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Register a new userImaging point sources with the gravitational lens of an extended Sun
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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.
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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.
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 Maxwells 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 SGLs 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.
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 Maxwells 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.
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.
We review current best estimates of the strength and detectability of the gravitational waves from a variety of sources, for both ground-based and space-based detectors, and we describe the information carried by the waves.
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