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arXiv 0809.4122astro-phgr-qc
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无对称N个质点引力透镜系统的微扰理论：小质量比近似

Perturbation theory of N point-mass gravitational lens systems without symmetry: small mass-ratio approximation

Hideki Asada

备注：
14 pages, 8 figures, 3 tables; abstract and text improved; accepted for publication in MNRAS
发表信息：
Mon.Not.Roy.Astron.Soc.394:818-830,2009
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摘要：

本文首次系统地尝试用微扰理论确定由任意数目的平面上不对称的共面质量引起的引力透镜的象的位置，它是透镜和源参数的函数。在小质量比近似下，提出了一种泰勒级数展开法求解透镜方程。首先，我们研究了通常使用的单复变多项式的微扰结构。找到了微扰根，有些根代表透镜像的位置，有些根则不满足透镜方程，因而是非物理的。这与以下事实一致：如果N=3（或更大），则多项式的次数（即零的数量）超过透镜图像的最大数量。这个定理从来没有告诉过哪些根是物理的（或非物理的）。在本文中，非物理的识别。其次，为了避免非物理的根源，我们重新审查的镜头方程。我们的方法的优点是，它允许系统的迭代分析。我们确定的二元透镜系统的图像位置的质量比的第三顺序和任意N点质量的第二顺序。这说明了对参数的依赖性。第三，允许小质量比限制的图像的数量小于最大数量。结果表明，额外像的位置不能用质量比的麦克劳林级数表示。最后讨论了放大率。

英文摘要：

This paper makes the first systematic attempt to determine using perturbation theory the positions of images by gravitational lensing due to arbitrary number of coplanar masses without any symmetry on a plane, as a function of lens and source parameters. We present a method of Taylor-series expansion to solve the lens equation under a small mass-ratio approximation. First, we investigate perturbative structures of a single-complex-variable polynomial, which has been commonly used. Perturbative roots are found. Some roots represent positions of lensed images, while the others are unphysical because they do not satisfy the lens equation. This is consistent with a fact that the degree of the polynomial, namely the number of zeros, exceeds the maximum number of lensed images if N=3 (or more). The theorem never tells which roots are physical (or unphysical). In this paper, unphysical ones are identified. Secondly, to avoid unphysical roots, we re-examine the lens equation. The advantage of our method is that it allows a systematic iterative analysis. We determine image positions for binary lens systems up to the third order in mass ratios and for arbitrary N point masses up to the second order. This clarifies the dependence on parameters. Thirdly, the number of the images that admit a small mass-ratio limit is less than the maximum number. It is suggested that positions of extra images could not be expressed as Maclaurin series in mass ratios. Magnifications are finally discussed.

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