Structure of Strange Dwarfs with Color Superconducting Core
Masayuki Matsuzaki∗ and Etsuchika Kobayashi
Department of Physics, Fukuoka University of Education,
Munakata, Fukuoka 811-4192, Japan
Abstract
We study effects of two-flavor color superconductivity on the structure of strange dwarfs, which
are stellar objects with similar masses and radii with ordinary white dwarfs but stabilized by the
strange quark matter core. We find that unpaired quark matter is a good approximation to the
core of strange dwarfs.
PACS numbers: 95.30.-k
∗matsuza@fukuoka-edu.ac.jp
http://arxiv.org/abs/0704.0844v1
mailto:matsuza@fukuoka-edu.ac.jp
Witten made a conjecture that the absolute ground state of quantum chromodynamics
(QCD) is not 56Fe but strange quark matter, which is a plasma composed of almost equal
number of deconfined u, d, and s quarks [1]. Although this conjecture has been neither
confirmed nor rejected, if this is true, since deconfinement is expected in high density cores
of compact stars, there could exist stars that contain strange quark matter converted from
two-flavor quark matter via weak interaction. Strange quark stars whose radii are about 10
km, with or without thin nuclear crust, have long been investigated.
Glendenning et al. proposed a new class of compact stars containing strange quark matter
and thick nuclear crust ranging from a few hundred to ten thousand km [2, 3, 4]. They named
them the strange dwarfs because their radii correspond to those of white dwarfs. Alcock et
al. discussed the mechanism that the strange quark core supports the cruct [5]. Since the
mass of s quark is larger than those of u and d, strange quark matter is positively charged.
In order to electrically neutralize the core, electrons are bound to the surface of the core.
They estimated that the thickness of this electric dipole layer is a few hundred fm. Then
this layer can support a nuclear crust. Although Alcock et al. considered only thin crusts,
Glendenning et al. considered thick crusts up to about ten thousand km. Very recently,
Mathews et al. identified eight candidates of strange dwarfs from observed data [6].
A theoretical facet whose importance in nuclear physics was recognized later is color
superconductivity in quark matter. At asymptotically high density, the color-flavor locking
(CFL) is believed to be the ground state [7]. At realistic densities, however, the two-flavor
color superconductivity (2SC) is thought to be realized even when electric neutrality is
imposed if the coupling constant is strong [8]. Thus, in the present paper, we discuss effects
of the 2SC phase in the strange quark matter core on the structure of strange dwarfs.
In order to determine the structure of compact stars, we solve the general relativistic
Tolman-Oppenheimer-Volkoff (TOV) equation,
dp(r)
Gǫ(r)M(r)
4πr3p(r)
M(r)c2
2GM(r)
, (1)
M(r) = 4π
ǫ(r′)
r′2dr′, (2)
for the pressure p(r), the energy density ǫ(r), and the mass enclosed within the radius r,
M(r). Here G is the gravitational constant and c is the speed of light. The equation is closed
when an equation of state (EOS), a relation between p and ǫ, is specified. In the present
case, strange dwarfs are composed of the strange quark matter core and the nuclear crust.
Accordingly two parameters, the pressure at the center and at the core-crust boundary,
must be specified to integrate the TOV equation. The latter must be equal or less than that
corresponds to the nucleon drip density ǫdrip. Otherwise neutrons drip and gravitate to the
core. In the present calculation we take a pcruct calculated from ǫcrust = ǫdrip.
We assume zero temperature throughout this paper. As for the EOS of the quark core,
we adopt the MIT bag model without any QCD corrections (see Ref. [4], for example). For
unpaired free quark matter,
p = −B +
µfkFf
µ2f −
m4f ln
µf + kFf
, (3)
ǫ = B +
µfkFf
µ2f −
m4f ln
µf + kFf
, (4)
where mf , kFf , and µf =
m2f + k
Ff are the mass, the Fermi momentum, and the chemical
potential of quarks of each flavor, respectively, and f runs u, d, and s. Hereafter we put
c = h̄ = 1. The quantity B is the bag constant. The effect of color superconductivity is
incorporated as a chemical potential dependent effective bag constant. In the 2SC case [9],
Beff = B −
∆2(µ)µ2, (5)
where ∆(µ) is the quark pairing gap as a function of a chemical potential µ, whose relation
to µf is specified later.
The pairing gap is obtained as a function of the Fermi momentum by solving the gap
equation [10]
∆(kF) = −
v̄(kF, k)
E ′(k)
k2dk, (6)
E ′(k) =
(Ek − EkF)
2 + 3∆2(k), (7)
with kF = kFu = kFd, Ek =
k2 +m2q , and mq = mu = md. The one gluon exchange pairing
interaction is given by
v̄(p, k) = −
pkEpEk
2EpEk + 2m
q + p
2 + k2 +m2E
(p+ k)2 +m2E
(p− k)2 +m2E
6EpEk − 6m
q − p
2 − k2
m2E =
2, (8)
where p and k are the magnitudes of 3-momenta. The running coupling constant is given
by [11]
q2max+q
q = p− k,
qmax = max{p, k}. (9)
As for the EOS of the crust, we adopt the tabulated one for β-equilibrium nuclear matter
of Baym, Pethick, and Sutherland [12] (BPS) conforming to Refs. [2, 3, 4].
The positively charged strange quark matter in the core is simply approximated by µ =
µu = µd = µs. Quark masses are given by mu = md = 10 MeV, ms = 150 MeV. The bag
constant is chosen to be B1/4 = 160 MeV. Parameters entering into the pairing interaction
are q2c = 1.5Λ
QCD and ΛQCD = 400 MeV. The nucleon drip density is ǫdrip = 4.3×10
11 g/cm3.
26 28 30 32 34 36 38
log� (J/m
free quark
2SC quark
FIG. 1: Equations of state of free and 2SC quark matter and β-equilibrium nuclear matter. The
latter is tabulated in Refs. [12] and [4].
The adopted EOS is displayed in Fig. 1. The logarithm is to base 10 throughout this
paper. The quark matter EOS describes the core and the BPS EOS describes the crust. At
the boundary, the pressure is common whereas the energy density jumps discontinuously.
In order to obtain the EOS for 2SC matter, the pairing gap must be calculated at each
kF beforehand. This is shown in Fig. 2 left. The effective bag constant determined by the
pairing gap is shown in Fig. 2 right. The resulting 2SC EOS is included in Fig. 1.
Figure 3 presents the mass-radius relation obtained by integrating the TOV equation with
a fixed pcruct, determined from ǫcrust = ǫdrip, and various central pressures. This result can
200 300 400 500 600 700 800
µ (MeV)
200 300 400 500 600 700 800
µ (MeV)
FIG. 2: Left: color superconducting pairing gap and right: effective bag constant, as functions of
the quark chemical potential.
0 1 2 3 4 5
logR (km)
FIG. 3: Mass-radius relation of strange dwarfs and white dwarfs.
be classified into three regions. The first region (larger central pressures), almost vertical
curve at around R ∼ 10 km, describes strange stars with thin crusts. In this region, color
superconductivity makes the maximum mass and radius larger because the pairing gap
reduces the bag constant and consequently the energy density decreases and the pressure
increases. This is consistent with another calculation with the CFL phase [13]. The second
region, horizontal at around M/Msun ∼ 10
−2, and the third region, vertical at around R ∼
104 km up to the maximum mass, correspond to strange dwarfs. In the second region, color
superconducting quark cores support slightly larger masses than unpaired free quark cores.
In the third region, effect of color superconductivity is negligible. In Fig. 3, The mass-radius
relation of ordinary white dwarfs without quark matter cores calculated by adopting the
BPS EOS is also shown although it is known that the BPS EOS is not very suitable for
white dwarfs. As the central pressure decreases, the quark matter core shrinks (Fig. 4 left)
and eventually strange dwarfs reduce to ordinary white dwarfs. When their masses are the
same, the former is more compact than the latter (see also Fig. 5 right) because of the
gravity of the core. Mathews et al. paid attention to this difference in the mass-radius
relation and classified the observed data of dwarfs [6]. According to their work, eight of
them are classified into strange dwarfs.
28 29 30 31 32 33 34 35
logp0 (J/m
28 29 30 31 32 33 34 35
logp0 (J/m
FIG. 4: Left: core radius and right: mass of strange dwarfs, as functions of the central pressure.
14 15 16
log�0 (g/cm
-1 0 1 2 3 4
logr (km)
SD(free)
FIG. 5: Left: mass of strange dwarfs as a function of the central energy density. Right: energy
profile of a strange dwarf with M/Msun = 0.465 and that of a white dwarf with M/Msun = 0.466.
Figure 4 right indicates that strange dwarfs, in particular those of 103 km < R < 104 km,
are realized in a very narrow range of the central pressure. This is reflected in the density of
calculated points. During this rapid structure change from the second to the third region,
the core radius almost does not change, see Fig. 4 left. Figure 5 left also graphs M/Msun as
Fig. 4 right but as a function of the central energy density. The difference between these two
figures at the low pressure/energy density side can be understood from the quark matter
EOS in Fig. 1 such that the pressure decreases steeply at the lowest energy density. Figure 5
left indicates that strange dwarfs have central energy densities just below the lowest stable
compact strange stars and several orders of magnitude larger than those of ordinary white
dwarfs. This is clearly demonstrated in Fig. 5 right.
To summarize, we have solved the Tolman-Oppenheimer-Volkoff equation for strange
dwarfs with ǫcrust = ǫdrip and a wide range of the central pressure. We have examined effects
of the two-flavor color superconductivity in the strange quark matter core in a simplified
manner. The obtained results indicate that, aside from a slight increase of the minimum
mass, effect of color superconductivity is negligible in the mass-radius relation. This is
consistent with the conjecture given in Ref. [6]. As a function of the central energy density,
however, strange dwarfs are realized at slightly lower energy densities than the unpaired
free quark case reflecting the effect on the equation of state. Recently Usov discussed that
electric fields are also generated on the surface of the color-flavor locked matter [14]. This
suggests that strange dwarfs with color-flavor locked cores might also be possible although
this is expected only at relatively high densities. Since the pairing gap enters into the
calculation only through the effective bag constant, aside from a possible slight change in
chemical potentials, it can surely be expected that the effect of color-flavor locking does
not differ much from that of the two-flavor color superconductivity. In conclusion, unpaired
quark matter is a good approximation to the core of strange dwarfs. Another aspect that
might be affected by color superconductivity is the cooling [15]. This is beyond the scope of
the present study.
[1] E. Witten, Phys. Rev. D 30 (1984), 272.
[2] N. K. Glendenning, Ch. Ketter and F. Weber, Phys. Rev. Lett. 74 (1995), 3519.
[3] N. K. Glendenning, Ch. Ketter and F. Weber, Astrophys. J. 450 (1995), 253.
[4] N. K. Glendenning, Compact Stars (Springer, New York, 1996).
[5] C. Alcock, E. Farhi and A. Olinto, Astrophys. J. 310 (1986), 261.
[6] G. J. Mathews, I. -S. Suh, B. O’Gorman, N. Q. Lan, W. Zech, K. Otsuki and F. Weber, J.
Phys. G 32 (2006), 747.
[7] K. Rajagopal and F. Wilczek, Phys. Rev. Lett. 86 (2001), 3492.
[8] H. Abuki and T. Kunihiro, Nucl. Phys. A 768 (2006), 118.
[9] M. Alford and K. Rajagopal, J. High Energy Phys. 06 (2002), 031.
[10] M. Matsuzaki, Phys. Rev. D 62 (2000), 017501.
[11] K. Higashijima, Prog. Theor. Phys. Suppl. 104 (1991), 1.
[12] G. Baym, C. Pethick and P. Sutherland, Astrophys. J. 170 (1971), 299.
[13] G. Lugones and J. E. Horvath, Astron. and Astrophys. 403 (2003), 173.
[14] V. V. Usov, Phys. Rev. D 70 (2004), 067301.
[15] O. G. Benvenuto and L. G. Althaus, Astrophys. J. 462 (1996), 364.
	References

  We study effects of two-flavor color superconductivity on the structure of
strange dwarfs, which are stellar objects with similar masses and radii with
ordinary white dwarfs but stabilized by the strange quark matter core. We find
that unpaired quark matter is a good approximation to the core of strange
dwarfs.

Structure of Strange Dwarfs with Color Superconducting Core
Masayuki Matsuzaki∗ and Etsuchika Kobayashi
Department of Physics, Fukuoka University of Education,
Munakata, Fukuoka 811-4192, Japan
Abstract
We study effects of two-flavor color superconductivity on the structure of strange dwarfs, which
are stellar objects with similar masses and radii with ordinary white dwarfs but stabilized by the
strange quark matter core. We find that unpaired quark matter is a good approximation to the
core of strange dwarfs.
PACS numbers: 95.30.-k
∗matsuza@fukuoka-edu.ac.jp
http://arxiv.org/abs/0704.0844v1
mailto:matsuza@fukuoka-edu.ac.jp
Witten made a conjecture that the absolute ground state of quantum chromodynamics
(QCD) is not 56Fe but strange quark matter, which is a plasma composed of almost equal
number of deconfined u, d, and s quarks [1]. Although this conjecture has been neither
confirmed nor rejected, if this is true, since deconfinement is expected in high density cores
of compact stars, there could exist stars that contain strange quark matter converted from
two-flavor quark matter via weak interaction. Strange quark stars whose radii are about 10
km, with or without thin nuclear crust, have long been investigated.
Glendenning et al. proposed a new class of compact stars containing strange quark matter
and thick nuclear crust ranging from a few hundred to ten thousand km [2, 3, 4]. They named
them the strange dwarfs because their radii correspond to those of white dwarfs. Alcock et
al. discussed the mechanism that the strange quark core supports the cruct [5]. Since the
mass of s quark is larger than those of u and d, strange quark matter is positively charged.
In order to electrically neutralize the core, electrons are bound to the surface of the core.
They estimated that the thickness of this electric dipole layer is a few hundred fm. Then
this layer can support a nuclear crust. Although Alcock et al. considered only thin crusts,
Glendenning et al. considered thick crusts up to about ten thousand km. Very recently,
Mathews et al. identified eight candidates of strange dwarfs from observed data [6].
A theoretical facet whose importance in nuclear physics was recognized later is color
superconductivity in quark matter. At asymptotically high density, the color-flavor locking
(CFL) is believed to be the ground state [7]. At realistic densities, however, the two-flavor
color superconductivity (2SC) is thought to be realized even when electric neutrality is
imposed if the coupling constant is strong [8]. Thus, in the present paper, we discuss effects
of the 2SC phase in the strange quark matter core on the structure of strange dwarfs.
In order to determine the structure of compact stars, we solve the general relativistic
Tolman-Oppenheimer-Volkoff (TOV) equation,
dp(r)
Gǫ(r)M(r)
4πr3p(r)
M(r)c2
2GM(r)
, (1)
M(r) = 4π
ǫ(r′)
r′2dr′, (2)
for the pressure p(r), the energy density ǫ(r), and the mass enclosed within the radius r,
M(r). Here G is the gravitational constant and c is the speed of light. The equation is closed
when an equation of state (EOS), a relation between p and ǫ, is specified. In the present
case, strange dwarfs are composed of the strange quark matter core and the nuclear crust.
Accordingly two parameters, the pressure at the center and at the core-crust boundary,
must be specified to integrate the TOV equation. The latter must be equal or less than that
corresponds to the nucleon drip density ǫdrip. Otherwise neutrons drip and gravitate to the
core. In the present calculation we take a pcruct calculated from ǫcrust = ǫdrip.
We assume zero temperature throughout this paper. As for the EOS of the quark core,
we adopt the MIT bag model without any QCD corrections (see Ref. [4], for example). For
unpaired free quark matter,
p = −B +
µfkFf
µ2f −
m4f ln
µf + kFf
, (3)
ǫ = B +
µfkFf
µ2f −
m4f ln
µf + kFf
, (4)
where mf , kFf , and µf =
m2f + k
Ff are the mass, the Fermi momentum, and the chemical
potential of quarks of each flavor, respectively, and f runs u, d, and s. Hereafter we put
c = h̄ = 1. The quantity B is the bag constant. The effect of color superconductivity is
incorporated as a chemical potential dependent effective bag constant. In the 2SC case [9],
Beff = B −
∆2(µ)µ2, (5)
where ∆(µ) is the quark pairing gap as a function of a chemical potential µ, whose relation
to µf is specified later.
The pairing gap is obtained as a function of the Fermi momentum by solving the gap
equation [10]
∆(kF) = −
v̄(kF, k)
E ′(k)
k2dk, (6)
E ′(k) =
(Ek − EkF)
2 + 3∆2(k), (7)
with kF = kFu = kFd, Ek =
k2 +m2q , and mq = mu = md. The one gluon exchange pairing
interaction is given by
v̄(p, k) = −
pkEpEk
2EpEk + 2m
q + p
2 + k2 +m2E
(p+ k)2 +m2E
(p− k)2 +m2E
6EpEk − 6m
q − p
2 − k2
m2E =
2, (8)
where p and k are the magnitudes of 3-momenta. The running coupling constant is given
by [11]
q2max+q
q = p− k,
qmax = max{p, k}. (9)
As for the EOS of the crust, we adopt the tabulated one for β-equilibrium nuclear matter
of Baym, Pethick, and Sutherland [12] (BPS) conforming to Refs. [2, 3, 4].
The positively charged strange quark matter in the core is simply approximated by µ =
µu = µd = µs. Quark masses are given by mu = md = 10 MeV, ms = 150 MeV. The bag
constant is chosen to be B1/4 = 160 MeV. Parameters entering into the pairing interaction
are q2c = 1.5Λ
QCD and ΛQCD = 400 MeV. The nucleon drip density is ǫdrip = 4.3×10
11 g/cm3.
26 28 30 32 34 36 38
log� (J/m
free quark
2SC quark
FIG. 1: Equations of state of free and 2SC quark matter and β-equilibrium nuclear matter. The
latter is tabulated in Refs. [12] and [4].
The adopted EOS is displayed in Fig. 1. The logarithm is to base 10 throughout this
paper. The quark matter EOS describes the core and the BPS EOS describes the crust. At
the boundary, the pressure is common whereas the energy density jumps discontinuously.
In order to obtain the EOS for 2SC matter, the pairing gap must be calculated at each
kF beforehand. This is shown in Fig. 2 left. The effective bag constant determined by the
pairing gap is shown in Fig. 2 right. The resulting 2SC EOS is included in Fig. 1.
Figure 3 presents the mass-radius relation obtained by integrating the TOV equation with
a fixed pcruct, determined from ǫcrust = ǫdrip, and various central pressures. This result can
200 300 400 500 600 700 800
µ (MeV)
200 300 400 500 600 700 800
µ (MeV)
FIG. 2: Left: color superconducting pairing gap and right: effective bag constant, as functions of
the quark chemical potential.
0 1 2 3 4 5
logR (km)
FIG. 3: Mass-radius relation of strange dwarfs and white dwarfs.
be classified into three regions. The first region (larger central pressures), almost vertical
curve at around R ∼ 10 km, describes strange stars with thin crusts. In this region, color
superconductivity makes the maximum mass and radius larger because the pairing gap
reduces the bag constant and consequently the energy density decreases and the pressure
increases. This is consistent with another calculation with the CFL phase [13]. The second
region, horizontal at around M/Msun ∼ 10
−2, and the third region, vertical at around R ∼
104 km up to the maximum mass, correspond to strange dwarfs. In the second region, color
superconducting quark cores support slightly larger masses than unpaired free quark cores.
In the third region, effect of color superconductivity is negligible. In Fig. 3, The mass-radius
relation of ordinary white dwarfs without quark matter cores calculated by adopting the
BPS EOS is also shown although it is known that the BPS EOS is not very suitable for
white dwarfs. As the central pressure decreases, the quark matter core shrinks (Fig. 4 left)
and eventually strange dwarfs reduce to ordinary white dwarfs. When their masses are the
same, the former is more compact than the latter (see also Fig. 5 right) because of the
gravity of the core. Mathews et al. paid attention to this difference in the mass-radius
relation and classified the observed data of dwarfs [6]. According to their work, eight of
them are classified into strange dwarfs.
28 29 30 31 32 33 34 35
logp0 (J/m
28 29 30 31 32 33 34 35
logp0 (J/m
FIG. 4: Left: core radius and right: mass of strange dwarfs, as functions of the central pressure.
14 15 16
log�0 (g/cm
-1 0 1 2 3 4
logr (km)
SD(free)
FIG. 5: Left: mass of strange dwarfs as a function of the central energy density. Right: energy
profile of a strange dwarf with M/Msun = 0.465 and that of a white dwarf with M/Msun = 0.466.
Figure 4 right indicates that strange dwarfs, in particular those of 103 km < R < 104 km,
are realized in a very narrow range of the central pressure. This is reflected in the density of
calculated points. During this rapid structure change from the second to the third region,
the core radius almost does not change, see Fig. 4 left. Figure 5 left also graphs M/Msun as
Fig. 4 right but as a function of the central energy density. The difference between these two
figures at the low pressure/energy density side can be understood from the quark matter
EOS in Fig. 1 such that the pressure decreases steeply at the lowest energy density. Figure 5
left indicates that strange dwarfs have central energy densities just below the lowest stable
compact strange stars and several orders of magnitude larger than those of ordinary white
dwarfs. This is clearly demonstrated in Fig. 5 right.
To summarize, we have solved the Tolman-Oppenheimer-Volkoff equation for strange
dwarfs with ǫcrust = ǫdrip and a wide range of the central pressure. We have examined effects
of the two-flavor color superconductivity in the strange quark matter core in a simplified
manner. The obtained results indicate that, aside from a slight increase of the minimum
mass, effect of color superconductivity is negligible in the mass-radius relation. This is
consistent with the conjecture given in Ref. [6]. As a function of the central energy density,
however, strange dwarfs are realized at slightly lower energy densities than the unpaired
free quark case reflecting the effect on the equation of state. Recently Usov discussed that
electric fields are also generated on the surface of the color-flavor locked matter [14]. This
suggests that strange dwarfs with color-flavor locked cores might also be possible although
this is expected only at relatively high densities. Since the pairing gap enters into the
calculation only through the effective bag constant, aside from a possible slight change in
chemical potentials, it can surely be expected that the effect of color-flavor locking does
not differ much from that of the two-flavor color superconductivity. In conclusion, unpaired
quark matter is a good approximation to the core of strange dwarfs. Another aspect that
might be affected by color superconductivity is the cooling [15]. This is beyond the scope of
the present study.
[1] E. Witten, Phys. Rev. D 30 (1984), 272.
[2] N. K. Glendenning, Ch. Ketter and F. Weber, Phys. Rev. Lett. 74 (1995), 3519.
[3] N. K. Glendenning, Ch. Ketter and F. Weber, Astrophys. J. 450 (1995), 253.
[4] N. K. Glendenning, Compact Stars (Springer, New York, 1996).
[5] C. Alcock, E. Farhi and A. Olinto, Astrophys. J. 310 (1986), 261.
[6] G. J. Mathews, I. -S. Suh, B. O’Gorman, N. Q. Lan, W. Zech, K. Otsuki and F. Weber, J.
Phys. G 32 (2006), 747.
[7] K. Rajagopal and F. Wilczek, Phys. Rev. Lett. 86 (2001), 3492.
[8] H. Abuki and T. Kunihiro, Nucl. Phys. A 768 (2006), 118.
[9] M. Alford and K. Rajagopal, J. High Energy Phys. 06 (2002), 031.
[10] M. Matsuzaki, Phys. Rev. D 62 (2000), 017501.
[11] K. Higashijima, Prog. Theor. Phys. Suppl. 104 (1991), 1.
[12] G. Baym, C. Pethick and P. Sutherland, Astrophys. J. 170 (1971), 299.
[13] G. Lugones and J. E. Horvath, Astron. and Astrophys. 403 (2003), 173.
[14] V. V. Usov, Phys. Rev. D 70 (2004), 067301.
[15] O. G. Benvenuto and L. G. Althaus, Astrophys. J. 462 (1996), 364.
	References
