From Decay Times to Charge States:
A curious character’s journey through the lands of models and data analysis
Luke Sollitt
In my analysis of the ACE/SIS data, I have noticed several solar particle events
where the characteristic e-folding timescale of the decay phase depends on energy and
particle species. This is not a new phenomenon: similar energy dependencies were seen
by Lupton (1973), Lupton and Stone (1973), and a boatload of others over the years (lots
of references). It is possible that this energy and species dependence – really a velocity
and rigidity dependence – could be used as a tool to study properties of solar energetic
particles (SEPs) and the interplanetary medium. In particular, one might be able to use
this dependence to determine average ionic charge states of SEPs. The purpose of this
essay is to explore how one might be able to use the velocity/rigidity dependence of SEP
decay timescales to determine average ionic charge states.
For the last 40 years or so, various efforts have been made to accurately describe
the time intensity and anisotropy profiles of solar particle events. Large SEP events
generally follow similar time intensity profiles: a rapid (~ hours) rise, followed by a slow
(~ days) exponential or quasi-exponential decay (references). There are several different
models to describe this phenomenon (Dalla et al, 2002). I am, for reasons that escape me
at the moment (as well as the fact that it describes the phenomenon fairly well and has
done so for decades), concentrating on the model of interplanetary diffusion. What is
sought in examining diffusion models is a dependence of the decay timescale τ on some
parameter that depends on the velocity and rigidity of the particle. It will be found (from
quasi-linear theory) that this dependence will show up in the diffusion coefficient κ.
Early efforts, such as that by Parker (1963) described the phenomenon with an
impulsive injection, followed by diffusion out through a medium. As the years have gone
by, refinements have been made to this model, with such additions as an outer boundary
to the diffusion cavity, convection in the solar wind, and adiabatic cooling (references).
Later refinements are more sophisticated, introducing such notions as focused transport
(references). During the decay phase of a solar particle event, the older models are still
reasonably accurate. Detailed effects, such as those due to focused transport, may be
ignored in this phase of the event: these effects are important only in the event onset.
Two simple models describe the equilibrium decay phase of a solar particle event
reasonably accurately: those of Forman (1971) and Lupton (1973, from his thesis, and
Lupton and Stone, 1973). Both models solve a Fokker-Planck equation for SEP particle
density:
n
1
 

   (  n)    (nV )    V  ( (T )Tn
t
3
 T

Here, κ is the diffusion coefficient, V is the solar wind speed, assumed to be radial, and T
is the particle energy. In this equation, the terms on the right are the diffusive term, the
convection term, and the adiabatic energy loss term. This particular form of the Parker
equation is from the Lupton model.
In both models, many simplifying assumptions are made: all quantities (except for
particle density n) are assumed to be independent of energy; the solar wind speed V is
assumed to be radial and constant with time; a perfectly absorbing boundary exists at r =
L such that the particle density vanishes at L; some assumptions are made about the
diffusion tensor κ; and Forman assumes that the magnetic field is radial, where it is more
appropriately described by a Parker spiral (reference). The chief difference between the
models is in the form of the diffusion coefficient assumed. Forman assumed a diffusion
coefficient of the form
   0r .
Lupton assumed a diffusion coefficient that was constant with radius. Both models also
assumed that the diffusion coefficient was constant with energy. However, in applying
their models, both Lupton and Forman applied different κ to different energy particles.
As perpendicular (cross-field) diffusion has been shown to be small (reference), only the
radial solution of each model need be considered.
In the Forman model, the dependence on the diffusion coefficient of the
characteristic decay time is given by:
 Forman 
4L
j  
,
2
 ,1
0
where L is the boundary of the diffusive cavity (taken by Forman to be about 2.3 AU), jη,1
is the first nonzero value of x where the Bessel function Jη(x) goes to zero, and η is given
by
1
2
2

V  2CV 
 
 .
  21 
0 
 2 0 

Here, V is the solar wind speed, and C is the Compton-Getting factor, which is
related to the spectral index.
The similar expression in the Lupton model is given by:
 Lupton 

,
4   V 2
2
where once again, V is the solar wind speed, and α solves the boundary condition
F0  
,  L   0 ,
2



where F0 is a Coulomb wave function, and β (which is not particle speed) is given by

V 2C  1

.
Once again, C is the Compton-Getting factor.
A third and earlier model, by Burlaga (1967), neglected the terms associated with
the solar wind speed: that is, convection and adiabatic cooling were neglected. In this
model, one finds
 Burlaga 
L2
 2
.
At high κ, the terms in η in the Forman model involving κ become small, and η
tends to a value of 2. Thus, the jη,1 turn into j2,1 = 5.135, and the decay time is inversely
proportional to the diffusion coefficient. Likewise, in the Lupton model, at high κ, β
(which is inversely proportional to κ) tends to zero, and α tends to the constant value of
2.0944. For large enough κ, the solar wind speed V can be neglected, and once again, the
decay time will be inversely proportional to the diffusion coefficient. This is also the
result from the Burlaga model.
At lower energies, these solutions break down. The Burlaga model neglects
convection and adiabatic cooling altogether. Both the Lupton and Forman models predict
small diffusion timescales at low energies, which conflicts with observations (various
references – notably the Reames stuff). Instead, in both models, the diffusion term must
be dropped from the Parker equation, and a new solution obtained. For both models, the
convection/cooling timescale τC obtained this way is given by:
C 
r
2CV
where once again, V is the solar wind speed, and C is the Compton-Getting factor. Here, r
is the distance of observer from the Sun: for my data, this is 1 AU.
The work of Reames and friends (references) at lower energies with invariant
spectra supports the notion of a single convective time scale, though various other
mechanisms are invoked to account for it. (need more on this)
In effect, one sees two superposed decays for the flux f: a convective and/or
cooling decay that dominates at low energies (or rigidities), and a diffusive decay that
dominates at high energies (rigidities).