Developing a model to describe the position of the ring found in the simulations

Model 1 - Normal shock (head on collision case), ignore magnetic fields

-Ignore one side of the flow
-M=1.5 flow creates a shock
-Incoming flow passes through this shock and acquires higher pressure, temperature, density, while slowing down
-The equations for this are derived by considering mass, momentum, and energy conservation on either side of the shock
-Making some simplifications (i.e. flow is adiabatic, etc.) yields normal shock conditions
-These give post shock variables as functions of upstream mach and gamma
-Assuming gamma=1.4 (which note, represents air at sea level), can look up values in normal shock table for ease of calculation. These are to good approximation for our case, gamma = 5/3

Thus, for M=1.5, get the following ratios of upstream (1) to downstream (2) fluid variables:

P_2/P_1	2.45
\rho_2/\rho_1	1.86
T_2/T_1	1.3
M_2	0.7
a_2 = \sqrt{(\gamma K_B T_2)/mH}
v_2= a_2 M_2

From this can get the post shock velocity of the flow. We can approximate this as the radial velocity of the splash. From this we can get a ram pressure of the splashed material:

P_{ram} = \rho_2 v_2^2

We can then look to where this outward ram pressure equals the pressure of the ambient medium. The ambient medium is not in HSE, so it will begin to fall inward toward the cylinder. We can approximate this as uniform collapse, which will specify density and velocity as a function of distance away from the cylinder, thus giving a ram pressure of the ambient medium. At that radius where these ram pressures are equal, we can expect a ring will build up.

Here are the shock parameters (1 is pre-shock, 2 is post-shock), in physical units (cgs):

gamma	5/3
kB	1.38*10-16
G	6.673*10-8
mH	1.67*10-24
M1	1.5
M2	0.7
T1	4931
T2	7370
rho1	1.67*10-24
rho2	2.86286 * 10-24
Cs1	733530
Cs2	1.00754*106
v1	1 100 376
v2	641 920
pram2	1.48*10-12

Uniform Collapse of the ambient:

Needed to revisit these solutions to develop the model of the ambient infall. This can be followed ​here.

Assuming the ambient is a uniform density sphere of radius = half the largest box size in the shear 0 case,

r0_{max} = 37.5 ~pc = 1.16*10^{20} ~cm

and density,

\rho0 = 1.67*10^{-24} ~g/cm^3

Here is how the different shells of this sphere fall in over time:

Here I only cared about the shells with radii between the collision region and the simulation box. The collision region was 40 pc across, so decided to track interior shells down to 20 pc in radius. From this graph, you can see the freefall time of the ambient is about t=51 Myr. We only ran the simulation until 27 Myr, which is the red vertical line.

Here is how the different shells speed up over time, in this velocity plot:

These curves are the velocity of a given shell (i.e. the outer radius of the various concentric spheres that make up the ambient medium). If that is confusing, ​see this page. The shell that has the highest velocity is the largest radius sphere (i.e. r=box size), and the shell with the smallest velocity is the sphere with radius = ½ collision region.

Here is the density of the sphere over time as it collapses (the density is increasing everywhere in the sphere at the same rate, hence the sphere continues to have a uniform density over time):

With these last two data sets, here are ram pressure radial profiles. Now the different curves represent time, and are only plotted from t = 0 to t = tsim:

Now, the post shock density is comparable to the density of the infalling ambient in this time range. However, the velocities are very different, as they should be in retrospect. This leads to large deviations between the ram pressure of the infalling ambient and the ram pressure of the splashed material. So large in fact (3 orders of magnitude), that the ram pressure of the splashed material doesn't make it on this plot; Pram = 1.5*10^-12 for the post-shock flow.

The incoming flows have a very high speed, and once shocked, though decreased, is still high. The ambient, however, is starting from rest. This means, that by the time the entire simulation has completed (i.e. long after the ring has formed), the ambient has still not acquired enough momentum to make the ram pressures balance.

In hindsight, this was always going to be the case, as the ring is not present in the hydro case, and so the ambient's own dynamics are too weak to constrain outflowing material from the collision region. Next need to include magnetic fields in the ambient pressure.

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Model 2 - Assume ambient is stationary, and outflowing material is a parker type wind

*Note - can choose which is dominant by considering time scales — freefall time compared to radius of splashed/ v_2, for instance

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Model 3 - Reflected shock, hydro case

-Same as above, but now treat the shock as being reflected off of a wall (i.e. the opposing flow)

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Model 4 - Include magnetic pressures