INSTABILITIES OF ROTATING RELATIVISTIC STARS John Friedman University of Wisconsin-Milwaukee Center 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) outli ne

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=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 by Gondek-Rosinska and Gourgoulhon Shibata, Karino, Eriguchi, Yoshida Watts, Andersson, Beyer, Schutz Centrella, New, Lowe and Brown Imamura, Durisen, Pickett New, Centrella and Tohline Shibata, Baumgarte and Shapiro

GROWTH OF AN l=m=1 INSTABILITY IN 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 cfs1

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 the star's rotation: A mode with behavior that moves backward relative to the star is dragged forward relative to, when m  The mode still has J < 0, because J star + J mode < J star. This backward mode, with J < 0, radiates positive J to. Thus J becomes increasingly negative, and THE AMPLITUDE OF THE MODE GROWS cfs4

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 surprise 2

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. th_surprise

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 These surprises led to an explosion of interest:

Stergioulas, Font, Kokkotas Kojima, Hosonuma Yoshida, Lee Rezania, Jahan-Miri Yoshida, Karino, Yoshida, Eriguchi Rezania, Maartens Andersson, Lockitch, JFLindblom, Mendell Andersson, Kokkotas, StergioulasAndersson Ushomirsky, Cutler, BildstenBildsten, Ushomirsky Andersson, Jones, Kokkotas, Brown, Ushomirsky Stergioulas Lindblom,Owen,UshomirskyRieutord Wu, Matzner, ArrasHo, Lai Levin, Ushomirsky Madsen Lindblom, Tohline, VallisneriStergioulas, Font Arras, Flanagan, Schenk, JF, Lockitch Sa Teukolsky,Wasserman MorsinkJones Lindblom,Owen Ruoff, Kokkotas, Andersson,Lockitch,JF STILL MORE RECENT

Karino, Yoshida, EriguchiHosonuma Watts, AnderssonRezzolla,Lamb,Markovic, Arras, Flanagan, Morsink, Shapiro Wagoner, Hennawi, Liu Shenk, Teukolsky, Wasserman Morsink Jones, Andersson, Stergioulas Haensel, Lockitch, Andersson Prix, Comer, Andersson Hehl Gressman, Lin, Suen, Stergioulas, JF Lin, Suen Xiaoping, Xuewen, Miao, Shuhua, Nana Reisnegger, Bonacic Yoon, Langer Drago, Lavagno, Pagliara Drago, Pagliara, Berezhiani Gondek-Rosinska, Gourgoulhon, Haensel Brink, Teukolsky, Wasserman AND MORE

ELECTROMAGNETIC RADIATION MASS QUADRUPOLECHARGE DIPOLE ENERGY RADIATED C

ELECTROMAGNETIC RADIATION MASS QUADRUPOLECHARGE QUADRUPOLE ENERGY RADIATED

GRAVITATIONAL RADIATION MASS QUADRUPOLE ENERGY RADIATED

AXIAL GRAVITATIONAL RADIATION MASS QUADRUPOLECURRENT QUADRUPOLE ENERGY RADIATED

AXIAL GRAVITATIONAL RADIATION MASS QUADRUPOLECURRENT QUADRUPOLE ENERGY RADIATED

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 Y lm AND Y lm 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 = 1P = -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 ROTATION ENERGY AT INSTABILITY 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-MODE Newtonian: Papaloizou & Pringle, Provost et al, Saio et al, Lee, Strohmayer The mode is a current that is odd under parity  v = r 2 r [ sin 2  e i(2  t   Frequency relative to a rotating observer:  R = 2/3  COUNTERROTATING R Frequency relative to an inertial observer:  R =  t  v = r 2 r [ sin 2  e i(2  t    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 K, beta decay and inverse beta decay n Below 10 9 K, 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 viscosity  BULK = CT 6 e p VISCOUS DAMPING

(From Lindblom-Owen-Morsink Figure) Temperature (K)  crit  max    Bulk viscosity kills instability at high temperature Shear viscosity kills instability at low 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

h c = (      20 Mpc/D)  AMPLITUDE,  v/R  h c = h[t(f)] / f 2 /|df/dt| Owen, Lindblom, Cutler, Schutz, Vecchio, Andersson Brady, Creighton Owen Lindblom GRAVITATIONAL WAVES FROM SPIN-DOWN

h c = (      20 Mpc/D)  AMPLITUDE,  v/R  100 Hz1000 Hz h c hchc LIGO I LIGO II IF ONE HAD A PRECISE TEMPLATE, SIGNAL/NOISE WOULD LOOK LIKE THIS FOR WAVES FROM A GALAXY 20 Mpc AWAY:

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

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 5x10 6 yr 4 months! T  critical  max SPIN DOWN TIME < 1/10 6 SPIN UP TIME IS A STAR YOU NOW OBSERVE SPINNING DOWN? PROBABILITY < 1/10 6

POSSIBLE WAY OUT (Wagoner, Andersson, Heyl) AS WE MENTION LATER, DISSIPATION IN A QUARK OR HYPERON CORE CAN INCREASE AS TEMPERATURE INCREASES: T  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 2 nd order 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 GR Cowling 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 2 nd 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,Wasserman Brink, 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 0 th order in rotation. 50x axial correction r/R 50x polar parts 10 Lockitch

Lockitch Lockitch, Andersson, JF

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)]  h rr + Ah r +Bh  IF THE COEFFICIENT OF h rr 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 MIX THE 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, Ruoff BUT THAT 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, UshomirskyRieutord Wu, Matzner, ArrasLindblom, Owen, Ushom. Levin, UshomirskyAndersson, Jones, Kokkotas, Yoshida Stergioulas

VISCOUS BOUNDARY LAYER NEAR CRUST VISCOSITY MUCH HIGHER NEAR CRUST: SHEAR DISSIPATION: dE/dt = -2  dV (   Boundary width d Bulk  v/d  v/R dE/dt (boundary) ~ (R/ d ) dE/dt(core) But, we don’t know to what extent the crust moves with the modes.

(Prepared by Ben Owen) Red curves show the much higher shear viscosity when a crust is present (for upper red curve, a superfluid has formed)

DOES NONLINEAR EVOLUTION LEAD TO DIFFERENTIAL ROTATION THAT DISSIPATES r-MODE ENERGY IN A MAGNETIC FIELD? SpruitRezzola, Lamb, Shapiro Levin, UshomirskyR, Markovic, L, S A computation of the 2 nd 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 G to G allows instability for 2 days to 15 minutes Rezzolla, Lamb, Markovic, Shapiro

GRAVITATIONAL WAVES FROM SPIN-DOWN WITH DAMPING BY MAGNETIC FIELD WINDUP 100 Hz1000 Hz h c hchc LIGO I LIGO II 10 8 G10 10 G10 12 G10 14 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: dd ud du n n W u s p

d u d d n n u s p

p

dd ud uu n p n W 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 The work is the energy lost by the fluid element in one oscillation

(From Lindblom-Owen-Morsink Figure) Temperature (K)      Shear viscosity kills instability at low temperature Bulk viscosity from hyperons cuts off instability below a few x 10 9 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 interactions or a crust to damp the instability before 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

T  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 amplitude and the more efficient damping mechanisms allowed by hyperon or quark cores enhance the probability of seeing gravitational waves from r-mode unstable LMXBs