matter in compact stars

• Low-density nuclear matter in the crust of neutron stars or core of supernovae

• High-density matter in the core of neutron stars

Toshiki Maruyama (JAEA)

Nobutoshi Yasutake (Chiba Inst. of Tech.) Minoru Okamoto (Univ. of Tsukuba & JAEA) Toshitaka Tatsumi (Kyoto Univ.)

Structure and properties of nuclear

matter in compact stars

Low-density nuclear matter  Inhomogeneous structures.

There are two ways of understanding inhomogeneous matter.

1. From inhomogeneous to uniform

Compression of low-density matter.

 crystal of atoms in degenerate electrons.

 crystal of nuclei in degenerate neutrons.

 uniform nuclear matter.

2. From uniform to inhomogeneous

Instability of uniform matter below the saturation density.

 phase transition & mixed phase (clustering).

Low−density nuclear matter

Neutron- rich nuclei in electron

sea Neutron-

rich nuclei in neutron sea

compression

uniform First, electrons degenerate.

 electron energy depends on the density.

 𝑌_𝑒 and 𝑌_𝑝 decrease. Baryon energy is not directly dependent on the density.

As 𝑌_𝑝 decreases ,neutrons begin to drip. (∎)

 Neutrons have found large space to escape.

 𝑌_𝑒 and 𝑌_𝑝 decrease rapidly.

As neutrons degenerate, increase of

𝑌_𝑛

(

decrease of 𝑌_𝑝

)

is suppressed.

 at last, 𝑌_𝑝 increases and protons degenerate. (∎)

 Uniform matter.

1. From inhomogeneous to uniform

𝑌

𝑒

& 𝑌

𝑝

4

Baryon partial pressure has a

density region of negative gradient (Note that the total pressure and its gradient are positive due to electron pressure.)

For symmetric matter at T=0, Maxwell constr leads:

negative pressure (at 𝜌_𝐵 < 𝜌₀) is not favored.

 mixed phase (clustering)

However, at some density just below 𝜌₀, uniform matter is

favored due to finite size effects (surface and Coulomb).

2. From uniform to inhomogeneous

 

 

 

1

2 * 2 2 * 2

,

1

3

, , , 0 3

( ; , ) 1 exp ( ) ( ) (

For Fermions, we employ Thomas-Fermi approx. with fini

) ,

( ; , ) 1 exp ( ( )) ,

( ) 2 ( ; ,

te

(2 ,

(

) )

i n p i i Fi i

e e e C

i p n e i i

n Fn

f p m p m T

f p V T

d p f

T

p





 

 















  

      

 

     







r p r r r

r p r

r r p

2 * 2

0 0

2 * 2

0 0

3 3 3

) ( ) ( ) ( ), ,

( ) ( ) ( ) ( ) ( ),

( ) ( ) const, ( ) ( ),

N N N n p e

p Fp N N N C

p n p e

V V V

m g g R

p m g g R V

d r d r d r

 

 

   

 

   

    

    

   

 

  

r r r r

r r r r r

r r r r

2 2 ( ) ( )

2 2

0 0

2 2

0 0

2 2

ch

( ) ( ) ( ( ) ( )) ( )

L (

,

( ) ( )

) L 0,

( , ,

( ( ) ( )),

( ) ( ) ( ( ) ( )),

( ) 4

, ,

( From

)

,

,

)

s s

N n p

N p n

N p n

C

m g dU

d

m g

R m R g

V

R

e

V

 

 

 

 

  

   



   

 









  

    



 

       

 



 

   









r r r r r

r r r r

r r r r

r r

* 3

2 2 2 2 2

*

,

1 Lagrangi

2

1 1 1 1 1 1

( ) ( ) ,

2 2 4 2 4 2

1 n

, )

a

4 (

N M e

N N N N

M

e e e e

N N N

L L L L

L i m g g b e V

L m U m R R m R R

L V V i m e V F F F

m m g

   

     

   

       

  

       



      

      

 



  

  

       

       

 

            

  1 ³ 1 ⁴

, ( ) ( ) ( )

3 ^{N N} 4 ^N

U   bm g_   c g_ 

RMF + Thomas-Fermi model

Nucleons interact with each other via coupling with 𝜎, 𝜔, 𝜌 mesons.

Simple but feasible!

Saturation property of symmetric nuclear matter : minimum energy

𝐸/𝐴 ≈ −16 MeV at 𝜌_𝐵 ≈ 0.16 fm⁻³.

Bulk properties of nuclei, such as binding energies, proton fractions, and density profiles, are well reproduced.

Choice of parameters

Matter properties

Properties of nuclei

7

• Total pressure is positive. Monotonically increases with density and temperature.

Pressure of uniform nuclear matter

• Baryon pressure exhibits van der Waals type behavior  mechanically unstable region ( 𝑑𝑃 𝑑𝜌 < 0 ).

 First order phase transition.

8

EOS of mixed phase

• Single component congruent _{(e.g. water)} Maxwell construction satisfies the Gibbs cond. T^I=T^II, P^I=P^II, ^I=^II.

• Many components non-congruent (e.g. water+ethanol)

Gibbs cond.T^I=T^II, P_i^I=P_i^II, _i^I=_i^II. No Maxwell construction !

• Many charged components

(nuclear matter)

Gibbs cond. T^I=T^II, _i^I=_i^II. No Maxwell construction ! No constant pressure !

 ^; 

i i

i U r

dP

dr r



  



EOS of mixed phase

What is necessary?

Satisfying the Gibbs conditions, we have to look for inhomogeneous density distribution of nucleons and electrons, which minimize the free-energy density.

10

Numerical calculation of mixed-phase structure

• Assume regularity in structure: divide whole space into equivalent and neutral cells with a geometrical symmetry (3D: sphere, 2D : cylinder, 1D: plate).

 Wigner-Seitz approx.

• Give a baryon density 𝜌_𝐵 and a geometry (Unif/Dropl/Rod/...).

• Solve the field equations numerically. Optimize the cell size (choose free-energy-minimum).

• Choose the free-energy-minimum geometry among 7 cases (Unif (I), Droplet, Rod, Slab, Tube, Bubble, Unif (II)),

some of which are called

“pasta” structures

.

WS-cell

Nuclear “pasta” structures

• Ravenhall et al 1983 &

Hashimoto et al 1984

Concept of “pasta” structures.

Minimizing free-energy of the inhomogeneous structure, i.e., achieving the balance between surface tension and the Coulomb repulsion

 nuclear pasta

Figure from K. Oyamatsu, NPA561, 431 (1993)

• Baym, Bethe, Pethick, 1971

“Nuclei inside-out”

uniform

uniform

Density profiles in WS cells

Pasta structures in matter (case of fixed Y

_p

)

Y_p=0.5

T=0 Y_p=0.1

T=0

12

13

EOS with pasta structures in

nuclear matter at 𝑇 ≥ 0 Symmetric matter Y_p=0.5

Asymmetric matter Y_p=0.3

Pasta structures appear at 𝑇 ≤ 10 MeV

coexistence region (Maxwell for Y_p=0.5 and bulk Gibbs for Y_p<0.5) is meta-

stable.

Uniform matter is allowed in some coexistence region due to finite-size effects.

• We discuss roles of neutrinos trapped in nuclear matter at

supernova cores or proto neutron stars.

• The relevant density is around or below the normal nuclear density (saturation density 𝜌₀ ≅ 0.16 fm⁻³).

• Mean free path of neutrino

𝜆_𝜈 = ¹

𝜌𝜎, 𝜌 ≈ 0.1 fm⁻³, 𝜎=10⁻³⁸cm²

 𝜆 ≈ 1cm ≪ size of compact stars  neutrinos are trapped.

• The ingredients of matter are 𝑝, 𝑛, 𝑒⁻, 𝜈 (trapped neutrino).

More realistic case for supernova core:

Neutrino-trapped matter

Preceding studies:

・Ogasawara & Sato PTP68(1982)222

Effective interaction + Thomas-Fermi, T = 0, Y_l = fixed

・Ogasawara & Sato PTP70(1983)1569

Effective interaction + Thomas-Fermi, T > 0, Y_l = fixed  Enhancement of inhomogeneous

structures (droplet & bubble)

・ Watanabe, Iida, Sato NPA687(2001)512

Effective interaction + flat density , T = 0, Y_l = fixed  Enhancement of pasta phases.

Our present study:

Relativistic mean field + Thomas-Fermi , T > 0, Y_l = fixed Fully consistent density distribution.

Neutrino degenerate and inhomogeneous matter.

 

 

 

 

 

1

2 * 2 2 * 2

1

3 ,

1 ,

, ,

( ; , ) 1 exp ( ) ( ) ( )

For Fermions, we employ Thomas-Fermi a

,

( ; , ) 1 exp ( ( )) ,

( ) 2 (2 ( , ) 1 ex

pprox. at finite

p

i n p i N Fi N

e e e C

i p n e

f p m p m T

f p V T

d

T p

T

f_{ } p _





 

 

 











  

      

 





    





    

p

r p r r r

r p r

r

 

0 3

2 * 2

0 0

2 * 2

0 0

3 3

3

3

( ; , ), )

( ) ( ) ( ) ( ) ( ),

( ) ( ) ( ) ( ),

( ) ( ) const, (baryon number) ( (total

( ) ( )

ch

) ( ), arge)

i i

p Fp N N N C

n Fn N N N n p e

p n p e

V

e

V V

f

p m g g R V

p m g g R

d

d r d r d r

r _

 

 



 

    

  

 





    

     

   

 







  

r p

r r r r r

r r r r

r r r

r

r

r



(lepton

const number)

V





Equations to be solved

Density distribution of particles in the WS cells.

Looking for free-energy minimum states (solving the coupled equations for fields and chemical potentials), we obtain density distribution in a cell. The optimum geometry (dimensionality of the cell) changes from “Droplet” to

“Rod”, “Slab”, “Tube”, “Bubble”, and to

“Uniform” with increasing the density.

The electron fraction is larger than the neutrino fraction 𝑌_𝑒 > 𝑌_𝜈 due to the attraction between 𝑒 ⁻ and 𝑝.

Proton fraction of matter depends on the density and the appearance of non-uniform structures.

Constant lepton fraction Y_l gives proton fraction Y_p about 3/4 of Y_l, which is much larger than the case of beta-equilibrium without neutrinos.

Higher temperature suppresses the appearance of “pasta” (rod, slab, etc) and enlarges the region of droplet.

Trapped neutrino enhances the appearance of inhomogeneous structure.

0.1 MeV 3 MeV

T=5 MeV 𝝂 trapped, 𝒀_𝒍 = 𝟎. 𝟑 EOS (pressure) of nuclear matter at finite

temperature with a fixed proton fraction Y_p=0.3 (below) and a fixed lepton fraction Y_l=0.3 (right panels).

The appearance of inhomogeneous structure softens the EOS for both cases.

They show similar dependence on the density.

No 𝝂, 𝒀_𝒑 = 𝟎. 𝟑

4 MeV T=0

8 MeV

Fully 3 dimensional (3D) calculation

Use of the Wigner-Seitz cell

dumbbell Gyroid double-

diamond ^network

bcc lattice

“droplet” “bubble” “rod” “tube”

Honeycomb

3D

2D 1D

Complex structures were

missing

Crystalline structures could not be

discussed.

 Limitation of emerging structures.

Results of our 3D RMF calculations

“Droplet”

[fcc_]

ρ_B = 0.012 fm^-3

“Rod”

[honeycomb]

0.024 fm^-3

“Slab”

0.05 fm^-3

“Tube”

[honeycomb]

0.08 fm^-3

“Bubble”

[fcc]

0.094 fm^-3

Y_p = Z/A = 0.5

proton

electron

EOS has a similar behavior to that of the conventional studies.

Novelty:

fcc lattice of droplets can be the ground state at some density.

 Not the Coulomb interaction among “point particles” but the change of the droplet size is

relevant.

Y_p = Z/A = 0.5

“droplet”

[fcc]

ρ_B=0.020 fm^-3

“rod”

[simple]

0.040fm^-3

“slab”

0.05 fm^-3

“tube”

[simple]

0.066 fm^-3

“bubble”

[fcc]

0.070 fm^-3 proton

neutron

“droplet”

[fcc]

ρ_B = 0.016 fm^-3

“rod”

[simple]

0.030 fm^-3

“slab”

0.05 fm^-3

“tube”

[simple]

0.066 fm^-3

“bubble”

[fcc]

0.080 fm^-3 proton

neutron

𝑌_𝑝 = 0.1 𝑌_𝑝 = 0.3

Some complex metastable states in our 3D RMF calc.

Mixture of Droplet and Rod

Mixture of Slab and Tube

Dumbbell Network

25

 

 

   

   

3 1

2

2 0

4 2

1 1 1

4

, 0

r

dP Gm r P P Gm

dr r m r

P P r

m m r s s ds

M m R R R

 





 

 



   

        





 

  



Density at position 𝑟

mass inside the position 𝑟 total mass and radius.

Pressure (EOS --- the input)‏

TOV equation

Structure of compact stars

Mass–radius relation and the maximum mass is determined by the EOS of matter.

High−density matter

26

Hyperon mixture at 2 − 3𝜌₀  EOS gets soft

 too small maximum mass

However, mixed phase may soften the EOS.

A bulk calculation suggests wide

region of mixed phase. [Glendenning, PRD46,1274].

Improve the maximum mass by introducing transition into quark matter (?)

[Maieron et al, PRD70 (2004) ⁰⁴³⁰¹⁰].

Schulze et al, PRC73 (2006) 058801

}hyperon included nucleon

Softening of EOS by hyperon

 

   

 

2

,

, 2

3 1

( ) ; ( ) ( )

5 2 2

; ;

( ) ( )

( ) Re A ; ( ) ( )

2

: AV18 UIX NSC89

F

a b

F

F

k k k

a b a b

k k a b

k k

k kk AG e k e k kk

m

k k Q k k

G w v v G w

w e k e k

e k k kk G e k e k kk

m v

   

 







  

  

   

  

  

 







27

Hadron phase

Brueckner Hartree Fock model

   

 

kin Fock

kin 4 2

2

Fock 4 4 2

3

( ) 2

( ) ( : bag constant)

3 2 1 log

8

3 log

8 2

( ) / , 1

f f

f

f f f f f f f

S

f f f f f f f

f

f F f f f f

B B

m x x x

m x x x

x p m x

   

  



   



 

 

   

 

     

   

       

  



Quark phase

MIT bag model

Hadron-Quark mixed-phase structure and EOS (T=0)

• Assume regularity in structure: divide whole space into equivalent neutral cells with a geometrical symmetry (3D: sphere, 2D : cylinder, 1D: plate).  Wigner- Seitz approx.

• Give a sharp boundary between H and Q phases and a geometry (Unif/Dropl/...) .

• Solve the field equations numerically and get density profiles. Optimize the cell size and H-Q boundary position (choose the energy-minimum).

• Choose energy-min geometry (Unif H, droplet, rod, slab, tube, bubble, Unif Q).

28

EOS of matter

Full calculation is

close to the Maxwell construction (local charge neutral).

Far from the bulk Gibbs calculation

(neglects the surface and Coulomb).

29

Mass-radius relation of a cold neutron star Full calculation with

pasta structures yields similar result to the Maxwell construction.

Maximum masses are almost the same for 3 cases.

We need to improve largely the quark EOS or hadron EOS to get

~2𝑀

_⨀

_surf=40

 RMF + Thomas Fermi calculation for low-density nuclear matter

 “Pasta” structure appears and affects on the EOS.

 For neutrino-trapped matter, pasta structures are enhanced by neutrinos

 The EOS depends largely on the structure and the existence of neutrinos.

 Fully 3D calculation （ Okamoto et al ）is developing.

 Hadron-quark mixed phase is important for structure and mass of compact stars.

 We have developed a method to calculate EOS of mixed phase. But each EOS should be improved to sustain neutron stars

Summary

Thank you for your attention!

END

32

Dependence of E/A on R.

Strong surface tension and weak Coulomb large R

extreme case

 no minimum.

(pasta unstable)

[Voskresensky et al, PLB541(2002)93;

NPA723(2003)291;

“amorphous” PRD(2012)]

Finite-size effects

33

 

 

   

   

3 1

2

2 0

4 2

1 1 1

4

, 0

r

dP Gm r P P Gm

dr r m r

P P r

m m r s s ds

M m R R R

 





 

 



   

        





 

  



Density at position 𝑟

mass inside the position 𝑟 total mass and radius.

Pressure (input of TOV eq.)‏

TOV equation

Structure of compact stars

Bulk Gbbs Full calc _surf=40 MeV/fm² Maxwell const.

Density profile of a compact star (𝑀 = 1.4 𝑀_⨀)‏

34

Hadron-Quark mixed-phase structure and EOS (T=0)

• Assume regularity in structure: divide whole space into equivalent and neutral cells with a geometrical symmetry (3D: sphere, 2D : cylinder, 1D: plate).

 Wigner-Seitz cell approx.

• Divide a cell into hadron phase and quark phases.

• Give a geometry (Unif/Dropl/Rod/...) and a baryon density _B.

• Solve the field equations numerically. Optimize the cell size and H-Q boundary position (choose the energy-minimum).

• Choose an energy-minimum geometry among 7 cases (Unif H, droplet, rod, slab, tube, bubble, Unif Q).

WS-cell

35

2-3 _でハイペロン混合。  EOS軟化  中性子星最大質量 < 1.4M_sol

 観測値 >1.5 M_{sol と}矛盾。

Schulze et al, PRC73 (2006) 058801

しかし 混合相の存在で再び軟化

する可能性。

バルク（構造を無視）なGibbs計算 では広い範囲で混合相。

[Glendenning, PRD46,1274].

クォーク物質への相転移で最大質量 の問題が解決(?) [Maieron et al, PRD70 (2004) 043010 etc].

核子のみ

}含ハイペロン

36

Beta equil.

Beta-equilibrium case

Only‏“droplet”‏structure‏appears.

The change of EOS due to the non-uniform structure is small.

T=0

37

phase

coexistence mechanical

instability

formation‏of‏“pasta”

Instability of uniform matter at finite T

Due to the surface tension and the Coulomb interaction, the region of inhomogeneous matter is limited ( “pasta” < coexistence ).

But Mechanical instability is not crucial for pasta formation!

EOS has a similar behavior to that of the conventional studies.

Novelty:

fcc lattice of droplets can be the ground state at some density.

 Not the Coulomb interaction among “point particles” but the change of the droplet size is

relevant.

Y_p = Z/A = 0.5

39

Kaonic pasta structure

40

ハドロン相、クォーク相それ ぞれでの荷電中性の条件が 無いためにハイペロンが抑制 される。

(バルクGibbs計算よりは局所中性

に近い _が、厳密には中性でない。)‏

中性物質  _th=0.34 fm^

電子や中性子を減らすためには ハイペロンが出た方が得。

荷電物質  _th=1.15 fm^

重いハイペロンは出なくても良い。

ハイペロン抑制機構

Neutral matter

Charged matter

混合相は負荷電のクォーク相と 正荷電のハドロン相とからなる ためハイペロンが抑制される。