text: distributions of these bursts: the differential distribution of Guidorzi 2015). Another method ﬁrst ﬁnds rapidly varying ﬂuence (integrated ﬂux) and the cumulative distribution of time segments in the data using a Bayesian Blocks approach waiting times are similar to those observed in earthquakes and (Scargle et al. 2013), which are subsequently used as ﬁrst solar ﬂares (Cheng et al. 1996; Go¨gˇu¨s¸ et al. 1999; Go¨gˇu¨s¸ guessed for ﬁtting a set of pulses to the data, removing the et al. 2000; Prieskorn & Kaaret 2012). The ﬂuence distribution smallest pulses based on some signiﬁcance criterion (Scargle in particular can be well-modelled by a power law with an et al. 1998; Hakkila & Preece 2011). index of ∼1.7, believed to be a typical signature for a system All these approaches have in common that they set a hard obeying self-organised criticality (SOC; Bak et al. 1987, 1988; limit (usually in peak amplitude, but sometimes in total inte- for a recent introduction and review on SOC in astrophysics grated counts) to what they consider to be a burst, resulting see Aschwanden et al. 2014). In the SOC framework, the in a set of pulses above this threshold that is then used for the physical system in question continuously drives itself towards subsequent sample inference. Many of these algorithms are a critical state, without requiring ﬁne tuning. When the critical additionally tuned deliberately towards rejecting weak features state is reached, relaxation occurs via a catastrophic release to keep the number of false positives low. This is equivalent to of energy. The system returns to a subcritical state, and the applying a step-function prior to the relevant pulse parameters cycle begins anew. One advantage of describing a system in (amplitude, or a combination of amplitude and duration). How- the SOC framework is that while the details of the process ever, because subsequent analyses are based on the ensemble leading to the critical state and subsequent relaxation depend of pulses above the threshold only, this prior is never taken into on the physical processes speciﬁc to the system, many of the account in the inferences derived from the sample (for a dis- overall statistical properties are universal. cussion of this problem applied to the waiting time distribution At present, it is not clear whether magnetar bursts are of pulses in GRBs, see Baldeschi & Guidorzi 2015). smaller-scale versions of the processes believed to produce The purpose of this paper is twofold: in the ﬁrst part, we pro- a giant ﬂare: either a rapid rupturing of the neutron star due pose a similar approach to model magnetar bursts as a linear to an internal re-structuring of the magnetic ﬁeld, with the combination of simple shapes, but without enforcing a speciﬁc energy of the cracking process released in the magnetosphere, limit on the signiﬁcance of the detected features. Instead, we or a slow untwisting of the magnetic ﬁeld leading to ductile allow the model to explore a parameter space (in practice, us- deformations in the crust and explosive reconnection in the ing Markov Chain Monte Carlo [MCMC]) that includes both magnetosphere. the pulse parameters for each included model component, as Because of the complex temporal morphology, which differs well as the hyper parameters determining the prior distribu- from burst to burst, it is difﬁcult to extract information from tions of the pulse parameters. We make no attempt to state our burst light curves. Many light curves show a pattern of one or conﬁdence in the presence of any given individual model com- multiple spikes, which are sometimes sitting on top of each ponent, but instead work with the posterior distribution rather other. To extract parameters that could be related to physical than a subset of it, and thus include the inherent uncertainty quantities such as the rise time of each spike and the waiting caused by noisy data in our inference over the sample. time distribution between spikes, we need to be able to infer Secondly, we present a ﬁrst, exploratory analysis of the kind both the number of spikes per burst as well as the individual of results that may be derived from a model such as the one we parameters of each spike: this requires efﬁcient sampling over present here. In a simple, naive way, we characterize differen- a large parameter space. At the same time, this process needs tial distributions of key quantities, and ﬁnd them inconsistent to operate automatically without the user’s intervention, both with predictions from self-organized criticality. We search for to avoid biases introduced by manual ﬁtting, as well as to and ﬁnd correlations between key physical quantities. This allow for an analysis of the large numbers of magnetar bursts work is only a ﬁrst step toward a generative model for magne- observed from various sources. tar bursts, and will be extended to samples of magnetar bursts Comparable studies abound in the literature on γ-Ray Bursts (and indeed, even over the population of magnetars as a whole) (GRBs), where variability has been used to study the details of in future work. the physical processes involved in the creation of GRB prompt In Section , we present our sample of bursts from SGR emission in γ-rays. GRBs have morphologically similar light J1550-5418, observed with the Gamma-ray Burst Monitor curves, usually ﬁt with a superposition of pulses, each with an (GBM) on board the Fermi Gamma-Ray Space Telescope. exponential rise and decay as well as a peakedness parameter The high sensitivity of the instrument as well as the large that parametrizes the roundness of the pulse peak. These number of observed bursts make this source an excellent target studies are generally split into two steps: a pulse-detection to demonstrate the power of the proposed methods, which we step, and an inference step, where detected pulses from many lay out in detail in Section . In Section , we test our model GRBs are combined into an ensemble to infer properties of on simple simulations of single-spiked bursts to test how well the entire population of bursts. Many of such studies rely on the model recovers the burst parameters, while in Section , we the visual identiﬁcation of pulses in the data (e.g. Norris et al. show an example of a model for a single burst. Finally, in 1996, 1999; Kocevski et al. 2003) Section , we apply our method to a large data set of bursts, One popular set of pulse detection algorithms involves the which allows us to extract a wealth of physically relevant time identiﬁcation of time bins with counts that exceed some ﬁxed scales from magnetar bursts. We place these timescales in multiple of the standard deviation of the background and the the context of the SOC framework, and tie them to physical DRAFT VERSION MAY 27, 2020 idePnretpirﬁinct taytpieosent uosifngtLrAoTEuXgshtysleoemfuelamteaipsjsvi.o1n2/1o6/n11either side of such parameters in the system in Section . a peak (see e.g. Li & Fenimore 1996; Quilligan et al. 2002; DISSECTING MAGNETAR VARIABILITY WITH BAYESIAN HIERARCHICAL MODELS

text: distributions

text: of

text: these

text: bursts:

text: the

text: differential

text: distribution

text: of

text: Guidorzi

text: Another

text: method

text: ﬁrst

text: ﬁnds

text: rapidly

text: varying

text: ﬂuence

text: (integrated

text: ﬂux)

text: and

text: the

text: cumulative

text: distribution

text: of

text: time

text: segments

text: in

text: the

text: data

text: using

text: a

text: Bayesian

text: Blocks

text: approach

text: waiting

text: times

text: are

text: similar

text: to

text: those

text: observed

text: in

text: earthquakes

text: and

text: (Scargle

text: et

text: al.

text: which

text: are

text: subsequently

text: used

text: as

text: ﬁrst

text: solar

text: ﬂares

text: (Cheng

text: et

text: al.

text: Go¨gˇu¨s¸

text: et

text: al.

text: Go¨gˇu¨s¸

text: guessed

text: for

text: ﬁtting

text: a

text: set

text: of

text: pulses

text: to

text: the

text: data,

text: removing

text: the

text: et

text: al.

text: Prieskorn

text: Kaaret

text: The

text: ﬂuence

text: distribution

text: smallest

text: pulses

text: based

text: on

text: some

text: signiﬁcance

text: criterion

text: (Scargle

text: in

text: particular

text: can

text: be

text: well-modelled

text: by

text: a

text: power

text: law

text: with

text: an

text: et

text: al.

text: Hakkila

text: Preece

text: index

text: of

text: believed

text: to

text: be

text: a

text: typical

text: signature

text: for

text: a

text: system

text: All

text: these

text: approaches

text: have

text: in

text: common

text: that

text: they

text: set

text: a

text: hard

text: obeying

text: self-organised

text: criticality

text: (SOC;

text: Bak

text: et

text: al.

text: limit

text: (usually

text: in

text: peak

text: amplitude,

text: but

text: sometimes

text: in

text: total

text: inte-

text: for

text: a

text: recent

text: introduction

text: and

text: review

text: on

text: SOC

text: in

text: astrophysics

text: grated

text: counts)

text: to

text: what

text: they

text: consider

text: to

text: be

text: a

text: burst,

text: resulting

text: see

text: Aschwanden

text: et

text: al.

text: In

text: the

text: SOC

text: framework,

text: the

text: in

text: a

text: set

text: of

text: pulses

text: above

text: this

text: threshold

text: that

text: is

text: then

text: used

text: for

text: the

text: physical

text: system

text: in

text: question

text: continuously

text: drives

text: itself

text: towards

text: subsequent

text: sample

text: inference.

text: Many

text: of

text: these

text: algorithms

text: are

text: a

text: critical

text: state,

text: without

text: requiring

text: ﬁne

text: tuning.

text: When

text: the

text: critical

text: additionally

text: tuned

text: deliberately

text: towards

text: rejecting

text: weak

text: features

text: state

text: is

text: reached,

text: relaxation

text: occurs

text: via

text: a

text: catastrophic

text: release

text: to

text: keep

text: the

text: number

text: of

text: false

text: positives

text: low.

text: This

text: is

text: equivalent

text: to

text: of

text: energy.

text: The

text: system

text: returns

text: to

text: a

text: subcritical

text: state,

text: and

text: the

text: applying

text: a

text: step-function

text: prior

text: to

text: the

text: relevant

text: pulse

text: parameters

text: cycle

text: begins

text: anew.

text: One

text: advantage

text: of

text: describing

text: a

text: system

text: in

text: (amplitude,

text: or

text: a

text: combination

text: of

text: amplitude

text: and

text: duration).

text: How-

text: the

text: SOC

text: framework

text: is

text: that

text: while

text: the

text: details

text: of

text: the

text: process

text: ever,

text: because

text: subsequent

text: analyses

text: are

text: based

text: on

text: the

text: ensemble

text: leading

text: to

text: the

text: critical

text: state

text: and

text: subsequent

text: relaxation

text: depend

text: of

text: pulses

text: above

text: the

text: threshold

text: only,

text: this

text: prior

text: is

text: never

text: taken

text: into

text: on

text: the

text: physical

text: processes

text: speciﬁc

text: to

text: the

text: system,

text: many

text: of

text: the

text: account

text: in

text: the

text: inferences

text: derived

text: from

text: the

text: sample

text: (for

text: a

text: dis-

text: overall

text: statistical

text: properties

text: are

text: universal.

text: cussion

text: of

text: this

text: problem

text: applied

text: to

text: the

text: waiting

text: time

text: distribution

text: At

text: present,

text: it

text: is

text: not

text: clear

text: whether

text: magnetar

text: bursts

text: are

text: of

text: pulses

text: in

text: GRBs,

text: see

text: Baldeschi

text: Guidorzi

text: smaller-scale

text: versions

text: of

text: the

text: processes

text: believed

text: to

text: produce

text: The

text: purpose

text: of

text: this

text: paper

text: is

text: twofold:

text: in

text: the

text: ﬁrst

text: part,

text: we

text: pro-

text: a

text: giant

text: ﬂare:

text: either

text: a

text: rapid

text: rupturing

text: of

text: the

text: neutron

text: star

text: due

text: pose

text: a

text: similar

text: approach

text: to

text: model

text: magnetar

text: bursts

text: as

text: a

text: linear

text: to

text: an

text: internal

text: re-structuring

text: of

text: the

text: magnetic

text: ﬁeld,

text: with

text: the

text: combination

text: of

text: simple

text: shapes,

text: but

text: without

text: enforcing

text: a

text: speciﬁc

text: energy

text: of

text: the

text: cracking

text: process

text: released

text: in

text: the

text: magnetosphere,

text: limit

text: on

text: the

text: signiﬁcance

text: of

text: the

text: detected

text: features.

text: Instead,

text: we

text: or

text: a

text: slow

text: untwisting

text: of

text: the

text: magnetic

text: ﬁeld

text: leading

text: to

text: ductile

text: allow

text: the

text: model

text: to

text: explore

text: a

text: parameter

text: space

text: (in

text: practice,

text: us-

text: deformations

text: in

text: the

text: crust

text: and

text: explosive

text: reconnection

text: in

text: the

text: ing

text: Markov

text: Chain

text: Monte

text: Carlo

text: [MCMC])

text: that

text: includes

text: both

text: magnetosphere.

text: the

text: pulse

text: parameters

text: for

text: each

text: included

text: model

text: component,

text: as

text: Because

text: of

text: the

text: complex

text: temporal

text: morphology,

text: which

text: differs

text: well

text: as

text: the

text: hyper

text: parameters

text: determining

text: the

text: prior

text: distribu-

text: from

text: burst

text: to

text: burst,

text: it

text: is

text: difﬁcult

text: to

text: extract

text: information

text: from

text: tions

text: of

text: the

text: pulse

text: parameters.

text: We

text: make

text: no

text: attempt

text: to

text: state

text: our

text: burst

text: light

text: curves.

text: Many

text: light

text: curves

text: show

text: a

text: pattern

text: of

text: one

text: or

text: conﬁdence

text: in

text: the

text: presence

text: of

text: any

text: given

text: individual

text: model

text: com-

text: multiple

text: spikes,

text: which

text: are

text: sometimes

text: sitting

text: on

text: top

text: of

text: each

text: ponent,

text: but

text: instead

text: work

text: with

text: the

text: posterior

text: distribution

text: rather

text: other.

text: To

text: extract

text: parameters

text: that

text: could

text: be

text: related

text: to

text: physical

text: than

text: a

text: subset

text: of

text: it,

text: and

text: thus

text: include

text: the

text: inherent

text: uncertainty

text: quantities

text: such

text: as

text: the

text: rise

text: time

text: of

text: each

text: spike

text: and

text: the

text: waiting

text: caused

text: by

text: noisy

text: data

text: in

text: our

text: inference

text: over

text: the

text: sample.

text: time

text: distribution

text: between

text: spikes,

text: we

text: need

text: to

text: be

text: able

text: to

text: infer

text: Secondly,

text: we

text: present

text: a

text: ﬁrst,

text: exploratory

text: analysis

text: of

text: the

text: kind

text: both

text: the

text: number

text: of

text: spikes

text: per

text: burst

text: as

text: well

text: as

text: the

text: individual

text: of

text: results

text: that

text: may

text: be

text: derived

text: from

text: a

text: model

text: such

text: as

text: the

text: one

text: we

text: parameters

text: of

text: each

text: spike:

text: this

text: requires

text: efﬁcient

text: sampling

text: over

text: present

text: here.

text: In

text: a

text: simple,

text: naive

text: way,

text: we

text: characterize

text: differen-

text: a

text: large

text: parameter

text: space.

text: At

text: the

text: same

text: time,

text: this

text: process

text: needs

text: tial

text: distributions

text: of

text: key

text: quantities,

text: and

text: ﬁnd

text: them

text: inconsistent

text: to

text: operate

text: automatically

text: without

text: the

text: user’s

text: intervention,

text: both

text: with

text: predictions

text: from

text: self-organized

text: criticality.

text: We

text: search

text: for

text: to

text: avoid

text: biases

text: introduced

text: by

text: manual

text: ﬁtting,

text: as

text: well

text: as

text: to

text: and

text: ﬁnd

text: correlations

text: between

text: key

text: physical

text: quantities.

text: This

text: allow

text: for

text: an

text: analysis

text: of

text: the

text: large

text: numbers

text: of

text: magnetar

text: bursts

text: work

text: is

text: only

text: a

text: ﬁrst

text: step

text: toward

text: a

text: generative

text: model

text: for

text: magne-

text: observed

text: from

text: various

text: sources.

text: tar

text: bursts,

text: and

text: will

text: be

text: extended

text: to

text: samples

text: of

text: magnetar

text: bursts

text: Comparable

text: studies

text: abound

text: in

text: the

text: literature

text: on

text: γ-Ray

text: Bursts

text: (and

text: indeed,

text: even

text: over

text: the

text: population

text: of

text: magnetars

text: as

text: a

text: whole)

text: (GRBs),

text: where

text: variability

text: has

text: been

text: used

text: to

text: study

text: the

text: details

text: of

text: in

text: future

text: work.

text: the

text: physical

text: processes

text: involved

text: in

text: the

text: creation

text: of

text: GRB

text: prompt

text: In

text: Section

text: we

text: present

text: our

text: sample

text: of

text: bursts

text: from

text: SGR

text: emission

text: in

text: γ-rays.

text: GRBs

text: have

text: morphologically

text: similar

text: light

text: observed

text: with

text: the

text: Gamma-ray

text: Burst

text: Monitor

text: curves,

text: usually

text: ﬁt

text: with

text: a

text: superposition

text: of

text: pulses,

text: each

text: with

text: an

text: (GBM)

text: on

text: board

text: the

text: Fermi

text: Gamma-Ray

text: Space

text: Telescope.

text: exponential

text: rise

text: and

text: decay

text: as

text: well

text: as

text: a

text: peakedness

text: parameter

text: The

text: high

text: sensitivity

text: of

text: the

text: instrument

text: as

text: well

text: as

text: the

text: large

text: that

text: parametrizes

text: the

text: roundness

text: of

text: the

text: pulse

text: peak.

text: These

text: number

text: of

text: observed

text: bursts

text: make

text: this

text: source

text: an

text: excellent

text: target

text: studies

text: are

text: generally

text: split

text: into

text: two

text: steps:

text: a

text: pulse-detection

text: to

text: demonstrate

text: the

text: power

text: of

text: the

text: proposed

text: methods,

text: which

text: we

text: step,

text: and

text: an

text: inference

text: step,

text: where

text: detected

text: pulses

text: from

text: many

text: lay

text: out

text: in

text: detail

text: in

text: Section

text: In

text: Section

text: we

text: test

text: our

text: model

text: GRBs

text: are

text: combined

text: into

text: an

text: ensemble

text: to

text: infer

text: properties

text: of

text: on

text: simple

text: simulations

text: of

text: single-spiked

text: bursts

text: to

text: test

text: how

text: well

text: the

text: entire

text: population

text: of

text: bursts.

text: Many

text: of

text: such

text: studies

text: rely

text: on

text: the

text: model

text: recovers

text: the

text: burst

text: parameters,

text: while

text: in

text: Section

text: we

text: the

text: visual

text: identiﬁcation

text: of

text: pulses

text: in

text: the

text: data

text: (e.g.

text: Norris

text: et

text: al.

text: show

text: an

text: example

text: of

text: a

text: model

text: for

text: a

text: single

text: burst.

text: Finally,

text: in

text: Kocevski

text: et

text: al.

text: Section

text: we

text: apply

text: our

text: method

text: to

text: a

text: large

text: data

text: set

text: of

text: bursts,

text: One

text: popular

text: set

text: of

text: pulse

text: detection

text: algorithms

text: involves

text: the

text: which

text: allows

text: us

text: to

text: extract

text: a

text: wealth

text: of

text: physically

text: relevant

text: time

text: identiﬁcation

text: of

text: time

text: bins

text: with

text: counts

text: that

text: exceed

text: some

text: ﬁxed

text: scales

text: from

text: magnetar

text: bursts.

text: We

text: place

text: these

text: timescales

text: in

text: multiple

text: of

text: the

text: standard

text: deviation

text: of

text: the

text: background

text: and

text: the

text: the

text: context

text: of

text: the

text: SOC

text: framework,

text: and

text: tie

text: them

text: to

text: physical

text: DRAFT

text: VERSION

text: MAY

text: idePnretpirﬁinct

text: taytpieosent

text: side

text: of

text: such

text: parameters

text: in

text: the

text: system

text: in

text: Section

text: a

text: peak

text: (see

text: e.g.

text: Li

text: Fenimore

text: Quilligan

text: et

text: al.

text: DISSECTING

text: MAGNETAR

text: VARIABILITY

text: WITH

text: BAYESIAN

text: HIERARCHICAL

text: MODELS


text: 21 the neutron star interior, the appropriate timescale is set by the To derive this distribution, we uniformely distributed a num- √ Alfve´n speed v = B/ 4πρ where B is the magnetic ﬁeld ber of weak points across the crust of a neutron star with typical A strength, giving (cid:18) (cid:19) (cid:32) (cid:33) radius following Equation 22, and then assumed failure at one initial point. We then calculated the time it would take for the information about that failure to propagate through the neutron 1/2 B 1015g/cm3 star interior. It is assumed that the impact of the original failure vA = 108cm/s 1016G ρ . (17) is sufﬁcient to trigger subsequent failure events in other weak points; the waiting time distribution is then the time for the information to travel between these weak points. This yields timescales for transmission of information In Figure 13 (top panel), we show the expected distribution through the neutron star core via torsional Alfve´n waves of (cid:46) of waiting times for transmission of information about failure 2R/vA = 0.02(R/10 km) s. Note however that the value points through the neutron star interior. We show both the of the ﬁeld strength B in magnetar cores is highly uncertain, analytical solution (from Equantion 23) as well as the result as is the appropriate value of the density ρ. In principle only of 106 Monte Carlo simulations of the process. The expected the charged component (∼ 5–10% of the core mass) should distribution is clearly at odds with what is observed: while it participate in Alfve´n waves, reducing ρ, however there are extends to values high enough to explain the long waiting times, mechanisms associated with superﬂuidity and superconduc- the distribution is dominated by a power law with a positive tivity that can couple the charged and neutral components exponent and a sharp drop-off at ∼ 0.04 s. This is clearly (van Hoven & Levin 2008; Andersson et al. 2009). As such not observed in the data, where the inter-spike waiting times this value is also consistent with the inter-spike timescale resemble much more a log-normal distribution. However, the distribution. In Figure 13 (top panel panel), we show the model assumed here does not include seismic waves traveling expected distribution of waiting times for propagation through through the star multiple times. Instead, it is assumed that the the neutron star interior. energy transferred in the ﬁrst instance is sufﬁcient to trigger The distance between two random points on a sphere of a follow-up event, such that the waiting times are directly radius R∗ may be written in terms of the colatitude angle related to the distance through the star between two points on φ ∈ [0, π]. Here we adopt spherical coordinates (r, φ, θ) and the surface. The sharp drop-off at long waiting times in this orientate our coordinate system such that one point is located at case corresponds to the travel time between the furthest two (R∗, 0, 0) and the other point at (R∗, φ, 0) – note that due to ro- points on the star (i.e. 2R). It is possible that seismic waves tational symmetry the distance is independent of the azimuthal could be reﬂected and travel through the interior multiple angle θ ∈ [0, 2π). The distance between the two points is then times, transferring small amounts of energy every time, and that then the waiting time is determined by the time it takes s(φ) = R∗(2 − 2 cos φ)1/2. (18) for the cumulative energy transmitted to a weak point to be sufﬁcient to trigger a starquake. However, including these effects would require much more detailed knowledge about the Next we choose to write φ in terms of x ∈ (0, 1], energy threshold for crust failure than is available, and beyond the scope of this simple toy model. φ = arccos(2x − 1), (19) For the second toy model, where information propagates through the neutron star crust only, the governing factor is the such that we obtain shear speed in the crust, v = (µ /ρ)1/2 where µ is the shear s s s modulus and ρ the density. The shear modulus is of the order s(x) = 2R∗(1 − x)1/2. (20) of the Coulomb potential energy ∼ Z2e2/r per unit volume − r3, where r ∼ (ρ/Am ) 1/3 is the inter-ion spacing, while p Z and A are the effective atomic number and mass number, Dividing the above equation by the appropriate velocity, i.e. respectively, of the ions in the crust. Using the shear modulus the core Alfve´n speed vA, gives us the following transforma- computed by Strohmayer et al. (1991) and scaling by typical tion function values for the inner crust (Douchin & Haensel 2001), the shear 2R∗ velocity is: (cid:32) (cid:33) (cid:18) (cid:19) t(x) = (1 − x)1/2, (21) v A 1/6 which describes the waiting time t in terms of the variable x. v =1.1(cid:18)× 10(cid:19)8cm/(cid:18)s ρ(cid:19) Z (24) Consequently, we assume(cid:26)that the random variable X obeys a s 1014g/cm3 38 uniform density distribution U(0, 1): 2/3 2/3 302 1 − X n × , 1 if 0 < x ≤ 1, A 0.25 u (x) = (22) X 0 otherwise. where X is the fraction of neutrons (see also Piro (2005)). n Then together with Eq. (21), we ﬁnd that the probability The timescales for transmission of information via shear waves (cid:46) density functi(cid:26)on (PDF) for the variable T is given by around the crust are therefore πR/vs = 0.03 (R/10 km) s. This is certainly consistent the values that we found for the wait times between individual spikes. fT (t) = 012 (vA/R∗)2 t iofth0e<rwtis≤e.2R∗/vA, (23) thaTthise mdiesatasnucreedbaectwroesesnthtewsouprfoaicnetsofonthae sspphheerree,omfaryadeiquusaRlly∗ be represented solely in terms of the colatitude angle φ ∈ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: the

text: neutron

text: star

text: interior,

text: the

text: appropriate

text: timescale

text: is

text: set

text: by

text: the

text: To

text: derive

text: this

text: distribution,

text: we

text: uniformely

text: distributed

text: a

text: num-

text: Alfve´n

text: speed

text: v

text: B/

text: 4πρ

text: where

text: B

text: is

text: the

text: magnetic

text: ﬁeld

text: ber

text: of

text: weak

text: points

text: across

text: the

text: crust

text: of

text: a

text: neutron

text: star

text: with

text: typical

text: A

text: strength,

text: giving

label: equation

text: radius

text: following

text: Equation

text: and

text: then

text: assumed

text: failure

text: at

text: one

text: initial

text: point.

text: We

text: then

text: calculated

text: the

text: time

text: it

text: would

text: take

text: for

text: the

text: information

text: about

text: that

text: failure

text: to

text: propagate

text: through

text: the

text: neutron

text: B

text: star

text: interior.

text: It

text: is

text: assumed

text: that

text: the

text: impact

text: of

text: the

text: original

text: failure

text: vA

text: ρ

text: is

text: sufﬁcient

text: to

text: trigger

text: subsequent

text: failure

text: events

text: in

text: other

text: weak

text: points;

text: the

text: waiting

text: time

text: distribution

text: is

text: then

text: the

text: time

text: for

text: the

text: information

text: to

text: travel

text: between

text: these

text: weak

text: points.

text: This

text: yields

text: timescales

text: for

text: transmission

text: of

text: information

text: In

text: Figure

text: (top

text: panel),

text: we

text: show

text: the

text: expected

text: distribution

text: through

text: the

text: neutron

text: star

text: core

text: via

text: torsional

text: Alfve´n

text: waves

text: of

text: of

text: waiting

text: times

text: for

text: transmission

text: of

text: information

text: about

text: failure

text: km)

text: s.

text: Note

text: however

text: that

text: the

text: value

text: points

text: through

text: the

text: neutron

text: star

text: interior.

text: We

text: show

text: both

text: the

text: of

text: the

text: ﬁeld

text: strength

text: B

text: in

text: magnetar

text: cores

text: is

text: highly

text: uncertain,

text: analytical

text: solution

text: (from

text: Equantion

text: as

text: well

text: as

text: the

text: result

text: as

text: is

text: the

text: appropriate

text: value

text: of

text: the

text: density

text: ρ.

text: In

text: principle

text: only

text: of

text: Monte

text: Carlo

text: simulations

text: of

text: the

text: process.

text: The

text: expected

text: the

text: charged

text: component

text: of

text: the

text: core

text: mass)

text: should

text: distribution

text: is

text: clearly

text: at

text: odds

text: with

text: what

text: is

text: observed:

text: while

text: it

text: participate

text: in

text: Alfve´n

text: waves,

text: reducing

text: ρ,

text: however

text: there

text: are

text: extends

text: to

text: values

text: high

text: enough

text: to

text: explain

text: the

text: long

text: waiting

text: times,

text: mechanisms

text: associated

text: with

text: superﬂuidity

text: and

text: superconduc-

text: the

text: distribution

text: is

text: dominated

text: by

text: a

text: power

text: law

text: with

text: a

text: positive

text: tivity

text: that

text: can

text: couple

text: the

text: charged

text: and

text: neutral

text: components

text: exponent

text: and

text: a

text: sharp

text: drop-off

text: at

text: s.

text: This

text: is

text: clearly

text: (van

text: Hoven

text: Levin

text: Andersson

text: et

text: al.

text: As

text: such

text: not

text: observed

text: in

text: the

text: data,

text: where

text: the

text: inter-spike

text: waiting

text: times

text: this

text: value

text: is

text: also

text: consistent

text: with

text: the

text: inter-spike

text: timescale

text: resemble

text: much

text: more

text: a

text: log-normal

text: distribution.

text: However,

text: the

text: distribution.

text: In

text: Figure

text: (top

text: panel

text: panel),

text: we

text: show

text: the

text: model

text: assumed

text: here

text: does

text: not

text: include

text: seismic

text: waves

text: traveling

text: expected

text: distribution

text: of

text: waiting

text: times

text: for

text: propagation

text: through

text: through

text: the

text: star

text: multiple

text: times.

text: Instead,

text: it

text: is

text: assumed

text: that

text: the

text: the

text: neutron

text: star

text: interior.

text: energy

text: transferred

text: in

text: the

text: ﬁrst

text: instance

text: is

text: sufﬁcient

text: to

text: trigger

text: The

text: distance

text: between

text: two

text: random

text: points

text: on

text: a

text: sphere

text: of

text: a

text: follow-up

text: event,

text: such

text: that

text: the

text: waiting

text: times

text: are

text: directly

text: radius

text: R∗

text: may

text: be

text: written

text: in

text: terms

text: of

text: the

text: colatitude

text: angle

text: related

text: to

text: the

text: distance

text: through

text: the

text: star

text: between

text: two

text: points

text: on

text: φ

text: π].

text: Here

text: we

text: adopt

text: spherical

text: coordinates

text: (r,

text: φ,

text: θ)

text: and

text: the

text: surface.

text: The

text: sharp

text: drop-off

text: at

text: long

text: waiting

text: times

text: in

text: this

text: orientate

text: our

text: coordinate

text: system

text: such

text: that

text: one

text: point

text: is

text: located

text: at

text: case

text: corresponds

text: to

text: the

text: travel

text: time

text: between

text: the

text: furthest

text: two

text: (R∗,

text: and

text: the

text: other

text: point

text: at

text: (R∗,

text: φ,

text: note

text: that

text: due

text: to

text: ro-

text: points

text: on

text: the

text: star

text: (i.e.

text: It

text: is

text: possible

text: that

text: seismic

text: waves

text: tational

text: symmetry

text: the

text: distance

text: is

text: independent

text: of

text: the

text: azimuthal

text: could

text: be

text: reﬂected

text: and

text: travel

text: through

text: the

text: interior

text: multiple

text: angle

text: θ

text: The

text: distance

text: between

text: the

text: two

text: points

text: is

text: then

text: times,

text: transferring

text: small

text: amounts

text: of

text: energy

text: every

text: time,

text: and

text: that

text: then

text: the

text: waiting

text: time

text: is

text: determined

text: by

text: the

text: time

text: it

text: takes

text: s(φ)

text: cos

text: for

text: the

text: cumulative

text: energy

text: transmitted

text: to

text: a

text: weak

text: point

text: to

text: be

text: sufﬁcient

text: to

text: trigger

text: a

text: starquake.

text: However,

text: including

text: these

text: effects

text: would

text: require

text: much

text: more

text: detailed

text: knowledge

text: about

text: the

text: Next

text: we

text: choose

text: to

text: write

text: φ

text: in

text: terms

text: of

text: x

text: energy

text: threshold

text: for

text: crust

text: failure

text: than

text: is

text: available,

text: and

text: beyond

text: the

text: scope

text: of

text: this

text: simple

text: toy

text: model.

text: φ

text: For

text: the

text: second

text: toy

text: model,

text: where

text: information

text: propagates

text: through

text: the

text: neutron

text: star

text: crust

text: only,

text: the

text: governing

text: factor

text: is

text: the

text: such

text: that

text: we

text: obtain

text: shear

text: speed

text: in

text: the

text: crust,

text: v

text: (µ

text: where

text: µ

text: is

text: the

text: shear

text: s

text: s

text: s

text: modulus

text: and

text: ρ

text: the

text: density.

text: The

text: shear

text: modulus

text: is

text: of

text: the

text: order

text: s(x)

text: of

text: the

text: Coulomb

text: potential

text: energy

text: per

text: unit

text: volume

text: where

text: r

text: (ρ/Am

text: is

text: the

text: inter-ion

text: spacing,

text: while

text: p

text: Z

text: and

text: A

text: are

text: the

text: effective

text: atomic

text: number

text: and

text: mass

text: number,

text: Dividing

text: the

text: above

text: equation

text: by

text: the

text: appropriate

text: velocity,

text: i.e.

text: respectively,

text: of

text: the

text: ions

text: in

text: the

text: crust.

text: Using

text: the

text: shear

text: modulus

text: the

text: core

text: Alfve´n

text: speed

text: vA,

text: gives

text: us

text: the

text: following

text: transforma-

text: computed

text: by

text: Strohmayer

text: et

text: al.

text: and

text: scaling

text: by

text: typical

text: tion

text: function

text: values

text: for

text: the

text: inner

text: crust

text: (Douchin

text: Haensel

text: the

text: shear

text: velocity

text: is:

text: t(x)

text: v

text: A

text: which

text: describes

text: the

text: waiting

text: time

text: t

text: in

text: terms

text: of

text: the

text: variable

text: x.

text: v

text: Z

text: Consequently,

text: we

text: the

text: random

text: variable

text: X

text: obeys

text: a

text: s

text: uniform

text: density

text: distribution

text: X

text: n

text: if

text: x

text: A

text: u

text: (x)

text: X

text: otherwise.

text: where

text: X

text: is

text: the

text: fraction

text: of

text: neutrons

text: (see

text: also

text: Piro

text: n

text: Then

text: together

text: with

text: Eq.

text: we

text: ﬁnd

text: that

text: the

text: probability

text: The

text: timescales

text: for

text: transmission

text: of

text: information

text: via

text: shear

text: waves

text: density

text: (PDF)

text: for

text: the

text: variable

text: T

text: is

text: given

text: by

text: around

text: the

text: crust

text: are

text: therefore

text: πR/vs

text: km)

text: s.

text: This

text: is

text: certainly

text: consistent

text: the

text: values

text: that

text: we

text: found

text: for

text: the

text: wait

text: times

text: between

text: individual

text: spikes.

text: fT

text: (t)

text: t

text: thaTthise

text: sspphheerree,omfaryadeiquusaRlly∗

text: be

text: represented

text: solely

text: in

text: terms

text: of

text: the

text: colatitude

text: angle

text: φ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##