Lecture #6b-- supernova blast waves



1)Phase I: ballistic trajectory of ejecta through ISM
Here the speed is essentially constant, and the ejecta is not driven or forced, it just glides under its own inertia, plowing through the ISM and picking up ISM material. The particle velocities are essentially purely radial.
Transition: as ISM material is picked up, momentum is shared across a velocity difference (the difference between the supernova ejecta at about 10,000 km/s and the stationary ISM). Whenever that happens, bulk flow kinetic energy is lost (work it out). That energy either goes into some potential energy storage, or more often, into heat. The heat raises the temperature of the ejecta until it picks up a significant nonradial velocity component. That happens when a significant fraction of the kinetic energy is thermalized, which happens when the ISM mass swept up is of order the mass of the ejecta. This is the transition into the "energy conserving" or "Sedov" phase.
2)Phase 2: Sedov phase
By this point, the velocity of the ejecta particles is no longer radial. As such, a significant fraction of the supernova energy is thermalized, and the particles can move across the expanding bubble, depositing momentum in the swept-up ISM material at several places along the shell boundary. This means that the ejecta momentum, treated as a scalar flux, is not conserved, and is essentially a useless quantity. However, the supernova energy is still conserved, because the high temperature of the shocked gas keeps it from radiating effectively (it is highly stripped of the effective coolant: bound electrons). So the thermalized energy is retained in the bubble, and generates a pressure. This pressure is primary responsible for the continued expansion of the bubble.
(The bulk flow kinetic energy of the ejecta continues to be important as well, but in the Sedov phase, the bubble expands in a self-similar way, so the fractional importance of the driving pressure in the bubble, compared to the bulk flow kinetic energy of the ejecta, remains the same. Thus for the purposes of getting the scaling laws, we do not need to know what that fraction is, it will scale with the pressure driving -- and it turns out the pressure driving is weakly dominant anyway.)
To determine the scaling laws for the characteristic parameters (the "outer scales" of the bubble), we simply assert that the pressure times the bubble volume is constant, as that conserves energy (pressure being proportional to energy density). Further, we connect the shell speed to the pressure by going into the frame of the shell and noting that there is a flux of momentum coming in from the swept-up ISM equal to the ram pressure, which is density times the square of the speed. Equating that to the pressure gives that the shell speed is the square root of the pressure over the ISM density. The latter is given, the former is inversely proportional to the bubble volume, so the shell speed is inversely proportional to the square root of the volume. Given that speed is the volume to the -2/3 power times the rate of change of volume, this means that the rate of change of volume scales with volume to the 1/6 power. That can be integrated to give that the volume scales with time to the 6/5 power-- the key scaling of the Sedov phase.
Transition: in the Sedov phase, the shell speed scales like time to the -3/5 power (work it out). So it constantly slows, being at about 500 km/s at a time of about 10,000 years post supernova. As the speed drops further, the shocked temperatures get less and less, and the shocked ISM gas does not get stripped of bound electrons and cools more and more effectively. Eventually the thermalized bulk flow kinetic energy is dissipated into radiation, which escapes and does not get fed back into the energy density in the bubble.
3) The "snowplow" (or adiabatic) phase
When the shell is radiative, the bubble that is driving it becomes effectively cut off from heating by the thermalized bulk flow energy. At this point, the bubble expands adiabatically, which for some reason gets called the "snowplow" phase. This is just like the Sedov phase, and is also expected to unfold self-similarly, but instead of the pressure being inversely dependent on the volume, it is inversely proportional to volume to the gamma power, where gamma is the ratio of specific heats (5/3 for monatomic ideal gas). This is the hallmark of adiabatic expansion, and note that it does not conserve energy, even though there is no heat exchange, because the expanding gas does work and the work turns into radiation rather than being fed back into the bubble as heat (as in the Sedov phase). Making this small adjustment to the behavior of the pressure, and taking gamma=5/3, then implies that the rate of change of bubble volume scales like volume to the -1/6, not 1/6 as in Sedov, so the volume acquisition starts to slow down for the first time. This also means that the volume scales like time to the 6/7 power.
Transition: the bubble pressure drops like time to the -10/7 power, until eventually it is no longer a significant push on the shell.
4) The scalar momentum-conserving phase
The bubble has been expanding and cooling all this time, and when it gets cool enough to have lots of bound electrons, it will start to radiatively cool, like the shell has been doing by this point. This will cause the bubble temperature to also drop, like the shell did, and the internal pressure in the bubble will become negligible. That means the ejecta particles are again effectively streaming radially, as they were originally, but now there will never be any appreciable thermalization of the kinetic energy-- it will just radiate away. Thus we can analyze each solid-angle sector independently, simply conserving momentum like blocks on an air track that stick together. In this phase, the speed is inversely proportional to the swept-up mass, so is inversely proportional to the volume of the bubble. That means the rate of change of volume scales like volume to the -1/3 power, or the volume scales like time to the 3/4 power.
5) Energetics
After the Sedov phase ends, the remnant begins to lose energy via radiation, and this is the final conversion channel for the majority of the supernova energy. It takes tens of thousands of years to radiate away, rather than the few weeks of the initial supernova, so even though it represents perhaps 100 times more energy than the initial light pulse, the radiating shell will have a total luminosity that is thousands of times less than the peak luminosity of the supernova. However, this radiation will mark the end of the supernova kinetic energy. Hence the net impact on the ISM is primarily through the creation of this ionizing radiation, and through the copious scalar momentum (i.e., not vectorially canceled) it has deposited as "stirring" of the ISM at the level of the sound speed.