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Building upon earlier work, we explore the limits of using a configuration of satellites to measure the trace of the gravitational gradient tensor using intersatellite laser ranging and timing observables without relying on high-precision external observables such as deep space radio navigation or astrometry with unrealistic accuracy. A refined model, calculated with extended numerical precision, confirms that exceptional sensitivity is possible, placing within reach observational tests of certain modified gravity theories (e.g., Yukawa terms, galileons) using heliocentric orbits in the vicinity of the Earth. The sensitivity of the experiment improves at larger heliocentric distances. A constellation placed at 30 astronomical units, still well within the domain of feasibility using available propulsion and deep space communication technologies, may approach sensitivities that are sufficient to detect not just the gravitational contribution of the interplanetary medium but perhaps even cosmological dark matter and dark energy constituents., Comment: 17 pages, 6 figures

The usual derivation of Einstein's field equations from the Einstein--Hilbert action is performed by silently assuming the metric tensor's symmetric character. If this symmetry is not assumed, the result is a new theory, such as Einstein's attempted Unified Field Theory or Moffat's Nonsymmetric Gravitational Theory. Explicitly enforcing the constraint by means of a Lagrange-multiplier term restores Einstein's field equations, but the multiplier appears as an additional, unconstrained antisymmetric term. We briefly discuss the possible significance of this term with respect to a nonvanishing cosmological angular momentum, a sourced spin current, the nonsymmetric nature of the Einstein pseudotensor characterizing the energy-momentum of the gravitational field, and possible implications on attempts to obtain a quantum theory of gravity., Comment: 6 pages, accepted for publication in CQG

Direct detection of gravitons in gravitational experiments, including gravitational wave observatories, has been all but ruled out given the weak coupling between the gravitational field and matter. Here we propose an alternative: looking not for the presence but for the absence of graviton shot noise in gravitational wave data. Gravitational wave experiments detect very weak signals that correspond to a surprisingly small number of gravitons even at the relatively low frequencies that characterize signals from gravitational wave events. A detailed calculation, which also yields results that are consistent with the existing literature, demonstrates that graviton shot noise may be present at detectable levels in gravitational wave observations. The absence of elevated noise levels due to graviton shot noise, in turn, would indicate that gravity is not a quantum field theory with a conventional perturbative expansion at low energies., Comment: 4 pages, 1 figure, additional calculation included

This paper describes in detail how erasure codes are implemented in the Swarm system. First, in Section 1, we introduce erasure codes, and show how to apply them to files in Swarm (Section 2). In Section 3, we introduce security levels of data availability and derive their respective parameterisations. In Section 4, we describe a construct that enables cross-neighbourhood redundancy for singleton chunks and which completes erasure coding. Finally, in 5, we propose a number of retrieval strategies applicable to erasure-coded files., Comment: 14 pages, 7 figures, 4 tables

We investigate the propagation of light signals across multiple gravitational lenses, with particular emphasis on the "communication bridge" scenario of two lenses with collinear source and observer. The lenses are assumed to be non-coplanar, far enough from one another for each lens to be treated independently as thin lenses in the limit of weak gravity. We analyze these scenarios using several different tools, including geometric optics, photon mapping, wave optics and ray tracing. Specifically, we use these tools to assess light amplification and image formation by a two-lens system. We then extend the ray tracing analysis to the case of multiple non-coplanar lenses, demonstrating the complexity of images that are projected even by relatively simple lens configurations. We introduce a simple simulation tool that can be used to analyze lensing by non-coplanar gravitational monopoles in the weak gravity limit, treating them as thin lenses., Comment: 21 pages, 12 figures. Final version before acceptance

We investigate the utility of a constellation of four satellites in heliocentric orbit, equipped with accurate means to measure intersatellite ranges, round-trip times and phases of signals coherently retransmitted between members of the constellation. Our goal is to reconstruct the measured trace of the gravitational gradient tensor as accurately as possible. Intersatellite ranges alone are not sufficient for its determination, as they do not account for any rotation of the satellite constellation, which introduces fictitious forces and accelerations. However, measuring signal round-trip time differences among the satellites supplies the necessary observables to estimate, and subtract, the effects of rotation. Utilizing, in addition, the approximate distance and direction from the Sun, it is possible to approach an accuracy of $10^{-24}~{\rm s}^{-2}$ for a constellation with typical intersatellite distances of 1,000 km in an orbit with a 1 astronomical unit semi-major axis. This is deemed sufficient to detect the presence of a galileonic modification of the solar gravitational field., Comment: 24 pages, 5 figures, final submitted version after referee comments

We investigate imaging point sources with a monopole gravitational lens, such as the Solar Gravitational Lens in the geometric optics limit. We compute the light amplification of the lens used in conjunction with a telescope featuring a circular aperture that is placed in the focal region of the lens, compared to the amount of light collected by the same telescope unaided by a gravitational lens. We recover an averaged point-spread function that is in robust agreement with a wave-theoretical description of the lens, and can be used in practical calculations or simulations., Comment: 6 pages, 6 figures, final submitted version after referee comments

NGC1277 is a compact but massive lenticular galaxy that shows no signs of the presence of dark matter. We find that this galaxy's behavior is consistent not only with Newtonian dynamics, but also with the predictions of Scalar--Tensor--Vector--Gravity, also known as MOG (MOdified Gravity). The compact size of the galaxy, in combination with its large mass, ensures that there are no observable deviations between the predictions of Newtonian and MOG orbital velocities within the galaxy's visible radius., Comment: 5 pages, 2 figures, expanded Discussion

The Maxima computer algebra system, the open-source successor to MACSYMA, the first general-purpose computer algebra system that was initially developed at the Massachusetts Institute of Technology in the late 1960s and later distributed by the United States Department of Energy, has some remarkable capabilities, some of which are implemented in the form of add-on, "share" packages that are distributed along with the core Maxima system. One such share package is itensor, for indicial tensor manipulation. One of the more remarkable features of itensor is functional differentiation. Through this, it is possible to use itensor to develop a Lagrangian field theory and derive the corresponding field equations. In the present note, we demonstrate this capability by deriving Maxwell's equations from the Maxwell Lagrangian, and exploring the properties of the system, including current conservation., Comment: 6 pages

We utilize the well-established properties of the solar gravitational lens (SGL) to consider realistic observational scenarios. Actual exoplanets, which may be the target of an SGL observational campaign, are not stationary. Their appearance changes as a result of their diurnal rotation and varying illumination due to their orbital motion around their host star. The nature of the SGL is such that imaging with one telescope is accomplished with a cadence of one pixel at a time, with substantial per-pixel integration times. Therefore, capturing a single snapshot of the target planet with a realistically-sized telescope is not possible. Instead, the planetary surface must be reconstructed by inverting the combined effect of the SGL's point-spread function and temporal changes induced by the planetary dynamics. Using the Earth as a stand-in, we demonstrate practical feasibility of this approach, by simulating a dynamical system and then recovering topographic images of acceptable quality. The dynamics-induced temporal variability of the exoplanet represents an added challenge, but even in the presence of such dynamics, use of the SGL for exoplanet imaging remains feasible., Comment: 14 pages, 4 figures

Recently, we witnessed how the synergy of small satellite technology and solar sailing propulsion enables new missions. Together, small satellites with lightweight instruments and solar sails offer affordable access to deep regions of the solar system, also making it possible to realize hard-to-reach trajectories that are not constrained to the ecliptic plane. Combining these two technologies can drastically reduce travel times within the solar system, while delivering robust science. With solar sailing propulsion capable of reaching the velocities of ~5-10 AU/yr, missions using a rideshare launch may reach the Jovian system in two years, Saturn in three. The same technologies could allow reaching solar polar orbits in less than two years. Fast, cost-effective, and maneuverable sailcraft that may travel outside the ecliptic plane open new opportunities for affordable solar system exploration, with great promise for heliophysics, planetary science, and astrophysics. Such missions could be modularized to reach different destinations with different sets of instruments. Benefiting from this progress, we present the "Sundiver" concept, offering novel possibilities for the science community. We discuss some of the key technologies, the current design of the Sundiver sailcraft vehicle and innovative instruments, along with unique science opportunities that these technologies enable, especially as this exploration paradigm evolves. We formulate policy recommendations to allow national space agencies, industry, and other stakeholders to establish a strong scientific, programmatic, and commercial focus, enrich and deepen the space enterprise and broaden its advocacy base by including the Sundiver paradigm as a part of broader space exploration efforts., Comment: 35 pages, 12 figures, 3 tables

We investigate the general relativistic phase of an electromagnetic wave as it propagates in the gravitational field of the Earth, which is modeled as an isolated, weakly aspherical gravitating body. We introduce coordinate systems to describe light propagation in the Earth's vicinity along with the relevant coordinate transformations, and discuss the transformations between proper and coordinate times. We represent the Earth's gravitational field using Cartesian symmetric trace-free (STF) mass multipole moments. The light propagation equation is solvable along the trajectory of a light ray to all STF orders $\ell$. Although we focus primarily on the quadrupole ($\ell=2$), octupole ($\ell=3$), and hexadecapole ($\ell=4$) cases, our approach is valid to all orders. We express the STF moments via spherical harmonic coefficients of various degree and order, $C_{\ell k}, S_{\ell k}$. The result is the gravitational phase shift expressed in terms of the spherical harmonics. These results are new. We also consider contributions due to tides and the Earth's rotation. We estimate the characteristic magnitudes of each term of the resulting overall gravitational phase shift. The terms assessed are either large enough to impact current-generation clocks or will become significant as future-generation clocks offer greater sensitivity. Our formulation is useful for many practical and scientific applications, including space-based time and frequency transfers, relativistic geodesy and navigation, as well as quantum communication links and space-based tests of fundamental physics., Comment: 36 pages, 2 figures, 1 table

We consider resolved imaging of faint sources with the solar gravitational lens (SGL) while treating the Sun as an extended gravitating body. We use our new diffraction integral that describes how a spherical electromagnetic wave is modified by the static gravitational field of an extended body, represented by series of multipole moments characterizing its interior mass distribution. Dominated by the solar quadrupole moment, these deviations from spherical symmetry significantly perturb the image that is projected by the Sun into its focal region, especially at solar equatorial latitudes. To study the optical properties of the quadrupole SGL, we develop an approximate solution for the point spread function of such an extended lens. We also derive semi-analytical expressions to estimate signal levels from extended targets. With these tools, we study the impact of solar oblateness on imaging with the SGL. Given the small value of the solar quadrupole moment, the majority of the signal photons arriving from an extended target still appear within the image area projected by the monopole lens. However, these photons are scrambled, thus reducing the achievable signal-to-noise ratio during image recovery process (i.e., deconvolution). We also evaluate the spectral sensitivity for high-resolution remote sensing of exoplanets with the extended SGL. We assess the impact on image quality and demonstrate that despite the adverse effects of the quadrupole moment, the SGL remains uniquely capable of delivering high-resolution imaging and spectroscopy of faint, small and distant targets, notably terrestrial exoplanets within ~30--100 parsec from us., Comment: 23 pages, 2 figures

We consider strong gravitational lensing by nearby stars. Using our wave-optical treatment of lensing phenomena, we study Einstein rings that may form around nearby stellar lenses. It is remarkable that these rings are bright and large enough to be detected and resolved by existing instruments. Such lensing events have durations of hours or days, with peak light amplification lasting for several minutes. Many such events may be predicted using the Gaia astrometric catalogue. Serendipitous discoveries are also possible. Fortuitous alignments can be used to confirm or discover and study exoplanets. For lenses that have dense stellar regions in their background, these events may occur several time a year, warranting their continuous or recurrent monitoring. Resolved imaging and spectroscopy of the evolving morphology of an Einstein ring offers knowledge about both the lens and the source. The angular size of the Einstein ring amounts to a direct measurement of the lens mass. The changing orientation of the major and minor images of the source offers astrometric information related to the mutual orientation of the objects. The event duration, when the full ring is present, helps to determine the source's size. The sky position of planetary lensing events constrains the planet's orbit. Spectroscopy of the ring allows for direct investigations of the source. The frequency and predictability of these events and the wealth of information that can be obtained by imaging motivate observational campaigns using existing facilities and/or construction of new instruments dedicated to the search and study of Einstein rings that are forming around nearby stars. As a specific example, we consider a predicted 2028 lensing of a red giant by $\alpha$ Centauri A and discuss the relevant science campaign., Comment: 26 pages, 10 figures, 2 tables

We present initial results of an ongoing engineering study on the feasibility of a space mission to the focal region of the solar gravitational lens (SGL). The mission goal is to conduct exoplanet imaging operations at heliocentric distances in the range ~548-900 astronomical units (AU). Starting at 548 AU from the Sun, light from an exoplanet located behind the Sun is greatly amplified by the SGL. The objective is to capture this light and use it for multipixel imaging of an exoplanet up to 100 light years distant. Using a meter-class telescope one can produce images of the exoplanet with a surface resolution measured in tens of kilometers and to identify signs of habitability. The data are acquired pixel-by-pixel while moving an imaging spacecraft within the image. Given the long duration of the mission, decades to 900 AU, we address an architecture for the fastest possible transit time while reducing mission risk and overall cost. The mission architecture implements solar sailing technologies and in-space aggregation of modularized functional units to form mission capable spacecraft. The study reveals elements of such a challenging mission, but it is nevertheless found to be feasible with technologies that are either extant or in active development., Comment: 43 pages, 8 figures, and 9 tables

We consider the optical properties of the solar gravitational lens (SGL) treating the Sun as a massive compact body. Using our previously developed wave-optical treatment of the SGL, we convolve it with a thin-lens representing an optical telescope, and estimate the power spectral density and associated photon flux at individual pixel locations on the image sensor at the focal plane of the telescope. We also consider the solar corona, which is the dominant noise source when imaging faint objects with the SGL. We evaluate the signal-to-noise ratio at individual pixels as a function of wavelength. To block out the solar light, we contrast the use of a conventional internal coronagraph with a Lyot-stop to an external occulter (i.e., starshade). An external occulter, not being a subject to the diffraction limit of the observing telescope, makes it possible to use small telescopes (e.g., $\sim 40$~cm) for spatially and spectrally resolved imaging with the SGL in a broad range of wavelengths from optical to mid-infrared (IR) and without the substantial loss of optical throughput that is characteristic to internal devices. Mid-IR observations are especially interesting as planets are self-luminous at these wavelengths, producing a strong signal, while there is significantly less noise from the solar corona. This part of the spectrum contains numerous features of interest for exobiology and biosignature detection. We develop tools that may be used to estimate instrument requirements and devise optimal observing strategies to use the SGL for high-resolution, spectrally resolved imaging, ultimately improving our ability to confirm and study the presence of life on a distant world., Comment: 16 pages, 11 figures, 1 table

We discuss the feasibility of direct multipixel imaging of exoplanets with the solar gravitational lens (SGL) in the context of a realistic deep space mission. For this, we consider an optical telescope, placed in the image plane that forms in the strong interference region of the SGL. We consider an Earth-like exoplanet located in our immediate stellar neighborhood and model its characteristics using our own Earth. We estimate photon fluxes from such a compact, extended, resolved exoplanet. This light appears in the form of an Einstein ring around the Sun, seen through the solar corona. The solar corona background contributes a significant amount of stochastic noise and represents the main noise source for observations utilizing the SGL. We estimate the magnitude of this noise. We compute the resulting signal-to-noise ratios (SNRs) and related integration times that are needed to perform imaging measurements under realistic conditions. It is known that deconvolution, removing the blur due to the SGL's spherical aberration substantially decreases the SNR. Our key finding is that this "penalty" is significantly mitigated when sampling locations in the image plane (image pixels) remain widely spaced. Consequently, we conclude that an imaging mission is challenging but feasible, using technologies that are either already available or in active development. Under realistic conditions, high-resolution imaging of Earth-like exoplanets in our galactic neighborhood requires only weeks or months of integration time, not years as previously thought: a high quality 1000x1000 pixel image of an Earth-like planet at Proxima Centauri could be obtained with SNR>10 using approximately 14 months of integration time., Comment: 12 pages, 4 figures, 2 tables

Constraint-based causal discovery methods leverage conditional independence tests to infer causal relationships in a wide variety of applications. Just as the majority of machine learning methods, existing work focuses on studying $\textit{independent and identically distributed}$ data. However, it is known that even with infinite i.i.d.$\ $ data, constraint-based methods can only identify causal structures up to broad Markov equivalence classes, posing a fundamental limitation for causal discovery. In this work, we observe that exchangeable data contains richer conditional independence structure than i.i.d.$\ $ data, and show how the richer structure can be leveraged for causal discovery. We first present causal de Finetti theorems, which state that exchangeable distributions with certain non-trivial conditional independences can always be represented as $\textit{independent causal mechanism (ICM)}$ generative processes. We then present our main identifiability theorem, which shows that given data from an ICM generative process, its unique causal structure can be identified through performing conditional independence tests. We finally develop a causal discovery algorithm and demonstrate its applicability to inferring causal relationships from multi-environment data. Our code and models are publicly available at: https://github.com/syguo96/Causal-de-Finetti, Comment: camera-ready NeurIPS 2023

The solar gravitational lens (SGL) offers unique capabilities for high-resolution imaging of faint, distant objects, such as exoplanets. In the near future, a spacecraft carrying a meter-class telescope with a solar coronagraph would be placed in the focal region of the SGL. That region begins at ~547 astronomical units from the Sun and occupies the vicinity of the target-specific primary optical axis - the line that connects the center of the target and that of the Sun. This axis undergoes complex motion as the exoplanet orbits its host star, as that star moves with respect to the Sun, and even as the Sun itself moves with respect to the solar system's barycenter due to the gravitational pull of planets in our solar system. An image of an extended object is projected by the SGL into an image plane and moves within that plane, responding to the motion of the optical axis. To sample the image, a telescope must follow the projection with precise knowledge of its own position with respect to the image. We consider the dominant motions that determine the position of the focal line as a function of time. We evaluate the needed navigational capability for the telescope to conduct a multiyear exoplanet imaging mission. We show that even in a rather conservative case, when an Earth-like exoplanet is in our stellar neighborhood at $\sim10$ light years, the motion of the image is characterized by a small total acceleration $\sim 6.1\,\mu {\rm m/s}^2$ that is driven primarily by the orbital motion of the exoplanet and by the reflex motion of our Sun. We discuss how the amplified light of the host star allows establishing a local reference frame thus relaxing navigational requirements. We conclude that the required navigation in the SGL's focal region, although complex, can be accurately modeled and a $\sim 10$-year imaging mission is achievable with the already available propulsion technology., Comment: 28 pages, 16 figures

Symmetric trace-free tensors are used in many areas of physics, including electromagnetism, relativistic celestial mechanics and geodesy, as well as in the study of gravitational radiation and gravitational lensing. Their use allows integration of the relevant wave propagation equations to arbitrary order. We present an improved iterative method for the trace-free decomposition of symmetric tensors of arbitrary rank. The method can be used both in coordinate-free symbolic derivations using a computer algebra system and in numerical modeling. We obtain a closed-form representation of the trace-free decomposition in arbitrary dimensions. To demonstrate the results, we compute the coordinate combinations representing the symmetric trace-free (STF) mass multipole moments for rank 5 through 8, not readily available in the literature., Comment: 14 pages, no figures

Beyond the Newtonian approximation, gravitational fields in general relativity can be described using a formalism known as gravitoelectromagnetism. In this formalism a vector potential, the gravitomagnetic potential, arises as a result of moving masses, in strong analogy with the magnetic force due to moving charges in Maxwell's theory. Gravitomagnetism can affect orbits in the gravitational field of a massive, rotating body. This raises the possibility that gravitomagnetism may serve as the dominant physics behind the anomalous rotation curves of spiral galaxies, eliminating the need for dark matter. In this essay, we methodically work out the magnitude of the gravitomagnetic equivalent of the Lorentz force and apply the result to the Milky Way. We find that the resulting contribution is too small to produce an observable effect on these orbits. We also investigate the impact of cosmological boundary conditions on the result and find that these, too, are negligible., Comment: 5 pages, no figures (accepted for publication in Int. J. Mod. Phys. D)

We investigate the possibility of determining the mass distribution of a gravitational lens via lensing observations. We consider an extended, compact gravitational lens, representing its static external gravitational potential via an infinite set of symmetric trace free (STF) multipole moments. Within the wave-optical treatment, we evaluate the caustics formed in the lens's point spread function (PSF). We study the only quantity that is available in astronomical lensing observations: the image of that PSF formed by an imaging telescope. This observable may be used to recover some physical characteristics of the lens, including its shape, orientation and composition. Illustrating this, we study exotic gravitational lenses formed by several well-known solids with uniform density. We show that when moments beyond the quadrupole are observed, some of the symmetry properties of the lens can be recovered. The presence of an octupole moment implies breaking the "north-south" symmetry of the mass distribution in the lens. The presence of a rotated hexadecapole moment implies breaking axial symmetry. As such, if observations of lensed images allow the reconstruction of these moments, important information about the mass distribution and dynamics of the lens can be obtained. This may help with choosing the most appropriate mass profile that is used to characterize the mass distribution of astrophysical lenses, such as the dark matter halos that are presumed to contain most of the mass of galaxies and clusters of galaxies. Our results are novel and offer new insight into gravitational lensing by realistic astrophysical systems., Comment: 21 pages, 11 figures, 1 table

We study gravitational lensing by a generic extended mass distribution. For that, we consider the diffraction of electromagnetic (EM) waves by an extended, weakly aspherical, gravitating object. We account for the static gravitational field of such a lens by representing its exterior potential in the most generic form, expressed via an infinite set of symmetric trace free (STF) tensor multipole mass moments. This yields the most general form of the gravitational phase shift, which allows for a comprehensive description of the optical properties of a generic gravitational lens. We found that at each order of the STF moments, the gravitational phase shift is characterized by only two parameters: a magnitude and a rotation angle that characterize the corresponding caustics, which form in the point spread function (PSF) of the lens. Both of these parameters are uniquely expressed in terms of the transverse-trace free (TT) projections of the multipole moments on the lens plane. Not only does this result simplify the development of physically consistent models of realistic lenses, it also drastically reduces the number of required parameters in the ultimate model. To help with the interpretation of the results, we established the correspondence of the gravitational phase shift expressed via the TT-projected STF multipole mass moments and its representation via spherical harmonics. For axisymmetric mass distributions, the new results are consistent with those that we obtained in previous studies. For arbitrary mass distributions, our results are novel and offer new insight into gravitational lensing by realistic astrophysical systems. These findings are discussed in the context of ongoing astrophysical gravitational lensing investigations as well as observations that are planned with the solar gravitational lens (SGL)., Comment: 21 pages, 1 figure

We continue our investigation of the optical properties of the solar gravitational lens (SGL). We treat the Sun as an extended axisymmetric body and model its gravitational field using zonal harmonics. We consider a point source that is positioned at a large but finite distance from the Sun and, using our new angular eikonal method, we established the electro-magnetic (EM) field on the image plane in the focal region behind the SGL and derive the SGL's impulse response in the form of its point-spread function (PSF). The expression that we derive describes the extended Sun in all regions of interest, including the regions of strong and weak interference and the region of geometric optics. The result is in the form of a single integral with respect to the azimuthal angle of the impact parameter, covering all lensing regimes of the SGL. The same expression can be used to describe gravitational lensing by a compact axisymmetric mass distribution, characterized by small deviations from spherical symmetry. It is valid in all lensing regimes. We also derive results that describe the intensity of light observed by an imaging telescope in the focal region. We present results of numerical simulations showing the view by a telescope that moves in the image plane toward the optical axis. We consider imaging of both point and extended sources. We show that while point sources yield a number of distinct images consistent with the caustics due to zonal harmonics of a particular order (e.g., Einstein cross), extended sources always result in the formation of an Einstein ring. These results represent the most comprehensive wave-theoretical treatment of gravitational lensing in the weak gravitational field of a compact axisymmetric gravitating object., Comment: 22 pages, 5 figures

We study imaging of point sources with a quadrupole gravitational lens while focusing on the formation and evolution of the Einstein cross formed on the image sensor of an imaging telescope. We use a new type of a diffraction integral that we developed to study generic, opaque, weakly aspherical gravitational lenses. To evaluate this integral, we use the method of stationary phase that yields a quartic equation with respect to a Cartesian projection of the observer's position vector with respect to the vector of the impact parameter. The resulting quartic equation can be solved analytically using the method first published by Cardano in 1545. We find that the resulting solution provides a good approximation of the electromagnetic (EM) field almost everywhere in the image plane, yielding the well-known astroid caustic of the quadrupole lens. The sole exception is the immediate vicinity of the caustic boundary, where a numerical treatment of the diffraction integral yields better results. We also convolve the quartic solution for the EM field on the image plane with the point-spread function of a thin lens imaging telescope. By doing so, we are able to explore the direct relationship between the algebraic properties of the quartic solution for the EM field, the geometry of the astroid caustic, and the geometry and shape of the resulting Einstein-cross that appear on the image plane of the thin lens telescope. The new quartic solution leads to significant improvements in numerical modeling as evaluation of this solution is computationally far less expensive than a direct numerical treatment of the new diffraction integral. In the case of the solar gravitational lens (SGL), the new results drastically improve the speed of numerical simulations related to sensitivity analysis performed in the context of high-resolution imaging of exoplanets., Comment: 23 pages, 10 figures

We study the optical properties of the solar gravitational lens (SGL) while treating the Sun as an extended, axisymmetric and rotating body. The gravitational field of the Sun is represented using a set of zonal harmonics. We develop an analytical description of the intensity of light that is observed in the image plane in the strong interference region of a realistic SGL. This formalism makes it possible to model not only the point-spread function of point sources, but also actual observables, images that form in the focal plane of an imaging telescope positioned in the image plane. Perturbations of the monopole gravitational field of the Sun are dominated by the solar quadrupole moment, which results in forming an astroid caustic on the image plane. Consequently, an imaging telescope placed inside the astroid caustic observes four bright spots, forming the well-known pattern of an Einstein cross. The relative intensities and positions of these spots change as the telescope is moved in the image plane, with spots merging into bright arcs when the telescope approaches the caustic boundary. Outside the astroid caustic, only two spots remain and the observed pattern eventually becomes indistinguishable from the imaging pattern of a monopole lens at greater distances from the optical axis. We present results from extensive numerical simulations, forming the basis of our ongoing study of prospective exoplanet imaging with the SGL. These results are also applicable to describe a large class of gravitational lensing scenarios involving axisymmetric lenses that can be represented using zonal harmonics., Comment: 13 pages, 8 figures

We continue to study the optical properties of the solar gravitational lens (SGL). The aim is prospective applications of the SGL for imaging purposes. We investigate the solution of Maxwell's equations for the electromagnetic (EM) field, obtained on the background of a static gravitational field of the Sun. We now treat the Sun as an extended body with a gravitational field that can be described using an infinite series of gravitational multipole moments. Studying the propagation of monochromatic EM waves in this extended solar gravitational field, we develop a wave-optical treatment of the SGL that allows us to study the caustics formed in an image plane in the SGL's strong interference region. We investigate the EM field in several important regions, namely i) the area in the inner part of the caustic and close to the optical axis, ii) the region outside the caustic, and iii) the region in the immediate vicinity of the caustic, especially around its cusps and folds. We show that in the first two regions the physical behavior of the EM field may be understood using the method of stationary phase. However, in the immediate vicinity of the caustic the method of stationary phase is inadequate and a wave-optical treatment is necessary. Relying on the angular eikonal method, we develop a new approach to describe the EM field accurately in all regions, including the immediate vicinity of the caustics and especially near the cusps and folds. The method allows us to investigate the EM field in this important region, which is characterized by rapidly oscillating behavior. Our results are new and can be used to describe gravitational lensing by realistic astrophysical objects, such as stars, spiral and elliptical galaxies., Comment: 25 pages, 10 figures

We continue our study of the optical properties of the solar gravitational lens (SGL). Taking the next step beyond representing it as an idealized monopole, we now characterize the gravitational field of the Sun using an infinite series of multipole moments. We consider the propagation of electromagnetic (EM) waves in this gravitational field within the first post-Newtonian approximation of the general theory of relativity. The problem is formulated within the Mie diffraction theory. We solve Maxwell's equations for the EM wave propagating in the background of a static gravitational field of an extended gravitating body, while accounting for multipole contributions. Using a wave-theoretical approach and the eikonal approximation, we find an exact closed form solution for the Debye potentials and determine the EM field at an image plane in the strong interference region of the lens. The resulting EM field is characterized by a new diffraction integral. We study this solution and show how the presence of multipoles affects the optical properties of the lens, resulting in distinct diffraction patterns. We identify the gravitational deflection angle with the individual contributions due to each of the multipoles. Treating the Sun as an extended, axisymmetric, rotating body, we show that each zonal harmonics causes light to diffract into an area whose boundary is a caustic of a particular shape. The appearance of the caustics modifies the point-spread function (PSF) of the lens, thus affecting its optical properties. The new wave-theoretical solution allows the study gravitational lensing by a realistic lens that possesses an arbitrary number of gravitational multipoles. This {\em angular eikonal method} represents an improved treatment of realistic gravitational lensing. It may be used for a wave-optical description of many astrophysical lenses., Comment: 36 pages, 6 figures. arXiv admin note: text overlap with arXiv:1810.06627

We report on the initial results obtained with an image convolution/deconvolution computer code that we developed and used to study the image formation capabilities of the solar gravitational lens (SGL). Although the SGL of a spherical Sun creates a greatly blurred image, knowledge of the SGL's point-spread function (PSF) makes it possible to reconstruct the original image and remove the blur by way of deconvolution. We discuss the deconvolution process, which can be implemented either with direct matrix inversion or with the Fourier quotient method. We observe that the process introduces a ``penalty'' in the form of a reduction in the signal-to-noise ratio (SNR) of a recovered image, compared to the SNR at which the blurred image data is collected. We estimate the magnitude of this penalty using an analytical approach and confirm the results with a series of numerical simulations. We find that the penalty is substantially reduced when the spacing between image samples is large compared to the telescope aperture. The penalty can be further reduced with suitable noise filtering, which can yield ${\cal O}(10)$ or better improvement for low-quality imaging data. Our results confirm that it is possible to use the SGL for imaging purposes. We offer insights on the data collection and image processing strategies that could yield a detailed image of an exoplanet within image data collection times that are consistent with the duration of a realistic space mission., Comment: 11 pages, 5 figures

We discuss the in-flight autonomous assembly as the means to build advanced planetary science payloads to explore the outer regions of the solar system. These payloads are robotically constructed from modular parts delivered by a group of smallsats (< 20 kg) which are placed on fast solar system transfer trajectories while being accelerated by solar sail propulsion to velocities of ~10 AU/yr. This concept provides the planetary science community with inexpensive, frequent access to distant regions of the solar system with flexible, reconfigurable instruments and systems that are assembled in flight. It permits faster revisit times, rapid replenishment and technology insertions, longer mission capability with lower costs. It also increases the science capabilities of smallsats via the use of modular, redundant architectures and allows for proliferation of sensing instrumentation throughout the solar system., Comment: A White Paper to the National Academy of Sciences Planetary Science and Astrobiology Decadal Survey 2023-2032. 7 pages

We examined the solar gravitational lens (SGL) as the means to produce direct high-resolution, multipixel images of exoplanets. The properties of the SGL are remarkable: it offers maximum light amplification of ~1e11 and angular resolution of ~1e-10 arcsec. A probe with a 1-m telescope in the SGL focal region can image an exoplanet at 30 pc with 10-kilometer resolution on its surface, sufficient to observe seasonal changes, oceans, continents, surface topography. We reached and exceeded all objectives set for our study: We developed a new wave-optical approach to study the imaging of exoplanets while treating them as extended, resolved, faint sources at large but finite distances. We properly accounted for the solar corona brightness. We developed deconvolution algorithms and demonstrated the feasibility of high-quality image reconstruction under realistic conditions. We have proven that multipixel imaging and spectroscopy of exoplanets with the SGL are feasible. We have developed a new mission concept that delivers an array of optical telescopes to the SGL focal region relying on three innovations: i) a new way to enable direct exoplanet imaging, ii) use of smallsats solar sails fast transit through the solar system and beyond, iii) an open architecture to take advantage of swarm technology. This approach enables entirely new missions, providing a great leap in capabilities for NASA and the greater aerospace community. Our results are encouraging as they lead to a realistic design for a mission that will be able to make direct resolved images of exoplanets in our stellar neighborhood. It could allow exploration of exoplanets relying on the SGL capabilities decades, if not centuries, earlier than possible with other extant technologies. The architecture and mission concepts for a mission to the strong interference region of the SGL are promising and should be explored further., Comment: Final Report for the NASA's Innovative Advanced Concepts (NIAC) Phase II proposal. 91 pages, 50 figures, 5 tables

We study the image formation process with the solar gravitational lens (SGL) in the case of an extended, resolved source. An imaging telescope, modeled as a convex lens, is positioned within the image cylinder formed by the light received from the source. In the strong interference region of the SGL, this light is greatly amplified, forming the Einstein ring around the Sun, representing a distorted image of the extended source. We study the intensity distribution within the Einstein ring observed in the focal plane of the convex lens. For any particular telescope position in the image plane, we model light received from the resolved source as a combination of two signals: light received from the directly imaged region of the source and light from the rest of the source. We also consider the case when the telescope points away from the extended source or, equivalently, it observes light from sources in sky positions that are some distance away from the extended source, but still in its proximity. At even larger distances from the optical axis, in the weak interference or geometric optics regions, our approach recovers known models related to microlensing, but now obtained via the wave-optical treatment. We then derive the power of the signal and related photon fluxes within the annulus that contains the Einstein ring of the extended source, as seen by the imaging telescope. We discuss the properties of the deconvolution process, especially its effects on noise in the recovered image. We compare anticipated signals from realistic exoplanetary targets against estimates of noise from the solar corona and estimate integration times needed for the recovery of high-quality images of faint sources. The results demonstrate that the SGL offers a unique, realistic capability to obtain resolved images of exoplanets in our galactic neighborhood., Comment: 37 pages, 15 figures

We present a statistical study on the orientation of outflows with respect to large-scale filaments and the magnetic fields. Although filaments are widely observed toward Galactic star-forming regions, the exact role of filaments in star formation is unclear. Studies toward low-mass star-forming regions revealed both preferred and random orientation of outflows respective to the filament long-axes, while outflows in massive star-forming regions mostly oriented perpendicular to the host filaments, and parallel to the magnetic fields at similar physical scales. Here, we explore outflows in a sample of 11 protoclusters in HII regions, a more evolved stage compared to IRDCs, using ALMA CO (3-2) line observations. We identify a total of 105 outflow lobes in these protoclusters. Among the 11 targets, 7 are embedded within parsec-scale filamentary structures detected in $^{13}$CO line and 870 $\mu m$ continuum emissions. The angles between outflow axes and corresponding filaments ($\gamma_\mathrm{Fil}$) do not show any hint of preferred orientations (i.e., orthogonal or parallel as inferred in numerical models) with respect to the position angle of the filaments. Identified outflow lobes are also not correlated with the magnetic fields and Galactic plane position angles. Outflows associated with filaments aligned along the large-scale magnetic fields are also randomly orientated. Our study presents the first statistical results of outflow orientation respective to large-scale filaments and magnetic fields in evolved massive star-forming regions. The random distribution suggests a lack of alignment of outflows with filaments, which may be a result of the evolutionary stage of the clusters., Comment: 18 pages, 2 tables, 6 figures (3 additional figures in Appendix); Accepted for publication in the Astrophysical Journal

We study image formation with the solar gravitational lens (SGL). We consider a point source that is positioned at a large but finite distance from the Sun. We assume that an optical telescope is positioned in the image plane, in the focal region of the SGL. We model the telescope as a convex lens and evaluate the intensity distribution produced by the electromagnetic field that forms the image in the focal plane of the convex lens. We first investigate the case when the telescope is located on the optical axis of the SGL or in its immediate vicinity. This is the region of strong interference where the SGL forms an image of a distant source, which is our primary interest. We derive analytic expressions that describe the progression of the image from an Einstein ring corresponding to an on-axis telescope position, to the case of two bright spots when the telescope is positioned some distance away from the optical axis. At greater distances from the optical axis, in the region of weak interference and that of geometric optics, we recover expressions that are familiar from models of gravitational microlensing, but developed here using a wave-optical treatment. We discuss applications of the results for imaging and spectroscopy of exoplanets with the SGL., Comment: 13 pages, 6 figures

We discuss the optical properties of the solar gravitational lens (SGL). We estimate the power of the EM field received by an imaging telescope. Studying the behavior of the EM field at the photometric detector, we develop expressions that describe the received power from a point source as well as from an extended resolved source. We model the source as a disk with uniform surface brightness and study the contribution of blur to a particular image pixel. To describe this process, we develop expressions describing the power received from the directly imaged region of the exoplanet, from the rest of the exoplanet and also the power for off-image pointing. We study the SGL's amplification and its angular resolution in the case of observing an extended source with a modest size telescope. The results can be applied to direct imaging of exoplanets using the SGL., Comment: 11 pages, 4 figures

We investigate the optical properties of the solar gravitational lens (SGL) with respect to an extended source located at a large but finite distance from the Sun. The static, spherically symmetric gravitational field of the Sun is modeled within the first post-Newtonian approximation of the general theory of relativity. We consider the propagation of monochromatic electromagnetic (EM) waves near the Sun. We develop, based on a Mie theory, a vector theory of diffraction that accounts for the refractive properties of the solar gravitational field. The finite distance to a point source can be accounted for using a rotation of the coordinate system to align its polar axis with the axis directed from the point source to the center of the Sun, which we call the optical axis. We determine the EM field and study the key optical properties of the SGL in all four regions formed behind the Sun by an EM wave diffracted by the solar gravity field: the shadow, geometric optics, and weak and strong interference regions. Extended sources can then be represented as collections of point sources. We present the power density of the signal received by a telescope in the image plane. Our discussion concludes with considering the implications for imaging with the SGL., Comment: 25 pages, 11 figures

Tens of millions of people live blind, and their number is ever increasing. Visual-to-auditory sensory substitution (SS) encompasses a family of cheap, generic solutions to assist the visually impaired by conveying visual information through sound. The required SS training is lengthy: months of effort is necessary to reach a practical level of adaptation. There are two reasons for the tedious training process: the elongated substituting audio signal, and the disregard for the compressive characteristics of the human hearing system. To overcome these obstacles, we developed a novel class of SS methods, by training deep recurrent autoencoders for image-to-sound conversion. We successfully trained deep learning models on different datasets to execute visual-to-auditory stimulus conversion. By constraining the visual space, we demonstrated the viability of shortened substituting audio signals, while proposing mechanisms, such as the integration of computational hearing models, to optimally convey visual features in the substituting stimulus as perceptually discernible auditory components. We tested our approach in two separate cases. In the first experiment, the author went blindfolded for 5 days, while performing SS training on hand posture discrimination. The second experiment assessed the accuracy of reaching movements towards objects on a table. In both test cases, above-chance-level accuracy was attained after a few hours of training. Our novel SS architecture broadens the horizon of rehabilitation methods engineered for the visually impaired. Further improvements on the proposed model shall yield hastened rehabilitation of the blind and a wider adaptation of SS devices as a consequence.

We investigate the optical properties of the solar gravitational lens (SGL) in the presence of the solar corona. For this, we consider the combined influence of the static spherically symmetric gravitational field of the Sun---modeled within the first post-Newtonian approximation of the general theory of relativity---and of the solar corona---modeled as a generic, static, spherically symmetric free electron plasma. We study the propagation of monochromatic electromagnetic (EM) waves through the solar system and develop a Mie theory that accounts for the refractive properties of the gravitational field of the Sun and that of the free electron plasma in the extended solar system. We establish a compact, closed-form solution to the boundary value problem and demonstrate that the presence of the solar plasma affects all characteristics of an incident unpolarized light. The affected properties include the direction of the EM wave propagation, its amplitude and its phase, leading to a reduction of the light amplification of the SGL and to a broadening of the corresponding point spread function. The wavelength-dependent plasma effect is critically important at radio frequencies, where it drastically reduces both the amplification factor of the SGL and also its angular resolution. However, for optical and shorter wavelengths, the plasma's contribution to the EM wave leaves the plasma-free optical properties of the SGL practically unaffected. We discuss the applicability of the SGL for direct high-resolution multipixel imaging and spatially-resolved spectroscopy of exoplanets, Comment: 16 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1810.06627

We study the optical properties of the solar gravitational lens (SGL) under the combined influence of the static spherically symmetric gravitational field of the Sun---modeled within the first post-Newtonian approximation of the general theory of relativity---and of the solar corona---modeled as a generic, steady-state, spherically symmetric free electron plasma. For this, we consider the propagation of monochromatic electromagnetic (EM) waves near the Sun and develop a Mie theory that accounts for the refractive properties of the gravitational field of the Sun and that of the free electron plasma in the extended solar system. We establish a compact, closed-form solution to the boundary value problem, which extends previously known results into the new regime where gravity and plasma are both present. Relying on the wave-optical approach, we consider three different regions of practical importance for the SGL, including the shadow region directly behind the Sun, the region of geometrical optics and the interference region. We demonstrate that the presence of the solar plasma affects all characteristics of an incident unpolarized light, including the direction of the EM wave propagation, its amplitude and its phase. We show that the presence of the solar plasma leads to a reduction of the light amplification of the SGL and to a broadening of its point spread function. We also show that the wavelength-dependent plasma effect is important at radio frequencies, where it drastically reduces both the amplification factor of the SGL and also its angular resolution. However, for optical and shorter wavelengths, the plasma's contribution to the EM wave is negligibly small, leaving the plasma-free optical properties of the SGL practically unaffected., Comment: 54 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1805.00398

We study the shadow cast by a large gravitating sphere, similar to our Sun. For this, we consider the gravitational field produced by a static mass monopole within the first post-Newtonian approximation of the general theory of relativity. We study the propagation of a monochromatic electromagnetic wave in the vicinity of a large, opaque, gravitating sphere. To treat the opaque nature of the body and its physical size, we implement fully absorbing boundary conditions and develop a wave-optical treatment of the shadow formed by this object. Based on this approach, we demonstrate that the structure of the shadow is determined by the Schwarzschild radius of the body and its physical size. The shadow's boundary has the shape of a concave rotational hyperboloid bent inward due to the refractive properties of the curved spacetime. We show that there is no light in the shadow. However, even in the presence of gravity and related gravitational bending of photon trajectories there is the bright spot of Arago that is formed behind a perfectly spherical obscuration., Comment: 16 pages, 3 figures

We study the propagation of electromagnetic (EM) waves in the solar system and develop a Mie theory that accounts for the refractive properties of the free electron plasma in the extended solar corona. We use a generic model for the electron number density distribution and apply the eikonal approximation to find a solution in terms of Debye potentials, which is then used to determine the EM field both within the inner solar system and at large heliocentric distances. As expected, the solution for the EM wave propagating through the solar system is characterized by a plasma-induced phase shift and related change in the light ray's direction of propagation. Our approach quantitatively accounts for these effects, providing a wave-optical treatment for diffraction in the solar plasma. As such, it may be used in practical applications involving big apertures, large interferometric baselines or otherwise widely distributed high-precision astronomical instruments., Comment: 33 pages, 1 figure