The Classifications of Double Neutron Stars and their Correlations with the Binary Orbital Parameters

Mass is one of the most important parameters of a double neutron star (DNS) system. An accurate measurement of a neutron star (NS) mass in a pulsar binary system requires determination of two of five Post-Keplerian (PK) parameters, i.e., periastron advance, time dilation, gravitational redshift, orbital decay rate, and Shapiro delay. Measuring three or more PK parameters allows for general relativity (GR) self-consistency tests (Freire et al. 2008a, 2008b; Lorimer 2008). The DNS systems are valuable physics laboratories that can be employed to precisely test GR and alternative gravitational theories (Burgay et al. 2003; Martinez et al. 2015; Abbott et al. 2016). In addition, accurate determination of DNS masses and merger rates are important in gravitational-wave (GW) sources that are being measured by ground-based GW detectors, such as LIGO (Weisberg et al. 2010; Fonseca et al. 2014; Abbott et al. 2017a, 2017b). DNS systems played a significant role in the observation of GWs and various GR effects (Ferdman et al. 2013), the successful cases of which can be found in the first-discovered DNS system PSR B1913+16 (Hulse & Taylor 1975) and the double-pulsar system PSR J0737−3039 (Burgay et al. 2003; Lyne et al. 2004; Kramer et al. 2006; Lyne & Smith 2012).

Since the discovery of the first pulsar in 1967, over 2700 pulsars have been detected. However, NS masses have not been measured in many systems because careful timing observations are needed to accurately determine binary parameters. At present, the masses of over 70 NSs in various types of NS binary systems have been measured (Zhang et al. 2011; Miller & Miller 2015; Özel & Freire 2016). The total masses of 14 DNS systems (excluding those in a globular cluster) have been measured, and seven DNSs have precisely measured masses of both the recycled NS and non-recycled companion; these are listed in Table 1 with references. The NS mass distribution has been obtained and its relationship to NS evolution has been inferred. For example, the least-massive DNSs (e.g., PSR J0737−3039 (Lyne et al. 2004; Kramer et al. 2006) and heavier millisecond pulsars (MSPs; e.g., PSR J0751+1807 (Desvignes et al. 2016) possess different histories, with the latter experiencing more time in an accretion phase than the former. Thus, it may be expected that each sample should share different nuclear matter compositions, e.g., soft versus stiff Equation of state (EOS) (Haensel et al. 2007; Miller 2002), or even quark matter (Menezes et al. 2006).

Table 1.  The Parameters of 18 Double Neutron Star Systems

System	Mp	Mc	Mtot	ΔM	Porb	Ps	e		B	FM
	(M⊙)	(M⊙)	(M⊙)	(M⊙)	(day)	(ms)		(s/s)	(G)
J1518+49041	⋯	⋯	2.7183(7)	⋯	8.634	40.935	0.249	2.72 × 10−20	1.07 × 109	SN
J1811−17362	⋯	⋯	2.57(10)	⋯	18.779	104.2	0.828	9.01 × 10−19	9.8 × 109	SN
B1534+123	1.3330(2)	1.3454(2)	2.678463(4)	−0.0124	0.421	37.904	0.274	2.42 × 10−18	9.7 × 109	SN
B1913+164	1.438(1)	1.390(1)	2.8284(1)	0.0512	0.323	59.031	0.617	8.63 × 10−18	2.28 × 1010	SN
J1930−18525	⋯	⋯	2.59(4)	⋯	45.06	185.52	0.399	9.01 × 10−19	5.85 × 1010	SN
J1753−22406	⋯	⋯	⋯	⋯	13.638	95.138	0.303	0.97 × 10−18	9.72 × 109	SN
J0737−3039AB7	1.3381(7)	1.2489(7)	2.58708(16)	0.0892	0.102	22.699	0.088	1.76 × 10−18	6.4 × 109	ECS
J1906+07468	1.322(11)	1.291(11)	2.6134(3)	0.0310	0.166	144.073	0.085	2.027 × 10−14	1.73 × 1012	ECS
J1913+11029	⋯	⋯	2.875(14)	⋯	0.206	27.285	0.089	1.61 × 10−19	2.1 × 109	ECS
J1829+245610	⋯	⋯	2.59(2)	⋯	1.760	41.009	0.139	5.25 × 10−20	1.48 × 109	USS
J0453+155911	1.559(5)	1.174(4)	2.734(3)	0.3850	4.07	45.782	0.113	1.85 × 10−19	2.9 × 109	USS
J1756−225112	1.341(7)	1.230(7)	2.56999(6)	0.0110	0.320	28.462	0.181	1.018 × 10−18	5.45 × 109	USS
J1411+255113	⋯	⋯	2.538(22)	⋯	2.61	62.4	0.17	9.6 × 10−20	<2.6 × 109	USS
J1755−255014	⋯	⋯	⋯	⋯	9.696	315.2	0.089	2.43 × 10−15	2.7 × 1011	USS
J1946+205215	⋯	⋯	2.50(4)	⋯	0.078	16.960	0.064	9 × 10−19	4 × 109	ECS?
J1757−185416	1.3384(9)	1.3946(9)	2.73295(9)	−0.056	0.183	21.497	0.606	2.63 × 10−18	7.61 × 109	SN
GCS
J1807−2500B17	1.3655(21)	1.2064(20)	2.57190(73)	0.1591	9.957	4.186	0.747	⋯	⋯	SN
B2127+11C18	1.358(10)	1.354(10)	2.71279(13)	0.0040	0.335	30.529	0.681	4.987 × 10−18	1.25 × 1010	SN

Note. (1) The magnetic field and spin period PSR J0737-3039B are 1.59 × 10¹²G and 2773 ms; (2) GCS = globular cluster systems; (3) FM = formation mechanism; SN, ECS, and USS represent, respectively, the iron core collapse, electron capture, and ultra-stripped supernova explosion.

Reference: (1) Janssen et al. (2008); (2) Corongiu et al. (2007); (3) Fonseca et al. (2014); (4) Hulse & Taylor (1975); Weisberg et al. (2010); Weisberg & Huang (2016); (5) Swiggum et al. (2015); (6) Keith et al. (2009); (7) Kramer et al. (2006); Lyne et al. (2004); (8) van Leeuwen et al. (2015); (9) Lazarus et al. (2016); (10) Champion et al. (2004); (11) Martinez et al. (2015); (12) Ferdman et al. (2014); (13) Martinez et al. (2017); (14) Ng et al. (2015, 2018); Tauris et al. (2017); (15) Stovall et al. (2018); (16) Cameron et al. (2018); (17) Lynch et al. (2012); (18) Weisberg et al. (2010).

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The pioneering work on NS mass statistics was initiated with 19 systems in 1999 (Thorsett & Chakrabarty 1999), which found all the measurements are consistent with a remarkably narrow underlying Gaussian mass distribution. Since then, more statistical studies have arisen; e.g., Nice et al. (2008) calculated the unweighted average masses of DNSs to be 1.35 M_⊙ and 1.30 M_⊙ for the recycled and unrecycled NSs in six pairs of systems, respectively, a difference of about 0.05 M_⊙. Nice et al. (2008) also found that the recycled pulsars are more commonly massive than their unrecycled companions; However, in NS-WD binaries their NS masses are broadly distributed (Zhang et al. 2011; Miller & Miller 2015; Özel & Freire 2016). Schwab et al. (2010) noticed that the underlying NS mass distribution for DNSs is bimodal, with narrow peaks at ∼1.25 M_⊙ for a lower-mass population and ∼1.35 M_⊙ for a higher-mass population. These authors interpret these two populations to be the result of distinct evolutionary formation scenarios. In detail, the statistical analysis of 18 DNS masses indicated that the average DNS mass is 1.32 ± 0.14 M_⊙ is systematically lower than the typical mass value of the other type pulsars (such as NS-WD). This suggests that the formation or evolutionary history of DNSs should differ from those of the other binary systems (Zhang et al. 2011). Kiziltan et al. (2013), on the other hand, presented an analysis in which only DNSs and NS-WD were selected, and found that NSs that have evolved through different evolutionary paths reflect distinctive signatures through different distribution peak and mass cutoff values. Moreover, Wong et al. (2010) investigated the core-collapse mechanisms in eight Galactic DNS systems by inferring progenitor masses of the second-born NS and the magnitude of the supernova kick it received at birth from measured DNS orbital parameter. Andrews et al. (2015) analyzed the binary population models of DNSs and compared the results with the precisely measured orbital periods and eccentricities of the eight DNS systems(J0737−3039, J1518+4904, B1534+12, J1756−2251, J1811−1736, J1829+2456, J1906+0746, and B1913+16) and put forward three dominant evolutionary channels, which correspond to the two DNS formation mechanisms, such as core-collapse supernova (SN) and electron-capture supernova (ECS). Furthermore, due to the relative low orbital eccentricity, small associated kick and the low system mass, Tauris et al. (2013, 2015) and Martinez et al. (2017) suggested that the second-formed NS in such close binary systems (e.g., PSR J1411+2551) experienced an ultra-stripped supernova (USS), which differs from the above two types of formation mechanisms.

To understand the nature of DNS formation, as well as the binary stellar evolution and their interaction, it is important to probe the distribution of DNS masses. The accretion of the high mass X-ray binary (HMXB) with the lifetime of several million years leads to DNS formation. The HMXB consists of a first-born NS and a massive star, where the star transfers matter to the NS by the Roche lobe overflow or stellar winds accretion (Podsiadlowski et al. 2004), to spin-up the NS while the surface magnetic field strength of DNS will drop to ∼10^9–10 G (Bhattacharya & van den Heuvel 1991; Zhang & Kojima 2006). At the final stage of massive binary evolution, the companion star evolves to a NS through various routes, e.g., SN, ECS, and USS (Podsiadlowski et al. 2004; Tauris et al. 2013).

The organization of this work is as follows. In Section 2, we present a statistical analysis of the total masses of 14 DNS systems, and the mass distributions of seven pairs of DNSs with the precisely measured masses of recycled NSs and their companions. Furthermore, we investigate the classifications of DNS formation and evolutionary scenarios (e.g., SN, ECS, and USS) by their masses and binary orbital parameters, and infer the DNS formation history by using the Anderson−Darling (A−D) and Mann−Whitney−Wilcoxon (M−W−W) tests on these samples. A summary of our results along with conclusions are given in Section 3.

It is well known that if the well-measured recycled and non-recycled NSs masses in close DNS systems fall within a narrow range (Faulkner et al. 2005; Zhang et al. 2011; Özel & Freire 2016), then the mass of the recycled pulsar in the resulting DNS is usually only slightly heavier than its mass at birth (Zhang & Kojima 2006; van den Heuvel 2004; Tauris et al. 2015). To investigate the evolutionary history of DNSs, we study the details of the DNS mass distribution, including the first-born (recycled) pulsar and the second-born (non-recycled) pulsar. In doing so, we try to uncover clues about DNS formation mechanisms. Thus, the motivations of this section are centered at employing the updated data of 18 pairs of DNSs and applying the statistics tests to classify their origins; the details are shown in the following.

The A−D test (Corder & Foreman 2009) is a statistical test of whether a given sample of data is drawn from a particular probability distribution. When applied to testing whether a normal distribution adequately describes a set of data, it is one of the most powerful statistical tools in adjudging the characteristic of sample distribution. In our case, we exploit the A−D test to determine if the samples of recycled NSs or non-recycled NSs follow a normal distribution. When we want to know whether the two population distributions are identical, the powerful tool for it is the M−W−W test (Corder & Foreman 2009), which is also called the Wilcoxon rank-sum test as a nonparametric test to make a comparison of the distribution characteristics of the two data populations. As for our application to DNS systems, we employ the M−W−W test to detect if two samples of DNS masses have the same origin.

2.1. The DNS Samples and Statistical Tests

2.2. DNS Mass Distribution and Classification by the A−D Test and the M−W−W Test

In Table 1, only seven pairs of DNSs have precisely measured masses for both component NSs. We notice that the mass of all recycled, the first-born NSs which have been spun up by accreting matter from its companion star (Bhattacharya & van den Heuvel 1991; Phinney & Kulkarni 1994; van den Heuvel 1995; Zhang & Kojima 2006; Nice et al. 2008), have measured masses in the range 1.322 M_⊙–1.559 M_⊙, while the masses of the second-formed non-recycled NSs, ranges within 1.174 M_⊙–1.395 M_⊙. Thus, it is clear that the mass average of recycled NS is higher than that of the non-recycled ones, and we can explore this in more detail by means of the cumulative distribution function (CDF). A CDF of recycled and non-recycled DNS masses is plotted in Figure 1, from which we find the following. The mass average and standard deviation of 14 NSs in seven DNS systems, is obtained as 1.34 ± 0.02 M_⊙, whereas the mass averages and standard deviation of the seven recycled and non-recycled NSs are, respectively, 1.38 ± 0.02 M_⊙ and 1.29 ± 0.02 M_⊙.

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First, by exploiting the A−D test, which is a statistical test of whether a given sample of data is drawn from a given probability distribution (Anderson & Darling 1952), the statistical parameters h and p represent a logical value for hypothesis test result and the probability of observing a test statistic. These two parameters are listed in Table 2 to test whether the masses of recycled NSs and their companions are drawn from the same distribution. The test results show that the masses of recycled NSs are indeed drawn from a normal probability distribution with mean and standard deviation σ = 0.09 M_⊙. On the contrary, the masses of seven non-recycled NSs fail to follow a normal distribution, although it presents a mean and standard deviation σ = 0.08 M_⊙. Investigation of various formation mechanisms of DNSs would be very interesting; for instance, all recycled NSs are formed in core-collapse supernova (Ferdman et al. 2014; Andrews et al. 2015). However, the later studies have also confirmed that the several non-recycled NSs are also produced by the same mechanism, such as the companions of PSR B1534+12 and PSR B1913+16 (Fonseca et al. 2014; Hulse & Taylor 1975; Weisberg et al. 2010; Weisberg & Huang 2016). Once again, we proceed with the A-D test for 10 NSs (seven recycled and three non-recycled with M > 1.3 M_⊙), and the test results are shown in Figure 2 and Table 2. Here, we find and σ = 0.07 M_⊙.

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Table 2.  A−D Test Result (α = 0.05)

Parameter	Value
Recycled NSs (7)
h⋯	1
p⋯	0.02
cv⋯	0.65
Non-recycled companions (7)
h⋯	0
p⋯	0.63
cv⋯	0.65
Recycled NSs (7) and non-recycled companions (3)a
h⋯	1
p⋯	0.009
cv⋯	0.68

Note.

^aThe three non-recycled companions are from B1534+12, B1913+16, and J1757-1854. A CDF is given in Figure 2. The logical value h = 0 indicates to reject the hypothesis that the data is from a population with a normal distribution, while the logical output h = 1 indicates a rejection of the null hypothesis at the significance level of 5%. The alternative hypothesis is that the mass is not from a population with a normal distribution. p is the P_value of the A−D test. cv represents the critical value for the A−D test at the significance level α = 0.05.

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To further explore the DNS population, a M−W−W test (Fay & Proschan 2010) is applied for the two samples of the seven recycled and seven non-recycled NSs. The M−W−W test is a nonparametric test of the null hypothesis that it is equally likely that a randomly selected value from one sample will be less than or greater than a randomly selected value from a second sample. The purpose of the M−W−W test is to distinguish if the two groups of samples come from a same origin.

As shown in Table 3, we find the logical value h of M−W−W test to be 0 for seven recycled and seven non-recycled NSs, indicating that the two groups of samples do not show the significant difference. That is to say, this test shows that the two group samples with the seven recycled and seven non-recycled NSs cannot be distinguished as the two different origins. Furthermore, we apply the M−W−W test to the above 10 and four non-recycled NSs, and find that the logical value h is 1, implying that the two groups of samples have different statistical properties. We propose that this difference is a result of the 10 NSs come from the core SN explosions while the other four non-recycled ones share the different origins from the above 10 NSs.

Table 3.  M−W−W Test Result (α = 0.05)

Parameter	Value
Recycled NSs (7) and non-recycled companions (7)
h⋯	0
p⋯	0.42
NSs (10)a and non-recycled companions (4)
h⋯	1
p⋯	0.002

Note.

^aNSs (10) represent the seven recycled ones plus three non-recycled companions of PSR B1534+12, PSR B1913+16 and J1757-1854. The logical value h = 0 indicates to reject the hypothesis that the data is from a population, while the logical output h = 1 indicates a rejection of the null hypothesis at the significance level of 5%. The alternative hypothesis is that the mass is not from a population. The symbol p stands for the P_value of the M−W−W test.

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Moreover, based on the statistical analysis of 14 pairs of DNS systems with the measured total masses, narrowly distributed within the range of 2.50 M_⊙–2.875 M_⊙, we obtain their average to be 2.65 ± 0.03 M_⊙, the CDF of which can be seen in Figure 3.

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As a contrast of mass distribution of DNSs to that of millisecond pulsars (MSPs), we find that the standard deviation of DNS mass distribution is as small as in the magnitude order of ∼0.02 M_⊙, which is about ten times smaller than that of MSPs, ∼0.2 M_⊙ (Zhang et al. 2011). The recent measured NS mass statistics with ∼70 samples shows that the all measured NS masses have a broad distribution ranging between 1 M_⊙ to >2 M_⊙ (Zhang et al. 2011; Özel & Freire 2016; Antoniadis et al. 2017). Therefore, we conclude that the particular evolutionary history of DNSs should significantly contribute to their mass formations, while the classifications of the distinctive DNSs should reveal their progenitor properties.

As for the MSP-WD systems, there exists a wide range of evolutionary scenarios, from low-mass binary pulsar (LMBP) systems, which generally seem to show a long, stable period of evolution, to intermediate-mass binary pulsar (IMBP) systems, some of which have undergone common envelope phases. Therefore, the mass distribution of MSP-WD systems is very broad compared with the DNS systems that are originated from high mass binary pulsar (HMBP). Thus, it is interesting to explore the narrow distribution of DNS masses, and investigate their origins.

2.3. The Eccentricity versus Orbital Period of DNSs and the Fisher Discriminant Classification

To understand the formation of DNS systems, we investigate the correlations between the eccentricity and orbital period, to which much attention has been paid by a number of researchers (Dewi et al. 2002; Faulkner et al. 2005; Dewi et al. 2005; Tauris et al. 2017). Indeed, by simulating the dynamical effects, the observed spread in eccentricities for a given value of P_orb can be understood if the second SNs in these binaries are even slightly asymmetric (Tauris et al. 2015, 2017).

We plotted a diagram of eccentricity versus orbital period (e-P_orb, see Figure 4), and found that the e-P_orb values of ECS and SN+USS systems are scattered in the different regions. To obtain a classification line to distinguish them, we employed the Fisher discriminant method, also called as the linear discriminant analysis (LDA; Fisher (1936); also see Raichoor et al. (2016)), which looks for the Fisher discriminant aiming to separate two known classes of samples (the detail can be seen in the Appendix).

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The simulation, LDA is presented in the following procedure. First, we have the normal distribution for both group samples of ECS and SN+USS, in which we make a simulation by selecting 1000 data samples, produced with the same mean and standard deviation as that of normal distributions. Second, by means of the LDA functions shown in the Appendix, we obtained the linear discriminant function and the discriminant value are f(x₁, x₂) = −0.0045x₁–0.001x₂ and f₀ = −0.0011, respectively. By the replacement of orbital period P_orb (in the unit of day) by x₁ and eccentricity e by x₂, the LDA function is written as below:

We have plotted a diagram of the eccentricity versus orbital period in Figure 4, where 16 pairs of observed DNS systems are labeled, which can be divided into two types (SN, ECS and USS) separated by the linear discriminant function of Equation (1).

For instance, PSR J1411+2551 is a subgroup of DNS systems thought to have undergone an USS with small ejecta mass and small kick (Tauris et al. 2013, 2015; Martinez et al. 2017). Tauris et al. (2017) extrapolate the pre-SN stellar properties and probe the second supernova events, and it is thought that all known close-orbit and small eccentricity (e < 0.2) DNS systems are consistent with the ultra-stripped exploding stars. In addition, for the ECS systems (e.g., PSR 0737-3039AB), they share the small orbit periods (P_orb < 0.25 days), slight kick velocity (v_p ≃ 0) together with the slight small eccentricity (e < 0.1).

2.4. Three Types of DNSs in the Magnetic Field versus the Spin Period Diagram

To advance the understanding of DNS evolution and classification, we demonstrate the magnetic field and spin period (B-P) diagram of pulsars, at where the positions of 18 DNSs of various types (also see Table 1) are shown in Figure 5. As can be seen, the magnetic field and spin period of 18 NSs range from 1.07 × 10⁹ G to 1.73 × 10¹² G, and from 22.699 ms to 2773 ms, respectively.

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Due to the higher magnetic fields (> 10¹¹ G), three pulsars (PSR J0737-3039B, J1906+0746, and J1755-2550) are considered to be the non-recycled NSs. The DNSs forming in ECS are located on the left of Figure 5, and these types of DNSs may originate from the intermediate-mass X-ray binaries (IMXBs ), some of which have undergone the CE phases, because the long interaction time between the CE and recycled NS, leading to the circularized orbit. As a result, the NS is spun up and becomes a faster recycled pulsar, with the very small eccentricities. The DNSs forming in USSs are located on the middle of B-P diagram, and the eccentricity is slightly bigger than that of DNS by ECS. However, the DNSs forming in SNs have the greater eccentricities. The DNSs by SNs are believed to be the descendants of HMXBs, so their spin periods and eccentricities are broadly diffused with the higher values than those of ECSs and USSs.

The average eccentricity of three kinds of DNSs is 0.0815, 0.1384, and 0.364, respectively, and their average spin period gradually increases in the same sequence, from 20 to 44ms and 75ms. Therefore, from an average point of view, there may exist a positive correlation between the spin period and eccentricity, say: the bigger the spin period, the greater the eccentricity, which can be also noticed from Figure 4.

From the evolutionary history of DNSs, it is approximately inferred that the three types of DNSs lie in the different positions (though there exist the overlaps) in B-P diagram, while the DNSs absorb about ∼0.001–0.01 M_⊙ from their companions, as inferred from the observation statistics and B-P evolution simulations (Bhattacharya & van den Heuvel 1991; Zhang & Kojima 2006).

In this paper, we studied the narrow mass distribution of DNSs, shown by the statistics as 1.33 ± 0.02 M_⊙, with a lower mean value of NS mass than that of all known NSs. As a comparison, the distribution regime of all measured NSs is as broad as (1–2 M_⊙), and the reasons for the particular mass ranges of DNSs are ascribed to their special evolution history (Bhattacharya & van den Heuvel 1991; van den Heuvel 2017), such as the interaction between two progenitor stars, the ECS and USS formation processes.

The conclusions and arguments for the special DNS mass distribution and its relation with the orbital parameters are summarized below.

• The A−D test and M−W−W test for the measured DNS masses show that the masses of recycled NSs share the same statistical origins and the non-recycled NSs have the different origins, which can be understood as a fact that the recycled NSs are all formed in SN processes and the non-recycled NSs follow the different formation mechanisms: for instance, SN, ECS, and USS. Based on the above two tests, the DNSs by SN processes are selected, and the further classifications by the conditions of eccentricities, orbital periods, and kick velocities can proceed.
• By using the Fisher discriminant method for two types of DNS systems, ECS and SN+USS, the simulation data of eccentricity and orbital period can infer a linear discriminant function as described by Equation (1), which has successfully separated the DNSs formed by ECS and SN+USS DNSs into two regions in e-P_orb diagram. In case the new DNS systems are measured and/or the DNS samples are doubled, the similar procedures could be also executed to test if the newly samples belong to what types of DNS systems.

Finally, the evolution history of various DNSs can be understood by their positions in the pulsar B-P diagram, and we notice that the DNSs by ECS, USS and SN seem to follow, on average, a sequence of decreasing the magnetic field and spin period.

This research has been supported by the National Program on Key Research and Development Project (grant No. 2016YFA0400801), the Strategic Priority Research Program of the Chinese Academy of Sciences, (grant No. XDB23000000), and CAS Interdisciplinary Innovation Team, the National Natural Science Foundation of China NSFC (11173034, 11673023, 11364007, and U1731238), the fundamental research funds for the central university. the Innovation Talent Team (grant No. (2015)4015) and the High Level Creative Talents program (grant No. (2016)-4008) of Guizhou provincial depart of Science and Technology. Special thanks are given to Dr. Wynn Ho for the English proofreading of our paper, and to the anonymous referee for the many critical comments and suggestions that greatly improved the quality of the paper.

Fisher discriminant analysis (LDA) is a widely used method for classification (Fisher 1936; McLachlan 2004; Chowdhury et al. 2018). LDA has been successfully applied to astronomy in discriminating the statistical samples (Itoh et al. 2007; Raichoor et al. 2016), and its details are described as below. We assume that we have a collection of events, where each event X(x₁, x₂) (x₁ and x₂ stand for the orbital period and eccentricity, respectively, in our case) is known to belong to one of the two classes, (A) and (B).

The linear discriminant function is defined as

Here, c_i(i = 1, 2) is an undetermined coefficient. If the discriminant value is f₀, for any unknown data point X(x₁, x₂), substituted in the discriminant function, f(x₁, x₂) can be compared with f₀ value to determine which kind of point X belongs to.

In order to find the coefficient c_i, we have listed the following equation:

Here, , and each of these sample mean vectors:

Then, the within-class scatter matrix,

To determine the discriminant value f₀,

Here, n_i is the number of samples, and is defined as

A threshold value f₀ is used to associate the events with y < f₀ to the (A) class and the events with y > f₀ to the (B) class.