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Appendix: Alfvénic Travel Time in the Magnetosphere

This travel time of an Alfvén wave from one point to another on the same field line is
\begin{displaymath}
\tau = \int_{s_{1}}^{s_{2}} \frac{ds}{v_{A}}
= \int_{s_{1}}^{s_{2}} \frac{\sqrt{\mu_{0} \rho (s)}}{B(s)} ds\end{displaymath} (2)

In the following calculations we will adopt the model of a hydrogen plasma with a density distribution that varies as
\begin{displaymath}
\rho = m_{p} n = m_{p} n_{0} \left( \frac{r_{0}}{r} \right)^{m}\end{displaymath} (3)
where r0 is the geocentric distance to the equatorial crossing point of the field line under consideration, and n0 is the proton number density at r0. Since $ r = r_{0} \cos^{2} \theta $for a dipole field line, the number density varies along the field line as $ n = n_{0} \sin^{-2 m} \theta $.The background magnetic field varies along the field line as
\begin{displaymath}
B(r, \theta) = \frac{mu_{0} M}{4 \pi {r_{0}}^{3}}
 \frac{\left( 1 + 3 \sin^2 \theta \right)^{1/2}}{\cos^{6} \theta}\end{displaymath} (4)
where M is the Earth's magnetic moment $ 8 \times 10^{22} \rm\;A\,m^2 $.

Consider an Alfvén wave that travels from the spacecraft location following the field line to the ionosphere and travels back through the same path. The latitude at the foot of the field line on the ionosphere is $ \theta_N = \cos^{-1} \sqrt{1/L} $ or $ \theta_S = - \cos^{-1} \sqrt{1/L} $depending on the ionosphere of concern. Therefore the travel time is

where $ ds = \sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2} d\theta $, or
\begin{displaymath}
\tau_S = 1.9 \times 10^{-5} {n_0}^{1/2} L^4
 \int_{\theta_S}^{\theta_{SC}} \cos^{7-m} \theta d\theta\end{displaymath} (5)

It should be noted that the above calculation can only be considered as a rough estimate of the Alfvénic travel time. First of all, the magnetospheric field at larger L shells deviates from the dipole model. Another reason, which might be even more serious, is that the power law density model as used in the calculation is a crude estimation. The m value in the density model can range from 0 to 6 [cf. Cummings et al., 1969]. For m = 3, (A4) or (A5) can also be written as
\begin{displaymath}
\tau = 1.9 \times 10^{-5} {n_0}^{1/2} L^4
 \left[ \frac{3 \t...
 ...)}{4}
 + \frac{\sin(4\theta)}{32} \right]_{\theta_1}^{\theta_2}\end{displaymath} (6)

We are indebted to C. A. Cattell at University of Minnesota, Minneapolis, for providing the ISEE 1 electric field data. One of us (P. J. C.) is grateful to M. G. Kivelson, R. J. Strangeway, and G. Le at IGPP, UCLA for their helpful comments. This work is supported by the National Aeronautics and Space Administration under grants NAGW-3974 and NAG 5-3171, and by the National Science Foundation under grant ATM 96-23163.

The Editor thanks W. J. Hughes and K. Takahashi for their assistance in evaluating this paper.




 
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\begin{planotable}
{ccc}
\tablecaption{Number of Poynting Flux Samples
in the No...
 ...pace & 59 & 51 \nl
Magnetic latitude $ < 0\deg $\space & 17 & 14\end{planotable}


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Next: About this document ... Up: Phase skipping and Poynting Previous: Conclusions