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Astronomy & Astrophysics manuscript no.
(will be inserted by hand later)
Formation of massive skyrmion stars
S.B. Popov 1,2 and M.E. Prokhorov 1
1 Sternberg Astronomical Institute, Universitetski pr. 13, 119992 Moscow, Russia
e­mail: polar@sai.msu.ru; mike@sai.msu.ru
2 Universit‘a di Padova, Dipartimento di Fisica, via Marzolo 8, 35131, Padova, Italy
e­mail: popov@pd.infn.it
Abstract. As it is well known for sti# equations of state an existence of neutron stars with masses >
# 2 M
# is
possible. Especially interesting possibility is opened if the equation of state based on the Skyrme theory is realized
in nature. This equation of state was proposed recently by Ouyed and Butler. We discuss di#erent channels of
formation of massive rapidly rotating neutron stars. We use a population synthesis code to estimate numbers
of massive neutron stars on di#erent evolutionary stages. A neutron star increases its mass by accretion from
a secondary companion. Significant growth of a neutron star mass due to accretion is possible only for certain
values of initial parameters of the binary. In this paper we show that significant part of massive neutron stars
with M >
# 2 M
# can be observed as millisecond radio pulsars, as X­ray sources in pair with white dwarfs, and as
accreting neutron stars with very low magnetic fields.
Key words. stars: neutron -- stars: evolution -- stars: statistics -- stars: binary -- X­ray: stars
1. Introduction
Mass is one of the key parameter for neutron star (NS)
physics and astrophysics. It can be measured with high
precision in binary radio pulsar systems. Up to very re­
cent time obtained results fell in a very narrow region
1.35­1.45 M
# (Thorsett & Chakrabarty 1999). These val­
ues lie very close to the Chandrasekhar limit for white
dwarfs. Thus, for years M = 1.4 M # was considered to
be a standard value of a NS mass. Recently the range
widened towards lower masses thanks to the discovery of
the double pulsar J0737­3039 (Burgay et al. 2003). One
of the NSs in this system has M = 1.25 M# (Lyne et
al. 2004). Also the mass range expanded towards higher
masses, though this result is less certain. There is only one
NS in a binary radio pulsar system with mass significantly
higher than the canonical value 1.4 M # . It is the pulsar
J0751+1807 with the mass 2.1 +0.4
-0.5 (95% confidence level)
(Nice & Splaver 2004). All others are consistent with the
standard mass value on the 1­2­# level. However, small
number of massive radio pulsars can be a result of a se­
lection e#ect(s).
There are reasons to suspect an existence of signif­
icant number of NSs with higher masses. Evidence for
such a proposal comes both from theory and observations.
Calculations of cooling curves of NSs suggest that some of
these objects might be more massive than known sources
in radio pulsar systems (see for example, Kaminker et al.
Send o#print requests to: S. Popov
2001) with M up to 1.8 M # and probably more. Modeling
of supernova (SN) explosions also suggest the existence of
NSs with higher masses (Woosley et al. 2002). Still models
of NS thermal history and SN explosions do not requier
masses M >
# 2 M # , but there are observational indications
for their existence.
Observationally high masses of NSs are mainly sup­
ported by data on X­ray binaries (we do not discuss
here data based on quasi­periodic oscillations measure­
ments as they are model dependent). Estimates for sev­
eral systems give very high values: 1.8--2.4 M # for Vela
X­1 1 (Quaintrell et al. 2003), 2.4±0.27 M # for 4U 1700­
37 (Clark et al. 2002; see also Heineke et al. 2003; van
Kerkwijk 2004). Very recently Shahbaz et al. (2004) pre­
sented observations of a low­mass X­ray binary 2S 0921­
630/V395 Car for which 1­# mass range for the compact
object is 2--4.3 M # . Still it is necessary to note that such
measurement are not as precise as the radio pulsar ones
(for example, at the 3­# level the mass of the NS in Vela
X­1 is still compatible with the standard value).
The existence of NSs with M # 2--2.4 M # is not in
contradiction with the present day theory of NS interiors.
There are several models with sti# equation of state (EOS)
which allow an existence of NSs with masses >
# 2 M # (see
a review and references in Haensel 2003). Here we will
focus on so called skyrmion stars (SkyS) as they are ex­
1 This range is based on the two estimates given in
(Quaintrell et al. 2003): 1.88 ± 0.13 and 2.27 ± 0.17 M # .

2 S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars
pected to be a kind of NSs with the highest value of max­
imum mass (Mmax ).
In 1999 Ouyed and Butler discussed an EOS based
on the model of Skyrme (1962). A NS with such EOS
has Mmax=2.95 M
# even for a non­rotating configura­
tion. Usualy maximum rotation can increase the limit
by # 15--20%. Rapidly rotating SkyS were discussed by
Ouyed (2002, 2004), and for this case Mmax=3.45 M#
and R = 23 km (this model also has relatively large radii
of NSs). Such model is very interesting from the astrophys­
ical point of view, and it is important to discuss scenarios
of formation of compact objects with such high masses.
Our goal in this note is to pick out evolutionary tracks
of binary systems which can lead to the formation of NSs
with high masses, and to discuss possible observational
appearences of such systems and their relative and abso­
lute numbers in the Galaxy. As we do not use explicitly
any EOS in our calculations, then our results can be ap­
plied to other sti# equation of state and even to low­mass
black holes (BHs).
In the next section we discuss evolutionary paths at the
end of which a massive NS can be formed. Then we give
an estimate of the number of massive NSs in the Galaxy.
Finally we discuss our results and propose systems which
are more favorable to host massive NSs.
2. Possible channels of massive neutron star
formation
As mass determination for NSs is possible only in binary
systems 2 we focus on potentially observable stages of evo­
lution of binary systems in which a massive NS can form.
Below we discuss possible ways of massive NS formation.
Since we are mostly interested in compact objects with
rapid rotation (because they can have higher maximum
masses) it is necessary to follow evolution in a binary as
such objects cannot form from single stars (Heger et al.
2003), so its necessary to study evolution of close binary
systems. Except evolutionary tracks which lead to a for­
mation of a massive NS in a binary we follow paths at
the end of which an isolated massive NS can form. An
appearence of a rapidly rotating single massive NS due
to a binary evolution can be a result of a coalescence of
two compact objects (NSs or white dwarfs --- WDs), or a
result of a more slowly merging process in which a normal
star is involved, or a result of an evaporation of a low­mass
secondary companion by an active pulsar. At some stages
during its evolution a binary which finally is going to pro­
duce an object of our interest can be observed as an X­ray
source, that is why it is important to select evolutionary
paths also for them.
The main output of a collapse of cores of massive stars
are NSs with M # 1.2­1.5 M # . This conclusion is sup­
ported both observationally (van Kerkwijk 2004) and the­
2 Note, that in principle there is a possibility to determine
an isolated NS mass by microlensing e#ects, however, we do
not touch this issue here.
oretically (Timmes et al. 1996; Fryer & Kalogera 2001;
Woosley et al. 2002). Numerical models of collapse are
not as precise as necessary to determine the exact shape
of a NS mass spectrum (for example the amount of fall­
back is not well known), however, calculations show that
the formation of NSs with high masses is not favourable
and most of them should have M # 1.3­1.4 M # .
A discovery of a NS with M >
# 1.8 M# should mean
that the mass was increased after formation of the com­
pact object during its evolution (if the mass is significantly
higher than 1.8 M# then such a conclusion seems to be
inevitable). Based on this proposition we call below as
massive NSs with M > 1.8 M
# .
A NS can increase its mass due to fallback, coalescence
with another NS, or accretion from a secondary compan­
ion. As we note above, the first way is not well studied,
and we do not discuss it below. Oppositely coalescence of
NSs is well understood (see Rosswog et al. 2003 and refer­
ences therein). The rate of NSs coalescence in the Galaxy
is about 1 per 10 4 yrs. As a result a rapidly rotating mas­
sive isolated NS (or a BH) can form. This way of evolution
also will not be discussed below. In the following only bi­
nary evolution of a NS in pair with a normal star or a WD
will be studied.
At first for an illustration let us assume an isotropic
collapse, ie. zero kick. Such an assumption is not realis­
tic as most part of NS -- nearly all radio pulsars -- ob­
tain at birth high additional velocity #100--1000 km s -1
(Arzoumanian et al. 2002). However it is much easier to
understand main processes in a binary evolution if one
neglects kick. In addition, if a binary was not unbounded
after a SN explosion then an orbital eccentricity quickly
decays after the secondary fills its Roche lobe. So, if at the
moment we are not interested in the question of the bi­
nary survival then it is possible to neglect kick to simplify
the explanation.
Let us start with a qualitative discussion (below in
sec. 2.1 a more detailed consideration is given). The most
obvious channel to form a rapidly rotating massive NS is
an evolution in a low­mass or intermediate mass binary
(see, for example, recent calculations by Podsiadlowski et
al. 2002). This path includes, for example, millisecond pul­
sars (however it is not the only possible output).
As we are interested here in systems with high mass
ratio (a massive primary produces a NS and the secondary
star has a low mass) it is necessary to consider three di#er­
ent situations after the NS formation when the secondary
fills its Roche lobe: i.) a normal star can fill its Roche lobe
without a common envelope formation; ii.) a normal star
can fill its Roche lobe with a common envelope formation;
iii.) a WD fills its Roche lobe.
To fill the Roche lobe a normal secondary star has to
evolve further the main sequence stage. During its evolu­
tion prior to the Roche lobe overflow the mass of the star
is nearly constant (see detailed tracks below). A common
envelope is not formed if the normal star is not signifi­
cantly heavier than the NS. In this regime mass is not lost
from the binary system. For more massive secondaries for­

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 3
mation of a common envelope is inevitable, mass transfer
is unstable. In this regime significant fraction of the mass
flow is lost from the system, so the mass of the NS grows
less e#ectively. It is only partly compensated by higher
mass of the donor.
After the common envelope stage an orbital separation
becomes smaller, so later on even a degenerated core of the
secondary -- a WD -- can fill the Roche lobe.
2.1. Evolutionary tracks
For our calculations we use the ``Scenario Machine'' code
developed at the Sternberg Astronomical Institute. 3
Description of most of parameters of the code can be found
in (Lipunov et al. 1996). Below we mention those which
are the most important for us here:
-- All NSs are born with M = 1.4 M # .
-- At the common envelope stage a hypercritical accre­
tion (with —
M much larger than the Eddington value)
is possible.
-- During accretion the magnetic field of a NS decays
down to the value which cannot prevent rapid (maxi­
mum) rotation of the NS.
-- Oppenheimer­Volko# mass of a rapidly rotating NS
(the critical mass of a BH formation) is assumed to be
3.45 M
# according to Ouyed (2004).
For zero kicks we distinguish two groups of tracks
which produce massive NSs. A typical track from the first
group has initial value of the semimajor axis a = 290 R #
and star masses M 1 = 10.5 M# , M 2 = 2 M# (fig. 1 left) 4 .
After the massive component leaves the main sequence it
expands and fills its Roche lobe. As a result the common
envelope stage sets on. During this stage the orbit shrinks
by more than an order of magnitude, and the primary
looses about 3/4 of its mass and becomes a low­mass he­
lium SN progenitor. After the SN explosion the orbit has
low eccentricity and a # 7--8 R # . Mass of the secondary
is not changed during these stages of the evolution.
Till the secondary fills its Roche lobe the NS is at the
stages of ejector and propeller (see for example Lipunov
1992 for stages descriptions). During these stages the mag­
netic field is assumed to be constant. Stage durations can
be found in (Lipunov et al. 1996) 5 .
After the secondary fills the Roche lobe the NS starts
to accrete. At that moment the mass ratio is about 0.7
(the NS is lighter) and a mass transfer is stable with nearly
zero mass loss from the system. Up to equalizing of com­
ponents masses matter transfer goes on a thermal time
3 Online materials are available at
http://xray.sai.msu.ru/sciwork/scenario.html and
http://xray.sai.msu.ru/ #mystery/articles/review/.
4 Colored version of the figure in
high resolution is avalable on the Web:
http://xray.sai.msu.ru/#polar/html/publications/ouyed/
5 The subsonic propeller stage is not taken into account as
for binaries with big accretion rates it is very short.
scale, after equalizing -- on a nuclear. Process of accre­
tion can be stopped because of a switching on of a mil­
lisecond radio pulsar. It happens when the donor's mass
is # 0.1 M # . The remnant of the secondary companion
then can be evaporated completely, while the evaporation
is proceeding the systems looks like the famous ``Black
widow'' pulsar 1957+20 (and its twin PSR J2051­0827).
If accretion is not stopped then it continues till a planet­
like (Jupiter mass) companion remains. As we see the final
stage of such an evolution is a ``single'' massive rapidly ro­
tating NS. In both cases the final mass of a NS can reach
3.2--3.3 M
# . We can observe such a system at the stage
of accretion which lasts 90% of the evolution. Masses of
NSs in these accreting systems can be in the range from
the initial mass (1.4 M
# in our case) up to 3.2--3.3 M
# .
Orbits can be relatively wide.
The described evolutionary channel appears to be nar­
row in a sense that small changes in the initial conditions
do not allow a massive NS formation. Also uncertain pa­
rameters of the common envelope stage can significantly
influence this path.
Ranges of initial parameters of evolutionary tracks
from the second group are given in the table 1. We give
maximal and minimal values for two types of tracks (2a
and 2b) which di#er by the final stages of evolution. The
orbital period, P orb , is given in the table 1 just for an illus­
tration. In our calculations we use masses and semimajor
axes. So the values of P orb given in the table are simply
calculated using maximum masses and minimum semima­
jor axes for the shortest period, and, oppositely, minimum
masses and maximum axes for the longest period. Due to
that ranges for P orb for tracks 2a and 2b intersect.
A typical representative of the 2a subgroup has the
following initial parameters: a = 300 R
# , M 1 = 12 M
# ,
M 2 = 4 M# . The main di#erence form the first group of
tracks is a more massive secondary companion. Because
of that the common envelope during the first mass trans­
fer is less e#ective, and after a SN a system with a =
170 R
# and low eccentricity is formed (the mass of the
secondary is not changed). Later the secondary fills the
Roche lobe. Mass ratio is high, mass transfer is unstable
and the common envelope forms. At the end of the com­
mon envelope stage the secondary becomes a WD with
M # 0.8 M # , and the orbital separation diminishes down
to 5 R # . During the common envelope stage the NS in­
creases its mass up to # 2.3 M # (for more massive donors
mass loss from the system is more e#ective, so in such
cases the NS mass can be lower: # 1.9 M # ).
After the formation of a binary consisting of a NS and
a WD the evolution in the second group can take one of
two di#erent paths. For some tracks (2a) from the sec­
ond group the time of rapprochement of the components
due to gravitational wave emission is too long, so there
is no Roche lobe overflow. Systems with smaller orbital
separation have enough time to approach to each other
close enough for the beginning of WD overflow. This sit­
uation corresponds to the initial parameters a = 200 R
# ,
M 1 = 12 M
# , M 2 = 4 M
# (track 2b in the table 1).

4 S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars
Table 1. Parameters for tracks from the second group
parameter min max width
Track 2a
a 279R # 670R # 0.20
M1 10.3M # 12.8M # 0.054
M2 3.9M # 6.7M # 0.13
P orb
(#) 123 d 537 d
Track 2b
a 135R
# 279R
# 0.17
M1 10.3M
# 12.4M
# 0.046
M2 3.9M
# 7.4M
# 0.15
P orb
(#) 41 d 144 d
(#) P orb is given just as an illustration, see
the text.
The main di#erence between tracks 2a and 2b is
smaller orbital separation in the latter case. Track 2b is
similar to the one on the right panel of fig.1, but after the
common envelope semiaxis of the system is just # 3 R
# .
A WD has enough time to fill the Roche lobe and com­
pletely transfer its mass to the NS. At the end we have a
single rapidly rotating NS. The NS mass for this case is
increased up to # 3 M
# . Stages with a WD are shown in
the box as they distinguish the track 2b from 2a.
When a semimajor axis is larger than a # 670 R
#
the second common envelope results in NS--star merging,
so the Thorne­Zytkow object is formed. Its evolutionary
path is not very clear. A formation of a massive NS and
a formation of a BH are both possible. We do not include
this possibility into our calculation.
2.2. Evolutionary tracks with kicks
Above we discuss two families of tracks with zero kicks
which result in massive NSs formation. However, it is nec­
essary to include kicks as they are a general property of a
NS formation. A kick can change orbital parameters after
a SN explosion, it can even make the system unbounded.
If after a SN (and after a brief period of circularization
of an orbit) we obtain in our calculations a system with
parameters in the range which was obtained above for the
zero kick, then the following history of the system should
be the same as described in sec. 2.1.
An additional velocity which a NS obtains at birth can
change the range of initial parameters that are necessary
for a massive NS formation. Especially it is important to
estimate if ranges for M 1 , M 2 and a are changed signifi­
cantly or not. As a kick velocity and a NS mass in our cal­
culations are assumed to be independent on a mass of an
exploding star (see below sec. 4.1) we do not expect that
a range of masses of primaries should be modified. The
same can be said about the range of initial masses of sec­
ondaries because these stars do not su#er any important
evolutionary changes before a SN expolsion. Since a kick
can dramatically change the orbital parameters the situ­
ation is di#erent for the initial orbital separation range.
NS "E"
Evaporation
NS "E"
NS "A"
NS "A"
NS "P"
NS "E"
SN Ib
3.22
3.22
1.66
1.40
1.40
1.40
WR RLO
RLO
Post MS
MS
NS "A"
NS "E" WD
NS "E" WD
NS "A"
NS "A"
NS "P"
NS "E"
SN Ib
WR RLO
3.09
2.28
2.28
1.40
1.40
1.40
1.40
RLO
Post MS
MS
Fig. 1. Evolutionary tracks for massive NS formation. In the
left panel we show a typical track from the first group. The first
mass transfer (from the primary) results in a common envelope
formation due to high mass ratio. Accretion onto a NS from
the secondary companion proceeds stably without a common
envelope. In the right panel we show an evolutionary path of
a system from the second group. This track di#ers by a higher
mass of a secondary companion. Because of this di#erence the
first mass transfer goes on without a common envelope. A NS
gathers an additional mass during one or two episodes of accre­
tion. If the orbital separation is not very large (# 200 R
# , see
text) then at first the NS accretes from a normal secondary fill­
ing its Roche lobe, and then from a WD (this stage is shown in
the dashed frame). For wider systems the evolution stops after
the mass transfer from the normal secondary star (ie. before
the frame). On the left from each track we indicate evolution­
ary stage (in notation from Lipunov, Postnov & Prokhorov
1996) and NS masses.
For example, with a kick systems wider then the ones dis­
cussed in sec. 2.1 can still form massive NSs.
In the next section we present results of our calcu­
lations of population synthesis of massive NSs for both
scenarios.
3. Estimate of observable number of massive
neutron stars in the Galaxy
To estimate the number of massive NSs in the Milky Way
we run several sets of population synthesis calculations for

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 5
Table 2. Fractions of massive NSs at di#erent stages
Stage with kick without kick
Ejector 0.32 0.39
Propeller + Georotator 0.02 0.08
Accretor 0.66 0.53
Hypercritical stages 5 · 10 -6 0
the ranges of initial parameters which correspond to the
two groups of tracks described above. Each run includes
calculations of 10 6 individual binary evolutionary tracks.
We run the model for zero kick velocities and for non­
zero ones. For the latter case we use the distribution simi­
lar to the one suggested in (Arzoumanian et al. 2002). We
use bimodal distribution with equal fraction of objects in
each mode. An average velocity in the first mode is 175
km s -1 and in the second it is 750 km s -1 , distribution
in each mode is maxwellian.
For the scenario without kick we proceed as follows.
For the second group of tracks we used ranges indicated
in the table 1. Width given in the table is calculated as
0.5(max­min)/(max+min). For the first family of tracks
we used the range for a from 230 to 346 R # , for M 1 from
8.4 to 12.6 M# , and for M 2 from 1.6 to 2.4 M# .
For the scenario which takes into account an additional
velocity gained by a NS at birth we used wider range of
initial semimajor axis: from 200 to 2000 R # . Masses are
chosen in the same way as for the zero kick variant.
The results of the calculations for non­zero kick are the
following (we assume the total number of all NSs in the
Galaxy as 10 9 , and the galactic age as 1.5 10 10 yrs). In the
first channel (fig. 1 left panel) we do not obtain significant
number of massive NSs. Most of these objects are formed
in the second channel. Formation rate of massive NSs was
found to be 6.7 10 -7 yrs -1 . This corresponds to # 10 000
of these compact stars in the Galaxy. For zero kick the
formation rate is larger 4 10 -6 yrs -1 , so the total number
is # 60 000.
Certainly only a fraction of massive NSs at any given
moment passes through stages which are observable, ie.
the accretor stage and the stage of radio pulsar. Some
of these objects are at stages of ejector and propeller or
georotator. All three of them are not favourable for detec­
tion 6 . In the table 2 we give fractions of massive NSs on
each stage. It is clear that accretors are more numerous
(but the number of massive NSs at the stage of superEd­
dington accretion is negligible).
For the non­zero kick model about 25% of accreting
massive NSs have normal stars as secondaries, the rest
75% have WD companions. For zero kick nearly all mas­
sive NSs accrete from WDs which fill their Roche lobes.
6 We note, that the ejector stage does not coinside with the
radio pulsar stage, but includes it as a substage. So here we
are speaking about non­detectability of ejectors which are not
active as radio pulsars. See for example (Lipunov 1992) or
(Lipunov et al. 1996) for more details.
1,8 2 2,2 2,4 2,6 2,8 3 3,2 3,4
M(NS)/Mo
0
0,05
0,1
0,15
0,2
0,25
Fig. 2. Mass distribution of NSs. As we are interested only in
the massive population we do not show the results for compact
objects with M < 1.8 M # . Upper mass limit corresponds to
SkyS with maximum rotation (Ouyed 2004). The dashed line
represents results for the scenario with zero kick. The solid line
-- non­zero kick. Left peaks for both distributions correspond to
NSs with a single episode of accretion. Right peaks are formed
by NSs which also increased their masses via accretion from
WDs. Distribution were normalized to unity, ie. an area below
each line is equal to one.
Mass distributions for both scenarios are shown in the
fig. 2. Note, that all small details in the figure are due to
statistical noise (for example, the first peak on the rising
part of the dashed curve, or the middle peak on the solid
one). The only important details are the two peaks at M #
2.3 M # and M # 3.1 M # , which correspond to tracks
2a and 2b (see the right panel of Fig. 1 and table 1).
As we found only two groups of tracks which lead to the
formation of massive NS only results obtained for these
groups are shown. All contributions from other types of
tracks are not considered here and in fig. 3.
Finally, in the last figure we represent luminosity dis­
tributions. For the scenario with non­zero kick about
1/2 of massive NSs have M > 2.5 M
# . Taking all to­
gether we can conclude that in the Galaxy there are sev­
eral thousand of accreting massive NSs with luminosities
10 34 <
# L <
# 10 36 erg s -1 .
4. Discussion and additional comments
Here at first we notice some uncertainties of the scenario.
Then we briefly discuss a possibility of massive NS for­
mation in globular clusters, low­mass BHs, and types of
sources which can host massive NSs.
4.1. Correlations between initial parameter