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MASSES

To test whether any evolutionary model is correct, reliable tests are required. These include a detailed comparison between models which predict evolution tracks and surface composition for a given stellar mass and accurate observations of stellar compositions and dimensions. Stellar compositions, temperatures and gravities can be measured relatively simply, but measuring mass is less straightforward. There are three principal approaches.

Spectroscopic Mass $M_{\rm S}$. Suppose some physical mechanism connects the mass $M$ of the star to its luminosity $L$, such as the mass-luminosity relation for main-sequence stars or a core-mass shell-luminosity relation for shell-burning stars. From spectroscopy and model atmospheres, the effective temperature $T$ and surface gravity $g$ of the star can be measured. Since $L/M\propto T^4/g$, then the spectroscopic mass $M_{\rm S}$ of the star may be deduced using an appropriate $M-L$ relation (e.g. Jeffery 1988).

Pulsation Mass $M_{\rm P}$. Stellar pulsations provide much more powerful tools for determining stellar masses. Fortuitously, pulsations appear to be common amongst EHes (see next section). The most straightforward approach is provided by pulsation periods $\Pi$ obtained from photometry. Linear theory provides theoretical pulsation periods for stellar models of given $M, T$ and $L$. In conjunction with spectroscopic measurements of $T$ and $g$, $\Pi$ can provide an estimate of the pulsation mass $M_{\rm P}$.

Direct Mass $M_{\rm D}$. In some cases, it may be possible to measure the angular radius ($\theta$) and radial velocity ($v$) of a pulsating stars throughout the pulsation cycle. $v$ may be integrated to yield the total radius change $\delta R$. $\delta \theta/\theta$ gives the relative radius change. The stellar radius $R$ is then given by $R = \delta R / (\delta \theta/\theta)$. Following Baade (1926) and combining the radius with $g$ from spectroscopy and model atmospheres yields the direct mass $M_{\rm D}\propto g/R^2$.