SOLAR STORM INDEX
Jul 27 2002
Clifford E Carnicom
An index method has been developed to estimate the expected level of solar activity. The method depends upon knowledge of the speed of the solar wind and the proton density. Both of these values are readily available from the home page of www. spaceweather.com, now linked into this site. The index is hopefully relatively easy to use, and should be able to provide the user with an estimate of solar activity and the potential subsequent influence upon the earth in terms of solar storms, auroras, and magnetic disturbance. The effects upon the earth would be expected approximately 3-4 days after solar flares and coronal mass ejections occur and as they may be indicated through this index value. A value of zero or less indicates solar activity is normal and a value of approaching 100+ indicates high solar activity.
The index given at this time is:
SSI = ((v^{2} * n) - 64) * ((((-sinh^{-1}(Bz)) / 3) + 1) / 2) (note : recently added term of Bz to be explained further at a later date).
where v is the solar wind speed in hundreds of thousands of kilometers per second (e.g., 451 km/sec has v as 4.51) and n is the proton density in protons per cm^{3}and Bz is the component of the interplanetary magnetic field in nT.
Representative values of the solar wind speed and the proton density would be 420 km/sec and 4 protons / cm^{3}. This would lead to a solar storm index (SSI) value of:
SSI = ((4.2^{2} * 4) - 64)
SSI = 7 (lower solar storm threat)
As a counter example, a solar wind speed of 450 km/sec and a proton density of 8 protons /cm3 would lead to an index value of:
SSI = ((4.5^{2} * 8) - 64)
SSI = 98 (high solar storm threat)
The first example will be indicative of low solar activity and the latter of much higher activity, where alertness for solar effects upon the earth is warranted.
It is repeated that the values of the solar wind speed and the proton density are readily available at the top left of the home page of www.spaceweather.com.
As an example of an application of the index, on Jul 27 2002, the solar wind was reported at 458 km/sec. and the proton density was 7.8 protons /cm^{3}, leading to an index value of 100. A solar flare of Class M (see spaceweather.com for descriptions and hazards) was reported on Fri Jul 26 2002 at 2100 UT. Auroras are expected to occur from this event.
This work is subject to revision as it is examined under varying conditions.
The work above is based upon the balance, or equilibrium point between the force of the solar wind against the extent of the magnetosphere that surrounds earth. This balance point can be represented by the equation:
.5 * n * m * v^2 * 2 * u = (3E-5 / R^3)^2
(See Plasma Dynamics by R.O. Dendy, Oxford Science Publications, 2000, Problem 4.2 for a further discussion on this result.)
where n is the proton density per cubic meter, v is the speed of the solar wind in meters per sec, m is the mass of a proton, u is the magnetic permeability of free space, and R is the number of earth radii where the balance point is achieved.
Given:
m = 1.67E-27kg
u = 4E-7 * pi H / m
and given representative values for v and n of:
m = 4E6 protons / m^3
v = 4E5 meters / sec
we can solve for R, and expected value of the extent of the magnetosphere under normal solar conditions.
This leads to a R value of:
R = 3.107E-2 / (.5 * n * m * v^2 * 2 * u) ^ (1 / 6)
or
R = 9.35 (approx).
Given that the approximate diameter of the earth is 4000 miles, we can estimate the extent of the magnetosphere under normal solar conditions as approximately (9.35 * 4000) = 37,400 miles. A value given in The New Solar System, Sky Publishing, 1999 is 64000km, or 39,700 miles. This represents a reasonable agreement on the expected value of the normal range of the magnetosphere. A commonly reported value is approximately 10 earth radii.
Next, let us look at a case of extreme solar activity that is reported in the literature. In the book entitled Storms from the Sun, by Carlowicz 2002, reference is given a series of extreme solar events that took place in April and May of 1998. It is reported that sensors detected that the extent of the magnetosphere had been reduced to 15,300 miles as a result of these solar storms. For the time being, taking the above as an example of an extreme case, we can evaluate the reduction in R and suggest a corresponding range of combined solar wind and proton density to produce this result.
Using the above example of extreme solar activity, this results in an approximate R value of:
15300 miles / 4000 miles = 3.82
We are therefore interested in the conditions of the ratio of 3.82 to 10, and to examine the combined values of v and n that are expected to occur under the extreme conditions.
Forming this ratio:
3.107E-2 / (.5 * n2 * m * v2^2 * 2 *u)^(1 / 6)
______________________________________ = .38
3.107E-2 / (.5 * n1 * m * v1^2 * 2 *u)^(1 / 6)
where n2 and v2 represent the proton density and solar wind speed under normal conditions, and n1 and v1 represent the proton density and solar wind speed under an extreme condition, we have:
[ 3.107E-2 / (.5 * n2 * m *v2^2 * 2 * u)^ (1/6) ] * [ (.5 * n1 * m * v1^2 * 2 *u)^ (1/6) / 3.107E-2]
or
(n1 * v1^2) / (n2 * v2^2) = .38
or
(v1 * n1^(1 / 2) ) / (v2 * n2^(1 / 2)) = .62
or v2 * n2^(1 /2) = (v1 * n1^(1 / 2)) / .62
and since
v1 * n1^(1 / 2) = 4E5 * (4E6)^(1 / 2)
then
v1 * n1^( 1 / 2) = 8E8 (approx.)
then v2 * n2^ (1 / 2) = 8E8 / .62 = 1.3E9 (approx.)
We can now form two constants that set limits between the extreme and normal solar conditions:
Extreme |
Normal |
vE5 * (nE6)^(1 / 2) = 1.3E9 v^2E10 * nE6 = (1.39E9)^2 v^2 * nE16 = (1.3E9)^2 v^2 * n = (1.3E9)^2 / 1E16 v^2 * n = 169 |
vE5 * (nE6)^(1 / 2) = 8E8 v^2E10 * nE6 = (8E8)^2 v^2 * nE16 = (8E8)^2 v^2 * n = (8E8)^2 / 1E16 v^2 * n = 64 |
where v of the solar wind is in hundreds of thousands of meters /sec and n is the number of protons / cm^3. These last modifications in units have been made to simply the final form of the index equation.
The controlling variables in the final results, therefore, involve the product of the solar wind and the proton density. If we rescale the index to range between 0 and 100 for normal to extreme condtiions, we can create an linear relationship by forming the equation:
SSI = ((v^2 * n) - 64) / (169 - 64) (solar storm index)
or
SSI = (((v^2 * n) - 64 ) / 105) * 100 (in percent terms)
and since 105 / 100 = approximately 1 accomodating a 5% error term
we can simplify the estimate of the solar storm index to:
SSI = ((v^2 * n) - 64 ) as stated originally.
This relationship is understood to be an approximation only and it is for the purpose of consolidating primary solar activity measurements into a single index for evaluation.
The additional term of Bz will be explained further in the future.
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