Mass-Ratio Distribution of Binaries From the LAMOST-MRS Survey¹¹1Released on , , 2022

Li et al. (2021) developed an automatic method to search for binaries using observations from LAMOST-MRS data. The method is based on calculating the Cross-Correlation Function (CCF) between a star’s spectrum and a template spectrum. They applied a template matching and a Gaussian smoothing to find the number, the location, and the amplitudes of peaks of the CCF. The template Li et al. (2021) adopted is the observed solar spectrum with a resolution of 100 000 (Kurucz et al., 1984). They have altogether found 3 133 SB2 candidates from 952 747 blue arm spectra with SNR$\geq 10$ from LAMOST-MRS. Above 95% of the candidates are new detections. The authors showed that radial velocity difference between the two components of a binary system ($\Delta RV=|RV_{1}-RV_{2}|$) is the key factor for SB2 candidate identification. The lower limit of $\Delta RV$ is found to be $\sim 50$ km/s, which mainly depends on the resolution of the observed spectra.

The CCF approach of Li et al. (2021) identifies a system to be a binary if there are two peaks in its CCF, and detection mainly depends on the radial velocity differences of the SB2s. However, as can be imagined, the amplitudes of the peaks should depend on the mass ratio of the secondary to the primary ($q={m_{2}/m_{1}}$). To test the hypothesis, we constructed synthetic spectra for a binary grid with different temperatures, mass ratios, RV differences, and signal-to-noise ratio (SNR) levels. We described the procedures as follows.

We first employed the radiative transfer code SPECTRUM (Gray & Corbally, 1994) to generate a grid of synthetic stellar spectra from the $ATLAS$9 stellar atmospheric model (Kurucz, 1993). The model is for the plane-parallel atmosphere with the assumptions of local thermodynamic equilibrium (LTE). The model spectra have a range of effective temperatures between 4 000 K and 8 000 K, corresponding to K-, G-, F-, and A-type stars, with a step size of 100 K. We adopted a standard value of surface gravity for MS star with ${\mathrm{log}}g$=4.5 ($g$ is in ${\rm cm\,s^{-2}}$), and the metallicities [Fe/H] are set to be 0, i.e. a solar metallicity. We constructed the model spectra covering the wavelength regime from 4 900 Å to 6 900 Å, and downgraded the resolution from 51 000 to 7 500 to match that of the LAMOST-MRS spectra via convolution with the $laspec$ code (Zhang et al., 2020, 2021). In Figure 2, we show three examples of model spectra constructed from the ATLAS9 grids. In the top panel, we plotted the original high resolution model spectrum (R $\sim$ 51 000) with $T_{\mathrm{eff}}$= 8 000 K, ${\mathrm{log}}g$ = 4.5, and [M/H] = 0 in gray, and overplotted the degraded spectrum with R $\sim$ 7 500 in red. The model spectra constructed with $T_{\mathrm{eff}}$= 6 000 K and $T_{\mathrm{eff}}$= 4 000 K with the same ${\mathrm{log}}g$ and [M/H] are shown in the middle and the bottom panels, respectively.

We then constructed a grid of synthetic spectra for MS binaries of solar metallicities from the $ALTAS$9 stellar spectra above. For an MS binary system, both components are MS stars. Given an effective temperature of $T_{\mathrm{eff}\,1}$ and a gravity of ${\mathrm{log}}g$ for the primary, we obtain its age $t$, its mass $M_{1}$ and its luminosity $L_{1}$ from the evolutionary grids of the binary population synthesis (BPS) of Han et al. (2002). Given a mass ratio $q=M_{2}/M_{1}$, we obtain the mass of the secondary star $M_{2}$. Using the BPS model grid of Han et al. (2002) again, we obtain the luminosity $L_{2}$ and the associated $T_{\mathrm{eff}\,2}$ of the secondary star of age $t$. Based upon the atmospheric parameters of the primary star $M_{1}$ and the secondary star $M_{2}$ with age $t$, we obtain the model spectra for the primary and the secondary from ATLAS9, respectively. We shift the secondary’s spectrum with a radial velocity $\Delta RV$ and combine the spectrum with that of the primary. The minimal radial velocity difference is set to be $50\,{\rm km/s}$ since the minimal detected $\Delta RV$ is around $50\,{\rm km/s}$ from LAMOST-MRS. The largest $\Delta RV$ added on the spectrum is $225\,{\rm km/s}$ because it is the maximum that Li et al. (2021) detected from observations. Beyond this range, the possibility of detecting the binary is set to zero. The combined spectra are then injected with Gaussian noise with a given signal-to-noise (SNR) ratio. The ratio is larger than 10 to be consistent with the observational spectra. If the ratio is larger than 100, there is no significant influence on the detected efficiency of the spectra.

We constructed binary synthetic spectra to find a way to derive mass ratios of observed SB2s from LAMOST-MRS data. To make things simple, we set primary stars’ gravity to be ${\mathrm{log}}g=4.5$, and adopt $T_{\mathrm{eff}\,1}=8\,000{\,\rm K}$ for primary stars of spectral type A, 6 400 K for type F, and 5 600 K for type G, respectively. We chose such temperature values as they can be found from the BPS model grids of Han et al. (2002), but still be typical for stars of spectral type A, F, and G. For each spectral type of the primary, we adopt various mass ratios $q$ from 0.4 to 1.0. $q$ is chosen in such a way that $L_{2}/L_{1}$ changes by a constant step size, and $q=0.4$ corresponds to $L_{2}/L1=0.1$, which is the lower limit for SB2 detection. For binaries with given mass ratios $q$ and given spectral types, we adopt various $\Delta RV$ and SNR (see Table 2) to construct their synthetic spectra.

We employ the code from Li et al. (2021) to calculate a CCF from the constructed synthetic spectra of binaries. Figure 3 shows an example of CCF calculation. We see there are two peaks for the CCF, with the large peak for the primary star ($A_{1}$) and the small peak for the secondary star ($A_{2}$). In the upper panel, we show a template binary spectrum with an effective temperature of $T_{\mathrm{eff\,1}}=6\,400$ K and a mass ratio of $q=0.79$. The spectrum has a resolution of 7 500 with a wavelength range from 4 900 Å to 5 400 Å. We show the CCF of the spectrum in the bottom panel. There are two peaks of $A_{1}=0.73$ and $A_{2}=0.61$. The radial velocity ranges from $-500$ to $500$ $\mathrm{km/s}$. We identify the positions of $A_{1}$ and $A_{2}$ as the radial velocities of the primary and the secondary ($V_{1}=-83$ $\mathrm{km/s}$ and $V_{1}=17$ $\mathrm{km/s}$ The peak amplitude ratio (PAR), $A_{2}/A_{1}$, where $A_{1}$ and $A_{2}$ are the peak amplitudes for the primary and the secondary in the CCF, mainly depends on the mass ratio $q$, but also on the temperature $T_{{\rm eff}\,1}$ of the primary and the radial velocity difference $\Delta RV$. By calculating the CCFs for all the synthetic spectra constructed for binaries, we obtained the relation between $A_{2}/A_{1}$ and $q$ for different spectral types of primaries in Figure 4. The error bars are the scatter resulting from various $\Delta RV$ adopted. We fitted the relation between $A_{2}/A_{1}$ and mass ratio $q$ using a quadratic function, shown in Equation 1 to Equation 3. With these equations, we can convert the PAR of observational SB2s to their mass ratio.

As pointed out by Li et al. (2021), the radial velocity differences $\Delta RV$ between the two components of binaries are the key factor for SB2 candidate identification. The identification also depends on mass ratio $q$, as shown in the previous section (Section 3.2). A binary with a large $q$ can be more easily detected. To understand how the CCF method works and how the bias of PAR derived $q$ distribution can be corrected, we have carried out an investigation on the SB2 detection efficiency, and its dependence on both $\Delta RV$ and $q$.

We have constructed a very dense grid²²2The grid is somewhat unnecessarily dense for the current purpose of synthetic spectra for binaries with various $q$ and $\Delta RV$ to investigate the SB2 detection efficiency. The effective temperature of the primaries $T_{{\rm eff}\,1}$ is taken to be 4 000 K to 8 000 K, with a step size of 100 K, The radial velocity difference $\Delta RV$ is taken from 50 to 225 ${\rm km/s}$ with a step size of 5 ${\rm km/s}$. For a binary with a given set of parameters of $T_{{\rm eff}\,1}$, $\Delta RV$ and $q$, we generate 10 spectra with a signal-to-noise ratio of SNR=20³³3We have also tested other SNRs from 10 to 100, and the results do not change., which is a typical value for our LAMOST-MRS data. We then apply the CCF method to the spectra to see we identify them as SB2s or single stars. We define the detection efficiency as the fraction of synthetic spectra that are identified as SB2s with the CCF method. If $\Delta RV$ is below 50 ${\rm km/s}$ or over 225 ${\rm km/s}$, the efficiency is set to zero. For $q<0.4$, the secondary is far too dimmer than the primary that the efficiency is zero.

Figure 5 shows the detection efficiency of SB2s for various $\Delta RV$. Panel (a) is for the primary’s temperature between 6 000 K and 8 000 K, panel (b) for the primary’s temperature between 4 000 K and 6 000 K， panel (c) for all the temperatures from 4 000 K to 8 000 K. Panel (d) shows the residuals of panel (a) minus panel (b). From panels (a), (b) and (d), we see that the SB2 detection efficiency are nearly the same for different temperatures. Therefore, we use panel (c), which is for all the temperatures, for the detection efficiency analysis. From panel (c), we see that the efficiency is high (close to 1) for a large $q$ and low for a small $q$. The low detection efficiency for a small $q$ is due to that the secondary’s contribution to the synthetic spectra becomes insignificant when $q$ decreases. We also see that SB2s can most probably be detected when the RV difference is $\sim 150\,{\rm km/s}$ for $q>0.7$ (see the right part of the panel (c)), and we explain this below. A spectral line in a component star manifests itself as two lines in the combined spectra due to an adopted RV difference between the two-component stars. A large $\Delta RV$ makes the separation between the two lines bigger, and the synthetic spectra are more obvious to be an SB2. Therefore large $\Delta RV$ are favored for the detection of SB2s. However, this is not always true. $\Delta RV$ of $\sim 150\,{\rm km/s}$ corresponds to a separation of lines of $\sim 3$ Å. Our template is the Sun spectrum of high-resolution (Kurucz, 1993) which Li et al. (2021) used, and we have many absorption lines of iron in the Sun spectrum with slight wavelength difference. The lines with a large RV difference are easy to be identified as a new line of the single star. As a result, the efficiency decreases with increasing RV differences if RV differences are larger than $\sim 150\,{\rm km/s}$.

To have a better look at how the detection efficiency depends on $q$, we plotted Figure 6. If we assume that the SB2s have $\Delta RV$ uniformly distributed between 50 and 225 ${\rm km/s}$, we sum up the efficiencies of all the $\Delta RV$s for every $q$ in panel (c), divide the sum by the number of data points, and we then obtain the efficiency dependence on $q$. We show the dependence curve by the red dashed line in the lower panel. However, $\Delta RV$ of our observed SB2s is not uniformly distributed between 50 and 225 ${\rm km/s}$, and the distribution of observed $\Delta RV$ peaks at $\sim 60\,{\rm km/s}$ (see the upper panel of Figure 6). We again average the efficiencies of various $\Delta RV$s for every $q$ in panel (c), but with a weight of observed $\Delta RV$ distribution. The result is shown as the black solid line in the lower panel. We see that the detection efficiencies with the observational $\Delta RV$ distribution are higher than the one with a uniform $\Delta RV$ distribution for $q$ from 0.4 to 1.0.

We have constructed binary synthetic spectra in order to find a way to derive mass ratios of observed SB2s from LAMOST-MRS data. To make things simple, we have assumed ${\mathrm{log}}g=4.5$ for the primary. We show below that such an assumption is suitable for our purpose. For a binary system of spectral type F, the effective temperature of the primary $T_{\mathrm{eff}\,1}$ is set to be 6 400K. We set $t=$ 0 yr, $9.3\times 10^{8}$ yr, and $1.8\times 10^{9}$ yr, corresponding to the zero-age MS (ZAMS), the middle age MS (MAMS), and terminal age MS (TAMS) of the primary star, respectively. The gravity, ${\mathrm{log}}g$, of the primary decrease from ZAMS to TAMS, but its value is not far from 4.5. We then constructed synthetic spectra of the binary with various mass ratios for three ages $t$. Employing the code from Li et al. (2021), we calculated CCFs from the constructed synthetic spectra, and Figure 8 shows the result for the relation between mass ratio $q$ and PAR $A_{2}/A1$. We see very little difference among ZAMS, MAMS, and TAMS primaries. Adopting ${\mathrm{log}}g=4.5$ would not noticeably affect our result.

In our analysis for mass ratio distributions, we have corrected the bias resulting from the dependence of SB2 detection efficiency on $q$ and $\Delta RV$. The detection also depends on the orbital period and the number of observation epochs for an SB2, and such a bias needs to be discussed. We show the detection fraction of SB2s versus orbital period in Figure 9.

In the figure, we assume the total mass of the primary and the secondary to be $2\,M_{\odot}$, the orbital eccentricity is zero, the mass ratio is one, the inclination ranges from 0 to $2\pi$ (uniformly distributed in solid angles), and the observational epoch is uniformly distributed at orbital phase between 0 and $2\pi$. For a binary with a given orbital period, we can then calculate the radial velocity difference $\Delta V$ of the observational epoch. We did such a calculation for 10 000 binaries with a given number of observational epochs and a given orbital period. We define the detection fraction as the ratio of the number of binaries with $\Delta RV$ between 50 and 225 ${\rm km/s}$ over the number of the total binaries (i.e. 10 000). We see that the CCF method we adopted is able to detect SB2s with orbital periods less than 155 d, and most of the detected SB2s are with orbital periods less than 50 d.

The LAMOST-MRS is a magnitude-limited survey in which the detection limitation is around 14.5 apparent magnitude in G band (Liu, 2019). For binaries with the same effective temperature, a binary with a large mass ratio is brighter than the one with a small mass ratio. It may cause a bias that the $\gamma$ of the observed mass ratio distribution could be higher than the true value. The distance of our candidates is below 3 000 pc. For considering the completeness of our result, we make three volume-limited samples with a distance to the Sun of 500, 1 000, and 2 000 pc, respectively. Using the number of identified SB2 stars in Table 1 for different distances to the Sun and the same procedure described in Section 4, we plot the mass ratio distribution for binaries of spectral type A, F, and G in Figure 10. Note that the distributions are not cumulative ones as in Figure 7. We see no significant differences between the distributions for different distances (see the bottom panel in Figure 10). Therefore, we conclude that our results are affected little by the bias of the magnitude-limited sample.

In Section 5.2, we have discussed how the detection efficiency depends on the orbital period and the number of observational epochs of an SB2. The upper panel of Figure 11 shows the distribution of observational epoch numbers of A, F, G type SB2s in our sample. Our SB2s have 1 to 15 observations. More than 55% SB2s have only one observation, and $\sim 96\%$ SB2s have been observed less than 5 times. With the detection fraction shown in Figure 9 and the detection efficiency shown in Figure 5, we convert the detected number of SB2s to the intrinsic one, and obtain the binary fraction by dividing the intrinsic number of binaries by the total number of stars observed.

The lower panel of Figure 11 shows the fraction of close binaries with an orbital period less than 155 d and a mass ratio between 0.6 and 1 for spectral type A, F, and G. The close binary fractions for A, F and G binaries are $7.6\pm 0.5\%$, $4.9\pm 0.2\%$ and $3.7\pm 0.1\%$, respectively. The close binary fractions we derived are consistent with that of Moe & Di Stefano (2017), in which the multiplicity of solar-type primaries $f_{q>0.3}=0.017\pm 0.007$ with $\mathrm{log}(P/\mathrm{d})=0.5\pm 0.5$ and $f_{q>0.3}=0.020\pm 0.007$ with $\mathrm{log}(P/\mathrm{d})=1.5\pm 0.5$. Also, in Moe & Di Stefano (2017), they showed that binary fraction increase with stellar mass, and estimated $f_{P<20\,{\mathrm{d}}}\propto M_{1}^{0.7}$, where A-dwarfs with $M_{1}=2.45\pm 0.75$ $M_{\odot}$ would have $\sim$ 1.96 times the close binary fraction of G-dwarfs with $M_{1}=0.94\pm 0.16M_{\odot}$.

By using the binary samples of Raghavan et al. (2010), Shatsky & Tokovinin (2002) and Kouwenhoven et al. (2007), Duchêne & Kraus (2013) found an increasing trend of the power index $\gamma$ of mass ratio distribution towards lower primary mass, shown as gray circles in Figure 12. In our study, the index $\gamma$ is marked by black circles in Figure 12, and they display a similar trend. However, from Figure 12, we see that the variation of index $\gamma$ with the primary mass in our results is much larger than their results. This may be caused by two reasons. Firstly, the mass ratio distribution may depend on the orbital periods. In our sample of SB2 candidates, the binaries have an orbital period of less than 155 days as shown in Figure 9, while the orbital periods of binaries from Duchêne & Kraus (2013) are mainly larger than 100 days. Actually, there is a compelling discrepancy between the close binaries and wide binaries both on the statistical distribution and binary formation (Moe & Di Stefano, 2017). Secondly, our mass ratio ranges from 0.6 to 1 while the mass ratio in Duchêne & Kraus (2013) ranges from 0 to 1. The range of mass ratio may lead to different mass ratio distribution (Moe & Di Stefano, 2017), which may indicate the variation of the mechanism for binary formation and evolution.

Figure 10 shows the mass ratio distributions of binaries of spectral type A, F, and G, and we see that type-G binaries tend to have comparable component masses. Such a phenomenon is also shown by the much bigger $\gamma$ we obtained for G-type binaries in Figure 12. Such an excess of ”twin” close binaries provides a valuable constraint on star formation mechanism and is helpful for the explanation of the origin of double white dwarfs (DWDs) with $q\sim 1$, which are potential progenitors of type Ia supernovae (SNe Ia) and also gravitational wave sources for space gravitational wave detectors, e.g. LISA, TianQin, and Taiji.

There are two major channels for the formation of binaries, i.e. fragmentation and accretion. Fragmentation tends to produce binaries with very different component masses, while accretion tends to produce binaries with more equal component masses. G-type binaries are low mass binaries, and the secondaries of the pre-MS binaries (seed binaries) accrete matter from the infalling gaseous material (molecular cloud). The preference growth of the secondary is due to the fact that the infalling matter has bigger specific angular momentum than that of the pre-MS binary, and the pre-MS binary accumulates matter to reach $q\sim 1$. For a low mass binary from such a scenario, the secondary star is younger than the primary star by a few million years. For a thorough discussion of the formation of twin binaries, we refer to Bate & Bonnell (1997); Bate (1998, 2018); Tokovinin (1992, 2000); Tokovinin & Moe (2020); Simon & Obbie (2009); Offner et al. (2016, 2022)

Recent observations of DWDs (Kruckow et al., 2021) show that a large fraction of DWDs has a mass ratio of $q\sim 1$, posing a challenge to the binary evolution theory. Conventional binary population synthesis (BPS) is difficult to explain the excess of DWDs of $q\sim 1$ as most of the DWDs are expected to have mass ratios around $\sim 0.5$ (Han (1998)). To get out of the dilemma, Li et al. (2022, in preparation) proposed a model. In their model, the primary of a binary system evolves to a giant and fills its Roche lobe, leading to a stable mass transfer from the primary to the secondary and producing a binary with a WD and an accreted secondary. The accreted secondary continues to evolve and becomes a giant, filling its Roche lobe and the mass transfer from the giant to the WD results in the formation of a common envelope (CE). The ejection of the CE results in the formation of a DWD with $q\sim 1$. To make the model work, they need to make the mass transfer of a giant binary stabilized. For this purpose, they adopted the adiabatic mass loss model of Ge et al. (2010, 2015, 2020b), which shows the mass transfer of a giant binary (a giant plus an MS star) can be stable if the mass ratio (primary to secondary) is larger than 1. This is contrary to what the widely adopted polytropic model Hjellming & Webbink (1987) has predicted that the mass transfer is dynamically unstable. On the other hand, a large mass ratio (primary to secondary) tends to make the mass transfer of a giant binary more unstable. An excess of binaries of $q\sim 1$ (secondary to primary) will make the mass transfer later on more stable, which is a great help for the formation of DWDs with $q\sim 1$.