Poynting Robertson effects due to a logarithmic correction to the Newtonian potential

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.