Dynamics and Gravitational Wave Signatures of Magnetized Neutron - PowerPoint PPT Presentation

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 equations 3 Methods for solving 4 Plots 5 Poynting flux 6 Further avenues of inquiry 7 References 8 Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 2 / 24

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

Geometry Stationary, precessing magnetic dipole in Schwarzschild space-time. Metric: ✡ ✁ 1 ✂ ✡ ✂ 1 ✁ 2 1 ✁ 2 ds 2 ✏ ✁ dt 2 � dr 2 � r 2 d θ 2 � r 2 sin 2 θ d φ 2 r r Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 4 / 24

Electric dipole Electric dipole: ✂ sin θ 0 cos ♣ ω t q ✡ , ✁ cos θ 0 , sin θ 0 sin ♣ ω t q � p ✏ ♣ p r , p θ , p φ q ✏ p ❄ g rr . r 0 r 0 Use EM duality: E Ñ B , B Ñ ✁ E , p Ñ m Dipole tensor: Q αµ ♣ τ q ✏ V α p µ ✁ p α V µ Four-current: ➺ Q αµ ♣ τ q δ ♣ 4 q r x ✁ x S ♣ τ qs dt p µ ✁ p α dx µ ✄ ♣ dx α ☛ dt q δ ♣ 3 q r x ✁ x S ♣ t qs J α ✏ ∇ µ ❄✁ g d τ ✏ ∇ µ ❄✁ g Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 5 / 24

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 ♣✁ 1 q ℓ , and even, which transform like ♣✁q ℓ � 1 . Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 6 / 24

J µ ☎ ✔ ✜ ✔ Ψ ℓ m ♣ r , t q Y ℓ m ✜ ☞ 0 η ℓ m ♣ r , t q Y ℓ m 0 ✝ ✖ ✣ ✖ ✣ ✍ ➳ 4 π J µ ✏ ✢ � ✝ ✖ ✣ ✖ ✣ ✍ α ℓ m ♣ r , t q χ ℓ m ♣ r , t q ❇ Y ℓ m ❇ Y ℓ m ✝ ✖ ✣ ✖ ✣ ✍ ❇ θ sin θ ❇ φ ✆ ✕ ✕ ✢ ✌ ℓ, m χ ℓ m ♣ r , t q ❇ Y ℓ m ✁ α ℓ m ♣ r , t q sin θ ❇ Y ℓ m ❇ φ ❇ θ ✒ ✂ δ r r ✁ R s ✡ Y ✝ ✁ i 1 ✚ g 00 r ❇ φ Y ✝ δ ♣ r ✁ R q e ✁ i ω t ψ ✏ p sin θ 0 ❇ r ❄ g rr r 2 ❄ g rr r 2 ωδ ♣ r ✁ R q Y ✝ e ✁ i ω t η ✏ ip sin θ 0 r ω ❇ Y ✝ 1 1 ❇ θ δ ♣ r ✁ R q e ✁ i ω t α ✏ p sin θ 0 ℓ ♣ ℓ � 1 q Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 7 / 24

A µ ☎ ✔ ✜ ✔ f ℓ m ♣ r , t q Y ℓ m ✜ ☞ 0 0 h ℓ m ♣ r , t q Y ℓ m ✝ ✖ ✣ ✖ ✣ ✍ ➳ A µ ✏ ✢ � ✝ ✖ ✣ ✖ ✣ ✍ a ℓ m ♣ r , t q χ ℓ m ♣ r , t q ❇ Y ℓ m ❇ Y ℓ m ✝ ✖ ✣ ✖ ✣ ✍ ❇ θ sin θ ❇ φ ℓ, m ✆ ✕ ✕ ✢ ✌ χ ℓ m ♣ r , t q ❇ Y ℓ m ✁ a ℓ m ♣ r , t q sin θ ❇ Y ℓ m ❇ φ ❇ θ Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 8 / 24

Maxwell’s equations We are interested in solving Maxwell’s equations in curved space-time ♣❄✁ gF µν q , ν ✏ ❄✁ g 4 π J µ , where g ✏ det g αβ and F µν ✏ ❇ µ A ν ✁ ❇ ν A µ , which reduces to solving a ✁ ℓ ♣ ℓ � 1 q ♣ g rr a ✶ q ✶ ✁ g rr ✿ a ✏ α r 2 b ✁ ℓ ♣ ℓ � 1 q 1 ♣ g rr b ✶ q ✶ ✁ g rr ✿ ℓ ♣ ℓ � 1 qr♣ r 2 Ψ q ✶ ✁ r 2 ✾ b ✏ η s , r 2 r 2 ℓ ♣ ℓ � 1 q ♣ ✾ h ✁ f ✶ q . Working in frequency space, where b ✏ ✂ ✡ g rr ω 2 ✁ ℓ ♣ ℓ � 1 q ♣ g rr a ✶ q ✶ � a ✏ α r 2 ✂ ✡ g rr ω 2 ✁ ℓ ♣ ℓ � 1 q 1 ♣ g rr b ✶ q ✶ � ℓ ♣ ℓ � 1 qr♣ r 2 Ψ q ✶ � i ω r 2 η s b ✏ r 2 Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 9 / 24

Parameters of simulation In units of G ✏ M BH ✏ c ✏ 1: r 0 ✏ 25 ω ✏ . 2 p sin θ 0 ✏ 1 Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 10 / 24

Method for solving 1 Due to delta function nature of source, separate region into two regions, r ➔ r 0 and r → r 0 . Let u L be solution for r ➔ r 0 , and u R be solution for r → r 0 . 2 Numerically solve, applying approprate boundary conditions for both a and b . 3 Apply junction conditions at r ✏ r 0 (taking into account delta function source terms). Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 11 / 24

Boundary conditions Tortoise coordinate r ✝ ✑ r � 2 log ♣ r ④ 2 ✁ 1 q . Ingoing wave conditions: r ✝ Ñ✁✽ u L ♣ r ✝ q ✒ e ✁ i ω r ✝ lim r ✝ Ñ✁✽ u ✶ L ♣ r ✝ q ✒ ✁ i ω e ✁ i ω r ✝ lim Outgoing wave conditions: r ✝ Ñ✽ u R ♣ r ✝ q ✒ e i ω r ✝ lim r ✝ Ñ✽ u ✶ R ♣ r ✝ q ✒ i ω e i ω r ✝ lim Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 12 / 24

Junction conditions For a : u R ♣ r 0 q ✁ u L ♣ r 0 q ✏ 0 ❇ Y ✝ 1 p ω sin θ 0 u ✶ R ♣ r 0 q ✁ u ✶ L ♣ r 0 q ✏ ❇ θ , ℓ ♣ ℓ � 1 q r 0 For b : u R ♣ r 0 q ✁ u L ♣ r 0 q ✏ ✁ i p sin θ 0 g rr ❇ φ Y ✝ ℓ ♣ ℓ � 1 q r 0 L ♣ r 0 q ✏ ✁ 1 ❄ g rr Y ✝ u ✶ R ♣ r 0 q ✁ u ✶ p sin θ 0 r 2 0 Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 13 / 24

EM components Odd parity: 1 a ❇ Y r 2 sin θ g rr a ✶ ❇ Y 1 r 2 sin θ g 00 ✾ E θ ✏ ✁ B θ ✏ ✁ ❇ φ ❇ θ 1 a ❇ Y r 2 sin θ g rr a ✶ ❇ Y 1 r 2 sin θ g 00 ✾ E φ ✏ B φ ✏ ✁ ❇ θ ❇ φ Even parity: E θ ✏ 1 r 2 b ✶ ❇ Y 1 b ❇ Y ✾ B θ ✏ r 2 sin 2 θ ❇ θ ❇ φ r 2 sin 2 θ b ✶ ❇ Y 1 B φ ✏ ✁ 1 b ❇ Y r 2 ✾ E φ ✏ ❇ φ ❇ θ Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 14 / 24

E r Plots (a) front (b) back Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 15 / 24

B r Plots (c) front (d) back Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 16 / 24

E tangential Plots (e) front (f) back Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 17 / 24

B tangential Plots (g) front (h) back Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 18 / 24

Poynting flux ✒ 1 ✚ � E ✂ � � B ✝ S ✏ Re 8 π ✔ ✜ ➺ A ✏ 1 a ✶ ✁ ✾ S ☎ d � � ✕➳ b ¯ b ✶ q P ✏ 8 π Re ℓ ♣ ℓ � 1 q♣ ✾ a ¯ ✢ ℓ, m Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 19 / 24

Poynting flux Power from precessing, magnetic dipole: P ✏ ♣ sin θ 0 p q 2 ω 4 . 3 Slope of line ✓ 4. Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 20 / 24

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

Further avenues of inquiry Plunging dipole Introduction of plasma Kerr geometry Farzan Vafa, Yanbei Chen (LIGO SURF) LIGO, Caltech August 19, 2014 22 / 24

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

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 of 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