Dynamics of superfluid-superconducting neutron stars - PowerPoint Presentation

Slide1

Dynamics of superfluid-superconducting neutron stars

Mikhail E. Gusakov, Vasiliy A. Dommes

Ioffe

Institute

Saint-Petersburg, RussiaSlide2

Introduction

It is generally accepted that baryons (neutrons and protons) in the internal layers of neutron stars undergo transition into superfluid/superconducting state at .

Thus, to study dynamics of neutron stars at sufficiently low temperatures one has to develop a system of equations describing superfluid-superconducting mixtures.

Generally, such mixture can be magnetized, relativistic, and can contain both neutron (Feynman-Onsager) and proton (

Abrikosov

) vortices

. Slide3

Introduction

Dynamics of superfluid-superconducting mixtures has been studied, both in the non-relativistic (e.g., Vardanyan

& Sedrakyan’81; Holm & Kupershmidt’87; Mendell

& Lindblom’91; Mendell’91; Sedrakyan

& Sedrakyan’95;

Glampedakis

,

Andersson

& Samuelsson’11

) and in the relativistic framework (Lebedev & Khalatnikov’81; Carter & Langlois’95; Carter & Langlois’98; Langlois, Sedrakyan & Carter’98;Kantor & Gusakov’11; Dommes & Gusakov’15; Andersson, Wells & Vickers’16). “State of the art” paper:Glampedakis, Andersson & Samuelsson’11 (GAS11)essentially nonrelativistic formulationapproximation of vanishing temperaturesuperfluid-superconducting mixture; type-II proton superconductivity

vortices; mutual friction; correct treatment of the magnetic field ( );Slide4

Introduction

So, initially, our aim was to extend the results of GAS11 to relativistic framework and to include into consideration the finite-temperature effects.

Eventually, the equations that we derived turn out to be more general than those of GAS11 (even in the non-relativistic limit)

We have also found that our equations

differ

from MHD of GAS11Slide5

Gusakov M.E. , PRD (2016)“Relativistic formulation of the Hall-Vinen-Bekarevich

-Khalatnikov

superfluid hydrodynamics”

Gusakov

M.E.,

Dommes

V.A. ,

arXiv

: 1607.01629 (submitted to PRD)“Relativistic dynamics of superfluid-superconducting mixtures in the presence of topological defects and the electromagnetic field, with application to neutron stars” Dommes V.A., Gusakov M.E. (in preparation)“Vortex buoyancy in superfluid and superconducting neutron stars”IntroductionAll these results will be discussed in my talk, which is based on the following works Slide6

The result:

Particle and energy-momentum conservation:

Second law of thermodynamics:Slide7

The result:

“Superfluid” equations for neutrons and protons:

vorticity tensor

Maxwell’s equations in the medium:Slide8

Idea of derivation

Initial idea [Bekarevich

& Khalatnikov’61]: consistency between conservation laws and entropy equation.

Then it is possible to constrain the system energy-momentum tensor from the requirement that

the

entropy

is

not produced in the system

(which means that the entropy density is subject to continuity equation) Consider a system in the absence of dissipation Assume that we know the form of the expressions for particle current densities as well as the form of the second law of thermodynamics

entropy density

four-velocity of normal excitationsSlide9

That is, by specifying, for example, vortex contribution

one finds the correction to the energy-momentum tensorSlide10

What physics is included (brief account)?

npe-composition (additional particle species can be easily included)

fully relativistic formulation

neutrons are superfluid, protons are superconducting

entrainment and finite temperature effects

both types (I and II) of proton superconductivity

electromagnetic effects

neutron and proton vortices

(or magnetic domains for type-I proton SP)

dissipation (e.g., mutual friction)Slide11

In what follows I will discuss some of these

physical “ingredients” in more detailSlide12

Importance of finite-temperature effects

superfluid density is a strong

function of temperature!

Zero-temperature approximation is justified only if

everywhere in the star.

In many interesting situations (e.g., in

magnetars

, LMXBs) this is not the case.

Note that the condition

does

not

justify the use of the zero-temperature hydrodynamics.Slide13

What physics is included: Type I/II proton superconductivity

Credit

:

Glampedakis

et al.’11

Coherence length:

London penetration depth:

=>

I type

=>

II typeSlide14

What is the difference between neutron star interiors

with type-I and type-II superconductors? Transition to superconducting state occurs at constant magnetic flux

(Baym et al. 1969

; typical cooling timescale is much shorter than the magnetic flux expulsion timescale)

Intermediate state of type-I superconductor

:

consists of alternating domains of superconducting (field-free ) regions and normal regions hosting magnetic field

Under these conditions type-I superconductor undergoes transition into an “intermediate” state, while type-II superconductors – into mixed state.

Mixed state of type-II

superconductor

: consists of Abrikosov vortices (fluxtubes)Slide15

Intermediate

vs mixed state

normal regions are dark

Huebener

’

00

Typical “open topology ”

intermediate state domain

structure

Distance between neighboring flux tubes:Flux tube radius:

Number of flux quanta in a flux tube:

Hess et al’89

Mixed state:

Abrikosov

vortices

Distance between neighboring vortices:

“Vortex radius”:

Number of flux quanta in a vortex:

(Huebener’13, Sedrakian’05, DeGennes’66)Slide16

What physics is included: vortices

neutron vortices

proton vortices

Total number of vortices

Vortex density =

Intervortex

spacing

P

is the neutron star period in seconds.

Neutron vortices appear in neutron stars in order to imitate solid-body rotation with a non-superfluid component.

Magnetic flux

Magnetic flux

Vortex density=

Total number of vortices

Intervortex

spacing

(assuming proton SP of type-II)Slide17

The suggested dynamic equations naturally account for:

vortex energy per unit

length divided by

curvature radius

R

Both neutron and proton vortex energies

Mutual friction (

as well as Magnus force etc

.)

Vortex tension (

appears when vortex is bent

)

Vortex buoyancySlide18

Vortex buoyancy in more detail

acts to push a vortex out into the region

with

smaller

superfluid density

usually it is either ignored (as in the Hall-

Vinen

hydrodynamics) or

introduced “by hands” in the form

(

e.g.,

Muslimov

& Tsygan’85,

Elfritz

et al.’16, …

)

which is popular in studies of the magnetic flux expulsion.

The latter expression reduces to the correct one

only

for a one-component liquid

at zero temperature

.

gravitation acceleration

speed of sound

It should be noted that the correct buoyancy force is con-

tained

implicitly in the

Bekarevich

&

Khalatnikov

superfluid

hydrodynamics and its

multifluid

extensions.Slide19

SP-SFL mixture as a medium with and

Maxwell’s equations in the medium:

magnetic induction

magnetic field

electric field

electric displacement

The next interesting feature of the dynamic equations that we propose is that they consider a superfluid-superconducting mixture as a medium in which and .

Thus they are coupled with the standard Maxwell’s equations

in the medium

.Slide20

Why ?

These “molecular” currents contribute to magnetization (magnetic moment of the unit volume)

Each vortex has a magnetic field supported by superconducting currents

It is straightforward to show:

(textbook result)

vortex magnetic flux

areal vortex density

Carter, Prix, Langlois’00;

Glampedakis

et al.’11

SP-SFL mixture as a medium with and

Short answer: Because

andSlide21

Using , one can calculate the electric polarization vector (or the electric dipole moment of a unit volume) and find:

Why ?

Each moving

vortex induces an electric field

and electric charge:

SP-SFL mixture as a medium with and

Gusakov

& Dommes’16

(textbook result)

Short answer: Because

andSlide22

EM + vortex energy density

By specifying the energy density we specify the energy-momentum tensor

What is the contribution to the system energy density from the electromagnetic field and vortices?Slide23

EM + vortex energy density

By specifying the energy density we specify the energy-momentum tensor

What is the contribution to the system energy density from the electromagnetic field and vortices?

“electromagnetic” contribution

has a standard form

in the

comoving

frame ,

moving with the normal liquid component

depends on the four-vectors which reduce to Slide24

EM + vortex energy density

By specifying the energy density we specify the energy-momentum tensor

What is the contribution to the system energy density from the electromagnetic field and vortices?

“electromagnetic” contribution

Generally, one can say that it depends on two tensors:

electromagnetic tensor

complementary tensorSlide25

EM + vortex energy density

By specifying the energy density we specify the energy-momentum tensor

What is the contribution to the system energy density from the

electromagnetic field and vortices?

“vortex” contribution

Depends on the vorticity tensor

Depends on a complementary tensor

superfluid velocity

which is related to the density of vortices

(non-relativistic analogue: )Slide26

EM + vortex energy-momentum tensor

Electromagnetic and vortex contributions to the second law of thermodynamics

induce corrections to the energy-momentum tensor

electromagnetic

correction

vortex

correction

Related to Abraham tensor

of ordinary electrodynamics Slide27

“Closing” the system of equations

We have found that the second law of thermodynamics and energy-momentum

tensor depend on the electromagnetic and vorticity tensors , as well as

on the complementary tensors

To close the system of equations we need to express the tensors

through . The relation between these tensors will

depend on a detailed

microphysics model of a mixture.

This is in full analogy with the ordinary electrodynamics where, in order to close

the system one needs to specify the relation between the tensors and

e.g.,

andSlide28

“Closing” the system of equations

In the next slides we will discuss the simplified dynamic equations in the

so called “MHD” approximation. In that case the complementary tensors

can be expressed as:

energy of neutron vortices is neglectedSlide29

“MHD” approximation

In neutron stars:

magnetic field stored in proton vortices

Assumptions:

protons form type-II superconductor

vortex interactions are neglected

diffusion of normal thermal excitations is suppressed

This allows one to simplify substantially general equations describing

superfluid-superconducting mixtureSlide30

In the absence of entrainment this means that

protons approximately co-move with electrons.1. One can discard the Maxwell’s equations:

and set to zero the four-current density of free charges in other equations:

“MHD” approximationSlide31

2. The electromagnetic + vortex contribution to the second law of

thermodynamics simplifies

“MHD” approximation

(neglect contribution from neutron vortices)

which is simply the statement:

vortex energy per unit length

vortex areal density

, because

NOTE

: This expression for

EM+vortex

energy density

corresponds to the following choice of complementary tensorsSlide32

Full system of MHD equations

Second law of thermodynamics:

Particle and energy-momentum conservation:

“Superfluid” equations for neutrons and protons:

Electromagnetic sectorSlide33

Evolution equation for the magnetic field

The MHD approximation discussed above allows one to obtain

a simple nonrelativistic evolution equation for the magnetic field

(see also

Konenkov

& Geppert’01

):

magnetic field transport

by vortices

velocity of proton vortices

drag coefficient

normal velocity

correct at

vanishes in the absence

of vortex tension and buoyancySlide34

Evolution equation for the magnetic field

This equation

differs

from the similar equation derived in

Glampedakis

et al.’11

,

Graber et al.’15

under the same assumptions:

Magnetic field here is not transported with the velocity of vortices

(although it is the magnetic field of flux tubes) – puzzling result.

In the

weak-drag limit

, , magnetic field is transported with velocity:

Our result:

Glampedakis

et al.’11, Graber et al.’15:Slide35

This result is easy to understand; it follows from the balance of forces

acting upon vortex in the weak-drag regime:

Our result:

The second term vanishes only if: Slide36

Conclusions and some comments

A set of fully relativistic finite-temperature equations is derived for superfluid-superconducting

npe

-mixture.

Neutron and proton vortices, both types of proton SP and various dissipative corrections are allowed

for;

buoyancy force (

i

) is contained in our equations (no need to introduce it “by hands”); (ii) differ from the “standard” usually used expression.

In comparison to MHD of Glampedakis et al’11 we: (i) take into account the relativistic and finite-temperature effects; (ii) provide a general framework allowing one to incorporate new physics into the existing dynamic equations (relation between and ); (iii) demonstrate that the displacement field is not equal to the electric field; and (iv) obtain a different evolution equation for the magnetic field in the MHD limit.Our equations does not reduce to those of GAS11 in the nonrelativistic limitSlide37

The proposed dynamic equations can be used, e.g., to study evolution of the NS magnetic field.

However, for sufficiently hot neutron stars, for which the effects of particle diffusion (more precisely, diffusion of thermal excitations) may become important.

These effects are ignored in the proposed MHD

.

Now we work to take them into account properly.

(

Dommes

& Gusakov’’16, in preparation

).

More details:

Gusakov

M.E. ,

PRD

(2016) Gusakov M.E., Dommes

V.A. , arXiv: 1607.01629 (submitted to PRD)Slide38

Magnetic field evolution equation will remain formally unchanged

But vortex velocity

will be different:

diffusion-induced term

Preliminary result:

Magnetic field evolution equation in the presence of diffusion

Dommes

& Gusakov’’16, in preparationSlide39
Slide40

Diffusion

Normal particles (electrons as well as neutron and proton thermal excitations) may move with different velocities.

No diffusion:

With diffusion:

kinetic coefficients

Dommes

& Gusakov’16, in preparationSlide41

Effect of diffusion

The four-current density of particle charges is still zero but now it has

an additional “diffusion” contribution:

Protons do not co-move with electrons anymore (even neglecting entrainment)

Magnetic field evolution equation will remain formally unchanged

But vortex velocity

will be different:

diffusion-induced termSlide42

diffusion terms

Zero-temperature equations with diffusion which can be found in the literature:

Diffusion terms cannot appear in the equation

describing superfluid (superconducting) part of the liquid!

(e.g., in the absence of rotation such terms would

violate the potentiality condition for neutrons)

Glampedakis

, Jones &Samuelsson (2011)

Passamonti

, Akgün, Pons & Miralles (2016)Slide43

Type-I and type-II superconductors

I type

Problem

: superconductor in an external field

II type

normal state

normal state

entrance of

Abrikosov

vortices

surface current

,

screeningSlide44

Arbitrary

antisymmetric

tensor:

(written in the

comoving

frame)

Dual tensor:

“electric” vector

“magnetic” vector Slide45

One-component neutral superfluid

; vanishing temperature, ;vortices are allowed for; no dissipation; vortex energy density is taken into account; the resulting equations of Carter & Langlois’95 slightly differ from those of Lebedev & Khalatnikov’81 (the reason is left unexplained)

Comparison with previous works:

Relativistic dynamics

Lebedev

& Khalatnikov’81; Carter & Langlois’95

Non-relativistic limit of their equations does not reproduce that of HVBK-hydrodynamics!Slide46

Comparison with previous works:

Non-relativistic dynamicsVardanyan

& Sedrakyan’81; Holm & Kupershmidt’87;

Mendell & Lindblom’91; Mendell’91;

Basic work:

Mendell

& Lindblom’91

superfluid-superconducting mixtures;

finite-temperature effects,

mutual friction; electromagnetism “State of the art”:Glampedakis, Andersson & Samuelsson’11superfluid-superconducting mixture; type-II proton superconductivityvortices; mutual friction; correct treatment of the magnetic field ( );approximation of vanishing temperature (although the earlier formulations of Mendell & Lindblom’91 and

Sedrakyan & Sedrakyan’95 allow for finite temperature effects).

Sedrakyan

& Sedrakyan’95;

Glampedakis

,

Andersson

& Samuelsson’11;

many othersSlide47

V. Intermediate state

Intermediate state appears if protons are type-I superconductor

Closed topology assumption: Normal domains are completely surrounded by the superconducting phase

normal regions are dark

“closed”

topology

Not unreasonable, since the magnetic field of a typical neutron star, , is much smaller than the critical thermodynamic field, , while it is well known (e.g.,

Huebener’13

) that it is advantageous for a relatively weak field to penetrate the superconductor in the form of flux tubes, each containing many flux quanta.

Strong-drag regime assumption:

Normal domains move with

the normal matterSlide48

Mixture?n and p vortices?Vortex energy?Dissipation?Nonrel. limit OK?

Lebedev & Khalat-nikov’81

Carter & Langlois’95

Carter & Langlois’98

not checked

Langlois

,

Sedrakyan

& Carter’98not checkedKantor & Gusakov’11Dommes & Gusakov’15

Andersson, Wells & Vickers’16Gusakov & Dommes’16Slide49

What physics is included: Temperature effects

From hydrodynamic point of view superfluidity leads to the presence of a few independent velocity fields. In the most simple case of one superfluid particle species these are the velocity of normal excitations and superfluid velocity (i.e., momentum of a Cooper pair divided by the particle mass m

) .

Then, for example, the particle current density is:

superfluid density

normal density

superfluid density is a strong

function of temperature!Slide50

Conclusion:

Zero-temperature approximation is justified only if

everywhere in the star.

In many interesting situations (e.g., in

magnetars

, LMXBs) this is not the case.

Note that the condition

does

not

justify the use of the zero-temperature hydrodynamics.

What physics is included: Temperature effectsSlide51

What physics is included: Entrainment

In the superfluid mixture superfluid flow of one particle species (e.g., neutrons) may contribute to the mass flow of another particle species (e.g., protons).

Relativistic analogue:

entrainment

matrix

particle 4-current

number density

4-velocity of normal excitations

relativistic

entrainment

matrix

roughly a difference between

the superfluid and normal velocitiesSlide52

For neutron star conditions is very small, so that almost all

magnetic induction is due to vortices, .

rotation induces a weak magnetic field (the so called “London field”)

rotation frequency

SP-SFL mixture as a medium with and

Why is small? Because it is generated by rotational effects.

Consider, for example a uniformly rotating vortex-free superconductor.

Then, as follows from the London equationSlide53

EM + vortex energy density

By specifying the energy density we specify the energy-momentum tensor

What is the contribution to the system energy density from the electromagnetic field and vortices?

“electromagnetic” contribution

has a standard form

in the

comoving

frame ,

moving with the normal liquid component

depends on the four-vectors which reduce to