Phenomenology of Compact Stars

Jurgen Schaffner-Bielich

Institute for Theoretical Physics and

Heidelberg Graduate School for Fundamental Physics and

ExtreMe Matter Institute EMMI

Blockcourse on Aspects of QCD at Finite Density,

Bielefeld, September 22+23, 2011

– p.1

Outline

Introduction: Observations of neutron stars

Modelling compact stars

Composition of neutron stars and quark starsMass-radius relation of compact starsProperties of quark starsCompact star cooling

Summary

– p.2

Introduction: Observations of neutron stars

– p.3

Supernova Explosions

stars with a mass of more than 8solar masses end in a (corecollapse) supernova (type II)

Supernova of AD 1054 was visiblefor three weeks during daytime(crab nebula)!

supernovae are several thousandtimes brighter than a whole galaxy!

last supernova explosion for the last400 years in our local group:SN1987A

most prominent candidate in the uni-verse for producing the heavy ele-ments (r-process)

Animation of a supernova explosion (Chandra, NASA)

– p.4

Stellar Evolution (Credit: NASA/CXC/M.Weiss)

– p.5

Neutron Stars

Movie (seven still images in 11/2000–04/2001)

produced in core collapsesupernova explosions

compact, massive objects:radius ≈ 10 km, mass1− 2M⊙

extreme densities, severaltimes nuclear density:n ≫ n0 = 2.5 · 1014 g/cm3

in the middle of the crabnebula: a pulsar, a rotatingneutron star!

– p.6

The Sounds of Pulsars

PSR B0329+54: typical pulsar with a period of 0.7145519 s (1.4pulses per second)

PSR B0833-45 (Vela pulsar): in Vela supernova remnant, periodof 89 ms (11 pulses per second)

PSR B0531+21 (crab pulsar): youngest known pulsar, in crabnebula (M1), period: 33 ms (30 pulses per second)

PSR J0437-4715: recently discovered pulsar, period of 5.7 ms(174 pulses per second)

PSR B1937+21: second fastest known pulsar with a period of1.56 ms (642 pulses per second)

(Jodrell Bank Observatory, University of Manchester)– p.7

The Dipole Model for Pulsars

pulsar: uniform rotation with magnetic dipole moment m at angle α

relative to rotation axis

~m =1

2Bpole ·R

3 (cosα, sinα cosΩt, sinα sinΩt)

energy radiation from time-varying dipole moment seen from infinity:

E = −2

3c2|m|2 = −

B2poleR

6Ω4 sin2 α

6c3

emitted energy originates from rotational energy:

E =1

2IΩ2 −→ E = IΩΩ

take typical values for the moment of inertia of a neutron star(usually with M = 1.4M⊙, R = 10 km)

– p.8

The Dipole Model for Pulsars II

set equal and integrate (initially Ω = Ωi):

Ω = Ωi

(

1 +2Ω2

i

Ω20

t

T

)−1/2

with the characteristic age at present time:

T = −

(

Ω

Ω

)

0

=6Ic3

B2poleR

6 sin2 αΩ2o

present age at Ω = Ω0:

t =T

2

(

1−Ω20

Ω2i

)

≈P

2Pfor Ω0 ≪ Ωi

magnetic dipole strength:

Bpole ≈

(

3I

8π2R6PP

)1/2

≈ 3.2 · 1019(

PP)1/2

Gauss

– p.9

The Pulsar Diagram

(ATNF pulsar catalog)

the diagram for pulsars: periodversus period change (P-P)

dipole model for pulsars:characteristic age: τ = P/(2P )

and magnetic fieldB = 2 · 1019(P · P )1/2 Gauss

anomalous x-ray pulsars: AXP,soft-gamma ray repeaters:SGR, young pulsars insupernova remnants: SNR

rapidly rotating pulsars (millisec-ond pulsars): mostly in binarysystems (old recycled pulsars!)

– p.10

How to create binary pulsars (Lorimer 2008)

mildly recycled pulsar

X-rays

runaway star

young pulsar

primary

millisecond pulsar - white dwarf binary

binary disrupts

double neutron star binary

binary disrupts

young pulsar

secondary

binary survives

secondary evolves(Roche Lobe overflow)

binary surviveslow-mass system

Woomph!

Woomph!high-mass system

start with two stars, one

with at least 8M⊙

first supernova creates a

neutron star

neutron star is spun up

by accreting matter from

the companion star

companion star might be-

come a white dwarf or an-

other neutron star

– p.11

The Double Pulsar PSR J0737-3039

sensational discovery of two pulsars orbiting each other (Lyne et al. 2004)

measured five post-Keplerian parameters: Shapiro delay r and s, redshift γ,periastron advance ω, decrease in orbital period Pb (Kramer et al. 2006)

all in agreement with the prediction of GR to within 0.05% !

fundamental tests of General Relativity in STRONG fields

animation (credit: Michael Kramer)– p.12

Masses of Pulsars (Stairs 2006)

more than 1800 pulsars known with140 binary pulsars

best determined mass:M = (1.4414± 0.0002)M⊙

Hulse-Taylor pulsar(Weisberg and Taylor, 2004)

mass of PSR J0751+1807 corr.from M = (2.1± 0.2)M⊙ toM = (1.14− 1.40)M⊙

(Nice et al. 2008)

extremely rapid rotations: up to 716Hz (1.397 ms)(PSR J1748-2446ad)

– p.13

Mass measurement of pulsar PSR J1903+0327 (Freire 2009)

ω.

.

s

r r

s

ω

measure post-Keplerian parameters from pulsar timing

Shapiro delay parameters r and s alone constrain M = (1.67± 0.11)M⊙

combined with periastron advance ω: M = (1.67± 0.01)M⊙

rotation of the companion star could influence ω

(follow-up observation with Hubble planned)– p.14

Mass measurement of pulsar PSR J1614-2230 (Demorest et al. 2010)

-40

-30

-20

-10

0

10

20

30

-40

-30

-20

-10

0

10

20

30

Tim

ing

resi

dual

(µs

)

-40

-30

-20

-10

0

10

20

30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Orbital Phase (turns)

89.1

89.12

89.14

89.16

89.18

89.2

89.22

89.24

0.48 0.49 0.5 0.51 0.52

Incl

inat

ion

Ang

le (

deg)

Companion Mass (solar)

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15

Pro

babi

lity

Den

sity

Pulsar Mass (solar)

extremely strong signal forShapiro delay

Shapiro delay parameters r

and s alone giveM = (1.97± 0.04)M⊙ - newrecord!

by far the highest preciselymeasured pulsar mass!

considerable constraints onneutron star matter properties!

– p.15

Constraints on the Mass–Radius Relation (Lattimer and Prakash 2004)

spin rate from PSR B1937+21 of 641 Hz: R < 15.5 km for M = 1.4M⊙

Schwarzschild limit (GR): R > 2GM = Rs

causality limit for EoS: R > 3GM– p.16

Constraints on the Mass–Radius Relation (Lattimer and Prakash 2004)

7 8 9 10 11 12 13 14 15Radius (km)

0.0

0.5

1.0

1.5

2.0

2.5

Mass

(so

lar)

AP4

MS0

MS2

MS1

MPA1

ENG

AP3

GM3PAL6

GS1

PAL1

SQM1

SQM3

FSU

GR

P <

causality

rotation

J1614-2230

J1903+0327

J1909-3744

Double NS Systems

Nucleons Nucleons+ExoticStrange Quark Matter

spin rate from PSR B1937+21 of 641 Hz: R < 15.5 km for M = 1.4M⊙

Schwarzschild limit (GR): R > 2GM = Rs

causality limit for EoS: R > 3GM

mass limit from PSR J1614-2230 (red band): M = (1.97± 0.04)M⊙

– p.16

How To Measure Masses AND Radii of Compact Stars

mass from binary systems (pulsar with a companion star)

radius and mass from thermal emission, for a blackbody:

F∞ =L∞

4πd2= σSBT

4eff,∞

(

R∞

d

)2

with Teff,∞ = Teff/(1 + z) and R∞ = R/(1 + z)

redshift:

1 + z =

(

1−2GM

R

)−1/2

need to know distance and effective temperature to get R∞

radius measured depends on true mass and radius of the star

additional constraint from redshift measurement from e.g.redshifted spectral lines fixes mass and radius uniquely

– p.17

Isolated Neutron Star RX J1856 (Drake et al. (2002))

closest known neutron star

perfect black–body spectrum, no spectral lines!

for black-body emission: T = 60 eV and R∞ = 4− 8 km!

– p.18

A Quark Star? (NASA press release 2002)

NASA news release 02-082:“Cosmic X-rays reveal evidence for new form of matter”

— a quark star?

– p.19

A Quark Star? (NASA press release 2002)

neutron star: built out of nucleons (neutrons which consists of quarks)quark star: built out of quarks (neutrons are dissolved!)

– p.20

RXJ 1856: Neutron Star or Quark Star? (Trümper et al. (2003), Ho et al. (2007))

101

102

103

104

10−20

10−19

10−18

10−17

10−16

10−15

10−14

10−13

10−12

Wavelength (A)

Fλ (

ergs

s−

1 cm

−2 A

−1 )

XM

M/R

GS

CX

O/L

ET

G

EU

VE

HS

T/S

TIS

VLT

/FO

RS

1

VLT/FORS1HST/STISCXO/LETGXMM/RGSEUVEVLT/FORS2HST/WFPC2

two-component blackbody: small soft temperature, so as not to spoil the x-ray

this implies a rather LARGE radius so that the optical flux is right!

lower limit for radiation radius: R∞ = R/√

1− 2GM/R = 17 km (d/140 pc)

redshift zg ≈ 0.22: R ≈ 14 km and M ≈ 1.55M⊙

largest uncertainty in distance d– p.21

X-Ray burster

binary systems of a neutron star with an ordinary star

accreting material on the neutron star ignites nuclear burning

explosion on the surface of the neutron star: x-ray burst

red shifted spectral lines measured! (z = 0.35 → M/M⊙ = 1.5 (R/10 km))(Cottam, Paerels, Mendez (2002))

– p.22

Supernova remnant 3C58 from 1181 AD (Slane et al. 2004)

CHANDRA press release 04-13:“Going to Extremes: Pulsar Gives Insight on Ultra Dense Matter and Magnetic

Fields” — rapid cooling due to unexpected conditions in the neutron star!– p.23

Cooling of Supernova Remnants (Kaplan et al. (2004))

102

103

104

105

106

1029

1030

1031

1032

1033

1034

1035

1036

B1757−24

J1811−1925

B1853+01

SS 433

B1951+32

Cas A

B2334+61

CTA 1

3C 58

B0538+2817

Crab

IC 443 B0656+14

Puppis A

Vela

1207.4−5209

RCW 103 B1706−44 B1727−33

G093.3+6.9G315.4−2.3

G084.2−0.8G127.1+0.5

Age (yr)

L X,0

.5−

2 ke

V (

ergs

s−

1 )

1.35 Msun

1.395 Msun

1.5 Msun

normal

5.7

5.9

6.1

6.3

6.5

6.7ThermalNon−thermalLimitX−ray PWN

newest data from four neutron stars suggest fast cooling (direct URCA)

standard cooling curves are too high!

large nuclear asymmetry energy generates fast cooling!

strange particles (exotic matter) generate fast cooling!– p.24

Cooling processes with neutrinos

modified URCA process (slow):

N + p+ e− → N + n+ νe N + n → N + p+ e− + νe

direct URCA process (fast):

p+ e− → n+ νe n → p+ e− + νe

can only proceed for ppF + peF ≥ pnF ! Charge neutrality implies:

np = ne → ppF = peF → 2ppF = pnF → np/n ≥ 1/9

nucleon URCA only for large proton fractions, but hyperon URCA process:

Λ → p+ e− + νe , Σ− → n+ e− + νe , . . .

happens immediately when hyperons are present!

only suppressed by hyperon pairing gaps!

– p.25

Basic cooling of neutron stars (Page and Reddy (2006))

slow standard cooling via themodified URCA processversus fast neutrino cooling(emissivities ofǫν = 10n × T 6

9 erg cm−3 s−1)

normal neutron matter: N,superfluid neutron matter: SF

fast cooling due to ’exotic’processes as nucleon directURCA or kaon condensation

– p.26

Cooling evolution for neutron stars (Lattimer and Prakash 2001)

– p.27

Future Telescopes and Detectors in Space

Constellation-X (Photo: NASA) XEUS (Photo: ESA)

James Webb Space Telescope (Photo: ESA) LISA (Photo: NASA)

– p.28

Future Probes Using Gravitational Waves

(Thorne (1997))

sources of gravitational waves: nonspherical rotating neutron stars, collidingneutron stars and black-holes

gravitational wave detectors are running now (LIGO,GEO600,VIRGO,TAMA)

future: LISA, satellite detector! – p.29

Future: Square Kilometer Array (SKA)

receiving surface of 1 million square kilometers

1 billion dollar international project

potential to discover:

10,000 to 20,000 new pulsars

more than 1,000 millisecond pulsars

at least 100 compact relativistic binaries!

probing the equation of state at extreme limits!

cosmic gravitational wave detector by usingpulsars as clocks!

design and location not fixed yet (maybe in SouthAfrica!)

movie– p.30

Modelling Compact Stars

– p.31

Structure of Neutron Stars — the Crust (Dany Page)

n ≤ 104 g/cm3:

atmosphere

(atoms)

n = 104 − 4 · 1011 g/cm3:

outer crust or envelope

(free e−, lattice of nuclei)

n = 4 · 1011 − 1014 g/cm3:

Inner crust

(lattice of nuclei with free

neutrons and e−)

– p.32

Composition of the crust of a neutron star

lattice of nuclei surrounded by free electrons

Wigner–Seitz–cell, lattice structure is bcc

minimize E = Enuclei +Elattice + Eelectrons

loop over all particle stable nuclei (up to 14.000)

use atomic mass evaluation of 2003 by Audi, Wapstra, and Thibault

extrapolate to the drip–line with various models

=⇒ sequence of nuclei AZ as a function of density

– p.33

Sequence to the Dripline (Hempel, Rüster, JSB 2005)

104 105 106 107 108 109 1010 1011 1012

in g/cm3

1019

1020

1021

1022

1023

1024

1025

1026

1027

1028

1029

1030

Pin

dyne

/cm

2

TMANL3defSLy4BSk8PCF1npNL3pBPS

56Fe

62Ni

64Ni

66Ni

86Kr

84Se

outer crust starts with iron (56Fe) up to ρ ≈ 107 g/cm3

continues along nickel isotopes (Z = 28), then Kr, Se (N = 50)

initial sequence at low densities independent of parameter set (data)!

equation of state (nearly) independent of parameter set!

– p.34

Sequence to the Dripline II (Hempel, Rüster, JSB 2005)

10102 5 1011

2 5 1012

in g/cm3

20

24

28

32

36

40

44

48

Z

FRDMTMANL3defBSk8SLy4

84Se82Ge

80Zn

selection of state-of-the-art mass tables (deformed calculations)

initial sequence of nuclei: Se, Ge, Zn (data)

overall narrow range in Z

neutron drip around 5 · 1011 g/cm3

– p.35

Nuclei in the crust (Hempel, Rüster, JSB 2005)

sequence of nuclei: along N = 50 then along N = 82 with Z = 46− 34

common endpoint around N = 82 and Z = 36 (!)

common location of the dripline at N= 82 (!)

updates classic work of Baym, Pethick, Sutherland from 1971!

– p.36

Structure of a Neutron Star — the Core (Fridolin Weber)

absolutely stable strange quark

matter

µm

s

M ~ 1.4 M

R ~ 10 km

quarksu,d,s

n,p,e, µ

neutron star withpion condensate

quark−hybridstar

hyperon star

g/cm3

1011

g/cm3

106

g/cm3

1014

Fe

n,p,e, µs ue r c n d c t

gp

oni

u

p

r ot o

ns−π

K−

u d s

crust

N+e

H

traditional neutron star

strange star

nucleon star

N+e+n

Σ,Λ,

Ξ,∆

n superfluid

– p.37

Neutron Star Matter for a Free Gas

(Ambartsumyan and Saakyan, 1960)

Hadron p,n Σ− Λ othersappears at: ≪ n0 4n0 8n0 > 20n0

but the corresponding equation of state results in amaximum mass of only

Mmax ≈ 0.7M⊙ < 1.44M⊙

(Oppenheimer and Volkoff, 1939)

=⇒ effects from strong interactions are essential todescribe neutron stars!

– p.38

Equation of State of Degenerate Gases (Non-Relativistic)

degeneracy pressure in the non-relativistic limit (kF ≪ me) for acomposite Fermi gas (electrons and nucleons):

P ∝

∫

d3k (k · v/c) , v/c = k/E ≈ k/me

integrate:

P ∝

∫

dk k2(k · k/me) =

∫

dk k4/me ∝ k5F/me

the number density is given by n ∝∫

dk3 ∝ k3F =⇒ kF ∝ n1/3.

P ∝k5F

me

∝1

me

· n5/3

the energy density is ǫ = ρ = mN · n, so that: P ∝ ρ5/3 = ǫ5/3

– p.39

Equation of State of Degenerate Gases: Relativistic Limit

degeneracy pressure in the relativistic limit (kF ≫ me):

P ∝

∫

d3k (k · v/c) , v/c = k/E ≈ 1

integrate:

P ∝

∫

dk k2k =

∫

dk k3 ∝ k4F ∝ n4/3

Assuming that the heavy fermion (nucleon) is still non-relativisticǫ = ρ = mN · n (kF ≪ mN ), one finds

P ∝ ρ4/3 = ǫ4/3

which is applicable for white dwarfs but not for neutron stars!

– p.40

Equation of State: Ultra-Relativistic Limit

For a gas of a single uncharged fermion (pure neutron gas), thedegeneracy pressure in the non-relativistic limit (kF ≪ mn) is

P ∝k5F

mn

∝1

mn

· n5/3 ∝ ǫ5/3

as ǫ = mN · n. Now, in the ultra-relativistic limit (kF ≫ mn)

P ∝ k4F ∝ n4/3

as n ∝ k3F . However, the energy density changes to

ǫ ∝

∫

d3k E(k) ∝

∫

dk k2 · k ∝ k4F ∝ n4/3

so that the equation of state approaches now P = ǫ/3.

– p.41

Summary for the Equations of State

General parameterization of the equations of state in the form

P ∝ nΓ = ρΓ or P ∝ ǫΓ

is called a polytropic equation of state or simply a polytrope.Non-relativistic limit:

P ∝ ρ5/3 =⇒ Γ = 5/3

Relativistic limit (for composite fermion gases):

P ∝ ρ4/3 =⇒ Γ = 4/3

Ultra-relativistic limit (for a single fermion gas):

P = ǫ/3 =⇒ Γ = 1

Note: ρ stands for the mass density of the non-relativistic species, ǫ for the more general energy density.– p.42

Spheres in Hydrostatic Equilibrium

Mass within a radius r (mass conservation):

Mr(r) = 4πr2∫ r

0

dr′ ρ′(r)

Pressure balances gravity in a shell of thickness dr:

4πr2dP (r) = −GMr(r) · dρ(r)

r2, dρ(r) = 4πr2ρ(r)dr

so that the equation of hydrostatic equilibrium reads

dP (r)

dr= −G

Mr(r) · ρ(r)

r2

At the center P (0) = Pc and M(0) = 0, while at the surface P (R) = 0

and Mr(R) = M .

– p.43

Relativistic Hydrostatic Equilibrium

General Relativity: add three relativistic correction factors

dP

dr= −G

Mrǫ

r2

(

1 +P

ǫ

)(

1 +4πr3P

Mr

)(

1−2GMr

r

)−1

with the mass conservation equation

dM

dr= 4πr2ǫ

these are called the Tolman–Oppenheimer–Volkoff equations(Tolman (1934), Oppenheimer and Volkoff (1939)).

The Schwarzschild radius is defined as Rs = 2GM :for the sun Rs = 3 km and for earth Rs = 9 mm.For a neutron star: R ≈ 10 km, M = 1–2M⊙, close to Rs = 3–6 km!

Note: the mass density ρ is replaced by the energy density ǫ !– p.44

Mass–radius relation for Polytropes

Approximate hydrostatic equilibrium (Pc: central pressure):

dP

dr≈

Pc

R∝ G

Mρ

R2

with the average mass density ρ ∝ M/R3 gives

Pc ∝ GM

R·M

R3= G

M 2

R4

Use the equation of state for polytropes P ∝ ρΓ:

(

M

R3

)Γ

∝ GM 2

R4

to arrive at the mass–radius relation for polytropes:

M 2−Γ ·R3Γ−4 = const.

– p.45

Mass–radius relation for Polytropes II

Full treatment: the Lane–Emden equation

1

ξ2d

dξ

(

ξ2dθ

dξ

)

= −θn

with dimensionless radius ξ: r = r0 · ξ, dimensionless density θ:ρ(r) = ρ0θ

n(r), and polytropic index n: Γ = (n+ 1)/n.Mass-radius relation:

M 2−Γ ·R3Γ−4 = const.

Γ = 4/3: M = Mch = const. (Chandrasekhar mass limit!)

Γ = 5/3: M · R3 = const. (M–R relation for compact stars)

Γ = 2: R = const. (interaction dominated EoS)

Γ = 2 + δ: M · R−(2/δ+3) = const. (slope of M-R curve has opposite sign)

Γ → ∞: M · R−3 = const. (constant density, selfbound stars)– p.46

Mass-radius relation for free fermions

(Narain et al., 2006)

dimensionless mass and radius in terms of the Landau mass and radius

M ·R3 = const. for large radii

maximum mass for neutrons: Mmax = 0.71M⊙ and R = 9.1 km– p.47

Landau’s Argument for a Maximum Mass

A relativistic particle at the radius R of the compact star has theenergy

E(R) = Eg + Ekin = −GMm

R+ kF

with kF ∝ n1/3 ∝ N 1/3/R and M ≈ N ·m, N being the number offermions:

E(R) = −GNm2

R+

(

9π

4

)1/3N 1/3

R

Stability analysis leads to

For E < 0: collapse to R → 0 for small R!

For E > 0: R → ∞ until kF < m, then Ekin ∝ k2F/mf ∝ R−2 and a

stable minimum exists

For E = 0: marginally stable

– p.48

Landau’s Argument for a Maximum Mass II

For the stability limit E = 0 one gets

Nmax =

(

9π

4

)1/2 M 3p

m3

with G = 1/M 2p (Planck mass Mp). Hence, the maximum mass is

Mmax ≈ M 3p /m

2 ≈ Mch

The corresponding radius for kF = m is

Rcrit ≈ N 1/3max/m ≈ Mp/m

2

For a neutron star M 3p/m

2 = 1.8M⊙ and Mp/m2 = 2.7 km.

Note: for white dwarfs, the momentum is provided by the electrons sothat Rwd ≈ Mp/(mn ·me) and Rwd/Rns ∼ mn/me ∼ 2000, whileMwd ≈ Mns as the mass is provided in both cases by the nucleons!

– p.49

Interacting Fermions

Add interaction to the EoS of the form ρint = c · nΓ, Γ > 1, so that

Pint = −∂E

∂V

∣

∣

∣

∣

N,T=0

= n2 (∂ρint/n)

∂n= (Γ− 1)c · n2 = (Γ− 1)ǫ

As c2s = P/ǫ, the EoS will be acausal at high-densities for Γ > 2

(standard interactions: Γ = 2). Non-relativistic low density limit:

ǫ = m · n+ c · n2 ≈ m · n p = c′ · n5/3 + c · n2 ≈ c′ · n5/3 ∝ ǫ5/3

At some intermediate density and strong interactions:

ǫ ≈ m · n p ≈ c · n2 ∝ ǫ2

a polytrope of Γ = 2 =⇒ constant radii!Note: the limit of n → ∞ is p = ǫ = c′ · n2!

– p.50

Mass-radius relation for interacting fermions

(Narain et al., 2006)

dimensionless interaction strength y = mf/mI , for strong interactionsy = mN/fπ ≈ 10

for small interaction strengths (y < 1): M ·R3 = const., Mmax = const.

for large interactions strengths (y > 1): R ≈ const. and Mmax ∝ y!– p.51

Impact of hyperons on the maximum mass of neutron stars

(Glendenning and Moszkowski 1991)

neutron star with nucleons

and leptons only:

M ≈ 2.3M⊙

substantial decrease of the

maximum mass due to

hyperons!

maximum mass for “giant

hypernuclei”: M ≈ 1.7M⊙

noninteracting hyperons

result in a too low mass:

M < 1.4M⊙ !

– p.52

Quarks in Neutron Stars!

– p.53

Phase Transitions in Quantum Chromodynamics QCD

Hadrons

µN

neutron starsnuclei

T

Tc

cm / 3

RHIC, LHC

FAIR

Plasma

Quark−Gluon

early universe

SuperconductivityColor

µEarly universe at zero density and high temperature

neutron star matter at small temperature and high density

first order phase transition at high density (not deconfinement)!

probed by heavy-ion collisions at GSI, Darmstadt (FAIR!)– p.54

Nuclear Equation of State as Input in Astrophysics

supernovae simulations: T = 1–50 MeV, n = 10−10–2n0

proto-neutron star: T = 1–50 MeV, n = 10−3–10n0

global properties of neutron stars: T = 0, n = 10−3–10n0

neutron star mergers: T = 0–100 MeV, n = 10−10–10n0

– p.55

Quark matter in the NJL model

p =1

2π2

18∑

i=1

∫ Λ

0

dk k2|ǫi|+ 4Kσuσdσs −1

4GD

3∑

c=1

|∆c|2

−2GS

3∑

α=1

σ2α +

1

4GV

ω20 + pe

use Nambu–Jona-Lasinio model for describing quark matter

describes both dynamical quark masses (quark condensates σ)and the color-superconducting gaps ∆ (Rüster et al. (2005))

parameters: cutoff, scalar and vector coupling constants GS, GV ,diquark coupling GD, ’t Hooft term coupling K

fixed to hadron masses, pion decay constant, free: GD and GV

– p.56

Phases in Quark Matter (Rüster et al. (2005))

0

10

20

30

40

50

60

320 340 360 380 400 420 440 460 480 500

T [M

eV]

µ [MeV]

NQ

χSB

NQ

g2SC

2SC

CFL

←gCFL

uSC

guSC→

first order phase transition based on symmetry arguments!

phases of color superconducting quark matter in β equilibrium:

normal (unpaired) quark matter (NQ)

two-flavor color superconducting phase (2SC), gapless 2SC phase

color-flavor locked phase (CFL), gapless CFL phase, metallic CFL phase

(Alford, Rajagopal, Wilczek, Reddy, Buballa, Blaschke, Shovkovy, Drago, Rüster, Rischke,

Aguilera, Banik, Bandyopadhyay, Pagliara, . . . ) – p.57

Mass-radius and maximum density of pure quark stars

0 5 10 15Radius (km)

0

0.5

1

1.5

2

M/M

sun

B1/4

=200 MeV

B1/4

=145 MeV

Λ=2µ

Λ=3µ

green curves: MIT bag model

blue curves: perturbative QCDcalculations(Fraga, JSB, Pisarski 2001)

case Λ = 2µ: Mmax = 1.05M⊙, Rmax = 5.8 km, nmax = 15n0

case Λ = 3µ: Mmax = 2.14M⊙, Rmax = 12 km, nmax = 5.1n0

other nonperturbative approaches: Schwinger–Dyson model (Blaschke et al.),massive quasiparticles (Peshier, Kämpfer, Soff), NJL model (Schertler,Steiner, Hanauske, . . . ), HDL (Andersen and Strickland), . . .

– p.58

A Simple Model of Dense QCD

star made of a gas of u, d and s quarks

interaction taken into account perturbatively up toα2s ; αs = g2/4π

αs runs according to the renormalization groupequation (u = ln(Λ2/Λ2

MS)):

αs(Λ) =4π

β0u

[

1−2β1

β20

ln(u)

u+

4β21

β40u

2

(

(

ln(u)−1

2

)2

+β2β0

8β21

−5

4

)]

Particle Data Group: αs(2 GeV) = 0.3089 −→ΛMS = 365 MeV for Nf = 3

No bag constant is introduced!

– p.59

A Simple Model of Dense QCD

star temperature ≪ typical chemical potentials−→ zero temperature

ms = 100 MeV ≪ µmin = mN/3−→ three flavor massless quarks

charge neutrality and β equilibrium

µs = µd = µu ≡ µ

as u+ d → u+ s and u+ e− → d+ νe(no electrons for three-flavor massless quarks)

– p.60

Equation of State in pQCD

The thermodynamic potential (Freedman and McLerran (1978))

Ω(µ) = −Nfµ

4

4π2

1− 2(αs

π

)

−

[

G+Nf lnαs

π+

(

11−2

3Nf

)

lnΛ

µ

]

(αs

π

)2

G is scheme dependent and in MS scheme:G = G0 − 0.536Nf +Nf lnNf , (G0 = 10.7)

From Ω(µ) we have

pressure: p(µ) = −Ω(µ)

number density: n(µ) = (∂p/∂µ)

energy density: ǫ = −p+ µn

– p.61

Equation of State in pQCD II

0 1 2 3Energy density (GeV/fm

3)

0.0

0.5

1.0

Pre

ssur

e (G

eV/fm

3 )

free gas

Λ=3µ

Λ=2µ

Nearly linear behaviour of the pressure with the energy density⇒ approximation with an effective nonideal bag model:

Ω(µ) = −Nf

4π2aeff µ4 +Beff

case 2: B1/4eff = 199 MeV, aeff = 0.628 (≤ 4% )

case 3: B1/4eff = 140 MeV, aeff = 0.626 (≤ 2%)

– p.62

Transition from nuclear matter

Matching the low-density equation of stateLarge N arguments à la Rob:

Quark loops suppressed in usual large Nc

Generalized large Nc limit: Nf , Nc → ∞ with Nf/Nc fixed

Baryon mass: mB = Ncmq, chemical potential: µB = Ncµ

Fermi momenta for free particles: kF,B =√

µ2B −m2

B = Nc · kF,q

pB = dB ·k5F,B

mB

= dB ·N 5

c k5F,q

Ncmq

=

(

dBNf

·N 3c

)

pq

⇒ baryonic models have to break down for small µ !

– p.63

A Model For Cold And Dense QCD

masslessquarks

hadrons /

massive quarks

µmin µ χ µ

Two possibilities for a first-order chiral phase transition:

A weakly first-order chiral transition (or no true phasetransition),=⇒ one type of compact star (neutron star)

A strongly first-order chiral transition=⇒ two types of compact stars:a new stable solution with smaller masses and radii

– p.64

Matching the two phases: two possible scenarios

µmin µχstrong µχ

weak0

0.2

0.4

0.6

Pre

ssur

e p/

p free

masslessquarks

massivebaryons/quarks

Weak: phase transition is weakly first order or a crossover → pressure inmassive phase rises strongly

Strong: transition is strongly first order → pressure rises slowly with µ

asymmetric matter up to ∼ 2n0: suggest a slow rise with density!(Akmal,Pandharipande,Ravenhall (1998))

– p.65

Gibbs Phase Construction (Glendenning (1992))

10001200

1400

0

50100

150200

0

100

200

300

0

100

200

300

p[MeV/fm3 ]

pHP (n; e)

pQP (n; e)

QP

MPHP

n [MeV]

e [MeV]

1

22 2(Schertler et al. (2000))

two conserved charges in β-equilibrium: baryon number and charge!

Gibbs criterium for phase equilibrium: equal pressure for equal chemicalpotentials PI(µB, µe) = PII(µB, µe)

globally charge neutral matter: mixed phase with charged bubbles forms!– p.66

Quark Matter in Cold Neutron Stars

2 4 6 8 10165

170

175

180

185g=2g=1g=0

=0

B1=4 [MeV]TM2

MP QPHP

central densityM = 1:3M(Schertler, C. Greiner, JSB, Thoma (2000))

phase transition to quark matter in the MIT bag model

onset of mixed phase appears between (1− 2)n0 even for large values of thebag constant

sufficiently high densities reached in the core for a 1.3M⊙ neutron star to havequark matter

– p.67

Hybrid Stars (Schertler et al. (2000))

0 2.5 5 7.5 10 12.5 15165

170

175

180

185

190g=0.

0 2.5 5 7.5 10 12.5 15165

170

175

180

185

190g=2.

R [km] R [km]

B1=4 [MeV]TM2

MPQP

HPMPQP HP

M=1:3M M=1:3M

hybrid star: consists of hadronic and quark matter

three phases possible: hadronic, mixed phase and pure quark phase

composition depends crucially on the parameters as the bag constant B(and on the mass!)

– p.68

Quark star twins? (Fraga, JSB, Pisarski (2001))

0 5 10 15Radius (km)

0

0.5

1

1.5

2

M/M

sun

weak transition

strong transition

Λ=2µ

Λ=3µ

nc=3n0

2.5n0

hadronic EoS

2n0

stablebranch

new

Weak transition: ordinary neutron star with quark core (hybrid star)

Strong transition: third class of compact stars possible with maximum massesM ∼ 1M⊙ and radii R ∼ 6 km

Quark phase dominates (n ∼ 15n0 at the center), small hadronic mantle

– p.69

Third Family of Compact Stars (Gerlach 1968)

(Glendenning, Kettner 2000; Schertler, Greiner, JSB, Thoma 2000)

R

M=M stable modesinstable modes

third family neutron stars white dwarfsAB

CDEFG H

Ithird solution to the TOV equations besides white dwarfs and neutron stars,solution is stable!

generates stars more compact than neutron stars!

possible for any first order phase transition!

– p.70

Signals for a Third Family/Phase Transition?

mass-radius relation: rising twins (Schertler et al.,2000)

spontaneous spin-up of pulsars (Glendenning, Pei,Weber, 1997)

delayed collapse of a proto-neutron star to a blackhole (Thorsson, Prakash, Lattimer, 1994)

bimodal distribution of pulsar kick velocities (Bombaciand Popov, 2004)

collapse of a neutron star to the third family?(gravitational waves, γ-rays, neutrinos)

secondary shock wave in supernova explosions?

gravitational waves from colliding neutron stars?

– p.71

Difference between quark stars, hybrid stars, etc?

hybrid stars: neutron stars mixed with quark matter inthe core

quark star twins: special hybrid stars with a purequark matter core

strange stars or selfbound stars: consists of stablequark matter only, purely hypothetical!

– p.72

Hypothetical Selfbound Star versus Ordinary Neutron Star

(Hartle, Sawyer, Scalapino (1975!))

selfbound stars:vanishing pressure at a finiteenergy density

mass-radius relation starts at theorigin (ignoring a possible crust)

arbitrarily small masses and radiipossible

neutron stars:bound by gravity, finite pressure forall energy density

mass-radius relation starts at largeradii

minimum neutron star mass:M ∼ 0.1M⊙ with R ∼ 200 km

– p.73

Signals for Strange Stars?

similar masses and radii, cooling, surface (crust), . . . butlook for

extremely small mass, small radius stars (includesstrangelets!)

strange dwarfs: small and light white dwarfs with astrange star core (Glendenning, Kettner, Weber, 1995)

super-Eddington luminosity from bare, hot strangestars (Page and Usov, 2002)

conversion of neutron stars to strange stars (explosiveevents!)

. . .

– p.74

Summary

equation of state (EoS) determines the maximummass and its radius

cooling is sensitive to the population

new hadronic degrees of freedoms normally softenthe EoS

but quark matter can also stiffen the EoS!

strong chiral phase transition leads to a third family ofcompact stars

sensitive to mass-radius relation, cooling, neutrinos,gravitational waves!

– p.75