The objective of this paper is to discuss anisotropic solutions representing static spherical self-gravitating systems in f(R) theory. We employ the extended gravitational decoupling approach and transform temporal as well as radial metric potentials which decomposes the system of non-linear field equations into two arrays: one set corresponding to seed source and the other one involves additional source terms. The domain of the isotropic solution is extended in the background of f(R) Starobinsky model by employing the metric potentials of Krori–Barua spacetime. We determine two anisotropic solutions by employing some physical constraints on the extra source. The values of unknown constants are computed by matching the interior and exterior spacetimes. We inspect the physical viability, equilibrium and stability of the obtained solutions corresponding to the star Her X-I. It is observed that one of the two extensions satisfies all the necessary physical requirements for particular values of the decoupling parameter.

Thanks for the releasing image of supermassive black holes (BHs) by the event horizon telescope (EHT) at the heart of the M87 galaxy. After the discovery of this mysterious object, scientists paid attention to exploring the BH shadow features under different gravitational backgrounds. In this scenario, we study the light rings and observational properties of BH shadow surrounded by different accretion flow models and then investigate the effect of model parameters on the observational display and space-time structure of BHs in the framework of our considering system. Under the incompatible configuration of the emission profiles, the images of BHs comprise that the observed luminosity is mainly determined by direct emission, while the lensing ring will provide a small contribution of the total observed flux and the photon ring makes a negligible contribution due to its exponential narrowness. More importantly, the observed regions and specific intensities of all emission profiles are changed correspondingly under variations of parameters. For optically thin accreting matters, we analyze the profile and specific intensity of the shadows with static and infalling accretions models, respectively. We find that with an infalling motion the interior region of the shadows will be darker than the static case, due to the Doppler effect of the infalling movement. Finally, it is concluded that these findings support the fact that the change of BH state parameters will change the way of space-time geometry, thus affecting the BH shadow dynamics.

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The light passing near the black hole (BH) is deflected due to the gravitational effect, producing the BH shadow, a dark inner region that is often surrounded by a bright ring, whose optical appearance comes directly from BH’s mass and its angular momentum. We mainly study the shadow and observable features of non-commutative (NC) charged Kiselev BH, surrounded by various profiles of accretions. To obtain the BH shadow profile, we choose specific values of the model parameters and concluded that the variations of each parameter directly vary the light trajectories and size of BH. For thin disk accretion, which includes direct lensing and photon rings emissions, we analyze that the profile of BH contains the dark interior region and bright photon ring. However, their details depends upon the emissions, generally, direct emission plays significant role in the total observed luminosity, while lensing ring has a small contribution and the photon ring makes a negligible contribution, as usual, the latter can be ignored safely. Moreover, we also consider the static and infalling accretion matters and found that the location of the photon sphere is almost the same for both cases. However, the specific intensity which is observed from BH profile found to be darker for infalling accretion case due to the Doppler effect of the infalling motion as compared to the static one.