Phase-Space Synchronization Driven by Moon-Magnetosphere Coupling in Gas Giants

Adnane Osmane; Elias Roussos; Peter Kollmann

REVIEW 3 major objections 6 minor 78 references

Phase-Space Synchronization Driven by Moon-Magnetosphere Coupling in Gas Giants

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that localized, fast moon absorption can make the Fourier modes of a radiation-belt distribution linearly unstable, so microsignature refilling is phase-space synchronization rather than radial diffusion.

desk verdict The central synchronization instability is an artifact of a factor-of-two error in the von Mises Fourier coefficients; with the correct coefficients the loss term is strictly dissipative and the claimed linear growth disappears. read the letter →

arxiv 2507.07739 v1 pith:ERNJ6VPP submitted 2025-07-10 physics.plasm-ph physics.space-ph

Adnane Osmane , Elias Roussos , Peter Kollmann This is my paper · ORCID

classification physics.plasm-phphysics.space-ph

keywords radiationbeltsmicrosignaturesphase-spacesynchronizationKuramotomodeldrift-kineticequationmoon-magnetospherecouplingradialdiffusiongasgiants

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the rapid refilling of radiation-belt microsignatures—sharp depletions in energetic particle flux carved out by moons—does not require radial diffusion. The authors derive a drift-kinetic equation in which a moon acts as a loss region localized in magnetic local time, with absorption fast enough to compete with the azimuthal drift period. Decomposing the distribution into azimuthal Fourier modes turns the system into linearly coupled oscillators, and for sufficiently localized and fast losses the modes become linearly unstable. The apparent recovery of the depleted flux is then a synchronization of Fourier phases, mathematically equivalent to a generalized Kuramoto model, rather than transport of particles inward or outward. If the claim holds, microsignature observations become a direct window onto synchronized phase-space dynamics instead of a measure of radial diffusion rates.

What carries the argument

The load-bearing object is the linear system $da_m/dt + (i\omega_m + \sigma)a_m = \eta_m - 2\sigma\sum_{m'\neq m}\psi_{m'-m}a_{m'}$, where $a_m$ is the normalized azimuthal Fourier mode of the distribution, $\omega_m=m\Omega_d$ is the drift harmonic, $\sigma$ is the normalized root-mean-square loss rate, and $\psi$ are coupling coefficients derived from a von Mises (circular-Gaussian) loss profile of width $1/\kappa$. The von Mises loss term localizes the moon's absorption in magnetic local time, and the off-diagonal coupling matrix it produces is what converts independent damped drift echoes into synchronized modes. The eigenvalues of the matrix $M=-\Omega-2\sigma\Psi$ determine stability, and in the unstable regime the phase equation for each mode is the classical Kuramoto equation with time-dependent coupling. This linear algebra is the device that carries the argument from a localized sink to apparent refilling.

What would settle it

Compute the eigenvalues of the linear operator in Eq. (30) using a strictly positive loss profile, for example $\nu(\varphi)=\sigma e^{\kappa\cos\varphi}/I_0(\kappa)$ with Fourier coefficients $c_0=\sigma$ and $c_m=\sigma\psi_m$ for $m\neq 0$, at $\kappa\geq 1$ and $\Omega_d/\sqrt{\langle\nu^2\rangle}\leq 1$; if the largest real part of the eigenvalues remains negative, the claimed growth disappears. A simpler version is to plot the paper's Eq. (24) as a function of $\varphi$ and check whether $\nu(\varphi)$ is negative anywhere on the drift orbit.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a linear instability in a dissipative kinetic system: when the absorbing moon covers a limited span of magnetic local time ($\kappa \geq 1$ in the von Mises width parameter) and the root-mean-square loss rate is comparable to or faster than the azimuthal drift frequency ($\Omega_d/\sqrt{\langle\nu^2\rangle} \leq 1$), the Fourier modes $\delta f_m$ of the distribution function grow or resist damping instead of decaying as ordinary drift echoes. Particle number is still conserved and the system is still dissipative, so the growth is a redistribution of amplitude among azimuthal harmonics that makes the phase-space density appear to refill within a few drift periods. The same equations, written in amplitude-phase variables, are shown to be a generalized Kuramoto system, which identifies the mechanism as phase-space synchronization. Because co-rotation lengthens the drift period of electrons more than protons at gas giants, the authors conclude that electrons enter this synchronized regime more readily, matching where fast microsignature refilling is observed.

Load-bearing premise

The load-bearing premise is that the Fourier representation of the moon's absorption still behaves as a pure particle sink at every magnetic local time after rescaling; if the truncated series has a negative lobe that acts as a source, the instability and the synchronization that follows from it are artifacts of the representation rather than physics.

Editorial extensions

If this is right

At Jupiter and Saturn, energetic electrons with corotation-lengthened drift periods should show microsignatures that refill by synchronization, while protons with comparable adiabatic invariants on the same shell stay damped.
A measured microsignature that fills in within one drift period no longer implies a large radial diffusion coefficient, so published diffusion coefficients inferred from refilling may need reinterpretation.
Near marginal stability the low-order azimuthal modes should remain phase-coherent over several drift periods, so a spacecraft downstream of a moon should see the depletion followed by localized enhancements that reappear at different magnetic local times.
Because the mode equations are a generalized Kuramoto model, sufficiently strong coupling could produce quasiperiodic or chaotic mode dynamics, which would surface as irregular microsignature morphology.
The same formalism applies to Earth's magnetopause shadowing, where the loss region is wider ($\kappa\sim 1$), predicting a slower-onset version of the same synchronization effect.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A decisive cross-check is to recompute the stability eigenvalues with the loss rate written as a strictly positive von Mises profile, using the exact Fourier coefficients $c_0=\sigma$ and $c_m=\sigma\psi_m$ for $m\neq 0$; if no eigenvalue of the coupling matrix then has positive real part, the claimed instability is an artifact of a negative lobe in the truncated series acting as a particle source
If the sign issue is repaired and the instability survives, the synchronization picture predicts a distinctive observable signature: apparent refilling accompanied by phase-locked oscillation of several azimuthal harmonics, distinguishable from diffusive broadening by the depletion reappearing at the same magnetic local time after integer drift periods.
A direct particle-tracing simulation with a moon-shaped absorbing region could settle the mechanism: bin particles by magnetic local time, Fourier-analyze the surviving distribution, and compare mode growth or damping with the eigenvalues of the linear operator.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing you should know: the paper's main result is not supported. The instability that drives the claimed phase-space synchronization comes from a factor-of-two error in the Fourier expansion of the loss rate. In Eq. (20) the authors correctly define c_m = (1/2π)∫β W e^{-imφ} dφ, but Eq. (21) gives c_m = β/(2π)(δ_{m0}+2ψ_m). The correct value for m≠0 is βψ_m/(2π), not 2βψ_m/(2π). That extra factor enters Eq. (24), so the reconstructed ν(φ) contains a negative lobe: for κ=3.3, ν(π) < 0, meaning the 'loss' term acts as a particle source over part of the drift orbit. That source is what produces the positive eigenvalues in Figs. 4-6. With the correct coefficients, the homogeneous equation is dissipative: d/dt ∫δf² dφ = -2∫ν δf² dφ ≤ 0, and the drift term is conservative. No exponential growth is possible. This is a load-bearing error, not a typo.

What the paper does well: the observational motivation is real and clearly laid out—microsignatures do seem to refill faster than quasi-linear radial diffusion allows. The derivation of the drift-kinetic equation with MLT-localized losses is self-contained, and the mapping to a generalized Kuramoto model is a genuinely nice observation for the coupled-mode system. The appendix generalizing Van Allen's radial diffusion solution to large absorbers is a useful pedagogical piece.

The soft spots beyond the coefficient error: the 'apparent refilling' in the marginally stable regime is just slow decay of an initial perturbation, not a new transport mechanism. In the driven case, the response simply balances forcing against dissipation. So even with the math fixed, the physical claim would need to be reframed as transient mode coupling, not instability. The paper also leans on the authors' earlier work for the baseline equations, but that's not a problem here.

Who should read it: someone interested in the microsignature puzzle might enjoy the motivation and the Kuramoto analogy, but the central result should not be cited. I would not send this to a referee in its current form; the arithmetic error is clear and the conclusion collapses. If the authors correct the coefficients and reposition the paper as a study of phase mixing and transient coupling under localized losses, there could be a worthwhile paper in there. As it stands, it should be rejected.

Referee Report

3 major / 6 minor

Summary. The paper proposes a drift-kinetic model for energetic particles trapped in gas giant radiation belts, with particle losses localized in magnetic local time (MLT) modeled by a von Mises profile. Fourier decomposition of the distribution function in azimuth leads to a linearly coupled system for the mode amplitudes. The authors claim that for sufficiently localized (kappa >= 1) and sufficiently fast (Omega_d / sqrt(<nu^2>) <= 1) losses, the mode system becomes linearly unstable, and that this instability produces apparent microsignature refilling through phase-space synchronization rather than radial diffusion. They further map the amplitude-phase dynamics onto a generalized Kuramoto model and argue that electrons, having longer effective drift periods in a corotating magnetosphere, should be more susceptible than protons. An appendix re-derives and generalizes the Van Allen et al. radial-diffusion refilling solution.

Significance. If the central instability were valid, the paper would present a genuinely novel, non-diffusive mechanism for microsignature refilling on sub-drift-period timescales, together with a testable electron/proton asymmetry. The derivation is largely self-contained and does not fit parameters to data, and the authors explicitly acknowledge several limitations (single drift shell, equatorially trapped particles, passive tracers). The Kuramoto analogy is pedagogically suggestive. However, the central claim is invalidated by an error in the Fourier representation of the loss term: with the correct, positive von Mises coefficients the linear system is dissipative and the reported exponential instability disappears. The remaining contribution is therefore mainly diagnostic: it sharpens the case against quasi-linear radial diffusion, but it does not provide the advertised replacement mechanism.

major comments (3)

[II.D, Eqs. (20)-(24)] The Fourier coefficients of the von Mises loss profile are incorrect. The standard expansion W(phi)=(1/2pi)(1+2 sum_{n>=1} psi_n cos n phi) gives c_0 = beta/(2pi) and c_m = beta psi_m/(2pi) for m != 0, whereas Eq. (21) gives c_m = beta(delta_{m0}+2 psi_m)/(2pi). This overestimates every nonzero harmonic by a factor of two and also mishandles the m=0 term (with psi_0=1 it would give 3 beta/(2pi)). As a result, Eq. (24) reconstructs nu = sigma(1+4 sum_{m>0} psi_m cos m phi) rather than the positive von Mises profile; for kappa=3.3 this function is negative near phi=pi, so the "loss rate" acts as a particle source over part of the drift orbit. This negative lobe, not localized damping, is what produces the positive eigenvalues in Figs. 4-5 and the growing modes in Figs. 6-7(c), 8(d), 9(d), and 12. With the correct positive nu, the homogeneous delta-f equation satisfies d/dt integral(delta f)^2 = -2 integral nu (delta f)^2 <= 0, so no exponential instability is possible. The central claim of Sec. III.A therefore rests on an unphysical sign of the loss term.
[III.A and III.C, Eqs. (30)-(32) and (42)-(43)] The same factor of two enters the coupling matrix M and the Kuramoto coupling constants. With the corrected coefficients, the coupling term in Eq. (30) should be -sigma sum_{m' != m} psi_{m'-m} a_{m'}, not -2 sigma times that sum, and the phase coupling in Eq. (43) is halved. Since the reported instability threshold and the synchronization criterion depend directly on this coupling strength, the numerical results in Figs. 4-12 and the quantitative electron/proton asymmetry prediction must be re-evaluated. The stable and marginally stable cases may still show slow decay or transient non-normal mode coupling, but the paper's stated mechanism, exponential growth feeding apparent refilling, is not supported by the corrected equations.
[II.D, Eq. (23)] The RMS rescaling in Eq. (23) is internally inconsistent with the reconstructed nu in Eq. (24). For the nu defined in Eq. (24), the drift average is <nu^2> = sigma^2(1+8 sum psi_m^2), not sigma^2(1+4 sum psi_m^2), because the factor of 2 in the complex sum doubles the cosine amplitudes. With the correct von Mises coefficients the denominator should be (1+2 sum psi_m^2)^{1/2}. This affects the normalization of sigma and hence all reported growth rates, decay rates, and instability thresholds.

minor comments (6)

[II.D, Eqs. (22) and (24)] The summation range in Eqs. (22) and (24) is never specified; if m=0 is included, Eq. (22) contradicts Eq. (21), and if it is excluded, the notation should say so explicitly.
[II.D, p. 8] The word "frequncy" in the paragraph after Eq. (28) should be "frequency."
[III.B, definition of A(0)] In the definition A(0) = [a_1(0), a_20t), ..., a_N(0)]^T, "a_20t)" appears to be a typo for "a_2(0)."
[Fig. 9 caption] The caption contains "coeffficients," which should be "coefficients."
[Fig. 14 caption] The caption uses "radiuses," which should be "radii."
[Eq. (28)] The normalization line "sigma => sigma / <nu^2>^{1/2}" is confusing because the rescaled sigma in Eq. (29) is written without an overbar or any other distinguishing notation; please use a consistent notation for normalized variables.

Circularity Check

1 steps flagged · score 8.0 of 10

The claimed instability is an artifact of an incorrect factor of two in the von Mises Fourier coefficients: Eq. (24) reconstructs a loss rate with a negative source lobe, so the synchronization result is forced by the Fourier representation rather than derived from the localized sink.

other [Section II.D, Eqs. (19)-(24); Section III.A, Eq. (30) and Figs. 4-7]
"we choose to represent MLT localised losses in terms of the von Mises distribution W(phi, kappa)... nu = beta W(kappa, phi)... which gives the following Fourier coefficients for a fixed azimuthal number m: cm = beta/(2pi)(delta_{m0} + 2 psi_m)... nu(kappa) = sigma(kappa)(1 + 2 sum_m psi_m e^{im phi})"

Starting from the stated nu=beta W with W>=0, the correct m!=0 Fourier coefficient is c_m=beta psi_m/(2pi), not beta(delta_{m0}+2 psi_m)/(2pi). The extra factor of two makes Eq. (24) reconstruct nu=sigma(1+4 sum_{m>0} psi_m cos m phi) instead of sigma(1+2 sum_{m>0} psi_m cos m phi). For kappa>=1 the former is negative over part of the orbit (e.g., near phi=pi for kappa=3.3), so the 'loss rate' contains a particle source. With the correct positive nu, the homogeneous operator in Eq. (30) is dissipative: d/dt integral delta f^2 d phi = -2 integral nu delta f^2 d phi <= 0 (the drift term is conservative), ruling out exponential growth. The positive eigenvalues in Figs.

full rationale

The paper's derivation is self-contained and does not fit parameters to data, so it does not suffer from fitted-input-called-prediction circularity. Its use of prior work by the same authors (e.g., Osmane et al. 2023, 2025) is limited to deriving the drift-kinetic equation and the RMS-rescaling procedure; those results are not the target claim of the paper and do not by themselves force the instability result. However, the central claim of linear instability of azimuthal Fourier modes (Section III.A) is not independent of the model's stated input. The Fourier representation of the loss term is derived from the von Mises distribution, but the paper's Eq. (21) gives coefficients that are twice the true nonzero Fourier coefficients of the stated von Mises density. Equation (24) therefore reconstructs a function that is not non-negative: it contains a negative lobe that acts as a particle source. It is this artificial source that produces the positive eigenvalues reported in Figs. 4-7. With the mathematically correct positive coefficients, the homogeneous equation is dissipative, so no exponential growth is possible and the synchronization mechanism as presented is unsupported. This is a structural reduction of the prediction to an error in the input representation, not a correct derivation from the stated localized sink. The self-citations are not load-bearing for this flaw; the flaw is internal to the paper's own equations.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The model contains several adjustable parameters (κ, ⟨ν²⟩^{1/2}, Ω_d/⟨ν²⟩^{1/2}, forcing amplitude, initial mode amplitudes) that are chosen for the numerical scans, not fitted to observational data. The central stability result depends on the ad hoc Fourier representation of the loss term in Eq (24), which is where the sign error enters. No new physical entities are introduced; the 'synchronization' is a description of mode dynamics, not a new force or particle.

free parameters (5)

kappa (κ)
Controls the MLT width of the loss region via the von Mises distribution; scanned from 0 to 50 in the figures, not fitted to data.
RMS loss rate ⟨ν²⟩^{1/2}
Normalizes time in the equations and sets the instability threshold via Ω_d/⟨ν²⟩^{1/2}; chosen per scenario, not fitted.
drift-to-loss ratio Ω_d/⟨ν²⟩^{1/2}
Key control parameter for stability; varied from 0.5 to 100 in the numerical scans.
driving amplitude η_m = -0.1 for m=1 (driven case)
Represents radial E×B forcing in the driven examples; set to zero in undriven cases.
initial mode amplitudes δf_m(0) = δf1=0.2, δf2=0.1 (or δf1=0.1 in later examples)
Initial perturbation size in the numerical solutions; an arbitrary choice.

assumptions (5)

domain assumption Equatorially trapped guiding centers in a dipolar field with electrostatic poloidal perturbations
The drift-kinetic equation (Eqs 6-13) is restricted to 90 degree pitch angle and ignores latitude and bounce motion; stated as a limitation in Section IV.
ad hoc to paper Losses are modeled as -ν(f - f0) with f0 the instantaneous drift average
This relaxation term is a modeling choice; a more complete treatment would couple to energy and pitch angle (Section IV).
ad hoc to paper The loss profile is represented by the Fourier coefficients in Eq (21), c_m = β/(2π)(δm0 + 2ψ_m)
This coefficient set is used to build the coupling matrix M; it differs by a factor of 2 from the standard von Mises expansion and creates a negative lobe, enabling the instability.
domain assumption Background distribution f0 is time-independent on drift-period timescales
Stated in note [56]; used to linearize and solve for δf_m with constant coefficients.
domain assumption Higher-order wave-particle interactions (term 4 in Eq 28) are negligible
The paper ignores this term to focus on mode coupling from losses; reasonable for small perturbations, but limits the regime of validity.

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Cite this review

Pith. "Pith review of Phase-Space Synchronization Driven by Moon-Magnetosphere Coupling in Gas Giants." pith.science (2026). https://pith.science/paper/ERNJ6VPP

@misc{pith2026250707739,
  author       = {Pith},
  title        = {Pith review of: Phase-Space Synchronization Driven by Moon-Magnetosphere Coupling in Gas Giants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ERNJ6VPP}},
  note         = {Machine review of arXiv:2507.07739}
}

read the original abstract

We present a new theoretical framework to describe the rapid and spatially localized loss of energetic particles in planetary radiation belts, focusing on interactions between gas giant magnetospheres and their moons. Observations show that flux depletions--known as microsignatures--often refill on timescales comparable to a single drift period, which conflicts with traditional quasi-linear radial diffusion models that assume slow, gradual transport and predict refilling only over many drift periods. To resolve this inconsistency, we develop a drift-kinetic model that explicitly captures localized losses occurring on timescales similar to the azimuthal drift period. We demonstrate that such localized loss regions can synchronize the azimuthal Fourier modes of the particle distribution function, producing apparent refilling through phase-space synchronization rather than diffusion. The resulting governing equations are mathematically equivalent to a generalized Kuramoto model, widely used to describe synchronization phenomena. This framework provides a first-principles, non-diffusive explanation for the evolution of microsignatures near moons, highlighting synchronization as a fundamental yet overlooked mechanism in magnetized plasma environments.

Figures

Figures reproduced from arXiv: 2507.07739 by the authors.

**Figure 2.** FIG. 2: Example of micro- and macro-signatures in the stronger magnetic field regions of Saturn, seen in integral [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: Number of modes [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: FIG. 4: Dependence of the fastest growing linear mode as a function of the [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: FIG. 5: Dependence of the fastest growing mode [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: FIG. 6: Time evolution of the Fourier coefficients [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: FIG. 7: Same as Figure (6) with the time evolution of the Fourier coefficients [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: FIG. 8: Surface plot of the perturbed distribution function [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: FIG. 9: Surface plot of the perturbed distribution function [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: FIG. 10: Total distribution function [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: FIG. 11: Same as for Figure (10) but for a system much closer to marginal stability with [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]

**Figure 12.** Figure 12: FIG. 12: Same as for Figure (11) but for a driven linearly unstable regime with [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

**Figure 13.** Figure 13: FIG. 13: Refilling of the absorption holes for electrons as per Equation (A6). The figure is identical in its content to [PITH_FULL_IMAGE:figures/full_fig_p029_13.png]

**Figure 14.** Figure 14: FIG. 14: Refilling of the distribution function for moons located at [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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⟨ν⟩1/2 which quantifies the root mean square loss rate over a drift orbit

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κ which quantifies the MLT localization of our loss region. After inserting Equation (24) into the right-hand side of the kinetic equation (13) we find : ∂f ∂t + 3µ qγr 2 + ωr ∂f ∂φ − X m δEφ,m BP r2 R2 E eimφ ∂f ∂r = −σ(κ) 1 + 2 X m ψmeimφ ! (f − f0) (25) If we now apply a Fourier decomposition of the distribution function along the magnetic local time t...

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Van Allen et al. [4] solution for small absorption regions withb/Rp ≪ 1 If the absorption region is very small in size compared to planetary radius, and the losses downstream of the moon are very deep, then the radial diffusion Equation (A1) can be simplified to: ∂f ∂t ≃ DLL ∂2f ∂L2 , (A2) The reason for this simplification is that the gradient termL2 ∂f ...

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General solution for arbitrarily sized absorption regions In this section, we solve the radial diffusion for the case where the dominant fluctuations are electrostatic but where the absorption region can be comparable to the planetary radius, i.e.b ≃ Rp. While the moons are small in size, and satisfy the limitb ≪ Rp, rings can be widespread in the equator...

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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⟨ν⟩1/2 which quantifies the root mean square loss rate over a drift orbit

work page

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κ which quantifies the MLT localization of our loss region. After inserting Equation (24) into the right-hand side of the kinetic equation (13) we find : ∂f ∂t + 3µ qγr 2 + ωr ∂f ∂φ − X m δEφ,m BP r2 R2 E eimφ ∂f ∂r = −σ(κ) 1 + 2 X m ψmeimφ ! (f − f0) (25) If we now apply a Fourier decomposition of the distribution function along the magnetic local time t...

work page

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Van Allen et al. [4] solution for small absorption regions withb/Rp ≪ 1 If the absorption region is very small in size compared to planetary radius, and the losses downstream of the moon are very deep, then the radial diffusion Equation (A1) can be simplified to: ∂f ∂t ≃ DLL ∂2f ∂L2 , (A2) The reason for this simplification is that the gradient termL2 ∂f ...

work page

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erf   L2Rp/b L0− b Rp − L Rp b √τ   −erf   L2Rp/b L0+ b Rp − L Rp b √τ   # . = 1 − 1 2

General solution for arbitrarily sized absorption regions In this section, we solve the radial diffusion for the case where the dominant fluctuations are electrostatic but where the absorption region can be comparable to the planetary radius, i.e.b ≃ Rp. While the moons are small in size, and satisfy the limitb ≪ Rp, rings can be widespread in the equator...

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