-
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
!!
"
!
!!#$
!!
!"#$
"
"#$
!
!#$
!!#$
!!
!"#$
"
"#$
!
!#$
!!
"
!
!!
"
!
!!#$
!!
!"#$
"
"#$
!
!#$
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.
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
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
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
1
1.5
y/R
- 1.5
- 1
- 0.5
0
0.5
1
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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Preprint typeset using LATEX style emulateapj v. 11/12/01
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
Saijo, Baumgarte, Shapiro.
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emulateapj v. 6/22/04
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
Preprint typeset using LATEX style emulateapj v. 11/12/01
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  = 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)
FIG. 6: The behavior of the instability for model U3. 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
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)
FIG. 6: The behavior of the instability for model U3. 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
9
m=4
m=3
m=2
ln(P
)
m=1
m
time
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)
FIG. 6: The behavior of the instability for model U3. 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
! =
√
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
FIG. 7: The behavior of the instability for model U13. 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)
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