A compact star rotating at 1122 Hz and the r-mode instability
Alessandro Drago1, Giuseppe Pagliara2 and Irene Parenti1
ABSTRACT
We show that r-mode instabilities severely constraint the composition of a
compact star rotating at a sub-millisecond period. In particular, the only vi-
able astrophysical scenario for such an object, present inside the Low Mass X-
ray Binary associated with the x-ray transient XTE J1739-285, is that it has
a strangeness content. Since previous analysis indicate that hyperonic stars or
stars containing a kaon condensate are not good candidates, the only remaining
possibility is that such an object is either a strange quark star or a hybrid quark-
hadron star. We also discuss under which conditions sub-millisecond pulsars are
rare.
Subject headings: star: neutron — star: rotation — r-mode — quark star —
submillisecond pulsar — gravitational waves
Very recently Kaaret et al. (2007) reported the evidence of a X-ray transient with a
pulsed component of the emission having a frequency f = 1122 ± 0.3 Hz. This signal is
interpreted as due to the rotation of the central neutron star. As such this object would be
the most rapidly rotating compact star discovered up to now. This single observation clearly
needs to be confirmed, maybe by the analysis of future X-ray transients of the same object.
The implications of this rapid rotation on the Equation of State (EOS) and in particular on
the allowed values of the mass and the radius have been discussed in Lavagetto et al. (2006)
and in Bejger et al. (2007). Here we discuss the problem of the stability respect to r-modes of
such a rapidly rotating object (for a review on r-modes see Andersson & Kokkotas (2001)).
It has been widely discussed in the literature the possibility that r-mode instabilities
can very efficiently drag angular momentum from a rotating compact star if its temporal
evolution in the Ω−T plane (angular velocity and temperature) enters the r-mode instability
window, see e.g. Andersson (1998) and Friedman & Morsink (1998). Therefore huge regions
of the Ω− T plane are excluded. Moreover, the size and position of that window is strictly
1Dipartimento di Fisica, Università di Ferrara and INFN, Sezione di Ferrara, 44100 Ferrara, Italy
2Inst. Theoretische Physik, Goethe Universität, D-60438, Frankfurt am Main, Germany and INFN, Italy
http://arxiv.org/abs/0704.1510v3
– 2 –
related to the composition of the star, since it is strongly dependent on the value of bulk and
shear viscosity. It is particularly important to recall that for stars containing strangeness,
as hyperon stars (Lindblom & Owen 2002), hybrid stars (Drago et al. 2005) and strange
quark stars (Madsen 2000) there is also a contribution to the bulk viscosity associated with
the formation of strangeness. Due to this, the instability window splits into two parts:
one which starts at temperatures larger than (7 ± 3) 109 K (High Temperature Instability
Window HTIW) and a lower temperature window at temperatures smaller than (5±4) 108 K
(Low Temperature Instability Window, LTIW) 1. Concerning the left border of the LTIW,
its position is regulated by the shear viscosity and by the so called viscous boundary layers
located at the interface between the crust and the fluid composing the inner part of the
star (Bildsten & Ushomirsky 2000). In particular, in bare quark stars, either composed by
normal or by superconducting quark matter, due to the absence of a significant crust, the
left border of the LTIW extends to much lower temperatures and the LTIW has a minimum
corresponding to a significantly lower temperature than in the case of stars containing a
crust.
In this Letter we discuss in which region of the Ω−T plane the compact star originating
XTE J1739-285 is most likely located, due to its composition. Let us start by discussing
the simplest possibility, i.e. that the object is a neutron star. In principle a neutron star
can rapidly rotate in two cases, either if its temperature is very large, above a few MeV,
or if it is recycling, spinning up due to mass accretion (see Fig. 1). Concerning the first
possibility, a hot neutron star would be a newly born one, since the time needed to cool
below one MeV is of the order of one minute. This is clearly not the case of the stellar
object under discussion. Concerning recycling, it should take place on the left side of the
instability window, located at lower temperatures (see Fig. 1). The main result of the analysis
of Andersson et al. (2000), revisiting previous analysis (Levin 1999; Bildsten & Ushomirsky
2000) is that a neutron star can never spin-up to a rotational period shorter than ∼ 1.5 ms.
This result is based on the estimates of the temperature and of the mass accretion rates
of Low Mass X-ray Binaries (Brown & Bildsten 1998; Bhattacharyya 2002) indicating that
temperatures lower than ∼ 108 K cannot be reached. Therefore a sub-millisecond neutron
star cannot be present at the center of a Low Mass X-ray Binary (LMXB). It is important
to remark that this conclusion is confirmed by more recent analysis, taking into account the
composition stratification of the rigid crust (Glampedakis & Andersson 2006) and discussing
the nonlinear development of the r-mode instability (Bondarescu et al. 2007).
Let us now discuss the case in which the compact star contains strangeness. It has been
1Here and in the following we are discussing the temperature of the region of the star where r-mode
instabilities can develop. Typically it is located at a depth of a few kilometers below the surface.
– 3 –
shown, in the case of hyperon stars (Wagoner 2002; Reisenegger & Bonacic 2003), and for
hybrid stars (Drago et al. 2006) (strange quark stars also have a similar behavior), that due
to the large value of the bulk viscosity associated with the non-leptonic weak decays, there
are two windows of instability, the LTIW and the HTIW introduced above. The HTIW does
not affect significantly the angular velocity of the star because the cooling of a newly born
star is so fast that there is not enough time for the r-mode instability to drag a significant
fraction of the angular momentum. Therefore, the star exits the HTIW with an angular
velocity close to the initial one. When the temperature drops down to a few 108 K the star
reaches the LTIW and it starts to lose angular momentum due to r-mode instability. In
Fig. 1 we show examples of the instability windows in the case of a pure quark star and of a
hybrid star. Notice that the position of the LTIW depends rather strongly on the mass of the
strange quark ms. For large values of ms the instability windows shrink considerably. In our
analysis we have considered two possibilities concerning the value of ms: a small value ∼ 100
MeV, of the order of the strange quark current mass, and a large value ∼ 300 MeV, similar
to what has been obtained in NJL-like models of quark matter in which a density-dependent
constituent mass has been computed (Buballa 2005).
Concerning the left side of the LTIW, it is easy to see that it cannot be used to accom-
modate a submillisecond pulsar. Indeed, in the case of hybrid or hyperonic stars the left side
of the LTIW is similar to the one of neutron stars, discussed above, and we can apply here
the same analysis done for a neutron star and concerning the minimal temperature of the
compact object at the center of a LMXB. Therefore also hybrid or hyperonic stars cannot
recycle to frequencies significantly exceeding 700–800 Hz if they are to the left of the LTIW.
Moreover, in the case of quark stars the absence of viscous boundary layers implies that they
can rotate rapidly only on the right side of the LTIW (a very light crust suspended over
the surface of a quark star is irrelevant from the viewpoint of r-mode instability, although
it can be important for the production of x-ray bursts, see Page & Cumming (2005)). In
conclusion, stars containing strangeness can rotate at submillisecond periods only if they are
to the right of the LTIW.
Let us now discuss the results of our analysis also considering the constraints posed
onto the EOS by the mass shedding limit. The main result of Lavagetto et al. (2006) and of
Bejger et al. (2007) is that soft EOSs, and in particular the ones based on hyperonic matter
or on matter in which kaon condensation takes place, are rather unlikely. In fact, not only
the range of allowed masses is rather small, but moreover the only configurations satisfying
the stability constraint for mass shedding turn out to be supra massive (Bejger et al. 2007),
making it very difficult to use these stellar structures in a mass accretion scenario. This result
confirms previous discussions of other astrophysical objects ruling out soft EOSs (Ozel 2006).
On the other hand, stars containing deconfined quark matter are not excluded if the quark
– 4 –
EOS is stiff enough (Alford et al. 2007). We can therefore conclude that the object under
discussion is either a quark or a hybrid star.
We can now study the temporal evolutions of the angular velocity of the star Ω, of
its temperature T and of the amplitude of the r-modes α which are calculated by solving
the set of differential equations given in Andersson et al. (2002) (Eqs. 15-23-24). The only
technical difference in our calculation is the inclusion of the reheating associated with the
dissipation of r-modes by bulk viscosity, as discussed in various papers (Wagoner 2002;
Reisenegger & Bonacic 2003; Drago et al. 2006). Due to the reheating, the time needed for
the star to cool down along the right border of the LTIW is rather long. In the equation
regulating the thermal evolution (Andersson et al. 2002) we have added a term:
Ėbv = 2E/τbv (1)
where E ∼ 1051α2M1.4R
erg is the energy of the r-modes (the mass, radius and ro-
tation period of the star are here expressed in units of 1.4 M⊙, 10 km and in millisecond,
respectively) and τbv is the bulk viscosity. Due to the large value of the bulk viscosity for
strange matter at temperatures of a few 108 K, the related dissipation of r-modes strongly
reheats the star. The trajectory in the Ω−T plane describing the time evolution of the star
follows essentially the border of the LTIW and the star keeps rotating as a sub-millisecond
pulsar for a very long time, strongly dependent on the value of ms and ranging from ∼ 10
years when ms = 100 MeV to ∼ 10
8 years when ms = 300 MeV, as shown in Fig. 2. In Fig. 3
we show the temporal dependence of the r-mode amplitude. Notice that, in the absence of
reheating the time needed to slow-down to frequencies below 1 kHz is at maximum of the
order of ∼ 104 years, and can be much shorter than a year if a small value of ms is adopted.
In the past this branch of the instability window was only discussed in connection with very
young pulsars because reheating due to bulk viscosity was not taken into account.
Before discussing the possible astrophysical scenarios for a submillisecond pulsar inside
a LMXB, it is important to remark two constraints that a realistic model should fulfil. First,
a LMBX is an old object, with a typical age of 108–109 years which is the time needed by the
companion (having a low mass) to fill its Roche lobe and to start accreting on the neutron
star. Therefore we need to provide a mechanism allowing the central object inside the LMXB
to rotate rapidly while being so old. Second, submillisecond pulsars are certainly rather rare,
and in particular there is no evidence of a uniform distribution of pulsar rotational frequencies
extending from a few hundred Hz up to more than a kHz. Therefore a realistic model should
also indicate why most of the compact stars will not be detected rotating at a submillisecond
period. As we will show, these two constraints can be satisfied in two different astrophysical
scenarios: a first scenario in which a old hybrid or quark star is accelerated up to frequencies
exceeding one kHz by mass accretion and a second scenario in which the quark or hybrid
– 5 –
star is born with a submillisecond period and it is now spinning down by r-mode instability
(see Fig. 4).
The possible realization of these scenarios depends on two main ingredients:
• the value ofms, which, as we have already shown, regulates the magnitude and position
of the LTIW;
• the cooling rate of the central region of the star which determines the inner temperature
T of a star which is accreting material from the companion (Miralda-Escudé et al.
1990).
Concerning the cooling rate, if strange quark matter is present in a compact star, direct
URCA processes are possible and therefore the cooling is (generally) fast. It turns out from
Miralda-Escudé et al. (1990) that in this case the inner temperature is ∼ 5×107 K for a mass
accretion rate of Ṁ = 10−10M⊙/year and that the temperature scales as T ∝ Ṁ
1/6. Another
possibility recently proposed is that, due to the formation of diquark condensates, URCA
processes are strongly suppressed and the cooling turns out to be slow (Blaschke et al. 2000).
At the same time, bulk viscosity can still be large enough to suppress r-mode instability
(Blaschke & Berdermann 2007). In the following we discuss both possibilities, either of a
fast or of a slow cooling.
Let us consider first the case in which ms = 300 MeV.
The first scenario, i.e. the star span up by accretion, can indeed be realized if T ≥ 108 K
(see lower panel of Fig. 1). Such a temperature can be reached via reheating due to mass
accretion. In the case of fast cooling a rather large value of mass accretion rate is needed,
Ṁ ∼ 10−8M⊙/year, and this stringent request can explain why submillisecond pulsars are
rare. A model in which the cooling is slow is instead excluded, because it would be extremely
easy to re-accelerate the star to very large frequencies and therefore submillisecond pulsars
would not be rare.
Also the second scenario in which the star is spinning down due to r-mode instabilities is
possible. As shown in Fig. 2, the time spent by the star above 1122 Hz is of order of 108 years
and it is therefore compatible with the typical age of LMXBs. In this case submillisecond
pulsars are rare because only a (small) fraction of newly born compact stars can rotate
with submillisecond periods (see e.g. the discussions in Andersson & Kokkotas (2001) and
in Perna et al. (2007)). In this scenario the distribution of millisecond and submillisecond
pulsars is determined by the mass of the star at birth: if the mass is large enough, strange
quark matter can form in the core of the star, r-modes are efficiently damped by the large
bulk viscosity and the star can be a submillisecond pulsar at birth; if the mass of the star
is small, strange quark matter can not form, the bulk viscosity of nucleonic matter can not
– 6 –
damp r-modes and therefore in a short period the star loses almost all its angular momentum
(Lindblom et al. 1998).
We consider now the case in which ms = 100 MeV.
The first scenario in which the star is spinning up by accretion can be realized but T must be
at least ∼ 5×108 K for the star to be accelerated up to 1122 (see upper panel of Fig. 4). Such
a temperature can be reached but it needs strong reheating due to a large mass accretion
rate and slow cooling.
The second scenario, i.e. the star is spinning down by r-mode instability, is ruled out since,
as shown in Fig. 2, the time spent by the star above 1122 Hz is of order of 100 years and
this is incompatible with the typical age of a LMXB.
In conclusion, the two astrophysical scenarios proposed to explain a submillisecond
pulsar inside a LMXB can both be realized by a quark or a hybrid star if ms ∼ 300 MeV.
Instead, if the ms is small, only the first scenario is possible and it requires a slow cooling.
Our results are summarized in Table 1.
The main uncertainties in our analysis are due to the possible existence of other damping
mechanisms, taking place on the left side of the LTIW. For instance magnetic fields can be
important to suppress r-mode instabilities (Rezzolla et al. 2000), but their effect is probably
negligible for frequencies exceeding ∼ 0.35 ΩK , if the internal magnetic field is not larger than
∼ 1016 G. Obviously, even larger uncertainties exist concerning quark matter. As discussed
above, the bulk viscosity of quark matter strongly depends on the strange quark mass and
on the possible formation of a diquark condensate (Alford & Schmitt 2007). Clearly, our
analysis can provide much needed constraints on the EOS of quark matter.
Finally, let us stress that the outcome of our analysis is that a compact star rotating at
a submillisecond period inside a LMXB can only be a quark or a hybrid star. Future obser-
vations will be important to clarify if the object at the center of XTE J1739-285 constitutes
indeed an example.
It is a pleasure to thanks Carmine Cuofano for many useful discussions.
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– 9 –
Table 1: Microphysics and astrophysical scenarios
ms [MeV] cooling scenarios (I or II) conclusions
300 fast I: O.K., Ṁ ≥ 10−8M⊙/y O.K. for
II: O.K., needs rapid rotation at birth both scenarios
300 slow I: no, submillisecond pulsars too frequent excluded
II: O.K. due to scenario I
100 fast I: no, T is too low not possible
II: no, slow down too rapid in both scenarios
100 slow I: O.K., needs large Ṁ O.K. for
II: no, slow down too rapid scenario I
Table 2: Connection between microphysics (mass of the strange quark and cooling rate)
and astrophysical scenarios for a submillisecond pulsar inside a LMXB. The first scenario
corresponds to a compact star which is spinning up due to mass accretion; the second scenario
to a compact star born as a submillisecond pulsar and still in the process of slowing down.
– 10 –
7 8 9 10 11
Log10T@KD
f@HzD
ms=300 MeV
7 8 9 10 11
Log10T@KD
f@HzD
7 8 9 10 11
Log10T@KD
f@HzD
ms=100 MeV
7 8 9 10 11
Log10T@KD
f@HzD
Fig. 1.— r-mode instability windows. The star is unstable and loses angular momentum
by emitting gravitational waves in the regions above the instability lines displayed in figure.
Thin solid lines correspond to neutron stars, for two extreme values of masses and radii
allowed by the analysis done by Bejger et al. (2007). The shaded area in between is repre-
sentative of intermediate values of masses and radii. Dashed lines delimit the LTIW and
the HTIW for strange quark stars. Thick solid lines correspond to hybrid stars. In both
cases the mass and radius of the star are M = 2M⊙ and R = 13 km. In the upper panel
ms = 100 MeV and in the lower panel ms = 300 MeV. The bulk and shear viscosities are
in general from Andersson & Kokkotas (2001). Bulk viscosity of strange quark matter at
high temperatures is from Madsen (2000). The arrow indicates the minimal temperature
considered possible in LMXBs. The dot indicates the actual position of the compact star
associated with XTE J1739-285, as it results from our analysis.
– 11 –
4 6 8 10 12 14 16
Log10@t@sDD
f@HzD
100 300
300100
Fig. 2.— Time dependence of the rotational frequency of the compact star. The thin lines
take into account only the reheating due to shear viscosity, the thick lines take into account
also the effect of reheating due to bulk viscosity. We show results for two different values
of the strange quark mass, ms =100 MeV and ms =300 MeV. The dashed horizontal line
corresponds to a frequency of 1122 Hz.
– 12 –
4 6 8 10 12
Log10@t@sDD
4 6 8 10 12
Log10@t@sDD
Fig. 3.— Temporal evolution of the amplitude α of the r-modes. Results
without reheating in panel (a) and taking into account reheating in panel
(b). The amplitude of the r-mode is significantly damped when reheating is
taken into account, the maximum amplitude is reduced by roughly 2 orders of
magnitude. Therefore non linear effects should be suppressed.
– 13 –
7 7.5 8 8.5 9
Log@T@KDD
f@HzD
HbL 108 yr
ms=300 MeV
HaL 10 yr
7.5 8 8.5 9
Log@T@KDD
f@HzD
ms=100 MeV
HdL 108yr
HbL 109yr
Fig. 4.— Upper panel: first scenario (see text). The star is born with a low angular velocity.
After a cooling period at a constant angular velocity (line (a)), the star reaches the instability
window and loses angular momentum during a long period, ∼ 109 years (line (b)). After
this long period the mass accretion process starts and the star is heated (line (c)) and re-
accelerated (line (d)) to a frequency of 1122 Hz (indicated by the thick dot). Lower panel:
second scenario (see text). The star is born with a large angular velocity. After a short
period of cooling at a constant frequency (line (a)) the star enters the instability window
and loses angular momentum via r-mode (line (b)). The frequency of the star can be larger
than 1122 Hz (indicated by the thick dot) for a period of ∼ 108 years. The mass and radius
of the star are M = 1.4M⊙ and R = 10 km, respectively.

  We show that r-mode instabilities severely constraint the composition of a
compact star rotating at a submillisecond period. In particular, the only
viable astrophysical scenario for such an object, present inside the Low Mass
X-ray Binary associated with the x-ray transient XTE J1739-285, is that it has
a strangeness content. Since previous analysis indicate that hyperonic stars or
stars containing a kaon condensate are not good candidates, the only remaining
possibility is that such an object is either a strange quark star or a hybrid
quark-hadron star. We also discuss under which conditions sub-millisecond
pulsars are rare.

A compact star rotating at 1122 Hz and the r-mode instability
Alessandro Drago1, Giuseppe Pagliara2 and Irene Parenti1
ABSTRACT
We show that r-mode instabilities severely constraint the composition of a
compact star rotating at a sub-millisecond period. In particular, the only vi-
able astrophysical scenario for such an object, present inside the Low Mass X-
ray Binary associated with the x-ray transient XTE J1739-285, is that it has
a strangeness content. Since previous analysis indicate that hyperonic stars or
stars containing a kaon condensate are not good candidates, the only remaining
possibility is that such an object is either a strange quark star or a hybrid quark-
hadron star. We also discuss under which conditions sub-millisecond pulsars are
rare.
Subject headings: star: neutron — star: rotation — r-mode — quark star —
submillisecond pulsar — gravitational waves
Very recently Kaaret et al. (2007) reported the evidence of a X-ray transient with a
pulsed component of the emission having a frequency f = 1122 ± 0.3 Hz. This signal is
interpreted as due to the rotation of the central neutron star. As such this object would be
the most rapidly rotating compact star discovered up to now. This single observation clearly
needs to be confirmed, maybe by the analysis of future X-ray transients of the same object.
The implications of this rapid rotation on the Equation of State (EOS) and in particular on
the allowed values of the mass and the radius have been discussed in Lavagetto et al. (2006)
and in Bejger et al. (2007). Here we discuss the problem of the stability respect to r-modes of
such a rapidly rotating object (for a review on r-modes see Andersson & Kokkotas (2001)).
It has been widely discussed in the literature the possibility that r-mode instabilities
can very efficiently drag angular momentum from a rotating compact star if its temporal
evolution in the Ω−T plane (angular velocity and temperature) enters the r-mode instability
window, see e.g. Andersson (1998) and Friedman & Morsink (1998). Therefore huge regions
of the Ω− T plane are excluded. Moreover, the size and position of that window is strictly
1Dipartimento di Fisica, Università di Ferrara and INFN, Sezione di Ferrara, 44100 Ferrara, Italy
2Inst. Theoretische Physik, Goethe Universität, D-60438, Frankfurt am Main, Germany and INFN, Italy
http://arxiv.org/abs/0704.1510v3
– 2 –
related to the composition of the star, since it is strongly dependent on the value of bulk and
shear viscosity. It is particularly important to recall that for stars containing strangeness,
as hyperon stars (Lindblom & Owen 2002), hybrid stars (Drago et al. 2005) and strange
quark stars (Madsen 2000) there is also a contribution to the bulk viscosity associated with
the formation of strangeness. Due to this, the instability window splits into two parts:
one which starts at temperatures larger than (7 ± 3) 109 K (High Temperature Instability
Window HTIW) and a lower temperature window at temperatures smaller than (5±4) 108 K
(Low Temperature Instability Window, LTIW) 1. Concerning the left border of the LTIW,
its position is regulated by the shear viscosity and by the so called viscous boundary layers
located at the interface between the crust and the fluid composing the inner part of the
star (Bildsten & Ushomirsky 2000). In particular, in bare quark stars, either composed by
normal or by superconducting quark matter, due to the absence of a significant crust, the
left border of the LTIW extends to much lower temperatures and the LTIW has a minimum
corresponding to a significantly lower temperature than in the case of stars containing a
crust.
In this Letter we discuss in which region of the Ω−T plane the compact star originating
XTE J1739-285 is most likely located, due to its composition. Let us start by discussing
the simplest possibility, i.e. that the object is a neutron star. In principle a neutron star
can rapidly rotate in two cases, either if its temperature is very large, above a few MeV,
or if it is recycling, spinning up due to mass accretion (see Fig. 1). Concerning the first
possibility, a hot neutron star would be a newly born one, since the time needed to cool
below one MeV is of the order of one minute. This is clearly not the case of the stellar
object under discussion. Concerning recycling, it should take place on the left side of the
instability window, located at lower temperatures (see Fig. 1). The main result of the analysis
of Andersson et al. (2000), revisiting previous analysis (Levin 1999; Bildsten & Ushomirsky
2000) is that a neutron star can never spin-up to a rotational period shorter than ∼ 1.5 ms.
This result is based on the estimates of the temperature and of the mass accretion rates
of Low Mass X-ray Binaries (Brown & Bildsten 1998; Bhattacharyya 2002) indicating that
temperatures lower than ∼ 108 K cannot be reached. Therefore a sub-millisecond neutron
star cannot be present at the center of a Low Mass X-ray Binary (LMXB). It is important
to remark that this conclusion is confirmed by more recent analysis, taking into account the
composition stratification of the rigid crust (Glampedakis & Andersson 2006) and discussing
the nonlinear development of the r-mode instability (Bondarescu et al. 2007).
Let us now discuss the case in which the compact star contains strangeness. It has been
1Here and in the following we are discussing the temperature of the region of the star where r-mode
instabilities can develop. Typically it is located at a depth of a few kilometers below the surface.
– 3 –
shown, in the case of hyperon stars (Wagoner 2002; Reisenegger & Bonacic 2003), and for
hybrid stars (Drago et al. 2006) (strange quark stars also have a similar behavior), that due
to the large value of the bulk viscosity associated with the non-leptonic weak decays, there
are two windows of instability, the LTIW and the HTIW introduced above. The HTIW does
not affect significantly the angular velocity of the star because the cooling of a newly born
star is so fast that there is not enough time for the r-mode instability to drag a significant
fraction of the angular momentum. Therefore, the star exits the HTIW with an angular
velocity close to the initial one. When the temperature drops down to a few 108 K the star
reaches the LTIW and it starts to lose angular momentum due to r-mode instability. In
Fig. 1 we show examples of the instability windows in the case of a pure quark star and of a
hybrid star. Notice that the position of the LTIW depends rather strongly on the mass of the
strange quark ms. For large values of ms the instability windows shrink considerably. In our
analysis we have considered two possibilities concerning the value of ms: a small value ∼ 100
MeV, of the order of the strange quark current mass, and a large value ∼ 300 MeV, similar
to what has been obtained in NJL-like models of quark matter in which a density-dependent
constituent mass has been computed (Buballa 2005).
Concerning the left side of the LTIW, it is easy to see that it cannot be used to accom-
modate a submillisecond pulsar. Indeed, in the case of hybrid or hyperonic stars the left side
of the LTIW is similar to the one of neutron stars, discussed above, and we can apply here
the same analysis done for a neutron star and concerning the minimal temperature of the
compact object at the center of a LMXB. Therefore also hybrid or hyperonic stars cannot
recycle to frequencies significantly exceeding 700–800 Hz if they are to the left of the LTIW.
Moreover, in the case of quark stars the absence of viscous boundary layers implies that they
can rotate rapidly only on the right side of the LTIW (a very light crust suspended over
the surface of a quark star is irrelevant from the viewpoint of r-mode instability, although
it can be important for the production of x-ray bursts, see Page & Cumming (2005)). In
conclusion, stars containing strangeness can rotate at submillisecond periods only if they are
to the right of the LTIW.
Let us now discuss the results of our analysis also considering the constraints posed
onto the EOS by the mass shedding limit. The main result of Lavagetto et al. (2006) and of
Bejger et al. (2007) is that soft EOSs, and in particular the ones based on hyperonic matter
or on matter in which kaon condensation takes place, are rather unlikely. In fact, not only
the range of allowed masses is rather small, but moreover the only configurations satisfying
the stability constraint for mass shedding turn out to be supra massive (Bejger et al. 2007),
making it very difficult to use these stellar structures in a mass accretion scenario. This result
confirms previous discussions of other astrophysical objects ruling out soft EOSs (Ozel 2006).
On the other hand, stars containing deconfined quark matter are not excluded if the quark
– 4 –
EOS is stiff enough (Alford et al. 2007). We can therefore conclude that the object under
discussion is either a quark or a hybrid star.
We can now study the temporal evolutions of the angular velocity of the star Ω, of
its temperature T and of the amplitude of the r-modes α which are calculated by solving
the set of differential equations given in Andersson et al. (2002) (Eqs. 15-23-24). The only
technical difference in our calculation is the inclusion of the reheating associated with the
dissipation of r-modes by bulk viscosity, as discussed in various papers (Wagoner 2002;
Reisenegger & Bonacic 2003; Drago et al. 2006). Due to the reheating, the time needed for
the star to cool down along the right border of the LTIW is rather long. In the equation
regulating the thermal evolution (Andersson et al. 2002) we have added a term:
Ėbv = 2E/τbv (1)
where E ∼ 1051α2M1.4R
erg is the energy of the r-modes (the mass, radius and ro-
tation period of the star are here expressed in units of 1.4 M⊙, 10 km and in millisecond,
respectively) and τbv is the bulk viscosity. Due to the large value of the bulk viscosity for
strange matter at temperatures of a few 108 K, the related dissipation of r-modes strongly
reheats the star. The trajectory in the Ω−T plane describing the time evolution of the star
follows essentially the border of the LTIW and the star keeps rotating as a sub-millisecond
pulsar for a very long time, strongly dependent on the value of ms and ranging from ∼ 10
years when ms = 100 MeV to ∼ 10
8 years when ms = 300 MeV, as shown in Fig. 2. In Fig. 3
we show the temporal dependence of the r-mode amplitude. Notice that, in the absence of
reheating the time needed to slow-down to frequencies below 1 kHz is at maximum of the
order of ∼ 104 years, and can be much shorter than a year if a small value of ms is adopted.
In the past this branch of the instability window was only discussed in connection with very
young pulsars because reheating due to bulk viscosity was not taken into account.
Before discussing the possible astrophysical scenarios for a submillisecond pulsar inside
a LMXB, it is important to remark two constraints that a realistic model should fulfil. First,
a LMBX is an old object, with a typical age of 108–109 years which is the time needed by the
companion (having a low mass) to fill its Roche lobe and to start accreting on the neutron
star. Therefore we need to provide a mechanism allowing the central object inside the LMXB
to rotate rapidly while being so old. Second, submillisecond pulsars are certainly rather rare,
and in particular there is no evidence of a uniform distribution of pulsar rotational frequencies
extending from a few hundred Hz up to more than a kHz. Therefore a realistic model should
also indicate why most of the compact stars will not be detected rotating at a submillisecond
period. As we will show, these two constraints can be satisfied in two different astrophysical
scenarios: a first scenario in which a old hybrid or quark star is accelerated up to frequencies
exceeding one kHz by mass accretion and a second scenario in which the quark or hybrid
– 5 –
star is born with a submillisecond period and it is now spinning down by r-mode instability
(see Fig. 4).
The possible realization of these scenarios depends on two main ingredients:
• the value ofms, which, as we have already shown, regulates the magnitude and position
of the LTIW;
• the cooling rate of the central region of the star which determines the inner temperature
T of a star which is accreting material from the companion (Miralda-Escudé et al.
1990).
Concerning the cooling rate, if strange quark matter is present in a compact star, direct
URCA processes are possible and therefore the cooling is (generally) fast. It turns out from
Miralda-Escudé et al. (1990) that in this case the inner temperature is ∼ 5×107 K for a mass
accretion rate of Ṁ = 10−10M⊙/year and that the temperature scales as T ∝ Ṁ
1/6. Another
possibility recently proposed is that, due to the formation of diquark condensates, URCA
processes are strongly suppressed and the cooling turns out to be slow (Blaschke et al. 2000).
At the same time, bulk viscosity can still be large enough to suppress r-mode instability
(Blaschke & Berdermann 2007). In the following we discuss both possibilities, either of a
fast or of a slow cooling.
Let us consider first the case in which ms = 300 MeV.
The first scenario, i.e. the star span up by accretion, can indeed be realized if T ≥ 108 K
(see lower panel of Fig. 1). Such a temperature can be reached via reheating due to mass
accretion. In the case of fast cooling a rather large value of mass accretion rate is needed,
Ṁ ∼ 10−8M⊙/year, and this stringent request can explain why submillisecond pulsars are
rare. A model in which the cooling is slow is instead excluded, because it would be extremely
easy to re-accelerate the star to very large frequencies and therefore submillisecond pulsars
would not be rare.
Also the second scenario in which the star is spinning down due to r-mode instabilities is
possible. As shown in Fig. 2, the time spent by the star above 1122 Hz is of order of 108 years
and it is therefore compatible with the typical age of LMXBs. In this case submillisecond
pulsars are rare because only a (small) fraction of newly born compact stars can rotate
with submillisecond periods (see e.g. the discussions in Andersson & Kokkotas (2001) and
in Perna et al. (2007)). In this scenario the distribution of millisecond and submillisecond
pulsars is determined by the mass of the star at birth: if the mass is large enough, strange
quark matter can form in the core of the star, r-modes are efficiently damped by the large
bulk viscosity and the star can be a submillisecond pulsar at birth; if the mass of the star
is small, strange quark matter can not form, the bulk viscosity of nucleonic matter can not
– 6 –
damp r-modes and therefore in a short period the star loses almost all its angular momentum
(Lindblom et al. 1998).
We consider now the case in which ms = 100 MeV.
The first scenario in which the star is spinning up by accretion can be realized but T must be
at least ∼ 5×108 K for the star to be accelerated up to 1122 (see upper panel of Fig. 4). Such
a temperature can be reached but it needs strong reheating due to a large mass accretion
rate and slow cooling.
The second scenario, i.e. the star is spinning down by r-mode instability, is ruled out since,
as shown in Fig. 2, the time spent by the star above 1122 Hz is of order of 100 years and
this is incompatible with the typical age of a LMXB.
In conclusion, the two astrophysical scenarios proposed to explain a submillisecond
pulsar inside a LMXB can both be realized by a quark or a hybrid star if ms ∼ 300 MeV.
Instead, if the ms is small, only the first scenario is possible and it requires a slow cooling.
Our results are summarized in Table 1.
The main uncertainties in our analysis are due to the possible existence of other damping
mechanisms, taking place on the left side of the LTIW. For instance magnetic fields can be
important to suppress r-mode instabilities (Rezzolla et al. 2000), but their effect is probably
negligible for frequencies exceeding ∼ 0.35 ΩK , if the internal magnetic field is not larger than
∼ 1016 G. Obviously, even larger uncertainties exist concerning quark matter. As discussed
above, the bulk viscosity of quark matter strongly depends on the strange quark mass and
on the possible formation of a diquark condensate (Alford & Schmitt 2007). Clearly, our
analysis can provide much needed constraints on the EOS of quark matter.
Finally, let us stress that the outcome of our analysis is that a compact star rotating at
a submillisecond period inside a LMXB can only be a quark or a hybrid star. Future obser-
vations will be important to clarify if the object at the center of XTE J1739-285 constitutes
indeed an example.
It is a pleasure to thanks Carmine Cuofano for many useful discussions.
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– 9 –
Table 1: Microphysics and astrophysical scenarios
ms [MeV] cooling scenarios (I or II) conclusions
300 fast I: O.K., Ṁ ≥ 10−8M⊙/y O.K. for
II: O.K., needs rapid rotation at birth both scenarios
300 slow I: no, submillisecond pulsars too frequent excluded
II: O.K. due to scenario I
100 fast I: no, T is too low not possible
II: no, slow down too rapid in both scenarios
100 slow I: O.K., needs large Ṁ O.K. for
II: no, slow down too rapid scenario I
Table 2: Connection between microphysics (mass of the strange quark and cooling rate)
and astrophysical scenarios for a submillisecond pulsar inside a LMXB. The first scenario
corresponds to a compact star which is spinning up due to mass accretion; the second scenario
to a compact star born as a submillisecond pulsar and still in the process of slowing down.
– 10 –
7 8 9 10 11
Log10T@KD
f@HzD
ms=300 MeV
7 8 9 10 11
Log10T@KD
f@HzD
7 8 9 10 11
Log10T@KD
f@HzD
ms=100 MeV
7 8 9 10 11
Log10T@KD
f@HzD
Fig. 1.— r-mode instability windows. The star is unstable and loses angular momentum
by emitting gravitational waves in the regions above the instability lines displayed in figure.
Thin solid lines correspond to neutron stars, for two extreme values of masses and radii
allowed by the analysis done by Bejger et al. (2007). The shaded area in between is repre-
sentative of intermediate values of masses and radii. Dashed lines delimit the LTIW and
the HTIW for strange quark stars. Thick solid lines correspond to hybrid stars. In both
cases the mass and radius of the star are M = 2M⊙ and R = 13 km. In the upper panel
ms = 100 MeV and in the lower panel ms = 300 MeV. The bulk and shear viscosities are
in general from Andersson & Kokkotas (2001). Bulk viscosity of strange quark matter at
high temperatures is from Madsen (2000). The arrow indicates the minimal temperature
considered possible in LMXBs. The dot indicates the actual position of the compact star
associated with XTE J1739-285, as it results from our analysis.
– 11 –
4 6 8 10 12 14 16
Log10@t@sDD
f@HzD
100 300
300100
Fig. 2.— Time dependence of the rotational frequency of the compact star. The thin lines
take into account only the reheating due to shear viscosity, the thick lines take into account
also the effect of reheating due to bulk viscosity. We show results for two different values
of the strange quark mass, ms =100 MeV and ms =300 MeV. The dashed horizontal line
corresponds to a frequency of 1122 Hz.
– 12 –
4 6 8 10 12
Log10@t@sDD
4 6 8 10 12
Log10@t@sDD
Fig. 3.— Temporal evolution of the amplitude α of the r-modes. Results
without reheating in panel (a) and taking into account reheating in panel
(b). The amplitude of the r-mode is significantly damped when reheating is
taken into account, the maximum amplitude is reduced by roughly 2 orders of
magnitude. Therefore non linear effects should be suppressed.
– 13 –
7 7.5 8 8.5 9
Log@T@KDD
f@HzD
HbL 108 yr
ms=300 MeV
HaL 10 yr
7.5 8 8.5 9
Log@T@KDD
f@HzD
ms=100 MeV
HdL 108yr
HbL 109yr
Fig. 4.— Upper panel: first scenario (see text). The star is born with a low angular velocity.
After a cooling period at a constant angular velocity (line (a)), the star reaches the instability
window and loses angular momentum during a long period, ∼ 109 years (line (b)). After
this long period the mass accretion process starts and the star is heated (line (c)) and re-
accelerated (line (d)) to a frequency of 1122 Hz (indicated by the thick dot). Lower panel:
second scenario (see text). The star is born with a large angular velocity. After a short
period of cooling at a constant frequency (line (a)) the star enters the instability window
and loses angular momentum via r-mode (line (b)). The frequency of the star can be larger
than 1122 Hz (indicated by the thick dot) for a period of ∼ 108 years. The mass and radius
of the star are M = 1.4M⊙ and R = 10 km, respectively.
