Pulsating low-mass white dwarf stars are white dwarfs with stellar masses between 0.30~M$_{ \odot}$ and 0.45~M$_{ \odot}$ that show photometric variability due to gravity-mode pulsations. Within this mass range, they can harbour both a helium- and hybrid-core, depending if the progenitor experienced helium-core burning during the pre-white dwarf evolution. SDSS J115219.99$+$024814.4 is an eclipsing binary system where both components are low-mass white dwarfs, with stellar masses of 0.362$\pm$0.014~M$_{ \odot}$ and 0.325$\pm$0.013~M$_{ \odot}$. In particular, the less massive component is a pulsating star, showing at least three pulsation periods of $\sim$1314 s, $\sim$1069 s and $\sim$582.9 s. This opens the way to use asteroseismology as a tool to uncover its inner chemical structure, in combination with the information obtained using the light-curve modelling of the eclipses. To this end, using binary evolutionary models leading to helium- and hybrid-core white dwarfs, we compute adiabatic pulsations for $\ell=1$ and $\ell=2$ gravity modes with \texttt{ Gyre}. We found that the pulsating component of the SDSS J115219.99$+$024814.4 system must have a hydrogen envelope thinner that the value obtained from binary evolution computations, independently of the inner composition. Finally, from our asteroseismological study, we find a best fit model characterised by T$_{ \rm e ff}=10\, 917$ K, M=0.338~M$_{ \odot}$, M$_{ \rm H}=10^{ -6}$~M$_{ \odot}$ with the inner composition of a hybrid WD.
Categories: astro-ph.SR