Toward Interpreting the IBEX Ribbon with Mirror Diffusion in Interstellar Turbulent Magnetic Fields

We investigate the role of the magnetohydrodynamic (MHD) turbulence measured by Voyager in the very local interstellar medium (VLISM) in modeling the Interstellar Boundary Explorer ribbon. We demonstrate that the mirroring by compressible modes of MHD turbulence dominates over that by the mean magnetic field. Based on the new mirror diffusion mechanism identified by Lazarian & Xu for particles with large pitch angles in MHD turbulence, we find that the mirror diffusion can both confine pickup ions and preserve their initial pitch angles, and thus it accounts for the enhanced intensity of energetic neutral atoms that return to the heliosphere. The ribbon width is determined by both the range of pitch angles for effective turbulent mirroring and the field line wandering induced by Alfvénic modes. It in turn provides a constraint on the amplitude of magnetic fluctuations of fast modes. The field line wandering also affects the coherence of the ribbon structure across the sky. By extrapolating the magnetic energy spectrum measured by Voyager, we find that the injection scale of the turbulence in the VLISM must be less than ∼500 au for the ribbon structure to be coherent.


INTRODUCTION
The "ribbon" of enhanced energetic neutral atom (ENA) emissions discovered by the Interstellar Boundary Explorer (IBEX) (McComas et al. 2009) provides valuable information about the local interstellar magnetic field (Schwadron et al. 2009;Pogorelov et al. 2011) and the interaction of pickup ions with the pre-existing turbulent magnetic fluctuations outside the heliopause (Giacalone & Jokipii 2015;Zirnstein et al. 2020).Turbulent magnetic fluctuations following a Kolmogorov spectrum are observed by Voyager 1 and 2 in the outer heliosheath (Burlaga et al. 2018;Zhao et al. 2020;Burlaga et al. 2022;Lee & Lee 2020;Fraternale & Pogorelov 2021).With a higher amplitude than that of the interstellar turbulent spectrum (Burlaga et al. 2018;Lee & Lee 2020;Ocker et al. 2021) and ion-neutral collisional damping of the interstellar turbulence in the partially ionized very local interstellar medium (VLISM) (Xu & Li 2022), the turbulence detected by Voyager is more likely to be of heliospheric origin (Zank et al. 2019).Among the numerous models for explaining the origin of the IBEX ribbon, the magnetic mirroring of pickup ions in the turbulent magnetic fields in the VLISM is found to be important in determining the ribbon structure (Giacalone & Jokipii 2015;Zirnstein et al. 2020).
The magnetic mirrors created by summing over a large number of randomly directed plane waves with random polarizations and phases (Giacalone & Jokipii 1999) can trap particles as particles are reflected back and forth between two mirror points (Cesarsky & Kulsrud 1973).Unlike compressible magnetohydrodynamic (MHD) waves, MHD turbulence contains fast and slow (or pseudo-Alfvénic) modes that generate magnetic mirrors and Alfvénic modes that cause a NASA Hubble Fellow perpendicular superdiffusion of magnetic field lines (Cho & Lazarian 2003;Beresnyak 2013).Following turbulent magnetic fields lines, particles also undergo superdiffusion in the direction perpendicular to the magnetic field (Xu & Yan 2013;Lazarian & Yan 2014;Hu et al. 2022) while interacting stochastically with different magnetic mirrors along the magnetic field.As a result, rather than being trapped, particles exhibit the mirror diffusion (Lazarian & Xu 2021) parallel to the magnetic field.The mirror diffusion is usually a much slower diffusion process compared to that associated with pitch-angle scattering (Lazarian & Xu 2021;Xu 2021).
The MHD turbulence measured by Voyager contains both incompressible and compressible modes (Zank et al. 2019;Lee & Lee 2020;Fraternale & Pogorelov 2021).In this work, we will investigate the importance of the recently identified mirror diffusion of particles in the MHD turbulence in the VLISM in explaining the origin of the IBEX ribbon and affecting its structure.In Section 2, we first examine the mirroring effect of the compressed interstellar mean magnetic field.In Section 3, we move to the mirroring effect of turbulent magnetic fields.The mirror diffusion resulting from the perpendicular superdiffusion of turbulent magnetic fields and its effect on the ribbon width is discussed in Section 4. The effect of field line wandering on ribbon width and structure is studied in Section 5. Further discussion is presented in Section 6.A summary of our main results follows in Section 7.

MIRRORING EFFECT OF THE INTERSTELLAR MEAN MAGNETIC FIELD
The compression of the large-scale mean field in the outer heliosheath gives rise to the magnetic mirroring effect.It can cause the concentration of pickup protons in a narrow region where the radial component of the compressed interstellar arXiv:2310.06032v1[astro-ph.GA] 9 Oct 2023 mean magnetic field is close to zero (McComas et al. 2009;Chalov et al. 2010).
For a particle with r g < L m , where r g is the particle gyroradius, and L m is the size of the magnetic mirror, i.e., the variation scale of the mean magnetic field, the particle preserves its first adiabatic invariant, with a constant magnetic moment Here v ⊥ is the particle velocity perpendicular to the magnetic field, m is the particle mass, and B is the magnetic field strength.When a particle with constant energy and magnetic moment moves into a region with converging magnetic fields, its velocity along the magnetic field v ∥ decreases, and the angle between the particle velocity v and the magnetic field B, i.e., the pitch angle, increases.The deceleration toward the mirror point where the mean magnetic field has the maximum strength and is perpendicular to the heliocentric radial direction can potentially lead to the accumulation of the pickup ions with large pitch angles (Chalov et al. 2010).
The magnetic mirror force is (Cesarsky & Kulsrud 1973) where µ is cosine of the pitch angle, p is the particle momentum, t is time, B 0 is the mean magnetic field strength, and δB m is the parallel magnetic fluctuation over L m .It follows that the rate of change in µ due to mirroring, i.e., the mirroring rate, is where v is the particle speed.Under the consideration of keV protons (v ≈ 438 km s −1 ), δB m /B 0 ≈ 1/4 and L m ≈ 100 au (Chalov et al. 2010), Γ m as a function of µ is presented in Fig. 1.It depends on the longitudinal gradient in the mean magnetic field.The maximum µ for mirroring is determined by which is derived from the first adiabatic invariant The turbulent magnetic fluctuations detected by Voyager 1 and 2 in the outer heliosheath (e.g., Burlaga et al. 2018) affects the transport of pickup protons via the field line wandering (see Section 5), turbulent mirroring (see Section 3), and pitch-angle scattering.As efficient scattering causes breaking of the first adiabatic invariant of particles, an effective magnetic mirroring imposes constraints on the amplitude of turbulent fluctuations.Here we focus on the gyroresonant scattering (Jokipii 1966;Kulsrud & Pearce 1969;Schlickeiser 2002) of pickup protons by fast modes of MHD turbulence.Compared with Alfvénic modes, fast modes usually have a smaller energy fraction (Cho & Lazarian 2002;Hu et al. 2022).However, their isotropic energy scaling (Cho & Lazarian 2002) leads to a more efficient gyroresonant scattering (Yan & Lazarian 2004;Xu & Yan 2013;Xu & Lazarian 2020) compared to the anisotropic Alfvénic modes (Chandran 2000; Yan & Lazarian 2002;Beresnyak et al. 2011).
The rate of scattering is defined as (Jokipii 1966;Cesarsky & Kulsrud 1973) where δµ is the change of µ due to scattering, and D µµ is the pitch-angle diffusion coefficient of fast modes (Voelk 1975) In the above expression, J ′ 1 (x) is the derivative of the Bessel function, and I(k) is the magnetic energy spectrum of fast modes (Cho & Lazarian 2002), k is the wavenumber, L is the injection scale of turbulence, δB f is the rms strength of magnetic fluctuations of fast modes at L, R(k) is the resonance function for gyroresonance, Ω is the particle gyrofrequency, ω k = kV A is the wave frequency, V A = B 0 / √ 4πm H n i is the Alfvén speed in ions, m H is the hydrogen atomic mass, n i is the ion number density, k ∥ is the parallel component of k, q is the electric charge, m is the proton mass, c is the light speed, and k ⊥ is the perpendicular component of k.The approximate expression of Γ sc in the limit of a small x is (Xu & Lazarian 2020), which depends on δB f .As discussed above, for the mirroring of the mean magnetic field to be effective, there should be Γ sc < Γ m .In Fig. 1, we present Γ sc for keV protons calculated using Eqs.( 6) and ( 7), with B 0 = 5 µG, L = 100 au, δB f /B 0 = 0.04, and n i = 0.07 cm −3 (Slavin & Frisch 2008;Swaczyna et al. 2020).We adopt 5 × 10 7 cm as the cutoff scale of turbulent spectrum, which is comparable to the ion inertial length (Fraternale & Pogorelov 2021) and smaller than r g ≈ 9 × 10 8 cm at µ = 0. Due to the lack of turbulent magnetic fluctuations at small scales, the scattering is absent at small µ.Given the above parameters, the mirroring by the mean field and the scattering by turbulence are approximately in balance over the µ values where they are both present.The condition Γ sc < Γ m can be used to constrain δB f and L for the MHD turbulence in the outer heliosheath.

MIRRORING EFFECT OF TURBULENT MAGNETIC FIELDS
The mirroring of the interstellar mean magnetic field alone may not be able to account for the ribbon.Moreover, Zirnstein et al. (2018) argued that the mean field mirror acts to push away pickup ions from the mirror point and reduces the ribbon flux.In addition to the mirroring effect of the interstellar mean magnetic field, the compressible MHD turbulence in the outer heliosheath can also contribute to the magnetic mirroring.Both fast and slow modes create multi-scale magnetic mirrors within the inertial range of turbulence.Isotropic fast modes are more efficient in mirroring than anisotropic slow modes (Xu & Lazarian 2020), as the latter have the magnetic fluctuations decrease more rapidly with decreasing scales in the direction parallel to the local magnetic field.
Similar to the analysis in Section 2, one easily finds that the mirroring rate of fast modes is where b f k is the magnetic fluctuation of fast modes at k.Among the mirrors at different wavenumbers, the ones that are most effective in reflecting the particles at µ have (Cesarsky & Kulsrud 1973) Smaller mirrors would have insufficient magnetic fluctuation to reflect the particles, and larger mirrors would have a lower mirroring rate.Given the scaling relation of fast modes (Cho & Lazarian 2002), there is (Xu & Lazarian 2020), We note that unlike the mirroring of the mean field, for the mirroring of turbulent magnetic fields, particles with a given µ predominantly interact with the mirrors at the corresponding k.Γ tm has a much stronger dependence on µ compared to Γ m .Despite the much smaller fluctuation of turbulent magnetic fields, Γ tm is significantly larger than Γ m due to the large magnetic field gradients at small scales (see Figs. 1 and 2).It suggests that the mirroring effect of the turbulent magnetic fields dominates over that of the mean field.Eq. ( 16) stands for the case with r g < 1/k < L, that is (Eqs.( 14) and ( 15)) (Xu & Lazarian 2020), and which is derived from the first adiabatic invariant, For 1/k < r g , i.e., µ < µ rg , the mirrors at 1/k ≈ r g dominate the mirroring.Therefore, there is (Xu & Lazarian 2020) (20) In Fig. 2, we present Γ tm and Γ sc for the mirroring and scattering of keV protons by fast modes.The same parameters as in Fig. 1 are used but for different δB f /B 0 values.If Γ tm and Γ sc can reach balance, the critical µ, µ c , is defined at their balance.With Γ tm > Γ sc at µ < µ c , the diffusion of protons (parallel to the local magnetic field) is dominated by mirroring (see Section 4).At larger µ, scattering becomes more important than mirroring.As an approximation, by using Eqs.( 12) and ( 16), we find (Xu & Lazarian 2020) which is close to the exact value of µ c at the intersection between Γ tm and Γ sc (see Fig. 2).The actual µ c is slightly larger than the above estimate due to the cutoff of turbulent spectrum that was not taken into account in Xu & Lazarian (2020).If the balance between Γ tm and Γ sc cannot be reached (see the case with δB f /B 0 = 0.01 in Fig. 2), µ c is determined by µ max in Eq. ( 18).By using Eqs.( 14) and ( 15), we find that the corresponding length scale is as the maximum size of the mirrors in turbulent magnetic fields for mirroring keV protons, which is much larger than r g .16) and ( 20)) and the rate of scattering Γsc (Eqs.( 6) and ( 7)) by the turbulent magnetic fluctuations in the outer heliosheath for keV protons.Same parameters as in Fig. 1 are used except for δB f /B0 values.The vertical dashed lines denote µc.

MIRROR DIFFUSION AND RIBBON WIDTH
Based on the mirroring by fast modes and perpendicular superdiffusion and wandering of magnetic fields induced by Alfvén modes, our model for interpreting the IBEX ribbon is illustrated in Fig. 3. Unlike the trapping of particles between the mirror points within static magnetic bottles, in MHD turbulence with the perpendicular superdiffusion of magnetic fields (Lazarian & Vishniac 1999;Eyink et al. 2013), particles that follow turbulent magnetic field lines also undergo the perpendicular superdiffusion and thus cannot be spatially trapped (Xu & Yan 2013;Lazarian & Yan 2014;Hu et al. 2022).During the perpendicular superdiffusion, particles always encounter different mirrors, rather than bouncing back and forth between two mirror points.The corresponding diffusion in the direction parallel to the magnetic field is termed mirror diffusion in Lazarian & Xu (2021).It is known that the diffusion arising from pitch-angle scattering faces the so-called 90 • problem, that is, a particle cannot be scattered through 90 • (i.e., µ = 0) according to the quasi-linear theory (Jokipii 1966) (see Fig. 2), leading to an infinitely large parallel mean free path (however, see e.g., Xu et al. (2016); Xu & Lazarian (2018) for gyroresonant scattering with turbulence-broadened resonance).The mirror diffusion does not depend on the pitch angle scattering.For a particle with a given µ, the step size of its random walk along the magnetic field is the size of the mirrors that are most effective in reflecting the particle (see Section 3).Therefore, its spatial diffusion coefficient along the turbulent field line is (Eqs.( 14) and ( 15)) (Lazarian & Xu 2021) which decreases drastically with decreasing µ.For µ < µ rg , there is Near the region with B • r = 0, where r is the radial line of sight direction, the pickup ions from neutral solar wind (Zirnstein et al. 2019) are subject to the mirroring effect of the mean field, provided that the turbulent magnetic fluctuations are sufficiently low for the mirroring to dominate over the turbulent scattering (see Section 2).The pitch-angle distribution of the pickup ions is expected to become more anisotropic toward the mirror point, with an excess at large pitch angles.The pickup ions accumulated near the mirror point of the mean field with µ < µ c and the pickup ions with initial µ's less than µ c can interact with the mirrors in turbulent magnetic fields (see Section 3) and undergo the mirror diffusion.The largest diffusion distance over their ∼ 1 year lifetime (Giacalone & Jokipii 2015) can be estimated as With the strong dependence of S D (µ) on µ, the pickup ions with smaller µ's have much more suppressed diffusion and are accumulated near the point with B • r = 0, causing the more enhanced ENA flux toward the center of the ribbon.With Γ m ≪ Γ tm (see Section 3), the effect of the mean field mirror on reflecting particles away from the mirror point (Zirnstein et al. 2018) is insignificant.
The width of the ribbon is determined by µ c , With µ c approximately proportional to (δB f /B 0 ) 4/11 (Eq.( 21)), W increases with δB f .For µ c = 0.18 in the case with δB f /B 0 = 0.04 (see Fig. 2), the corresponding W ≈ 21 • is consistent with the observed ribbon width of ∼ 20 • at ∼ 1 keV (Schwadron et al. 2011a).The observed ribbon width provides a constraint on the maximum δB f .As shown in Fig. 3, the shaded region indicates the ribbon width W .The pickup ions with initial µ's smaller than µ c are subject to the mirroring of the interstellar turbulent magnetic fields and undergo the µ-dependent mirror diffusion, resulting in the enhanced pickup-ion intensity.The pickup ions with larger initial µ's are scattered by interstellar turbulent magnetic fields.The diffusion associated with scattering is much faster than the mirror diffusion (Lazarian & Xu 2021), and the particles lose their memory of the initial pitch angles.Therefore, the pickup ions with initial µ's larger than µ c , after being neutralized, are unlikely to return to the inner heliosphere to be observed by IBEX.We note that as δB f of fast modes may change with the distance from the heliopause, W corresponding to different distances beyond the heliopause can be different.Future observations by Interstellar Mapping and Acceleration Probe (IMAP, Schwadron et al. 2016) that will resolve the substructure of the ribbon might provide more information on the variation of W along the line of sight to test our theory.

THE EFFECT OF FIELD LINE WANDERING ON THE RIBBON
The confinement of pickup ions near the region with B • r = 0 is attributed to their mirroring interaction with the compressible modes of MHD turbulence on length scales less than k −1 c (Eq. ( 22)).The incompressible Alfvénic modes do not contribute to mirroring and have a negligible contribution to scattering due to their anisotropy, but they dominate the wandering of turbulent magnetic field lines (Lazarian & Vishniac 1999).Next we will discuss the effect of field line wandering on broadening of the ribbon and distortion of the ribbon structure.
The ribbon source region stretches out from the heliopause to a radial distance ∼ 100 au that is determined by the mean free path λ of neutral solar wind atoms beyond the heliopause (Heerikhuisen et al. 2016).We extrapolate the observed magnetic fluctuation δB obs ≈ 0. We assume that the measured magnetic fluctuation is mainly induced by Alfvénic modes based on the observed Kolmogorov magnetic energy spectrum.Different from the mirroring effect of compressible fast and slow modes, incompressible Alfvénic modes cause wandering of magnetic field lines.The degree of wandering over a perpendicular length scale λ is determined by the corresponding magnetic fluctuation, with The wandering of field lines within the ribbon source causes shift of the points with B • r = 0 and thus shift of the ribbon center (see Fig. 3).Given the above estimate of δB λ and B 0 = 5 µG, we find that the additional broadening of the ribbon due to the field line wandering along the line of sight which provides the upper limit for the broadening caused by Alfvénic modes.Combining the broadening effects of both Alfvénic and fast modes, we have the ribbon width (Eqs.( 26) and ( 29)) Both broadening effects are illustrated in Fig. 3.In addition to the above broadening effect, the field line wandering induced by Alfvénic modes can also cause distortion of the ribbon structure across the sky.The turbulent magnetic fluctuation at L in the VLISM is Under the assumption that δB L is mainly associated with Alfvénic modes, the field line wandering over L in the direction perpendicular to the radial line of sight and perpendicular to the interstellar mean magnetic field can give rise to the misalignment of ribbon centers across the sky by θ L , with For the ribbon structure to remain spatially coherent across the sky, θ L should be smaller than half of the average ribbon width, i.e., θ L < 10 • .This constraint leads to (34) The above estimate is consistent with the simulation in Zirnstein et al. (2020), where they found that given L = 500 au, the ribbon at large scales changes drastically, with the peak of the ribbon meandering around the sky.
In Xu & Li (2022), an upper limit of L was found to be ∼ 194 au under the ion-neutral decoupling condition in the partially ionized VLISM.It is more stringent than that imposed by the spatial coherence of the ribbon (Eq.( 34)).
6. DISCUSSION Earlier studies on magnetic mirroring (Cesarsky & Kulsrud 1973) and its application to trapping of pickup ions (Giacalone & Jokipii 2015;Zirnstein et al. 2020) adopt a model of MHD turbulence as a superposition of MHD waves.In the MHD turbulence model with strong nonlinear interactions between oppositely directed wave packets, the turbulent energy cascade mainly happens in the direction perpendicular to the local magnetic field (Goldreich & Sridhar 1995;Lazarian & Vishniac 1999;Cho & Lazarian 2002).The anisotropic nature of MHD turbulence was not taken into account in Giacalone & Jokipii (2015); Zirnstein et al. (2020), so it is unclear what types of magnetic fluctuations are most efficient in interacting with the ∼ keV ions in their model.In nonlinear MHD turbulence, due to the stochasticity and perpendicular superdiffusion of turbulent magnetic fields (Eyink et al. 2011;Beresnyak & Lazarian 2019), particles simultaneously undergo the perpendicular superdiffusion (Xu & Yan 2013;Lazarian & Yan 2014;Hu et al. 2022;Lazarian et al. 2023) by following the turbulent field lines and mirror reflection by magnetic compressions (Zhang & Xu 2023).The former process enables encounters of a particle with multiple magnetic mirrors and its diffusive motion along the magnetic field.Therefore, in both parallel and perpendicular directions, we expect that particles are not trapped.More detailed comparison with earlier studies will be carried out by using test particle simulations in MHD turbulence in our future work.
In addition, the relation between the characteristic MHD turbulence parameters, e.g., the energy fraction and scaling of compressible modes, the injection scale of turbulence, and the ribbon width was not clearly established in earlier studies.In this work, we quantified the range of pitch angles for effective turbulent mirroring with µ < µ c , where µ c is determined by the balance between mirroring and scattering (Eq.( 21)) or by µ max (Eq.( 18)), the corresponding range of scales r g < k −1 < k −1 c (Eq. ( 22)), and the diffusion coefficient and distance of mirror diffusion (Eqs.( 23), ( 24), and ( 25)).The ribbon width depends on µ c and thus can provide a constraint on the magnetic fluctuation of compressible modes, whose energy fraction may change with the radial distance away from the heliopause.Analysis of Voyager 1 data reveals that the magnetic fluctuations in the VLISM are of mixed compressible and transverse nature (Burlaga et al. 2018;Fraternale & Pogorelov 2021).Studies by, e.g., Zank et al. (2017); Matsukiyo et al. (2020) suggest that the turbulent fluctuations in the solar wind are transmitted into the VLISM as fast modes.A significant level of compressible magnetic fluctuations is still seen in the VLISM in late 2018 (Fraternale & Pogorelov 2021).Based on these studies, we assume that magnetic compressions are present in an extended region spanning over ∼ 100 au beyond the heliopause and account for the turbulent mirroring.The mirror diffusion is sufficiently slow to account for the confinement of pickup ions near the point with B • r = 0.In addition, it preserves the initial pitch angles of particles for the ENAs to travel back to 1 au and generate the ribbon.
With λ of neutral atoms increasing with increasing energies (Lindsay & Stebbings 2005), the ribbon source region extends farther into the VLISM at higher energies (Zirnstein et al. 2016).With more significant field line wandering within the ribbon source, a broader ribbon is expected at higher energies.This is consistent with the observed ribbon width that increases with increasing ENA energies (Schwadron et al. 2011b).The multi-scale turbulent motions of magnetic field lines, including both turbulent compression of magnetic fields and field line wandering, can also give rise to fine structure seen in the ribbon (McComas et al. 2009) and temporal variations.
We note that the turbulent spectrum in the VLISM on scales less than ∼ 6 × 10 11 cm becomes shallower than the Kolmogorov scaling (Lee & Lee 2020).Despite the un-

Figure 1 .
Figure1.The rate of mirroring Γm by the interstellar mean magnetic field and the rate of scattering Γsc by the turbulent magnetic fluctuations in the outer heliosheath for keV protons.The turbulence parameters are adopted (see text) to reach an approximate balance between Γm and Γsc.For the magnetic mirroring to be effective, the turbulence parameters should satisfy Γm > Γsc.

Figure 2 .
Figure 2. The rate of mirroring Γtm (Eqs.(16) and (20)) and the rate of scattering Γsc (Eqs.(6) and (7)) by the turbulent magnetic fluctuations in the outer heliosheath for keV protons.Same parameters as in Fig.1are used except for δB f /B0 values.The vertical dashed lines denote µc.

Figure 3 .
Figure 3. Illustration for the ribbon width.The shaded regions, after projection along the line of sight, correspond to the ENA flux distribution transverse to the ribbon structure in the sky.The width W corresponds to the region in the outer heliosheath where pickup ions undergo the slow mirror diffusion in interstellar turbulent magnetic fields.The misalignment between red and green shaded regions shows the additional broadening of the ribbon ∆W due to the field line wandering caused by Alfvénic modes over a radial distance ∼ λ.
3 µG at length scale l obs ≈ 20 au by Voyager 1 in the VLISM (Lee & Lee 2020) by following the Kolmogorov scaling (Burlaga et al. 2018; Lee & Lee 2020) and find the magnetic fluctuation at length scale λ, δB λ = δB obs l obs λ