Vol.0 (20xx) No.0, 000–000

11institutetext: School of Physics and Electronic Engineering, Qilu Normal
University, Jinan 250200, China<EMAIL_ADDRESS>
22institutetext: School of Physics and Electronic Sciences, Guizhou Education
University, Guiyang 550018, China
33institutetext: Astronomy Department, Beijing Normal University, Beijing
100875, China
44institutetext: Research Institute for Astronomy and Astrophysics of
Maragha(RIAAM)-Maragha, IRAN
Received 20xx month day; accepted 20xx month day

# Simulation of the orbit and spin period evolution of the double pulsars PSR
J0737-3039 from their birth to coalescence induced by the gravitational wave
radiation

Peng Liu 11 Yi-Yan Yang 22 Jian-Wei Zhang 33 Maria Rah 44

###### Abstract

The complete orbital and sin period evolutions of the double neutron star (NS)
system PSR J0737-3039 are simulated from the birth to coalescence, which
include the two observed radio pulsars classified as primary NS PSR
J0737-3039A and companion NS PSR J0737-3039B. By employing the characteristic
age of PSR J0737-3039B to constrain the true age of the double pulsar system,
the initial orbital period and initial binary separation are obtained as 2.89
hrs and $1.44\times 10^{6}$ km, respectively, and the coalescence age or the
lifetime from the birth to merger of PSR J0737-3039 is obtained to be
$1.38\times 10^{8}$ yr. At the last minute of coalescence, corresponding to
that the gravitational wave frequency changes from 20 Hz to 1180 Hz, we
present the binary separation of PSR J0737-3039 to be from 442 km to 30 km,
while the spin periods of PSR J0737-3039A and PSR J0737-3039B are 27.10 ms and
4.63 s, respectively. From the standard radio pulsar emission model, before
the system merged, the primary NS could still be observed by the radio
telescope, while the companion NS has crossed the death line in the pulsar
magnetic-field versus period ($B-P$) diagram which is usually considered to
cease the life as a pulsar. It is for the first time that the whole life
evolutionary simulation of the orbit and spin periods for double NS system is
presented, which provides the useful information for observing the primary NS
at the coalescence stage.

###### keywords:

PSR J0737-3039: double neutron star: pulsar: gravitational wave: simulation:

In 1974, Taylor and Hulse discovered the first double neutron star (DNS)
system PSR J1913+16, using the Arecibo telescope (Hulse & Taylor 1975), by
which, the existence of gravitational waves (GW) as a prediction of general
relativity was indirectly verified by studying its orbital contraction
(Einstein & Sitzungsber 1916; Taylor & Weisberg 1982). More than 40 years
later, for the first time, LIGO and Virgo detected GW directly in the merge of
DNS system of an old elliptical galaxy, named as GW170817. Together with a
serials of discoveries of the coalescence between the stellar black holes and
NSs, the GW theory has been confirmed and the multi-messenger astronomy is out
of birth (Abbott et al. 2017; Troja et al. 2017; Blanchard et al. 2017).
Therefore, the investigation of the whole evolutionary process of the DNS
system from its birth to coalescence and clarifying the details of its
evolution is a matter of concern to some scientists working on NS and pulsar
astrophysics.

Among the 19 discovered DNS systems until now, PSR J0737-3039 is the only
known DNS system in which primary and companion NSs have been detected as
pulsars (Burgay et al. 2003; Lyne et al. 2004), which can not only provide the
best direct test for the correctness of general relativity, but also can be
employed as a natural lab for studying the plasma physics and strong field
gravitation (Kramer et al. 2004, 2006; Kramer & Wex 2009). Only PSR J0737-3039
in 19 DNSs can provide more information to let us know the initial orbital and
initial NS spin periods, therefore the PSR J0737-3039 system can be studied as
a complete evolutionary sample of the DNS systems from the birth to the
merger. PSR J0737-3039 has a very compact orbit with the orbital period of
only 2.45 hrs, and the eccentricity is as small as 0.088 (approximately a
circular orbit) (Lyne et al. 2004). From the DNS formation model (Bhattacharya
& van den Heuvel 1991; van den Heuvel 2004), the primary NS (PSR J0737-3039A)
was the first NS to be formed, generated by a massive star directly through a
supernova explosion and experienced accretion induced spun-up and magnetic
field decay, with the spin period of 22.7 ms and the dipole magnetic field of
$6.4\times 10^{9}$ G (Lyne et al. 2004); the companion NS (PSR J0737-3039B)
was the second NS produced by a electron capture supernova explosion
(Podsiadlowski et al. 2004; Nomoto 1984) and ultra-stripped supernova
explosion two methods (Tauris et al. 2013, 2015, 2017), with the spin period
and the dipole magnetic field of 2773.46 ms and $6.4\times 10^{9}$ G,
respectively (Lyne et al. 2004).

As expected, the orbits of the DNSs are contracted continuously by GW
radiation and eventually go to merge, the direct coalescence observation and
calculations of which was pointed out by Schutz (Schutz 1986), and by Cutler
(Cutler et al. 1992). Now, the numerical relativity is a very powerful tool
for studying DNS orbital evolution, when the DNS system enter the merger
(Maione et al. 2016; Shibata & Uryu 2000). In order to study the complete
evolution of PSR J0737-3039, we deduced the variation of the orbital period
given by Peters (Peters 1964) to obtain a new evolution formula and applied it
to simulate the orbital evolution of DNS. Since PSR J0737-3039B is a non-
recycled pulsar like Crab pulsar (Lyne & Graham-Smith 2012; Yang et al. 2006),
and the current spin period is much larger than its initial spin period,
therefore we can use its characteristic age to approximate the true age of PSR
J0737-3039 system (Camilo et al. 1994; Lorimer et al. 2007). Next, we derived
the initial orbital period of the system and the initial spin periods of the
two pulsars, based on which the simulating of complete evolution of the system
is presented.

The paper is organized as bellow: In section 2, we introduce both of the
orbital evolution formula of DNS caused by gravitational radiation, then the
simulation of the orbital complete evolutions of PSR J0737-3039 system was
given in section 3. We simulate the entire spin period evolution of PSR
J0737-3039A and PSR J0737-3039B in section 4. Finally, we simulate the
gravitational wave frequency of the last minute before the DNS merge, and then
discuss the results.

Figure 1: 3-D diagram of PSR J0737-3039 evolutionary trajectory by GW
radiation, where the vertical axis stands for the time in the units of
$10^{7}$ yrs and two horizontal axes of the orbital plane stands for the DNS
orbit, with the units of radius of Sun $R_{\odot}=7\times 10^{10}$ cm.

##  1 Orbital evolution of double neutron stars

In this section, we derive the contraction evolution of DNS orbit by GW
radiation, based on the variation of the orbit radius ($a$) of DNS system,
which is given below (Peters 1964; Lyne & Graham-Smith 2012; Shapiro &
Teukolsky 1983; Lightman et al. 1975; Lightman & Shapiro 1975; Ohanian &
Ruffini 1994):

$\frac{1}{a}\frac{da}{dt}=-\frac{64}{5}\frac{G^{3}M^{2}\mu}{c^{5}a^{4}}f(e),$
(1)

where $M$ is the total mass of DNS, expressed as $M=M_{p}+M_{c}$ with $M_{p}$
and $M_{c}$ the mass of the primary and companion NS, respectively; $G$ ($c$)
is the gravitational constant (speed of light); $\mu$ is the reduced mass of
DNS, expressed as: $\mu=M_{p}M_{c}/(M_{p}+M_{c})$; $f(e)$ is a function of the
orbital eccentricity $e$ of DNS, described in the following (Peters & Mathews
1963):

In this paper, we only consider the case of circular orbit $e=0$ because the
eccentricity of PSR J0737-3039 is 0.088, which arises the eccentricity
function to be $f(e=0)=1$.

$f(e)=(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4})(1-e^{2})^{-\frac{7}{2}}.$ (2)

According to the Keplerian motion, the relationship between the DNS orbital
period and the binary separation is:

$P=(\frac{a^{3}{4\pi}^{2}}{GM})^{\frac{1}{2}},$ (3)

thus the orbital evolution can be also described by the orbital period
equivalently. We bring Eq.(2) and Eq.(3) into Eq.(1) and solve the evolution
of DNS binary separation with time, and obtain the following equation:

$a=a_{0}(1-\frac{t}{T_{gw}})^{\frac{1}{4}},$ (4)

where $a_{0}$ is initial binary separation of DNS, and $T_{gw}$ is a
characteristic time of GW induced coalescence, expressed as:

$T_{gw}=\frac{5}{32}\frac{Ma_{0}^{4}}{\mu cR_{s}^{3}},$ (5)

with the Schwarzschild radius $R_{s}=2GM/c^{2}$.

From Eq.(3) and Eq.(4), we can also deduce the relationship between orbital
period and time as follows:

$P=P_{0}(1-\frac{t}{T_{gw}})^{\frac{3}{8}},$ (6)

with $P_{0}$ the initial orbital period.

Now that we have the evolutionary equations of orbital scale and period
induced by the GW radiation, the detailed simulations of double pulsars can be
performed. PSR J0737-3039 is a DNS system where two pulsars were discovered by
the pulsar search of Parkes 64 m radio telescope in 2003, lied in a distance
of $1.15^{+0.22}_{-0.15}$ kpc (Burgay et al. 2003; Lyne et al. 2004). The
parameters of two NSs are noted below: PSR J0737-3039A is an old, fast-
spinning, recycled pulsar, with mass of 1.3381 $M_{\odot}$, the spin period of
22.7 ms, and the spin period derivative is $1.76\times 10^{-18}$ s/s (Lyne et
al. 2004); PSR J0737-3039B is a slow-spinning, non-recycled normal pulsar with
a mass of 1.2489 $M_{\odot}$, the spin period of 2773.46 ms, and the spin
period derivative is $8.29\times 10^{-16}$ s/s (Burgay et al. 2003). This
system has also the shortest orbital period of 2.448 hrs among 19 pairs of
DNSs, and the orbital eccentricity is as small as 0.088, an almost circular
orbit (Burgay et al. 2003). Moreover, the system presents a strong
relativistic effect, orbit of which is confirmed to shrink by gravitational
wave radiation during the evolution process (Kramer & Wex 2009; van Leeuwen et
al. 2015; Abbott et al. 2009).

Here, we simulate the orbital evolution of PSR J0737-3039 before the two NSs
merge, where we assume that the two NSs are the point particles with the
nearly circular orbit. Then, during the simulation, we regard that the DNS
system begins to merge when the orbit radius decays to the scale of two NS
radii 30 km with the conventional NS radius of 15 km. Our simulation found
that the merge age of PSR J0737-3039 system is $8.83\times 10^{7}$ yr, and the
maximum orbital frequency of the system before the merge is 590 Hz,
corresponding to the GW frequency as 1180 Hz. A schematic diagram of orbital
evolution of PSR J0737-3039 is plotted in Fig.1, where we draw a complete
cycle of trajectory motion in every time interval $10^{7}$ yr. in-between
$0-8\times 10^{7}$ yr, and also include $8.8\times 10^{7}$ yr and $8.83\times
10^{7}$ yr, and each trajectory gives the corresponding orbital period.

Table 1: Parameters of double pulsar PSR J0737-3039(Lyne et al. (2004); Burgay et al. (2003))  System | $M_{p}(M_{\odot})$ | $M_{c}(M_{\odot})$ |  $P$(h) |  $P_{s}$(ms) | d (kpc) | eccentricity |  $\tau$ (yr) |  $\dot{P_{s}}$(s/s) |  $B$(G)
---|---|---|---|---|---|---|---|---|---
J0737-3039A | 1.3381(7) | $..........$ | 2.448 | 22.699 | $1.15^{+0.22}_{-0.15}$ | 0.088 | $2.04\times 10^{8}$ | $1.76\times 10^{-18}$ | $6.4\times 10^{9}$
J0737-3039B | $..........$ | 1.2489(7) | $..........$ | 2773.46 | $..........$ | $..........$ | $4.92\times 10^{7}$ | $8.29\times 10^{-16}$ | $1.59\times 10^{12}$

##  2 The evolution of PSR J073-3039 system in complete coalescence age

Figure 2: The orbit evolution of double pulsar system PSR J0737-3039. As a
comparison, PSR J0737-3039 and PSR J1906+0749 are indicated.

Based on the magnetic dipole radiation model of pulsar (braking index n = 3)
(Lorimer et al. 2007; Shapiro & Teukolsky 1983), we make a simple integral
calculation for the pulsar spin-down model ($\dot{P_{s}}\varpropto
P_{s}^{2-n}$), deriving the relationship among the true age (t), spin period
($P_{s}$), spin period derivative ($\dot{P_{s}}$), and initial spin period
($P_{s0}$) of the pulsar, which can be written as (Lorimer et al. 2007;
Shapiro & Teukolsky 1983; Zhang et al. 2016):

$P_{s}^{2}=P_{s0}^{2}+2t[P_{s}\dot{P_{s}}].$ (7)

Since the normal pulsar (e.g., PSR J0737-3039B) has no experience of the
accretion precess, its initial spin period is much smaller than its current
spin period ($P_{s0}\ll P_{s}$), we can approximately deform Eq.(7) as:

$t\simeq\frac{P_{s}}{2\dot{P_{s}}}\equiv\tau,$ (8)

where $\tau$ is the current characteristic age of the pulsar, expressed as
$\tau\equiv P_{s}/(2\dot{P_{s}})$. The above formula indicates that we can use
the characteristic age of pulsar to approximately replace its true age (when
$P_{s0}\ll P_{s}$) (Lorimer et al. 2007; Camilo et al. 1994). This means that
the age of PSR J0737-3039 system is approximately equal to the current
characteristic age of PSR J0737-3039B.

Compared to PSR J0737-3039A that experienced the accretion recycled process,
PSR J0737-3039B is a relatively young normal pulsar with the spin period of
2773.46 ms, with the characteristic age of $4.92\times 10^{7}$ yr (Burgay et
al. 2003). Similar to Crab pulsar that has the constant magnetic field dipole
radiation (Staelin & Reifenstein 1968), we assume that the initial spin period
of PSR J0737-3039B is 20 ms (Zhang et al. 2016), because the current spin
period is much larger than the birth spin period ($20~{}\rm ms\ll
2773.46~{}\rm ms$), therefore, consequently the PSR J0737-3039B true age
($t_{B}$) can be approximated by the characteristic age ($\tau_{B}$), namely:

$t_{B}\simeq\tau_{B}.$ (9)

Lorimer et al. (2007) pointed out that when the PSR J0737-3039A ceased to
spin-up and the PSR J0737-3039B was birth occurred almost at the same time
(Lorimer et al. 2007), thus the spin-down age of PSR J0737-3039A (time since
the accretion induced spin-up stopped: $t_{A}$) can be approximately replaced
by the true age of PSR J0737-3039B, that is:

$t_{A}=t_{B}\simeq\tau_{B}.$ (10)

According to the standard DNS formation model (van den Heuvel 2007), the first
NS of the system was formed by a massive star undergoing main sequence
evolution and supernova explosion. The first NS will accrete matter from the
progenitor star of the companion star (B), which will spin-up the period of
the first NS; If the system survives in the second supernova explosion, the
DNS system will be born at this moment. Hence the true age of the DNS system
is equal to the true age of the second NS, and the true age of PSR J0737-3039
system ($t_{AB}$):

$t_{AB}=t_{B}\simeq\tau_{B},$ (11)

from the above formula we can constrain the real age of PSR J0737-3039 system
to be $4.92\times 10^{7}$ yr, and the system will complete coalescence age
(time from birth to merger) in approximately $1.38\times 10^{8}$ yr (the
characteristic age of PSR J0737-3039B to be $4.92\times 10^{7}$ yr, the
coalescence age of system to be $1.38\times 10^{8}$ yr).

We bring the age of PSR J0737-3039 system $t_{AB}$ into Eq.(4) and Eq.(6), and
obtain the initial binary separation of the system to be $1.44\times 10^{6}$
km, corresponding to be initial orbital period of 2.89 hrs, which means that
the binary separation and orbital period of system is only contracted by
$0.15\times 10^{6}$ km and 0.44 hrs, respectively (the present binary
separation and orbital period are $1.29\times 10^{6}$ km and 2.45 hrs,
respectively). Next, based on the initial information of PSR J0737-3039
system, we simulated the evolution of the orbital period and binary separation
of PSR J0737-3039 within the complete coalescence age, as shown in Fig.2.

###  2.1 The spin evolution of PSR J0737-3039A and PSR J0737-3039B in complete
coalescence age

Figure 3: Spin period evolutions of PSR J0737-3039A and PSR J0737-3039B.

We bring the observational data of PSR J0737-3039A and $t_{A}$ into Eq.(7),
obtaining that the birth spin period is 19.78 ms when it starts to spin-down,
which means that its spin period is only increased by 2.92 ms (the present
spin period is 22.70 ms). From the above, the initial spin period of PSR
J0737-3039B is assumed to be similar to that of the Crab-pulsar of 20 ms,
compared to its current period of 2773.46 ms, thus the spin period of PSR
J0737-3039B evolves relatively quickly.

In order to simulate the evolution of the pulsar spin-down, we assume that the
dipole magnetic field is invariant during the evolution, therefore we can
transform Eq.(7) into:

$P_{s}^{2}=P_{0}^{2}+2t[P_{n}\dot{P_{n}}],$ (12)

where $P_{n}$ and $\dot{P_{n}}$ are the current measured data of the pulsar
spin period and spin period derivative, respectively.

According to Eq.(12), we simulated the spin period evolution of PSR
J0737-3039A and PSR J0737-3039B within the complete merger age, and stopped
the simulation when the evolution time going to $1.38\times 10^{8}$ yr. Our
simulation found that: the spin period of PSR J0737-3039A and PSR J0737-3039B
are 27.10 ms and 4.63 s, respectively, when the system was merged. Based on
the simulation results, we plot the spin period evolution of PSR J0737-3039A
and PSR J0737-3039B in Fig.3.

In order to acquire a complete scenario of the evolution of both pulsars in
the diagram of spin period and magnetic field (the magnetic fields of PSR
J0737-3039A and PSR J0737-3039B are $6.4\times 10^{9}$ G and $1.59\times
10^{12}$ G, respectively (Burgay et al. 2003; Lyne et al. 2004), we plot their
complete simulation results in Fig.4.

In the Figure 4, the small box is a close-up of the evolution track of PSR
J0737-3039A, and the arrows indicate the direction of evolution (the evolution
from left to right means that from birth to merger). It can be seen from the
figure that from the system birth to merger, PSR J0737-3039A would be always
seen by the radio telescope, while PSR J0737-3039B will cross the death line
of pulsar and may be not observed (Bhattacharya & van den Heuvel 1991).

Figure 4: The evolution track of PSR J0737-3039A and PSR J0737-3039B along the
$B-P$ diagram (data from ATNF pulsar catalog (Manchester et al. 2005). The
cyan dots, red circles, black lines, black dotted lines, pentagrams and
squares are represent normal pulsars, pulsed binary, acceleration lines, death
lines, PSR J0737-3039A and PSR J0737-3039B, respectively.

##  3 The results and Discussions

In this paper, based on the observational data of the double pulsar system PSR
J0737-3039, we simulated its orbital evolution induced by the GW radiation,
and found that the system will merge after $8.83\times 10^{7}$ yr. By the
magnetic dipole model of pulsar, we obtained that PSR J0737-3039 system exists
about $4.92\times 10^{7}$ yr, thus the complete coalescence age (from its
birth to merge) of PSR J0737-3039 system should be be $1.38\times 10^{8}$ yr.
Next, we simulated the complete orbital evolution of PSR J0737-3039 system and
obtained the initial orbital period and radius as 2.89 hrs and 1.44 km,
respectively. In addition, the GW frequency generated within last minute
before the system merge ranges from 20 Hz to 1180 Hz, and the corresponding
binary separation decays from 442 km to 30 km. We compared the GW frequency
evolution of PSR J10737-3039 with that of the observation data of GW170817 by
LIGO, as shown in Fig.5, and found that both curves have no much difference,
and a meager bias is on account of the mass difference of both systems (the
primary NS and companion NS masses of GW170817 are:
$1.46_{-0.10}^{+0.12}M_{\odot}$ and $1.27_{-0.09}^{+0.09}$, respectively
(Abbott et al. 2018).

Furthermore, we calculated the spin period evolution of PSR J0737-3039A and
PSR J0737-3039B within the complete merger age, and obtained that their spin
periods are 27.10 ms and 4.63 s at coalescence. Through the evolution
trajectories of PSR J0737-3039A and PSR J0737-3039B in the pulsar magnetic
field and spin period $B-P$ diagram, we concluded that PSR J0737-3039A can
always be observed by radio telescopes, but PSR J0737-3039B will cross the
death line of radio pulsar and could not be observed in general.

It is remarked that we employed the point masses of both NSs to perform the
simulation. However, Kuznetsov et al. (1998) studied the orbital evolution of
DNS by the GW radiation while the internal structure of NSs have been taken
into account, and pointed out that there exists little effect on their merging
time (only deviation of 10.5 ms) (Kuznetsov et al. 1998), which is much less
than the merging time of 30 s as observed in GW170817 by LIGO.

Figure 5:  The evolution of GW frequency at the last minute of DNS merge,
where the dashed (solid) curve represents the case of PSR J0737-3039
(GW170817).

###### Acknowledgements.

This research is supported by the National Natural Science Foundation of China
NSFC (11988101, 11773005, U1631236, 11703001, U1731238, U1938117, 11725313,
11721303), the International Partnership Program of Chinese Academy of
Sciences grant No. 114A11KYSB20160008, the National Science Foundation of
China No. 11721303, and the National Key R&D Program of China No.
2016YFA0400702. And, this work is also supported by the National Basic
Research Program (973 Program) (no. 2015CB857100), National Key R&D Program of
China (no. 2017YFA0402600), and the Guizhou Provincial Science and Technology
Foundation (Grant no. [2020]1Y019).

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