Magnetic fields generation in the core of pulsars

Luca BonannoBordeaux, 15/11/2010

Goethe Universität – Frankfurt am Main

Introduction

Introduction to r-modes

The generation of toroidal magnetic fields

Results for LMXBs

Conclusions

R-modes

A rapidly rotating star produces particular fluid motions or currents, called r-modes. R-modes are similar to hurricanes or ocean currents on the Earth. R-modes are very efficient in emitting gravitational waves

Chandrasekar-Friedman-Schutz instability

The r-moves move backwards relative to the rotation of the star.

If the star is sufficiently highly rotating the r-modes move backwards in the corotating frame and forward in the inertial frame.

Jtot I Jc

Gravitational wave emission remove positive angular momentum, increasing Jc and thus the amplitude of the r-modes.

The r-modes are unstable under emission of gravitational waves.

The mode energy increases by gravitational wave emission and decreases due to the viscosity (shear+bulk):

e it t /R-modes evolve as where 1/τ is the imaginary part of the frequency:

1

1

2E

dE

dt

E 1

2 22R 2l 2 r2l 2dr

0

R

1

1

2E

dE

dt

1

2E

dE

dt

GW

1

2E

dE

dt

shear

1

2E

dE

dt

bulk

1

GW

1

shear

1

bulk

1

0

1

GW

1

shear

1

bulk

the mode is suppressed

the mode grows exponentially

R-modes evolution

1

0

1

GW

1

shear

1

bulk

where is the energy of the l-th mode.

˜ x (r,,t) 2

3

r

R

1/2 7

(5!/ )(sin2 2cos2 ) 2(t ')(t ')dt'O( 3)t0

t

At second order in α, a velocity drift in the azimutal direction appear (l=m=2):

v (r,, t) 2

3R

r

R

2

1/2 7(5!/ )(sin2 2cos2 ) 2(t)(t)

Nonlinear motions of fluids elements

L. Rezzolla, F. K. Lamb, D. Markovic, and S. L. Shapiro,Phys. Rev. D64, 104013 (2001).L. Rezzolla, F. K. Lamb, D. Markovic, and S. L. Shapiro,Phys. Rev. D64, 104014 (2001).

If the star has an initial poloidal magnetic field, the azimutal secular motion generates a toroidal magnetic component, winding up the poloidal component.

From the induction equation:

B Br 0

The energy of the r-modes is then converted into magnetic energy

dEM

dt

dE

dt

GW

the r-modes are damped when

Generation of a toroidal magnetic field

t

tdttBtB

0

')'()( 0

A new damping term, due to the toroidal magnetic, field must be included:

1

M

1

E

dEM

dt

4 d k22()sind r4B2(r,)

0

R0

0

2 dr

9 (8.2 10 3)MR4

2(t')(t')dt'4A

9 (8.2 10 3)

Bp2R 2(t ')

0

t (t')dt'

M0

t

brakingMbulkshearj

GWjC

brakingMbulkshearj

GWjC

MbulkshearGW

MR

G

I

M

M

MKKK

dt

d

MR

G

I

MKKK

dt

d

2/1

32

2/1

33

~111

)1(1

2

2~

2

111)1(

11111

accrviscurcaVTCdt

d

2

1

)()()102.8(9

4 23

2

ttM

RAB

dt

d p

M

4 coupled first order differential equations

A≈O(1)

Equations with the magnetic dampingL. Rezzolla, F. K. Lamb, D. Markovic, and S. L. Shapiro,Phys. Rev. D64, 104013 (2001).L. Rezzolla, F. K. Lamb, D. Markovic, and S. L. Shapiro,Phys. Rev. D64, 104014 (2001).

Buoyancy instability: a strong magnetic field reduces the gas pressure and density in it, so that loops of the azimutal field tend to float upward against the direction of gravity.

A toroidal field is generally unstable. After the damping of the r-modes, the toroidal component previously wound up will be unwound again, if any instability sets in.

Relevant MHD instabilities:

Tayler (or pinch type) instability: after a critical value of the toroidal field, it is energetically convenient to produce a new poloidal component, which can be wound up itself, closing the dynamo loop. The star then reaches a stable magnetic configuration where the new poloidal component has a strenght comparable with the toroidal one.The instability sets in when:

B

(4)1/ 2 R

N

1/ 2 R2

1/ 4

Btorcr 1012G for both neutron and quark stars

Btorcr 1017G

for quark starsGBcrtor

1510At T≈108 K

for neutron stars

Stability of the toroidal field

Diffusive processes

tohmic 2 1011 L52

T82

nucl

3

yr

tambip 3109 T82L5

2

B122 yr

tHall 5 108 L52

B12

nucl

yr

H. C. Spruit, Astron. Astrophys. 349, 189 (1999)H. C. Spruit, Astron. Astrophys. 381, 923 (2002)

Results for LMXBs

The instability window goes up!The r-modes are damped before the saturation is reached!

B=108GyrMM /10 8

Neutron stars inside LMXBsC. Cuofano and A. Drago, Phys. Rev. D82, 084027 (2010)

Neutron stars inside LMXBs

•) R-modes can generate strong toroidal fields in the core of accreting millisecond neutron stars.

•) Toroidal field influences the growth rate of r-mode instability.

•) Tayler instabilities sets in for strenghts of the generated fields of the order of 1012 G and stabilizes the toroidal component by producing a new poloidal field of similar strenght. This stable configuration evolves on a time scale regulated by the diffusive processes.

•) The present results imply that in the core of accreting neutron stars in LMXBs, rotating at frequencies larger than 200 Hz, there are strong magnetic fields with strenghts B≥1012 G.

Neutron stars inside LMXBs

10-10

10-9

10-8

10-10

10-9

10-8

10-10

10-9

10-8

Differently from neutron stars, quark stars enter the instability window from the right-hand side.

Differently from neutron stars, quark stars stay in the border of the instability window.

Differently from the case of neutron stars, the instability window is not modified and r-modes are not damped!

ms=100 MeV ms=200 MeV ms=300 MeV

B=108G

Ý M 10 8 M / yr

Quark stars inside LMXBs

ms=100 MeV

ms =200 MeV

ms =300 MeV

Quark stars inside LMXBs

•) After Tayler instability sets in a large poloidal component is generated (Bp≈1012G).

•) If the crust is small or non-existent (as it is supposed to be for a pure quark star) such a large poloidal field is not screened and appears outside of the star, stopping mass accretion.

•) The typical timescales for the magnetic field to reach the critical value for the Tayler instability depends on the mass of the strange quark and on the accretion rate, ranging from 104 yrs for ms=100 MeV to 108 yrs for ms=300 MeV.

Quark stars inside LMXBs

•) The frequency at which the instability takes place ranges from about 400 Hz for ms=100 MeV up to 1400 Hz for ms=300 MeV.

•) After accretion stops, the star should slow down in a few thousends years, due to the large magnetic braking, going to the region of pulsars. This means that the quark star becomes a pulsar with a large magnetic field.

Conclusions

LMXBs

Neutron stars: large magnetic fields (B≥1012 G) in the core of accreting stars in LMXBs, rotating at frequencies larger than 200 Hz.

Quark stars: large magnetic fields (B≥1012 G) generated in the core of accreting stars in LMXBs. Due to the absence of crust, the quark star stops accreting, and becomes a pulsar with a large magnetic field.

Generation of toroidal fields is a very efficient mechanism to damp the r-modes.

Outlooks: Study of hybrid and hyperonic stars including superconductivity.

MHD calculations are needed!

R-modes

A rapidly rotating star produces particular fluid motions or currents, called r-modes. R-modes are similar to hurricanes or ocean currents on the Earth. Solution of the perturbed hydrodynamic equations, having Eulerian velocity perturbation of “axial type” at first order in Ω and in the amplitude α:

v1(r,,,t) Rr

R

l

YlmBe it

Magnetic spherical harmonic functions:

YlmB (,)

1

l(l 1)r(rYlm )

tl

lcR

rt

tl

lcR

rt

tr

ll

l

ll

l

1

2sin)(sincos),,(

1

2cos)(sinsin),,(

0),,(

21

21

c l 1

2

ll

(l 1)!

(2l 1)!

(l 1)(l 2)

l 1

2ll(l 1)

In the corotating frame

In the inertial frame

In the corotating frame

Frequency of the r-modes:

Jtot I (1 K j )JC

JC KC 2I

1

2JC

dJC

dt

1

GW

1

shear

1

bulk

K j 1

KC 9.4 10 2

˜ I 0.261

n=1 polytropebrakingbulkshear

jGW

jC

brakingbulkshearj

GWjC

bulkshearGW

MR

G

I

M

M

MKKK

dt

d

MR

G

I

MKKK

dt

d

2/1

32

2/1

33

~11

)1(1

2

2~

2

11)1(

1111

temperature evolutionaccrviscurcaVTCdt

d

2

1

Evolution of compact stars (no toroidal field) where is the angular momentum of the r-modes.

brakinga

GW

Ctot IJ

J

dt

dJ

2Equations for the angular momenta

3 coupled first order differential equations:

R-modes emit gravitational waves, where the lowest order contributions comes from the current multipole moment:

Jll 2cR l 1

l

l 1r2l 2dr

0

R the most relevant mode is l=m=2

Jtot I Jc

If the star is sufficiently highly rotating the r-modes move backwards in the corotating frame and forward in the inertial frame.

Gravitational wave emission remove positive angular momentum, increasing Jc and thus the amplitude of the r-modes.

The r-modes are unstable under emission of gravitational waves.

The r-moves move backwards relative to the rotation of the star.

Chandrasekar-Friedman-Schutz instability