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Filippov Sliding Dynamics of Cosmic Dust Atmospheric Entry: Survival Boundaries, Asymptotic Mass Loss, and Inverse Problem Limits

Md Shahrier Islam Arham, Prasun Panthi, Min Heo

Abstract

We develop a mathematically rigorous framework for modelling the atmospheric entry of micrometeoroids with radii $r_{0}\in[0.5,1000]μm$ at hypervelocity speeds $v_{0}\in[11.2,72]km/s$. The governing four-state ODE system coupling altitude, speed, temperature, and radius has a discontinuous right-hand side at the ablation threshold $T=T_{melt}$, making it a Filippov dynamical system. We prove three original results. First, the empirical survival boundary $r_{0}^{crit}\propto v_{0}^{-3}$ (known since Love & Brownlee 1991) is the sliding bifurcation locus of the Filippov system: the locus in parameter space where particle trajectories first touch the attracting sliding region on the switching surface. This gives the $v_{0}^{-3}$ scaling its first rigorous dynamical-systems derivation. Second, a matched perturbation expansion with small parameter $δ_ε=3Λρ_{0}H/(8Q^{*}ρ_{p}r_{0}\sin γ)$ yields a closed-form mass-loss formula with numerically confirmed $\mathcal{O}(δ_ε^{2})$ error bound. Third, the transfer matrix K mapping entry distributions to stratospheric surface outcomes has a null space in its surviving-particle submatrix $K_{surv}$ determined by the full-ablation manifold A. For iron and cometary compositions (high vo, large ro), A is non-empty and contains particles permanently invisible to stratospheric collectors regardless of sample size. Fate maps for silicate and iron compositions are validated against Love & Brownlee (1991). Global Sobol analysis confirms entry speed as the dominant parameter. A regularised nonnegative inversion demonstrates partial recovery of the pre-atmospheric flux distribution.

Filippov Sliding Dynamics of Cosmic Dust Atmospheric Entry: Survival Boundaries, Asymptotic Mass Loss, and Inverse Problem Limits

Abstract

We develop a mathematically rigorous framework for modelling the atmospheric entry of micrometeoroids with radii at hypervelocity speeds . The governing four-state ODE system coupling altitude, speed, temperature, and radius has a discontinuous right-hand side at the ablation threshold , making it a Filippov dynamical system. We prove three original results. First, the empirical survival boundary (known since Love & Brownlee 1991) is the sliding bifurcation locus of the Filippov system: the locus in parameter space where particle trajectories first touch the attracting sliding region on the switching surface. This gives the scaling its first rigorous dynamical-systems derivation. Second, a matched perturbation expansion with small parameter yields a closed-form mass-loss formula with numerically confirmed error bound. Third, the transfer matrix K mapping entry distributions to stratospheric surface outcomes has a null space in its surviving-particle submatrix determined by the full-ablation manifold A. For iron and cometary compositions (high vo, large ro), A is non-empty and contains particles permanently invisible to stratospheric collectors regardless of sample size. Fate maps for silicate and iron compositions are validated against Love & Brownlee (1991). Global Sobol analysis confirms entry speed as the dominant parameter. A regularised nonnegative inversion demonstrates partial recovery of the pre-atmospheric flux distribution.

Paper Structure

This paper contains 39 sections, 9 theorems, 43 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Physical Model and Governing Equations
  3. Geometry and Kinematics
  4. Momentum Equation
  5. Energy Balance
  6. Ablation Law
  7. The Closed System
  8. Local Existence and Uniqueness
  9. Global Existence
  10. Filippov Sliding Analysis
  11. The Filippov Framework
  12. Crossing vs Sliding Classification
  13. Theorem 1: Survival Boundary
  14. Reduced Sliding Dynamics
  15. Nondimensional Analysis
  16. ...and 24 more sections

Key Result

Theorem 3.1

On $\Omega \setminus \Sigma$, the vector field $\mathbf{f}$ defined by Eq. eq:full_system is locally Lipschitz. Therefore, for any initial condition $\mathbf{y}_0 \in \Omega \setminus \Sigma$, there exists a unique solution on some interval $[0, \tau)$.

Figures (6)

  • Figure 1: Global Sobol first-order ($S_i$) and total-order ($S_{Ti}$) indices for $T_{\max}$. Computed over $N = 8192$ Saltelli samples. The analytical high-temperature limit $(S_{v_0})^2 = 9/16 \approx 0.56$ (dashed line) exceeds the full-physics value $S_{v_0} = 0.31$, quantifying the variance carried by radiation and ablation feedback.
  • Figure 2: Fate map for silicate micrometeoroids as a function of entry speed $v_0$ and initial radius $r_0$ (vertical entry, $\gamma = 90^\circ$). Colours: blue --- unmelted survival; cyan --- scoriaceous; orange --- cosmic spherule; red --- complete ablation. The dashed curve is the analytical survival boundary $r_0^{\mathrm{crit}} \propto v_0^{-3}$ from Theorem \ref{['thm:filippov']}; the solid curve is the Filippov sliding locus $\partial\Sigma^s$ from which Theorem \ref{['thm:filippov']} is derived. The agreement between these two curves confirms that the survival boundary is the sliding bifurcation locus of the Filippov system.
  • Figure 3: Fate map for iron micrometeoroids. The upper-right quadrant (red) shows complete ablation for $v_0 > 50km\per s$ and $r_0 > 200µm$. This non-empty full-ablation set $\mathcal{A}$ corresponds to the kernel of $K_{\mathrm{surv}}$ characterised in Theorem \ref{['thm:nullspace']}.
  • Figure 4: Singular value spectrum of $K$ and $K_{\mathrm{surv}}$. The full transfer matrix has condition number $\kappa(K) \approx 3.5\times 10^{16}$ (near machine precision); the surviving-particle submatrix has $\kappa(K_{\mathrm{surv}}) \approx 1.4\times 10^2$, indicating moderate ill-posedness for the recoverable component.
  • Figure 5: L-curve for Tikhonov regularisation. The optimal regularisation parameter $\lambda^* = 7.2$ is at the corner of maximum curvature.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Remark 2.5: Knudsen correction
  • Remark 2.6: Rocket term
  • Remark 2.9: Biot number
  • Theorem 3.1: Local Existence and Uniqueness
  • proof
  • Proposition 3.2: Carathéodory Extension
  • Remark 3.3: Sigmoid Regularisation
  • Theorem 4.1: Global Existence
  • proof
  • Lemma 5.1: Simplified Sliding Condition
  • ...and 16 more