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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 2
1 Universita di Padova, Dipartimento di Fisica, via Marzolo 8, 35131, Padova,
Italy
e-mail: popov@pd.infn.it
2 Sternberg Astronomical Institute, Universitetski pr. 13, 119992 Moscow, Russia
e-mail: polar@sai.msu.ru; mike@sai.msu.ru
Abstract. We discuss dierent 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 dierent evolutionary stages. A neu-
tron star increases its mass by accretion from a secondary companion. Signicant
growth of a neutron star mass due to accretion is possible only for particular val-
ues of initial parameters of the binary. In this paper we show that signicant part
of massive neutron stars with M >  2 M  can be observed as millisecond radio
pulsars, as X-ray sources in pair with white dwarfs, and as accreting neutron stars
with very low magnetic elds.
Key words. stars: neutron { stars: evolution { stars: statistics { stars: binary {
X-ray: 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 recent
time obtained results fall in a very narrow region 1.35-1.45 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
Send oprint requests to: S. Popov

2 S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars
no known NSs in binary radio pulsar systems with masses signicantly higher than the
canonical value 1:4 M  . However, this can be a result of a selection eect(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.
Observationally high masses of NSs are supported by data on X-ray binaries.
Estimates for several systems give very high values: 1.8-2.2 M  for Vela X-1 (Quaitrell
et al. 2003), 2.4 M  for 4U 1700-37 (Clark et al. 2002; see also Heineke et al. 2003, van
Kerkwijk 2004). Very recently Shahbaz et al. (2004) presented observations of a low-mass
X-ray 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.
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 expected to be
a kind of NSs with the highest value of maximum 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 non-rotating conguration. 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 astrophysical 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 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 low-mass 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.

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 3
2. Possible channels of massive neutron star formation
As mass determination for NSs is possible only in binary systems 1 we focus on potentially
observable stages of evolution 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 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 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 low-mass secondary companion by active pulsar. At some stages
during its evolution a binary which is going to produce nally an object of our interest
can be observed as an X-ray 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  1:2-
1.5 M  . This conclusion is supported both observationally (van Kerkwijk 2004) and
theoretically (Timmes et al. 1996, Frayer & Kalogera 2001, Woosley, Heger & Weaver
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 fallback 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  1:3-1:4 M  .
A discovery of a NS with M >  1:8 M  should mean that the mass was increased after
formation of the compact object during its evolution (if the mass is signicantly 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 companion. As we note above the rst 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 massive isolated NS (or a BH) can form. This
way of evolution also will not be discussed below. In the following only binary evolution
of a NS in pair with a normal star or a WD will be studied.
At rst for an illustration let us assume an isotropic collapse, ie. zero kick. Such
an assumption is not realistic as most part of NS { nearly all radio pulsars { obtain
1 Note, that in principle there is a possibility to determine an isolated NS mass by microlensing
eects, however we do not touch this issue here.

4 S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars
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 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 lls 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 consid-
eration is given). The most obvious channel to form a rapidly rotating massive NS is
an evolution in a low-mass or intermediate mass binary (see for example recent calcula-
tions by Podsiadlowski et al. 2002). This path includes for example millisecond pulsars
(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 dierent
situations after the NS formation when the secondary lls its Roche lobe: i.) a normal
star can ll its Roche lobe without a common envelope formation; ii.) a normal star can
ll its Roche lobe with a common envelope formation; iii.) a WD lls its Roche lobe.
To ll the Roche lobe a normal secondary star has to evolve further the main sequence
stage. During its evolution prior to the Roche lobe over ow the mass of the star is nearly
constant (see detailed tracks below). A common envelope is not formed if the normal star
is not signicantly heavier than the NS. In this regime mass is not lost from the binary
system. For more massive secondaries formation of a common envelope is inevitable, mass
transfer is unstable. In this regime signicant fraction of the mass ow is lost from the
system, so the mass of the NS grows less eectively. Which is only partly compensated
by higher mass of the donor.
After the common envelope stage orbital separation becomes smaller, so later on even
a degenerated core of the secondary { a WD { can ll the Roche lobe.
2.1. Evolutionary tracks
For our calculations we use the \Scenario Machine" code developed in the Sternberg
Astronomical Institute. 2 Description of most of parameters of the code can be found
in Lipunov, Postnov & Prokhorov (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 accretion (with _
M much larger than
the Eddington value) is possible.
2 Online materials are available at http://xray.sai.msu.ru/sciwork/scenario.html and
http://xray.sai.msu.ru/ mystery/articles/review/.

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 5
{ During accretion the magnetic eld of a NS decays down to the value which cannot
prevent rapid (maximum) rotation of the NS.
{ Oppenheimer-Volko mass of a rapidly rotating NS (the critical mass of a BH forma-
tion) 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 rst group has initial value of the semimajor axis a = 290 R  and
star masses M 1 = 10:5 M  , M 2 = 2 M  (g. 1 left) 3 . After the massive component leaves
the main sequence it expands and lls 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 low-mass helium 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 lls its Roche lobe the NS is at the stages of ejector and propeller
(see for example Lipunov 1992 for stages descriptions). After the secondary lls the
Roche lobe the NS starts to accrete. At that moment mass ratio 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 switching 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 evaporation
is proceeding the systems looks like the famous \Black widow" pulsar 1957+20 (and
its twin PSR J2051-0827). If accretion is not stopped then it continues till a planet-like
(Jupiter mass) companion remains. As we see the nal stage of such an evolution is a
\single" massive rapidly rotating NS. In both cases the nal 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 narrow in a sense that small changes
in the initial conditions do not allow a massive NS formation. Also uncertain parameters
of the common envelope stage can signicantly in uence 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 dieres by the nal stages of evolution. 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
dierence form the rst group of tracks is a more massive secondary companion. Because
of that the common envelope during the rst mass transfer is less eective, and after a SN
3 Colored version of the gure in high resolution is avalable on the Web:
http://xray.sai.msu.ru/polar/html/publications/ouyed/

6 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
Track 2b
a 135R  279R  0.17
M1 10.3M  12.4M  0.046
M2 3.9M  7.4M  0.15
a system with a = 170 R  and low eccentricity is formed (the mass of the secondary is
not changed). Later the secondary lls the Roche lobe. Mass ratio is high, mass transfer is
unstable and the common envelope forms. At the end of the common 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 increases its mass up to  2:3 M 
(for more massive donors mass loss from the system is more eective, 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 dierent 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 over ow. Systems with smaller orbital separation
have enough time to approach to each other close enough for the beginning of WD
over ow. This situation corresponds to the initial parameters a = 200 R  , M 1 = 12 M  ,
M 2 = 4 M  (track 2b in the table 1).
The main dierence 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 g.1, but after the common
envelope semiaxis of the system is just  3 R  . A WD has enough time to ll the Roche
lobe and completely 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 g. 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 common envelope results in NS-star
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.

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 7
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 rst group. The rst 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 diers by a higher mass of a secondary companion. Because
of this dierence the rst mass transfer goes on without a common envelope. A NS gathers an
additional mass during one or two episodes of accretion. If the orbital separation is not very
large ( 200 R  , see text) then at rst the NS accretes from a normal secondary lling 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.
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 necessary 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

8 S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars
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 pa-
rameters 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 signicantly or not. As a kick velocity
and a NS mass in our calculations are assumed to be independent on a mass of an ex-
ploding star we do not expect that a range of masses of primaries should be modied.
The same can be said about the range of initial masses of secondaries because these stars
do not suer any important evolutionary changes before a SN expolsion. Since a kick
can dramatically change the orbital parameters the situation is dierent for the initial
orbital separation range. For example, with a kick systems wider then the ones discussed
in sec. 2.1 can still form massive NSs.
In the next section we present results of our calculations 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 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 similar 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 rst 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(max-
min)/(max+min). For the rst 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 non-zero 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 rst channel (g. 1 left panel) we do not obtain signicant 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

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 9
Table 2. Fractions of massive NSs at dierent 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 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 detection 4 . 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 supereddington accretion is negligible).
For the non-zero kick model about 25% of accreting massive NSs have normal stars
as secondaries, the rest 75% have WD companions. For zero kick nearly all massive NSs
accrete from WDs which ll their Roche lobes.
Mass distributions for both scenarios are shown in the g. 2. Finally in the last gure
we represent luminosity distributions. For the scenario with non-zero 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 we brie y discuss a possibility of massive NS formation in globular clusters, low-
mass BHs, and types of sources which can host massive NSs.
4.1. Globular clusters
All evolutionary tracks that we present above correspond to binary evolution in the
Galaxy, and they cannot be directly applied to globular clusters. However, we are mainly
interested in systems which after a formation of a NS appear 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 over ow by a WD. We can conclude that a dynamical in uence of the globular
4 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 non-detectability of ejectors which are not active
as radio pulsars. See for example Lipunov (1992) or Lipunov, Postnov & Prokhorov (1996) for
more details.

10 S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars
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 { non-zero 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.
cluster stellar population should not destroy systems under discussion. Still duration of
various evolutionary stages can be dierent 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 because massive NSs from globular
clusters can enrich the disc population of these objects. In our opinion millisecond radio
pulsars and X-ray sources in globular clusters can be good candidates for a search of
massive NSs.
4.2. Low-mass 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
6-10 M (Ziolkowski 2004). So there is an indication on a gap in intermediate mass
range.
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 nd NSs. Otherwise the gap above
 2M  should be lled only by BHs. Even in the case of the EOS discussed by Ouyed

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 11
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 {
non-zero 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.
& Butler (1999) low-mass BHs can form from rapidly rotating massive NSs as they slow
down.
There are several examples of systems with an estimate of a mass of a compact
object  3-4 M  (Orosz et al. 2004, Shahbaz et al. 2004). In principle such objects can
be formed in the frame of the scenario discussed above after mass of a NS exceeds the
Oppenheimer-Volko limit.
4.3. Possible candidates
Main astrophysical manifestations of massive NSs are the same as for normal NSs: X-ray
sources and radio pulsars. However, there are dierences. Very massive NSs should have
short spin periods as they get an additional mass by accretion which spin-up them and
provoke magnetic eld decay. 5 Of course a given millisecond pulsar can contain a NS
with a normal mass. Presence of a low-mass degenerated 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 eld.
5 If a NS has a very short spin period then pulsations in an X-ray source can be undetectable
as it is observed for many low-mass X-ray binaries.

12 S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars
If the magnetic eld is very small, then the accretion disk can nearly approach the
NS surface. 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.
{ X-ray sources with weak pulsations with signatures of a boundary layer;
{ millisecond X-ray pulsars with WD companions;
{ millisecond radio pulsars with WD companions;
{ other kinds of millisecond X-ray 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 suфcient. NSs with
very low-mass companions like the millisecond accreting pulsar SAX J1808.4-3658 or
like \black widow"-like radio pulsars can be produced in our scenario via evaporating
degenerated or non-degenerated 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 connected with in uence of population of sources from globular clusters
and with uncertainties of the scenario itself. For example we absolutely do not take into
account in uence of rotation on the evolution of normal stars (see Langer et al. 2003).
Ouyed (2002, 2004) discussed three binary systems as possible candidates to massive
SkyS: 4U 0614+09, 4U 1636-53, 4U 1820-30. From the point of view of evolutionary sce-
narios discussed above all three really can contain a massive NS. 4U 1820-30 is especially
interesting. The orbital period of the system is only 11 minutes which means that the
secondary is a low-mass 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

S.B. Popov and M.E. Prokhorov: Formation of massive skyrmion stars 13
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 estimated magnetic eld. If no of so massive NSs are found in these systems then
the SkyS EOS has to be rejected.
Acknowledgements. We thank dr. R. Ouyed for comments on the text of the manuscript. SP
thanks prof. J.L. Zdunik for discussions on the EOS. This work was supported by the RFBR
grants 04-02-16720 and 03-02-16068.
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