Evolution of AM CVn Binaries with White Dwarf Donors

In this work, we adopted the He WD models from our previous work about the evolution of NS+He binaries (Chen et al. 2022b). Here we briefly describe how these He WD models are made.

First, the evolution of a grid of low-mass X-ray binaries was computed with the stellar evolution code mesa (Paxton et al. 2011, 2013, 2015, 2018, 2019). The initial neutron star (NS) and donors are assumed to be point mass and zero-age main-sequence stars, respectively. Their masses are assumed to be 1.30 and 1.20 M_⊙, respectively. The initial orbital periods range from 1.0 to 600 days. In this grid, some donors can evolve into He WDs. Then we extract the WD models when the central temperatures of He WDs are around 10⁷ K. In these models, the He WDs have H envelopes with masses smaller than 0.01 M_⊙. The initial He WD masses are 0.17, 0.21, 0.25, 0.30, 0.35, 0.40, 0.45 M_⊙. Compared with the He WD models in Li et al. (2019) who modeled the formation of CO+He WDs, our He WD models may have different temperatures and H envelope masses. But from the following discussion, we can find these factors do not influence our results significantly.

To model the evolution of AM CVn binaries, we make use of the star_plus_point_mass test suite of mesa code (version 12115). In our simulation, the WD accretor is assumed to be a point mass. The initial masses of WD accretors are assumed to be 0.50, 0.60, 0.70, 0.80, 0.90, 1.00, 1.10, 1.20, 1.30 M_⊙. These WDs with masses larger than 1.10 M_⊙ are oxygen–neon (ONe) WDs and others are CO WDs. The initial orbital periods are assumed to be 0.05 days.

In our calculation, we consider two kinds of mechanisms of angular momentum loss: GW radiation and angular momentum loss due to mass loss. The angular momentum loss due to GW radiation can be computed with the following formula (Landau & Lifshitz 1971):

$$\begin{eqnarray}&&\displaystyle \frac{{{dJ}}_{\mathrm{gw}}}{{dt}}=-\displaystyle \frac{32}{5}\displaystyle \frac{{G}^{7/2}}{{c}^{5}}\displaystyle \frac{{M}_{{\rm{a}}}^{2}{M}_{{\rm{d}}}^{2}{\left({M}_{{\rm{a}}}+{M}_{{\rm{d}}}\right)}^{1/2}}{{a}^{7/2}},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant and c is the speed of light in a vacuum; a is the binary separation; M_a and M_d are the masses of the WD accretor and the WD donor, respectively.

We compute the mass transfer rate with the Ritter scheme (Ritter 1988). The mass transfer in our calculation is not conservative and the isotropic reemission model is adopted (Tauris & van den Heuvel 2006). The mass retention efficiency is computed as follows.

Given the H shell of the He WD, the accreted material is H-rich at the early phase of mass transfer. We simply assume that the retention efficiency is zero in this phase. We have tested that this assumption has little impact on our results. As for He burning, we adopt the optical thick wind model (Kato & Hachisu 1994; Hachisu et al. 1999a) and the prescription of Kato & Hachisu (2004). If the mass transfer rate (${\dot{M}}_{\mathrm{tr}}$) is larger than a threshold value ${\dot{M}}_{\mathrm{up}}$, we assume that He burns steadily at a rate of ${\dot{M}}_{\mathrm{up}}$ and the excess material is lost in the form of optically thick wind. The threshold mass transfer rate is

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{up}}=7.2\times {10}^{-6}\left({M}_{{\rm{a}}}/{M}_{\odot }-0.6\right){M}_{\odot }\,{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 2 }$$

for accretor masses M_a ≥ 0.75 M_⊙ (Nomoto 1982).

If the mass transfer rate is larger than the minimum rate of stable burning (${\dot{M}}_{\mathrm{cr}}$) and smaller than ${\dot{M}}_{\mathrm{up}}$, we assume that He burns stably on the surface of the WD and there is no mass loss. If the mass transfer rate is smaller than ${\dot{M}}_{\mathrm{cr}}$ but larger than ${\dot{M}}_{\mathrm{low}}$, we assume that the He burns unstably, triggering He flashes. In this regime, the retention efficiency is computed following Kato & Hachisu (2004). If the mass transfer rate is smaller than ${\dot{M}}_{\mathrm{low}}$, we assume that the He flashes are too strong to retain any material. Therefore the retention efficiency for He burning is

$$\begin{eqnarray*}{\eta }_{\mathrm{He}}=\left\{\begin{array}{cc}{\dot{M}}_{\mathrm{up}}/{\dot{M}}_{\mathrm{tr}} & {\dot{M}}_{\mathrm{tr}}\geqslant {\dot{M}}_{\mathrm{up}}\\ 1 & {\dot{M}}_{\mathrm{up}}\gt {\dot{M}}_{\mathrm{tr}}\geqslant {\dot{M}}_{\mathrm{cr}}\\ {\eta }_{\mathrm{KH}04} & {\dot{M}}_{\mathrm{cr}}\gt {\dot{M}}_{\mathrm{tr}}\geqslant {\dot{M}}_{\mathrm{low}}\\ 0 & {\dot{M}}_{\mathrm{tr}}\lt {\dot{M}}_{\mathrm{low},}\end{array}\right.\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&\begin{array}{l}{\dot{M}}_{\mathrm{cr}}={10}^{-5.8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\\ {\dot{M}}_{\mathrm{low}}={10}^{-7.4}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}.\end{array}\end{eqnarray*}$$

For WDs with masses M_a < 0.75 M_⊙, Equation (2) is not validated any more and we simply assume that the mass transfer is completely nonconservative, i.e., no mass is retained by the accretor. This material not accreted by the accretor leaves the system and takes away the specific angular momentum of the accretor.

In addition, we also take the Eddington limit into consideration. The Eddington limit can be given by Tauris & van den Heuvel (2023) as

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{Edd}}=4.4\times {10}^{-6}({M}_{{\rm{a}}}/{M}_{\odot }){M}_{\odot }\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 3 }$$

Compared with the Eddington limit of Han & Webbink (1999), the Eddington limit in our calculation is lower. This can be understood as follows. In the calculation of the Eddington limit, Han & Webbink (1999) only considered the gravitational energy released by the accreted material. However, Tauris & van den Heuvel (2023) suggest that the nuclear burning energy from this accreted material should be also considered, which leads to a lower Eddington accretion rate. Following Han & Webbink (1999), we assume that the binary will merge in a common envelope if the mass transfer rate is larger than the Eddington limit. But we do not stop the calculation in order to know if the binary system can have dynamically stable mass transfer.

The inlist files for our simulations can be made available on request by contacting the corresponding author.

In Figure 1, we present an example of binary evolution of AM CVn binaries. The initial masses of the accretor and donor are 0.90 M_⊙ and 0.21 M_⊙, respectively. The initial orbital period is 0.05 days. From this plot, we can find that the mass transfer rate in this example is always below the stable burning regime of He (indicated by the green and red dashed lines in the upper panel). The mass of the WD accretor increases when the mass transfer rate is between ${\dot{M}}_{\mathrm{low}}$ (see the purple dashed line) and ${\dot{M}}_{\mathrm{cr}}$ (see the red dashed line). At the early phase of evolution, the orbital period decreases because of GW radiation. When the H envelope is stripped, the mass transfer rate is around its maximum value, and the orbital period reaches its minimum. Afterwards, the mass transfer leads to the increase of the orbital period.

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In Figure 2, we show the evolution of mass transfer rate, orbital period, and accretor mass for binaries with different WD accretor masses and the same donor mass. From the upper and middle panels, we can find that the evolution of mass transfer rate and orbital period are very similar for these binaries. The minimum orbital periods during the evolution are almost the same for these binaries. This is because the evolution of the orbital period during the mass transfer mainly depends on the donor mass (Chen et al. 2022b). In the binary system with an accretor mass of 1.30 M_⊙, the accretor mass reaches 1.40 M_⊙ during its evolution. The WD accretor can collapse into a NS and we do not stop the calculation at that point. We have a further discussion on this kind of system in Section 4.3. For the systems with initial accretor masses of 0.50 and 0.70 M_⊙, the accretor masses do not change during their evolution. This is because we assume that the mass transfer is completely nonconservative for these systems with accretor masses M_a ≤ 0.75 M_⊙.

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Figure 3 presents the evolution of mass transfer rate as a function of the orbital period and He WD masses for binaries with different He WD masses and the same accretor mass. From this plot, we can also find that these tracks converge to a single branch after the peaks of mass transfer rate. Compared with these systems with smaller He WD masses, the systems with massive He WDs have a larger maximum mass transfer rate and a smaller minimum orbital period. This is mainly because massive He WDs have a smaller radius.

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Given the short orbital periods of AM CVn binaries, they are expected to be important sources of low GW frequency. In Figure 4, we present an example of the evolution of GW frequency, chirp mass, and signal-to-noise ratio (S/N) as a function of time for a binary system. In addition, the evolution of donor mass as a function of GW frequency is also shown. Here the S/N for LISA and TianQin is computed with the Python package LEGWORK (Wagg et al. 2022). The initial binary parameters are M_a = 0.90 M_⊙, M_d = 0.21 M_⊙, and P_orb = 0.05 days. From these plots, we can find that the AM CVn binaries have a strong GW emission in the mHZ regime. The S/N can be up to ∼800 (∼50) if the source is located at 1 kpc (15 kpc). If we adopt the critical S/N = 7, above which the source becomes detectable, then we can find that this source can be detected by LISA and TianQin. The chirp mass for the detectable source is between 0.11 M_⊙ and 0.36 M_⊙ (0.19 M_⊙ and 0.36 M_⊙) if the source is located at a distance of 1 kpc (15 kpc). The donor mass for the detectable source is between 0.03 M_⊙ and 0.21 M_⊙ (0.07 M_⊙ and 0.21 M_⊙) if the source is located at a distance of 1 kpc (15 kpc). From panel (d), we can find there is a tight relation between the donor mass and GW frequency after the maximum frequency.

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In Figure 5, we present the evolution of WD donor mass as a function of GW frequency for all binaries. In the plot, we do not show these binaries with donor masses larger than 0.30 M_⊙. This is because the mass transfer rates of these systems exceed the Eddington limit and these systems are assumed to merge in our calculation. These systems with the same donor mass have almost the same evolutionary track in this plot. Therefore, it appears that there are only three lines in the plot. From this plot, we can find that all evolutionary tracks converge to the same branch after the maximum peak in frequencies are reached. In other words, there is a linear relation between the donor mass and GW frequency. This relation has been analytically derived by Chen et al. (2022b) and can be described with their Equations (7) and (13). A fitting formula for this relation has been proposed by Breivik et al. (2018) (see their Figure 1). With this relation, we can infer the He WD mass if the GW frequency is measured. Then the accretor mass can also be derived if the chirp mass is observed.

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The stability of mass transfer of double WDs is still under debate (e.g. Han & Webbink 1999, Nelemans et al. 2001a, Marsh et al. 2004, Shen 2015). This problem could strongly influence the number of AM CVn and GW sources predicted by the binary population synthesis model.

In Figure 6, we present the stability limits for mass transfer of AM CVn binaries and compare our results with previous works. Following Chen & Han (2008), if a binary system has a runaway mass transfer, we assume that the binary system will have dynamically unstable mass transfer. With this criteria, we can find that all binaries in our calculation have dynamically stable mass transfer. In addition, following Marsh et al. (2004) and Kremer et al. (2017), if we adopt a critical mass transfer rate of $\dot{M}=0.01\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ as the limit for dynamically stable mass transfer, we can also find that all binaries in our calculation have dynamically stable mass transfer. This is very different from Nelemans et al. (2001a) and Marsh et al. (2004) (see the solid lines in Figure 6). The mass transfer of these binaries with a smaller accretor and larger donor masses are dynamically stable in our calculation and unstable in Nelemans et al. (2001a) and Marsh et al. (2004). This is partially due to us having nonconservative mass transfer in our calculation, while they assumed conservative mass transfer in their calculation (see Section 4.4 for detailed discussion). In addition, a zero temperature is assumed for the WD donors in Nelemans et al. (2001a) and Marsh et al. (2004), which may lead to higher mass transfer rate (see Section 4.5 for more discussion).

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Han & Webbink (1999) suggested that a common envelope may form if the mass transfer rate in double WD binaries is larger than the Eddington limit. With this restriction, we can find the threshold for Eddington-limited mass transfer and show these binaries below the threshold with crosses in Figure 6. From the plot, we can find that the threshold in our calculation is slightly below that found by Han & Webbink (1999) and Nelemans et al. (2001b). This is because the Eddington limit we adopted is lower than that in Han & Webbink (1999) and Nelemans et al. (2001b).

In our calculation, we assume that the optically thick wind occurs if the mass transfer rate is larger than the maximum stable burning rate of He. But there is another possibility in this regime. For example, Piersanti et al. (2014) found that the accreting WD may evolve into a red giant. In this scenario, the binary may enter a common envelope and merge eventually. With this scenario in mind, we can find all binaries with maximum mass transfer rates smaller than the maximum stable burning rates of He, ³ which are shown as empty squares in Figure 6. This may provide a new criterion for double WDs surviving mass transfer, which is slightly below the threshold for the Eddington limit.

Carter et al. (2013) found the AM CVn space density to be (5 ± 3) × 10⁻⁷ pc⁻³, which is 50 times smaller than the predicted value by the optimistic population synthesis model from the Nelemans et al. (2001a). Recently, van Roestel et al. (2022) found a space density of ${6}_{-2}^{+6}\times {10}^{-7}\,{\mathrm{pc}}^{-3}$ and confirmed this discrepancy. Our new stability limits may have an important implication for the formation of AM CVn binaries in a binary population synthesis study. Since the stability limits in our calculation are lower than that in Nelemans et al. (2001a), we expect that less binaries will have stable mass transfer and less AM CVn binaries will be produced in the population synthesis models. This will be helpful to mitigate this discrepancy between the theoretical model and observation.

It is also worth noting that we do not consider the rotation of the accretors in our calculation. During the mass transfer, the accretor may be spun up due to accretion. The coupling between the accretor's spin and orbit may lead to extra orbital angular momentum loss during binary evolution. Marsh et al. (2004) showed that the coupling may have an important effect on the stability of mass transfer depending on the synchronization timescale. On the other hand, Kupfer et al. (2016) have shown that the accretor velocities of two AM CVn systems, GP Com and V396 Hya, are much slower than the critical.

As we show in Figure 2, some ONe WDs in our calculation can increase their masses to the Chandrasekhar mass limit (∼1.40 M_⊙) and will collapse into NSs. At this point, the He WD masses are around 0.09–0.15 M_⊙. After the WD collapses into a NS, the binary system will become eccentric because of the mass loss during collapse and the natal kick of the NS (e.g., Tauris et al. 2013). Because of mass loss during the collapse, the Roche-lobe radius of the donors increases. Then these systems may evolve into detached binaries consisting of NSs and extreme low-mass He WDs. Due to the GW radiation, the He WDs in these systems may fill their Roche lobe again at some point. Then these binaries will evolve into ultracompact X-ray binaries.

In order to understand the influence of accretion efficiency, we compute the evolution of a binary system assuming the mass transfer is conservative, i.e., all the material lost by the donor is accreted by the accretor. In Figure 7, we show the comparison of the evolution of mass transfer rate for binaries with different prescriptions of retention efficiency. From this plot, we can find that their evolution is very similar. But the mass transfer rate is relatively higher in the model assuming conservative mass transfer. Binaries with high mass transfer rates are more likely to be dynamically unstable. This may partially explain the difference of dynamical stability of mass transfer between our calculation and previous studies.

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In our calculation, the initial temperatures of He WDs and orbital periods are assumed to be the same for all binaries. This may be not realistic, given that the double WD systems can be produced from stable mass transfer and common envelope ejection channels (Li et al. 2019). The He WDs in these binaries with long (short) orbital periods have a long (short) time to cool down, leading to low (high) temperatures at the onset of mass transfer. Therefore, the influence of initial effective temperature and orbital period should be similar.

In order to understand their influence, we make a new 0.25 M_⊙ He WD model with a high temperature following the method described in Section 2.1. The central and effective temperatures in this model are T_c = 3.24 × 10⁷ K and T_eff = 2.74 × 10⁴ K, respectively. With this model, we compare the evolution of mass transfer rate for binaries with different initial orbital periods, which is shown in Figure 8. From this plot, we can find that the mass transfer rate is slightly higher for binaries with larger orbital periods at an early phase of mass transfer. At later times, the difference is very small.

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