Introduction to Astrophysical Gas Dynamics

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Introduction to Astrophysical Gas Dynamics
Part 5

• Bram Achterberg
• a.achterberg_at_astro.uu.nl
• http//www.astro.uu.nl/achterb/aigdppt

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Shocks non-linear fluid structures
Shocks occur whenever a flow hits an obstacle at
a speed larger than the sound speed
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Shock properties

• Shocks are sudden transitions in flow properties
• such as density, velocity and pressure
• In shocks the kinetic energy of the flow is
  converted
• into heat, (pressure)
• Shocks are inevitable if sound waves propagate
  over
• long distances
• Shocks always occur when a flow hits an obstacle
• supersonically
• In shocks, the flow speed along the shock normal
• changes from supersonic to subsonic

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The marble-tube analogy for shocks
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• What do we learn from this analogy?
• The marble density behind the shock is larger
• than the density in front of the shock
• there is compression!
• 2. The transition is sudden
• 3. The shock speed is larger than the piston speed

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Wave Breaking
High-pressure/density regions move faster
Shock must form
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Effect of a sudden transition on the conservation
law
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Change of amount in layer
flux in - flux out
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Infinitely thin layer
What goes in must come out!
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Infinitely thin layer
What goes in must come out!
Formal proof limiting process
Flux in Flux out
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Simplest case normal shock in 1D flow
Starting point 1D fluid equations in
conservative form
Mass conservation
Momentum conservation
Energy conservation
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Shock conditions what goes in must come out!(1
in front of shock, 2 behind shock)
Three conservation laws means three conserved
fluxes!
Mass flux
Momentum flux
Energy flux
Three equations for three unknowns
post-shock state (2) is uniquely determined by
pre-shock (1) state!
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New variables specific volume
The three conserved fluxes
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From momentum conservation
From energy conservation
You can combine these two relations!
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From momentum conservation
From energy conservation
Shock Adiabat
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Shock compression ratio
Definition compression ratio
Shock jump condition
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Limiting cases weak and strong shocks
Weak shock pressure density change by small
amount
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Limiting cases weak and strong shocks
Weak shock pressure density change by small
amount
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Limiting cases weak and strong shocks
Weak shock pressure density change by small
amount
Weak shock (strong) sound wave!
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Limiting cases weak and strong shocks
Strong shock pressure jump is large!
Density- and velocity jump both remain finite!
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Jump conditions in terms of Mach Numberthe
Rankine-Hugoniot relations
Shocks all have ?S gt 1
Compression ratio density contrast
Pressure jump
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From normal shock to oblique shocks
All relations remain the same if one makes
the replacement

• is the angle between upstream velocity and
  normal
• on shock surface

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From normal shock to oblique shocks
All relations remain the same if one makes
the replacement

• is the angle between upstream velocity and
  normal
• on shock surface

Tangential velocity along shock surface is
unchanged
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Examples of Astrophysical shocks
Cometary bow-shocks
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Earths bow shock
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Heliosphere
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Supernova Remnant Cassiopeia A
Supernova blast waves
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Radio galaxy Cygnus A
Radio picture
Hot spots are shocks!
X-ray picture
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Knots in jet of Galaxy M87 are shocks!
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Relativistic Jet
Vjet 0.99c
Calculation Courtesy Jeroen Bergmans
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Summary shock physics
Across an infinitely thin steady shock you have
in the shock frame where the shock is at
rest Mass-flux conservation Momentum-flux
conservation Energy-flux conservation
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Summary Rankine-Hugoniot relations(for normal
shock)
Fundamental parameter Mach Number
R-H Jump Conditions relate the up- and
downstream quantities at the shock
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Theory of Supernova Blast Waves
Supernovae Type Ia Subsonic deflagration wave
turning into a supersonic
detonation wave in outer
layers. Mechanism explosive carbon burning
in a mass-accreting white
dwarf Type Ib-Ic Core collapse of
massive star Type II
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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 lt 1000 yr)
  R ? t
• - Sedov-Taylor stage (1000 yr lt t lt 10,000
  yr) R ? t 2/5
• - Pressure-driven snowplow (10,000 yr lt t
  lt 250,000 yr) R ? t 3/10

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Remnant of Tychos supernova of 1572 AD
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Radio map Cassiopeia A (VLA)
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An old supernova remnant (age 10,000 years)
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Free-expansion phase
Energy budget
Expansion speed
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Sedov-Taylor stage

• Expansion starts to decelerate due to swept-up
  mass
• Interior of the bubble is reheated due to
  reverse shock
• Hot bubble is preceded in ISM by strong blast
  wave

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Shock relations for strong (high-Mach
number) shocks
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Pressure behind strong shock (blast wave)
Pressure in hot SNR interior
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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!
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At contact discontinuity equal pressure on
both sides!
Relation between velocity and radius gives
expansion law!
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Step 1 write the relation as difference equation
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Step 2 write as total differentials and
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integrate to find the Sedov-Taylor solution
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Alternative derivations
1. Energy conservation
Deceleration radius
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Alternative derivations
2. Force balance on a thin moving shell
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Stellar Wind Bubbles

• Bubble blown due to high-speed wind of massive,
• evolved star
• Continuous energy input over gt 106 years
• More complicated internal structure there are
• two shocks involved, the outer Blast Wave and
• an inner termination shock, which ends the
• supersonic stellar wind

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Simple stellar wind bubble
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Mechanical luminosity wind Mass loss x kinetic
energy per unit mass
Rts
RS
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Mechanical luminosity wind
Energy in expanding shell
Mass in expanding shell
Rts
RS
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Mechanical luminosity wind
Energy in expanding shell
Mass in expanding shell
Rts
RS
Expansion law analogy with Sedov-Taylor
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Mechanical luminosity wind
Pressure behind termination shock
Rts
RS
Pressure behind Blast Wave
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Pressure balance across hot bubble/contact
discontinuity
subsonic!
Rts
RS
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Pressure balance
Energy law
Rts
RS
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Pressure balance
Energy law
Pressure balance condition determines term.
shock radius
Rts
RS
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Ring Nebula
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Eskimo Nebula
Helix Nebula
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Hourglass Nebula
Eta Carinae
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