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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. We discuss di#erent channels of formation of massive rapidly rotating neutron stars. 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 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 particular 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 fall in a very narrow region
1.35­1.45 M # (Thorsett & Chakrabarty 1999). This value
lies very close to the Chandrasekhar limit, and so 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 J0737­
3039 (Burgay et al. 2003). One of the NSs in this system
has M = 1.25 M
# (Lyne et al. 2004). There are no known
NSs in binary radio pulsar systems with masses signifi­
cantly higher than the canonical value 1.4 M
# . However,
this can be a result of a selection e#ect(s).
There are reasons to suspect an existence 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 should be
more massive than known sources in radio pulsar systems
(see for example, Kaminker, Haensel & Yakovlev 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, Heger & Weaver 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.
Send o#print requests to: S. Popov
Observationally high masses of NSs are supported by
data on X­ray binaries. Estimates for several systems give
very high values: 1.8­2.4 M# for Vela X­1 1 (Quaitrell
et al. 2003), 2.4±0.27 M # for 4U 1700­37 (Clark et al.
2002; see also Heineke et al. 2003, van Kerkwijk 2004).
Very recently Shahbaz et al. (2004) presented observations
of a low­mass X­ray binary 2S 0921­630/V395 Car for
which 1­# mass range for the compact object is 2­4.3 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 # 2­2.4 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
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 non­rotating configuration.
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 =
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
23 km (this model also has relatively large radii of NSs).
Such model is very interesting from the astrophysical point
of view, and it is important to discuss scenarios of forma­
tion 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 absolute numbers
in the Galaxy. As we do not use explicitly any EOS in our
calculations, then our results can be applied 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 masses) it
is necessary to follow evolution in a binary as such ob­
jects 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 formation of
a massive NS in a binary we follow the 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 sec­
ondary companion by active pulsar. At some stages during
its evolution a binary which is going to produce finally 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
oretically (Timmes et al. 1996, Frayer & Kalogera 2001,
Woosley, Heger & Weaver 2002). Numerical models of col­
lapse are not as precise as necessary to determine the
exact shape of a NS mass spectrum (for example the
amount of fallback is not well known), however calcula­
tions show that the formation of NSs with high masses is
not favourable and most of them should have M # 1.3­
1.4 M
# .
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.
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 the 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 references
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 high additional velocity (#100--1000 km s -1 ) at birth
(Arzoumanian, Cherno# & Cordes 2002). However it is
much easier to understand main processes in a binary evo­
lution if one neglects kick. In addition, if a binary was not
unbounded after a SN explosion then an orbital eccentric­
ity quickly decays after the secondary fills its Roche lobe.
So, if we are not interested at the moment in the question
of the binary 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 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
mation of a common envelope is inevitable, mass trans­
fer 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. Which is only partly compensated
by higher mass of the donor.

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 3
After the common envelope stage 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 in the Sternberg Astronomical Institute. 3
Description of most of parameters of the code can be found
in Lipunov, Postnov & Prokhorov (1996). Below we men­
tion 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
magnetic field is assumed to be constant. Stage
durations can be found in Lipunov, Postnov &
Prokhorov (1996). 5 After the secondary fills the Roche
lobe the NS starts to accrete. At that moment mass ra­
tio is about 0.7 (the NS is lighter) and mass transfer is
stable with nearly zero mass loss from the system. Up
to equalizing of components masses matter transfer goes
on a thermal time scale, after equalizing -- on a nuclear.
Process of accretion can be stopped because of a switch­
ing on of a millisecond 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
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.
evaporation is proceeding the systems looks like the fa­
mous ``Black widow'' pulsar 1957+20 (and its twin PSR
J2051­0827). If accretion is not stopped then it continues
till a planet­like (Jupiter mass) companion remains. As we
see the final stage of such an evolution is a ``single'' mas­
sive rapidly rotating NS. In both cases the final mass of a
NS can reach 3.2­3.3 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
3.2­3.3 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.
The range 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#eres by the final stages of evolution.
The orbital period, P orb , is given in the table 1
just for an illustration. In our calculations we use
masses and semimajor axes. So the values of P orb
given in the table are simply calculated using max­
imum masses and minimum semimajor 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 second
group the time of rapprochement of the components due
to gravitational wave emission is too long, so that 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).
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
# .

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.
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# . This track is shown in the right
panel of fig. 1. Stages with a WD are shown in the box as
they distinguish the track 2b from 2a.
For semimajor axis a > 670 R
# the second com­
mon envelope results in NS­star merging, so the Thorne­
Zytkow 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 a
zero kick, then the following history of the system should
be the same as described in sec. 2.1.
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 signif­
icantly or not. As a kick velocity and a NS mass in our
calculations are assumed to be independent on a mass
of an exploding star 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 secondaries
because these stars do not su#er any important evolu­
tionary changes before a SN expolsion. Since a kick can
dramatically change the orbital parameters the situation
is di#erent for the initial orbital separation range. For ex­
ample, with a kick systems wider then the ones discussed
in sec. 2.1 can still form massive NSs.
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 "P"
NS "E" WD "E"
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 evolutionary
stage (in notation from Lipunov, Postnov & Prokhorov 1996)
and NS masses. In the frame we also indicate WD evolutionary
stages.
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 calculation for
the ranges of initial parameters which correspond to the

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
two groups of tracks described above. Each run includes
calculation 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 sim­
ilar to the one suggested in Arzoumanian, Cherno# &
Cordes (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 , what 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.
Mass distributions for both scenarios are shown in the
fig. 2. Note, that all small details in the figure are
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, Postnov & Prokhorov (1996) for more details.
1,5 2 2,5 3 3,5
M(NS)/Mo
0
0,05
0,1
0,15
0,2
0,25
0,3
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.
due to statistical noise (for example, the first peak
on the rising part of the dashed curve, or the mid­
dle peak on the solid one). The only important
details are the two peaks which corresponds to
tracks 2a and 2b (see the right panel of Fig. 1 and
table 1).
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 together we
can conclude that there are several 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 parameters of neutron
stars
The scenario of binary evolution we used has dif­
ferent types of uncertainties. Here we touch just
on of them -- possible correlations between param­
eters of the scenario.
We assume that such parameters of NSs as ini­
tal spin period, magnetic field, mass, velocity are
uncorrelated with each other. The reason for this
assumption is trivial: there is no direct indication
on such correlations. However, theorists suggested

6 S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars
30 32 34 36 38
Luminosity, [erg/s]
0
0,2
0,4
0,6
0,8
Fig. 3. Luminosity distribution of accreting massive NSs. The
left bin includes all sources with L < 10 30 erg s -1 . The dashed
line corresponds to the scenario with zero kick. The solid line
-- non­zero kick. In the ranges 10 30 <
# L <
# 10 34 erg s -1 and
10 36 <
# L <
# 10 37 erg s -1 the number of systems is not equal to
zero however it is very small. All distributions are normalized
to unity.
plethora of of them. We just give a list (incom­
plete) of possibilities and corresponding references
to original papers.
-- Spin -- magnetic field Thompson & Duncan
(1993).
-- Magnetic field -- mass Popov et al. (2002),
Heger et al. (2004).
-- Internal structure -- velocity Bombaci & Popov
(2004).
-- Binarity -- velocity Podsiadlowski et al. (2004).
-- Core mass -- velocity Scheck et al. (2004).
As here we mainly speak about masses and kick
velocities, let us make a short comment on last
two items. Our calculations may not be strongly
influenced by such correlations. The reason is that
the mass added during accretion is much larger
then the di#erence in initial masses. I.e., as we
need to accrete nearly two solar masses to obtain
the most massive SkyS at maximum rotation, we
can safely forget about the range of initial masses.
That's why in our calculations we even assume,
that all NSs have the same initial mass. For the
same reason we can neglect the second item in the
list above.
4.2. Globular clusters
All evolutionary tracks that we present above correspond
to binary evolution in the Galaxy, and they cannot be di­
rectly applied to globular clusters. However, we are mainly
interested in systems which after a formation of a NS ap­
pear to be sti#, ie. orbital velocities in binaries are larger
than a velocity dispersion in a cluster. It is also true during
the following evolution of a system, and it can be violated
only at the stage of Roche lobe overflow by a WD. We can
conclude that a dynamical influence of the globular cluster
stellar population should not destroy systems under dis­
cussion. Still duration of various evolutionary stages can
be di#erent in clusters and in the galactic disc, and so our
estimates of relative fractions cannot be valid for globular
clusters.
It is possible to speculate that as the formation rate of
millisecond pulsars is enhanced in globular clusters then
the formation rate of massive NSs can also be higher than
in the disc of the Galaxy. It is an important question be­
cause massive NSs from globular clusters can enrich the
disc population of these objects. In our opinion millisec­
ond radio pulsars and X­ray sources in globular clusters
can be good candidates for a search of massive NSs.
4.3. Low­mass black holes
As it was described in the Introduction at present all
well determined values of NS masses lie below #1.5 M # ,
on the other hand most of BH mass determinations lie
around values 6­10 M# (Ziolkowski 2004). So there is an
indication on a gap in intermediate mass range. Briefly
we can say, that accretion cannot fill this gap if,
as it is standardly assumed, NSs are formed with
M <
# 2 M # and BHs are formed with M >
# 5 M # .
If an EOS of NSs with a very high Mmax is realized in
nature, then up to # 3 M
# or even further, in the case of
maximum rotation, we can find NSs. Otherwise the gap
above # 2M
# should be filled only by BHs. Even in the
case of the EOS discussed by Ouyed & Butler (1999) low­
mass BHs can form from rapidly rotating massive NSs as
they slow down.
Fig. 2 (the solid line) clearly shows that the
number of low­mass BHs (or any other type of
compact objects) with 3.2 <
# M in our scenarion is
small. However, if Mmax is # 2 M
# , and if a binary is
not significantly influenced during a BH formation
(i.e. accretion continues), then the number of BHs
with Mmax <
# M <
# 3.2 M
# is significant. 7
There are several examples of binary systems with an
estimate of a mass of a compact object # 3­4 M
# (Orosz
et al. 2004, Shahbaz et al. 2004). These objects are con­
sidered as BH candidates. In principle such objects can be
formed in the frame of the scenario discussed above after
mass of a NS exceeds the Oppenheimer­Volko# limit.
4.4. Possible candidates
Main astrophysical manifestations of massive NSs are the
same as for normal NSs: X­ray sources and radio pulsars.
However, there are di#erences. Very massive NSs should
7 Mass growth of NSs and BHs in close binaries is also dis­
cussed in Bogomazov et al. (2005) and Bogomazov et al. (in
preparation).

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 7
have short spin periods as they get an additional mass by
accretion which spin­up them and provoke magnetic field
decay. 8 Of course a given millisecond pulsar can contain
a NS with a normal mass. Presence of a low­mass degen­
erated companion (a WD) can be an indication that the
system can hide a massive NS. One can mention another
additional signature of a massive NS -- very low magnetic
field.
If the magnetic field is very small, then the
Alfven radius becomes less than the NS radius,
and the accretion disk can nearly approach the
NS surface. This situation takes place when
B <
# 2 · 10 9 G —
M 1/2
-8 m 1/4 r -5/4 10 ,
here —
M-8 # —
M/10 -8 M# /yr, r 10 # r NS /10 km and
m --- mass of NS in Solar units. Thus, the mag­
netic field strenght can be <
# 10 9 G for Eddington
accretion rate. In that case a formation of a boundary
layer is favorable, and in the NS spectrum an additional
thermal component can be present (Imogamov & Sunyaev
1999). For massive NSs including Ouyed EOS radius of a
star is smaller than the distance to the last stable orbit,
so the disc cannot actually smoothly approach the surface
but qualitative properties of a spectrum will remain the
same.
All these consideration can be summarized in a list of
types of objects which can contain massive NSs.
-- X­ray sources with weak pulsations with signatures of
a boundary layer;
-- millisecond X­ray pulsars with WD companions;
-- millisecond radio pulsars with WD companions;
-- other kinds of millisecond X­ray pulsars;
-- other kinds of millisecond radio pulsars.
By ''other kinds'' we mean millisecond pulsars with other
types of companions or isolated (but old) ones. Note,
that we do not include into our calculations secondary
companions with very low initial mass (brown dwarfs).
However, such systems cannot produce massive rapidly
rotating SkyS as the total amount of accreted matter is
not su#cient. NSs with very low­mass companions like
the millisecond accreting pulsar SAX J1808.4­3658 or like
``black widow''­like radio pulsars can be produced in our
scenario via evaporating degenerated or non­degenerated
secondaries (see discussions on the evolution of this source
in Ergma & Antipova 1999, Bildsten & Chakrabarty 2001
and references therein).
Unfortunately our calculations cannot provide exact
numbers of objects of each type. Uncertainties are con­
nected with influence of population of sources from glob­
ular clusters and with uncertainties of the scenario itself.
For example we absolutely do not take into account in­
fluence of rotation on the evolution of normal stars (see
Langer et al. 2003).
8 If a NS has a very short spin period then pulsations in an
X­ray source can be undetectable as it is observed for many
low­mass X­ray binaries.
Ouyed (2002, 2004) discussed three binary systems as
possible candidates to massive SkyS: 4U 0614+09, 4U
1636­53, 4U 1820­30. From the point of view of evolu­
tionary scenarios discussed above all three really can con­
tain a massive NS. 4U 1820­30 is especially interesting.
The orbital period of the system is only 11 minutes which
means that the secondary is a low­mass helium star (see
Ballantyne & Strohmayer 2004 and references therein).
However this sources is situated in a globular cluster, and
so our considerations should be applied with care.
5. Conclusions
We discussed possible channels of massive NS formation. If
the EOS based on the Skyrme model suggested by Ouyed
& Butler (1999) is realized in nature then these objects
can be SkyS with masses up to 3.45 M # for maximum
rotation. The estimated numbers of these sources in the
Galaxy is high enough. Most favourable candidates are X­
ray binaries with WDs as donors, millisecond radio pulsars
in pair with WDs and accreting NSs with very low esti­
mated magnetic field. If no of so massive NSs are found
in these systems then the SkyS EOS has to be rejected.
Acknowledgements. We thank drs. I. Bombaci, R. Ouyed,
and the unknown referee for comments on the text of the
manuscript. SP thanks prof. J. Zdunik for discussions on the
EOS. This work was supported by the RFBR grants 04­02­
16720 and 03­02­16068.
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