NEUTRON STAR DYNAMICS

Transcription

1 NEUTRON STAR DYNAMICS Kostas Kokkotas Theoretical Astrophysics, IAAT, Eberhard Karls University of Tübingen Erice 23/09/10 1

2 Gravitational Wave Asteroseismology We can estimate their masses, radii, equations of state by analysing the seismic data via the emitted gravitational waves Neutron Stars oscillate wildly during the very first seconds of their life Rotation is responsible for a number of instabilities which emit copious amounts of GWs Erice 23/09/10 2

3 A Laboratory for Theoretical Physics NS modelling involves the very extremes of physics: General Rela)vity Rota)on Equa)on of State Magne)c Fields Crust Core interface Alternative theories, Boson, Q-stars, Slow, Fast, Differential Instabilities Exotic nuclear physics, strange quarks, hyperons, Superfluidity, Cold vs Warm Magnetars, Slowdown, Suppression of Instabilities, Uniform Rotation Supression of Instabilities, Can GW, x-ray, γ-ray observations constrain the theoretical models? Erice 23/09/10 3

4 Neutron Star ringing p-modes: main restoring force is the pressure (f-mode) (>1.5 khz) σ M R 3 Inertial modes: (r-modes) main restoring force is the Coriolis force w-modes: pure space-time modes (only in GR) (>5kHz) σ Ω σ 1 R Torsional modes (t-modes) (>20 Hz) shear deformations. Restoring force, the weak Coulomb force of the crystal ions. σ v S R and many more Erice 23/09/10 4

5 GW Asteroseismology Oscillation patters can reveal the internal structure of neutron stars : mass, radius, EoS, rotation, B-field, crust, M σ f (khz) R 1/2 f-mode frequency 1 M3 M τ f (s) R R σ w (khz) f-mode frequency 1 M R R w-mode frequency Andersson,KK 1996,1998, Erice /2 (M/R ) Rωw -mode ωf-mode A B C D E F G I L G240 G300 WFF Q_1 Q_2 Q_ A B C D E F G I L WFF G240 G M (Solar Masses) A B C D E F G I L G240 G300 WFF Q_1 Q_2 Q_ /09/10 M/R Radius (Km)

6 Effect of Rotation & Magnetic Fields ROTATION Frame dragging Quadrupole deformation Rotational instabilities The degeneracy in m is removed and the nonrotating mode of index is split into 2 +1 different (,m) modes Shifting of the frequencies and damping times Coupling of polar -term to an axial ±1 term and v-v MAGNETIC FIELD No significant effect in the fluid frequencies and damping/growth times magnetic energy gravitational energy ~ B2 R 3 GM 2 / R ~ 10 4 B G For magnetars we may observe Alfvén oscillations 2 Erice 23/09/10 6

7 Stability of Rotating Stars Non-Axisymmetric Perturbations A general criterion is: T : rotational kinetic energy W : gravitational binding energy β = T W 2 15 e Dynamical Instabilities Driven by hydrodynamical forces (bar-mode instability) Develop at a time scale of about one rotation period Secular Instabilities Driven by dissipative forces (viscosity, gravitational radiation) Develop at a time scale of several rotation periods. Chandrasekhar-Friedman-Schutz (CFS) e > or β > e > or β > GR predicts considerably lower β β~0.24 for the onset of the dynamical instabilities β~0.07 for the onset of the secular instabilities Erice 23/09/10 7

8 Bar-mode dynamical instability For rapidly (differentially!) rotating stars with: β = T W ~ 1 R > β 0.27 dyn GR enhances the onset of the instability ( dyn 0.24) and decreases with increasing M/R. The bar-mode grows on a dynamical timescale. If the bar persists for many (~10-100) rotation periods, the signal will be easily detectable from at least Virgo cluster. ε f h khz Typical Frequencies ~ kHz 2 15 Mpc d M R Erice 23/09/10 8

9 Bar Mode Dynamical Instability Bars can be also created during the merging of NS-NS, BH-NS, BH-WD and Collapsars (type II). Bar-mode instability might happen for much smaller if centrifugal forces produce a peak in the density off the source s rotational center. f h eff Hz 1/ 2 LOW T/ W Instability R eq M 30km 1.4M Highly differentially rotating stars are shown to be dynamically unstable for significantly lower (even when 0.01). 1/ Mpc d Bars can be also create during the collapse of a SMS before the creation of a SMBH. Ideal sources for LISA. Rezzola et al Erice 23/09/10 9

10 The Excitation of Secular Instabilities SATURATION AMPLITUDE REACHED Mode is damped via shocks or mode coupling OSCILLATION AMPLITUDE UNSTABLE GROWING PHASE Dura)on 20 sec 20 min Rota)on Temperature EoS, UNKNOWNS Critical rotation Maximum Amplitude Duration Differential Rotation Instability Window TIME Erice 23/09/10 10

11 INSTABILITY WINDOW ROTATION Ω/Ω K Shear Viscosity Magnetic field Hyperon viscosity Crust Quark matter 0.05 Mutual fric)on r mode f mode 10 7 K 10 8 K 10 9 K K Kepler limit Stergioulas+Friedman 1998 Gaertig+KK 2008 Krüger+Gaertig+KK 2009 Zink+Stergioulas Gaertig+KK 2010 Gaertig+Glampedakis+KK +Zink 2010 Bulk Viscosity Andersson+KK+Schutz 1999 KK+Stergioulas 1999 Bildsten+Ushomirsky 2000 Lindblom+Ipser 2002 TEMPERATURE Erice 23/09/10 11

12 f-mode Instability Onset of instability Unknowns (?): Duration (width of the instability window) Amplitude (saturation) 10Hz 100Hz 1000Hz Erice 23/09/10 12

13 LMXBs & r-modes Limiting Period : ~ ms Fastest known pulsar: 1.4ms UNSTABLE 1.5ms 5ms Period clustering of ms pulsars STABLE Andersson, KK, Stergioulas 1999, Levin 2000 Erice Andersson, Jones, KK, Stergioulas 2000, Heyl 2002 Andersson, Jones, KK /09/10 13

14 10Kpc h( t) 10 α 1 khz d α 20 Ω R-modes GW amplitude depends on the saturation amplitude Mode coupling might not allow the growth of instability to high amplitudes (Cornell group `04-`07) The existence of crust, hyperons in the core, magnetic fields, affects the efficiency of the instability. For newly born neutron stars might be quite weak ; unless we have the creation of a strange star Old accreting neutron (or strange) stars, probably the best source! Erice 23/09/10 14

15 Fast Rotating NS in GR: f-mode Frequency 2l(l 1) GM ω 2 = 2l +1 R 3 Gaer8g+KK 2008,09,10 Krueger,,Gaer8g, KK 2009 Zink etal 2010 Damping/Growth time t GW f (l)r R M l+1 ~ M R M 10km 3 4 sec In GR the m=2 mode becomes unstable for >0.85 Kepler Differential rotation affects the onset of the instability Up to 10% of energy and angular momentum will be dissipated by GWs. Major uncertainties: 1. Relativistic growth times 2. Nonlinear saturation 3. Initial rotation rates of protoneutron stars f-modes Gaertig+KK 2008 LIGO/Virgo/GEO-HF band g & r-modes Passamonti et al 2008 Gaertig+KK 09 Erice 23/09/10 15

16 f-mode : non-linear results B. Zink, N. Stergioulas, O. Korobkin, P. Diener, E. SchneIer (2010) Top: Frequencies of l = m = 2 f modes in rapidly rota8ng Γ= 2 polytropes BoWom: Comparison between Γ= 2 and Γ= 2.5 Erice 23/09/10 16

17 f-modes: Asteroseismology We can trace the effect of rotation of the f-mode (and any p, i or g-mode) frequency and the onset of the CFS instability We can produce empirical relation relating the parameters of the neutron stars to the observed frequencies. Gaer)g KK 2008, 2010 σ Ω σ 0 Ω K 0.32 Ω Ω K σ Ω σ 0 Ω K 0.53 Ω Ω K (m = 2) +... (m = -2) Ω K 0.67 M R 3 1/2 C(χ) = M R M or Ω K C(χ) R 3 1/2 Erice 23/09/10 17

18 E = 1 2 de dt = σ i f-mode: Damping/Growth time ρδu a δu * a + δ p ρ δρ* d 3 x E σ 2 4 ( σ i + mω) N δ D m σ i de dt σ 6 i 1 = 1 τ GR 2E de dt σ 4 Gaer)g KK /4 τ 0 τ sgn(σ i )0.71 σ i σ σ i σ σ i σ 0 2 τ 0 τ σ c σ σ c σ σ c σ 0 3 Erice 23/09/10 18

19 Instability Window For the first time we have the window of f-mode instability in GR Gaertig, Glampedakis, KK, Zink (2010) 33 min typical Growth Time Mutual friction ~28 sec l=2 ~48 sec l=3 ~90 sec l=4 Erice 23/09/10 19

20 Animation of the l=m=2 f-mode Kastaun, Willburger, KK (2010) Erice 23/09/10 20

21 Detectability (10Mpc) Kastaun, Willburger, KK (2010) h Mpc r 10Mpc h eff = h N r N Erice 23/09/10 21 N

22 Magnetized Stars Observations Watts & Strohmayer (2006) Giant flares in SGRs Up to now, three giant flares have been detected. SGR in 1979, SGR in 1998, SGR in 2004 Peak luminosities : erg/s A decaying tail for several hundred seconds follows the flare. QPOs in decaying tail (Israel et al. 2005; Watts & Strohmayer 2005, 2006) SGR : 28, 54, 84, and 155 Hz SGR : 18, 26, 29, 92.5, 150, 626.5, & 1837 Hz (possible additional frequencies : 720 & 2384 Hz) Erice 23/09/10 22

23 Piro 2005 Glampedakis, Samuelsson, Andersson 2006 Sotani, KK, Stergioulas 2007 Samuelsson, Andersson 2007 Levin 2007 Sotani, KK, Stergioulas 2008 Vavoulidis, Stavridis, KK 2008 Sotani, Colaiuda, KK 2008 Colaiuda,Beyer,KK 2009 Cerda Duran, Stergioulas, Font 2009 Sotani, KK 2009 Steiner, WaWs 2009 Lander, Jones, Passamon) 2009 Van Hoven, Levin 2010 Cerda Duran etal 2010 Colaiuda+ KK 2010 (i) Erice 23/09/10 23 (ii)

24 Alfven Con)nuum + Discrete oscilla)ons APR 14 44Hz 8 y x 10 APR 14 _66Hz 8 Without Crust Levin 2007 Sotani,KK, Stergioulas 2008 Colaiuda, Beyer, KK 2009 Cerda Duran, Stergioulas, Font 2009 With Crust Van Hoven, Levin 2010 Cerda Duran, Stergioulas, Font 2010 Colaiuda, KK 2010 Erice 23/09/10 y x 24

25 Axial type of perturbations described by discrete modes (crust) and a continuum (core) and they can explain the lower frequencies observed Polar type of perturbations only discrete modes with higher frequencies Alfvén modes f n even (2n + 1) f 0 f odd n (n + 1) f 0 B f C C t n 1 + a n n (n + 1) f 0 On the other hand the observed frequencies of QPOs in SGRs SGR : 28, 54, 84, 155 Hz B µ 2 1/2 t 0 Crust modes Sotani,KK,Stergioulas 2007 crust torsional oscillation? or polar oscillation? SGR : 18, 26, 30, 92.5, 150 Hz 0.6 crust torsional oscillation? EoS =>APR+DH, M=1.4 M, R=12.1Km ΔR/R=0.93km, B=4x10 15 Gauss Erice 23/09/10 25

26 SGR Colaiuda, KK crust APR 14 B = Gauss 10 discrete Alfvén modes Frequency(HZ) Identification of the frequencies of SGR We show that a stellar model APR with mass M = 1.4M can explain all the frequencies observed. Erice 23/09/10 26

27 SGR Colaiuda, KK discrete Alfvén mode crust WFF 14 B = Gauss continuum 73!91 Hz continuum 84 60Hz!76Hz Frequency(Hz) Identification of the frequencies of SGR We show that a stellar model WFF with mass M = 1.4M can explain all the frequencies observed. Erice 23/09/10 27

28 Magnetars & GWs Many mode frequencies in the op)mal band of GW detectors Ideal Source for Mul) messenger Astronomy It will be hard to excite density perturba)ons E mode E burst Amplitude of GWs UNKNOWN (Only Galac)c Sources) x10-4 f-mode? 10Hz 100Hz 1000Hz Erice 23/09/10 28

29 Conclusions Rotational Instabilities of Neutron Stars Are potential sources for GW beyond our galaxy Many open issues (growth time, non-linear coupling, ) are resolved one after the other. Dynamics of magnetars Offers the possibility to understand their structure Most probably a weak source for GW with the present generation detectors Erice 23/09/10 29