Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone

In a recent study of the self-adjoint extensions of the Hamiltonian of a particle confined to a finite region of space, in which we generalized the Heisenberg uncertainty relation to a finite volume, we encountered bound states localized at the wall of the cavity. In this paper, we study this situation in detail both for a free particle and for a hydrogen atom centered in a spherical cavity. For appropriate values of the self-adjoint extension parameter, the bound states localized at the wall resonate with the standard hydrogen bound states. We also examine the accidental symmetry generated by the Runge–Lenz vector, which is explicitly broken in a spherical cavity with general Robin boundary conditions. However, for specific radii of the confining sphere, a remnant of the accidental symmetry persists. The same is true for an electron moving on the surface of a finite circular cone, bound to its tip by a 1/r1/r potential.

Energetic neutral atom imaging of the lunar surface

Since the Moon is not shielded by a global magnetic field or by an atmosphere, solar wind plasma impinges onto the lunar surface almost unhindered. Until recently, it was assumed that almost all of the impinging solar wind ions are absorbed by the surface. However, recent Interstellar Boundary Explorer, Chandrayaan-1, and Kaguya observations showed that the interaction process between the solar wind ions and the lunar surface is more complex than previously assumed. In contrast to previous assumptions, a large fraction of the impinging solar wind ions is backscattered as energetic neutral atoms. Using the complete Chandrayaan-1 Energetic Neutral Analyzer data set, we compute a global solar wind reflection ratio of 0.16 ± 0.05 from the lunar surface. Since these backscattered neutral particles are not affected by any electric or magnetic fields, each particle's point of origin on the lunar surface can be determined in a straight-forward manner allowing us to create energetic neutral atom maps of the lunar surface. The energetic neutral atom measurements recorded by the Chandrayaan-1 Energetic Neutral Analyzer cover ˜89% of the lunar surface, whereby the lunar farside is almost completely covered. We analyzed all available energetic neutral atom measurements recorded by the Chandrayaan-1 Energetic Neutral Analyzer to create the first global energetic neutral hydrogen maps of the lunar surface.

Lunar energetic neutral atom (ENA) spectra measured by the interstellar boundary explorer (IBEX)

The solar wind continuously flows out from the Sun, filling interplanetary space and directly interacting with the surfaces of small planetary bodies and other objects throughout the solar system. A significant fraction of these ions backscatter from the surface as energetic neutral atoms (ENAs). The first observations of these ENA emissions from the Moon were recently reported from the Interstellar Boundary Explorer (IBEX). These observations yielded a lunar ENA albedo of ˜10% and showed that the Moon reflects ˜150 metric tons of neutral hydrogen per year. More recently, a survey of the first 2.5 years of IBEX observations of lunar ENAs was conducted for times when the Moon was in the solar wind. Here, we present the first IBEX ENA observations when the Moon is inside the terrestrial magnetosheath and compare them with observations when the Moon is in the solar wind. Our analysis shows that: (1) the ENA intensities are on average higher when the Moon is in the magnetosheath, (2) the energy spectra are similar above ~0.6* solar wind energy but below there are large differences of the order of a factor of 10, (3) the energy spectra resemble a power law with a "hump" at ˜0.6 * solar wind energy, and (4) this "hump" is broader when the Moon is in the magnetosheath. We explore potential scenarios to explain the differences, namely the effects of the topography of the lunar surface and the consequences of a very different Mach number in the solar wind versus in the magnetosheath.

Remote energetic neutral atom imaging of electric potential over a lunar magnetic anomaly

The formation of electric potential over lunar magnetized regions is essential for understanding fundamental lunar science, for understanding the lunar environment, and for planning human exploration on the Moon. A large positive electric potential was predicted and detected from single point measurements. Here, we demonstrate a remote imaging technique of electric potential mapping at the lunar surface, making use of a new concept involving hydrogen neutral atoms derived from solar wind. We apply the technique to a lunar magnetized region using an existing dataset of the neutral atom energy spectrometer SARA/CENA on Chandrayaan-1. Electrostatic potential larger than +135 V inside the Gerasimovic anomaly is confirmed. This structure is found spreading all over the magnetized region. The widely spread electric potential can influence the local plasma and dust environment near the magnetic anomaly.