using kappa functions to characterize outer heliosphere ... - IOPscience

The Astrophysical Journal, 815:31 (13pp), 2015 December 10

doi:10.1088/0004-637X/815/1/31

© 2015. The American Astronomical Society. All rights reserved.

USING KAPPA FUNCTIONS TO CHARACTERIZE OUTER HELIOSPHERE PROTON DISTRIBUTIONS IN THE PRESENCE OF CHARGE-EXCHANGE 1

E. J. Zirnstein1 and D. J. McComas1,2

Southwest Research Institute, San Antonio, TX 78228, USA; [email protected], [email protected] 2 Department of Physics and Astronomy, University of Texas at San Antonio, San Antonio, TX 78249, USA Received 2015 August 28; accepted 2015 November 3; published 2015 December 3

ABSTRACT Kappa functions have long been used in the analysis and modeling of suprathermal particles in various space plasmas. In situ observations of the supersonic solar wind show its distribution contains a cold ion core and powerlaw tail, which is well-represented by a kappa function. In situ plasma observations by Voyager, as well as observations of energetic neutral atom (ENA) spectra by the Interstellar Boundary Explorer (IBEX), showed that the compressed and heated inner heliosheath (IHS) plasma beyond the termination shock can also be represented by a kappa function. IBEX exposes the IHS plasma properties through the detection of ENAs generated by chargeexchange in the IHS. However, charge-exchange modifies the plasma as it flows through the IHS, and makes it difficult to ascertain the parent proton distribution. In this paper we investigate the evolution of proton distributions, initially represented by a kappa function, that experience losses due to charge-exchange in the IHS. In the absence of other processes, it is no longer representable by a single kappa function due to the energydependent, charge-exchange process. While one can still fit a kappa function to the evolving proton distribution over limited energy ranges, this yields fitting parameters (pseudo-density, pseudo-temperature, pseudo-kappa index) that depend on the energy range of the fit. We discuss the effects of fitting a kappa function to the IHS proton distribution over limited energy ranges, its dependence on the initial proton distribution properties at the termination shock, and implications for understanding the observations. Key words: ISM: atoms – methods: analytical – solar wind – Sun: heliosphere 1. INTRODUCTION

4 MeV with a kappa function for the proton distribution revealed a kappa index (κ) of 1.63 (Decker et al. 2005). In situ plasma observations by Voyager 2 (V2) showed that the solar wind remained fairly cold after crossing the termination shock (∼105 K), suggesting that most of the energy was transferred to accelerating the PUIs, while the core solar wind was only moderately heated (Richardson et al. 2008). This also supports the applicability of a kappa function, which often appears as a cool core and suprathermal tail, to the IHS plasma. Starting with the application of the kappa function to describe the distributions of electrons in the magnetosphere (Olbert 1968; Vasyliunas 1968), the kappa function has been used to describe other space plasma environments besides the solar wind, such as particle distributions in planetary magnetospheres and the local interstellar medium. For a review of the application of kappa functions in describing various space plasma environments, see Pierrard & Lazar (2010), Livadiotis & McComas (2013), Livadiotis (2015), and references therein, as well as the recent JGR special issue Origins and Properties of Kappa Distributions. It is important to distinguish the role of the kappa “function” in characterizing space plasma distributions from kappa distributions themselves. Kappa distributions have been shown to arise naturally from non-extensive statistical mechanics (e.g., Milovanov & Zelenyi 2000; Leubner 2002; Livadiotis & McComas 2009), and can describe a physical, stationary state for systems out of thermal equilibrium. The distributions are defined by the kappa index which describes the correlation between the particles and their distance from thermal equilibrium (e.g., Livadiotis & McComas 2013 and references therein). “Kappa-like” distributions have been derived from various physical phenomena, such as (to name a few) a plasma immersed in a suprathermal radiation field that experiences

The solar wind plasma emanating from the Sun interacts with the local interstellar medium plasma, creating the heliosphere (e.g., Parker 1961). While the solar wind and interstellar plasmas are separated by a tangential discontinuity called the heliopause, cold interstellar neutral atoms (mainly hydrogen) traverse through the heliosphere and chargeexchange with solar wind ions. The neutralized solar wind ions become energetic neutral atoms (ENAs), and the ionized interstellar neutrals are picked up by the motional electric field of the solar wind, forming a population of pickup ions (PUIs) that are convected outward with the solar wind (e.g., Vasyliunas & Siscoe 1976; Isenberg 1986). As the solar wind and incorporated PUIs continue to flow away from the Sun at supersonic speeds, they eventually must be turned back by the interstellar plasma at the heliopause. This creates a termination shock, where the plasma slows, compresses, and is heated. Between the termination shock and heliopause, the inner heliosheath (IHS) plasma is a region of extreme interest for missions such as NASAʼs Interstellar Boundary Explorer (IBEX; McComas et al. 2009a), since charge-exchange between IHS ions and interstellar neutral atoms can create ENAs that propagate inward toward Earth and be detected by IBEX. Thus, the analysis and modeling of IBEX ENA measurements exposes the properties of the IHS plasma. Observations of the supersonic solar wind by the Advanced Composition Explorer and Ulysses revealed that the solar wind distribution is composed of a cool core, PUI filled shell, and suprathermal tail (e.g., Gloeckler et al. 2000, 2008). Downstream of the termination shock, Voyager 1 (V1) LECP measurements showed that the scaled intensity of ions in the energy range ∼0.04–20 MeV produced a power-law spectral index of −1.67, and a model fit to observations from 0.04 to 1

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velocity space diffusion (Hasegawa et al. 1985), particles experiencing a series of random walk jumps in velocity space governed by a Lévy flight probability distribution (Collier 1993), a kinetic theory of particles in a stable, but turbulent, thermodynamic state out of equilibrium (Treumann 1999a, 1999b), or electrons coupled to Langmuir turbulence (Yoon 2014; see references in previous paragraphs for a more thorough review). The collisionless nature of space plasmas are not well-described by Maxwellian distributions, since the rarity of collisions between particles means their distribution cannot thermalize and the particles are highly correlated, which is quantified by the kappa index (Livadiotis & McComas 2011b). While kappa distributions are a physically meaningful state for collisionless particle distributions to reside in, the kappa function is usually, and understandably, used as an empirical function to fit to particle measurements over a limited energy range that show a power-law tail; but, this does not necessarily mean the distribution is kappa. If a particle distribution is truly a kappa distribution, then the derived kappa index must be the same over the entire energy range of the distribution (see also the use of more complex kappa functions, e.g., Summers & Thorne 1991; Štverák et al. 2009; Lazar et al. 2012; Livadiotis & McComas 2014, and references therein), which is practically impossible to prove with limited observations. Since its launch in 2008 (McComas et al. 2009a), IBEX ENA observations have shown the persistent “ribbon” of enhanced ENA intensity encircling the sky, as well as a globally distributed flux (McComas et al. 2009b, 2014a). While the origin of the ribbon is still under debate (see McComas et al. 2014b for a recent review), the globally distributed flux is assumed to originate largely from the IHS (e.g., Gruntman et al. 2001; Fahr et al. 2007; Heerikhuisen et al. 2008; Prested et al. 2008; Schwadron et al. 2011; for OHS sources, see e.g., Izmodenov et al. 2009; Prested et al. 2010; Opher et al. 2013; Desai et al. 2014; Zirnstein et al. 2014). Analyses of IBEX allsky ENA spectra revealed that the majority of the plasma producing ENAs observed over the IBEX energy range can be represented by kappa functions, with the kappa index varying between 1.5 and 2.5 (Livadiotis et al. 2011). This precludes spectra from high latitudes, which are not modeled well by a (single) kappa function, probably because of the inclusion of PUIs from the fast solar wind (see also Dayeh et al. 2012). The kappa function has already been used in several studies to model the IHS proton distribution and, thus, ENAs observable by IBEX. However, due to computational limitations and/or the lack of a complete understanding of the IHS plasma in the context of kappa distributions, most models assume the value of κ to be constant throughout the IHS (Heerikhuisen et al. 2008, 2015; Prested et al. 2008, 2010; Zank et al. 2010; Desai et al. 2012; Opher et al. 2013; Zirnstein et al. 2014) which is unlikely to be realistic. Another model assumes κ increases as a function of distance along a streamline from the termination shock in order to approximate the plasmaʼs evolution toward a Maxwellian over time (Heerikhuisen et al. 2014). Because the charge-exchange process, which produces the ENAs that are observed by IBEX, is energy dependent, it might significantly affect the parent proton distribution as energetic particles are preferentially removed over time. Schwadron et al. (2011) estimated that most of the energetic PUIs that create observable ENAs by IBEX at energies near 1 keV should come from distances within

∼120 AU of the termination shock (see also Malama et al. 2006; Zirnstein et al. 2014). If the proton distribution is initially kappa-like at the termination shock, the energydependent, charge-exchange removal process may significantly alter its distribution. Thus, understanding this process is important for interpreting IBEX observations. Here, our aim is to track the evolution of a proton distribution as it flows through the IHS, assuming it is initially kappa immediately downstream of the termination shock, while losing particles due to charge-exchange. We determine if the distribution can still be reasonably modeled by a kappa function, and if so, present the evolution of its derived parameters when fitting over different, limited energy ranges, which are a useful test to see if the evolving distribution can be described by a single kappa function. We refer to the fitted parameters of the kappa function as pseudo-density (p‐n p ), pseudo-temperature (p‐Tp ), and pseudo-kappa index (p-κ), since they may not represent the actual properties of the proton distribution (np, Tp, κ). We assume that the time it takes for protons to diffuse in velocity space is large compared to the flow time and charge-exchange time, such that diffusion can be ignored. Also, we ignore the injection of new PUIs into the IHS plasma. PUIs are injected initially in ring beams, and eventually scatter to nearly isotropic shells centered on the bulk flow speed, with speeds ranging from 0 to twice the bulk flow speed in the solar inertial frame. The evolution of those PUIs is not well-understood, especially as the distribution evolves to a new stationary state represented by a kappa distribution with different kappa index (Livadiotis & McComas 2011a). 2. METHODS 2.1. PUI Transport We model the evolution of the proton distribution using the transport equation for an isotropic PUI/suprathermal distribution, given by (e.g., Isenberg 1987) ¶fp ¶t

+ up · fp =

1 ¶ ⎛ 2 ¶fp ⎞ ⎜v D ⎟ v 2 ¶v ⎝ ¶v ⎠ +

v ¶fp ( · u) + S (s , v) , 3 ¶v

(1 )

where fp= fp (s, v ) is the proton distribution function in the plasma frame, up = up (s) is the bulk solar wind velocity, D is the velocity diffusion coefficient, and S is the source/sink of protons (spatial diffusion is ignored), and s is a position vector that will be defined later. The second term in Equation (1) describes the advection with the solar wind with velocity up , the third term describes diffusion of particles in velocity space, the fourth term describes adiabatic heating/cooling, and the last term describes a source (or sink) for the distribution due to, for example, charge-exchange with interstellar neutral atoms. Other ionization processes are ignored, such as photoionization (which is negligible at large distances from the Sun) and electron-impact ionization (the distribution of electrons in the IHS is not known, even though it has had recent attention, see e.g., Chalov & Fahr 2013; Chashei & Fahr 2013; Scherer et al. 2014; Fahr et al. 2015; Gruntman 2015). We can make some simplifying assumptions to Equation (1). First, we assume a steady-state solution of the proton 2

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distribution. Second, diffusion in velocity space in the IHS can be assumed to be negligible (see arguments by, e.g., Fahr & Lay 2000; Fahr & Scherer 2004). Third, the shocked IHS plasma downstream of the termination shock is approximately incompressible (e.g., Fahr et al. 1993), except in the presence of charge-exchange. However, as we show in Appendix A, the divergence term is small in the IHS, and can be ignored for small distances from the termination shock. The transport equation for the PUI distribution is now given by up · fp = S (s , v) .

the relative speed between the plasma and neutrals, we set uH = 0. Since the charge-exchange rate (Equation (6)) is a function of the individual PUI (v) and bulk plasma flow (up ) vectors, the PUI distribution will not evolve isotropically, depending on the angle between v and up and their relative magnitudes. This becomes more complicated considering that the speed and direction of the bulk plasma flow may change significantly over short distances from the termination shock. A proper calculation of this effect is out of the scope of this paper, therefore we set up = 0 in Equation (6). This is approximately valid for the high energy part of the distribution, where v  up . Assuming the proton distribution is isotropic, and simplifying the advection term along the flow vector (i.e., streamline) with distance s along the vector, Equation (2) can now be written as (e.g., Fahr & Lay 2000; Fahr & Scherer 2004)

(2 )

The source term (S) for PUIs due to charge-exchange with interstellar neutral atoms is given by S (s , v) = P - L = f * (s , v) h (s , v) - fp (s , v) b (s , v) ,

(3 )

where the function f* defines the injection of PUIs into the distribution. The proton distribution may charge-exchange with interstellar neutral atoms (with distribution fH), and can gain (P) or lose (L) PUIs at a production rate η and loss rate β, respectively, given by

ò sex ( v - vp ) fp ( s, vp ) v - vp dvp , b (s , v) = ò sex ( v - vH ) fH ( s , vH ) v - vH dvH ,

u p (s)

(4 )

⎛ fp (s , v) = fp ( sTS, v) exp ⎜ ⎝

(

(7 )

nH ( s ¢ )

TS

)

)

(8 )

where fp(sTS, v) is the initial kappa distribution of protons just downstream of the termination shock. The bulk plasma speed is taken from a flow model that includes terms for a point source at the origin (i.e., solar wind), uniform flow from infinity (i.e., LISM), and a term due to the presence of charge-exchange. This is described in more detail in Appendix B. The neutral density varies slowly through the IHS, and is assumed to be constant in our model, thus we arrive at

(5 )

⎞ ⎛ s s ( v ) v (v ) rel rel fp (s , v) = fp ( sTS, v) exp ⎜⎜ - nH ds¢⎟⎟ , s TS u p ( s ¢) ⎠ ⎝ ⎞ ⎛ L s ( v ) v (v ) rel rel fp (L, v) = fp (0, v) exp ⎜⎜ - nH dL¢⎟⎟ , 0 u p ( L¢ ) ⎠ ⎝

ò

ò

(9 )

where we define L=s − sTS as the distance from the termination shock along a flow streamline. Note that the calculation of fp(L, v) in Equation (9) is in the plasma frame. In this paper we assume that the protons immediately downstream of the termination shock are initially defined by an isotropic, single kappa distribution in the plasma frame, given by (e.g., Livadiotis & McComas 2009, 2013)

b (s , v) » nH sex ( vrel ) vrel , ⎡ exp ( - w 2 ) ⎤ ⎛ 1⎞ vrel = vth,H ⎢ + ⎜ w + ⎟ erf (w ) ⎥ , ⎝ ⎢⎣ ⎥⎦ p w⎠ vth,H

òs

s

´ sex vrel ( s¢ , v ) vrel ( s¢ , v ) u p ( s¢) ds¢ ,

and E is the kinetic energy of the charge-exchange interaction. We approximate the inflowing interstellar neutral atom distribution as Maxwellian, and ignore contributions from neutral atoms generated by charge-exchange in the supersonic solar wind, IHS, or outer heliosheath (OHS; e.g., Alexashov & Izmodenov 2005; Heerikhuisen et al. 2006). Neutral atoms generated in the hot OHS (hot relative to the LISM, but not the IHS) may propagate inside the IHS and constitute a significant percentage of the total neutral density. We do not expect that ignoring the contribution of OHS neutrals will significantly change the results since the relative speed of interaction is only a weak function of the neutral atom temperature. This simplifies the loss rate of Equation (4) to be (Ripken & Fahr 1983, with typos corrected)

v + up - uH

= - fp (s , v) b (s , v)

Solving for the proton distribution as a function of distance s and speed v yields

where the charge-exchange cross section σex is from Lindsay & Stebbings (2005),

w=

¶s

= - fp (s , v) nH (s) sex ( vrel (s , v) ) vrel (s , v) .

h (s , v ) =

sex (E ) = (4.15 - 0.531 ln (E ))2 ´ (1 - exp ( - 67.3 E ))4.5 10-16 cm2, 1 2 , E = mvrel 2

¶fp (s)

⎡ ⎛ 3 ⎞ ⎤ 2 G (k + 1 ) fp (v) = n p ⎢ pvth2 ⎜ k - ⎟ ⎥ ⎝ ⎛ ⎣ 1⎞ 2 ⎠⎦ G⎜k - ⎟ ⎝ 2⎠ k 1 ⎡ ⎤ ⎢ ⎥ 2 1 v ⎥ , ´ ⎢1 + ⎛ 3 ⎞⎟ vth2 ⎥ ⎢ ⎜k ⎢⎣ ⎝ 2 ⎠ ⎥⎦ 3

,

(6 )

where vth,H = 2kB TH m is the thermal speed of the interstellar neutral distribution, TH is the neutral atom temperature, and uH is the neutral bulk velocity. Note that we include the bulk plasma velocity up in the equation for ω, since we designate the velocity v in the plasma frame. Since the interstellar neutrals are relatively slow, and we are interested in 3

(10)

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Zirnstein & McComas

where np is the proton density, Γ is the gamma function, vth = 2kB Tp m is the thermal speed of protons, Tp is their temperature, κ is the kappa index, m is the proton mass, and kB is Boltzmannʼs constant. The kappa index can have values between 1.5 and ¥ , where the latter corresponds to thermal equilibrium (Maxwellian).

the values for the background neutral atom distribution are nH = 0.1 cm−3 (Bzowski et al. 2009), TH = 8000 K (McComas et al. 2015), and uH = 0, and the parameters defining the initial kappa distribution at the termination shock are n p,TS = 0.002 cm−3 (Richardson & Decker 2014), Tp,TS = 3×106 K (the total proton temperature; e.g., Heerikhuisen et al. 2008; Zank et al. 2010), and κ = 1.63 (Decker et al. 2005). We calculate up along a streamline from our flow model (Appendix B) starting at the termination shock (RTS = 90 AU) at an angle θ = 10° from the upwind direction. The flow speed as a function of distance L is shown in Figure 10. While the plasma temperature measured by V2 just downstream of the termination shock was unexpectedly low (∼105 K; Richardson et al. 2008), the temperature increase across the termination shock expected by MHD simulations (∼3×106 K) can be explained if the majority of the energy went to heating the PUI/suprathermal part of the distribution (e.g., Zank et al. 1996, 2010; Richardson 2008). While V2 measured the IHS plasma flow speed to be between ∼100 and 200 km s−1 (e.g., Richardson & Decker 2014), the flow speed may be different at other locations in the IHS. The bulk plasma flow speed largely controls the amount of time the energetic protons experience charge-exchange over a fixed distance. In Section 3.5 we demonstrate how the flow speed along different streamlines affects the PUI distribution. The value for κ from Decker et al. (2005) was found by a fit to scaled ion intensity measurements by the LECP instrument on board V1 in the energy range ∼0.04–4 MeV, after V1 crossed the termination shock (toward ∼(255°, 35°) in ecliptic J2000 coordinates). While the same value for κ should also apply to lower and higher energies if it is truly a kappa distribution, this may not necessarily be true, and cannot even be determined if particle measurements are taken over limited energy ranges. We will show that, at sufficient distances from the termination shock, the IHS proton distribution becomes highly distorted due to the effects of energy-dependent charge-exchange, and may no longer be a kappa distribution or even well-represented by a kappa function (assuming negligible velocity diffusion or acceleration). In Figure 1 we show the proton distribution as the plasma flows through the IHS using the parameters described above. We show the results over distances L = 0–150 AU, in steps of 25 AU along a streamline from the termination shock. As the plasma flows through the background neutral hydrogen, it preferentially loses higher energy particles due to chargeexchange. While the largest numbers of particles are lost at the lowest energies, a significant amount of the energy density is lost at ∼keV energies, where the σex(vrel)vrel term is greatest. Above ∼10 keV, the distribution is losing significantly fewer particles since the rate of charge-exchange rolls over and decreases significantly at higher energies. This can be seen in Figure 2, where we plot σex(vrel)vrel as a function of the interaction energy (also see Scherer et al. 2014), where the peak of σex(vrel)vrel occurs near ∼13 keV (we note that this may slightly vary between different determinations of the cross section). This roll-over may have significant implications for the interpretation of particle measurements above ∼10 keV, such as Cassini Ion and Neutral Camera (INCA) measurements, which detects ENAs at energies from ∼5 to >220 keV (e.g., Krimigis et al. 2004; Dialynas et al. 2013). This will also have important implications for the transfer of energy by

2.2. Fitting a Kappa Function We investigate the evolution of the transmitted proton distribution as it flows through the IHS, assuming it is initially well-represented by a kappa distribution immediately downstream of the termination shock. We aim to determine if the evolving distribution can still be represented by a kappa function (Equation (10)) while losing particles due to chargeexchange with a stationary, homogeneous background of neutral hydrogen. This act of loss by charge-exchange results in the majority of ENAs from the IHS observed by IBEX, which is a useful method for deriving the IHS plasma properties, since it is impractical to make in situ measurements at multiple locations in the IHS. For the majority of the results in this paper we fit over the energy range 0.1–10 keV, which brackets the IBEX observations of ENAs, and excludes the low energy range of PUIs injected at IHS bulk flow speeds (for up = 50–150 km s−1, EPUI∼mu 2p 2 ~ 0.01–0.12 keV). However, we will show results found from fits over other energy ranges to demonstrate how the energy-dependent, chargeexchange process can skew the fitting results when a kappa function is fit to a non-kappa distribution. Our approach is to propagate the proton distribution over distance L along a streamline from the termination shock using Equation (9), compute the new distribution fp(L, v) over a range of speeds v in the plasma frame, and then fit a kappa function to fp(L, v) assuming the pseudo-parameters p‐n p , p‐Tp, and p‐k p are (potentially) free fitting parameters. We fit a kappa function using Levenberg–Marquardt least-squares minimization (Markwardt 2009), where the unweighted variance between fp(L, v) and the fitting function is minimized. We present results in several different cases: (1) p‐n p and p-κ are free fitting parameters, and p‐Tp is assumed to be constant; (2) p‐n p and p‐Tp are free fitting parameters, and p-κ is assumed to be constant; and (3) p‐n p , p‐Tp, and p-κ are all free fitting parameters. In all of our results, we plot the uncertainties of p‐n p , p‐Tp, and p-κ derived from the fitting routine. In some cases, the uncertainties are very low and they are not visible on the plots; in other instances the solution clearly diverges. Since our fits are unweighted, the uncertainties computed from the covariance matrix may not represent the true uncertainties of the parameters. Therefore, the computed variances are scaled by the reduced chi-square value of the fit (e.g., Bevington & Robinson 2003). In addition, in order to maintain simplicity in the analysis of our results, we assume the interstellar neutral atom parameters (nH, TH, uH) are independent of distance, and that uH = 0. 3. RESULTS 3.1. The Evolving Distribution First we present a typical solution for the evolution of a proton distribution (initially kappa-like) through the IHS as it experience losses by charge-exchange. In this solution, 4

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Figure 2. Normalized charge-exchange rate (excluding the charge-exchange partner density) as a function of the kinetic energy of the interaction. Note that the cross section is limited to interaction energies ∼0.005–250 keV (see Lindsay & Stebbings 2005).

exchange, and the pseudo-temperature drops as the energydependent, charge-exchange process removes more high energy particles, but eventually p‐n p increases toward higher values as the fitting begins to diverge. Clearly, both the assumptions of constant p‐Tp and constant p-κ process are not applicable to this scenario. An important part of the fitting process is the energy range at which one fits the kappa function. In Figure 3, we fit over the energy range 0.1–10 keV, which is approximately the largest interval relevant to IBEX ENA observations. This range is below the “break” in the distribution above 10 keV where a (single) kappa function is obviously not applicable, and it is approximately above the energy of PUIs injected in the IHS plasma. However, the results change if we fit over different energies. This has important implications for in situ plasma measurements, as well as remote ENA measurements. In Figure 4 we show p‐n p , p‐Tp, and p-κ in the constant p‐Tp (left) and constant p-κ (right) cases, derived over different energy ranges: (1) 0.1–1 keV, (2) 1–10 keV, and (3) 0.1–10 keV, using the same parameters as in Figure 3. First, the fits yield the same initial results (distance = 0 AU), since the initial distribution was defined by a single kappa function. Farther from the termination shock the solutions clearly differ from each other. First, the fact that the derived kappa function parameters do not coincide at all energy intervals means the distribution is no longer a kappa distribution. In the case of constant p‐Tp (left column), the results for both cases (1) and (3) overlap and eventually the solution begins to diverge. The results from the fit over 1–10 keV, however, do not diverge, where p‐n p drops quickly, and p-κ increases over distance from the termination shock. Clearly there is a discrepancy in the distribution between low and high energies due to the energy-dependent, charge-exchange process. The fact that solutions (1) and (3) overlap suggest that the low energy portion of the fit (near ∼0.1 keV), which is excluded in (2), is controlling the minimization of the fit. If the distribution is more strongly distorted at higher energies by charge-exchange, then the lower energy part of the distribution will produce a better fit to a kappa function. Interestingly, in the case of constant p-κ (right column), there is no stable solution over any of these energy ranges, especially over 1–10 keV, which diverges very quickly. By ∼20 AU, p‐Tp reaches 0 K, and the solution for p‐n p diverges.

Figure 1. Evolution of fp(L, v) in the plasma frame as a function of distance through the IHS in the presence of charge-exchange. The black curve shows the initial proton kappa distribution immediately downstream of the termination shock, and the blue curves show the evolved distribution in steps of 25 AU, which decrease over time. The two panels show the same results, however with different plot ranges. Also note that the parameters used in the calculations are shown in the top right of the first panel.

charge-exchange between the IHS and OHS, which is limited by the energy-dependent, charge-exchange cross section (Heerikhuisen et al. 2015). In Figure 1, we can also see that the proton distribution above ∼100 keV remains nearly unchanged over time. It is clear that the entire distribution cannot be described by a single kappa distribution. However, at the energies applicable to IBEX ENA measurements (