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Reference Notes

Reference Notes

1. In general, the minute given is that in/during which the crossing is observed. For 'extended' crossings, the minute given generally corresponds to the beginning of the magnetopause interval. For multiple crossings in the early IMP data submitted by Fairfield, sometimes an 'average position' was selected by visually determining the point that equalized the magnetosphere time outside the point with the magnetosheath time inside the point; the minute given corresponds to the 'average position'.

2. The field intensity is [Bx**2+By**2+Bz**2]**0.5 (wds 13-15 --> wd 12)

3. The Xgse component of the solar wind velocity vector is negative. To avoid negative values for this component, we have chosen to give components for the vector anti-aligned with the velocity vector.

Further explanations of some equations below are given here

4. Flow pressure = (2.0/10**6) * N * V**2 (wds 23, 18 --> wd 24) (note that this 2.0 includes a 5% alpha contribution)

5. Electric field = V(km/s) * Bz (nT; GSM) * 10**-3

6. Plasma beta = [(T*4.16/10**5) + 5.34] * N / B**2 (wds 22, 23, 12 --> wd 26)

7. The Alfven speed = 20. * B / N**0.5 (wds 12, 23 --> wd 27)

8. The sound speed = 0.12 * [T + 1.28*10**5]**0.5 (wd 22 --> wd 28)

9. Magnetosonic speed = [(sound speed)**2 + (Alfv speed)**2]**0.5 (wds 28, 27 --> wd 29)

10. IMF cone angle = 57.3 * arc cos [|Bx|/Bt] (wds 13, 12 --> wd 33)

11. IMF clock angle = 57.3 * arc tan [|Bygsm|/Bzgsm] (wds 16, 17 --> wd 34)
Detailed Derivation of Parameters

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

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Magnetopause Database

List of Providers
The magnetopause data base was created using data from various providers. To distinguish among the various providers, we added an additional word (word 39) in each record.
Word 39 Provider
1D. Fairfield
2N. Crooker,
G. Paschman,
G. Haerendel
3J. Slavin
4T. Eastman
5A. T. Y. Lui
6R. Rijnbeck
7G. Zastenker
8T. Eastman,
S. Christon
9H. Hasegawa
10D. G. Sibeck
12S. Boardsen,
T. Eastman
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