Energy Dissipation in Hybrid Stars. Sophia Han. Washington University

Transkript

1 Energy Dissipation in Hybrid Stars Sophia Han Washington University Texas Symposium December 8-13, 2013 Mark Alford, Sophia Han, Kai Schwenzer 1

2 Context Nuclear Matter Crust Quark Matter Core -Sharp interface: -In reality: first-order transformation density oscillations form r-mode etc. -Periodic compression & decompression 2

3 Motivation -What are neutron stars made of? probes: different forms of matter have different dissipation behavior -Why is dissipation important? r-modes are subject to gravitationally driven instability without dissipation -Paradox: fast-rotating neutrons stars do exist! efficient damping mechanisms required to saturate r-modes at a fairly low amplitude -Novel scenario: compact star models with multi-components 3

4 Outline -One dimensional toy-model calculation of the dissipated energy in three different categories -Extension to r-mode oscillations multiple dissipation cases in the star; damping behavior and saturation amplitude in this mechanism -Discussion astrophysical observables and theoretical predictions -Future work 4

5 Toy Model: Piston Picture Newtonian gravitational field dp dr = gε -1-D simple model Assumptions -Incompressible matter -Harmonic pressure oscillation in NM -Work done by a piston in NM away from the interface (in reality: r-mode) 5

6 Toy Model: Piston Picture -Pressure at the piston p p (t) = p p gε N δ x p (t) + Δp N sin(ωt) -Baryon number conservation δ x p (t) = 1 n Q n N δ p N (t) harmonic δ x w(t) linear -Energy dissipation in one cycle out of phase! W = S τ 0 p p (t) dδ x p(t) dt dt the wall piston 6

7 Real wall and ideal wall ideal wall determined by pressure oscillation δ x iw (t) = Δp N gε N sin(ωt) harmonic! v iw max δ x iw (t) t real wall max maximal speed not always as fast as the ideal wall; lags behind sometimes real wall ideal wall 7

8 Three Dissipation Cases real wall -real wall moves with a constant speed whenever it lags behind -no acceleration/deceleration processes near turning points Assumptions stays on top (A) no dissipation partially overlapped (B) medium dissipation totally lags behind (C) large dissipation 8

9 Case A: No Dissipation -If the real wall always coincides with its ideal position δ x w (t) = δ x iw (t) > v iw max -Net contribution to dissipation is zero τ W = S p p (t) dδ x (t) p dt = 0 0 dt -Pressure oscillation and piston position are completely in phase 9

10 Case B: Medium Dissipation -Near the equilibrium position, real wall starts to lag behind (straight-line part of the solid curve) until it catches up with ideal wall again -In the limit W 2S n Q 1 n N v iw max, τ π Δp N dissipated energy η ( ) v iw dis < v iw max slope: where η = / v iw max. -Dissipated energy rises so fast that damping would be strong enough to saturate oscillation modes -There is a lower limit on below which the overlapped region (sinusoidal part of the solid curve) disappears 10

11 Case C: Large Dissipation -There is no overlap between real wall and ideal wall in one cycle 0 < v iw dis -In the limit 0, t 1 π / 2 and t 2 π, wall velocity is completely in phase with pressure oscillation, and dissipated energy W = 2S n Q 1 n N τ π Δp N -Dissipation vanishes if system barely responses to external driving force 11

12 R-mode Oscillations -Ideal wall oscillation amplitude function of r-mode amplitude the polar angle δ R iw (t) = C w α sin 2 θ cosθ sin(ωt) g w where frequency of the wall C w 1 4 and g w C w 105 2π θ Δx 3 8 Rw 189 Ω2 R, α Δx and is determined by rotation Ω Rw is a and equilibrium position is gravitational acceleration at equilibrium position of the wall. 12

13 Multiple Dissipation Cases -Given a fixed real wall speed, depending on the maximum of ideal wall speed v iw max we have different dissipation cases in the star. v iw max = C w g w αω sin 2 θ cosθ (A) no dissipation αω sin 2 θ cosθ < g w C w (B) medium dissipation g w C w αω sin 2 θ cosθ < g w 0.537C w (C) large dissipation αω sin 2 θ cosθ g w 0.537C w 13

14 Damping and Saturation -For small the star -If α is above the critical value v α crit = w g w C w ω sin 2 θ cosθ max then at [ θ 1,θ ] and [ 2 π θ 2,π θ ] 1 we have case B and other regions in the star case A -We found that r-mode is damped immediately after α rises above α crit, and the saturation amplitude α sat = (1+ ε sat )α crit,ε sat 10 3 ~ 10 4 α sat α crit α only case A exists in sensitive to wall speed and frequency! lower limit: ~10^(-4) 14

15 Concluding Remarks -A novel damping mechanism for hybrid stars generic for any compact star model with multiple components; different forms of quark matter etc. -Strong dissipation once above critical value Future Work a simple analytical prediction for the saturation amplitude; observation: gravitational waves; spin-down rate of pulsars -Interpretation of observational data frequency/temperature-dependence of the saturation amplitude; evolution of neutron stars 15

16 THANK YOU! Q & A 16

17 Misc.! crit =!! crit! f $ # & " 1kHz % '3! T # " 10 8 K $ & % '1 lower limit: 10^(-4) all! (7m/s) a 0 2! µ Q # 1! a " T 0 $ & % For T=10^8 K, the speed varies from 100km/s to 25m/s Olinto, Phys.Lett. B192 (1987) 71 R! 9km, Rw! 6.5km, M star! 1.9M solar ; n trans = 4n 0,"! /! trans = 0.4,c Q 2 = 1. parametrization of HS EoS arxiv:

18 Case B: Medium Dissipation -Dissipated energy W = 2S n Q 1 n N τ π Δp N η + cosβ 2 ; β where η = / v max iw, and is the solution of v iw dis < v iw max 1 η 2 = sinβ η[arccos( η) β] v max iw (η 1), (1 η) -In the limit expand in powers of W 2S n Q v 1 w τ n N π Δp N if we, then η ( ) -In this regime where dissipation begins to exist, dissipated energy rises so fast that damping would max be strong enough when is just slightly below v iw. 18

19 Case C: Large Dissipation -There is no overlap between real wall and ideal wall in one cycle Dissipated energy W = 2S n Q 1 n N where viw = 2 π v max iw > v dis iw. τ π Δp N 1 viw 2 -When 0, t 1 π / 2 and t 2 π, which means wall velocity is completely in phase with pressure oscillation W = 2S n Q v 1 w τ n N π Δp N -Dissipation vanishes if system barely responses to external driving force 0 < v iw dis 19

20 Case C: Large Dissipation -Small speed limit: v iw dis -Dissipated energy W = 2S n Q v 1 w τ n N π Δp N -Dissipation vanishes if system barely responses to external driving force 20