Self-Similar Gas Dynamics of Voids and Supernova Ejecta arxiv:1109.2682v2 [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
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
Some Easy-Reading Materials An Age-Old Joke... Spherical Cow A Cow A Spherical Cow Figure/COW.jpg Figure/SPHCOW.jpg
Some Easy-Reading Materials... and a Practical Simplification Spherical Nebula Crab by HST [5] Spherical Counterpart of Crab Figure/CRAB.jpg Figure/SNR0509.jpg
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
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
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: 10 51 erg Neutrinos: 10 53 erg; Scattering ineffective Photons and pair production (PP) products: 10 51 erg
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
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)...
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...
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)
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
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
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
Solutions to Self-Similar ODEs Numerical Results: Examples I Numerical Results: Voids, Shocks and EWCS Envelope 2 1.5 1 0.5 -v0 0.5 1 0.05 1.5 0 2 0.05 0.1 2.5 n=0.9, EWCS =1.1 Panel A Model E1 x Model E2 s1= 0.4 Model E3 cd 0.499 x cd= 1.45 x s1= 2 x s1= 2.5 Model E1 x cd= 0.0490 ZML 0.05 0.1 0.15 ZML SCC x cd, α cd : Values at Contact Discontinuity x s1 : x at upstream of shock 10 3 10 2 Model E1 cd 1810 EWCS Panel B 10 1 10 0 10 1 SCC Model E2 cd 2.51 Model E3 cd 1.42 SCC 10 2 x 0.5 1 1.5 2 2.5 3
Solutions to Self-Similar ODEs Numerical Results: Examples II Numerical Results: Voids, Shocks and Various Envelopes -v 2 Model 1 cd 1 0.809 x s1= 1.5 0 SCC 1 ZML 2 Model 4 cd 3 1.204 x s10 = 1 2.5 10 0 Model 1 cd 13.9 10 1 10 2 10 3 Model 3 cd 1.559 x s1= 2.5 n=0.9, =1.1 Model 3 cd 2.03 Model 2 cd 21.01 Model 4 0.5 1 1.5 2 cd 0.88 2.5 3 3.5 4 4.5 5 x Panel A Model 2 cd 2.509 x s1= 3.5 Panel B SCC -v 0.5 n0=0.67, =1.33 Panel A Model 5 x cd= 0.239 0.5 1 Model 7 x cd= 0.358 1.5 SCC 2 2.5 3 3.5 4 ZML Model 5 10 1 cd 59.6 x s1= 5.5 10 Model 6 0 cd 12.6 x s1= 7 10 1 10 2 10 3 x Model 6 x cd= 1.031 Model 7 cd 28.2 x = s1 9 Panel B SCC 2 4 6 8 10
Solutions to Self-Similar ODEs Numerical Results: Examples III Numerical Results: Voids, Crossing SCC Smoothly -v α 3 2 1 0 1 2 3 4 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Model S4 cd 1.516 x 0 4.571 v 0= 3.093 Model S4 cd 0.642 α 0= 0.66497 n=0.9, γ =1.1 x Model S5 cd 1.99 x 0 4.036 v 0= 2.608 SCC ZML SCC Model S5 cd 0.958 α 0= 0.67397 Panel A Panel B 1 2 3 4 5 6 7 0 0.5 1 1.5 -v 2 2.5 3 3.5 4 4.5 α 5 10 1 10 0 10 1 10 2 10 3 Model S1 cd 0.713 x 0 4.277 v 0= 2.254 Model S2 cd 1.373 x 5.010 Model S3 v 0= 2.736 cd 2.315 x 6.084 v 0= 3.442 SCC n=0.67, γ =1.33 Panel A Model S2 cd 9.081 α 0= 0.02358 Model S1 cd 17.01 α 0= 0.02157 x SCC ZML Panel B Model S3 cd 5.699 α 0= 0.02657 1 2 3 4 5 6 7
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
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 )
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
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 0 10 11 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 11 10 9 10 8 10 7 1 0.5 10 10 ρ 0 Radiation Field T 9 10 10 K Radiation Field Shock Shock 10 7 10 8 Shock r /cm Panel A Panel B Panel C Stellar Mass: 20M Density at Edge: 5 10 10 g cm 3 Pressure at Edge: 10 29 dyn cm 2 Initial Bubble Radius: 160 km Initial T in Bubble: 10 11 K Time before Invalid: 10 6 s
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 =2.4 10 cm 0.5 10 1 1 0 Shock 10 r =3.9 10 cm s,i Shock Accelerated Shock Panel A Panel B Stellar Mass: 17M Shock Radius: 0.85 mas (t/1 yr) 0.933 30% Bright Shocked Region -3 T /K /(g cm ) ρ 10 0 10 1 10 7.8 10 7.7 Radiation Field ρ 0 Radiation Field T 3 10 8 K Shock Compressed Shock Shock Heated Panel C Marcaid et al. (2009) [4] Figure/SN1993J O BN.pdf 10 7.6 r /cm 10 10 10 11
What s Next? 1 Background & Motivation 2 Technique: Model Construction 3 Application: Modeling SNe Ejecta 4 Summary Summary References Herein
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
References Herein References Herein H. A. Bethe and J. R. Wilson. ApJ, 295:14 23, August 1985. H.-T. Janka and E. Müller. A&A, 306:167 +, feb 1996. Yu-Qing Lou and Ren-Yu Hu. New Astronomy, 15(2):198 214, 2010. J. M. Marcaide, I. Martí-Vidal, A. Alberdi, and et al. A&A, 505:927 945, October 2009. HST Mission. Nasa website. http://www.nasa.gov/mission pages/hubble/science/ornament.html. T. Padmanabhan. Theoretical Astrophysics, volume 2. Cambridge University Press, 2001. Y. Suto and J. Silk. ApJ, 326:527 538, March 1988.