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We examine the Poynting-Robertson (PR) effect under the influence of a logarithmic predicted as a correction to the Newtonian gravitational potential in the solar system. Our motivation emanates from the fact that today’s gravitational theories predict various corrections such that exponential, logarithmic to describe and understand the existence of dark matter, postulated to resolve discrepancies in astrophysical observations in today’s accepted theories of gravity. By saying that we simply mean that our main interest is to calculate the times that dust particles take to reach the Earth’s orbit and derive analytical expressions for the time changes of basic orbital elements in the influence of the PR effect. We use dust particle starting at a distance $r = 2.7~\text{AU}$ or approximately in the asteroid belt. In a first order perturbation treatment we obtain time solutions for the orbital elements of semi major axis, eccentricity, and mean motion, as well as expressions for the time taken for these particles to reach Earth’s orbit. We find that for circular orbits in a logarithmic potential the time taken to reach Earth is slightly larger that the elliptical orbits. Newtonian potential circular orbits require more times when compared to the logarithmic ones. Finally elliptical orbits time to reach Earth are of the same order of magnitude with a dependence on the eccentricity, density, and particle diameter.

We study the motion of a secondary celestial body under the influence of a corrected gravitational potential in a modified Newtonian dynamics scenario. Furthermore we look within the Milky-way where the first correction to the potential results from a modified Poisson equation, and includes two mew terms one of which is of the form $\ln(r/r_{\max})$ and the other is associated with the cosmological constant lambda $\varLambda $ added to the Newtonian potential. The regions of influence of the two potentials are associated with regions of interested bounded by the conditions $r < r_{\max}$ for the Newtonian potential, $r > r_{\max}$ the logarithmic correction to the potential relating to the term $(\nabla \phi )^{2}$ in the Poisson equation for the gravitational field that has matter density $\rho $ , and finally, the domain where $r \gg r_{\max}$ the potential scales as $c^{2}\varLambda r^{2}$ and the cosmological constant lambda dominates. Next using an average disturbing potential we integrate Lagrange’s planetary equations and we obtain analytical expressions for the average time rates of change of the orbital elements using our sun as an example. We find that both dark matter and cosmological constant affect only the argument of the perigalaktikon point as well as the mean anomaly.

We explore the dynamics and stability of the two-body problem by modifying the Newtonian potential with the Yukawa potential. This model may be considered a theory of modified gravity; where the interaction is not simply the Kepler solution for large distance. We investigate stability by deriving the Jacobian of the linearized matrix equation and evaluating the eigenvalues of the various equilibrium points calculated during the analysis. The subcases of a purely Yukawa and purely Newtonian potential are also explored.