Evolution of a neutron stars magnetic inclination angle Sam Lander - - PowerPoint PPT Presentation

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Evolution of a neutron star’s magnetic inclination angle

Sam Lander

Nicolaus Copernicus Astronomical Centre, Warsaw

Warsaw

27th March 2018

1 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Overview

1

Motivation

2

Steady-state fluid motion with χ = 0

3

Dissipation and evolution of χ

4

Summary

2 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Pulsars and their spindown

Ω χ B χ = magnetic inclination angle between rotation Ω and magnetic B axes χ = 0 since pulsars pulse probably expect χ ≈ 0 at birth though strong hints from observations that χ not randomly distributed (Lyne & Manchester 1988, Tauris &

Manchester 1998, Rookyard, Weltevrede, Johnston 2015)

so, some mechanism causes evolution of χ

3 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

An exterior-only problem?

To understand neutron-star observables people have (naturally) started from the exterior. Pulsar emission problem not ‘solved’, but broad structure of magnetosphere (exterior field) probably accepted.

Spindown: function of χ (Deutsch 1955, Ostriker & Gunn 1969, Gruzinov 2005, Spitkovsky 2006)

˙ Ωvacuum = −R2

∗B2 pole

6c3I Ω3 sin2 χ , ˙ Ωmagnetosphere = −R2

∗B2 pole

6c3I Ω3(1 + sin2 χ). In both prescriptions, spindown causes χ → 0 over time but many χ values

• bserved to be large (χ ≈ π/2), or even increasing. What could drive this?

χ-evolving mechanisms

spindown? χ → 0 × crustal stresses reducing? χ → 0 × gravitational radiation reaction? χ → 0 × strong internal poloidal field? χ → 0 × strong internal toroidal field? χ → π/2

4 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Magnetic distortions

The two magnetic-field components of a star distort it in different ways:

(Lander & Jones 2009; Lander, Andersson, Glampedakis 2012; Lander 2014)

Poloidal field

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2

(|B|=colourscale, direction=black lines) ρ contours show the

• blate

distortion induced by B:

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2

Minimum energy state: χ = 0

Toroidal field

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.005 0.01 0.015 0.02 0.025 0.03

(|B|=colourscale, direction=⊥ screen) ρ contours show the prolate distortion induced by B:

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2

Minimum energy state: χ = π/2

5 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Simplest possible misaligned model

...this will already be hard enough!

χ α ω e(α)

z

e(B)

z

uniformly rotating, self-gravitating fluid star purely toroidal magnetic field barotropic EOS, no superconductivity rotational ǫα and magnetic ǫB distortions ≪ 1 B gives the star ‘rigidity’, since ǫB is a distortion misaligned with α-axis angular momentum conservation = ⇒ star must precess (Spitzer 1958, Mestel & Takhar 1972) angular velocity now Ω = αe(α)

z

+ ωe(B)

z

precession frequency ω = αǫB cos χ how do we account for non-rigidity of star?

6 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Non-rigid response of the star

What is fluid response? Look in frame moving with primary rotation α: ω e(B)

z

e(α)

z

χ if axes aligned (χ = 0), fluid elements would appear stationary if not, and the star is truly rigid, precession would cause slow dragging (rate ω) of elements across ρ contours continuity equation → get non-rigid response: additional velocity field ˙ ξ: ∂ρ ∂t = ∂ ∂t (ρ0 + δρα + δρB) = ∂ ∂t (δρα) = −∇ · (ρ0 ˙ ξ). Velocity of a fluid element seen from inertial frame is: V = Ω × r + ˙ ξ = (αe(α)

z

+ ωe(B)

z

) × r + ˙ ξ.

7 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Second-order perturbative model

Already argued for the form of the continuity equation. To complete the set of equations governing the stellar dynamics we have: ∂V ∂t + (V · ∇)V = −∇P ρ − ∇Φ + 1 4πρ(∇ × B) × B, ∂B ∂t = ∇ × (V × B), ∇2Φ = 4πGρ, P = P(ρ) = kρ2, ∇ · B = 0. Have to expand in the small parameters ǫα and ǫB. Then solve, successively, a series of perturbation problems: O(1) → spherical background model; ρ0(r) ∼ (sin r)/r for γ = 2 polytrope O(ǫB) → background magnetic field B0, ellipticity ǫB → can find ω O(ǫα) → centrifugal bulge δρα moving slowly around e(B)

z

Finally, all these provide input into the O(ǫαǫB) equations; this is the perturbative order at which we find the velocity field ˙ ξ (Lander & Jones 2017).

8 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Steady-state fluid motions

left: |˙ ξ| for near-aligned model (χ = π/16), right: near-orthogonal model eigenfunction rotates around z-axis at rate ω motions most rapid towards centre, each contour is √ 2 times the next below: direction field on spherical shells (blue: inward, white: outward)

9 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Neutron-star spindown

Neutron-star rotation rate Ω and inclination angle χ evolve due to dissipation: external: electromagnetic waves carry angular momentum J away; χ → 0 internal: viscous dissipation of precessional kinetic energy; χ → 0 or π/2 These both reduce the star’s kinetic energy E = E(J, χ): ˙ E = ˙ Eint + ˙ Eext and by the chain rule ˙ E = ∂E ∂J

• χ

˙ J + ∂E ∂χ

• J

˙ χ. Can combine these to find loss of precessional kinetic energy: ˙ χ = ˙ Eint(Ω, χ) + ǫB cos2 χ ˙ Eext(Ω, χ) ∂Eprec/∂χ , ˙ Ω = −R2

∗B2 pole

6c3 Ω3 sin2 χ.

10 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Damping and inclination-angle evolution

With our solutions for ˙ ξ, can now perform first ever quantitative study of coupled Ω and χ evolution (Lander & Jones, in prep.). External torque prescribed. For internal dissipation allow for two extra terms in the Euler equation: ∂V ∂t + (V · ∇)V = −∇P ρ − ∇Φ + 1 4πρ(∇ × B) × B + 2∇ · (ησ) + ∇(ζ∇ · v), where σ is the fluid stress tensor. This fluid physics will appear in ˙ Eint: shear viscosity η ∝ T −2 bulk viscosity ζ ∝ T 6 and ∝ precession frequency ω Follow evolution of χ as star cools from birth (when T ∼ 1011 K):

50 100 150 t[s] 0.5 1.0 1.5 χ 1 2 3 4 5 6 7 t[1010 yr] 0.5 1.0 1.5 χ

bulk viscosity only: shear viscosity only:

– bulk viscosity acts very quickly or not at all. Shear viscosity always very slow.

11 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Distribution of inclination angles (preliminary)

A A A A O O colour scale shows value of χ in radians χ(t = 0) = π/100 rad regions within π/360 rads of alignment/orthogonality omitted for clarity: A=virtually-aligned models O=virtually-orthogonal models left: distribution of χ after one minute, right: after a year

• nly a toroidal field allows χ → π/2

12 / 13

Motivation Steady-state fluid motion with χ = 0 Dissipation and evolution of χ Summary

Summary

This talk

rotation and magnetic field interact through the evolution of the angle between their axes dissipation causes the axes to align or orthogonalise

Outlook

basic details of neutron-star birth and interior physics not understood the observed distribution of inclination angles could give us a probe of: interior field strength rotation rate at birth

13 / 13