The relation between helium white dwarf mass and orbital period under two types of opacity

TL;DR

The paper addresses how low-temperature radiative opacity influences the helium white dwarf mass–orbital period relation ($M_{ m WD}$–$P_{ m orb}$) in close binaries. Using the MESA stellar evolution code, the authors compare two opacity prescriptions, Ferguson (2005) and Freedman (2008/2014), across three metallicities to quantify shifts in the $M_{ m WD}$–$P_{ m orb}$ relation and provide metallicity-dependent fits. They find that Freedman (2008/2014) yields a systematically lower relation than Ferguson (2005) and better matches observations of extremely low-mass WDs and binary millisecond pulsars, ultimately supplying fitting formulae for $Z=0.02,0.001,0.0001$. These results refine the interpretation of WD binaries and improve predictive modeling of binary evolution, with the data and MESA inlists publicly available for reproduction.

Abstract

Helium white dwarfs (He WDs) are end products of low-mass red giant donors in close binary systems via stable mass transfer or common envelope evolution. At the end of stable mass transfer, there is a well-known relation between the He WD mass and orbital period. Although this relation has been widely investigated, the influence of different types of opacity at low temperatures is ignored. In this work, we modeled the evolution of WD binaries with stellar evolution code MESA and two types of opacity at low temperatures from Ferguson et al. (2005) and Freedman et al. (2008, 2014). We investigated the relation between the WD mass and orbital period and compared these results with observations. We find that the relation derived from the opacity of Freedman et al. (2008, 2014) is below that from the opacity of Ferguson et al. (2005) and the relation derived from the opacity of Freedman et al. (2008, 2014) can better explain the observations. In addition, we provided fitting formulae for the relations derived from the opacity of Freedman et al. (2008,2014) at different metallicities.

Figures (5)

• Figure 1: Evolution of a binary system with an initial donor mass of $1\,M_\odot$, an initial carbon--oxygen white dwarf mass of $1\,M_\odot$, and an initial orbital period of $4\,\mathrm{days}$, calculated with two different low-temperature radiative opacity tables, i.e., Ferguson2005 and Freedman2008Freedman2014. Top left panel: Hertzsprung--Russell diagram; Top right panel: Evolution of mass-transfer rate as a function of age; Bottom left panel: Evolution of the radius (blue color), stellar mass (pink color) and He core mass (brown color) as a function of age. Bottom right panel: Evolution of orbital period as a function of donor mass. In all the panels, the solid lines are for the models with the opacity of Ferguson2005 and the dashed lines are for the models with the opacity of Freedman2008Freedman2014. The red stars denote the onset of the mass-transfer phase, while the black squares indicate the end of the second mass transfer phase. In the upper right panel, the very short mass transfer phases at the end of evolution are due to the H-shell flashes (see the HR diagram).
• Figure 2: $M_{\mathrm{WD}}$–$P_{\mathrm{orb}}$ relations obtained with binary models with different opacity tables at solar metallicity. Here the orbital period and WD mass represents the values at the end of mass transfer. The yellow squares are for the low-temperature opacity table from Ferguson2005, while the blue pentagrams are for the opacity tables from Freedman2008Freedman2014. The red dashed line shows the $M_{\mathrm{WD}}$–$P_{\mathrm{orb}}$ relation for solar metallicity from Lin2011, and the yellow dash-dotted line denotes the relation for population I given by Tauris1999.
• Figure 3: The $M_{\rm WD}-P_{\rm orb}$ relations computed with the opacity of Freedman2008Freedman2014. The yellow squares, blue pentagons, and purple upward triangles represent the theoretical results for metallicities of Z = 0.02, 0.001, and 0.0001, respectively. The black, pink, and light blue solid lines correspond to the fitted relations for each of these metallicities.
• Figure 4: Comparison of the $M_{\rm WD}-P_{\rm orb}$ relation derived from the models with the opacity of Freedman2008Freedman2014 with the observations of extremely low-mass WD binaries. The dark blue, purple, and light blue symbols represent the theoretical results for metallicities Z = 0.02, 0.001 and 0.0001, respectively. At the low mass end, for a given WD mass, there is an orbital period range which is due to gravitational wave radiation. The upper limit corresponds to the orbital period at the end of mass transfer, while the lower limit is taken as the orbital period either at the onset of the mass transfer from He WD to the CO WD or at the Hubble time, whichever occurs first. The observational systems (i.e. black points with error bars) are selected from the Brown2020 sample using the method described in Lizw2019, representing systems formed via stable RLOF.
• Figure 5: Comparison of the $M_{\rm WD}-P_{\rm orb}$ relation derived from models with Freedman2008Freedman2014 opacity at solar metallicity with observations of BMSPs with He WD companions listed in Tab. \ref{['tab:hewd_bmsp']}. The relations by Tauris1999 and Lin2011 are over-plotted.