Predictive modeling of a geomagnetically trapped strangelet population (the strangelet belt)

If strange quark matter is stable, small lumps of it (strangelets) should reach Earth as an exotic component of cosmic rays. Paulucci, Horvath and Medina-Tanco (Phys. Rev. D 77, 043003, 2008) showed that fully ionized strangelets with magnetic rigidities between the geomagnetic cutoff and an adiabaticity threshold occupy a narrow window for stable trapping, and estimated that a stationary trapped population — a 'strangelet belt' analogous to the trapped anomalous cosmic ray belt — could enhance the interstellar strangelet flux by one to two orders of magnitude at L-shells 1.3–2. The estimate rests on a Bohr-atom ionization model, fixed strange-matter parameters, an assumed fluctuation power spectral density $P_A(\nu)\propto\nu^{-2}$ with no pitch-angle diffusion, no numerical treatment of nonadiabatic quasistable orbits, no solar-cycle modulation, and crude atmospheric loss modeling. No later work has refined these estimates. The open problem is to convert the order-of-magnitude estimate into a predictive, uncertainty-quantified trapped strangelet flux — or to demonstrate that no trapping window survives realistic assumptions — in a form suitable for guiding experimental searches.

Background

Strange quark matter (SQM) is a hypothetical deconfined phase containing up, down and strange quarks in nearly equal proportions. The Bodmer–Witten–Terazawa conjecture holds that SQM, rather than nuclear matter, is the true ground state of hadronic matter; if so, 'strangelets' — stable or metastable lumps of SQM with baryon number $A$ up to a few thousand — could be produced in compact-star mergers or supernovae, contaminate the interstellar medium (ISM), and arrive at Earth as cosmic-ray primaries with an anomalously small charge-to-mass ratio (Madsen 2005). For unpaired ('normal') SQM the MIT bag model — a phenomenological description of quark matter as nearly free quarks confined within a region ('bag') of vacuum energy — gives a charge $Z\approx0.1$–$0.125\,A$; for SQM in the color-flavor-locked (CFL) state — a superconducting phase in which quarks of all colors and flavors form Cooper pairs, enforcing equal flavor numbers — the net charge is smaller, $Z\approx0.3\,A^{2/3}$. No strangelet has ever been confirmed; the strongest direct limits on their cosmic-ray abundance come from space spectrometers such as PAMELA (the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics satellite experiment; Adriani et al. 2015).

Whether charged particles can be trapped by Earth's magnetic field is governed by magnetic rigidity, $R=pc/(Ze)$, where $p$ is momentum and $Ze$ the particle charge. Stable trapping in the quasi-dipolar inner field requires rigidity above the local geomagnetic cutoff (so the particle can reach low magnetic shells at all) but low enough that gyration and field-line bounce are adiabatic, i.e. the gyroradius is small compared with the field-scale length, so that the first and second adiabatic invariants (magnetic moment and longitudinal invariant) are conserved and the particle mirrors between conjugate points while drifting around Earth. Position is conventionally labeled by the McIlwain $L$-shell, the equatorial crossing distance of the guiding field line in Earth radii. Slow radial transport across $L$-shells is modeled as diffusion with coefficient $D_{LL}$ driven by fluctuating electric and magnetic fields described by a power spectral density $P_A(\nu)$; losses arise from absorption in the residual atmosphere (particles whose mirror points reach the dense atmosphere fall into the 'loss cone' of equatorial pitch angles) and from pitch-angle diffusion, the scattering of velocity direction that moves particles into that cone. The established analogue is the anomalous cosmic ray (ACR) belt at $L\approx2$: interstellar neutral atoms are ionized and accelerated at the solar wind termination shock, injected with low charge state, and trapped; solar-cycle modulation of the incident ACR flux (e.g. the observed factor-of-up-to-100 suppression of trapped oxygen at solar maximum) is a central feature of that belt.

Paulucci, Horvath and Medina-Tanco (2008) applied this framework to strangelets ionized by ISM collisions. For $A\lesssim10^{3}$ they found a narrow rigidity window, between cutoff and adiabaticity threshold, in which fully ionized strangelets are stably trapped; solving a steady-state diffusion–loss equation with $P_A(\nu)\propto\nu^{-2}$ (which makes $D_{LL}$ energy-independent, $D_{LL}\propto L^{10}$), geometric strangelet–atmosphere cross sections, and loss-cone absorption, they estimated stationary trapped fluxes at rigidity $R=1$ GV enhanced by roughly a factor of $10$ at $L=2$ and up to $10^{2}$ at $L\approx1.3$ relative to the ISM strangelet flux, concluding that a 'strangelet belt' is plausible and that searching the magnetosphere for low-$Z/A$ particles is feasible but difficult. The calculation was deliberately conservative, and the authors enumerated what would be needed to make it predictive: (i) the ionization state was computed in a Bohr-atom approximation that omits many-electron, relativistic, spin–orbit and screening effects; (ii) the charge relations $Z(A)$ and the SQM parameters (energy per baryon $\epsilon_0\approx930$ MeV, strange-quark mass $m_s=150$ MeV, CFL pairing gap $\Delta=100$ MeV, bag constant) were held fixed, without screening or shell effects; (iii) the $\nu^{-2}$ power spectrum and the neglect of pitch-angle diffusion were untested assumptions rather than empirical inputs; (iv) trajectories violating the adiabaticity constraint but remaining temporarily confined (quasistable, nonadiabatic orbits) were flagged as requiring a numerical trajectory study; (v) solar-cycle modulation of the incident flux was omitted despite the known order-of-magnitude effects for ACRs; and (vi) atmospheric composition and strangelet–nucleus cross sections were treated crudely.

Problem Statement

Determine, with predictive and quantified uncertainties, whether a stationary strangelet population is trapped in the geomagnetic field, and at what flux — or demonstrate that no appreciable trapping window survives realistic assumptions. Starting from the narrow stable-trapping rigidity window and the conservative enhancement estimates ($\sim10$ at $L=2$, $\sim10^{2}$ at $L\approx1.3$) of Paulucci, Horvath and Medina-Tanco (2008), this requires replacing the acknowledged approximations of that calculation: (i) a realistic treatment of the strangelet ionization balance beyond the Bohr-atom model, giving the charge state as a function of baryon number $A$, energy, and environment (interstellar medium and magnetosphere), including many-electron, relativistic, spin–orbit, and screening effects; (ii) propagation of strange-matter parameter uncertainties ($\epsilon_0$, $m_s$, bag constant, color-flavor-locked (CFL) gap $\Delta$) and of the $Z(A)$ relations — normal $Z\approx0.1$–$0.125\,A$ and CFL $Z\approx0.3\,A^{2/3}$, including screening and shell effects — through the trapping windows; (iii) radial diffusion driven by empirical geomagnetic fluctuation power spectra in place of the assumed $P_A(\nu)\propto\nu^{-2}$, with energy-dependent diffusion coefficients and the inclusion of pitch-angle diffusion, yielding realistic residence times; (iv) a numerical trajectory study quantifying the additional trapped fraction from nonadiabatic (quasistable) orbits that violate the adiabaticity constraint but remain temporarily confined; (v) time-dependent injection coupling the ISM strangelet flux to solar-cycle modulation and geomagnetic activity; and (vi) atmospheric loss and loss-cone absorption using realistic atmospheric composition and strangelet–nucleus interaction cross sections. These components are joint conditions that a predictive treatment must incorporate together, and they jointly bear on a single objective: a predicted trapped strangelet flux, with uncertainty bands, as a function of $L$-shell (McIlwain shell label), rigidity, and baryon number, or a no-trapping result that holds across the full propagated parameter range, in a form that can guide experimental searches for this distinctive low-$Z/A$ component in Earth's magnetosphere.

References

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