Saturation of the f -mode instability in neutron stars Pantelis Pnigouras in collab. with K. D. Kokkotas 28th Texas Symposium on Relativistic Astrophysics Geneva, 17.12.2015
Outline 1 Oscillation modes 2 The CFS instability The instability window 3 Instability saturation Non-linear mode coupling Parametric resonance Saturation conditions 4 Results Supernova-derived neutron stars Merger-derived neutron stars Stochastic background
Oscillation modes ∞ l ∑ ∑ [ U m l ( r ) Y m e r + V m ( r ) ∇ Y m ( θ, φ ) + W m e r × ∇ Y m ξ ( r, θ, φ ) = ( θ, φ )ˆ l ( r )ˆ ( θ, φ )] l l l l l =0 m = − l W m • Polar modes: = 0 l as Ω → 0 U m = V m • Axial modes: = 0 l l l : degree m : order n : overtone Mode name Mode class Mode type Restoring force p -mode Polar Sound wave ( ω → ∞ as n → ∞ ) Pressure gradient } Low- ω sound wave f -mode Polar n = 0 High- ω gravity wave Buoyancy g -mode Polar Gravity wave ( ω → 0 as n → ∞ ) r -mode Axial Inertial wave Coriolis Hybrid mode Combination Zero-buoyancy limit or r - and g -modes • Only for non-zero rotation • Only for non-zero buoyancy 1 / 11
Oscillation modes ∞ l ∑ ∑ [ U m l ( r ) Y m e r + V m ( r ) ∇ Y m ( θ, φ ) + W m e r × ∇ Y m ξ ( r, θ, φ ) = ( θ, φ )ˆ l ( r )ˆ ( θ, φ )] l l l l l =0 m = − l 10 n W m • Polar modes: = 0 l as Ω → 0 U m = V m • Axial modes: = 0 1 l l ˜ ω l : degree n m : order 0.1 n : overtone p f g 2 3 4 5 6 7 8 9 10 11 l Mode name Mode class Mode type Restoring force p -mode Polar Sound wave ( ω → ∞ as n → ∞ ) Pressure gradient Low- ω sound wave } f -mode Polar n = 0 High- ω gravity wave Buoyancy g -mode Polar Gravity wave ( ω → 0 as n → ∞ ) r -mode Axial Inertial wave Coriolis Hybrid mode Combination Zero-buoyancy limit or r - and g -modes • Only for non-zero rotation • Only for non-zero buoyancy 1 / 11
The CFS instability Chandrasekhar (1970) first realised its existence for Maclaurin spheroids Friedman and Schutz (1978) proved that the instability is generic For any angular velocity Ω there is always a mode driven unstable by GW emission Multipole expansion of power radiated in GWs (Thorne, 1980): ( d E ∞ ) l | 2 + | δJ m N l ω ( ω − m Ω) 2 l +1 ( l | 2 ) ∑ | δD m = − d t GW l ≥ 2 ◦ If ω ( ω − m Ω) < 0, then (d E/ d t ) GW > 0 ◦ Polar (axial) modes emit through the mass (current) multipoles ◦ Most susceptible to the CFS instability are the f -modes and r -modes Octupole ( l = m = 3) f -mode [left, credit: Wolfgang ohler, GFZ Potsdam ] and K¨ r -mode [right, credit: Saio (1982) ]; density and velocity perturbations dominate, respectively. 2 / 11
The CFS instability – The instability window The instability is suppressed by viscosity (Ipser and Lindblom, 1991) ( d E ( d E ) ) Instability window : + > 0 d t d t GW V 1.00 Instability windows of the quadrupole, octupole, and Shear 0.98 hexadecapole f -modes, for a viscosity Newtonian star with p ∝ ρ 3 . ∼ T - 2 Ω / Ω K 0.96 Shear viscosity, due to particle l = m = 2 scattering, and bulk viscosity, Bulk viscosity 0.94 due to disturbance of l = m = 3 ∼ T 6 β -equilibrium by the l = m = 4 perturbation, dominate at low 0.92 and high temperatures, respectively. 10 8 10 9 10 10 T ( K ) ◦ The r -mode instability is favoured, because of i) much larger window, and ii) shorter growth time scales ◦ Significance : → Neutron star evolution [nascent (Bondarescu et al., 2009; Passamonti et al., 2013), LMXBs (Levin, 1999; Bondarescu et al., 2007)] → Gravitational wave asteroseismology (Andersson and Kokkotas, 1996, 1998) 3 / 11
Instability saturation – Mode coupling Non-linear mode coupling stops instability’s growth (Dziembowski, 1982) Linear amplitude evolution: ˙ Q α = γ α Q α 4 / 11
Instability saturation – Mode coupling Non-linear mode coupling stops instability’s growth (Dziembowski, 1982) Quadratic amplitude evolution: ˙ Q α = γ α Q α + iω α H Q β Q γ e − i ∆ ωt Modes couple ˙ Q β = γ β Q β + iω β H Q ∗ γ Q α e i ∆ ωt in triplets Q γ = γ γ Q γ + iω γ H Q α Q ∗ ˙ β e i ∆ ωt 4 / 11
Instability saturation – Mode coupling Non-linear mode coupling stops instability’s growth (Dziembowski, 1982) Quadratic amplitude evolution: ˙ Q α = γ α Q α + iω α H Q β Q γ e − i ∆ ωt Modes couple ˙ Q β = γ β Q β + iω β H Q ∗ γ Q α e i ∆ ωt in triplets Q γ = γ γ Q γ + iω γ H Q α Q ∗ ˙ β e i ∆ ωt ◦ Detuning ∆ ω ≡ ω α − ω β − ω γ ≈ 0 resonance condition 4 / 11
Instability saturation – Mode coupling Non-linear mode coupling stops instability’s growth (Dziembowski, 1982) Quadratic amplitude evolution: ˙ Q α = γ α Q α + iω α H Q β Q γ e − i ∆ ωt Modes couple ˙ Q β = γ β Q β + iω β H Q ∗ γ Q α e i ∆ ωt in triplets Q γ = γ γ Q γ + iω γ H Q α Q ∗ ˙ β e i ∆ ωt ◦ Detuning ∆ ω ≡ ω α − ω β − ω γ ≈ 0 resonance condition ◦ Coupling coefficient H ̸ = 0 if m α = m β + m γ l α + l β + l γ = even number coupling selection rules | l β − l γ | ≤ l α ≤ l β + l γ 4 / 11
Instability saturation – Mode coupling Non-linear mode coupling stops instability’s growth (Dziembowski, 1982) Quadratic amplitude evolution: ˙ Q α = γ α Q α + iω α H Q β Q γ e − i ∆ ωt Modes couple ˙ Q β = γ β Q β + iω β H Q ∗ γ Q α e i ∆ ωt in triplets Q γ = γ γ Q γ + iω γ H Q α Q ∗ ˙ β e i ∆ ωt ◦ Detuning ∆ ω ≡ ω α − ω β − ω γ ≈ 0 resonance condition ◦ Coupling coefficient H ̸ = 0 if m α = m β + m γ l α + l β + l γ = even number coupling selection rules | l β − l γ | ≤ l α ≤ l β + l γ 1 d E i ◦ Growth/damping rates γ i = ≷ 0 2 E i d t ( d E ( d E d E ) ) d t = + ≷ 0 d t d t GW V 4 / 11
Instability saturation – Parametric resonance Q α = γ α Q α + iω α H Q β Q γ e − i ∆ ωt ˙ Detuning ∆ ω Q β = γ β Q β + iω β H Q ∗ ˙ γ Q α e i ∆ ωt Coupling coefficient H Q γ = γ γ Q γ + iω γ H Q α Q ∗ ˙ β e i ∆ ωt Growth/damping rates γ i (Schenk et al., 2001; Parent mode: unstable r -mode ( γ α > 0) Morsink, 2002; Arras et al., 2003; Brink et al., Daughter modes: other (stable) axial modes ( γ β,γ < 0) 2004, 2005) 5 / 11
Instability saturation – Parametric resonance Q α = γ α Q α + iω α H Q β Q γ e − i ∆ ωt ˙ Detuning ∆ ω Q β = γ β Q β + iω β H Q ∗ ˙ γ Q α e i ∆ ωt Coupling coefficient H Q γ = γ γ Q γ + iω γ H Q α Q ∗ ˙ β e i ∆ ωt Growth/damping rates γ i Parent mode: unstable f -mode ( γ α > 0) (PP and Kokkotas, 2015) Daughter modes: other (stable) polar modes ( γ β,γ < 0) 5 / 11
Instability saturation – Parametric resonance Q α = γ α Q α + iω α H Q β Q γ e − i ∆ ωt ˙ Detuning ∆ ω Q β = γ β Q β + iω β H Q ∗ ˙ γ Q α e i ∆ ωt Coupling coefficient H Q γ = γ γ Q γ + iω γ H Q α Q ∗ ˙ β e i ∆ ωt Growth/damping rates γ i Parent mode: unstable f -mode ( γ α > 0) (PP and Kokkotas, 2015) Daughter modes: other (stable) polar modes ( γ β,γ < 0) No mode coupling: H = 0 or ∆ ω ≫ 0 α ◦ Modes evolve independently ◦ No non-linear interaction | Q | ˙ Q α = γ α Q α γ ˙ Q β = γ β Q β β ˙ Q γ = γ γ Q γ t 5 / 11
β α γ Instability saturation – Parametric resonance Q α = γ α Q α + iω α H Q β Q γ e − i ∆ ωt ˙ Detuning ∆ ω Q β = γ β Q β + iω β H Q ∗ ˙ γ Q α e i ∆ ωt Coupling coefficient H Q γ = γ γ Q γ + iω γ H Q α Q ∗ ˙ β e i ∆ ωt Growth/damping rates γ i Parent mode: unstable f -mode ( γ α > 0) (PP and Kokkotas, 2015) Daughter modes: other (stable) polar modes ( γ β,γ < 0) Parametric resonance: H ̸ = 0 and ∆ ω ≈ 0 | Q PT | ◦ Parent feeds daughters and makes them grow ◦ Parametric threshold: daughters grow when ) 2 ] [ ( γ β γ γ ∆ ω | Q | | Q α | 2 > | Q PT | 2 ≡ 1 + ω β ω γ H 2 γ β + γ γ ◦ | Q sat | ≈ | Q PT | α t PT t 5 / 11
γ α β Instability saturation – Saturation conditions ˙ Q α = γ α Q α + iω α H Q β Q γ e − i ∆ ωt γ α > 0 , γ β,γ < 0 Detuning ∆ ω Q β = γ β Q β + iω β H Q ∗ ˙ γ Q α e i ∆ ωt [ ) 2 ] Coupling coefficient H γ β γ γ ( ∆ ω | Q PT | 2 ≡ 1 + ˙ Q γ = γ γ Q γ + iω γ H Q α Q ∗ β e i ∆ ωt ω β ω γ H 2 γ β + γ γ Growth/damping rates γ i Saturation successful if: | γ β + γ γ | ≳ γ α and ∆ ω ≳ | γ α + γ β + γ γ | Saturation successful Saturation unsuccessful | Q PT | | Q PT | | Q | | Q | t PT t PT t t 6 / 11
Results – Supernova-derived neutron stars Saturation amplitude throughout the instability window (PP and Kokkotas, in prep.) Q PT 10 - 5 10 - 6 10 - 7 10 - 8 Model: M = 1 . 4 M ⊙ , R = 10 km, p ∝ ρ 3 Units: E mode = | Q | 2 Mc 2 { T − 1 , T ≲ 10 9 K | Q sat | ∝ for Ω = const . T ≳ 10 9 K T 3 , 7 / 11
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