球对称时空中的热点与光子环
Hotspots and Photon Rings in Spherically-Symmetric Spacetimes
AI总结：
本文研究球对称时空中热点的高阶图像，提出解析估计时间延迟与图像尺寸，并验证于史瓦西黑洞，为未来黑洞成像观测提供理论依据。
摘要：
未来的黑洞（BH）成像观测预计将解析出与热点及视界尺度吸积流的高阶图像相对应的更精细结构。在球对称时空中，图像阶次由形成该图像的光子所执行的半圈数决定。连续阶次的图像大约在延迟时间≈π倍黑洞阴影半径后到达。连续阶次图像的分数直径、宽度和流量密度被透镜化李雅普诺夫指数（透镜化Lyapunov指数）指数级缩小，该指数是时空的一个特征。本文研究了位于固定空间位置或在圆轨道上运动的简单点状热点的外观。其零阶与一阶图像出现之间的精确时间延迟与我们的解析估计相符，该估计考虑了观测者倾角，对于位于质量M的史瓦西黑洞约≤5M处的热点，误差≤20%。由于M87*和Sgr A*拥有几何厚吸积流，我们还探讨了其一阶图像的直径和宽度随盘标高（盘尺度高度）的变化。使用简单的锥形环状体模型，对于现实形态，我们估计一阶图像直径与阴影直径的偏差≤30%，其宽度≤1.3M。最后，当使用一阶和二阶图像的直径或宽度来恢复史瓦西透镜化指数（π）时，估计误差≤20%。不久之后，通过此类测量，将有可能稳健地了解更多关于天体物理黑洞的时空几何信息。
英文摘要：
Future black hole (BH) imaging observations are expected to resolve finer features corresponding to higher-order images of hotspots and of the horizon-scale accretion flow. In spherical spacetimes, the image order is determined by the number of half-loops executed by the photons that form it. Consecutive-order images arrive approximately after a delay time of $\approxπ$ times the BH shadow radius. The fractional diameters, widths, and flux-densities of consecutive-order images are exponentially demagnified by the lensing Lyapunov exponent, a characteristic of the spacetime. The appearance of a simple point-sized hotspot when located at fixed spatial locations or in motion on circular orbits is investigated. The exact time delay between the appearance of its zeroth and first-order images agrees with our analytic estimate, which accounts for the observer inclination, with $\lesssim 20\%$ error for hotspots located about $\lesssim 5M$ from a Schwarzschild BH of mass $M$. Since M87$^\star$ and Sgr A$^\star$ host geometrically-thick accretion flows, we also explore the variation in the diameters and widths of their first-order images with disk scale-height. Using a simple conical torus model, for realistic morphologies, we estimate the first-order image diameter to deviate from that of the shadow by $\lesssim 30\%$ and its width to be $\lesssim 1.3M$. Finally, the error in recovering the Schwarzschild lensing exponent ($π$), when using the diameters or the widths of the first and second-order images is estimated to be $\lesssim 20\%$. It will soon become possible to robustly learn more about the spacetime geometry of astrophysical BHs from such measurements.
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