Hadronic Matter and the Equation of State

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Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Hadronic Matter and the Equation of State

Jurgen¨ Schaffner-Bielich

Institut fur¨ Theoretische Physik

NewCompstar School 2017 – Neutron stars: theory, observations and gravitational wave emission University of Soﬁa, Bulgaria, September 11-15, 2017 Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Content

1 Introduction: Compact Objects

2 Matter in White Dwarfs

3 Equation of State of Degenerate Gases

4 Mass-Radius Relations

5 Maximum Masses of Compact Stars

6 Scaling Solutions for Compact Stars

7 Modelling Compact Stars

8 Quarks in Neutron Stars! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Compact Objects

compact objects: endpoint of stellar evolution, white dwarfs, neutron stars, and black holes differences to ordinary stars: do not burn nuclear fuel, not supported by thermal pressure against gravity white dwarfs: supported by degeneracy pressure of electrons neutron stars: supported by interaction pressure of nucleons black holes: completely collapsed, singularity! all are essentially static (except mini black holes) over the lifetime of the universe and (astronomically) small in size Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Comparison of compact objects

mass radius density (g/cm3) GM/R

−6 Sun M⊙ R⊙ 1.4 10

7 −4 white dwarf ≤ 1.44M⊙ ∼ 0.01R⊙ ≤ 10 ∼ 10

−5 15 −1 neutron star 0.1 ... 2 − 3M⊙ 10 R⊙ ≤ 10 ∼ 10

2 3 black hole MPlanck ... Muniverse 2GM/c ∼ M/R ∼ 1

33 10 (M⊙ = 1.989 · 10 g, R⊙ = 6.9599 · 10 cm)

huge range of densities, involves all four fundamental forces (strong, weak, electromagnetic, gravity) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Etymology of compact objects

white dwarfs: high effective temperatures, much ’whiter’ than ordinary stars neutron stars: dominantly neutrons in the interior, − inverse β-decay of e + p → n + νe, 57 giant nuclei (ρ ≈ ρnuclei) with 10 baryons! black holes: no light (matter) can escape, appears ’black’ to observer Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Observations

white dwarfs: by optical telescopes (Sirius B!) neutron stars: optical (ESO, Hubble Space Telescope), x-ray satellites, radio (pulsating source of radiation: pulsars!) black holes: indirectly by mass-radius constraint, infalling objects (x-rays), supermassive ones by observing active galactic nuclei (AGN) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Stellar Evolution (Credit: NASA/CXC/M.Weiss) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Formation of Compact Objects

endpoints of stellar evolution:

M < 0.08M⊙: no stable nuclear burning → brown dwarfs M < 0.4M⊙: hydrogen burning → H-He white dwarfs M < 8M⊙: helium burning → C-O white dwarfs (our Sun!) 8 < M < 10M⊙: carbon burning, degenerate Ne-O-Mg core, core collapse supernova → neutron star M > 10M⊙: Ne-O-Si burning, degenerate Fe core, core collapse supernova → neutron star M ≳ 25M⊙: core collapse supernova → black hole M ∼ 100M⊙: upper mass limit for stars

(formation of black holes also by supermassive stars (ﬁrst stars), primordial black holes (’mini black holes’ with

M > 1015g) or collision of compact stars in binary systems) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Matter in White Dwarfs

relation between pressure and energy density → the equation of state (EoS) global properties: large-scale dynamical response to matter to gravity, rotation, electromagnetic ﬁelds local properties: internal stress, dissipation, emissivity Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Thermodynamics I

ﬁrst law of thermodynamics: ( ) ( ) ϵ 1 ∑ d = −P · d + T · ds + µ · dY (1) n n i i i

for different species with a concentration Yi = ni /n n: number density of baryons (B is conserved!)

ni : number density of species i (not conserved!) ϵ: energy density ϵ/n: energy per baryon (E/A) s: entropy density

µi : chemical potential for species i

Yi : particle fraction of species i (Yi = ni /n) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Thermodynamics II

Thermodynamic relations for ϵ = ϵ(n, s, Yi ):

∂(ϵ/n) ∂(ϵ/n) P = − = n2 (2) ∂(1/n) ∂n ∂(ϵ/n) ∂(ϵ/n) ∂ϵ T = , µi = = (3) ∂s ∂Yi ∂ni in equilibrium reactions between particles are in detailed balance, i.e. each reaction is balanced by backreaction and ni is constant → Yi is not an independent quantity!

system close to equilibrium, thermally isolated (δQ = T ds = 0), constant volume (d(1/n) = 0), constant energy (no work, d(ϵ/n) = 0), hence ∑ µi · dYi = 0 (4) i Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Weak Reactions

look at weak reaction

− e + p ←→ n + νe (5) − − then dYe = dYp = dYn = dYνe and

µe + µp = µn + µνe (6)

composition determined by chemical potentials at ﬁxed values of n and ϵ

number of independent quantities required: consider mixture of baryons (neutrons, protons, . . . ) and − − leptons (e , µ , νe, νµ,...) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Independent quantities

all reactions conserve baryon number (nB), lepton number (nL) and charge (nQ) choose nB, nL, and nQ, then all chemical potentials are ﬁxed in terms of the corresponding chemical potentials µB, µL and µQ in general for a species i:

µi = Bi · µB + Li · µL + Qi · µQ (7)

thermodynamic quantities are a function of µi and T only

astrophysical matter is globally charge neutral: nQ = 0 ﬁxes µQ

no trapped neutrinos for cold matter: µνe = µνµ = 0 cold degenerate matter (the temperature is much smaller than the Fermi energy T ≪ EF ): T = 0

=⇒ everything given in terms of µB or nB! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Beta-equilibrium I

ideal cold gas of electrons, protons, and neutrons treat neutrons and protons as free particles ﬁrst (in white dwarfs they are bound in nuclei)

− e + p ←→ n + νe (8)

reaction proceeds, when e− has enough energy to overcome 2 the mass difference of (mn − mp)c = 1.29 MeV inverse reaction − n −→ e + p +ν ¯e (9) is blocked, if the density of e− is high enough, so that all available energy levels are occupied assume equilibrium (T = 0, µν = 0), then

µe + µp = µn (10) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Beta-equilibrium II

the Fermi energy equals the chemical potential, hence √ i 2 2 µi = EF = pF,i + mi (11)

with the Fermi momenta pF,i and the equilibrium condition implies √ √ √ 2 2 2 2 2 2 pF,e + me + pF,p + mp = pF,n + mn (12)

number densities are given by ∫ pF,i dp3 1 = · = 3 ni gi 3 gi 2 pF,i (13) 0 (2π) 6π

with the degeneracy factor gi = 2si + 1. Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Beta-equilibrium III

Charge neutrality implies −→ · 3 · 3 −→ ne = np ge pF,e = gp pF,p pF,e = pF,p (14)

determines everything in terms of µB or nB. Thermodynamic quantities:

P = Pe+Pp+Pn ϵ = ϵe+ϵp+ϵn nB = np+nn nQ = np−ne = 0 (15) neutrons appear at pF,n = 0, then eq. (12) reads √ √ 2 2 2 2 pF,e + me + pF,p + mp = mn (16)

or for the nonrelativistic limit for protons pF,p ≪ mp: √ 2 2 − pF,e + me = mn mp (17) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Beta-equilibrium IV

critical electron density (which is equal to the proton density) ( ) 1 3/2 n = (m − m )2 − m2 = 7.4 · 1030 cm−3 (18) c 3π2 n p e amounts to a mass density of

7 −3 ρc = mp · nc = 1.2 · 10 g cm (19)

the neutron fraction increases drastically then so that nn ≫ np. In the ultrarelativistic limit (pF ≫ mN ) it follows from the equilibrium condition eq. (12) and charge neutrality

pF,e + pF,p = 2pF,p = pF,n (20) ∝ 3 with n pF the neutron particle ratio approaches then

nn 8 nn = 8np −→ = (21) nB 9 Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Beta-equilibrium for nuclei and electrons

white dwarf material: nucleons in their lowest energy state −→ nuclei immersed within an electron gas recipe: ﬁnd combination of mass number A and charge Z that minimizes the total energy include lattice energy: important for composition, but not for the EoS low densities: 56Fe is the most stable nucleus relativistic e− transform protons to neutrons: nuclei become neutron-rich ’neutron-drip density’: free neutrons appear at ρ ∼ 4 · 1011 g/cm3 shell effects important: magic numbers for neutrons are N = 50, 82 Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Summary of white dwarf matter

white dwarf material consists of free e− and ions forming a lattice ρ < 104 g/cm3: atomic corrections ρ = 106 g/cm3: electrons become relativistic ρ < 107 g/cm3: only e− and 56Fe ρ < 4 · 1011 g/cm3: lattice of ions with free e−, nuclei become increasingly neutron-rich ρ = 4 · 1011 g/cm3: free neutrons appear, ’neutron drip’ determines structure of planets and white dwarfs Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

History for White Dwarfs

some history: 1862: discovery of Sirius B, son of lensmaker Alvan Clark by testing his father newest telescope (18 inch refractor), mass M ∼ 1M⊙ 1914: discovery that spectrum of Sirius B is ’white’ 1925: gravitational redshift measured → Sirius B is as small as earth, ρ¯ ∼ 106 g/cm3 1926: Dirac formulates Fermi-Dirac statistics, Fowler applies it to white dwarfs 1930: Chandrasekhar takes into account special relativity → maximum mass of white dwarfs

modern values for Sirius B: M = 1.000 0.016M⊙, R = 0.0084 0.0002R⊙, Teff = 24700 300 K Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Equation of State of Degenerate Gases (Non-Relativistic) I

degeneracy pressure in the non-relativistic limit (kF ≪ me) for a composite Fermi gas (electrons and nucleons): dominated by electron pressure ∫ 3 Pe ∝ d k (k · v/c) , v/c = k/E ≈ k/me (22)

integrate: ∫ ∫ ∝ 2 · 4 ∝ 5 Pe dk k (k k/me) = dk k /me kF /me (23) ∫ ∝ 3 ∝ 3 ⇒ ∝ 1/3 the number density is given by n dk kF = kF n . k 5 ∝ F,e ∝ 1 · 5/3 Pe ne (24) me me Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Equation of State of Degenerate Gases (Non-Relativistic) II

charge neutrality demands

Z n = n = n (25) e p A N the energy density is just the mass density and dominated by the nucleons ϵN = ρN = mN · nN so that ﬁnally: ( ) · 5/3 ∝ 1 5/3 1 Z ρN Pe ne = (26) me me A · mN and ﬁnally the equation of state in the non-relativistic limit reads: ( ) Z 5/3 P ∝ · ρ5/3 (27) A Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Equation of State of Degenerate Gases (Relativistic Limit)

degeneracy pressure in the relativistic limit (kF ≫ me): ∫ 3 Pe ∝ d k (k · v/c) , v/c = k/E ≈ 1 (28)

integrate: ∫ ∫ ∝ 2 3 ∝ 4 ∝ 4/3 Pe dk k k = dk k kF,e ne (29)

Assuming that the heavy fermion (nucleon) is still non-relativistic ϵ = ρ = mN · nN (kF ≪ mN ), one ﬁnds ( ) Z 4/3 P ∝ · ρ4/3 (30) A

which is applicable for white dwarfs but not for neutron stars! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Equation of State: Pure Neutron Gas

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

k 5 ∝ F,n ∝ 1 · 5/3 ∝ 5/3 Pn nn ϵ (31) mn mn ∝ 3 · using nn kF,n and ϵ = mn nn. Now, in the ultra-relativistic limit (kF ≫ mn)

∝ 4 ∝ 4/3 Pn kF,n nn (32)

However, the energy density changes to ∫ ∫ ∝ 3 ∝ 2 · ∝ 4 ∝ 4/3 ϵn d k En dk k k kF,n nn (33)

so that the equation of state approaches now P = ϵ/3. Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Summary for the Equations of State

General parameterization of the equations of state in the form ( ) ρ Γ P ∝ nΓ = or P ∝ ϵΓ (34) m is called a polytropic equation of state or simply a polytrope. Non-relativistic limit:

P ∝ ρ5/3 =⇒ Γ = 5/3 (35)

Relativistic limit (for composite fermion gases):

P ∝ ρ4/3 =⇒ Γ = 4/3 (36)

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

P = ϵ/3 =⇒ Γ = 1 (37)

Note: ρ stands for the mass density of the non-relativistic species, ϵ for the more general energy density. Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Spheres in Hydrostatic Equilibrium I

Mass within a radius r: ∫ r ′ ′2 ′ Mr (r) = 4π dr r ρ (r) (38) 0 expressing just mass conservation. For a mass element in a shell in differential form:

dm(r) = 4πr 2ρ(r)dr (39)

At the center M(0) = 0, while at the (Carroll and Ostlie: An Introduction to Modern surface Mr (R) = M. Astrophysics) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Spheres in Hydrostatic Equilibrium II

Pressure balances gravity (forces) in an area A of thickness dr: M (r) · dm(r) A · dP(r) = −G r (40) r 2 use dm = A · ρ · dr, then the area cancels out. It follows the equation of hydrostatic equilibrium:

dP(r) M (r) · ρ(r) = −G r (41) dr r 2

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

(Carroll and Ostlie: An Introduction P(R) = 0.

to Modern Astrophysics) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Mass–radius relation for Polytropes

Approximate hydrostatic equilibrium (Pc: central pressure): dP P Mρ¯ − ≈ c ∝ G (42) dr R R2 with the average mass density ρ¯ ∝ M/R3 it follows

M M M2 P ∝ G · = G (43) c R R3 R4 Use the equation of state for polytropes P = K · ρΓ: ( ) M Γ M2 P ∝ ρ¯Γ ∝ ∝ G (44) c R3 R4 to arrive at the mass–radius relation for polytropes:

M2−Γ · R3Γ−4 ∝ 1/G = const. (45) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Mass–radius relation for Polytropes II

Full Newtonian treatment: the Lane–Emden equation ( ) 1 d dθ ξ2 = −θn (46) ξ2 dξ dξ

n with dimensionless density θ: ρ(r) = ρcθ (r), the polytropic index n: Γ = (n + 1)/n, and the dimensionless radius ξ: r = rn · ξ, where √ − K ρ(1 n)/n r = (n + 1) c (47) n 4πG

Boundary conditions: θ(0) = 1, and as dP/dr → 0 for r → 0, it follows from the polytrope that θ′(0) = 0. Integrate outwards until the pressure vanishes θ(ξn) = 0 where ξn deﬁnes the radius of the star. Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Mass–radius relation for Polytropes II

Solutions for the radius and the mass for a given central mass density ρc:

R = rn · ξn (48) − 3 · 2 ′ M = 4πrn ρc ξnθ (ξn) (49)

eliminating ρc one arrives at the mass-radius relation:

M2−Γ · R3Γ−4 = const. (50)

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. (different slope of M-R curve) Γ → ∞: M · R−3 = const. (constant density, selfbound stars) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Relativistic Hydrostatic Equilibrium

General Relativity: add three relativistic correction factors

( )( )( )− dP M ϵ P 4πr 3P 2GM 1 = − r + + − r G 2 1 1 1 (51) dr r ϵ Mr r with the mass conservation equation

dM = 4πr 2ϵ (52) dr 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. Note: the mass density ρ is replaced by the energy density ϵ ! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Mass-radius relation for free neutrons

(Sagert, Hempel, Greiner, JSB 2006) numerical solution of the TOV equations for free neutrons (Oppenheimer and Volkoff 1939!) M · R3 = const. for large radii

maximum mass: Mmax = 0.71M⊙ at R = 9.1 km Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Landau’s Argument for a Maximum Mass

A relativistic particle at the radius R of the compact star has the energy Mm E(R) = E + E = −G + k (53) g kin R F 1/3 1/3 with kF ∝ n ∝ N /R and M ≈ N · m, N being the number of fermions: ( ) Nm2 9π 1/3 N1/3 E(R) = −G + (54) R 4 R

Stability analysis leads to For E < 0: collapse to R → 0 for small R! → ∞ ∝ 2 ∝ −2 For E > 0: R until kF < m, then Ekin kF /mf R and a stable minimum exists For E = 0: marginally stable Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Landau’s Argument for a Maximum Mass II

For the stability limit E = 0 one gets ( ) ( ) 9π 1/3 9π 1/2 M3 GN m2 = N1/3 ⇒ N = P (55) max 4 max max 4 m3

2 (G = 1/MP , MP : Planck mass). Hence, the maximum mass is M3 M ∼ P (56) max m2

The corresponding radius for kF = m is N1/3 M R ∼ max ∼ P (57) crit m m2 For a neutron star: 3 3 MP 57 MP MP ∼ ∼ ∼ = . ⊙ ∼ = . Nmax 3 10 Mmax 2 1 8M Rcrit 2 2 7 km mn mn mn Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Maximum mass for white dwarfs I

for white dwarfs, the momentum is provided by the electrons, the mass by nucleons for each nucleon, there are Z/A electrons around due to the charge neutrality condition: Z n = n = n (58) e p A N so the energy balance equation reads for each nucleon Mm Z E(R) = −G N + k · (59) R F,e A ( ) 1/3 ( ) N m2 9π 1/3 N Z 4/3 = −G N N + N (60) R 4 R A The critical condition E = 0 results in: ( ) Z 4/3 GN m2 ∼ N1/3 (61) max N A max Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Maximum mass for white dwarfs II

the maximum number of nucleons is then given by ( ) M3 2 ∼ P Z Nmax 3 (62) mN A and the maximum mass is the Chandrasekhar mass ( ) 3 2 MP Z ∼ ∼ = . ⊙ Mmax 2 MChandra 1 44M (63) mN A

the critical radius for kF,e = me is ( ) ( ) 1/3 1/3 1/3 Nmax Z MP Z Rcrit ∼ ∼ (64) me A mn · me A

so that Rwd ∼ MP /(mN · me) ∼ 5.000 km, about the size of earth! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Neutron star versus white dwarf

note that the maximum masses of white dwarfs and neutron stars are similar M3 ∼ ∼ P Mwd Mns 2 mN (modulo factors of Z /A) as the mass is provided in both cases by the nucleon mass! note that the radius corresponding to the maximum mass is related to the mass of the particle providing the pressure

R m wd ∼ N ∼ 2000 Rns me (modulo factors of Z /A) so that the radius of a white dwarf is larger by the ratio of the nucleon to electron mass Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Interacting Fermions

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

∂E 2 (∂ρint /n) Γ Pint = − = n = (Γ−1)c·n = (Γ−1)ϵ (65) ∂V N,T =0 ∂n

2 As cs = P/ϵ, the EoS will be acausal at high-densities for Γ > 2 (standard interactions: Γ = 2). Non-relativistic low density limit:

ϵ = m · n + c · n2 ≈ m · n p = c′ · n5/3 + c · n2 ≈ c′ · n5/3 ∝ ϵ5/3 (66) At some intermediate density and strong interactions:

ϵ ≈ m · n p ≈ c · n2 ∝ ϵ2 (67)

a polytrope of Γ = 2 =⇒ constant radii! Note: the limit of n → ∞ is p = ϵ = c′ · n2! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Mass-radius relation for interacting fermions

(Narain, JSB, Mishustin 2006)

dimensionless interaction strength y = mf /mI for 2 2 ≈ Pint = n /mI (strong interactions y = mN /fπ 10) 3 weakly interacting (y < 1): M · R = const., Mmax = const.

strongly interacting (y > 1): R ≈ const., Mmax ∝ y! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Scaling of the TOV equations I

consider dimensionsless quantities for the EoS:

′ ′ P = ϵo · P ϵ = ϵo · ϵ (68)

and then rescale radius and mass so that the TOV equation is scale-free · ′ · ′ r = a r Mr = b Mr (69) with coefﬁcents a and b to be determined. Rescaled TOV equation in terms of primed (dimensionless) quantities: ( )( ) ϵ · dP′ bM′ϵ ϵ′ ϵ · P′ 4πa3 · r ′3ϵ P′ 0 = −G r 0 1 + 0 1 + 0 · ′ 2 ′2 · ′ · ′ a dr a · r ϵ0 ϵ b Mr ( )− 2Gb · M′ 1 1 − r (70) a · r ′ Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Scaling of the TOV equations II

with the conditions:

3 −1/2 3 −1/2 G · b = a a · ϵ0 = b → a = (Gϵ0) b = (G ϵ0) (71) Then the dimensionless TOV equation reads

( )( )( )− dP′ M′ϵ′ P′ 4πr ′3P′ 2M′ 1 = − r + + − r ′ ′2 1 ′ 1 ′ 1 ′ (72) dr r ϵ Mr r

That means that quantities scale with the factor√ ϵ0 to some power, in particular Mmax und Rcrit scale as ϵ0! Note: Rescaling works also for the equation for mass conservation. Applications: Solve the dimensionless TOV equations for a given√ type of dimensionless EoS and then rescale the results with ϵ0 for physical values! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Scaling of the TOV equations III

Examples: EoS for a free (massive) Fermi gas ′ 4 ′ 4 rescale the EoS: P = P/mf and ϵ = ϵ/mf where mf is the Fermi mass. Then the maximum mass and the corresponding radius are given by

M3 = ′ · ( 3 · 4)−1/2 = ′ · P M M G mf M 2 (73) mf M = ′ · ( · 4)−1/2 = ′ · P R R G mf R 2 (74) mf where we recognize the Landau mass and Landau radius as solutions! Numerically one ﬁnds M′ = 0.384 and R′ = 3.367. Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Scaling of the TOV equations IV

EoS for a relativistic Fermi gas rescale the relativistic EoS of the form 1 P = (ϵ − ϵ ) (75) 3 vac

where ϵvac is an offset (vacuum energy density). Otherwise the relativistic EoS does not lead to stable conﬁgurations! Then the maximum mass and the corresponding radius are given by

2 ′ 3 −1/2 (145 MeV) M = M · (G · ϵvac) = 2.00M⊙ √ (76) B 2 ′ −1/2 (145 MeV) R = R · (G · ϵvac) = 10.9 km √ (77) B

1/4 where B = 145 MeV is the MIT bag model value (ϵvac = 4B). Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Scaling of the TOV equations IV

Limiting EoS from causality 2 speed of sound: cs = ∂P/∂ϵ 2 stiffest possible EoS has cs = 1 realized via

P = Pf + (ϵ − ϵf ) (78)

above some ﬁducial energy density (pressure) ϵf (Pf ). If P of the low-density EoS can be ignored, then the maximum mass is given by √ ′ 3 −1/2 ϵ0 M = M · (G · ϵf ) = 4.2M⊙ · (79) ϵf

where ϵ0 is the ground-state energy density of nuclear matter (Rhoades and Rufﬁni 1974). Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

History for Neutron Stars

some history: 1932: Landau postulates the existence of neutron stars 1934: Baade and Zwicky:

With all reserve we advance the view that supernovae represent the transition from ordinary stars to neutron stars, which in their ﬁnal stages consist of closely packed neutrons.

1939: papers of Tolman and Oppenheimer with Volkoff (’On neutron star cores’) 1942: translation of chinese records on ’guest stars’ 1965: Wheeler et al.: work on neutron star EoS 1967: Pacini: discusses rotating neutron stars 1967: discovery of pulsars by Jocelyn Bell (Nobel prize for her supervisor Anthony Hewish) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

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−) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

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

quark−hybrid traditional neutron star star N+e N+e+n

n,p,e, µ n superfluid hyperon n d u c o c t neutron star with star e r i n p g pion condensate u n,p,e, s p µ r o u,d,s t quarks o − n π s H crust Σ,Λ,Ξ,∆ Fe − absolutely stable K 6 3 strange quark 10 g/cm matter 11 10 g/cm 3 3 µ u d s 10 14 g/cm

m s

strange star nucleon star R ~ 10 km M ~ 1.4 M Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Neutron Star Matter for a Free Gas

(Ambartsumyan and Saakyan, 1960)

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

but the corresponding equation of state results in a maximum mass of only Mmax ≈ 0.7M⊙ < 1.44M⊙

(Oppenheimer and Volkoff, 1939) =⇒ effects from strong interactions are essential to describe neutron stars! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Pure Neutron Matter: State of the Art

(Tews, Kruger,¨ Hebeler, Schwenk 2012) chiral effective ﬁeld theory including N3LO terms with 3N and 4N forces comparison to Quantum Monte Carlo Simulations QMC and GCR band of uncertainty for the neutron equation of state up to saturation density Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Impact of hyperons on the maximum mass of neutron stars

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⊙ !

(Glendenning and Moszkowski 1991) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Phase Transitions in Quantum Chromodynamics QCD

T early universe

RHIC, LHC Quark−Gluon

Tc Plasma

FAIR

Hadrons Color Superconductivity nuclei neutron stars

µ mN / 3 c µ

Early 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) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Phase Transitions in Quantum Chromodynamics QCD

T early universe

RHIC, LHC Quark−Gluon

Tc Plasma

FAIR

Hadrons Color Superconductivity nuclei neutron stars

µ mN / 3 c µ

Nuclear Equation of State as Input in Astrophysics

−10 supernovae simulations: T = 1–50 MeV, n = 10 –2n0 −3 proto-neutron star: T = 1–50 MeV, n = 10 –10n0 −3 global properties of neutron stars: T = 0, n = 10 –10n0 −10 neutron star mergers: T = 0–100 MeV, n = 10 –10n0 Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

High-Density Side: perturbative QCD

(Fraga, Kurkela, Vuorinen, 2013) asymptotic freedom of strong interactions (QCD): weakly interacting at large scales (temperature and/or chemical potential) O 2 perturbative calculations up to (αs ): follows lattice data at nonzero temperature band of uncertainty for the pQCD equation of state at nonzero µ (from choice of renormalization scale) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

What we know about the EoS for neutron star matter

SB limit

) 1000 -3 pQCD matter inner 1 crust outer neutron crust matter ? 0,001 Pressure (MeV fm

1e-06 1 10 100 1000 µ − µ Quark Chemical Potential iron/3 (MeV) (Kurkela, Fraga, JSB, Vuorinen (2014)) pressure versus the quark chemical potential (above the one of 56Fe) low-density regime: outer crust (lattice of nuclei), inner crust (lattice of nuclei in a neutron ﬂuid), neutron matter high-density regime: pQCD, close to Stefan-Boltzmann limit Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Mass-radius and maximum density of pure quark stars

2 B1/4=145 MeV

Λ=3µ

1.5

1/4 green curves: MIT bag model B =200 MeV sun

1 Λ=2µ blue curves: perturbative QCD

M/M calculations

0.5 (Fraga, JSB, Pisarski 2001)

0 0 5 10 15 Radius (km)

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

case Λ = 3µ: Mmax = 2.14 M⊙, Rmax = 12 km, nmax = 5.1 n0 other nonperturbative approaches: Schwinger–Dyson model (Blaschke et al.), massive quasiparticles (Peshier, Kampfer,¨ Soff), NJL model (Schertler, Steiner, Hanauske, . . . ), HDL (Andersen and Strickland), ... Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

A Model For Cold And Dense QCD

hadrons / massless massive quarks quarks

µ µ min χ µ Two possibilities for a first-order chiral phase transition: A weakly first-order chiral transition (or no true phase transition), =⇒ 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 Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Matching the two phases: two possible scenarios

0.6

free 0.4 massive baryons/quarks massless quarks 0.2 Pressure p/p

0 µ µ strong µ weak min χ χ

Weak: phase transition is weakly ﬁrst order or a crossover → pressure in massive 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)) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Gibbs Phase Construction (Glendenning (1992))

MeV n 1400 1200

1000

n e 300 QP

200 MP MeVfm

2 100

200 150

100

n e

50 HP

MeV e 0 (Schertler et al. (2000)) two conserved charges in β-equilibrium: baryon number and charge! Gibbs criterium for phase equilibrium: equal pressure for equal chemical potentials PI (µB, µe) = PII (µB, µe) globally charge neutral matter: mixed phase with charged bubbles forms! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Hybrid Stars (Schertler et al. (2000))

g=0. g=2.

190 190

HP TM

185 185

MP HP 180 180 MeV 4 =

1 175 175 QP B

MP M M 170 170

QP M 165 165 M

0 2.5 5 7.5 10 12.5 15 0 2.5 5 7.5 10 12.5 15

R km R km

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!) Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Third Family of Compact Stars (Gerlach 1968)

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

third family neutron stars white dwarfs

I D

F H

M M

stable mo des

instable mo des R

third 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! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

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 black hole (Thorsson, Prakash, Lattimer, 1994) bimodal distribution of pulsar kick velocities (Bombaci and Popov, 2004) collapse of a neutron star to the third family? (gravitational waves, γ-rays, neutrinos) secondary shock wave in supernova explosions (Sagert, Fischer et al. 2009)? gravitational waves from colliding neutron stars? Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Difference between quark stars, hybrid stars, etc?

hybrid stars: neutron stars mixed with quark matter in the core quark star twins: special hybrid stars with a pure quark matter core strange stars or selfbound stars: consists of stable quark matter only, purely hypothetical! Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars! Hypothetical Selfbound Star versus Ordinary Neutron Star

(Hartle, Sawyer, Scalapino (1975!))

selfbound stars: neutron stars: vanishing pressure at a finite energy bound by gravity, finite pressure for density all energy density mass-radius relation starts at the mass-radius relation starts at large origin (ignoring a possible crust) radii arbitrarily small masses and radii minimum neutron star mass: possible M ∼ 0.1M⊙ with R ∼ 200 km Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Signals for Strange Stars?

similar masses and radii, cooling, surface (crust), . . . but look for extremely small mass, small radius stars (includes strangelets!) strange dwarfs: small and light white dwarfs with a strange star core (Glendenning, Kettner, Weber, 1995) super-Eddington luminosity from bare, hot strange stars (Page and Usov, 2002) conversion of neutron stars to strange stars (explosive events!) ... Introduction: Compact Objects Matter in White Dwarfs Equation of State of Degenerate Gases Mass-Radius Relations Maximum Masses of Compact Stars Scaling Solutions for Compact Stars Wipe Quarks in Neutron Stars!

Observing Quark Matter in the Sky

Two different types of modiﬁcation from high-density QCD matter equation of state, phase transition(s): mass-radius relation (third family of compact stars?) dynamics of core-collapse supernovae (second shock wave?) proto-neutron star evolution (collapse to a black hole?) neutron star merger (gravitational wave signal) transport properties: bulk and shear viscosity (r-mode instability) neutrino reactions (cooling of (proto-) neutron stars)

x-ray satellites (NuSTAR, eRosita, ATHENA, NICER!), optical (JWST), radio (SKA), gravitational wave (Advanced LIGO, LISA) and neutrino detectors (Super-K, IceCube)