Kicks of magnetized strange quarks stars induced by anisotropic - - PowerPoint PPT Presentation

Kicks of magnetized strange quarks stars induced by anisotropic emision of neutrinos

Daryel Manreza Paret, ICN-UNAM dmanreza@gmail.com Collaborators Alejandro Ayala, ICN-UNAM México,

• A. Perez Martinez, ICIMAF La Habana Cuba,
• G. Piccinelli, FES Aragón-UNAM, México,
• A. Sanchez, Facultad de Ciencias UNAM México,
• J. Salvador Ruiz Montaño, Universidad Autónoma de Sinaloa, México.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 1 / 22

Introduction

Compact Objects Figure: Artist’s illustration of an isolated

neutron star. Author: Casey Reed - Penn State University

Neutron Stars M ∼ 1.4M⊙ R ∼ 12 km ρ ∼ 1014 g/cm3 B ∼ 1012 − 1015 G

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 2 / 22

Introduction

Compact Objects

Hydrogen/He atmosphere

R ~ 10 km n,p,e, µ neutron star with pion condensate quark−hybrid star hyperon star g/cm 3 10 11 g/cm 3 10 6 g/cm 3 10 14 Fe

− π

K−

s u e r c n d c t g p

• n

i u

p r

• t
• n

s

color−superconducting strange quark matter (u,d,s quarks)

CFL−K+ CFL−K0 CFL−

π

n , p , e , µ

quarks u,d,s

2SC CSL gCFL LOFF crust N+e H traditional neutron star strange star N+e+n Σ , Λ , Ξ , ∆ n superfluid nucleon star

CFL

CFL

2SC †F. Weber. doi:10.1016/j.ppnp.2004.07.001.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 3 / 22

Introduction

Observational evidences for NSs kicks

Pulsar kicks refers to peculiar translational velocities observed on pulsars with respect to surrounding stars and with respect to their progenitors. Kicks can be natal or post-natal: a natal kick is imparted to the NS at birth while post-natal kicks is due to some inner process of the pulsar. Hobbs et al.† have studied the data from the proper motion of 233 pulsars, obtaining velocities as high as 1000 km s−1 and that the mean velocity of young pulsar is 400 km s−1.

†Hobbs, G., Lorimer, D., Lyne, A., & Kramer, M. 2005, Mon. Not. R. Astron. Soc., 360, 974.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 4 / 22

Introduction

Models to describe NSs kicks

Some mechanism to explain the kicks are:

1

Hydrodynamically Driven Kicks: This mechanism explain a natal kick during the core collapse and supernova explosion due to hydrodynamical perturbations that could lead to asymmetric matter ejection.

2

Electromagnetic rocket effect: Electromagnetic radiation from an

• ff-centered rotating magnetic dipole imparts a kick to the pulsar along its

spin axis.

3

Neutrino–Magnetic Field Driven Kicks: Asymmetric neutrino emission induced by strong magnetic fields.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 5 / 22

Introduction

Models to describe NSs kicks

Some mechanism to explain the kicks are:

1

Hydrodynamically Driven Kicks: This mechanism explain a natal kick during the core collapse and supernova explosion due to hydrodynamical perturbations that could lead to asymmetric matter ejection.

2

Electromagnetic rocket effect: Electromagnetic radiation from an

• ff-centered rotating magnetic dipole imparts a kick to the pulsar along its

spin axis.

3

Neutrino–Magnetic Field Driven Kicks: Asymmetric neutrino emission induced by strong magnetic fields.

Neutrino emissivity from the process d → u + e + ¯ νe, u + e → d + νe,

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 5 / 22

Introduction

Models to describe NSs kicks

Some mechanism to explain the kicks are:

1

Hydrodynamically Driven Kicks: This mechanism explain a natal kick during the core collapse and supernova explosion due to hydrodynamical perturbations that could lead to asymmetric matter ejection.

2

Electromagnetic rocket effect: Electromagnetic radiation from an

• ff-centered rotating magnetic dipole imparts a kick to the pulsar along its

spin axis.

3

Neutrino–Magnetic Field Driven Kicks: Asymmetric neutrino emission induced by strong magnetic fields.

Neutrino emissivity from the process d → u + e + ¯ νe, u + e → d + νe, The polarisation of the electron spin will fix the neutrino emission in one direction along the magnetic field

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 5 / 22

Introduction

Models to describe NSs kicks

Some mechanism to explain the kicks are:

1

Hydrodynamically Driven Kicks: This mechanism explain a natal kick during the core collapse and supernova explosion due to hydrodynamical perturbations that could lead to asymmetric matter ejection.

2

Electromagnetic rocket effect: Electromagnetic radiation from an

• ff-centered rotating magnetic dipole imparts a kick to the pulsar along its

spin axis.

3

Neutrino–Magnetic Field Driven Kicks: Asymmetric neutrino emission induced by strong magnetic fields.

Neutrino emissivity from the process d → u + e + ¯ νe, u + e → d + νe, The polarisation of the electron spin will fix the neutrino emission in one direction along the magnetic field The neutrinos work as a propulsion mechanism for the neutron star

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 5 / 22

Pulsar kick velocity

The idea that neutrinos can be the cause for the kick is easily understood with the following estimation: Energy released in the emission of neutrinos ∼ 1053 erg. Kinetic energy of a 1.4M⊙ NS moving at 1000 km s−1 ∼ 1049 erg. The momentum of neutrinos (pν) and the NS (pNS) are pν = Eν c ∼ 1043 erg · s cm pNS = MNS · vkick ∼ 2.8 × 1041 erg · s cm ∼ 0.03pν

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 6 / 22

Pulsar kick velocity

To compute the kick velocity of the NS we have to take into account that the acceleration of the NS is due to the luminosity of the fraction of asymmetric emitted neutrinos: dv dt MNS = χL c , L = 4 3πR3ǫ where χ is the electron polarization, L is the neutrino luminosity and ǫ the neutrino emissivity. The cooling equation give a relation between the emissivity and the specific heat CvdT = −ǫdt. In this way we have for the kick velocity dv = − χe MNS 4 3πR3CvdT. We have to compute the electron polarization and the specific heat in a magnetic field!

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 7 / 22

Pulsar kick velocity

Electron polarization in a magnetic field

The energy spectrum of electrons in a magnetic field is quantized by the so called Landau Levels E2

l = m2 e + p2 3 + 2leB,

and the number density reads ne = dem3

e

2π2 B Be

c ∞

• l=0

(2 − δl0) ∞ dp3 1 e(El−µe)/T + 1, where l = ν + 1

2 + s, are the Landau level quantum numbers,

ν = 0, 1, 2, . . . , s = ±1/2 and Be

c = m2 e/e = 4.41 × 1013 G.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 8 / 22

Pulsar kick velocity

Electron polarization in a magnetic field

The electron spin polarization χ is given by χ = n− − n+ n− + n+ , where n± are the number densities of electrons with spin parallel (s = +1) or anti–parallel (s = −1) to the magnetic field direction respectively, given by n− = dem3

e

2π2 B Be

c ∞

• ν=0

∞ dx3 1 e( me

T

√

x2

3+1+2νB/Be c−xe) + 1

, n+ = dem3

e

2π2 B Be

c ∞

• ν=1

∞ dx3 1 e( me

T

√

x2

3+1+2νB/Be c−xe) + 1

, where xe = µe/me and we have used the relation between l, ν and s, changing the summation over l by the summation over ν (its important to noticed that the change is ∞

l=0(2 − δl0) → s=±1

∞

ν=0).

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 9 / 22

Pulsar kick velocity

Electron polarization in a magnetic field

We can numerically compute the dependence of χ with the parameters B, T, and µ from the following expression χe =        1 + 2

∞

• ν=1

∞ dx3

1 e

( me T

√

x2 3+1+2νB/Be c −xe)+1

∞ dx3

1 e

( me T

√

x2 3+1−xe)+1

      

−1

.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 10 / 22

Pulsar kick velocity

Electron polarization in a magnetic field

10

• 1

10 10

1

10

2

10

3

10

4

10

5

0.0 0.2 0.4 0.6 0.8 1.0

T=0.1 MeV T=1.0 MeV T=10 MeV

χ

B/B

e c

Figure: Polarization of electrons χ as function of the magnetic field and temperature, for a fixed chemical potential xe = 10.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 11 / 22

Pulsar kick velocity

Electron polarization in a magnetic field

0.1 1 10 0.0 0.2 0.4 0.6 0.8 1.0

B/B

e c=10 5

B/B

e c=10 4

B/B

e c=10 3

B/B

e c=10 2

χ

T [MeV]

Figure: Polarization of electrons χ as function of the temperature for several values of the magnetic field and fixed chemical potential xe = 10.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 12 / 22

Pulsar kick velocity

Electron polarization in a magnetic field Figure: Polarization of electrons χ as function of the chemical potential for several values of the magnetic field and fixed temperature.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 13 / 22

Pulsar kick velocity

Specific heat capacity for SQM in a magnetic field

The thermodynamical potential of a magnetized Fermi gas is given by Ωf(B, µ, T) = −efdfB 2π2 ∞ dp3

∞

• l=0

(2 − δl0) 1 β ln

• 1 + e−β(Elf −µf )

, being f the fermion species. From the thermodynamical potential we can compute the specific heat as Cvf = T ∂Sf ∂T , Sf = −∂Ωf ∂T ,

• btaining

Cvf = efdfB 2π2 ∞ dp3

∞

• l=0

(2 − δl0) (Elf − µf)2 2T 2[1 + cosh Elf −µf

T

] .

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 14 / 22

Pulsar kick velocity

Stellar equilibrium equations

As we have seen, χe = χe(B, T, µi) and Cv = Cv(B, T, µi) =

i Cvi,

i = e, u, d, s. Strange quark matter consist of u, d, s quarks and electrons in weak equilibrium d → u + e + ¯ νe, u + e → d + νe, s → u + e + ¯ νe, u + d → u + s. If we impose stellar equilibrium conditions ( beta equilibrium, charge neutrality and barionic number conservation), then we will need to solve the system of equations µu + µe − µd = 0 , µd − µs = 0, 2nu − nd − ns − 3ne = 0, nu + nd + ns − 3nB = 0. Te solution of this system of equations will fix the chemical potentials as a function of temperature and baryon density (nB).

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 15 / 22

Pulsar kick velocity

Stellar equilibrium equations Figure: Chemical potentials of SQM as function of the temperature (B = 105Be

c).

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 16 / 22

Pulsar kick velocity

The pulsar kick velocity can be rewritten in the following form v = −803.925 km s 1.4 M⊙ MNS R 10 km 3 I MeV fm−3

• ,

where I = Tf

Ti

χe(B, T, µi) Cv(B, T, µi)dT, i = e, u, d, s. We can compare our result with the one obtained by I. Sagert and

• J. Schaffner-Bielich†

v ∼ 40 km s 1.4 M⊙ MNS R 10 km 3 µq 400 MeV 2 T0 MeV 2 , where µq = 400 MeV, MNS = 1.4 M⊙, χe = 1 and T0 = 10 MeV.

†I. Sagert and J. Schaffner-Bielich, Astron. Astrophys. Astron. Astrophys. 489, 281 (2008)

doi:10.1051/0004-6361:20078530 [arXiv:0708.2352 [astro-ph]]

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 17 / 22

Pulsar kick velocity

5 10 15 20 10

1

10

2

10

3

B=10

2 B e c

B=10

3 B e c

B=10

4 B e c

B=10

5 B e c

• I. Sagert

v [km/s] R [km]

Figure: Pulsar velocity as a function of the star radius for different values of the magnetic field and a baryon density of nB = 5 n0. We have taken Ti = 10 MeV and Tf = 0.1 MeV.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 18 / 22

Pulsar kick velocity

5 10 10

1

10

2

10

3

nB= 8 n0 nB= 5 n0 nB= 3 n0 B=10

5 B e c

v [km/s] R [km]

Figure: Pulsar velocity as a function of the star radius for different values of the baryon density and a magnetic field of B = 105Be

c.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 19 / 22

Conclusions

1

We have studied pulsar kicks by the emission of neutrinos taking into account the effects of a strong magnetic field.

2

The polarizations of electron was obtained exactly (numerically) as a function of all the parameters (µe, T, B).

3

The specific heat of an electron, u, d, s quark gas was obtained exactly (numerically) as a function of all the parameters (µi, T, B).

4

The velocity of NS was computed for SQM in presence of a magnetic field in stellar equilibrium.

We have obtained kick velocities vkick ∼ 1000 km s−1 for different values of magnetic fields and star radius. We have studied the dependence of the kick velocity with the central densities of the star obtaining that when the central density increases the stars can reach higher velocities for the same value of magnetic fields.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 20 / 22

Future work

1

Analyze the absorption and scattering of neutrinos in quark matter under strong magnetic fields.

2

Incorporate color superconducting phases.

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 21 / 22

MUCHAS GRACIAS

(ISMD2017 Daryel Manreza Paret, ICN/UNAM ) September 14, 2017 22 / 22