Introduction
Magnetically-stressed crusts
Summary
Magnetically-driven failure of neutron-star crusts
Sam Lander
Lander, Andersson, Antonopoulou, Watts (2015)
Lander (2016)
Warsaw
28th March 2017
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Introduction
Magnetically-stressed crusts
Summary
How and when will the crust break?
Electron MHD
Magnetars
crustal lattice totally static
and never breaks
only electrons move →
Hall+Ohmic evolution
versus:
widely used to model B-field
properties and evolution
(Pons,Miralles,Geppert 2009; Vigano+ 2013;
Gourgouliatos,Wood,Hollerbach 2016)
magnetic energy powers
activity: bursts and flares
interior field stresses crust,
which eventually fails and
twists exterior field (Parfrey,
Beloborodov, Hui 2013)
=⇒ crust must break!
How do we reconcile these?
Easy answer: Some critical value of B separates left and right-hand scenarios
But:
separation between magnetars and other NSs is disappearing
(Rea+ 2012, etc)
still need a quantitative picture of how magnetic field causes crustal failure
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Introduction
Magnetically-stressed crusts
Summary
Creating a magnetar giant flare
magnetic field evolves and decays over time (!)
crust resists adjustment of field, and stresses build
beyond a critical strain σel (or stress τel ) the crust fails (!)
magnetospheric footpoints move, exterior twists → giant flare
First (!): field evolution – especially in the core – poorly understood
Second (!): how does the crust fail? e.g. smallscale or collective failure?
Warning from history
σel estimates vary over six orders of magnitude...
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Introduction
Magnetically-stressed crusts
Summary
The elastic-limit magnetic field strength
2.5
2
Bel
1015 G
1.5
1
0.5
0
0.9
0.92
0.94
0.96
0.98
1
r /R∗
Magnetic energy budget must be & 1047 erg to power largest flares
translates into an average B & 1015 G
crust can hold out-of-equilibrium B in place as long as
1
∇·τ >
(∇ × B) × B
4π
√
beyond some ‘yield field’ Bel ∼ 4πτel crust fails
using a calculation for τel (Chugunov & Horowitz 2010) calculate depth-dependent Bel
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Introduction
Magnetically-stressed crusts
Summary
Large, sudden, collective failures
σel = 0.1
σel = 0.001
2.5
4.5
4
1
2
0.9
1
1.5
3.5
3
0.9
2.5
1
2
1.5
0.5
1
0.5
0
0
0
-0.5
0
0.9
0
1
-0.5
0
0.9
1
First assume a large-scale collective failure (Bel may be exceeded locally)
look at strain patterns based on von Mises criterion
can store enough energy to power giant flares in crust alone (upper limit
from our calculations ∼ 4 × 1046 erg)
von Mises criterion
1≤
1
σel
q
1
σ σ ij
2 ij
=
1
8πµσel
q
B 2 B02 + 23 B 4 + 32 B04 − 4(B · B0 )2
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Introduction
Magnetically-stressed crusts
Summary
Gradual failure through plastic flow
For localised failure, as soon as B > Bel we can model the viscous, plastic
motion vpl of the crust as follows:
ν∇2 vpl = −
1
(∇ × B) × B , ∇ · vpl = 0,
4π
where ν is the completely unknown viscosity of the crust in its plastic phase.
The plastic flow contributes a new term to the crustal field evolution
∇×B
∂B
(∇ × B) × B
−∇×
,
= ∇ × (vpl × B) − ∇ ×
∂t
4πρe
4πσ0
spoiling electron MHD.
Timescale
tpl ∼
LB ν
Lpl B 2
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Introduction
Magnetically-stressed crusts
Summary
Gradual failure through plastic flow
demand a plastic flow fast enough to sustain a magnetar corona
must beat decay timescale 1 − 10 yr
38
→ estimate ν ∼ 10
(Beloborodov&Thompson 2007)
poise
With this ν, we infer that:
if unbalanced B & 1015 G, plastic flow always dominates; characteristic
timescale . 3LB /Lpl yr
in the range B ∼ (2 − 7) × 1014 G, get failure for depth below
∼ 100 − 400m → timescale comparable with Hall drift
if B . 1013 G, plastic flow shuts off essentially everywhere
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Introduction
Magnetically-stressed crusts
Summary
Summary
Implications
crust can indeed store requisite flare energy
depending on how crust fails:
occasional large-scale failure
or persistent plastic flow in outer crust
could ‘wash out’ effect of Hall drift in field evolution
Open questions
mechanical properties of crust still poorly understood
need self-consistent evolutions leading to crust failure
magnetar activity could potentially help constrain crustal physics
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