Title: Pulsating low-mass white dwarfs in the frame of new evolutionary sequences: IV. The secular rate of period change,
Abstract: We present a theoretical assessment of the expected temporal rates of change
of periods ($\dot{\Pi}$) for low-mass ($M_{\star}/M_{\sun} \lesssim 0.45$) and
extremely low-mass (ELM, $M_{\star}/M_{\sun} \lesssim 0.18-0.20$) white-dwarf
stars, based on fully evolutionary low-mass He-core white dwarf and pre-white
dwarf models. Our analysis is based on a large set of adiabatic periods of
radial and nonradial pulsation modes computed on a suite of low-mass He-core
white dwarf and pre-white dwarf models with masses ranging from $0.1554$ to
$0.4352 M_{\sun}$. We compute the secular rates of period change of radial
($\ell= 0$) and nonradial ($\ell= 1, 2$) $g$ and $p$ modes for stellar models
representative of ELMV and pre-ELMV stars, as well as for stellar objects that
are evolving just before the occurrence of CNO flashes at the early cooling
branches. We found that the theoretically expected magnitude of $\dot{\Pi}$ of
$g$ modes for pre-ELMVs are by far larger than for ELMVs. In turn, $\dot{\Pi}$
of $g$ modes for models evolving before the occurrence of CNO flashes are
larger than the maximum values of the rates of period change predicted for
pre-ELMV stars. Regarding $p$ and radial modes, we found that the larger
absolute values of $\dot{\Pi}$ correspond to pre-ELMV models. We conclude that
any eventual measurement of a rate of period change for a given pulsating
low-mass pre-white dwarf or white dwarf star could shed light about its
evolutionary status. Also, in view of the systematic difficulties in the
spectroscopic classification of stars of the ELM Survey, an eventual
measurement of $\dot{\Pi}$ could help to confirm that a given pulsating star is
an authentic low-mass white dwarf and not a star from another stellar
population.