1. Shock waves; Supernova remnant evolution
ISM Lecture 9
Shock waves;
Supernova remnant evolution

2. 9.1 Shock waves
A shock wave is a pressure-driven compressive disturbance propagating faster than “signal speed”
Shock waves produce an irreversible change in the state of the fluid
Literature:
Landau & Lifshitz, Fluid mechanics, Ch. IV
Dyson & Williams 1997, Ch
Tielens, Ch. 11

3. Sound speed and Mach number
P = Kργ with γ = 5/3 for adiabatic flow, γ = 1 for isothermal flow 
C  1/3 for adiabatic flow  sound speed larger in denser gas
for isothermal gas in ISM
Mach number: M  ratio of velocity w.r.t. sound speed, M  v/C

4. Steepening of non-linear acoustic wave
Speed is larger in denser gas 
Speed of crest is larger than speed of trough 
Crests “catch up” with troughs 
Steepening of waveform 
Steepening halted by viscous forces 
Development of shock
speed of
crest
speed of trough

5. “Signal speed” in ISM
In the absence of magnetic fields, information travels with sound speed
M < 1 subsonic
M > 1 supersonic  shocks
If magnetic field is present, disturbances will travel along B at Alfvén speed vA:
Interstellar magnetic field (empirical):
for 10 < nH<106 cm-3

6. Examples of shock waves in the ISM
Cloud-cloud collisions
Expansion of H II regions
Fast stellar winds (“interstellar bubbles”)
Supernova blast waves
Accretion and outflows during star formation
Spiral shocks in Galactic disk

7. Cloud-cloud collisions
shock front v0 
Δv = 2 v0
shocked gas
v0  5-10 km/s
vs  v0

8. Expansion of H II region
compressed neutral gas
(cold, high density)
H II region
ambient neutral gas
(cold, medium density)
(hot, low
density)
vs  10 km/s

9. Fast stellar wind (“Interstellar Bubble”)
inner shock
outer shock
contact discontinuity
(wind meets ISM)
unshocked wind material
shocked wind material
shocked cloud material
unshocked cloud material

10. Terminal velocity of wind from hot stars

11. Supernova blast wave Early: Mswept < Mejecta  like stellar wind
Late: Mswept >> Mejecta  ignore Mejecta
shock front
unshocked ISM
hot interior
shocked ISM
(cool shell)
shocked ISM
(did not cool radiatively)

12. Star formation: infall and outflow
shock front
hot, shocked
accreted matter
infalling ambient
cold material (low
pressure, free fall)

13. Spiral shocks in Galactic disk
flow
shock front
compressed
material forms
spiral arm
material expands and
leaves spiral arm

14. 9.2 Shock jump conditions Adopt frame in which shock is stationary
Consider plane-parallel shock: fluid properties depend only on distance x from shock;
Neglect viscosity, except in shock transition zone:
large v-gradient  viscous dissipation  transform bulk kinetic energy into heat  irreversible change (entropy increases)

15. Shock profile (shock frame)

16. Shock jump conditions (cont’d)
Thickness of shock transition  mean free path of particles
This is always << thickness of radiative zone (need collisions to get radiative cooling)
Regard shock thickness as 0  discontinuity
Find physical conditions at 2 (immediately behind shock) or 3 (in post-radiative zone) given those at 1 (pre-shock) and shock velocity vs

17. Schematic depiction of a radiative shock
zone
radiative
zone
Undisturbed pre-shock material
Immediate post-shock conditions
Cold post-radiative conditions

18. Mass conservation across shock
Steady plane-parallel shock:
Integrate across shock:

19. Momentum conservation across shock
Note:
no change in gradient of gravitational potential F =>

20. Energy conservation across shock
Same procedure as before
Heat conduction
Heating
Cooling
~0 if 1 and 2 are close

21. Magnetic field equation
The electric field vanishes in the fluid frame
Under this condition the Maxwell Equations can be simplified to give
The last term describes diffusion of the magnetic field and vanishes if the conductivity    (ideal MHD)

22. Special (often used) case: simplifying assumptions
Planar shock:
No heat conduction:
No net heating / cooling:
No gravitational force:
Internal energy density: with g=cP/cV
B-field  flow velocity:
  :

23. Rankine-Hugoniot equations for shock jump conditions
Follow general practice and write vx=u1(=vS)
Mass conservation:
Momentum conservation:
Energy conservation:
Flux conservation:

24. Solution of shock jump equations
These are the jump conditions: 4 eqs. in 4 unknowns: r2, u2, P2, B2
Define
Now we have two equations for P2 and x (cubics in x):
Define Mach number M by M2=u12/vms2 , where vms is the magnetosonic speed given by

25. Solution of shock jump equations (cont’d)
The trivial solution 1 = 2 always exists
A second solution exists for M > 1
Strong shock: M >> 1 

26. Some properties of strong shocks
Compression ratio x = 4
If 16B12/8p<<r1u12 and P1<< r1u12 , then momentum conservation gives
P23/4 r1u12=3/4 . ram pressure
B2 = 4 B1  magnetic pressure increases by factor 16
Post-shock flow is at P  constant, and subsonic:

27. Adiabatic vs. radiative shocks
So far considered shocks without radiative cooling: “adiabatic shocks”
This is a bad terminology, since shocks are abrupt and irreversible
The term ‘adiabatic’ refers to the fact that no heat is removed during shock
If post-shock gas radiates lines  cools down  “radiative shock”
If temperature in region 3 is the same as in region 1  “isothermal shock”
This is also a bad terminology since T always rises at shock front

28. Isothermal shock structure
Solve jump conditions with T3 = T1 => r3>>r1
B = 0:
compression factor can become much larger than 4!
B  0:
where
 typical compression factor  100!

29. J vs. C shocks: MHD effects
So far considered only single-fluid shocks (“J-shocks”)
Normally interstellar gas consists of 3 fluids: (i) neutral particles, (ii) ions, (iii) electrons
Fluids can develop different flow velocities and temperatures
For B  0 information travels by MHD waves
Perpendicular to B the propagation speed is the magnetosonic speed

30. Magnetic Precursors: C-shocks
Alfvén speed for decoupled ion-electron plasma can be much larger than Alfvén speed for coupled neutral-ion-electron fluid
In many cases: CS<vA,n < vs < vA,ie
Ion-electron plasma sends information ahead of disturbance to “inform” pre-shock plasma that compression is coming: magnetic precursor
Compression is sub-sonic and transition is smooth and continuous  “C-shock”
Ions couple by collisions to neutrals

31. Comparison of J-shocks and C-shocks
J (“jump”)-shocks: vS50 km/s
Shock abrupt
Neutrals and ions tied into single fluid
T high: T40 (vS/km s-1)2
Most of radiation in ultraviolet
C (“continuous”)-shocks: vS50 km/s
Gas variables (T, ρ, v) change continuously
Ions ahead of neutrals; drag modifies neutral flow
Ti  Tn; both much lower than in J-shocks
Most of radiation in infrared

32. Schematic structure of J- and C-shocks

33. C-shock structure
Draine & Katz 1986

34. Post-shock temperature structure of fast J-shock
Three regions:
Hot, UV production
Recombination, Lyα absorption
Molecule formation
Grains are weakly coupled to gas, so that Tgr << Tgas

35. Computed fine structure line intensities for J-shock

36. Computed H2 near-IR line intensities for J-shock

37. 9.3 Single point explosion in uniform cold medium
Idealized treatment of SN explosion due to Sedov (1959) and Taylor
See also Chevalier (1974)
Equally applicable to atomic bombs
Assumptions:
E conserved  no radiative cooling  no charateristic tcool
Fluid treatment adequate (mean free path << R)
Neglect viscosity, heat conduction
Ambient pressure small
Then there is only 1 characteristic
Length: R=size SNR
Time: t=age
Velocity: R/t

38. Self-similar solution of blast-wave problem
Three characteristic quantities:
Explosion energy E
Ambient density ρ0
Time since explosion t
Self-similar solution: the space-time evolution of all relevant quantities (ρ, v, T) can be described by universal functions that depend only on one dimensionless parameter
v(r,t) = R/t fv(r/R)
ρ(r,t) = ρ0 fρ (r/R)
T(r,t)=m/k (R/t)2 fT(r/R)

39. Dimensional analysis
Use dimensional analysis to determine dependence of R(t) on E, ρ0, t:
=>

40. Calculation of A Expect A~1. To calculate, put in physics
Assume ‘adiabatic’: E=Ekin+Ethermal=const=E0
Assume ‘self-similar’: Ekin/Ethermal=const
Behind shock:

41. More detailed calculation of A
Assume most of mass concentrated just behind shock (O.K. from detailed solution)
Ekin~Ethermal for entire SNR
Ekin~1/2MvS2, M= 4/3pR3r0
Then E0=2Ekin=2.1/2. 4/3pR3r0.vS2
R=cst.ta =>vS=a.cst.ta-1
=> E0=4/3pr0(cst)5a2t5a-2=const. => a=2/5
=> (cst)5=3/(4p)(5/2)2(E0/r0)
=> (cst)=1.083 (E0/r0)1/5 => A=1.083
Exact: A= for g=5/3

42. Sedov solution 50% of mass in outer 6% of radius
T  P/ρ drops from center towards edge

43. 9.4 Evolution of SNR in homogeneous ISM 1. Early phase
Early: Mswept < Mejecta (‘Free expansion’)
RS=vSt
vSv0=ejecta velocity=(2E0/Mejecta)1/2 =10000 km/s (E0/1051 erg)1/2 (Msun/Mejecta)1/2
Phase ends at t=tM when Mswept=Mejecta
For density r0, Mejecta=Mswept=r04/3p(v0tM)3 =>

44. 2. Sedov phase
Mejecta<Mswept and t<trad

45. Cooling at end of Sedov phase
shock front
cool dense shell
hot, low density, pressure pi
radiative cooling negligible

46. Estimate of trad
Estimate t=trad when cooling becomes important and Sedov phase ends
Most cooling just behind shock where r highest and T lowest
Define trad when 50% of Ethermal radiated
almost constant!

47. 3. Post-Sedov or ‘radiative’ phase
Cold dense shell slows down due to sweeping of ISM; momentum is conserved
Naïve snowplow (neglect Pi): 
Rt1/4

48. Radiative phase (cont’d)
Pressure-modified snowplow (Oort snowplow):
Pi drops due to adiabatic expansion:

49. 4. Merging phase
SNR fades away when expansion velocity = velocity dispersion ambient gas (~10 km/s)
Use Oort snowplow:

50. Summary phases of supernova shell expansion
Early phase (mswept < mejecta): Free expansion, Rs = vs t
Sedov phase (mswept > mejecta and t < trad): Energy conservation, R  t2/5
Radiative “snowplow” phase (t > trad): Momentum conservation, R  t1/4 or R  t2/7
Merging phase: vs drops below velocity dispersion of ambient ISM

51. Overview of SNR evolution

52. Overview of SNR evolution