The study of the quiet Sun in gamma rays started over a decade ago, and rapidly gained a wide interest. Gamma rays from the quiet Sun are produced by Cosmic Rays (CRs) interacting with its surface (disk component) and with its photon field (spatially extended inverse-Compton component, IC). The latter component is maximum close to the Sun and it is above the background even at large angular distances, extending over the whole sky. First detected with EGRET, it is studied now with Fermi-LAT with high statistical significance. Observations of the IC component allow us to obtain information on CR electrons and positrons close to the Sun and in the heliosphere for the various periods of solar activity and polarity. They allow to learn about CR interactions and propagation close to stars, in the heliosphere and on the solar surface, and to understand the Sun itself, its environment, and its activity. Analyses of solar observations are usually model-driven. Hence advances in model calculations and constraints from precise CR measurements are timely and needed. Here we present our StellarICS code to compute the gamma-ray IC emission from the Sun and also from single stars. The code is publicly available and it is extensively used by the scientific community to analyze Fermi-LAT data. It has been used by the Fermi-LAT collaboration to produce the solar models released with the FSSC Fermi Tools. Our modeling provides the basis for analyzing and interpreting high-energy data of the Sun and of stars. After presenting examples of updated solar IC models in the Fermi-LAT energy range that account for the various CR measurements, we extend the models to keV, MeV, and TeV energies for predictions for future possible telescopes such as AMEGO, GECCO, e-ASTROGAM, HAWC, LHAASO, SWGO, and present X-ray telescopes. We also present predictions for some of the closest and most luminous stars.
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