THE EVOLUTION OF THE F-MODE INSTABILITY

Transcription

1 THE EVOLUTION OF THE F-MODE INSTABILITY Andrea Passamonti University of Tübingen In collaboration with E. Gaertig and K. Kokkotas 9 July 202, SFB Video Seminar

2 Motivation GW driven f-mode instability of relativistic star Time evolution of the instability GW signal of the f-mode and its detection prospectives

3 CFS instability Rotating NS are prone to CFS gravitational-wave instability Radiation drives a mode unstable if ω r (ω r mω) 0 = τ gw 0 and δρ e t/τ gw Viscous mechanisms limit the gravitational-wave instability δρ e iωt t/τ where τ = τ gw + τ b + τ s +... and τ = Ė 2E Instability condition: τ 0 where τ = τ (Ω,T)

4 CFS instability Rotating NS are prone to CFS gravitational-wave instability Radiation drives a mode unstable if ω r (ω r mω) 0 = τ gw 0!"! K M b =.6 M. l=m=4 f-mode and Viscous mechanisms limit the gravitational-wave instability δρ e iωt t/τ δρ e t/τ gw where T [K] τ = τ gw + τ b + τ s +... Mutual friction and T cn τ = Ė 2E Instability condition: τ 0 where τ = τ (Ω,T)

5 Unstable modes F-mode (2 m 4) Newtonian N= Star CFS unstable in rapidly rotating stars e+2 τ 44 gw s R-mode (m=2) CFS unstable for any Ω τ 22 gw 0s t gw (s) e l=m=4 f-mode l=m=2 r-mode Ω/Ω K General Relativity may strengthen the f-mode instability considerably

6 Equations

7 Method We calculate the mode-frequency and eigenfunctions from time simulations. With the energy volume integrals we determine the damping/growth times. We study the instability evolution with a set of evolution equations

8 Mode Frequency Evolution of the relativistic perturbation equations in Cowling approximation δ ( ν T µν )=0 δg µν =0 where Standard model N = M =.4M Ω K =4.229 khz Supramassive model N =2/3 M =.6M Ω K =.206 khz [ khz ] [ khz ] f-mode l = m = 4 N = f-mode l = m = 4 l = m = 3 l = m = !"! K

9 Viscous and GW timescales Assumption: dissipative timescales are much longer than the oscillation period Bulk and shear viscosity = dv ζδσδσ τ b 2E = dv ηδσ ij δσij τ s E ζ T 6 η T 2 δσ ij = 2 δσ = j δu j i δu j + j δu i 2 3 gij δσ GW radiation reaction = ω τ gw 2E N l (ω mω) 2l+ δd lm 2 + δj lm 2 l 2 ω (ω mω) 0 = τ gw 0

10 Viscous and GW timescales Assumption: dissipative timescales are much longer than the oscillation period Bulk and shear viscosity = dv ζδσδσ τ b 2E = dv ηδσ ij δσij τ s E GW radiation reaction = ω τ gw 2E δσ ij = i δu j + j δu i gij δσ N = δσ = j δu j N l (ω mω) 2l+ δd lm 2 + δj lm 2 l 2 ω (ω mω) 0 ζ T 6 η T 2 = τ gw 0 Log gw f-mode l = m = 3 l = m = !"! K

11 Viscous and GW timescales Assumption: dissipative timescales are much longer than the oscillation period Bulk and shear viscosity = dv ζδσδσ τ b 2E = dv ηδσ ij δσij τ s E GW radiation reaction = ω τ gw 2E δσ ij = i δu j + j δu i gij δσ N = δσ = j δu j N l (ω mω) 2l+ δd lm 2 + δj lm 2 l 2 ω (ω mω) 0 ζ T 6 η T 2 = τ gw 0!"! K Log gw f-mode l = m = 3 N = l = m = 4 l = m = 3 l = m = 4 l = m = T f-mode cn !"! T [K] K

12 Instability Evolution Basic equations de dt = 2E τ dj dt = dj gw dt + dj mag dt C v dt dt = L ν + H s E = αẽ(ω) J = J s + α J c (Ω) H s = 2E τ s Amplitude Normalization E = αe rot α == E 0 2 M c 2 Note: δρ α /2 Mode growth Non-linear saturation dα dt = 2α τ gw 2α dω dt = 2F D +αq τ v D + 2P D, α τ v τ mag α τ mag, dα dt =0 dω dt = 2F +αq α + τ gw τ mag

13 Results

14 l=m=4 f-mode Evolution N = polytrope Instability trajectory 0.99 Ω / Ω K Ω =.0 Ω K Ω = 0.99 Ω K!"! K 0.97 M b =.5 M. N = B = 0 G sat = 0-4 α 0-5 Ω = Ω N = K B = 0Ω = G0.97 Ω K N = B = 0 G t [yr] 6 - Amplitude: α 0 = 0 and - At Ω = Ω K the growth time is 0.95 τ gw 0 4 s T cn T [K] α sat = 0 4 = E 0 6 M c 2

15 l=m=4 f-mode Evolution polytrope Trajectory Ω / Ω K B = 0 G Ω / Ω K M b =.6 M. α sat = 0-4 α 0-5 Ω/Ω K = 0.9 B = 0 G With H s Ω/Ω K = t [yr] T [K] T cn - At Ω = Ω K the growth time is τ gw 0 2 s

16 GW signal Characteristic strain h c = h ν 2 dt dν t obs yr l=m=4 f-mode l=m=3 f-mode l = m = 4 f-mode α sat = 0-4 N =.0 Adv. LIGO ET l = m = 3 f-mode B = 0 G sat = 0-4 Adv. LIGO ET h c h c B = 0 G d = 20 Mpc d = 20 Mpc ν [ khz ] [ khz ]

17 Magnetic Torque N = polytrope Magnetic field accelerates the transition through the instability window α Ω / Ω K Dipole formula dj mag dt B 2 pr 6 Ω t [yr] N = B p = 0 2 G T [K] Mag. Torque becomes dominant for B p 0 2 G α t [yr] Ω / Ω K N = B p = 0 3 G T [K]

18 Magnetic Torque Magnetic field accelerates the transition through the instability window Dipole formula dj mag dt B 2 pr 6 Ω 3 Mag. Torque becomes dominant for B p 0 2 G α α t [yr] t [yr] polytrope Ω / Ω K Ω / Ω K T [K] T [K] B p = 0 2 G B p = 0 3 G

19 GW signal Characteristic strain 0-20 Adv. LIGO t obs yr For the more massive model with B p = 0 2 G the GW signal may be still detectable by ET. h c B = 0 2 G l = m = l = m = ET Source is 20 Mpc (Virgo Cluster) α sat = 0-4 d = 20 Mpc ν [ khz ]

20 F-mode versus R-mode model Non-linear saturation of the f-mode α sat f α sat r = 0 4 = E 0 6 M c 2 Non-linear mode coupling saturates r-mode = !"! K = 0-9 = 0-8 = 0-7 = 0-6 = 0-5 = 0-4 l=m=4 f-mode l=m=2 r-mode B p = 0 6 G t [yr]

21 F-mode versus R-mode α sat f α sat r model Non-linear saturation of the f-mode = 0 4 = E 0 6 M c 2 Non-linear mode coupling saturates r-mode = !"! K!"! K sat = = 0-9 = = 0-8 r = 0-8 l=m=4 f-mode = 0-7 r = = 0-6 l=m=2 r-mode = 0 r = 0-6 = 0-5 = 0-4 B p = 0 6 r = G 0-6 = 0-4 T cn t [yr] N = B p = 0 G T [K]

22 F-mode versus R-mode α sat f α sat r model Non-linear saturation of the f-mode = 0 4 = E 0 6 M c 2 Non-linear mode coupling saturates r-mode = !"! K h c 0-20!"! K sat = ET = 0-9 = = 0-8 r = l=m=4 f-mode = 0-7 r = = 0-6 l=m=2 r = r-mode = 0 r = 0-6 = 0-5 r = = 0-4 B p = 0 6 r = 0-5 = 0-7 G 0-6 = 0-4 T r = 0-6 cn t [yr] ! K = 0.94 d = 20 Mpc N = B p = 0 G T [ khz [K] ] Adv. LIGO

23 Conclusions The GW signal of very compact objects may be detectable from Virgo Cluster by ET. The magnetic torque affects the spin down when B p 0 2 G The r-mode may limit the f-mode instability, but we need to know the relative saturation amplitude more accurately. More ingredients in future work. The l=m=2 f-mode may become important if we abandon the Cowling approximation. Study realistic EoS and consider durca reactions. Include the Crust and the effects of Ekman layers.

24 This is the End