magnetic fields in axisymmetric neutron stars

IntroductionGoverning equations and formalism

ResultsWhat next?

Summary

Magnetic fields in axisymmetric neutron stars

Samuel Lander

University of Southampton

Thursday 5th March

Samuel Lander Magnetic fields in axisymmetric neutron stars


ResultsWhat next?

Summary

Overview

1 IntroductionThe magnetic fields of neutron starsMotivation

2 Governing equations and formalism

3 ResultsField configurationsMagnetically-induced distortionsConnection with perturbative work

4 What next?

5 Summary



ResultsWhat next?

Summary

The magnetic fields of neutron starsMotivation

The magnetic fields of neutron stars

Credit: NASA, CXC, M. Weiss

Among the strongest in the Universe: ∼ 1012 G for ordinaryneutron stars and ∼ 1015 G for magnetars

Both ordinary NSs (pulsars) and magnetars (SGRs) are visiblethanks to their magnetic fields (Duncan & Thompson 1992)

Observation difficult: need modelling to better understand NSphysics



ResultsWhat next?

Summary

The magnetic fields of neutron starsMotivation

Motivation

Highly magnetised NSs are a potential source of detectablegravitational radiation

Analytic work is restricted to weak fields and simple geometries

Need numerics for more sophisticated modelling of NS magneticfields



ResultsWhat next?

Summary

Basic MHD equations

The governing equations for our neutron star model are:

−∇Pρ

−∇Φg + ∇Φr +j × B

ρ= 0 (1)

∇2Φg = 4πGρ (2)

∇× B = 4πj (3)

P = kργ (4)

plus the constraint∇ · B = 0. (5)



ResultsWhat next?

Summary

MHD formalism in axisymmetry: I

We now work in cylindrical polars ($,φ, z). The solenoidal nature ofB, together with the additional assumption of axisymmetry, allows usto write B in terms of a streamfunction u:

B$ = −1$

∂u∂z

, Bz =1$

∂u∂$

(6)

Additionally, taking the curl of the Euler equation yields

∇×

(j × B

ρ

)

= 0. (7)

Using equations (6) and (7) and defining

∆∗≡∂2

∂$2 −1$

∂

∂$+

∂2

∂z2 , (8)

Ampère’s law in axisymmetry becomes

4πj =1$∇($Bφ) × eφ −

1$

∆∗ueφ. (9)



ResultsWhat next?

Summary

MHD formalism in axisymmetry: II

Splitting the Lorentz force up in terms of toroidal (φ) and poloidal($, z) pieces gives:

L = j × B = (jpol + jφeφ) × (Bpol + Bφeφ)

= jpol × Bpol︸︷︷︸

Ltor

+ jφeφ × Bpol + Bφjpol × eφ︸︷︷︸

Lpol

. (10)

From axisymmetry we require

Ltor = jpol × Bpol = 0 (11)

so there are two ways to proceed: either Bpol = 0 (purely toroidalfields) or Bpol and jpol are parallel (mixed-field).



ResultsWhat next?

Summary

MHD formalism in axisymmetry: III

When Bpol and jpol are parallel, Ampère’s law becomes the relation

j = α(u)B + $ρκ(u)eφ (12)

where α, κ are functions of u; we will use this later to write an integralequation for the magnetic field.When Bpol = 0, Ampère’s law constrains the magnetic field to be ofthe form

Bφ = λρ$ (13)

where λ is a constant.



ResultsWhat next?

Summary

Purely toroidal-field equations

We begin with the simpler of the two cases, when

B = Bφeφ. (14)

In this case the governing integral equations are

H = C − Φ +Ω2$2

2−

λ2ρ$2

4π(15)

Φ(r) = −G∫ r

0

ρ(r′)|r − r′|

dr. (16)

This case is the simpler of the two because the magnetic field isdirectly linked to the density distribution through Bφ = λρ$.



ResultsWhat next?

Summary

The mixed-field equations

For a mixed magnetic field the governing integral equations are

H = C − Φ +Ω2$2

2+

∫ $Aφ

0κ(u′) du′ (17)

Φ(r) = −G∫ r

0

ρ(r′)|r − r′|

dr. (18)

and a separate integral equation for the magnetic field (expressed interms of the vector potential A),

Aφ(r) sin φ =1

4π

∫ r

0

α($′Aφ′)

$′

∫$′Aφ′

0 α(u) du + κ(u)ρ′$′

|r − r′|sin φ′ dr′

(19)(Tomimura & Eriguchi 2005). Note that u is related to A by u = $Aφ.



ResultsWhat next?

Summary

Field configurationsMagnetically-induced distortionsConnection with perturbative work

The governing equations just discussed are solved iteratively to findstationary axisymmetric solutions for magnetised stars.Some terminology and parameters used in the next section:

‘axis ratio’ means polar radius divided by equatorial radius,rp/req . Oblate stars have rp/req < 1; for prolate stars rp/req > 1

we only use rigid rotation here

perfect MHD

physical quantities are rescaled to a star with M = 1.4M¯,R0 = 10 km, γ = 2

field strengths required to generate the distortions shown areunphysically high, ∼ 1017 G, in order to emphasise the effects



ResultsWhat next?

Summary


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(b)tor comp

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(c)mixed



ResultsWhat next?

Summary


Poloidal-field distortions

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ResultsWhat next?

Summary


Mixed-field and rotational distortions

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PSfrag replacements

(a): pure pol (b): 3.4% tor

(c): 5.5% tor (d): rotation



ResultsWhat next?

Summary


Toroidal-field distortions

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PSfrag replacements

nonrotating rotating



ResultsWhat next?

Summary


Connection with perturbative work

0

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PSfrag replacements

ε

−ε

(B

1017 G

)2

distortion

ε

1 −rp

req

(B

1017 G

)2

B2 is the total B2 = B · B averaged over the star’s volume

ε = (Ieq − Ip)/Ieq

the analytic result ε ∝ B2 holds to within 10% for B < 1.5 × 1017

G or ε < 0.15.Samuel Lander Magnetic fields in axisymmetric neutron stars


ResultsWhat next?

Summary

Magnetar flares and oscillations

Credit: NASA, CXC, M. Weiss

Giant flares observed from magnetars — powered by the star’shuge magnetic energy

QPOs seen in the X-ray spectrum of magnetar flares

Linked to oscillation modes of the star

Potential probe of the star’s physics



ResultsWhat next?

Summary

Studying magnetic modes numerically

Analytic calculations of oscillation modes in magnetised stars arevery difficult and necessarily simplistic. Better modelling of magnetaroscillations relies on numerical MHD evolutions.With an extant code for modes of rotating stars (Jones, Passamonti),we are able to make some modifications to study magnetised stars:

Use the stationary code discussed earlier to provide thebackground star

Add magnetic terms to the evolution equations for theperturbations

Construct initial data using available analytic work for magneticmodes

Work out what boundary conditions are best...



ResultsWhat next?

Summary

Summary

Modelled a NS as a perfectly conducting fluid ball

In axisymmetry the MHD equations reduce to two cases

We have used this formalism to find equilibrium solutions forstars with poloidal, toroidal and mixed magnetic fields

Analytic perturbative results should be accurate up to ∼ 1017 G

Hope to be studying modes of these magnetised stars soon



ResultsWhat next?

Summary

References

(1) Duncan R.C., Thompson C., 1992, ApJ, 392, L9(2) Tomimura Y., Eriguchi Y., 2005, MNRAS, 359, 1117(3) Glampedakis K., Samuelsson L., Andersson N., 2006, MNRAS,371, L74(4) Jones D.I., Andersson N., Stergioulas N., 2002, MNRAS, 334, 933(5) Passamonti A. et al., 2008, arXiv:0807.3457(6) Lander S.K., Jones D.I., 2009, accepted for publication in MNRAS


magnetic fields in axisymmetric neutron stars

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