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Perturbation theory for gravitational shadows in Kerr-like spacetimes

We present a fully analytical method for calculating the key parameters of a Kerr-like gravitational shadow, including its horizontal and vertical diameters, $D_X$ and $D_Y$, the coordinates of its center $X_{C}$, the average radius $\bar{R}$, the deviation from sphericity $δC$, and the mean deviation from the Kerr shadow $δK$. Developed within the framework of perturbation theory, this approach yields all characteristic parameters as simple polynomial expressions with an accuracy of $\sim a^5$, where $a$ is the Kerr spin parameter. This eliminates the need for repeated numerical integration of cumbersome parametric equations. Furthermore, our derived formulas account for the effects of a plasma medium - a feature of particular relevance given the prospect of multi-frequency astrophysical observations.