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Neutron Star Oscillations: Dynamics & Gravitational Waves

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Neutron Star Oscillations: Dynamics & Gravitational Waves Neutron star oscillations: dynamics & gravitational waves NewCompStar School 2016 Coimbra, Portugal Kostas Glampedakis Contents GWs from neutron stars Oscillations: basic formalism Taxonomy of oscillations modes Classical ellipsoids & unstable modes Neutron star The CFS instability: asteroseismology f-mode, r-mode & w-mode Theory assignments A note on Notation Abbreviations: NS = neutron star LMXB = low mass x-ray binary BH = black hole SGR = soft gamma repeater MSP = millisecond radio pulsar GW = gravitational waves EOS = equation of state EM = electromagnetic waves sGRB = short gamma-ray burst Basic parameters: M M = stellar (gravitating) mass M = 1.4 1.4M stellar radius R R = R = 6 106 cm ⇢ = density ⌦=rotational angular frequency T = stellar core temperature 1 f = = rotational frequency & period ! = mode’s angular frequency spin P A cosmic laboratory of matter & gravity • Supranuclear equation of state (hyperons, quarks) • Relativistic gravity • Rotation (oblateness, various instabilities) • Magnetic fields (configuration, stability) • Elastic crust (fractures) • Superfluids/superconductors (multi-fluids, vortices, fluxtubes) • Viscosity (mode damping) • Temperature profiles (exotic cooling mechanisms) [ figure: D. Page] Neutron stars as GW sources (I) “Burst” emission Binary neutron star mergers Magnetar flares Pulsar glitches (our safest bet for detection) (likely too weak) (likely too weak) Neutron stars as GW sources (II) Continuous emission Non-axisymmetric mass Fluid part (oscillations) quadrupole (“mountains”) Taxonomy of NS oscillation modes (I) • Pressure ( p ) modes: driven by pressure. • Fundamental ( f ) mode: (aka “Kelvin mode”) the first (nodeless) p-mode. • Gravity ( g ) modes: driven by buoyancy (thermal/composition gradients). • Inertial ( i ) modes: driven by rotation (Coriolis force). • Magnetic (Alfven) modes: driven by the magnetic force. • Spacetime (w) modes: akin to BH QNMs, need dynamical spacetime (non-existent in Newtonian gravity) Taxonomy of NS oscillation modes (II) More physics in stellar model richer mode spectrum ) • Shear (s,t) modes: driven by elastic forces in the crust. • Superfluidity: the system becomes a multi-fluid (i.e. relative motion of one fluid with respect to the others). Modes are “doubled”, due to the “co- moving” and “counter-moving” degrees of freedom. • Tkachenko modes: driven by tension of superfluid vortex array (never computed for NS, except from local plane waves). Taxonomy of NS oscillation modes (III) Typical spectra of non-rotating NSs without stratification w-modes A typical spectrum “doubling” in p-modes a non-rotating superfluid NS f-modes [ Andersson et al. 2002] [ Kokkotas et al. 2001] NS modes: geometry • The velocity perturbation associated with a mode can be decomposed in a standard way in radial and angular parts: δv(x,t)= [ W ˆr + V Y m + U (ˆr Y m)]ei!t ` `r ` ` ⇥ r ` X`,m polar part = parity ( 1)` axial part = parity ( 1)`+1 − − radial eigenfunctions: W`(r|{z}),V`(r),U`(r) |{z} • In spherical stars (i.e. up to ( ⌦ ) ), axial and polar sectors remain decoupled. O • Purely polar: f-mode, p-modes, g-modes, … • Purely axial: r-modes, t-crust modes, … δv =0 & flow “horizontal” ) r · • Coupling: with rotation ( ( ⌦ 2 ) and higher), B-field, … O • Similar decomposition in GR stars NS modes observed: magnetar flares • Quasi-periodic oscillations in the x-ray light curve of giant flares in SGRs. • These are believed to be global magnetic/magneto-elastic modes. SGR 1900+14 SGR 1806-20 [ Strohmayer & Watts 2005, 2006 ] [ Gabler et al. 2016] NS modes observed: bursting LMXBs • NSs in LMXBs frequently undergo x-ray burst whose light curves are oscillatory. Two main models: • Surface modes (r-modes, g-modes …) in fluid ocean, excited by infalling matter and burning. • Surface “hot spot” emission modulated by rotation. [ Watts et al. 2009] fspin fspin [ Chakrabarty et al. 2003] [ Piro & Bildsten 2005] NS modes: basic formalism (I) • Linearised equations, written in the stellar rotating frame. • Mass continuity equation: @t⇢ + (⇢v)=0 r · |{z} • Poisson equation: 2 δ Φ =4 ⇡ G δ⇢ r Euler (or Navier-Stokes) equation: GW radiation • reaction force δp 1 1 @tδv +2⌦ δv + + δΦe↵ = ( Fsv + Fbv + FGR ) + Fmag + ... ⇥ r ⇢ ⇢ ⇢ { } ✓ ◆ shear & bulk viscous forces magnetic force • A barotropic EOS p= p(ρ) was assumed (realistic NSs are not barotropes). |{z} |{z} • Superfluid NSs require a multi-fluid formalism, instead of a single-fluid one. • In the presence of a magnetic field, the Maxwell equations have to be added. NS modes: basic formalism (II) • The mode’s total energy E mode is conserved in the absence of dissipation. • In the presence of dissipation E mode is not conserved and the mode’s frequency ω becomes complex-valued. 2E where E˙ = mode 1 mode − ⌧ Im(!)= • The Navier-Stokes equation leads to: ⌧ ˙ ˙ ˙ ˙ { Emode = Esv + Ebv + EGRR shear viscosity damping rate: bulk viscosity damping rate: ij E˙ sv = 2 dV ⌘ δσ δσ¯ij − E˙ = dV ⇣ δσ 2 Z bv − | | 1 2 Z δσij = iδvj + jδvi gij δvk j 2 r r − 3 rk δσ = jδv ✓ ◆ r ˙ 1 E˙ 1 Ebv = sv = ⌧sv Emode ⌧bv Emode Calculating mode damping: basic strategy Solve directly the daunting task due to complexity Navier-Stokes equation of dissipative forces mind causality of viscosity in GR! realistic scenario: weak dissipation Re(!) Im(!) | | use inviscid mode eigenfunctions Solve the non-dissipative & frequencies in E ˙ , E ˙ , E ˙ Euler equation sv bv GRR to obtain approximate viscous and GW timescales. NS modes: GR formalism • The formalism is considerably more complicated in GR: δGµ⌫ =8⇡T µ⌫ δ( T µ⌫ )=0 r⌫ Cowling approximation: δgµ⌫ =0 switches off GWs but also metric = gµ⌫ + δgµ⌫ ) “contaminates” mode • Example: the symbolic form of polar perturbation equations in a spherical background star + constraint: f-mode: back of the envelope • The “minimal” (Newtonian) stellar model supporting f-modes: uniform density incompressible flow Cowling approximation ⇢ =0 δv =0 δΦ=0 r · 1 δv = @t⇠ = χ “potential flow” Euler: @ δv + δp =0 r t ⇢r ) 2δp =0 & 2χ =0 r r m i!t { ` χ = χ`(r)Y` e χ` = ↵`r β` Euler: i!↵` + =0 m i!t ) ` ⇢ δp = δp`(r)Y` e δp` = β`r background { { pressure gradient r Surface boundary condition (r=R): ∆p = δp + ⇠ @rp =0 2 4⇡G⇢ ` 2 4⇡` β` = ↵` ! = G⇢ ! G⇢ ) 3i! ) 3 ⇠ p A simple GR f-mode calculation • Ultracompact, uniform fluid ball (i.e. the Schwarzschild solution). • The system can only support w-modes and the fluid f-mode (only the latter in Newtonian gravity). /M • The figure provides a beautiful 3 w R example of mode avoided crossings. p ) ! Re( • At each crossing the two modes w “transmute” by exchanging properties. f • In this particular example, the avoided f crossings “produce” the f-mode. w w [ Andersson et al. 1996] GW asteroseismology • Key idea of asteroseismology: parametrise mode frequencies & decay rates (due to GW emission) in terms of the bulk stellar parameter: M,R, ⌦ { } Once an oscillation is observed, use the parametrisation to infer the stellar parameters. • A clever parametrisation can lead to “universal” (i.e. quasi EOS-independent) relations. • As an example, we consider f-mode asteroseismology. f-mode asteroseismology: no rotation • Fitting formulae for mode frequency and GW decay time. 1/2 4 1 M1.4 R6 M1.4 − !f (kHz) 0.78 + 1.635 3 ⌧f (s) 3 22.85 14.65 ⇡ R ⇡ M − R6 ✓ 6 ◆ 1.4 [ Andersson & Kokkotas 1998] f-mode asteroseismology: with rotation • For rotating NSs we need to consider the prograde (stable) and retrograde (potentially unstable) f-modes. ⌦K = Kepler frequency !u ⌦ ⌦ 2 r =1+0.402 0.406 !0 = non-rotating f-mode ! ⌦ − ⌦ 0 K ✓ K ◆ frequency s 2 !r ⌦ ⌦ =1 0.235 0.358 !i = !r m⌦ ! − ⌦ − ⌦ 0 K ✓ K ◆ − inertial frame rotating frame [ Doneva & Kokkotas 2013] f-mode asteroseismology: with rotation • Polynomial fitting formulae exist for the f-mode’s GW damping timescales. [ Doneva & Kokkotas 2013] Unstable modes & Ellipsoids (I) • A 300 years-old question: what is the equilibrium shape of a rotating self-gravitating fluid body? • We consider homogenous & incompressible bodies. • Maclaurin (1742): body is oblate and biaxial, the angular frequency Ω and ellipticity e are related as: 2 1/2 2 2 2 (1 e ) 2 1 3(1 e ) a3 ⌦ =2⇡G⇢ − 3 (3 2e )sin− e −2 e = 1 e − − e s − a1 ✓ ◆ • Jacobi (1834): equilibrium shape can be triaxial ellipsoidal. x2 y2 z2 2 + 2 + 2 =1 a1 a2 a3 • The fluid velocity for both configurations is a linear function of coordinates: v =⌦( y ˆx + x ˆy ) − Unstable modes & Ellipsoids (II) • The Maclaurin sequence bifurcates at: e 0.813 β 0.14 ⇡ ⇡ where T kinetic energy β = = W grav. potential energy Jacobi sequence ends in a “cigar”: a a e 1 3 0, 3 1 ! ) a1 ! a2 ! Maclaurin sequence ends in a “disk”: a e 1 3 0,J ! ) a1 ! !1 Unstable modes & Ellipsoids (III) • Dirichlet-Dedekind (1861): a new class of triaxial ellipsoids, with zero rigid body rotation and non-zero uniform vorticity ⇣ 2 2 ⌦=0 ⇣ = v =0 velocity: v = 2 2 ( a2 y ˆx + a1 x ˆy ) r ⇥ 6 a1 + a2 − • The previous solutions are special cases of the general Riemann family of ellipsoids. ⇣ ⌦ ⇣ along a principal axis of the ellipsoid, = const. • S-type ellipsoids: k ⌦ i ij ˆz { linear fluid flow v = A xj 0 Jacobi ⌦ =(0, 0, ⌦) ⇣ ˆy ˆz = k ⌦ ⇣ =(0, 0,⇣) Dedekind ˆx { {1 Unstable modes & Ellipsoids (IV) [ Andersson 2003] GW-driven instability Viscosity-driven instability JDed <JMac =0 ,⇣ 0 > EJac <EMac Ded = Mac ⌦ C C JJac = JMac angular momentum: J energy: E circulation: = v dl C · I Secular instability: with dissipation, J & D sequences branch out at β 0.14 s ⇡ Dynamical instability: the Maclaurin sequence ends at β 0.27 d ⇡ Unstable modes & Ellipsoids (V) • Ellipsoidal changes are achieved via unstable ` = m =2 (“bar”) f-modes. | | At : • βs prograde retrograde f-mode becomes prograde (dragged by stellar rotation) K • At β d : ⌦ the two f-modes merge and / i become complex-valued.
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