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.

中文翻译：

---

由暗物质和宇宙常数控制的区域中的修正牛顿动力学效应 $\varLambda $

我们研究了在修正的牛顿动力学场景中校正引力势影响下的次级天体的运动。此外，我们在银河系中观察，其中对潜在修正的第一次修正来自修正的泊松方程，并且包括两个 mew 项，其中一个是 $\ln(r/r_{\max})$ 形式，另一个是与添加到牛顿势的宇宙常数 lambda $\varLambda $ 相关联。两个势能的影响区域与受牛顿势条件 $r < r_{\max}$ 约束的感兴趣区域有关，$r > r_{\max}$ 是对与泊松方程中的项 $(\nabla \phi )^{2}$ 用于具有物质密度 $\rho $ 的引力场，最后，$r \gg r_{\max}$ 势能缩放为 $c^{2}\varLambda r^{2}$ 并且宇宙常数 lambda 占主导地位的域。接下来，我们使用平均干扰势对拉格朗日行星方程进行积分，并以太阳为例，获得轨道元素平均时间变化率的解析表达式。我们发现暗物质和宇宙常数都只影响 perigalaktikon 点的参数以及平均异常。