INSTABILITIES OF ROTATING RELATIVISTIC STARS
John Friedman
University of Wisconsin-MilwaukeeCenter for Gravitation and Cosmology
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 ANDERSSON’S TALK)
NONAXISYMMETRIC INSTABILITY
MINIMIZING ENERGY AT FIXED ANGULAR MOMENTUM:
GRAVITY BUT NO ROTATION: MINIMIZE ENERGY BY MAXIMIZING GRAVITATIONAL BINDING ENERGY
NONAXISYMMETRIC INSTABILITY
MINIMIZING ENERGY AT FIXED ANGULAR MOMENTUM:
ROTATION BUT NO GRAVITY, MINIMIZE KINETIC ENERGYAT FIXED J
BY PUSHING FLUID TO BOUNDARY
NONAXISYMMETRIC INSTABILITY
MINIMIZING ENERGY AT FIXED ANGULAR MOMENTUM:
RAPID ROTATION AND GRAVITY:COMPROMISE: SEPARATE FLUID INTO TWO
SYMMETRIC PARTS
DYNAMICAL INSTABILITY
GROWS RAPIDLY DYNAMICAL TIMESCALE
= TIME FOR SOUND TO CROSS STAR
SECULAR INSTABILITY
REQUIRES DISSIPATION – VISCOSITY OR GRAVITATIONAL RADIATION
SLOWER, DISSIPATIVE TIMESCALE
CONSERVATION LAWS BLOCK NONAXISYMMETRIC INSTABILITY IN
UNIFORMLY ROTATING STARS UNTIL STAR ROTATES FAST ENOUGH THAT
T ( ROTATIONAL KINETIC ENERGY ) . |W| ( GRAVITATIONAL BINDING ENERGY)
DYNAMICAL INSTABILITY
t= > 0.26
UNIFORMLY ROTATING STARS WITH NS EQUATIONS OF STATE HAVE MAXIMUM
ROTATION t < 0.12
Bar-mode instability of rotating disk (Simulation by Kimberly New)
BUT A COLLAPSING STAR WITH LARGE DIFFERENTIAL ROTATION MAY BECOME UNSTABLE AS IT CONTRACTS AND SPINS UP
• TWO SURPRISES FOR LARGE DIFFERENTIAL ROTATION
• m=2 (BAR MODE) INSTABILITY CAN SET IN FOR SMALL VALUES OF t
• m=1 (ONE ARMED SPIRAL) INSTABILITY CAN DOMINATE
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
GROWTH OF AN l=m=1 INSTABILITYIN A RAPIDLY DIFFERENTIALLY
ROTATATING MODEL
SAIJO,YOSHIDA
SECULAR INSTABILITY
If the pattern rotates forward relative to , it radiates positive J to
GRAVITATIONAL-WAVE INSTABILITY Chandrasekhar, F, Schutz
Outgoing nonaxisymmetric modes radiate angular momentum to
If the pattern rotates backward relative to , it radiates negative J to
cfs2
That is:A forward mode, with J > 0, radiates positive J to A backward mode, with J < 0, radiates negative J to
Radiation damps all modes of a spherical star
cfs3
But, a rotating star drags a mode in the direction of thestar's rotation:
A mode with behavior that moves)( tmie
backward relative to the star is dragged forward relative to , when
m
The mode still has J < 0, becauseJstar + J mode < J star .
Thisbackward mode, with J < 0, radiates positive J to .Thus J becomes increasingly negative, and THE AMPLITUDE OF THE MODE GROWS
OBSERVATIONAL SUPRISE:16 ms pulsar seen in a supernova remnant
surprise
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.
OBSERVATIONAL SUPRISE:16 ms pulsar seen in a supernova remnant
A nearly simultaneous THEORETICAL SURPRISE:
A 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.
Andersson JF, MorsinkKojima Lindblom,Owen, MorsinkOwen, Lindblom, Cutler, Andersson, Kokkotas,Schutz Schutz, Vecchio, Andersson MadsenAndersson, Kokkotas, Stergioulas Levin BildstenIpser, Lindblom JF, LockitchBeyer, Kokkotas Kojima, HosonumaHiscock Lindblom Brady, Creighton OwenRezzolla, Shibata, Asada, Lindblom, Mendell, Owen Baumgarte, Shapiro FlanaganRezzola,Lamb, Shapiro Spruit LevinFerrari, Matarrese,Schneider Lockitch RezaniaPrior work on axial modes: Chandrasekhar & Ferrari
These surprises led to an explosion of interest:
Stergioulas, Font, Kokkotas Kojima, HosonumaYoshida, Lee Rezania, Jahan-MiriYoshida, Karino, Yoshida, Eriguchi Rezania, MaartensAndersson, 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, LaiLevin, Ushomirsky MadsenLindblom, Tohline, Vallisneri Stergioulas, Font Arras, Flanagan, Schenk, JF, Lockitch Sa Teukolsky,Wasserman Morsink Jones Lindblom,OwenRuoff, Kokkotas, Andersson,Lockitch,JF
STILL MORE RECENT
Karino, Yoshida, Eriguchi HosonumaWatts, Andersson Rezzolla,Lamb,Markovic,Arras, Flanagan, Morsink, ShapiroWagoner, Hennawi, Liu Shenk, Teukolsky, Wasserman Morsink Jones, Andersson, Stergioulas Haensel, Lockitch, Andersson Prix, Comer, AnderssonHehlGressman, Lin, Suen, Stergioulas, JFLin, SuenXiaoping, Xuewen, Miao, Shuhua, NanaReisnegger, Bonacic Yoon, LangerDrago, Lavagno, Pagliara Drago, Pagliara, BerezhianiGondek-Rosinska, Gourgoulhon, HaenselBrink, Teukolsky, Wasserman
AND MORE
GRAVITATIONAL RADIATION
dVrYQ 222
MASS QUADRUPOLEMASS QUADRUPOLE
ENERGY RADIATED
2Qdt
dE
AXIAL GRAVITATIONAL RADIATION
dVrYJ 22222 v
MASS QUADRUPOLECURRENT QUADRUPOLE
ENERGY RADIATED
2Jdt
dE
22Yr
AXIAL GRAVITATIONAL RADIATION
dVrYJ 22222 v
MASS QUADRUPOLECURRENT QUADRUPOLE
ENERGY RADIATED
2Jdt
dE
PERTURBATIONS WITH ODINARY (POLAR) PARITY
l = 0
P = 1
l = 1
P = -1
l = 2
P = 1
l = 0
P = 1
l = 1
P = -1
l = 2
P = 1
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
l = 0
NONE
l = 1
P = 1
l = m = 2
View from pole View from equator
l = 0
NONE
l = 1
l = 2
Below equator
P = 1
P = -1
Above equator
GROWTH TIME:ENERGY PUMPED INTO MODE
= ENERGY RADIATED TO I+
THE QUADRUPOLE POLAR MODE (f-mode )
HAS FREQUENCY OF ORDER THE MAXIMUM ANGULAR VELOCITY MAX OF A STAR.
INSTABILITY OF POLAR MODES
CENTRAL DENSITY
MAX
RO
TA
TIO
N E
NE
RG
Y
AT
IN
ST
AB
ILIT
Y
1014 1015
THAT MEANS A BACKWARD MOVING POLAR MODE IS DRAGGED FORWARD, ONLY WHEN A STAR ROTATES NEAR ITS MAXIMUM ANGULAR VELOCITY, MAX
Stergioulas
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
THE UNSTABLE l = m = 2 r-MODENewtonian: Papaloizou & Pringle, Provost et al, Saio et al, Lee, StrohmayerThe mode is a current that is odd under parity
v = r2 r [ sin2ei(2t
Frequency relative to a rotating observer: R = 2/3 COUNTERROTATING
R
Frequency relative to an inertial observer: R = t
v = r2 r [ sin2ei(2t
R = - 2/3 COROTATING
FLOW PATTERN OF THE l = m = 2 r-MODE
Rotating Frame
Animation shows backward (clockwise) motion of pattern
and motion of fluid elements
Ben Owen’s animation
Inertial Frame Pattern moves forward
(counterclockwise)
Star and fluid elements rotate forward more rapidly
Above 1010K, beta decay and inverse beta decay
n
Below 109K, shear viscosity (free e-e scattering) dissipates the mode’s energy in heat SHEAR = CT-2
produce neutrinos that carry off the energy of the mode:bulk viscosityBULK = CT6
ep
VISCOUS DAMPING
105 107 109 1011 (From Lindblom-Owen-Morsink Figure) Temperature (K)
critmax
Bulk viscosity kills instability at high temperature
Shear viscosity kills instability atlow temperature
Star is unstable only when is larger than critical frequency set by bulk and shear viscosity
Star spins down as it radiates its angular momentum in gravitational waves
hc = 1024 (20 Mpc/D)
AMPLITUDE, v/R
hc = h[t(f)] / f2/|df/dt|
Owen, Lindblom, Cutler, Schutz, Vecchio, Andersson
Brady, Creighton
Owen
Lindblom
GRAVITATIONAL WAVES FROM SPIN-DOWN
GRAVITATIONAL WAVES FROM SPIN-DOWN
hc = 1024 (20 Mpc/D)
AMPLITUDE, v/R
100 Hz 1000 Hz
hc
10-20
10-21
10-22
10-23
hc
LIGO I
LIGO II
IF ONE HAD A PRECISE TEMPLATE, SIGNAL/NOISE
WOULD LOOK LIKE THISFOR WAVES FROM A GALAXY 20 Mpc AWAY:
INSTABILITY OF OLD ACCRETING STARS:
LMXBs
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%
FASTEST 3: 619 Hz, 622 Hz, 642 Hz
From Chakrabarty, Bildsten
VERY DIFFERENT MAGNETIC FIELDS – IMPLIES SPIN NOT LIMITED BY MAGNETIC FIELD
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
CAN GW FROM LMXBs BE OBSERVED?
IF WAGONER’S 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 IT’S UNSTABLE
LEVIN’S CYCLE
5x106 yr
4 months!
T107 108 109
criticalmax
SPIN DOWN TIME < 1/106 SPIN UP TIME IS A STAR YOU NOW OBSERVE SPINNING DOWN?
PROBABILITY < 1/106
POSSIBLE WAY OUT (Wagoner, Andersson, Heyl)
AS WE MENTION LATER, DISSIPATION IN A QUARK OR HYPERON CORE CAN INCREASE AS TEMPERATURE INCREASES:
T109 109.5108.5108
critical (Hz)
INSTABILITY GROWTH ENDS AS STAR HEATS UP AND VISCOSITY LIMITS INSTABILITY
DOES THE INSTABILITY SURVIVE THE PHYSICS OF A REAL NEUTRON STAR?
Will nonlinear couplings limit the amplitude to v/v << 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 production
kill the instability?
Will nonlinear couplings limit the amplitude to v/v << 1?
Fully nonlinear numerical evolutions show no evidence that nonlinear couplings limiting the amplitude to v/v < 1:
Nonlinear 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
GR Evolution Font, Stergioulas
Newtonian evolution with artificially enhanced radiation reaction
Lindblom, Tohline, Vallisneri
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)
Newtonian evolution with somewhat higher resolution, w/ and w/out enhanced radiation-driving force
Gressman, Lin, Suen, Stergioulas, JF
Catastrophic decay of r-mode
Fourier transform shows sidebands - apparent daughter modes.
RELATIVISTIC r-MODES
Andersson, Kojima, Lockitch, Beyer & Kokkotas, Kojima & Hosonuma, Lockitch, Andersson, JF, Lockitch&Andersson, Kokkotas & Ruoff
Newtonian axial mode
Relativistic corrections to the l=m=2 r-mode mix axial and polar parts to 0th order in rotation.
50x axial correction
r/R
50x polar parts
10
Lockitch
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), [– m)/l(l+1)]hrr + Ahr+Bh
IF THE COEFFICIENT OF hrr VANISHES IN THE STAR, THERE IS NO SMOOTH EIGENFUNCTION.
INSTEAD, THE SPECTRUM IS CONTINUOUS.
THIS COULD BE DUE TO THE APPROXIMATION’S 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
BUT
NEWTONIAN STARS WITH SOME DIFFERENTIAL ROTATION LAWS ALSO MAY HAVE A CONTINUOUS SPECTRUM Karino, Yoshida, Eriguchi
NUMERICAL 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
VISCOUS BOUNDARY LAYER NEAR CRUST(NILS ANDERSSON’S TALK)
Bildsten, Ushomirsky RieutordWu, Matzner, Arras Lindblom, Owen, Ushom.Levin, Ushomirsky Andersson, Jones, Kokkotas, Yoshida Stergioulas
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
A GROWING MAGNETIC FIELD DAMPS INSTABILITY WHEN
GW> 1
B-field 1010 G to 1012 G allows instability for 2 days to 15 minutes
Rezzolla, Lamb, Markovic, Shapiro
GRAVITATIONAL WAVES FROM SPIN-DOWNWITH DAMPING BY MAGNETIC FIELD WINDUP
100 Hz 1000 Hz
hc
10-20
10-21
10-22
10-23
hc
LIGO I
LIGO II
108 G1010 G1012 G1014 G
FINALLY
Will bulk viscosity from hyperon production
kill the instability?
P.B. Jones
Lindblom, Owen
Haensel, et al
If the core is dense enough ( ) to have
hyperons, nonleptonic weak interactions can greatly
increase the bulk viscosity:
14106
pnn
d d
u
d
du
n
n
W
u
s
p
If the core is dense enough ( ) to have
hyperons, nonleptonic weak interactions can greatly
increase the bulk viscosity:
14106
pnn
d
u
d
d
n
n
u
s
p
p
V
Equilibrium
With no neutrinos emitted, dissipation comes from the net p dV work done in an out-of-equilibrium cycle
As fluid element contracts, nucleons change to hyperons; if reactions are too slow to reach equilibrium, have more nucleons and higher pressure than in equilibrium
Fluid expands: if reactions too slow,more hyperons and lower pressure than in equilibrium
With no neutrinos emitted, dissipation comes from the net p dV work done in an out-of-equilibrium cycle
p
V
dVpThe work is the energy lost by the fluid element in one oscillation
105 107 109 1011 (From Lindblom-Owen-Morsink Figure) Temperature (K)
Shear viscosity kills instability atlow temperature
Bulk viscosity from hyperons cuts off instability below a few x 109 K
Standard cooling (modified URCA) still allows a one- day spin down, radiating most of the initial KE
Bulk viscosity kills instability at high temperature
But faster cooling is more likely:direct URCA cools in seconds, allowing hyperon interactionsor a crust to damp the instabilitybefore it has a chance to grow
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
T109 109.5108.5108
critical (Hz)
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
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