A new C-D-like diagram for SPB stars: The variations of period spacing as a signature of evolutionary status

The Slowly Pulsating B-type (SPB) stars are the upper main-sequence stars on the HR diagram. Their oscillations are high-order, low-degree g-mode and can be used to probe the structure of the radiative zone where is near the outer boundary of the convective core and constrain the chemical mixing in stellar interiors. In SPB stars, the period spacing periodically varies with periods. It has regarded as a signature of the chemical composition gradient beyond the convective core. Based on theoretical calculations, we find that the variation frequency of the period spacings ($f_{\Delta P}$) is related to the width of the $\mu$-gradient region on the buoyancy radius ($\Lambda_{\mu}$) with the relation of $f_{\Delta P}\sim0.5\Lambda_{\mu}$. This indicates that the variation frequency $f_{\Delta P}$ is sensitive to the central hydrogen mass fraction $X_{\rm C}$ (i.e., the evolutionary status). Finally, we find that the variation frequency $f_{\Delta P}$ and the means of the period spacings $\langle\Delta P\rangle$ can be used to construct a new C-D-like diagram ($f_{\Delta P}$ vs. $\langle\Delta P\rangle$) which can be used to roughly decide the stellar evolutionary stages and to approximately determine stellar mass for SPB stars.
Comment: 11 figures, 2 tables, accepted by ApJ