# Pulsating low-mass white dwarfs in the frame of new evolutionary sequences: IV. The secular rate of period change

## TL;DR

This paper provides theoretical predictions for the rates of period change in pulsating low-mass white dwarfs, aiding in their evolutionary classification and understanding stellar evolution in these objects.

## Contribution

It offers the first comprehensive calculation of secular period change rates for low-mass and extremely low-mass white dwarf models, including pre-white dwarf stages.

## Key findings

- Pre-ELMV $g$ modes have larger $\dot{\Pi}$ than ELMV.
- Pre-CNO flash models show even larger $\dot{\Pi}$ values.
- Measurement of $\dot{\Pi}$ can determine evolutionary status and confirm low-mass white dwarf nature.

## 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.