Instabilities in neutron stars and gravitational waves

Transcription

1 Instabilities in neutron stars and gravitational waves Andrea Passamonti INAF-Osservatorio di Roma 2014

2 Rotational instabilities Non-axisymmetric instabilities of a rotating fluid star What is their origin? Some references Shapiro & Teukolsky, The physics of Compact objects In which astrophysical systems? Andersson 2003, Class. Quantum Grav. 20 R105 Stergioulas 2003, Living Review in relativity What is their gravitational-wave signal?

3 Basic classification of Instabilities Rotation parameter β = T W T : rotational kinetic energy W : gravitational energy Bar-mode instability, when β 0.24 Low T/W instability, for and high differential rotation β 0.01 Ω (r, θ) Gravitational-wave driven instability β 0.14

4 Basic classification of Instabilities Rotation parameter β = T W T : rotational kinetic energy W : gravitational energy Bar-mode instability, when Low T/W instability, for and high differential rotation β 0.24 β 0.01 Ω (r, θ) } Dynamical instabilities τ ~ ms Gravitational-wave driven instability β 0.14

5 Basic classification of Instabilities Rotation parameter β = T W T : rotational kinetic energy W : gravitational energy Bar-mode instability, when Low T/W instability, for and high differential rotation β 0.24 β 0.01 Ω (r, θ) } Dynamical instabilities τ ~ ms Gravitational-wave driven instability β 0.14 } Secular instabilities τ > minutes CFS instability

6 Where do they can occur? protoneutron stars core collapse from a SN accretion induced collapse of a WD in a binary system recycled pulsars in Low Mass X-ray Binary systems (LMXB).

7 Oscillation modes The oscillation spectrum in neutron stars is very rich In principle any physical ingredient of a neutron star can be associated to a family of oscillations with different frequency and characteristics Fundamental mode (f-mode): M ν R 3 ν =1 4 khz r mode: ω 2mΩ l (l + 1) - rotationally restored by Coriolis force Harmonic decomposition Y lm P l (cos θ) e imφ

8 Dynamical instabilities

9 Bar mode instability Classical studies with incompressible self-gravitating rotating ellipsoids affirm that the instability sets in when β > 0.27 Relativistic simulations reduces slightly this value β > Persistence of the bar depends on the distance from the critical point, but typically lasts for ms τ > s (e.g. see De Pietri et al and reference therein)

10 Bar mode instability Classical studies with incompressible self-gravitating rotating ellipsoids affirm that the instability sets in when β > 0.27 Relativistic simulations reduces slightly this value β > Persistence of the bar depends on the distance from the critical point, but typically lasts for ms τ > s (e.g. see De Pietri et al and reference therein)

11 Gravitational waves GW strain estimate h ε 2 f 15 Mpc M khz d 1.4M 2 R 10 km GW frequency in incompressible spheroids (Newtonian) is twice the star s spin Detection prospectives for Adv. LIGO/Virgo: SNR ~ 1 for sources at τ 10 > Mpc s Magnetic field: suppresses the instability for a seed Bp > G <Bp < G modifies the growth time etc.. (Baiotti et al. 2007) Bp < G, no effects. (Camarda et al. 2009, Franci et al. 2013)

12 Low T/W instability Originally found in numerical simulations by Centrella et al. (2001), Shibata et al. (2002) It develops in highly differentially rotating stars even for β ~ 0.01 Several modes are excited during the instability the l=m=2 f-mode is typically the dominant Growth time ~ 1-4 ms Shibata et al. 2002

13 Low T/W instability Originally found in numerical simulations by Centrella et al. (2001), Shibata et al. (2002) It develops in highly differentially rotating stars even for β ~ 0.01 Several modes are excited during the instability the l=m=2 f-mode is typically the dominant Growth time ~ 1-4 ms Corvino et al. 2010

14 low T/W instability It seems to originate at the corotation points, where (Watts, Andersson and Jones, 2005) ω m = Ω Rotation law: j-constant Ω c A 2 Ω = A 2 + r 2 sin 2 θ Ω / Ω c J-constant rotation law δσ ij = 1 A 2 = 10 i δu j + j δu i gij δσ δσ = j δu j A 2 = A 2 = 0.01 A 2 = ϖ

15 low T/W instability σ p : pattern speed It seems to originate at the corotation points, where (Watts, Andersson and Jones, 2005) ω m = Ω Rotation law: j-constant Ω c A 2 Ω = A 2 + r 2 sin 2 θ Ω / Ω c J-constant rotation law δσ ij = 1 A 2 = 10 i δu j + j δu i gij δσ δσ = j δu j A 2 = A 2 = 0.01 A 2 = ϖ

16 low T/W instability σ p : pattern speed It seems to originate at the corotation points, where (Watts, Andersson and Jones, 2005) J-constant rotation law ω m = Ω patt. speed Rotation law: j-constant Ω c A 2 Ω = A 2 + r 2 sin 2 θ Ω / Ω c δσ ij = 1 A 2 = 10 i δu j + j δu i gij δσ δσ = j δu j A 2 = A 2 = 0.01 A 2 = ϖ

17 low T/W instability σ p : pattern speed It seems to originate at the corotation points, where (Watts, Andersson and Jones, 2005) ω m = Ω Rotation law: j-constant Ω c A 2 Ω = A 2 + r 2 sin 2 θ Ω / Ω c J-constant rotation law δσ ij = 1 A 2 = 10 i δu j + j δu i gij δσ δσ = j δu j A 2 = 1 patt. speed 0.2 A 2 = 0.01 A 2 = ϖ

18 low T/W instability σ p : pattern speed It seems to originate at the corotation points, where (Watts, Andersson and Jones, 2005) ω m = Ω Rotation law: j-constant Ω c A 2 Ω = A 2 + r 2 sin 2 θ Ω / Ω c J-constant rotation law δσ ij = 1 A 2 = 10 i δu j + j δu i gij δσ δσ = j δu j A 2 = 1 patt. speed 0.2 A 2 = 0.01 A 2 = ϖ

19 low T/W instability σ p : pattern speed It seems to originate at the corotation points, where (Watts, Andersson and Jones, 2005) ω m = Ω Rotation law: j-constant Ω c A 2 Ω = A 2 + r 2 sin 2 θ σ / ( G Ω ρ m / Ω ) 1/2 c l = m = 2 f-mode J-constant rotation law A 2 = 1 δσ ij = 1 A 2 = 10 i δu j + j δu i gij δσ δσ = j δu j corotation band A 2 = 1 patt. speed A 2 = 0.01 A 2 = ϖ Cowling Non-Cowling T / W

20 R-mode? The r-mode enters in corotation only marginally for highly differentially rotating stars The growth time should be ~ infinite ω / Ω c N=1 Polytrope m Ω eq / Ω c j-constant law l=m=2 r-mode δσ ij = 1 2 δσ = j δu j i δu j + j δu i 2 3 gij δσ r ax = 0.95 r ax = 0.7 r Corotation region ax = 0.6 No-corotation region log A Passamonti and Andersson (in preparation)

21 Gravitational waves GW strain: quasi-periodic oscillations after the saturation of instability h eff /2 f 100 Mpc Req 800 Hz d 30 km M 1.4M 1/2 Shibata τ et > al. s (2002) GW emitted by the initial growth of the instability: a SNR 20 for Adv. LIGO/Virgo from a source at 92 kpc. (Ott et al. 2014) Effects of magnetic field (Ott et al. 2014) For seed field with Bp < G no effects For seed field with Bp > G, magnetic instability can amplify a m=2 mode.

22 Secular Instabilities Chandrasekhar-Friedman-Schutz instability Chandrasekhar S 1970 Astrophys. J Friedman J L and Schutz B F 1978 Astrophys. J

23 Origin: CFS instability Inertial observer retrograde mode Comoving observer retrograde mode Star prograde mode Star prograde mode Ω > Ω c both modes are prograde prograde mode Star

24 Origin: CFS instability Inertial observer retrograde mode Comoving observer retrograde mode Star prograde mode Star prograde mode Ω > Ω c both modes are prograde prograde mode Star

25 Origin: CFS instability Inertial observer retrograde mode Comoving observer retrograde mode Star prograde mode Star prograde mode Ω > Ω c both modes are prograde prograde mode Star

26 Origin: CFS instability Inertial observer retrograde mode Comoving observer retrograde mode Star prograde mode Star prograde mode Ω > Ω c both modes are prograde prograde mode J gw >0 Star

27 Origin: CFS instability Inertial observer retrograde mode Comoving observer retrograde mode Star prograde mode Star prograde mode Ω > Ω c both modes are prograde prograde mode Star J gw >0 But the intrinsic J of the mode is negative. Thus it becomes increasing negative and the mode is unstable

28 Unstable modes Neutral point when the retrograde mode has zero frequency in the inertial frame R-mode (m=2) CFS unstable for any Ω l = m = 2 f-mode A 2 = 1 τ 22 gw 10s F-mode (2 m 4) CFS unstable in rapidly rotating stars τ 44 gw s ω / ( G ρ m ) 1/ Cowling Non-Cowling Ω c / ( G ρ m ) 1/2

29 Unstable modes Neutral point when the retrograde mode has zero frequency in the inertial frame R-mode (m=2) CFS unstable for any Ω l = m = 2 f-mode A 2 = 1 τ 22 gw 10s F-mode (2 m 4) CFS unstable in rapidly rotating stars τ 44 gw s ω / ( G ρ m ) 1/ Cowling Non-Cowling Ω c / ( G ρ m ) 1/2 Neutral point

30 Instability window Ω unstable Ω c stable

31 Instability window Bulk viscosity 1 τ b T 6 Ω unstable Ω c stable

32 Instability window Bulk viscosity 1 τ b T 6 Ω unstable Ω c stable T

33 Instability window Bulk viscosity 1 τ b T 6 Ω unstable bulk visc. Ω c stable T

34 Instability window Bulk viscosity 1 τ b T 6 Ω unstable Shear viscosity bulk visc. 1 τ s T 2 Ω c stable T

35 Instability window Bulk viscosity 1 τ b T 6 Shear viscosity 1 τ s T 2 Ω Ω c shear visc. unstable stable bulk visc. T

36 Instability window Bulk viscosity 1 τ b T 6 Shear viscosity (Hz) Ω shear visc. shear viscosity UNSTABLE rigid crust? unstable bulk visc. bulk viscosity 1 τ s T 2 g Ω c 200 STABLE stable log 10 T c T

37 LMXBs Observed fastest rotating NS in LMXB spins at 640 Hz (4U 1608) ~ 700 Hz rotational cut-off (< break-up limit) rigid crust? r-mode instability?

38 LMXBs Observed fastest rotating NS in LMXB spins at 640 Hz (4U 1608) ~ 700 Hz rotational cut-off (< break-up limit) rigid crust? r-mode instability? LMXBs

39 LMXBs Observed fastest rotating NS in LMXB spins at 640 Hz (4U 1608) ~ 700 Hz rotational cut-off (< break-up limit) rigid crust? r-mode instability? LMXBs paradox neutron stars should be experience fast spin-down. Not observed!! It is required additional damping (more physics in the models)? r-mode saturation amplitude much lower than current theoretical prediction? LMXBs

40 l=m=4 f-mode Evolution Model: M =1.98M N = 0.62 polytrope 1 Evolutionary path Population: 1% of core collapse from progenitors with M>10M Rate: per year from Virgo cluster Ω / Ω K M = 1.98 M. N = 0.62 α sat = 10-4 Shear viscosity heating prevents the star from entering the mutual friction region delaying the superfluid transition T cn At Ω = Ω K τ gw 10 2 s the growth time is T [K] Passamonti et al. 2013

41 l=m=4 f-mode Evolution Model: M =1.98M N = 0.62 polytrope 1 Evolutionary path Population: 1% of core collapse from progenitors with M>10M M = 1.98 M. N = 0.62 α sat = yr 0.1 yr Rate: per year from Virgo cluster Ω / Ω K yr Shear viscosity heating prevents the star from entering the mutual friction region delaying the superfluid transition T cn 100 yr At Ω = Ω K τ gw 10 2 s the growth time is T [K] Passamonti et al. 2013

42 GW signal Characteristic strain h c = h ν 2 dt dν l=m=4 f-mode l = m = 4 f-mode α sat = 10-4 d = 20 Mpc N = 0.62 Adv. LIGO N = 1 ET Integration time f mode h c t obs 1yr Source distance: Virgo cluster d = 20 Mpc ν [ khz ] Passamonti et al. 2013

43 GW signal Characteristic strain h c = h ν 2 dt dν l=m=3 f-mode l = m = 3 f-mode α sat = 10-4 N = 0.62 d = 20 Mpc Adv. LIGO Integration time f mode h c ET t obs 1yr Source distance: Virgo cluster d = 20 Mpc ν [ khz ] Passamonti et al. 2013

44 GW signal Characteristic strain h c = h ν 2 dt dν l=m=3 f-mode l = m = 3 f-mode α sat = 10-4 N = 0.62 d = 20 Mpc Adv. LIGO Integration time f mode h c ET t obs 1yr Source distance: Virgo cluster d = 20 Mpc l=m=2 r-mode d=20 Mpc ν [ khz ] 1.0 Passamonti et al. 2013

45 GW signal Characteristic strain h c = h ν 2 dt dν l=m=3 f-mode l = m = 3 f-mode α sat = 10-4 N = 0.62 d = 20 Mpc Adv. LIGO Integration time f mode t obs 1yr h c l=m=2 r-mode d=20 Mpc 1.0 ET Source distance: Virgo cluster d = 20 Mpc ν [ khz ] Passamonti et al. 2013

46 Dynamical instabilities Conclusions Bar-mode instability is a promising source of GW. Is it possible to reach the critical rotation rate in protoneutron stars? Low T/W instability is a potential strong GW source and it is worth investigating more from 3D numerical simulations. Secular instabilities (CFS) r-mode has a large instability window, but it still not complete. r-mode can be important in several astrophysical scenarios. f-mode requires high rotation rates and very compact objects (ET source) What is the saturation amplitude of the f and r modes?