Formation of Extremely Low-mass White Dwarf Binaries

Binaries are evolved using the "binary_donor_only" option in the Modules for Experiments in Stellar Astrophysics code (MESA, version 8845; (Paxton et al. 2011, 2013, 2015), which evolves the structure of the donor star and orbit in time, but treats the accretor as a point mass. MT is assumed to be fully non-conservative (MESA parameter $\mathrm{mass}\_\mathrm{transfer}\_\mathrm{beta}=1$), so the accretor mass M_a,i is a constant in time and mass loss from the binary is assumed to take place in a fast wind from the accretor. The physical basis for this assumption is that accretion disk winds may limit the mass that falls onto the accretor, and nova explosions may remove the accreted mass.

The mixing length parameter is set to α_ML = 1.9. The ZAMS metallicity of all stars is Z = 0.01, which is characteristic of the disk stars in the Galaxy (Bensby et al. 2014). As stars evolve faster for lower metallicity with the same star mass (Istrate et al. 2016), this metallicity choice helps accelerate the production of a WD within the age of the Galaxy. The nuclear burning network used is "pp_and_cno_extras," which includes ${}^{1}{\rm{H}},{}^{3}\mathrm{He},{}^{4}\mathrm{He},{}^{12}{\rm{C}},{}^{14}{\rm{N}},{}^{16}{\rm{O}},{}^{20}\mathrm{Ne},{}^{24}\mathrm{Mg}$, and extended networks that comprise the pp-chain and CNO cycle. Element diffusion is included over the entire evolution, starting from the ZAMS end extending through the WD cooling track. This setting is crucial for the critical mass at which H flashes occur, as well as for the number of flashes before the WD cooling as found by Istrate et al. (2016). The setting "diffusion_use_cgs_solver = .true." is used to allow for electron degeneracy in the diffusion physics. Five classes of elements, ${}^{1}{\rm{H}},{}^{3}\mathrm{He},{}^{4}\mathrm{He},{}^{16}{\rm{O}},{}^{56}\mathrm{Fe}$, are evolved. The helium-core masses, M_c, reported here are computed as the mass interior to the point where the mass fraction of ¹H is 1% that of ⁴He.

The total orbital angular momentum, J, evolves through torques due to magnetic braking (${\dot{J}}_{\mathrm{mb}}$), gravitational waves (${\dot{J}}_{\mathrm{gr}}$, Landau & Lifshitz 1975), and mass loss from the binary (${\dot{J}}_{\mathrm{ml}}$),

$$\begin{eqnarray}&&\dot{J}={\dot{J}}_{\mathrm{mb}}+{\dot{J}}_{\mathrm{gr}}+{\dot{J}}_{\mathrm{ml}}.\end{eqnarray} \tag{ 1 }$$

The angular momentum loss from the binary, modeled as a fast wind in the viscinity of the accretor, is (Tauris & van den Heuvel 2006)

$$\begin{eqnarray}&&{\dot{J}}_{\mathrm{ml}}=J\ \displaystyle \frac{{M}_{{\rm{d}}}{\dot{M}}_{{\rm{d}}}}{{M}_{{\rm{a}}}({M}_{{\rm{d}}}+{M}_{{\rm{a}}})}.\end{eqnarray} \tag{ 2 }$$

The mass-loss torque ${\dot{J}}_{\mathrm{ml}}$ is important during the thermal timescale mass transfer TTMT, when ${M}_{{\rm{d}}}\gtrsim {M}_{{\rm{a}}}$ and the mass-loss rate of the donor ${\dot{M}}_{{\rm{d}}}$ is high. Thereafter, ${\dot{J}}_{\mathrm{mb}}$ takes over until the convection zone thins. The gravitational wave torque ${\dot{J}}_{\mathrm{gr}}$ is important for short periods P_orb ≲ 3 hr, and is the dominant torque at the second phase of MT at P_orb ≲ 1 hr.

For thick convection zones with mass fraction q_conv > 0.02, the magnetic braking formula of Rappaport et al. (1983) is used. The MESA implementation is to set ${\dot{J}}_{\mathrm{mb}}=0$ when the fraction of mass in the convection zone q_conv > 0.75 to implement the above-mentioned reduction at low stellar mass. To take into account reduced magnetic braking when the surface convection zone is thin, the ansatz from Podsiadlowski et al. (2002) is that ${\dot{J}}_{\mathrm{mb}}$ is reduced by an exponential factor as the convection zone mass becomes small. The end result used in the simulations is then

$$\begin{eqnarray}{\dot{J}}_{\mathrm{mb}} & = & -3.8\times {10}^{-30}{M}_{{\rm{d}}}{R}_{\odot }^{4}{\left(\displaystyle \frac{{R}_{{\rm{d}}}}{{R}_{\odot }}\right)}^{\gamma }{\omega }^{3}\\ & & \times \,\left\{\begin{array}{rc}0, & 1\gt {q}_{\mathrm{conv}}\gt 0.75\\ 1, & 0.75\gt {q}_{\mathrm{conv}}\gt 0.02\\ {e}^{1-0.02/{q}_{\mathrm{conv}}}, & {q}_{\mathrm{conv}}\lt 0.02\end{array}\right.,\end{eqnarray} \tag{ 3 }$$

where ${\dot{J}}_{\mathrm{mb}}$ is in CGS units ${\rm{g}}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$, a magnetic braking index γ = 4 was used in the calculations, ω = 2π/P_orb is the orbital angular velocity in $\mathrm{rad}\,{{\rm{s}}}^{-1}$, and R_d is the donor star radius. The mass fraction q_conv = 0.02 is for the current solar convection zone and so magnetic braking is suppressed on the MS for more massive stars. Because the donors are evolved, their radii shrink less than for unevolved donors, and there is only a weak dependence on γ. Calculations with γ = 3 and 4 gave similar results.

A side effect of the reduced ${\dot{J}}_{\mathrm{mb}}$ at small q_conv is that donors with mass 1.3 ≲ M_d/M_⊙ ≲ 1.4, which have weak magnetic braking on the MS, can have sufficient magnetic braking as evolved donors, with thicker convection zones, so that they work well as the progenitors of ELM WDs. Their MS lifetime is much shorter than that of a 0.9 ≤ M_d/M_⊙ ≤ 1.1 donor, so this leaves more time for the WD to cool to low T_eff ≲ 9000 K and enter the blue edge of the instability strip. Simulations of donors with higher masses M_d ≳ 1.5 M_⊙ had difficulty forming an ELM WD because M_c ≳ 0.1 M_⊙ at the end of the MS, which when combined with core growth during the accretion phase, makes them too large to be the ELM WD with M < 0.18 M_⊙. Furthermore, the orbits are much wider than for the 1.0 ≤ M_d/M_⊙ ≤ 1.4 cases.

Simulations in which the MT rose sharply and exceeded $| {\dot{M}}_{{\rm{d}}}| \gt {10}^{-3}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ were stopped and labeled as exhibiting unstable MT. This occurs if the initial mass ratio ${q}_{{\rm{i}}}={M}_{{\rm{d}},{\rm{i}}}/{M}_{{\rm{a}},{\rm{i}}}$ is too high, where ${M}_{{\rm{d}},{\rm{i}}}$ is the initial donor star mass, M_a,i is the initial accretor mass, and is exacerbated by wider orbital separations such that the donor was well up the giant branch when MT commenced. As discussed in Appendix B, unstable MT and CE may lead to merging for the M_c ≲ 0.1 M_⊙ here.

In our model, the ELM WD progenitor is assumed to be the initially less massive star, and the initially more massive star becomes the ELM WD companion, itself a WD. The initially more massive star is assumed to form a WD through a CE phase, because short orbital periods from 1 to 3 days are required in the second phase of MT to form the ELM WD. Let ${M}_{1}$ be the mass of the initially more more massive star, M₂ the mass of the initially less massive star, and a_CE,i the initial semimajor axis before the CE. Note that the subscripts "1," "2" and "CE" are only used in this section, and indicate the star parameters before the stable RLOF phase. For a wide initial orbit, a core mass ${M}_{1,{\rm{c}}}$ is formed in star 1, and by removing the envelope, ${M}_{1,{\rm{c}}}$ is the mass of the ELM companion. Applying the CE energy equation (Equation (8) in the appendix), and expressing the answer in terms of the post-CE (but pre-magnetic braking) orbital period ${P}_{\mathrm{CE},\mathrm{orb},{\rm{f}}}$, gives

$$\begin{eqnarray}&&{\left(\displaystyle \frac{G({M}_{1,{\rm{c}}}+{M}_{2}){P}_{\mathrm{CE},\mathrm{orb},{\rm{f}}}^{2}}{4{\pi }^{2}}\right)}^{1/3}={R}_{1}({M}_{1,{\rm{c}}})\\ &&\quad \times \,\left(\displaystyle \frac{{M}_{1,{\rm{c}}}{M}_{2}}{(2/\alpha \lambda ){M}_{1}({M}_{1}-{M}_{1,{\rm{c}}})+{M}_{1}{M}_{2}{r}_{{\rm{L}}}({M}_{1}/{M}_{2})}\right),\end{eqnarray} \tag{ 4 }$$

with the appropriate ${R}_{1}({M}_{1,{\rm{c}}})$ relation for each core mass range, this equation can be solved for ${P}_{\mathrm{CE},\mathrm{orb},{\rm{f}}}$ during the evolution, where ${P}_{\mathrm{CE},\mathrm{orb},{\rm{f}}}$ is the post-common envelope (but pre-magnetic braking) orbital period, λ ≃ 1 is a mass-dependent factor describing the binding energy, α is the efficiency of tapping orbital energy to remove the envelope, and ${r}_{{\rm{L}}}a$ is the effective radius of the Roche lobe, and r_L is a parameter defined in Eggleton (1983). MESA models for ${M}_{1,{\rm{i}}}=1$, 2, 3, 4, 5 M_⊙ were used to find R₁, ${M}_{1,{\rm{c}}}$ and M₁ during the evolution. The radius grows non-monotonically, so this leads to gaps in ${M}_{1,{\rm{c}}}$ over regions where the radius decreases below its maximum value.

Figure 1 shows numerical solutions of Equation (4) for companion mass ${M}_{1,c}$ as a function of ${P}_{\mathrm{CE},\mathrm{orb},{\rm{f}}}$. The ELM WD progenitor mass has been fixed at M₂ = 1.3 M_⊙, and five different M₁ have been used to give the different lines. The product αλ is set to 2 for convenience. There is a general trend that ${M}_{1,{\rm{c}}}$ must be higher for larger orbital period or M₁ for the orbital energy release to be able to balance the binding energy. During the second phase of MT, the companion is the accretor and so ${M}_{{\rm{a}},{\rm{i}}}={M}_{1,{\rm{c}}}$, and the progenitor of the ELM WD is the donor, so ${M}_{{\rm{d}},{\rm{i}}}={M}_{2}$.

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The separation at the onset of the RLOF should be slightly greater than 5 R_⊙ to form an ELM WD. If ${M}_{1,{\rm{c}}}$ is fixed at 0.6 M_⊙ with M₂ = 1.3 M_⊙ and the separation after the CE ${a}_{\mathrm{CE},{\rm{f}}}=5{R}_{\odot }$, there are still two free (but not completely free) parameters M₁ and the orbital period before the CE ${P}_{\mathrm{CE},{\rm{i}}}$. Moreover, M₁ is greater than M₂ because the massive star evolves first. This can lead to a CE phase. In addition, M₁/M₂ is greater than one to have unstable MT followed by a CE phase (Woods & Ivanova 2011). For M₁ = 2 M_⊙, ${P}_{\mathrm{CE},{\rm{i}}}$ is 7.6 days.

Figure 2 displays evolution tracks in the ${\mathrm{log}}_{10}\,g$ versus T_eff plane for the fiducial case with ${M}_{{\rm{d}},{\rm{i}}}=1.3\,{M}_{\odot }$ and (constant) accretor mass ${M}_{{\rm{a}},{\rm{i}}}=0.6\,{M}_{\odot }$. The entire range of ELM WDs is covered by the initial orbital period ${P}_{\mathrm{orb},{\rm{i}}}=0.90$, 0.91, 0.93, 0.95, 0.97, 0.99, and 1.02 days, after the CE but before the RLOF phase. The narrow range of ${P}_{\mathrm{orb},{\rm{i}}}$ that produce ELM WD is similar to the result found by Smedley et al. (2017). The donor in the track with ${P}_{\mathrm{orb},{\rm{i}}}=1.03\,\mathrm{day}$ has a core mass high enough that diffusion-aided shell flashes occur on the WD cooling track. The color indicates M_c. The black points with error boxes represent the seven pulsating ELM WDs with parameters derived using 3D atmosphere models (Tremblay et al. 2015) except for J1618+3854 (since only the ${\mathrm{log}}_{10}\,g$ and T_eff from a 1D atmosphere model are given in other references), with the half-width of the box showing the measurement uncertainty.

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All runs begin at the ZAMS with ${\mathrm{log}}_{10}(g/\mathrm{cm}\,{{\rm{s}}}^{-2})=4.4$ and ${T}_{\mathrm{eff}}=6500\,{\rm{K}}$. Along the MS, and as the star evolves to the RGB, its radius increases with M_c, and so wider orbits come into contact with higher M_c. The ELM WD commences MT with 0.06 ≲ M_c/M_⊙ ≲ 0.1, and M_c increases during the MT phase. Figure 2 shows that models with higher M_c evolve to a higher maximum T_eff, the elbow in the curve that separates the pre-WD phase (increasing T_eff) from the WD cooling track (decreasing T_eff). This plot shows the same behavior between shell flashes: the loops in the $\mathrm{log}g\mbox{--}{T}_{\mathrm{eff}}$ plane become larger, evolving to higher maximum T_eff. As a result, when systems with shell flashes enter the WD cooling track, their evolution is more nearly horizontal, at constant ${\mathrm{log}}_{10}\,g$. This gives rise to a wedge in the ${\mathrm{log}}_{10}\,g\mbox{--}{T}_{\mathrm{eff}}$ plane that separates the ELM WD with M ≲ 0.18 M_⊙ without shell flashes from the slightly more massive WDs, with M ≳ 0.18 M_⊙, which do have shell flashes. The hydrogen-rich envelope is thinner after the shell flashes, so the residual hydrogen burning is smaller and the system evolves to lower T_eff more quickly. All runs were evolved to an age 13.7 Gyr, except for the one run in the figure that comes back into contact. Furthermore, the low-mass donor star evolves more slowly and cannot reach the WD cooling phase by the age of the Galactic disk (10 Gyr). We extended the evolution to 13.7 Gyr to see if the WD cooling phase can be reached within a Hubble time. The ELM WDs in Figure 2 have much longer cooling times, and only decrease to T_eff ≃ 8000 K, while the run with shell flashes in the lower panel reaches T_eff ≲ 4000 K. The observed systems evidently span the range of ELM WDs with thick envelopes as well as those that have undergone shell flashes.

For smaller P_orb,i, the donor comes into contact at core mass $0.01\lesssim {M}_{{\rm{c}}}/{M}_{\odot }\lesssim 0.07$, and stays in contact to short P_orb ≲ 1 hr. For systems that come into contact early on the MS, at very low M_c ≲ 0.01 M_⊙, standard CV evolution with a period gap at 2–3 hr is recovered. However, the radiative core is small or nonexistent in this case, and they are not expected to be g-mode pulsators. The core mass at contact for these cases is low, at roughly M_c ≲ 0.06 M_⊙, in agreement with Podsiadlowski et al. (2003).

Figure 3 shows ${\dot{M}}_{{\rm{d}}}$ versus P_orb (top panel), $\dot{J}$ contributions versus P_orb (second panel), donor R_d versus M_d (third panel), and P_orb versus age (bottom panel). The initial periods are ${P}_{\mathrm{orb},{\rm{i}}}=0.90$ (blue), 0.95 (green), 0.99 (red), and 1.02 days (cyan).

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In the top two panels, evolutionary tracks producing ELM WDs start at long periods and proceed to shorter periods on the whole. Magnetic braking is small for ${M}_{{\rm{d}},{\rm{i}}}=1.3\,{M}_{\odot }$ on the MS because of the small surface convection zone, so P_orb is nearly constant during that time. When the system first comes into contact, TTMT results in high mass-loss rates ${10}^{-8}\,\lesssim {\dot{M}}_{{\rm{d}}}/{M}_{\odot }\,{\mathrm{yr}}^{-1}\lesssim {10}^{-7}$. TTMT continues until the ratio M_d/M_a decreases to the critical value (1 for conservative transfer, see Woods et al. 2012) at which point TTMT ends, and the much slower nuclear or $\dot{J}$ timescale MT takes over. During TTMT, ${\dot{J}}_{\mathrm{ml}}$ dominates because of the high accretion rates (second panel). Shortly thereafter, the increased size of the convection zone removes the suppression of ${\dot{J}}_{\mathrm{mb}}$, and it subsequently dominates until MT turns off at $5\leqslant {P}_{\mathrm{orb}}/\mathrm{hr}\leqslant 11$. Subsequently, J_gr dominates and all systems undergo orbital decay. Only less evolved donors with the lowest mass undergo sufficient orbital decay to come back into contact at P_orb ≈ 1 hr. For higher M_c, evolution driven by the expansion of the star as it tries to ascend the RGB becomes more important than orbital shrinkage due to magnetic braking, leading to a period bifurcation separating the orbits that shrink from those that expand (Podsiadlowski et al. 2003).

The third panel in Figure 3 shows the donor radius as a function of mass during the evolution. Before contact, R_d increases with M_c. Smaller ${P}_{\mathrm{orb},{\rm{i}}}$ runs commence MT first, at smaller R_d, while larger ${P}_{\mathrm{orb},{\rm{i}}}$ allows R_d to grow further. During TTMT, the high MT rate causes the radius to be slightly inflated. As discussed in Appendix A (see Figure 22), the decrease in mass of the hydrogen-rich envelope leads to lower hydrogen shell-burning luminosity, accompanied by shrinkage of the radius. This is seen in the steep drop in radius in the third panel of Figure 3, leading the system to fall out of contact, as shown in the first panel. Models with thick hydrogen envelopes have modest shrinkage in radius beyond that point. The model with the lowest mass comes back into contact, evolving toward lower M_d.

The bottom panel in Figure 3 shows P_orb versus age. The evolution starts on the left, with tracks at different ${P}_{\mathrm{orb},{\rm{i}}}$ denoted by solid lines. When MT commences (dashed lines starting at 3 Gyr), rapid orbital decay occurs during TTMT. Then the slower orbit evolution on the $\dot{J}$ timescale lasts 2–3 Gyr. During this slow phase of MT, slight orbit expansion occurs for high M_c, while continued orbital decay occurs for low M_c. When MT ceases, the donor becomes an ELM WD near 5–6 Gyr (solid lines). The three largest ${P}_{\mathrm{orb},{\rm{i}}}$ and M_c tracks show only modest orbital decay due to ${\dot{J}}_{\mathrm{gr}}$, while the lowest line decays from P_orb = 5 hr to 1 hr, at which point MT re-commences (the magenta cross).

Figure 4 shows total donor mass M_d, helium-core mass M_c and envelope mass ${M}_{\mathrm{env}}={M}_{{\rm{d}}}-{M}_{{\rm{c}}}$ versus P_orb. The tracks start from the right of the plot near 20 ≲ P_orb/hr ≲ 25. The total donor mass M_d decreases downward during MT, and becomes constant when MT ceases. The envelope mass is seen to smoothly decrease, until the end of MT at 5 ≤ P_orb/hr ≤ 20. The envelope mass M_env continues to decrease due to residual hydrogen burning, while the orbit slowly decays due to ${\dot{J}}_{\mathrm{gr}}$. The M_c lines initially rise vertically, as M_c increases before MT. The TTMT phase is so short that there is no time for M_c to grow above 0.07 ≲ M_c/M_⊙ ≲ 0.10. The nuclear timescale MT is much longer, and M_c increases to 0.13 ≲ M_c/M_⊙ ≲ 0.15. After MT, an additional 0.02 ≲ M_c/M_⊙ ≲ 0.03 is converted from envelope to core by nuclear burning, during which time the orbit decays due to ${\dot{J}}_{\mathrm{gr}}$.

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From the discussion in Section 2 (see Equation (3)), the thickness of the convective envelope is an important parameter for the effectiveness of magnetic braking. Figure 5 shows the mass of the convective envelope M_conv as a function of M_d. The systems evolve from right to left during MT. The ${M}_{{\rm{d}},{\rm{i}}}=1.3\,{M}_{\odot }$ donor has a small convective envelope on the MS. The spike at ${M}_{{\rm{d}}}=1.3\,{M}_{\odot }$ is due to the convective core on the MS. As M_c grows and the shell-burning luminosity increases, M_conv increases. When nearly all the hydrogen-rich envelope has been lost, the luminosity drops and the convection zone shrinks again. The orange dashed line gives the threshold below which magnetic braking is exponentially suppressed. For the ELM WD, M_conv drops below the threshold and magnetic braking shuts off.

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Figure 6 shows the same $\mathrm{log}\,g$ versus T_eff as Figure 2. The circles are placed at 1 Gyr intervals. Systems covered by the tracks have ages 9–12 Gyr. The solid line shows phases where the system is out of contact, while the dashed lines show phases where MT is occurring. Following Steinfadt et al. (2010), phases for which the lowest order ℓ = 1 g-mode is unstable are estimated by Brickhill's criterion, $P({\rm{g}}1)\leqslant 8\pi {{\rm{t}}}_{\mathrm{th}}$, and are covered by red lines, where $P({\rm{g}}1)$ is the mode period of the lowest order g-mode and t_th is the thermal time at the base of the surface convection zone. The two data points at high $\mathrm{log}\,g$ would require ${M}_{{\rm{d}},{\rm{f}}}\gt 0.18\,{M}_{\odot }$ tracks, which are not shown on the plot. The estimate of the instability strip used here appears to give too cool a blue edge T_eff to explain the systems near 8500 ≤ T_eff/K ≤ 9,500.

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Figure 7 displays T_eff versus P_orb for ${P}_{\mathrm{orb},{\rm{i}}}=0.90$, 0.93, 0.95, 0.97, 0.99, and 1.02 days as well as the four observed systems with measured P_orb. The evolution starts from the right-hand side of the plot. The dashed, solid, and red lines have the same meaning as in Figure 6. The pre-WD phase starts at T_eff ≳ 6500 K. After the turning point, as the T_eff reaches the maximum, the WD enters its cooling phase. The two pulsators with the shortest P_orb have slightly higher T_eff than the models. The lines covered by red segments show the unstable g1 mode with Brickhill's criterion. In the following discussion section, we show that lower-mass donor and high-mass accretor can give better agreement.

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The runs in this section use the same ${M}_{{\rm{d}},{\rm{i}}}=1.3\,{M}_{\odot }$, but a heavier accretor mass ${M}_{{\rm{a}},{\rm{i}}}=0.9\,{M}_{\odot }$. The structures and the evolutionary tracks of the donor stars do not change significantly with companion mass, while the orbital period of the system can be different (Istrate et al. 2016).

Figure 8 shows M_d, M_c, and M_env versus P_orb, and should be compared to the fiducial case in Figure 4. After the onset of RLOF, the orbit first goes slightly outward and then shrinks until M_d ≈ M_a. The accretor is larger, so the orbit does not shrink as much as the fiducial case and the mass-loss rate is also lower. The donor star is slightly less evolved at the beginning of the RLOF because for the same separation, the larger accretor causes the Roche-lobe radius to shrink. As a result, smaller ${P}_{\mathrm{orb},{\rm{i}}}$ must be used to obtain the same ${M}_{{\rm{d}},{\rm{f}}}$.

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The main difference between Figures 9 and 7 occurs on the WD cooling track after maximum T_eff. During this long period, the higher ${M}_{{\rm{a}},{\rm{i}}}$ increases ${\dot{J}}_{\mathrm{gr}}$, causing the orbit to shrink faster. This is more evident for small P_orb. A specific example is given in Figure 21, in which the pre-WD evolution is similar but the heavier accretor causes more orbital decay on the WD cooling track.

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This section contains a comparison of evolutionary models for ${M}_{{\rm{d}},{\rm{i}}}=1.0\,{M}_{\odot }$ and ${M}_{{\rm{a}},{\rm{i}}}=0.45\,{M}_{\odot }$ to the fiducial case results in Section 3.1. The companion mass is near the upper end of the mass range for helium-core WDs. In addition, ${M}_{{\rm{a}},{\rm{i}}}$ is also low enough that long ${P}_{\mathrm{orb},{\rm{i}}}$ models exhibit unstable MT. Even lower ${M}_{{\rm{a}},{\rm{i}}}$ can lead to unstable MT at a broader range of ${P}_{\mathrm{orb},{\rm{i}}}$.

Figure 10 gives the evolutionary tracks for ${M}_{{\rm{d}},{\rm{i}}}=1.0\,{M}_{\odot }$ and ${M}_{{\rm{a}},{\rm{i}}}=0.45\,{M}_{\odot }$ with ${P}_{\mathrm{orb},{\rm{i}}}=2.3$, 2.35, 2.4, 2.45, 2.5, 2.55, 2.6, and 2.67 days. The axes are the same as in Figure 2. For ${P}_{\mathrm{orb},{\rm{i}}}\lt 2.3$ days, accretion never ceases and the orbit shrinks to ${P}_{\mathrm{orb}}\lt 1$ hr. For ${P}_{\mathrm{orb},{\rm{i}}}\gt 2.67$ days, MT commences with a sufficiently large convective envelope that unstable MT occurs, yielding an upper limit to the WD mass produced with these evolutionary sequences. This is to be compared with the fiducial case in Figure 2, where the ELM WD sequence joins on to the sequences of WDs at larger ${P}_{\mathrm{orb},{\rm{i}}}$ that have shell flashes. Hence the bottom track in Figure 2, which shows the WD cooling track after flashes have stopped, would not occur for this case because of the lower ${M}_{{\rm{a}},{\rm{i}}}$ used in this section.

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Figure 10 shows that most tracks have insufficient time to reach the T_eff of the data points. This is due to the long MS evolution. The $\mathrm{log}\,g$ at the elbow is slightly smaller than for the fiducial case.

Similar to Figure 4, Figure 11 shows ${M}_{{\rm{d}}}$, ${M}_{{\rm{c}}}$, and ${M}_{\mathrm{env}}$ as a function of ${P}_{\mathrm{orb},{\rm{i}}}$, now with ${M}_{{\rm{d}},{\rm{i}}}=1.0\,{M}_{\odot }$ and ${M}_{{\rm{a}},{\rm{i}}}=0.45\,{M}_{\odot }$. The selected ${P}_{\mathrm{orb},{\rm{i}}}$ are 2.3, 2.45, 2.55, and 2.67 days. Tracks enter from the right-hand side of the plot because of the strong magnetic braking. This is in contrast to the fiducial case, where RLOF began due to an increase in the stellar radius near the end of the MS. The tracks at smaller ${P}_{\mathrm{orb}}$ have incomplete burning of the envelope within the Hubble time.

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Figure 12 shows ${M}_{\mathrm{conv}}/{M}_{{\rm{d}}}$ versus ${M}_{{\rm{d}}}$ for ${M}_{{\rm{d}},{\rm{i}}}=1.0\,{M}_{\odot }$. The outer convection zone grows during the MS. As ${P}_{\mathrm{orb},{\rm{i}}}$ increases, the onset of RLOF occurs later, and with a larger surface convection zone. Since the outer convection zone has ${q}_{\mathrm{conv}}\gt 0.02$, magnetic braking is much stronger than for the fiducial case and is evident in Figure 11. Therefore, making an ELM WD needs longer ${P}_{\mathrm{orb},{\rm{i}}}$ for ${M}_{{\rm{d}},{\rm{i}}}=1.0\,{M}_{\odot }$, ${P}_{\mathrm{orb},{\rm{i}}}\gt 25$ hr. For even longer ${P}_{\mathrm{orb},{\rm{i}}}$, ${M}_{\mathrm{conv}}$ is sufficiently large for unstable MT to occur.

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Figure 13 shows ${T}_{\mathrm{eff}}$ versus ${P}_{\mathrm{orb}}$, and should be compared to the fiducial model in Figure 7. First note that there are no tracks that end at ${P}_{\mathrm{orb}}\gt 10$ hr due to unstable MT. In the fiducial case, the higher-mass WDs with shell flashes would end in that region. Given sufficient time, the tracks at ${P}_{\mathrm{orb}}\lt 5$ hr would have slightly higher maximum ${T}_{\mathrm{eff}}$ and would explain the data points better. However, there is insufficient time to reach the elbow.

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Given that more massive donors have a shorter MS phase, this leaves more time for the resultant ELM WDs to cool to ${T}_{\mathrm{eff}}\simeq 9000\,{\rm{K}}$ and become pulsators. However, sufficiently massive progenitors produce helium cores at terminal age MS that are larger than the maximum ELM WD to avoid shell flashes. Hence there is a maximum progenitor that can create an ELM WD. This section describes models with a donor mass ${M}_{{\rm{d}}}=1.5\,{M}_{\odot }$ that approaches this limit.

Figure 14 shows evolutionary tracks for ${M}_{{\rm{d}},{\rm{i}}}=1.5\,{M}_{\odot }$ and ${M}_{{\rm{a}},{\rm{i}}}=0.6\,{M}_{\odot }$ for the range of initial orbital periods ${P}_{\mathrm{orb},{\rm{i}}}=0.85$, 0.86, 0.87, 0.88, 0.89, and 0.9 day. Compared to the fiducial case in Figure 2, the ${M}_{{\rm{d}},{\rm{i}}}=1.5\,{M}_{\odot }$ case does not produce lower mass ${M}_{{\rm{d}},{\rm{f}}}$ that cover the small $\mathrm{log}\,g$ and ${T}_{\mathrm{eff}}$ part of the plot. In the ${M}_{{\rm{d}},{\rm{i}}}=1.3\,{M}_{\odot }$ case, $0.07\lesssim {M}_{{\rm{c}}}/{M}_{\odot }\lesssim 0.1$ before the onset of RLOF for cases that make an ELM WD with ${M}_{{\rm{d}},{\rm{f}}}\lesssim 0.17\,{M}_{\odot }$. By contrast, the runs with ${M}_{{\rm{d}},{\rm{i}}}=1.5\,{M}_{\odot }$ failed to produce an ELM WD (for which MT ceased) with mass lower than 0.168 M_⊙. The leftmost track in Figure 14 has ${P}_{\mathrm{orb},{\rm{i}}}=0.85$ days, and initial periods shorter than this value will have continuous MT and never emerge as an ELM pulsator.

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Figure 15 shows ${T}_{\mathrm{eff}}$ versus P_orb. The tracks start at the ZAMS with $20\lesssim {P}_{\mathrm{orb},{\rm{i}}}/\mathrm{hr}\lesssim 21$, and the MT starts roughly 1–2 Gyr into the MS evolution of the donor star. The final WD thus has more time to cool, and more time passes between the start of MT and the end of the 13.7 Gyr simulation than in the lower-mass donor case. After MT commences and enough mass has been lost that M_d ≲ M_a, the orbit expands dramatically and can exceed the initial separation a_i. The smallest ${P}_{\mathrm{orb},{\rm{f}}}$ is near 12 hr. The ${M}_{{\rm{d}},{\rm{i}}}=1.5{M}_{\odot }$ case is therefore unable to account for the three systems with ${P}_{\mathrm{orb}}\lt 12\,\mathrm{hr}$.

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A further increase of the donor mass above ${M}_{{\rm{d}},{\rm{i}}}=1.5\,{M}_{\odot }$ would lead to higher ${M}_{{\rm{c}}}$ at terminal age MS, and higher final WD mass. The upper limit for ${M}_{{\rm{d}},{\rm{i}}}$ that may produce an ELM WD is thus near $1.5\leqslant {M}_{{\rm{d}},{\rm{i}}}/{M}_{\odot }\leqslant 1.6$.

Adiabatic mode periods have been computed using the GYRE code (Townsend & Teitler 2013), which is part of the MESA distribution.

Figure 16 shows the propagation diagram and the composition versus radius fraction $r/{R}_{{\rm{d}}}$ during the post-MT evolution of the ${M}_{{\rm{d}},{\rm{i}}}=1.3\,{M}_{\odot }$, ${M}_{{\rm{a}},{\rm{i}}}=0.6\,{M}_{\odot }$, and ${P}_{\mathrm{orb},{\rm{i}}}=0.95\,\mathrm{day}$ model. Three different times are shown, where the color of the lines indicates the age, with the blue, green, and red lines representing models at 5.63 Gyr (right after the MT), 7.85 Gyr (at the elbow), and 13.7 Gyr (the termination of the simulation), respectively. The bump in the square of the Brunt–Väisälä frequency N² is caused by the composition switch from hydrogen to helium. After MT ends, the size of the helium core increases due to sinking of helium in the envelope and ongoing burning of hydrogen in the envelope.

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In order for the composition profile to be in diffusive equilibrium, the diffusion timescale must be shorter than the nuclear burning and cooling timescales. The composition profile in Figure 16 is far from diffusive equilibrium just after MT (blue line) and also at the elbow (green line). The red line is in diffusive equilibrium to a good approximation, and a range of ages (not shown here) were in diffusive equilibrium as well. A close examination of the composition profiles at different ages shows that the residual hydrogen burning ends at nearly the time that diffusive equilibrium is established. The age at which diffusive equilibrium is established was determined for each of the tracks in Figure 6, and their position marked by a black cross. The rightmost track with the shortest ${P}_{\mathrm{orb},{\rm{i}}}$ did not reach diffusive equilibrium before the second MT phase. The six pulsators with $8500\lesssim {T}_{\mathrm{eff}}/{\rm{K}}\lesssim 9000$ are in diffusive equilibrium to a good approximation.

Figure 17 shows the p-mode frequency spacing and g-mode period spacing for the fiducial model after MT has ceased. The different color lines represent different ${P}_{\mathrm{orb},{\rm{i}}}$. The blue and green lines appear shorter because they were terminated at the start of a second phase of MT. The g-mode period spacing is strongly dependent on the WD mass, so the lines differ by up to 30%. Furthermore, g-mode period spacing depends on the age, mainly through the thickness of the hydrogen envelope, so there can be about 15% differences in the period spacing along an individual evolutionary track. A minimum in the g-mode period spacing occurs near the elbow separating the pre-WD and the WD cooling track. The p-mode frequency spacing becomes nearly constant for the high-mass models, but for the lower-mass models, the spacing is slowly increasing in time over many Gyr.

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Figure 18 displays the lowest-order g-mode and p-mode for one evolutionary track (${M}_{{\rm{d}},{\rm{i}}}=1.3\,{M}_{\odot }$, ${M}_{{\rm{a}},{\rm{i}}}=0.6\,{M}_{\odot }$, and ${P}_{\mathrm{orb},{\rm{i}}}=0.95$ day). Most of the oscillation modes are mixed modes on the pre-WD track, meaning that near the radiative core, the mode behaves like a g-mode, and in the outer convection zone, the mode behaves like a p-mode with larger radial displacement. A sequence of avoided crossings is observed during the approximately 2 Gyr pre-WD phase. Starting at 7.5 Gyr, on the WD cooling track, the avoided crossings end, and the g-mode and p-mode are distinct, separated with gap in period. This period separation increases during the subsequent WD cooling phase.

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Figure 19 shows the ${\ell }=1$ p-modes (dashed) and g-modes (solid) at a fixed ${T}_{\mathrm{eff}}=9000$ K. The final WD masses are from six evolutionary tracks, with ${P}_{\mathrm{orb},{\rm{i}}}=0.93$, 0.95, 0.97, 0.99, 1.01, and 1.03 days and ${M}_{{\rm{d}},{\rm{i}}}=1.3\,{M}_{\odot }$ and ${M}_{{\rm{a}},{\rm{i}}}=0.6\,{M}_{\odot }$. The period gap between the p-modes and g-modes increases with the WD mass. The g-mode periods decrease slightly with the increased final mass in the mass range of the ELM WD. For WDs with masses above about 0.18 M_⊙, the gap between the p-modes and g-modes begins to increase even more rapidly. The mode periods decrease with increasing WD mass, which agrees with CA's result (Córsico & Althaus 2014).

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Figure 20 shows the stellar mass as a function of P_orb just after the MT phase has ended. Three different initial binary mass configurations (fiducial case, higher accretor mass, and lower donor mass) are plotted. The helium-core mass and hydrogen envelope mass are also plotted for each of these systems. We recall that P_orb continues to change in the pre-WD and the WD cooling phases due to gravitational wave losses. Moreover, there is continued burning of the envelope that adds to the core.

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For the fiducial case, P_orb,i smaller than 0.9 day results in continuous accretion and thus no ELM WD pulsator. The lowest ELM WD mass is ${M}_{{\rm{d}},{\rm{f}}}=0.146\,{M}_{\odot }$, with ${M}_{{\rm{c}}}=0.120\,{M}_{\odot }$ and P_orb = 3.72 hr. The envelope is 18% of the total mass in this case, much higher than for a standard 0.6 M_⊙ carbon/oxygen WD. A large fraction of the radius of the star is also taken up by the envelope in this case. All else being equal, longer ${P}_{\mathrm{orb},{\rm{i}}}$ results in higher pre-WD masses and M_c, but lower M_env, immediately post-MT. For ${P}_{\mathrm{orb},{\rm{i}}}$ above 1.03 days (in the fiducial case), the WD experiences hydrogen flashes prior to the cooling phase. At this upper boundary, the pre-WD mass is 0.179 M_⊙ and P_orb is 18.35 hr immediately post-MT, with an envelope containing only 11.5% of the total mass, which is lower than all the ELM WDs with no hydrogen flashes before cooling. For non-ELM WDs that experience hydrogen flashes prior to the cooling phase, the resulting envelope is much thinner than for the lower-mass ELM WD. The trend of the thinner envelope with an increasing total WD mass agrees with Istrate et al. (2016).

The results from a simulation with a high-mass companion, ${M}_{{\rm{a}},{\rm{i}}}=0.9\,{M}_{\odot }$, are plotted as green lines in Figure 20. The blue and green lines nearly overlap, producing ELM WDs with almost identical mass, composition, and orbits immediately post-MT. Similarly, the results for a lower-mass donor ${M}_{{\rm{d}},{\rm{i}}}=1.1\,{M}_{\odot }$ are shown in red; this setup can create ELM WDs with even lower masses and shorter ${P}_{\mathrm{orb}}$. The minimum ELM WD mass for this setup is 0.143 M_⊙, with ${P}_{\mathrm{orb}}=2.75$ hr immediately post-MT, and the range of ${P}_{\mathrm{orb},{\rm{i}}}$ that results in ELM WDs is considerably larger than for higher-mass donors. For low donor masses (e.g., ${M}_{{\rm{d}},{\rm{i}}}=1.0$ and 1.1 M_⊙), the thick convective envelope present during the MS phase causes magnetic braking to be much stronger, resulting in a wider accessible range of ${P}_{\mathrm{orb},{\rm{i}}}$. Chen et al. (2017) used a wider range for ${M}_{{\rm{d}}}$ and ${M}_{{\rm{a}}}$ and ${P}_{\mathrm{orb}}$ with different metallicities. Their mass-period relation is in agreement with Figure 20, and the difference in metallicity does not affect this relation at the low-mass WD range.

From Figure 7, two of the ELM WDs, J1840 and J1112, have $4\lesssim {P}_{\mathrm{orb}}/\mathrm{hr}\lesssim 5$ (a range that is accessible with our simulations), but with a higher ${T}_{\mathrm{eff}}$ that falls slightly above the theoretical tracks. This section is about making ELM WDs with ${T}_{\mathrm{eff}}\approx 9000$ K and short orbital periods ${P}_{\mathrm{orb}}\lesssim 5\,\mathrm{hr}$.

Figure 21 compares evolutionary tracks for simulations with different ${M}_{{\rm{d}},{\rm{i}}}$ and ${M}_{{\rm{a}},{\rm{i}}}$. Prior to the WD cooling phase, the donor star reaches a maximum ${T}_{\mathrm{eff}}$. The trend is for this maximum ${T}_{\mathrm{eff}}$ to decrease with decreasing ${P}_{\mathrm{orb},{\rm{i}}}$, and for the fiducial case, it appears that having ${T}_{\mathrm{eff}}\approx 9000$ K with $4\lesssim {P}_{\mathrm{orb}}/\mathrm{hr}\lesssim 5$ is inaccessible. For a higher accretor mass, and even more dramatically for a lower donor mass, the maximum ${T}_{\mathrm{eff}}$ is increased relative to the fiducial case. So to make a WD with ${T}_{\mathrm{eff}}\gt 9000$ K at short orbital periods, the preference is to have a low-mass donor and high-mass accretor.

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Instead of making ELM WDs models through binary evolution including magnetic braking, a simpler and cheaper alternative would be the following. Evolve a single ZAMS star until it reaches the desired ${M}_{{\rm{c}}}$. Then rapidly (on a timescale much shorter than the thermal and nuclear burning timescales) remove the envelope until the desired ${M}_{\mathrm{env}}$ is left. The resulting model would represent the start of the pre-ELM WD track seen in this work. The two parameters are ${M}_{{\rm{c}}}$ and ${M}_{\mathrm{env}}$ for a fixed composition. The star is then evolved through the pre-ELM WD and WD cooling tracks.

Although much simpler, the problem with this method is that it is not known a priori what to choose for ${M}_{{\rm{c}}}$ and ${M}_{\mathrm{env}}$. Furthermore, this method does not give the expected ${P}_{\mathrm{orb}}$ for the binary, or possible ranges of the mass of the companion WD. The former issue has been addressed in Figure 20, which shows how the total stellar mass is partitioned into core and envelope. This greatly restricts the range of allowed ${M}_{\mathrm{env}}$ because even for higher ${M}_{\mathrm{env}}$, up to half the mass of the star can be used for $M\lt 0.17\,{M}_{\odot }$ without incurring shell flashes. Such high ${M}_{\mathrm{env}}$ would have many Gyr of hydrogen burning until a more physically motivated envelope size would result.