Estimation of the SC propagation time

The following calculation only serves as a rough estimate of the SC propagation because of the many assumptions involved. Consider an impulse at the subsolar point of the magnetosphere results in a propagation of signals in the hydromagnetic fashion. The signal can either propagate across the field lines as the fast mode or along the field as the Alfven mode.

Introduced first are the density model and magnetic field model we use, followed by the calculation of Alfven speed and the signal travel time.
 

Density

Following the model used by Dong-Hun Lee (1996), we divide the magnetosphere into two regions: the outer part and the inner part (plasmasphere). For both regions the density is proportional to r^(-3). The dividing line is L = 5 (Lee uses 5.3).

n_eq(L) = n_mp (r_mp / r)^3        Outer magnetosphere
n_eq(L) = n_pp (r_pp / r)^3            Inner magnetosphere (plasmasphere)

where n_mp = 1 /c.c., r_mp = 10 Re, n_pp = 150/c.c., and r_pp = 5 Re. To avoid distraction, we assume that there are only protons.

The result is shown in the figure below. Black curves are the magnetic field lines for L = 2,4,6,8,10.

Magnetic Field

As tradition dictates a dipole field is used. See the figure below.

Alfven velocity

Having the above two quantities, we can calculate the Alfven velocity in space (the magnetosphere) and the result is shown in the following figure. Notice that the Alfven speed is slower inside the dense plasmasphere.

Signal Travel Time from the Magnetopause to the Earth

The SC signal is (or should be) observed at different latitudes on the Earth. At a particular latitude, or L-shell, we can assume that the signal comes to the Earth along the field line of the L-shell, and before that the signal should have traveled radially inward from the magnetopause to the L-shell radially inward. Therefore we can consider the path of signal propagation consists of two parts:
  1. From the magnetopause to the L-shell of consideration in the radial direction. Consider it takes place on the equatorial plane.

    2.  
  2. From the equator to the ionosphere/Earth along the field line on the same L-shell.
To further approximate the problem, we only consider the cold plasma limit, that is, the fast mode speed is equal to the Alfven speed. The signal travel time can therefore be easily calculated, and the result is shown in the following figure.


Last modified by Peter Chi on 04/21/99.