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|a The prospect of combining integral field spectroscopy with the solar gravitational lens (SGL) to spectrally and spatially resolve the surfaces and atmospheres of extrasolar planets is investigated. The properties of hyperbolic orbits visiting the focal region of the SGL are calculated analytically, demonstrating trade-offs between departure velocity and time of arrival, as well as gravity assist maneuvers and heliocentric angular velocity. Numerical integration of the solar barycentric motion demonstrates that navigational acceleration of dv 280ms+6.7mstyear is needed to obtain and maintain alignment. Obtaining target ephemerides of sufficient precision is an open problem. The optical properties of an oblate gravitational lens are reviewed, including calculations of the magnification and the point-spread function that forms inside a telescope. Image formation for extended, incoherent sources is discussed when the projected image is smaller than, approximately equal to, and larger than the critical caustic. Sources of contamination that limit observational signal-to-noise ratio (S/N) are considered in detail, including the Sun, the solar corona, the host star, and potential background objects. A noise mitigation strategy of spectrally and spatially separating the light using integral field spectroscopy is emphasized. A pseudo-inverse-based image reconstruction scheme demonstrates that direct reconstruction of an Earth-like source from single measurements of the Einstein ring is possible when the critical caustic and observed S/N are sufficiently large. In this arrangement, a mission would not require multiple telescopes or navigational symmetry breaking, enabling continuous monitoring of the atmospheric composition and dynamics on other planets. © 2022. The Author(s). Published by the American Astronomical Society.
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