# On the Formation of Double Neutron Stars in the Milky Way: Influence of Key
Parameters

Zhu-Ling Deng School of Astronomy and Space Science, Nanjing University,
Nanjing 210023, China<EMAIL_ADDRESS>Key Laboratory of Modern Astronomy and
Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023,
China  Xiang-Dong Li  School of Astronomy and Space Science, Nanjing
University, Nanjing 210023, China<EMAIL_ADDRESS>Key Laboratory of Modern
Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210023, China  Yong Shao School of Astronomy and Space Science,
Nanjing University, Nanjing 210023, China<EMAIL_ADDRESS>Key Laboratory of
Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210023, China  Kun Xu  School of Sciences, Qingdao University of
Technology, Qingdao 266525, China

###### Abstract

The detection of gravitational wave events has stimulated theoretical modeling
of the formation and evolution of double compact objects (DCOs). However, even
for the most studied isolated binary evolution channel, there exist large
uncertainties in the input parameters and treatments of the binary evolution
process. So far, double neutron stars (DNSs) are the only DCOs for which
direct observations are available through traditional electromagnetic
astronomy. In this work, we adopt a population synthesis method to investigate
the formation and evolution of Galactic DNSs. We construct 324 models for the
formation of Galactic DNSs, taking into account various possible combinations
of critical input parameters and processes such as mass transfer efficiency,
supernova type, common envelope efficiency, neutron star kick velocity, and
pulsar selection effect. We employ Bayesian analysis to evaluate the adopted
models by comparing with observations. We also compare the expected DNS merger
rate in the Galaxy with that inferred from the known Galactic population of
Pulsar-NS systems. Based on these analyses we derive favorable range of the
aforementioned key parameters.

Binary stars (154); Neutron stars (1108); Pulsars (1306); Stellar evolution
(1599)

##  1 Introduction

Since the discovery of the first chirp of gravitational waves (GWs) from a
binary black hole (BH) merger GW 150914 by the Laser Interferometer
Gravitational-wave Observatory (LIGO; Aasi et al., 2015; Abbott et al., 2016),
there have been 90 compact binary merger events detected (The LIGO Scientific
Collaboration et al., 2021). The first double neutron star (DNS) merger GW
event GW170817 (Abbott et al., 2017a) was found to be accompanied by a short
gamma-ray burst (SGRB, Abbott et al., 2017b) and a kilonova (Abbott et al.,
2017c), providing direct evidence for the connection between these phenomena.

The detection of GW events has also stimulated theoretical modelling of the
formation and evolution of double compact objects (DCOs). The proposed
formation channels for merging DCOs include isolated binary evolution,
chemically homogeneous evolution, dynamical interactions in dense
environments, hierarchical triple evolution, binary evolution in active galaxy
nucleus disks, etc (see Barack et al., 2019; Mandel & Broekgaarden, 2022;
Mapelli et al., 2022, for recent reviews). For the first time, it is possible
to quantitatively compare the merging rate predictions against observations.

However, ever for the most studied isolated binary evolution channel, there
exist big uncertainties in the input parameters and in the treatments of the
binary evolution processes. For example, a large fraction of merging DCOs have
experienced at least one common envelope (CE) evolution phase (Ivanova et al.,
2013, for a review). Depending on the CE efficiency $\alpha$ and the binding
energy parameter $\lambda$ of the donor’s envelope, this channel can produce a
large range of the merging rates. Since they can be broadly consistent with
the merger rate inferred from GW observations, it is difficult to put
stringent constraints on the above mentioned parameters. In particular, some
of the models still adopt a constant $\lambda$, which seems physically
unrealistic.

So far, DNSs are the only DCOs for which direct observations are available
through traditional electromagnetic astronomy. A viable model predicting the
properties of merging DNSs detectable by GW observatories should also be
simultaneously able to account for the Galactic DNS population, although the
known 21 DNS pulsars from radio observations represent a very small fraction
of all DNSs in the Galaxy. Phinney (1991) and Narayan et al. (1991) tried to
infer the DNS merger rates from binary pulsar observations. Since then, there
has been remarkable progress in both observation and theory including the
growing DNS data and increasingly accurate treatments of the selection
effects. And it has enabled comparing theoretical predictions with the
observed DNS characteristics, to judge the feasibility of the models and place
useful constraints on the model parameters (Tutukov & Yungelson, 1993;
Portegies Zwart & Yungelson, 1998; Belczyński & Bulik, 1999; Voss & Tauris,
2003; Wang et al., 2006; Belczynski et al., 2008; Wong et al., 2010; Osłowski
et al., 2011; Andrews et al., 2015; Beniamini & Piran, 2016; Belczynski et
al., 2017; Tauris et al., 2017; Vigna-Gómez et al., 2018; Chruslinska et al.,
2018; Andrews & Mandel, 2019; Andrews & Zezas, 2019; Chattopadhyay et al.,
2020; Chu et al., 2022).

In this work, we construct a large number of models for the formation of
Galactic DNSs, taking into account various possible combinations of the
critical input parameters and processes like mass transfer efficiency,
supernova (SN) type, CE efficiency, NS kick velocity, and selection effect of
pulsars. Through both qualitative and quantitative comparison with the
Galactic DNSs, we aim to obtain useful constraints on the parameters and
understanding of the binary evolution processes. Here we focus on isolated
binary evolution, since most DNSs are in the Galactic field (see below) and
dynamical formation of DNSs in star cluster environments does not contribute
significantly to the overall merger rate (Phinney, 1991; Belczynski et al.,
2018; Fong et al., 2019; Ye et al., 2019).

Since the first DNS system PSR B1913+16 was discovered by Hulse & Taylor
(1975), there are now 21 DNSs known in the Galaxy (see Table 1, part of data
are from the ATNF pulsar
catalog111https://www.atnf.csiro.au/research/pulsar/psrcat/, Manchester et
al., 2005). Among them, 18 DNSs are in the Galactic field and 3 are in
globular clusters (GCs). The Galactic field DNSs are thought to be
predominantly formed through isolated binary evolution (Tauris et al., 2017).

Most of the pulsars in DNSs are the first-born NS, and are observed as
(mildly) recycled pulsars. They have accreted more or less mass from the
progenitor star of the secondly born NS. Because of mass accretion-induced
field decay (Bhattacharya, 2002), the recycled pulsars have a weaker magnetic
field than normal pulsars, and a longer observable duration in radio (see Fig.
2 of Chattopadhyay et al., 2020). Therefore, we limit our study of the radio
characteristic of DNSs to the first-born pulsars.

In this paper, we combine binary population synthesis (BSE) with pulsar
evolution to explore the formation of Galactic DNSs, their radio
characteristics, and expected merger rates. The aim of this work is to
evaluate the influence of the key parameters by comparing theory with
observation. The remainder of this paper is organized as follows. We introduce
isolated binary evolution, the BSE code and the pulsar evolution model in
Section 2. We demonstrate the calculated results in Section 3, and explore the
influence of the radio selection effect in Section 4. We present our
discussion and conclusions in Section 4.

Table 1: Parameters of the detected Galactic DNSs. The spin period $P$, period
derivative $\dot{P}$, magnetic field $B$, orbital period $P_{\rm orb}$,
eccentricity $e$, mass $M_{\rm psr}$ of the pulsar, mass $M_{\rm com}$ of the
companion NS, the total mass $M_{\rm total}$, the characteristic age
$\tau_{\rm c}$, and the merging time $\tau_{\rm gwr}$ due to GW radiation.

|  | $P$ | $\dot{P}$ | $B$ | $P_{\rm orb}$ | $e$ | $M_{\rm psr}$ | $M_{\rm com}$ | $M_{\rm total}$ | $\tau_{\rm c}$ | $\tau_{\rm gwr}$
---|---|---|---|---|---|---|---|---|---|---|---
Name | Type | (ms) | ($10^{-18}$) | ($10^{9}$G) | (days) |  | ($M_{\odot}$) | ($M_{\odot}$) | ($M_{\odot}$) | (Myr) | (Gyr)
Subpopulation (i)
J0737-3039a | recycled | 22.7 | 1.76 | 2.0 | 0.102 | 0.088 | 1.338 | 1.249 | 2.587 | 204 | 0.086
B1534+12a | recycled | 37.9 | 2.42 | 3.0 | 0.421 | 0.274 | 1.333 | 1.346 | 2.678 | 248 | 2.7
J1756-2251a | recycled | 28.5 | 1.02 | 1.7 | 0.320 | 0.181 | 1.341 | 1.230 | 2.570 | 443 | 1.7
J1906+0746a,b | young | 144.1 | 20300 | 530 | 0.166 | 0.085 | 1.291 | 1.322 | 2.613 | 0.1 | 0.31
J1913+1102a | recycled | 27.3 | 0.161 | 0.63 | 0.206 | 0.090 |  $<$1.84 |  $>$1.04 | 2.88 | 2687 | 0.47
J1946+2052c | recycled | 17.0 | 0.9 | 1.2 | 0.078 | 0.06 |  $<$1.31 |  $>$1.18 | 2.50 | 299 | 0.047-0.049
Subpopulation (ii)
J0453+1559a | recycled | 45.8 | 0.186 | 0.92 | 4.072 | 0.113 | 1.559 | 1.174 | 2.734 | 3901 | 1500
J1411+2551d | recycled | 62.4 | 0.0956 | 0.77 | 2.616 | 0.1699 | $<1.62$ | $>0.92$ | 2.538 | 10342 | 490-510
J1518+4904a | recycled | 40.9 | 0.0272 | 0.29 | 8.634 | 0.249 | 1.41 | 1.31 | 2.718 | 23834 | 9200
J1753-2240a | recycled | 95.1 | 0.970 | 2.7 | 13.638 | 0.304 | - | - | - | 1553 | $>42940$
J1755-2550a,b | young | 315.2 | 2430 | 270 | 9.696 | 0.089 | - | $>0.4$ | - | 2 | $>25086$
J1811-1736a | recycled | 104.2 | 0.901 | 3.0 | 18.779 | 0.828 | $<1.64$ | $>0.93$ | 2.57 | 1832 | 1810-1870
J1829+2456a | recycled | 41.0 | 0.0525 | 0.46 | 1.176 | 0.139 | $<1.38$ | $>1.22$ | 2.59 | 12373 | 59-61
J1930-1852a | recycled | 185.5 | 18.0 | 18 | 45.060 | 0.399 | $<1.32$ | $>1.30$ | 2.59 | 163 | 554000-572000
J1325-6253b,e | recycled | 29.0 | 0.048 | 1.18 | 1.816 | 0.064 | $<1.59$ | $>0.98$ | 2.57 | 19180 | 212
Subpopulation (iii)
J0509+3801f | recycled | 76.5 | 7.93 | 7.8 | 0.380 | 0.586 | 1.34 | 1.46 | 2.805 | 153 | 0.58
J1757-1854g | recycled | 21.5 | 2.63 | 2.4 | 0.183 | 0.606 | 1.338 | 1.395 | 2.733 | 130 | 0.076
B1913+16a | recycled | 59.0 | 8.63 | 7 | 0.323 | 0.167 | 1.440 | 1.389 | 2.828 | 108 | 0.3
Globular Cluster
B2127+11Ca | recycled | 30.5 | 4.99 | 3.8 | 0.335 | 0.681 | 1.358 | 1.354 | 2.713 | 97 | 0.22
J1807-2500a,b | recycled | 4.2 | 0.0823 | 0.18 | 9.957 | 0.747 | 1.366 | 1.206 | 2.572 | 805 | 1100
J0514-4002h | recycled | 4.99 | 0.0007 | 0.019 | 18.8 | 0.89 | 1.25 | 1.22 | 2.473 | 113000 | 470

Note. — a Tauris et al. (2017) and references therein. b Not a confirmed DNS
system–could also be a WD+NS binary. c Stovall et al. (2018). d Martinez et
al. (2017). e Sengar et al. (2022). f Lynch et al. (2018). g Cameron et al.
(2018). h Ridolfi et al. (2019).

##  2 Methods

###  2.1 Binary Population Synthesis

We first briefly introduce the two dominant channels of DNS formation by
isolated binary evolution (Bhattacharya & van den Heuvel, 1991; Tauris & van
den Heuvel, 2006; Vigna-Gómez et al., 2018). In the first channel (Channel I),
the evolution begins from a primordial binary with both components more
massive than $8M_{\sun}$, and the mass ratio ($q=$ primary mass/secondary
mass) considerably larger than unity. The primary star evolves more rapidly to
overflow its Roche-lobe (RL) and transfers mass to the secondary. After its
envelope is exhausted, the primary evolves to be a helium star, and finally a
NS with a SN explosion. The subsequent evolution of the secondary star leads
to a CE phase, in which the NS spirals into the envelope of the secondary
star. If the NS is able to expel the envelope, the outcome is a NS-helium star
binary in a compact orbit. A DNS finally forms if the binary is not disrupted
during the second SN. In the second channel (Channel II), the initial mass
ratio $q\sim 1$, and the binary experiences a double-core CE phase. This may
leave a double helium star binary in a compact orbit. The DNS then evolves
from the double helium star binary (Brown, 1995; Dewi et al., 2006; Hwang et
al., 2015; Andrews et al., 2015).

For the isolated binary evolution, there are some critical physical processes
that have not been well understood, such as the stability of mass transfer
(MT), the fate of CE evolution, the SN mechanisms and the natal kicks to the
NS (e.g., Andrews et al., 2015; Vigna-Gómez et al., 2018; Shao & Li, 2018a;
Chu et al., 2022). They will be discussed in some detail below. Besides them,
before the second SN, there may exist a Case BB MT phase during which the
binary comprises a NS and a naked helium star companion (Habets, 1986). The
Case BB phase also plays an important role in the formation of DNSs (Vigna-
Gómez et al., 2018). During this phase, the first-born NS undergoes mass
accretion and is accelerated to short spin period ($\sim 10-100$ ms, depending
on the accretion rate and the amount of accreted mass). This may result in an
ultra-stripped helium star before the SN, which will finally evolve to be an
ONeMg WD or a NS depending on the initial orbital separation and the mass of
the helium star (Tauris et al., 2013, 2015, 2017).

After the birth of the DNS, its orbit keeps shrinking by GW radiation (Peters,
1964; Hulse & Taylor, 1975), finally producing a merger event that may be
associated with a SGRB (depending on the assumptions about the DNS masses and
other factors, Abbott et al., 2017a; Mösta et al., 2020; Mandel &
Broekgaarden, 2022). The merger time ($\tau_{\rm gwr}$) is determined by the
orbital separation, eccentricity and the NS masses. Andrews & Mandel (2019)
accordingly split the Galactic field DNSs into three categories: subpopulation
(i) have small eccentricities and $\tau_{\rm gwr}$ less than Hubble time;
subpopulation (ii) contain widely separated systems and will not merge within
Hubble time; and subpopulation (iii) have large eccentricities and $\tau_{\rm
gwr}$ less than Hubble time (see also Fig. 1).

We use the BSE code originally developed by Hurley et al. (2002) to calculate
the Galactic DNS distribution. We adopt a star formation rate of 3 $M_{\odot}$
yr-1, and follow the binary evolution over 12 Gyr. The primordial binaries
consist of a primary star with mass $M_{1}$ and a secondary star with mass
$M_{2}$ in a circular orbit with separation $a$ (or orbital period $P_{\rm
orb}$). For the distributions of $M_{1}$, $M_{2}$ and $a$, we set $n_{\chi}$
grid points of the parameter $\chi$ logarithmically spaced with

$\delta{\rm ln}\chi=\frac{1}{{\rm n}_{\chi}-1}({\rm ln}\chi_{\rm max}-{\rm
ln}\chi_{\rm min}).$ (1)

If a specific binary $i$ evolves to a DNS, then the binary contributes to the
DNS population with a rate

$R_{\rm i}=(\frac{f_{\rm bin}}{2})(\frac{SFR}{M_{\star}})\Phi({\rm
ln}M_{1})\phi({\rm ln}M_{2})\Psi({\rm ln}a)\delta{\rm ln}M_{1}\delta{\rm
ln}M_{2}\delta{\rm ln}a,$ (2)

where $f_{\rm bin}$ is the binary fraction (we assume all-stars are initially
in binaries, i.e., $f_{\rm bin}=1.0$), $SFR$ is the star formation rate, and
$M_{\star}=0.5M_{\odot}$ is the average mass for all stars; $\Phi$, $\phi$,
and $\Psi$ are the distribution functions.

We assume the distribution of the primary masses to follow the initial mass
function $\xi(M_{1})$ of Kroupa et al. (1993) in the range of $5-40M_{\odot}$.
Thus,

$\Phi({\rm ln}M_{1})=M_{1}\xi(M_{1}).$ (3)

For the secondary masses, we assume that they follow a flat distribution
between 0 and $M_{1}$ (Kobulnicky & Fryer, 2007), then,

$\phi({\rm ln}M_{2})=\frac{M_{2}}{M_{1}}.$ (4)

The initial orbital separations $a$ are taken to be uniformly distributed in
the range of $3-10^{4}R_{\odot}$ in the logarithm (Abt, 1983), so

$\Psi(\ln a)=k=0.12328.$ (5)

Besides the standard treatments of stellar and binary evolution in the BSE
code, we modify and update some key processes in the formation of DNSs. They
are described as follows.

####  2.1.1 Mass loss, mass transfer and CE evolution

We adopt the fitting formula of Nieuwenhuijzen & de Jager (1990) to model
stellar wind mass loss. For hot OB stars ($T_{\rm eff}>11000$ K) and stripped
helium stars, we replace it with the simulated relations by Vink et al. (2001)
and Vink (2017), respectively.

When the primary star evolves to fill its RL, the first MT begins. The
accretion efficiency play a vital role in determining the fate of the binary
evolution. If the MT is sufficiently rapid, the accreting secondary star will
be out of thermal equilibrium, and the responding expansion may lead the
secondary star to fill its own RL, resulting in a contact binary (Nelson &
Eggleton, 2001). In this situation, if the primary star is a main sequence
star, the binary will merge; otherwise, it will enter a CE phase (de Mink et
al., 2013).

Here we adopt three models for the accretion efficiency in the primordial
binaries.

Model I assumes that the mass accretion rate $\dot{M}_{2}$ of the secondary
star is modulated by its rotation rate:
$\dot{M}_{2}=\dot{M}_{1}\times(1-\frac{\Omega_{2}}{\Omega_{\rm cr}})$, where
$\dot{M}_{1}$, $\Omega_{2}$, and $\Omega_{\rm cr}$ are the MT rate, the
angular velocity of the secondary and its critical value (break-up rate),
respectively. The rotation of the star is set to be rigid, and its variation
is affected by both accretion and tidal synchronization222According to Shao &
Li (2016), for narrow binaries, tidal synchronization can prevent the
accreting star from reaching critical rotation, and the overall accretion
efficiency is $\sim 0.2-0.7$; for wide systems where tidal torques are no
longer effective, the accretor quickly spins up to critical rotation and stops
accreting, and the overall accretion efficiency is less than 0.2..

Model II assumes that half of the transferred mass is accreted by the
secondary.

Model III assumes that the transferred mass is always accumulated by the
secondary unless its thermal timescale becomes significantly shorter than the
mass transfer timescale. So $\dot{M}_{2}$ is set to be the MT rate multiplied
by a factor of min(10$\frac{\tau_{\dot{M}}}{\tau_{\rm KH,2}}$,1) where
$\tau_{\dot{M}}$ is the MT timescale and $\tau_{\rm KH,2}$ is the secondary’s
thermal timescale (Hurley et al., 2002). Rapid mass accretion may push the
accretor beyond its thermal equilibrium, causing it to expand and become over-
luminous. According to the results of Shao & Li (2014), $\tau_{\rm KH,2}$ is
usually less than $\tau_{\dot{M}}$, which means that the MT is almost
conservative.

In all the three models, the excess material is assumed to be ejected out of
the binary in the form of isotropic winds, taking away the accretor’s specific
angular momentum. Thus, the accretion efficiency affects not only the orbital
evolution of the binary system, but also the stability of MT (Soberman et al.,
1997). According to the calculation by Shao & Li (2014), the maxium $q_{\rm
cr}$ for stable MT are $\sim 6$, 2.5, and 2.2 for models I, II and III,
respectively. We adopt their results (Fig. 1) of $q_{\rm cr}$ to deal with the
MT processes in the primordial binaries.

In the case of NSs accreting from a main-sequence/giant donor star, we use the
Eddington limit to constrain the accretion rate, and adopt the maximal mass
ratio $q_{\rm cr}=3.5$ for stable MT following previous works (Tauris et al.,
2000; Podsiadlowski et al., 2002; Shao & Li, 2012; Deng et al., 2020). For NS-
Helium star binaries, we assume that the MT becomes dynamically unstable when
the orbital period is less than 0.06 day (Tauris et al., 2015).

To deal with CE evolution, we compare the orbital energy of the binary and the
binding energy of the donor’s envelope, to evaluate whether the orbital energy
is sufficient to expel the envelope (Webbink, 1984),

$\alpha_{\rm CE}\left(\frac{GM_{\rm 1c}M_{2}}{2a_{\rm
f}}-\frac{GM_{1}M_{2}}{2a_{\rm i}}\right)=\frac{G(M_{1}-M_{\rm
1c})M_{1}}{\lambda r_{\rm L}},$ (6)

where $\alpha_{\rm CE}$ is the CE efficiency, $M_{\rm 1c}$ is the mass of the
primary’s core, $a_{\rm i}$ and $a_{\rm f}$ are the orbital separations of the
binary before and after the CE stage respectively, $G$ is the gravitational
constant, $r_{\rm L}$ is the RL radius of the core, and $\lambda$ is the
binding energy parameter of the primary’s envelope. We let $\alpha_{\rm
CE}=0.1$, 0.5, 1, 2, 5 and 10 to evaluate its possible influence. In the case
with $\alpha_{\rm CE}>1$ it is assumed that there are extra energy sources
(such as recombination energy) used to expel the stellar envelope (values
significantly larger than 1 might not reflect real physical process but for
illustrative purposes). The binding energy parameter $\lambda$ depends on the
stellar evolutionary stage, and we adopt the calculated results of Xu & Li
(2010a, b) and Wang et al. (2016).

####  2.1.2 SN models and natal kicks

We use the rapid and delayed explosion models proposed by Fryer et al. (2012)
and the stochastic explosion model proposed by Mandel & Müller (2020) to
process the products of core-collapse supernovae (CCSN). In the rapid and
delayed SN models, the CO core masses at the SN determines the remnant masses,
which are further increased by accreting from the subsequent fallback
material. Stars with inital mass $\sim 9\,M_{\odot}$ are thought to produce
electron-capture supernovae (ECSN), and the helium core mass $M_{\rm core}$ is
traditionally used to discriminate CCSN from ECSN. Fryer et al. (2012)
suggested the following criteria: if $M_{\rm core}<1.83\,M_{\odot}$, there
will be a degenerate CO core, which ends up as a CO WD; if $M_{\rm
core}>2.25\,M_{\odot}$, a non-degenerate CO core is formed, and subsequent
thermonuclear burning leads to the formation of an FeNi core, which ultimately
collapses to be a NS or BH; if $1.83\,M_{\odot}<M_{\rm core}<2.25\,M_{\odot}$,
the star will form a partially degenerate CO core. If the CO core reaches a
critical mass $M_{\rm CO,crit}$($=1.08\,M_{\odot}$), it will non-explosively
burn into a degenerate ONe core. If the ONe core mass could increase to
$M_{\rm ecs}=1.38\,M_{\odot}$, the star eventually collapses into a NS through
an ECSN, or otherwise forms an ONe WD.

However, there are considerable uncertainties in the triggering condition of
ECSN, and MT in binaries makes it more complicated (Nomoto, 1984;
Podsiadlowski et al., 2004; Woosley & Heger, 2015; Poelarends et al., 2017;
Doherty et al., 2017). Bearing this in mind, Shao & Li (2018b) considered
three possible helium core mass ranges $M_{\rm ecs}=(1.83-2.25)\,M_{\odot}$,
$(1.83-2.5)\,M_{\odot}$, and $(1.83-2.75)\,M_{\odot}$, within which the star
may eventually explode in an ECSN. Their results seem to prefer the helium
core mass range $(1.83-2.75)\,M_{\odot}$ for ECSN for the formation of
Galactic DNSs. In this work, we use two possible helium core mass ranges
$M_{\rm ecs}=(1.83-2.25)\,M_{\odot}$, and $(1.83-2.75)\,M_{\odot}$ for ECSN.

In addition, Case BB MT may strip most of the envelope of stars before SN
explosions (Tauris et al., 2013, 2015, 2017). For such ultra-stripped
supernovae (USSN), we use the recipe in the the rapid model to estimate the
compact star mass.

Taking into account all the above factors, we adopt the following five models
for SN explosions:

(1) SN A - [rapid CCSN explosion] $+$ [$M_{\rm ecs}=(1.83-2.25)\,M_{\odot}$
for ECSN];

(2) SN B - [rapid CCSN explosion] $+$ [$M_{\rm ecs}=(1.83-2.75)\,M_{\odot}$
for ECSN];

(3) SN C - [delayed CCSN explosion] $+$ [$M_{\rm ecs}=(1.83-2.25)\,M_{\odot}$
for ECSN];

(4) SN D - [stochastic CCSN explosion] $+$ [$M_{\rm
ecs}=(1.83-2.25)\,M_{\odot}$ for ECSN];

(5) SN E - [rapid CCSN explosion] $+$ [$M_{\rm ecs}=(1.83-2.25)\,M_{\odot}$
for ECSN] $+$ [USSN].

The natal kick velocities imparted on the NSs are set as follows.

(1) In both rapid and delayed SN models, the kick velocity is assumed to
follow a Maxwellian distribution with the dispersion velocity $\sigma_{\rm
k}=150$ or 300 km s-1 for CCSN, and $\sigma_{\rm k}=20$ or 40 kms-1 for ECSN.

(2) The remnant masses and natal kicks in stochastic SN satisfy some specific
probability distributions depending on the masses of the CO core (Mandel &
Müller, 2020).

(3) For USSN, we use the formula given by Andrews & Zezas (2019) to evaluate
the kick velocity (the best fit model of the SN kicks in Bray & Eldridge,
2018):

$V_{\rm k}=|100\frac{M_{\rm He}-M_{\rm NS}}{M_{\rm NS}}-170|\,\rm kms^{-1},$
(7)

where $M_{\rm He}$ and $M_{\rm NS}$ are the masses of the pre-SN helium star
and the NS, respectively.

####  2.1.3 Binary evolution models

Adopting different MT models, SN models, kick velocity prescriptions, and CE
efficiency $\alpha$, we construct 324 binary evolution models, and present the
details in Table 2. The name of each model describes the properties of the key
parameters. For example, model MTISNAk20150$\alpha$0.1 means [MT Model I + (SN
A) $+$ ($\sigma_{\rm k}=20/150$ km$s^{-1}$ for EC/CCSN) $+$ ($\alpha_{\rm
CE}=0.1$)]. We evolve $10^{8}$ binary systems and obtain a few to several
hundred thousand newborn DNS systems in each run.

Table 2: Parameter settings for binary stellar evolution models

Model | Accretiona | CCSN | USSN | Kick (Maxwell) | Notes | $\alpha_{\rm CE}^{b}$
---|---|---|---|---|---|---
| efficiency | Mechanism |  |  $\sigma_{\rm ECSN},\sigma_{\rm CCSN}$ (km s-1) |  |
MTISNAk20150$\alpha$x | $1-\frac{\Omega_{2}}{\Omega_{\rm cr}}$ | rapid | no | 20, 150 |  | 0.1-10
MTISNAk20300$\alpha$x | rapid | no | 20, 300 |
MTISNAk40150$\alpha$x | rapid | no | 40, 150 |
MTISNAk40300$\alpha$x | rapid | no | 40, 300 |
MTISNBk20150$\alpha$x | rapid | no | 20, 150 | ECSN rangec
MTISNBk20300$\alpha$x | rapid | no | 20, 300 | ECSN rangec
MTISNBk40150$\alpha$x | rapid | no | 40, 150 | ECSN rangec
MTISNBk40300$\alpha$x | rapid | no | 40, 300 | ECSN rangec
MTISNCk20150$\alpha$x | delay | no | 20, 150 |
MTISNCk20300$\alpha$x | delay | no | 20, 300 |
MTISNCk40150$\alpha$x | delay | no | 40, 150 |
MTISNCk40300$\alpha$x | delay | no | 40, 300 |
MTISNDk20$\alpha$x | stochastic | no | $\sigma_{\rm ECSN}=20$ | kick for CCSNd
MTISNDk40$\alpha$x | stochastic | no | $\sigma_{\rm ECSN}=40$ | kick for CCSNd
MTISNEk20150$\alpha$x | rapid | yes | 20, 150 | kick for USSNe
MTISNEk20300$\alpha$x | rapid | yes | 20, 300 | kick for USSNe
MTISNEk40150$\alpha$x | rapid | yes | 40, 150 | kick for USSNe
MTISNEk40300$\alpha$x | rapid | yes | 40, 300 | kick for USSNe
MTIISNAk20150$\alpha$x | 0.5 | rapid | no | 20, 150 |  | 0.1-10
MTIISNAk20300$\alpha$x | rapid | no | 20, 300 |
MTIISNAk40150$\alpha$x | rapid | no | 40, 150 |
MTIISNAk40300$\alpha$x | rapid | no | 40, 300 |
MTIISNBk20150$\alpha$x | rapid | no | 20, 150 | ECSN rangec
MTIISNBk20300$\alpha$x | rapid | no | 20, 300 | ECSN rangec
MTIISNBk40150$\alpha$x | rapid | no | 40, 150 | ECSN rangec
MTIISNBk40300$\alpha$x | rapid | no | 40, 300 | ECSN rangec
MTIISNCk20150$\alpha$x | delay | no | 20, 150 |
MTIISNCk20300$\alpha$x | delay | no | 20, 300 |
MTIISNCk40150$\alpha$x | delay | no | 40, 150 |
MTIISNCk40300$\alpha$x | delay | no | 40, 300 |
MTIISNDk20$\alpha$x | stochastic | no | $\sigma_{\rm ECSN}=20$ | kick for CCSNd
MTIISNDk40$\alpha$x | stochastic | no | $\sigma_{\rm ECSN}=40$ | kick for CCSNd
MTIISNEk20150$\alpha$x | rapid | yes | 20, 150 | kick for USSNe
MTIISNEk20300$\alpha$x | rapid | yes | 20, 300 | kick for USSNe
MTIISNEk40150$\alpha$x | rapid | yes | 40, 150 | kick for USSNe
MTIISNEk40300$\alpha$x | rapid | yes | 40, 300 | kick for USSNe
MTIIISNAk20150$\alpha$x | min$(10\frac{\tau_{\dot{M}}}{\tau_{\rm KH,2}},1)$ | rapid | no | 20, 150 |  | 0.1-10
MTIIISNAk20300$\alpha$x | rapid | no | 20, 300 |
MTIIISNAk40150$\alpha$x | rapid | no | 40, 150 |
MTIIISNAk40300$\alpha$x | rapid | no | 40, 300 |
MTIIISNBk20150$\alpha$x | rapid | no | 20, 150 | ECSN rangec
MTIIISNBk20300$\alpha$x | rapid | no | 20, 300 | ECSN rangec
MTIIISNBk40150$\alpha$x | rapid | no | 40, 150 | ECSN rangec
MTIIISNBk40300$\alpha$x | rapid | no | 40, 300 | ECSN rangec
MTIIISNCk20150$\alpha$x | delay | no | 20, 150 |
MTIIISNCk20300$\alpha$x | delay | no | 20, 300 |
MTIIISNCk40150$\alpha$x | delay | no | 40, 150 |
MTIIISNCk40300$\alpha$x | delay | no | 40, 300 |
MTIIISNDk20$\alpha$x | stochastic | no | $\sigma_{\rm ECSN}=20$ | kick for CCSNd
MTIIISNDk40$\alpha$x | stochastic | no | $\sigma_{\rm ECSN}=40$ | kick for CCSNd
MTIIISNEk20150$\alpha$x | rapid | yes | 20, 150 | kick for USSNe
MTIIISNEk20300$\alpha$x | rapid | yes | 20, 300 | kick for USSNe
MTIIISNEk40150$\alpha$x | rapid | yes | 40, 150 | kick for USSNe
MTIIISNEk40300$\alpha$x | rapid | yes | 40, 300 | kick for USSNe

Note. — a The related critical mass ratios ($q_{\rm cr}$) for stable MT comes
from Shao & Li (2014) Fig. 1. b Specifically, all values of $\alpha_{\rm CE}$
are 0.1, 0.5, 1, 2, 5 and 10. c The ECSN mass range for SN B is 1.83-2.75
$M_{\odot}$. d Satisfy specific probability distributions described in (Mandel
& Müller, 2020). e See Eq. (8).

Using the formulae of Peters (1964), we trace the orbital evolution of each
DNS with time driven by GW radiation. We limit the evolutionary time to the
Galactic age of 12 Gyrs, and calculate the total number of the Galactic DNSs.
Each DNS spends specific time in the pulsar phase, and the number and
characteristics of observable Pulsar-NS systems are dependent on the pulsar’s
evolution, the sensitivity of the radio telescopes, and selection effects.
They are discussed in the following subsection.

###  2.2 Radio characteristics of DNSs

Faucher-Giguère & Kaspi (2006) performed Monte Carlo-based population
synthesis of the birth properties of Galactic isolated pulsars, their time
evolution, and their detection in the Parkes and Swinburne Multibeam surveys.
They presented a population model that appears generally consistent with the
observations, in which newborn NSs have normally distributed spin periods
(with $\mu_{\rm P}=300$ ms and $\sigma_{\rm P}=150$ ms), and log-normally
distributed magnetic fields (with $\mu_{\rm log(B/G)}=12.65$ and $\sigma_{\rm
log(B/G)}=0.55$). More recently, Chattopadhyay et al. (2020) and Sgalletta et
al. (2023) considered several different initial parameter distributions to
model the radio characteristics of Galactic DNSs. Their best fit models
indicate that the NSs’ initial spin periods and magnetic fields are uniformly
distributed in the range of $10-100$ ms and $10^{10}-10^{13}$ G, respectively.

The first-born NS in DNSs behaves as radio pulsars mainly after the birth of
the second NS. Because it had been (partially) recycled during the CE
evolution and the subsequent Case BB MT phase, its spin period and magnetic
field at the birth of the second NS (or at the birth of the PSR-NS binary)
should deviate from its initial values. Three-dimensional hydrodynamic
simulations by MacLeod & Ramirez-Ruiz (2015) show that NSs undergoing typical
CE episodes could accrete a few percent their own mass or less depending on
the density gradient inside the envelope and the assumption of NS’s
hypercritical accretion under neutrino-dominated cooling. The additional phase
of MT from the helium star to the NS after the CE stage would also be
responsible for the spin and magnetic field of the first-born pulsar (Jiang et
al., 2021, and references therein). However, the uncertainties in the accreted
mass during both processes and the mechanism(s) of accretion-induced magnetic
field decay make it difficult to evaluate their influence on the spin and
magnetic field evolution in a very reliable way. We instead use the current
spin periods and magnetic fields as references for the pulsars. The spin
periods of the recycled pulsars in Galactic DNSs are distributed in the range
of $17-185.5$ ms (see Table 1). Assuming that the NSs are spinning down due to
magnetic dipole radiation, from their spin periods and period derivatives we
obtain the magnetic fields $\sim 10^{8.5}-10^{10.3}$ G. Therefore, we
tentatively assume that, at the birth of the second NS, the spin periods of
the first-born NSs are uniformly distributed in $10-200$ ms, and the initial
magnetic fields (in the logarithm) are uniformly distributed in $8-11$. We
note these choices are somewhat arbitrary, and in section 4.3 we vary our
assumptions of the spin periods and magnetic fields and discuss their possible
influence.

We also assume that, the first NS’s surface magnetic field decays
spontaneously in an exponential form

$B=\left(B_{0}-B_{\min}\right)\times\exp\left(-t/\tau_{\mathrm{d}}\right)+B_{\min},$
(8)

where $B_{\rm min}$ is the lowest magnetic field strength taken to be $10^{8}$
G (Osłowski et al., 2011), and $\tau_{\rm d}$ is the magnetic field decay
timescale. We set it to be $10^{9}$ yr according to the best-fitting models of
Chattopadhyay et al. (2020) and Sgalletta et al. (2023).

We assume that the first-born NS acts as a pulsar and starts spinning down by
magnetic dipole radiation after the second NS’s birth. We adopt the fitting
formula in Faucher-Giguère & Kaspi (2006) to estimate the pulsar’s radio
luminosity

$\log L=\log\left(L_{0}P^{\epsilon_{\rm P}}{\dot{P}_{15}}^{\epsilon_{\dot{\rm
P}}}\right)+L_{\mathrm{corr}}.$ (9)

Here, $L_{0}=0.18\,\rm mJy\,kpc^{-2}$, $\dot{P}=(10^{-15}\,{\rm
ss}^{-1})\dot{P}_{15}$ is the period derivative, $\epsilon_{P}=-1.5$,
$\epsilon_{\dot{P}}=0.5$, and $L_{\mathrm{corr}}$ is a zero-centered normal
distribution with $\sigma_{L_{\rm corr}}=0.8$. The flux of the pulsar is

$F=\frac{L}{4\pi D^{2}},$ (10)

where $D$ is the distance between the pulsar and the Solar system. The pulsar
could be observed if $F$ is larger than the limiting flux $S_{\rm min}$ of the
radio telescope. Approximately half of the Pulsar-NS systems in the Galaxy
were detected by the Parkes Multi-beam Pulsar Survey. We set the limiting flux
of the Parkes telescope to be 0.2 mJy (Manchester et al., 2001).

The distance between a pulsar and the Solar system depend on the initial
location and the subsequent motion of the pulsar under the Galactic
gravitational potential. Following Faucher-Giguère & Kaspi (2006) we set the
birth distribution to be

$p(r_{\rm birth};{\rm Gauss})\propto\exp[-\frac{(r_{\rm birth}-R_{\rm
peak})^{2}}{2\sigma^{2}_{r_{\rm birth}}}],$ (11)

where $r_{\rm birth}$ is the pulsar’s birth distance from the Galactic center,
$R_{\rm peak}=7.04$ kpc and $\sigma_{\rm r_{birth}}=1.83$ kpc. Combining the
Galactic rotation curve and the kick velocity at the second SN, one can follow
the motion of the DNS. We use the NIGO code333Numerical Integrator of Galactic
Orbits, which was designed to predict the orbital evolution of test particles
moving within a fully-analytical gravitational potential generated by a multi-
component galaxy. NIGO has been included in the Astrophysics Source Code
Library and can be downloaded from https://github.com/LucaJRossi/NIGO (Rossi,
2015). to calculate the three-dimensional (3D) movement of the Pulsar-NS
system and the Solar system with time, and to estimate their distance
according to their 3D positions. The gravitational potential in the NIGO code
includes contributions from the Galactic bulge, the disk, and the dark matter
halo (Miyamoto & Nagai, 1975; Flynn et al., 1996; Navarro et al., 1997;
Prugniel & Simien, 1997).

The pulsar’s radiation is known to be anisotropic. The beaming effect
significantly influences the observable number of pulsars. We adopt the
formula of Tauris & Manchester (1998) to model the beaming fraction,

$f_{\text{beaming}}=0.09\left(\log\frac{P}{10}\right)^{2}+0.03,\quad 0\leq
f_{\text{beaming}}\leq 1$ (12)

where $P$ is the spin period of the pulsar in seconds.

Finally, when the pulsars cross a “death line” in the $P-\dot{P}$ diagram,
they stop emitting in the radio band (Rudak & Ritter, 1994),

$\log_{10}\dot{P}=3.29\times\log_{10}P-16.55,$ (13)
$\log_{10}\dot{P}=0.92\times\log_{10}P-18.65.$ (14)

Here, we adopt the second equation (14). Following Sgalletta et al. (2023), we
add a cut-off on the efficiency of radio emission as in Szary et al. (2014) to
avoid the accumulation of pulsars at the death lines. The radio efficiency is
defined as $\xi=L/\dot{E}$, where $\dot{E}$ is the pulsar’s spin-down power.
We set a threshold of $\xi_{\rm max}=0.01$. If $\xi>\xi_{\rm max}$, the pulsar
stops emitting radio waves (Szary et al., 2014).

##  3 Results

###  3.1 The Orbital period and eccentricity distributions of Galactic DNSs

####  3.1.1 Qualitative analysis

We plot the distribution of the Galactic DNSs in the $P_{\rm orb}-e$ plane
(Figs. 1-4). Fig. 1 shows the impact of $\alpha_{\rm CE}$ on the number and
orbit of DNSs in the SN A model. The number of observable DNSs increases by
about 5 times when $\alpha_{\rm CE}$ increases from 0.1 to 10. The modeled
DNSs are mostly distributed along the strip extending from lower bottom to
upper right (i.e., from near circular short orbits to eccentric wide orbits).
While quite a few observed DNSs are indeed on this strip, most of the model
distributions face difficulty in accounting for DNSs with relatively long
orbital periods and small eccentricities.

Figure 1: Distribution of the Galactic DNSs in the $P_{\rm orb}-e$ plane. Each
panel demonstrates a snapshot of the current DNS population in a specific
model (MTISNAk40300$\alpha$0.1, MTISNAk40300$\alpha$0.5,
MTISNAk40300$\alpha$1, MTISNAk40300$\alpha$2, MTISNAk40300$\alpha$5,
MTISNAk40300$\alpha$10). Different colors indicate the predicted DNS numbers,
as seen from the color bars on the right. The stars, triangles, and circles
represent the Galactic field DNS systems with subpopulation (i), (ii), and
(iii), respectively. The square, plus, and cross represent the GC DNSs PSRs
J1807-2500, B2127+11C, and J0514-4002, respectively. The gray symbols
represent the DNS systems that need further confirmation. The black dash lines
represent the merger time of $t_{\rm gwr}=10^{8},10^{9},$ and $1.2\times
10^{10}$ yrs, from left to right, respectively. Note that if the observable
number is less than 0.01, the color of the pixel grid ($\Delta P_{\rm
orb}-\Delta e$) is set to white.

Fig. 2 shows the effect of kick velocities in the SN A model with $\alpha_{\rm
CE}=1$. The larger kick velocity, the fewer DNSs formed (the observable DNS
number in the model with k20150 is approximately four times that in the model
with k40300). When the second SN is CCSN, most DNSs are short-period and high-
eccentricity systems, and it is difficult (with $\sigma_{\rm k}=150$ km s-1)
or even impossible (with $\sigma_{\rm k}=300$ km s-1) to form long-period
systems. When the second SN is ECSN, most DNSs are low-eccentricity and long-
period systems, but very few of them can merge within the Galactic age. In
summary, both high and low kicks are required for the formation of the
Galactic DNSs.

Figure 2: Distributions of Galactic DNS binaries in the $P_{\rm orb}-e$ plane
under the SN A model with $\alpha_{\rm CE}=1$. The top to bottom panels
correspond to models MTISNAk20150$\alpha$1, MTISNAk20300$\alpha$1,
MTISNAk40150$\alpha$1, and MTISNAk40300$\alpha$1\. The left, middle and right
panels represent the distributions of total DNSs, DNSs with the second SN
being CCSN and ECSN, respectively.

Fig. 3 demonstrates the influence of the initial mass ratio on the formation
of DNSs. Similar to Shao & Li (2018b), both channels I and II contribute to
the formation of DNSs. In the SN B model, $\sim 60\%$ DNSs form through
channel I (especially for near circular short orbit DNSs), while in the SN A
model, the numbers of DNSs formed through channels I and II are roughly
comparable.

Figure 3: Distributions of Galactic DNS systems. The top and bottom panels
correspond to models MTISNAk20150$\alpha$1 and MTISNBk20150$\alpha$1,
respectively. The left, middle, and right panels represent the distributions
of total DNSs, DNSs formed through channel I (high initial mass ratio), and
channel II (initial mass ratio close to 1), respectively.

Fig. 4 compares the results in different SN and MT models. SN B model produces
approximately two times DNSs than other models, and their distribution seems
more consistent with observations, especially with MT II. The main reason is
that in this model $M_{\rm ecs}=(1.83-2.75)M_{\odot}$, which substantially
increases the occurrence rate of ECSN. The relatively low kicks during the SN
reduces the disruption probability of the binaries, and helps form DNSs with
low-eccentricity. SN A and C models show similar distributions. The DNSs
formed in SN D model are mostly concentrated in the region of $\log P_{\rm
orb}(\rm day)\sim-1-0.5$ and $e\sim 0.2-0.6$. Compared with SN A model, SN E
model includes Case BB MT and USSN, but the results are similar. For given SN
prescription, we find that MT II model produces approximately two times DNSs
than MT I model, and approximately four times than MT III model.

Figure 4: Distributions of Galactic DNS systems in different SN and MT models.
The top to bottom rows correspond to SN A-E models, and the left, middle and
right columns MT I-III models, respectively.

####  3.1.2 Quantitative analysis

We use the Bayesian analysis method employed by Andrews et al. (2015) to
quantitatively compare the computed results with observations. We consider two
independent orbital parameters $P_{\rm orb}$ and $e$ for each model. According
to the Bayesian theory, the posterior probability related to model $M$ given
the observed data $D$ is

$P(M|D)=\frac{P(D|M)P(M)}{P(D)},$ (15)

where $D$ denotes $P_{\rm orb}$ and e, $P(D|M)$ is the likelihood of observed
data given that the model is true, and we set it to be $\Lambda(D)$, $P(M)$ is
the prior probability of a specific model, and $P(D)$ is a normalizing
constant which is independent of the model. Since all models have the same
prior probability, $P(M)/P(D)$ can be normalized as a constant $C$. Thus,
$P(M|D)$ is rewritten to be

$P(M|D)=C\Lambda(D).$ (16)

The relative probability of each model is determined by $\Lambda(D)$ because
$C$ is independent of the models. Since the observed DNS systems are
independent, the relative probability is equal to the probabilities of each
independent system,

$\Lambda(D)=\Lambda\left({\log}\,P_{\text{orb}},e\right)=\prod_{i}P\left({\log}\,P_{\text{orb},i},e_{i}\mid
M\right).$ (17)

Here $P({\rm log}\,P_{\rm orb,i},e_{\rm i}|M)$ is the probability density of a
model at a specific point in the ${\log}\,P_{\rm orb}-e$ parameter space.
Similar as Chu et al. (2022), for each pixel ($\log\,P_{\rm orb,i},e_{\rm
i}$), we adopt a 2D normal distribution $N({\log}\,P_{\rm orb},e|{\rm
log}\,P_{\rm orb,i},e_{\rm i},\sigma^{2}_{{\log}\,P_{\rm
orb,i}},\sigma^{2}_{e_{\rm i}})|_{M}$ to avoid zero probability at some
pixels. The standard deviations of $\log\,P_{\rm orb,i}$ and $e_{\rm i}$ are
constant and assigned to be $\sigma_{{\log}\,P_{\rm orb,i}}=0.3$ and
$\sigma_{e_{\rm i}}=0.03$.

So far, 18 DNSs have been observed in the Galactic field, of which 3 DNSs have
not been fully confirmed (see Table 1). Therefore, we consider two sets of
data. The first set of data $D_{1}$ consists of 15 fully confirmed DNSs, and
the second set $D_{2}$ includes all 18 DNSs. Fig. 5 shows the values of
$\Lambda(D_{1})$ as a function of the CE parameter $\alpha_{\rm CE}$. Larger
value of $\Lambda(D)$ means that the model predictions are more consistent
with observations.

Figure 5: The posterior probability for different models. The red, blue,
yellow, green and black colors correspond to SNA-E models, respectively. The
solid, dotted, dashed and dash-dotted lines represent the cases with
$\sigma_{\rm ECSN,CCSN}$=20/150, 20/300, 40/150 and 40/300 km s-1. Note that
for SN D and E models, the kick velocity distribution is not a single Maxwell
distribution (see Table 2). The cyan color corresponds to SNEs. Based on SNE
and assuming that the kicks of both USSN and ECSN follow the same distribution
(i.e., $\sigma_{\rm USSN}=20/40$ km s-1). The left, middle, and right panels
represent MT I, II, and III models, respectively. The best fit model is marked
by a red star.

We draw the following conclusions from Fig. 5:

  * •

The values of $\Lambda(D)$ vary in a wide range with different input
parameters. There is not apparent dependence of $\Lambda(D)$ on the CE
efficiency $\alpha_{\rm CE}$ and the kick velocity distribution.

  * •

Overall nonconservative MT I and II models are considerably better than nearly
conservative MT III models.

  * •

SN B model (rapid CCSN explosion and $M_{\rm ecs}=[1.83-2.75]\,M_{\sun}$ for
ECSN) is more preferred compared with other SN models.

Table A (in Appendix A) shows the results of all the adopted models. We also
present the ranking numbers (level1 and level2) of each model according to
their calculated $\log\,\Lambda(D_{1})$ and $\log\,\Lambda(D_{2})$, as well as
the birth rates and merger rates of DNSs. The top ten models in level1 are
MTIISNEk20150$\alpha$0.1, MTIISNBk20300$\alpha$0.5, MTIISNAk20150$\alpha$0.1,
MTIISNBk40300$\alpha$0.5, MTIISNAk20300$\alpha$0.1, MTISNBk20150$\alpha$0.1,
MTIISNBk40300$\alpha$10, MTIISNBk20150$\alpha$0.1, MTIISNBk20150$\alpha$0.5
and MTIISNCk20150$\alpha$0.1.

###  3.2 The merger rate of Galactic DNSs

Another way to calibrate the models of the formation and evolution of DNSs is
comparing the expected DNS merger rate in the Galaxy by first principles with
that inferred from the known Galactic population of Pulsar-NS systems. Figure
6 depicts the DNS merger rates calculated for different models. They are in
the range of $\sim 1-70$ Myr-1 depending on the model parameters. In general,
the merger rate increases with the CE efficiency $\alpha_{\rm CE}$ and
decreases with the kick velocity. The reason is that larger $\alpha_{\rm CE}$
results in more DNSs, and larger kick velocity causes more binary systems to
be disrupted. However, Figure 6 also shows that the DNS merger rate does not
change monotonically with $\alpha_{\rm CE}$. In some cases the merger rate
decreases with increasing $\alpha_{\rm CE}$. This is because larger
$\alpha_{\rm CE}$ results in wider post-CE binaries, which are more easily
disrupted after the SN. Additionally, the merger rate is critically dependent
on the MT models. The overall merger rates in MT I and II models are higher
than in MT III model.

On the observational side, Kim et al. (2003) developed a Bayesian statistical
method to derive the merger rate probability distribution by modelling the
Galactic DNS population as well as the selection effects based on the observed
properties of known binaries and survey characteristics. Based on this work,
Kalogera et al. (2004) estimated the merger rate $\sim 90$ Myr-1, considering
PSRs B1913+16, B1534+12, and J0737$-$3039A. Considering PSRs B1913+16,
B1534+12, J0737$-$3039A, J1756$-$5521, and J1906+0746 with estimated beaming
correction factors, O’Shaughnessy & Kim (2010) obtained the merger rate $\sim
60$ Myr-1. Taking into account the known DNS binaries with the best
observational constraints, including both PSRs J0737$-$3039A and B, Kim et al.
(2015) derived the merger rate to be $21^{+28}_{-14}$ Myr-1 at 95 per cent
confidence. The method was recently applied by Pol et al. (2019, 2020) with
updated DNS data, including the highly eccentric DNS system PSR J0509+3801
(Lynch et al., 2018), to estimate the merger rate to be $37^{+24}_{-11}$ Myr-1
at 90 per cent confidence. Grunthal et al. (2021) revisited this problem
taking account of recent observation of the longitudinal and latitudinal beam
shape of PSR J1906+0746 (Desvignes et al., 2019), and derived a Galactic DNS
merger rate of $32^{+19}_{-9}$ Myr-1 at 90 per cent confidence.

Fig. 6 compares the calculated and inferred DNS merger rates. For MT I and III
models, to be compatible with these estimates requires $\alpha_{\rm CE}\sim
5-10$, while for MT II models, the allowed values of $\alpha_{\rm CE}$ reside
in a wide range of $0.5-10$. Among the ten best models selected by the
Bayesian method in the Section 3.1.2, only models MTIISNBk20300$\alpha$0.5,
MTIISNBk40300$\alpha$0.5, and MTIISNBk40300$\alpha$10 match the derived DNS
merger rate. So, we conclude that the best fit model is
MTIISNBk20300$\alpha$0.5.

Figure 6: Same as Figure 5 but for DNS merger rates. The orange and cyan
horizontal lines and the shaded areas represent the merger rates of
$37^{+24}_{-11}$ Myr-1 (Pol et al., 2019, 2020) and $32^{+19}_{-9}$ Myr-1
(Desvignes et al., 2019), respectively. Note that the two red hollow stars
represent the best fit model obtained by adopting and not adopting the Doppler
shifting effect.

##  4 Influence of The Doppler Shifting Effect

In the above section we have neglected the possible influence of the Doppler
shifting effect and assumed that all Pulsar-NS binaries have the same
detection probability. Actually it is challenging to detect eccentric binary
pulsars with short orbital periods in pulsar searches considering the Doppler
modulation of the pulsed signal as the pulsar moves with respect to the center
of mass of the binary system during an observation (Johnston & Kulkarni, 1991;
Bagchi et al., 2013). Johnston & Kulkarni (1991) developed an analytical
framework for calculating the reduction in signal-to-noise ratio caused by
binary motion in the case of circular orbits. Bagchi et al. (2013) generalized
the work of Johnston & Kulkarni (1991) and presented analytical expressions to
calculate the sensitivity reduction for orbits of arbitrary eccentricity.
Based on the results of Bagchi et al. (2013), Chattopadhyay et al. (2021)
derived a fitting formula in the case of a 1.4 $M_{\odot}$ NS and an
observation duration of 1000 s at an inclination angle of 60∘, and fit
eccentricities via linear regression for $e=0.1$, 0.5, and 0.8. These authors
suggested that binary pulsars are detectable if

$P_{\rm orb}({\rm day})>m\times P({\rm s})+c,$ (18)

where

$m=m_{\rm m}\times e+c_{\rm m},$ (19)

and

$c=m_{\rm c}\times e+c_{\rm c},$ (20)

where $m_{\rm m}=-8.9$, $c_{\rm m}=-27.68$, $m_{\rm c}=-3.4$, and $c_{\rm
c}=-5.72$.

The above criterion depends on both the orbital evolution and the pulsar’s
spin evolution. The latter is determined by the NS spin period and magnetic
field when it behaves as a radio pulsar. In section 2.2 we simply adopt
uniform distribution for both $P_{\rm s}$ and $\log B$ inferred from their
current observational data. However, due to previous accretion during the CE
stage and the Case BB MT stage, both parameters should be related to the
orbital period and eccentricity of the DNS systems444The wider the initial
orbit of the NS-helium star system, the more evolved the helium star by the
time it fills its RL and the shorter Case BB MT duration. As a result, the
recycling of the first-born NS is less efficient in wider systems, and its
rejuvenated spin period may be longer (Tauris et al., 2015, 2017).. Moreover,
the birth555Here “birth” means the values at the birth of the DNSs. $P_{\rm
orb}$ and $P_{\rm s}$ of DNS systems are not be exactly reflected by their
currently observational data due to the loss of rotational energy and the
decay of orbits caused by GW radiation. Taking these effects into account in a
qualitative manner and placing particular weight on the very wide-orbit DNS
PSR J1930$-$1852 (Swiggum et al., 2015), Tauris et al. (2017) obtained a
simple empirical correlation between their birth $P_{\rm s}$ and $P_{\rm
orb}$,

$P_{\rm s}=(36\pm 14)\,{\rm ms}\,P_{\rm orb}(\rm{day})^{0.4\pm 0.1}.$ (21)

We adopt above equation to evaluate the birth spin periods of the first-born
pulsars.

As to the pulsar’s birth magnetic field, we select ten relatively young DNSs
from Table 1 with the characteristic ages less than $10^{9}$ yr, to ensure
that their magnetic fields have not substantially decayed since the birth of
the DNSs. Assuming that the characteristic age reflects the true age, we use
the GW-induced orbital evolution functions and a Monte Carlo method to obtain
the birth $P_{\rm orb}$ and $e$. Then we employ a nonlinear least squares
method to derive the correlation between the magnetic field and the orbital
period and eccentricity (see Fig. 7),

$\log B({\rm G})=1.47e^{0.051}+8.34P_{\rm orb}({\rm day})^{0.018}.$ (22)

The coefficient of determination of the above fit is 0.764.

Figure 7: The birth magnetic field ($B_{0}$) as a function of birth orbital
period ($P_{\rm orb,0}$) and birth eccentricity ($e_{0}$) for selected first-
born (recycled) NSs in DNS systems. The derived selected DNS birth data are
plotted with blue points. The blue line represents a fit correlation (Eq.
[22]).

Note also that the results of Bagchi et al. (2013) suggest a stiff cut-off of
radio pulsar detection, which means that no pulsar can be detected if Eq. (18)
is not satisfied. We instead revise it to be a soft cut-off: if a binary
crosses the cut-off line, the observable number is reduced by a factor

$f_{\rm d}=\frac{P_{\rm orb}({\rm day})}{m\times P({\rm s})+c}.$ (23)

According to the results of Bagchi et al. (2013), we find that $f_{\rm d}$ is
limited in the range of $0.3-0.55$.

Fig. 8 compares the cumulative distributions of the orbital period ($P_{\rm
orb}$), eccentricity ($e$), magnetic field strength ($B$), and spin period
($P_{\rm s}$) (from top left to bottom right) of DNSs for both observational
and modeled DNSs. We find that considering a stiff cut-off due to the Doppler
shifting effect significantly reduces the number of DNS systems with $P_{\rm
orb}<1$ day, inconsistent with observations, and the modeled distributions of
$e$, $P_{\rm s}$ and $B$ also deviate from observations. On the other hand,
the predicted orbital distribution of DNSs without considering the Doppler
shifting effect is incompatible with observations. In comparison, when soft
cut-off considered, the distributions of all the four parameters seem to
better match observations.

Figure 8: Cumulative distributions of the DNS parameters $P_{\rm orb}$, $e$,
$B$ and $P_{\rm s}$ (from top left to bottom right) for the model of
MTIISNBk20300$\alpha$5\. The blue line represents the observational data, The
orange, green, and red lines represents the calculated DNS population that
does not take into account the Doppler Shifting effect, adopts hard (Eq. [18])
and soft (Eq. [23]) cut-off, respectively. Figure 9: Similar to Figure 5, but
with the Doppler Shifting effect taken into account.

We then re-calculate $\Lambda(D_{1})$ taking into account the Doppler shifting
effect (see Table A), and present the results in Fig. 9. Comparing it with
Fig. 5, we see the following features.

  * •

Considering a soft cut-off improves most of the models in general.

  * •

Similar as in Fig. 5, MT I and II models are better than MT III model, and SN
B model is more preferred compared with other SN models.

  * •

In most cases, $\sigma_{\rm CCSN}=300$ km s-1 and $\sigma_{\rm ECSN}=20$ km
s-1 are preferred over other kick velocity combinations.

In addition, we find that adopting the prescription proposed by Bray &
Eldridge (2018) for the kick velocity of USSN (dark lines) does not
significantly change the properties of the DNS population. However, assuming
that USSN kicks follow the same distribution as ECSN kicks (i.e., $\sigma_{\rm
k}=20/40$ km s-1, cyan lines) can significantly improve the match between SN E
models and observation.

Combining with Fig. 6, we find that among the ten best models selected by the
Bayesian method, model MTIISNBk20300$\alpha$5 seems to best match the derived
DNS merger rate.

##  5 Discussion and Conclusions

In this work, we construct 324 binary evolution models to investigate the
formation of Galactic DNSs, considering the effects of MT stability, SN
mechanisms and related kick velocity distribution, and CE efficiency. We
evolve $10^{8}$ binary systems and obtain a few to several hundred newborn DNS
systems in each model. Using the formulae of Peters (1964), we trace the
orbital evolution of each DNS with time driven by GW radiation. Combining with
empirical relations for known Galactic pulsars, we predict the radio
characteristics and merger rates of the DNSs in the Galaxy. We aim to evaluate
the influence of the key parameters by comparing theory with observation.

###  5.1 CE evolution and MT stability

The evolution of the CE plays a key role in the formation of DNSs, but the
specific physical processes are still not well understood (Ivanova et al.,
2013). Regarding the formation of DNS in the Galaxy, different groups have
used different CE parameters in their studies. Andrews et al. (2015) combined
$\alpha_{\rm CE}$ with $\lambda$ and set $\alpha_{\rm CE}\lambda=0.01$, 0.05,
0.1, 0.2, 0.25, 0.3, and 1.0. Using the Bayesian analysis method, they
concluded that $\alpha_{\rm CE}\lambda=0.3-1$ is consistent with pulsar
observations, while $\alpha_{\rm CE}\lambda\leq 0.25$ can be effectively ruled
out, with the best value being 0.5. Considering that the value of $\lambda$
for supergiants is less than 0.1 (Dewi & Tauris 2000; Podsiadlowski et al.
2003; Wang et al. 2016), their results indicate that $\alpha_{\rm CE}$ should
be $\gtrsim 5$. Vigna-Gómez et al. (2018) used the values of $\lambda$
calculated by Xu & Li (2010b, a) and set $\alpha_{\rm CE}=0.1$, 1 and 10 to
simulate the formation of Galactic DNSs. Their results show that the value of
$\alpha_{\rm CE}$ mainly affects the birth rate and merger rate of DNSs, but
the Bayesian analysis neither clearly favours nor disfavours the $\alpha_{\rm
CE}$ variations with respect to the fiducial value $1$. Kruckow et al. (2018)
calculated the binding energy parameter $\lambda$ of star’s envelope based on
their evolutionary state and set $\alpha_{\rm CE}=0.2$, 0.5, and 0.8, and
showed that the birth rate and merger rate of DNSs increase with increasing
$\alpha_{\rm CE}$. Chu et al. (2022) took $\lambda$ to be 0.5 and $\alpha_{\rm
CE}=0.1$, 1, and 10. Their results of Bayesian analysis showed that the
formation of Galactic DNSs requires a high efficiency factor ($\alpha_{\rm
CE}=10$). Sgalletta et al. (2023) calculated the value of $\lambda$ based on
the model given by Claeys et al. (2014), and set $\alpha_{\rm CE}=0.5$, 1, 3,
and 5. Comparing their results with the predicted DNS merger rate
($37^{+24}_{-11}$Myr-1 in Pol et al., 2020), they concluded that only
$\alpha_{\rm CE}=3$ matches the observation.

The occurrence of CE evolution is critically dependent on the stability of MT
(Soberman et al., 1997; Shao & Li, 2014; Mandel & Broekgaarden, 2022). Packet
(1981) pointed out that, when a star accretes about 10%$\sim$15% of its
original mass, it can be spun up to the critical rotation, and the centrifugal
force would inhibit further accretion (Petrovic et al., 2005). However, in
close binaries tidal synchronization can prevent the star reaching the
critical rotation (de Mink et al., 2013). In addition, stars with strong
magnetic fields may decelerate their rotation under the effect of magnetic
braking (Dervişoǧlu et al., 2010; Deschamps et al., 2013; Shao & Li, 2014).

Be/X-ray binaries are often used to evaluate the efficiency of stellar
accretion (e.g., de Mink et al., 2013; Shao & Li, 2014; Vinciguerra et al.,
2020). Shao & Li (2014) investigated the binary interaction channel for the
formation of Be/X-ray binaries. They showed that, if adopting the rotation-
limited accretion model (Model I), the masses of the accreting stars are
mostly less than $6\,M_{\odot}$, which are inconsistent with the masses of the
Be stars. Vinciguerra et al. (2020) compared the observed population of
Be/X-ray binaries in the Small Magellanic Cloud with simulated populations of
Be/X-ray binaries. Their results suggest that at least 30 per cent of the mass
donated by the progenitor of the NS is typically accreted by the B-star
companion. Additionally both Shao & Li (2014) and Vinciguerra et al. (2020)
disfavor conservative MT. On the other hand, it has been suggested that the
rotation-limited accretion model may help form the Galactic Wolf-Rayet/O
binaries (Shao & Li, 2016), Be/black hole binaries like MWC 656 with an
orbital period of $\sim$ 60 days (Casares et al., 2014), and OB star-black
hole system binaries (Langer et al., 2020), without the occurrence of CE
evolution (Podsiadlowski et al., 2003; Shao & Li, 2014; Grudzinska et al.,
2015).

In this work, we adopt three MT models, the calculated $\lambda$ values by Xu
& Li (2010a, b) and Wang et al. (2016), and $\alpha_{\rm CE}=0.1-10$ to
simulate the formation of DNS in the Galaxy. We find that the Pulsar-NS
distribution does not effectively constrain the MT model and the value of
$\alpha_{\rm CE}$, but the derived DNS merger rate prefers MT II model and
$\alpha_{\rm CE}=0.5\sim 10$. However, our best fit models with and without
considering the Doppler shifting effect reveal $\alpha_{\rm CE}=5$ and 0.5,
respectively. The former is broadly compatible with the studies on the
formation of black hole low-mass X-ray binaries (e.g., Kalogera, 1999;
Podsiadlowski et al., 2003; Kiel & Hurley, 2006; Yungelson & Lasota, 2008),
while the latter is consistent with studies on the formation of white dwarf
binaries (e.g., Zorotovic et al., 2010; Davis et al., 2012; Nandez et al.,
2015; Ge et al., 2022; Zorotovic & Schreiber, 2022; Scherbak & Fuller, 2022).
Since the progenitors of black holes and white dwarfs are respectively mass
stars and intermediate-/low-mass stars, the difference could imply possible
dependence of $\alpha_{\rm CE}$ on stellar mass and mass ratio.

###  5.2 SN models and natal kicks

In this work, we adopt rapid, delay and stochastic CCSN models to estimate the
CO masses. We note that under the same initial conditions, there are no
significant difference between the predicted PSR-NS populations in the rapid
and delay SN models, while the performance of the stochastic SN model seems
not be as well as the other two. The main cause is associated with its
different prescriptions of the kick velocity distribution. There is also large
difference between the DNS merger rates calculated with the rapid/delay models
and with the stochastic SN models.

Besides CCSN, we also consider the contribution from ECSN and USSN. We assume
two possible helium core mass range $M_{\rm ecs}=(1.83-2.25)M_{\odot}$, and
$(1.83-2.75)M_{\odot}$ for ECSN. Our simulations show that using $M_{\rm
ecs}=(1.83-2.75)M_{\odot}$ better match the observations than using $M_{\rm
ecs}=(1.83-2.25)M_{\odot}$, indicating that a large portion of NSs in DNSs
have been formed via ECSN with a relatively small kick velocity (see also Shao
& Li, 2018a).

We adopt Maxwellian distributions for the kick velocities of CCSN and ECSN.
For the specific combinations of the $\sigma_{\rm k}$ values (20/150, 40/150,
20/300, 40/300 km s-1), our Bayesian analysis results indicate that
$\sigma_{\rm k}=20/300$ km s-1 is better than other combinations in most
cases. Furthermore, if taking $\sigma_{\rm k}=300$ km s-1 for CCSN, only MT II
model can match the merger rate of Galactic DNSs.

As to USSN, it is not certain whether the NS is exerted a significantly
smaller kick compared with CCSN. For example, Vigna-Gómez et al. (2018)
adopted a Maxwellian distribution with $\sigma_{\rm k}=30$ km s-1 for USSN
kick velocities following Podsiadlowski et al. (2004), but numerical
simulation by Schneider et al. (2021) suggested that stripped stars on average
give rise to lower-mass NSs, higher explosion energies, and higher kick
velocities ($\sigma_{\rm k}=315\pm 24$ km s-1). In this work, we use the
prescription proposed by Bray & Eldridge (2018) for the kick velocity of USSN,
and find that there are not significantly changes in the properties of the DNS
population and their merger rate. If we instead assume that USSN kicks follow
the same distribution as ECSN kicks (i.e., $\sigma_{\rm k}=20/40$ km s-1),
then the match between SN E models and observation is greatly improved, as
shown in Fig. 9 (in the cyan color and labeled as SN Es). Therefore, the USSN
kick distribution will be an important subject for further research.

###  5.3 Summary

In this work we construct 324 models for the formation of Galactic DNSs taking
into account various possible combinations of the key parameters. We try to
explore their influence by comparing the properties of modeled DNS population
with observation. Our work differs from previous studies in several ways. For
example, compared with Andrews et al. (2015) and Chu et al. (2022), we used
the $\lambda$ values calculated from stellar evolution and considered the
radio properties 666We adopt beamming effect, death line, and doppler shifting
to deal the pulsar select effect. Note that we do not account for CE accretion
when fitting $P_{\rm s}$ and $B$ of the fort-born NS. and merger rate of DNSs
to constrain models. Our results show that the double-core CE channel (Channel
II) also contribute to the formation of the Milky Way DNSs, which was not
considered by Andrews et al. (2015). Compared with Vigna-Gómez et al. (2018)
and Sgalletta et al. (2023), we considered more choices for the MT efficiency.
We systematically explore the effects of different MT models, CE efficiency
factor $\alpha_{\rm CE}$, SNs and NS kicks on the formation of Galactic DNSs,
and obtain useful estimates of the key parameters by comparing models
predictions with both observed Galactic DNSs and their inferred merger rates.
We list all the adopted models in Table 2 and use Bayesian analysis to
evaluate them (see Section 3.2). Our main results are summarized as follows:

1\. Comparing the predicted and derived Galactic DNSs merger rates reveals the
allowed CE efficiency factor $\alpha_{\rm CE}$ ranging from 0.5 to 10.
However, the Bayesian analysis indicates that there is no apparent dependence
of the Pulsar-NS distribution on $\alpha_{\rm CE}$. We find that $\alpha_{\rm
CE}=0.5$ and 5 in our best fit models when the Doppler Shifting effect is not
considered and when a soft cut-off related to the Doppler Shifting effect is
considered, respectively.

2\. Adopting MT II model (with $50\%$ accretion efficiency) would allow more
models to match the Galactic DNS merger rates, while Bayesian analysis
indicates that there is no apparent dependence of the Pulsar-NS distribution
on the MT mode.

3\. Model SN B (with $M_{\rm ecs}=[1.83-2.75]\,M_{\odot}$ for ECSN) seems to
significantly better match observation than other SN models, implying that
ECSN plays an important role in the formation of Galactic DNSs.

4\. In most cases, $\sigma_{\rm ECSN}=20$ km s-1 and $\sigma_{\rm CCSN}=300$
km s-1 combination are preferred compared with other kick velocity
combinations. The results are also sensitively dependent on the kick
prescriptions for USSN.

Considering the fact that the current DNS sample is rather limited and there
are many parameters involved in the formation of DNSs, we caution that our
results are preliminary and subject to several issues. For example, we assume
that the Milky Way has a constant metal abundance and a constant SFR, which
will affect the merger rate of DNSs, even though the impact on the current
merger rate is limited according to previous researches (Chu et al., 2022;
Sgalletta et al., 2023). We do not take into account the merger history of the
Milky Way when calculating the 3D movement of DNSs, which might have
significant influence on the spatial distribution of DNSs. In addition, we
assume that part of the accreting material is ejected out of the binary in the
form of isotropic wind, while the actual situation for mass loss may be more
complicated, which impacts the orbital evolution (Soberman et al., 1997;
Kruckow et al., 2018; Vinciguerra et al., 2020). Together with more detections
of DNSs in the Milky Way, a thorough investigation on the evolutionary history
of DNSs needs to incorporate the formation of massive binaries (e.g., Wang et
al., 2020), high-mass X-ray binaries (e.g., Misra et al., 2023), and binary
pulsars in a self-consistent way. This will be the subject of the future
study.

We thank an anonymous referee for valuable comments that helped improve the
manuscript. This work was supported by the National Key Research and
Development Program of China (2021YFA0718500), the Natural Science Foundation
of Chian under grant No. 12041301, 12121003, 12203051.

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## Appendix A: Results of Bayesian analysis

Table A: The posterior probabilities, ranking numbers, birth rates and merger
rates of Galactic DNSs in all adopted models. The second and third sets
correspond to the results without and with considering the Doppler shifting
effect, respectively.

Model | log$\Lambda(\rm D_{1})$ | level1 | log$\Lambda(\rm D_{2})$ | level2 | log$\Lambda(\rm D_{1})$ | level1 | log$\Lambda(\rm D_{2})$ | level2 | Birth rates | Merger rates
---|---|---|---|---|---|---|---|---|---|---
| No Doppler shifting effect | Doppler shifting effect | ($\rm Myr^{-1}$) | ($\rm Myr^{-1}$)
MTIISNEk20150$\alpha$0.1 | -23.84 | 1 | -28.33 | 1 | -24.9 | 36 | -29.92 | 38 | 8.39 | 5.97
MTIISNBk20300$\alpha$0.5 | -23.9 | 2 | -28.4 | 2 | -25.29 | 52 | -29.94 | 39 | 55.79 | 44.06
MTIISNAk20150$\alpha$0.1 | -24.25 | 3 | -29.61 | 7 | -25.1 | 45 | -30.09 | 47 | 9.33 | 6.58
MTIISNBk40300$\alpha$0.5 | -24.54 | 4 | -30.21 | 16 | -26.13 | 84 | -31.91 | 99 | 32.72 | 27.35
MTIISNAk20300$\alpha$0.1 | -24.69 | 5 | -29.24 | 5 | -24.49 | 23 | -28.77 | 18 | 5.55 | 3.92
MTISNBk20150$\alpha$0.1 | -24.81 | 6 | -28.92 | 3 | -24.05 | 12 | -28.23 | 9 | 13.44 | 6.91
MTIISNBk40300$\alpha$10 | -24.92 | 7 | -30.44 | 19 | -24.51 | 24 | -29.83 | 36 | 69.11 | 40.99
MTIISNBk20150$\alpha$0.1 | -24.96 | 8 | -29.68 | 9 | -24.22 | 17 | -28.48 | 14 | 14.41 | 8.5
MTIISNBk20150$\alpha$0.5 | -25.01 | 9 | -29.88 | 10 | -25.61 | 64 | -30.71 | 61 | 69.19 | 53.66
MTIISNCk20150$\alpha$0.1 | -25.05 | 10 | -30.81 | 27 | -25.21 | 48 | -30.04 | 46 | 10.05 | 7.1
MTIISNBk20300$\alpha$1 | -25.12 | 11 | -29.68 | 8 | -25.21 | 47 | -29.71 | 33 | 57.05 | 33.92
MTIISNBk20300$\alpha$0.1 | -25.2 | 12 | -29.19 | 4 | -23.98 | 9 | -27.7 | 2 | 10.78 | 5.99
MTIISNCk20300$\alpha$1 | -25.2 | 13 | -31.27 | 37 | -26.53 | 113 | -32.34 | 123 | 24.8 | 17.94
MTIIISNBk20150$\alpha$0.1 | -25.24 | 14 | -29.54 | 6 | -23.95 | 7 | -28.11 | 8 | 11.92 | 6.47
MTISNCk20300$\alpha$0.1 | -25.29 | 15 | -29.93 | 12 | -24.44 | 22 | -28.78 | 19 | 5.07 | 3.25
MTIISNBk40300$\alpha$5 | -25.3 | 16 | -30.81 | 26 | -24.32 | 19 | -29.53 | 29 | 59.39 | 38.26
MTIISNEk20300$\alpha$0.1 | -25.32 | 17 | -29.92 | 11 | -23.7 | 3 | -28.09 | 7 | 5.35 | 3.84
MTIISNBk20150$\alpha$1 | -25.45 | 18 | -30.25 | 17 | -25.77 | 70 | -30.48 | 57 | 72.45 | 41.92
MTIISNBk20300$\alpha$5 | -25.47 | 19 | -29.96 | 13 | -23.67 | 2 | -27.77 | 4 | 90.84 | 47.63
MTIISNAk20300$\alpha$5 | -25.48 | 20 | -29.98 | 14 | -25.36 | 55 | -30.44 | 54 | 26.24 | 17.89
MTIIISNEk20300$\alpha$0.1 | -25.51 | 21 | -30.64 | 23 | -25.47 | 58 | -30.03 | 45 | 4.67 | 3.21
MTISNBk40300$\alpha$0.5 | -25.54 | 22 | -31.3 | 39 | -24.66 | 29 | -30.1 | 48 | 10.94 | 6.29
MTISNBk40300$\alpha$5 | -25.59 | 23 | -31.38 | 44 | -24.76 | 32 | -30.16 | 51 | 31.04 | 19.02
MTIISNBk20150$\alpha$5 | -25.59 | 24 | -30.38 | 18 | -24.07 | 13 | -28.53 | 15 | 119.99 | 64.97
MTIISNBk40150$\alpha$10 | -25.59 | 25 | -31.49 | 50 | -25.51 | 60 | -31.24 | 77 | 109.69 | 68.38
MTIISNBk40150$\alpha$0.5 | -25.64 | 26 | -31.41 | 47 | -26.35 | 98 | -32.09 | 112 | 43.13 | 34.56
MTISNCk20150$\alpha$0.1 | -25.74 | 27 | -30.45 | 20 | -25.79 | 73 | -30.83 | 65 | 9.85 | 6.22
MTIIISNCk20150$\alpha$0.1 | -25.79 | 28 | -31.42 | 48 | -26.38 | 101 | -31.74 | 92 | 8.61 | 6.16
MTISNEk20150$\alpha$0.1 | -25.83 | 29 | -31.03 | 31 | -25.07 | 43 | -29.85 | 37 | 8.51 | 5.57
MTIISNBk40150$\alpha$5 | -25.83 | 30 | -31.69 | 54 | -24.93 | 37 | -30.45 | 56 | 81.95 | 52.75
MTIIISNBk20300$\alpha$0.1 | -25.84 | 31 | -30.18 | 15 | -23.83 | 5 | -27.52 | 1 | 9.26 | 4.75
MTISNBk40300$\alpha$10 | -25.88 | 32 | -31.35 | 42 | -24.63 | 28 | -29.97 | 42 | 35.51 | 19.77
MTIISNBk20150$\alpha$10 | -25.89 | 33 | -30.92 | 28 | -24.53 | 25 | -29.14 | 22 | 140.2 | 68.71
MTIISNCk40300$\alpha$0.1 | -25.92 | 34 | -32.5 | 77 | -26.27 | 92 | -32.03 | 109 | 3.99 | 3.15
MTIISNCk20300$\alpha$5 | -25.94 | 35 | -31.35 | 43 | -25.29 | 54 | -30.44 | 55 | 28.88 | 20.1
MTISNBk20150$\alpha$5 | -26.01 | 36 | -31.22 | 36 | -24.4 | 20 | -29.26 | 23 | 72.9 | 39.59
MTISNBk40150$\alpha$1 | -26.02 | 37 | -31.55 | 51 | -24.95 | 39 | -30.43 | 53 | 22.13 | 11.46
MTIISNCk20300$\alpha$10 | -26.03 | 38 | -31.1 | 34 | -25.0 | 41 | -29.77 | 34 | 27.96 | 17.99
MTIIISNBk40150$\alpha$0.1 | -26.05 | 39 | -31.29 | 38 | -26.27 | 91 | -31.88 | 97 | 7.43 | 5.13
MTIISNBk20300$\alpha$2 | -26.09 | 40 | -30.54 | 21 | -25.07 | 44 | -29.28 | 25 | 63.79 | 24.69
MTIISNBk20150$\alpha$2 | -26.09 | 41 | -30.95 | 29 | -25.14 | 46 | -29.66 | 32 | 81.28 | 31.11
MTIISNBk40150$\alpha$0.1 | -26.1 | 42 | -31.94 | 58 | -25.89 | 77 | -31.6 | 87 | 9.52 | 6.82
MTIISNBk20300$\alpha$10 | -26.11 | 43 | -30.8 | 25 | -24.16 | 16 | -28.38 | 13 | 94.37 | 40.4
MTISNBk40150$\alpha$0.5 | -26.11 | 44 | -32.06 | 65 | -25.56 | 63 | -31.15 | 74 | 20.54 | 11.48
MTIISNBk40300$\alpha$2 | -26.13 | 45 | -31.8 | 56 | -25.65 | 65 | -30.91 | 66 | 36.59 | 16.37
MTISNBk40150$\alpha$10 | -26.14 | 46 | -32.11 | 66 | -25.79 | 74 | -31.42 | 82 | 70.8 | 41.3
MTISNBk20150$\alpha$10 | -26.19 | 47 | -31.31 | 40 | -24.77 | 33 | -29.55 | 31 | 86.86 | 42.52
MTISNBk40300$\alpha$0.1 | -26.19 | 48 | -31.08 | 32 | -24.69 | 31 | -29.55 | 30 | 5.6 | 3.73
MTISNBk20150$\alpha$0.5 | -26.22 | 49 | -31.56 | 52 | -24.22 | 18 | -28.62 | 16 | 27.43 | 12.56
MTIISNAk40300$\alpha$5 | -26.23 | 50 | -32.11 | 67 | -26.76 | 130 | -32.82 | 147 | 20.77 | 16.15
MTIISNCk20300$\alpha$0.1 | -26.23 | 51 | -30.72 | 24 | -24.83 | 34 | -29.49 | 28 | 5.57 | 3.88
MTIISNEk20300$\alpha$2 | -26.24 | 52 | -31.32 | 41 | -27.06 | 145 | -32.3 | 120 | 28.52 | 18.32
MTISNEk20300$\alpha$0.1 | -26.26 | 53 | -30.95 | 30 | -24.42 | 21 | -28.9 | 20 | 4.96 | 3.19
MTIIISNBk40300$\alpha$0.1 | -26.27 | 54 | -31.62 | 53 | -24.94 | 38 | -29.96 | 41 | 4.98 | 3.46
MTIISNEk20150$\alpha$0.5 | -26.28 | 55 | -32.31 | 71 | -27.9 | 225 | -34.0 | 208 | 39.42 | 32.75
MTIISNEk20300$\alpha$10 | -26.32 | 56 | -31.08 | 33 | -26.24 | 90 | -30.97 | 67 | 35.1 | 21.44
MTISNBk20300$\alpha$0.1 | -26.32 | 57 | -30.55 | 22 | -23.99 | 10 | -27.78 | 5 | 9.66 | 4.64
MTIISNAk20300$\alpha$10 | -26.33 | 58 | -31.46 | 49 | -25.44 | 57 | -30.24 | 52 | 27.59 | 18.18
MTISNBk40150$\alpha$5 | -26.34 | 59 | -32.41 | 72 | -25.48 | 59 | -31.18 | 75 | 58.86 | 37.2
MTISNBk20150$\alpha$1 | -26.35 | 60 | -31.39 | 45 | -24.12 | 15 | -28.73 | 17 | 30.24 | 12.58
MTIISNBk40150$\alpha$2 | -26.36 | 61 | -31.96 | 60 | -25.77 | 72 | -31.1 | 73 | 48.89 | 20.86
MTIISNAk40300$\alpha$2 | -26.42 | 62 | -32.92 | 90 | -27.34 | 174 | -33.86 | 200 | 13.25 | 8.39
MTIISNAk20150$\alpha$5 | -26.44 | 63 | -31.97 | 61 | -26.34 | 96 | -31.98 | 105 | 55.95 | 34.77
MTIISNEk40300$\alpha$10 | -26.44 | 64 | -32.0 | 62 | -26.46 | 110 | -31.78 | 94 | 29.71 | 21.39
MTIISNCk40300$\alpha$5 | -26.47 | 65 | -32.78 | 82 | -26.89 | 138 | -32.86 | 149 | 23.1 | 18.2
MTIISNBk40300$\alpha$1 | -26.48 | 66 | -32.06 | 64 | -26.17 | 88 | -31.54 | 84 | 32.83 | 21.26
MTIISNBk40150$\alpha$1 | -26.52 | 67 | -32.47 | 75 | -26.56 | 116 | -32.19 | 117 | 44.47 | 27.05
MTISNAk20150$\alpha$0.1 | -26.6 | 68 | -32.18 | 70 | -25.66 | 66 | -30.72 | 62 | 9.39 | 5.78
MTIISNAk20150$\alpha$10 | -26.6 | 69 | -32.02 | 63 | -26.49 | 112 | -32.17 | 116 | 75.21 | 49.15
MTISNBk40300$\alpha$1 | -26.64 | 70 | -32.16 | 69 | -24.1 | 14 | -29.09 | 21 | 11.64 | 5.98
MTISNAk40300$\alpha$0.1 | -26.69 | 71 | -32.41 | 73 | -27.07 | 148 | -32.74 | 141 | 3.96 | 2.99
MTISNBk20300$\alpha$0.5 | -26.72 | 72 | -31.22 | 35 | -23.55 | 1 | -27.72 | 3 | 17.32 | 7.12
MTIISNBk40300$\alpha$0.1 | -26.77 | 73 | -31.39 | 46 | -24.89 | 35 | -30.55 | 60 | 6.36 | 4.66
MTISNBk40150$\alpha$2 | -26.79 | 74 | -32.92 | 91 | -25.75 | 69 | -31.42 | 81 | 32.19 | 14.48
MTIISNCk40150$\alpha$5 | -26.82 | 75 | -33.53 | 109 | -27.57 | 195 | -33.99 | 207 | 54.32 | 39.87
MTIISNCk40300$\alpha$2 | -26.82 | 76 | -32.88 | 87 | -27.61 | 202 | -33.39 | 174 | 14.06 | 9.1
MTISNCk20150$\alpha$2 | -26.82 | 77 | -32.91 | 88 | -27.31 | 172 | -33.42 | 177 | 27.21 | 15.71
MTIISNCk20150$\alpha$5 | -26.83 | 78 | -32.84 | 85 | -26.43 | 105 | -32.04 | 110 | 65.55 | 42.85
MTIISNAk40300$\alpha$10 | -26.87 | 79 | -33.26 | 98 | -26.96 | 142 | -33.07 | 158 | 25.36 | 19.48
MTIISNEk40150$\alpha$2 | -26.9 | 80 | -32.92 | 89 | -27.36 | 177 | -33.16 | 161 | 27.54 | 16.62
MTISNEk20150$\alpha$1 | -26.92 | 81 | -32.5 | 76 | -26.39 | 102 | -32.11 | 113 | 17.78 | 11.25
MTISNBk20300$\alpha$10 | -26.93 | 82 | -31.93 | 57 | -24.0 | 11 | -28.33 | 11 | 50.08 | 20.71
MTIISNEk20300$\alpha$1 | -26.95 | 83 | -31.74 | 55 | -26.48 | 111 | -31.81 | 95 | 26.14 | 19.02
MTIISNEk20150$\alpha$2 | -27.08 | 84 | -32.51 | 78 | -27.0 | 144 | -32.44 | 127 | 42.0 | 24.29
MTIISNEk20300$\alpha$0.5 | -27.11 | 85 | -32.88 | 86 | -27.82 | 217 | -33.97 | 205 | 27.31 | 23.46
MTISNEk40150$\alpha$1 | -27.14 | 86 | -33.68 | 116 | -27.1 | 152 | -32.86 | 152 | 15.78 | 10.77
MTISNBk20300$\alpha$5 | -27.14 | 87 | -32.16 | 68 | -23.86 | 6 | -28.34 | 12 | 44.33 | 21.22
MTISNBk20300$\alpha$1 | -27.15 | 88 | -31.94 | 59 | -23.74 | 4 | -27.86 | 6 | 19.39 | 7.02
MTIISNAk20300$\alpha$1 | -27.15 | 89 | -33.6 | 113 | -27.17 | 156 | -32.79 | 144 | 23.51 | 16.41
MTISNBk40300$\alpha$2 | -27.16 | 90 | -33.51 | 108 | -24.69 | 30 | -30.03 | 44 | 17.6 | 8.37
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MTIISNDk20$\alpha$0.1 | -27.18 | 92 | -33.59 | 111 | -27.78 | 214 | -34.09 | 215 | 12.98 | 9.22
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MTISNEk20150$\alpha$5 | -27.28 | 96 | -33.13 | 95 | -26.95 | 141 | -33.02 | 156 | 41.98 | 29.84
MTISNCk20150$\alpha$10 | -27.29 | 97 | -33.68 | 117 | -27.1 | 151 | -33.16 | 160 | 53.86 | 34.62
MTIISNCk40300$\alpha$10 | -27.29 | 98 | -33.27 | 99 | -26.62 | 120 | -32.58 | 134 | 25.42 | 19.16
MTISNCk20150$\alpha$5 | -27.33 | 99 | -33.64 | 115 | -27.51 | 192 | -33.68 | 189 | 51.0 | 36.73
MTIISNEk20150$\alpha$10 | -27.36 | 100 | -32.68 | 80 | -26.29 | 93 | -31.4 | 79 | 70.98 | 46.52
MTISNEk20150$\alpha$10 | -27.38 | 101 | -33.49 | 104 | -26.76 | 131 | -32.55 | 131 | 50.14 | 35.57
MTISNBk20300$\alpha$2 | -27.39 | 102 | -32.57 | 79 | -23.96 | 8 | -28.23 | 10 | 27.2 | 9.45
MTIISNCk20150$\alpha$2 | -27.4 | 103 | -32.84 | 84 | -27.08 | 150 | -32.86 | 150 | 44.44 | 22.93
MTISNEk40300$\alpha$1 | -27.41 | 104 | -33.01 | 92 | -26.53 | 114 | -32.39 | 125 | 6.51 | 4.96
MTIISNEk20150$\alpha$1 | -27.41 | 105 | -33.16 | 96 | -27.43 | 185 | -33.3 | 168 | 40.56 | 27.23
MTIISNAk40300$\alpha$0.1 | -27.41 | 106 | -34.83 | 162 | -26.34 | 95 | -32.33 | 122 | 3.86 | 3.1
MTISNEk40150$\alpha$10 | -27.43 | 107 | -34.06 | 126 | -27.37 | 179 | -33.73 | 192 | 48.2 | 35.54
MTIISNCk40150$\alpha$0.1 | -27.43 | 108 | -33.78 | 119 | -28.1 | 242 | -34.74 | 247 | 7.97 | 6.09
MTIISNEk40150$\alpha$10 | -27.44 | 109 | -33.35 | 101 | -26.78 | 132 | -32.48 | 128 | 63.31 | 46.43
MTIISNAk20300$\alpha$2 | -27.47 | 110 | -32.76 | 81 | -26.35 | 99 | -31.64 | 89 | 21.71 | 13.13
MTISNAk20150$\alpha$5 | -27.48 | 111 | -33.49 | 106 | -27.5 | 190 | -33.8 | 195 | 47.08 | 33.46
MTISNCk40300$\alpha$0.1 | -27.52 | 112 | -33.47 | 102 | -27.26 | 167 | -33.81 | 197 | 3.97 | 3.04
MTIISNCk20150$\alpha$1 | -27.56 | 113 | -33.59 | 112 | -27.45 | 187 | -33.44 | 178 | 43.53 | 28.26
MTIIISNEk20150$\alpha$10 | -27.58 | 114 | -34.13 | 128 | -28.31 | 258 | -35.26 | 276 | 34.19 | 28.6
MTIISNEk40150$\alpha$0.5 | -27.58 | 115 | -34.26 | 135 | -28.64 | 279 | -35.11 | 272 | 26.21 | 22.1
MTIISNCk20150$\alpha$10 | -27.62 | 116 | -33.27 | 100 | -26.42 | 104 | -31.96 | 103 | 77.88 | 48.92
MTIISNCk20150$\alpha$0.5 | -27.62 | 117 | -34.31 | 137 | -27.93 | 229 | -34.35 | 229 | 44.76 | 37.0
MTISNEk20300$\alpha$10 | -27.63 | 118 | -33.49 | 105 | -26.7 | 124 | -32.56 | 132 | 18.75 | 13.98
MTIISNCk20300$\alpha$2 | -27.64 | 119 | -32.82 | 83 | -26.82 | 136 | -32.08 | 111 | 23.19 | 14.19
MTISNEk40300$\alpha$10 | -27.64 | 120 | -33.71 | 118 | -27.24 | 165 | -33.18 | 162 | 16.86 | 13.68
MTISNCk40150$\alpha$1 | -27.64 | 121 | -34.03 | 125 | -27.45 | 188 | -33.8 | 196 | 17.1 | 10.48
MTISNAk20150$\alpha$1 | -27.68 | 122 | -33.91 | 123 | -26.34 | 97 | -32.19 | 118 | 18.03 | 10.7
MTIIISNAk20300$\alpha$0.1 | -27.7 | 123 | -33.11 | 94 | -25.29 | 53 | -30.0 | 43 | 4.94 | 3.43
MTISNAk40150$\alpha$10 | -27.71 | 124 | -34.22 | 133 | -27.94 | 231 | -34.68 | 244 | 52.4 | 36.55
MTIISNEk40150$\alpha$0.1 | -27.72 | 125 | -34.7 | 158 | -27.23 | 163 | -32.89 | 153 | 6.35 | 4.95
MTIISNAk20150$\alpha$0.5 | -27.75 | 126 | -34.15 | 129 | -28.44 | 265 | -34.88 | 255 | 42.89 | 35.11
MTIISNEk40300$\alpha$1 | -27.76 | 127 | -33.96 | 124 | -27.86 | 222 | -33.88 | 202 | 15.9 | 11.97
MTIIISNCk40150$\alpha$0.1 | -27.76 | 128 | -34.39 | 141 | -27.19 | 160 | -33.67 | 187 | 6.53 | 4.98
MTIIISNAk40150$\alpha$0.5 | -27.77 | 129 | -34.54 | 150 | -27.85 | 221 | -34.09 | 213 | 6.05 | 3.11
MTIISNEk20150$\alpha$5 | -27.79 | 130 | -33.19 | 97 | -27.88 | 223 | -33.54 | 182 | 58.53 | 37.62
MTIIISNAk20150$\alpha$0.1 | -27.83 | 131 | -33.62 | 114 | -26.07 | 81 | -31.03 | 70 | 8.02 | 5.54
MTIISNAk20150$\alpha$1 | -27.87 | 132 | -34.16 | 130 | -28.02 | 236 | -34.2 | 220 | 41.8 | 26.23
MTISNEk40150$\alpha$5 | -27.87 | 133 | -34.63 | 154 | -27.82 | 216 | -34.24 | 222 | 39.9 | 29.52
MTIIISNCk20150$\alpha$0.5 | -27.89 | 134 | -34.41 | 142 | -26.8 | 133 | -32.66 | 139 | 8.28 | 3.58
MTIISNAk40300$\alpha$1 | -27.89 | 135 | -34.99 | 172 | -28.46 | 268 | -35.0 | 265 | 14.2 | 10.28
MTIIISNEk40150$\alpha$10 | -27.92 | 136 | -34.98 | 170 | -28.69 | 281 | -35.38 | 280 | 32.13 | 27.88
MTISNDk20$\alpha$5 | -27.92 | 137 | -34.44 | 143 | -28.25 | 250 | -34.65 | 243 | 36.9 | 27.44
MTISNDk20$\alpha$10 | -27.93 | 138 | -34.32 | 138 | -27.61 | 201 | -33.65 | 186 | 37.4 | 22.56
MTIISNAk40150$\alpha$10 | -27.94 | 139 | -34.88 | 166 | -27.75 | 210 | -34.32 | 226 | 69.12 | 50.59
MTISNCk20150$\alpha$0.5 | -27.96 | 140 | -34.45 | 144 | -26.82 | 137 | -32.84 | 148 | 19.22 | 12.0
MTISNCk40150$\alpha$5 | -28.01 | 141 | -34.66 | 155 | -27.92 | 227 | -34.6 | 240 | 48.52 | 36.14
MTIISNCk40150$\alpha$0.5 | -28.01 | 142 | -35.12 | 181 | -28.82 | 286 | -35.37 | 279 | 30.11 | 24.79
MTISNAk20150$\alpha$10 | -28.02 | 143 | -34.46 | 145 | -27.25 | 166 | -33.32 | 169 | 54.88 | 36.6
MTIISNEk40300$\alpha$2 | -28.04 | 144 | -33.06 | 93 | -27.08 | 149 | -32.51 | 129 | 17.06 | 11.45
MTIISNCk40150$\alpha$1 | -28.06 | 145 | -34.79 | 160 | -28.59 | 275 | -35.55 | 283 | 28.53 | 18.88
MTIISNAk40150$\alpha$5 | -28.08 | 146 | -34.85 | 163 | -27.35 | 176 | -33.76 | 193 | 45.25 | 31.96
MTIIISNEk20150$\alpha$0.1 | -28.1 | 147 | -33.89 | 121 | -25.26 | 50 | -30.55 | 59 | 6.86 | 4.69
MTIIISNBk40150$\alpha$10 | -28.11 | 148 | -34.68 | 157 | -28.18 | 245 | -35.06 | 269 | 50.13 | 36.61
MTIIISNDk40$\alpha$2 | -28.12 | 149 | -35.28 | 187 | -27.4 | 183 | -33.51 | 181 | 3.34 | 1.61
MTISNEk40300$\alpha$0.1 | -28.13 | 150 | -34.76 | 159 | -26.73 | 125 | -32.02 | 108 | 3.72 | 2.92
MTISNAk40150$\alpha$0.5 | -28.14 | 151 | -35.03 | 176 | -27.58 | 196 | -34.21 | 221 | 16.1 | 10.77
MTISNDk40$\alpha$5 | -28.14 | 152 | -35.18 | 185 | -29.06 | 292 | -36.08 | 289 | 36.14 | 27.45
MTISNEk40150$\alpha$0.5 | -28.17 | 153 | -34.46 | 146 | -28.19 | 248 | -34.58 | 238 | 15.53 | 11.17
MTIISNCk40150$\alpha$10 | -28.17 | 154 | -34.97 | 168 | -28.35 | 261 | -34.97 | 262 | 71.2 | 50.35
MTIIISNBk40150$\alpha$5 | -28.18 | 155 | -34.99 | 171 | -28.28 | 257 | -34.9 | 259 | 28.49 | 21.83
MTIIISNBk40300$\alpha$10 | -28.18 | 156 | -35.53 | 199 | -28.14 | 243 | -35.15 | 274 | 29.45 | 22.06
MTISNDk20$\alpha$0.1 | -28.18 | 157 | -34.52 | 149 | -27.06 | 146 | -32.39 | 124 | 9.03 | 4.81
MTISNBk40150$\alpha$0.1 | -28.2 | 158 | -34.68 | 156 | -25.88 | 76 | -31.89 | 98 | 9.43 | 6.04
MTISNAk40150$\alpha$0.1 | -28.22 | 159 | -33.54 | 110 | -27.64 | 204 | -34.36 | 230 | 8.04 | 5.48
MTIISNAk40150$\alpha$2 | -28.23 | 160 | -35.17 | 184 | -28.35 | 262 | -34.62 | 241 | 27.22 | 14.38
MTIISNAk40150$\alpha$0.5 | -28.27 | 161 | -35.29 | 188 | -29.14 | 293 | -36.12 | 290 | 28.09 | 23.13
MTISNEk40150$\alpha$0.1 | -28.3 | 162 | -35.48 | 195 | -27.19 | 159 | -33.27 | 166 | 7.08 | 5.19
MTISNAk20150$\alpha$0.5 | -28.3 | 163 | -34.33 | 139 | -26.76 | 129 | -32.65 | 136 | 17.88 | 11.27
MTISNEk20150$\alpha$2 | -28.3 | 164 | -34.81 | 161 | -27.14 | 155 | -33.02 | 155 | 24.07 | 13.93
MTISNEk20150$\alpha$0.5 | -28.31 | 165 | -34.59 | 153 | -26.74 | 127 | -32.33 | 121 | 17.49 | 11.67
MTISNDk40$\alpha$2 | -28.32 | 166 | -35.79 | 213 | -28.19 | 246 | -34.56 | 237 | 15.2 | 7.26
MTIISNCk40150$\alpha$2 | -28.32 | 167 | -35.07 | 177 | -27.77 | 212 | -34.11 | 216 | 29.3 | 15.85
MTISNAk20300$\alpha$0.5 | -28.34 | 168 | -34.06 | 127 | -26.06 | 80 | -31.64 | 90 | 7.49 | 5.09
MTISNCk20300$\alpha$0.5 | -28.34 | 169 | -34.16 | 131 | -25.87 | 75 | -31.1 | 72 | 8.03 | 5.42
MTIIISNEk40150$\alpha$5 | -28.35 | 170 | -35.01 | 175 | -27.75 | 211 | -34.51 | 234 | 18.73 | 16.18
MTIISNDk20$\alpha$5 | -28.36 | 171 | -35.15 | 182 | -28.54 | 271 | -34.82 | 251 | 55.22 | 40.63
MTIISNEk20300$\alpha$5 | -28.4 | 172 | -33.9 | 122 | -28.79 | 285 | -34.27 | 223 | 40.68 | 25.66
MTIISNAk20150$\alpha$2 | -28.41 | 173 | -34.59 | 152 | -27.44 | 186 | -33.29 | 167 | 41.61 | 20.99
MTIIISNBk40150$\alpha$1 | -28.43 | 174 | -34.21 | 132 | -25.38 | 56 | -30.5 | 58 | 8.05 | 2.21
MTIIISNBk40150$\alpha$0.5 | -28.43 | 175 | -34.37 | 140 | -25.51 | 61 | -30.78 | 63 | 8.64 | 2.9
MTISNEk40150$\alpha$2 | -28.46 | 176 | -34.98 | 169 | -27.54 | 194 | -33.95 | 204 | 22.12 | 13.53
MTISNCk20300$\alpha$5 | -28.5 | 177 | -34.55 | 151 | -28.19 | 247 | -34.59 | 239 | 18.47 | 14.17
MTIISNAk20300$\alpha$0.5 | -28.53 | 178 | -34.87 | 165 | -28.15 | 244 | -34.18 | 219 | 26.72 | 23.03
MTISNEk40300$\alpha$2 | -28.54 | 179 | -35.43 | 194 | -27.94 | 233 | -34.56 | 236 | 8.77 | 6.35
MTIIISNAk40150$\alpha$0.1 | -28.54 | 180 | -35.0 | 173 | -27.83 | 220 | -34.28 | 224 | 6.09 | 4.63
MTIISNAk40150$\alpha$0.1 | -28.57 | 181 | -36.41 | 233 | -27.83 | 218 | -35.04 | 266 | 7.37 | 5.6
MTIISNEk40300$\alpha$5 | -28.61 | 182 | -35.01 | 174 | -28.74 | 283 | -34.84 | 253 | 29.9 | 21.15
MTISNDk40$\alpha$10 | -28.63 | 183 | -35.69 | 204 | -28.54 | 272 | -35.33 | 277 | 36.63 | 22.8
MTIISNEk40150$\alpha$1 | -28.64 | 184 | -35.15 | 183 | -28.72 | 282 | -35.06 | 270 | 26.44 | 18.32
MTIIISNDk20$\alpha$10 | -28.66 | 185 | -35.61 | 201 | -30.77 | 309 | -37.86 | 306 | 30.0 | 22.26
MTIIISNAk40300$\alpha$0.1 | -28.67 | 186 | -36.28 | 229 | -27.77 | 213 | -34.88 | 256 | 3.49 | 2.78
MTISNAk20150$\alpha$2 | -28.7 | 187 | -35.11 | 180 | -27.31 | 171 | -33.32 | 170 | 24.5 | 13.0
MTISNAk20300$\alpha$10 | -28.77 | 188 | -35.37 | 191 | -26.8 | 134 | -32.65 | 135 | 17.84 | 12.52
MTISNAk40150$\alpha$5 | -28.8 | 189 | -36.11 | 224 | -28.54 | 273 | -35.46 | 282 | 44.8 | 33.09
MTIIISNBk20150$\alpha$10 | -28.8 | 190 | -35.7 | 208 | -27.42 | 184 | -33.68 | 188 | 59.94 | 37.45
MTISNCk40150$\alpha$0.5 | -28.8 | 191 | -35.97 | 219 | -27.14 | 154 | -33.41 | 176 | 17.65 | 11.62
MTISNCk20300$\alpha$10 | -28.84 | 192 | -34.47 | 147 | -27.07 | 147 | -33.12 | 159 | 17.59 | 12.18
MTISNDk40$\alpha$0.5 | -28.86 | 193 | -36.53 | 237 | -28.41 | 264 | -35.06 | 268 | 14.57 | 8.33
MTISNDk40$\alpha$1 | -28.88 | 194 | -36.6 | 244 | -29.04 | 290 | -36.17 | 291 | 14.51 | 7.85
MTISNEk20300$\alpha$2 | -28.89 | 195 | -34.86 | 164 | -27.21 | 162 | -33.2 | 164 | 10.35 | 6.74
MTISNDk20$\alpha$2 | -28.91 | 196 | -35.09 | 178 | -27.58 | 198 | -33.36 | 172 | 16.07 | 7.16
MTISNAk40300$\alpha$5 | -28.92 | 197 | -35.71 | 209 | -28.27 | 255 | -34.83 | 252 | 15.33 | 12.64
MTIIISNCk20150$\alpha$1 | -28.93 | 198 | -35.61 | 202 | -26.61 | 119 | -32.16 | 115 | 7.03 | 2.64
MTISNEk40300$\alpha$5 | -28.93 | 199 | -36.53 | 238 | -28.44 | 266 | -34.81 | 249 | 15.91 | 13.16
MTISNAk40150$\alpha$1 | -28.97 | 200 | -36.56 | 240 | -27.97 | 234 | -34.48 | 233 | 16.1 | 10.23
MTIIISNEk40150$\alpha$0.1 | -28.97 | 201 | -36.33 | 231 | -27.65 | 205 | -33.83 | 199 | 5.13 | 3.95
MTIIISNCk20150$\alpha$2 | -28.98 | 202 | -35.09 | 179 | -25.75 | 68 | -31.04 | 71 | 6.38 | 2.4
MTIISNCk40300$\alpha$1 | -28.98 | 203 | -35.64 | 203 | -28.25 | 251 | -35.13 | 273 | 15.01 | 11.24
MTIISNDk20$\alpha$10 | -28.99 | 204 | -35.7 | 207 | -28.49 | 269 | -34.81 | 250 | 54.46 | 32.52
MTISNCk40300$\alpha$5 | -29.0 | 205 | -35.72 | 210 | -28.37 | 263 | -35.84 | 287 | 16.5 | 13.76
MTIISNDk20$\alpha$0.5 | -29.01 | 206 | -36.08 | 222 | -30.33 | 307 | -37.29 | 300 | 50.06 | 41.47
MTISNDk20$\alpha$0.5 | -29.09 | 207 | -35.8 | 215 | -27.38 | 182 | -33.7 | 191 | 15.48 | 8.46
MTIIISNAk40150$\alpha$2 | -29.13 | 208 | -35.96 | 218 | -26.46 | 108 | -32.66 | 138 | 4.62 | 2.18
MTIIISNAk40150$\alpha$10 | -29.14 | 209 | -36.66 | 246 | -30.17 | 303 | -37.96 | 308 | 39.18 | 31.91
MTIIISNDk20$\alpha$0.5 | -29.15 | 210 | -34.52 | 148 | -27.59 | 199 | -34.08 | 211 | 4.98 | 1.5
MTIIISNAk40150$\alpha$5 | -29.15 | 211 | -36.58 | 241 | -30.05 | 301 | -38.13 | 310 | 23.42 | 20.4
MTISNEk20300$\alpha$1 | -29.15 | 212 | -34.95 | 167 | -26.45 | 107 | -31.69 | 91 | 8.17 | 5.33
MTISNEk20300$\alpha$5 | -29.16 | 213 | -35.49 | 196 | -27.91 | 226 | -33.97 | 206 | 17.9 | 13.67
MTISNAk20300$\alpha$0.1 | -29.21 | 214 | -34.22 | 134 | -24.63 | 27 | -29.29 | 26 | 5.06 | 3.21
MTISNAk20300$\alpha$5 | -29.22 | 215 | -36.09 | 223 | -27.93 | 230 | -34.32 | 227 | 17.5 | 13.07
MTISNAk20300$\alpha$1 | -29.22 | 216 | -35.31 | 190 | -25.99 | 79 | -32.0 | 107 | 7.08 | 4.54
MTISNCk40150$\alpha$2 | -29.23 | 217 | -36.67 | 247 | -28.02 | 237 | -34.94 | 261 | 25.0 | 15.3
MTIISNAk40300$\alpha$0.5 | -29.24 | 218 | -36.58 | 243 | -28.78 | 284 | -36.22 | 293 | 16.56 | 14.6
MTIIISNCk20300$\alpha$0.1 | -29.25 | 219 | -35.22 | 186 | -26.46 | 109 | -31.41 | 80 | 5.34 | 3.8
MTIIISNEk20150$\alpha$5 | -29.28 | 220 | -35.39 | 192 | -26.75 | 128 | -32.72 | 140 | 22.26 | 18.39
MTIIISNEk20150$\alpha$1 | -29.29 | 221 | -35.51 | 197 | -26.56 | 117 | -31.95 | 102 | 6.74 | 2.95
MTIIISNDk20$\alpha$0.1 | -29.31 | 222 | -34.29 | 136 | -28.27 | 254 | -34.33 | 228 | 9.16 | 3.23
MTIIISNEk40150$\alpha$0.5 | -29.31 | 223 | -35.58 | 200 | -28.02 | 235 | -34.89 | 258 | 6.14 | 3.5
MTISNAk40150$\alpha$2 | -29.33 | 224 | -36.33 | 230 | -28.26 | 252 | -34.93 | 260 | 23.26 | 13.19
MTIIISNBk40150$\alpha$2 | -29.34 | 225 | -35.7 | 206 | -25.06 | 42 | -30.12 | 49 | 8.04 | 2.16
MTISNCk20150$\alpha$1 | -29.34 | 226 | -35.95 | 217 | -26.43 | 106 | -32.41 | 126 | 18.88 | 10.85
MTISNCk40300$\alpha$10 | -29.37 | 227 | -36.84 | 251 | -28.28 | 256 | -34.98 | 264 | 15.73 | 12.21
MTIISNEk40150$\alpha$5 | -29.38 | 228 | -36.47 | 234 | -28.49 | 270 | -34.86 | 254 | 45.3 | 32.18
MTISNEk40300$\alpha$0.5 | -29.47 | 229 | -35.4 | 193 | -27.73 | 208 | -33.89 | 203 | 6.85 | 5.29
MTISNAk40300$\alpha$10 | -29.54 | 230 | -36.95 | 254 | -28.62 | 278 | -35.4 | 281 | 15.72 | 12.38
MTISNCk40300$\alpha$0.5 | -29.54 | 231 | -37.51 | 268 | -27.66 | 206 | -34.7 | 246 | 6.53 | 5.07
MTIIISNBk40300$\alpha$0.5 | -29.57 | 232 | -35.89 | 216 | -25.77 | 71 | -31.0 | 68 | 5.29 | 1.81
MTISNCk20300$\alpha$1 | -29.6 | 233 | -35.3 | 189 | -26.13 | 85 | -31.56 | 86 | 7.53 | 4.8
MTISNCk40150$\alpha$0.1 | -29.62 | 234 | -37.08 | 257 | -27.37 | 180 | -33.86 | 201 | 8.4 | 5.84
MTIIISNCk40150$\alpha$2 | -29.65 | 235 | -36.86 | 253 | -26.9 | 139 | -32.65 | 137 | 5.04 | 2.32
MTIISNDk40$\alpha$0.5 | -29.71 | 236 | -37.85 | 281 | -31.53 | 315 | -38.85 | 313 | 46.89 | 39.16
MTIIISNEk40150$\alpha$2 | -29.72 | 237 | -35.8 | 214 | -26.82 | 135 | -32.19 | 119 | 4.59 | 2.19
MTIIISNEk40300$\alpha$10 | -29.72 | 238 | -38.86 | 293 | -29.0 | 289 | -36.61 | 296 | 15.94 | 14.41
MTISNCk20300$\alpha$2 | -29.77 | 239 | -35.76 | 212 | -26.97 | 143 | -32.56 | 133 | 10.46 | 6.89
MTISNAk40300$\alpha$2 | -29.8 | 240 | -36.58 | 242 | -28.06 | 238 | -34.89 | 257 | 8.57 | 6.14
MTIIISNCk40150$\alpha$10 | -29.8 | 241 | -37.44 | 266 | -29.77 | 298 | -36.82 | 299 | 40.06 | 31.6
MTIIISNAk20150$\alpha$5 | -29.82 | 242 | -36.84 | 252 | -28.82 | 287 | -35.62 | 285 | 28.09 | 23.54
MTIIISNBk40300$\alpha$5 | -29.84 | 243 | -37.05 | 256 | -28.07 | 239 | -34.97 | 263 | 18.87 | 14.5
MTIIISNEk40150$\alpha$1 | -29.84 | 244 | -36.2 | 226 | -27.36 | 178 | -33.19 | 163 | 5.28 | 2.76
MTIISNCk40300$\alpha$0.5 | -29.94 | 245 | -37.31 | 263 | -30.08 | 302 | -37.57 | 303 | 17.43 | 15.47
MTIIISNCk40150$\alpha$1 | -29.96 | 246 | -37.73 | 279 | -27.32 | 173 | -33.64 | 185 | 5.38 | 2.46
MTIISNAk40150$\alpha$1 | -29.96 | 247 | -37.03 | 255 | -28.61 | 277 | -35.36 | 278 | 27.2 | 17.56
MTISNAk40300$\alpha$1 | -29.97 | 248 | -37.96 | 283 | -27.5 | 191 | -34.75 | 248 | 5.65 | 4.27
MTIIISNEk20150$\alpha$0.5 | -29.97 | 249 | -36.37 | 232 | -27.14 | 153 | -32.78 | 143 | 7.61 | 3.69
MTISNAk20300$\alpha$2 | -30.01 | 250 | -36.48 | 235 | -27.62 | 203 | -34.05 | 210 | 10.19 | 6.3
MTIISNEk40300$\alpha$0.5 | -30.01 | 251 | -37.59 | 270 | -28.54 | 274 | -35.06 | 267 | 16.75 | 14.89
MTIIISNCk20150$\alpha$5 | -30.03 | 252 | -37.24 | 262 | -29.57 | 296 | -36.29 | 294 | 31.95 | 27.04
MTISNCk40300$\alpha$2 | -30.06 | 253 | -37.08 | 259 | -27.94 | 232 | -34.31 | 225 | 8.78 | 6.56
MTIISNDk40$\alpha$5 | -30.06 | 254 | -37.96 | 282 | -30.2 | 304 | -37.78 | 305 | 52.56 | 40.35
MTIIISNAk20150$\alpha$10 | -30.07 | 255 | -37.68 | 276 | -29.73 | 297 | -36.62 | 297 | 42.27 | 32.26
MTISNDk20$\alpha$1 | -30.07 | 256 | -36.74 | 249 | -27.29 | 169 | -33.37 | 173 | 15.53 | 8.02
MTIIISNAk20150$\alpha$0.5 | -30.09 | 257 | -36.22 | 227 | -26.29 | 94 | -31.98 | 104 | 7.52 | 3.36
MTIIISNCk20150$\alpha$10 | -30.09 | 258 | -37.18 | 260 | -29.21 | 294 | -36.21 | 292 | 43.53 | 32.3
MTISNDk40$\alpha$0.1 | -30.1 | 259 | -37.32 | 265 | -29.52 | 295 | -36.75 | 298 | 8.05 | 4.79
MTIISNDk40$\alpha$10 | -30.13 | 260 | -37.65 | 273 | -30.05 | 300 | -37.65 | 304 | 52.55 | 32.96
MTIIISNBk40300$\alpha$1 | -30.14 | 261 | -36.56 | 239 | -25.52 | 62 | -30.8 | 64 | 4.87 | 1.15
MTIIISNDk20$\alpha$1 | -30.14 | 262 | -36.16 | 225 | -26.64 | 121 | -31.93 | 101 | 3.57 | 0.96
MTIISNDk20$\alpha$1 | -30.14 | 263 | -37.08 | 258 | -31.02 | 311 | -38.06 | 309 | 51.21 | 29.24
MTIIISNAk20300$\alpha$1 | -30.16 | 264 | -36.04 | 220 | -26.57 | 118 | -32.16 | 114 | 2.97 | 1.13
MTIIISNBk20150$\alpha$0.5 | -30.17 | 265 | -35.69 | 205 | -25.24 | 49 | -29.78 | 35 | 13.84 | 3.06
MTISNEk20300$\alpha$0.5 | -30.19 | 266 | -36.08 | 221 | -27.23 | 164 | -32.81 | 146 | 8.57 | 5.78
MTIIISNAk40300$\alpha$1 | -30.21 | 267 | -37.19 | 261 | -26.42 | 103 | -31.62 | 88 | 1.89 | 1.05
MTIIISNBk20150$\alpha$1 | -30.21 | 268 | -35.73 | 211 | -24.98 | 40 | -29.48 | 27 | 13.83 | 2.46
MTIIISNCk40150$\alpha$5 | -30.25 | 269 | -38.15 | 286 | -29.86 | 299 | -37.34 | 301 | 26.66 | 23.25
MTIIISNBk20150$\alpha$5 | -30.38 | 270 | -36.65 | 245 | -27.37 | 181 | -33.5 | 180 | 38.84 | 25.39
MTIIISNAk20150$\alpha$1 | -30.4 | 271 | -35.51 | 198 | -25.89 | 78 | -31.48 | 83 | 6.46 | 2.45
MTIIISNBk20300$\alpha$10 | -30.42 | 272 | -37.49 | 267 | -26.74 | 126 | -32.8 | 145 | 38.38 | 23.41
MTIIISNEk20300$\alpha$10 | -30.44 | 273 | -37.63 | 271 | -28.45 | 267 | -34.52 | 235 | 17.69 | 14.93
MTISNCk40300$\alpha$1 | -30.44 | 274 | -38.14 | 285 | -27.88 | 224 | -34.64 | 242 | 6.09 | 4.61
MTIIISNAk40300$\alpha$0.5 | -30.65 | 275 | -37.72 | 278 | -27.17 | 157 | -34.11 | 217 | 2.59 | 1.64
MTIIISNAk40150$\alpha$1 | -30.71 | 276 | -37.71 | 277 | -28.34 | 260 | -34.09 | 214 | 5.01 | 2.29
MTIIISNEk40300$\alpha$5 | -30.74 | 277 | -38.67 | 291 | -28.61 | 276 | -35.17 | 275 | 11.46 | 10.49
MTIIISNDk20$\alpha$2 | -30.75 | 278 | -36.71 | 248 | -26.68 | 122 | -31.83 | 96 | 4.01 | 1.6
MTIIISNBk20300$\alpha$1 | -30.75 | 279 | -36.49 | 236 | -26.09 | 82 | -31.0 | 69 | 10.15 | 1.21
MTIIISNCk40300$\alpha$2 | -30.76 | 280 | -39.59 | 300 | -27.74 | 209 | -35.91 | 288 | 1.74 | 0.93
MTIIISNCk20300$\alpha$5 | -30.76 | 281 | -36.28 | 228 | -31.03 | 312 | -38.58 | 311 | 17.82 | 15.55
MTIIISNEk20300$\alpha$0.5 | -31.09 | 282 | -37.32 | 264 | -28.27 | 253 | -33.83 | 198 | 4.26 | 2.18
MTIIISNAk20150$\alpha$2 | -31.11 | 283 | -37.54 | 269 | -26.23 | 89 | -31.98 | 106 | 5.9 | 2.13
MTIIISNEk20150$\alpha$2 | -31.13 | 284 | -37.67 | 274 | -26.11 | 83 | -31.24 | 76 | 5.96 | 2.29
MTISNAk40300$\alpha$0.5 | -31.19 | 285 | -37.64 | 272 | -29.04 | 291 | -35.08 | 271 | 6.14 | 4.81
MTIIISNCk20300$\alpha$0.5 | -31.21 | 286 | -38.51 | 290 | -26.69 | 123 | -32.78 | 142 | 4.14 | 2.01
MTIISNDk20$\alpha$2 | -31.26 | 287 | -38.31 | 287 | -30.25 | 306 | -36.61 | 295 | 43.39 | 14.12
MTIIISNEk40300$\alpha$0.5 | -31.27 | 288 | -37.79 | 280 | -27.17 | 158 | -33.63 | 184 | 3.08 | 2.01
MTIIISNCk20300$\alpha$10 | -31.32 | 289 | -38.98 | 295 | -30.4 | 308 | -37.5 | 302 | 20.15 | 15.72
MTIIISNCk40300$\alpha$1 | -31.45 | 290 | -36.78 | 250 | -27.61 | 200 | -34.15 | 218 | 2.08 | 1.12
MTIIISNAk20300$\alpha$5 | -31.46 | 291 | -38.94 | 294 | -32.49 | 321 | -40.22 | 319 | 16.54 | 14.31
MTIIISNBk20150$\alpha$2 | -31.48 | 292 | -37.67 | 275 | -25.28 | 51 | -29.95 | 40 | 14.38 | 2.23
MTIIISNAk40300$\alpha$10 | -31.54 | 293 | -39.61 | 301 | -31.68 | 316 | -40.32 | 320 | 17.05 | 14.66
MTIIISNCk40150$\alpha$0.5 | -31.68 | 294 | -38.46 | 289 | -27.81 | 215 | -34.01 | 209 | 6.65 | 3.39
MTIIISNDk40$\alpha$10 | -31.73 | 295 | -40.03 | 305 | -30.21 | 305 | -37.89 | 307 | 29.09 | 22.41
MTIIISNDk20$\alpha$5 | -31.75 | 296 | -39.72 | 303 | -32.1 | 319 | -40.33 | 321 | 26.32 | 22.84
MTIIISNAk20300$\alpha$10 | -31.76 | 297 | -39.82 | 304 | -30.78 | 310 | -38.64 | 312 | 19.24 | 15.1
MTIIISNAk20300$\alpha$0.5 | -31.84 | 298 | -38.08 | 284 | -27.28 | 168 | -32.86 | 151 | 3.72 | 1.82
MTIIISNEk40300$\alpha$0.1 | -31.92 | 299 | -39.06 | 297 | -26.93 | 140 | -33.0 | 154 | 3.16 | 2.59
MTIISNDk40$\alpha$1 | -31.93 | 300 | -40.11 | 306 | -31.84 | 317 | -40.04 | 317 | 49.24 | 28.82
MTIIISNDk40$\alpha$1 | -32.01 | 301 | -40.35 | 309 | -27.83 | 219 | -34.38 | 231 | 2.96 | 0.94
MTIIISNCk40300$\alpha$0.5 | -32.1 | 302 | -39.11 | 299 | -28.09 | 240 | -34.09 | 212 | 2.84 | 1.81
MTIIISNAk40300$\alpha$2 | -32.19 | 303 | -39.07 | 298 | -27.35 | 175 | -33.48 | 179 | 1.59 | 0.82
MTIIISNBk20300$\alpha$2 | -32.4 | 304 | -38.45 | 288 | -27.29 | 170 | -31.93 | 100 | 10.9 | 1.06
MTIIISNDk40$\alpha$0.5 | -32.42 | 305 | -40.34 | 308 | -28.64 | 280 | -35.76 | 286 | 4.4 | 1.54
MTIISNDk40$\alpha$2 | -32.89 | 306 | -40.98 | 311 | -31.4 | 313 | -39.17 | 314 | 39.44 | 12.99
MTIIISNEk40300$\alpha$2 | -32.91 | 307 | -38.98 | 296 | -27.68 | 207 | -33.21 | 165 | 1.95 | 1.03
MTIIISNBk20300$\alpha$5 | -32.93 | 308 | -39.71 | 302 | -27.58 | 197 | -33.69 | 190 | 28.08 | 17.24
MTIIISNCk40300$\alpha$10 | -32.95 | 309 | -41.29 | 314 | -32.21 | 320 | -40.04 | 318 | 17.5 | 14.95
MTIIISNBk20300$\alpha$0.5 | -32.97 | 310 | -38.81 | 292 | -25.69 | 67 | -30.14 | 50 | 10.12 | 1.87
MTIIISNDk40$\alpha$0.1 | -33.15 | 311 | -42.87 | 319 | -32.51 | 322 | -40.01 | 316 | 8.38 | 3.19
MTIIISNEk20300$\alpha$1 | -33.33 | 312 | -40.75 | 310 | -28.1 | 241 | -33.4 | 175 | 3.55 | 1.53
MTIIISNCk40300$\alpha$0.1 | -33.35 | 313 | -40.15 | 307 | -26.16 | 87 | -31.78 | 93 | 3.69 | 2.96
MTIISNDk40$\alpha$0.1 | -33.55 | 314 | -41.24 | 312 | -31.45 | 314 | -39.37 | 315 | 12.31 | 9.3
MTIIISNDk40$\alpha$5 | -33.57 | 315 | -44.22 | 322 | -33.07 | 323 | -42.51 | 323 | 25.08 | 22.35
MTIIISNAk20300$\alpha$2 | -33.85 | 316 | -41.27 | 313 | -26.14 | 86 | -31.35 | 78 | 2.77 | 0.86
MTIIISNBk40300$\alpha$2 | -34.02 | 317 | -41.5 | 316 | -27.49 | 189 | -33.34 | 171 | 5.01 | 0.98
MTIIISNEk20300$\alpha$2 | -34.12 | 318 | -41.39 | 315 | -28.33 | 259 | -33.79 | 194 | 3.02 | 1.08
MTIIISNEk40300$\alpha$1 | -34.66 | 319 | -42.27 | 318 | -28.86 | 288 | -35.55 | 284 | 2.35 | 1.37
MTIIISNEk20300$\alpha$5 | -34.7 | 320 | -41.92 | 317 | -28.23 | 249 | -34.69 | 245 | 14.42 | 12.39
MTIIISNCk40300$\alpha$5 | -35.35 | 321 | -44.97 | 323 | -31.9 | 318 | -40.34 | 322 | 13.82 | 12.76
MTIIISNCk20300$\alpha$2 | -35.46 | 322 | -44.1 | 321 | -27.2 | 161 | -32.52 | 130 | 2.97 | 1.02
MTIIISNCk20300$\alpha$1 | -35.86 | 323 | -43.16 | 320 | -26.54 | 115 | -31.55 | 85 | 3.17 | 1.24
MTIIISNAk40300$\alpha$5 | -36.48 | 324 | -45.88 | 324 | -35.0 | 324 | -45.94 | 324 | 12.51 | 11.56

Note. — The first set of data $D_{1}$ consists 15 fully confirmed DNSs, and
the second set $D_{2}$ includes all 18 DNSs. Level1 and level2 of each model
when ordered by log$\Lambda(\rm D_{1})$ and log$\Lambda(\rm D_{2})$. The best
fit model in level1 is displayed in boldface.