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Thursday, May 7, 2020

Shock compression ratio

The derivation of the shock compression factor.

ρv=const
ρv2+P=const
ρv(12v2+γγ1Pρ)=const
Let 1(2) to indicates the shock upstream (downstream) quantity.
The compression factor is defined by 
r=ρ2ρ1=v1v2.
P2=ρ1v21+P1ρ2v22
P2ρ2=P1ρ2+ρ1v21ρ2v22=P1ρ1r+v21r(v1r)2

12v21+γγ1P1ρ1=12v22+γγ1(P1ρ1r+v21r(v1r)2)
multiply by 2v21,
r2(1+1γ12M2)2γ1(1M21+γ)r+2γγ11
It is easy to verify that r1=1 is one of the solution. Thus, we can write the equation as 
(r1)(rr2)=r2(r21)r+r2
Comparing with equation *, we can get
r2=(2γγ11)/(1+2(γ1)M21)=(γ+1)M21(γ1)M21+2

cosmic rays modified shock


Three regions:
(0) far upstream, the presence of accelerated particles can be neglected
(1) in the shock precursor just upstream of the subshock
(2) behind the shock
ρ0v0=ρ1v1=ρ2v2
v0=Vs is the velocity of the gas with respect to the shock. Vs is the shock velocity.

Momentum flux conservation
P0+ρ0V2s=P1+ρ1v21=P2+ρ2v22
P=Pth+Pcr is the pressure.

Mach number for the shock structure far upstream
M201γgρ0V2sP0

Compressive ratio across the different regions 0, 1, and 2.
ξ1=ρ1ρ0,ξ2=ρ2ρ1,ξ12=ξ1ξ2=ρ2ρ0

Assuming the thermal pressure in the precursor in only due to adiabatic heating in the precursor, P1,th=P0ξγg1 and across the sub-shock the pressure associated with the accelerated particles does not change.
The cosmic ray terms cancel each other when considering the momentum flux from region 1 to 2, P1,cr=P2,cr.

M21=ρ1v21γgP1=ρ0V2s/ξ1γgP0ξγg1=M20ξ(γg+1)

P2ρ0V2s=P2,cr+P2,thρ0V2s=1γgM20+11ξ12

P2,thρ1v21=ξγg+11γgM20+11ξ2

The compression ratio of the sub-shock is given by the standard shock relation:
ξ2=(γg+1)M21(γg1)M21+2

ξ12=M2γg+10[ξγg2(γg+1)ξγg+12(γg1)2]1γg+1


Pth1Pw1=pth1pth0Pth0Pw1
Pth0Pw1=Pth0ρ0u20/Pw1ρ0u20

I will update later...  
Burlaga, L. F., Ness, N. F., Gurnett, D. A., & Kurth, W. S. 2013, ApJL, 778, L3

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