Shocked Clumps

Common params

namb	250 cc-1
Tamb	100 K
Rclump	100 AU

Run Params

Run	vwind (km/s)	\chi	final time (yr)	Tbow (K)	Ttrans (K)	Lbow (Rc)	Ltrans (Rc)
A	50	10	568.9 yr	5.64e+04	5.64e+03	2.73e-01	7.86e+00
B	100	10	248.4 yr
C	200	10	142.2 yr
D	400	10	71.1 yr	3.61e+06	3.61e+05	1.88e+03	2.11e-01
E	800	10	35.6 yr	1.44e+07	1.44e+06	1.40e+04	2.30e+01
F	200	100	142.2 yr	9.02e+05	9.02e+04	3.93e+01	1.41e+00

Shock Structure

Bow Shock

For the bow shock temperature, we assume that this shock is stationary with respect to the flow in which case the material flowing into the shock will see a shock travelling at v_{wind} so T_{bow}=\frac{3 \mu v_{wind}^2}{16 k_b}

Wind Shock

However, before the wind reaches the clump it first shocks against the stationary ambient and produces a wind shock. To solve for the speed and temperature of this wind shock we first switch to a reference frame in which both the wind and the ambient are colliding at v_{wind}/2

If we shift to this reference frame, we then know from symmetry that the postshock velocity must be 0. Using this along with the pre-shock conditions we can solve for the post-shock density, pressure, and the shock speed by Applying the Rankine-Hugoniot jump conditions at the interface

begin{eqnarray} s(\rho_2-\rho_1)= \rho_2 u_2 - \rho_1 u_1 \\ s(\rho_2u_2-\rho_1u_1)=(\rho_2u_2^2+p_2) - (\rho_1u_1^2+p_1) \\ e_1+\frac{1}{2}u_1^2+\frac{p_1}{\rho_1}=e_2+\frac{1}{2}u_2^2+\frac{p_2}{\rho_2} \end{eqnarray}

Or we can assume that the shock is adiabatic with a high mach number which then gives us that \rho_2 = 4\rho_1 in which case s=-1/3u_1 or in this case s=+1/6v_{wind} or in the ambient frame of reference s=2/3v_{wind}. The backward facing shock would then have velocity s=1/3v_{wind} so the forward shock would be travelling at twice the speed as the backward shock and the shocked region would be as large as the unshocked inflow region - which is visible in the non-cooling bow runs. In any event the velocity of the wind shock is then T_{windshock}=4/9T_{bow}… On a side note, if you run material into a wall at a high supersonic velocity v, the shocked temperature ends up being \frac{16}{9} \times \frac{3\mu v^2}{16k_b}.

Eventually this wind shock slams into the shock and should produce a reflected shock. Using the same analysis we get that the reflected shock temperature within the windshock should be \frac{16}{9} \frac{T_bow}{4}=\frac{4}{9}T_{bow} however we have ignored the fact that this windshock material is already shocked, so this reflected shock would be fairly weak and some of the assumptions used above break down. Eventually this reflected shock breaks out of the windshock and into the ambient where it now sees cold material flowing in at v_wind and forms the bow shock.

Transmitted Shock

And finally we have

T_{trans}=T_{bow}/\chi

Temperature lineout for non-cooling case

Below is a snapshot from this movie​ that shows the evolution of the shock structure along the axis. Also plotted are the temperatures for the bow, wind, and transmitted shocks calculated above.

Cooling Lengths

A naive calculation of L_shock using the instantaneous cooling rate at T_shock would implies that run A is bow only, run D is trans only, run E is neither, and that run F is marginally trans only. However it does appear that run F has both a cooling bow shock and transmitted shock. To understand the discrepancy we should consider the following figure where we have plotted the cooling lengths in units of clump radius as a function of wind velocity for the bow shock as well as for the transmitted shocks with \chi = 10 & 100

Not here we have calculated the cooling length by integrating along the cooling curve until the temperature has dropped by half. To see the same curve but using the instantaneous cooling rate see this image​

If we consider where the blue curve intersects with a wind speed of 200 km/s, we see that the slope of the cooling length as a function of velocity (or temperature) is very high. As the bow shock cools it effectively slides to the left and the cooling length becomes around .1 R_c. This explains the dramatic behavior of the ambient shock in run F. The transmitted shock also follows a similar behavior transitioning from marginally cooling to a cooling length of approx .1 R_c as well.

Run E which is also very curious because while neither the bow shock nor the transmitted shock initially is cooling, the intersection of the transmitted shock with the wrap around shock triggers dramatic cooling and fragmentation. From the intersection of the red line with 800 km/s we see that the transmitted shock is on the same region of the cooling curve as the bow shock was for run F and once it begins to cool the cooling length drops from 10 R_c to .01 R_c.

Images

	A 50 10 (bow only)	D 400 10 (clump only)	E 800 10 (neither )	F 200 100 (both)
	movie​	movie​ zoom​	movie​	movie​ zoom​
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