FORMULATIONS AND A NUMERICAL METHOD FOR MAGNETIZED...

FORMULATIONS AND A NUMERICALMETHOD FOR MAGNETIZED NEUTRON STARS

Koji Uryu

Department of Physics

University of the Ryukyus

Talk at the University of Tokyo – Komaba 5/Nov/2009

PLAN FOR A TALK

Introduction

Formulations for magnetized compact objects in

equilibriums.

A brief introduction to COCAL code.

Recent results of computations for compact

objects in quasi-equilibriums.

Triaxially deformed rapidly rotating neutron star.

BNS solutions computed from the waveless formulation.

R 10km

M1.35Msol

NEUTRON STAR VS OKINAWA

Sugimura (2008)

STRONG MAGNETIC FIELD IN COMPACT OBJECTS

Comparing the energy of strong magnetic field (B-field) to

that of gravitation in NS,

M

B-field may change the structure of NS for M /|W|～0.01.

Then, for NS

Magnetar

Observation – anomalous X-ray pulsar, soft g-ray

repeater may contain NS with B » 1014 – 1015 G at the

NS surface.

Toroidal B-field of NS interior may be much stronger.


Proto-Neutron star

B-field of the collapsing massive core is amplified by magnetic

winding (toroidal), and magneto-rotational instability (poloidal).

e.g. NR simulation for a core collapse shows M /T～0.1.

Shibata, Liu, Shapiro, Stephens ,PRD (2006)

Different resolutions Different initial models

(Intial radius

～2000km )


Binary neutron stars – inspiral to merger How strong is the B-field of the inspiraling BNS?

The poloidal B-field (seen as the surface B-field) may decay but the toroidal field may stay strong??

Recall, even for B ～1014 G, M /|W|～10–6.

Motivations

B-field may be amplified ～10 times by the magnetic winding and MRI in a post-merger pre-collapse object.

An important candidate for a source of short g-ray burst.

Initial data calculated from a consistent system of equations is desirable for merger simulations.

To be honest, we also have some academic interests – why not!

MAGNETIZED COMPACT OBJECTS IN

(QUASI-) EQUILIBRIUMS

Our project is to develop a method for computing compact objects with strong magnetic field in (quasi-) equilibriums.

Magnetar model

Quasi-equilibrium initial data of magnetized BNS for realistic merger simulations.

For a successful numerical computation, we want to derive a formulation which include

sufficient conditions for hydrostationary equilibriums with magnetic field, (integrability conditions for the relativistic MHD-Euler equation),

an initial value formulation for Einstein’s equation,

(waveless BNS initial data : Uryu, Limousin, Gourgoulhon, Friedman, Shibata PRL 2006, in preparation 2009),

a formulation for Maxwell’s equation, (one analogous to Einstein’s equation may work).

A MAGNETIZED ROTATING NEUTRON STARIN COLLABORATION WITH FUJISAWA, ERIGUCHI (TOKYO)

Theory for stationary and axi-symmetirc spacetime with electro-magnetic fields and perfect fluids.

– Long history including several articles by Brandon.

NS with poloidal magnetic fields– Silvano and a company at Meudon (1993, 1995,1996). Unique result for a

rapidly rotating star with strong magnetic fields including Newtonian

gravity for several years.

Recently, rapidly rotating NS with troidal B-field is calculated. Tomimura & Eriguchi (2005) for Newtonian rotating star.

Integrability conditions by Ferraro (1937) are used.

Kiuchi, Yoshida, and a company purely toroidal B-field (2008,2009).

Our formulation is GR version of Yoshida & Eriguchi (2007) –mathematically the same as Beckenstein & Oron, Ioka & Sasaki. Solutions may have both toroidal and poloidal B-fields.

A FORMULATION FOR A MAGNETIZED RNS

Perfect fluid Einstein-Maxwell spacetime with

ideal MHD condition.

Assume time and axi-symmetries.

Integrability conditions for ideal MHD condition

and MHD-Euler equations are derived.

Solutions involve flow in the meridional plane.

In this case, a stationary axisymmetric spacetime is

not circular. The metric components may

not be zero.

Waveless formulation is exact for such a general

stationary axi-symmetric spacetime.

BRIEF LOOK AT THE DERIVATION OF THE

INTEGRABILITY CONDITIONS

Basic equations

Rest mass density conservation

Maxwell’s equations

Current conservation

MHD-Euler equation

Ideal MHD condition

Einstein’s equation – waveless formulation

One-parameter EOS is assumed.

BRIEF LOOK AT THE DERIVATION OF FIRST

INTEGRALS

Assuming time- and axi-symmetries and one-parameter EOS, ideal MHD condition and MHD-Euler equation have a system of integrabilityconditions, which are written in algebraic relations including and 4 arbitrary functions of .

Each component of them has one of following forms; for f either a scalar or scalar density,

(xA-components)

d is the 2D exterior derivative in meridional plane.

BRIEF LOOK AT THE DERIVATION OF FIRST

INTEGRALS

Rest mass conservation

Introduce a flow function in the meridional plane.

Ideal MHD condition – component by component.

t and f components

hence, in a fixed gauge, , , and , are functions of each other. We choose as a variable;

xA components

SUMMARY OF THE INTEGRABILITY

Ideal MHD conditon

MHD-Euler eq.

METHOD OF SOLUTION

Using the relations, current is rewritten

Einstein’s eq., Maxwell’s eq with the above source, are solved for the field variables, and the integrabilityconditions for the fluid variables. These are iteratively solved until solution is converged.

Maxwell’s eq. may be directly solved or solved using Grad-Shafranov eq.

QUASI-EQUILIBRIUM INITIAL DATA FOR

MAGNETIZED BINARY NEUTRON STARS

IN COLLABORATION WITH ERIC GOURGOULHON, HARRIS MARKAKIS (UWM)

New formulation for the helically symmetric

irrotational BNS with magnetic field is derived.

Bekenstein & Betschart (PRD ) derived a Lagrangian

for the ideal MHD introducing several potentials for

the 4-velocity and electric current.

It is found that the current in the ideal MHD can be

written , using a vector potential ,

First integral of the ideal MHD-Euler equation is

derived using this form of the current.

First law for the MHD-BNS is written with .

THERMODYNAMICS OF MAGNETIZED BINARY

BLACK HOLES AND NEUTRON STARS


Helically symmetric perfect-fluid Einstein-Maxwell

spacetime is a model for magnetized binary black holes and

neutron stars.

P

c (P)T

k = t + W f

S

0

T

f

t

y(P)tc (P)t

y-t c (P)tc(t) =

S

a

a

aa a




Same as the non-magnetized binary (Friedman, Uryu, Shibata

PRD2002), thermodynamic laws are derived.

Consider a family of spacetime

Noether charge of the helical Killing field (Wald & Iyer)

Noether charge is associated with the Lagrangian




Horizon is defined as the boundary of the domain of outer

communication of a history of each spacelike sphere.

Exsistence of the global helical symmetry assures that the

horizon is a Killing horizon.

The surface gravity and the electric potential in the rotating

frame are constant on the horizon.

tangent to horizons

Variation of the Noether charge is written (Carter 1973)

CONSERVATION LAWS

We assume rest mass, entropy are conserved.

Ideal MHD implies conservation of the magnetic flux.

Circulation is not conserved in MHD flow in general.

Oron, Oron, Bekenstein & Betschart found a circulation of

conservation law for the magnetized flow when the current

is written

The current is rewritten, using

Substitute this to the MHD-Euler eq.

the Lorenz force becomes

hence

BRIEF LOOK AT THE DERIVATION OF THE FIRST

LAW, AND FIRST INTEGRAL OF MHD-EULER EQ.

First law

First integral of the MHD-Euler equation

Write 4 velocity using the helical vector

MHD-Euler is rewritten

BRIEF LOOK AT THE DERIVATION OF THE

FIRST INTEGRAL

Helically symmetric irrotational BNS with B-field.

Applying helical symmetry,

Irrotational ideal MHD flow is defined by

Using a freedom to choose the vector potential qa, we assume and as a function of , then we have the first integral

Stationarity of the current further restricts

, but it does not have to vanish.

METHOD OF SOLUTION

Equations for MHD fluid

The first integral and are solved for h and

The rest mass conservation eq. is written elliptic eq.

for F with Neumann type boundary condition.

Maxwell’s eq. and Einstein eq. are solved using the

same mehtod.

Should check how to impose ideal MHD condition

during the iteration.

The first law for helically symmetric irrotational

BNS with B-field is useful to construct a

sequence of solutions to model a inspiral .

CODING ON COCAL

Cocal (Compact Object CALculator) “Cocal” means a “seagull” in Triestino (Trieste dialect of Italian).

Plan to develop magnetic RNS/BNS codes on the Cocal code.

Cocal code project is aim to develop a minimalistic code for computing compact objects, including A rapidly rotating neutron star

Binary neutron stars

Black hole – neutron star binary

Binary black holes

And all the above with magnetic fields

Cocal – as simple as ever Cocal code is simple, minimalistic, and robust.

No symmetry is assumed for the 3D computational domain.

2nd order finite difference scheme on the spherical grids is used.

Multipole expansion of Green’s function is used in the Poisson solver.

A set of subroutines written in Fortran 90 are reasonably structured.

COCAL CODE STATUS

Rotating neutron star code is working for

IWM formulation

Waveless/Near-zone helically symmetric formulations

Parametrized EOS (piecewise polytropic EOS) is implemented.

Also several tools are available including,

1D spherical NS solver

TOV solver – computation for the model parameters.

NS solver in isotropic coordinates – for the initial guess.

Subroutines for the mass, the angular momentum and the other physical quantities.

Quadrupole formula to estimate GW amplitude.

COCAL GRID FOR

BINARY SYSTEM – EXAMPLE

Case for BH–NS binary Two spherical domains for each

compact object are extended to

asymptotics (or larger radii).

In each domain, a spherical region

surrounding a companion is excised.

A contribution from the companion is

calculated through the surface integral.

Surface fitted coordinate is used for NS.

Excision or puncture is used for BH.

Green’s formula

NS

BH

PARAMETRIZED EQUATIONS OF STATE(PEOS)

fitting

Read et.al. PRD(2009)

In each interval, polytropic EOS is assumed pi = KiriGi

3 intervals above nuclear density with 4 parameters fit the nuclear EOS table with rms error O(0.1%), and 4.3% for the worst case.

EXAMPLE OF A SOLUTION FROM COCAL CODE:

RELATIVISTIC JACOBI ELLIPSOID

Huang et.al.

PRD (2008)

Triaxially

deformed

rapidly

rotating NS

G = 4.33

M/R = 0.2

SOLUTION SEQUENCE OF RELATIVISTIC

JACOBI ELLIPSOID

Triaxially deformed solutions exsist for Γ > 2.24

for Newtonian polytrope, (Γ = 1+1/n)

In GR(IWM formulation), triaxial solution

disappear for larger M/R.

Huang, Markakis, Sugiyama, Uryu, PRD (2008)

GW FROM A TRIAXIALLY DEFORMED

NEUTRON STAR

Type II SN may produce

rapidly rotating triaxially

deformed proto-NS.

GW amplitude is estimated

using quadrupole formula,

and GW signal is integrated

for 30sec.

Type II SN rate

in the local group »1×10–4 /yr Mpc3

» 10/yr for 30Mpc

LIGO hasn’t found it. Not all of SNII produce

proto-NS. Not all of

proto-NS rotate rapid enough.

Gravitational wave form may not be modeled well.

Effective Γ of high density EOS may not be large enough to have triaxial solutions.

RECENT RESULTS FOR BNS COMPUTED FROM

WL/NHS FORMULATION

An orbit of BNS initial data is not exactly a circular orbit (nor an inspiral orbit without an ellipticity), because Radial velocity (radiation reaction) is neglected.

Spacelike hypersurface St is assumed to be conformally flat.

In WL/NHS formulations, all components of Einstein’s eq. are solved including non-conformal flat part of the metric.

(Formulation : Bonazzola et.al. (2003), Schafer et.al.(2003), Shibata et.al.(2004))

With maximal slicing K = 0, Dirac gauge , and a condition for conformal decomposition.


WL/NHS FORMULATION

Waveless condition is applied to a time derivative term of the conformal metric.

For the other time derivatives, the helical symmetry (stationarity in a rotating frame) is applied, e.g.,

Spin of NS is likely to be negligible, and therefore an irrotational flow field is assumed for the neutron stars.

(Kochanek (1992), Bildsten & Cutler (1992))

A sequence of solution is calculated assuming the rest mass, the entropy (T=0), and the circulation are fixed constant. The sequence models the final several orbits of an inspiral.


WL/NHS FORMULATION

When the non-conformal flat part of the metric is

solved from , Dirac gauge and a

condition are not satisfied

accurately.

A gauge transformation

and a correction are made at each

iteration cycle to enforce these conditions.

A gauge vector is solved from

or more concretely


WL/NHS FORMULATION

Comparison with

2PN asymptoricsolution in the same gauge.

(Asada et.al.1996)

Left : (hxx-hyy)/2

Right : hzz

M/R = 0.172

d/R0 = 1.5


WL/NHS FORMULATION

Selected components of hij for the different compactnessM/R = 0.131, 0.172, 0.205.

RECENT RESULTS FOR

BNS COMPUTED FROM

WL/NHS FORMULATION

Constant rest mass

sequences for different

M/R = 0.131, 0.172, 0.205.

Effacing of tidal effect

for increasing M/R is

much smaller in WL/NHS

sequences than in IWM.

Differences in the binding

energy between WL/NHS and

IWM amounts to a half orbit of

overestimation in IWM.


WL/NHS FORMULATION

Is the circularity of the orbits

really improved in WL/NHS

sequences?

For two point-particle dynamics

in Post-Newtonian gravity,

a quantity Eb – WJ becomes

minimum for the circular orbits.

It is expected that this may also

true for the extended body.

SUMMARY

Computations of magnetized compact object in (quasi) equilibriums is an useful tool for various problems.

Magnetar model.

Proto-NS model.

Initial data for numerical relativity simulations of various kinds – BNS merger, BH-NS merger, dynamics of rapidly rotating NS.

Modeling for the gravitational waveforms of various sources.

Formulation of the problem and coding are in progress.

Many problems in relativistic astrophysics are left to be solved by Cocal code.

FORMULATIONS AND A NUMERICAL METHOD FOR MAGNETIZED...

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