accurate simulations of the dynamical bar-mode instability...

Accurate simulations of the dynamical BAR-mode instability in General relativity

Gian Mario Manca(Parma University)In collaboration with:

Roberto De Pietri (Parma)Luciano Rezzolla (AEI)Luca Baiotti (AEI)

Using CACTUS and WHISKY

The ILIAS 3rd Annual MeetingLNGS - February 27th / March 3rd, 2006

Secretariat:Fax +39 0862 437 559e-mail: [email protected]://ilias2006.lngs.infn.it

[email protected]

mailto:[email protected]:[email protected]://ilias2006.lngs.infn.ithttp://ilias2006.lngs.infn.itmailto:[email protected]:[email protected]

CACTUS: (www.cactuscode.org)Mainly developed at AEI (Golm, Germany) and LSU (USA)

WHISKY: (http://www.aei-potsdam.mpg.de/~hawke/Whisky.html)Whisky is a code to evolve the equations of hydrodynamics on curved space. It is being written by and for members of the EU Network on Sources of Gravitational Radiation and is based on the Cactus Computational Toolkit.

http://www.cactuscode.orghttp://www.cactuscode.orghttp://www.aei-potsdam.mpg.de/~hawke/Whisky.htmlhttp://www.aei-potsdam.mpg.de/~hawke/Whisky.html

BAR MODE

Rapidly rotating compact object develop the rotationally induced “bar mode” instability

Global rotational instability arise from non-axis-symmetric modes of the fluid

If mode for l=2, m=2 “bar mode” is the fasten growing unstable mode it is a good candidate for being a good source of gravitational radiation

!!

"

!

!!#$

!!

!"#$

"

"#$

!

!#$

!!#$

!!

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"

"#$

!

!#$

!!

"

!

!!

"

!

!!#$

!!

!"#$

"

"#$

!

!#$

z z

a1

a2a2

a3

a2

a1

a1 != a2 = a3

a1 != a2 != a3

NON-Axisymmetric configuration

Ellipsoidal figures of equilibrium

Topical Review R119

β=0.14

β=0.27

Mac

laur

in

1.00.60.2 0.20.6a /a2 1

JacobiΩ>0, ζ=0

DedekindΩ=0, ζ>0

Figure 4. A schematic summary of the instability results for rotating ellipsoids (a2/a1 representsthe axis ratio, i.e., the ellipticity of the configuration). For values of β greater than 0.14the Maclaurin spheroids are secularly unstable. Viscosity tends to drive the system towards atriaxial Jacobi ellipsoid, while gravitational radiation leads to an evolution towards a Dedekindconfiguration. Indicated in the figure is an evolution of this latter kind. Above β ≈ 0.27 theMaclaurin spheroids are dynamically unstable, as there exists a Riemann-S ellipsoid with lower(free) energy. (For more details, see [54, 56].)

when β > βs . The gravitational-wave instability tends to drive the system towards theDedekind sequence (the members of which do not radiate gravitationally)5.

These classical secular instabilities set in through the quadrupole f-modes of the ellipsoids.In figure 5 we show the frequencies of the l = |m| = 2 Maclaurin spheroid f-modes. Thesemodes are usually referred to as the ‘bar-modes’. The figure illustrates several general featuresof the pulsation problem for rotating stars. In particular, we note that (i) the rotational splittingof modes that are degenerate in the non-rotating limit, i.e., the m = ±2 modes become distinctin the rotating case, and (ii) the symmetry with respect to ω = 0, which reflects the fact thatthe governing equations are invariant under the change [ω,m] → [−ω,−m]. In figure 5 wealso show the pattern speed for the two modes that have positive frequency in the non-rotatinglimit, cf (18). From this figure we see that the l = −m = 2 mode, which is always progrademoving in the inertial frame, has zero pattern speed in the rotating frame at βs (σp = $). Atthis point, the mode becomes unstable to the viscosity driven instability. That the instabilityshould set in at this point is natural since the perturbed configuration is ‘Jacobi-like’ whenthe mode is stationary in the rotating frame. Meanwhile, the gravitational-wave instabilitysets in through the originally retrograde moving l = m = 2 modes. At βs these modes havezero pattern speed in the inertial frame (σp = 0). At this point, the perturbed configuration is‘Dedekind-like’ since the mode is stationary according to an inertial observer.

The evolution of the secular instabilities depends on the relative strength of thedissipation mechanisms. This tug-of-war is typical of these kinds of problems. Since the

5 Recent results concerning the stability of the Riemann-S ellipsoids complicate this picture considerably. Theseresults, due to Lebovitz and Lifschitz [57], show that the Riemann-S ellipsoids suffer a ‘strain’ instability in most ofthe parameter space. In particular, the Dedekind ellipsoids are always unstable due to this new instability.

σ = Ω(e)±√

4B11(e)− Ω2(e)

B11 =3 e− 5 e3 + 2 e5 +

√1− e2

(−3 + 4 e2

)arcsin(e)

4 e5

Ω2 =−6

(1− e2

)

e2+

2(3− 2 e2

) √1− e2 arcsin(e)e3

β(e) =T

|W| = −1 +3

2e2− 3

√1− e2

2e arcsin(e)Eigenvalue of the m=2 mode

Axisymmetricconfiguration

β

eβ = 0.14 β = 0.27

σR120 Topical Review

0 0.2 0.4-2

-1

0

1

2

Re

ω/Ω

K a

nd I

m ω

/ΩK

βs βd

0 0.2 0.4ββ

-1

-0.5

0

0.5

1

σ i/Ω

K

l=m=2

l=-m=2

βs βd

Figure 5. Results for the l = |m| = 2 f-modes of a Maclaurin spheroid. In the left frame we showthe oscillation frequencies (solid lines) and imaginary parts (dashed lines) of the modes, whilethe right frame shows the mode pattern speed σi for the two modes that have positive frequencyin the non-rotating limit (the pattern speeds for the modes which have negative frequency in thenon-rotating limit are obtained by reversing the sign of m). All results are according to an observerin the inertial frame. The dashed curves in the right frame represent a vanishing pattern speed (i) inthe inertial frame (the horizontal line), and (ii) in the rotating frame (the circular arc, which shows"/"K as a function of β). The points where the Maclaurin ellipsoid becomes secularly (βs ) anddynamically (βd ) unstable are indicated by vertical dotted lines.

gravitational-wave driven mode involves differential rotation it is damped by viscosity, andsince the viscosity driven mode is triaxial it tends to be damped by gravitational-waveemission. A detailed understanding of the dissipation mechanisms is therefore crucial forany investigation into secular instabilities of spinning stars.

Given the competition between gravitational radiation and viscosity, one would expecta ‘realistic’ star to be stabilized beyond the point βs . Also, the secular instabilities are nolonger realized in the extreme case of a perfect fluid which conserves both angular momentumand circulation6. Then the Maclaurin sequence remains stable up to the point βd ≈ 0.27. Atthis point, there exists a bifurcation to the x = +1 Riemann-S sequence. These equilibriahave lower ‘free energy’ [56] than the corresponding Maclaurin spheroid for the same angularmomentum and circulation. This means that a dynamical transition to a lower energy statemay take place without violating any conservation laws. In other words, at βd the Maclaurinspheroids become dynamically unstable to m = 2 perturbations. This instability is usuallyreferred to as the dynamical bar-mode instability.

In terms of the pulsation modes, the dynamical instability sets in at a point where tworeal-frequency modes merge, cf figure 5. At the bifurcation point βd the two modes haveidentical oscillation frequencies and their angular momenta will vanish. Given this, one of thedegenerate modes can grow without violating the conservation of angular momentum. Thephysical conditions required for the dynamical instability are easily understood. The instabilityoccurs when the originally backward moving f-mode (which has δJ < 0 for β < βd) hasbeen dragged forwards by rotation so much that it has ‘caught up’ with the originally forward

6 Note that in general relativity all non-axisymmetric modes of oscillation radiate gravitational waves. Hence, thisargument is only relevant in Newtonian gravity.

Real part

Imaginarypart

Ratio of the axes on the xy plane

Ω

2π= fc + f

′

c(β − βc) +

1

2f ′′

c(β − βc)

2

1

τ=

√

k(β − βc)INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class. Quantum Grav. 20 (2003) R105–R144 PII: S0264-9381(03)17654-X

TOPICAL REVIEW

Gravitational waves from instabilities in relativisticstars

Nils Andersson

Department of Mathematics, University of Southampton, Southampton SO17 1BJ, UK

Received 7 May 2002, in final form 5 February 2003Published 12 March 2003Online at stacks.iop.org/CQG/20/R105

AbstractThis paper provides an overview of stellar instabilities as sources ofgravitational waves. The aim is to put recent work on secular and dynamicalinstabilities in compact stars in context, and to summarize the current thinkingabout the detectability of gravitational waves from various scenarios. As anew generation of kilometre length interferometric detectors is now comingonline this is a highly topical theme. The review is motivated by two keyquestions for future gravitational-wave astronomy: are the gravitational wavesfrom various instabilities detectable? If so, what can these gravitational-wavesignals teach us about neutron star physics? Even though we may not have clearanswers to these questions, recent studies of the dynamical bar-mode instabilityand the secular r-mode instability have provided new insights into many ofthe difficult issues involved in modelling unstable stars as gravitational-wavesources.

PACS numbers: 04.40.Dg, 04.30.Db, 97.10.Sj, 97.60.Jd

1. Introduction

Neutron stars may suffer a number of instabilities. These instabilities come in differentflavours, but they have one general feature in common: they can be directly associatedwith unstable modes of oscillation. A study of the stability properties of a relativistic staris closely related to an investigation of the star’s various pulsation modes. Furthermore,non-axisymmetric stellar oscillations will inevitably lead to the production of gravitationalradiation. Should these waves turn out to be detectable, they would provide a fingerprintthat could be used to put constraints on the interior structure of the star [1]. This would beanalogous to the recent success story of helioseismology, where the detailed spectrum of solaroscillation modes has been matched to theoretical models of the interior to provide insightsinto, for example, the sound speed at different depths in the Sun. In order for ‘gravitational-wave asteroseismology’ to be a realistic proposition, one must find scenarios which lead to astar pulsating wildly. The most obvious situation where this may be the case is when a newly

0264-9381/03/070105+40$30.00 © 2003 IOP Publishing Ltd Printed in the UK R105

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class. Quantum Grav. 20 (2003) R105–R144 PII: S0264-9381(03)17654-X

TOPICAL REVIEW

Gravitational waves from instabilities in relativisticstars

Nils Andersson

Department of Mathematics, University of Southampton, Southampton SO17 1BJ, UK

Received 7 May 2002, in final form 5 February 2003Published 12 March 2003Online at stacks.iop.org/CQG/20/R105

AbstractThis paper provides an overview of stellar instabilities as sources ofgravitational waves. The aim is to put recent work on secular and dynamicalinstabilities in compact stars in context, and to summarize the current thinkingabout the detectability of gravitational waves from various scenarios. As anew generation of kilometre length interferometric detectors is now comingonline this is a highly topical theme. The review is motivated by two keyquestions for future gravitational-wave astronomy: are the gravitational wavesfrom various instabilities detectable? If so, what can these gravitational-wavesignals teach us about neutron star physics? Even though we may not have clearanswers to these questions, recent studies of the dynamical bar-mode instabilityand the secular r-mode instability have provided new insights into many ofthe difficult issues involved in modelling unstable stars as gravitational-wavesources.

PACS numbers: 04.40.Dg, 04.30.Db, 97.10.Sj, 97.60.Jd

1. Introduction

Neutron stars may suffer a number of instabilities. These instabilities come in differentflavours, but they have one general feature in common: they can be directly associatedwith unstable modes of oscillation. A study of the stability properties of a relativistic staris closely related to an investigation of the star’s various pulsation modes. Furthermore,non-axisymmetric stellar oscillations will inevitably lead to the production of gravitationalradiation. Should these waves turn out to be detectable, they would provide a fingerprintthat could be used to put constraints on the interior structure of the star [1]. This would beanalogous to the recent success story of helioseismology, where the detailed spectrum of solaroscillation modes has been matched to theoretical models of the interior to provide insightsinto, for example, the sound speed at different depths in the Sun. In order for ‘gravitational-wave asteroseismology’ to be a realistic proposition, one must find scenarios which lead to astar pulsating wildly. The most obvious situation where this may be the case is when a newly

0264-9381/03/070105+40$30.00 © 2003 IOP Publishing Ltd Printed in the UK R105

central region. This feature is more outstanding for modelM7c3C in which the increase of !0 from the initial value isnot seen. Thus, we conclude that the bar-mode perturbationis amplified only in the central region. This is reasonablesince in the models with A ! 0:1, the outcomes are rapidlyrotating only in the central region.

B. Criterion for the onset of nonaxisymmetricdynamical instabilities

Models M7c2C and M7c3C are dynamically unstableagainst the bar mode and m ! 1 mode deformation, whilemodel M7c4C is stable for both modes. This implies thatfor the onset of dynamical nonaxisymmetric instabilities,

FIG. 11. The same as Fig. 10 but for model M7c3C.

FIG. 12. The same as Fig. 10 but for model M5c2C.

THREE-DIMENSIONAL SIMULATIONS OF STELLAR . . . PHYSICAL REVIEW D 71, 024014 (2005)

024014-23

Three-dimensional simulations of stellar core collapse in full general relativity:Nonaxisymmetric dynamical instabilities

Masaru Shibata and Yu-ichirou SekiguchiGraduate School of Arts and Sciences, University of Tokyo, Tokyo, 153-8902, Japan

(Received 1 October 2004; published 18 January 2005)

We perform fully general relativistic simulations of rotating stellar core collapse in three spatialdimensions. The hydrodynamic equations are solved using a high-resolution shock-capturing scheme. Aparametric equation of state is adopted to model collapsing stellar cores and neutron stars followingDimmelmeier et al. The early stage of the collapse is followed by an axisymmetric code. When the stellarcore becomes compact enough, we start a three-dimensional simulation adding a bar-mode nonaxisym-metric density perturbation. The axisymmetric simulations are performed for a wide variety of initialconditions changing the rotational velocity profile, parameters of the equations of state, and the total mass.It is clarified that the maximum density, the maximum value of the compactness, and the maximum valueof the ratio of the kinetic energy T to the gravitational potential energy W (! ! T=W) achieved during thestellar collapse and bounce depend sensitively on the velocity profile and the total mass of the initial coreand equations of state. It is also found that for all the models with a high degree of differential rotation, afunnel structure is formed around the rotational axis after the formation of neutron stars. For selectedmodels in which the maximum value of ! is larger than "0:27, three-dimensional numerical simulationsare performed. It is found that the bar-mode dynamical instability sets in for the case that the followingconditions are satisfied: (i) the progenitor of the stellar core collapse should be rapidly rotating with theinitial value of 0:01 & ! & 0:02, (ii) the degree of differential rotation for the velocity profile of the initialcondition should be sufficiently high, and (iii) a depletion factor of pressure in an early stage of collapseshould be large enough to induce a significant contraction to form a compact stellar core for which anefficient spin-up can be achieved surmounting the strong centrifugal force. As a result of the onset of thebar-mode dynamical instabilities, the amplitude of gravitational waves can be by a factor of "10 largerthan that in the axisymmetric collapse. It is found that a dynamical instability with the m # 1 mode is alsoinduced for the dynamically unstable cases against the bar mode, but the perturbation does not growsignificantly and, hence, it does not contribute to an outstanding amplification of gravitational waves. Noevidence for fragmentation of the protoneutron stars is found in the first few 10 msec after the bounce.

DOI: 10.1103/PhysRevD.71.024014 PACS numbers: 04.25.Dm, 04.30.–w, 04.40.Dg

I. INTRODUCTION

One of the most important issues of hydrodynamicsimulations in general relativity is to clarify stellar corecollapse to a neutron star or a black hole. The formation ofneutron stars and black holes is among the most promisingsources of gravitational waves. This fact has stimulatednumerical simulations for the stellar core collapse [1–12].However, most of these works have been done in theNewtonian framework and in the assumption of axialsymmetry. As demonstrated in [10,12], general relativisticeffects modify the dynamics of the collapse and the gravi-tational waveforms significantly in the formation of neu-tron stars. Thus, the simulation should be performed in theframework of general relativity. The assumption of axialsymmetry is appropriate for the case that the rotatingstellar core is not rapidly rotating. However, for the suffi-ciently rapidly rotating cases, nonaxisymmetric instabil-ities may grow during the collapse and the bounce [7]. As aresult, the amplitude of gravitational waves may be in-creased significantly.

To date, there has been no general relativistic work forthe stellar core collapse in three spatial dimensions. Three-dimensional simulations of the stellar core collapse have

been performed only in the framework of Newtonian grav-ity [4,7]. Hydrodynamic simulations for gravitational col-lapse or for the onset of nonaxisymmetric instabilities ofrotating neutron stars in full general relativity have beenperformed so far [13–17], but no simulation has been donefor the rotating stellar core collapse to a neutron star or ablack hole. In this paper, we present the first numericalresults of three-dimensional simulations for rapidly rotat-ing stellar core collapse in full general relativity.

Three-dimensional simulation is motivated by two ma-jor purposes. One is to clarify the criterion for the onset ofnonaxisymmetric dynamical instabilities during the col-lapse, and the outcome after the onset of the instabilities.So far, a number of numerical simulations have illustratedthat rapidly rotating stars in isolation and in equilibriumare often subject to nonaxisymmetric dynamical instabil-ities not only in Newtonian theory [18–28], but also inpost-Newtonian approximation [29], and in general rela-tivity [15]. These simulations have shown that the dynami-cal bar-mode instabilities set in (i) when the ratio of thekinetic energy T to the gravitational potential energy W(hereafter ! ! T=W) is larger than "0:27 or (ii) when therotating star is highly differentially rotating, even for ! $0:27 [28]. As a result of the onset of the nonaxisymmetric

PHYSICAL REVIEW D 71, 024014 (2005)

1550-7998=2005=71(2)=024014(32)$23.00 024014-1 2005 The American Physical Society

Putting ad-hocperturbations.






I. INTRODUCTION












I. INTRODUCTION







m=2 m=1

Interaction betweenm=2 and m=1 mode?

Mass 1.2M!

value of ! is large enough ( * 0:14) for a collapsed star,m ! 1 modes may grow faster than the m ! 2 mode[55,56]. In the formation of neutron stars in which "max >"nuc, the equation of state is stiff, and hence, the m ! 1mode may not be very important. On the other hand, in theformation of oscillating stars, equations of state can be softfor "max < "nuc. However, the values of ! in such a phaseof subnuclear density are not very large. Thus, it is ex-pected that even if the m ! 1 mode becomes unstable, theperturbation may not grow as significantly as found in[55,56]. Hence, we do not pay particular attention to thismode in this paper. Since nonaxisymmetric numericalnoises are randomly included at t ! 0, in some models,the m ! 1 mode grows as found in Sec. V. However, theamplitude of the perturbation is indeed not as large as thatfor m ! 2.

Since we assume the conformal flatness in spite of thefact that the conformal three-metric is slightly differentfrom zero in reality, a small systematic error is introducedin setting the initial data. Moreover, we discard the matterlocated in the outer region of the collapsing core accordingto Eq. (52). This could also introduce a systematic error. Toconfirm that the magnitude of such error induced by theseapproximate treatments is small, we compare the results inthe three-dimensional simulations with those in the axi-symmetric ones. We have found that the results agree welleach other and the systematic error is not very large. Thiswill be illustrated in Sec. V (cf. Fig. 13).

Simulations for each model with the grid size (441, 441,221) (N ! 220) were performed for about 15 000 timesteps. The required CPU time for computing one modelis about 30 h using 32 processors of FACOM VPP 5000 at

the data processing center of the National AstronomicalObservatory of Japan.

IV. NUMERICAL RESULTS OFAXISYMMETRIC SIMULATIONS

A. Outcomes

In the last column of Table II, we summarize the out-comes of stellar core collapse in the axisymmetric simula-tions for !1 ! 1:3 and !2 ! 2:5. They are divided intothree types: black hole, neutron star, and oscillating star forwhich the maximum density inside the star is not alwayslarger than "nuc. For given values of K0"# 7$ 1014 cgs%and A, a black hole is formed when the initial value of !(hereafter !init) is smaller than critical values that dependon A. As described in Sec. III A, ! in the collapse is definedby

! & TT 'U : (54)

In the dynamical spacetime with M( ) M for ! ! 4=3, Wwould be approximately written as

W ) U' T ' Tother; (55)where Tother denotes a partial kinetic energy obtained bysubtracting the rotational kinetic energy from the total.Thus, T=W should be approximated by T="U' T 'Tother%, but we do not know how to appropriately defineTother. Fortunately, it would be much smaller than T at theinitial state, at the maximum compression at which the spinof the collapsing star becomes maximum, and in a latephase when the outcome relaxes to a quasistationary state.

(a) (b)

FIG. 1. Evolution of (a) #c and "max and (b) ! ! T="T 'U% for models M7c1 (solid curves), M7c2 (dotted curves), M7c3 (dashedcurves), M7c5 (long-dashed curves), and M7c6 (dotted-dashed curves). The dotted horizontal line denotes ! ! 0:27.

MASARU SHIBATA AND YU-ICHIROU SEKIGUCHI PHYSICAL REVIEW D 71, 024014 (2005)

024014-10

β

COLLAPSE

2005

8 SAIJO, BAUMGARTE, & SHAPIRO

- 1.5 - 1 - 0.5 0 0.5 1 1.5x / R

- 1.5

- 1

- 0.5

0

0.5

1

1.5y/R

- 1.5 - 1 - 0.5 0 0.5 1 1.5x / R

- 1.5

- 1

- 0.5

0

0.5

1

1.5

y/R

- 1.5

- 1

- 0.5

0

0.5

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1.5

y/R

- 1.5

- 1

- 0.5

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1.5

y/R

II(a)-i II(a)-ii

II(b)-i II(b)-ii

II(c)-i II(c)-ii

II(d)-i II(d)-ii

Fig. 11.— Intermediate and final density contours in the equatorial plane for Models II. Snapshots are plotted at (t/Pc, ρmax/ρ(0)max, d) =

(a)-i (23.8, 1.14, 0.220), (b)-i (20.6, 1.90, 0.220), (c)-i (17.3, 4.98, 0.287), (d)-i (16.3, 3.63, 0.287), (a)-ii (34.7, 1.24, 0.220), (b)-ii (30.1, 2.92,0.220), (c)-ii (25.2, 7.41, 0.287), and (d)-ii (23.3, 11.5, 0.287). The contour lines denote densities ρ/ρmax = 10−(16−i)d(i = 1, · · · , 15).

3

bounces is 2τdyn ∼ 4 ms (consistent with a mean coredensity ρ̄ ∼ (πGτ2dyn)

−1 = 1.2 × 1012 g cm−3) and theaxisymmetric oscillations persist for ∼ 50ms. The curvesof h+(t) for model W5 also signal that the dynamics isessentially axisymmetric: as viewed along the x-axis, theGW strain exhibits oscillations of diminishing amplitude,as reported in Ott et al. (2004), but for the first ∼ 40 msafter tb, essentially no GW radiation is emitted alongthe z-axis. In model Q15, fewer “radial” bounces occur,they damp out somewhat more rapidly, and the resultingh+(t) signal is weaker as viewed along the x-axis. This is,in part, because the postbounce core configuration wasintroduced into FLOW•ER at a later time (t−tb = 15 msfor model Q15 instead of t − tb = 5 ms for model W5)and, in part, because the effects of numerical dampingare inevitably more apparent when a simulation is runon a grid having lower spatial resolution. As is illus-trated by the solid h+(t) curves in the bottom panels ofFigs. 1 and 2, at early times the amplitude of the gravi-tational radiation that would be emitted along the z-axisis larger in model Q15 than in model W5. This reflectsthe fact that the nonaxisymmetric perturbation that wasinitially introduced into model Q15 was larger and it hadan entirely m = 2 character.

Although in model Q15 the postbounce core was sub-jected to a pure, m = 2 bar-mode perturbation when itwas mapped onto the FLOW•ER grid, the amplitude ofthe model’s mass-quadrupole distortion did not grow per-ceptibly during the first 100 ms (∼ 50 dynamical times)of the model’s evolution (Fig. 1). However, as the solidcurve in the same figure panel shows, the model spon-taneously developed an m = 1 “dipole” distortion eventhough the initial density perturbation did not containany m = 1 contribution. As early as t − tb ≈ 70 ms, aglobally coherent m = 1 mode appeared out of the noiseand grew exponentially on a timescale τgrow ≈ 5 ms. Att − tb ≈ 100 ms, the amplitude of this m = 1 distor-tion surpassed the amplitude of the languishing m = 2structure and, shortly thereafter, it became nonlinear.At t− tb ≈ 100 ms, the quadrupole distortion also beganto amplify, but it appears to have only been followingthe exponential development of the m = 1 mode. Ananalysis of the oscillation frequency of both modes re-veals them to be harmonics of one another. As the toppanel of Fig. 2 illustrates, the same m = 1 mode devel-oped spontaneously out of the 0.02% amplitude, randomperturbation that was introduced into model W5. Themode reached a nonlinear amplitude somewhat earlier inmodel W5 than in model Q15, presumably because theinitially imposed random perturbation included a finite-size contribution to an m = 1 distortion whereas the den-sity perturbation introduced into model Q15 containedno m = 1 component. The growth timescale of the in-stability is τgrow ≈ 4.8 ms for model W5. Although wehave described the unstable m = 1 mode as a “dipole”mass distortion, this is somewhat misleading because inneither model did the lopsided mass distribution producea shift in the location of the center of mass of the system.Instead, as is illustrated in Fig. 3, the mode developedas a tightly wound, one-armed spiral, very similar to them = 1 - dominated structures that have been reportedby Centrella et al. (2001) and Saijo et al. (2003).

After the spiral pattern reached its maximum ampli-

Fig. 3.— The equatorial-plane structure of model W5 is shownat time t− tb = 71 ms. Left: Two-dimensional isodensity contourswith velocity vectors superposed; contour levels are (from the in-nermost, outward) ρ/ρmax =0.15,0.01,0.001,0.0001. Right: Spiralcharacter of the m = 1 distortion as determined by a Fourier anal-ysis of the density distribution; specifically, the phase angle φ1(#)of the m = 1 Fourier mode is drawn as a function of #.

Fig. 4.— Equatorial-plane profiles of the azimuthally averagedangular velocity Ω(#) (top frame) and the mass density ρ(#) (bot-tom frame) are shown at four different times during the evolutionof model W5. Changes in these profiles at late times illustrate theeffects of angular momentum redistribution by the m = 1 spiralmode: angular momentum migrates radially outward while massmigrates radially inward. A horizontal line drawn in the top panelat ωCR = 2.5×103 rad s

−1 identifies the corotation radius for thisone-armed spiral mode. The “kink” seen in ρ(#) at late times atabout 8 km is connected to the discontinous switch of the EOS Γat ρnuc.

tude in both of our model evolutions, the maximum den-sity began to slowly increase and β started decreasing(Figs. 1 & 2). Following Saijo et al. (2003), we inter-pret this behavior as resulting from angular momentumredistribution that is driven by the spiral-like deforma-tion and by gravitational torques associated with it. Asangular momentum is transported outward, the centrifu-gal support of the innermost region is reduced, a largerfraction of the core’s mass is compressed to nuclear den-sities and, in turn, β decreases because the magnitudeof the gravitational potential energy correspondingly in-creases. Fig. 4 supports this interpretation. As theproto-NS evolves, we see that the outermost layers spinfaster and the innermost region becomes denser. (Wenote that throughout the evolution our models conservedtotal angular momentum to within a few parts in 104.)Also, as is shown in the top panel of Fig. 4, through-out most of the model’s evolution there is a radius in-side the proto-NS (%CR ≈ 12 km) at which the angularvelocity of the fluid matches the angular eigenfrequency(ωCR = 2.5×103 rad s

−1) of the spiral mode. Hence, it isentirely reasonable to expect that resonances associatedwith this “corotation” region are able to effect a redis-

ONE-ARMED SPIRAL INSTABILITY IN DIFFERENTIALLY ROTATING STARS 7

0 5 10 15 20 25

t / Pc

100

101

!m

ax /

!m

ax

(0)

Fig. 7.— Maximum density ρmax as a function of t/Pc for ModelI (a) (solid) and Model I (b) (dotted). We terminate our simulationat t ∼ 20Pc or when the maximum density of the star exceeds about

10 times its initial value ρ(0)max.

-1 0 1

x / R10-1

100

101

! /

!m

ax

I (a)

-1 0 1

x / R

I (b)

(0)

Fig. 8.— Density profiles along the x-axis during the evolution forModels I (a) and I (b). Solid, dotted, dashed, dash-dotted lines de-note times t/Pc = (a) (1.16×10−3, 6.99, 14.0, 21.0), (b) (7.36×10−4,6.63, 13.3, 19.9), respectively. Note that the density distribution de-velops asymmetrically in the presence of the m = 1 mode instability,and that this instability destroys the toroidal structure.

0 5 10 15 20 25

t / Pc

-5

0

5

r h

+ R

/ M

2

Fig. 9.— Gravitational waveforms as seen by a distant observerlocated on the z-axis for Model I (a) (solid line) and Model I (b)(dashed line).

In Fig. 9 we show the gravitational wave signal emittedfrom this instability. Gravitational radiation couples toquadrupole moments, and the emitted radiation thereforescales with the quadrupole diagnostic Q, which we alwaysfind excited along with the m = 1 instability. We con-sistently find that the pattern period of the the m = 2modes is very similar to that of the m = 1 mode, suggest-ing that the former is a harmonic of the latter (see Table3). Since the diagnostic Q does not remain at its maximumamplitude after saturating, we find that the gravitationalwave amplitude is not nearly as persistent as for the barmode instability. We also find that the gravitational waveperiod, here PGW ∼ 0.7Pc ∼ Ω−1c , is different from thevalue PGW ∼ 3.3Pc ∼ Ω−1eq we found for the bar mode in§ 3.1, which points to a difference in the generation mecha-nism. Characteristic wave frequencies fGW correspond tothe central rotation period of the star.

-0.1

-0.05

0

0.05

0.1

Re[D

, Q

]

0 10 20 30 40

t / Pc

-0.1

-0.05

0

0.05

0.1

Re[D

, Q

]

0 10 20 30 40

t / Pc

II (c)

II (a) II (b)

II (d)

Fig. 10.— Diagnostics D and Q as a function of t/Pc for Models II (see Table 4). Solid and dotted lines denote D and Q. We terminateour simulation at t ∼ 25Pc or when the maximum density of the star exceeds about 10 times its initial value.

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THE ASTROPHYSICAL JOURNAL, 595, 2003 September 20

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ONE-ARMED SPIRAL INSTABILITY IN DIFFERENTIALLY ROTATING STARS

Motoyuki [email protected]

Department of Physics, Kyoto University, Kyoto 606-8502, Japan

Thomas W. [email protected]

Department of Physics and Astronomy, Bowdoin College,8800 College Station Brunswick, ME 04011

Stuart L. Shapiro2,[email protected]

Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801Received 2003 January 22; accepted 2003 June 4

ABSTRACT

We investigate the dynamical instability of the one-armed spiral m = 1 mode in differentially rotatingstars by means of 3 + 1 hydrodynamical simulations in Newtonian gravitation. We find that both asoft equation of state and a high degree of differential rotation in the equilibrium star are necessary toexcite a dynamical m = 1 mode as the dominant instability at small values of the ratio of rotationalkinetic to gravitational potential energy, T/|W |. We find that this spiral mode propagates outward fromits point of origin near the maximum density at the center to the surface over several central orbitalperiods. An unstable m = 1 mode triggers a secondary m = 2 bar mode of smaller amplitude, andthe bar mode can excite gravitational waves. As the spiral mode propagates to the surface it weakens,simultaneously damping the emitted gravitational wave signal. This behavior is in contrast to wavestriggered by a dynamical m = 2 bar instability, which persist for many rotation periods and decay onlyafter a radiation-reaction damping timescale.

Subject headings: Gravitation — hydrodynamics — instabilities — stars: neutron — stars: rotation

1. introduction

Stars in nature are usually rotating and may be sub-ject to nonaxisymmetric rotational instabilities. An exacttreatment of these instabilities exists only for incompress-ible equilibrium fluids in Newtonian gravity, (e.g. Chan-drasekhar 1969; Tassoul 1978). For these configurations,global rotational instabilities may arise from non-radialtoroidal modes eimϕ (where m = ±1,±2, . . . and ϕ is theazimuthal angle).

For sufficiently rapid rotation, the m = 2 bar modebecomes either secularly or dynamically unstable. The on-set of instability can typically be identified with a criticalvalue of the non-dimensional parameter β ≡ T/|W |, whereT is the rotational kinetic energy and W the gravitationalpotential energy. Uniformly rotating, incompressible starsin Newtonian theory are secularly unstable to bar-modeformation when β ≥ βsec # 0.14. This instability cangrow only in the presence of some dissipative mechanism,like viscosity or gravitational radiation, and the associ-ated growth timescale is the dissipative timescale, whichis usually much longer than the dynamical timescale of thesystem. By contrast, a dynamical instability to bar-modeformation sets in when β ≥ βdyn # 0.27. This instabil-ity is independent of any dissipative mechanisms, and thegrowth time is the hydrodynamic timescale.

Determining the onset of the dynamical bar-mode in-stability, as well as the subsequent evolution of an unsta-

ble star, requires a fully nonlinear hydrodynamic simu-lation. Simulations performed in Newtonian gravity, (e.g.Tohline, Durisen, & McCollough 1985; Durisen et al. 1986;Williams & Tohline 1988; Houser, Centrella, & Smith1994; Smith, Houser, & Centrella 1995; Houser & Cen-trella 1996; Pickett, Durisen, & Davis 1996; Toman et al.1998; New, Centrella, & Tohline 2000) have shown thatβdyn depends only very weakly on the stiffness of the equa-tion of state. Once a bar has developed, the formation ofa two-arm spiral plays an important role in redistribut-ing the angular momentum and forming a core-halo struc-ture. Both βdyn and βsec are smaller for stars with high de-gree of differential rotation (Tohline & Hachisu 1990; Pick-ett, Durisen, & Davis 1996; Shibata, Karino, & Eriguchi2002, 2003). Simulations in relativistic gravitation (Shi-bata, Baumgarte, & Shapiro 2000; Saijo et al. 2001) haveshown that βdyn decreases with the compaction of the star,indicating that relativistic gravitation enhances the barmode instability. In order to efficiently use computationalresources, most of these simulations have been performedunder certain symmetry assumptions (e.g. π-symmetry),which do not affect the growth of the m = 2 bar mode,but which suppress any m = 1 modes.

Recently, Centrella et al. (2001) reported that suchm = 1 “one-armed spiral” modes are dynamically un-stable at surprisingly small values of T/|W |. Centrellaet al. (2001) found this instability in evolutions of highly

1 Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 618012 Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 618013 NCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801

1

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ONE-ARMED SPIRAL INSTABILITY IN A LOW T/|W | POSTBOUNCE SUPERNOVA CORE

Christian D. Ott1, Shangli Ou2, Joel E. Tohline2, and Adam Burrows3

Accepted by ApJL; AEI-2005-021

ABSTRACT

A three-dimensional, Newtonian hydrodynamic technique is used to follow the postbounce phase of astellar core collapse event. For realistic initial data we have employed post core-bounce snapshots of theiron core of a 20 M! star. The models exhibit strong differential rotation but have centrally condenseddensity stratifications. We demonstrate for the first time that such postbounce cores are subject to aso-called low-T/|W | nonaxisymmetric instability and, in particular, can become dynamically unstableto an m = 1 - dominated spiral mode at T/|W | ∼ 0.08. We calculate the gravitational wave (GW)emission by the instability and find that the emitted waves may be detectable by current and futureGW observatories from anywhere in the Milky Way.Subject headings: hydrodynamics - instabilities - gravitational waves - stars: neutron - stars: rotation

1. INTRODUCTION

Rotational instabilities are potentially important in theevolution of newly-formed proto-neutron stars (proto-NSs). In particular, immediately following the pre-supernova collapse – and accompanying rapid spin up –of the iron core of a massive star, nonaxisymmetric insta-bilities may be effective at redistributing angular momen-tum within the core. By transferring angular momentumout of the centermost region of the core, nonaxisymmet-ric instabilities could help explain why the spin periods ofnewly formed pulsars are longer than what one would ex-pect from standard stellar evolutionary calculations thatdo not invoke magnetic field action for angular momen-tum redistribution and generation of very slowly rotatingcores (Heger et al. 2000,Hirschi et al. 2004,Heger et al.2004). Alternatively, in situations where the initial col-lapse “fizzles” and the proto-NS is hung up by centrifugalforces in a configuration below nuclear density, a rapidredistribution of angular momentum would facilitate thefinal collapse to NS densities. The time-varying massmultipole moments resulting from nonaxisymmetric in-stabilities in proto-NSs may also produce GW signalsthat are detectable by the burgeoning, international ar-ray of GW interferometers. The analysis of such signalswould provide us with the unprecedented ability to di-rectly monitor the formation of NSs and, perhaps, blackholes.

In this Letter, we present results from numerical sim-ulations that show the spontaneous development of aspiral-shaped instability during the postbounce phaseof the evolution of a newly formed proto-NS. Theseare the most realistic such simulations performed, todate, because the pre-collapse iron core has been drawnfrom the central region of a realistically evolved 20M! star, and the collapse of the core as well as thepostbounce evolution has been modeled in a dynami-cally self-consistent manner. Starting from somewhat

1 Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Golm, Germany;[email protected]

2 Center for Computation & Technology, Department of Physics& Astronomy, Louisiana State University, Baton Rouge, LA 70803

3 Steward Observatory & the Department of Astronomy, TheUniversity of Arizona, Tucson, AZ 85721

simpler initial states, other groups have followed thedevelopment of bar-like structure in postbounce coresusing Newtonian (Rampp et al. 1998) and relativistic(Shibata & Sekiguchi 2005) gravity, but their analyseshave been limited to cores having a high ratio of rota-tional to gravitational potential energy, β ≡ T/|W | !0.27. We demonstrate that a one-armed spiral (not thetraditional bar-like) instability can develop in a proto-NS even if it has a relatively low T/|W | ∼ 0.08. This issignificant, but perhaps not surprising given the recentstudies by Centrella et al. (2001), Shibata et al. (2002,2003), and Saijo et al. (2003).

2. NUMERICAL SIMULATION

The results presented in this Letter are drawn fromthree-dimensional hydrodynamic simulations that followapproximately 130 ms of the “postbounce” evolution of anewly forming proto-NS. Before presenting the details ofthese simulations, however, it is important to emphasizethe broader evolutionary context within which they havebeen conducted and, specifically, from what source(s) theinitial conditions for the simulations have been drawn.The two simulations presented here cover the final por-tion (Stage 3) of a much longer, three-part evolution thatalso included: (Stage 1) the main-sequence and post-main-sequence evolution of a spherically symmetric, 20M! star through the formation of an iron core that isdynamically unstable toward collapse; and (Stage 2) theaxisymmetric collapse of this unstable iron core throughthe evolutionary phase at which “bounces” at nucleardensities.

Stage 1 of the complete evolution was originally pre-sented as model “S20” by Woosley & Weaver (1995).The initial configuration for this model was a chemicallyhomogeneous, spherically symmetric, zero-age main-sequence star with solar metallicities. Evolution up tothe development of an unstable iron core took some2 × 107 yr of physical time. In Stage 2 the spheri-cally symmetric model from Stage 1 was mapped ontothe two-dimensional, axisymmetric grid of the hydrody-namics code “VULCAN/2D” (Livne 1993) and evolvedas model “S20A500β0.2” by Ott et al. (2004). Rotationwas introduced into the core with a radial angular ve-locity profile Ω(") = Ω0[1 + ("/A)2]−1 (where " is thecylindrical radius). The scale length in the initial rota-

Ott, Ou, Tohline, Burrows

Astrophys.J.625:L119-L122,2005

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THE ASTROPHYSICAL JOURNAL, 595, 2003 September 20

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ONE-ARMED SPIRAL INSTABILITY IN DIFFERENTIALLY ROTATING STARS

Motoyuki [email protected]

Department of Physics, Kyoto University, Kyoto 606-8502, Japan

Thomas W. [email protected]

Department of Physics and Astronomy, Bowdoin College,8800 College Station Brunswick, ME 04011

Stuart L. Shapiro2,[email protected]

Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801Received 2003 January 22; accepted 2003 June 4

ABSTRACT

We investigate the dynamical instability of the one-armed spiral m = 1 mode in differentially rotatingstars by means of 3 + 1 hydrodynamical simulations in Newtonian gravitation. We find that both asoft equation of state and a high degree of differential rotation in the equilibrium star are necessary toexcite a dynamical m = 1 mode as the dominant instability at small values of the ratio of rotationalkinetic to gravitational potential energy, T/|W |. We find that this spiral mode propagates outward fromits point of origin near the maximum density at the center to the surface over several central orbitalperiods. An unstable m = 1 mode triggers a secondary m = 2 bar mode of smaller amplitude, andthe bar mode can excite gravitational waves. As the spiral mode propagates to the surface it weakens,simultaneously damping the emitted gravitational wave signal. This behavior is in contrast to wavestriggered by a dynamical m = 2 bar instability, which persist for many rotation periods and decay onlyafter a radiation-reaction damping timescale.

Subject headings: Gravitation — hydrodynamics — instabilities — stars: neutron — stars: rotation

1. introduction

Stars in nature are usually rotating and may be sub-ject to nonaxisymmetric rotational instabilities. An exacttreatment of these instabilities exists only for incompress-ible equilibrium fluids in Newtonian gravity, (e.g. Chan-drasekhar 1969; Tassoul 1978). For these configurations,global rotational instabilities may arise from non-radialtoroidal modes eimϕ (where m = ±1,±2, . . . and ϕ is theazimuthal angle).

For sufficiently rapid rotation, the m = 2 bar modebecomes either secularly or dynamically unstable. The on-set of instability can typically be identified with a criticalvalue of the non-dimensional parameter β ≡ T/|W |, whereT is the rotational kinetic energy and W the gravitationalpotential energy. Uniformly rotating, incompressible starsin Newtonian theory are secularly unstable to bar-modeformation when β ≥ βsec # 0.14. This instability cangrow only in the presence of some dissipative mechanism,like viscosity or gravitational radiation, and the associ-ated growth timescale is the dissipative timescale, whichis usually much longer than the dynamical timescale of thesystem. By contrast, a dynamical instability to bar-modeformation sets in when β ≥ βdyn # 0.27. This instabil-ity is independent of any dissipative mechanisms, and thegrowth time is the hydrodynamic timescale.

Determining the onset of the dynamical bar-mode in-stability, as well as the subsequent evolution of an unsta-

ble star, requires a fully nonlinear hydrodynamic simu-lation. Simulations performed in Newtonian gravity, (e.g.Tohline, Durisen, & McCollough 1985; Durisen et al. 1986;Williams & Tohline 1988; Houser, Centrella, & Smith1994; Smith, Houser, & Centrella 1995; Houser & Cen-trella 1996; Pickett, Durisen, & Davis 1996; Toman et al.1998; New, Centrella, & Tohline 2000) have shown thatβdyn depends only very weakly on the stiffness of the equa-tion of state. Once a bar has developed, the formation ofa two-arm spiral plays an important role in redistribut-ing the angular momentum and forming a core-halo struc-ture. Both βdyn and βsec are smaller for stars with high de-gree of differential rotation (Tohline & Hachisu 1990; Pick-ett, Durisen, & Davis 1996; Shibata, Karino, & Eriguchi2002, 2003). Simulations in relativistic gravitation (Shi-bata, Baumgarte, & Shapiro 2000; Saijo et al. 2001) haveshown that βdyn decreases with the compaction of the star,indicating that relativistic gravitation enhances the barmode instability. In order to efficiently use computationalresources, most of these simulations have been performedunder certain symmetry assumptions (e.g. π-symmetry),which do not affect the growth of the m = 2 bar mode,but which suppress any m = 1 modes.

Recently, Centrella et al. (2001) reported that suchm = 1 “one-armed spiral” modes are dynamically un-stable at surprisingly small values of T/|W |. Centrellaet al. (2001) found this instability in evolutions of highly

1 Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 618012 Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 618013 NCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801

1

Something about m=1 modes?

2003

2005

Initial states: diff-rotating NS

Standard axisymmetric metric

Differential rotation law

Polytropic EOSK=100Γ=2

ds2 = −eγ+ρdt2 + eγ−ρr2 sin2 θ(dϕ− ωdt)2 + e2α(dr2 + r2dθ2)

4

to the so called conserved variables q ≡ (D, S i, τ) via the re-lations

D ≡ ρ∗ = √γWρ ,Si ≡ √γρhW 2vi , (2.19)τ ≡ √γ

(ρhW 2 − p

)− D ,

where h ≡ 1+ $+p/ρ is the specific enthalpy and αu0 = W ,αui = W

(αvi − βi

), W ≡ (1 − γijvivj)−1/2. Note that

only five of the seven primitive variables are independent.

In order to close the system of equations for the hydrody-

namics an EOS which relates the pressure to the rest-mass

density and to the energy density must be specified.

In the case of the polytropic EOS (2.14), Γ = 1 + 1/N ,where N is the polytropic index and the evolution equationfor τ needs not be solved. In the case of the ideal-fluid EOS(2.16), on the other hand, non-isentropic changes can take

place in the fluid and the evolution equation for τ needs tobe solved.

Note that polytropic EOSs (2.14), do not allow any transfer

of kinetic energy to thermal energy, a process which occurs in

physical shocks (shock heating). However, we have verified,

by performing simulations with the more general EOS (2.16),

on some selected cases, that for the physical systems treated

here, shock heating is indeed not so important in the dynamics

of the bar.

Additional details of the formulation we use for the hydro-

dynamics equations can be found in [47]. We stress that an

important feature of this formulation is that it has allowed to

extend to a general relativistic context the powerful numer-

ical methods developed in classical hydrodynamics, in par-

ticular HRSC schemes based on linearized Riemann solvers

(see [47]). Such schemes are essential for a correct represen-

tation of shocks, whose presence is expected in several as-

trophysical scenarios. Two important results corroborate this

view. The first one, by Lax and Wendroff [48], states that

a stable scheme converges to a weak solution of the hydrody-

namical equations. The second one, by Hou and LeFloch [49],

states that, in general, a non-conservative scheme will con-

verge to the wrong weak solution in the presence of a shock,

hence underlining the importance of flux-conservative formu-

lations. For a full introduction to HRSC methods the reader is

also referred to [50–52].

III. INITIAL DATA

The initial configuration were generated on a code based

on the 2D Komatsu-Eriguch-Hachisu (KEH) method for con-

structing models of rotating neutron stars and details on the

code can be found in ref. [53]. Most of these models used are

described in [54]. The data are then transformed to Cartesian

coordinates using standard coordinate transformations. The

same initial data routines have been used in previous 3D sim-

ulations [53] and details on the accuracy of the code can be

found in [53].

In generating these equilibrium models the metric describ-

ing an axisymmetric relativistic star is assumed to have the

usual form:

ds2 = −eγ+ρdt2 + eγ−ρr2 sin2 θ(dϕ − ωdt)2

+e2α(dr2 + r2dθ2)(3.1)

and an angular velocity distribution of the form:

Ωc − Ω =r2eÂ2

[(Ω − ω)r2 sin2 θe−2ρ

1 − (Ω − ω)2r2 sin2 θe−2ρ

](3.2)

with Â = 1. On the the xy-plane the expression forΩ in termsof used 3+1 variable is given by:

Ω =uϕ

u0=

uy cosϕ − ux sin ϕu0

√x2 + y2

P =2πΩ

. (3.3)

In order to determine the characteristic group time and fre-

quency of the bar-mote instability and a precise measurement

of the critical value βd we have also considered density per-turbations of the type:

δρ(x, y, z) = ρ × δ × x2 − y2r2e

(3.4)

that have the effect of creating initials data with an already big

enough (m = 2) bar-mode perturbations already active.In order to test the effect of a pre-existing mode 1 perturba-

tion we instead used a density perturbation of the type:

δρ(x, y, z) = ρ × δm=1 × sin(

ϕ ± n2π,re

). (3.5)

Notice that for n = 0 this is just

δρ(x, y, z) = ρ × δm=1 ×y

re. (3.6)

In this cases we have first generated the unperturbed model

with the methods described above, and we have then super-

imposed a perturbation of the type of Eq. (3.4) we have then

solved the super-Hamiltonian and super-momentums con-

straints using the standard technique described in... This is

exactly the same initial state density perturbation used in [19]

and [21].

On the initial (axisymmetric) condition one can compute

the barioninicmass (M0), the gravitational mass (M ), the an-gular momentum (J), the rotational kinetic energy (T) and thegravitational binding energy (W) as [55]:

M0 =∫

ρ∗d3x (3.7)

M =∫ (

−2T 00 + T µµ)α√

γd3x (3.8)

J =∫

T 0ϕα√

γd3x (3.9)

T =12

∫ΩT 0ϕα

√γd3x (3.10)

W =∫

ρ∗$d3x + M0 + T − M (3.11)

β = T/|W| (3.12)

3

In particular we are using the BSSN variant of the ADM

evolution [35–37] which is conformal traceless reformulation

of the above system of evolution equation where the evolved

variable are the conformal factor (φ), the trace of the extrinsiccurvature (K), the conformal 3-metric (γ̃ ij), the conformaltraceless extrinsic curvature (Ãij) and the conformal connec-tion functions (Γ̃i) defined as:

φ =14

log( 3√

γ) (2.5)

K = γijKij (2.6)γ̃ij = e−4φγij (2.7)

Ãij = e−4φ(Kij − γijK) (2.8)Γ̃i = γ̃ij,j (2.9)

The code is designed to handle arbitrary shift and lapse con-

ditions, which can be chosen as appropriate for a given space-

time simulation. More information about the possible families

of space-time slicings which have been tested and used with

the present code can be found in [38? ]. Here, we limit our-

selves to recalling details about the specific foliations used in

the present evolutions. In particular, we have used hyperbolic

K-driver slicing conditions of the form

(∂t − βi∂i)α = −f(α) α2(K − K0), (2.10)

with f(α) > 0 andK0 ≡ K(t = 0). This is a generalizationof many well known slicing conditions. For example, setting

f = 1 we recover the “harmonic” slicing condition, while,by setting f = q/α, with q an integer, we recover the gener-alized “1+log” slicing condition [39]. In particular, all of thesimulations discussed in this paper are done using condition

(2.10) with f = 2/α. This choice has been made mostly be-cause of its computational efficiency, but we are aware that

“gauge pathologies” could develop with the “1+log” slic-ings [40, 41].

As for the spatial gauge, we use one of the “Gamma-driver”

shift conditions proposed in [38] (see also [42]), that essen-

tially act so as to drive the Γ̃i to be constant. In this re-spect, the “Gamma-driver” shift conditions are similar to the

“Gamma-freezing” condition ∂tΓ̃k = 0, which, in turn, isclosely related to the well-knownminimal distortion shift con-

dition [43]. The differences between these two conditions in-

volve the Christoffel symbols and are basically due to the fact

that the minimal distortion condition is covariant, while the

Gamma-freezing condition is not.

In particular, all of the results reported here have been ob-

tained using the hyperbolic Gamma-driver condition,

∂2t βi = F ∂tΓ̃i − η ∂tβi, (2.11)

where F and η are, in general, positive functions of spaceand time. For the hyperbolic Gamma-driver conditions it is

crucial to add a dissipation term with coefficient η to avoidstrong oscillations in the shift. Experience has shown that by

tuning the value of this dissipation coefficient it is possible to

almost freeze the evolution of the system at late times. We

typically choose F = 34α and η = 2 and do not vary them intime.

B. Evolution of the hydrodynamics equations

In this work we have considered the space time described

in the standard 3+1 metric decomposition variables γ ij , α, βi

andmatter is assumed described by a perfect fluid EnergyMo-

mentum tensor:

T µν = ρhuµuν + pgµν (2.12)

h = 1 + ( +p

ρ(2.13)

and an equation of state of type p = p(ρ, (). The code hasbeen written to use any EOS, but all of the simulation per-

formed so far have been performed using either a (isoentropic)

polytropic EOS

p = KρΓ , (2.14)

e = ρ +p

Γ − 1 , (2.15)

or an “ideal fluid” (Γ-law) EOS

p = (Γ − 1)ρ ( . (2.16)

Here, e = ρ(1+() is the energy density in the rest-frame of thefluid,K the polytropic constant and Γ the adiabatic exponent.In the case of the polytropic EOS (2.14), Γ = 1+1/N , whereN is the polytropic index (we have always used N = 1, i.e.,Γ = 2 that is a good approximation for a quite stiff equation ofstate) and the evolution equation for τ needs not be solved. Inthe case of the ideal-fluid EOS (2.16), on the other hand, non-

isentropic changes can take place in the fluid and the evolution

equation for τ (see below) needs to be solved. This means thatmatter is described by the five dynamical variables ρ, (, uµ

(where uµuµ = −1) with the equation of motions

!µT µν = 0 ,!µ(ρuµ) = 0 .

(2.17)

An important feature of the Whisky code is the imple-

mentation of a conservative formulation of the hydrodynam-

ics equations [44–46], in which the set of equations (2.17) is

written in a hyperbolic, first-order and flux-conservative form

of the type

∂tq + ∂if (i)(q) = s(q) , (2.18)

where f (i)(q) and s(q) are the flux-vectors and source terms,respectively [47]. Note that the right-hand side (the source

terms) depends only on the metric, and its first derivatives,

and on the stress-energy tensor. Furthermore, while the sys-

tem (2.18) is not strictly hyperbolic, strong hyperbolicity is

recovered in a flat space-time, where s(q) = 0.As shown by [45], in order to write system (2.17) in the

form of system (2.18), the primitive hydrodynamical variables

(i.e. the rest-mass density ρ and the pressure p (measured inthe rest-frame of the fluid), the fluid three-velocity v i (mea-sured by a local zero-angular momentum observer), the spe-

cific internal energy ( and the Lorentz factor W ) are mapped

A=1

Simulated models .....

[2] Stergioulas, Apostolatos, Font: Mon. Not. R. Astron. Soc. 352(2004) 1089--1101

2.64M⊙

K=100Γ=2

[1] Shibata, Baumgarte, Shapiro, ApJ. 542,(2000)453.

2.1M⊙

2.77M⊙

1.5M⊙

6

FIG. 1: Characteristic of the studied stellar models. On the bottom

they are reported asM vs !c. On the top they are reported as T/|W |vsM/R.

FIG. 2: Initial profiles of the rest mass density ρ of the angular veloc-ity Ω for models A8,A9,A10,S2,U1,U3,U11 andU13. Indicated witha thick dashed line is the profile for the first unstable model (U1) with

β = 0.255.

tified in terms of the distortion[65] parameters [21]:

η+ =Ixx − IyyIxx + Iyy

η× =2 Ixy

Ixx + Iyy(4.2)

η =√

η2+ + η2×

Clearly these quantities present the advantage that for their

evaluation isn’t needed the execution of any numerical deriva-

tive and for this reasons they are usually preferred to the al-

most equivalent size of h for determination of the parametersof the BAR instability. It is, in fact, possible to quantify the

growth time τB and the oscillation frequency fB of the un-stable BAR mode by a non linear least square fit to the trial

form:

η×(t) = η0 et/τB cos(2π fB t + φ0) . (4.3)

Notice that in Eq. 4.3, and in the whole paper, t is the coordi-nated time, and so the frequancies we obtain are approximate

frequencies with an expected error of the order of the relative

deviation of the lapse from 1. In our simulation the lapse (α) isof order 1 within a few % at the border of the simulation grid

while at the center of the grid is of order 0.68 for simulation

of D’s models and of order 0.85 for all the other models.

In our evoultion scheme the simulated stars are not con-

strained to be centered at the origin of the coordinate system.

To monitor the movement of the star with respect to the Carte-

sian grid used the first momentum of the density distribution:

X icm =1M̃

∫d3x ρ(x) xi (4.4)

where M̃ =∫

d3x ρ(x). These quantities represent a sortof center-of-mass of the Star but, since they are coordinate

dependent, they do not represent the physical position of the

star at the given time and there is no reason to be conserved.

This definition of the center-of-mass may be defined, in prin-

ciple, using either ρ, ρ∗ or the energy density T00 but there isno reason to prefer one with respect to the other. We didn’t

noticed any real difference between these possible alternative

definition and this support the idea that this definition of the

center-of-mass is a good indicator on how well the coordinate

system is centered with respect to the star.

In order to make meaningful comparison we have indeed to

normalize this effect chosing a suitable comparison time-shift

and and angle. Indeed to make a superposition of η+(t) be-tween two different simulation of the same model we should

chose a time shift ∆t and and angular shift∆φ in such a wayto have a maximal superposition of the two distortion param-

eters:

η(R)+ (t) " αη+(t + ∆t) + βη×(t + ∆t) (4.5)

where α = cos(∆φ), β = sin(∆φ) and ηR+(t) is the distortionpatameter of the reference model.

6

FIG. 1: Characteristic of the studied stellar models. On the bottom

they are reported asM vs !c. On the top they are reported as T/|W |vsM/R.

FIG. 2: Initial profiles of the rest mass density ρ of the angular veloc-ity Ω for models A8,A9,A10,S2,U1,U3,U11 andU13. Indicated witha thick dashed line is the profile for the first unstable model (U1) with

β = 0.255.

tified in terms of the distortion[65] parameters [21]:

η+ =Ixx − IyyIxx + Iyy

η× =2 Ixy

Ixx + Iyy(4.2)

η =√

η2+ + η2×

Clearly these quantities present the advantage that for their

evaluation isn’t needed the execution of any numerical deriva-

tive and for this reasons they are usually preferred to the al-

most equivalent size of h for determination of the parametersof the BAR instability. It is, in fact, possible to quantify the

growth time τB and the oscillation frequency fB of the un-stable BAR mode by a non linear least square fit to the trial

form:

η×(t) = η0 et/τB cos(2π fB t + φ0) . (4.3)

Notice that in Eq. 4.3, and in the whole paper, t is the coordi-nated time, and so the frequancies we obtain are approximate

frequencies with an expected error of the order of the relative

deviation of the lapse from 1. In our simulation the lapse (α) isof order 1 within a few % at the border of the simulation grid

while at the center of the grid is of order 0.68 for simulation

of D’s models and of order 0.85 for all the other models.

In our evoultion scheme the simulated stars are not con-

strained to be centered at the origin of the coordinate system.

To monitor the movement of the star with respect to the Carte-

sian grid used the first momentum of the density distribution:

X icm =1M̃

∫d3x ρ(x) xi (4.4)

where M̃ =∫

d3x ρ(x). These quantities represent a sortof center-of-mass of the Star but, since they are coordinate

dependent, they do not represent the physical position of the

star at the given time and there is no reason to be conserved.

This definition of the center-of-mass may be defined, in prin-

ciple, using either ρ, ρ∗ or the energy density T00 but there isno reason to prefer one with respect to the other. We didn’t

noticed any real difference between these possible alternative

definition and this support the idea that this definition of the

center-of-mass is a good indicator on how well the coordinate

system is centered with respect to the star.

In order to make meaningful comparison we have indeed to

normalize this effect chosing a suitable comparison time-shift

and and angle. Indeed to make a superposition of η+(t) be-tween two different simulation of the same model we should

chose a time shift ∆t and and angular shift∆φ in such a wayto have a maximal superposition of the two distortion param-

eters:

η(R)+ (t) " αη+(t + ∆t) + βη×(t + ∆t) (4.5)

where α = cos(∆φ), β = sin(∆φ) and ηR+(t) is the distortionpatameter of the reference model.

ρ

Ω

β

The evolutions of 3 models....

They shows very similar behaviors...... but* different time scales* different size of the BAR

β = 0.2812β = 0.2743β = 0.2596

9

m=4

m=3

m=2

ln(P

)

m=1

mtime

FIG. 4: Schematic evolution of the collective modes (Eq. 4.6) of the

matter density ρ.

FIG. 5: The behavior of the instability for model U11. In the top

part of the figure it is shown the time behavior of the η+ and η×quadrupole distortion parameters of Eq. (4.2). In the bottom part is

report the time behavior of the logarithm of the modulus of the modes

1, 2 and 4 and the phase of the mode 2. The Pn defined in Eq. (4.6)





to the crossing between mode 1 and mode 2. The mass start

going away from the grid in the saturation stage when the bar

is at the maximum extension and become almost contant after

the bar disappear.

Notice that the center of mass start moving when the matter

go away from the grid. This mouvement so is not meaningfull,

and in any case do not influence the modes evolution up to the

the crossing between mode 1 and mode 2.

The global phase of the mode 2 is well defined after the first

crossing between modes 2 and 4 up to the end of the saturation

stage, which is another indication of a well formed bar.

More compact stars like D2, as espected, show a smooth

descrease of the mode 2 independently of the growth of the

mode 1. Less compact stars like U3, U11 and U13 show in-

stead a plateau in the mode 2.

C. Methodology of the comparisons

If we change resolution or we impose a mode 2 density

perturbation the initial ratio between ln(P2) and ln(P4) will

General behavior of the instability

Best way.... look at the dynamics of the global modes

Model U11 β = 0.2743m=2

m=3

m=1

m=4

time

ln(|P_m|)

Pm =

∫d3x ρ(x) eimϕ General scheme

Are modes good indicators ?Pm =

∫d3x ρ(x) eimϕ Pm(!) =

∫dϕ ρ(!,ϕ, z = 0) eimϕ

9

m=4

m=3

m=2

ln(P

)

m=1

m

time


matter density ρ.












the bar disappear.















9

m=4

m=3

m=2

ln(P

)

m=1

m

time


matter density ρ.












the bar disappear.















! =

√

x2 + y2 z = 0

arg(Pm(!))/m pattern speed

Importance of the mode m=1

β = 0.2812β = 0.2596

High BETA: the bounces destroy the bar. Low BETA: the BAR is NOW persistent !!!

f(r, z, φ) = f(r, z, φ + π)10





be different and indeed will be different the time for which we

will have that mode 2 will be greater than mode 4. In the same

way we may have a different orientation of the bar in the xyplane at the same stage of the instability.

We consistently chose as referencemodels the simulation at

grid resolution dx = 0.5, unperturbed, and with an ideal fluidequation of state., i.e., the model whose dynamics are shown

in Fig. 6, Fig. 5, Fig. 7 and Fig. 12.

D. Mode-1 and persistence of the BAR

Among the 1.5 solar masses stars only in the most unstable

mode U13 the bar after the plateau can disappear in a dynam-

ical timescale without the interaction with mode 1.

In Fig. 4 the blue dashed line represent the evolution of

the mode 2 in simulation where mode 1 is rouled-out by the

symmetry.

There is a very efficient a conclusive way to prove that the

non-persistence is due the growing of odd-modes. In fact is we

perform the same simulation imposing that all the dynamical

FIG. 8: Effect of teh π-symmetry on the dynamics of the deformationparameter η(t) and modes P2(t) P1(t) for model U3

FIG. 9: Effect of teh π-symmetry on the dynamics of the deformationparameter η(t) and modes P2(t) P1(t) for model U11

variable have the symmetry f(xi) = f(!, ϕ, z) = f(! +π, ϕ, z) all the oddmodes will not be allowed. We have indeedrepeated the simulation of modelsU3, U11 andU13 imposing

this symmetry (π-sym).

Le compatte vanno gi e le meno compatte hanno un plateau,

riusciamo ad avere una stima di quale sarebbe il tempo di

scomparsa della barra dovuto alla sola gravit?

11

FIG. 10: Effect of teh π-symmetry on the dynamics of the deforma-tion parameter η(t) and modes P2(t) P1(t) for model U13

Model dx notes dt ti tf η τB fBms ms ms (max) (ms) Hz

D2 0.500 9 10.5 0.59 0.90 1053

D2 0.500 δ = .01 -6.54 9 10.5 0.67 0.78 1052

D2 0.500 δ = .04 -7.57 9 10.5 0.67 0.77 1056

D3 0.500 9 12 0.10 —– 1098

D3 0.500 δ = .04 9 12 0.38 1.54 1086

D7 0.500 δ = .04 11 14 0.48 1.74 821

U3 0.500 21 26 0.47 2.69 547

U3 0.500 δ = .01 -15.94 21 26 0.53 2.42 548

U3 0.625 π-sym -1.00 21 26 0.43 2.82 543

U3 0.625 δ = .01 π-sym -16.28 21 26 0.54 2.52 547

U3 0.500 adiab 0.76 21 26 0.48 2.79 552

U11 0.500 11 14 0.78 1.15 494

U11 0.500 δ = .01 -8.55 11 14 0.79 1.11 494

U11 0.375 1.64 11 14 0.79 1.11 492

U11 0.625 1.79 11 14 0.78 1.15 492

U11 0.625 large 2.54 11 14 0.78 1.15 492

U11 0.625 π-sym 1.39 11 14 0.77 1.12 494

U11 0.750 3.79 11 14 0.76 1.17 493

U11 0.500 adiab 5.35 11 14 0.78 1.12 497

U13 0.500 10 13 0.85 0.95 454

U13 0.500 δ = .01 -8.55 10 13 0.86 0.93 454

U13 0.625 π-sym -0.16 10 13 0.86 0.96 453

U13 0.625 δ = .01 π-sym -8.71 10 13 0.86 0.94 454

U13 0.500 adiab 1.69 10 13 0.86 0.94 457

TABLE II: Maximum distortion, grow rate and frequency of the Bar

mode during the initial part of the instability for models uti

accurate simulations of the dynamical bar-mode instability...

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