  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 $\alpha$.

Earth and Planetary Astrophysics;General Relativity and Quantum Cosmology