Observational signatures of Schwarzschild-MOG black holes in scalar-tensor-vector gravity: shadows and rings with different accretions

Abstract We study the influence of parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> on the optical features of Schwarzschild-MOG black holes with different thin accretions in scalar-tensor-vector gravity. As $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> increases from 0, the radii of the event horizon, photon sphere, and observed shadow increase in comparison with the Schwarzschild black hole. We constrain the parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> with the experimental data reported by the Event Horizon Telescope Collaboration for M87 $$^{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow/><mml:mrow><mml:mrow/><mml:mo>∗</mml:mo></mml:mrow></mml:msup></mml:math> and Sagittarius A $$^{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow/><mml:mrow><mml:mrow/><mml:mo>∗</mml:mo></mml:mrow></mml:msup></mml:math> . In the situation of spherical accretions, we unveil that the parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> has a positive effect on the shadow size but a negative effect on the observed specific intensities. Considering that the Schwarzschild-MOG black hole is surrounded by an optical and geometrically thin accretion disk, we find that the total observed specific intensities are mainly contributed by the direct emissions, while the photon rings and lensed rings provide small contributions. It is also found that with the increase of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> , the black hole shadow expands, the photon rings and lensed rings become larger and thicker. Besides, we emphasize that the boundary of the observed shadow cast by the aim black hole in the disk accretion scenario is determined by the direct emissions rather than the photon ring emissions. Consequently, we unveil that there is a linear relationship involving the critical impact parameter and the starting point of the direct emissions. This finding helps to use the experimental results of the Event Horizon Telescope to infer the critical impact parameter and to test General Relativity.