Dynamics and stability of the two-body problem with Yukawa correction to Newton's gravity, revisited and applied numerically to the solar system

In this manuscript, we review the motion of two-body celestial system (planet-sun) for a Yukawa-type correction on Newton's gravitational potential using Hamilton's formulation. We reexamine the stability using the corresponding linearization Jacobian matrix, and verify that the Bertrand's theorem conditions are met for radii $\ll 10^{15} m$, and so bound closed orbits are expected. Applied to the solar system, we present the equation of motion of the planet, then solve it both analytically and numerically. Making use of the analytical expression of the orbit, we estimate the Yukawa strength $\alpha$, and find it larger than the nominal value ($10^{-8}$) adopted in previous studies, in that it is of order ($\alpha = 10^{-4}-10^{-5}$) for terrestrial planets (Mercury, Venus, earth, Mars and Pluto) whereas it is even larger ($\alpha = 10^{-3}$) for the Giant planets (Jupiter, Saturn, Uranus and Neptune). Taking as inputs ($r_{min}, v_{max}, e$) observed by NASA, we analyze the orbits analytically and numerically for both the estimated and nominal values of $\alpha$, and determine the corresponding trajectories. For each obtained orbit we recalculate the characterizing parameters ($r_{min}, r_{max}, a, b, e $) and compare their values according to the used potential (Newton with/without Yukawa correction) and to the method used (analytical and/or numerical). When compared to the observational data, we conclude that the correction on the path due to Yukawa correction is of order of and up to 80 million km (20 million km) as a maximum deviation occurring for Neptune (Pluto) for nominal (estimated) value of $\alpha$.