centimeter and B o in nanotesla), which were chosen to represent the IHS conditions as available from the observations of Voyager and previous reports (the representative cases described in Section 2 in the text).(b) Results for different parametric choices while the other parameters remain the same as for the blue dashed curve in (a): the black curve is the case with reduced hot proton density by 50% (and increased cold protons accordingly), the magenta curve is the case with reduced temperatures by 1 order for all of the hot populations, and the olive and orange curves are the cases with κ = 4 and 5, respectively.f is defined by the ratio between the electron plasma and gyrofrequencies, ω pe /Ω e .

Figure 2 .
Figure 2. Example of the ion cyclotron wave solution of the kinetic dispersion equation.

Figure 3 .
Figure 3. Expanded view of the unstable EMIC wave solution taken from Figure 2 and the corresponding minimum resonant energies E min for interaction with protons and helium ions.

Figure 4 .
Figure 4. Demonstration of pitch-angle scattering effects for 48 protons launched with the same initial energy (KE o = 5 keV) and pitch angle (α o = 135°) but different gyro-phases.(a) Local pitch angles over time shown only for six protons for visual clarity.The average pitch angle a á ñ estimated for all 48 protons is also shown as a thick line.(b) The diffusive (red) and advective (blue) pitch-angle changes.The average pitch angle (black) is repeated for reference.The scales of the diffusive and advective changes are shown on the right axis, and the scale of the average pitch angle is shown on the left axis.

Figure 5 .
Figure 5. Diffusive and advective changes in the pitch angle ((Δα) dif , (Δα) adv in (a) and (b), respectively) and in the kinetic energy ((Δ KE) dif /KE o , (Δ KE) adv /KE o ) in (c) and (d), respectively) for protons shown in the space defined by the initial pitch angle α o versus the initial kinetic energy KE o .Note that the pitch-angle changes are in degrees, and the kinetic energy changes are normalized to the corresponding initial kinetic energy.The ellipsoidal contours refer to the main scattering domains to help visual identification.

(
Figure 4(b) shows the advective (blue) and diffusive (red) pitch-angle changes computed from the local pitch angles of 48 protons (the average pitch angle a á ñ (black) is repeated here for reference).Note that the scales for the advective and diffusive changes are shown on the right axis and that for the average pitch angle is shown on the left axis.The advective change adv ( ) a D implies the extent to which the average pitch angle a á ñ is shifted from the initial pitch angle (α o = 135°).Note that the average pitch angle a á ñ differs from the initial pitch angle α o = 135°by ∼±10 o or less, so adv ( ) a D lies within ∼±10 o in this example ((Δα) adv =〈α〉 − 135 o ).The diffusive change (red) indicates the extent to which the pitchangle distribution of individual particles at a given time is "diffusive relative to the local average."This diffusive change dif ( ) a D increases (up to ∼20°) quickly in time initially until ∼20 gyro-periods, followed by an overall saturated level although with some fluctuations.The early stage of the growth of the diffusive pitch-angle change is not quite linear; nevertheless, it may be approximately related to the widely known quasi-linear resonant diffusion coefficient viaLiu et al. 2012).
) and (b) show the diffusive and advective pitchangle changes, respectively ( We identify the main domain in the space defined by the initial pitch angle α o and initial kinetic energy KE o where notably large pitch-angle changes occur (marked by ellipsoidal contours to help visual inspection).The pitch-angle scattering in that domain is significant (mostly a few tens of degrees or larger), mainly between KE o = ∼2 keV and ∼10 keV.The details of the scattering are dependent on the initial pitch angle α o such that with an increasing initial energy KE o , the scattering effect in both the diffusive and advective changes becomes limited toward a more perpendicular pitch angle (closer to α o = 90°) to meet the resonance condition.Figures 5(c) and (d) show the diffusive and advective energy changes ( KE dif ( corresponding initial kinetic energies KE o , respectively.They indicate that in the same pitch angle-energy domain (the ellipsoidal contour regime), the relative energy changes are mostly ∼10% or less.

Figure 6 .
Figure 6.Example of the particle positions in velocity space for a proton launched with (KE o = 5 keV, α o = 135°).The start point is marked by the solid blue circle, the particle positions at numerous selected times are in open black circles, the constant kinetic energy contour is in green, and the single wave characteristics curve is in red.V Res || is the parallel velocity of resonant kinetic energy.
(i) diffusive pitchangle change of several degrees at KE o < ∼0.05 keV and in the initial pitch-angle range from α o ∼ 120°to ∼170°(the domain below the line with arrows in Figure 7(a)); (ii) advective pitchangle change of a few tens of degrees at KE o < ∼ 0.05 keV and in the pitch-angle range α o > ∼160°(the blue domain approximately bounded by the dotted line in Figure 7(b)); (iii) diffusive energy change up to above 30% at KE o < ∼ 0.03 keV and in the initial pitch-angle range of α o < ∼130°(the yellow/ green domain bounded by the dotted line in Figure 7(c)); and (iv) advective energy change up to above 10% at KE o < ∼ 0.04 keV and in the nearly entire pitch-angle range (mainly the domain below the dotted line in Figure 7(d)).

Figure 7 .
Figure 7. Diffusive (upper panels) and advective (lower panels) changes in the pitch angle (left) and energy (right) for He 2+ shown in the same format as Figure 5.
) and (b) indicate that the main changes in the pitch angle by up to ∼25°occur at energies between KE o = ∼0.02keV and ∼0.5 keV (the ellipsoidal contours).This is consistent with the resonant interactions according to Figure9.In the same energy domain, the energy changes are mostly ∼30% or above (the ellipsoidal contours in

Figure 8 .
Figure 8. Diffusive (upper panels) and advective (lower panels) changes in the pitch angle (left) and energy (right) for He + shown in the same format as Figure 5.
Figures11(c) and (d)), which are more significant than for He2+ with the smaller wavenumber model in Figure7.Additionally, the pitch-angle changes at higher energies (KE o > ∼0.5 keV) are non-negligible although with limitations to specific pitch-angle domains (the thin arrows in Figures11(a) and (b)).Below KE o = ∼0.02keV, the scattering process should be nonresonant (according to Figure 9), which is quite notable, particularly in the advective changes (thick arrows in Figures 11(b) and (d): mostly a few tens of degrees in the advective pitch-angle change and up to ∼50% in the advective energy change.However, a precise distinction in energy between resonant and nonresonant scatterings is not practical, particularly in the advective pitch-angle and energy changes (Figures 11(b) and (d)).The results for He + are shown in Figure12, which clearly reveals the main pitch angle-energy domains for the largest scatterings.They are dramatically different from those for He + with the smaller wavenumber model in Figure8.The scattering effect here is nonresonant.First, Figure12 (a) indicates that the diffusive pitch-angle change of > ∼10°occurs mainly below KE o = ∼0.5 keV, and the largest changes (up to ∼25°) occur at the lowest energy domain of KE o < ∼0.05 keV and at the intermediate pitch-angle domain of α o ∼ 120°-155°(near the white arrow).Figure 12(b) shows that the advective pitch-angle change is notable at even broader energy ranges: negative advection by a few tens of degrees from KE o = ∼0.01keV up to tens of keV but becoming limited to more perpendicular pitch-angle domains with increasing energy and another one above KE o = ∼5 keV and at a pitch-angle range of α o ∼ 120°-160°(the black arrow).In similar domains, Figures 12(c) and (d) reveal significant energy scattering in terms of diffusive and advective changes.Specifically, the diffusive energy change is most significant below the KE o = ∼0.05keV domain, mostly ∼100% or larger, up to ∼200% (near the white arrow in Figures11(c) and (d)), which are more significant than for He2+ with the smaller wavenumber model in Figure7.Additionally, the pitch-angle changes at higher energies (KE o > ∼0.5 keV) are non-negligible although with limitations to specific pitch-angle domains (the thin arrows in Figures11(a) and (b)).Below KE o = ∼0.02keV, the scattering process should be nonresonant (according to Figure 9), which is quite notable, particularly in the advective changes (thick arrows in Figures 11(b) and (d): mostly a few tens of degrees in the advective pitch-angle change and up to ∼50% in the advective energy change.However, a precise distinction in energy between resonant and nonresonant scatterings is not practical, particularly in the advective pitch-angle and energy changes (Figures 11(b) and (d)).The results for He + are shown in Figure12, which clearly reveals the main pitch angle-energy domains for the largest scatterings.They are dramatically different from those for He + with the smaller wavenumber model in Figure8.The scattering effect here is nonresonant.First, Figure12 (a) indicates that the diffusive pitch-angle change of > ∼10°occurs mainly below KE o = ∼0.5 keV, and the largest changes (up to ∼25°) occur at the lowest energy domain of KE o < ∼0.05 keV and at the intermediate pitch-angle domain of α o ∼ 120°-155°(near the white arrow).Figure 12(b) shows that the advective pitch-angle change is notable at even broader energy ranges: negative advection by a few tens of degrees from KE o = ∼0.01keV up to tens of keV but becoming limited to more perpendicular pitch-angle domains with increasing energy and another one above KE o = ∼5 keV and at a pitch-angle range of α o ∼ 120°-160°(the black arrow).In similar domains, Figures 12(c) and (d) reveal significant energy scattering in terms of diffusive and advective changes.Specifically, the diffusive energy change is most significant below the KE o = ∼0.05keV domain, mostly ∼100% or larger, up to ∼200% (near the white arrow in Figures11(c) and (d)), which are more significant than for He2+ with the smaller wavenumber model in Figure7.Additionally, the pitch-angle changes at higher energies (KE o > ∼0.5 keV) are non-negligible although with limitations to specific pitch-angle domains (the thin arrows in Figures11(a) and (b)).Below KE o = ∼0.02keV, the scattering process should be nonresonant (according to Figure 9), which is quite notable, particularly in the advective changes (thick arrows in Figures 11(b) and (d): mostly a few tens of degrees in the advective pitch-angle change and up to ∼50% in the advective energy change.However, a precise distinction in energy between resonant and nonresonant scatterings is not practical, particularly in the advective pitch-angle and energy changes (Figures 11(b) and (d)).The results for He + are shown in Figure12, which clearly reveals the main pitch angle-energy domains for the largest scatterings.They are dramatically different from those for He + with the smaller wavenumber model in Figure8.The scattering effect here is nonresonant.First, Figure12 (a) indicates that the diffusive pitch-angle change of > ∼10°occurs mainly below KE o = ∼0.5 keV, and the largest changes (up to ∼25°) occur at the lowest energy domain of KE o < ∼0.05 keV and at the intermediate pitch-angle domain of α o ∼ 120°-155°(near the white arrow).Figure 12(b) shows that the advective pitch-angle change is notable at even broader energy ranges: negative advection by a few tens of degrees from KE o = ∼0.01keV up to tens of keV but becoming limited to more perpendicular pitch-angle domains with increasing energy and another one above KE o = ∼5 keV and at a pitch-angle range of α o ∼ 120°-160°(the black arrow).In similar domains, Figures 12(c) and (d) reveal significant energy scattering in terms of diffusive and advective changes.Specifically, the diffusive energy change is most significant below the KE o = ∼0.05keV domain, mostly ∼100% or larger, up to ∼200% (near the white arrow in Figures11(c) and (d)), which are more significant than for He2+ with the smaller wavenumber model in Figure7.Additionally, the pitch-angle changes at higher energies (KE o > ∼0.5 keV) are non-negligible although with limitations to specific pitch-angle domains (the thin arrows in Figures11(a) and (b)).Below KE o = ∼0.02keV, the scattering process should be nonresonant (according to Figure 9), which is quite notable, particularly in the advective changes (thick arrows in Figures 11(b) and (d): mostly a few tens of degrees in the advective pitch-angle change and up to ∼50% in the advective energy change.However, a precise distinction in energy between resonant and nonresonant scatterings is not practical, particularly in the advective pitch-angle and energy changes (Figures 11(b) and (d)).The results for He + are shown in Figure12, which clearly reveals the main pitch angle-energy domains for the largest scatterings.They are dramatically different from those for He + with the smaller wavenumber model in Figure8.The scattering effect here is nonresonant.First, Figure12 (a) indicates that the diffusive pitch-angle change of > ∼10°occurs mainly below KE o = ∼0.5 keV, and the largest changes (up to ∼25°) occur at the lowest energy domain of KE o < ∼0.05 keV and at the intermediate pitch-angle domain of α o ∼ 120°-155°(near the white arrow).Figure 12(b) shows that the advective pitch-angle change is notable at even broader energy ranges: negative advection by a few tens of degrees from KE o = ∼0.01keV up to tens of keV but becoming limited to more perpendicular pitch-angle domains with increasing energy and another one above KE o = ∼5 keV and at a pitch-angle range of α o ∼ 120°-160°(the black arrow).In similar domains, Figures 12(c) and (d) reveal significant energy scattering in terms of diffusive and advective changes.Specifically, the diffusive energy change is most significant below the KE o = ∼0.05keV domain, mostly ∼100% or larger, up to ∼200% (near the white arrow in

Figure 9 .
Figure 9. Unstable EMIC wave solution in a larger wavenumber domain and the corresponding minimum resonant energies for interaction with protons and He 2+ .

Figure 10 .
Figure10.Diffusive (upper panels) and advective (lower panels) changes for protons shown in the same format as in Figure5but with a different EMIC wave model covering a larger wavenumber regime (as shown in Figure9).

Figure 11 .
Figure 11.Diffusive (upper panels) and advective (lower panels) changes for He 2+ shown in the same format as Figure 5 but with a different EMIC wave model covering a larger wavenumber regime (Figure 9).
, which has not yet been attempted for IHS conditions.Ultimately, the specific evolution of ion distribution functions under EMIC wave influence needs to be quantified to relate them to ENA observations by IBEX (McComas et al. 2009), (Krimigis et al. 2009), and the future IMAP (McComas et al. 2018).

Figure 12 .
Figure 12.Diffusive (upper panels) and advective (lower panels) changes for He + shown in the same format as Figure 5 but with a different EMIC wave model covering a larger wavenumber regime (Figure 9).Note that the larger ranges of color scales are used for the energy changes in (c) and (d) than those used for protons and He 2+ in Figures 10 and 11.

Figure 13 .
Figure13.Schematic demonstration of two cases where the velocity vector v of ions along the line of sight (LOS) toward an ENA observing spacecraft makes different angles with the background magnetic field B o at the time when the ions are supposed to charge exchange with neutral atoms.In one case (LOS 0 ), the angle is 90°, and in the other case (LOS 1 ), it is > 90°.The waves (magenta) are superposed, which may scatter ions (see text in Section 3.3 for details).