Self-Similar Gas Dynamics of Voids and Supernova Ejecta

Transcription

1 Self-Similar Gas Dynamics of Voids and Supernova Ejecta arxiv: v2 [astro-ph.sr] Li-Le Wang Director: Prof. Yu-Qing Lou Department of Physics and Tsinghua Center for Astrophysics (THCA) Tsinghua University October 12th, 2011

2 What s Next? 1 Background & Motivation Some Easy-Reading Materials Self-Similar Bubbles in Astrophysics Motivations 2 Technique: Model Construction 3 Application: Modeling SNe Ejecta 4 Summary

3 Some Easy-Reading Materials An Age-Old Joke... Spherical Cow A Cow A Spherical Cow Figure/COW.jpg Figure/SPHCOW.jpg

4 Some Easy-Reading Materials... and a Practical Simplification Spherical Nebula Crab by HST [5] Spherical Counterpart of Crab Figure/CRAB.jpg Figure/SNR0509.jpg

5 Self-Similar Bubbles in Astrophysics Bubbles and Self-Similarity I Astrophysical Voids SNR 0509 by HST [5] Voids: Bubble-like structures PNe bubbles Jet-induced bubbles Stellar wind bubbles SNe bubbles: the Cows herein Figure/SNR0509.jpg

6 Self-Similar Bubbles in Astrophysics Bubbles and Self-Similarity II Self-Similarity in SNe Explosions Simulation of Janka & Müller (1996) [2] Fitting of Janka & Müller Results by [3] Figure/JANKA S IM H U F IT.pdf Figure/JANKA S IM.png

7 Motivations Motivations: Modeling the SNe Bubbles Figure/CHUINIU.pdf Literal: To Puff the Cow; Extension: To Boast Self-Similar Cows (Ejecta of SNe) Later: Sedov stage [6] Earlier: Simulations indicated [2] Who Can Puff a Cow? Required by explosion: erg Neutrinos: erg; Scattering ineffective Photons and pair production (PP) products: erg

8 What s Next? 1 Background & Motivation 2 Technique: Model Construction Self-Similar ODEs and Add-Ons Solutions to Self-Similar ODEs 3 Application: Modeling SNe Ejecta 4 Summary

9 Self-Similar ODEs and Add-Ons From Euler Equation to Self-Similar ODEs I Euler Eq. and Continuity Eq. with Spherical Symmetry u t + u u r = 1 p ρ r GM r 2, ρ t + 1 (ρur 2 ) r 2 = 0. r Nonlinear Partial Differential Equation 1 Analytic solution: Formidable, almost impossible 2 Numerical simulation: Straightforward; harsh coding 3 Much simpler but with essentials: Self-similar Assuming Polytropic EoS: p = κρ γ (κ is constant)...

10 Self-Similar ODEs and Add-Ons From Euler Equation to Self-Similar ODEs II Self-Similar Transformation from Suto & Silk (1988) [7] x = r k 1/2 t, u(r, t) = n k1/2 t n 1 v(x), ρ(r, t) = α(x) 4πGt 2, p(r, t) = kt2n 4 4πG [α(x)]γ, M(r, t) = k3/2 t 3n 2 (3n 2)G m(x). Reduced Dimensionless Variables: x, v, α and m Constant κ in p = κρ γ : n + γ = 2 Under This Transformation We Have...

11 Self-Similar ODEs and Add-Ons From Euler Equation to Self-Similar ODEs III Self-Similar Hydrodynamic ODEs dα dx = α (nx v) 2 γα γ 1 [ (n 1)v + (nx v) (x v)(nx v) α 2 (3n 2) x dv dx = 1 (nx v) 2 γα γ 1 (nx v)2 [(nx v)(n 1)v + (3n 2) α 2γ x v ] x αγ 1. ], Nonlinear ODEs: Possible to Solve Numerically Specific Asymptotic Behavior Near Infinity (Later)

12 Self-Similar ODEs and Add-Ons Sonic Critical Behaviours Self-Similar ODEs Have a Singular Surface Flow speed exceeds local sound speed Eigensolutions : Going through Singular Surface Smoothly Numerators must vanish simultaneously Qualitative analysis of ODEs required (omitted here) Shock Solutions: Going through by Shock(s) Assuming spherical symmetry and self-similarity Entropy increases across shock fronts Energy, momentum and mass conservation

13 Self-Similar ODEs and Add-Ons Central Void and Contact Discontinuity Reduced Enclosed Mass: m = αx 2 (nx v) Surface on Which nx = v: Special Features Zero enclosed mass: (Almost) massless bubble inside ρ can have a jump from zero to finite dr/dt = u: No mass flow across the surface Modeling Dynamic Gas with Central Massless Voids Massless : Gravity negligible (compared with self-gravity) Mechanical and other requirements: See later

14 Self-Similar ODEs and Add-Ons Asymptotic Behaviours near Infinity Asymptotic Behavior near Infinity v = Ax (n 2)/n [ 2(2 n) na n α = Ax 2/n. ] n (n 1)B2 + Bx (n 1)/n, (3n 2) na Critical A: A e = [ 2(2 n)(3n 2)/n 2] 1/n Classification by Asymptotic Behavior of Envelope 1 B = 0, A A + e : Expansion-Wave Collapse Solution (EWCS) 2 B = 0: v > 0 Breeze, v < 0 Contraction 3 B 0: B > 0 Outflow, B < 0 Inflow

15 Solutions to Self-Similar ODEs Numerical Results: Examples I Numerical Results: Voids, Shocks and EWCS Envelope v n=0.9, EWCS =1.1 Panel A Model E1 x Model E2 s1= 0.4 Model E3 cd x cd= 1.45 x s1= 2 x s1= 2.5 Model E1 x cd= ZML ZML SCC x cd, α cd : Values at Contact Discontinuity x s1 : x at upstream of shock Model E1 cd 1810 EWCS Panel B SCC Model E2 cd 2.51 Model E3 cd 1.42 SCC 10 2 x

16 Solutions to Self-Similar ODEs Numerical Results: Examples II Numerical Results: Voids, Shocks and Various Envelopes -v 2 Model 1 cd x s1= SCC 1 ZML 2 Model 4 cd x s10 = Model 1 cd Model 3 cd x s1= 2.5 n=0.9, =1.1 Model 3 cd 2.03 Model 2 cd Model cd x Panel A Model 2 cd x s1= 3.5 Panel B SCC -v 0.5 n0=0.67, =1.33 Panel A Model 5 x cd= Model 7 x cd= SCC ZML Model cd 59.6 x s1= Model 6 0 cd 12.6 x s1= x Model 6 x cd= Model 7 cd 28.2 x = s1 9 Panel B SCC

17 Solutions to Self-Similar ODEs Numerical Results: Examples III Numerical Results: Voids, Crossing SCC Smoothly -v α Model S4 cd x v 0= Model S4 cd α 0= n=0.9, γ =1.1 x Model S5 cd 1.99 x v 0= SCC ZML SCC Model S5 cd α 0= Panel A Panel B v α Model S1 cd x v 0= Model S2 cd x Model S3 v 0= cd x v 0= SCC n=0.67, γ =1.33 Panel A Model S2 cd α 0= Model S1 cd α 0= x SCC ZML Panel B Model S3 cd α 0=

18 What s Next? 1 Background & Motivation 2 Technique: Model Construction 3 Application: Modeling SNe Ejecta Scenario and Context Near or Inside the Contact Discontinuity Self-Similar Model in SNe Scenario 4 Summary

19 Scenario and Context General Overview: Scenario and Context Wilson Model of SNe Explosion [1] 1 Core-collapse; Rebound shock ignited 2 Neutrino flow: Rebound shock revitalized 3 Bubble or void, r 100km 4 Neutrinos escape; Photons and PP products are left After the Bubble is Shaped Up Neutrino: Transparent, λ R at ρ 10 8 g cm 3 Photons and PP: Opaque, λ 5 cm at ρ 1 g cm 3 Central compact star: Negligible gravity ( 1M )

20 Near or Inside the Contact Discontinuity Near the Void Boundary Mechanical: Pressure Balance Required Possible when considering central power input Photons and PP, rather than neutrinos Attenuation: Optically thin after 1 yr Diffusion: Smoothing sharp edges Molecular Dynamics: Non equilibrium of chemical potential Molecules diffuse into contact discontinuity Not severe: 1% as radius doubles Physical Quantities: Being Realistic Recover dimensions from dimensionless models

21 Self-Similar Model in SNe Scenario Toy Model: Strongly Decelerating SN Ejecta Numerical Results: Voids, Shocks and EWCS Envelope -1 u/(cm s ) -3 ρ /(g cm ) T/K 2.5 x 109 Contact 2 Discontinuity 1.5 Surface ρ 0 Radiation Field T K Radiation Field Shock Shock Shock r /cm Panel A Panel B Panel C Stellar Mass: 20M Density at Edge: g cm 3 Pressure at Edge: dyn cm 2 Initial Bubble Radius: 160 km Initial T in Bubble: K Time before Invalid: 10 6 s

22 Self-Similar Model in SNe Scenario SN1993J Numerical Results: Voids, Shocks and EWCS Envelope -1 u/(cm s ) 3 x 109 Contact 2.5 Discontinuity 2 Surface 10 r 1.5 cd,i = cm Shock 10 r = cm s,i Shock Accelerated Shock Panel A Panel B Stellar Mass: 17M Shock Radius: 0.85 mas (t/1 yr) % Bright Shocked Region -3 T /K /(g cm ) ρ Radiation Field ρ 0 Radiation Field T K Shock Compressed Shock Shock Heated Panel C Marcaid et al. (2009) [4] Figure/SN1993J O BN.pdf r /cm

23 What s Next? 1 Background & Motivation 2 Technique: Model Construction 3 Application: Modeling SNe Ejecta 4 Summary Summary References Herein

24 Summary Summary Self-Similar Model of SN Ejecta Describe an spherically expanding cow Neutrinos: Not a sustainable cow puffer Photons and pair production products Potential undermining effects A More Realistic Cow : SN 1993J Possible version of profiles of radius Plausible expansion profiles of time Although a Specific Cow Radiation-driven expansion of SN Ejecta Figure/SPHCOW.jpg

25 References Herein References Herein H. A. Bethe and J. R. Wilson. ApJ, 295:14 23, August H.-T. Janka and E. Müller. A&A, 306:167 +, feb Yu-Qing Lou and Ren-Yu Hu. New Astronomy, 15(2): , J. M. Marcaide, I. Martí-Vidal, A. Alberdi, and et al. A&A, 505: , October HST Mission. Nasa website. pages/hubble/science/ornament.html. T. Padmanabhan. Theoretical Astrophysics, volume 2. Cambridge University Press, Y. Suto and J. Silk. ApJ, 326: , March 1988.