AI 中文总结
针对有效半光半径$R_{\rm e}$的标度关系随比例变化的问题，提出$R_5\equiv R_{-2}$作为替代尺度半径，推导其与Sérsic模型的关系，构建改进的Wolf型质量估计量，修正了原半光半径替代带来的系统性偏差。
AI 中文摘要
有效半光半径$R_{\rm e}$是一个人为设定的50%光边界，当比例变化时，涉及这类有效半径及其相关表面亮度$μ_{\rm e}$的标度关系会出现系统性（且不符合预期）的变化。本文提出将投影半径$R_5\not\to R_{-2}$作为替代量，该半径处表面亮度和强度轮廓的对数斜率分别等于$5.00\text{mag\\,dex}^{-1}$和-2，且每个对数径向间隔贡献的光度达到最大。它可以通过非参数方法或参数化拟合进行测量。对于Sérsic $R^{1/n}$族，推导得到精确关系$R_5=(2n/b_n)^n\\,R_{\rm e}$，当$n\rightarrow\infty$时，$R_5/R_{\rm e}\rightarrow{\rm e}^{1/6}\approx1.181$。用可观测对$(R_5,μ_5)$对（不含$b_n$的）$R^{1/n}$模型重新参数化，消除了$R_{\rm e}$与$n$之间的非线性耦合；由于$R_5$处的局部斜率为$5\text{mag\\,dex}^{-1}$，在推导推断的总星等时，$R_5$和$μ_5$的相关测量误差会大幅抵消。此外，本文提供了一个精确的单积分恒等式，可将任意投影光分数与相同半径的球内光分数联系起来。研究表明，可直接观测的$R_5$通过一个弱依赖于$n$的因子与各向异性不敏感的内禀半径$r_{-3}$相关联，由此得到改进的、依赖于$n$的Wolf型质量估计量$M_{-3}$和空间质光比$(M_{\rm dyn}/L)_{-3}$。具体而言，$M_{-3}\approx4\\,G^{-1}\langleσ_{\rm los}^2\rangle\\,R_{-2}\approx4.72\\,G^{-1}\langleσ_{\rm los}^2\rangle\\,R_{\rm e}$。过去在动力学质量估计中使用半光半径替代，会导致包络质量出现12%--18%的、依赖于Sérsic指数的系统性偏差，质光比的偏差幅度则超过20%。
英文摘要
The effective half-light radius $R_{\rm e}$ marks an arbitrary 50-per-cent light boundary, and scaling relations involving such {\it effective} radii and their associated surface brightnesses, $μ_{\rm e}$, systematically (and undesirably) vary as the percentage changes. Here, the projected radius $R_5\equiv R_{-2}$, where the logarithmic slope of the surface-brightness and intensity profile equals $5.00\,\text{mag\,dex}^{-1}$ and $-2$, respectively, and where the luminosity contributed per logarithmic radial interval is maximal, is developed as an alternative. It can be measured non-parametrically or with a parametrized fit. For the Sérsic $R^{1/n}$ family, the exact relation $R_5=(2n/b_n)^n\,R_{\rm e}$ is derived, with $R_5/R_{\rm e}\rightarrow{\rm e}^{1/6}\approx1.181$ as $n\rightarrow\infty$. Reparameterizing the (now $b_n$-free) $R^{1/n}$ model in terms of the observable pair $(R_5,μ_5)$ removes the non-linear $R_{\rm e}$--$n$ coupling, and because the local slope is $5\,\text{mag\,dex}^{-1}$ at $R_5$, correlated measurement errors in $R_5$ and $μ_5$ largely cancel when deriving the inferred total magnitude. Additionally, an exact single-integral identity is provided to relate any projected light fraction to the fraction within a sphere of the same radius. The directly observable $R_5$ is shown to be connected, through a weakly $n$-dependent factor to the anisotropy-insensitive intrinsic radius $r_{-3}$, yielding a refined $n$-dependent Wolf-type mass estimator $M_{-3}$ and spatial mass-to-light ratio $(M_{\rm dyn}/L)_{-3}$. Specifically, $M_{-3}\approx4\,G^{-1}\langleσ_{\rm los}^2\rangle\,R_{-2}\approx4.72\,G^{-1}\langleσ_{\rm los}^2\rangle\,R_{\rm e}$. Past half-light substitutions in dynamical mass estimators introduce systematic Sérsic-dependent offsets of 12--18 per~cent in enclosed mass and offsets spanning $>20$ per~cent in the mass-to-light ratio.
Comments20 pages, 3 figures, 5 tables. Submitted to MNRAS. Companion Python package R5Tools available at https://github.com/A-Graham/R5Tools
