Relativistic Self-similar Solutions: Explosions, implosions and shock breakouts

Transcription

1 Relativistic Self-similar Solutions: Explosions, implosions and shock breakouts Re em Sari Best & Sari 00, Sari 06, Pan & Sari 06, 08 Caltech & HUJI Margaret Pan IAS ESA

2 Supernovae & GRBs A. MacFadyen Galama et al. BeppoSAX

3 COMPACTNESS PROBLEM γ + γ e + + e - dt ~ 1ms R < cm E ~ ergs photons high photon density (many above 500 kev). Optical depth σ T n R~10 15 >>1 Inconsistent with the non thermal spectrum! Spectrum: Optically thin? Paradox? Size & Energy: Optically thick

4 The Solution: Relativistic Motion Due to Relativistic Motion: R = γ 2 c dt E ph (emitted) = E ph (obs) / γ τ γγ = γ (4+2α) nσ T R ~ /γ 4+2α γ > 100 (Goodman; Paczynski; Krolik & Pier; Fenimore; Woods & Loeb;Baring &Harding; Piran & Shemi; Lithwick & RS)

5 Exploding stars Polytropic envelopes: P ρ q, R-r«R ρ x -k degenerate (k=-3,-1.5), convective (-3<k<-1.5) Planar scale-free problem r x =R-r R

6 Exploding stars Evolution of a single fluid element: γ shocked to relativistic temps & speeds cools; acc. stops expands, accelerates unshocked fluid t

7 Self-similarity For scale-free problems far from initial conditions Profiles of hydrodynamic variables Decouple t and x evolution: x t PDEs in, ODEs in d dχ

8 Relativistic Self Similar Equations Energy equation Momentum equation PDEs γ(t,r) = Γ(t) g(χ)/ 2 p(t,r) = p(t) f(χ) χ = (t-x)/(t-r) k - external density profile ρ~r -k. α - 0,1,2 - system s dimension. m - temporal evolution, γ~t -m/2 ODEs

9 Type I " # Evolution follows from global energy conservation Sedov-Taylor (1940s): const E ~ ρr 3 v 2 ~ ρr 5 t -2 R ~ t 2/5 v ~ t -3/5 Blandford & McKee 1976: const E ~ ρr 3 γ 2 c 2 1 1$ v 2 /c 2 >>1 γ ~ R -3/2 ~ T -3/8 NRAO

10 Type II Implosions LLNL Shock accelerates Boundary conditions from sonic point, not global energy conservation Sonic point protects the shock from non self-similar region.

11 Solutions f(1)=g(1)=1 First type Energy conservation: m=1+α-k Sonic point: Second type Smooth solution at sonic point: Calculate the energy in the solution Solve: Finite for: Explosions k<(7+5α)/4 Implosions k>(7+5α)/4 Sonic point exists exists if: Explosions k>5-(3/4) 1/2 Implosions k<4

12 First & Second Type Solutions Explored all three geometries Overlaps & gaps - mysteries: Is there a 3rd type self similar solutions to fill the gaps? χ sonic What selects between the two solution in the overlap regions? k

13 Pre-breakout (Sari 06) t<0, R<0 Shock accelerates toward edge: Type II R = position of shock 10 3 γ x

14 Post-breakout (Pan & Sari 06) t>0, R> No new scales: still self-similar! γ 10 3 R = position of fluid element which has expanded by factor of ~few x Same time evolution as t<0

15 10 3 Time evolution 10 3 t>0 Γ t< P N R 10-6 R

16 Summary - pre & post breakout fluid element trajectories t=0 (breakout) t>0 profiles t<0 profiles

17 Fluid elements at late times p=e/3: assumed hot fluid shocked to relativistic temps & speeds γ cools; acc. stops To get final γ, cut off the flow when p/n 1 unshocked fluid expands, accelerates t p/n p/n=1 time back front

18 How to deal with cold fluid? Cold fluid breaks self-similarity later γ hot earlier cold R x

19 How to deal with cold fluid? Cold fluid not self-similar in the hot solution Transition from hot to cold fluid is self-similar: t-δ cooling solution earlier later γ hot cold R x

20 Hot solution vs cooling solution p=e/3 R: fluid which has expanded by factor of few χ=(t-x)/(t-r) Valid at early times & for fluid near the front p=(e-nmc 2 )/3 t-δ: fluid which has just become cold (p/n=1) ξ=(t-x)/δ Valid at late times, when t-r» δ

21 e,p,n conservation Cooling solution (g,f,h) ODE solver new EOS self-similar form similarity

22 Fluid elements at late times log 10 γ time cold 2 solutions together give γ final =1.95γ , k=-3 hot log 10 (p/n+1) Tan, Matzner, & McKee: γ final =2.6γ

23 2 self-similar solutions for postbreakout flow which time evolve with different power laws Summary γ shocked to relativistic temps & speeds unshocked fluid hot expands, accelerates Together give accurate expression for final Lorentz factors of fluid elements cools; acc. stops cooling t