标题： 伪牛顿势的一般形式，模仿施瓦茨希尔德测地线 显示英文标题

标题： General form for Pseudo-Newtonian Potentials, imitating Schwarzschild geodesics

摘要： 我们提出了一种新的、广义的伪牛顿引力势（PNP）形式，它表示为一系列类似Paczyński-Wiita的函数，并增加了任意系数的$r$的负幂次项。我们提出了一种确定这些系数的过程，以构建一个自定义的PNP，该势能能够再现测试粒子在黑洞附近的施瓦茨希尔德测地线的关键特性。作为示例，我们构建了势能以重现（I）在$r=6$（几何单位）处存在一个最内稳定圆轨道，并具有小偏差的正确下落速度（测地线通用下落），（II）在远距离处的近日点进动，以及（III）具有特定角动量$L=4$的边缘束缚圆轨道的存在，以及接近它的抛物线轨道的近日点进动。我们将我们的示例性能与Paczyński-Wiita势和其他现有势进行了比较。最后，我们讨论了我们公式的局限性和优势。 显示英文摘要

摘要： We propose a new, general form for a pseudo-Newtonian gravitational potential (PNP), expressed as a series of Paczy\'nski-Wiita-like functions with the addition of increasing negative powers of $r$ with arbitrary coefficients. We present a procedure for determining these coefficients to construct a custom PNP that replicates key features of Schwarzschild geodesics for a test particle near a black hole. As an example, we construct potentials set to reproduce (I) the presence of an innermost stable circular orbit at the $r=6$ (geometric units), with the correct infall velocity for small deviations (on the geodesic universal infall), (II) the periapsis advance at large distances, and (III) the presence of a marginally bound circular orbit with specific angular momentum $L=4$, and the periapsis advance of parabolic orbits close to it. We compare the performance of our examples against the Paczy\'nski-Wiita potential and other existing potentials. Finally, we discuss the limitations and advantages of our formulation.