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Since last two decades $f(R)$ gravity theory has been extensively used as a serious alternative of general relativity to mimic the effects of dark energy. The theory presents a Yukawa correction to Newtonian gravitational potential, acting as a fifth force of Nature. Generally speaking, this new force is mediated by a scalar field known as scalaron. It affects orbital dynamics of test bodies around a central mass. When the scalaron becomes extremely massive $f(R)$ gravity reduces to Newtonian theory in the weak field limit. In this paper we test $f(R)$ gravity theory in the solar system by constraining scalaron mass through existing measurements of perihelion shift of planets, cassini's measurement of the Parameterised Post Newtonian parameter and measurement of the Brans-Dicke coupling constant. We calculate acceleration due to gravity in the theory for planets, Trans Neptunian Objects (TNOs), Centaurs, Scattered Disk Objects (SDOs) and Oort cloud objects and compare it with the values predicted by Newtonian and Modified Newtonian Dynamics (MOND). It is found that the theory reproduces to MOND like acceleration in the outer solar system ($r_p \sim 2000$ au - $3000$ au) for available interpolating functions of the MOND paradigm. From its MOND equivalence we constrain the parameters of the theory. Our results are consistent with existing constraints on the theory arising from the environment of the Galactic Centre black hole. Scalarons realized in the solar system are reproduced in the radiation era of the universe with a time varying mass., Comment: 17 pages, 9 figures, 4 tables, Comments are welcomed

Astronomical tests of spacetime metric and gravitation theory near the Galactic Center (GC) black hole, Sgr A* have gained momentum with the observations of compact stellar orbits near the black hole and measurement of the black hole shadow. Deviation from the Kerr metric is a potential signature of modified gravity theory. In this work, we use Newman-Janis algorithm to construct an axially symmetric and asymptotically flat metric in f(R) scalaron gravity theory. We call it as Kerr-scalaron metric. For studying astronomical consequences of the new metric we use the compact stellar orbits and the black hole shadow. We use the observed size of the emission ring of the GC black hole shadow for estimating deviation of the new metric from general relativity. It has been found that scalarons with mass within $10^{-17}$ eV - $10^{-16}$ eV are compatible with the observed emission ring size for black hole spin $\chi=0.9$. Schwarzschild limit of the pericenter shift is estimated for compact stellar orbits near the black hole. General relativistic pericenter shift in wider orbits including S-stars such as S4716 and S2 has been reproduced with these scalarons. The parameter $f_{SP}$ measuring deviation from Schwarzschild pericenter shift has been found as $f_{SP}=1.00-1.04$ within stellar orbits having semi-major axes $45$ au - $100$ au. Scalarons have the capability to dominate Schwarzschild precession for orbits much below $45$ au. Lense-Thirring (LT) precession with the new metric is estimated for the compact orbits. The massive scalarons produce LT precession with magnitude ($12.25-24.5$) $\mu$as/yr in the orbit of S2. The LT precession time scale is within $0.1$% of the age of the S-stars., Comment: 22 pages, 14 figures, Accepted for publication in ApJ