# Mass-Ratio Distribution of Hierarchical Triple Systems From the LAMOST-MRS
Survey

Tongyu He  College of Physics Science and Technology, Hebei University,
Baoding 071002, China  Department of Physics, College of Physical Sciences and
Technology, Hebei University, Baoding, China  Key Laboratory of High-precision
Computation and Application of Quantum Field Theory of Hebei Province, Hebei
University, Baoding, China  Research Center for Computational Physics of Hebei
Province, Baoding, 071002, China  Jiangdan Li  Yunnan Observatories, Chinese
Academy of Sciences (CAS), 396 Yangfangwang, Guandu District, Kunming 650216,
P.R. China  Xuefei Chen  Yunnan Observatories, Chinese Academy of Sciences
(CAS), 396 Yangfangwang, Guandu District, Kunming 650216, P.R. China  Rong-jia
Yang  College of Physics Science and Technology, Hebei University, Baoding
071002, China  Department of Physics, College of Physical Sciences and
Technology, Hebei University, Baoding, China  Key Laboratory of High-precision
Computation and Application of Quantum Field Theory of Hebei Province, Hebei
University, Baoding, China  Research Center for Computational Physics of Hebei
Province, Baoding, 071002, China  Lin Xiao  College of Physics Science and
Technology, Hebei University, Baoding 071002, China  Department of Physics,
College of Physical Sciences and Technology, Hebei University, Baoding, China
Key Laboratory of High-precision Computation and Application of Quantum Field
Theory of Hebei Province, Hebei University, Baoding, China  Research Center
for Computational Physics of Hebei Province, Baoding, 071002, China  Zhanwen
Han  Yunnan Observatories, Chinese Academy of Sciences (CAS), 396
Yangfangwang, Guandu District, Kunming 650216, P.R. China  College of Physics
Science and Technology, Hebei University, Baoding 071002, China

###### Abstract

Hierarchical triple-star systems consists of three components organised into
an inner binary ($M_{1}$,$M_{2}$) and a more distant outer tertiary ($M_{3}$)
star. The LAMOST Medium-Resolution Spectroscopic Survey (LAMOST-MRS) has
offered a great sample for the study of triple system populations. We used the
Peak Amplitude Ratio (PAR) method to obtain the mass ratio
($q_{\mathrm{{in}}}$, $q_{\mathrm{{out}}}$) of a triple system from its
normalised spectrum. By calculating Cross-Correlation Function (CCF), we
determined the correlation between the mass ratio $q_{\mathrm{{out}}}$
($M_{3}$/($M_{1}$+$M_{2}$)) and the amplitude ratio
($A_{3}$/($A_{1}$+$A_{2}$)). We derived $q_{\mathrm{{in}}}$ of $0.5-1.0$ and
$q_{\mathrm{{out}}}$ between 0.2 and 0.8. By fitting a power-law function of
the corrected $q_{\mathrm{{in}}}$ distribution, the $\gamma_{\mathrm{{in}}}$
are estimated to be $-0.654\pm 2.915$, $4.304\pm 1.125$ and $11.371\pm 1.309$
for A, F and G type stars. The derived $\gamma_{\mathrm{{in}}}$-values
increase as the mass decrease, indicating that less massive stars are more
likely to have companion stars with similar masses. By fitting a power-law
function of the corrected $q_{\mathrm{{out}}}$ distribution, the
${\gamma_{\mathrm{{out}}}}$ are estimated to be $-2.016\pm 0.172$, $-1.962\pm
0.853$ and $-1.238\pm 0.141$ for G, F and A type stars, respectively. The
${\gamma_{\mathrm{{out}}}}$-values show a trend of growth toward lower primary
star masses.

Triple stars (1714); Radial velocity (1332)

††software: LASP pipeline (Wu et al., 2011; Luo et al., 2015) ULySS (Koleva et
al., 2009) SPECTRUM (Gray & Corbally, 1994) laspec (Zhang et al., 2020, 2021)

##  1 Introduction

Stellar multiplicity is a ubiquitous result of the star formation process,
systems with more than two components are quite common in the Universe
(Duchêne & Kraus, 2013). According to Raghavan et al. (2010a), roughly half of
the stars in our Milky Way galaxy are part of binary, triple, or higher order
systems. Multi-star systems can provide important information on topics
ranging from the formation of a star system, stellar evolution, and galaxy
dynamics. Research on stellar multiplicities greater than two, such as triple-
star systems, is relatively limited compared to binary systems. Understanding
the period distribution, mass ratio and other characteristics of triple-star
systems can provide valuable insights into the stellar formation process and
stellar evolution. Exploring these systems can help explain the dynamics and
interactions within multiple star systems, and contribute to a clearer
understanding of stellar evolution and the formation of complex stellar
systems (Eggleton & Tokovinin, 2008). In the past few decades, there has been
an abundance of interesting research on the origin of triple systems. However,
exploring the dynamics of triple systems requires solving one of the questions
of celestial mechanics: the general three-body problem. It is a triple-star
system consisting of three celestial bodies with arbitrary masses and
spacings. These bodies can be generally considered as point sources of gravity
(Harrington, 1977).The study of triple systems is a fascinating subject that
involves the interaction and dynamics between three celestial bodies. The
gravitational forces between these three bodies can lead to complex and
unpredictable motions, making it a challenging problem to solve. Ascertaining
the motion in such systems is challenging and hence, researchers rely on
observations to discover more triple-star systems. Based on the increasing
sophistication of observational techniques, more and more triple systems are
being discovered. Using the Hipparcos catalogue, Eggleton and Tokovinin found
that at least 10% of bright stars belong to triple-star (or Multiple-Star)
systems (Eggleton & Tokovinin, 2008, 2010). Czavalinga et al. (2023) found 376
newly compact hierarchical triple system candidates in their study. They
considered that the orbital period is the most trustworthy orbital parameter
and that other parameters should be used cautiously. Borkovits et al. (2022)
carried out a detailed study of three triple systems in the northern region of
TESS and determined their mass, period and RV characteristics. Six new compact
triple-star systems were discovered and analyzed by the TESS mission,
according to the description in Rappaport et al. (2022). 201 compact
hierarchical triple (CHT) systems with outer orbital periods of under 1,000
days were identified in Tokovinin (2018). Kounkel et al. (2021) identified 813
triple candidates in the APOGEE spectra. In additation, Tokovinin (2021) and
Toonen et al. (2020, 2022) have reviewed the formation and evolutionary
scenarios of the triple system, respectively.

Triple-stars are referred to hierarchical or non-hierarchical structure. Non-
hierarchical systems refer to stellar systems that do not have a clear
hierarchy or dominance among the triple stars. In these systems, each star is
influenced by the gravitational pull of the other stars, but none of them
dominate the system. These systems are thought dynamically unstable by Nasab
et al. (2023), therefore we only consider hierarchical triple star systems.
For example, during the initial mission of the Kepler satellite, Gaulme et al.
(2022) discovered KIC 7955301. This is a hierarchical triple system with clear
depth and temporal differences in the eclipses. The two white dwarfs are
closely connected to each other, orbiting each other. This binary system is
joined by a distant red giant orbiting the centre of mass of all three of
them.

The hierarchical triple star system consists of three components that are
organised into an inner binary pair and a more distant outer star (Czavalinga
et al., 2023). To facilitate the application of Kepler’s law, we can
approximate the triple as a large orbit (considering the inner binary as a
whole and the third star as an outer binary) over a small orbit. For
hierarchical triple star systems, we need to take account the stability
condition for the existence of the triple systems, which serve as a criterion
for the system’s stability. We have picked the most basic formulas through
various numerical analyses of the outer to inner binary periods ratio
$P_{\mathrm{{out}}}$/$P_{\mathrm{{in}}}$ 111$P_{\mathrm{{out}}}$ is the
external binary period and $P_{\mathrm{{in}}}$ is the internal binary period
of a triple system  (Eggleton & Kiseleva, 1995). Equation (1) and equation (2)
are pretty straightforward and quite robust.

$\left(X_{0}^{\min}\right)^{2/3}=\left(\frac{1}{1+q_{\text{out}}}\right)^{1/3}\frac{1+e_{\text{in}}}{1-e_{\text{out}}}Y_{0}^{\min}\\\
$ (1)

$Y_{0}^{\min}\approx
1+\frac{3.7}{q_{\text{out}}^{-1/3}}+\frac{2.2}{1+q_{\text{out}}^{-1/3}}+\frac{1.4}{q_{\text{in}}^{-1/3}}\frac{q_{\text{out}}^{-1/3}-1}{q_{\text{out}}^{-1/3}+1}\\\
$ (2)

$Y_{0}^{\min}$ is the critical initial ratio of the outer orbit’s periastron
distance to the inner orbit’s apastron distance. The value of $Y_{0}^{\min}$
is linked to the critical initial period ratio $X_{0}^{\min}$ through the
relation shown in equation (1). $e_{\text{in}}$ and $e_{\text{out}}$ are the
eccentricities of the inner and outer binary in a triple system, respectively.
$q_{\text{in}}$ and $q_{\text{out}}$ are the inner and outer mass ratios,
$q_{\mathrm{in}}=M_{2}/M_{1}\leq$ 1 222$M_{1}$ is the primary star and $M_{1}$
mass is greater than $M_{2}$ . We defined $M_{1}$, $M_{2}$ as the inner binary
and $M_{3}$ as the third star. Due to the limitation of our detection method,
our detection efficiency will be very low if the mass of $M_{3}$ is greater
than the the mass of $(M_{1}+M_{2})$, so we restrict
$q_{\mathrm{out}}=M_{3}/(M_{1}+M_{2})\leq 1$
333$q_{\mathrm{out}}=M_{3}/(M_{1}+M_{2})\leq 1$, because our detection
efficiency will be very low if the mass of $M_{3}$ is greater than the the
mass of $(M_{1}+M_{2})$. . Moe & Kratter (2018) had considered a triple system
with $q_{\text{out}}>$ 1, the tertiary star ($M_{3}$) is 1 $\mathrm{M}\odot$
and the inner binary $M_{1}$ = 0.5 $\mathrm{M}\odot$, $M_{2}$ =0.4
$\mathrm{M}\odot$, but we do not consider this case. In section 5 we
demonstrated why this is assumed and demonstrated that the radial velocity
(RV) of the third star in most triple systems likely to be located between the
RVs of the inner binary stars. These principles are most applicable to triple
systems that are nearly co-planar, and our research focuses exclusively on
such triple systems. The upper limit for the outer-to-inner period ratio of a
stable hierarchical triple star system is around five, according to our
estimations thanks to Eggleton’s methods. The smallest outer-to-inner period
ratio of the known compact triple system is 5.4 (Xia et al., 2019). Due to the
dynamical stability issues in the above equation it is unlikely to find a
triple system with a period ratio limit less than 5, so the minimum value of
the outer to inner binary period ratio is taken to be 5 in our paper (Offner
et al., 2022).

We can study some triple systems by combining information on their orbital
parameters, spectra, and photometric measurements. For example, single-line
star spectra can be identified because of their variable radial velocities. We
can determine if an object is a multi-star system or not because the spectral
lines of its distinct components separate at different radial velocities
(RVs). If a target has a triple-line spectrum, it can be considered as a
candidate for a spectroscopic triple-star (Li et al., 2021).

In this paper, we have selected main sequence spectroscopic triple (ST)
candidaties from the LAMOST-MRS spectra given by Li et al. (2021) and
investigate the mass-ratio distribution of triple stellar systems. The
structure of this paper is as follows. In section 2 We shortly present how the
LAMOST-MRS data were selected and classified. In section 3, we introduce the
approaches and frameworks applied to the data processing. The results about
triple-star systems are shown in Section 4. In section 5 we discuss the
completeness of our investigation, comparisons with previous researches, and
the significance of the results, and conclusions are given in Sections 6.

##  2 THE DATA

Figure 1: The color-magnitude graph of 67 spectroscopic triple (ST) candidates
(black dots) determined from LAMOST DR7 data (blue dots). The x-axis shows the
colour difference between $G_{\mathrm{BP}}$ and $G_{\mathrm{RP}}$, and the
y-axis displays the absolute value of the $M_{\mathrm{G}}$. All photometric
data are from $Gaia$ DR3, and our samples were chosen between the two lines
according to their position on the diagram.

LAMOST is a 4-meter quasi-meridian reflector Schmidt telescope, sited in Hebei
province of China. 4,000 fibers are mounted on its 5-degree FoV focal plane,
allowing simultaneous observation of 4,000 targets within a 20-square-degree
field of view. The LAMOST-MRS red and blue arms have respective wavelength
ranges of 6,300 to 6,800Å and 4,950 to 5,350Å. We chose the spectra of the
blue arm to detect triple-star candidates and to measure their radial velocity
(RV), as they have more absorption lines (Li et al., 2021, 2022). Fernandez et
al. (2017) found that spectra with low signal-to-noise ratios posed a
significant challenge for detecting spectroscopic triple (ST) candidates and
Li et al. (2021) chose the blue arm spectra with SNR larger than 10. As a
result, the blue arm spectra in LAMOST with SNR higher than 10 were also
chosen in our sample.

We applied observations from the LAMOST-MRS spectra of Li et al. (2021) to
obtain the mass ratio distribution of spectroscopic triple (ST) candidates. Li
et al. (2021) used a cross-correlation function (CCF) and obtained 132 ST
candidates from LAMOST-MRS DR7. We cross-matched $Gaia$ DR3 with the STs and
obtained their magnitudes in the Bp-band, G-band, and Rp-band (Li et al.,
2022; Gaia Collaboration et al., 2018). Since the diameter of the LAMOST fibre
is about 3 arcsec, we used the 3 arcsec search cone to cross-match the
spectrum of Gaia (Cui et al., 2012). Figure 1 illustrates the colour magnitude
map of the ST candidates (black dots). We have corrected the photometric
results for the effects of interstellar reddening. We calculated the error in
parallax and found that the probability of the error being less than 10% is
86.494%. And all of our samples are in the main-sequence phase and close to
the Sun based on Table 1. We used the three-dimensional map of dust reddening
from Green et al. (2019). The value of the reddening factor is taken from Wang
& Chen (2019), they derived a table of extinction values of different bands.
444All EBVs are smaller than 0.6, so we can consider that reddening will have
little effect on these triples.. And we plotted two lines on the figure after
visually analyzing the stars’ photometry in the picture. The lower line stands
for zero-age MS and the upper line stands for terminal-age MS. We then
selected the ST candidates at the position between these two lines in Figure 1
and got a sample of main sequence stages. Finally we got 67 spectroscopic
triple (ST) candidates for the main sequence stage.

The essential atmosphere parameters of A and FGK-type stars detected by the
LAMOST survey can be automatically determined by LAMOST Stellar Parameter
(LASP) (Wu et al., 2014). The main database (MDB) for the LAMOST data
processing system is centrally versioned and supported by MySQL. Utilizing the
CFI and ULySS algorithms, LASP can perform a two-stage spectral fitting
process that efficiently invert the stellar parameters, thereby extracting the
stellar parameters of the LAMOST-MRS spectra (Koleva et al., 2009). We
utilized the LASP to determine the effective temperature and divide the ST
candidates into various spectral types according to the Morgan-Keenan
classification (Wu et al., 2011; Luo et al., 2015; Li et al., 2022). Since the
spectra of A-type stars are not very sensitive to their temperature, only
temperatures below 8,500 K are given. The error values for estimating the
effective temperature are typically smaller than 100 K (Li et al., 2022; Wang
et al., 2020).

Table 1: For each spectral type, the number of spectroscopic triple (ST)
candidates has been determined

$T_{\mathrm{eff}}$(K) | SPEC | $N_{\mathrm{all}}$ | $N_{\mathrm{ST}}$ |  $N_{\mathrm{ST}}$(500 $\mathrm{pc}$) |  $N_{\mathrm{ST}}$(1000 $\mathrm{pc}$) |  $N_{\mathrm{ST}}$(2000 $\mathrm{pc}$)
---|---|---|---|---|---|---
7200 - 8000 | A | 9669 | 7 | 2 | 5 | 7
6000 - 7200 | F | 58586 | 35 | 6 | 27 | 34
5150 - 6000 | G | 76066 | 20 | 8 | 18 | 20
Total | All | 144321 | 62 | 16 | 50 | 61

Note. — $N_{\mathrm{ST}}$ is the amount of spectroscopic triple (ST)
candidates from the LAMOST-MRS DR7. These ST candidates are sorted into three
segments on the basis of $T_{\mathrm{eff}}$ ranges. $N_{\mathrm{ST}}$,
$N_{\mathrm{ST}}$ and $N_{\mathrm{ST}}$ display the quantity of spectroscopic
triple (ST) candidates within 500 pc, 1000 pc and 2000 pc from the sun,
respectively.

Then we represented the amount of stars for each spectral type and the amount
of spectroscopic triple (ST) candidates for each type according to Table 1.
Since the effective temperature provided by LASP for late-type stars are
unreliable, we excluded stars with temperature below 5,150K. Moreover, we
excluded stars with temperature above 8000K to reduce the probability of
mixing early-type stars in our ST sample. Therefore, we picked 62 ST
candidates of various spectral types. We can see the number of ST candidates
at different distance in Table 1.

##  3 METHOD

In this section, we have taken the LAMOST-MRS data and extract the mass ratios
of the triple systems. First we generated synthetic spectra by creating
simulated triple samples. We then used Merle’s technique to identify
multilinear spectral candidates from the spectral samples (Merle et al.,
2017). The method that smooth the continuous derivatives of the cross-
correlation function (CCF) using a Gaussian kernel. The CCF code is from Zhang
et al. (2021). Following this approach, we can identify spectroscopic triple
(ST) candidates from synthetic spectra, which exhibit three peaks in their
CCF, and we can determine their radial velocities (RVs). Li et al. (2022)
discovered a correlation between the mass ratios and Peak Amplitude Ratio
(PAR) of spectroscopic binaries (SBs). In the referenced work, the PAR is
defined as the ratio between the peak amplitudes of the primary star and
secondary star in the CCF, denoted as $A_{1}$ and $A_{2}$ respectively.
However, in our paper, we defined the PAR as the ratio between the peak
amplitude of the tertiary star ($A_{3}$) and the combined peak amplitudes of
the inner binary ($A_{2}+A_{1}$). We can apply this method and find a similar
relationship between the mass ratios of STs and PAR. By utilizing this
relationship, from the PAR displayed in LAMOST-MRS, we can deduce the mass
ratios of STs and determine their distribution.

###  3.1 The CCF method

The classical definition of a Cross-Correlation Function(CCF) function about
the stellar spectra is:

$\mathrm{CCF}(h)=\int_{-\infty}^{+\infty}f(x)g(x+h)\mathrm{d}x$ (3)

where $f$ is a normalised spectrum, $g$ is a normalised template spectrum and
$h$ is the lag expressed in $\mathrm{km/s}$ (Merle et al., 2017). The
normalized range of the CCF is between -1 and +1, where +1 indicates perfect
correlation and -1 indicates perfect anti-correlation.

In this paper, we followed Li et al. (2022) and utilized the normalized solar
spectrum of Kurucz (1993) as the template spectrum. The RV range was set from
-500 to +500 $\mathrm{km/s}$ with a step size of 1 $\mathrm{km/s}$. Some of
the peaks in the CCF were too small to be detected, so only the spectra with a
peak minimum in the CCF greater than 0.2 were considered. The spectra also
need to be normalized prior to analysis.

###  3.2 Construct spectroscopic triples (STs) synthetic spectra

Li et al. (2022) found a relationship between PAR and the mass ratios of SBs.
Based on this approach, we found that there should also be some relationships
between the mass ratios of spectroscopic triple (ST) candidates and PAR. To
test this hypothesis, we have constructed synthetic spectra of triple systems
with different temperatures, inner binary mass ratios $q_{\mathrm{{in}}}$,
outer binary mass ratios $q_{\mathrm{{out}}}$, radial velocity ($\Delta RV$)
and signal-to-noise ratio (SNR) (The stable hierarchical triple stellar system
can be thought of as a system of two binary systems, which we here approximate
as a large orbit over a small orbit).

To match the LAMOST-MRS spectra, we first created model spectra with a
wavelength range of 4,900 Å to 5,400 Å and lowered the resolution from 51,000
to 7,500 by convolution with the laspec code (Zhang et al., 2020, 2021). Then
we used the standard value of log$g$ = 4.5 ($\mathrm{cm\ s^{-2}}$) for the
surface gravity of the main sequence stars, and the metal abundance [Fe/H] was
set to 0, the solar metal abundance. Specific information can be found in here
Li et al. (2022).

We next built a synthesis spectral lattice of the main-sequence triple system
of the solar metal abundance. In the case of MS triple system, all three
components are MS stars. By utilizing the effective temperature
$T_{\mathrm{eff1}}$ and logarithmic gravitational acceleration log$g$ of the
primary star, we have identified a set of data for the main-sequence stars,
which include age t, mass M, temperature T, luminosity L, and other
parameters. In order to select simulated spectral ages, it is crucial to
control the age range from the zero-age main sequence (t = 0 yr) to the
terminal age of the main sequence (t = 1.8$\times 10^{9}$yr). Additionally, it
is important to ensure that all three components of a triple system share the
same age. These data can be find from the PARSEC stellar evolution program Li
et al. (2022); Bressan et al. (2012).

For a hierarchical triple stellar system, fixing the primary star as $M_{1}$
(a constant), we can give the inner binary mass ratio $q_{\mathrm{{in}}}$ =
$M_{2}$/$M_{1}\leq 1$ 2. We have defined $M_{1}$ as the primary star because
it is the more massive of the inner binary of the triple system.
$q_{\mathrm{{out}}}=M_{3}/(M_{1}+M_{2})\leq 1$, here we set $M_{1}$, $M_{2}$
as the inner binary and $M_{3}$ as the third star 3. When we identify a
primary star as $M_{1}$, it is important to ensure that all three components
are main sequence stars of equal age.

Then we interpolated the spectra of the secondary and the third by moving them
with a radial velocity difference ($\Delta RV$) and combined the spectra of
the secondaries with those of the primary star. We know that identifying
spectroscopic triple (ST) candidates relies heavily on the $\Delta RV$ among
the three components. The $\Delta RV$ between any two components of the three
components is a key factor in the identification of ST candidates. It was
discovered that the lower limit of the $\Delta RV$ is 50 $\mathrm{km/s}$, and
that this limit is heavily determined by the resolution of the observed
spectrum (Li et al., 2022). According to observations made in Li et al.
(2021), the maximum $\Delta RV$ ever detected was 300 $\mathrm{km/s}$.
Therefore, any value outside this range has a zero probability of detecting ST
candidates. To create a more realistic simulation of the spectra, we
introduced Gaussian noise with a targeted signal-to-noise ratio (SNR).

Table 2: Parameters for our synthesis spectra grid

Parameter | Value
---|---
Wavelength ( Å) | 4900 - 5300
Temperature (K) | 4000, 4100, 4200, … , 8000
SNR | 10, 20, 50, 100
$\Delta{RV}$ (km/s) | 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150
$q_{\mathrm{in}}$ | 0.5, 0.6, 0.7, 0.8, 0.9, 1.0
$q_{\mathrm{out}}$ | 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0

Note. — We have build a lattice of synthetic triple stellar spectra with a
resolution of 7,500. Rows 1st, 2nd and 3rd show the wavelength range,
effective temperature and signal-to-noise ratio, respectively. Row 4th shows
the RV difference of any two components, and rows 5th and 6th show the mass
ratios of the inner and outer binaries.

We constructed triple stellar synthetic spectra and hoped to derive the mass
ratios distribution ($q_{\mathrm{{in}}}$ and $q_{\mathrm{{out}}}$) of ST
candidates from LAMOST-MRS data. To simplify the calculation, we set the
gravity of all the components of one triple sysytem to be log$g$ = 4.5. Then,
with 100 K intervals, we chose the $T_{\mathrm{eff}}$ of the A-, F- and
G-primary stars from 8,000 K to 4,000 K, respectively (Li et al., 2022). These
temperature values were chosen because they are characteristic of stars with
spectral types of A, F, and G, and are available in the BPS model of Han et
al. (2002). Corresponding to the temperature of a primary star, the
temperature of these secondary and tertiary stars are also taken at 100 K
intervals in the range 4,000-8,000K, but they are always lower than the
temperature of the primary star. Here we have used the different temperatures
of the three components at the same age in the main sequence phase to
correspond to their mass ratios. For each of the primary star types, we
experimented with mass ratios $q_{\mathrm{{in}}}$ between 0.5 and 1.0, as well
as $q_{\mathrm{{out}}}$ between 0.2 and 1.0 3. Synthetic spectra were then
constructed for triple stars with a given mass ratio and spectral type, using
different $\mathrm{\Delta RV}$ and SNR values (refer to Table 2).

Figure 2: An illustration of a triple system’s CCF. In the bottom panel, we
display a spectrum of a normalized triple-star system (SNR=20). The effective
temperature of three components are $T_{\mathrm{eff1}}$ = 6,400 K,
$T_{\mathrm{eff2}}$= 6,000 K, and $T_{\mathrm{eff3}}$ = 5,600 K. In the lower
part of the figure, we present the generated CCF, A1 = 0.53, V1= - 131km/s, A2
= 0.48, V2= 134 km/s, and A3 = 0.41, V3= 19 km/s. The mass ratios are
$q_{\mathrm{in}}$= 0.96 and $q_{\mathrm{out}}$ = 0.441, respectively.

We computed the cross-correlation function (CCF) using triple stellar
synthetic spectra, Figure 2 displays an example of the CCF calculation. The
top portion of Figure 2 displays a triple stellar synthetic spectrum that has
been normalized, with effective temperatures of $T_{\mathrm{eff1}}$ = 6,400 K,
$T_{\mathrm{eff2}}$ = 6,000 K and $T_{\mathrm{eff3}}$ = 5,600 K. In the lower
part of Figure 2 is the generated CCF, $A_{1}$ = 0.53, $V_{1}$= - 131
$\mathrm{km/s}$, $A_{2}$ = 0.48, $V_{2}$= 134 $\mathrm{km/s}$, and $A_{3}$ =
0.41, $V_{3}$= 19 $\mathrm{km/s}$. $A_{3}$ is the peak amplitude of the
tertiary star, $A_{1}$ and $A_{2}$ are the peak amplitudes of the inner
binary, respectively. The mass ratios are $q_{\mathrm{{in}}}$ = 0.96 and
$q_{\mathrm{{out}}}$ = 0.441. The wavelength range considered is between 4,900
Å and 5,300 Å, and the spectral resolution is 7,500. Three peaks are
observable, with the left peak representing the amplitude ($A_{1}$) of the
primary star $M_{1}$, the middle peak indicating the amplitude ($A_{3}$) of
the third star $M_{3}$, and the right peak representing the amplitude
($A_{2}$) of the secondary star $M_{2}$. $M_{1}$ is the primary star because
it is the more massive of the inner binary ($M_{1}$,$M_{2}$) of the triple
system. We can derive $A_{1}$ is larger than $A_{2}$, $A_{3}$ is the lowset of
the three components.

In Figure 3, when the 1st derivative crosses zero in the falling phase or the
3rd derivative crosses zero in the rising phase, we can determine the number
of the three RV components and their values from the triple system (Merle et
al., 2017). We noticed that there is more noise in the higher order
derivatives of the CCF, minor bumps can occasionally be distinguished as peaks
in the CCF. It was necessary to filter the RV range and use the Gaussian
filter function gaussian filter1d in Python’s scipy.ndimage package to smooth
the derivatives (Virtanen et al., 2020). To select the RV range, two
thresholds were employed. The percentile of the CCF is used as the first
threshold, while the smoothed second order derivative is used as the second.
At the top of Figure 3, we selected 75th percentile of the maximum peak among
these three peaks as the threshold. A red line is drawn below this threshold
to indicate that peaks below it will not be selected. Similarly, in the third
panel of Figure 3, we chose 6th percentile of the maximum peaks among these
three peaks as the threshold. A red line is drawn above this threshold to
indicate that peaks above it will not be selected. These threshold values were
derived from Li et al. (2021).

Figure 3: CCF and derivatives of one spectroscopic triple (ST) candidate
(SNR=20). All the selected range of CCF and smoothed derivatives are shown as
black solid lines, while the original CCF and derivatives are shown as gray
dashed lines. Each candidate’s thresholds are shown as red horizontal lines,
their RVs are shown as black vertical lines.

###  3.3 Relationship between mass ratios and amplitude ratios

After constructed synthetic spectra of hierarchical triple systems, we can
find the relationship between mass ratios and amplitude ratios. By computing
the CCF for all synthetic spectra built for triple systems, we have
established the relationship between
$A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{\mathrm{2}})$ and $q_{\mathrm{out}}$, as
well as the relationship between $A_{\mathrm{2}}/A_{\mathrm{1}}$ and
$q_{\mathrm{in}}$ in Figure 4 555$A_{\mathrm{1}}$, $A_{\mathrm{2}}$ and
$A_{\mathrm{3}}$ are the primary, secondary and tertiary peak amplitudes . For
convenience, we have divided these spectroscopic triple (ST) candidates into
different types of main sequence stars of G,F,A according to temperatures of
5600K, 6400K and 8000K. The error bars are the dispersion due to various
$\Delta RV$s used. In Figure 4 we can see the $A_{\mathrm{2}}/A_{\mathrm{1}}$
are from 0.37 to 1.0 and the $q_{\mathrm{{in}}}$ are from 0.5 to 1.0, the
$A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{\mathrm{2}})$ are from 0.22 to 0.95 and the
$q_{\mathrm{{out}}}$ are from 0.2 to 1.0. When the
$A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{\mathrm{2}})$ is 0.5, the
$q_{\mathrm{out}}$ is also 0.5. We have fitted the relationship between
$A_{\mathrm{2}}/A_{\mathrm{1}}$ and $q_{\mathrm{in}}$, as well as the
relationship between $A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{\mathrm{2}})$ and
$q_{\mathrm{out}}$ using some functions, as shown in equation (4-9). Using
these formulas, we can translate the observed amplitude ratios of the
spectroscopic triple candidates into their mass ratios.

Figure 4: Relationship between $A_{\mathrm{2}}/A_{\mathrm{1}}$ and
$q_{\mathrm{in}}$, as well as the relationship between
$A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{\mathrm{2}})$ and $q_{\mathrm{out}}$. The
blue dashed line indicates the result at 6,400 K, the red solid line indicates
the relationship at 5,600 K, and the black dashed line indicates the result at
8,000 K.

$q_{\mathrm{in}}=0.348\pm 0.044\times(\frac{A_{2}}{A_{1}})^{2}-0.051\pm
0.064\times\frac{A_{2}}{A_{1}}+0.702\pm
0.021\quad(5200\mathrm{K}\leq\mathrm{T}_{\mathrm{eff}}\leq 6000\mathrm{K})\\\
$ (4) $q_{\mathrm{in}}=0.495\pm 0.054\times(\frac{A_{2}}{A_{1}})^{2}-0.061\pm
0.082\times\frac{A_{2}}{A_{1}}+0.566\pm
0.028\quad(6000\mathrm{K}\leq\mathrm{T}_{\mathrm{eff}}\leq 7200\mathrm{K})\\\
$ (5) $q_{\mathrm{in}}=1.256\pm 0.046\times(\frac{A_{2}}{A_{1}})^{2}-0.894\pm
0.068\times\frac{A_{2}}{A_{1}}+0.638\pm
0.023\quad(7200\mathrm{K}\leq\mathrm{T}_{\mathrm{eff}}\leq 8000\mathrm{K})\\\
$ (6) $q_{\mathrm{out}}=0.650\pm 0.011\times\frac{A_{3}}{A_{1}+A_{2}}+0.172\pm
0.006\quad(5200\mathrm{K}\leq\mathrm{T}_{\mathrm{eff}}\leq 6000\mathrm{K})\\\
$ (7) $q_{\mathrm{out}}=0.753\pm 0.023\times\frac{A_{3}}{A_{1}+A_{2}}+0.123\pm
0.012\quad(6000\mathrm{K}\leq\mathrm{T}_{\mathrm{eff}}\leq 7200\mathrm{K})\\\
$ (8) $q_{\mathrm{out}}=1.214\pm 0.018\times\frac{A_{3}}{A_{1}+A_{2}}-0.109\pm
0.008\quad(7200\mathrm{K}\leq\mathrm{T}_{\mathrm{eff}}\leq 8000\mathrm{K})\\\
$ (9)

###  3.4 The spectroscopic triples (STs) detection efficiency

We investigated the spectroscopic triple (ST) detection efficiency to better
comprehend how the CCF approach operates and how to rectify the bias in the
PAR-derived $q_{\mathrm{out}}$ distribution. And the detection efficiency is
apparent rely on $q_{\mathrm{out}}$ and $q_{\mathrm{in}}$. A triple system
with $q_{\mathrm{in}}$ $>$ 0.5 and $q_{\mathrm{out}}$ from 0.3 to 0.8 can be
more easily detected.

Figure 5: Detection efficiency versus mass ratio $q_{\mathrm{out}}$ and
$q_{\mathrm{in}}$ of spectroscopic triples (STs). The color scale on the right
side represents the detection efficiency.

To test the efficiency of the ST detection, we have created a grid of
synthetic triplet spectra containing various $q_{\mathrm{out}}$ and
$q_{\mathrm{in}}$ values. The primaries’ effective temperature
$T_{\mathrm{eff1}}$ is assumed to range from 4,000 K to 8,000 K with a step
size of 100 K. For a triple system, e.g. Figure 2, there are three radial
velocity differences ($\Delta V$s), they are $\Delta V_{1-2}$, $\Delta
V_{1-3}$ and $\Delta V_{3-2}$. According to the data we obtained from LAMOST
in Figure 8, the (left-right) maximum value does not exceed 250
$\mathrm{km/s}$, so we can restrict $\Delta V_{1-2}\leq$ 250 $\mathrm{km/s}$.
In addition, we defined $\Delta V_{\text{min}}$ as the smaller value between
$\Delta V_{1-3}$ (left-middle) and $\Delta V_{3-2}$ (middle-right), which
takes the range of 50 to 125 $\mathrm{km/s}$ with a step size of 5
$\mathrm{km/s}$. And we defined $\Delta V_{\text{max}}$ as the larger value
between $\Delta V_{1-3}$ and $\Delta V_{3-2}$, which is larger than $\Delta
V_{\text{min}}$. With guaranteed $\Delta V_{1-2}$ = $\Delta V_{1-3}$ \+
$\Delta V_{3-2}\leq$ 250 $\mathrm{km/s}$, $\Delta V_{\text{max}}$ is also
taken with a step size of 5 $\mathrm{km/s}$.

For a triple system, with some parameters such as $T_{\mathrm{eff1}}$, $\Delta
V_{1-2}$, $\Delta V_{2-3}$, $\Delta V_{1-3}$, $q_{\mathrm{in}}$ and
$q_{\mathrm{out}}$, we can generate 20 spectra with SNR=20, which is a
classical value for LAMOST-MRS data. We have examined these spectra using the
CCF approach to determine if we can recognize them as STs. The percentage of
synthetic spectra that are recognized as STs using the CCF approach is how we
determine the detection efficiency. Figure 5 illustrates the detection
efficiency of STs for different $q_{\mathrm{in}}$ and $q_{\mathrm{out}}$. We
can find the efficiency is large if $q_{\mathrm{in}}$ is close to 1 and
$q_{\mathrm{out}}$ is from 0.3 to 0.8. The low detection efficiency when
$q_{\mathrm{out}}$ $<$ 0.3 is due to that the third’s contribution to the
triple-star synthetic spectra becomes unimportant when $q_{\mathrm{out}}$
declines. The low detection efficiency when $q_{\mathrm{out}}$ $>$ 0.8 is due
to that the third’s contribution to the triple-star synthetic spectra becomes
so large that the contribution of the other two components becomes
unimportant.

##  4 RESULTS

In section 3.3 we established the relationships between mass ratios
($q_{\mathrm{out}}$,$q_{\mathrm{in}}$) and amplitude ratios
($A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{\mathrm{2}}$),$A_{\mathrm{2}}/A_{\mathrm{1}}$)
through several functions. These relationships allowed us to extract
information about one component in triple system from LAMOST-MRS spectra.
Initially, we have employed the CCFs of the triple stellar spectra to identify
some spectroscopic triple (ST) candidates and obtain the amplitude ratios
$A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{\mathrm{2}})$ and
$A_{\mathrm{2}}/A_{\mathrm{1}}$ for the three peaks of the ST candidates
(refer to Figure 2). Subsequently, combining the amplitude ratio
$A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{\mathrm{2}})$ and
$A_{\mathrm{2}}/A_{\mathrm{1}}$ with these equations derived previously, we
estimated the mass ratio $q_{\text{out}}$ =
$M_{\mathrm{3}}/(M_{\mathrm{1}}+M_{\mathrm{2}})$ and
$q_{\text{in}}$=$M_{\mathrm{2}}/M_{\mathrm{1}}$.

Figure 6: The cumulative $q_{\mathrm{in}}$ and $q_{\mathrm{out}}$ distribution
of triple systems of different spectral types. The $q_{\mathrm{in}}$
distributions are shown in (a,b), the bias-corrected $q_{\mathrm{in}}$
distributions and fits are shown in (b). The $q_{\mathrm{out}}$ distributions
are shown in (c,d), and the bias-corrected $q_{\mathrm{out}}$ distributions
and fits are shown in (d).

Based on our assumption that $M_{\mathrm{1}}$ and $M_{\mathrm{2}}$ represent
the inner binary, $M_{\mathrm{3}}$ is the third star, we can determine some
properties of the triple system. Based on equation (4-9), we have obtained the
relation between $A_{\mathrm{2}}/A_{\mathrm{1}}$ and $q_{\mathrm{in}}$(in
triple system), and have further obtained the relation between
$q_{\mathrm{out}}$ and the $A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{2})$ of the
triple system. We can also convert the amplitude ratio
$A_{\mathrm{2}}/A_{\mathrm{1}}$ into $q_{\mathrm{{in}}}$ =
$M_{\mathrm{2}}/M_{\mathrm{1}}$ for inner binary and convert the amplitude
ratio $A_{\mathrm{3}}/(A_{\mathrm{1}}+A_{2})$ into $q_{\mathrm{{out}}}$ =
$M_{\mathrm{3}}/(M_{\mathrm{1}}+M_{2})$. Finally we obtained
$q_{\mathrm{{in}}}$=$M_{\mathrm{2}}/M_{\mathrm{1}}$, the mass ratio
$q_{\mathrm{{out}}}$=$M_{\mathrm{3}}/(M_{\mathrm{1}}+M_{\mathrm{2}})$, as well
as their respective RVs for the triple systems.

Figure 7: The distribution of inner binaries (red squares, black circles, and
blue triangles) in triple systems of types A, F, and G are shown in (a,b), the
bias-corrected $q_{\mathrm{in}}$ distributions and fits are shown in (b). And
(c,d) show the distribution of outer binaries in triple systems in our sample,
(d) shows the bias-corrected $q_{\mathrm{out}}$ distributions and fits. In
order to clearly show the error bars, we have shifted the mass ratio
distribution of A-type stars to the left by 0.1 and the mass ratio
distribution of G-type stars to the right by 0.1.

We can derive the cumulative $q_{\mathrm{{in}}}$ and $q_{\mathrm{{out}}}$
distribution of their inner and outer binaries in Figure 6 by investigating
the spectra of the 62 spectroscopic triple (ST) candidates. In (a) of Figure
6, we have received the $q_{\mathrm{{in}}}$ distribution around the inner
binaries. (b) depicts the bias-corrected $q_{\mathrm{{in}}}$ distributions
using the efficiency of Figure 5. Each triple candidate corresponds to a set
of $q_{\mathrm{{in}}}$ and $q_{\mathrm{{out}}}$ values, which can be found in
Figure 5, allowing us to determine the detection efficiency for each
candidate. Then, dividing by the detection efficiency can give us the total
number of triples, which is the corrected triple count. Based on their
different types, they are then categorized as A, F, and G type stars, and
finally, their mass ratio distributions are obtained. The bias-corrected
$q_{\mathrm{{in}}}$ and $q_{\mathrm{{out}}}$ distribution, we hypothesized,
are all follow the power-law function of $q^{\gamma}$. We then fitted a power-
law function to the distribution of corrected $q_{\mathrm{{in}}}$
($\mathrm{dN}/\text{d}q_{\mathrm{{in}}}\propto
q_{\mathrm{{in}}}^{\gamma_{\mathrm{{in}}}}$), the $\gamma_{\mathrm{{in}}}$ of
$q_{\mathrm{{in}}}$ distributions of ST candidate systems are estimated to be
$-0.654\pm 2.915$, $4.304\pm 1.125$ and $11.371\pm 1.309$ for A, F and G type
stars, respectively666Given the limited sample size of only 7 A-type stars and
the presence of excessively high $\mathrm{vsini}$ values, caution must be
exercised when interpreting the results of the mass ratio distribution for
A-type stars(Section 5.4). We found that there is a large uncertainty in the
$\gamma_{\mathrm{{in}}}$ values for A-type stars. This could be attributed to
a smaller sample size, which led to a rough estimation of the mass ratio
distribution. The range of $q_{\mathrm{{in}}}$ is from 0.5 to 1.0, and the
range of $q_{\mathrm{{out}}}$ is from 0.2 to 0.8. The power law exponent
$\gamma_{\mathrm{{in}}}$ increases as the mass decreases, indicating that less
massive stars are more probably to have partner stars with similar masses.
Using the efficiency of Figure 5, we also received the $q_{\mathrm{{out}}}$
distribution about the triple-stars in (b,d) of Figure 6. The (d) depicts the
bias-corrected $q_{\mathrm{{out}}}$ triple distributions. We fitted a power-
law distribution to the distribution of the corrected $q_{\mathrm{{out}}}$
($\mathrm{dN}/\text{d}q_{\mathrm{{out}}}\propto
q_{\mathrm{{out}}}^{\gamma_{\mathrm{{out}}}}$), the $\gamma_{\mathrm{{out}}}$
of $q_{\mathrm{{out}}}$ distributions of ST candidate systems are estimated to
be $-2.016\pm 0.172$, $-1.962\pm 0.853$ and $-1.238\pm 0.141$ for G, F and A
type stars6. The $\gamma_{\mathrm{{out}}}$-values show a tendency to increase
towards lower main star masses.

The distribution of the inner and outer binaries’ $q_{\mathrm{{in}}}$ and
$q_{\mathrm{{out}}}$ are shown in Figure 7. The vertical axis is the density,
it is $dn/dq$, $n$ is the number of the triples, $q$ is the bin of x-axis,
bin=0.1. In (a,b), we displayed the distribution of inner binaries (red
squares, black circles and blue triangles) in triple systems of types A, F and
G, the bias-corrected $q_{\mathrm{{in}}}$ distributions and the fits are shown
in (b). The (c,d) show the distribution of outer binaries in triple systems in
our sample, the bias-corrected $q_{\mathrm{{out}}}$ distributions and the fits
are shown in (d) (For detailed information, please see Figure 6). The range of
$q_{\mathrm{{in}}}$, is from 0.5 to 1.0, and the range of
$q_{\mathrm{{out}}}$, is from 0.2 to 0.8. In Figure 7, (a,b) display that for
triple systems the inner binaries tend to be more twin distributions with
similar masses, and we can see that the overall trend of the number of ST
candidates increase with increasing $q_{\mathrm{{in}}}$, and this trend is
most obvious in G-type primary stars. While (c,d) show that the number of the
third star decreases as $q_{\mathrm{{out}}}$ increases. However, our method
are limited by the mass ratio $q_{\mathrm{{in}}}$ and $q_{\mathrm{{out}}}$.
Triple systems can only be efficiently detected in the range
$q_{\mathrm{{in}}}$ of $0.5-1.0$ and $q_{\mathrm{{out}}}$ of $0.2-0.8$.

Figure 8: Distribution of the number of $\Delta V$ between any two components
in the spectroscopic triple (ST) spectra.

As a final addition, in Figure 8, we statistically obtained images of the
$\Delta V$ distribution between the any two components of one triple system in
the sample. The black dashed line indicates the distance between the left peak
and the middle peak ($\Delta V_{3-2}$), the red dashed line indicates the
distance between the right peak and the middle peak($\Delta V_{1-2}$), and the
blue dashed line indicates the distance between the left peak and the right
peak ($\Delta V_{1-2}$) (refer to 3.4 about the definition of different
$\Delta V$). In Figure 8, we found the detected $\Delta V$ limit is about 50
$\mathrm{km/s}$ for ST candidates, which meets the resolution of the LAMOST-
MRS spectra. The largest $\Delta V$ are about 250 $\mathrm{km/s}$ for ST
candidates. These observational data can confirm the validity of our selection
criteria for radial velocity in section 3.2, which specifies a minimum of 50
$\mathrm{km/s}$ and a maximum of 300 $\mathrm{km/s}$. Most of our
observational results fall within these ranges. Additionally, the large
$\Delta V$ values represented by the blue line in the graph suggest that inner
binary stars in triple systems are all discriminable in our sample using the
CCF method (Almost all of the $\Delta Vs$ are large enough). (refer to section
5.1 for more details).

Figure 9: Proportion of observed spectroscopic triple (ST) candidates
displayed against orbital period for different
$P_{\mathrm{out}}$/$P_{\mathrm{in}}$ values for a given number of
observational epochs. X-axis are the inner binary periods of the triple
systems. For 1, 2, and 4 observational epochs, thick solid, dashed, and dotted
lines are shown; for 8, 16, and 32 observational epochs, thin solid, dashed,
and dotted lines are shown.

##  5 DISCUSSION

###  5.1 The dependence of spectroscopic triple (ST) candidates detection on
orbital periods

Figure 10: Number of the detected spectroscopic triple (ST) candidates versus
number of observations.

The orbital period and the number of spectroscopic triple (ST) candidate
observed epochs also affect the detection results, and this bias has to be
investigated. In Figure 9, we displayed the detection percentage of STs versus
orbital period. We assumed that the masses of the primary and the other two
components in the triple system are 1$\mathrm{M}\odot$, 1$\mathrm{M}\odot$,
and 1$\mathrm{M}\odot$ 777We can choose different qualities, which does not
affect the result.. The observational epochs are uniformly distributed in
orbital phase between 0 and 2 $\pi$, the orbital eccentricity is 0. The
inclination range is from 0 to 2$\pi$ and is evenly distributed at solid
angles. For a triple system, applying the conditional limit for the existence
of a stable hierarchical triple stellar system
$P_{\mathrm{out}}$/$P_{\mathrm{in}}$ = 5 (Eggleton & Kiseleva, 1995), we can
roughly calculate the $\Delta V$ between the individual stars of the triple
systems by means of Kepler’s law 1. We gave orbital period $P_{\mathrm{in}}$
and a number of observational epochs, then we did this calculation for 10,000
triple systems. Li et al. (2022) defined the minimal $\Delta RV$ about
binaries is 50 $\mathrm{km/s}$ because the minimal detected $\Delta RV$ is
around 50 $\mathrm{km/s}$ from LAMOST-MRS, for triple systems we estimate the
minimal $\Delta V_{1-2}$ is 100 $\mathrm{km/s}$. The detection efficiency is
the probability of finding the triple systems with CCF from 10000 simulated
samples, corresponding to the vertical coordinate of Figure 9. We defined the
detection fraction ratio as the number of (observable) triple systems (the the
distance of $\Delta V_{1-3}$ and $\Delta V_{3-2}$ are all greater than 50
$\mathrm{km/s}$ and less than 300 $\mathrm{km/s}$ , $\Delta V_{1-2}$ is
greater than 100 $\mathrm{km/s}$ and less than 300 $\mathrm{km/s}$ ) divided
by the number of total triple systems.

From Figure 9 we can see that we are able to detect STs orbiting with inner
period $P_{\mathrm{in}}$ of less than 20 days. From the figure we can see that
the maximum values of detection efficiency for different number of
observational epochs are all in the range of $P_{\mathrm{in}}$ less than 10
days. In addition the detection efficiency increases slightly as the
$P_{\mathrm{out}}$/$P_{\mathrm{in}}$ becomes larger. In Figure 9, under the
condition where the inner binary period in a triple system is less than 20
days, we can estimate the detection probabilities of the triple systems types
A, F, and G to be 7.9 ‰, 5.0 ‰and 2.6 ‰, respectively 888 We estimated the
probability of A, F, G-type stars in triples at different outer-to-inner
period ratios ($P_{\mathrm{out}}$/$P_{\mathrm{in}}$) and found that the
results did not differ much, we finally chose the result with
$P_{\mathrm{out}}$/$P_{\mathrm{in}}$ = 50.. However there are some limitations
about our estimates. With only seven A-type stars in our sample and their
rapid axial rotation, the detection efficiency in identifying A-type star
triple systems may be significantly lower than our previous predictions, with
a reduction by a factor of approximately four. For more details, see 5.4. The
majority number of observations in Figure 10, are less than 5, indicating that
the triple systems we are currently able to detect represent (13.95%) is a
very small fraction of the triple systems.

###  5.2  Probability that the RV of the third star is in the middle of the
inner binary (a triple stellar system)

Figure 11: The probability that the RV of the third star ($M_{3}$) is in the
middle of the inner binary ($M_{1}$, $M_{2}$) for hierarchical triple systems.
The black line represents the value of the inner to outer period ratio
($P_{\mathrm{out}}$/$P_{\mathrm{in}}$) is greater than 50.

Our previous argument was based on the assumption that $V_{3}$ is located
between $V_{1}$ and $V_{2}$ (the inner binary) for a stable hierarchical
triple system with $M_{1}$ and $M_{2}$ as the inner binary and $M_{3}$ as the
tertiary star. We can prove that $V_{3}$ is mostly located between $V_{1}$ and
$V_{2}$. We assumed that the masses of the primary star, secondary star, and
third star are all 1$\mathrm{M}\odot$, which is consistent with section 5.1.
We randomly selected 10000 triple systems, we defined the detection fraction
as the number of cases where $V_{3}$ is in the middle of $V_{1}$ and $V_{2}$
(and the distance of $\Delta V_{1-3}$ and $\Delta V_{3-2}$ are all greater
than 50 $\mathrm{km/s}$ and less than 300 $\mathrm{km/s}$ , $\Delta V_{1-2}$
is greater than 100 $\mathrm{km/s}$ and less than 300 $\mathrm{km/s}$ )
divided by the total number of cases where the distance between any two of
$\Delta V_{1-2}$, $\Delta V_{3-2}$, and $\Delta V_{1-3}$ is greater than 50
$\mathrm{km/s}$ and less than 300 $\mathrm{km/s}$ (refer to section 5.1 for
more details).

Figure 11 shows the probability of $V_{3}$ being located between $V_{1}$ and
$V_{2}$ for different values of the outer-to-inner period ratio
$P_{\mathrm{out}}$/$P_{\mathrm{in}}$ in triple systems. We found that as the
value of $P_{\mathrm{out}}$/$P_{\mathrm{in}}$ increased, the probability that
$V_{3}$ is located between $V_{1}$ and $V_{2}$ gradually approached 1. The
black line represents the minimum value of the outer-to-inner period ratio
($P_{\mathrm{out}}$/$P_{\mathrm{in}}$) obtained from our simulations with the
62 ST candidates.

First, we utilized Kepler’s laws to establish a relationship between the
outer-to-inner period ratio ($P_{\mathrm{out}}$/$P_{\mathrm{in}}$) and the
three radial velocities (RVs). To simplify the triple system problem, we
treated it as two binary systems. Let $m$ represent the total mass of the
inner binary, $M$ denote the total mass of all three components, $t$ represent
the period of the inner binary, and $T$ indicate the outer orbital period.
Additionally, $r_{2}$ signifies the distance of the inner binary, while
$r_{1}$ is the distance of the tertiary star from the center of the inner
binary (approximation). $\phi$ is the phase and $i$ is the inclination. The
formulas used were as follows:

$M=m_{1}+m_{2}+m_{3},\\\ m=m_{1}+m_{2}\\\ $ (10)
$r_{1}=\sqrt[3]{\frac{\mathrm{G}MT^{2}}{4\pi^{2}}},r_{2}=\sqrt[3]{\frac{\mathrm{G}mt^{2}}{4\pi^{2}}}\\\
$ (11) $V_{1}=\frac{2\pi}{t}r_{2}\frac{m_{2}}{m_{1}+m_{2}}\mathrm{sin\phi
sin}i,V_{2}=\frac{2\pi}{t}r_{2}\frac{m_{1}}{m_{1}+m_{2}}\mathrm{sin\phi
sin}i\\\ $ (12)
$V_{3}=\frac{2\pi}{T}r_{1}\frac{m_{1}+m_{2}}{m_{1}+m_{2}+m_{3}}\mathrm{sin\phi
sin}i\\\ $ (13)

We can finally derive:

$P_{\mathrm{out}}/P_{\mathrm{in}}=\frac{T}{t}\propto\frac{\Delta
V_{1-2}^{3}}{(V_{3}-\frac{V_{1}+V_{2}}{2})^{3}}\\\ $ (14)

In section 3.2, we already obtain the radial velocities of each component.
Bringing the different $V$s into the formulas, we can roughly estimate to get
the ratio $P_{\mathrm{out}}$/$P_{\mathrm{in}}$ of these 62 triple candidates.
We put a black line in Figure 11, which means most of the triples in our
sample have $P_{\mathrm{out}}$/$P_{\mathrm{in}}$ greater than 50. And
corresponding to Figure 11, we can find that with
$P_{\mathrm{out}}$/$P_{\mathrm{in}}$ greater than 50, the tertiary star’s
$V_{3}$ has a 95 % probability of being in the middle of $V_{1}$ and $V_{2}$.
This suggests that our previous hypothesis is reasonable. In recent years,
Czavalinga et al. (2023) summarized the inner and outer periods of some
discovered triple systems and found that most of the inner binary periods are
less than 10 days while most of the outer periods are larger than 300 days,
which further confirms the validity of our hypothesis for most triple systems.

###  5.3 Multiple observations

Figure 12: The coordinates of the triple candidate depicted in the upper part
are RA=134.1496 and DEC=12.59047, while the coordinates of the triple
candidate illustrated in the lower part are RA=292.718 and DEC=41.922. The
x-axis is the $\mathrm{RV}_{1}$ and the y-axis is the $\mathrm{RV}_{2}$. The
black line represents the fitted equation, with the parameters listed in the
title. The intersection points of the two black dashed lines represent the
systemic velocities $\gamma_{\mathrm{dyn}}$.

In this section, we discussed the results obtained by analyzing multiple
observations of several triple candidates that had a high number of
observations in Figure 10. We specifically counted these multiple observations
in order to extract information about their mass ratios from the changes
observed in their radial velocities (RVs).

Figure 12 illustrates two examples where we obtained multiple measurements of
$\mathrm{RV}_{1}$ and $\mathrm{RV}_{2}$. Utilizing this equation:
$\mathrm{RV}_{1}=\gamma_{\mathrm{dyn}}(1+q_{\mathrm{in}})-q_{\mathrm{in}}\mathrm{RV}_{2}$,
we can determine the mass ratio ($q_{\mathrm{in}}$) and systemic velocity
($\gamma_{\mathrm{dyn}}$) of the inner binary (Wilson, 1941; Kovalev et al.,
2022).The parameters obtained from the linear fitting in the upper part of
Figure 12 are as follows: $q_{\mathrm{in}}=0.902\pm
0.030,\gamma_{\mathrm{dyn}}=39.586\pm 0.594$$\mathrm{km/s}$, and the results
obtained from the CCF is $q_{\mathrm{in}}=0.881\pm 0.138$ 999We obtained the
results of the CCF analysis based on one selected spectrum.. The parameters
obtained from the lower part are as follows: $q_{\mathrm{in}}=0.891\pm
0.044,\gamma_{\mathrm{dyn}}=-0.822\pm 1.623$$\mathrm{km/s}$, and the results
obtained from the CCF is $q_{\mathrm{in}}=0.872\pm 0.136$ 9. The fact that
these results fall within the range of error indicates that our method is
valid and reliable, and our method does not rely on multiple observations.
101010The mass ratio results we obtained are slightly smaller, which may be
due to the fact that as the signal-to-noise ratio increases, the peak obtained
from the CCF becomes higher.

###  5.4 Comparison with previous Work

In Li et al. (2022)’s work, they came up with a PAR method to calculate mass
ratios from PARs of CCFs between the spectra of binaries and the solar
spectrum. Type-G binaries are more likely to be twins, according to their
analysis of the mass ratio distribution. For orbital periods less than 155 d
and mass ratios greater than 0.6, the binary fractions for A, F and G stars
are $7.6\pm 0.5\%$, $4.9\pm 0.2\%$ and $3.7\pm 0.1\%$, respectively.

In our work, we estimated the fractions of the hierarchical triple systems A,
F, and G to be 7.9 ‰, 5.0 ‰ and 2.6 ‰, respectively. Note that the derived
mass ratio distribution and the proportion of triple systems mentioned here
specifically apply to triple systems with an inner period
($P_{\mathrm{{in}}}$) shorter than 20 days, an inner mass ratio
($q_{\mathrm{{in}}}$) higher than 0.5, and an outer mass ratio
($q_{\mathrm{{out}}}$) higher than 0.2. The rapid axial rotation of the A-type
stars can indeed have an impact on spectral analysis. Royer et al. (2007) used
the whole profile of Fourier transform of spectral lines to derive the
$v\mathrm{sin}i$, and the corresponding $v\mathrm{sin}i$ values for different
classes of stars are extensively presented in their work. Our sample of A-type
stars all have temperatures below 8,000K and we can find that the median value
of $v\mathrm{sin}i$ is 125$\mathrm{km/s}$ in Royer et al. (2007)’s work.

We constructed a batch of triple spectra with $v\mathrm{sin}i$ values of 125
$\mathrm{km/s}$ (the temperature of A-type stars to be 8,000K, log$g$ = 4.5,
metal abundance [Fe/H] set to 0). And we made a rough estimate of the mass
ratio distribution ($q_{\mathrm{{in}}}$ and $q_{\mathrm{{out}}}$). The results
are very similar to those in Figure 4 for the temperature of 8,000 K. We found
that the rotation of A-type stars has little effect on the mass ratio
distribution of $q_{\mathrm{{in}}}$ and $q_{\mathrm{{out}}}$. This is because
we are primarily interested in the amplitude ratios
$\mathrm{A}_{2}/\mathrm{A}_{1}$ and
$\mathrm{A}_{3}/(\mathrm{A}_{1}+\mathrm{A}_{2})$ within the triple systems,
rather than the individual amplitudes of each component. And in our sample of
A-type stars, almost all the $\Delta RV$ are less than 125 $\mathrm{km/s}$.
The lower the rotational velocity of A-type stars, the higher our detection
efficiency. Furthermore, we utilized Kepler’s law to roughly estimate the
$v\mathrm{sin}i$ of A-type binary stars with periods less than ten days and
found that the most values are lower than 125 $\mathrm{km/s}$. This indicates
that the $v\mathrm{sin}i$ values of A-type stars in multiple star systems are
generally lower than those of single A-type stars.

$\log{P}$ is the simplest and most commonly used form, assuming $\log{P}$ can
conveniently describe and analyze the multiplicity of stars. Raghavan et al.
(2010b) provided observational evidence, and later Duchê ne & Kraus (2013) and
Moe & Kratter (2018) also used the $\log{P}$ assumptions. We compared the
inferences we estimated with Li et al. (2022)’s binary fraction, and we found
that our estimated proportion of triples is about 1/5 of the proportion of
their binaries. The $P_{\mathrm{in}}$ in our triples are all less than 20
days, which indicates that inner binaries are relatively close together and
mass transfer are likely to occur in them. Based on Toonen et al. (2020), the
triple system’s inner binary’s initially most massive star fills its Roche
lobe and begins transferring mass to its companion. In their sample, 63–74% of
the triple systems experienced this, while some triples (2–4.5%) experienced
dynamic instability during their evolution. Furthermore, according to Kummer
et al. (2023), mass transfer occurs in the inner binary of the vast majority
of massive triple stars (65–77%), which is different from isolated binary
development, where mass transfer occurs less frequently. Our study of
$q_{\mathrm{out}}$ suggests that the tertiary stars in triple systems have
little to do with inner binaries. In addition, because the inner binaries of
triples tend to be more compact compared to isolated binaries, the probability
of mass transfer occurring in triples are greater than in isolated binaries
(Toonen et al., 2020). By studying triple-star systems, we can gain a deeper
understanding of the evolutionary processes, internal structures, and
behavioral characteristics of stars in the universe. This knowledge is
significant in helping us comprehend the mechanisms involved in star formation
and evolution.

##  6 CONCLUSIONS

In this paper, we utilized the peak amplitude ratio (PAR) approach to
determine mass ratios from the cross-correlation function (CCF) between triple
stellar spectra and the binary spectra (Li et al., 2022). We subsequently
investigated the mass ratios distribution using this approach with the LAMOST-
MRS data.

We obtained some equations that describe the relationships between mass ratios
and amplitude ratios of triple stellar systems. Using these equations, we can
derive information about the masses and radial velocities of triple systems,
which can aid in more thorough investigations of spectroscopic triple (ST)
candidates. We identified 62 main-sequence ST candidates and categorized them
as A, F and G-type stars to further analyze their distribution
characteristics.

We fitted a power-law function to the distribution of corrected
$q_{\mathrm{{in}}}$ ($\mathrm{dN}/\text{d}q_{\mathrm{{in}}}\propto
q_{\mathrm{{in}}}^{\gamma_{\mathrm{{in}}}}$), the $\gamma_{\mathrm{{in}}}$ of
$q_{\mathrm{{in}}}$ distributions of ST candidate systems are estimated to be
$-0.654\pm 2.915$, $4.304\pm 1.125$ and $11.371\pm 1.309$ for A, F and G type
stars, respectively. The power law exponent $\gamma_{\mathrm{{in}}}$ increases
as the mass decreases, indicating that less massive stars are more likely to
have partner stars with similar masses. We have found that the inner binaries
in triple systems tend to have more similar mass combination, and this trend
is most obvious in G-type primary stars. We also fitted a power-law
distribution to the distribution of the corrected $q_{\mathrm{{out}}}$
($\mathrm{dN}/\text{d}q_{\mathrm{{out}}}\propto
q_{\mathrm{{out}}}^{\gamma_{\mathrm{{out}}}}$), the $\gamma_{\mathrm{{out}}}$
are estimated to be $-2.016\pm 0.172$, $-1.962\pm 0.853$ and $-1.238\pm 0.141$
for G, F and A type stars, respectively. The $\gamma_{\mathrm{{out}}}$-values
show a tendency to increase towards lower main star masses. Under the
condition where the inner binary period in a triple system is less than 20
days($\log{P}$ is uniformly distributed) and the masses of the three
components do not vary significantly($\mathrm{q}_{\mathrm{in}}>0.5$,
$0.8>\mathrm{q}_{\mathrm{out}}>0.2$), we can estimate (imprecise) the
probabilities of the triple systems A, F, and G to be 7.9 ‰, 5.0 ‰and 2.6 ‰,
providing information for further study of the distribution of triple
systems6.

## Acknowledgments

We thank Lihuan Yu, Jiao Li, Qida Li, Qiyuan Cheng, and Zhenwei Li for their
helpful insights. This work was in part supported by the National Natural
Science Foundation of China under Grant Nos.12288102, 12090040/3, the National
Key R&D Program of China under Grant No.2021YFA1600400/1, and the
International Center of Supernova, Yunnan Key Laboratory under Grant
No.202201BC070003. Guoshoujing Telescope (LAMOST) is a National Major
Scientific Project built by the Chinese Academy of Sciences. Funding for the
project has been provided by the National Development and Reform Commission.
LAMOST is operated and managed by the National Astronomical Observatories,
Chinese Academy of Sciences. This work has made use of data from the European
Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed
by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium).

## References

  * Borkovits et al. (2022) Borkovits, T., Mitnyan, T., Rappaport, S. A., et al. 2022, MNRAS, 510, 1352, doi: 10.1093/mnras/stab3397
  * Bressan et al. (2012) Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, 427, 127, doi: 10.1111/j.1365-2966.2012.21948.x
  * Cui et al. (2012) Cui, X.-Q., Zhao, Y.-H., Chu, Y.-Q., et al. 2012, Research in Astronomy and Astrophysics, 12, 1197, doi: 10.1088/1674-4527/12/9/003
  * Czavalinga et al. (2023) Czavalinga, D. R., Mitnyan, T., Rappaport, S. A., et al. 2023, A&A, 670, A75, doi: 10.1051/0004-6361/202245300
  * Duchêne & Kraus (2013) Duchêne, G., & Kraus, A. 2013, ARA&A, 51, 269, doi: 10.1146/annurev-astro-081710-102602
  * Duchê ne & Kraus (2013) Duchê ne, G., & Kraus, A. 2013, Annual Review of Astronomy and Astrophysics, 51, 269, doi: 10.1146/annurev-astro-081710-102602
  * Eggleton & Kiseleva (1995) Eggleton, P., & Kiseleva, L. 1995, ApJ, 455, 640, doi: 10.1086/176611
  * Eggleton & Tokovinin (2008) Eggleton, P. P., & Tokovinin, A. A. 2008, MNRAS, 389, 869, doi: 10.1111/j.1365-2966.2008.13596.x
  * Eggleton & Tokovinin (2010) —. 2010, VizieR Online Data Catalog, J/MNRAS/389/869
  * Fernandez et al. (2017) Fernandez, M. A., Covey, K. R., De Lee, N., et al. 2017, PASP, 129, 084201, doi: 10.1088/1538-3873/aa77e0
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1, doi: 10.1051/0004-6361/201833051
  * Gaulme et al. (2022) Gaulme, P., Borkovits, T., Appourchaux, T., et al. 2022, A&A, 668, A173, doi: 10.1051/0004-6361/202244373
  * Gray & Corbally (1994) Gray, R. O., & Corbally, C. J. 1994, AJ, 107, 742, doi: 10.1086/116893
  * Green et al. (2019) Green, G. M., Schlafly, E., Zucker, C., Speagle, J. S., & Finkbeiner, D. 2019, ApJ, 887, 93, doi: 10.3847/1538-4357/ab5362
  * Han et al. (2002) Han, Z., Podsiadlowski, P., Maxted, P. F. L., Marsh, T. R., & Ivanova, N. 2002, MNRAS, 336, 449, doi: 10.1046/j.1365-8711.2002.05752.x
  * Harrington (1977) Harrington, R. S. 1977, Rev. Mexicana Astron. Astrofis., 3, 139
  * Koleva et al. (2009) Koleva, M., Prugniel, P., Bouchard, A., & Wu, Y. 2009, A&A, 501, 1269, doi: 10.1051/0004-6361/200811467
  * Kounkel et al. (2021) Kounkel, M., Covey, K. R., Stassun, K. G., et al. 2021, AJ, 162, 184, doi: 10.3847/1538-3881/ac1798
  * Kovalev et al. (2022) Kovalev, M., Chen, X., & Han, Z. 2022, MNRAS, 517, 356, doi: 10.1093/mnras/stac2513
  * Kummer et al. (2023) Kummer, F., Toonen, S., & de Koter, A. 2023, arXiv e-prints, arXiv:2306.09400, doi: 10.48550/arXiv.2306.09400
  * Kurucz (1993) Kurucz, R. 1993, SYNTHE Spectrum Synthesis Programs and Line Data. Kurucz CD-ROM No. 18. Cambridge, 18
  * Li et al. (2021) Li, C.-q., Shi, J.-r., Yan, H.-l., et al. 2021, ApJS, 256, 31, doi: 10.3847/1538-4365/ac22a8
  * Li et al. (2022) Li, J., Li, J., Liu, C., et al. 2022, The Astrophysical Journal, 933, 119, doi: 10.3847/1538-4357/ac731d
  * Luo et al. (2015) Luo, A. L., Zhao, Y.-H., Zhao, G., et al. 2015, Research in Astronomy and Astrophysics, 15, 1095, doi: 10.1088/1674-4527/15/8/002
  * Merle et al. (2017) Merle, T., Van Eck, S., Jorissen, A., et al. 2017, A&A, 608, A95, doi: 10.1051/0004-6361/201730442
  * Moe & Kratter (2018) Moe, M., & Kratter, K. M. 2018, ApJ, 854, 44, doi: 10.3847/1538-4357/aaa6d2
  * Nasab et al. (2023) Nasab, H. H., Pazhouhesh, R., & Roobiat, K. Y. 2023, New A, 103, 102051, doi: 10.1016/j.newast.2023.102051
  * Offner et al. (2022) Offner, S. S. R., Moe, M., Kratter, K. M., et al. 2022, arXiv e-prints, arXiv:2203.10066, doi: 10.48550/arXiv.2203.10066
  * Raghavan et al. (2010a) Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010a, ApJS, 190, 1, doi: 10.1088/0067-0049/190/1/1
  * Raghavan et al. (2010b) —. 2010b, ApJS, 190, 1, doi: 10.1088/0067-0049/190/1/1
  * Rappaport et al. (2022) Rappaport, S. A., Borkovits, T., Gagliano, R., et al. 2022, MNRAS, 513, 4341, doi: 10.1093/mnras/stac957
  * Royer et al. (2007) Royer, F., Zorec, J., & Gómez, A. E. 2007, A&A, 463, 671, doi: 10.1051/0004-6361:20065224
  * Tokovinin (2018) Tokovinin, A. 2018, ApJS, 235, 6, doi: 10.3847/1538-4365/aaa1a5
  * Tokovinin (2021) —. 2021, Universe, 7, 352, doi: 10.3390/universe7090352
  * Toonen et al. (2022) Toonen, S., Boekholt, T. C. N., & Portegies Zwart, S. 2022, A&A, 661, A61, doi: 10.1051/0004-6361/202141991
  * Toonen et al. (2020) Toonen, S., Portegies Zwart, S., Hamers, A. S., & Bandopadhyay, D. 2020, A&A, 640, A16, doi: 10.1051/0004-6361/201936835
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Wang et al. (2020) Wang, R., Luo, A. L., Chen, J.-J., et al. 2020, ApJ, 891, 23, doi: 10.3847/1538-4357/ab6dea
  * Wang & Chen (2019) Wang, S., & Chen, X. 2019, ApJ, 877, 116, doi: 10.3847/1538-4357/ab1c61
  * Wilson (1941) Wilson, O. C. 1941, ApJ, 93, 29, doi: 10.1086/144239
  * Wu et al. (2014) Wu, Y., Du, B., Luo, A., Zhao, Y., & Yuan, H. 2014, in Statistical Challenges in 21st Century Cosmology, ed. A. Heavens, J.-L. Starck, & A. Krone-Martins, Vol. 306, 340–342, doi: 10.1017/S1743921314010825
  * Wu et al. (2011) Wu, Y., Luo, A. L., Li, H.-N., et al. 2011, Research in Astronomy and Astrophysics, 11, 924, doi: 10.1088/1674-4527/11/8/006
  * Xia et al. (2019) Xia, F., Fu, Y., & Wang, X. 2019, ApJ, 882, 147, doi: 10.3847/1538-4357/ab32de
  * Zhang et al. (2020) Zhang, B., Liu, C., & Deng, L.-C. 2020, ApJS, 246, 9, doi: 10.3847/1538-4365/ab55ef
  * Zhang et al. (2021) Zhang, B., Li, J., Yang, F., et al. 2021, ApJS, 256, 14, doi: 10.3847/1538-4365/ac0834