The SNR of idealized radial velocity signals: RM curve grid based on Ohta+2005 and Hirano+2011 models

To derive the SNR equation for an ideal RM curve without second-order effects, including limb darkening, micro-turbulence, and macro-turbulence, we created an RM curve grid based on the Ohta+2005 and Hirano+2011 models. In the grid generation process, We fixed the projected stellar rotational velocity, vsini, to 3 km/s and transformed the limb-darkening coefficients, q1 and q2, to 0. For the Hirano+ model, the micro-turbulent and macro-turbulent velocities are set to 0. The grid spans sky-projected spin-orbit angles, lambda, from [0, 350] degrees in steps of 10 degrees, impact parameters, b, from [0,1] in steps of 0.1, and the base-2 logarithm of the ratio of the semi-major axis to the stellar radius, log_2(a/R_star) from [1, 8] in steps of 0.25. Thus, a total of 11,484 unique grid positions were defined. In each case, a/R_star and b are converted to T_14 and T_23 using the Seager & Mallen-Ornelas (2003) equations, assuming a circular orbit and a Solar bulk stellar density to obtain orbital period.

为推导不含临边昏暗、微湍流与宏湍流等二阶效应的理想罗斯特-麦克劳林（Rossiter-McLaughlin, RM）曲线信噪比方程，我们基于Ohta等人2005年与Hirano等人2011年的模型构建了RM曲线网格。在网格生成过程中，我们将恒星投影自转速度（vsini）固定为3 km/s，并将临边昏暗系数q1与q2均设为0。针对Hirano等人的模型，我们将微湍流与宏湍流速度均设为0。该网格的参数涵盖：天球投影自旋轨道夹角λ，取值范围为[0, 350]度，步长为10度；冲击参数b，取值范围为[0, 1]，步长为0.1；半长轴与恒星半径之比的以2为底的对数log₂(a/R_star)，取值范围为[1, 8]，步长为0.25。综上，该网格共定义了11484个唯一的网格节点。在每组参数组合下，我们均假设轨道为圆形，并采用太阳本体恒星密度计算轨道周期，随后通过Seager与Mallen-Ornelas（2003）提出的公式将a/R_star与b转换为T_14和T_23。