We discuss and characterise the power spectral density properties of a model aimed at describing pulsations in stars from the main-sequence to the asymptotic giant branch. We show that the predicted limit of the power spectral density for a pulsation mode in the presence of stochastic noise is always well approximated by a Lorentzian function. While in stars predominantly stochastically driven the width of the Lorentzian is defined by the mode lifetime, in stars where the driving is predominately coherent the width is defined by the amplitude of the stochastic perturbations. In stars where both drivings are comparable, the width is defined by both these parameters and is smaller than that expected from pure stochastic driving. We illustrate our model through numerical simulations and propose a well defined classification of stars into predominantly stochastic (solar-like) and predominately coherent (classic) pulsators. We apply the model to the study of the Mira variable U Per, and the semiregular variable L2 Pup and, following our classification, conclude that they are both classical pulsators. Our model provides a natural explanation for the change in behaviour of the pulsation amplitude-period relation noted in several earlier works. Moreover, our study of L2 Pup enables us to test the scaling relation between the mode line width and effective temperature, confirming that an exponential scaling reproduces well the data all the way from the main sequence to the asymptotic giant branch, down to temperatures about 1000 K below what has been tested in previous studies.

中文翻译：

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从类太阳星到米拉星：存在随机噪声的恒星脉动器的统一描述

我们讨论并表征了一个模型的功率谱密度特性，该模型旨在描述恒星从主序到渐近巨星分支的脉动。我们表明，在存在随机噪声的情况下，脉动模式的功率谱密度的预测极限总是可以通过洛伦兹函数很好地近似。虽然在主要随机驱动的恒星中，洛伦兹的宽度由模式寿命定义，但在驱动主要是相干的恒星中，宽度由随机扰动的幅度定义。在两种驱动都具有可比性的恒星中，宽度由这两个参数定义，并且小于纯随机驱动的预期值。我们通过数值模拟来说明我们的模型，并提出将恒星分类为主要是随机的（类太阳）和主要是相干的（经典）波轮。我们将该模型应用于 Mira 变量 U Per 和半规则变量 L2 Pup 的研究，并根据我们的分类得出结论，它们都是经典波轮。我们的模型为早期几部作品中提到的脉动幅度-周期关系的行为变化提供了自然的解释。此外，我们对 L2 Pup 的研究使我们能够测试模式线宽度和有效温度之间的比例关系，证实指数比例可以很好地再现从主序到渐近巨星分支的数据，低至约 1000 摄氏度的温度K 低于先前研究中测试过的值。