THE PAST AND FUTURE HISTORY OF REGULUS

Regulus (α Leo; HD 87901) was recently discovered to be a spectroscopic binary which has a likely white dwarf companion of mass M_wd ≃ 0.3 M_☉ and an orbital period of P_orb ≃ 40.11 d (Gies et al. 2008; hereafter "the inner binary system"). Regulus itself has an inferred mass of ∼3.4 ± 0.2 M_☉. This very bright star has been known for many years to have at least two other companions (BC) which together form a binary system (McAlister et al. 2005). This BC subsystem is located 177'' from Regulus, too great a distance to have ever directly interacted with the star. The B component (α Leo B; HD 87884) is an ∼0.8 M_☉ star of spectral type K2 V; the C component is a very faint M4 V star with a mass of ∼0.2 M_☉. The Washington Double Star Catalog (Mason et al. 2001) lists a D component of the system, also having a common proper motion with the system and a separation of 217'' from component A.

Located at a distance of 24.3 ± 0.2 pc, Regulus is a star of spectral type B7 V with an apparent magnitude of 1.36, making it one of the brightest stars in the sky. The star rotates very rapidly, with a rotational period of 15.9 hr, constituting ∼86% of its break-up speed (McAlister et al. 2005). Its rapid rotation causes it to be highly flattened, with an equatorial diameter 32% greater than its polar diameter (4.16 R_☉ and 3.14 R_☉, respectively; McAlister et al. 2005).

There is some debate regarding the age of Regulus. Gerbaldi et al. (2001) estimate it to be ∼150 Myr old; however, a comparison of its age with that of component B, assumed to be coeval, results in a 100 Myr discrepancy. It is worth noting that the age estimate for Regulus is based largely on the effective temperature of the star. However, the rapid rotation causes a difference of ∼5000 K between the poles and the equator (McAlister et al. 2005), making this assessment somewhat suspect. Moreover, this estimate assumes that Regulus, with M ≃ 3.4 M_☉, has evolved as a single star. As we will show in this paper, the original mass of Regulus was almost certainly much lower, and Regulus has attained its present mass by mass transfer from its close companion. If this scenario is correct, it is difficult to escape the conclusion that the entire Regulus system has a probable age of ≳1 Gyr.

In this paper, we use the fact that Regulus has a white dwarf companion of close to 0.3 M_☉ in a 40 day orbit to reconstruct an approximate past history for the system (Section 2). Based on the current state of the Regulus inner binary, we discuss various possible future channels for the system (Section 3) with one of the most interesting possibilities that it will become an AM CVn binary with a period as short as a few minutes.

It has been known for a long time that accretion from a companion star can result in very high rotation rates for the accretor. However, in the case of Regulus, no close companion star was known to exist until the discovery by Gies et al. (2008) that Regulus has a companion in a 40 day orbit that is likely a white dwarf. Gies et al. (2008) suggested that matter accreted by Regulus from the envelope of the progenitor of the putative white dwarf was responsible for the high rotation rate. Since only a mass function is measured for the Regulus inner binary, the mass of the unseen companion star is formally limited to M ≳ 0.3 M_☉. Gies et al. (2008) argue against a neutron star companion on the grounds that a ∼1.4 M_☉ companion would require an orbital inclination of i ≲ 15° which has only a small a priori probability of ∼3%. No formal limit on the orbital eccentricity is given by Gies et al. (2008), but an inspection of their Figure 1 suggests that e ≲ 0.05. This makes it even more unlikely that the unseen companion is a neutron star considering that the loss of even a small amount of matter during the supernova explosion that gave birth to the neutron star would undoubtedly have led to a sizable eccentricity. We present another argument below that the mass of the unseen companion is almost certainly near 0.3 M_☉.

Zoom In Zoom Out Reset image size

Assuming that the unseen companion to Regulus is a white dwarf, we can show why the 40 day orbit is just what is expected on theoretical grounds. For stars with an initial mass ≲2.5 M_☉ there is a nearly unique relation between its properties on the giant branch and its He core mass. Since, for Roche-lobe filling donor stars that are lower in mass than the accretor (or even slightly higher), the orbital period depends only on the mass and radius of the donor star, the orbital period at the end of mass transfer from a giant star depends only on the core mass of the donor star. This period−white dwarf mass relation has been studied in detail by Rappaport et al. (1995). They found the following approximate analytic expression:

$$\begin{equation} P_{\rm orb} \simeq 1.3 \times 10^5 M_{\rm wd}^{6.25} \big/ \big(1+4M_{\rm wd}^4\big)^{3/2}, \end{equation} \tag{ 1 }$$

where the orbital period is expressed in days and the white dwarf mass is in units of M_☉. Rappaport et al. (1995) estimated a theoretical uncertainty in the white dwarf mass of ∼±18% that takes into account uncertainties in the chemical composition, the initial mass of the parent star, and the mixing length parameter. If we solve this expression for M_wd with P_orb = 40.1 d, we find

$$\begin{equation} M_{\rm wd} \simeq 0.28 \pm 0.05 \;M_\odot \end{equation} \tag{ 2 }$$

(full uncertainty). This is entirely consistent with the measured value of M_wd ≳ 0.3 M_☉.

If we assume that the rotation axis of Regulus, which is measured to be ≳75° with respect to the line of sight (McAlister et al. 2005), coincides with the normal to the orbital plane, then the orbital inclination angle of the Regulus inner binary should be i ≳ 75°. That this is the case follows from the assumption that the matter transferred from the progenitor of the white dwarf companion was responsible for spinning up Regulus (see Section 2.4). Therefore, the angular momentum vector of the orbit and that of the rapidly rotating Regulus should coincide. For i ≳ 75°, the mass function for Regulus (Gies et al. 2008) yields a mass for the white dwarf of M_wd = 0.302 ± 0.017. Again, this is highly consistent with the theoretical value expected for a white dwarf remnant in a 40 day circular orbit (see Equation (2); Rappaport et al. 1995). Finally, we note that, according to the theoretical scenario for forming the Regulus inner binary, the orbital eccentricity is expected to be e ≲ 10⁻⁴ (see Rappaport et al. 1995 and references therein), which can possibly be falsified in the future.

In the evolutionary scenario for the current Regulus inner binary, the progenitor of the white dwarf is the more massive star, while the secondary—the progenitor of the current Regulus—is somewhat less massive. The initial orbital period must be less than 40 days in order for the He core of the primary not to exceed ∼0.3 M_☉ (see Equation (1)). When the primary overfills its Roche lobe, its envelope is transferred stably, though not necessarily conservatively, to the secondary. At first, the orbit shrinks due to the transfer of mass from the more to the less massive star. Once the mass ratio reaches unity, the orbit will start to expand. The ratio of the final-to-initial orbital period is given by

$$\begin{equation} \frac{P_{\rm orb,f}}{P_{\rm orb,i}} = \frac{M_{\rm b,f}}{M_{\rm b,i}} \left(\frac{M_{\rm 1,f}}{M_{\rm 1,i}}\right)^{C_1} \left(\frac{M_{\rm 2,f}}{M_{\rm 2,i}}\right)^{C_2}, \end{equation} \tag{ 3 }$$

where C₁ = 3α(1 − β) − 3 and C₂ = −3α(1 − β)/β − 3 (Podsiadlowski et al. 1992). The parameter β is the fraction of the transferred matter retained by the secondary, and α is the specific angular momentum in units of the specific orbital angular momentum of the system carried away by any matter lost from the system. The notation is that "1," "2," and "b" stand for the primary, secondary, and the binary, respectively, while "f" and "i" indicate final and initial states, respectively. The derivation of Equation (3) involves the assumption that the parameters α and β are constant throughout the mass-transfer phase; this approximation seems justified in light of the other theoretical uncertainties for this phase of the evolution.

In order to study the range of possible progenitor stars for the Regulus inner binary system, we considered primary stars of mass M₁₀ ≲ 3 M_☉, and secondaries of any mass less than the primary mass. For primaries with M₁₀ ≳ 2.5–3 M_☉ the core mass that develops would be M_core ≳ 0.5 M_☉ and would not leave a remnant consistent with the 0.3 M_☉ white dwarf that is inferred to be orbiting Regulus. For each point in the M₁₀–M₂₀ plane we estimated the ratio of thermal timescales, τ_KH, and nuclear timescales, τ_nuc, for the primary and secondary. Contours of constant timescale ratios are shown in the bottom two panels ((c) and (d)) of Figure 1. We take τ_KH ∝ M²/(RL) and τ_nuc ∝ M/L, with stellar luminosity and radius scaling like M^3.7 and M^0.8, respectively. We then find τ_KH ∝ M^−2.5 and τ_nuc ∝ M^−2.7.

In the top left panel of Figure 1 (panel (a)), we show contours of constant β, the fraction of transferred mass retained by the secondary, in the M₁₀–M₂₀ plane. These values are inferred by comparing the initial mass of the primordial binary with the current observed mass of M_b,f ≃ 3.4 + 0.3 M_☉ = 3.7 M_☉. Finally, the top right panel (b) in Figure 1 shows contours of constant initial orbital period that would lead to the observed current value of P_orb = 40 days. To compute P_orb we utilized Equation (3) with β taken from Figure 1(a), and α, the specific angular momentum associated with mass loss, taken to be a constant throughout the mass loss process and to have a typical fiducial value of unity (but see Section 2.5).

If we require that the ratio of thermal timescales for the progenitor stars, τ_KH,1/τ_KH,2 be not too different from unity, so that the secondary can retain a sizable fraction of the transferred matter, we can restrict the allowed range of progenitor masses to lie above a particular contour in Figure 1(c). We somewhat arbitrarily choose τ_KH,1/τ_KH,2 ≳ 0.4 in order that the thermal timescales not be too disparate. Similarly, we require that the ratio of nuclear timescales for the progenitor stars, τ_nuc,1/τ_nuc,2 be sufficiently different from unity, so that the current Regulus has not already evolved up the giant branch. Since, as we show in Section 2.5, the initial secondary star will spend ∼200 Myr with sufficient accreted mass to exceed the mass of the original primary, and subsequently, another ∼50 Myr with much more mass than the primary, as it approaches its current mass of 3.4 M_☉, the original mass of the secondary cannot be too close to that of the primary. In order for Regulus not to be more evolved than its current state, we require that τ_nuc,1/τ_nuc,2 ≲ 0.6, and we adopt this as a rough upper boundary in the bottom right panel of Figure 1.

Combining these constraints yields the allowed range of M₁₀ and M₂₀ (see Figure 2). The most likely primordial masses are M₁₀ ≃ 2.3 ±  0.2 M_☉ and M₂₀ ≃ 1.7 ±  0.2 M_☉. The initial orbital period could have been anywhere in the range of P_orb ≃ 1–15 days. Calculations of grids, and even entire populations, of such mass transfer binaries are given, e.g., in Nelson & Eggleton (2001) and Willems & Kolb (2004). These authors considered only conservative mass transfer, but their detailed evolutionary calculations are nonetheless quite instructive.

Zoom In Zoom Out Reset image size

It has long been argued that one way to produce a very rapidly rotating star (e.g., a Be star) is by the accretion of mass and angular momentum from a companion star (e.g., Pols et al. 1991), similar to the origin of the Be phenomenon in Be-/X-ray binaries (Rappaport & van den Heuvel 1982). Generally, a star has to accrete ≲10% of its initial mass to be spun up to near critical rotation⁵ (Packet 1981). But, if the companion is a hard-to-detect degenerate object, it is generally difficult to verify such a mass-transfer scenario (Pols et al. 1991). The discovery of the close degenerate companion to Regulus confirms the mass-transfer hypothesis in this case. Considering that this a nearby, bright star, it also demonstrates how difficult it is to test this scenario in individual cases. Indeed, the majority of intermediate-mass stars rotating near breakup may have such an unseen companion, suggesting that further observational scrutiny is warranted.

In order to check more quantitatively our proposed evolutionary scenario for the formation of the current Regulus system, we carried out a number of binary evolution calculations using a Henyey stellar evolution code. All calculations were carried out with an up-to-date, standard Henyey-type stellar evolution code (Kippenhahn et al. 1967), which uses OPAL opacities (Rogers & Iglesias 1992) complemented with those from Alexander & Ferguson (1994) at low temperatures. We use solar metallicity (Z = 0.02), take a mixing length of 2 pressure scale heights, and assume 0.25 pressure scale heights of convective overshooting from the core, following the calibration of this parameter by Schröder et al. (1997) and Pols et al. (1997; for more details, see Podsiadlowski et al. 2002).

We ran binary evolution sequences for a substantial number of initial mass points lying inside the white region of Figure 2. In most, but not all, cases we adopted a value for the angular momentum loss parameter of α = 1. The mass retention fraction, β, is fixed from the initial masses and the current binary mass. Since the values of β typically lie between ∼0.7 and 1.0, and therefore not much mass is lost, our results are not highly sensitive to the exact choice for α. We have also verified this by running a number of evolutionary models where α = 0.5 and α = 1.5. The main effect of adopting higher values of α is that the initial orbital period can be longer, and this tends to push the onset of mass transfer more into the regime of early case B mass transfer.

The model which most closely matches the current-day properties of the Regulus system has M₁₀ = 2.1 M_☉, M₂₀ = 1.74 M_☉, and P_orb = 40 hr (1.7 d). This particular choice of initial system parameters is shown as a filled square symbol within the white region of Figure 2. The first of two sets of results from the binary evolution calculation for this model is shown in Figure 3. The four panels show (a) the evolution of P_orb, (b) the progenitor masses of Regulus (long-dashed curve) and the white dwarf progenitor (solid curve), (c) the radius of the primary (WD progenitor), and (d) the mass transfer rate, $\dot{M}$, onto the progenitor of Regulus. Note that as P_orb approaches 40 days (∼1000 hr), the primary has transferred most of its envelope mass to the progenitor of Regulus, which by then has a mass of ∼3.4 M_☉. At the same time, the primary has developed a ∼0.3 M_☉ degenerate He core (the short-dashed curve in Figure 3, panel (b))—the progenitor of the current white dwarf in the system.⁶ Figure 4 shows the evolution of the radius of Regulus during the epoch when it accretes mass from the primary. Note that the radius of Regulus, even during the interval when it is accreting at ∼10⁻⁸ M_☉ yr⁻¹, never exceeds 4 R_☉ because the mass transfer timescale is substantially longer than the Kelvin–Helmholtz timescale.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

If the ∼0.3 M_☉ companion to Regulus, originally the core of the primordial primary in the system, has a mass below ∼0.31 M_☉, then the star will not burn He to CO and will cool to become a degenerate dwarf. After a time of several hundred Myr, the maximum time allowed since the current Regulus binary has been in existence, such a ∼0.3 M_☉ white dwarf would have cooled to T_eff ≲ 15, 000 K and a luminosity of ≲0.02 L_☉ (see, e.g., Althaus et al. 2001). This is consistent with the limits on the optical and UV emission from such a companion in the presence of Regulus, with a temperature nearly this high and a luminosity ∼10⁴ times higher.

If, on the other hand, the companion to Regulus has a mass just slightly in excess of that required for a He star to burn He to CO in thermal equilibrium, i.e., M ≃ 0.32 M_☉, then it would still be undergoing nuclear burning at the current epoch. The properties of such a low-mass He star are R ≃ 0.056 R_☉, L ≃ 1.4 L_☉, and T_eff ≃ 26, 000 K; and the nuclear lifetime is in excess of 1 Gyr (see, e.g., Hurley et al. 2000). Even though such a companion would have a substantially larger luminosity than its degenerate counterpart, it would still not contribute very much light to the Regulus system (i.e., ≲1%) for wavelengths longward of 1700 Å. It even seems unlikely that such a He star would have been detected in the study of B stars (including Regulus) by Morales et al. (2001) using the IUE satellite and EURD spectrograph on MINISAT-01.

One consequence of our scenario is that the age of the Regulus system, including any members bound to the inner binary, must exceed ∼900 Myr, the time required for the ∼2.1 M_☉ progenitor of the white dwarf to evolve. Thus, all previous age estimates of ∼50–200 Myr (see, e.g., Gerbaldi et al. 2001) are far too young. The reason for these age discrepancies, in essence, is that Regulus was rejuvenated by the accretion of nearly 1.7 M_☉ of hydrogen-rich material.

In about 100–200 Myr, Regulus will complete its main sequence evolution and begin its ascent of the giant branch (see Figure 4 for times after 10^9.1 yr). The exact time to this phase will depend, of course, upon its current core development. The present orbital separation of the Regulus binary (at P_orb ≃ 40 d) is a ≃ 74 R_☉, and the corresponding Roche-lobe radius of Regulus is R_L ≃ 44 ±  4.7 R_☉. By the time Regulus fills its Roche lobe, starts mass transfer to its white dwarf companion, and the common-envelope phase commences, it will have developed a ∼0.48 M_☉ He core.⁷ This is the core that is unveiled after the common envelope phase (see the following section).

Note, however, that since a 3.4 M_☉ star will ignite helium in its core, its post-CE remnant will continue to burn helium. In the case of a successful common-envelope ejection (see Section 3.2) it will appear as an sdB star in a very short-period binary for ∼10⁸ yr (e.g., Maxted et al. 2001; Han et al. 2002), and ultimately produce a hybrid HeCO white dwarf (see Section 3.3.2).

Once mass transfer from the evolved Regulus to its white dwarf companion commences, it will quickly become dynamically unstable (due to the extreme mass ratio of the two stars), and a common envelope phase will ensue. The relation between the initial orbital separation of a_i ≃ 76 R_☉, and the post-common envelope separation, a_f is given by

$$\begin{equation} \left(\frac{a_{\rm f}}{a_{\rm i}}\right) \simeq \frac{M_{\rm core,Reg} M_{\rm wd}}{M_{\rm Reg}} \left(M_{\rm wd}+\frac{2M_{\rm env,Reg}}{\lambda \eta r_{\rm L1}}\right)^{-1}, \end{equation} \tag{ 4 }$$

where λ is the inverse binding energy of the core of Regulus with its envelope, in units of

$$\begin{equation*} R_{\rm Reg}/(GM_{\rm core,Reg}M_{\rm env,Reg}), \end{equation*}$$

η is the fraction of the orbital binding energy that goes into ejecting the common envelope, and r_L1 is the size of the Roche lobe of Regulus in units of the initial orbital separation (see, e.g., Webbink 1984; Pfahl et al. 2003). For the parameters of the Regulus–white dwarf binary, this expression yields a_f/a_i ≃ 0.0046λη. The corresponding post-common envelope orbital period of the binary, now consisting of a ∼0.5 M_☉ HeCO dwarf and a 0.3 M_☉ white dwarf, is P_orb ≃ 40(λη)^3/2 minutes. For plausible values of λη ≃ 0.1–1 (see, e.g., Dewi & Tauris 2000, 2001; Podsiadlowski et al. 2003), the post-CE orbital periods would likely range between ∼2 and 40 minutes, respectively, if both of the compact stars were degenerate. Given that the HeCO star is expected to still be burning He at the end of the common-envelope phase, and that its radius would be no smaller than ∼0.1 R_☉, the actual range of post-CE orbital periods is probably limited to between ∼40 and ∼20 minutes if a merger of the cores is to be avoided (see Section 3.3.1 for more details). If the two dwarfs do effectively merge, they would produce a rapidly rotating giant with unusual properties (see Section 3.3.1).

3.3.1. Possible Outcomes of the CE Spiral-in

After the common envelope phase, the final orbital separation should be a ≃ 1/3 ηλ R_☉, where, as discussed above, the product ηλ reflects the efficacy of the common-envelope process, and is expected to be in the range of 0.1–1. What happens to the binary pair consisting of the 0.3 M_☉ He white dwarf companion of Regulus and the ∼0.48 M_☉ core of Regulus depends critically on the uncertain value of ηλ. If the 0.3 M_☉ dwarf is degenerate, it will have a radius of 0.017 R_☉, and would fit well inside its Roche lobe, nearly independent of ηλ. (If it is still undergoing He burning, its radius would be ∼3 times larger.) The 0.48 M_☉ core of Regulus, on the other hand, is still burning He to CO in its core, and will have a radius of ∼0.1 R_☉ until the He burning phase has ended, and the HeCO remnant becomes degenerate (see Figures 4 and 6). This phase lasts for ∼100 Myr.

This implies that there is a minimum period of ∼20 minutes for the post-CE system. If the ejection efficiency (ηλ) is too low, the orbital energy released in the spiral-in will not be able to eject the common envelope, and the two compact objects, the 0.48 M_☉ core and the 0.3 M_☉ white dwarf, will merge inside the common envelope. This would likely produce a rapidly rotating single giant with possibly some very unusual chemical properties (perhaps similar to the unusual, rapidly rotating carbon star V Hydrae (Kahane et al. 1996); also see FK Comae stars (Bopp & Stencel 1981) for related earlier-type counterparts).

If the common envelope is ejected, the subsequent evolution depends on the post-CE orbital period, since it determines the timescale on which either of the two compact components will achieve contact and start to transfer mass again. Immediately after a successful common-envelope ejection, both components (by definition) will underfill their Roche lobes, but the binary orbit will continue to decay because of gravitational wave emission. The time for orbital decay is

$$\begin{equation} t = 68 \big[P_{\rm orb,CE}^{8/3} - P_{\rm orb}^{8/3}\big]{\rm Myr}, \end{equation} \tag{ 5 }$$

where P_orb,CE is the orbital period (in hours) immediately after the common envelope, and P_orb is the orbital period at time t later. The system remains a detached binary until either of the two components starts to fill its Roche lobe.

If the immediate post-CE orbital period is ≲80 minutes, the 0.48 M_☉ helium-burning component will start to fill its Roche lobe while it is still burning helium in the center, and mass transfer will start from the more massive helium-burning star to the 0.3 M_☉ white dwarf. The system will now become an AM CVn star with a helium-star donor (e.g., Nelemans et al. 2001). The stability of the mass transfer is addressed in Section 3.3.3.

On the other hand, if the post-CE orbital period is longer than ∼80 minutes,⁸ the helium-burning star will have completed helium-core-burning and become a degenerate, much smaller HeCO white dwarf (see Figures 4 and 6) before the system becomes semi-detached again. In this case, the lighter and physically larger white dwarf will fill its Roche lobe first, and mass transfer will take place from the lighter white dwarf to the more massive HeCO white dwarf. The system will again appear as an AM CVn binary (see, e.g., Nelemans et al. 2001), but in this case with either a degenerate donor star with P_orb ≃ 2 minutes and M ≃ 0.3 M_☉, or with a He-burning star with M ≃ 0.32 M_☉ and P_orb ≃ 13 minutes.

Figure 5 illustrates these different evolutionary paths in a "scenario" diagram.

Zoom In Zoom Out Reset image size

3.3.2. Evolution of the Exposed Core of Regulus

Once the core of Regulus has been exposed after the common-envelope phase, it will evolve as an isolated He star of mass 0.475 M_☉ until it or its companion white dwarf fill its respective Roche lobe. As mentioned above, which star first fills its Roche lobe depends on the orbital period immediately following the common-envelope phase. To help understand this, we show in Figure 6 the evolution of an isolated He star of mass 0.475 M_☉. The first three panels show the evolution with time of the radius, T_eff, and luminosity. After ∼140 Myr, the star has completed its He burning, and contracts from ∼0.1 R_☉ to its final degenerate radius of ∼0.015 R_☉. The mass of the CO core and the central He abundance of the star are plotted as functions of time in the bottom right panel. The CO core reaches a final mass of ∼0.34 M_☉.

Zoom In Zoom Out Reset image size

Thus, the critical timescale for the exposed core of Regulus to complete its nuclear burning and contract to a degenerate state is ∼140 Myr. From Equation (5) above we see that the original (i.e., post-CE) orbital period would have to be in excess of ∼80 minutes in order for the HeCO star to avoid filling its Roche lobe at P_orb ≃ 20 minutes, and for the system to shrink via gravitational wave loses to P_orb ≃ 2 minutes when the He white dwarf would first overflow its Roche lobe (see Sections 3.3.1 and 3.3.3). If the lower-mass He star is still undergoing He burning (e.g., for a mass of ∼0.32 M_☉) then P_orb would be ∼13 minutes rather than 2 minutes.

3.3.3. Illustrative Calculations of an AM CVn Phase

In Figure 7 we show the evolution that results when the post-CE orbital period is 90 minutes. This allows sufficient time for the HeCO dwarf to complete its He burning before gravitational radiation losses bring the system into Roche-lobe contact. The first star to fill its Roche lobe and thereby become the donor star is the 0.3 M_☉ He white dwarf. The initial orbital period at the start of mass transfer is ∼2 minutes (13 minutes if the low-mass He star is a bit more massive and is still undergoing He burning). The mass transfer rate from the lower to the higher mass white dwarf is stable. It starts at a rate of $\dot{M} \simeq 3 \times 10^{-6}\;M_\odot \;{\rm yr}^{-1}$ and steadily declines over the ensuing billion years to ≲10⁻¹² M_☉ yr⁻¹. The details of the evolution of the constituent masses, P_orb, and $\dot{M}$ can be seen in Figure 7. The later parts of the binary evolution can, in fact, be computed semianalytically (see, e.g., Rappaport et al. 1983; Eggleton 2007). We find P_orb ≃ 30 t^3/11₈ minutes, and $\dot{M} \simeq 6 \times 10^{-11} t_8^{-14/11}\;M_\odot$ yr⁻¹, where t₈ is the time in units of 10⁸ years from the start of mass transfer. Such a system would be observed as an AM CVn star (see, e.g., Nelemans 2001).

Zoom In Zoom Out Reset image size

In Figure 8 we show the evolution that results when the post-CE orbital period is 40 minutes. In this case, the orbital decay time is sufficiently short that the post-CE ∼0.48 M_☉-helium star does not have sufficient time to complete He burning and cool to a degenerate state where its radius is smaller than that of the 0.3 M_☉ He white dwarf. In this case, mass transfer starts when the helium star is still burning helium in its core (i.e., in its hot subdwarf phase), and mass is transferred from the hot subdwarf to the 0.3 M_☉ He white dwarf. Initially, mass transfer proceeds on the thermal timescale of the helium star due to the fact that the donor is the more massive star. However, after an initial phase of very high $\dot{M}$, the mass ratio quickly inverts, and the subsequent mass transfer is driven entirely by the angular momentum loss due to gravitational radiation. Note that, nuclear burning turns off soon after the helium star has lost a significant amount of mass (by this stage, the central helium abundance has been reduced from 0.98 to 0.74, causing the shrinking of the star and the associated dip in the mass-transfer rate). The subsequent evolution of the binary is similar to that in Figure 7, except that the now degenerate donor star is helium-depleted and oxygen- and carbon-enriched.

Zoom In Zoom Out Reset image size

3.3.4. The Final Fate of the System