The Impact of Pickup Ion Thermal Spread on Pickup Ion Ring-beam-driven Instabilities and Scattering in the Outer Heliosheath

The present study investigates the unstable waves driven by the pickup ions of ring-beam distributions with various pickup angles and pickup ion temperatures in the outer heliosheath, using both linear instability analysis and hybrid simulations. While previous papers have generally assumed specific pickup ion temperatures, this study takes a more comprehensive approach by examining a wide range of pickup ion temperatures that may occur in the outer heliosheath. Our one-dimensional simulations demonstrate that regardless of the initial pickup ion temperature, the pitch-angle scattering of the pickup ions at small pickup angles saturates before the pickup ions can reach the hemisphere of negative parallel velocities with respect to the background magnetic field in velocity space, while at a 90° pickup angle, the pickup ions of ring distributions attain a significant level of isotropy. In contrast, the two-dimensional simulation results show that at all pickup angles, increasing the pickup ion temperature only slightly reduces the pickup ion scattering rate, but does not prevent the pickup ions from reaching the hemisphere of negative parallel velocities. Overall, the results do not align with the requirements of the earlier proposed scenarios for the secondary energetic neutral atom (ENA) mechanism of the Interstellar Boundary Explorer ENA ribbon, which demand either weak pickup ion scattering in the outer heliosheath or at least incomplete pickup ion scattering in the off-ribbon directions.


Introduction
Observed by NASA's Interstellar Boundary Explorer (IBEX) spacecraft, the IBEX energetic neutral atom (ENA) ribbon is one of the most fascinating features of our solar system (McComas et al. 2004(McComas et al. , 2009b)).Observations show that the narrow ENA ribbon is brighter than the globally distributed ENA flux and extends across the entire sky (Fuselier et al. 2009;McComas et al. 2009aMcComas et al. , 2011;;Schwadron et al. 2011).Understanding the characteristics of the IBEX ENA ribbon is vital for scientists to gain insights into the dynamic interactions between the Sun, the heliosphere, and the interstellar medium.It is believed that the outer heliosheath pickup ions are the origin of the IBEX ENA ribbon, based on several lines of evidence.The pickup ions are generated in the outer heliosheath, due to the charge-exchange interaction between the solar wind neutrals and the interstellar medium ions.The energy spectrum of the pickup ions in the outer heliosheath is not known, but it is expected to match the energy range of the ENAs observed in the ribbon.It can be deduced from ENAs, but this depends on the model of the origin of the ribbon neutrals.In addition, according to global MHD simulations of the heliosphere, the orientation of the IBEX ENA ribbon is aligned with the directions where the local interstellar magnetic field (B 0 ) is perpendicular to the line of sight that is radial to the Sun (Funsten et al. 2009;McComas et al. 2009a;Schwadron et al. 2009).However, the magnetic field direction in the models does not match the Voyager observations.The models in question assume that the pristine interstellar field is close to the ribbon center direction, but the actual direction is not known.The ribbon directions are consistent with the expected ENA flow produced by the pickup ions in the outer heliosheath after they charge exchange with the local interstellar neutrals.Other independent observations are required to support the ribbon origin hypothesis (e.g., the delay in the response to the changing solar activity; Zirnstein et al. 2023) The origin of the ribbon has been a subject of scientific inquiry, and the explanation that appears most viable is the secondary ENA mechanism, as it is well aligned with the observed features of the ribbon.This mechanism proposes that the ENAs in the ribbon are created through a three-step process.First, the solar wind ions charge exchange with the interstellar neutral atoms after they have entered the heliosphere, leading to the production of primary ENAs (also called solar wind neutrals).Second, the primary ENAs travel across the heliopause into the outer heliosheath and undergo charge exchange with the local interstellar plasma ions, creating energetic ions.The energetic ions inherit the velocities of their parental primary ENAs in the interstellar plasma reference frame (v p ) and are then picked up by the interstellar magnetic field.Consequently, the pickup ions in the outer heliosheath are expected to exhibit a ring-beam velocity distribution defined by the pickup angle (α), the angle between v p and B 0 .Finally, the pickup ions undergo charge exchange with the neutral atoms in the interstellar medium, resulting in the production of secondary ENAs.The ENAs can travel back toward the Earth without being influenced by the solar wind magnetic field and form a ribbon-like structure, due to their preferential emission perpendicular to the local interstellar magnetic field direction (Chalov et al. 2010;Heerikhuisen et al. 2010;Schwadron & McComas 2013;McComas et al. 2014).
Related to the secondary ENA mechanism, two opposing scenarios have been proposed on the degree of pickup ion scattering in the outer heliosheath.One hypothesis, known as the weak-scattering scenario, suggests that the pickup ions in the ribbon directions undergo only minor scattering and retain their anisotropic ringlike distribution before being neutralized into the earthward ENAs of the IBEX ribbon.This scenario was supported by a three-dimensional MHD-plasma/kineticneutral model of the heliosphere, which assumed a narrow partial shell distribution for the outer heliosheath pickup ions and was able to reproduce an ENA ribbon similar to the observed one (Heerikhuisen et al. 2010).In contrast, another hypothesis called the spatial retention scenario, proposed by Schwadron & McComas (2013), argues that strong scattering of the pickup ions in the ribbon directions is necessary.According to this scenario, the fast scattering of pickup ions by Alfvén-cyclotron waves in the vicinity of the ribbon directions (α ∼ 90°) leads to a reduced ion scattering mean free path, ultimately restricting the pickup ions within the ribbon region.However, in the directions away from the ribbon (i.e., the offribbon directions), where α deviates from 90°, it is postulated that pickup ion scattering by Alfvén-cyclotron waves is confined to a single hemisphere in velocity space, allowing the pickup ions to drift away from the ribbon directions.The ribbon structure arises due to the pickup ion retention near the ribbon directions and a depletion of the pickup ions in the offribbon directions.
Numerous investigations of the dynamics of the outer heliosheath pickup ions have suggested that the weakscattering scenario of the secondary ENA mechanism may not be feasible, as there is substantial pickup ion scattering induced by electromagnetic instabilities or turbulence in the outer heliosheath.Hybrid simulations performed by Florinski et al. (2010) revealed considerable scattering of ringlike pickup ion distributions at different pickup angles.Liu et al. (2012) further took into account the gradual injection of pickup ions with a ring distribution in their hybrid simulations.While the timescale for pickup ion scattering estimated in Liu et al. (2012) is longer than the one reported in Florinski et al. (2010), it remains significantly shorter than the charge-exchange timescale of a few years for the outer heliosheath pickup ions to transform into secondary ENAs.Later, Summerlin et al. (2014) and Florinski et al. (2016) suggested that a pickup ion ring distribution with a finite thermal spread is stable to the parallel-propagating Alfvén-cyclotron waves.However, the linear instability analysis and hybrid simulations in Min & Liu (2018) demonstrated that oblique unstable mirror modes contribute actively to the scattering of the pickup ions with a thermal spread inside the so-called Alfvén-cyclotron stability gap suggested by Summerlin et al. (2014) and Florinski et al. (2016).The significant pickup scattering observed in the mentioned studies suggests that the weak-scattering assumption may not be applicable to the secondary ENA mechanism.As a result, the spatial retention scenario has emerged as a more reliable alternative that can overcome the challenge of fast pickup ion scattering.
The spatial retention scenario, on the other hand, poses challenges of its own, as the isotropic pickup ion velocity distribution that arises from fast scattering within the retention region results in an ENA flux lower than previously estimated (e.g., Heerikhuisen et al. 2014;Isenberg 2014;Zirnstein et al. 2019), unless the spatial retention sufficiently enhances the pickup ion density in the ribbon directions.The "dominant turbulence" model proposed by Isenberg (2014) represents an alternative version of the spatial retention scenario.According to this model, the strong pickup ion scattering in the retention region is mainly caused by turbulent fluctuations.Later on, this model was developed further by taking into account the neutral atoms from the inner heliosheath and expanding the twodimensional model to three dimensions (Isenberg 2015).Nevertheless, the maximum ENA fluxes estimated in both Isenberg (2014) and Isenberg (2015) are lower than the observed values, which could be attributed to oversimplified assumptions made in the models, such as assuming a uniform and straight interstellar magnetic field.
Besides the ENA flux deficiency challenge, the spatial retention scenario involves an important assumption of incomplete pickup ion scattering in the off-ribbon directions.To assess this assumption, our recent studies have investigated the electromagnetic instabilities driven by ring-beam velocity distributions of pickup ions at various pickup angles.Linear instability analysis conducted by Mousavi et al. (2021) demonstrated that the left-helicity waves are the most unstable ring-beam-driven waves at parallel or antiparallel propagation relative to the background magnetic field.Additionally, it was shown that the parallel-propagating left-helicity waves are unstable in two distinct frequency and wavenumber ranges, one below and the other far above the proton cyclotron frequency.The linear instability analysis results in Mousavi et al. (2021) were subsequently confirmed by one-dimensional hybrid and particle-in-cell simulations in Mousavi et al. (2022a) and Mousavi et al. (2022b), respectively.The evaluation of pickup ion scattering in these studies demonstrated that the outer heliosheath pickup ions at small pickup angles (in the offribbon directions) cannot be scattered to the hemisphere of negative parallel velocities by either low-frequency or highfrequency left-helicity waves (of parallel propagation).This is in line with the requirement of the spatial retention scenario that the pickup ions in the off-ribbon directions undergo an incomplete scattering and stream away from the retention region.However, the linear instability analysis of the oblique waves in Mousavi et al. (2022c) revealed that the outer heliosheath pickup ions can excite quasi-parallel and oblique left-helicity modes as well as mirror mode.The corresponding two-dimensional hybrid simulations in Mousavi et al. (2022d) demonstrated that the pickup ions are scattered toward the hemisphere of negative parallel velocities when the contributions of oblique unstable waves are included.So the pickup ions in the off-ribbon directions do not maintain their magneticfield-aligned streaming required by the spatial retention scenario, conflicting with its incomplete scattering assumption.
In our previous studies, the parallel temperature of the pickup ions was chosen to make the system lie in the Alfvéncyclotron stability gap to exclude the well-studied Alfvéncyclotron instability at α ∼ 90°.However, this value may not represent realistic pickup ion temperatures in the outer heliosheath.Lacking direct in situ observations, the exact plasma parameters in the outer heliosheath are unknown.The stability of the outer heliosheath pickup ions of more realistic multicomponent velocity distributions (Heerikhuisen et al. 2016) was examined by Roytershteyn et al. (2019) and Mousavi et al. (2020) through particle-in-cell and hybrid simulations, respectively.However, the distribution only applies to pickup ions in the ribbon directions and cannot be easily generalized to pickup ions in the off-ribbon directions.The uncertainty about the pickup ion distributions in the outer heliosheath underscores the necessity of investigating various parameters that have the potential to influence the pickup ion scattering, including the thermal spread of pickup ions.
In this paper, we evaluate the impact of pickup ion thermal spread on the electromagnetic instabilities driven by ring-beam velocity distributions of pickup ions and the associated pickup ion scattering.The parallel-and antiparallel-propagating as well as oblique waves driven by ring-beam pickup ion distributions of various thermal spreads in the outer heliosheath are studied using linear instability analysis.Several onedimensional and two-dimensional hybrid simulations are also performed to assess the effect of the pickup ion thermal spread on pickup ion scattering in both the ribbon and off-ribbon directions.Two-dimensional simulations are particularly carried out to investigate how the pickup ion thermal spread affects pickup ion scattering in the presence of oblique unstable modes at different pickup angles.The paper is organized as follows.Section 2.1 describes the theoretical model and notation.Section 2.2 presents the linear instability analysis  results and discussion.Section 3.1 describes the simulation setup, and Sections 3.2 and 3.3 present the one-and twodimensional hybrid simulation results, respectively.Finally, Section 4 concludes by summarizing the results.

Theoretical Model
The study is performed for a three-component magnetized plasma in the outer heliosheath.The plasma comprises pickup ions, background ions, and electrons, denoted by the subscripts p, b, and e, respectively.The plasma components are described by general ring-beam velocity distribution functions of ere, v ∥ and v ⊥ are the parallel and perpendicular velocity components; θ s∥ and θ s⊥ are the parallel and perpendicular thermal spreads of component s, respectively.The drift and ring speeds of the pickup ions for a given pickup velocity, v p , and pickup angle, α, are given by pr p a = respectively.The drift speed of the background protons can be determined by applying the zero net current condition, which gives v bd = − n p v pd /n b , and the charge neutrality condition yields n b /n e = 1 − n p /n e , where n e , n p , and n b are the electron, pickup ion, and background ion densities, respectively.While the pickup ion thermal spread varies in a wide range in the present study, the ) and growth rate ( ( ) ) using Muller's method (Press et al. 2007).These quantities are functions of the wavenumber where k ∥ and k ⊥ represent the parallel and perpendicular wavenumbers, respectively, with respect to the background magnetic field B 0 .The linear analysis is performed for various pickup angles and pickup ion temperatures = ^in the present study).Consistent with our previous linear analysis studies (e.g., Mousavi et al. 2021Mousavi et al. , 2022c)), the unstable waves are divided into four categories:

Linear Analysis Results
Figure 1 shows the linear analysis results for the parallelpropagating right-helicity waves at 0 < ω r < Ω p (left column) and ω r Ω p (right column) at a wide range of pickup ion temperatures and different pickup angles.The maximum growth rate of the right-helicity waves occurs at α = 90°, and its value decreases with decreasing α.For 0 < ω r < Ω p , the growth rate of the most unstable mode varies with T p and reaches the maximum at a T p value that increases with decreasing α, whereas for ω r > Ω p , the maximum growth rate occurs when T p → 0. Figure 1 also reveals that the wavenumber of the most unstable right-helicity wave shifts to smaller values as α decreases.
It can be seen from Figure 1 that there are upper and lower limits of T p for unstable high-and low-frequency right-helicity waves.These limits vary with the pickup angle, as further illustrated in Figure 2. In Figure 2, the hatched portions represent the parameter regions where the right-helicity waves exhibit positive growth rates (γ/Ω p > 10 −5 ).The orange and blue regions correspond to the unstable right-helicity waves at ω r Ω p and 0 < ω r < Ω p , respectively.At α = 90°, the righthelicity waves are stable in the range 0.004 < T p < 0.13, which is consistent with the so-called Alfvén-cyclotron stability gap (Florinski et al. 2016).Given the low values of T p in the orange region, which do not accurately represent the pickup ion temperatures in the outer heliosheath, the forthcoming sections of hybrid simulations will not investigate the unstable waves within this particular region.Moreover, the high-frequency waves are unlikely to have a significant impact on the pickup ion scattering (Mousavi et al. 2022b).
Figure 3 presents the growth rates and real frequencies of the parallel-and antiparallel-propagating left-helicity waves as functions of wavenumber and T p at various pickup angles.The growth rates are color coded, and the solid and dashed black contours correspond to the real frequencies of the parallel-(ω r > 0) and antiparallel-propagating (ω r < 0) left-helicity waves, respectively.The logarithmic color scale is employed to better visualize the changes in growth rates.At α = 90°, the unstable left-helicity waves propagate against the background magnetic field direction.These waves are antiparallel-propagating Alfvén-cyclotron waves whose parallel-propagating counterparts are shown in Figure 1(a).The symmetry between Figures 1(a) and 3(a) is due to the symmetric pickup ion ring distribution with respect to the background magnetic field at α = 90°.Figures 3(b)-(f) then demonstrate that the growth rates of the left-helicity waves decrease with the increase of T p .Furthermore, at T p = 0.05, the growth rates and real frequencies for all pickup angles are consistent with those in Mousavi et al. (2021).
Mirror waves are the fastest-growing oblique mode according to the previous linear instability analysis results in Mousavi et al. (2022c).Figure 4 shows the real frequencies (solid contours) and growth rates (color scale) of the mirror waves in k ∥ − k ⊥ space for three selected pickup angles (α = 80°, 70°, and 45°) and five selected pickup ion temperatures (T p = 0.2, 1, 2, 5, and 10).The maximum growth rate of mirror waves decreases as T p increases for all pickup angles.However, the propagation direction of the most unstable mirror wave is not significantly influenced by T p , but rather by the pickup angle.
In particular, the unstable mirror modes propagate more perpendicularly at smaller pickup angles, regardless of the pickup ion temperature.

Simulation Setup
Corresponding to the linear analysis in Section 2, one-and two-dimensional hybrid simulations are conducted, in which both the pickup ions and background ions are treated as simulation particles, while the electrons are described as a massless fluid (Winske & Omidi 1993).The one-dimensional simulations have the simulation boxes along the background magnetic field direction and, therefore, contain only the parallel-and antiparallel-propagating waves.The simulation box includes N = 800 cells and 10 5 simulation particles per cell for each ion component.The simulation domain size is L = 1000λ p , where λ p = v A /Ω p is the proton inertial length and Ω p is the proton cyclotron frequency.In the two-dimensional simulations, the simulation box is in the x − y plane, and the background magnetic field is oriented to the x-axis.The domain size is 1000λ p and divided into 800 grids in each direction.There are 1000 simulation particles per cell for each ion component.In all the simulations, periodic boundary conditions are applied, and the simulation time step is t 0.01 p 1 D = W -.

One-dimensional Hybrid Simulations
Figure 5 shows the temporal evolution of the wave magnetic field energy densities in six one-dimensional hybrid simulations for two different pickup angles and three different pickup ion temperatures, as listed in the legends as well as in Table 1.The exponential growth rates estimated at the early stage of the simulations are presented in Table 1 and compared with the growth rates of the parallel-and antiparallelpropagating modes from the linear instability analysis results shown in Figures 1 and 3.The comparison indicates that the growth rates obtained from the simulations align closely with, albeit slightly lower than, the linear analysis results.The minor difference should be due to the evolution of the pickup ion distributions during the simulations, as the linear analysis is strictly for the initial pickup ion distributions.Although the growth rate of the most unstable wave decreases with increasing T p , the saturation level of the fluctuating magnetic field energy is more affected by the pickup angle than the initial pickup ion temperature.
For each simulation, the magnetic power spectra of the enhanced waves in frequency and wavenumber space during two different simulation periods are plotted in Figure 6, to confirm the wavenumber and frequency ranges of the unstable waves predicted by the linear instability analysis.While the enhanced waves in the early stage of the simulations appear in the wavenumber and frequency ranges predicted by the linear analysis, the right panels of Figure 6 also reveal that the enhanced waves continue to grow at later times in the simulations, maintaining similar wavenumbers and frequencies, but over broader ranges.
Figure 7 illustrates the temporal evolution of the pickup ion distributions in the one-dimensional hybrid simulations of various pickup ion temperatures and two different pickup angles.At α = 45°, the pickup ion scattering in the v || − v ⊥ space is confined to one hemisphere, while at α = 90°, the pickup ions are scattered toward an isotropic shell.To more quantitatively compare the pickup ion scattering in the onedimensional simulations shown in Figure 7, Figure 8 further presents the standard deviations of the pickup ion pitch angles in the simulations.The initial growth of the standard deviation in a simulation with α = 45°is faster than that in the simulation with α = 90°and the same pickup ion temperature.However, by the end of the simulations, the standard deviations for α = 90°surpass those for α = 45°, regardless of T p .For α = 45°(red curves), the standard deviations saturate below 20°in all three simulations of different pickup ion temperatures, consistent with the pickup ion scattering being limited to the hemisphere of positive parallel velocities in Figures 7(c), (f), and (i).However, at α = 90°(black curves), the standard deviations increase until they tend to the level of an isotropic pitch-angle distribution (39.2°).This result confirms the results in Mousavi et al. (2022a) that when only parallel-and antiparallel-propagating waves are concerned, the pickup ion scattering is incomplete in the off-ribbon directions, as required by the spatial retention scenario of the secondary ENA mechanism.

Two-dimensional Hybrid Simulations
Several two-dimensional simulations are also performed to investigate pickup ion scattering at various pickup angles and pickup ion thermal spreads in the presence of oblique unstable modes.Since the two-dimensional simulations consume much more computation time than the one-dimensional simulations, the number of simulation particles per cell for each ion component has been reduced (from 10 5 ) to 1000 in the twodimensional simulations.In addition, the pickup ion concentration is increased by a factor of 2.5 to n p /n e = 2 × 10 −4 in comparison with the one-dimensional simulations and the linear analysis.The slight increase of n p /n e is to boost the instability growth in the simulations in order to save computation times.According to our linear analysis, it does not much affect the frequency and wavenumber ranges of the unstable waves, let alone that n p /n e = 2 × 10 −4 is still a reasonable value of the pickup ion concentration in the outer heliosheath (Florinski et al. 2010).
Figure 9 shows the wave magnetic field power spectra in k ∥ − k ⊥ space at three different times in the two-dimensional simulations for α = 45°and T p = 1, 5, and 10.Just like in the one-dimensional simulations, the parallel and perpendicular wavenumber ranges of the enhanced waves in the early stage of the simulations are consistent with those predicted by the linear instability analysis as shown in Figures 1, 3, and 4. The wavenumber ranges of the enhanced waves extend to larger extents as the simulations proceed.
The Fourier decomposition is then used to separate the waves of right and left helicities and opposite propagation directions with respect to the background magnetic field (Kodera et al. 1977).The resulting wave magnetic field wavenumber-frequency power spectra are shown in Figure 10 for the three two-dimensional simulations of α = 45°and different T p in two time intervals.The waves are divided into four quadrants, labeled as R + , R − , L + , and L − (as indicated in Figure 10(a)).Here, R and L denote the right and left helicities, and the superscripts + and − indicate the parallel and antiparallel propagation directions, respectively.The spectra at early times in the simulations (the left column in Figure 10) indicate that the wavenumbers and frequencies of the enhanced left-helicity waves are consistent with the linear analysis results.For example, early in the simulation of T p = 5, the strongest left-helicity wave corresponds to |k ∥ λ p | ∼ 0.06 and | ω| ∼ 0.05 at antiparallel propagation (Figure 10(c)).These values are in good agreement with the linear analysis prediction shown in Figure 3(e).However, the right-helicity waves that arise in the simulations are not predicted by the linear analysis.As illustrated in Figure 2, the linear instability analysis reveals that at α = 45°, the right-helicity waves are stable at 1 < T p < 10.Interestingly, Figure 10 shows that the absolute frequencies and wavenumbers of the right-helicity waves that appear in the simulations are identical to those of the lefthelicity waves excited.This suggests that the enhanced righthelicity waves in the simulations may be the result of energy transfer from their left-helicity counterparts through wavewave coupling, rather than being directly driven by the ringbeam pickup ions.
Besides the three two-dimensional simulations for α = 45°d iscussed above, two extra groups of two-dimensional hybrid simulations are carried out for α = 70°and α = 80°, respectively, with T p = 1, 5, and 10.The wave magnetic field spectra (not shown) demonstrate similar features as Figures 9 and 10, in that the left-helicity waves excited in the simulations have wavenumbers and frequencies consistent with the prediction of the linear analysis and the enhanced right-helicity waves (that are not predicted by the linear analysis) arise with the same absolute wavenumbers and frequencies as the left-helicity waves excited in the simulations.Figure 11 depicts the temporal evolution of the wave magnetic field energy densities from these different two-dimensional simulations.It shows that at all three pickup angles, the exponential growth rates during the early stages in the simulations decline with the increase of the pickup ion temperature, consistent with the linear instability analysis results.In addition, Figure 11 indicates that at all three pickup ion temperatures, the saturation level of the fluctuating magnetic field in the simulation for α = 45°is larger than that for α = 70°and 80°.
The pickup ion velocity distributions at two different times in all nine two-dimensional simulations are compared in Figure 12.The simulations of different pickup ion temperatures and pickup angles are arranged by the rows and columns, respectively.The two panels within each block depict the velocity distributions of the pickup ions at the beginning and late stages of the simulations.Figure 12 demonstrates that for all pickup angles, the pickup ion distributions with an initial pickup ion temperature of T p = 1 are more scattered compared with those with larger initial pickup ion temperatures.In all cases, the pickup ions extend beyond the hemisphere of positive parallel velocities late in the simulations and are scattered toward an isotropic shell.
Figure 13 shows the standard deviations of the pickup ion pitch angles in the two-dimensional hybrid simulations of various pickup ion temperatures and pickup angles.In contrast to the standard deviations in the one-dimensional simulations of α = 45°saturating below 20°, they continue to grow toward the level of an isotropic pitch-angle distribution (39.2°) in the two-dimensional simulations, regardless of the initial pickup ion temperature and pickup angle.Although the pickup ion distributions of larger T p initially have larger standard deviations, their scattering rates in the simulations are lower than those of smaller T p .Subsequently, the overall scattering timescales are similar in the simulations of different T p .Note that the larger scattering rates in the simulations of smaller T p are consistent with the temporal evolution of the wave magnetic energy densities shown in Figure 11, which indicates that the wave growth rate increases with decreasing T p .

Summary and Conclusion
Linear instability analysis and hybrid simulations are carried out to study the pickup ion ring-beam-driven instabilities at various pickup angles and pickup ion thermal spreads in the outer heliosheath.For each pickup angle, the growth rates, frequencies, and wavenumbers of parallel-and antiparallelpropagating, left-and right-helicity waves as well as oblique mirror waves under varying pickup ion thermal spreads are investigated using linear instability analysis.The results show that the maximum growth rate of the left-helicity waves decreases with the increase of the pickup ion temperature.Also, the right-helicity waves exhibit a stability gap in the pickup ion temperature, which becomes wider when the pickup angle decreases.
Based on the linear instability analysis, several onedimensional hybrid simulations are carried out for different pickup ion temperatures and at 45°and 90°pickup angles, representing the off-ribbon and ribbon directions, respectively.
The one-dimensional simulations contain only the parallel-and antiparallel-propagating waves.The results that even though increasing the pickup ion temperature reduces the  maximum growth rates of the parallel-and antiparallelpropagating modes, it does not significantly alter the saturation levels of the wave magnetic field energy densities at the end of the simulations.The evolution of the pickup ion velocity distributions and the standard deviations of the pickup ion pitch angles indicates that in the off-ribbon directions, the pitchangle scattering of the pickup ions is limited to the hemisphere of positive parallel velocities in velocity space for all the pickup ion temperatures examined.Conversely, in the ribbon directions, ring distributions with different thermal spreads can approach a full isotropy within a timescale much shorter than the charge-exchange time of the outer heliosheath pickup ions.Increasing the pickup ion temperature by a factor of 10 (from T p∥ = 1 to T p∥ = 10) does not significantly affect the isotropization timescale of the pickup ions in the ribbon directions.
Several two-dimensional hybrid simulations are performed for T p∥ = 1, 5, 10 and α = 45°, 70°, and 80°.In comparison with the one-dimensional simulations, the two-dimensional simulations allow the oblique unstable waves to grow.The comparison between the linear instability analysis and hybrid simulation results indicates that the unstable left-helicity and mirror waves predicted by the linear instability analysis are indeed excited in the simulations at the expected frequencies and wavenumbers.The simulation results also demonstrate the excitation of the right-helicity waves that propagate in the same direction, with the same frequencies and wavenumbers as the left-helicity waves excited through the linear instabilities driven by the pickup ion ring-beam distributions.These right-helicity waves are not expected to arise according to the linear instability analysis.They are likely the result of energy transfer from their left-helicity counterparts through wave-wave  coupling.Furthermore, the evolution of the pickup ion distributions in the two-dimensional simulations shows that, with the help of the oblique waves excited, the pickup ions are scattered toward an isotropic shell in all the simulations.This is different from the incomplete pickup ion scattering in the onedimensional simulations of α = 45°, which only contain the enhanced waves of parallel and antiparallel propagation.
Since the pickup ions are scattered toward an isotropic distribution within a relatively short timescale in all the twodimensional simulations of different pickup angles and pickup ion temperatures, the results do not support the assumptions underlying the weak-scattering or spatial retention scenarios of the secondary ENA mechanism.These scenarios demand either weak pickup ion scattering or at least incomplete pickup ion scattering in the off-ribbon directions.Therefore, the previously proposed scenarios for the secondary ENA mechanism may require further investigation, or other factors not considered in the present study need to be taken into account to slow down the pickup ion scattering, especially in the off-ribbon directions.
Voyager 1 measurements have revealed the presence of a large-scale magnetic turbulence background (Burlaga et al. 2015).The turbulence spectrum has a Kolmogorov slope of −5/3.Florinski et al. (2016) estimated the turbulence magnetic field energy density in the wavenumber range contained in their simulation box to be B B 10 2 0 2 7 d ~-.This value is much smaller than the self-excited waves due to the pickup ions in their simulation.A more recent investigation by Burlaga et al. (2018) introduced updated parameters for the turbulence, suggesting a magnetic field energy density about 100 times stronger.Given this correction, and taking into account that the spatial resolution and domain size in our simulations are similar to those in Florinski et al. (2016), the turbulence magnetic field energy density in our simulation is estimated to be on the scale of B B 10 d ~-) are still higher than those for the larger pickup angles.Understanding the connection between the pickup-ion-generated waves and the observed turbulence presents several challenges that need to be carefully considered in the future.

Figure 1 .
Figure1.Linear instability analysis results for the parallel-propagating right-helicity waves at 0 < ω r < Ω p (left column) and ω r Ω p (right column) at a wide range of T p and different pickup angles, as labeled.The growth rate is indicated by the color scale, while the black contours represent the real frequency.

Figure 2 .
Figure 2. The pickup ion temperature range of the unstable parallelpropagating right-helicity waves at frequencies below (blue region) and above (orange) the proton cyclotron frequency as a function of the pickup angle.The hatched area indicates where γ/Ω p > 10 −5 .

Figure 3 .
Figure 3. Linear instability analysis results for the parallel-and antiparallel-propagating left-helicity waves at a wide range of T p and different pickup angles, as labeled.The growth rate is indicated by the color scale, while the black contours represent the real frequency.

Figure 4 .
Figure 4.The growth rates and real frequencies of the mirror waves for α = 80°, 70°, and 45°and at different pickup ion temperatures.Each column corresponds to a different pickup angle and each row corresponds to a different pickup ion temperature, as labeled.

Figure 6 .
Figure 6.Magnetic power spectra of the enhanced waves in frequency and wavenumber space in two different intervals in the one-dimensional simulations of different pickup angles and pickup ion temperatures.

Figure 7 .
Figure 7.The pickup ion velocity distributions at tΩ p = 0, 3000, and 30,000 in the one-dimensional simulations of different pickup angles and pickup ion temperatures, as labeled on the top of each panel.

Figure 8 .
Figure 8.The evolution of the standard deviations of the pickup ion pitch angles in the one-dimensional hybrid simulations of different pickup angles and pickup ion temperatures.

Figure 9 .
Figure 9.The wave magnetic field power spectra in k ∥ − k ⊥ space at three different times in the three two-dimensional simulations for α = 45°and different pickup ion temperatures, as labeled.

Figure 10 .
Figure10.The wave magnetic field wavenumber-frequency power spectra given by the Fourier decomposition analysis in two different intervals in the three twodimensional simulations of α = 45°and different pickup ion temperatures, as labeled.

Figure 11 .
Figure11.Temporal evolution of the wave magnetic field energy densities normalized to the background magnetic field energy density in the twodimensional hybrid simulations corresponding to different pickup angles and pickup ion temperatures, as labeled.

Figure 12 .
Figure 12.The pickup ion velocity distributions at tΩ p = 0 and tΩ p = 2400 from the two-dimensional hybrid simulations of various pickup angles and pickup ion temperatures, as labeled.

Figure 13 .
Figure 13.The evolution of the standard deviations of the pickup ion pitch angles in the two-dimensional hybrid simulations of different pickup angles and pickup ion temperatures.
is at least 1 order of magnitude larger than the estimated magnetic turbulence level.Later,Fraternale & Pogorelov (2021) used high-resolution measurements to estimate the turbulence in a wide range of wavenumbers, including the ion inertial length scale at the Voyager 1 location in the very local interstellar medium.Interestingly, the smallscale magnetic field fluctuations in their results seem to be of the same order of magnetic field fluctuations in our present paper for α = 70°and 80°( which are close to the Voyager 1 direction.However, the excited wave magnetic fluctuations for α = 45°(

Table 1
Growth Rates of the Most Unstable Parallel-and Antiparallel-propagating Modes at Different Pickup Angles and Pickup Ion Temperatures