INSTABILITIES OF ROTATING RELATIVISTIC STARS

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INSTABILITIES OF ROTATING RELATIVISTIC STARS
John Friedman University of Wisconsin-MilwaukeeCe
nter for Gravitation and Cosmology
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outline
I. NONAXISYMMETRIC INSTABILITY II.
DYNAMICAL INSTABILITY III. GW-DRIVEN (CFS)
INSTABILITY R-MODES IV. SPIN-DOWN AND
GRAVITATIONAL WAVES FROM A NEWBORN
NEUTRON STAR V. INSTABIILTY OF OLD NEUTRON
STARS SPUN-UP BY ACCRETION VI. DOES
THE INSTABILITY SURVIVE THE PHYSICS
OF A REAL NEUTRON STAR? (MUCH OF THIS
LAST PART TO BE COVERED BY NILS
ANDERSSONS TALK)
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NONAXISYMMETRIC INSTABILITY
MINIMIZING ENERGY AT FIXED ANGULAR MOMENTUM
GRAVITY BUT NO ROTATION MINIMIZE ENERGY BY
MAXIMIZING GRAVITATIONAL BINDING ENERGY
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NONAXISYMMETRIC INSTABILITY
MINIMIZING ENERGY AT FIXED ANGULAR MOMENTUM
ROTATION BUT NO GRAVITY, MINIMIZE KINETIC
ENERGYAT FIXED J BY PUSHING FLUID TO BOUNDARY
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NONAXISYMMETRIC INSTABILITY
MINIMIZING ENERGY AT FIXED ANGULAR MOMENTUM
RAPID ROTATION AND GRAVITYCOMPROMISE SEPARATE
FLUID INTO TWO SYMMETRIC PARTS
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DYNAMICAL INSTABILITY
GROWS RAPIDLY DYNAMICAL TIMESCALE TIME FOR
SOUND TO CROSS STAR
SECULAR INSTABILITY
REQUIRES DISSIPATION VISCOSITY OR
GRAVITATIONAL RADIATIONSLOWER, DISSIPATIVE
TIMESCALE
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DYNAMICAL INSTABILITY

• CONSERVATION LAWS BLOCK NONAXISYMMETRIC
  INSTABILITY IN UNIFORMLY ROTATING STARS UNTIL
  STAR ROTATES FAST ENOUGH THAT
• T ( ROTATIONAL KINETIC ENERGY ) . W
  ( GRAVITATIONAL BINDING ENERGY)

t
gt 0.26
UNIFORMLY ROTATING STARS WITH NS EQUATIONS OF
STATE HAVE MAXIMUM ROTATION t lt 0.12
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BUT A COLLAPSING STAR WITH LARGE DIFFERENTIAL
ROTATION MAY BECOME UNSTABLE AS IT CONTRACTS AND
SPINS UP
Bar-mode instability of rotating disk (Simulation
by Kimberly New)
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Recent studies of dynamical instability
byGondek-Rosinska and GourgoulhonShibata,
Karino, Eriguchi, YoshidaWatts, Andersson,
Beyer, SchutzCentrella, New, Lowe and
BrownImamura, Durisen, PickettNew, Centrella
and TohlineShibata, Baumgarte and Shapiro

• TWO SURPRISES FOR LARGE DIFFERENTIAL ROTATION
• m2 (BAR MODE) INSTABILITY CAN SET IN FOR SMALL
  VALUES OF t
• m1 (ONE ARMED SPIRAL) INSTABILITY CAN DOMINATE

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SAIJO,YOSHIDA
GROWTH OF AN lm1 INSTABILITYIN A RAPIDLY
DIFFERENTIALLY ROTATATING MODEL
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SECULAR INSTABILITY
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cfs1
GRAVITATIONAL-WAVE INSTABILITY
Chandrasekhar, F, SchutzOutgoing
nonaxisymmetric modes radiate angular momentum
to
If the pattern rotates forward
relative to , it radiates positive J to
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cfs2
If the pattern rotates backward
relative to , it radiates negative J to
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cfs3
That is A
forward mode, with J gt 0, radiates positive J
to A backward mode, with J lt 0, radiates
negative J to Radiation damps all modes of a
spherical star
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cfs4
But, a rotating star drags a mode in the
direction of thestar's rotation A mode with
behavior that moves
backward relative to the star is dragged forward
relative to , when m W gt s The mode still
has J lt 0, because Jstar J mode lt J
star . This backward mode, with J lt 0, radiates
positive J to . Thus J becomes increasingly
negative, and THE AMPLITUDE OF THE MODE GROWS
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surprise
OBSERVATIONAL SUPRISE 16 ms pulsar seen in a
supernova remnant
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surprise2
OBSERVATIONAL SUPRISE 16 ms pulsar seen in a
supernova remnant
In a young (5000 yr old) supernova remnant in the
Large Magellanic Cloud, Marshall et al found a
pulsar with a 16 ms period and a spin-down time
lifetime of the remnant This, for the first
time, implies A class of neutron stars have
millisecond periods at birth.
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th_surprise
A nearly simultaneous THEORETICAL
SURPRISEA new variant of a gravitational-wave
driven instability of relativistic stars may
limit the spin of newly formed pulsars and of old
neutron stars spun up by accretion. The newly
discovered instability may set the initial
spin of pulsars in the newly discovered class.
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These surprises led to an explosion of interest
Andersson JF, Morsink Kojima
Lindblom,Owen, Morsink Owen, Lindblom, Cutler,
Andersson, Kokkotas,Schutz Schutz, Vecchio,
Andersson Madsen Andersson, Kokkotas,
Stergioulas Levin Bildsten Ipser, Lindblom
JF, Lockitch Beyer, Kokkotas Kojima,
Hosonuma Hiscock Lindblom Brady, Creighton
Owen Rezzolla, Shibata, Asada, Lindblom,
Mendell, Owen Baumgarte, Shapiro
Flanagan Rezzola,Lamb, Shapiro Spruit
Levin Ferrari, Matarrese,Schneider Lockitch
Rezania Prior work on axial modes Chandrasekhar
Ferrari
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STILL MORE RECENT
Stergioulas, Font, Kokkotas Kojima,
Hosonuma Yoshida, Lee Rezania,
Jahan-Miri Yoshida, Karino, Yoshida, Eriguchi
Rezania, Maartens Andersson, Lockitch,
JF Lindblom, Mendell Andersson, Kokkotas,
Stergioulas AnderssonUshomirsky, Cutler,
Bildsten Bildsten, Ushomirsky Andersson, Jones,
Kokkotas, Brown, Ushomirsky Stergioulas
Lindblom,Owen,Ushomirsky Rieutord Wu,
Matzner, Arras Ho, Lai Levin, Ushomirsky
Madsen Lindblom, Tohline, Vallisneri Stergioul
as, Font Arras, Flanagan, Schenk, JF,
Lockitch Sa Teukolsky,Wasserman
Morsink Jones Lindblom,Owen Ruoff, Kokkotas,
Andersson,Lockitch,JF
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AND MORE
Karino, Yoshida, Eriguchi Hosonuma Watts,
Andersson Rezzolla,Lamb,Markovic,Arras,
Flanagan, Morsink, ShapiroWagoner, Hennawi,
Liu Shenk, Teukolsky, Wasserman Morsink Jones,
Andersson, Stergioulas Haensel, Lockitch,
Andersson Prix, Comer, Andersson Hehl Gressman
, Lin, Suen, Stergioulas, JF Lin, SuenXiaoping,
Xuewen, Miao, Shuhua, NanaReisnegger, Bonacic
Yoon, LangerDrago, Lavagno,
Pagliara Drago, Pagliara,
BerezhianiGondek-Rosinska, Gourgoulhon,
HaenselBrink, Teukolsky, Wasserman
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ELECTROMAGNETIC RADIATION
MASS QUADRUPOLE
CHARGE DIPOLE
ENERGY RADIATED
C
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ELECTROMAGNETIC RADIATION
MASS QUADRUPOLE
CHARGE QUADRUPOLE
ENERGY RADIATED
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GRAVITATIONAL RADIATION
MASS QUADRUPOLE
MASS QUADRUPOLE
ENERGY RADIATED
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AXIAL GRAVITATIONAL RADIATION
MASS QUADRUPOLE
CURRENT QUADRUPOLE
ENERGY RADIATED
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AXIAL GRAVITATIONAL RADIATION
MASS QUADRUPOLE
CURRENT QUADRUPOLE
ENERGY RADIATED
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PERTURBATIONS WITH ODINARY (POLAR) PARITY
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l 0
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l 0
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PERTURBATIONS WITH AXIAL PARITY
BECAUSE ANY SCALAR IS A SUPERPOSITION OF Ylm AND
Ylm HAS, BY DEFINITION, POLAR PARITY, EVERY
SCALAR HAS AXIAL PARITY
BUT VECTORS ( TENSORS) CAN HAVE AXIAL PARITY
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l 0
NONE
l 1
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l m 2
View from pole View from equator
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l 0
NONE
l 1
l 2
Below equator
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GROWTH TIMEENERGY PUMPED INTO MODE ENERGY
RADIATED TO I
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(No Transcript)
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INSTABILITY OF POLAR MODES
THE QUADRUPOLE POLAR MODE (f-mode
) HAS FREQUENCY s OF ORDER THE MAXIMUM
ANGULAR VELOCITY WMAX OF A STAR.
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THAT MEANS A BACKWARD MOVING POLAR MODE IS
DRAGGED FORWARD, ONLY WHEN A STAR ROTATES NEAR
ITS MAXIMUM ANGULAR VELOCITY, WMAX
Stergioulas
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BECAUSE AN AXIAL
PERTURBATION OF A SPHERICAL STAR HAS NO RESTORING
FORCE ITS FREQUENCY VANISHES. IN A ROTATING
STAR IT HAS A CORIOLIS-LIKE RESTORING FORCE,
PROPORTIONAL TO W
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THE UNSTABLE l m 2 r-MODE
Newtonian Papaloizou Pringle, Provost et al,
Saio et al, Lee, Strohmayer
The mode is a current that is odd under
parity dv r2 r sin2 q ei(2f 2/3 Wt)
Frequency relative to a rotating observer
sR 2/3 W COUNTERROTATING
R
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FLOW PATTERN OF THE l m 2 r-MODE
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• Rotating Frame
• Animation shows backward (clockwise) motion
  of pattern and motion of fluid elements

Ben Owens animation
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• Inertial Frame
• Pattern moves forward(counterclockwise)
• Star and fluid elements rotate forward more
  rapidly

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VISCOUS DAMPING
Above 1010K, beta decay and inverse beta decay
n
produce neutrinos that carry off the energy of
the modebulk viscositytBULK CT6
Below 109K, shear viscosity (free e-e scattering)
dissipates the modes energy in heat tSHEAR
CT-2
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Star is unstable only when W is larger than
critical frequency set by bulk and shear
viscosity
Wcrit/Wmax 1 0.5 0.1
Star spins down as it radiates its angular
momentum in gravitational waves
105 107 109
1011 (From Lindblom-Owen-Morsink
Figure) Temperature (K)
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GRAVITATIONAL WAVES FROM SPIN-DOWN
hc 1024 (W/WK)3 (20 Mpc/D) a
AMPLITUDE, dv/RW
Owen, Lindblom, Cutler, Schutz, Vecchio,
Andersson Brady, Creighton Owen Lindblom
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GRAVITATIONAL WAVES FROM SPIN-DOWN
hc 1024 (W/WK)3 (20 Mpc/D) a
AMPLITUDE, dv/RW
IF ONE HAD A PRECISE TEMPLATE, SIGNAL/NOISE
WOULD LOOK LIKE THISFOR WAVES FROM A GALAXY 20
Mpc AWAY
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GRAVITATIONAL WAVES FROM SPIN-DOWN
With sensitivity 10 times greater, one may be
able to detect a cosmological
background of waves emitted by past GW-driven
spin-downs Ferrari, Matarrese, Schneider
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INSTABILITY OF OLD ACCRETING STARSLMXBs
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BINARIES WITH A NEUTRON STAR AND A SOLAR-MASS
COMPANION CAN BE OBSERVED AS LOW-MASS X-RAY
BINARIES (LMXBs), WHEN MATTER FROM THE COMPANION
ACCRETES ONTO THE NEUTRON STAR.MYSTERY THE
MAXIMUM PERIODS CLUSTER BELOW 642 HZ, WITH THE
FASTEST 3 WITHIN 4
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From Chakrabarty, Bildsten
FASTEST 3 619 Hz, 622 Hz, 642 Hz
VERY DIFFERENT MAGNETIC FIELDS IMPLIES SPIN
NOT LIMITED BY MAGNETIC FIELD
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PAPALOIZOU PRINGLE, AND WAGONER (80s)ACCRETION
MIGHT SPIN UP A STAR UNTIL J LOST IN GW J
GAINED IN ACCRETION
FOR POLAR MODES, VISCOSITY OF SUPERFLUID DAMPS
THE INSTABILITY AND RULES THIS OUT
BUT AXIAL MODES CAN BE UNSTABLE Andersson,
Kokkotas, Stergioulas Bildsten Levin Wagoner
Heyl Owen Reisenegger Bonacic R-MODE
INSTABILITY IS NOW A LEADING CANDIDATE FOR LIMIT
ON SPIN OF OLD NSs
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CAN GW FROM LMXBs BE OBSERVED?
IF WAGONERS PICTURE IS RIGHT, R-MODES ARE AN
ATTRACTIVE TARGET FOR OBSERVATORIES WITH THE
SENSITIVITY OF ADVANCED LIGO WITH NARROW BANDING
BUT YURI LEVIN POINTED OUT THAT IF THE VISCOSITY
DECREASES AS THE UNSTABIILITY HEATS UP THE STAR,
A RUNAWAY GROWTH IN AMPLITUDE RADIATES WAVE TOO
QUICKLY TO HOPE TO SEE A STAR WHEN ITS UNSTABLE
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LEVINS CYCLE
Wcritical .Wmax
T
108
109
107
SPIN DOWN TIME lt 1/106 SPIN UP TIME IS A STAR
YOU NOW OBSERVE SPINNING DOWN? PROBABILITY lt
1/106
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POSSIBLE WAY OUT (Wagoner, Andersson, Heyl)AS WE
MENTION LATER, DISSIPATION IN A QUARK OR HYPERON
CORE CAN INCREASE AS TEMPERATURE INCREASES
Wcritical (Hz)
T
109
109.5
108.5
108
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DOES THE INSTABILITY SURVIVE THE PHYSICS OF A
REAL NEUTRON STAR?
Will nonlinear couplings limit the amplitude to
dv/v ltlt 1?Will a continuous spectrum from GR
or differential rotation eliminate the
r-modes? Will a viscous boundary layer near a
solid crust windup of magnetic-field from
2ndorder differential rotation of the
mode bulk viscosity from hyperon productionkill
the instability?
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Will nonlinear couplings limit the amplitude to
dv/v ltlt 1?
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Fully nonlinear numerical evolutions show no
evidence that nonlinear couplings limiting the
amplitude to dv/v lt 1Nonlinear fluid
evolution in GRCowling approximation (background
metric fixed) Font, Stergioulas Newtonian
approximation, with radiation-reaction term GRR
enhanced by huge factor to see growth in 20
dynamical times. Lindblom, Tohline, Vallisneri
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GR Evolution Font, Stergioulas
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Newtonian evolution with artificially enhanced
radiation reaction Lindblom, Tohline,
Vallisneri
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BUT Work to 2nd order in the perturbation
amplitude shows TURBULENT CASCADE The
energy of an r-mode appears in this approximation
to flow into short wavelength modes, with the
effective dissipation too slow to be seen in the
nonlinear runs. Arras, Flanagan, Morsink,
Schenk, Teukolsky,WassermanBrink, Teukolsky,
Wasserman (Maclaurin)
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Newtonian evolution with somewhat higher
resolution, w/ and w/out enhanced
radiation-driving force
Catastrophic decay of r-mode
Gressman, Lin, Suen, Stergioulas, JF
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Fourier transform shows sidebands - apparent
daughter modes.
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RELATIVISTIC r-MODES Andersson, Kojima, Lockitch,
Beyer Kokkotas, Kojima Hosonuma, Lockitch,
Andersson, JF, LockitchAndersson, Kokkotas
Ruoff
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Relativistic corrections to the lm2 r-mode mix
axial and polar parts to 0th order in rotation.
Newtonian axial mode
Lockitch
1
0
r/R
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LockitchLockitch, Andersson, JF
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Will a continuous spectrum from GR or
differential rotation eliminate the r-modes?
IN A SLOW-ROTATION APPROXIMATION, AXIAL
PERTURBATIONS OF A NON-BARATROPIC STAR SATISFY A
SINGULAR EIGENVALUE PROBLEM (Kojima), s mW
(W-w)/l(l1) hrr AhrBh 0. IF THE
COEFFICIENT OF hrr VANISHES IN THE STAR, THERE IS
NO SMOOTH EIGENFUNCTION.
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INSTEAD, THE SPECTRUM IS CONTINUOUS. THIS COULD
BE DUE TO THE APPROXIMATIONS ARTIFICIALLY REAL
FREQUENCY AND WHEN THE STAR IS NEARLY
BARATROPIC, AXIAL AND POLAR PERTURBATIONS MIXTHE
KOJIMA EQUATION IS NOT VALID. Lockitch,
Andersson, JF Andersson, Lockitch
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BUT NEWTONIAN STARS WITH SOME DIFFERENTIAL
ROTATION LAWS ALSO MAY HAVE A CONTINUOUS SPECTRUM
Karino, Yoshida, EriguchiNUMERICAL EVOLUTION
OF SLOWLY ROTATING MODELS SEEM TO SHOW THAT AN
r-MODE IS APPROXIMATELY PRESENT, EVEN WHEN NO
EXACT MODE EXISTS. Kokkotas,
RuoffBUTTHAT APPROXIMATE MODE DISAPPEARS WHEN
THE SINGULAR POINT IS DEEP IN THE STAR. STILL
HAVE UNSTABLE INITIAL DATA (JF, Morsink) BUT
GROWTH TIME MAY BE LONG
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VISCOUS BOUNDARY LAYER NEAR CRUST(NILS
ANDERSSONS TALK) Bildsten, Ushomirsky Rieutord W
u, Matzner, Arras Lindblom, Owen, Ushom. Levin,
Ushomirsky Andersson, Jones, Kokkotas, Yoshida
Stergioulas
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VISCOUS BOUNDARY LAYER NEAR CRUST
VISCOSITY MUCH HIGHER NEAR CRUST SHEAR
DISSIPATION dE/dt -2h dV (ds)2
Boundary width d Bulk ds dv/d ds
dv/R dE/dt (boundary) (R/ d )
dE/dt(core) But, we dont know to what extent
the crust moves with the modes.
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Red curves show the much higher shear viscosity
when a crust is present (for upper red curve, a
superfluid has formed)
(Prepared by Ben Owen)
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DOES NONLINEAR EVOLUTION LEAD TO DIFFERENTIAL
ROTATION THAT DISSIPATES r-MODE ENERGY IN A
MAGNETIC FIELD? Spruit Rezzola, Lamb,
ShapiroLevin, Ushomirsky R, Markovic, L,
S A computation of the 2nd order
r-mode of rapidly rotating Newtonian (Maclaurin)
models and slowly rotating polytropes shows
growing differential rotation Sa JF,
Lockitch
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A GROWING MAGNETIC FIELD DAMPS INSTABILITY WHEN
tGW /tB gt 1
B-field 1010 G to 1012 G allows instability
for 2 days to 15 minutes
Rezzolla, Lamb, Markovic, Shapiro
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GRAVITATIONAL WAVES FROM SPIN-DOWNWITH DAMPING
BY MAGNETIC FIELD WINDUP
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FINALLY Will bulk viscosity from hyperon
productionkill the instability? P.B.
JonesLindblom, OwenHaensel, et al
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n
n
77
n
s
d
u
n
p
u
d
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p
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p
n
n
80
With no neutrinos emitted, dissipation comes from
the net p dV work done in an out-of-equilibrium
cycle
p
Equilibrium
V
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With no neutrinos emitted, dissipation comes from
the net p dV work done in an out-of-equilibrium
cycle
p
The work is the energy lost by the fluid element
in one oscillation
V
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Bulk viscosity from hyperons cuts off instability
below a few x 109 K
W/WK 1 0.5 0.1
Bulk viscosity kills instability at high
temperature
105 107 109
1011 (From Lindblom-Owen-Morsink
Figure) Temperature (K)
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Density above which the core has hyperons is not
well understood. Few hyperons at low density
implies few hyperons at low mass
Critical angular velocities for an EOS allowing
hyperons above 1.25 M .
Lindblom-Owen
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OLD ACCRETING NEUTRON STARS HIGHER MASSES, SO
MORE LIKELY TO HAVE HYPERON OR QUARK CORES.
REACTION RATES IN CORE INCREASE WITH T. BULK
VISCOSITY INCREASES WITH T
Wcritical (Hz)
T
109
109.5
108.5
108
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Young stars Nothing yet definitively kills
the r-mode instability in nascent NSs, but
there are too many plausible ways it may be
damped to bet in favor of its existence.
Old stars Surprisingly, the nonlinear limit on
amplitudeand the more efficient damping
mechanisms allowed by hyperon or quark cores
enhance the probability of seeing gravitational
waves from r-mode unstable LMXBs