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