2.1. BSG merger products via fast accretion

In Henneco et al. (Reference Henneco, Schneider, Hekker and Aerts2024a), we used an approximate 1D method to emulate a merger between a post-MS star and its companion. This method, commonly referred to as ‘fast accretion’, consists of accreting a certain amount of mass onto a 1D single-star model at a given time during its evolution, in this case right after the MS. Since mergers are dynamical processes, the aim is to make the accretion timescale as short as possible, while still safeguarding the numerical convergence of the model. To this end, a timescale of 10% of the star’s global thermal timescale was chosen. The resulting stellar model is then further evolved until the end of core helium burning. These evolutionary calculations and the fast accretion method were all conducted using the <momospace>MESA</momospace> stellar evolution code (Paxton et al. Reference Paxton, Smolec and Schwab2019, and references therein). For more details, we refer to Sect. 3 of Henneco et al. (Reference Henneco, Schneider, Hekker and Aerts2024a).

2.2. MS merger product from a 3D MHD simulation

As detailed in Henneco et al. (Reference Henneco, Schneider, Heller, Hekker and Aerts2025), we used the outcome of the 3D magnetohydrodynamic (MHD) stellar merger simulation from Schneider et al. (Reference Schneider, Ohlmann and Podsiadlowski2019) as a starting point for our stellar evolution calculations with <momospace>MESA</momospace>. Following the methodology of Schneider et al. (Reference Schneider, Ohlmann and Podsiadlowski2020), the 3D merger product structure was mapped to 1D and relaxed in <momospace>MESA</momospace>. After this relaxation, we modelled the MS evolution of the 1D merger product model until the terminal-age main sequence (TAMS). For more details on the setup of these calculations, we refer the reader to Sect. 3 of Henneco et al. (Reference Henneco, Schneider, Heller, Hekker and Aerts2025).

Since the goal of both works was to compare the asteroseismic properties of merger products with those of genuine single stars, we computed suites of genuine single-star evolution models with <momospace>MESA</momospace> and selected those models that crossed (in the case of the BSG merger product) or overlapped with (in the case of the MS merger product) the HRD evolution tracks of the merger products. This was done with the following question in mind: If a genuine single star and a merger product have exactly the same surface properties, will their asteroseismic properties allow us to distinguish them?

2.3. Stellar pulsation predictions

One of the main asteroseismic diagnostics for the type of stars under consideration, is the spacing between the periods of their gravity-mode (g-mode) pulsations. These g modes have the buoyancy force as their restoring force and, as a result, become evanescent in regions where the stellar matter is unstable against convection. The power of this spacing is that, in the simplest, non-rotating, non-magnetic, chemically homogeneous case, for modes with a large number of radial nodes (asymptotic regime), it is constant and related to the size, shape and chemical composition inside of the mode cavity, which roughly corresponds to the radiative region(s) in the star (Aerts et al. Reference Aerts, Christensen-Dalsgaard and Kurtz2010). The asymptotic period spacing $\Pi_{\ell}$ can be computed directly from the stellar structure models through (Tassoul Reference Tassoul1980)

(1) \begin{equation} \Pi_{\ell} = \frac{\Pi_{0}}{\sqrt{\ell(\ell+1)}}\,,\quad \text{with} \quad \Pi_{0} = \pi\left(\int_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}\frac{\tilde{N}}{r}\mathrm{d}r\right)^{-1} \, .\end{equation}

Here, $\tilde{N}$ is the Brunt-Väisälä (BV) or buoyancy frequency, $\ell$ the spherical degree of the modes in question, r the radial coordinate, and $r_{\mathrm{i}}$ and $r_{\mathrm{o}}$ are the inner and outer boundaries of the g-mode cavity, respectively.

What makes the period spacing even more powerful, is that any deviations from the conditions posed above, such as chemical gradients, internal magnetic fields and rotation, cause deviations from its constant behaviour. These deviations become apparent in so-called ‘period spacing patterns’, which show the spacings between the periods of modes of the same degree but of consecutive radial orders in terms of the radial order or mode period, and can be used to constrain the interiors of stars (see, e.g. Miglio et al. Reference Miglio, Montalbán, Noels and Eggenberger2008, Bouabid et al. Reference Bouabid, Dupret and Salmon2013, Van Reeth et al. Reference Van Reeth, Tkachenko and Aerts2016, Pápics et al. Reference Pápics, Tkachenko and Van Reeth2017, Cunha et al. Reference Cunha, Avelino and Christensen-Dalsgaard2019, Li et al. Reference Li, Van Reeth and Bedding2020, and Pedersen et al. Reference Pedersen, Aerts and Pápics2021). To construct theoretical period spacing patterns, we require predictions for the individual pulsation modes, which we computed using the stellar pulsation code <momospace>GYRE</momospace> (Townsend & Teitler Reference Townsend and Teitler2013; Townsend et al. Reference Townsend, Goldstein and Zweibel2018). This code takes the stellar structure profiles from <momospace>MESA</momospace> (and other stellar evolution codes) as input.