Towards neutron star asteroseismology

Pablo Cerdá-Durán University of Valencia

Trento 9/10/2017

Collaborators: M. Gabler (MPA) A. Torres-Forné (U. Valencia) A. Passamonti (U. Alicante) E. Müller (MPA) J.A. Font (U. Valencia) N. Stergioulas (U. Thessaloniki)

Outline

•  Magnetar QPOs in giant flares

•  GWs from proto-neutron star oscillations in core-collapse supernovae

Magnetar QPOs in giant flares

Magnetars •  Anomalous X-ray pulsars (AXP)

and Soft Gamma repeaters (SGRs)

•  Spin-down-inferred magnetic field ~1014 – 1015 G

•  Magnetically powered emission àLX ~1033-1035 erg s-1

•  Recurrent gamma/X-ray activity (bursts, giant flares and outbursts)

February 26, 2016 1:22 WSPC/INSTRUCTION FILE kaspi˙kramer˙v4

Radio Pulsars: The Neutron Star Population & Fundamental Physics 5

Fig. 3. P -P diagram. Red dots indicate known radio pulsars as of April 2014. Blue circles repre-sent binaries. Stars represent associated supernova remnants. Magnetars are represented by greentriangles (see §5). The XDINS and CCOs (see §4,5) are pink and yellow triangles, respectively.RRATs (§9) are in cyan. Solid grey lines are of constant magnetic field (Eq. 1) and dotted linesare of constant characteristic age (Eq. 2). From Tauris et al. (2014).

Finally, a pulsar’s spin-down luminosity Lsd (also known as E, where E ≡ 12Iω

2

with ω ≡ 2π/P is the stellar rotational kinetic energy) can be estimated from Pand P and is given by

Lsd =d

dt

!

1

2Iω2

"

= Iωω = 4π2IP

P 3= 4× 1031

#

˙P−15

P1

$

erg/s, (3)

where P−15 is P in units of 10−15 and P1 is the period in units of seconds. Lsd

represents the power available for conversion into electromagnetic radiation, anupper limit on the (non-thermal; see §2.1) radiation a pulsar can produce. For thisreason, radio pulsars are also known as ‘rotation-powered pulsars.’

A traditional way of summarizing the pulsar population is via the P -P diagram(Fig. 3). Here the spin periods of pulsars are plotted on the x-axis and P on they-axis. The swarm of conventional radio pulsars clearly has its P peak near ∼500ms, with typical B ≃ 1011 G and characteristic age τc ≃ 107 yr. The youngest radio

Kaspi & Kramer 2016

•  Theoretical model (Duncan & Thompson 1996, Lyutikov 2003, Beloborodov & Levin 2014… ): 1 : Stresses build in the crust à Hall drift in the crust (Vigano et al 2013) 2a : Crust breaks (crustquake) and disturbs magnetosphere 2b : Crust deforms plastically à magnetospheric reconnection event 3 : Fast energy release à soft γ-ray spike 4 : Magnetically trapped fireball à x-ray emission

Quasi-periodic oscillations (QPOs)

Soft gamma-ray flash (~0.1 s) à saturated detectors

Strohmayer & Watts 2006

X-ray tail (~100 s) à fireball evaporation

QPOs in giant flares SGR 0526-66 giant flare on March 5, 1979 : à 43 Hz ? (Barat et al 1983) SGR 1900+14 giant flare on Aug. 27, 1998 : à 28, 56, 84, 155 Hz (Strohmayer & Watts 2005) SGR 1806-20 giant flare on Dec 27, 2004 : à 18, 26, 30, 92, 150, 625, 1840 Hz (Israel et al 2005; Watts & Strohmayer 2006, Strohmayer & Watts 2006) SGR J1550-5418 intermediate flare storm on Jan 2009 à 93, 127, 260 Hz (Huppenkothen et al 2014)

Crust shear oscillations model (Schomaker & Thorne 1983, Piro 2005, Samuelsson & Andersson 2007)

Torsional shear oscillations of the crust

+ Explains low/high freq. QPOs + Explains similar freq. At different B-field -  Cannot explain all QPOs at once -  Neglects B-field

Samuelsson & Andersson 2007

RC R

Δ

Magneto-elastic model At B~1015 G: •  Alfvén speed ~ shear speed •  Magneto-elastic waves in the crust •  Alfvén waves in the core •  Long-lived QPOs by resonances

between continuum and crust

Gabler et al 2012

cm

Thin-crust approx: Levin 2006, 2007, Sotani et al 2006, 2008, 2009, CD et al 2009, Colaiuda et al 2009, Lander & Jones 2011, Passamonti & Lander 2012 Extended crust: Glampedakis et al. 2006, Gabler et al 2011,2012, 2013a, 2013b, 2016, Colaiuda et al 2011 &

2012, Van Hoven & Levin 2011 & 2012, Passamonti & Lander 2014, Asai & Lee 2014, Link 2015, Sotani 2015

Numerical code •  2D axisymmetric simulations of torsional modes (MCoCoA code) •  Magneto-elastic equations in General relativity (Ideal MHD +linear elasticity) •  Realistic EoS and shear modulus •  Self-consistent magnetic field configurations •  Simplified treatment for superfluidity Explored parameter space (so far): •  Mass •  Equation of state •  Shear modulus •  Magnetic field strength •  Magnetic field configuration (dipolar, toroidal, crustal field, tangled field) •  Superfluid neutrons in the core

Numerical simulations

Oscillations outbreak field

Pure crustal oscillations cannot exist in magnetars

Observational constraints •  Outbreak (minimum) field à bellow no QPOs at the surface

•  Maximum field à above Alfvén continuum (short-lived QPOs)

•  Lowest frequency QPO (~30 Hz)

•  High frequency QPO (~600 Hz)

Bayesian analysis

Parameters ε*Xc : effective mass x proton fraction (neutron-proton coupling) B : average magnetic field at the crust M/R : compactness (mass / radius) cs : average shear speed at the crust Δr : crust thickness Priors -  Magnetic field: Boutbreak > B > Bmax -  Other: flat prior

Observations (SGR 1806-20) -  2U2 = 29 Hz -  2t1 = 625 Hz

Bayes theorem

Marginalized 1D posterior PDF G

able

r et a

l 201

7

ε*Xc compatible with a large fraction of the core being superfluid B : similar to spin-down estimates (2x1015G) à dipolar component dominant

Marginalized 1D posterior PDF

Gabler et al 2017 M/R : favours smaller values à stiff EOS or low mass (<1.5 Msun) cs : relatively large values à upper end of theoretical expectation Δr : compatible with theoretical expectation of ~1 km

Conclusions (magnetar QPOs)

•  It is possible to extract information about the properties of NS interiors with the observation of magnetar QPOs

•  The main limitation is observational, not the theoretical modelling:

•  Lack of observations à this may change in the future with new observatories (XTP, LOFT?, NICER, XIPE, IXPE, Athena...), which could look at QPOs in intermediate flares.

•  Difficult to identify the fundamental frequency (e.g. recent claim of 9.2 Hz QPO, Pumpe et al 2017)

•  For the future •  Effect of superconductivity •  Non-axisymmetric modes (3D) •  Presence of a strong toroidal field •  Link between oscillation and emission à magnetospheric emission

Gravitational waves from proto-neutron star oscillations in core collapse supernova

Fate of massive stars (8-100 Msun)

Collapse of iron core

Type II SN

black hole Failed SN (unnovae)

Long GRB +

Type Ib/c SN

neutron star

black hole by fallback

?

Core-collapse supernovae

§ Proto-neutron star (PNS) § Cooling by neutrinos § Trapped neutrinos escape in ~1 s § R~30 km à 10 km in ~1 s §  Increases mass with time

§ Hot bubble § Between PNS and shock §  Instabilities: convection & SASI §  If ~1% neutrino energy deposited à SNe E~1051 erg

Janka

18

Gravitational wave signal (BH formation)

0

1

2

3

4

0 200 400 600 800 1000 1200 1400 1600

f [

kH

z]

t - tbounce [ms]

A

BC

C

CC

C

CC

D

(b)

-23

-22

-21

-20

A

BC

C

CC

C

CC

D

(b)

-50

0

50

D h

x [c

m]

AB CC CCCCCD(b)

bounce accretion onto PNS BH formation(a)

Cerda-Duran et al 2013 SASI

g-modes

g-modes 8 H. Andresen et al.

r = 10 km

fm = 0.1fmax

mfm = 0

⇢ = 10

10

g/cm

3

Standing accretion shock

A

A1

A2B

C

Figure 4. Schematic overview of the regions of hydrodynamical activity. InSection 3.4 we investigate the contribution to the total GW signal from threedi↵erent layers. The PNS, indicated by the shaded red area, is divided intotwo layers: Layer A includes the convectively unstable region in the PNS(layer A1) and the overshooting layer A2 directly above it. The boundarybetween the convective layer and the overshooting layer is indicated by adashed curve within layer A. The second layer, layer B, extends from the topof the overshooting region and out to the PNS surface, defined by a fiducialdensity of 1010 g cm�3. Layer C extends from the PNS surface to the outerboundary of our simulation volume. Layer C therefore includes the post-shock region, the standing accretion shock (indicated by the blue line), andthe pre-shock region. Formal definitions of the boundaries between layersare given on the right hand side, see Section 3.4 for details.

0 5 10 15 20 25 30 35 40 45

r [km]

�6

�4

�2

0

1

f m[10

18

gcm/s]

fmax

m

Overshootinglayer(A2)

Convectivelayer(A1)

Figure 5. Turbulent mass flux fm (blue curve) for model s27, calculated192 ms after core bounce. The shaded region indicates the convectivelyunstable region and the overshooting layer, which are lumped together aslayer A (see Fig. 4).

analysed with some care. Since we plot the square of the Fouriercoe�cients one can not add the values of layers A, B, and C to-gether and recover the value for the total signal. In addition, arte-facts can arise due to e↵ects at the boundaries between layers, asin the case of model s20 (top right panel of Fig. 6). There is anartificially strong peak at 160 Hz, particularly from layer B. Wehave confirmed that shifting the boundary between layers A and Binwards reduces this peak significantly. The exact values of the low-frequency amplitudes are sensitive to the boundary definition, butthe fact that all three layers contribute to emission below 250 Hzis robust. The high-frequency component is less a↵ected by suchartefacts since the high-frequency emission is mostly confined tolayer A.

The results of this dissection of the contributions to the in-tegral in Eq. (6) are somewhat unexpected. The high-frequencyemission mostly stems from aspherical mass motions in layer Aand there is only a minor contribution from layer B, which hasbeen posited as the crucial region for GW emission during the pre-explosion phase in works based on 2D simulations (Marek et al.2009; Murphy et al. 2009; Müller et al. 2013). Aspherical massmotions in layer C hardly contribute to this component at all.

By contrast, all three regions contribute to the low-frequencysignal (i.e. emission at frequencies lower than 250 Hz) to a sim-ilar degree. This is also surprising if the dominant frequency ofthis component appears to be set by the SASI as speculated before.In this case, one might expect that the fluid motions responsiblefor GW emission are propagating waves in layer C and perhapslayer B, where the conversion of vorticity perturbations into acous-tic perturbations occurs in the SASI feedback cycle.

3.5 Origin of High-Frequency Emission

What do these findings imply about the physical mechanisms thatgive rise to GW emission and determine their frequency? Let usfirst address the high-frequency signal. Recent 2D studies have con-nected GW emission at &500 Hz to oscillatory modes (g-modes)excited either in the PNS surface (layer B) from above by down-flows impinging onto the PNS (Marek et al. 2009; Murphy et al.2009; Müller et al. 2013), or from below by PNS convection(Marek et al. 2009; Müller et al. 2012a, 2013). Prior to shock re-vival, the excitation of oscillations by mass motions in the gainlayer was found to be dominant, with PNS convection taking overas the dominant excitation mechanism only after the onset of theexplosion (Müller et al. 2012a, 2013). The typical angular fre-quency of such processes is roughly given by the Brunt-Väisäläfrequency, N, in the convectively stable region between the gainregion and the PNS convection zone,

N2 =1⇢

@�

@r

"1c2

s

@P@r� @⇢@r

#, (17)

where cs is the sound speed. Müller et al. (2013) further investi-gated the dependence of this frequency on the mass M, the radiusR, and the surface temperature T of the PNS to explain the secu-lar increase of N during the contraction of the PNS and a tendencytowards higher frequencies for more massive neutron stars.

Our results confirm that the peak frequency of the high-frequency GW emission is still set by the Brunt-Väisälä frequencyin 3D and therefore point to a similar role of buoyancy forces in de-termining the spectral structure of the high-frequency component.As shown in Fig. 7 for model s27, we find very good agreementbetween the peak GW frequency, fpeak, and the Brunt-Väisälä fre-quency, N, calculated at the outer boundary of the overshooting

MNRAS 000, 000–000 (0000)

Shock Hot bubble àconvection/SASI PNS surface (stable) à g-modes PNS interior àconvection Inner core (stable)àg-modes

Stability criterion •  N2<0 :

unstable (convection)

•  N2>0 : stable (g-modes)

Brunt Väisälä frequency (N2)

Shock PNS surface Inner core Andresen et al 2016

20

Gravitational wave signal (Supernovae) The Astrophysical Journal, 766:43 (21pp), 2013 March 20 Muller, Janka, & Marek

Figure 10. Comparison of the wavelet spectra (identical to those in Figures 2 and 3, respectively) of model G11.2 (left) and G15 (right) and the prediction ofEquation (17) for the typical GW frequency in terms of the PNS mass and radius and the mean energy of electron antineutrinos measured by an observer at infinity.Equation (17) describes the evolution of the typical frequency fairly well and only overestimates fpeak somewhat at late times.

4.4. Explosion Phase—Proto-neutron StarConvection and Late-time Bursts

As established by Murphy et al. (2009) and Yakunin et al.(2010), the emission of high-frequency GWs subsides to areduced level once the shock accelerates outward becausethe excitation of oscillations in the PNS surface by violentconvective motions in the hot-bubble region largely ceases.This is well reflected in the GW signals of models G11.2 andG15 (Figures 2 and 3) in the initial phase of the explosion, asthe GW amplitude drops noticeably after 400 ms and 600 ms,respectively. However, the surface g-mode oscillations remainthe dominant source of high-frequency GWs during this phase:the wavelet spectra clearly show that the late-time signal comesfrom the same emission band as the signal from the accretionphase (which shifts continuously to higher frequencies as theneutron star contracts). An analysis of the integrand ψ inthe quadrupole formula (1) as in Section 4.2.1 also confirmsthat aspherical motions in the PNS surface region are still thedominant source of GWs. PNS convection now provides anexcitation mechanism for the oscillations (Muller & Janka 1997;Marek et al. 2009; Murphy et al. 2009; Muller et al. 2012c), butdue to the subsonic character (with Mach numbers !0.05) ofthe convective motions, the GW amplitude AE2

20 remains rathersmall.

However, model G11.2 contradicts the established picture ofsubsiding GW emission during the explosion. For this progeni-tor, we observe several “bursts” of stronger GW emission later inthe explosion phase on at least three occasions (630 ms, 790 ms,and 820 ms), during which the amplitude can become compa-rable to that coming from the phase of hot-bubble convection.About 40% of the GW energy is emitted later than 600 ms afterbounce (Figure 11), which is in stark contrast to the more vigor-ous explosion in model G15 with faster shock expansion. Thesebursts occur when matter falls back onto the PNS through anewly developing downflow, which excites g-mode oscillationsin the PNS surface region (Figure 12). The formation of newdownflows in the explosion phase is a consequence of the smallexplosion energy of model G11.2 (see Paper II), and one couldspeculate that low-energy fallback supernovae may generallyreveal themselves through such multiple GW burst episodes.While the development of such an accretion downflow may befacilitated by the constraint of axisymmetry, it is conceivablethat funnel- or plume-shaped downdrafts may impinge on the

0 0.2 0.4 0.6 0.8 1time after bounce [s]

0

1

2

3

4

5

EG

W [1

045 e

rg]

G15G11.2

Figure 11. Energy EGW radiated in GWs as a function of time for the explosionmodels G11.2 (black solid line) and G15 (red dashed line) for the entire durationof the simulation. For model G15, most of the energy is radiated around theonset of the explosion between 400 ms and 600 ms after bounce. By contrast,late-time GW bursts carry a sizable fraction of the radiated energy in the caseof model G11.2.(A color version of this figure is available in the online journal.)

PNS in a similar manner in a weak explosion in 3D (cf. some 3Dresults of Muller et al. 2012c, where long-lasting accretion af-ter the explosion was associated with time-dependent accretiondowndrafts). Given the fact that Takiwaki et al. (2012) do notobtain faster shock propagation in 3D than in 2D for this par-ticular progenitor in their simulations using the IDSA scheme,it is not unlikely that the explosion energies will remain lowas in our 2D model (a few 1049 erg; see Muller et al. 2012b)and that late-time bursts seen in our simulations will survive in3D. Without detailed 3D simulations of this sort, however, it isunclear how massive the downdrafts can become and what GWburst amplitudes they can cause.

Several interesting features of these late-time bursts should benoted: between the few instances where new accretion downflowdevelops, aspherical mass motions above the PNS surface areapparently not strong enough to produce significant noise inthe GW signal, and the g-mode frequency therefore emergesmuch more cleanly from the spectrum during the late-time GWbursts than in the accretion phase. The narrowband characterof the spectrum (see Figure 2 and the red curve in Figure 14)is potentially helpful both for the detection of the GW signaland for the interpretation of the data (e.g., by allowing for

11

Murphy et al 2009 Müller et al 2013 Yakunin et al 2015

Kuroda et al 2016 Andresen et al 2017

Asteroseismology

•  All features can be understood in terms of different modes present in the PNS-shock system

•  Frequencies depend on PNS properties •  Mass •  Radius •  Temperature (or in general thermal structure) •  Rotation (small correction ~20%, even for this model)

•  Amplitude of each mode depends on •  Detailed microphysics •  Excitation of modes due to instabilities (convection/SASI) •  2D vs 3D modelling

Is parameter estimation possible?

First step: can we model waveforms as PNS oscillations?

Linear perturbation analysis

•  Linear perturbations of a spherical background à 1D average of 2D simulations •  Hydrostatic equilibrium à the PNS evolves in time-scales of 100 ms •  Cowling approximation •  Model 35OC of CD et al 2013

Torres-Forné et a 2017

Brunt-Väisälä frequency Lamb frequency

σ: eigenfrequency

Eigenfrequencies Torres-Forné et a 2017

p-modes

g-modes

f-modes

GW spectrograms vs eigenfrequencies Torres-Forné et a 2017

p-modes and SASI Spectrogram of the shock location p-modes rescaled by number of nodes (n)

•  Oscillations of the shock (SASI) trace the p-mode frequencies •  Two low frequency bellow the lowest frequency p-mode à integer fractions of p-

modes •  SASI theory (Foglizzo et al 2007): advective-acoustic cycle (acoustic = p-mode)

GW emission timescale Torres-Forné et a 2017

•  p-modes emit much more efficiently than g-modes àexplains why we see many p-modes but not high order g-modes

•  l=4 highly suppressed because we consider a spherical background

Less efficient emitters

More efficient emitters

Conclusions (PNS)

•  GW emission in core-collapse SNe can be explained to a large degree by the excitation of PNS modes

•  This will allow in the future to do parameter estimation (M, R, …) from GW observations

•  We need accurate methods to compute the eigenfrequencies and estimate the systematic uncertainties of the model

•  Future work: •  Remove cowling approximation •  Deformated background •  Quasi-radial oscillations (l=0)