Charles Danforth
Intermediate Seminar
September 30, 1997
To gain a better understanding of shocks, we will start with a quick review of sound waves. In a fluid medium with a pressure P
, we can define a perturbation 
. Using the wave equation
we see that the perturbation produces waves which travel at speed c. From thermodynamics, we know that the sound speed is related to temperature and density
At terrestrial temperatures and pressures, the sound speed tends to be a fraction of a kilometer per second. However, in space, the temperatures can be very high and the densities very low. Thus a typical interstellar sound speed is of the order 10 km/s.
is not small. If we imagine a fluid filled tube with a piston at one end. As the piston is moved into the fluid, the fluid starts to compress. Information about this rise in pressure propagates away from the piston at the sound speed of the fluid. If the piston speed 
is greater than the sound speed, then the pressure continues to build in front of the piston with the gradient in pressure becoming steeper and steeper. A good analogy to this is a snow plow which pushes an increasingly long mass of snow in front of it.
The edge of the pressure hump (the shock) moves down the tube at speed 
. We can define the Mach number as 
In the parlance of shock physics, different mach numbers correspond to different regimes. For M<0.3, we have an incompressible flow. This is the limit where 
. 0.3<M<0.8 is referred to as subsonic. 0.8<M<1.5 is transonic while 1.5<M<5.0 is supersonic. Anything higher is hypersonic.
We can also talk about isothermal shocks which differ from adiabatic shocks discussed above in that the sound speed is lower and the jump conditions are different. All real life shocks are a compromise between adiabatic and isothermal shocks.
Another is the sonic boom we hear from high performance aircraft. Streamlined as they are, these machines act the same way as the piston in our tube and create a shock front. The exact shape and separation of the shock front and the leading surfaces of the aircraft depend on velocity and the subtleties of the shape. In the case of very strong shocks, such as those produced by hypersonic spacecraft returning to earth, the shock can be so strong that ionization of the air is produced. This blanket of ionization causes the famous radio blackout experienced by astronauts.
In condensed matter, shock waves may result during earthquakes, meteor strikes, atomic bomb tests and mining explosions. It is thought that in regions where the ideal gas law REALLY doesn't apply that shocks go through three stages. First is the strong shock period where 
and 
. Next is a transition period where the spherical shock wave is converted into a sound wave. Finally is the acoustic decay period where 
and 
. It has been theorized that craters are related to the radius at which the transition occurs from supersonic to subsonic.
One interesting application of shocks in a terrestrial application is that of ramjet engines and ram accelerators. In the case of a ramjet, the engine is shaped like a simple tube with a conical body slung inside. Fuel is injected into the air stream ahead of the engine. The conical body causes a series of shock waves which compresses and ignites the fuel. The thrust then pushes on the back of the cone pushing the engine forward.
A ram accelerator works in a similar way except that it is the conical projectile which moves inside a tube of fuel and oxidizer. Ram accelerators are more efficient than conventional gas guns because the thrust is maintained at the projectile instead of at the beginning of the barrel. Current ram accelerators can achieve muzzle velocities of 4 km/s. This technology presents a promising area of research for space launching.
In addition to sound waves, magnetohydrodynamic (MHD) fluids support Alfven or Magnetosonic waves. Since the magnetic flux is strongly coupled to heavy ions in the gas, the situation is much like masses on strings. MHD shocks are similar to the adiabatic shocks examined above, but with the addition of a magnetic field term and one more conserved quantity.
One stunning observational example of a cosmic shock is the Herbig-Haro class of objects. These objects are believed to be protostellar objects with strong bipolar jets. The jets are composed of high-velocity particles which cause shock waves as they interact with interstellar material. This is but one example of shocks caused by jets. Indeed, jets come in all sizes from the relatively small HH Objects to those from active galactic nuclei (AGN) which produce galactic-scale jets.
Colliding clouds in interstellar space produce unspectacular but very important shocks typically only slightly supersonic. Still, these shocks cause mixing of processed and unprocessed materials from stars and serve to heat the material up. They can also become the catalyst which condenses clouds prior to star formation.
Related to colliding clouds are the tremendous shock waves resulting from supernovae. After a star explodes, the shock passes through three distinct phases. First is the free expansion phase in which the blast wave moves away from the stellar remnant at a constant velocity of roughly 5000 km/s sweeping up material. After it has accumulated a mass of material roughly equal to the mass of the original ejecta (about 700 years), it enters the adiabatic/Sedov phase. In this phase, which lasts another 40,000 years, the material cools from expansion. The shock begins to slow down as 
and 
. Finally, the temperature drops low enough to allow significant line radiation to cool the shocked material. This is the isothermal phase and will last until the shock velocity drops below ambient sound speed-a few hundred thousand years or more. During this phase the temperature remains constant at 
K, 
and 
.
These shock waves are made spectacularly visible to observers in the form of supernova remnants. Young SNRs are typified by a smooth spherical shell of hot gas expanding outward. What we see is a ring (the optical depth is highest through the edges of the shell with the longest lines of sight). As these shells expand, cool and age, the Wardle Instability starts to play a major role in distorting the smooth surface into bumps and ripples. Volumes of different density will cause the shock to slow at different rates and we end up with a twisted, knotted filamentary structure. The filaments seen are probably just the portions of the shock front seen edge-on from our line of sight.
Shocks add random velocities and bulk motion to the ISM. This contributes strongly to the kinetic energy and heating in the galaxy and is, if you will, a sort of tilling of the cosmic soil. Enriched material is dispersed from supernovae and molecules from clouds. Without this kind of mixing, life on earth would undoubtedly never have arrisen.
Shock waves cause the creation and destruction of chemical species in the ISM. Through the condensing effects of a weak shock, material becomes dense enough to cool and form molecules. Slightly stronger shocks will ionize certain species facilitating ion-neutral pathways to molecule formation. This denser material on a larger scale also leads to gravitational collapse and star formation. Stretching the analogy a bit further, this is the planting of cosmic seeds.
Stronger shocks, of course break up molecules and cause ionization and background radiation across the entire spectrum (some of which will cause further damage). Tracing the chemical evolution of a dense molecular cloud, we see that current conditions match those predicted for an age of about 100,000 years but no further. If the cloud is shocked with about that period, it will reset the evolution so that we never see older evolutionary stages. This is the cosmological equivalent to weeding and pruning.
Hence, Cosmic Gardeners seems like an appropriate title for shocks, anthropomorphic as it is. They are partially responsible for all stages and scales of development within the ISM.
The command line arguments were:
latex2html shock.tex.
The translation was initiated by Charles Danforth on Fri May 15 11:19:40 EDT 1998
