Gas Dynamics, Lecture 7 (Shocks & Point Explosions) see: astro.ru.nl/~achterb/

1. Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, RadboudUniversiteit Gas Dynamics, Lecture 7(Shocks & Point Explosions)see: www.astro.ru.nl/~achterb/

2. 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

3. 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)!

4. “Bernoulli” in shock jump:

5. Shock strength and Mach Number 1D case: Shocks can only exist if Ms>1 ! Weak shocks:Ms=1+ with << 1; Strong shocks: Ms>> 1.

6. Weak shock:

7. From jump conditions:

8. Weak shock ~ strong sound wave! Sound waves:

9. Very strong normal shock

10. Strong shock: P1<< 1V12 Approximate jump conditions: put P1 = 0!

12. Conclusion for a strong shock:

13. Jump conditions in terms of Mach Number:the Rankine-Hugoniot relations Shocks all have S > 1 Compression ratio: density contrast Pressure jump

15. Oblique shocks: four jump conditions! (1) (2) (3) (4)

16. Oblique shocks: tangential velocity unchanged!

18. 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

19. 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

20. Example from Jet/Rocket engines:over-expanded jet exhaust

21. Under-expanded jet exhaust

22. Bell X1 Rocket Plane

24. “Diamond” shocks in Jet Simulation

25. Summary: Fundamental parameter of shock physics: Mach Number Rankine-Hugoniot jump conditions: Strong shock limit

26. Application: point explosions Trinity nuclear test explosion, New Mexico, 1945 Supernova remnant Cassiopeia A

27. Tycho’s Remnant (SN 1572AD)

28. 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!

29. Sedov scaling law for point explosions (2) Dimensional analysis: Sedov: fireball radius ~ Sedov radius RS

30. 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

31. Evolution of a massive star (25 solar masses) Core collapse: t ~ 0.2 s (!) Collapse onset: photo-dissociation of iron

32. Processes around collapsed core

34. Available energy: Gravitational binding energy:

36. Lots of things happen………

37. Where does the energy go? neutronization core:

38. 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

40. Free-expansion phase: R=Vexpt Energy budget: Expansion speed:

41. 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.

43. Shock relations for strong (high-Mach number) shocks:

44. Pressure behind strong shock (blast wave) Pressure in hot SNR interior

45. 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!

46. At contact discontinuity: equal pressure on both sides! Relation between velocity and radius gives expansion law!

47. Step 1: write the relation as difference equation

48. Step 2: write as total differentials and………

49. ……integrate to find the Sedov-Taylor solution

50. Alternative derivation: Energy Conservation shock speed = expansion speed Deceleration radius Rd: