14/03/2014
9
Shocks1. Shocks are sudden transitions in flow properties
such as density, velocity and pressure;
2. In shocks the kinetic energy of the flow is convertedinto heat, (pressure);
3. Shocks are inevitable if sound waves propagate overlong distances;
4. Shocks always occur when a flow hits an obstacle supersonically
5. In shocks, the flow speed along the shock normalchanges from supersonic to subsonic
14/03/2014
10
Wave BreakingHigh-pressure/densityregions move faster
( 1)/2
0
0
0
21
1
s
s
cu
c
Shock must form
14/03/2014
12
Chelyabinsk Meteorite (Feb. 2013):Sonic Boom
Examples of Astrophysical shocks
Cometary bow-shocks
14/03/2014
17
Hiroshima, the Shockwave
Radio galaxyCygnus A
Radio picture
X-ray picture
Hot spotsare shocks!
14/03/2014
19
Summary :Shock Physics
Across an infinitely thin steady shock you have, in the shock frame where the shock is at rest, the following Rankine-Hugoniot Jump conditions:
Mass-flux conservation
Momentum-flux conservation
Energy-flux conservation
1 1 2 2n nV V
2 2
1 1 1 2 2 2n nV P V P
1 2t tV V
2 21 21 11 22 2
1 2( 1) ( 1)n n
P PV V
14/03/2014
20
Summary: Rankine-Hugoniot relations
(for normal shock)
22
21
22
1
1 1
1 2 1
2 1
1
s
s
sP
P
1
1
shock speed
sound speedss
V
c Fundamental parameter:
Mach Number
R-H Jump Conditionsrelate the up- and downstreamquantities at the shock:
From normal shock to oblique shocks:
All relations remain the same if one makes the replacement:
1 n1 1 1
n n1 1 1
cos ,
/ cosS s S
V V V
V c
is the angle between upstream velocity and normal on shock surface
Tangential velocity along shock surface is unchanged
t1 1 1 t2 2 2sin sinV V V V
14/03/2014
24
Cas A SNR flythrough
Theory of Supernova Blast Waves
Supernovae:
Type Ia Subsonic deflagration wave turning into a supersonic detonation wave in outerlayers.
Mechanism: explosive carbon burning ina mass-accreting white dwarf
Type Ib-Ic& Core collapse of massive star
Type II
14/03/2014
26
Supernova Ia Explosion
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
14/03/2014
28
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
14/03/2014
31
Free-expansion phaseEnergy budget:
2533
grav 5
99% into neutrino's10 erg
1% into mechanical energyc
c
GME
R
-1/21/2
ejmech mechexp 51
ej
23000 km/s
10 erg 10
ME EV
M M
Expansion speed:
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
14/03/2014
33
Shock relationsfor strong
(high-Mach number)shocks:
22
21
2 22 1 1 12
221 1
1
22 1 1
1 1
1 2 1
as 2 1 2
1 +1
2
1
s
s
ssss
V VP c PP
P V
2 22 1 ism
2 2
1 1s sP P V
SNRi i
3S
4
3
1 1E
P eR
Pressure behind strongshock (blast wave)
Pressure in hot SNR interior
14/03/2014
34
At contact discontinuity:equal pressure on bothsides!
2 SNRism
3s
4
3
21
1 s
EV
R
This procedure is allowed because of high sound speedsin hot interior and in shell of hot, shocked ISM:No large pressure differences are possible!
At contact discontinuity:equal pressure on bothsides!
2 SNRism
3s
4
3
21
1 s
EV
R
1/ 2
3/ 2snr2
ism
8
3 1s
s s
dR EV R
dt
Relation betweenvelocity and radiusgives expansion law!
14/03/2014
35
1/ 2
3/ 2 snrs s 2
ism
8
3 1
ER dR dt
Step 1: write the relationas difference equation
1/ 2
3/ 2 snrs s 2
ism
1/ 2
5/2 snrs 2
ism
8
3 1
2 8
5 3 1
ER dR dt
Ed R dt
Step 2: write as totaldifferentials and………