Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit
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1 Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit
2 Shocks occur in supersonic flows; Shocks are sudden jumps in velocity, density and pressure; Shocks satisfy flux in = flux out principle for - mass flux - momentum flux - energy flux
3 Three conservation laws means three fluxes for flux in = flux out! ( ρv) ( ρv) = 1 Mass flux ( ) ( ρv P ρv P) + = + 1 Momentum flux V γp V γp ρv + = ρv + ( γ 1) ρ ( γ 1) ρ 1 Energy flux Three equations for three unknowns: post-shock state () is uniquely determined by pre-shock state (1)!
4 = 1 V γ P + = = ( 1) γ ρ V γp V γp ρv ρv + = + ( γ 1) ρ ( γ 1) ρ ( ρv) ( ρv) 1 constant
5 1D case: Mach Number s pre-shock flow speed V = = pre-shock sound speed C s 1 Shocks can only exist if s >1! Weak shocks: s =1+e with e<< 1; Strong shocks: s >> 1.
6 V 1 S = = Cs 1 upstream flow speed upstream sound speed Shocks all have S > 1 r ρ ρ V V 1 = = = 1 γ + 1 ( ) γ 1 + ( ) Compression ratio: density contrast P P 1 γ = 1+ 1 γ + 1 ( ) Pressure jump
7
8 ( ρv ) ( ρv ) n = 1 n ( ) ( ρv P ρv P) + = + n 1 n n V = V t1 t ( ρvv) ( ρvv) = n t 1 n t t V γp V γp ρvn + = ρvn + ( γ 1) ρ ( γ 1) ρ 1
9
10 All relations remain the same if one makes the replacement: V V = V cos θ, = V / C = cosθ 1 n1 1 1 S n n1 s1 S 1 θ is the angle between upstream velocity and normal on shock surface Tangential velocity along shock surface is unchanged V = V sinθ = V = V sinθ t1 1 1 t
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12
13 Bell X1 Rocket Plane
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15 Diamond shocks in Jet Simulation
16 Fundamental parameter of shock physics: Mach Number normal shock speed n = sound speed V C n1 s1 Rankine-Hugoniot jump conditions: ρ ( γ + ) = = ρ ( ) 1 γ 1 + γ 1 P P 1 1 n γ + 1 ρ 1 n γ ( 1) n γ = P = ρ V γ + 1 γ + 1 ρ 1 n1 Strong shock limit
17 Trinity nuclear test explosion, New Mexico, 1945 Supernova remnant Cassiopeia A
18 Tycho s Remnant (SN 157AD)
19 Assumptions: 1. Explosion takes place in uniform medium with density ρ;. spherical expanding fireball! 3. Total available energy: E. Point explosion + uniform medium: no EXTERNAL scale imposed on the problem!
20 Dimensional analysis: [ E] [ m][ ] [ t] [ ρ] =, = [ m] [ ] 3 Et ρ 1/5 R S is a length! Sedov: fireball radius ~ Sedov radius R S rt () R() t t S /5
21 Steps: 1. Photo dissociation of Iron in hot nucleus star: loss of (radiation) pressure!. Collapse of core under its own weight formation of proto-neutron star when ρ ~ g/cm 3 3. Gravitational binding energy becomes more negative: positive amount of energy is lost from the system! 4. Core Bounce shock formation and ejection envelope
22 Evolution of a massive star (5 solar masses) Core collapse: t ~ 0. s (!) Collapse onset: photo-dissociation of iron
23 Processes around collapsed core
24
25 Gravitational binding energy: E gr GM core mass potential Mcore = R GM R core
26 1 1 GM core 46 Egr GM core 10 J Rinit Rfinal Rfinal M 1 M ~ 10 kg, R ~ 10 km =10 m core 30 4 final
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28 neutronization core: 99% into neutrinos p+ e n+ ν e E sn = E = gr J 1% into explosion E snr 44 = 0.01Esn = 10 J
29 Main properties: 1. Strong shock propagating through the Interstellar Medium; (or through the wind of the progenitor star). Different expansion stages: - Free expansion stage (t < 1000 yr) R t - Sedov-Taylor stage (1000 yr < t < 10,000 yr) R t /5 - Pressure-driven snowplow (10,000 yr < t < 50,000 yr) R t 3/10
30
31 Energy budget: E GM 3 c 46 grav = 5 10 J Rc 99% into neutrino's 1% into mechanical energy Expansion speed: V 1/ -1/ ESNR ESNR M ej exp = 3000 km/s 44 Mej 10 J 10 M
32 - Expansion decelerates due to swept-up mass; - Interior of the bubble is reheated due to reverse shock; - Hot bubble is preceded in ISM by strong shock: - the supernova blast wave.
33
34 Shock relations for strong (high-mach number) shocks: ( γ ) ( ) ρ + 1 s γ + 1 = ρ1 γ 1 s + γ 1 P P 1 V1 ρ1v1 s cs 1 γ P1 γ ( 1) as s γ γ = = s γ + 1 γ+1 P= ρ1v1 γ + 1
35 P γ P = ρ V γ + 1 γ + 1 s 1 ism s Pressure behind strong shock (blast wave) P i = γ 1 e γ 1 E ( ) ( ) SNR i 4π 3 R 3 S Pressure in hot SNR interior
36 At contact discontinuity: equal pressure on both sides! ρ γ + 1 ism V γ 1 ( ) E SNR s 4π 3 Rs 3 This procedure is allowed because of high sound speeds in hot interior and in shell of hot, shocked ISM: No large pressure differences are possible!
37 At contact discontinuity: equal pressure on both sides! ρ γ + 1 ism V γ 1 ( ) E SNR s 4π 3 Rs 3 V s dr 8π s snr = dt 3 1 ρism ( γ ) E 1/ R 3/ s Relation between velocity and radius gives expansion law!
38 1/ 3/ 8π E snr Rs drs dt ( 3 γ 1) ρism Step 1: write the relation as difference equation
39 1/ 3/ 8π E snr Rs drs dt ( 3 γ 1) ρism ( 5/ ) 8π E snr d Rs ( 5 3 γ 1) ρism 1/ dt Step : write as total differentials and
40 1/ 3/ 8π E snr Rs drs dt ( 3 γ 1) ρism ( 5/ ) 8π E snr d Rs ( 5 3 γ 1) ρism 1/ dt integrate to find the Sedov-Taylor solution 1/5 R t C E t C snr /5 s( ) γ, ρism γ /5 5 8π = ( 3 γ 1) 1/5 1.96
41 1 MsnrVs = E snr V = 4π 3 Msnr = Mej + 3 ρismr s shock speed = expansion speed E snr s 4π 3 Mej + 3 ρismrs Deceleration radius R d : R 1/3 1/3 1/3 3Mej Mej nism d = -3 4πρism M 1 cm pc
42 V s 1/ 1/ E 1 1 snr = = V M ej 1 + ( R/ Rd) 1 + ( R/ Rd) /5 t R 3/ t
43 1. Energy is put in gradually: E(t)=L wind t L wind = mechanical luminosity 1 MV w M = r rv r = 4 π ρw( ) w( ) mass loss = constant
44 1. Energy is put in gradually: E(t)=L wind t L wind = mechanical luminosity 1 MV w M = r rv r = 4 π ρw( ) w( ) mass loss = constant. Dimensional analysis: R () t S 1/5 3 1/5 Ett () Lwindt = = ρ ρ t 3/5
45 View from rest frame FW Shock for V w >> V S Towards Star
46 ρ V = ρ V ism S w w V S Sedov: 1/5 dr 3 L 3R = = t = dt 5 ρism 5t S wind /5 S ρ () r Wind properties: M = = L wind w 3 4πrVw πrvw
47 ρ V = ρ V ism S w w r = RTS t /5 ρ ism V Sedov: 9ρ L = /5 ism wind 4/5 S 5 ρism t Wind properties: ρ () rv = L wind w w π rvw
48 Ring Nebula
49 Helix Nebula Eskimo Nebula
50 Eta Carinae Hourglass Nebula
51
Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit
Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit Shocks occur in supersonic flows; Shocks are sudden jumps in velocity, density and pressure; Shocks satisfy flux in = flux out principle
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