Probing the Schwarzschild black hole in modified gravity by precessing motion of S2 star and OJ 287

Abstract We investigate the precessing motion around the Schwarzschild black hole in modified gravity (MOG), an alternative gravitational theory that introduces a dimensionless coupling parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> to modify general relativity and describe gravitational correction effects. We obtain the first-order post-Newtonian quasi-Keplerian motion of a test particle in this spacetime and find its relativistic periastron advance. Then, we test this model with the precessing motion of the star S2 and of OJ 287 and determine the upper limits of the dimensionless coupling parameter as $$\alpha \sim 10^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Compared with the best-fitting result given by the galaxy rotation curves as $$\alpha =8.89\pm 0.34$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8.89</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.34</mml:mn> </mml:mrow> </mml:math> , our bound on $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> is reduced by one order of magnitude.