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.
We consider scattering of electromagnetic waves from a distant point source by the gravitational field of the sun, taking the field oblateness due to the quadrupole moment of the sun into account. Effects of the field oblateness can play an important role in the high resolution solar gravitational lens imaging in the sub-micrometer wavelength range of the electromagnetic spectrum.
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 res…
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 …
We present an analysis of high-resolution imaging of an exoplanet by a meter-class telescope positioned at a real image of the exoplanet created by the solar gravity lens. We assume an exoplanet viewed in full phase and a simple deconvolution method to correct for the intrinsic blur caused by aberrations in the solar gravity lens, and account for the foreground light of the solar corona. We derive equations for the measurement times required for the telescope to produce such a high-resolution image, and find that due to shot noise considerations these times are large compared to human lifespan…
Can a gravitational lens magnify gravity? Leading theories of gravity likely indicate "no", but the possibility might be testable by using our Sun as a gravitational lens and carefully aligning a satellite past the minimum transparent focal distance of about 25 AU. Here the magnitude of a maximal effect is estimated and found potentially observable.
In this talk, we review the empirical status for modern gravitational theories with emphases on (i) Equivalence Principles; (ii) Lense-Thirring effects and the implications of Gravity Probe B experiment; (iii) Solar-System Tests of Cosmological Models.
We discuss the propagation of electromagnetic (EM) waves in the post-Newtonian approximation of the general theory of relativity. We consider diffraction of EM waves in the static gravitational field of a massive monopole. We develop a wave-theoretical description of the solar gravitational lens (SGL) and show that with its enormous magnifying power of $\sim 10^{11}$ (for $λ=1~μ$m) and angular resolution of $\lesssim 10^{-10}$ arcsec, the SGL may be used for direct megapixel imaging of an exoplanet.
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 ac…
Solar-Limb Effect is an observational phenomena connected to the solar gravitational red-shift. It shows a variation of the magnitude of the gravitational red-shift from the center to the limb of the solar disc. In the present work an attempt, for interpreting this phenomena using a general relativistic red-shift formula, is given . This formula takes into account the effect of the the Sun's gravitational field, the effect of the solar rotation, the effect of inclination of the line of sight and the motion of the observer. In this study the gravitational field of the Sun is assumed to be given…
We estimate the changes in solar system tests of general relativity when "dark energy" arises from a Weyl scaling invariant action, using the modified Schwarzschild-like spherically symmetric solution obtained in this case by Adler and Ramazano\vglu. We show that the standard results of general relativity are modified by amounts that are far below current experimental errors. However, because the metric is not conformally flat when the central mass $M$ vanishes, there is a contribution to light deflection proportional to the cosmological constant $Λ$, with opposite sign to the one proportional…
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 gravitation…
The Solar Gravitational Lens (SGL) allows for major brightness amplification ($\sim 10^{11}$ at wavelength of $1~μ$m) and extreme angular resolution ($\sim10^{-10}$ arcsec) within a narrow field of view. A meter-class telescope, with a modest coronagraph to block solar light with 1e-6 suppression placed in the focal area of the SGL, can image an exoplanet at a distance of 30 parsec with few kilometer-scale resolution on its surface. Notably, spectroscopic broadband SNR is $\sim 10^{-6}$ in two weeks of integration time, providing this instrument with incredible remote sensing capabilities. A m…
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, w…
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 wavelengt…
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 exoplanet…
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 s…
The remarkable optical properties of the solar gravitational lens (SGL) include major brightness amplification (~1e11 at wavelength of 1 um) and extreme angular resolution (~1e-10 arcsec) in a narrow field of view. A mission to the SGL carrying a modest telescope and coronagraph opens up a possibility for direct megapixel imaging and high-resolution spectroscopy of a habitable Earth-like exoplanet at a distance of up to 100 light years. The entire image of such a planet is compressed by the SGL into a region with a diameter of ~1.3 km in the vicinity of the focal line. The telescope, acting as…
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 ev…
MOND predicts that the asymptotic gravitational potential of an isolated, bounded (baryonic) mass, M, is phi(r)=(MGa0)^{1/2}ln(r); a0 is the MOND constant. Relativistic MOND theories predict that the lensing effects of M are dictated by phi(r) as general-relativity lensing is dictated by the Newtonian potential. Thus, MOND predicts that the asymptotic Newtonian potential deduced from galaxy-galaxy gravitational lensing will have: (1) a logarithmic r dependence, and (2) a normalization (parametrized standardly as 2s^2) that depends only on M: s=(MGa0/4)^{1/4}. I compare these predictions with r…
We propose a class of graded coronagraphic "amplitude" image masks for a high throughput Lyot-type coronagraph that transmits light from an annular region around an extended source and suppresses light, with extremely high ratio, from elsewhere. The interior radius of the region is comparable with its exterior radius. The masks are designed using an idea inspired by approach due M.J. Kuchner and W.A. Traub ("band-limited" masks) and approach to optimal apodization by D.Slepian. One potential application of our masks is direct high-resolution imaging of exo-planets with the help of the Solar Gr…
We investigate the weak lensing effects of line-of-sight structures on quadruple images in quasar-galaxy strong lens systems based on N-body and ray-tracing simulations that can resolve halos with a mass of 10^5 solar mass. The intervening halos and voids disturb the magnification ratios of lensed images as well as their relative positions due to lensing. The magnification ratios typically change by O(10%) when the shifts of relative angular positions of lensed images are constrained to <0.004 arcsec. The constrained amplitudes of projected density perturbations due to line-of-sight structures…
The mystery of dark energy suggests that there is new gravitational physics on long length scales. Yet light degrees of freedom in gravity are strictly limited by Solar System observations. We can resolve this apparent contradiction by adding a Galilean-invariant scalar field to gravity. Called Galileons, these scalars have strong self-interactions near overdensities, like the Solar System, that suppress their dynamical effect. These nonlinearities are weak on cosmological scales, permitting new physics to operate. In this Letter, we point out that a massive gravity inspired coupling of Galile…
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. Perturbat…
Many applications and algorithms in the field of gravitational lensing make use of meshes with a finite number of nodes to analyze and manipulate data. Specific examples in lensing are astronomical CCD images in general, the reconstruction of density distributions from lensing data, lens-source plane mapping or the characterization and interpolation of a point-spread-function. We present a numerical framework to interpolate and differentiate in the mesh-free domain, defined by nodes with coordinates that follow no regular pattern. The framework is based on radial basis functions (RBFs) to smoo…
We demonstrate how to construct purely metric modifications of gravity which agree with general relativity in the weak field regime appropriate to the solar system, but which possess an ultra-weak field regime when the gravitational acceleration becomes comparable to $a_0 \sim 10^{-10} {\rm m/s}^2$. In this ultra-weak field regime, the models reproduce the MOND force without dark matter and also give enough gravitational lensing to be consistent with existing data. Our models are nonlocal and might conceivably derive from quantum corrections to the effective field equations.
The quadruple image configurations of gravitational lenses with vanishing ellipticity are examined. Even though such lenses asymptotically approach circularity, the configurations are stable if the position of the source relative to the vanishing diamond caustic is held constant. The configurations are the solutions of a quartic equation, an "Asymptotically Circular Lens Equation" (ACLE), parameterized by a single complex quantity. Several alternative parameterizations are examined. Relative magnifications of the images are derived. When a non-vanishing quadrupole, in the form of an external s…
