This note addresses the derived parameters found in the
bowshock records and magnetopause records for home pages:

http://ftpbrowser.gsfc.nasa.gov/bowshock.html

http://ftpbrowser.gsfc.nasa.gov/magnetopause.html 

Consider first the multi-species nature of the solar wind
plasma: protons, alphas, electrons.  We use subscripts p, a and
e for these.  N is density, T temperature, V flow speed, m mass

Let Na = f*Np 
    Ne = Np + 2*Na = Np*(1+2f)

Mass density   = mp*Np + ma*Na + me*Ne
               = mp*Np + 4*mp*f*Np
               = mp*Np * (1+4f)

Thermal pressure  = k * (Np*Tp + Na*Ta + Ne*Te)
                  = k * (Np*Tp + f*Np*Ta + (1+2f)*Np*Te)
                  = k*Np*Tp * [1 + (f*Ta/Tp) + (1+2f)*Te/Tp]

Flow pressure  = Np*mp*Vp**2 + Na*ma*Va**2 + Ne*me*Ve**2
               = Np*mp*Vp**2 + f*Np*4*mp*Va**2
               = Np*mp*Vp**2 * [l + 4f*(Va/Vp)**2]

Rewrite:

Mass density         = C*mp*Np
Thermal pressure     = D*Np*k*Tp
Flow pressure        = E*Np*mp*Vp**2

Where
                   C = 1+ 4f
                   D = 1 + (f*Ta/Tp) + (1+2f)*Te/Tp
                   E = 1 + 4f*(Va/Vp)**2

Now, some issues.

1. f is typically in the range 0.04-0.05, although there are
   significant differences for different flow types.

2. Ta/Tp is typically in the range 4-6.

3. What about Te?  Feldman et al, JGR, 80, 4181, 1975 says that
   Te is almost always in the range 1-2*10**5 deg K. Te rises and
   falls with Tp, but with a much smaller range of variability.
   Kawano et al (JGR, 105, 7583, 2000) cites Newbury et al (JGR,
   103, 9553, 1998) recommending Te = 1.4E5 based on 1978-82 ISEE
   3 data. So we'll use Te = 1.4E5 deg K for our analysis.

4. What about (Va/Vp)**2? We should probably let this be unity always.

If  we let f=0.05, Ta=4*Tp, Va=Vp, and Te=1.4*10**5, we'd have

C = 1.2
D = 1.2 + 1.54E5/Tp
E = 1.2

Characteristic speeds:

Sound speed = Vs = (gamma * thermal pressure / mass density)**0.5
                 = gamma**O.5 * [D*Np*k*Tp /C*mp*Np]**0.5
                 = gamma**0.5 * (D/C)**0.5 *(k*Tp/mp)**0.5

With the above assumptions for f, Ta, Va, and Te, and with gamma = 5/3, we'd get
             Vs (km/s) = 0.12 * [Tp (deg K) + 1.28*10**5]**0.5

Alfven speed = VA   = B/(4pi*mass_density)**0.5
                    = B/(4pi*C*mp*Np)**0.5

With the above assumptions, we'd get
         VA (km/s) = 20 * B (nT)/Np**0.5

Magnetosonic speed Vms = [(VA**2 + Vs**2)/(1+(VA/C)**2)]**0.5
Since C=speed of light in this expression, VA/C <<< 1,
So Vms**2 = VA**2 + Vs**2
But please check here also!

Mach numbers: 

Sonic: V/Vs
Alfven: V/VA
Magnetosonic: V/Vms

Plasma beta:

Plasma beta = thermal energy density (= thermal pressure) /magnetic energy density
    = D*Np*k*Tp*8pi/B**2

With above assumptions, we'd get

Beta = [(4.16*10**-5 * Tp) + 5.34] * Np/B**2 (B in nT)

Flow pressure

The flow (ram) pressure is E*Np*mp*Vp**2

With above assumptions, we'd get
        FP = (2*10**-14)*Np*Vp**2 (N in cm**-3, Vp in km/s; FP in
dynes/cm**2)

Converting units, this becomes
        FP = (2*10**-6)*Np*Vp**2 nPa (N in cm**-3, Vp in km/s)

Shock strength

Shock strength is defined as N (downstream) / N(upstream)

IMF Clock and Cone Angles

We'll provide the cone angle as the arc-cotan of the abs value of Bx over Btotal.
This assumes the cone angle's value is just in measuring the extent of non-radialness of
the IMF. We'll provide the clock angle as the arc cotan of Bz
over Bt, or clock angle = 0 for IMF due north and 180 for IMF due south.

Joe King, 2002