Modified Newtonian dynamics effects in a region dominated by dark matter and a cosmological constant Λ$\varLambda $

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.