GLE definition
The Ground Level Enhancements (GLE) events are caused by the relativistic solar cosmic ray effect
on the ground based detectors (mainly neutron monitors). The solar cosmic rays consist basically of
protons. And the particles responsible for the GLE are the relativistic solar protons (RSP).
Their energy usually is of order of 1-10 GeV. In rare occasions it can exceed 20 GeV.
Our real-time GLE modeling technique
The parameters of RSP can be determined from the data of the ground based neutron monitor network,
with the help of a GLE modeling technique. Such a modeling includes the next steps:
1. Definition of asymptotic viewing cones of the NM stations under study by the particle trajectory
computations in a model magnetosphere.
2. Calculation of the NM responses at variable primary solar proton flux parameters.
3. Application of a least square procedure for determining primary solar proton parameters
(namely, energy spectrum, anisotropy axis direction, pitch-angle distribution) outside the magnetosphere
by comparison of computed ground based detector responses with observations.
The GLE modeling technique was presented in [1]. The improved version of [1] was presented in [2],
which took into account the contribution in the NM response not only vertical but oblique incident particles.
The account of oblique incident particles is made in our version of the GLE modeling technique [3] too.
It is necessary to note, that for real-time of definition of RSP parameters it is required as much as
possible simplified technique, which, nevertheless, would give exact enough results. Taking into account
such limiting factors, as the limited number of available stations at the moment (~20 versus 35-40 in case
of [1-3]), limiting time of calculations, etc., we suggest a simplifying version of our modeling technique
[3] which can be used for real-time conditions. First of all, we are limiting computations with accounts
only vertically incident particles. And the asymptotic directions of viewing for them are calculated in real
time with the Tsyganenko 1989 magnetosphere model with the Kp index of a current geomagnetic activity
as a parameter. The time of one set of 25 NMBD stations asymptotic cones (AC) computations takes about 3 min.
The file containing asymptotic directions of viewing for available NMDB stations is forming as one of the
input files of the model.
The task to determine parameters of the RSP flux outside the Earth's magnetosphere using data of the ground
based network of NMs is an 'inverse problem'. The corresponding 'direct problem' (to determine the GLE increase
on a given NM provided primary RSP flux parameters are known) can be expressed by the relation:
(1) |
where ΔN/N is a relative increase of the NM count rate; R is a particles' rigidity; R_{
C} is a cutoff rigidity; J(R) is a rigidity spectrum; F(Θ) is a pitch-angle
distribution. Pitch-angle Θ depends on R because given NM accepts those particles with given R
which had specific Θ outside the magnetosphere. Finally, S(R), Specific Yield Function (SYF,
taken from [4]) is a function determining a NM response to RSP flux at a given R.
For computer calculations we substitute the formulum (1) with its discrete analogue:
(2) |
Here a factor A(R) is introduced which is a discrete function A(R)=1 for allowed trajectory (proton with
such rigidity can reach the station) and A(R)=0 for forbidden trajectory. Parameter A is determined
at the asymptotic cone calculations.
In our model we set:
R_{C}=1 GV (atmospheric cutoff rigidity), R_{UP}=10 GV is the arbitrary upper limit of solar
cosmic ray spectrum.
J(R) = J_{0}R^{-γ*} is a modified power rigidity spectrum [2] with variable slope
γ* = γ + Δγ ∙ (R-1); γ is a power-law spectral exponent at R = 1 GV,
Δγ is a rate of γ increase per 1 GV.
F(θ(R)) = exp(-θ^{2}/2σ^{2}) is a Gaussian pitch-angle distribution. Anisotropy
axis is defined by two angles φ and λ which are GSE latitude and longitude.
J_{0} | intensity of the flux, (m^{2} s ster GV)^{-1} | |
γ | power-law spectral exponent at R = 1 GV | |
Δγ | rate of γ increase per 1 GV | |
σ | Gaussian parameter of the pitch-angle distribution, deg | |
φ, λ | GSE latitude and longitude of anisotropy axis, deg |
(3) |
where P is vector of parameters {J_{0}, γ, Δγ, σ, φ, λ}; (ΔN/N)_{i}
^{mod} is a model relative increase at a given NM station i; (ΔN/N)_{i}^{obs} is an
observed increase at the station. The summation is taken over all NM stations whose data are available.
The optimization quality may be characterized by the value | |
As our practice shows good convergence of optimization process is at D_{Σ} ≤ 5% [4,5]. |