Following the discovery of short-period oscillations in the low-mass pre-white-dwarf component of an eclipsing binary, we have explored the radial (p-mode) stability of extremely low mass stars across a range of composition, effective temperature and luminosity. We have identified the instability boundaries associated with low- to high-order radial oscillations (0 ≤ n ≤ 13) and show that these are a strong function of both composition and radial order (n). The blue edge shifts to higher effective temperature and luminosity with decreasing hydrogen abundance. In addition, high-order modes are more easily excited, and small islands of highradial- order instability develop, some of which may correspond to real stars. The excitation mechanism is discussed with reference to these instability islands. If the envelope of the lowmass pre-white dwarf 1SWASP J024743.37-251549.2 (J0247-25B) is depleted in hydrogen, it is unstable to high-order p modes. Driving is by the classical κ mechanism operating in the second helium-ionization zone, with some support from the first helium-ionization zone. The instability is very sensitive to composition, temperature and luminosity. The reported periods, temperature and luminosity of J0247-25B require the envelope hydrogen abundance to lie in the range 0.2 less-than or approximate X less-than or approximate 0.3, in agreement with current evolution models.
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