Distinguishing Core and Shell Helium-burning Subdwarf B Stars by Asteroseismology

The stellar models were calculated with version 9575 of the Modules for Experiments in Stellar Astrophysics (MESA) code (Paxton et al. 2011, 2013, 2015). MESA provides abundant applications in computational stellar astrophysics including evolving stars through the He-flash stage, which is considered to be a crucial part of the evolutionary path to sdB stars. Details of the input physics used in this work are described in Guo & Li (2018). By default, we have not considered diffusion and overshoot mixing in our models unless stated otherwise.

Our procedure of constructing initial canonical sdB (ZAEHB) models is the same as that of Guo & Li (2018). First, we used the MESA code to evolve a model with an initial mass of 1 M_⊙ and initial abundances of X = 0.7 and Z = 0.02 from the pre-main-sequence phase until the beginning of the He-flash stage. Then, we imposed an enhanced mass-loss wind with η_Reimers ∼ 1000 to strip matter from the hydrogen envelope until the beginning of stable core helium-burning. We generated seven initial canonical sdB models with specified values of hydrogen envelope masses, which are in the range of ${M}_{H}={10}^{-4}\mbox{--}{10}^{-2}\,{M}_{\odot }$. After constructing these initial sdB models, we evolved them through the core helium-burning phase until the shell helium was exhausted. The input physics we chose in this stage remains the same as in the previous stage. But the stellar wind is canceled because it is weak during the sdB stage.

These canonical sdB models have hydrogen-rich envelopes that are too thin to sustain shell hydrogen-burning. Their evolutionary tracks have a distinctive shape as shown in Figure 1. In this figure, we computed seven sdB models with a constant helium core mass of 0.4619 M_⊙ and different envelope masses of 3 × 10⁻⁴, 6 × 10⁻⁴, 1 × 10⁻³, 2 × 10⁻³, 3 × 10⁻³, 4.5 × 10⁻³, and 6 × 10⁻³ M_⊙. Their evolution spans from ZAEHB to the shell helium-burning phase. They were chosen to cover the whole $\mathrm{log}g\mbox{--}{T}_{\mathrm{eff}}$ band where sdB stars are normally found.

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The sdB evolution in our models includes the core and shell helium-burning phases. We found that there are some clear differences between the early and late evolution of the core helium-burning phase and the shell helium-burning phase. For the convenience of explanation, we divided the whole sdB evolution into four phases here: (i) the stable core helium-burning phase, (ii) the phase of core helium exhaustion, (iii) the short-lived phase of shell helium ignition, and (iv) the stable shell helium-burning phase. In Figure 1, the third evolutionary track is divided into four segments that represent the four phases. The solid, dashed, crossed, and dashed–dotted lines in the evolutionary trace correspond to phases (i)–(iv), respectively.

Figure 2 reflects the varying radius for the model with a hydrogen-rich envelope mass of $1\times {10}^{-3}\,{M}_{\odot }$ during the sdB evolution. We can see how the stellar radius is changing. In the stable core helium-burning phase (solid line), the core helium-burning luminosity is gradually increasing and so is the stellar radius. When the central helium content reduces to about 0.2, the sdB star goes to the phase of core helium exhaustion (dashed line). In this phase, the core helium-burning luminosity begins to reduce and the sdB star begins to shrink. When the central helium is exhausted, the C/O core begins to degenerate and the star goes to the short-lived phase of shell helium ignition (crossed line). During this phase, the shell helium-burning luminosity increases quickly and the star expands rapidly. After that, the star moves to the stable shell helium-burning phase (dashed–dotted line).

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In the core helium-burning phase, which includes the stable core helium-burning phase and the phase of core helium exhaustion, the sdB model has a convective core about 0.1 M_⊙. However, in the shell helium-burning phase, which include the short-lived phase of shell helium ignition and the stable shell helium-burning phase, the central convective core has vanished. This is one of the main differences between core helium-burning sdB stars and shell helium-burning sdB stars.

The pulsation periods of the short-period p-mode sdB pulsators are inversely proportional to the square root of the average density:

$$\begin{eqnarray}&&P\propto \displaystyle \frac{1}{\sqrt{G\bar{\rho }}}\propto {R}^{3/2}{({GM})}^{-1/2}.\end{eqnarray} \tag{ 1 }$$

So, the rates of change of period ($\dot{P}$ or dP/dt) of p modes are mainly determined by the change in the stellar radius. In Figure 3, we computed the pulsation periods of the p modes for the model with a hydrogen-rich envelope mass of $1\times {10}^{-3}\,{M}_{\odot }$ during the sdB evolution. We can see the changing trend of their periods in this figure, which is the same as the changing trend of their radius as shown in Figure 2. On the left of the vertical dashed line, i.e., in the stable core helium-burning phase, the p-mode periods are gradually increasing. That means that $\dot{P}$ is positive and small in this phase. Between the vertical dashed line and the vertical dotted line, i.e., in the phase of core helium exhaustion, their periods are rapidly decreasing. That means that $\dot{P}$ is negative and large in this phase. Between the vertical dotted line and the vertical dashed–dotted line, i.e., in the short-lived phase of shell helium ignition, their periods are rapidly increasing. That means that $\dot{P}$ is positive and large in this phase. On the right of the vertical dashed–dotted line, i.e., in the stable shell helium-burning phase, their periods are slowly changing. That means that $\dot{P}$ is small in this phase as in the stable core helium-burning phase.

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In our models, we have adjusted the time step and allowed the rates of change of period to be accurately calculated in any phase of the sdB stage. In Figure 4, we show the rates of change of period for p modes during the sdB evolution. We can see that the values of $\dot{P}$ during the stable core helium-burning phase and the stable shell helium-burning phase are all at a magnitude of 10⁻¹⁴. However, the values of $\dot{P}$ during the phase of core helium exhaustion and the short-lived phase of shell helium ignition are remarkably bigger. They could reach a magnitude of 10⁻¹², which is about two orders of magnitude bigger than in the stable core and shell helium-burning phases. In addition, the values of $\dot{P}$ during the short-lived phase of shell helium ignition are all positive, while those during the phase of core helium exhaustion are all negative.

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In our work, we just displayed the pulsation properties of the model with a hydrogen-rich envelope mass of $1\times {10}^{-3}\,{M}_{\odot }$ during the sdB stage. Actually, the other models with different hydrogen-rich envelope masses display the same pulsation properties in the four phases. That is because all the models have the same physics processes (the varying trend of the burning luminosity and radius during the sdB stage) as described in Section 2.3. In the other models, the values of $\dot{P}$ during the stable shell helium-burning phase are also all at a magnitude of 10⁻¹⁴ and the values of $\dot{P}$ during the short-lived phase of shell helium ignition are also positive and could reach a magnitude of 10⁻¹². In addition, the values of the $\dot{P}$ during the phase of core helium exhaustion in the other models are also negative and significantly bigger than the values of $\dot{P}$ in the stable core and shell helium-burning phases.

Given that the values of $\dot{P}$ are clearly different in the four phases of the sdB evolution, we may distinguish sdB stars in the phase of core helium exhaustion and the short-lived phase of shell helium ignition from those in the stable core and shell helium-burning phases by the measurements of $\dot{P}$. Because some sdB pulsators have stable pulsation properties, their pulsation periods change linearly in time as they evolve, which could be presented in an O–C (observed-minus-calculated) diagram. It is therefore possible to measure their $\dot{P}$ as has been done for white dwarf pulsators (Costa et al. 1999; Mukadam et al. 2003; Kepler et al. 2005). Fortunately, there are two p-mode sdB pulsators—Feige 48 and V391 Pegasi—that have had their $\dot{P}$ measured (Reed et al. 2004; Silvotti et al. 2007). Reed et al. (2004) have measured $\dot{P}$ to constrain the evolutionary timescale for Feige 48. They used about three years of phase-stable data to place an upper limit on the magnitude of $\dot{P}$ (about 3.5 × 10⁻¹³). Thus, they provided a lower limit on the evolutionary timescale (P/$\dot{P}$) of 3.1 × 10⁷ yr, which is consistent with the canonical evolutionary timescale for sdB stars in the phase of steady core helium-burning. Silvotti et al. (2007) have measured $\dot{P}$ for two main pulsation modes of V391 Pegasi and found that their periods are increasing at rates of ${\dot{P}}_{1}=(1.46\pm 0.07)\times {10}^{-12}$ and ${\dot{P}}_{2}=(2.05\pm 0.26)\times {10}^{-12}$. The two $\dot{P}$ correspond evolutionary timescales for increasing radius of about 7.6 × 10⁶ yr and 5.5 × 10⁶ yr, respectively. These are much faster than the canonical evolutionary timescale for sdB stars in the phase of steady core helium-burning. However, $\dot{P}$ and the evolutionary timescale of the two modes fit well with the modes of the short-lived phase of shell helium ignition. Only sdB stars in this phase have positive $\dot{P}$ and reach a magnitude of 10⁻¹², according to our calculation. So we thought that the p-mode sdB pulsator V391 Pegasi is likely in the short-lived phase of shell helium ignition. Kawaler (2010) has used some full evolutionary models to fit the periods and their $\dot{P}$ of the two modes, within spectroscopic constraints. Kawaler was surprised to find a model that is burning helium in the shell and has both periods and $\dot{P}$ close to the observation. So, he suggested that the star may be a shell helium-burning star. In our work, we found it is only sdB stars in the short-lived phase of shell helium ignition (i.e., the initial phase of the shell helium-burning evolution) that could have their $\dot{P}$ of p modes at a magnitude of 10⁻¹² and positive. In this phase, they are beginning to burn helium in the shell. Their shell helium-burning luminosity is quickly increasing, which results in rapid expansion of their stellar radius and a very large $\dot{P}$. In other phases, the $\dot{P}$ of p modes is all negative or is all at a magnitude of 10⁻¹⁴.

In addition to the scaling relationship of Equation (1), the values of $\dot{P}$ also depend on the eigenfunctions. The shapes of the eigenfunctions relate to the distribution of oscillation energy. They determine the pulsation periods, and the changing of the eigenfunctions determines the rates of change of the periods. The eigenfunctions of the trapped modes that experience mode bumping and the mixed modes that experience avoided crossing may change relatively quickly with evolution. So the mode bumping and avoided crossing will influence the measurements of $\dot{P}$. The pulsation properties of the mixed modes could change from p modes to g modes or from g modes to p modes during the avoided crossing. Their eigenfunctions are normally changing more quickly than those of trapped modes, and so do their $\dot{P}$. However, we thought that it would be difficult for them to influence $\dot{P}$ by two orders of magnitude. We will show this in Figure 6 of Section 3.2.

Normally, short-period sdB variables are pure p-mode pulsators. A few of them may have mixed character (Kawaler 1999; Charpinet et al. 2000; Hu et al. 2009). That means they display g-mode character in the core and p-mode character near the surface. Charpinet et al. (2000) have shown that mixed modes are generally absent in sequences of models with relatively thin hydrogen-rich envelopes during the core helium-burning phase. Even models with relatively thick hydrogen-rich envelopes may have just a slight avoided crossing between the low-order p modes and the low-order g modes during this phase. However, the p-mode pulsators usually have a high temperature (more than 28,000 K) or relatively thin hydrogen-rich envelopes. So it is impossible to find many mixed modes during the core helium-burning phase.

Figure 5 indicates that the model with a hydrogen-rich envelope mass of $1\times {10}^{-3}\,{M}_{\odot }$ has no avoided crossing during the core helium-burning phase, while it displays frequent avoided crossings during the shell helium-burning phase. We can see that the p-mode periods are all shorter than the g-mode periods during the core helium-burning phase. However, the g-mode periods decrease rapidly during the late phase of core helium-burning evolution, where the convective core disappears. In the shell helium-burning phase, the periods of the low-order g modes begin to overlap with those of the low-order p modes, which causes many avoided crossings. That means that many mixed modes will appear in the shell helium-burning phase. This phenomenon is also widely presented in other canonical models with different hydrogen-rich envelope masses. In addition, we can see from Figure 5 that the periods of all the low-order p modes and g modes change gradually with evolution, even in the time of avoided crossings during the shell helium-burning phase. Figure 6 indicates the values of $\dot{P}$ for the two modes p₂ and p₃ as labeled in Figure 5 that experience avoided crossings during the shell helium-burning phase. This figure shows that the appearance of the avoided crossings cannot influence $\dot{P}$ by two orders of magnitude.

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Searching for mixed modes in the shell helium-burning phase requires accurate mode identification. However, neither the second-generation (2G) nor third-generation (3G) models (Charpinet et al. 2001; Brassard & Fontaine 2008) are suitable for identifying the mixed modes in the shell helium-burning phase. The 2G models (Charpinet et al. 2001) could in principle fit sdB stars in the shell helium-burning phase. But the mixed modes have g-mode character in the core, which the 2G models cannot properly simulate. The 3G models (Brassard & Fontaine 2008) are suitable for both p modes and g modes, but are limited to sdB pulsators in the core helium-burning phase.

The special sdB pulsator PG 1605+072 has a large amplitude and an abnormally rich pulsation spectrum. Its unusual location on the T_eff–$\mathrm{log}g$ diagram may be explained in two different ways (Van Grootel et al. 2010). The first explanation invokes a more evolved status, which is beyond the exhaustion of core helium. The second explanation considers a very high mass, which implies a noncanonical formation channel for PG 1605+072. Many attempts have been made to derive its seismic references (Van Spaandonk et al. 2008; Van Grootel et al. 2010). But they failed to derive a convincing seismic model or their results conflicted with each other. However, all these works—whether 2G models (Van Spaandonk et al. 2008) or 3G models (Van Grootel et al. 2010)—supported that the idea that the star has many mixed modes. Although the 2G and 3G models cannot properly simulate sdB stars in the shell helium-burning phase, the abnormally dense pulsation spectrum of PG 1605+072 seems to indicate that it cannot be explained by p modes but rather by many mixed modes. So we guessed that PG 1605+072 is likely a shell helium-burning sdB star. But improved models are still required to affirm this.

Recently, more than a dozen g-mode sdB pulsators have been found by the space missions CoRoT and Kepler. They showed very rich and accurate frequency spectra, which led to some important discoveries. Almost all of them displayed regular period spacings and their period spacings have been measured (Reed et al. 2011; Østensen et al. 2012; Bachulski et al. 2016; Baran et al. 2017; Ketzer et al. 2017).

Figure 7 shows the period spacings of l = 1 g modes for the model with a hydrogen-rich envelope mass of $1\times {10}^{-3}\,{M}_{\odot }$. We can see that the period spacings are significantly different between the core and shell helium-burning phases. In the core helium-burning phase, the period spacings are about 220 s. However, in the shell helium-burning phase, they drop rapidly to about 60 s. This is because the central convective core has vanished in the shell helium-burning phase. This property, i.e., the clear changing of period spacings during the two phases, is not related to the masses of the hydrogen-rich envelope and core. It provides a way to distinguish between core and shell helium-burning sdB stars. This method is the same as that which Bedding et al. (2011) have used to distinguish between hydrogen- and helium-burning red giant stars.

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Actually, all observed g-mode sdB pulsators have period spacings of l = 1 modes spanning from 230 to 275 s (Reed et al. 2011; Østensen et al. 2012; Bachulski et al. 2016; Baran et al. 2017; Ketzer et al. 2017), which is bigger than the computed values of the canonical models. Constantino et al. (2015) and Ghasemi et al. (2017) have matched some trapped modes (Østensen et al. 2014) in KIC 10553698A by considering core overshooting in their models. They found that the period spacings of their models fit well with the observed value of KIC 10553698A. In our model with a hydrogen-rich envelope mass of $1\times {10}^{-3}\,{M}_{\odot }$ in Figure 8, we added a small core overshoot with ${f}_{\mathrm{ov}}={10}^{-5}$. In the core overshooting model, the convective core grows more than twice as much as in the previous canonical model during the core helium-burning phase. The development of the convective core in our model is similar to the case shown in Ghasemi et al. (2017). The convective zone is periodically divided into an inner convective core and an outer convective shell. In Figure 9, the period spacings of the models during the core helium-burning phase fit well with the observations except for the initial 20 million years. In the shell helium-burning phase, the period spacings are also rapidly decreasing to about 50 s as shown in Figure 7.

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Unfortunately, for current observations, there is no one g-mode sdB pulsator that remains in the shell helium-burning phase.