Quark matter and the astrophysics of neutron stars
M Prakash
Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA
E-mail: prakash@harsha.phy.ohiou.edu
Abstract. Some of the means through which the possible presence of nearly
deconfined quarks in neutron stars can be detected by astrophysical observations of
neutron stars from their birth to old age are highlighted.
1. Introduction
Utilizing the asymptotic freedom of QCD, Collins and Perry [1] first noted that the dense
cores of neutron stars may consist of deconfined quarks instead of hadrons. The crucial
question is whether observations of neutron stars from their birth to death through
neutrino, photon and gravity-wave emissions can unequivocably reveal the presence of
nearly deconfined quarks instead of other possibilities such as only nucleons or other
exotica such as strangeness-bearing hyperons or Bose (pion and kaon) condensates.
2. Neutrino signals during the birth of a neutron star
The birth of a neutron star is heralded by the arrival of neutrinos on earth as confirmed
by IMB and Kamiokande neutrino detectors in the case of supernova SN 1987A. Nearly
all of the gravitational binding energy (of order 300 bethes, where 1 bethe ≡ 1051 erg)
released in the progenitor star’s white dwarf-like core is carried off by neutrinos and
antineutrinos of all flavors in roughly equal proportions. The remarkable fact that the
weakly interacting neutrinos are trapped in matter prior to their release as a burst is due
to their short mean free paths in matter, λ ≈ (σn)−1 ≈ 10 cm, (here σ ≈ 10−40 cm2 is the
neutrino-matter cross section and n ≈ 2 to 3 ns, where ns ≃ 0.16 fm−3 is the reference
nuclear equilibrium density), which is much less than the proto-neutron star radius,
which exceeds 20 km. Should a core-collapse supernova occur in their lifetimes, current
neutrino detectors, such as SK, SNO, LVD’s, AMANDA, etc., offer a great opportunity
for understanding a proto-neutron star’s birth and propagation of neutrinos in dense
matter insofar as they can detect tens of thousands of neutrinos in contrast to the tens
of neutrinos detected by IMB and Kamiokande.
The appearence of quarks inside a neutron star leads to a decrease in the maximum
mass that matter can support, implying metastability of the star. This would occur
if the proto-neutron star’s mass, which must be less than the maximum mass of the
http://arxiv.org/abs/0704.0207v1
Quark matter and the astrophysics of neutron stars 2
Figure 1. Evolutions of the central baryon density nB, ν concentration Yν , quark
volume fraction χ and temperature T for different baryon masses MB. Solid lines
show stable stars whereas dashed lines showing stars with larger masses are metastable.
Diamonds indicate when quarks appear at the star’s center, and asterisks denote when
metastable stars become gravitationally unstable. Figure after Ref. [2]
hot, lepton-rich matter is greater than the maximum mass of hot, lepton-poor matter.
For matter with nucleons only, such a metastability is denied (see, e.g., [3]). Figure 1
shows the evolution of some thermodynamic quantities at the center of stars of various
fixed baryonic masses. With the equation of state used (see [2] for details), stars with
MB ∼< 1.1 M⊙ do not contain quarks and those with MB ∼ 1.7 M⊙ are metastable.
The subsequent collapse to a black hole could be observed as a cessation in the neutrino
signals well above the sensitivity limits of the current detectors (Figure 2).
3. Photon signals during the thermal evolution of a neutron star
Multiwavelength photon observations of neutron stars, the bread and butter affair of
astronomy, has yielded estimates of the surface tempeartures and ages of several neutron
stars (Fig. 3). As neutron stars cool principally through neutrino emission from their
cores, the possibility exists that the interior composition can be determined. The
star continuosly emits photons, dominantly in x-rays, with an effective temperature
Teff that tracks the interior temperature but that is smaller by a factor ∼ 100. The
dominant neutrino cooling reactions are of a general type, known as Urca processes
[4], in which thermally excited particles undergo beta and inverse-beta decays. Each
Quark matter and the astrophysics of neutron stars 3
Figure 2. The evolution of the total neutrino luminosity for stars of indicated baryon
masses. Shaded bands illustrate the limiting luninosities corresponding to a count
rate of 0.2 Hz in all detectors assuming 50 kpc for IMB and and Kamioka, 8.5 kpc
for SNO, SuperK, and UNO. Shaded regions represent uncertanities in the average
neutrino energy from the use of a diffusion scheme for neutrino transport in matter.
Figure after Ref. [2].
reaction produces a neutrino or anti-neutrino, and thermal energy is thus continuously
lost. Depending upon the proton-fraction of matter, which in turn depends on the
nature of strong interactions at high density, direct Urca processes involving nucleons,
hyperons or quarks lead to enhanced cooling compared to modified Urca processes in
which an additional particle is required to conserve momentum. However, effects of
superfluidity abates cooling as sufficient thermal energy is required to break paired
fermions. In addition, the poorly known envelope composition also plays a role in the
inferred surface temperature (Fig. 3). The multitude of high density phases, cooling
mechanisms, effects of superfluidity, and unknown envelope composition have thus far
prevented definitive conclusions to be drawn (see, e.g., [5]).
4. Mesured masses and their implications
Several recent observations of neutron stars have direct bearing on the determination of
the maximum mass. The most accurately measured masses are from timing observations
of the radio binary pulsars. As shown in Fig. 4, which is compilation of the measured
neutron star masses as of November 2006, observations include pulsars orbiting another
Quark matter and the astrophysics of neutron stars 4
Figure 3. Observational estimates of neutron star temperatures and ages together
with theoretical cooling simulations forM = 1.4 M⊙. Models and data are described in
[6]. Orange error boxes (see online) indicate sources from which both X-ray and optical
emissions have been observed. Simulations are for models with Fe or H envelopes,
with and without the effects of superfluidity, and allowing or forbidding direct Urca
processes. Models forbidding direct Urca processes are relatively independent of M
and superfluid properties. Trajectories for models with enhanced cooling (direct Urca
processes) and superfluidity lie within the yellow region, the exact location depending
upon M as well as superfluid and Urca properties. Figure adapted from Ref. [7].
neutron star, a white dwarf or a main-sequence star.
One significant development concerns mass determinations in binaries with white
dwarf companions, which show a broader range of neutron star masses than binary
neutron star pulsars. Perhaps a rather narrow set of evolutionary circumstances conspire
to form double neutron star binaries, leading to a restricted range of neutron star masses
[9]. This restriction is likely relaxed for other neutron star binaries. A few of the
white dwarf binaries may contain neutron stars larger than the canonical 1.4 M⊙ value,
including the intriguing case [10] of PSR J0751+1807 in which the estimated mass with
1σ error bars is 2.1 ± 0.2 M⊙. In addition, to 95% confidence, one of the two pulsars
Ter 5 I and J has a reported mass larger than 1.68 M⊙ [11].
Whereas the observed simple mean mass of neutron stars with white dwarf
companions exceeds those with neutron star companions by 0.25 M⊙, the weighted
means of the two groups are virtually the same. The 2.1 M⊙ neutron star, PSR
J0751+1807, is about 4σ from the canonical value of 1.4 M⊙. It is furthermore the
case that the 2σ errors of all but two systems extend into the range below 1.45 M⊙, so
caution should be exercised before concluding that firm evidence of large neutron star
masses exists. Continued observations, which will reduce the observational errors, are
necessary to clarify this situation.
Masses can also be estimated for another handful of binaries which contain an
Quark matter and the astrophysics of neutron stars 5
Figure 4. Measured and estimated masses of neutron stars in binarie pulsars (gold,
silver and blue regions online) and in x-ray accreting binaries (green). For each region,
simple averages are shown as dotted lines; error weighted averages are shown as dashed
lines. For labels and other details, consult Ref. [8].
accreting neutron star emitting x-rays. Some of these systems are characterized by
relatively large masses, but the estimated errors are also large. The system of Vela X-1
is noteworthy because its lower mass limit (1.6 to 1.7M⊙) is at least mildly constrained
by geometry [12].
Raising the limit for the neutron star maximum mass could eliminate entire families
of EOS’s, especially those in which substantial softening begins around 2 to 3ns. This
could be extremely significant, since exotica (hyperons, Bose condensates, or quarks)
generally reduce the maximum mass appreciably.
Ultimate energy density of observable cold baryonic matter
Measurements of neutron star masses can set an upper limit to the maximum possible
energy density in any compact object. It has been found [13] that no causal EOS has
a central density, for a given mass, greater than that for the Tolman VII [14] analytic
solution. This solution corresponds to a quadratic mass-energy density ρ dependence
Quark matter and the astrophysics of neutron stars 6
Figure 5. Model predictions are compared with results from the Tolman IV and
VII analytic solutions of general relativistic stucture equations. NR refers to non-
relativistic potential models, R are field- theoretical models, and Exotica refers to NR
or R models in which strong softening occurs, due to hyperons, a Bose condensate, or
quark matter as well as self-bound strange quark matter. Constraints from a possible
redshift measurement of z = 0.35 is also shown. The dashed lines for 1.44 and 2.2 M⊙
serve to guide the eye. Figure taken from Ref. [13].
on r, ρ = ρc[1− (r/R)2], where the central density is ρc. For this solution,
ρc,T V II = 2.5ρc,Inc ≃ 1.5× 1016
g cm−3 . (1)
A measured mass of 2.2 M⊙ would imply ρmax < 3.1× 1015 g cm−3, or about 8ns.
Figure 5 displays maximum masses and accompanying central densities for a wide
wariety of neutron star EOS’s, including models containing significant softening due
to “exotica”, such as strange quark matter. The upper limit to the density could be
lowered if the causal constraint is not approached in practice. For example, at high
densities in which quark asymptotic freedom is realized, the sound speed is limited to
3. Using this as a strict limit at all densities, the Rhoades & Ruffini [15] mass
limit is reduced by approximately 1/
3 and the compactness limit GM/Rc2 = 1/2.94
is reduced by a factor 3−1/4 to 1/3.8 [16]. In this extreme case, the maximum density
would be reduced by a factor of 3−1/4 from that of Eq. (1). A 2.2 M⊙ measured mass
would imply a maximum density of about 4.2ns.
5. Gravitational wave signals during mergers of binary stars
Mergers of compact objects in binary systems, such as a pair of neutron stars (NS-
NS), a neutron star and a black hole (NS-BH), or two black holes (BH-BH), are
expected to be prominent sources of gravitational radiation [17]. The gravitational-
wave signature of such systems is primarily determined by the chirp mass Mchirp =
Quark matter and the astrophysics of neutron stars 7
Figure 6. Physical and observational variables in mergers between low-mass black
holes and neutron stars or self-bound quark stars. The total system mass is 6 M⊙ and
the initial mass ratio is q = 1/3 in both cases. The initial radii of the neutron star
and quark star were assumed to be equal. The time scales have arbitrary zero points.
Upper panel displays semi-major axis a (thick lines) and component mass MNS,MQS
(thin lines) evolution. Lower panel displays orbital frequency ν (thick lines) and strain
amplitude |h+r| evolution. Solid curves refer to the neutron star simulation and dashed
curves to the quark star simulations. Figure taken from Ref. [8].
(M1M2)
3/5(M1 +M2)
−1/5, where M1 and M2 are the masses of the coalescing objects.
The radiation of gravitational waves removes energy which causes the mutual orbits to
decay. For example, the binary pulsar PSR B1913+16 has a merger timescale of about
250 million years, and the pulsar binary PSR J0737-3039 has a merger timescale of
about 85 million years [18], so there is ample reason to expect that many such decaying
compact binaries exist in the Galaxy. Besides emitting copious amounts of gravitational
radiation, binary mergers have been proposed as a source of the r-process elements [19]
and the origin of the shorter-duration gamma ray bursters [20].
Observations of gravity waves from merger events can simultaneously measure
masses and radii of neutron stars, and could set firm limits on the neutron star maximum
mass [21, 22]. Binary mergers for the two cases of a black hole and a normal neutron
star and a black hole and a self-bound strange quark matter star (Fig. 6) illustrate
the unique opportunity afforded by gravitational wave detectors due to begin operation
over the next decade, including LIGO, VIRGO, GEO600, and TAMA.
A careful analysis of the gravitational waveform during inspiral yields values for
not only the chirp mass Mchirp, but for also the reduced mass MBHMNS/M , so that
both MBH and MNS can be found [23]. The onset of mass transfer can be determined
by the peak in ω, and the value of ω there gives a. A general relativistic analysis of
mass transfer conditions then allows the determination of the star’s radius [22]. Thus
a point on the mass-radius diagram can be estimated [24]. The combination h+ω
depends only on a function of q, so the ratio of that combination and knowledge of qi
Quark matter and the astrophysics of neutron stars 8
should allow determination of qf . From the Roche condition and knowledge of af from
ωf , another mass-radius combination can be found.
The sharp contrast between the evolutions during stable mass transfer of a normal
neutron star and a strange quark star should make these cases distinguishable. For
strange quark matter stars, the differences in the height of the frequency peak and the
plateau in the frequency values at later times are related to the differences in radii of
the stars at these two epochs. It could be an indirect indicator of the maximum mass
of the star: the closer is the star’s mass before mass transfer to the maximum mass, the
greater is the difference between these frequency values, because the radius change will
be larger. Together with radius information, the value of the maximum mass remains the
most important unknown that could reveal the true equation of state at high densities.
Acknowledgments
This work was supported in part by the U.S. Department of Energy under the grant
DOE/DE-FG02-93ER40756.
References
[1] Collins J C and Perry M J 1975 Phys. Rev. Lett. 30, 1353
[2] Pons J A, Steiner A W, Prakash M and Lattimer J A 2001 Phys. Rev. Lett. 86, 5223
[3] Ellis P J, Lattimer J M and Prakash M 1996 Comments in Nuclear and Particle Physics 22, 63
[4] Lattimer J M, Pethick C J, Prakash M and Haensel P, 1991, Phys. Rev. Lett. 66, 2701
[5] Page D, Prakash M, Lattimer J M, and Steiner A W, 2000 Phys. Rev. Lett. 85, 2048
[6] Page D, Lattimer J M, Prakash M and Steiner A W 2004 Astrophys. Jl 155, 623
[7] Lattimer J M and Prakash M 2004 Science 304,536
[8] Lattimer J M and Prakash M 2006, astro-ph/0612440
[9] Bethe H A and Brown G E 1998 Astrophys. Jl 506, 780
[10] Nice D J et al. 2005 Astrophys. Jl. 634, 1242
[11] Ransom S M 2005 Science 307, 892
[12] Quaintrell et al. 2003 Astron. Astrophys. 401, 303
[13] Lattimer J M and Prakash M Phys. Rev. Lett. 94, 111101
[14] Tolman R C 1939 Phys. Rev. 55, 364
[15] Rhoades C E and Ruffini R 1974 Phys. Rev. Lett. 32, 324
[16] Lattimer J M, Prakash M, Masak D, and Yahil A 1990 Astrophys. Jl. 355, 241
[17] Thorne K S, 1973 Three Hundred Years of Gravitation, ed. S. W. Hawking and W. Israel,
Cambridge Univ. Press, Cambridge, Ch. 9
[18] Lyne A. G. et al., 2004 Science 303, 1153
[19] Lattimer J M and Schramm D, 1976 Astrophys. Jl. 210, 549
[20] Eichler D et al., 1989 Science 340, 126
[21] Prakash M and Lattimer J M 2003, J. Phys. G. Nucl. Part. Phys. 30, S451
[22] Ratkovic S, Prakash M and Lattimer J M 2005, astro-ph/0512136
[23] Cutler C and Flanagen E E 1994, Phys. Rev. D49, 2658
[24] Faber J A et al., 2002 Phys. Rev. Lett. 89, 231102
http://arxiv.org/abs/astro-ph/0612440
http://arxiv.org/abs/astro-ph/0512136
	Introduction
	Neutrino signals during the birth of a neutron star
	Photon signals during the thermal evolution of a neutron star
	Mesured masses and their implications
	Gravitational wave signals during mergers of binary stars

  Some of the means through which the possible presence of nearly deconfined
quarks in neutron stars can be detected by astrophysical observations of
neutron stars from their birth to old age are highlighted.

Introduction
Utilizing the asymptotic freedom of QCD, Collins and Perry [1] first noted that the dense
cores of neutron stars may consist of deconfined quarks instead of hadrons. The crucial
question is whether observations of neutron stars from their birth to death through
neutrino, photon and gravity-wave emissions can unequivocably reveal the presence of
nearly deconfined quarks instead of other possibilities such as only nucleons or other
exotica such as strangeness-bearing hyperons or Bose (pion and kaon) condensates.
2. Neutrino signals during the birth of a neutron star
The birth of a neutron star is heralded by the arrival of neutrinos on earth as confirmed
by IMB and Kamiokande neutrino detectors in the case of supernova SN 1987A. Nearly
all of the gravitational binding energy (of order 300 bethes, where 1 bethe ≡ 1051 erg)
released in the progenitor star’s white dwarf-like core is carried off by neutrinos and
antineutrinos of all flavors in roughly equal proportions. The remarkable fact that the
weakly interacting neutrinos are trapped in matter prior to their release as a burst is due
to their short mean free paths in matter, λ ≈ (σn)−1 ≈ 10 cm, (here σ ≈ 10−40 cm2 is the
neutrino-matter cross section and n ≈ 2 to 3 ns, where ns ≃ 0.16 fm−3 is the reference
nuclear equilibrium density), which is much less than the proto-neutron star radius,
which exceeds 20 km. Should a core-collapse supernova occur in their lifetimes, current
neutrino detectors, such as SK, SNO, LVD’s, AMANDA, etc., offer a great opportunity
for understanding a proto-neutron star’s birth and propagation of neutrinos in dense
matter insofar as they can detect tens of thousands of neutrinos in contrast to the tens
of neutrinos detected by IMB and Kamiokande.
The appearence of quarks inside a neutron star leads to a decrease in the maximum
mass that matter can support, implying metastability of the star. This would occur
if the proto-neutron star’s mass, which must be less than the maximum mass of the
http://arxiv.org/abs/0704.0207v1
Quark matter and the astrophysics of neutron stars 2
Figure 1. Evolutions of the central baryon density nB, ν concentration Yν , quark
volume fraction χ and temperature T for different baryon masses MB. Solid lines
show stable stars whereas dashed lines showing stars with larger masses are metastable.
Diamonds indicate when quarks appear at the star’s center, and asterisks denote when
metastable stars become gravitationally unstable. Figure after Ref. [2]
hot, lepton-rich matter is greater than the maximum mass of hot, lepton-poor matter.
For matter with nucleons only, such a metastability is denied (see, e.g., [3]). Figure 1
shows the evolution of some thermodynamic quantities at the center of stars of various
fixed baryonic masses. With the equation of state used (see [2] for details), stars with
MB ∼< 1.1 M⊙ do not contain quarks and those with MB ∼ 1.7 M⊙ are metastable.
The subsequent collapse to a black hole could be observed as a cessation in the neutrino
signals well above the sensitivity limits of the current detectors (Figure 2).
3. Photon signals during the thermal evolution of a neutron star
Multiwavelength photon observations of neutron stars, the bread and butter affair of
astronomy, has yielded estimates of the surface tempeartures and ages of several neutron
stars (Fig. 3). As neutron stars cool principally through neutrino emission from their
cores, the possibility exists that the interior composition can be determined. The
star continuosly emits photons, dominantly in x-rays, with an effective temperature
Teff that tracks the interior temperature but that is smaller by a factor ∼ 100. The
dominant neutrino cooling reactions are of a general type, known as Urca processes
[4], in which thermally excited particles undergo beta and inverse-beta decays. Each
Quark matter and the astrophysics of neutron stars 3
Figure 2. The evolution of the total neutrino luminosity for stars of indicated baryon
masses. Shaded bands illustrate the limiting luninosities corresponding to a count
rate of 0.2 Hz in all detectors assuming 50 kpc for IMB and and Kamioka, 8.5 kpc
for SNO, SuperK, and UNO. Shaded regions represent uncertanities in the average
neutrino energy from the use of a diffusion scheme for neutrino transport in matter.
Figure after Ref. [2].
reaction produces a neutrino or anti-neutrino, and thermal energy is thus continuously
lost. Depending upon the proton-fraction of matter, which in turn depends on the
nature of strong interactions at high density, direct Urca processes involving nucleons,
hyperons or quarks lead to enhanced cooling compared to modified Urca processes in
which an additional particle is required to conserve momentum. However, effects of
superfluidity abates cooling as sufficient thermal energy is required to break paired
fermions. In addition, the poorly known envelope composition also plays a role in the
inferred surface temperature (Fig. 3). The multitude of high density phases, cooling
mechanisms, effects of superfluidity, and unknown envelope composition have thus far
prevented definitive conclusions to be drawn (see, e.g., [5]).
4. Mesured masses and their implications
Several recent observations of neutron stars have direct bearing on the determination of
the maximum mass. The most accurately measured masses are from timing observations
of the radio binary pulsars. As shown in Fig. 4, which is compilation of the measured
neutron star masses as of November 2006, observations include pulsars orbiting another
Quark matter and the astrophysics of neutron stars 4
Figure 3. Observational estimates of neutron star temperatures and ages together
with theoretical cooling simulations forM = 1.4 M⊙. Models and data are described in
[6]. Orange error boxes (see online) indicate sources from which both X-ray and optical
emissions have been observed. Simulations are for models with Fe or H envelopes,
with and without the effects of superfluidity, and allowing or forbidding direct Urca
processes. Models forbidding direct Urca processes are relatively independent of M
and superfluid properties. Trajectories for models with enhanced cooling (direct Urca
processes) and superfluidity lie within the yellow region, the exact location depending
upon M as well as superfluid and Urca properties. Figure adapted from Ref. [7].
neutron star, a white dwarf or a main-sequence star.
One significant development concerns mass determinations in binaries with white
dwarf companions, which show a broader range of neutron star masses than binary
neutron star pulsars. Perhaps a rather narrow set of evolutionary circumstances conspire
to form double neutron star binaries, leading to a restricted range of neutron star masses
[9]. This restriction is likely relaxed for other neutron star binaries. A few of the
white dwarf binaries may contain neutron stars larger than the canonical 1.4 M⊙ value,
including the intriguing case [10] of PSR J0751+1807 in which the estimated mass with
1σ error bars is 2.1 ± 0.2 M⊙. In addition, to 95% confidence, one of the two pulsars
Ter 5 I and J has a reported mass larger than 1.68 M⊙ [11].
Whereas the observed simple mean mass of neutron stars with white dwarf
companions exceeds those with neutron star companions by 0.25 M⊙, the weighted
means of the two groups are virtually the same. The 2.1 M⊙ neutron star, PSR
J0751+1807, is about 4σ from the canonical value of 1.4 M⊙. It is furthermore the
case that the 2σ errors of all but two systems extend into the range below 1.45 M⊙, so
caution should be exercised before concluding that firm evidence of large neutron star
masses exists. Continued observations, which will reduce the observational errors, are
necessary to clarify this situation.
Masses can also be estimated for another handful of binaries which contain an
Quark matter and the astrophysics of neutron stars 5
Figure 4. Measured and estimated masses of neutron stars in binarie pulsars (gold,
silver and blue regions online) and in x-ray accreting binaries (green). For each region,
simple averages are shown as dotted lines; error weighted averages are shown as dashed
lines. For labels and other details, consult Ref. [8].
accreting neutron star emitting x-rays. Some of these systems are characterized by
relatively large masses, but the estimated errors are also large. The system of Vela X-1
is noteworthy because its lower mass limit (1.6 to 1.7M⊙) is at least mildly constrained
by geometry [12].
Raising the limit for the neutron star maximum mass could eliminate entire families
of EOS’s, especially those in which substantial softening begins around 2 to 3ns. This
could be extremely significant, since exotica (hyperons, Bose condensates, or quarks)
generally reduce the maximum mass appreciably.
Ultimate energy density of observable cold baryonic matter
Measurements of neutron star masses can set an upper limit to the maximum possible
energy density in any compact object. It has been found [13] that no causal EOS has
a central density, for a given mass, greater than that for the Tolman VII [14] analytic
solution. This solution corresponds to a quadratic mass-energy density ρ dependence
Quark matter and the astrophysics of neutron stars 6
Figure 5. Model predictions are compared with results from the Tolman IV and
VII analytic solutions of general relativistic stucture equations. NR refers to non-
relativistic potential models, R are field- theoretical models, and Exotica refers to NR
or R models in which strong softening occurs, due to hyperons, a Bose condensate, or
quark matter as well as self-bound strange quark matter. Constraints from a possible
redshift measurement of z = 0.35 is also shown. The dashed lines for 1.44 and 2.2 M⊙
serve to guide the eye. Figure taken from Ref. [13].
on r, ρ = ρc[1− (r/R)2], where the central density is ρc. For this solution,
ρc,T V II = 2.5ρc,Inc ≃ 1.5× 1016
g cm−3 . (1)
A measured mass of 2.2 M⊙ would imply ρmax < 3.1× 1015 g cm−3, or about 8ns.
Figure 5 displays maximum masses and accompanying central densities for a wide
wariety of neutron star EOS’s, including models containing significant softening due
to “exotica”, such as strange quark matter. The upper limit to the density could be
lowered if the causal constraint is not approached in practice. For example, at high
densities in which quark asymptotic freedom is realized, the sound speed is limited to
3. Using this as a strict limit at all densities, the Rhoades & Ruffini [15] mass
limit is reduced by approximately 1/
3 and the compactness limit GM/Rc2 = 1/2.94
is reduced by a factor 3−1/4 to 1/3.8 [16]. In this extreme case, the maximum density
would be reduced by a factor of 3−1/4 from that of Eq. (1). A 2.2 M⊙ measured mass
would imply a maximum density of about 4.2ns.
5. Gravitational wave signals during mergers of binary stars
Mergers of compact objects in binary systems, such as a pair of neutron stars (NS-
NS), a neutron star and a black hole (NS-BH), or two black holes (BH-BH), are
expected to be prominent sources of gravitational radiation [17]. The gravitational-
wave signature of such systems is primarily determined by the chirp mass Mchirp =
Quark matter and the astrophysics of neutron stars 7
Figure 6. Physical and observational variables in mergers between low-mass black
holes and neutron stars or self-bound quark stars. The total system mass is 6 M⊙ and
the initial mass ratio is q = 1/3 in both cases. The initial radii of the neutron star
and quark star were assumed to be equal. The time scales have arbitrary zero points.
Upper panel displays semi-major axis a (thick lines) and component mass MNS,MQS
(thin lines) evolution. Lower panel displays orbital frequency ν (thick lines) and strain
amplitude |h+r| evolution. Solid curves refer to the neutron star simulation and dashed
curves to the quark star simulations. Figure taken from Ref. [8].
(M1M2)
3/5(M1 +M2)
−1/5, where M1 and M2 are the masses of the coalescing objects.
The radiation of gravitational waves removes energy which causes the mutual orbits to
decay. For example, the binary pulsar PSR B1913+16 has a merger timescale of about
250 million years, and the pulsar binary PSR J0737-3039 has a merger timescale of
about 85 million years [18], so there is ample reason to expect that many such decaying
compact binaries exist in the Galaxy. Besides emitting copious amounts of gravitational
radiation, binary mergers have been proposed as a source of the r-process elements [19]
and the origin of the shorter-duration gamma ray bursters [20].
Observations of gravity waves from merger events can simultaneously measure
masses and radii of neutron stars, and could set firm limits on the neutron star maximum
mass [21, 22]. Binary mergers for the two cases of a black hole and a normal neutron
star and a black hole and a self-bound strange quark matter star (Fig. 6) illustrate
the unique opportunity afforded by gravitational wave detectors due to begin operation
over the next decade, including LIGO, VIRGO, GEO600, and TAMA.
A careful analysis of the gravitational waveform during inspiral yields values for
not only the chirp mass Mchirp, but for also the reduced mass MBHMNS/M , so that
both MBH and MNS can be found [23]. The onset of mass transfer can be determined
by the peak in ω, and the value of ω there gives a. A general relativistic analysis of
mass transfer conditions then allows the determination of the star’s radius [22]. Thus
a point on the mass-radius diagram can be estimated [24]. The combination h+ω
depends only on a function of q, so the ratio of that combination and knowledge of qi
Quark matter and the astrophysics of neutron stars 8
should allow determination of qf . From the Roche condition and knowledge of af from
ωf , another mass-radius combination can be found.
The sharp contrast between the evolutions during stable mass transfer of a normal
neutron star and a strange quark star should make these cases distinguishable. For
strange quark matter stars, the differences in the height of the frequency peak and the
plateau in the frequency values at later times are related to the differences in radii of
the stars at these two epochs. It could be an indirect indicator of the maximum mass
of the star: the closer is the star’s mass before mass transfer to the maximum mass, the
greater is the difference between these frequency values, because the radius change will
be larger. Together with radius information, the value of the maximum mass remains the
most important unknown that could reveal the true equation of state at high densities.
Acknowledgments
This work was supported in part by the U.S. Department of Energy under the grant
DOE/DE-FG02-93ER40756.
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http://arxiv.org/abs/astro-ph/0612440
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	Introduction
	Neutrino signals during the birth of a neutron star
	Photon signals during the thermal evolution of a neutron star
	Mesured masses and their implications
	Gravitational wave signals during mergers of binary stars
