Detailed Derivation of Parameters

This note addresses the derived parameters found in the OMNI data at http://omni.sci.gsfc.nasa.gov//ow.html and http://omniweb.sci.gsfc.nasa.gov/ow_min.html, and in the bowshock records and magnetopause records at: http://omniweb.sci.gsfc.nasa.gov/ftpbrowser/bowshock.html and http://omniweb.sci.gsfc.nasa.gov/ftpbrowser/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)**2Now, 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 = 1.2Characteristic 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 see special note on magnetosonic speed below.Mach numbers:Sonic: V/Vs Alfven: V/VA Magnetosonic: V/VmsPlasma 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 pressureThe 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 strengthShock strength is defined as N (downstream) / N(upstream)IMF Clock and Cone AnglesWe'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, 2002Special 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, 6/2/2005Special note on Mac numbers (added 2016)The OMNI data set does not have the benefit of individual bow shock crossing fit results. Nor it is possible as the bow shock changes its speed rapidly while the spacecraft are away from it. In general, the shock speed should go into the Mach number calculation, necessity forces us to make an assumption. Since the bow shock generally stays in front of Earth, assuming that it is stationary is correct for long-term averages. With a stationary bow shock, the Mach number simplifies to the one used as it is described above for OMNI. Adam Szabo, 02/04/2016

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