On the parameters of the spherically symmetric parametrized Rezzolla-Zhidenko spacetime through solar system tests, orbit of S2 star about Sgr A$^\star$, and quasiperiodic oscillations

In this paper, we find the higher order expansion parameters $\alpha$ and $\lambda$ of spherically symmetric parametrized Rezzolla--Zhidenko (PRZ) spacetime by using its functions of the radial coordinate. We subject the parameters of this spacetime to classical tests including weak gravitational field effects in Solar System, observations of the S2 star located in the star cluster close to the Sgr A$^{\star}$ and of the frequencies of selected microquasars. Based on this spherically symmetric spacetime we perform the analytic calculations for Solar System effects like perihelion shift, light deflection, and gravitational time delay so as to determine limits on the parameters by using observational data. We restrict our attention to the limits on the two higher order expansion parameters $\alpha$ and $\lambda$ that survive at the horizon or near the horizon of spherically symmetric metrics. The properties of these two small parameters expansion in PRZ parametrization are discussed. We further apply the Monte Carlo Markov Chain (MCMC) simulations to analyze and obtain the limits on the expansion parameters by using observations of phenomena of the S2 star. Finally, we consider the epicyclic motions and derive analytic expressions of the epicyclic frequencies. Applying these expressions to the quasiperiodic oscillations (QPOs) of selected microquasars allows us to set further limits on the parameters of the PRZ spacetime. Our results demonstrate that the higher order expansion parameters can be given in the range $\alpha\, ,\lambda=(-0.09\, , 0.09)$ and of order $\sim 10^{-2}$ as a consequence of three various tests and observations.