Inferring lunar wake potentials from electron phase space densities

This paper introduces a Hamiltonian inversion method that overcomes challenges posed by solar wind strahl asymmetry and ion acoustic shocks to accurately infer spatial electric potential profiles in the lunar wake from electron phase space density measurements, a technique validated by simulations and applied to ARTEMIS observations.

The Gist

The Big Picture: The Moon's "Shadow"

Imagine the Moon is a giant, invisible wall floating in a river of invisible wind. This wind is the solar wind—a constant stream of charged particles (mostly protons and electrons) blowing from the Sun.

Because the Moon has no atmosphere and no magnetic shield, the solar wind hits it head-on and stops. This creates a giant, empty "shadow" behind the Moon called the lunar wake. It's like the calm, empty water behind a boat moving through a lake.

The problem? This empty shadow doesn't stay empty for long. The solar wind rushes in to fill it back up. As it does, it creates invisible electric fields (like invisible slopes or hills) that control how the particles move. Scientists want to map these invisible hills to understand how the wake works, but they can't measure the hills directly with their instruments because the fields are too weak.

The Old Way: The "Energy Shift" Method

Previously, scientists tried to guess the shape of these invisible hills by looking at the energy of the electrons.

The Analogy: Imagine you are hiking up a mountain. You know what your energy level was at the bottom (the solar wind). If you measure your energy at the top of a hill, you can guess how high the hill is by seeing how much energy you "lost" climbing it.

The Problem: This old method had two major flaws in the lunar wake:

1. The One-Way Street: The solar wind has a "beam" of fast electrons (called the strahl) that only flows one way. This makes the left side of the wake look totally different from the right side. The old method assumed the mountain was symmetrical, so it got confused.
2. The Traffic Jam: In the center of the wake, the wind from the left and right sides crash into each other, creating a shockwave (like a traffic jam). This traps electrons in a "flat-top" distribution. The old method couldn't read these trapped electrons at all, leaving a blind spot right in the middle of the map.

The New Solution: The "Hamiltonian Inversion" Method

The authors of this paper developed a new, smarter way to map these invisible hills. They call it the Hamiltonian Inversion Method.

Think of this method as a smart detective that doesn't try to solve the whole mystery at once. Instead, it breaks the crime scene into three different zones and uses a different clue for each one.

Zone 1 & 2: The Left and Right Sides (The "Separate Maps" Strategy)

On the left and right sides of the wake, the electron populations are different because of that one-way beam (the strahl).

• The Strategy: The new method says, "Let's ignore the other side for a moment." It splits the wake in half.
• The Analogy: Imagine trying to figure out the shape of a valley where the wind blows hard on the left but is calm on the right. Instead of trying to draw one perfect line for the whole valley, you draw a separate map for the left side and a separate map for the right side. You then stitch them together later.
• The Result: This solves the "asymmetry" problem. It accurately maps the invisible hills on both sides, even though they look different.

Zone 3: The Middle (The "Flat-Top" Strategy)

In the very center, where the shockwaves happen, the electrons get trapped and form a "flat-top" distribution (like a plateau).

• The Strategy: The old method failed here because it couldn't use the "energy shift" trick. The new method looks at the width of that flat plateau.
• The Analogy: Imagine a trampoline with a heavy ball sitting in the middle. The ball sinks down, creating a dip. If you look at the edge of the dip, the width of the flat area tells you exactly how deep the hole is. The wider the flat top, the deeper the electric "hole" (potential well).
• The Result: By measuring the width of this electron "plateau," the method can calculate the depth of the electric potential right in the center, where the old method was blind.

Putting It All Together

The researchers tested this new detective method in two ways:

1. Computer Simulations: They created a virtual Moon and solar wind in a supercomputer. They knew the "true" shape of the invisible hills because they built them. When they ran their new method, it perfectly matched the truth, whereas the old method missed the details in the center.
2. Real Space Data: They applied the method to data from the ARTEMIS spacecraft, which orbits the Moon.
   • Early Stage: When the wake was just forming, they found a huge electric potential drop (about 800 Volts) and saw the clear asymmetry between the two sides.
   • Late Stage: When the wake was older and shockwaves had formed, they found a smaller potential drop (about 200 Volts) but successfully mapped the "bumps" caused by the shockwaves in the center.

Why This Matters

This new method is like upgrading from a blurry, black-and-white photo to a high-definition, 3D color map. It allows scientists to finally see the invisible electric forces that control how the Moon's wake refills.

Because this method works on any object that doesn't have a magnetic field (like asteroids or comets), it's a universal tool. It helps us understand how the solar wind interacts with the "empty" spaces in our solar system, revealing the hidden physics of our cosmic neighborhood.

Technical Summary

1. Problem Statement

The lunar wake is a plasma void formed downstream of the Moon due to the lack of a global magnetic field and atmosphere. Understanding the electric potential structure ($\phi$) within this wake is crucial for modeling plasma refilling dynamics. However, direct measurement of these potentials is generally infeasible because:

• Weak Fields: The average parallel electric field is on the order of $0.06$ mV/m, well below the sensitivity threshold of spacecraft instruments.
• Instrument Limitations: Only localized, intense potential enhancements near electrostatic shocks are directly measurable.

Consequently, potentials are typically inferred from electron phase space density (PSD) measurements under the assumption of quasi-static Vlasov equilibrium ($f = f(H)$, where $H$ is the electron Hamiltonian). The existing standard, the Hamiltonian shift method, faces two critical limitations in the lunar wake:

1. Strahl-Driven Asymmetry: The solar wind contains a field-aligned beam of suprathermal electrons (strahl). This creates an inherent asymmetry between the sunward and anti-sunward sides of the wake, making the global relationship $f(H)$ multi-valued and invalidating single-domain inversion.
2. Shock-Induced Trapping: In the central wake, counter-streaming ion beams generate ion acoustic shocks. These shocks trap electrons via nonlinear Landau resonance, creating "flat-top" velocity distributions where $f(H)$ becomes ill-defined. The Hamiltonian shift method struggles here because it relies on high-energy "passing" electrons, which often have low phase space densities and poor statistics in these regions.

2. Methodology: The Hamiltonian Inversion Method

The authors propose a Hamiltonian inversion method that infers the full spatial electric potential profile $\phi(x)$ by exploiting the quasi-static condition $f = f(H)$ through a domain-decomposition strategy. The method assumes negligible variation in the magnetic field magnitude along the field line, reducing the problem to 2D phase space $(x, p_x)$.

The algorithm proceeds in five steps:

1. Initial Guess & Split Point:

   • An initial potential guess $\tilde{\phi}(x)$ is derived from the Boltzmann relation using measured electron density.
   • The potential minimum $x^*$ (coinciding with the density minimum) is identified as the preliminary split point between the left (sunward) and right (anti-sunward) domains.

2. Domain Decomposition:

   • Left/Right Domains: Defined as regions where the electron distribution is dominated by passing electrons.
   • Middle Region: Identified by calculating the effective electron thermal velocity $v_{th}(x)$. A sharp, localized enhancement in $v_{th}$ near $x^*$ indicates the presence of ion acoustic shocks and trapped electrons. This region is bounded by $x_L$ and $x_R$.

3. Independent Inversion (Left & Right):

   • For each side, the method assumes $f = f(H)$ is single-valued locally.
   • It constructs a functional relationship $f_{interp}(\tilde{H})$ by binning observed phase space pixels $(x, p_x)$ based on their Hamiltonian values.
   • An error functional $L(\tilde{\phi})$ is minimized using the L-BFGS-B algorithm (a quasi-Newton method). The objective is to find the potential $\tilde{\phi}$ that minimizes the misfit between the observed $\ln f$ and the reconstructed $\ln f_{interp}$.
   • A multi-grid approach (coarse to fine) is used to avoid local minima.

4. Direct Inversion (Middle Region):

   • In the shock region, $f(H)$ is ill-defined. Instead, the method utilizes the flat-top structure of trapped electrons.
   • The cutoff velocity $v_{cut}(x)$ is identified as the velocity where the PSD drops to 50% of its plateau value.
   • The potential is inferred directly via energy conservation relative to the boundary:
     $$\tilde{\phi}(x) = \tilde{\phi}(x_L) + \frac{m_e}{2e} [v_{cut}^2(x) - v_{cut}^2(x_L)]$$

5. Stitching:

   • The three solutions are combined into a single continuous profile. The middle solution is anchored to the left domain, and the right domain is shifted by a constant offset to ensure continuity at $x_R$.

3. Key Contributions

• Novel Algorithm: Development of a domain-decomposition inversion technique that explicitly handles the multi-valued nature of $f(H)$ caused by strahl asymmetry and the ill-defined nature of $f(H)$ caused by shock trapping.
• Utilization of Reflected/Trapped Electrons: Unlike the shift method, which discards reflected electrons or struggles with low-statistics passing electrons in shocks, this method leverages the well-measured flat-top distributions of trapped electrons to infer potential drops.
• Robust Validation: The method is rigorously validated against Vector Particle-in-Cell (VPIC) simulations at two distinct evolutionary stages and applied to real spacecraft data.

4. Results

A. Simulation Validation

• Early Stage (No Shocks, $r/R_L \approx 1.65$):
  • Characterized by strong strahl asymmetry but no central shocks.
  • Result: The Hamiltonian inversion method accurately recovered the asymmetric potential profile. The Hamiltonian shift method failed near the wake center due to low counting statistics for penetrating electrons.
• Late Stage (With Shocks, $r/R_L \approx 5.5$):
  • Characterized by ion acoustic shocks and flat-top electron distributions.
  • Result: The domain decomposition successfully identified the middle region. The direct $v_{cut}$-based inversion accurately captured the potential enhancements associated with shocks, a feature the shift method could only approximate with high uncertainty.

B. ARTEMIS Spacecraft Application

The method was applied to two ARTEMIS lunar wake crossings:

1. Early Stage Crossing:
   • Inferred normalized potential drop: $e\Delta\phi/T_e \sim 15$ ($\Delta\phi \approx 800$ V).
   • Captured strong asymmetry driven by the strahl.
   • Results agreed with the shift method in open regions but provided better resolution in the center.
2. Late Stage Crossing:
   • Inferred normalized potential drop: $e\Delta\phi/T_e \sim 5$ ($\Delta\phi \approx 200$ V).
   • Captured shock-associated potential enhancements in the central wake.
   • Asymmetry was less pronounced, consistent with simulation trends.
   • Discrepancies with the shift method occurred near shock boundaries, attributed to the difficulty of precisely defining the flat-top edge.

5. Significance

• Scientific Advancement: This method provides the first robust, self-consistent framework to reconstruct the full electric potential profile of the lunar wake, including the complex central shock regions that were previously difficult to characterize.
• Broader Applicability: The technique is not limited to the Moon. It is applicable to any unmagnetized or weakly magnetized body (e.g., asteroids, comets, Martian moons) where plasma refilling creates similar wake structures and electrons are in quasi-static equilibrium.
• Future Utility: The authors plan to apply this method to a statistical sample of ARTEMIS data to map the evolution of lunar wake potentials as a function of radial distance and solar wind conditions, offering new insights into fundamental plasma processes like electrostatic shock formation and particle trapping.