THE MASS-LOSS-INDUCED ECCENTRIC KOZAI MECHANISM: A NEW CHANNEL FOR THE PRODUCTION OF CLOSE COMPACT OBJECT–STELLAR BINARIES

When we evolve a triple system, we break its evolutionary phases into four distinct stages: (1) primary on the MS, (2) primary evolved into a giant but prior to significant mass loss, (3) during a period of significant mass loss, and (4) after the end of mass loss when the primary has become a compact object. However, because of the prohibitive computational time required to integrate close and highly eccentric triples for the full MS lifetime of the primary, we truncate our integrations after 10³ Kozai–Lidov times (t_K) defined by (Innanen et al. 1997; Holman et al. 1997)

$$\begin{eqnarray} {t}_{{\rm K}}&=&\frac{4}{3}\left(\frac{a_{{\rm 1}}^3 (m_{{\rm 0}}+ m_{{\rm 1}})}{Gm_{{\rm 2}}^2}\right)^{1/2}\left(\frac{b_{{\rm 2}}}{a_{{\rm 1}}}\right)^3 \nonumber \\ &\simeq & 6.4 \times 10^{4} \,\hbox{yr} \left(\frac{a_{{\rm 1}}}{10 \,\hbox{AU}}\right)^{3/2} \left(\frac{m_{{\rm 0}}+ m_{{\rm 1}}}{13.5 \,{M}_\odot }\right)^{1/2}\nonumber\\ &&\times \left(\frac{6\,{M}_\odot }{m_{{\rm 2}}}\right) \left(\frac{1}{25}\frac{b_{{\rm 2}}}{a_{{\rm 1}}}\right)^{3}, \end{eqnarray} \tag{ 3 }$$

where b₂ = a₂(1 − e₂)^1/2. Thus, we implicitly assume that the triple systems have no interesting behavior after this time.

We account for the MS lifetime of the primary by only integrating the triple systems for a time (t_MS) defined by

$$\begin{equation} {t}_{{\rm MS}}= \hbox{min}({t}_{{\rm ms, 0}}, 10^{3} {t}_{{\rm K}}) \times (0.95 + 0.1 \xi), \end{equation} \tag{ 4 }$$

where ξ is a uniformly distributed random number on the interval [0, 1) and t_ms,0 is the MS lifetime of the primary. We determine the MS lifetimes by interpolating the logarithmic lifetimes on a grid of stellar models run with the Yale Rotating Evolution Code with input physics described in van Saders & Pinsonneault (2012). We multiply by the second factor in Equation (4) in order to randomize exactly when the primary evolves off the MS to avoid any correlations of the start of mass loss between different triple systems in our sample.

To simulate the giant phase of the primary, we continue to integrate the triple for a time (t_RG) defined by

$$\begin{equation} {t}_{{\rm RG}}= \rm {min}(0.1 {t}_{{\rm ms, 0}}, 10^{3} {t}_{{\rm K}}). \end{equation} \tag{ 5 }$$

Then, to simulate mass loss, we linearly decrease m₀ over a mass-loss timescale (t_ml) to the WD mass as specified by the initial–final mass relation of Kalirai et al. (2008) given by

$$\begin{equation} m_{0,\hbox{f}} = (0.109 \pm 0.007) \, m_{{\rm 0}}+ (0.394 \pm 0.025) \, {M}_\odot, \end{equation} \tag{ 6 }$$

where m_{0, f} is final mass of the primary. To test our mass-loss prescription, we integrated binaries where one member underwent mass loss and tested that the change in the semimajor axis and eccentricity of the binary matched that for instantaneous mass loss and adiabatic mass loss (i.e., Equations (4)–(6) in Baribault & Pineault 1994) when t_ml ≪ P and when t_ml ≫ P, respectively, where P is the orbital period of the orbit. We found good agreement.

After mass loss has finished we integrate the evolved triple for a time (t_after) given by

$$\begin{equation} {t}_{{\rm after}}= \rm {min}\left({t}_{{\rm ms, 1}}- 1.1 {t}_{{\rm ms, 0}}- {t}_{{\rm ml}}, 10^{3} {t}_{{\rm K}}\right), \end{equation} \tag{ 7 }$$

where t_ms,1 is the MS lifetime of the next most massive star in the triple system and t_K is redefined by Equation (3) with the orbital parameters of the triple right after the conclusion of mass loss.

A detailed treatment of stellar tides and tidal dissipation is beyond the scope of this paper. However, to determine when tidal effects are likely to be important, we define an ad hoc minimal separation between the inner two binary members, r_tide, below which strong tidal contact or a collision is assumed to have occurred. That is, if the separation between the primary and secondary is r < r_tide, we assume either (1) that tidal dissipation will remove energy from the inner orbit decreasing a₁ or (2) if the separation between the inner binary members changes dramatically in a single inner orbital period (P₁), a physical collision may occur before tides become important, as in the recent work of Katz & Dong (2012). In either case, our simulations are not valid for r < r_tide.

When choosing a tidal criterion we considered two tidal effects which will alter the evolution of the system. The first is the nondissipative contribution that tidal bulges on the members of the inner binary have on apsidal motion (Sterne 1939). This additional apsidal motion may detune the Kozai mechanism (e.g., Wu & Murray 2003; Fabrycky & Tremaine 2007). We find that the timescale of this apsidal motion becomes comparable to t_K when r_peri ∼ 4 R_{ms, 0}, assuming the typical tidal Love number valid for n = 3 polytropes, k = 0.028 (Eggleton & Kiseleva-Eggleton 2001), and the system parameters for the fiducial system presented in Section 3. The second effect is eccentricity damping due to dissipative tides (Mazeh & Shaham 1979; Wu et al. 2007). For our fiducial system, we derive a critical inner separation by equating the tidal and Kozai forcing terms in the equation for the secular evolution of e₁. We find a critical inner separation of

$$\begin{equation} r_{{\rm crit}} \sim 3 \,R_{{\rm ms, 0}}\left(\frac{10^4}{Q} \ \frac{k}{0.028} \right)^{2/13}, \end{equation} \tag{ 8 }$$

where Q is the tidal dissipation factor. As can be seen in Equation (8), r_crit is very insensitive to the exact values chosen for k and Q. For simplicity, and to allow a broad, albeit incomplete, survey of parameter space, we take r_tide = 3 × max (R_{ms, 0}, R_{ms, 1}) for the bulk of this study, but we investigate the effect of changing r_tide on our results in Section 4.3.

GR periastron precession can "detune" the normal Kozai mechanism and truncate the maximum eccentricity attainable for a triple system (Blaes et al. 2002; Miller & Hamilton 2002; Thompson 2011). FEWBODY does not contain post-Newtonian corrections, so we must check if it is permissible to ignore GR precession. The GR precession timescale (e.g., Equation (23) of Fabrycky & Tremaine 2007) is given by

$$\begin{eqnarray} {t}_{{\rm GRp}}&=& \frac{1}{3}\frac{a_{{\rm 1}}}{c}\left(\frac{a_{{\rm 1}}c^2}{G(m_{{\rm 0}}+ m_{{\rm 1}})}\right)^{3/2}\hspace{-5.69046pt}\left(1-e_{{\rm 1}}^2\right) \nonumber \\ &\simeq &3.4\times 10^7{\rm \,\,yr}\,\, \left(\frac{13.5 \,{M}_\odot }{(m_{{\rm 0}}+ m_{{\rm 1}})}\right)^{3/2}\left(\frac{a_1}{10\,{\rm AU}}\right)^{5/2}\left(1-e_1^2\right).\nonumber \\ \end{eqnarray} \tag{ 9 }$$

As discussed in Blaes et al. (2002), the Kozai mechanism only operates if t_K ≲ t_GRp. We see that for the masses and semimajor axes of our fiducial case presented in Section 3, t_K ≲ t_GRp for any choice of e₂ until the inner eccentricity is increased such that r_peri < R_{ms, 1}. Thus, the normal Kozai mechanism is free to operate in our triple integrations.

However, we also see in Section 3 that the eccentric Kozai timescale (t_EK), the timescale for "flips" and extreme eccentricity maxima, is much longer than t_K, t_EK ∼ 10⁶ yr. Furthermore, we see that t_EK>t_GRp for values of e₁ where r_peri > R_☉, which would seem to imply that GR precession might detune the EKM before tidal effects become important. This is not clear, however, because the majority of the GR precession occurs when e₁ is large, which is only a small fraction of the eccentric Kozai cycle. To test if GR precession will suppress the maximum e₁ reached, we take the fiducial triple system after mass loss that exhibits the eccentric Kozai flip, and compare this to the equivalent octupole-order calculation that includes GR precession using the code of Thompson (2011; Blaes et al. 2002). We found good agreement between the two calculations, implying that GR precession does not shut off the EKM, and is thus not important for the current study. However, these effects should be more carefully considered in future studies where a₁ is much smaller, and when the periastron distance can be significantly smaller than R_☉ if both the primary and secondary are compact objects.

To explore the effects of inclination and eccentricity, we integrate ∼10⁵ triple systems over an equally spaced grid with 41, 10, 20, and 4 values of cos  i, e₁, e₂, and g₁, respectively, while keeping the initial masses and initial semimajor axes from our example system. Because the outcomes of triple systems affected by MIEK may differ depending on the initial phase angles, we integrate three triple systems at each grid point in cos  i, e₁, e₂, and g₁with randomized initial phase angles.

In order to determine if the inner binary of each system becomes tidally affected or collides by our ad hoc tidal criterion (Section 2.2), we record the minimum inner separation (r_min) in each of the four evolutionary phases described in Section 2. We integrate these triple systems assuming t_ml = 10⁴ yr and posit that tides will affect the inner binary if r_min is smaller than some ad hoc tidal radius (r_tide). We define this ad hoc tidal radius differently for the different evolutionary phases of the primary, in an attempt to capture the different strength of tidal forces due to the varying physical size and mass of the primary. Before the primary evolves off the MS we take r_tide = 3 R_{ms, 0}, where R_{ms, 0} in the MS radii of the primary. When the primary is a giant and during the mass-loss phase, we take r_tide = 1 AU. Finally, when the primary becomes a WD we take r_tide = 3 R_{ms, 1}, where R_{ms, 1} is the MS radii of the secondary in the inner binary. We determine R_{ms, 0} and R_{ms, 1} from the Torres et al. (2010) sample by taking the median of the observed radii for stars with similar masses (|log M − log M'| < 0.1).

Additionally, we would like to determine when the EKM is operating. Because orbital flips (i.e., a change of sign in cos  i) only occur during the EKM, we monitor each triple systems for flips during each of the aforementioned evolutionary phases. We then use the two above criteria to loosely define systems that are strongly affected by the MIEK mechanism, as systems that come into tidal contact and exhibit a flip after the onset of mass loss.

The results of the parameter space search in inclination and eccentricity are presented in Table 1 and Figures 3 and 4. In Table 1 we break down the fraction of the triple systems that flip and become tidally affected or collide during the various evolutionary stages presented in Section 2. We highlight any asymmetries about cos  i₀ = 0 by presenting the fractions for prograde and retrograde systems separately. We now describe each of the evolutionary phases of the primary in turn.

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When the primary is on the MS ("Primary MS" in Table 1), 13% of triple systems become tidally affected or collide (r_min ⩽ r_tide). We show the initial inclinations and eccentricities of these triple systems in the left panel of Figure 3. Each contour plot displays the parameter space projected onto a two-parameter plane. Each histogram marginalizes over all parameters except one, and then displays the fraction of systems in each bin that become tidally affected or collide. From the left panel of Figure 3, we see that when the primary is on the MS the majority of systems which become tidally affected or collide are initially highly inclined. These systems undergo normal Kozai–Lidov oscillations, which bring the inner binary to tidal contact with a maximum pericenter distance given by

$$\begin{equation} r_{{\rm peri}}= a_{{\rm 1}}(1-e_{\rm {1, max}})=a_{{\rm 1}}\left[1- \left(1 - \frac{5}{3} \cos ^{2}i_0 \right)^{1/2} \right] \end{equation} \tag{ 13 }$$

with e_{1, max} given by Equation (1) and i₀ the initial eccentricity. Then, setting the periastron distance equal to r_tide,

$$\begin{equation} r_{{\rm peri}}= r_{{\rm tide}}= 3 \,R_{{\rm ms, 0}}\sim 12 \,{R}_\odot \sim 0.05 \,\hbox{AU} \sim 0.005 \ a_{{\rm 1}}, \end{equation} \tag{ 14 }$$

we see that the required critical initial inclination to interact tidally when the primary is on the MS is

$$\begin{equation} \cos i_{0,\rm crit}\sim 0.08, \end{equation} \tag{ 15 }$$

which is in good agreement with the prograde orbits in the left panel of Figure 3. Note, however, that the distribution is asymmetric about cos i₀ = 0, with 22% of the retrograde systems becoming tidally affected or colliding while only 3.9% of the prograde systems do. This asymmetry is not captured by Equations (1) and (13) because of the approximations under which Equation (1) is derived. A more complete treatment of the quadrupole-order Hamiltonian shows that e_{1, max} does not occur at cos i₀ = 0, but instead at retrograde mutual inclination (see Figure 1 of Miller & Hamilton 2002).

Importantly, we find that 6.0% of the triple systems, with cos i₀ ≃ 0, become tidally affected and exhibit an orbital flip, which is characteristic of the EKM. These systems comprise almost half of the total number of systems that become tidally affected or collide when the primary is on the MS, showing that the EKM is also important. The implications that the eccentric Kozai has for previous studies involving the Kozai–Lidov mechanism and tidal friction is discussed further in Section 5, but here we note that the systems that tidally interact or collide on the MS are analogous to the systems investigated by Fabrycky & Tremaine (2007), except that these authors only included quadrupole-order terms in the secular Hamiltonian, and thus may have missed the systems that undergo the EKM and execute a "flip" with e₁ → 1 (Naoz et al. 2011a, 2011b).

Additionally, 11% of triple systems become unbound when the primary is on the MS. The systems that are unbound have large e₂ and do not satisfy the standard, empirically derived, stability criterion for triple systems given by (Mardling & Aarseth 2001)

$$\begin{equation} \frac{a_{{\rm 2}}}{a_{{\rm 1}}} \ge C \: f \left[ \left(1 + \frac{m_{{\rm 2}}}{m_{{\rm 0}}+ m_{{\rm 1}}} \right) \frac{1 + e_{{\rm 2}}}{(1 - e_{{\rm 2}})^{3}} \right] ^{0.4}, \end{equation} \tag{ 16 }$$

where C = 2.8 and f = 1 − (0.3/π)i is an ad hoc inclination term. These systems are present in our study because we uniformly sample the eccentricities without regard for the initial dynamical stability of the system. We see from Equation (16) and our initial assumed semimajor axes that if cos  i = 0 then e₂ ⩽ 0.83 must hold for the system to satisfy the stability criterion. It is interesting to note that in the cos  i₀ versus e_{2, 0} contour plot in the left panel of Figure 3, many triple systems that are unstable given the stability criteria become tidally affected or collide before they are unbound. It is possible that tidal friction will quickly shrink a₁ of these systems, causing them to become stable as the ratio a₂/a₁ increases.

After the primary has evolved off the MS but before significant mass loss has begun ("Primary Giant" in Table 1), 28% of triple systems become tidally affected or collide (r_min ⩽ 1 AU) for the first time. We show the initial inclinations and eccentricities of these triple systems in the right panel of Figure 3. Such a large fraction of the triple systems become tidally affected or collide during this stage of evolution because the physical size of the primary dramatically increases, so we impose an ad hoc increase to r_tide = 1 AU. Then from Equation (13) we find that

$$\begin{equation} \cos i_{0,\,\rm crit} \simeq 0.34, \end{equation} \tag{ 17 }$$

encompassing about a third of our parameter space. Additionally, we see that only a small fraction of the systems exhibit the orbital flip characteristic of the EKM when the primary is a giant, showing that the EKM is not important during this phase of the primary's evolution. Lastly, only 0.037% of the systems are unbound during this evolutionary phase, emphasizing that initially dynamically unstable systems are either quickly unbound, when the primary is still on the MS, collide, or become tidally affected.

During the t_ml = 10⁴ yr mass-loss phase ("During Mass-Loss" in Table 1) only a small fraction, 0.016%, of the triple systems become tidally affected or collide for the first time. This is because previous stages of evolution have removed the majority of triple systems that would tidally interact or collide with r_tide = 1 AU, and because t_ml < t_K, so the systems do not have time to complete a Kozai cycle during the primary's mass-loss episode. We find that the fraction of systems that are unbound increases to 0.27% from the previous phase of evolution. More systems become unstable because, as shown by Equation (11), the outer to inner semimajor axis ratio decreases during mass loss. Then, as a result of Equation (16), this decrease in the semimajor axis ratio causes more triple systems to be unstable.

Finally, after the end of mass loss ("After Mass-Loss" in Table 1) we see that 2.2% of triple systems spanning a broad range in orbital parameters become tidally affected or collide for the first time. We note that the lion's share, 1.9%, of these systems exhibit an orbital flip characteristic of the EKM and thus the MIEK mechanism because it is induced by mass loss of the primary. We now discuss the distribution of tidally affected or colliding systems after mass loss for the inclination and eccentricities individually.

The distribution of these tidally affected or colliding systems is asymmetric about cos  i₀ = 0, with 3.0% of prograde systems and only 1.5% of retrograde systems becoming tidally affect of colliding. At least some of this asymmetry arises because, like previous stages of evolution of the primary, systems with cos i₀ near 0 are strongly preferred. However, earlier stages of evolution have removed a disproportionate fraction of the prograde systems, 33%, compared to the fraction of retrograde systems, 49%. This causes the reverse asymmetry, more prograde than retrograde systems, in the distribution of systems that become tidally affected or collide after the end of mass loss.

The distribution in e_{2, 0} for the systems that become tidally affected or collide after the end of mass loss shows a strong preference for larger values, peaking at e_{2, 0} = 0.75. This preference is easily understood when considering Equation (2), which also shows a strong dependence on e_{2, 0}. Thus, the octupole-order terms and consequently the EKM are expected to become more important with larger e_{2, 0}. Even though there is a strong preference for large e_{2, 0}, the MIEK mechanism continues to operate for systems with modest e_{2, 0} ∼ 0.4. We also note that the e_{2, 0} distribution is truncated at very large values, which arises because most of these systems became tidally affected, collided, or were unbound when the primary was on the MS.

Lastly, the distribution of e_{1, 0} for the systems that become tidally affected or collide shows a preference for both large and small values. The minimum in the distribution occurs at e_{1, 0} ∼ 0.5, with ∼3 times fewer systems becoming tidally affected or colliding then at e_{1, 0} = 0. The preference for large values of e_{1, 0} is easily understood because these systems do not need to increase their inner eccentricity as much to collide or come into tidal contact. The preference for smaller values of e_{1, 0}, however, is not easily understood. We also note that the majority of systems with e_{1, 0} = 0.9 become tidally affected or collide in earlier stages of the primary's evolution, so these systems are absent in the current e_{1, 0} distribution.

We now briefly evaluate the sensitivity of the results presented in Section 4.2 on the ad hoc tidal criterion chosen (Section 2.2).

First, we evaluate the dependence of the fraction of systems that become tidally affected or collide on our choice of tidal criterion. We both decreased and increased the tidal criterion from r_tide = 3 R_* to r_tide = 2.5 R_*, r_tide = 3.5 R_*, r_tide = 5 R_*, and r_tide = 10.0 R_*. R_* = R_{ms, 0} when the primary is on the MS and R_* = R_{ms, 1} when the primary is a WD. The fractions of systems that become tidally affected or collide after the end of mass loss with these new tidal criteria are 0.022, 0.023, 0.023, and 0.025, respectively. When these fractions are compared to those presented in Table 1, we see that the fraction of systems affected by the MIEK mechanism is relatively insensitive to the exact choice of tidal criterion. This insensitivity results because an increase in the tidal radius on the MS affects systems that will already be affected by the large tidal radius when the primary becomes a giant, and does not affect regions of parameter space shown in Figure 4 where the MIEK mechanism operates. In fact, increasing the tidal radius marginally increases the fraction of systems affected by MIEK.

Additionally, we emphasize that by eliminating all systems that collide or were tidally affected at a prior stage of evolution, 1.9% is a conservative estimate of the fraction of triple systems that will undergo the MIEK mechanism. This is because many of the systems we remove from the sample as the evolutionary stages progress may still be viable candidates for the MIEK mechanism. One possibility is that tides on the giant primary only detune the normal Kozai–Lidov mechanism, and do not allow e₁ to increase so dramatically. Then, our removal of these systems from the sample would unrealistically decrease the number of systems that experience MIEK. These systems would survive in our sample to the stage after mass loss is complete. Another possibility is that the combination of the Kozai–Lidov mechanism and tides on a giant will work to shrink a₁ to a few AU. These systems will then have a₂/a₁ ∼ 100, but they can still be affected by the EKM after mass loss, which will shrink this semimajor axis ratio according to Equation (11).

Evaluating these effects would involve a realistic calculation of the three-body dynamics with tides since cos i will evolve secularly during the tidal interaction (Fabrycky & Tremaine 2007). We save this for a future work. Here, to roughly quantify the number of systems that could potentially be affected by the MIEK mechanism, we changed the tidal criterion when the primary star is a giant and during mass loss to be the same as it was when the primary star was on the MS, r_tide = 3 R_{ms, 0}. Under this new tidal criterion, the analogue of Figure 4 is presented in Figure 5. As expected, we see a sharp decrease, from 28% to 0.011%, in the fraction of triple systems that are tidally affected or collide when the primary is a giant. There are then many additional systems available after mass loss to increase the fraction of systems that are tidally affected or collide after the primary becomes a WD. The total change in the fraction is from 2.2% to 8.7%. Similarly, there is a significant increase in the fraction of MIEK systems, systems that collide or become tidally affected and flip the sign of cos  i, from 1.9% to 6.7%. For the example system considered, and for a uniform distribution in eccentricities and mutual inclination, we consider 1.9% and 6.7% to be lower and upper limits, respectively, to the fraction of all systems that undergo MIEK.

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We have shown in our fiducial parameter search that systems that interact tidally while the primary is on the MS undergo both normal Kozai–Lidov oscillations and the EKM in approximately equal proportion. Those systems that undergo normal Kozai–Lidov oscillations are analogous to the systems considered by Fabrycky & Tremaine (2007), who calculated the long-term evolution of a distribution of triple systems using the quadrupole-order secular equations of Eggleton & Kiseleva-Eggleton (2001). Fabrycky & Tremaine (2007) found that the combination of normal Kozai–Lidov cycles and tidal friction produces a substantial population of close binaries, and that their calculations matched the observations of Tokovinin et al. (2006), which show that essentially all close spectroscopic binaries are actually triple systems. Given these results, we anticipate that our systems with cos i₀ ∼ 0 will also tidally circularize with small a₁ and P₁, with the caveat that the MS lifetime of the primary we consider here is much shorter than for the systems considered by Fabrycky & Tremaine (2007), and thus a fraction of our systems may not circularize before WD formation.

Importantly, though, we find that half of the systems that tidally interact or collide on the MS do so as a result of the EKM, which, as Lithwick & Naoz (2011), Katz et al. (2011), and Naoz et al. (2011a, 2011b) have shown, cannot be captured by the quadrupole-order calculations of Fabrycky & Tremaine (2007). What is the fate of these systems? Unfortunately, no study has yet been performed to investigate the EKM with the inclusion of tidal dissipation for triple stellar systems. However, Naoz et al. (2011a) demonstrate with an octupole-order calculation that the EKM and tidal friction can possibly explain the occurrence of retrograde hot Jupiters. They show that the extremely high eccentricities obtained by the inner binary when cos  i flips signs can lead to a rapid capture of a planet into a short-period retrograde orbit ("Kozai capture"). We suppose that a similar mechanism operates for stellar triple systems, and that those that are tidally affected on the MS due to the EKM rapidly dissipate energy from the inner binary and circularize with a semimajor axis of a₁ ≈ 2r_peri ∼ 2r_tide. If so, then the Fabrycky & Tremaine (2007) study may be missing of the order of half of the systems that become close binaries.

The action of the EKM for MS binaries may also affect the distribution of binary eccentricities as a function of inner period as the population is circularized (Socrates et al. 2012; Dong et al. 2013).

An example observed (probable) intermediate-mass triple star system that has likely already tidally circularized is MT429 in the Cygnus OB2 association, where the inner P₁ ≃ 3 d B3V+B6V binary is orbited by a B0V tertiary at 138 AU (Kiminki et al. 2012). An example of an even more massive system that might give rise to an NS–MS triple system as discussed in Section 5.4 is HD 150136. It consists of O3-3.5V+O5.5-6V (∼64 + 40 M_☉) inner binary with P₁ ≃ 2.7 d and an O6.5-7V (∼35 M_☉) tertiary with P₂ ∼ 3000–5000 d (Mahy et al. 2012).

Finally, after the inner binary orbit has shrunk due to either normal Kozai–Lidov oscillations or the EKM, the binary evolves like other close binaries, potentially leading to the progenitors of single-degenerate (SD) supernovae (SNe; Whelan & Iben 1973), double-degenerate (DD) SNe (Webbink 1984; Iben & Tutukov 1984), barium stars (BA; McClure et al. 1980), and asymmetric planetary nebulae (PNe; Morris 1987).

We find that ∼30% of the triple systems we consider interact tidally or collide as the primary evolves off the MS, but before mass loss, mostly as a result of normal Kozai–Lidov oscillations, which bring the pericenter of the secondary orbit inside ∼AU (see the right panel of Figure 3 and Equation (17)). Since the post-MS phase is short (particularly for the intermediate- and high-mass stars we consider here), and since the efficiency of tidal friction is uncertain, it is unclear if the system will circularize before mass loss, or whether the Kozai–Lidov oscillation will simply be detuned such that pericenter ∼AU for the duration of the giant phase. The interaction of Kozai–Lidov oscillations as the primary evolves to the giant phase may also initiate mass transfer and complex secular dynamics, perhaps analogous to the system considered by Prodan & Murray (2012). Another possibility is that this phase initiates common envelope evolution, perhaps circularizing the inner orbit at smaller semimajor axis. If the inner binary maintains significant eccentricity, or if e₁ is pumped after the giant phase and during mass loss via normal Kozai–Lidov oscillations or MIEK, such systems may form off-center or bipolar PNe (e.g., Soker et al. 1998; Soker & Rappaport 2000; Witt et al. 2009). On the other hand, if the inner binary is circularized, it might also participate in the shaping of PNe (e.g., Mastrodemos & Morris 1998, 1999). An example of some of these mechanisms in action may be ring PN SuWt 2 (Exter et al. 2010). Since a very large fraction of triple stars will be first affected by Kozai–Lidov cycles when the MS star evolves, this phase should be investigated in detail.

We find that ∼2% of our triple systems undergo the MIEK mechanism after WD formation (see Figure 2). We again emphasize that by removing all systems from our sample that were tidally affected during the giant phase, the fraction of triples that go through MIEK presented in Table 1 ("After Mass-Loss") is conservative since many of these systems may be available to go through MIEK. The reason for this is that as the primary enters its post-MS evolution and begins to interact tidally with the secondary, the orbits will be affected by processes that we do not model (e.g., Prodan & Murray 2012). We attempt to provide an upper limit on the fraction of triple systems that could be tidally affected or collide after mass loss due to the MIEK mechanism in Section 4.3 by assuming a small value for r_tide during the giant phase (compare Figures 4 and 5). This increases the fraction of triple systems that undergo MIEK from 1.9% to 6.7%. A calculation of the three-body dynamics with tides appropriate to an MS star interacting with a red giant is clearly needed to address this issue in more detail and to address the issues raised in Section 5.2.

Systems that undergo the MIEK mechanism also have many possible outcomes. One possibility is that the extreme eccentricities obtained during a "flip" might lead to physical collisions (Katz & Dong 2012). This may occur if, in a single orbit of the inner binary, r_peri changes from r_peri > r_tide to r_peri < R_*. This is possible because the angular momentum of the inner binary at such large e₁ is small. Thus, the change in angular momentum required for r_peri ∼ 0 is also small. Another possible outcome for the system is a tidal Kozai capture as presented in Naoz et al. (2011a). The inner semimajor axis a₁ will strongly decrease because of tides, leaving a close MS–WD binary. When the secondary subsequently evolves off the MS, the WD will either accrete or go through common envelope evolution. Both of these scenarios could lead to SD SNe Ia or DD SNe Ia, cataclysmic variables, or other types of WD accretors such as AM CVn stars (Warner 1995), which may produce faint ".Ia" SNe (Bildsten et al. 2007).

One interesting consequence of this evolutionary scheme for producing close WD-MS binaries is that it skips the initial common envelope evolutionary phase usually required in binary evolution to produce a close WD-MS pair. In the MIEK mechanism the close binary is produced without common envelope, and thus the WD produced should have a mass appropriate to a single star with the mass of the primary (e.g., Kalirai et al. 2008). This stands in contrast to the conventional picture for the production of WD-MS and WD–WD binaries relevant for SD and DD Ia progenitors, where the WD growth is truncated by common envelope interaction with the close secondary and the mass of the resulting WD is smaller than one would naively estimate from the single-star initial–final mass relation (Iben & Tutukov 1984). For SD SN Ia models, this means that less mass needs to be accreted onto the WD from the secondary, leading to a shorter delay time between production of the system and explosion. For DD SN models, if the model presented in Pakmor et al. (2012) is accurate, then the luminosity of the SN resulting from the merger of a WD–WD binary is primarily determined by the mass of the primary. Thus, all else being equal, if a WD–WD binary results from an MIEK triple system then its primary WD mass should be larger and thus result in a more luminous DD SN than the analogous close binary system that is produced through normal common envelope evolution.

The MIEK mechanism may also play an important role in triple systems after NS formation, particularly since the NS mass, ≃ 1.4 M_☉, is so much less than its progenitor massive star, leading to a large increase in _oct at the time of the associated SN (Equation (2)). However, several effects differ in the NS case versus the case of WD formation considered throughout this paper. First, NSs receive a "kick" at birth that may unbind the inner binary or the outer tertiary, depending on the masses of the constituents, their eccentricities, and their semimajor axes (Hills 1983; Kalogera 1996). Second, mass loss on the MS can affect the mass ratios as a function of time (e.g., Ekström et al. 2012). Third, even in the absence of an NS kick, the large and instantaneous mass loss in the SN explosion from the primary provides a kick to the center of mass of the inner binary which may unbind the tertiary, and shift the mutual inclination. Even so, we expect a population of NS–MS binaries to survive with massive tertiaries and large e₂ due to the aforementioned kicks. Thus, _oct will be large for the surviving triple systems that will cause many to exhibit the MIEK mechanism. This is a qualitatively new way to produce close NS–MS binaries.

As a test, we integrated the orbit of a system with m₀ = 11 M_☉, m₁ = 10 M_☉, m₂ = 10 M_☉, e₁ = 0.7, e₂ = 0, cos i = 0.25, g₁ = 0°, g₂ = 270°, a₁ = 15 AU, and a₂ = 300 AU. At a time ∼7 Myr after the start of the simulation we instantaneously decrease the mass of the primary to 1.4 M_☉ and apply no kick to the NS. Figure 6 shows the resulting evolution. The system flips immediately after the SN and evolves to an inner eccentricity of 1 − e₁ > 10⁻³, bringing the NS to tidal contact with the secondary. Note the large increase in e₂ from ∼0 to ∼0.3 immediately after the SN. This calculation is meant only to be illustrative since it does not include an NS kick. A full population study, with kicks and tides, is clearly needed to assess the viability of this mechanism for forming close NS–MS binaries.

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A further aspect of this type of evolution is that the ratio of a₂/a₁ can decrease during the strong MS mass loss and the SN explosion, which can cause the triple to become dynamically unstable and eventually disrupt, or potentially initiate collisions (the TEDI mechanism of Perets & Kratter 2012).

In this study we have explored a single set of m₀ and m₁; however, different combinations of masses will affect the fraction of systems that become tidally affected or collide during each of the evolutionary phases. For example, if m₀ ≈ m₁ the system will have little or no time after the primary becomes a compact object before the secondary enters its giant phase. This will reduce the fraction of systems that will flip and become tidally affected or collide due to the MIEK mechanism. However, if the difference between the masses of the inner binary members is significantly greater, then a larger fraction of systems will flip before the primary goes through mass loss. Additionally, the difference in the masses after the primary has become a compact object will be smaller, further decreasing the fraction of systems that will be affected by the MIEK mechanism. Determining the optimal balance between these two effects is beyond the scope of the current study.

In a future study we will present a more general study of mass loss in a full population of triples, quantifying the number of systems affected by Kozai–Lidov oscillations and the EKM, an accounting of those that are unbound and those that are tidally affected or collide at each evolutionary stage, and those that undergo the MIEK mechanism.

Any study that evolves a full population of triple systems with distributions in a₁ and a₂ will have a significant fraction of systems that will be affected by the TEDI. Perets & Kratter (2012) demonstrate that systems that start with a small a₂/a₁ ratio become unstable during mass loss and evolve chaotically, leading to close encounters, collisions, and exchanges between the stellar components. Additionally, Perets & Kratter (2012) also raise the issue of Kozai cycles and stellar evolution as important. Future studies will be needed to quantitatively evaluate the importance of both the TEDI and MIEK mechanisms.