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HERMES Pathfinder is an in-orbit demonstration consisting of a constellation of six 3U cubesats hosting simple but innovative X-ray/gamma-ray detectors for the monitoring of cosmic high-energy transients. HERMES-PF, funded by ASI and by the EC Horizon 2020 grant, is scheduled for launch in Q1 2025. An identical X-ray/gamma-ray detector is hosted by the Australian 6U cubesat SpIRIT, launched on December 1st 2023. The main objective of HERMES-PF/SpIRIT is to demonstrate that high energy cosmic transients can be detected efficiently by miniatured hardware and localized using triangulation techniques. The HERMES-PF X-ray/gamma-ray detector is made by 60 GAGG:Ce scintillator crystals and 12 2x5 silicon drift detector (SDD) mosaics, used to detect both the cosmic X-rays directly and the optical photons produced by gamma-ray interactions with the scintillator crystals. This design provides a unique broad band spectral coverage from a few keV to a few MeV. Furthermore, the use of fast GAGG:Ce crystals and small SDD cells allows us to reach an exquisite time resolution better than a microsecond. We present a progress report on the missions focusing the discussion on the scientific innovation of the project and on the main lessons learned during the project development including: the importance and the challenges of using distributed architectures to achieve ambitious scientific objectives; the importance of developing critical technologies under science agreements for the realization of high-performing but low-cost payloads; best use of COTS technologies in scientific missions. We finally discuss the prospects of applying these concepts for the creation of an all-sky, all-time monitor to search for the high-energy counterparts of gravitational wave events that Advanced LIGO/Virgo/Kagra will find at the end of this decade and the Einstein Telescope during the 2030s., Comment: proceedings of the 75th International Astronautical Congress (IAC), Milan, Italy, 14-18 October 2024

HERMES (High Energy Rapid Modular Ensemble of Satellites) Pathfinder is a space-borne mission based on a constellation of six nano-satellites flying in a low-Earth orbit (LEO). The 3U CubeSats, to be launched in early 2025, host miniaturized instruments with a hybrid Silicon Drift Detector/GAGG:Ce scintillator photodetector system, sensitive to X-rays and gamma-rays in a large energy band. HERMES will operate in conjunction with Australian Space Industry Responsive Intelligent Thermal (SpIRIT) 6U CubeSat, launched in December 2023. HERMES will probe the temporal emission of bright high-energy transients such as Gamma-Ray Bursts (GRBs), ensuring a fast transient localization in a field of view of several steradians exploiting the triangulation technique. HERMES intrinsically modular transient monitoring experiment represents a keystone capability to complement the next generation of gravitational wave experiments. In this paper we outline the scientific case, development and programmatic status of the mission, Comment: 13 pages, 16 figures. Proceedings of SPIE "Astronomical Telescopes and Instrumentation" 2024. arXiv admin note: text overlap with arXiv:2210.13866

Gamma Ray Bursts (GRBs) bridge relativistic astrophysics and multi-messenger astronomy. Space-based gamma/X-ray wide field detectors have proven essential to detect and localize the highly variable GRB prompt emission, which is also a counterpart of gravitational wave events. We study the capabilities to detect long and short GRBs by the High Energy Rapid Modular Ensemble of Satellites (HERMES) Pathfinder (HP) and SpIRIT, namely a swarm of six 3U CubeSats to be launched in early 2025, and a 6U CubeSat launched on December 1st 2023. We also study the capabilities of two advanced configurations of swarms of >8 satellites with improved detector performances (HERMES Constellations). The HERMES detectors, sensitive down to ~2-3 keV, will be able to detect faint/soft GRBs which comprise X-ray flashes and high redshift bursts. By combining state-of-the-art long and short GRB population models with a description of the single module performance, we estimate that HP will detect ~195^{+22}_{-21} long GRBs (3.4^{+0.3}_{-0.8} at redshift z>6) and ~19^{+5}_{-3} short GRBs per year. The larger HERMES Constellations under study can detect between ~1300 and ~3000 long GRBs per year and between ~160 and ~400 short GRBs per year, depending on the chosen configuration, with a rate of long GRBs above z>6 between 30 and 75 per year. Finally, we explore the capabilities of HERMES to detect short GRBs as electromagnetic counterparts of binary neutron star (BNS) mergers detected as gravitational signals by current and future ground-based interferometers. Under the assumption that the GRB jets are structured, we estimate that HP can provide up to 1 (14) yr^{-1} joint detections during the fifth LIGO-Virgo-KAGRA observing run (Einstein Telescope single triangle 10 km arm configuration). These numbers become 4 (100) yr^{-1}, respectively, for the HERMES Constellation configuration., Comment: 13 pages, 6 figures, 4 tabels. Accepted for publication by Astronomy & Astrophysics

We studied the orbit of an electron revolving around an infinitely massive nucleus of a large classical Hydrogen atom subject to an AC electric field oscillating perpendicular to the electron's circular orbit. Using perturbation theory in geometric algebra, we show that the equation of motion of the electron perpendicular to the unperturbed orbital plane satisfies a forced simple harmonic oscillator equation found in Lorentz dispersion law in Optics. We show that even though we did not introduce a damping term, the initial orbital position and velocity of the electron results to a solution whose absorbed energies are finite at the dominant resonant frequency $\omega=\omega_0$; the electron slowly increases its amplitude of oscillation until it becomes ionized. We computed the average power absorbed by the electron both at the perturbing frequency and at the electron's orbital frequency. We graphed the trace of the angular momentum vector at different frequencies. We showed that at different perturbing frequencies, the angular momentum vector traces epicyclical patterns., Comment: 12 pages, 8 figures

We used Geometric Algebra to compute the paths of skew rays in a cylindrical, step-index multimode optical fiber. To do this, we used the vector addition form for the law of propagation, the exponential of an imaginary vector form for the law of refraction, and the juxtaposed vector product form for the law of reflection. In particular, the exponential forms of the vector rotations enables us to take advantage of the addition or subtraction of exponential arguments of two rotated vectors in the derivation of the ray tracing invariants in cylindrical and spherical coordinates. We showed that the light rays inside the optical fiber trace a polygonal helical path characterized by three invariants that relate successive reflections inside the fiber: the ray path distance, the difference in axial distances, and the difference in the azimuthal angles. We also rederived the known generalized formula for the numerical aperture for skew rays, which simplifies to the standard form for meridional rays., Comment: 11 pages, 7 figures. This paper is currently being processed in Applied Optics. The Early Posting Version of the paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/ao/upcoming_pdf.cfm?id=231285. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law

We perform an analysis of the combined effects of geometry and a magnetic field for the case of a charged particle on a helicoid. The effective quantum potentials for a charged spinless particle confined on a helicoid for two simple magnetic field configurations are derived. These potentials depend nontrivially on the surface curvature and the external magnetic field. We find that the qualitative behavior of the effective potential can be altered by changing the strength of the applied magnetic field. The application of a magnetic field results in effective potentials that are either repulsive or attractive, depending on the magnitude of the magnetic field and the angular momentum of the particle. Finally, for the case of effective potentials that have a minimum, we also obtain approximate expressions for the energy levels valid when the particle is near a minimum, and these are found to be similar in form to the energy levels of a particle in a harmonic oscillator potential., Comment: 9 pages, 4 figures; fixed Fig. 2

We use the vector wedge product in geometric algebra to show that Poisson commutator brackets measure preservation of phase space areas. We also use the vector dot product to define the Poisson anticommutator bracket that measures the preservation of phase space angles. We apply these brackets to the paraxial meridional complex height-angle ray vectors that transform via a 2x2 matrix, and we show that this transformation preserves areas but not angles in phase space. The Poisson brackets here are expressed in terms of the coefficients of the ABCD matrix. We also apply these brackets to the distance-height ray vectors measured from the input and output sides of the optical system. We show that these vectors obey a partial Moebius transformation, and that this transformation preserves neither areas nor angles. The Poisson brackets here are expressed in terms of the transverse and longitudinal magnifications., Comment: 10 pages, 9 figures

We propose that the height-angle ray vector in matrix optics should be complex, based on a geometric algebra analysis. We also propose that the ray's 2x2 matrix operators should be right-acting, so that the matrix product succession would go with light's left-to-right propagation. We express the propagation and refraction operators as a sum of a unit matrix and an imaginary matrix proportional to the Fermion creation or annihilation matrix. In this way, we reduce the products of matrix operators into sums of creation-annihilation product combinations. We classify ABCD optical systems into four: telescopic, inverse Fourier transforming, Fourier transforming, and imaging. We show that each of these systems have a corresponding Lagrange theorem expressed in partial derivatives, and that only the telescopic and imaging systems have Lagrange invariants., Comment: 10 pages, 6 figures

We represent vector rotation operators in terms of bras or kets of half-angle exponentials in Clifford (geometric) algebra Cl_{3,0}. We show that SO_3 is a rotation group and we define the dihedral group D_4 as its finite subgroup. We use the Euler-Rodrigues formulas to compute the multiplication table of D_4 and derive its group algebra identities. We take the linear combination of rotation operators in D_4 to represent the four Fermion matrices in Sakurai, which in turn we use to decompose any 2x2 matrix. We show that bra and ket operators generate left- and right-acting matrices, respectively. We also show that the Pauli spin matrices are not vectors but vector rotation operators, except for \sigma_2 which requires a subsequent multiplication by the imaginary number i geometrically interpreted as the unit oriented volume., Comment: 11 pages, 3 figures, 1 table

We derive the paraxial meridional ray tracing equations from the unified reflection-refraction law using geometric algebra. This unified law states that the normal vector to the interface is a rotation of the incident ray or of the refracted ray or of the reflected ray by an angle equal to the angle of incidence or of refraction. We obtain the finite meridional ray tracing equations by simply equating the arguments of the exponential rotation operators. We then derive the paraxial limits of these equations with the help of sign function identities. We show that by embedding the sign functions in the ray tracing equations, we explicitly declare our chosen sign conventions in symbols and not in prose., Comment: 5 pages, 2 figures

We list the subgroups of the basis set of Cl_{3,0} and classify them according to three criteria for construction of universal Clifford algebras: (1) each generator squares to +1 or -1, (2) the generators within the group anticommute, and (3) the order of the resulting group is 2^{n+1}, where n is the number of nontrivial generators. Obedient groups we call choirs; disobedient groups, bands. We classify choirs by modes and bands by rhythms, based on canonical equality. Each band generator has a transposition (number of other generators it commutes with). The band's transposition signature is the band's chord. The sum of transpositions divided by twice the number of generator pair combinations is the band's beat. The band's order deviation is the band's disorder. For n less than or equal 3, we show that the Cl_{3,0} basis set has 21 non-isomorphic subgroups consisting of 9 choirs and 12 bands., Comment: 9 pages, 10 tables

We derive Copernicus's epicycles from Newton's gravitational force law by assuming that a planet's orbit is a perturbed circular orbit, with the perturbation defined to be co-rotating with the said orbit. We substitute this orbit expression into Newton's gravitation law and showed that the perturbation satisfies the linear part of Hill's oscillator equation for lunar motion. We solve this oscillator equation using an exponential Fourier series and impose the boundary conditions at the aphelion and perihelion to derive Copernicus's formulas for the eccentric, deferent, and epicycle. We show that for small eccetricity, the Copernican orbit expression also leads to Kepler's law of areas for planetary motion. The formalism we use is the Clifford (geometric) algebra Cl_{2,0}., Comment: 7 pages, 3 figures

In this paper, we define energy-momentum density as a product of the complex vector electromagnetic field and its complex conjugate. We derive an equation for the spacetime derivative of the energy-momentum density. We show that the scalar and vector parts of this equation are the differential conservation laws for energy and momentum, and the imaginary vector part is a relation for the curl of the Poynting vector. We can show that the spacetime derivative of this energy-momentum equation is a wave equation. Our formalism is Dirac-Pauli-Hestenes algebra in the framework of Clifford (Geometric) algebra Cl_{4,0}., Comment: 5 pages, no figures. The following article has been submitted to the American Journal of Physics. After it is published, it will be found at http://scitation.aip.org/ajp