论文标题
偏移微透析变性的数学处理
A Mathematical Treatment of the Offset Microlensing Degeneracy
论文作者
论文摘要
Zhang等人最近提出的偏移微透析变性。 （2022）已显示出在解释2型行星微晶观测的解释中，将近范围内和内部的腐蚀性变性概括为统一的放大元素退化制度。虽然内部的堕落性期望源轨迹将其等距与退化镜头配置的行星苛性遗传量相等，但偏移二分化指出，相同的数学表达适用于近距离，宽和共振的苛刻拓扑的任何组合，而预计的星形平面分离，在其中neq a a ofs a offs a offs a offs a offsect（$ s___________________i s _ {\ rm b} $）取决于源轨迹穿越星形行动轴的位置。一个重要的含义是，$ s _ {\ rm a} = 1/s _ {\ rm b} $近距离退化的溶液从未严格体现在观测值中，除非源在主距离附近的一个奇异点。然而，根据数值计算提出了偏移，并且没有给出理论上的理由。在这里，我们提供了对偏移式的理论处理，这表明其本质是数学堕落。从第一原则来看，我们表明偏移退化形式主义在两种情况下以质量比（$ q $）精确：当源在苛性碱内部的镜头轴时，以及$（s _ {\ rm a} -s _} -s _ {\ rm b} \ rm b}）^6 \ ll1 $ caustics cassics coctics coctics cosics。在数值上探索了偏向源轨迹中偏移量的持续程度的程度。最后，结果表明，叠加原理可以直接概括为$ n $ body microlenses，其中$ n-1 $ $ $ n-1 $ planetary镜头组件（$ q \ ll1 $），这会导致$ 2^{n-1} $ - fold demeneracy。
The offset microlensing degeneracy, recently proposed by Zhang et al. (2022), has been shown to generalize the close-wide and inner-outer caustic degeneracies into a unified regime of magnification degeneracy in the interpretation of 2-body planetary microlensing observations. While the inner-outer degeneracy expects the source trajectory to pass equidistant to the planetary caustics of the degenerate lens configurations, the offset degeneracy states that the same mathematical expression applies to any combination of the close, wide, and resonant caustic topologies, where the projected star-planet separations differ by an offset ($s_{\rm A}\neq s_{\rm B}$) that depends on where the source trajectory crosses the star-planet axis. An important implication is that the $s_{\rm A}=1/s_{\rm B}$ solution of the close-wide degeneracy never strictly manifests in observations except when the source crosses a singular point near the primary. Nevertheless, the offset degeneracy was proposed upon numerical calculations, and no theoretical justification was given. Here, we provide a theoretical treatment of the offset degeneracy, which demonstrates its nature as a mathematical degeneracy. From first principles, we show that the offset degeneracy formalism is exact to zeroth-order in the mass ratio ($q$) for two cases: when the source crosses the lens-axis inside of caustics, and for $(s_{\rm A}-s_{\rm B})^6\ll1$ when crossing outside of caustics. The extent to which the offset degeneracy persists in oblique source trajectories is explored numerically. Lastly, it is shown that the superposition principle allows for a straightforward generalization to $N$-body microlenses with $N-1$ planetary lens components ($q\ll1$), which results in a $2^{N-1}$-fold degeneracy.
