Motivated by the prospect of measuring a black hole photon ring, in previous work we explored the interferometric signature produced by a bright, narrow curve in the sky. Interferometric observations of such a curve measure its "projected position function" $\mathbf{r}\cdot\hat{\mathbf{n}}$, where $\mathbf{r}$ parameterizes the curve and $\hat{\mathbf{n}}$ denotes its unit normal vector. In this paper, we show by explicit construction that a curve can be fully reconstructed from its projected position, completing the argument that space interferometry can in principle determine the detailed photon ring shape. In practice, near-term observations may be limited to the visibility amplitude alone, which contains incomplete shape information: for convex curves, the amplitude only encodes the set of projected diameters (or "widths") of the shape. We explore the freedom in reconstructing a convex curve from its widths, giving insight into the shape information probed by technically plausible future astronomical measurements. Finally, we consider the Kerr "critical curve" in this framework and present some new results on its shape. We analytically show that the critical curve is an ellipse at small spin or inclination, while at extremal spin it becomes the convex hull of a Cartesian oval. We find a simple oval shape, the "phoval", which reproduces the critical curve with high fidelity over the whole parameter range.

中文翻译：

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黑洞光子环的可观察形状

受测量黑洞光子环前景的启发，在之前的工作中，我们探索了天空中明亮而狭窄的曲线产生的干涉特征。对这种曲线的干涉观测测量其“投影位置函数”$\mathbf{r}\cdot\hat{\mathbf{n}}$，其中 $\mathbf{r}$ 参数化曲线，$\hat{\mathbf {n}}$ 表示它的单位法向量。在本文中，我们通过显式构造表明曲线可以从其投影位置完全重建，完成了空间干涉测量原则上可以确定详细光子环形状的论点。在实践中，近期观测可能仅限于能见度振幅，其中包含不完整的形状信息：对于凸曲线，振幅仅编码投影直径（或“宽度”）的集合 ) 的形状。我们探索了从宽度重建凸曲线的自由度，深入了解技术上合理的未来天文测量所探测的形状信息。最后，我们考虑了这个框架中的克尔“临界曲线”，并展示了一些关于其形状的新结果。我们分析表明，临界曲线在小自旋或倾斜时是椭圆，而在极值自旋时它变成笛卡尔椭圆的凸包。我们找到了一个简单的椭圆形状，即“phoval”，它在整个参数范围内以高保真度再现了临界曲线。