Wide binaries with white dwarf or neutron star companions discovered from Gaia DR3 and LAMOST

The $nss\_two\_body\_orbit$ table includes 443,205 binaries or multiple star systems. For spectroscopic binaries (flagged as “SB1”, “SB2”, or “AstroSpectroSB1”), it provides orbital parameters such as period $P$, eccentricity $e$, and semi-amplitude $K$ of primary, etc. For “AstroSpectroSB1” binaries, the $K$ value can be calculated from the astrometric solutions ($c\_thiele\_innes$ and $h\_thiele\_innes$). First, we selected single-lined systems (“SB1” and “AstroSpectroSB1”) and cross-matched them with the LAMOST DR9 low-resolution and med-resolution general catalog¹¹1http://www.lamost.org/dr9/v1.0/catalogue. Second, we chose sources with more than two LAMOST spectral observations with signal-to-noise ratio ($SNR$) greater than 5 and clear RV variation. Third, we calculated the binary mass function of those systems using orbital parameters from the $nss\_two\_body\_orbit$, and the gravitational masses of visible stars with multi-band magnitudes and atmospheric parameters from LAMOST parameter catalog. Finally, we estimated the minimum masses of the unseen stars using the mass functions and the evolutionary masses of the visible stars.

Following these steps, three binaries with possible compact objects were selected, namely Gaia ID 4031997035561149824 (R.A. = 176.46978^o; Decl. = 35.29075^o; hereafter G4031), 3431326755205579264 (R.A. = 90.61903^o; Decl. = 28.13287^o; hereafter G3431), and 844176650958726144 (R.A. = 169.12464^o; Decl. = 55.72840^o; hereafter G8441). The positions on the Hertzsprung–Russell diagram (Figure 1) indicate that the visible stars of G4031 and G3431 are main-sequence stars while the visible star of G8441 is a giant.

We obtained all available LRS and MRS observations from the LAMOST archive for our candidate sources. These spectra have undergone various reduction steps with the LAMOST 2D pipeline, including bias and dark subtraction, flat field correction, spectrum extraction, sky background subtraction, and wavelength calibration, etc.(See Luo et al., 2015, for detalis). Figure 1 displays the LRS observations for the three targets. The cross-correlation function (CCF) was used to calculate the RV values using the red band (6300–7000 Å) of each MRS spectrum with $SNR>5$. Furthermore, for each spectrum, we derived a calibration factor ($\Delta rv=rv\_r1-rv\_r0$) with the $rv\_r1$ and $rv\_r0$ from the LAMOST MRS catalog. Here $rv\_r1$ represents the calibrated velocity of $rv\_r0$ by using RV standard stars (Huang et al., 2018). The final RV values were obtained by summing the measured value from the CCF method and the calibration factors (Table LABEL:lamost3.tab).

In addition, we applied for spectral observations using the Beijing Faint Object Spectrograph and Camera (BFOSC) mounted on the 2.16 m telescope at the Xinglong Observatory and the Double Spectrograph (DBSP) mounted on the Palomar’s 200 in. telescope (P200). For G8441, we carried out eight observations using the 2.16 m telescope from Feb. 20th, 2019 to Apr. 21th, 2019, with the E9/G10 grism and a 1.6^′′ slit configuration, and four observations using the P200 telescope on Mar. 14th, 2019. For G3431, we applied for two observations using the 2.16 m telescope on Oct. 4th, 2022. The observed spectra were reduced using the IRAF v2.16 software (Tody, 1986, 1993) following standard steps, and the reduced spectra were then corrected to vacuum wavelength. The CCF method was used to calculate RV, and the barycentric corrections to the observation time and RV were made using the light_travel_time and radial_velocity_correction functions provided by the Python package Astropy (Astropy Collaboration et al., 2013, 2018, 2022).

Each of our targets has been observed multiple times by LAMOST. Their atmospheric parameters can be estimated following (Zong et al., 2020):

and

where the index $k$ is the $k_{th}$ epoch of the measurements of parameter $P$ (i.e., $T_{\rm eff}$, log$g$, and [Fe/H]) for each star, and the weight $w_{k}$ is square of SNR of each spectrum corresponding to the $k_{th}$ epoch. Table 1 lists stellar parameters of G4031 measured by different methods, such as the LAMOST Stellar Parameter Pipeline (LASP) (Luo et al., 2015), DD-Payne (Xiang et al., 2019; Ting et al., 2019), and CNN (Wang et al., 2020a). In the following analysis, we preferred to use the LASP LRS results followed by the LASP MRS results. The stellar parameters of G4031 are $T_{\rm eff}=5953{\pm 17}K$, log$g$ $=4.05{\pm 0.02}$ dex, and [Fe/H] $=-0.37{\pm 0.01}$ (Table 2).

The distance from Gaia DR3 is approximately 720 pc. The Pan-STARRS DR1 3D dust map²²2http://argonaut.skymaps.info does not provide an extinction estimation, instead we used the SFD value with $E(B-V)=0.02$.

The stellar parameters can also be obtained by fitting the spectral energy distribution (SED). We used the Python module ARIADNE³³3https://github.com/jvines/astroARIADNE to estimate the stellar parameters of visible star, which can automatically fit the broadband photometry by using different stellar atmosphere models, such as Phoenix⁴⁴4ftp://phoenix.astro.physik.uni-goettingen.de/, BTSettl⁵⁵5http://osubdd.ens-lyon.fr/phoenix/, Castelli & Kurucz⁶⁶6http://ssb.stsci.edu/cdbs/tarfiles/synphot3.tar.gz, and Kurucz 1993⁷⁷7http://ssb.stsci.edu/cdbs/tarfiles/synphot4.tar.gz. We constructed the SED using magnitudes from various surveys: Gaia DR3 ($G$, $G_{\rm BP}$, and $G_{\rm RP}$), 2MASS ($J$, $H$, and $K_{\rm S}$), APASS ($B$, $V$, $g$, $r$, and $i$) and WISE ($W$1 and $W$2). The Gaia DR3 parallax and extinction $A_{V}$ were also used as input priors. From the SED fitting (Figure 2), we obtained stellar parameters for a G-type star with an effective temperature of $5910^{+36}_{-29}K$, surface gravity of $4.05^{+0.01}_{-0.01}$ dex, metallicity of $-0.37^{+0.01}_{-0.01}$ (Table 3), which are in good agreement with spectroscopic results.

Furthermore, we calculated stellar mass using two methods: one method involved the use of stellar evolution models, referred to as evolutionary mass; the other method utilized observed photometric and spectroscopic parameters, referred to as gravitational mass.

First, we used the isochrones, a Python module (Morton, 2015), to calculate evolutionary mass of G4031 by fitting the spectroscopic and photometric data. We used the atmospheric parameters ($T_{\rm eff}$, log$g$, [Fe/H]), Gaia parallax (Gaia Collaboration et al., 2018), multi-band magnitudes ($G$, $G_{\rm BP}$, $G_{\rm RP}$, $J$, $H$, and $K_{\rm S}$), and extinction $A_{V}$ ($=3.1\times E(B-V)$) as the input parameters. The evolutionary mass and radius of visible star from isochrones are $1.46^{+0.02}_{-0.02}M_{\odot}$ and $2.04^{+0.03}_{-0.03}R_{\odot}$, respectively.

Second, we used Gaia and 2MASS magnitudes ($G$, $G_{\rm BP}$, $G_{\rm RP}$, $J$, $H$, and $K_{\rm S}$) and the effective temperature to calculate the gravitational mass following $M=gR^{2}/G$ (See details in Li et al., 2022). The final mass and corresponding uncertainty were calculated as the average values and standard deviations of the masses derived from different bands. The gravitational mass of G4031 is $1.62{\pm 0.02}M_{\odot}$.

We averaged the Barycentric Julian Dates (BJDs) and RVs taken on the same day (Appendix A) for the purpose of RV fitting. The Joker (Price-Whelan et al., 2017), a custom Markov chain Monte Carlo sampler, was employed to fit the Keplerian orbit. Figure 2 displays the phase-folded RV data along with the best-fit RV curve. Table 4 presents the orbital parameters of G4031 obtained through The Joker fitting, including orbital period $P$, eccentricity $e$, argument of periastron $\omega$, mean anomaly at the first exposure $M_{\rm 0}$, RV semi-amplitude $K$, and systematic RV $\nu_{\rm 0}$. The fitting results differ slightly from those obtained from the table $nss\_two\_body\_orbit$ of Gaia DR3.

G4031 has been observed by Zwicky Transient Facility (ZTF; Bellm et al., 2019) in $g$ band and All-Sky Automated Survey for Supernovae (ASAS-SN; Kochanek et al., 2017) in $V$ band. No clear variation caused by ellipsoidal deformation can be seen in the folded light curves (Figure 3). Therefore, no further information such as the orbital inclination can be constrained by the light curve.

We calculated the size of Roche-lobe by using the standard Roche-lobe approximation (Eggleton, 1983),

where $a=(1+q)a_{1}=(1+q)P(1-e^{2})^{1/2}K_{1}/(2\pi{\rm\sin}i)$ is the separation of binary system. The filling factor is $\approx 3.2\%$ assuming $i=90^{\circ}$, indicating this system is a detached binary system.

For the visible star, the absolute magnitude in the $G$ band is $G=3.04$ mag; while for a main-sequence star with a mass of $0.77M_{\odot}$ or $0.98M_{\odot}$, the absolute magnitude in the $G$ band would be around 6.20 mag or 4.80 mag. By comparing the flux-calibrated Gaia XP spectrum (Huang et al. 2023, in preparation) with the Phoenix template, no flux excess is observed (Figure 2), indicating the companion of G4031 is a compact object if the secondary has a mass of $0.98M_{\odot}$. However, if the secondary has a mass of $0.77M_{\odot}$, it cannot be distinguished based on the SED or spectral continuum. Given this uncertainty, we decided to investigate the binary nature of G4031 with the LAMOST MRS observations by using the spectral disentangling method.

We employed the algorithm of spectral disentangling proposed by Simon & Sturm (1994). Before applying this algorithm, we performed preliminary tests to determine the detection limits for our targets. We derived the effective temperature and surface gravity of the secondary (assuming a main-sequence star) using the minimum mass⁸⁸8http://www.pas.rochester.edu/ẽmamajek/EEM_dwarf_UBV-IJHK_colors_Teff.txt following different mass ratio (Table 5). The corresponding Phoenix⁹⁹9https://phoenix.astro.physik.uni-goettingen.de model was used as the theoretical spectra of the secondary. Then, we combined the observed spectra and theoretical spectra, with a reduced resolution of $R=$ 7500 and different rotational broadening ($v{\rm sin}i=$ 10 km/s, 50 km/s, 100 km/s and 150 km/s), to produce synthetic binary spectra. For the simulated binary spectra, the wavelength range of 6400 Å to 6600 Å was used for spectral disentangling. The results by visual check are presented in Table 5. It can be seen that for G4031, when $v{\rm sin}i$ is less than 100 km/s, this method can successfully separate the spectra of primary and secondary components.

The MRS observations within the wavelength range of 6400 Å to 6600 Å were utilized for spectral disentangling. As an example, Figure 4 shows the results for a mass ratio of 1.4 and 1.8. No feature of absorption spectrum is apparent for an additional component (red lines in Figure 4), implying that the component star is a compact object.

We also used single- and binary-star spectral models (Liu et al. 2023, in preparation) to perform fitting for the LRS observations. In brief, the spectra from LAMOST LRS and the stellar parameters from the Apache Point Observatory Galactic Evolution Experiment (APOGEE) DR16 were used as the training set to develop a model using the neural network version of Stellar LAbel Machine (SLAM) (Zhang et al., 2020a, b) for single-star spectra. The binary-star model was then created by combining two single-star models with appropriate radial velocities. Spectral observations were fitted with both the single- and binary-star models, and their corresponding $\chi^{2}$ values were compared to determine the best fit. The results of the semi-empirical spectroscopic experiments show that one system is a single star if the difference in $\chi^{2}$ (i.e., $\Delta\chi^{2}$) between the single-star and binary-star fitting is less than 0.4. For G4031, the $\Delta\chi^{2}$ value is about $-$0.04, further evidencing that it includes one compact object.

The visible star of G3431 is a G-type main-sequence star. The stellar parameters from the LASP (LRS) method are used in the following analysis, with an effective temperature of $T_{\rm eff}=6402{\pm 13}K$, surface gravity of log$g$ $=4.23{\pm 0.02}$ dex, and metallicity of [Fe/H] $=-0.06{\pm 0.01}$ (Table 2). According to Gaia DR3, the estimated distance to G3431 is approximately 338 pc. The Pan-STARRS DR1 3D dust map indicates an extinction with $E(B-V)=0.03$ at this distance. SED fitting returns stellar parameter estimations for the visible star, with $T_{\rm eff}=6156^{+69}_{-39}K$, log$g$ $=4.22^{+0.04}_{-0.03}$ dex, [Fe/H] $=-0.06^{+0.01}_{-0.01}$ (Table 3 and Figure 5).

The isochrones fitting based on evolutionary models yields an estimated evolutionary mass and radius for G3431 of $1.42^{+0.03}_{-0.03}M_{\odot}$ and $1.83^{+0.03}_{-0.03}R_{\odot}$, respectively. On the other hand, the gravitational mass of G3431, obtained by averaging six gravitational mass estimations, is determined to be $2.06{\pm 0.13}M_{\odot}$, which differs from the evolutionary mass estimation. It is important to note that even within the gravitational mass measurements, there are discrepancies between the results derived from Gaia bands ($\approx 1.88\ M_{\odot}$) and those from 2MASS bands ($\approx 2.24\ M_{\odot}$). By reviewing the calculation process, we found that the bolometric magnitudes from Gaia bands ($BP$, $G$, $RP$) are 3.13, 3.09, 3.04, while the bolometric magnitudes from 2MASS bands ($J$, $H$, $K_{\rm S}$) are 2.93, 2.89, 2.86. This gradual variation with wavelength can be attributed to the measurement accuracy of stellar parameters, such as $T_{\rm eff}$ and $E(B-V)$. We noticed that the difference in temperature between different methods is about 250 K (Table 1), and the temperature from SED fitting is lower than spectral results. In this case, the mass measurement from stellar model fitting is more reliable than the mass calculated with atmospheric parameters. Additionally, the evolutionary mass of $\approx$1.42 $M_{\odot}$ is more reasonable for a late-F main-sequence star. Taking all these factors into consideration, we adopted the evolutionary mass measurement for further analysis.

The Barycentric Julian Dates (BJDs) and RV values taken on the same day (Appendix A) were averaged to improve the accuracy of the RV fitting. The phase-folded RV data and the best-fit RV curve from The Joker are shown in Figure 5. Based on the RV fitting results, we obtained a mass function of $\approx$0.32 $M_{\odot}$ and a minimum mass of $1.36\pm 0.09M_{\odot}$ for the unseen star, which are slightly lower than the values calculated from Gaia DR3 solution. As G4031, no clear variation can be seen in the light curve of G3431 (Figure 6).

Using the Roche-lobe approximation (Eq. 4), we estimated that the filling factor of G3431 is approximately $3.4\%$ (assuming $i=90^{\circ}$), suggesting this system is a detached binary system. The possibility that G3431 contains two main-sequence stars can be ruled out according to the luminosity ratio. For instance, the absolute $G$-band magnitude of the visible star is $G=$ 3.06 mag, while the absolute magnitude of a main-sequence star with a mass of $1.36M_{\odot}$ is $G=$ 3.10 mag.

We also applied the spectral disentangling technique to the observed spectra of G3431. The results of the spectral disentangling tests (Table 5) indicate that for binary systems with two normal stars having a mass set similar to G3431, this method can successfully separate the binary components including two normal stars with high significance when $v{\rm sin}i$ is less than 150 km/s. No additional component with an optical absorption spectrum can be detected (Figure 7), excluding the possibility of a normal binary system. Furthermore, the $\Delta\chi^{2}$ values of G3431 from single-star and binary fitting are about $-$0.18 and $-$0.24 for its two LRS observations, indicating that these spectra are more likely from a single star.

The stellar parameters of G8441 obtained from LASP (LRS) are as follows: $T_{\rm eff}=4194{\pm 2}\ K$, log$g$ $=1.83{\pm 0.03}$ dex, and [Fe/H] $=-0.74{\pm 0.01}$ (Table 2). According to Gaia DR3, the estimated distance to G8441 is approximately 857 pc, and the Pan-STARRS DR1 3D dust map suggests an extinction of $E(B-V)\approx 0.01$ at this distance. SED fitting using the ARIADNE package (Figure 8a) yields an effective temperature of $4144^{+23}_{-20}\ K$, surface gravity of $1.86^{+0.11}_{-0.09}$ dex, and metallicity of $-0.77^{+0.10}_{-0.10}$, which are in good agreement with the spectroscopic estimation.

Unlike G4031 and G3431, the light curves of G8441 exhibit significant ellipsoidal variations (Section 5.2). The large amplitude of variation in $V$-band light curve ($\gtrsim$ 0.4 mag) implies that the visible star of G8441 is (close to) Roche lobe-filling. As a result, the single-star evolution model is unsuitable for assessing the mass of the visible star. For a stripped star with Roche lobe overflow, its density can be calculated following (Frank et al., 2002),

Using the radius from SED fitting ($R=$ 16.55 $R_{\odot}$; Table 3), we calculated the mass of the visible star to be $M=4\pi\bar{\rho}R^{3}/3\approx 0.28{\pm 0.03}\ M_{\odot}$.

The phase-folded RV data and the best-fit RV curve from The Joker are shown in Figure 8. The RV fitting returns a mass function of $\approx$0.84 $M_{\odot}$ and a minimum mass of $1.25\pm 0.09\ M_{\odot}$ for the unseen star, while using the orbital parameters of Gaia DR3 (Table 4), the mass function is $\approx$0.67 $M_{\odot}$ and the minimum mass of the unseen star is $1.07\pm 0.10\ M_{\odot}$.

The Lomb-Scargle algorithm (Lomb, 1976; Scargle, 1982) was used to calculate the orbital period ($\approx$46.885 days) with the ASAS-SN $g$-band light curve (Figure 9). To obtain a more precise period estimation, we further folded the light curves with periods ranging from 46.5 to 47 days, in a step of 0.001 day. Through visual inspection of the folded light curves, we determined a period of $\approx$46.895 days. Figure 10 displays the folded light curves in multiple bands.

We employed the software Physics of eclipsing binaries (Prša et al., 2016; Horvat et al., 2018; Conroy et al., 2020) (PHOEBE) to fit the ASAS-SN $V$- and $g$-band light curves. The ellipsoidal modulation (Figure 10) and the substantial variation amplitude indicate the visible star is a stripped star. During the LC fitting, we used the eclipse_method $=$ only_horizon as the eclipse model and specified a semi-detached system for the binary configuration. Two scenarios were considered: one with a stripped star and a compact object, and the other with a stripped star and a standard main-sequence star. In the first case, a cold ($T_{\rm eff}=$ 300 $K$) and small ($R=3\times 10^{-6}R_{\odot}$) blackbody was used as the secondary. We used two different sets of $a$sin$i$ values: one from The Joker fitting and the other from the Gaia DR3 solution (Table 4). The bolometric gravity darkening coefficients was adopted as $\beta=0.58$, which was derived from Claret & Bloemen (2011) in the $V$ band for stars with similar atmospheric parameters. Table 6 presents the parameter estimates from the PHOEBE fitting. For the scenario of a stripped star and a compact object, the fitting results show that the unseen star is a neutron star or a massive white dwarf, while for the scenario of a stripped star and a main-sequence star, the best-fit models indicate the unseen dwarf is an A- or F-type star.

G8441 has been reported as a possible compact object candidate in previous studies (Gu et al., 2019; Zheng et al., 2019). In this section, we will try to investigate the nature of the unseen object by examining the optical spectra, and the UV and X-ray emission. Due to the limited number of spectral observations, we did not perform the spectral disentangling.

No emission line is detectable in the optical spectra (Figure 1). It is evident that the $H_{\alpha}$ lines in all spectral observations are absorption lines (Figure 11), indicating the absence of an accretion disk. This contrasts with most binaries containing a stripped giant star, such as V723 Mon and 2M04123153+6738486 (Jayasinghe et al., 2021, 2022), which display clear $H_{\alpha}$ emission lines. G8441 is likely similar to a few binaries without emission lines, explained by temporary halted or slowed accretion (El-Badry & Rix, 2022).

G8441 has been observed in FUV and NUV bands by GALEX. By using the UV magnitudes of $f_{\rm FUV}=23.28$ $\mu$Jy and $f_{\rm NUV}=1142.51$ $\mu$Jy, we calculated the luminosities as $L_{\rm FUV}=1.16\times 10^{31}$ erg s^-1 and $L_{\rm NUV}=6.19\times 10^{32}$ erg s^-1. This luminosity can be attributed to the giant star, given that K-type giants typically exhibit FUV and NUV luminosities within the ranges of $\approx 5\times 10^{29}$–$5\times 10^{32}$ erg s^-1 and $\approx 3\times 10^{29}$–$7\times 10^{33}$ erg s^-1, respectively (Wang et al., 2020b). However, it is uncertain whether a stripped star has the same magnetic dynamo mechanism as a normal giant. If the the unseen star is an A-type star ($T_{\rm eff}\approx$ 8200 $K$ from PHOEBE fitting), its FUV and NUV emission would exceed the observed levels significantly (Figure 12). Therefore, the A-type star scenario can be ruled out. If the unseen star is an F-type star ($T_{\rm eff}\approx$ 7200 $K$ from PHOEBE fitting), its UV emission are compatible with the observation. On the other hand, if the unseen object is a white dwarf, the upper limit of its surface temperature can be constrained with the FUV emission. White dwarfs with effective temperatures greater than 20,000 $K$ can be ruled out since models predict higher fluxes than the GALEX FUV observation (Figure 12). Thus, the UV luminosities (especially the high NUV luminosity) can be either totally from the stripped star, or from a combination of the stripped star and the unseen object.

G8441 was observed by XMM-Newton on May 4, 2020. No X-ray counterpart was detected from the EPIC instruments, including M1, M2, and PN. Using a radius of 10^′′, an upper limit of the net count rate in the 0.3–-8 keV range was calculated with the PN instrument, yielding approximately 0.0004 counts/sec. We utilized the PIMMS tool¹⁰¹⁰10https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl to convert the count rate into flux. Applying the standard linear relation between the hydrogen column density $N_{\rm H}$ and the reddening $N_{\rm H}=5.8\times 10^{21}/E(B-V)$ (Schlegel et al., 1998), the hydrogen column density $N_{\rm H}$ was calculated to be $5.8\times 10^{19}$ cm^-2 with the optical reddening of $E(B-V)=0.01$ mag. By using a power-law photon index of 1.5 or 2, the upper limit of the unabsorbed flux in the 0.3–8 keV band was derived to be $1.1\times 10^{-15}$ erg cm^-2 s^-1 or $7.9\times 10^{-16}$ erg cm^-2 s^-1, respectively. At the adopted distance of 857 pc, the upper limit of the emitted luminosity was estimated to be $L_{X}\lesssim$ $9.7\times 10^{28}$ erg s^-1 or $6.9\times 10^{28}$ erg s^-1.

To sum up, the substantial variation amplitude of light curves indicates the visible star is most likely a stripped star, although there is no emission feature in the optical spectra. The unseen object could be a compact object (white dwarf or neutron star) or a main-sequence F star. Unfortunately, present observational data (e.g., UV and X-ray data, SED, and optical spectra) do not provide sufficient evidence to distinguish between these scenarios.

The Binary Population And Spectral Synthesis code (BPASS) (Eldridge et al., 2017; Stanway & Eldridge, 2018) is a tool to calculate the evolutionary tracks of single stars or binary systems. We used the hoki package (Stevance et al., 2020) to search for binary models consistent with the observed properties of G8441 based on the results from BPASS. The selection criteria included $P_{\rm orb}=46.88\pm 5$ days, $V=0.935\pm 1$ mag, $J=-1.385\pm 1$ mag, $K=-2.23\pm 1$ mag, $\log T_{\rm eff}(K)=3.62\pm 0.04$, $M_{1}=0.28\pm 0.1\ M_{\odot}$, and $M_{2}$ in the range of 1.1–1.5 $M_{\odot}$, where $M_{1}$ and $M_{2}$ represent the masses of the giant and the invisible star, respectively. We identified four systems including an F- or G-type star and one system with a compact object (Table 7). The parameters of these systems are not entirely consistent with the analysis in Section 5.2 and 5.3. For example, the masses of the stripped star in these models are higher than our calculation ($\approx$0.28 $M_{\odot}$), and for the four models containing normal stars, the temperatures of these normal star are lower than the PHOEBE fitting results (Table 6).

One interesting finding is that the mass of the stripped star ($\approx$0.28 $M_{\odot}$) follows the relationship between the mass and orbital period ($M_{\rm WD}/M_{\odot}=(\frac{P_{\rm orb}/{\rm day}}{1.1\times 10^{5}})^{1/4.75}+0.115$) of white dwarfs or the core mass of low-mass giants (Tauris & Savonije, 1999). This suggests that the mass transfer in G8441 might have been close to termination and most of the hydrogen envelope of the stripped star has likely been stripped away. This donor star could eventually evolve into an extremely low mass white dwarf.