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 Motivation 1 Steady-state fluid motion with χ � = 0 2 3 Dissipation and evolution of χ Summary 4 2 / 13
Motivation Steady-state fluid motion with χ � = 0 Dissipation and evolution of χ Summary Pulsars and their spindown Ω B χ 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) Ω vacuum = − R 2 ∗ B 2 Ω magnetosphere = − R 2 ∗ B 2 Ω 3 sin 2 χ , Ω 3 (1 + sin 2 χ ) . ˙ pole ˙ pole 6 c 3 I 6 c 3 I In both prescriptions, spindown causes χ → 0 over time but many χ values observed 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) Toroidal field Poloidal field 1 0.03 1.2 1.2 0.035 0.025 0.03 1 1 0.8 0.025 0.02 0.8 0.8 0.6 0.02 0.015 0.6 0.6 0.015 0.4 0.4 0.4 0.01 0.01 0.2 0.2 0.2 0.005 0.005 0 0 0 0 0 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2 0 0.2 0.4 0.6 0.8 1 ( | B | =colourscale, direction= ⊥ screen) ( | B | =colourscale, direction=black lines) 1.2 1.2 1 ρ contours ρ contours 1 show the 0.8 show the 0.8 prolate 0.6 0.6 oblate distortion 0.4 distortion 0.4 induced by B : 0.2 induced by B : 0.2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Minimum energy state: χ = π/ 2 Minimum energy state: χ = 0 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! e ( α ) z α uniformly rotating, self-gravitating fluid star purely toroidal magnetic field e ( B ) ω z barotropic EOS, no superconductivity rotational ǫ α and magnetic ǫ B distortions ≪ 1 B gives the star ‘rigidity’, since ǫ B is a χ distortion misaligned with α -axis angular momentum conservation = ⇒ star must precess (Spitzer 1958, Mestel & Takhar 1972) angular velocity now Ω = α e ( α ) + ω e ( B ) z z precession frequency ω = αǫ B 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 e ( α ) e ( B ) z z ω What is fluid response? Look in frame moving with primary χ rotation α : 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 ( ρ 0 + δρ α + δρ B ) = ∂ ∂ t ( δρ α ) = −∇ · ( ρ 0 ˙ ξ ) . Velocity of a fluid element seen from inertial frame is: V = Ω × r + ˙ ξ = ( α e ( α ) + ω e ( B ) ) × r + ˙ ξ . z z 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 1 − ∇ Φ + 4 πρ ( ∇ × B ) × B , ρ ∂ B ∂ t = ∇ × ( V × B ) , ∇ 2 Φ = 4 π G ρ, P = P ( ρ ) = k ρ 2 , ∇ · B = 0 . Have to expand in the small parameters ǫ α and ǫ B . Then solve, successively, a series of perturbation problems: O (1) → spherical background model; ρ 0 ( r ) ∼ (sin r ) / r for γ = 2 polytrope O ( ǫ B ) → background magnetic field B 0 , ellipticity ǫ B → can find ω O ( ǫ α ) → centrifugal bulge δρ α moving slowly around e ( B ) z Finally, all these provide input into the O ( ǫ α ǫ B ) 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 = ˙ ˙ E int + ˙ E ext and by the chain rule � � E = ∂ E J + ∂ E � � ˙ ˙ χ. ˙ � � ∂ J ∂χ � � � � J χ Can combine these to find loss of precessional kinetic energy: E int (Ω , χ ) + ǫ B cos 2 χ ˙ ˙ Ω = − R 2 ∗ B 2 E ext (Ω , χ ) Ω 3 sin 2 χ. ˙ pole χ = ˙ , ∂ E prec /∂χ 6 c 3 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 1 − ∇ Φ + 4 πρ ( ∇ × B ) × B + 2 ∇ · ( η σ ) + ∇ ( ζ ∇ · v ) , ρ where σ is the fluid stress tensor. This fluid physics will appear in ˙ E int : 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 ∼ 10 11 K): bulk viscosity only: shear viscosity only: χ χ 1.5 1.5 1.0 1.0 0.5 0.5 7 t [ 10 10 yr ] t [ s ] 1 2 3 4 5 6 50 100 150 – 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 O A O A 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 only 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
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