The Formation of Blue Large-amplitude Pulsators from White-dwarf Main-sequence Star Mergers

Pulsating stars have been identified in diverse groups across the Hertzsprung–Russell (H-R) diagram (Jeffery & Saio 2016; Kurtz 2022). Using periodic light variations, asteroseismology is used to measure basic parameters and to infer the internal structure of pulsating stars (Bedding et al. 2011; Huber et al. 2011; Chaplin & Miglio 2013). Due to the large-scale surveys, more and more pulsating stars have been discovered. Many new classes have been identified in recent years, including the blue large-amplitude pulsators (BLAPs) discovered by Pietrukowicz et al. (2017).

BLAPs have several interesting properties:

• (1)  
  BLAPs have periods in the range of 2–60 minutes. As their name implies, their light variations show a larger amplitude than other early-type pulsators with similar periods, i.e., from 0.05–0.4 mag at optical wavelengths. In general, it is difficult for short-period stars to drive a high amplitude; e.g., the pulsating Ap stars have periods in the range of 5–24 minutes and only have relatively lower amplitudes from 0.001–0.02 mag.
• (2)  
  BLAPs have higher effective temperatures (25,000–34,000 K) and gravities ($\mathrm{log}(g/\mathrm{cm}\,{{\rm{s}}}^{-2})$ = 4.2–5.7) than classical Cepheids and RR Lyrae-type stars but have similar sawtooth light curves.
• (3)  
  The periods of BLAPs are not constant; a key observation is the detection of period changes with both positive and negative signs observable on timescales of years (Pietrukowicz et al. 2017). These imply that BLAPS are observed in both rapid expansion and contraction phases, which evolution models must be capable of explaining.
• (4)  
  Most BLAPs have an enrichment of surface helium ($\mathrm{log}n(\mathrm{He})/n({\rm{H}})$ in the range of −2.4 to −0.4), which is similar to intermediate helium-rich (iHe-rich) hot subdwarfs.
• (5)  
  The observed BLAPs have been divided into two groups: (a) low-gravity BLAPs (classical BLAPs) with pulsation periods in the range of 20–60 minutes, $\mathrm{log}(g/\mathrm{cm}\,{{\rm{s}}}^{-2})$ = 4.2–4.7, and amplitudes 0.2–0.4 mag; (b) high-gravity BLAPs, with periods in the range of 2–8 minutes, $\mathrm{log}(g/\mathrm{cm}\,{{\rm{s}}}^{-2})$ = 5.3–5.7, and amplitudes 0.05–0.2 mag. The high-gravity BLAPs reported by Kupfer et al. (2019) have spectral properties and pulsation periods similar to p-mode hot subdwarfs (Østensen et al. 2010). Unlike most BLAPs, TMTS-BLAP-1 is located in the period gap between low-gravity and high-gravity BLAPs (Lin et al. 2022). BLAP OW-BLAP-1 is also found in the gap (Ramsay et al. 2022). Thus, the gap between low-gravity and high-gravity BLAPs may not be real.
• (6)  
  They are rare, and only HD 133729 has been clearly identified as binary (Pigulski et al. 2022). Whether any other BLAPs are binaries is not known.

Thus, a successful model of BLAPs must explain the formation channel, the driving mechanism of pulsation, the enrichment of helium abundance, the relation between low-gravity and high-gravity BLAPs, period changes, and space density. In the standard stellar evolution theory, obtaining a single model to explain all of these features is difficult.

Based on stellar structure, two principal scenarios to explain BLAPs were proposed by Pietrukowicz et al. (2017); several authors have tried to reproduce the pulsational properties using models consistent with one or both scenarios. (1) The shell-hydrogen-burning model: low-mass pre-white dwarfs (∼0.3 M_⊙) with hydrogen-burning shells pass through the BLAP instability zone, where the opacity of iron group elements drives their pulsations (Byrne & Jeffery 2018, 2020; Romero et al. 2018; Wu & Li 2018; Byrne et al. 2021). (2) The helium core burning model: BLAPs have similar effective temperatures but surface gravities lower than those associated with the extended horizontal branch. Hence, BLAPs could be evolving toward or away from the extended horizontal branch, either as pre- or post-hot subdwarfs (Wu & Li 2018; Kupfer et al. 2019; Meng et al. 2020; Lin et al. 2022; Xiong et al. 2022). Both models can represent some features of BLAPs, but not all.

On the H-R diagram, BLAPs are closely associated with the area occupied by hot subdwarfs. Hot subdwarf stars can be roughly divided into subdwarf B (sdB), subdwarf O (sdO), and subdwarf OB (sdOB) by spectrum (Heber 2009, 2016). Most hot subdwarfs have a nearly pure hydrogen surface. Some 10% of hot subdwarfs have a surface helium abundance >90% by number (Drilling et al. 2013; Luo et al. 2016, 2021; Lei et al. 2019, 2023). The formation channel of most hot subdwarfs is well explained by the interaction of binaries (Han et al. 2002, 2003; Zhang & Jeffery 2012). Between the H-rich and He-rich type of stars, a small number of hot subdwarfs have a surface helium number fraction of 10%–90% and are referred to as iHe-rich hot subdwarfs (Ahmad & Jeffery 2003; Naslim et al. 2010; Jeffery et al. 2021). It is not known whether these represent an intermediate state of either H-rich or He-rich subdwarfs, some other evolution channel, or a combination of several channels. For example, it is possible that they formed from a helium white dwarf (HeWD) merged with a main-sequence (MS) companion (Zhang et al. 2017). BLAPS show a similar surface abundance to the iHe-rich hot subdwarfs. It is therefore helpful to investigate the properties and formation channels of iHe-rich subdwarfs as a possible channel for the formation of BLAPs.

One such channel is the white dwarf MS merger. Many short-period detached binary systems consist of a HeWD with an MS companion (Zorotovic et al. 2011). For example, the Sloan Digital Sky Survey J121010.1+334722.9 is a cool 0.4 M_⊙ HeWD with a 0.16 M_⊙ M dwarf companion in a 3 hr eclipsing binary. Owing to a combination of gravitational-wave radiation, tidal interaction, and magnetic braking, the orbital period and separation of such a binary can decrease over time, and as a consequence, the MS star may fill its Roche lobe. If the MS star has a low mass, M_MS ≤ 0.7 M_⊙, the mass transfer is expected to be dynamically unstable and lead to a merger (Hurley et al. 2002; Shen et al. 2009). The immediate products of these HeWD+MS mergers are expected to be red giant branch-like (RGB-like) stars (Hurley et al. 2002). Some such remnants are expected to ignite helium with a low envelope mass and thus become hot subdwarfs (Clausen & Wade 2011). Zhang et al. (2017) found that some of the mergers result in the formation of hot subdwarfs with iHe-rich surfaces.

In the Zhang et al. (2017) HeWD+MS merger model, the star takes a few tens of megayears following the merger before reaching the He-burning MS (or zero-age extended horizontal branch). The evolutionary tracks pass through the region of the H-R diagram occupied by BLAPs, and the surface abundance of helium, representing a mixture of hydrogen and helium, approximately matches that of BLAPs for which measurements exist. This paper therefore examines the products of HeWD+MS mergers in more detail in order to assess their candidacy as BLAPs. In Section 2, we introduce the post-merger evolution models. The comparison of theory with the observation of BLAPs is shown in section Section 3. The conclusions are given in Section 4.

Zhang et al. (2017) presented 29 HeWD+MS post-merger models. Some of the evolutionary tracks of post-merger pass through the region of the H-R diagram occupied by BLAPs. Zhang et al. (2017) did not analyze the models for pulsation stability. An opacity bump due to iron group element abundances enhanced by radiation levitation is required to drive BLAP pulsations (e.g., Byrne & Jeffery 2020). It is therefore worth investigating the pulsation properties of the Zhang et al. (2017) post-merger models with full chemical diffusion. We used the stellar evolution code mesa version 23.05.11 (Modules for Experiments in Stellar Astrophysics; Paxton et al. 2011, 2013, 2015, 2018, 2019) to calculate similar models to Zhang et al. (2017). We analyzed the pulsation stability of each model using GYRE (Townsend & Teitler 2013).

Zhang et al. (2017) describe how, after the merger, an RGB-like star forms in which the structure is a very degenerate He core surrounded by an extended hydrogen envelope. Following an evolution similar to that of normal RGB stars, hydrogen burns in a shell, and helium ash adds to the helium core. Meanwhile, mass is lost from the surface through a stellar wind. Fresh helium continues to be produced during RGB-like evolution to compress and heat the He core. Once the helium core is massive enough, the helium ignites in a shell, followed by a series of helium flashes propagating inward toward the star center. The first helium shell flash is the strongest. Its position near the surface of the degenerate He core drives a strong convection zone upward that reaches the surface, enriching the surface in helium forced upward from beneath to create an iHe-sdB star. The flash forces the star to expand, and then shrink as the shell luminosity drops again, so the evolution track performs loops in the H-R diagram (Sweigart 1997; Saio & Jeffery 2000; Brown et al. 2001). After a few megayears, the He-burning flame reaches the center and the star commences true core He burning, i.e., on the zero-age extended horizontal branch. While helium burns in the center, heavier elements near the surface diffuse downward and finally produce an almost pure hydrogen atmosphere. Thus, the remnants evolve to become hydrogen-rich single hot subdwarfs (H-sdB). The entire evolutionary track might be represented as RGB → iHe − sdB(BLAP) → H − sdB → WD.

As identified by Zhang et al. (2017), the evolution of HeWD+MS mergers can be divided into two paths corresponding to an early hot flasher or a late hot flasher (Lanz et al. 2004; Heber 2009, 2016): (1) late hot flasher: the hydrogen envelope is of very low mass, so flash-driven convection can yield a maximum helium surface abundance Y = 0.954 (mass fraction); (2) early hot flasher: the hydrogen shell re-ignites after the helium shell flash, the star expands, and initiates deep opacity-driven surface convection. In the combination of flash-driven convection followed by opacity-driven convection some helium and other newly produced elements are dredged to the surface, yielding a maximum surface abundance of Y = 0.636.

Analyzing each model in detail, we selected seven early hot flasher models as possible BLAPs, by having tracks that pass through a similar volume of gravity–temperature space and $\mathrm{log}n(\mathrm{He})/n({\rm{H}})\leqslant 0$. These models are, 0.250+0.650, 0.275+0.660, 0.300+0.660, 0.325+0.660, 0.350+0.650, 0.375+0.650, and 0.400+0.640 M_⊙, where the first and second quantity in each pair refers to the progenitor HeWD and MS mass, respectively. The masses for each model as they cross the BLAP region are 0.484, 0.492, 0.499, 0.511, 0.517, 0.527, and 0.543 M_⊙. Thus, the masses of BLAPs are in the range of 0.484–0.543 M_⊙.

Figure 1 shows a detailed post-merger evolutionary track of an early hot flasher model, i.e., a 0.250 + 0.650 M_⊙ remnant. The flash-driven loops span the region where BLAPs are observed and then reach the zone of sdB stars. Figure 2 shows an expanded section of the same evolutionary track. Part of the flash loop is marked by points (1)–(3), which represent key phases during the loop including (1) the peak of the helium flash, (2) the minimum radius, and (3) the He-shell luminosity minimum. The duration of the helium shell flash is short. It is initially accompanied by a halt in the envelope contraction (1) and a drop in total luminosity (1) and (2), followed by envelope expansion as heat generated in the flash is transmitted outward on a thermal timescale (2) and (3). Once the envelope reaches radiative equilibrium after the flash, contraction will resume at a total luminosity slightly less than before the flash.

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Having found that model HeWD+MS merger remnants can become BLAPs, we compare their properties to observed examples of such stars in more detail. Table 1 shows a sample of 24 confirmed BLAPs (Pietrukowicz et al. 2017; Kupfer et al. 2019; Lin et al. 2022; Pigulski et al. 2022; Ramsay et al. 2022). Since our model is for merged stars, which, unless they were originally in a triple or higher multiplicity system, should now be single, we do not include the binary HD 133729 for comparison (Pigulski et al. 2022). Figure 3 shows these observed BLAPs and the tracks for the seven possible BLAP models in both the $\mathrm{log}{T}_{\mathrm{eff}}$–$\mathrm{log}L$ and $\mathrm{log}{T}_{\mathrm{eff}}$–$\mathrm{log}g$ planes. Figure 4 compares models and observations in the $\mathrm{log}g$ surface helium abundance plane. This figure shows that the model HeWD+MS merger remnants can explain the surface helium abundance of BLAPs, with a strong preference for the lowest mass models: 0.250+0.650, 0.275+0.660, 0.300+0.660, 0.325+0.660, and 0.350+0.650 M_⊙. Figures 3 and 4 show that the spaces occupied by low-gravity and high-gravity BLAPs represent different stages of similar tracks, which indicates a possible evolutionary relation between both types of BLAPs. The TMTS-BLAP-1 and OW-BLAP-1 could also also associated with the evolution of merger models. At least, our theoretical models have no gap between low-gravity and high-gravity BLAPs.

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Table 1. List of BLAPs

No.	Name	p(minutes)	$\dot{p}/p({10}^{-7}\,{\mathrm{yr}}^{-1})$	Teff	$\mathrm{log}g$	$\mathrm{log}n(\mathrm{He})/n({\rm{H}})$	References
1	OGLE-BLAP-001	28.26	2.90 ± 3.70	30,800 ± 500	4.61 ± 0.07	−0.55 ± 0.05	(1)
2	OGLE-BLAP-002	23.29	−19.23 ± 8.05	⋯	⋯	⋯	(1)
3	OGLE-BLAP-003	28.46	0.82 ± 0.32	⋯	⋯	⋯	(1)
4	OGLE-BLAP-004	22.36	−5.03 ± 1.57	⋯	⋯	⋯	(1)
5	OGLE-BLAP-005	27.25	0.63 ± 0.26	⋯	⋯	⋯	(1)
6	OGLE-BLAP-006	38.02	−2.85 ± 0.31	⋯	⋯	⋯	(1)
7	OGLE-BLAP-007	35.18	−2.40 ± 0.51	⋯	⋯	⋯	(1)
8	OGLE-BLAP-008	34.48	2.11 ± 0.27	⋯	⋯	⋯	(1)
9	OGLE-BLAP-009	31.94	1.63 ± 0.08	31,800 ± 1400	4.40 ± 0.18	−0.41 ± 0.13	(1)
10	OGLE-BLAP-010	32.13	0.44 ± 0.21	⋯	⋯	⋯	(1)
11	OGLE-BLAP-011	34.87	6.77 ± 8.87	26,200 ± 2900	4.20 ± 0.20	−0.45 ± 0.11	(1)
12	OGLE-BLAP-012	30.90	0.03 ± 0.15	⋯	⋯	⋯	(1)
13	OGLE-BLAP-013	39.33	7.65 ± 0.67	⋯	⋯	⋯	(1)
14	OGLE-BLAP-014	33.62	4.82 ± 0.39	30,900 ± 2100	4.42 ± 0.26	−0.54 ± 0.16	(1)
15	High-gravity-BLAP-1	3.34	⋯	34,000 ± 500	5.70 ± 0.05	−2.1 ± 0.2	(2)
16	High-gravity-BLAP-2	6.05	⋯	31,400 ± 600	5.41 ± 0.06	−2.2 ± 0.3	(2)
17	High-gravity-BLAP-3	7.31	⋯	31,600 ± 600	5.33 ± 0.05	−2.0 ± 0.2	(2)
18	High-gravity-BLAP-4	7.92	⋯	31,700 ± 500	5.31 ± 0.05	−2.4 ± 0.4	(2)
19	OW-BLAP-1	10.8	⋯	30,600 ± 2500	4.67 ± 0.25	−2.1 ± 0.2	(3)
20	OW-BLAP-2	23.0	⋯	27,300 ± 1500	4.83 ± 0.20	−0.7 ± 0.1	(3)
21	OW-BLAP-3	28.9	⋯	29,900 ± 3500	4.16 ± 0.40	−0.8 ± 0.3	(3)
22	OW-BLAP-4	32.0	⋯	27,300 ± 2000	4.20 ± 0.20	−0.8 ± 0.2	(3)
23	TMTS-BLAP-1	18.9	22.3 ± 0.9	31,780 ± 350	4.90 ± 0.06	−0.66 ± 0.05	(4)
24	HD 133729	32.27	−11.5	29,000	4.5	⋯	(5)

References. (1) Pietrukowicz et al. (2017), (2) Kupfer et al. (2019), (3) Ramsay et al. (2022), (4) Lin et al. (2022), (5) Pigulski et al. (2022).

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We use the oscillation code GYRE to analyze the stability of the models. The input model for GYRE is obtained from the MESA calculation and then uses a nonadiabatic analysis to investigate the stability of each mode. We only calculate the stability (η) and period (P) for radial modes (l = 0) as the BLAPs are known to be large-amplitude pulsators. Modes with η > 0 are unstable. Figure 5 shows the logarithm of opacity (log(κ)) and the work function (dW/dx) as a function of interior temperature for a model around point 2 of Figure 2. The mass fractions of iron and nickel are also included. The location of the peak in dW/dx coincides with the area of maximum opacity, which is related to the ionization of iron and nickel around $\mathrm{log}(T/{\rm{K}})=5.35$. Thus, this unstable mode is driven by the κ-mechanism related to the opacity bump due to the accumulation of iron group elements in the envelope. We also check the models without radiative levitation, which modes are always stable. These results agree with previous studies in which the modes are only excited when radiative levitation is included (Byrne & Jeffery 2018, 2020; Romero et al. 2018). Figure 6 shows the pulsation stability of the seven models. The locations of unstable models are shown by colored circles. Sections of the tracks shown in Figure 3 that are stable against pulsation are shown as dotted lines. The instability region satisfactorily coincides with the region where BLAPS are observed and indicates why BLAPS might not be observed outside this region.

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BLAPs are considered to be rapidly evolving stars. Both the pulsation period and the rates of period change are important features to compare with observation. We calculate the period P and the corresponding rates of period change r for the fundamental radial mode from our models. Comparing P, M, and R, the classical period mean–density relation (Eddington 1918) for radial-mode pulsations,

$$\begin{eqnarray}&&P\approx {Q}_{{\rm{F}}}/\sqrt{\displaystyle \frac{M}{{M}_{\odot }}{\left(\displaystyle \frac{{R}_{\odot }}{R}\right)}^{3}}\,\mathrm{minutes},\end{eqnarray} \tag{ 1 }$$

holds with the pulsation constant for the fundamental radial mode Q_F ≈ 47 minutes.

Figure 7 shows the observed periods compared with models in the $P\mbox{--}\mathrm{log}{T}_{\mathrm{eff}}$ and $P\mbox{--}\mathrm{log}g$ planes. In the $P\mbox{--}\mathrm{log}{T}_{\mathrm{eff}}$ plane, the locations of all observed BLAPs are in the region of our seven proposed possible merger models. Meanwhile, the same models can explain all of the BLAPs in the $P\mbox{--}\mathrm{log}g$ plane; 6 out of 13 lie within 1σ (68% confidence interval), 12 lie within 2σ (95%), and all lie within 3σ (99%) of the model predictions.

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Using the same models, we calculate the expected period change r_i from

$$\begin{eqnarray}&&{r}_{i}\equiv \displaystyle \frac{\dot{P}}{P}=\displaystyle \frac{{\rm{\Delta }}P}{{\rm{\Delta }}t}\displaystyle \frac{1}{P}=\displaystyle \frac{{P}_{i+1}-{P}_{i}}{{t}_{i+1}-{t}_{i}}\displaystyle \frac{1}{{P}_{i+1}},\end{eqnarray} \tag{ 2 }$$

where i refers to each model along a track. Figure 8 shows the rates of period change in the P–r plane. It is a feature of post-merger models that, like BLAPs, show both positive and negative values of r as a consequence of the r–T_eff loops associated with helium shell flashes. The 15 BLAPs with measured r all lie within the range covered by the theoretical values for the seven possible BLAP models: ∼ −10⁻⁵ to ∼10⁻⁵ yr⁻¹. Figure 8 highlights the loop covering one helium shell flash for the 0.250 + 0.650 M_⊙ post-merger model already shown on the $\mathrm{log}{T}_{\mathrm{eff}}\mbox{--}\mathrm{log}g$ plane in Figure 2. This demonstrates how the period decreases (negative $\dot{P}$) as the star contracts, but briefly increases (positive $\dot{P}$) during the shell flash itself.

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From Figures 3–8, most properties of BLAPs can be reasonably well reproduced by our merger model. Furthermore, the TMTS-BLAP-1 was identified as the "Hertzsprung gap" of Hot Subdwarfs (Lin et al. 2022). Here, we suggest that TMTS-BLAP-1 also formed from a merger and is currently in the pre-sdB stage, as shown in our models.

From an analysis of the HeWD+MS post-merger models, we have found that some of the early flasher models provide a possible channel for the origin of BLAPs. Such mergers had previously been identified as a channel to form iHe-rich hot subdwarfs, and that is confirmed here. Some of these models transit the parameter space occupied by BLAPs during their helium shell flash phase prior to full helium core burning. At this point a star represented by these models will become an iHe-rich sdB star and eventually, the heavier elements will sink to leave an H-rich surface as the star approaches the He-burning MS as a single H-rich sdB star.

Analysis of the distribution of post-merger evolution tracks shows that predictions for HeWD+MS mergers are consistent with observations of almost all recent BLAPs in terms of surface effective temperature (T_eff), surface gravity ($\mathrm{log}\,g$), surface luminosity ($\mathrm{log}\,L$), surface helium abundance, period (P), and rates of period change (r). Significantly, because of the cyclic expansion, and contraction of helium shell-flashing stars, the models predict both negative and positive values for the period change, as observed.

BLAPS that evolve from mergers are likely to be single stars. It may be that BLAPs, especially BLAPs in binaries, can also be formed in other channels or formed in triple systems and leave behind a merged star in a binary (Preece et al. 2022). Recently, the BLAP HD 133729 was found to be binary. We hope that further observations will identify whether any other BLAPs have companions.

We thank the referee for the helpful suggestions and comments that improved the manuscript. This work is supported by grant Nos. 12073006, 12090040, 12090042, 12288102, 12133011, and 11833006 from the National Natural Science Foundation of China, the Joint Research Fund in Astronomy (U2031203) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and Chinese Academy of Sciences (CAS) and the CAS "Light of West China" Program. We also acknowledge the science research grant from the China Manned Space Project with No. CMS-CSST-2021-A10. Armagh Observatory and Planetarium is supported by a grant from the Northern Ireland Department for Communities. X.Z. thanks Jie Lin and Tao Wu for the helpful conversations.