Prof dr A Achterberg Astronomical Dept IMAPP Radboud
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; �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)!
“Bernoulli” in shock jump:
Shock strength and Mach Number 1 D case: Shocks can only exist if Ms>1 ! Weak shocks: s=1+�with � << 1; Strong shocks: s>> 1.
Jump conditions in terms of Mach Number: the Rankine-Hugoniot relations Shocks all have S > 1 Compression ratio: density contrast Pressure jump
Oblique shocks: tangential velocity unchanged!
From normal shock to oblique shocks: All relations remain the same if one makes the replacement: θ is the angle between upstream velocity and normal on shock surface Tangential velocity along shock surface is unchanged
Example from Jet/Rocket engines: over-expanded jet exhaust
Under-expanded jet exhaust
Bell X 1 Rocket Plane
“Diamond” shocks in Jet Simulation
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
Tycho’s Remnant (SN 1572 AD)
Sedov scaling law for point explosions (1) Assumptions: 1. Explosion takes place in uniform medium with density ρ; 2. → spherical expanding fireball! 3. 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: 1. Photo dissociation of Iron in hot nucleus star: loss of (radiation) pressure! 2. Collapse of core under its own weight formation of proto-neutron star when ρ ~ 1014 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
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:
Lots of things happen………
Where does the energy go? neutronization core:
Supernova Blast Waves Main properties: 1. Strong shock propagating through the Interstellar Medium; (or through the wind of the progenitor star) 2. 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 ~ t 2/5 - 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:
Sedov’s argument for a stellar wind bubble 1. Energy is put in gradually: E(t)=Lwindt
Sedov’s argument for a stellar wind bubble 1. Energy is put in gradually: E(t)=Lwindt 2. Dimensional analysis:
More complicated structure Towards Star View from rest frame FW Shock for Vw >> VS
Pressure balance across contact discontinuity: Sedov: Wind properties:
Pressure balance across contact discontinuity: fixes location Wind Termination Shock! Sedov: Wind properties:
Ring Nebula
Eskimo Nebula Helix Nebula
Hourglass Nebula Eta Carinae
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