Analytical solutions of the geodesic equation in the space-time of a black hole surrounded by perfect fluid in Rastall theory.

In this paper, we investigate the geodesic motion of massive and massless test particles in the vicinity of a black hole space-time surrounded by perfect fluid (quintessence, dust, radiation, cosmological constant and phantom) in Rastall theory. We obtain the full set of analytical solutions of the geodesic equation of motion in the space-time of this black hole. For all cases of perfect fluid, we consider some different values of Rastall coupling constant $k\lambda$ so that the equations of motion have integer powers of $\tilde{r}$ and also can be solved analytically. These analytical solutions are presented in the form of elliptic and also hyperelliptic functions. In addition, using obtained analytical solution and also figures of effective potential and $L-E^2$ diagrams, we plot some examples of possibles orbits. moreover we use of the angular momentum, conserved energy, electrical charge and also Rastall parameter, to classify the different types of the possible gained orbits. Moreover, we show that when Rastall field structure constant becomes zero ($N=0$) our results are consistent with the analysis of a Reissner-Nordstr\"om black hole, however when both Rastall geometric parameter and electric charge vanish $(N=Q=0)$, the metric and results are same as analysis of a Schwarzschild black hole.