We consider possible signatures for Yukawa gravity within the Galactic Central Parsec, based on our analysis of the S2 star orbital precession around the massive compact dark object at the Galactic Centre, and on the comparisons between the simulated orbits in Yukawa gravity and two independent sets of observations. Our simulations resulted in strong constraints on the range of Yukawa interaction Λ and showed that its most probable value in the case of S2 star is around 5000 -7000 AU. At the same time, we were not able to obtain reliable constrains on the universal constant δ of Yukawa gravity, because the current observations of S2 star indicated that it may be highly correlated with parameter Λ in the range (0 < δ < 1). For δ > 2 they are not correlated. However, the same universal constant which was successfully applied to clusters of galaxies and rotation curves of spiral galaxies (δ = 1/3) also gives a satisfactory agreement with the observed orbital precession of the S2 star, and in that case the most probable value for the scale parameter is Λ ≈ 3000 ± 1500 AU. Also, the Yukawa gravity potential induces precession of S2 star orbit in the same direction as General Relativity for δ > 0 and for δ < −1, and in the opposite direction for −1 < δ < 0. The future observations with advanced facilities, such as GRAVITY or/and European Extremely Large Telescope, are needed in order to verify these claims.

Using an algebraic condition of vanishing discriminant for multiple roots of fourth degree polynomials we derive an analytical expression of a shadow size as a function of a charge in the Reissner -Nordström (RN) metric [1,2]. We consider shadows for negative tidal charges and charges corresponding to naked singularities q = Q 2 /M 2 > 1, where Q and M are black hole charge and mass, respectively, with the derived expression. An introduction of a negative tidal charge q can describe black hole solutions in theories with extra dimensions, so following the approach we consider an opportunity to extend RN metric to negative Q 2 , while for the standard RN metric Q 2 is always non-negative. We found that for q > 9/8 black hole shadows disappear. Significant tidal charges q = −6.4 (suggested by Bin-Nun [3-5]) are not consistent with observations of a minimal spot size at the Galactic Center observed in mm-band, moreover, these observations demonstrate that a Reissner -Nordström black hole with a significant charge q ≈ 1 provides a better fit of recent observational data for the black hole at the Galactic Center in comparison with the Schwarzschild black hole.