horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a...

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Crust breaking on accreting stars C. J. Horowitz, Indiana U, Astro-solids, INT, Apr. 2018 0 2 4 6 8 10 12 0.8 0.4 0.6 IU-FSU crust Ω r (km) ε/σ = 1.0 core φ φ

Transcript of horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a...

Page 1: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Crust breaking on accreting stars

C. J. Horowitz, Indiana U, Astro-solids, INT, Apr. 2018

0 2 4 6 8 10 12

0.8

0.4

0.6

IU-FSU

crustΩ

r (km)

ε/σ = 1.0core

φ φ

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Crust Breaking on Accreting stars

• Review MD simulations of crust breaking. —> Crust is very strong!

• Crust breaking as a star spins down.

• Crust breaking as a star spins up.

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5 Chandra

2 XMM-Newton

KS 1731-260: observations

KS 1731-260: did it cool?

Wijnands et al. 2002

YES!

Cooling of crust of KS 1731-260

X-ray observations of a NS cooling after long outburst. --Ed. Cackett

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Cooling of KS 1730-260 Surface After Extended Outburst

Neutron star cooling in KS 1731–260 5

Figure 2. Theoretical cooling curves for (a) M = 1.6 M⊙ and (b) 1.4 M⊙ neutron stars, and (c) for stars with both M compared withobservations. The curves are explained in Table 1 and in text.

idity can change the core heat capacity and neutrino lu-minosity, but the principal conclusions will be the same.Our calculations are not entirely self-consistent. For in-stance, the surface temperature was inferred from observa-tions (Cackett et al. 2006), assuming neutron star massesand radii different from those used in our cooling models.This inconsistency cannot affect our main conclusions, butit would be desirable to infer T∞

s for our neutron star mod-els. The thermal relaxation in the quiescent state has beenobserved also (Cackett et al. 2006) for another neutron starX-ray transient, MXB 1659–29. We hope to analyse thesedata in the next publication.

5 ACKNOWLEDGMENTS

We are grateful to A. I. Chugunov, O. Y. Gnedin, K. P.Levenfish and Yu. A. Shibanov for useful discussions, andto the referee, Ulrich Geppert, for valuable remarks. Thework was partly supported by the Russian Foundation forBasic Research (grants 05-02-16245, 05-02-22003), by theFederal Agency for Science and Innovations (grant NSh9879.2006.2), and by the Polish Ministry of Science andHigher Education (grant N20300632/0450). One of the au-thors (P.S.) acknowledges support of the Dynasty Founda-tion and perfect conditions of the Nicolaus Copernicus As-tronomical Center in Warsaw, where this work was partlyperformed. P.H. and A.P. thank the Institute for NuclearTheory at the University of Washington for hospitality andthe U.S. Department of Energy for partial support duringthe completion of this work.

References

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Baiko D. A., Haensel P., Yakovlev D. G., 2001, A&A, 374,151

Brown E. F., 2000, ApJ, 531, 988Brown E. F., Bildsten L., Rutledge R. E., 1998, ApJ, 504,L95

Cackett E. M., Wijnands R., Linares M., Miller J. M.,Homan J., Lewin W. H. G., 2006, MNRAS, 372, 479

Gnedin O. Y., Yakovlev D. G., Potekhin A. Y., 2001, MN-RAS, 324, 725

Gupta S., Brown E. F., Schatz H., Moller P., Kratz K.-L.,2007, ApJ, 662, 1188

Gusakov M. E., Kaminker A. D., Yakovlev D. G., GnedinO. Y., 2005, MNRAS, 363, 555

Haensel P., Zdunik J. L., 1990, A&A, 227, 431Haensel P., Zdunik J. L., 2003, A&A, 404, L33Haensel P., Zdunik J. L., 2007, A&A, in preparationHaensel P., Potekhin A. Y., Yakovlev D. G., 2007, NeutronStars. 1. Equation of State and Structure, Springer, New-York

Heiselberg H., Hjorth-Jensen M., 1999, ApJ, 525, L45Jones P. B., 2004, MNRAS, 351, 956Lombardo U., Schulze H.-J., 2001, Lecture Notes Phys, 578,30

Page D., Geppert U., Weber F., 2006, Nucl. Phys. A, 777,497

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Schatz H. et al., 2001, Phys. Rev. Lett., 86, 3471Shternin P. S., Yakovlev D. G., 2006, Phys. Rev. D, 74,043004.

Shternin P. S., Yakovlev D. G., 2007, Phys. Rev. D, 75,103004

Sunyaev R. A. et al., 1990, SvA Lett., 16, 59Ushomirsky G., Rutledge R. E., 2001, MNRAS, 325, 1157Wambach J., Ainsworth T. L., Pines D., 1993, Nucl. Phys.A, 555, 128

Wijnands R., Miller J. M., Markwardt C., Lewin W.H.G.,van der Klis M., 2001, ApJ, 560, L159

Yakovlev D. G., Pethick C. J., 2004, ARA&A, 42, 169Yakovlev D. G., Levenfish K. P., Haensel P., 2003, A&A,407, 265

Yakovlev D. G., Levenfish K. P., Shibanov Yu. A., 1999,Physics–Uspekhi, 42, 737

Yakovlev D. G., Kaminker A. D., Haensel P., Gnedin O. Y.,2001, Physics Reports, 354, 1

Rutledge et al. suggested cooling would measure crust properties.

Also calculations by E. Brown and A. Cumming.

Curves 1-4 use high crust thermal conductivity (regular lattice) while 5 uses low conductivity (amorphous)

Data favor high conductivity! Crust is observed to be crystalline with few impurities.

P S Shternin et al. 2007

Page 5: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

MD Simulation of Breaking Strain• Slowly shear square

simulation volume with time.

• Calculate force from nearest periodic image.

• If particle leaves simulation volume have it enter simulation volume from other side.

• CJH, Kai Kadau, PRL 102, 191102

Simulation volume

Periodic copy

Periodic copy

Periodic copy

Periodic copy

Periodic copy

Periodic copy

Periodic copy

Periodic copy

Page 6: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Shear Stress vs Strain

• Shear stress versus strain for strain rates of (left to right) 0.125 (black), 0.25 (red), 0.5(green), 1(blue), 2(yellow), 4(brown), 8(gray), 16(violet), and 32(cyan) X10-8 c/fm.

N=9826 pure bcc lattice

• Stress tensor is force per unit area resisting strain (fractional deformation).

• Hook’s law: slope of stress vs strain is shear modulus.

• Very long ranged tails of screened coulomb interactions between ions important for strength.

V (r) =Z2e2

re−r/λ

Page 7: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Size dependence of Stress vs Strain

• Shear stress vs. system size at a rate of 4 X 10-7 c/fm as calculated with the Scalable Parallel Short ranged Molecular dynamics (SPaSM) code on up to 512 processors.

Page 8: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Breaking of NS Crust• Fracture in brittle material such as

silicon involves propagation of cracks that open voids.

• Crack propagating in MD simulation of Silicon. Swadener et al., PRL89 (2002) 085503.

The dislocations and other damage on the crack surfacesresulted in atoms displaced from their lattice positions inthe wake of the crack as shown in Fig. 4. The excess energyassociated with these displaced atoms and surface atomswas calculated by completely relaxing the H ! 147 !Amodel at 0 K after the crack had run completely throughit. Then the energy of the atoms in ten bilayers above andbelow the crack faces was determined for a 100 A longregion where steady crack propagation had occurred. In theabsence of any other dissipation mechanism, the energy inexcess of the bulk potential energy is equal to the dynamicfracture toughness (Jd). A least squares regression of thedata from the H ! 147 !A model determined that Jd in-creased linearly with Js according to: Jd ! 1:15 J=m2 "0:337 Js (regression coefficient, r! 0:99). Extrapolationof this linear relation to large values indicates Jd thatapproaches #1=3$ Js. Inserting Jd ! #1=3$ Js into Eq. (1)predicts a limiting value of the crack speed equal to#2=3$cR, as observed in the simulations and in agreementwith experimental results [1].

In order to test whether Eq. (1) holds for the stripgeometry, the results for v=cR are plotted versus Jd=Js inFig. 5. Plotted this way, the prediction from Eq. (1) is astraight line, as shown by the solid line in Fig. 5. The datumpoint for the lowest crack speed does not agree with theprediction, because the crack has probably not reachedsteady state. The rest of the data shows approximate agree-ment with Eq. (1).

The strain energy that is not consumed as Jd is convertedinto phonon vibrations. Using the modified SW potential,Hauch et al. [1] determined that most of the strain energywas converted into phonon vibrations during fracture.Gumbsch et al. [19] propose that each bond breaking eventleads to phonon vibrations. For a wide range of dynamicfracture toughness, the current results show that the phononvibration energy is approximately equal to the elastic waveenergy predicted by continuum theory for an infinite body.

This research was funded by the U.S. Department ofEnergy Office of Basic Energy Sciences. The authors

would like to acknowledge helpful discussions with R. G.Hoagland and M. Marder. Computer assistance fromF. Cherne is also acknowledged.

[1] J. A. Hauch, D. Holland, M. P. Marder, and H. L. Swinney,Phys. Rev. Lett. 82, 3823 (1999).

[2] T. Cramer, A. Wanner, and P. Gumbsch, Phys. Rev. Lett.85, 788 (2000).

[3] D. Holland and M. Marder, Phys. Rev. Lett. 80, 746(1998).

[4] D. Holland and M. Marder, Phys. Rev. Lett. 81, 4029(1998).

[5] F. F. Abraham, N. Bernstein, J. Q. Broughton, and D. Hess,Mater. Res. Soc. Bull. 25, 27 (2000).

[6] N. Bernstein and D. Hess, Mater. Res. Soc. Symp. Proc.653, 37 (2001).

[7] M. I. Baskes, Phys. Rev. Lett. 59, 2666 (1987).[8] M. I. Baskes, Phys. Rev. B 46, 2727 (1992).[9] J. J. Gilman, J. Appl. Phys. 31, 2208 (1960).

[10] R. J. Jacodine, J. Electrochem. Soc. 110, 524 (1963).[11] C. P. Chen and M. H. Liepold, Am. Ceram. Soc. Bull. 59,

469 (1980).[12] L. B. Freund, J. Mech. Phys. Solids 20, 129 (1972);

J. Appl. Mech. 39, 1027 (1972); Dynamic FractureMechanics (Cambridge University Press, Cambridge,United Kingdom, 1990).

[13] J. W. Hutchinson and Z. Suo, Adv. Appl. Mech. 29, 63(1992).

[14] S. Nose, Mol. Phys. 52, 255 (1984).[15] W. G. Hoover, Phys. Rev. A 31, 1695 (1985).[16] A. N. Darinskii, Wave Motion 25, 35 (1997).[17] W. G. Knauss, J. Appl. Mech. 33, 356 (1966).[18] J. R. Rice, J. Appl. Mech. 34, 248 (1967).[19] P. Gumbsch, S. J. Zhou, and B. L. Holian, Phys. Rev. B 55,

3445 (1997).

FIG. 5. Normalized crack speed (v=cR) as a function of Jd=Jsfor dynamic fracture in the H ! 147 !A silicon model. Solid lineis the prediction from Eq. (1).

[111]

[211]

FIG. 4. Detail of a crack propagating in the H ! 147 !A siliconmodel for Js ! 13:1 J=m2.

VOLUME 89, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 19 AUGUST 2002

085503-4 085503-4

1.7 million ion crystal with cylindrical defect in center. Red color indicates

deformation.

• Neutron star crust is under great pressure which prevents formation of voids. Crust does not fracture!

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Effects of impurities

27648 ion simulation with complex rp ash composition (MCP) that includes many impurities is only slightly weaker than pure crystal (OCP).

Page 10: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Role of Grain Boundaries

• Grain boundaries may weaken crust.

• Expect grain size to be larger than we can simulate.

• However we find strength only grows with grain size.

• Example of poly-crystalline sample with 8 grains and 13 million ions.

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Neutron Star Crust is Very Strong

• Each ion has long range Coulomb interactions with thousands of neighbors. The system is still strong even if several of these redundant bonds are broken.

• The great pressure suppresses the formation of dislocations, voids, and fractures. This inhibits many failure mechanisms.

• We find neutron star crust is the strongest material known. It is ten billion times stronger than steel (has 1010 the breaking stress)!

• The breaking strain σ (fractional deformation at failure) is very large, of order σ=0.1 even including the effects of impurities, defects, and grain boundaries.

• Ushomirsky et al. speculate on implications of σ=0.01, but this is a guess. Our σ is ten times bigger. But more importantly, our result is based on detailed MD simulations.

Page 12: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Star quakes and glitches• Crust is stressed as isolated NS spins down, reducing

centrifugal support of equatorial bulge. When crust breaks, moment of inertia changes producing glitch.

• Not enough angular momentum in deformation of crust for star quakes to explain all glitches.

• Strong crust could increase time between star quakes and size of produced glitch.

• Strain tensor uij =1/2(dui/drj+duj/dri) where u is displacement field of crust.

• Strain angle is difference between largest and smallest eigenvalue of strain tensor. Crust fails when strain angle exceeds breaking strain.

Page 13: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Strain in crust• Assume uniform density crust over

incompressible core [L. M. Franco, B. Link, and R. I. Epstein, ApJ. 543, 987 (2000)]. Strain tensor is

• Boundary conditions at surface R and crust core interface R’.

0 2 4 6 8 10 12

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Page 14: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Crust breaking in region ɸ

φ φ

Spinning up accreting star Spinning down star

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EOS and crust thickness

8 9 10 11 12 13 14 15 160.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2 HLPSstiff HLPSsoft IU-FSU IU-FSUmax

M (M

)

R (km)

0.01

0.02

0.03

0.04

0.05

(b)

Mcr

ust (M

)

(a)

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.40.02

0.04

0.06

0.08

0.10

0.12

HLPSstiff HLPSsoft IU-FSU IU-FSUmax

M (M )

ΔR

crus

t/Rto

t

Page 16: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Crust breaking on spinning down NS

• If breaking strain is 0.1 there is a minimum initial rotational frequency for crust to break before it stops spinning.

• Most isolated NS likely born spinning too slowly for crust ever to brake.

• If star born spinning very fast then asymmetric breaking of crust can produce significant ellipticity. 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

300

400

500

600

700

800

900

1000

σ = 0.1

HLPSstiff HLPSsoft IU-FSU IU-FSUmax

f max

(H

z)

M (M )

Page 17: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Spinning up accreting NS: Strength so crust breaks at 716 Hz

• Assume crust starts at 0 strain at a frequency so that it reaches 716 Hz by time crust is replaced.

• What is breaking strain so that crust then fails?

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.40.00

0.02

0.04

0.06

0.08

0.10

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σ = 0.008

HLPSstiff HLPSsoft IU-FSU IU-FSUmax

ε =

σ

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σ = 0.089

Page 18: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Limiting rotational freq. of NS • May be set by the strength of the

crust.

• If crust breaks asymmetrically —> get nonzero ellipticity ϵ and GW radiation.

• Crust breaking gives an ϵ related to maximum ϵ [10-6 to 10-5] because crust is maximally stressed when it breaks.

• Very roughly, ϵ is maximum ϵ times fraction of crust that breaks. ~1% break could give torque balance.

• F. J. Fattoyev, C. J. H., and Hao Lu, ArXiv:1804.04952

φ φ

Page 19: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Some open questions• What happens after the crust breaks?

• Shear modulus, breaking strain of pasta?

• Crust breaking in nonuniform crust?

• Role of strong magnetic fields for shear modulus, breaking strain, mode of crust failure…?

• Can there be hybrid crust / magnetic mountains? Can strong B field lines act as “rebar” to reinforce crust?

Page 20: horowitz crust breaking - Institute for Nuclear TheoryFIG. 5. Normalized crack speed (v=c R) as a function of J d=J s for dynamic fracture in the H! 147 A! silicon model. Solid line

Crust breaking on accreting stars

C. J. Horowitz, [email protected], Astro-solids, INT, Apr. 2018

0 2 4 6 8 10 12

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IU-FSU

crustΩ

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ε/σ = 1.0core

• Limiting rotational speed of NS may be set by strength of the crust, ArXiv:1804.04952.

• Kai Kadau, Andrey Chugunov, Farrukh Fattoyev

• Graduate students: Joe Hughto, Andre Schneider, Matt Caplan, Hao Lu

• Support from DOE DE-FG02-87ER40365 (Indiana University) and DE-SC0018083 (NUCLEI SciDAC-4 Collaboration).