The very slow (~350 km/s) 'stealth' coronal mass ejection (CME) launched on 2020 April 15 was observed in situ by Solar Orbiter at ~0.8 AU on April 19 and near Earth the next day, producing Forbush decreases (short-term depressions of the galactic cosmic-ray flux) of about 3% in low-energy space-based channels and about 1% at neutron-monitor and 10 GV rigidity. The analytical ForbMod model reproduces the low-energy amplitudes only with flux-rope expansion parameters unsupported by the multi-point observations, and with any observationally consistent input it underpredicts the high-energy amplitudes (at most 0.44% at the South Pole neutron monitor, below 0.01% at 10 GV). The open problem is twofold and linked: determine the radial expansion behavior of this flux rope — constant versus time-varying power-law indices nan_a and nBn_B, including compression by the following stream interaction region and its effect on the flux-rope radius and central magnetic field — and establish whether the high-energy underprediction is a systematic deficiency of ForbMod's empirical energy dependence of the diffusion coefficient or of its galactic-cosmic-ray spectrum inputs, or is specific to this event. No statistical validation of ForbMod against neutron-monitor catalogs and no multi-point time-dependent expansion solution for this event are known in the literature; the problem remains open.

Background

A coronal mass ejection (CME) is a large eruption of magnetized plasma from the Sun; when its interplanetary counterpart (ICME) contains an ordered helical magnetic structure it is called a magnetic flux rope or magnetic cloud. A 'stealth' CME is one that leaves only very weak signatures in extreme-ultraviolet observations of the low corona, making it hard to detect and forecast from Earth's vantage point. As ICMEs propagate, they shield and modulate galactic cosmic rays (GCRs), producing a Forbush decrease (FD): a short-term depression of the GCR count rate, typically a few percent at spacecraft energies near 1 AU, with two contributions — a shock/sheath step and a deeper, recovering depression inside the magnetic ejecta. GCRs at roughly 1–20 GV rigidity (rigidity is momentum per unit charge) are monitored by ground-based neutron monitors (NM), which detect the secondary particles produced when cosmic rays enter the atmosphere; the worldwide NM network can be inverted by the Global Survey Method to obtain the isotropic GCR density variation at a fixed rigidity of 10 GV, close to the effective response energy of NMs.

The source work (von Forstner et al. 2021) studied the first flux-rope CME observed in situ by Solar Orbiter (SolO). This stealth CME erupted on 2020 April 15, was tracked in coronagraph images by STEREO-A out to ~0.6 AU, passed SolO at 0.8 AU on April 19 (less than 4 degrees in heliolongitude from Earth) and Earth on April 20, after more than five days of propagation at below 350 km/s. Despite its slow speed it carried a strong magnetic field (peak ~21 nT at SolO, ~16 nT at Wind) and caused a weak geomagnetic storm. It was followed by a stream interaction region (SIR), the compressed boundary between slow and fast solar wind streams. The FD amplitudes were ~3% in the low-energy C-detector counter of SolO's High Energy Telescope (HET), ~2% in the D2 counter of the CRaTER telescope on the Lunar Reconnaissance Orbiter, and ~1% in NM data and in the 10 GV Global Survey Method reconstruction.

The authors applied ForbMod, an analytical physics-based model of flux-rope FDs (Dumbović et al. 2018; energy-dependent extension by Dumbović et al. 2020). ForbMod assumes the flux rope expands self-similarly: with R(t)R(t) the heliocentric distance of the rope center, its radius aa and central magnetic field BcB_c follow power laws

a(t)=a0(R(t)R0)na,Bc(t)=B0(R(t)R0)−nB,a(t) = a_0 \left(\frac{R(t)}{R_0}\right)^{n_a}, \qquad B_c(t) = B_0 \left(\frac{R(t)}{R_0}\right)^{-n_B},

defining the expansion indices nan_a and nBn_B; the combination x=nB−2nax = n_B - 2 n_a (the 'expansion type') measures whether magnetic flux is conserved (x=0x = 0) or changes with distance, and is assumed constant during propagation. Inside the expanding rope, GCRs diffuse inwards with a diffusion coefficient DD taken proportional to 1/Bc1/B_c and with an empirical energy dependence; folding the resulting GCR spectrum with an instrument response function predicts the FD measured by that detector. Near-Sun inputs (a0a_0, R0R_0) came from a Graduated Cylindrical Shell (GCS) reconstruction, an empirical 3D geometric model of the CME white-light structure; the in-situ configuration was fitted with the 3DCORE flux-rope model, which uses approximate Bayesian computation (a likelihood-free statistical fitting technique) to constrain flux-rope orientation and size from magnetic-field time series (Weiss et al. 2021).

Three independent estimates of the size-expansion index were mutually inconsistent: na≈0.37n_a \approx 0.37 from comparing flux-rope durations at SolO and Wind, na≈0.90n_a \approx 0.90 from the local in-situ velocity profile at Wind, and na≈0.70n_a \approx 0.70 from the near-Sun GCS geometry compared with the in-situ size at Earth; only one estimate of nB≈1.30n_B \approx 1.30 (SolO-to-Wind central field ratio) exists. With any of these observationally derived parameter sets, ForbMod underestimates even the low-energy FD amplitudes (by more than a factor of two at best); a 'best fit' matching the SolO HET and CRaTER amplitudes requires a larger nan_a (stronger expansion) not supported by the observations. More seriously, the high-energy FDs are drastically underpredicted: at most 0.44% at the South Pole NM versus ~1% observed, and below 0.01% at the 10 GV Global Survey Method rigidity. The authors offer two candidate explanations: (a) the empirical energy dependence of the diffusion coefficient or the GCR spectrum inputs (which lacked a contemporaneous value of the solar modulation potential Φ\Phi — the force-field-approximation parameter describing how solar activity suppresses the GCR spectrum inside the heliosphere — for 2020) is deficient; (b) the constant-index, constant-xx self-similar assumption breaks down for this slow CME, plausibly because it was intermittently compressed by the following SIR — an interaction shown for a similar event by He et al. (2018), where a slow stealth CME bracketed by fast wind was compressed and became unusually geoeffective. Instrumental response uncertainties (the spacecraft-interaction model built with the Geant4 Monte Carlo particle-transport toolkit simulated protons only; the CRaTER lunar-albedo correction retains residual periodicities) are acknowledged as limitations of the measurement chain, but are not advanced by the authors as an explanation of the high-energy deficit. No quantified compression solution (magnitude, timing, radial location, and its effect on Bc(R)B_c(R) and a(R)a(R)) exists for this event.

Problem Statement

For the 2020 April 15 stealth coronal mass ejection observed in situ at Solar Orbiter (~0.8 AU, 2020 April 19) and near Earth (2020 April 20) — a very slow (~350 km/s) flux-rope CME followed by a stream interaction region — the problem is to close the physical account of its radial expansion and of its Forbush decrease from the outer corona to 1 AU, by resolving two linked questions:

(1) Expansion behavior. Determine whether constant power-law expansion indices nan_a and nBn_B (with constant expansion type x=nB−2nax = n_B - 2 n_a) are consistent with all available constraints — near-Sun Graduated Cylindrical Shell geometry, STEREO-A imaging, and the Solar Orbiter and Wind in-situ observations, which currently give mutually inconsistent estimates (na≈0.37n_a \approx 0.37, 0.700.70, 0.900.90; nB≈1.30n_B \approx 1.30) — or whether the expansion varied with heliocentric distance. If time-dependent expansion is required, quantify it: in particular, establish the magnitude, timing, and radial location of the hypothesized compression of the flux rope by the following SIR (and any subsequent re-expansion) and its direct effect on the radial profiles of the flux-rope radius a(R)a(R) and central magnetic field Bc(R)B_c(R), for example through a simultaneous multi-point 3D flux-rope fit (such as 3DCORE constrained by both in-situ datasets) combined with a propagation/interaction model validated against the observations.

(2) The high-energy deficit. Establish whether the failure of ForbMod, with observationally supported inputs, to reproduce the high-energy Forbush-decrease amplitudes (South Pole neutron monitor and 10 GV Global Survey Method data, observed ~1% versus modeled at most 0.44% and below 0.01%) is (a) a systematic deficiency of the model's empirical energy dependence of the diffusion coefficient or of its GCR-spectrum inputs for near-solar-minimum 2020 conditions, or (b) a breakdown of the self-similar constant-index assumptions under non-self-similar expansion or CME–SIR interaction specific to this event — for example through a statistical validation of ForbMod's energy dependence against Global Survey Method and neutron-monitor Forbush-decrease catalogs, or an equivalent demonstration that event-specific physics accounts for the discrepancy.

The two questions share one objective: a self-consistent, observationally constrained account of this CME's radial evolution and of its Forbush decrease that determines the expansion behavior (question 1) and establishes which candidate cause underlies the high-energy deficit (question 2). Reproducing the observed high-energy amplitudes is not itself a required deliverable. Improved GCR-modulation inputs (solar modulation potential, energy-dependent diffusion coefficient for 2020 conditions) and refined instrument response models (heavy ions, spacecraft mass models, albedo corrections) may be needed as inputs or by-products, but are not separate deliverables.

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