We have studied, in a series of papers, the properties of the $M_{\bullet}$ versus $M_{G}\sigma^2$ relation and we have found that it is useful to describe the evolution of galaxies in the same way as the HR diagram does for stars and to predict the masses of Supermassive Black Holes that are difficult to be guessed using other scaling relations. In this paper, analyzing five samples of galaxies, we find that this relation has intrinsic scatter similar to the $M_{\bullet} - \sigma$, but follows the theoretical models much better than the $M_{\bullet} - \sigma$. Furthermore, we analyze the role of the bulge mass in the behavior of $M_{\bullet}$ versus $M_{G}\sigma^2$ relation because the difference with the $M_{\bullet} - \sigma$ is often determined by the choice of the right sample of galactic masses.

中文翻译：

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$M_{\bullet}$ vs $M_{G}\sigma ^{2}$ 与 $M_{\bullet}$ vs $\sigma$ 关系的更新比较以及星系质量问题

我们在一系列论文中研究了 $M_{\bullet}$ 与 $M_{G}\sigma^2$ 关系的性质，我们发现描述星系的演化是有用的。就像 HR 图对恒星所做的那样，并预测使用其他比例关系难以猜测的超大质量黑洞的质量。在本文中，分析五个星系样本，我们发现这种关系具有类似于 $M_{\bullet} - \sigma$ 的内在散射，但比 $M_{\bullet} - \sigma$ 更符合理论模型$. 此外，我们分析了膨胀质量在 $M_{\bullet}$ 与 $M_{G}\sigma^2$ 关系中的作用，因为与 $M_{\bullet} - \sigma$ 的差异通常是由选择正确的星系质量样本决定。