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Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit. Gas Dynamics, Lecture 7 (Shocks & Point Explosions) see: www.astro.ru.nl/~achterb/. Summary of shock physics. Shocks occur in supersonic flows; Shocks are sudden jumps in velocity, density and pressure;
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Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, RadboudUniversiteit Gas Dynamics, Lecture 7(Shocks & Point Explosions)see: www.astro.ru.nl/~achterb/
Summary of shock physics • 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
Flux in = flux out: three jump conditions;Case of a normal shock Three conservation laws means three fluxes for flux in = flux out! Mass flux Momentum flux Energy flux Three equations for three unknowns: post-shock state (2) is uniquely determined by pre-shock state (1)!
Shock strength and Mach Number 1D case: Shocks can only exist if Ms>1 ! Weak shocks:Ms=1+ with << 1; Strong shocks: Ms>> 1.
Weak shock ~ strong sound wave! Sound waves:
Strong shock: P1<< 1V12 Approximate jump conditions: put P1 = 0!
Jump conditions in terms of Mach Number:the Rankine-Hugoniot relations Shocks all have S > 1 Compression ratio: density contrast Pressure jump
Oblique shocks: four jump conditions! (1) (2) (3) (4)
From normal shock to oblique shocks: All relations remain the same if one makes the replacement: θis the angle between upstream velocity and normal on shock surface
From normal shock to oblique shocks: All relations remain the same if one makes the replacement: θis the angle between upstream velocity and normal on shock surface Tangential velocity along shock surface is unchanged
Summary: Fundamental parameter of shock physics: Mach Number Rankine-Hugoniot jump conditions: Strong shock limit
Application: point explosions Trinity nuclear test explosion, New Mexico, 1945 Supernova remnant Cassiopeia A
Sedov scaling law for point explosions (1) Assumptions: Explosion takes place in uniform medium with density ρ; → spherical expanding fireball! Total available energy: E. Point explosion + uniform medium: no EXTERNAL scale imposed on the problem!
Sedov scaling law for point explosions (2) Dimensional analysis: Sedov: fireball radius ~ Sedov radius RS
Supernova explosions Steps: 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 ~ 1014 g/cm3 Gravitational binding energy becomes more negative: positive amount of energy is lost from the system! 4. Core Bounce shock formation and ejection envelope
Evolution of a massive star (25 solar masses) Core collapse: t ~ 0.2 s (!) Collapse onset: photo-dissociation of iron
Processes around collapsed core
Available energy: Gravitational binding energy:
Where does the energy go? neutronization core:
Supernova Blast Waves • Main properties: • 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 2/5 • - Pressure-driven snowplow (10,000 yr < t < 250,000 yr) R t 3/10
Free-expansion phase: R=Vexpt Energy budget: Expansion speed:
Sedov-Taylor stage: R ~ RS ~ t2/5 • - 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.
Shock relations for strong (high-Mach number) shocks:
Pressure behind strong shock (blast wave) Pressure in hot SNR interior
At contact discontinuity: equal pressure on both sides! 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!
At contact discontinuity: equal pressure on both sides! Relation between velocity and radius gives expansion law!
Step 1: write the relation as difference equation
Step 2: write as total differentials and………
……integrate to find the Sedov-Taylor solution
Alternative derivation: Energy Conservation shock speed = expansion speed Deceleration radius Rd: