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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 theparameters of this spacetime to classical tests including weak gravitationalfield effects in Solar System, observations of the S2 star located in the starcluster close to the Sgr A$^{\star}$ and of the frequencies of selectedmicroquasars. Based on this spherically symmetric spacetime we perform theanalytic calculations for Solar System effects like perihelion shift, lightdeflection, and gravitational time delay so as to determine limits on theparameters by using observational data. We restrict our attention to the limitson the two higher order expansion parameters $\alpha$ and $\lambda$ thatsurvive at the horizon or near the horizon of spherically symmetric metrics.The properties of these two small parameters expansion in PRZ parametrizationare discussed. We further apply the Monte Carlo Markov Chain (MCMC) simulationsto analyze and obtain the limits on the expansion parameters by usingobservations of phenomena of the S2 star. Finally, we consider the epicyclicmotions and derive analytic expressions of the epicyclic frequencies. Applyingthese expressions to the quasiperiodic oscillations (QPOs) of selectedmicroquasars allows us to set further limits on the parameters of the PRZspacetime. Our results demonstrate that the higher order expansion parameterscan 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.

astrophysical journal/the astrophysical journal

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