Unicorns and Giraffes in the binary zoo: stripped giants with subgiant companions

The Giraffe (Jayasinghe et al., 2022) is a binary in Camelopardalis containing a luminous red giant ($L_{\rm giant}\approx 200\,L_{\odot}$) and a companion in an 81-day circular orbit. The spectroscopic mass function, $f(M_{2})\equiv\bigl{(}M_{2}\sin{i}\bigr{)}^{3}/(M_{\rm giant}+M_{2})^{2}=0.58\,M_{\odot}$, implies a relatively massive companion. By modeling the system’s SED, light curves, and RVs, Jayasinghe et al. (2022) inferred a giant mass of $\approx 0.6\,M_{\odot}$, inclination of $\approx 41$ degrees, and unseen companion mass of $\approx 3\,M_{\odot}$.¹¹1The Jayasinghe et al. (2022) model for the Giraffe to which we refer in this paper corresponds to the preprint dated January 28, 2022. As this is a preprint, the final version of that paper may ultimately present different models. A main-sequence star of this mass would overwhelm the observed SED at blue wavelengths, so they concluded that the companion is a BH.

When they subtracted the best-fit spectral model for the giant from the observed optical spectra, Jayasinghe et al. (2022) found residuals resembling the narrow-lined absorption spectrum of a star. The residual spectrum moves in anti-phase with the giant, but with an RV amplitude 5 times smaller. Assuming these RVs trace the reflex motion of the companion’s center of mass, they imply a mass ratio $M_{\rm giant}/M_{2}\approx 0.2$. Ascribing this secondary spectrum to a stellar companion was noted as the most natural explanation by Jayasinghe et al. 2022). But they found that a normal star of the required mass would produce a UV flux inconsistent with the observed limits. This led to their proposal that the secondary spectrum traces an accretion disk around a BH companion.

No X-rays were detected, and the observed upper limit is $L_{X}\lesssim 10^{32}\,\rm erg\,s^{-1}$. Relatively strong, double-peaked H$\alpha$ emission was observed, suggesting that some sort of mass transfer is likely ongoing. Jayasinghe et al. (2022) found that the giant does not fill its Roche lobe but nearly does, with $R/R_{\rm Roche\,lobe}\approx 0.93$. This, and the inferred inclination, is sensitive to the amount of dilution from the secondary. The distance to the system is of order $3-4$ kpc; as we discuss in Section 3.3, the exact value is sensitive to the Gaia parallax zeropoint and assumptions about the nature of the system.

V723 Mon (“the Unicorn”) is a bright, nearby giant in Monoceros. It was identified as a 60-day spectroscopic binary by Griffin (2010), with the orbit refined by Strassmeier et al. (2012) and Griffin (2014). All works found an orbital period $P_{\rm orb}\approx 60$ days, RV semi-amplitude for the giant $K_{\rm giant}\approx 65\,\rm km\,s^{-1}$, and an eccentricity near 0, yielding a mass function $f(M_{2})\approx 1.71\,M_{\odot}$. Strassmeier et al. (2012) found small ($\approx 1-2\,\rm km\,s^{-1}$) but significant residuals when fitting the giant’s RVs with a Keplerian orbit. They also found evidence of a luminous secondary in their spectra, which moved approximately in anti-phase with the giant. The RVs of this secondary component were poorly fit with a Keplerian orbit. Strassmeier et al. (2012) concluded that the system is a hierarchical triple, with a period of $\approx 20$ days for the inner binary and a total inner binary mass of 3-4 times the giant mass.

V723 Mon was re-analyzed by Jayasinghe et al. (2021), who argued that the system contains a BH. They found that the triple model proposed by Strassmeier et al. (2012) was not viable because (a) the system would not be dynamically stable and (b) variations in the tidal force on the giant due to the orbit of the inner binary would make the outer orbit eccentric and lead to observable variations in its tidal deformation. Jayasinghe et al. (2021) also noted that tidal deformation of the giant is expected to cause changes in the giant’s line profiles with orbital phase. They proposed that this tidal deformation is responsible for both the apparent deviations of the giant’s RVs from a Keplerian orbit (as explored quantitatively by Masuda & Hirano 2021) and for the apparent evidence of a luminous secondary in the spectra.

Jayasinghe et al. (2021) found that the giant’s absorption lines are shallower than expected at blue wavelengths, which they interpreted as “veiling” by another light source. The ellipsoidal variability amplitude also becomes weaker toward bluer wavelengths, suggesting dilution by a non-variable object that is warmer than the giant. They did not find the spectrum of this diluting component to have absorption lines or other star-like features and therefore attributed it to continuum from a diffuse circumbinary component.

The depth and width of the Unicorn’s Balmer lines varies with orbital phase, with the lines appearing deepest at phase $\phi\approx 0.5$. Jayasinghe et al. (2021) attributed this to emission from an accretion disk surrounding the putative BH, which would be eclipsed at phase $\phi=0.5$. Weak X-rays were detected in Swift XRT data, with a luminosity $L_{X}\sim 10^{30}\,\rm erg\,s^{-1}$, but Hare et al. (2021) showed that this detection is likely spurious. The X-ray luminosity is less than $10^{-5}\times$ the inferred optical luminosity of the veiling component.

After correcting for the Gaia parallax zeropoint (Lindegren et al., 2021) and underestimated parallax uncertainties, the inferred parallax of V723 Mon is $2.20\pm 0.04$ mas, corresponding to a distance of $454\pm 8$ pc. Because the system is nearby, its inferred properties are only weakly sensitive to assumptions about the parallax zeropoint.

The Giraffe was observed by the APOGEE survey (Majewski et al., 2017), which obtained spectra at 4 epochs with resolution $R\approx 22,500$ in the $H-$band (15,000-17,000 Å). We fit these spectra with a binary spectral model, as described in detail in Appendix A.

We find that two luminous components contribute to the APOGEE spectra: a giant with effective temperature of $4050\pm 100$ K, and a subgiant with effective temperature $5200\pm 200$. The giant dominates the $H-$band light, contributing 80% of the flux, but the 20% contribution from the secondary component, appearing to be a subgiant, is unambiguous. This is somewhat unexpected, because Jayasinghe et al. (2022) found that the light contributions of the secondary detected in the optical became negligible at $\lambda\gtrsim 7,000$ Å. We find the subgiant to move in anti-phase with the giant, with an implied dynamical mass ratio $M_{\rm giant}/M_{2}=0.21\pm 0.02$. This is in good agreement with our findings from the optical spectra and implies that we detect the same secondary in the near-infrared and optical, and derive consistent characteristics.

We analyzed 8 epochs of Keck/HIRES spectra of the Giraffe, which are sampled roughly uniformly in orbital phase and are described in detail in Jayasinghe et al. (2022). These data provide useful coverage of most of the wavelength range between 3900 and 8000 Å, with resolution $R\approx 60,000$ and typical per-epoch SNR of 20 per pixel at 5000 Å.

We first fit the optical spectra using a binary spectral model, following the approach outlined in Appendix A and El-Badry et al. (2018b). We fit all usable orders simultaneously, self-consistently accounting for the wavelength-dependent flux ratio. We fit separate visits independently, estimating uncertainties based on the epoch-to-epoch scatter. The spectral model was trained on synthetic spectra from the BOSZ grid (Bohlin et al., 2017) calculated with ATLAS12 and SYNTHE (Kurucz, 1970, 1979, 1993). In good agreement with the results from the APOGEE spectra, we found a giant with effective temperature of $4000\pm 100$ K, and a subgiant with effective temperature $5150\pm 200$ K. The best-fit metallicity (assumed the same for both components) is $[\rm Fe/H]\approx-0.5$. The subgiant’s inferred flux contributions increase toward the blue orders: we find that it contributes $\approx 35\%$ of the light at 6,000 Å, but $\approx 70\%$ at 4,000 Å.

We analyzed the Giraffe’s $g-$ and $r-$ band light curves from the 8th data release of the Zwicky Transient Facility (ZTF; Bellm et al., 2019); these are shown in Figure 4. We focus on ZTF data because it has the smallest empirical photometric uncertainty of the available ground-based light curves. Photometry from TESS is also available, but it does not span a full orbital period and thus is not well suited for determining the full ellipsoidal variability amplitude and the amount of gravity darkening.

We phased the ZTF data to the RV ephemeris calculated by Jayasinghe et al. (2022) and compared them to a range of models calculated with PHOEBE (Prša & Zwitter, 2005; Prša et al., 2016). We focus on two main features of the light curves: (a) the total variability amplitude, which provides a joint constraint on inclination, the amount of dilution from the secondary, and the giant’s Roche lobe filling factor, and (b) the strength of gravity darkening, as manifest in the different depths of the light curve minima at phase 0 and 0.5.

The min-to-max variability amplitude of the phased light curves is $14\pm 1$% in both $g-$ and $r-$ bands. One might be tempted to infer that this means there is no more dilution from the secondary in $g$ than in $r$. However, we find that with no dilution, the predicted variability amplitude from PHOEBE is $\approx 20\%$ larger in $g$ than $r$ due to wavelength-dependent limb and gravity darkening. This implies that there is 15-20% more dilution from the secondary in $g$ than in $r$.

We fix the mass ratio to $q=0.2$ based on the RVs and account for dilution from the secondary by modeling it as a star in PHOEBE, with temperature and radius consistent with the SED and spectral fit. The observed ellipsoidal variability amplitude then limits possible solutions to a one-dimensional slice in the inclination vs. Roche lobe filling factor plane. ⁴⁴4The light curve and mass function alone do not strongly constrain the inclination or mass ratio, because solutions with high inclination and lower filling factor produce nearly identical light curves to solutions with lower inclination and higher filling factor. One can always get a formally tight constraint with enough data, but systematics (e.g. unaccounted-for spots, and especially contamination from the secondary) will drastically limit the utility of these constraints. If we assume that the giant fills its Roche lobe – which seems reasonable given that it clearly is close (given the ellipsoidal variability) and there is an accretion disk around the companion seen in H$\alpha$ – then the observed variability amplitude and RV-inferred mass ratio imply an inclination of $i=50\pm 3$ degrees (Figure 5; this is basically independent of the giant’s mass). The uncertainty on this inclination is mainly due to uncertainty in how much dilution there is from the companion, and uncertainty in the observed variability amplitude itself. For an inclination of 50 degrees, the mass function $f(M_{2})=0.58\,M_{\odot}$ implies a companion mass of $M_{2}=1.83\,M_{\odot}$ if $M_{1}=0.35\,M_{\odot}$ (this value is justified in Section 3.5). This yields a mass ratio $M_{1}/M_{2}=0.197$, in agreement with the RV-inferred mass ratio.

The model favored by Jayasinghe et al. (2022) has a lower inclination of 41 degrees, and thus would predict a lower ellipsoidal variability amplitude without the contributions of the companion. However, their modeling included significantly less dilution from a luminous secondary (e.g., they inferred a dilution factor of only $7\%$ in the $r-$ band, whereas in our model, it is $\approx 35\%$).

We proceed under the assumption that the giant fills its Roche lobe. A scenario in which it does not fill its Roche lobe seems a priori less likely for two reasons. First, it requires more fine tuning, because we would have to observe the giant just after detachment, rather than during the much longer period where there is ongoing mass transfer (Section 5). This is true irrespective of the nature of the companion. Second, the observed double-peaked H$\alpha$ emission suggests that there is a sizable accretion disk, which is not expected in a detached system with only wind mass transfer.

Figure 4 compares the ZTF $r-$ and $g-$ band light curves of the Giraffe to PHOEBE model predictions for a luminous subgiant companion (left) and a BH companion (right). The models make similar predictions, because the amplitude of the main signal – near-sinusoidal variation on half the orbital period due to tidal deformation of the giant – is covariant between inclination, Roche lobe filling factor, and dilution. There is, however, a key difference between the two models: the predicted asymmetry due to gravity darkening is stronger – in agreement with the data – in the case with a luminous companion. This is a result of the higher inclination and Roche lobe filling factor in the subgiant model, and another piece of evidence for the luminous companion.

Although both light curve models explain most of the observed variability, there are some systematic difference between the shape of the observed light curve and both models. Most obviously, the observed light curve maximum at phase $\approx 0.25$ is higher than the one at phase $\approx 0.75$. The scatter in the phased data is also somewhat larger than expected given the photometric uncertainties, suggesting some long-term variability. Both of these features could be due to spots on the giant, or changes in the disk around the secondary.

Limits on the masses of both components are summarized in Figure 5. Given the inferred giant radius of $R_{\rm giant}=(25\pm 1.5)R_{\odot}$, the constraint that it fills its Roche lobe translates to $M_{\rm giant}=(0.33\pm 0.06)\,M_{\odot}$ (e.g. El-Badry & Burdge, 2022). The mass ratio from the companion’s reflex motion then implies $M_{2}$ between $1.38\,M_{\odot}$ and $2\,M_{\odot}$. The inclination required to match the observed mass function over this range ranges for 48 to 58 deg, with higher masses implying lower inclinations. The inclination constraint from the observed ellipsoidal variability amplitude is slightly more stringent (lower panel of Figure 5), excluding inclinations above $\approx 56$ deg. We thus conclude (before considering possible evolutionary scenarios) that giant masses ranging from 0.29 to 0.39 are most plausible, corresponding to companion masses ranging from 1.5 to 2 $M_{\odot}$.

We analyze the 7 Keck/HIRES spectra of the Unicorn that were also analyzed by Jayasinghe et al. (2021). These provide useful coverage of most of the range between 3500 and 8000 Å, with typical per-epoch SNR of 100 per pixel at 5000 Åand resolution $R=60,000$. The phase coverage samples most of the dynamic range in the giant’s RVs, but does not include observations near $\phi=0.5$ (when the companion would be eclipsed for sufficiently high inclinations).

The qualitative appearance of the spectra changes dramatically from the blue to red orders. Blueward of 3800 Å, the spectra have broad absorption lines, indicative of rapid rotation (e.g. Figure 10). They are well-fit by a single-star model with $T_{\rm eff}\approx 5800\,\rm K$, $\log g\approx 3$, and $v\sin i\approx 70\,\rm km\,s^{-1}$. The source here appears to be almost stationary, with epoch-to-epoch RV variations of $\lesssim 20\,\rm km\,s^{-1}$. On the other hand, the redder regions of the spectra are dominated by narrow lines from a cooler and much more slowly rotating star ($v\sin i\approx 15\,\rm km\,s^{-1}$). This component is much more RV variable, with epoch-to-epoch RV variations exceeding 120 $\rm km\,s^{-1}$. Jayasinghe et al. (2021) focused their analysis on red wavelengths, so the properties of the giant they report correspond to the object that dominates there.

We infer the radii of both components from the SED, following the same approach used for the Giraffe. We do not include the SkyMapper photometry, which is likely saturated. The results are shown in Figure 11. The inferred giant radius is $R\approx 22.5\pm 1\,R_{\odot}$, and the effective temperatures of both components are consistent with our spectroscopic fits. The subgiant is predicted to contribute $\approx 20\%$ of the light at $\lambda>16,000$ Å, also similar to the Giraffe.

The large-amplitude ellipsoidal variability observed in the Unicorn implies that the giant must nearly fill its Roche lobe, once dilution from the companion is accounted for. For example, the observed min-to-max variability amplitude in the $V-$band is $\approx 17\%$, but a (presumably) barely-variable companion contributes $\approx 60\%$ of the $V-$band light. This implies that the min-to-max variability amplitude of the giant is $\approx 37\%$, which implies a Roche lobe filling factor $R_{\rm giant}/R_{\rm Roche\,lobe}\gtrsim 0.97$. As in the Giraffe, we interpret this and the time-variability of the Balmer lines as evidence for ongoing mass transfer, and proceed under the assumption that the giant fills its Roche lobe.

In this case, the orbital period and observed giant radius constrain the giant mass to $M_{\rm giant}=0.44\pm 0.06\,M_{\odot}$. The inclination is constrained by the amplitude of ellipsoidal variability and the lack of eclipses. For plausible giant and subgiant parameters, we find that inclinations $i\lesssim 62\,\rm$ deg are ruled out because they would underpredict the observed ellipsoidal variability amplitude even with a Roche lobe-filling giant. Inclinations $i>72\,\rm deg$ would produce detectable eclipses and are similarly ruled out.

Jayasinghe et al. (2021) interpreted the increased Balmer absorption at $\phi\approx 0.5$ (Section 4.1.2) as evidence for eclipses of the companion and a near edge-on inclination. Such a scenario is difficult to reconcile with the secondary detected in the spectra. Irrespective of the nature of the secondary, the spectra imply that it is an optically thick source with $T_{\rm eff}\approx 5800\,\rm K$ and surface area similar to an $R\approx 8\,R_{\odot}$ sphere. Such a companion would cause eclipses in the light curve for edge-on inclinations, even if it were e.g. a disk around a BH. Given our constraints on the secondary $T_{\rm eff}$ and optical luminosity, it also cannot be a larger circumbinary structure: this would have much larger luminosity and would outshine the giant at all wavelengths.

We adopt a fiducial inclination of $i=70$ deg, which matches the observed variability amplitude for our fiducial parameters, but we consider values from 62 to 72 deg plausible. For $i=70$ deg, our PHOEBE models predict ellipsoidal variability amplitudes of 9.5% in the $B$ band and $17.2\%$ in the TESS T band; both of these are in good agreement with the observed values.

Given the observed mass function, $f(M_{2})=1.71\,M_{\odot}$, these masses and inclinations correspond to companion masses $2.60\lesssim M_{2}/M_{\odot}\lesssim 3.3$, with $M_{2}\approx 2.8\,M_{\odot}$ for our fiducial model. This is not far from the range inferred by Jayasinghe et al. (2021), because our lower assumed $M_{\rm giant}$ and lower assumed inclination move the implied companion mass in opposite directions.

We calculated a small grid of binary evolution models using MESA (Modules for Experiments in Stellar Astrophysics, version r12778; Paxton et al., 2011; Paxton et al., 2013, 2015, 2018, 2019). MESA simultaneously solves the 1D stellar structure equations for both stars, while accounting for mass and angular momentum transfer using simplified prescriptions.

We assumed an initial composition of $X=0.712,Y=0.28,Z=0.006$ for both stars. Opacities are taken from OPAL (Iglesias & Rogers, 1996) at $\log(T/{\rm K})\geq 3.8$ and from Ferguson et al. (2005) at $\log(T/{\rm K})<3.8$. Nuclear reaction rates are from Angulo et al. (1999) and Caughlan & Fowler (1988). We use the photosphere atmosphere table, which uses atmosphere models from Hauschildt et al. (1999) and Castelli & Kurucz (2003). The MESAbinary module is described in Paxton et al. (2015). We used the evolve_both_stars inlists in the MESA test suite as a starting point for our calculations, and most inlist parameters are set to their default values. Roche lobe radii are computed using the fit of Eggleton (1983). Mass transfer rates in Roche lobe overflowing systems are determined following the prescription of Kolb & Ritter (1990). The orbital separation evolves such that the binary’s total angular momentum is conserved, as described in Paxton et al. (2015).

We first highlight models that approximately reproduce the observed parameters of the Giraffe. A wide range of initial binary parameters can produce a $0.3-0.4\,M_{\odot}$ Roche-lobe filling giant with an orbital period of $\sim$80 days (e.g. Kalomeni et al., 2016). Here we explore only a subset of the possible parameter space. We initialize the calculation with a detached binary containing two zero-age-main sequence stars on a circular orbit. We experimented with initial periods ranging from 7 to 20 days, initial primary masses ranging from 1 to 2 $M_{\odot}$, and initial mass ratios ranging from 0.8 to 1. We model mass transfer as being fully conservative for simplicity. This is a reasonable assumption during most of the calculation, but it is unlikely to hold during the initial period of thermal-timescale mass transfer, when the mass transfer rate briefly exceeds the thermal limit by a factor of $\sim$10.

We present one model in detail in Figure 13 and 14, and comment on the effects of varying initial parameters in Section 5.2. The model we highlight has initial primary mass $1.15\,M_{\odot}$, initial mass ratio 0.997, and initial period of 13 days. (We comment on the assumption of initial mass ratio $q\approx 1$ in Section 6.2.) The evolution of the primary, which is today the giant, is shown in Figure 13. Mass transfer begins as the primary is on its way up the giant branch, when its radius is $11\,R_{\odot}$. Because of the nearly-equal mass ratio, the secondary has also already left the main-sequence at that time, with a radius of $6.5\,R_{\odot}$.

Mass transfer is at first rapid, proceeding on the donor’s thermal timescale and reaching $\dot{M}\approx 3\times 10^{-4}\,M_{\odot}\,\rm yr^{-1}$, but slows once the giant has become the less-massive component, such that the mass transfer leads to orbit widening. Subsequent evolution is governed by the donor’s nuclear evolution, and the system gradually expands as a semi-detached binary for $\approx$50 Myr. It reaches an orbital period of 81 days $\approx 45$ Myr after the onset of mass transfer, when the giant’s mass and radius are $M_{\rm giant}=0.37$ and $R_{\rm giant}=24.3\,R_{\odot}$. At this time the secondary mass and radius are $M_{2}=1.93\,M_{\odot}$ and $R_{2}=10\,R_{\odot}$, yielding a mass ratio $M_{\rm giant}/M_{2}=0.19$. The predicted surface carbon-to-nitrogen ratio is $\rm[C/N]=-1.0$, in good agreement with the observed value. This is a consequence of material that was inside the giant’s convective core during its main-sequence evolution becoming exposed at the surface once the envelope is stripped: without mass transfer, the surface [C/N] ratio for a $\approx 1\,M_{\odot}$ giant would be near 0 (e.g. Hasselquist et al., 2019).

A qualitatively similar scenario can produce the Unicorn, but the total initial mass must be larger in order to produce the observed final mass. For near-conservative mass transfer, the current system masses imply an initial primary mass between about 1.6 and 1.9 $M_{\odot}$; values up to $\approx 3\,M_{\odot}$ are possible if mass transfer is highly nonconservative. We refer to Rappaport et al. (2009) for a detailed exploration of the initial masses and periods required to produce such a system, which are quite similar to those that can produce the Regulus system.

Here we consider what range of initial parameters could produce systems like the Giraffe and Unicorn.

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  Initial primary mass: A relatively wide range of initial primary masses can produce a stripped, Roche-lobe filling $0.3-0.4\,M_{\odot}$ giant in a 60- or 81-day orbit. This is because the radii of giants with degenerate cores is set mainly by core mass (e.g. Rappaport et al., 1995). Primaries with larger initial masses begin mass transfer earlier in their evolution up the giant branch, but their masses and radii subsequently follow similar tracks to lower-mass stars at a given orbital period.

  The total mass of the Giraffe today is constrained to $2\lesssim M_{\rm tot}/M_{\rm odot}\lesssim 2.5$, and that of the Unicorn is constrained to $3.1\lesssim M_{\rm tot}/M_{\rm odot}\lesssim 3.8$. This limits the initial primary mass to $1\lesssim M_{1}/M_{\rm odot}\lesssim 1.3$ (Giraffe) and $1.6\lesssim M_{1}/M_{\rm odot}\lesssim 1.9$ (Unicorn), unless mass transfer is significantly non-conservative. Highly nonconservative mass transfer is not implausible: many of the models we explore go through an initial period of very rapid thermal-timescale mass transfer, with $\dot{M}$ up to $10^{-3}\,M_{\odot}\,\rm yr^{-1}$ when a subgiant donor first overflows its Roche lobe.

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  Initial mass ratio: If the mass ratio at the onset of Roche lobe overflow is too unequal, the giant will be vulnerable to runaway dynamical mass transfer and a common envelope event. The critical mass ratio beyond which this occurs depends on the structure of the star and remains imperfectly understood (e.g. Hjellming & Webbink, 1987; Soberman et al., 1997; Pavlovskii & Ivanova, 2015). We found mass transfer in our calculations to be stable for initial mass ratios $M_{\rm giant}/M_{2}\lesssim 1.15$ at the onset of RLOF, with some dependence on initial period. However, only initial mass ratios very near unity allow the companion to evolve off the main sequence prior to the onset of mass transfer (Section 6.2).

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  Initial orbital period: Longer initial periods cause mass transfer to begin when the donor is more evolved and has had time to form a larger helium core. This leads to a somewhat larger giant mass and radius when the giant reaches a period of 60 or 81 days. If the efficiency of mass transfer is modeled as constant, there are simple analytic relations between the current and initial periods and mass ratios (e.g. Soberman et al., 1997, their Equation 29). These allow us to model the systems’ period evolution backward in time given constraints on their current orbital periods and mass ratios. The results are shown in Figure 15 for both systems. For the Giraffe, we assume $M_{\rm giant}/M_{2}=0.198$, as is observationally well-constrained. For the Unicorn, we assume $M_{\rm giant}/M_{2}=0.16$; this is consistent with our constraints but less certain (Section 4.1.3). We expect that the initial mass ratios of both systems were near 1 in order for mass transfer to remain stable and the companion to have time to leave the main sequence. For conservative mass transfer, this implies initial periods of 13 and 6 days in the Giraffe and Unicorn. Non-conservative mass transfer leads to shorter initial periods, down to a few days.

An interesting difference between the Unicorn and the Giraffe is that the companion in the Unicorn is rotating rapidly (for a subgiant), almost certainly a consequence of recent accretion, while the companion to the Giraffe is rotating slowly. Rapid rotation is a natural prediction of the evolutionary scenario we propose, and so the Giraffe’s slow rotation seems more puzzling than the Unicorn’s rapid rotation. There are, however, other cases of mass transfer binaries with highly unequal mass ratios (implying the donor has lost a large fraction of its mass) and slowly-rotating accretors (e.g. Whelan et al., 2021).

Even in the Giraffe, the subgiant is rotating too rapidly to be tidally synchronized. Tides may, however, play a roll in slowing both subgiants’ rotation. Both are well inside their Roche lobes, with $R/R_{\rm Roche\,lobe}\approx 0.17$. In the theory of Zahn (1977), the tidal synchronization timescale due to turbulent friction in the stars’ convective envelopes is $t_{{\rm sync}}\approx\beta/\left(6q^{2}k_{2}\right)\left(MR^{2}/L\right)^{1/3}\left(a/R\right)^{6}$. Here $q=M_{\rm giant}/M$, $\beta=I/(MR^{2})$, where $I$, $M$, $R$, and $L$ are the moment of inertia, mass, radius, and luminosity of the subgiant, $k_{2}$ is it apsidal motion constant, and $a$ is the binary semi-major axis. From MESA models of subgiants, we find $\beta\approx 0.025$ in the Unicorn and $\beta\approx 0.14$ in the Giraffe. From Claret (2019), we estimate $k_{2}\approx 0.0015$ for the Unicorn and $k_{2}\approx 0.07$ for the Giraffe. This leads to $t_{\rm sync}\approx 10\,\rm Myr$ for the Giraffe (shorter than the duration of mass transfer; Figure 14), and $t_{\rm sync}\approx 100\,\rm Myr$ for the Unicorn (longer).

It thus seems plausible that tides have played a significant roll in spinning-down the Giraffe, but do not yet play a significant roll in the Unicorn. The difference in the timescales estimated for the two systems is, however, driven mainly by differences in $k_{2}$ and $\beta$, which are calculated purely from evolutionary models.

In Figure 16, we compare the Giraffe and Unicorn to several binaries in related evolutionary stages. HR 6819 (Bodensteiner et al., 2020; El-Badry & Quataert, 2021) and LB-1 (Shenar et al., 2020) are recently-detached systems in which the stripped star is still bloated. NGC-1850 BH1 (El-Badry & Burdge, 2022) and HD 15124 (El-Badry et al., 2022a) are still mass-transferring. HZ 22 (Greenstein, 1973) and 59 Cyg (Peters et al., 2013) contain stripped stars that have already contracted and heated significantly. These objects are chosen as a representative, but not exhaustive, subsample of the stripped binary population; see e.g. Schootemeijer et al. (2018) and Wang et al. (2021) for other similar systems.

Figure 16 also shows the MESA model from Figures 13 and 14, as well as a model for a more massive donor with initial mass of $4\,M_{\odot}$. This model is described in El-Badry & Quataert (2021) and approximately reproduces the observed properties of the HR 6819 system. The qualitative evolution of the two models is similar. The most important difference is that a more massive donor ignites helium burning in its core while mass transfer is still ongoing and subsequently becomes a (non-degenerate) sdOB star rather than a He WD. Even higher-mass donors are expected to follow similar trajectories at higher luminosity (e.g. Götberg et al., 2018). The Unicorn and Giraffe have the coolest and most diffuse donors of the highly-stripped objects shown here. Irrespective of whether their future evolution involves core helium ignition, the models shown in Figure 16 imply that they will soon contract and heat up.

There are several reasons to believe that the secondaries we detect in optical and IR spectra are not accretion disks around BHs, as was proposed for the Giraffe by Jayasinghe et al. (2022). There are abundant observations of astrophysical disks in similar contexts to the one proposed for the Giraffe – in low-mass X-ray binaries, cataclysmic variables, Be stars, and T Tauri stars – and their spectra generally do not look like slowly-rotating stellar photospheres. They are usually dominated by emission lines, not absorption lines (e.g. Shahbaz et al., 1996; Warner, 2003). If absorption lines are present in a disk around a compact object, they should be broadened by at least the Keplerian velocity at the radius where most of the light is emitted. The observed $v\sin i\lesssim 20\,\rm km\,s^{-1}$ for the secondary in the Giraffe – if it traced material in orbit around a $\approx 3\,M_{\odot}$ BH – would correspond to material on AU scales – significantly larger than the binary separation. Such material could exist in a circumbinary disk, but it would then not be expected to trace the secondary’s center of mass, as it is observed to do.

An accretion disk that emits significantly in the optical but not in X-rays is also unexpected. Irrespective of whether accretion is radiatively efficient or not, we expect $L_{X}\gtrsim L_{\rm opt}$ for accretion disks around compact objects at the relevant accretion rates (e.g. Quataert & Narayan, 1999). For example, the LMXB studied by Strader et al. (2016) – a rare system with an absorption spectrum from the disk at some epochs – has $L_{X}\approx 100L_{\rm opt}$. For the Unicorn and Giraffe, the X-ray upper limits imply $L_{X}\lesssim 10^{-5}L_{\rm opt}$ and $L_{X}\lesssim 10^{-3}L_{\rm opt}$. The implied radiative efficiencies in both systems are also much lower than expected for realistic models of accretion flows (e.g. Sharma et al., 2007).

The basic evolutionary scenario we propose for the giants in the Giraffe and Unicorn can occur for a relatively broad range of initial binary masses and separations, and the existence of helium WDs as wide companions to main-sequence stars implies that it is not uncommon in nature. There is one aspect, however, which had to be fine-tuned in our MESA models: in order for the secondary to be observed as a subgiant today, the initial mass ratio has to be very close to 1 (initial $q\gtrsim 0.995$ in the Giraffe, and $q\gtrsim 0.98$ in the Unicorn). Otherwise, the two stars would not terminate their main-sequence evolution at the same time, and the secondary would be observed as a main-sequence star rather than a subgiant. Unless the secondary has already formed a helium core and started to expand when mass transfer begins, accretion will rejuvenate it.

On the other hand, there is a strong selection effect against systems with non-subgiant accretors: they will have blue/UV excess, making the companions straightforward to detect. That is, if the companions in the Giraffe and Unicorn were still on the main-sequence, these systems would not have been identified as BH candidates in the first place. Nearly-equal mass binaries are not uncommon (e.g. Tokovinin, 2000; El-Badry et al., 2019) and are required to produce any observed binary containing two evolved stars. At least one similar system with a subgiant accretor has been observed (Miller et al., 2021).

One possibility that could solve the fine-tuning “problem" (if indeed it exists) is that the accreting star may not actually be near the end of its main-sequence evolution, but could be only temporarily inflated due to recent accretion on a timescale short enough that its envelope did not have time to thermally adjust. Such temporary expansion does indeed occur in several of the MESA models we explored with main-sequence accretors and high initial mass transfer rates. In our models, these accretors reach equilibrium again before the giant is stripped down to $\approx 0.4\,M_{\odot}$, but it is worth exploring whether there is a plausible model in which the recent mass transfer rate was higher and the accretors are only temporarily inflated. We defer investigation of such a scenario to future work.