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Dynamics and Gravitational Wave Signatures of Magnetized Neutron Stars Farzan Vafa, Yanbei Chen LIGO SURF August 19, 2014 Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 1 / 24 Overview Introduction 1 Set-up 2 Set-up

SLIDE 1

Dynamics and Gravitational Wave Signatures of Magnetized Neutron Stars

Farzan Vafa, Yanbei Chen

LIGO SURF

August 19, 2014

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 1 / 24

SLIDE 2

Overview

1

Introduction

2

Set-up

3

Set-up equations

4

Methods for solving

5

Plots

6

Poynting flux

7

Further avenues of inquiry

8

References

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 2 / 24

SLIDE 3

Introduction

As neutron star falls into black hole, precession of magnetic dipole creates EM waves. The induced electric field drives a current, establishing a circuit between the neutron star, black hole, and the plasma surrounding the black hole. Electromagnetic waves are emitted that can be detected. Black-hole neutron star binary can serve as source of electromagnetic waves.

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 3 / 24

SLIDE 4

Geometry

Stationary, precessing magnetic dipole in Schwarzschild space-time. Metric: ds2 ✏ ✁ ✂ 1 ✁ 2 r ✡ dt2 ✂ 1 ✁ 2 r ✡✁1 dr2 r2dθ2 r2 sin2 θdφ2

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 4 / 24

SLIDE 5

Electric dipole

Electric dipole:

p ✏ ♣pr, pθ, pφq ✏ p

✂sin θ0 cos♣ωtq ❄grr , ✁cos θ0 r0 , sin θ0 sin♣ωtq r0 ✡ . Use EM duality: E Ñ B, B Ñ ✁E, p Ñ m Dipole tensor: Qαµ♣τq ✏ V αpµ ✁ pαV µ Four-current: Jα ✏ ∇µ ➺ Qαµ♣τqδ♣4qrx ✁ xS♣τqs ❄✁g dτ ✏ ∇µ ✄ ♣ dxα

dt pµ ✁ pα dxµ dt qδ♣3qrx ✁ xS♣tqs

❄✁g ☛

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 5 / 24

SLIDE 6

Vector harmonics expansions

Similar to solving the hydrogen atom in quantum mechanics, separate solution into angular and radial part. Spherical harmonics are convenient basis for angular part. Vector harmonics are generalization to vectors. Vector harmonics have two parities: odd, which transform like ♣✁1qℓ, and even, which transform like ♣✁qℓ1.

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 6 / 24

SLIDE 7

Jµ

4πJµ ✏ ➳

ℓ,m

☎ ✝ ✝ ✝ ✆ ✔ ✖ ✖ ✖ ✕

αℓm♣r,tq sin θ ❇Y ℓm ❇φ

✁αℓm♣r, tq sin θ ❇Y ℓm

❇θ

✜ ✣ ✣ ✣ ✢ ✔ ✖ ✖ ✖ ✕ Ψℓm♣r, tqY ℓm ηℓm♣r, tqY ℓm χℓm♣r, tq ❇Y ℓm

❇θ

χℓm♣r, tq ❇Y ℓm

❇φ

✜ ✣ ✣ ✣ ✢ ☞ ✍ ✍ ✍ ✌ ψ ✏ p sin θ0 g00 r2 ✒ ❇r ✂δrr ✁ Rs ❄grr ✡ Y ✝ ✁ i 1 r ❇φY ✝δ♣r ✁ Rq ✚ e✁iωt η ✏ ip sin θ0 ❄grr r2 ωδ♣r ✁ RqY ✝e✁iωt α ✏ p sin θ0 1 ℓ♣ℓ 1q 1 r ω❇Y ✝ ❇θ δ♣r ✁ Rqe✁iωt

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 7 / 24

SLIDE 8

Aµ

Aµ ✏ ➳

ℓ,m

☎ ✝ ✝ ✝ ✆ ✔ ✖ ✖ ✖ ✕

aℓm♣r,tq sin θ ❇Y ℓm ❇φ

✁aℓm♣r, tq sin θ ❇Y ℓm

❇θ

✜ ✣ ✣ ✣ ✢ ✔ ✖ ✖ ✖ ✕ f ℓm♣r, tqY ℓm hℓm♣r, tqY ℓm χℓm♣r, tq ❇Y ℓm

❇θ

χℓm♣r, tq ❇Y ℓm

❇φ

✜ ✣ ✣ ✣ ✢ ☞ ✍ ✍ ✍ ✌

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 8 / 24

SLIDE 9

Maxwell’s equations

We are interested in solving Maxwell’s equations in curved space-time ♣❄✁gF µνq, ν ✏ ❄✁g4πJµ, where g ✏ det gαβ and Fµν ✏ ❇µAν ✁ ❇νAµ, which reduces to solving ♣grra✶q✶ ✁ grr ✿ a ✁ ℓ♣ℓ 1q r2 a ✏ α ♣grrb✶q✶ ✁ grr ✿ b ✁ ℓ♣ℓ 1q r2 b ✏ 1 ℓ♣ℓ 1qr♣r2Ψq✶ ✁ r2 ✾ ηs, where b ✏

r2 ℓ♣ℓ1q♣✾

h ✁ f ✶q. Working in frequency space, ♣grra✶q✶ ✂ grrω2 ✁ ℓ♣ℓ 1q r2 ✡ a ✏ α ♣grrb✶q✶ ✂ grrω2 ✁ ℓ♣ℓ 1q r2 ✡ b ✏ 1 ℓ♣ℓ 1qr♣r2Ψq✶ iωr2ηs

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 9 / 24

SLIDE 10

Parameters of simulation

In units of G ✏ MBH ✏ c ✏ 1: r0 ✏ 25 ω ✏ .2 p sin θ0 ✏ 1

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 10 / 24

SLIDE 11

Method for solving

1 Due to delta function nature of source, separate region into two

regions, r ➔ r0 and r → r0. Let uL be solution for r ➔ r0, and uR be solution for r → r0.

2 Numerically solve, applying approprate boundary conditions for both a

and b.

3 Apply junction conditions at r ✏ r0 (taking into account delta

function source terms).

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 11 / 24

SLIDE 12

Boundary conditions

Tortoise coordinate r✝ ✑ r 2 log♣r④2 ✁ 1q. Ingoing wave conditions: lim

r✝Ñ✁✽ uL♣r✝q ✒ e✁iωr✝

lim

r✝Ñ✁✽ u✶ L♣r✝q ✒ ✁iωe✁iωr✝

Outgoing wave conditions: lim

r✝Ñ✽ uR♣r✝q ✒ eiωr✝

lim

r✝Ñ✽ u✶ R♣r✝q ✒ iωeiωr✝

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 12 / 24

SLIDE 13

Junction conditions

For a: uR♣r0q ✁ uL♣r0q ✏ 0 u✶

R♣r0q ✁ u✶ L♣r0q ✏

1 ℓ♣ℓ 1q pω sin θ0 r0 ❇Y ✝ ❇θ , For b: uR♣r0q ✁ uL♣r0q ✏ ✁i p sin θ0 ℓ♣ℓ 1q grr r0 ❇φY ✝ u✶

R♣r0q ✁ u✶ L♣r0q ✏ ✁ 1

r2 p sin θ0 ❄grrY ✝

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 13 / 24

SLIDE 14

EM components

Odd parity: Eθ ✏ ✁ 1 r2 sin θg00 ✾ a❇Y ❇φ Bθ ✏ ✁ 1 r2 sin θgrra✶ ❇Y ❇θ Eφ ✏ 1 r2 sin θg00 ✾ a❇Y ❇θ Bφ ✏ ✁ 1 r2 sin θgrra✶ ❇Y ❇φ Even parity: Eθ ✏ 1 r2 b✶ ❇Y ❇θ Bθ ✏ 1 r2 sin2 θ ✾ b❇Y ❇φ Eφ ✏ 1 r2 sin2 θb✶ ❇Y ❇φ Bφ ✏ ✁ 1 r2 ✾ b❇Y ❇θ

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 14 / 24

SLIDE 15

Er Plots

(a) front (b) back

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 15 / 24

SLIDE 16

Br Plots

(c) front (d) back

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 16 / 24

SLIDE 17

E tangential Plots

(e) front (f) back

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 17 / 24

SLIDE 18

B tangential Plots

(g) front (h) back

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 18 / 24

SLIDE 19

Poynting flux

S ✏ Re

✒ 1 8π

E ✂

B✝ ✚ P ✏ ➺

S ☎ d

A ✏ 1 8πRe ✔ ✕➳

ℓ,m

ℓ♣ℓ 1q♣ ✾ a¯ a✶ ✁ ✾ b¯ b✶q ✜ ✢

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 19 / 24

SLIDE 20

Poynting flux

Power from precessing, magnetic dipole: P ✏ ♣sin θ0pq2ω4

3

. Slope of line ✓ 4.

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 20 / 24

SLIDE 21

Poynting flux

Flux at infinity is 5.3 ✂ 10✁4 Flux through horizon is 8.6 ✂ 10✁7. Flux at infinity in flat space-time is 5.3 ✂ 10✁4.

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 21 / 24

SLIDE 22

Further avenues of inquiry

Plunging dipole Introduction of plasma Kerr geometry

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 22 / 24

SLIDE 23

Acknowledgements

I would like to thank Prof. Yanbei Chen for mentoring me, as well as for the suggestion of this very interesting project. I have learned a lot about what physics research is. I’d also like to thank Profs. Weinstein and Ooguri for helpful duscussions, as well as Chad Galley, Yiqui Ma, Zach Marks, Bassam Helou. And finally, I’d like to thank my peers for their moral support. This research was funded by LIGO and NSF.

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 23 / 24

SLIDE 24

References I

D. J. D’Orazio and J. Levin, “Big Black Hole, Little Neutron

Star:Magnetic Dipole Fields in the Rindler Spacetime,”.

S. T. McWilliams and J. Levin, “Electromagnetic Extraction of Energy

from Black-Hole-Neutron-Star Binaries,” The Astrophysical Journal 742 (2011) 6.

H. Yang and F. Zhang, “Stability of Force-Free Magnetospheres,”

ArXiv e-prints (June, 2014) , arXiv:1406.4602 [astro-ph.HE].

R. Ruffini, J. Tiomno, and C. V. Vishveshwara, “Electromagnetic field
f a particle moving in a spherically symmetric black-hole

background.,” Nuovo Cimento Lettere 3 (1972) 211–215.

Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 24 / 24