This note addresses the derived parameters found in footnotes 4-11.

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 see the Special note on magnetosonic speed below.

Mach numbers:

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

Plasma beta:

Plasma beta = thermal energy density /magnetic 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-cocos 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 the abs value of By over Bz, or clock angle = 0
for IMF due north and 180 for IMF due south.

Joe King, 2002

**Special note on magnetosonic speed (added 2012)**
The definition of magnetosonic speed (Vms) used above is not the
most generic definition thereof. The generic (non-relativistic)
definition of Vms (for "fast mode") is given by (e.g., Merka et al,
JGR, Feb 2003).
Vms**2 = 0.5 * {VA**2 + Vs**2 + [(VA**2 + Vs**2)**2 -
4 * VA**2 * Vs**2 * (cos(theta))**2]**0.5}
(Note to reader: For "slow mode," replace the "+" immediately preceding
the [...] term in the above expression with a "-".)
In this expression, VA and Vs are the Alfven and sound speeds,
and theta is the angle between the wave propagation direction and
the ambient magnetic field. In the case of wave propagation normal to
the magnetic field vector, cos(theta) = 0, and the expression reduces
to Vms**2 = VA**2 + Vs**2, which, as indicated above is what we have
used.
Looked at another way, under the frequent assumption that the
wave propagation direction and the solar wind flow direction are
nearly aligned, theta may be taken to be the angle between
the magnetic field vector and the solar wind flow vector. In this
context, it clear that our databases' "magnetosonic speed" is
actually the magnetosonic speed for the fast-mode wave for the case
of magnetic field vector normal to the solar wind flow vector.
Users may compute "true" magnetosonic speed from the parameters
contained in the data records of this database.
JHK, 6/2/2005