对牛顿引力进行汤川修正的二体问题的动力学和稳定性，再研究和数值应用于太阳系
Dynamics and stability of the two-body problem with Yukawa correction to Newton's gravity, revisited and applied numerically to the solar system
摘要：
在这篇手稿中，我们回顾了二体天体系统（行星-太阳）的运动，并使用汉密尔顿公式对牛顿引力势进行汤川式修正。利用相应的线性化雅可比矩阵重新检验了系统的稳定性，并验证了半径$\ll 10^{15} m$满足Bertrand定理条件，因此期望有界闭轨道.应用于太阳系，我们提出了行星的运动方程，然后解决它的解析和数值。利用轨道的解析表达式，我们估计汤川强度$α$，并发现它大于标称值（$10^{-8}$）在以前的研究中采用，因为它是阶（$α= 10^{-4}-10^{-5}$）的类地行星（水星，金星，地球，火星和冥王星），而它甚至更大（$α= 10^{-3}$）的巨行星（木星，土星，天王星和海王星）。以NASA观测到的（$r_{min}, v_{max}, e$）为输入，对$α$的估计值和标称值的轨道进行了解析和数值分析，并确定了相应的轨道。对于每个获得的轨道，我们重新计算的特征参数（$r_{min}, r_{max}, a, b, e $），并比较它们的值根据所使用的潜力（牛顿与/不汤川修正）和所使用的方法（分析和/或数值）。当与观测数据相比，我们得出结论，由于汤川修正的路径上的修正是的顺序和高达80万公里（20万公里）的最大偏差发生在海王星（冥王星）的名义（估计）值$α$。
英文摘要：
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 $α$, and find it larger than the nominal value ($10^{-8}$) adopted in previous studies, in that it is of order ($α= 10^{-4}-10^{-5}$) for terrestrial planets (Mercury, Venus, earth, Mars and Pluto) whereas it is even larger ($α= 10^{-3}$) 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 $α$, 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 $α$.
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