GREAT FEP EVENTS AND SPACE WEATHER ,
3. AUTOMATICALLY DETERMINATION OF DIFFUSION
COEFFICIENT IN THE INTERPLANETARY SPACE, TIME OF
EJECTION AND ENERGY SPECTRUM IN SOURCE; FORECASTING
OF FEP DEVELOPMENT IN CR NEUTRON INTENSITY
L.I. Dorman 1,2, N. Iucci 3 , M. Murat 4, L.A. Pustil’nik 1, A. Sternlieb 1, G. Villoresi 3, I.G. Zukerman 1
1
Israel Cosmic Ray/Space Weather Center and Emilio Segre’ Observatory, affiliated to Tel Aviv University,
Technion and Israel Space Agency, P.O.Box 2217, Qazrin 12900, ISRAEL
2
Cosmic Ray Department of IZMIRAN, Russian Academy of Science, Troitsk 142092, Moscow Region, RUSSIA
3
Dipartimento di Fisica “E. Amaldi”, Università “Roma Tre”, Rome, Italy;
4Space Environment Division, Soreq NRC, Yavne 81800, Israel
ABSTRACT
In Paper 2 (Dorman et al., 2003b) was described how works automatically the program FEP-Research/Spectrum,
determined the FEP spectrum on the Earth on the basis of on-line one-minute data for intensities of different CR
components (for example  NM data of total intensity and different multiplicities). In the present paper we
consider the next step  more complicated inverse problem: how on the basis of determined FEP spectrum on the
Earth in the different moments in the beginning of FEP event  to determine the diffusion coefficient in the
interplanetary space, time of FEP ejection into solar wind, and energy spectrum in the source of FEP. We consider
several possibilities: 1) only one of these three parameters is unknown, 2) two of these three parameters are
unknown, 3) all these three parameters are unknown. We show that in the first case is necessary to determine
energy spectrum of FEP on the Earth at least in two different moments of time and from one equation automatically
can be determined the unknown parameter (energy spectrum in source or diffusion coefficient, or time of ejection;
determination is made from one equation, and other is used for control of used model). In the second case is
necessary to determine energy spectrum of FEP on the Earth in three different moments of time and from three
equations automatically can be determined two parameters (for example, the energy spectrum in source and
diffusion coefficient in the interplanetary space). In the third case by using data for four different moments of time
can be determined all three unknown parameters (time of ejection, diffusion coefficient in the interplanetary space
and energy spectrum in source of FEP), and one equations can be used for control of model. We describe in details
the algorithms of the programs FEP-Research/Ejection Time, FEP-Research/Source and FEP-Research/Diffusion.
We check obtained results by forecasting of FEP development in total neutron intensity and comparison with
observations. We show how worked these programs on example of historical FEP event of 29 September 1989.
This research is partly supported by EU INTAS grant 00810.
ON-LINE DETERMINATION OF FEP SPECTRUM IN SOURCE WHEN DIFFUSION COEFFICIENT
IN THE INTERPLANETARY SPACE AND TIME OF EJECTION ARE KNOWN
According to observation data of many events for about 60 years (see review in Dorman, 1957, 1978; Dorman
and Miroshnichenko, 1968; Duggal, 1979; Dorman and Venkatesan, 1993; Stoker, 1995) the time variation of FEP
flux and energy spectrum can be described in the first approximation by the solution of isotropic diffusion from the
pointing instantaneous source described by function QR, r ' , t '  N o R  r ' t '. Let us suppose that the time of
ejection and diffusion coefficient are known. In this case the expected FEP rigidity spectrum on the distance r
from the Sun in the time t after ejection will be
N R, r , t   N o R   2 1 2 K R  t 3 2


1
2 

,
 exp  r
 4 K R  t 


(1)
where N o R  is the rigidity spectrum of total number of FEP in the source, t is the time relative to the time of
ejection and K R  is the known diffusion coefficient in the interplanetary space in the period of FEP event. At
r  r1  1 AU and at some moment t1 the spectrum determined in Dorman et al. (2003b) will be described by the
function
N R, r , t   bt  R   t1  D R  ,
(2)
1 1
1
o
where b t1  and  t1  are parameters determined the observed rigidity spectrum in the moment t1 , and Do R  is
the spectrum of galactic cosmic rays before event (see in Dorman et al., 2003b). From other side, the FEP spectrum
will be determined at r  r1 , t  t1 according to Eq. 1; so we have equation for determining N o R 
bt1  R   t1  D o R = N o R  2  1 2 K R t1 3 2


1


 exp  r12 / 4K R t1  .
(3)
If the diffusion coefficient K R  , is known function from R from some other investigations, that from Eq. 3 we
obtain

.
2
3/ 2
N o R   2 1/ 2 bt1  R   t1  D o R  K R t1  exp r1 / 4K R t1 
(4)
Therefore, if the diffusion coefficient in the interplanetary space and time of ejection are known, for determining
of the spectrum in source it is enough to determine energy spectrum outside of magnetosphere in some moment of
time t1 after ejection of FEP into solar wind (by using coupling functions, as it was described in Dorman et al.,
2003). The measurements of FEP spectrum in some next moment t 2 will be useful for checking previous result:
determining of spectrum in source N o R  on the basis of Eq. 4 at t= t 2 must give the same result. If it will be
sufficient difference, it means that something is wrong: diffusion coefficient or time of FEP ejection are not
correct, or the model of propagation is far from reality. Let us note that FEP spectrum measurements in several
moments of time will be very useful for correcting and more exact determination of diffusion coefficient for FEP
propagation in the interplanetary space and the time of FEP ejection (see below), as well as for checking of FEP
propagation model.
ON-LINE DETERMINATION SIMULTANEOUSLY OF FEP DIFFUSION COEFFICIENT AND FEP
SPECTRUM IN SOURCE, IF THE TIME OF EJECTION IS KNOWN
Let us suppose that K R  is unknown function, but the time of ejection is known. In this case we need to use data
at least for two moments of time t1 and t 2 relative to the time of ejection (what is supposed as known value). In
this case we will have system from two equations:

 1 exp r /4K Rt ,
 1 exp r /4K Rt ,
bt1 R  t1 Do R   N o R  2 1 2 K R t1 3 2

bt 2 R  t 2 Do R   N o R  2 1 2 K R t 2 3 2
2
1
1
2
1
2
(5)
(6)
By dividing Eq. (5) on Eq. (6), we obtain
2 


bt1 
t1 t2 3 2 R   t1    t2  exp  r1  1  1   ,
bt 2 
 4 K R   t1 t 2  
from what follows
(7)
1
 r 2  1 1     bt 

K R     1       ln  1 t1 t 2 3 2 R   t1    t 2   .
 4  t1 t 2  


   bt 2 
(8)
The found result for K R  will be controlled and made more exactly on the basis of Eq. 8 by using the next data in
moments t 2 and t 3 , as well as for moments t1 and t 3 , then by data in moments t 3 , and t 4 , and so on. By
introducing result of determining of diffusion coefficient K R  on the basis of Eq. 8, we determine immediately by
Eq. 4 the expected flux and spectrum FEP in the source on the basis of data for each moment of time t1 , t 2 , and so
on (results must be equal).
ON-LINE DETERMINATION SIMULTANEOUSLY TIME OF EJECTION AND FEP SPECTRUM IN
SOURCE IF THE DIFFUSION COEFFICIENT IS KNOWN
Let us suppose that diffusion coefficient K R  is known, but the time of ejection Te is unknown. Let us suppose
that in two moments T1 and T2 were made measurements of energetic spectrum (all times T are in UT scale). In
this case in Eq. 5 and Eq. 6
t1  T1  Te  x, t 2  T2  Te  T2  T1  x ,
(9)
where T2 - T1 is known value and x is unknown value what can be determined from Eq. 5 and Eq. 6 with taking
into account Eq. 9. After dividing of Eq. (5) on Eq. (6), we obtain
2 

 
bT1 
1
 ,
x T2  T1  x 3 2 R   T1    T2  exp  r1  1 

bT2 
 4 K R   x T2  T1  x  
(10)
From Eq. 10 unknown value x can be found by iteration:

x  T2  T1  2 3 1   2 3
1 ,
(11)
where
x  

 
bT2   T    T 
r12  1
1

 
 .
1
2

exp

R
 4 K R   x T2  T1  x  
bT1 


(12)
As the first approximation we can use x1  T1  Te  500 sec what is a minimum time of relativistic particles
propagation from the Sun to the Earth’s orbit. Then by Eq. 12 we determine   x1  and by Eq. 11 determine the
second approximation x 2 . To put x 2 in Eq. 12 we compute  x 2  , and then by Eq. 11 determine the third
approximation x3 , and so on.
ON-LINE DETERMINATION SIMULTANEOUSLY TIME OF EJECTION, DIFFUSION COEFFICIENT
AND FEP SPECTRUM IN SOURCE
Let us suppose that the time of ejection Te and diffusion coefficient K R  are unknown. In this case for
determining on-line simultaneously time of ejection Te , diffusion coefficient K R  and FEP spectrum in source
N o R  we need information on FEP spectrum at least in three moments of time T1 , T2 and T3 (all times T are in
UT scale). In this case instead of Eq. 9 we will have for times after FEP ejection into solar wind:
t1  T1  Te  x, t 2  T2  T1  x, t 3  T3  T1  x ,
(13)
where T2 - T1 and T3 - T1 are known values and x is unknown value what we need to determine in the first. From
Eq. 10 with taking into account Eq. 13 we obtain
 bT 

T2  T1
4 K R 

 ln  1 x T2  T1  x 3 2 R   T1    T2  ,
2
xT2  T1  x 
r1
 bT2 

(14)
 bT 

T3  T1
4 K R 

 ln  1 x T3  T1  x 3 2 R   T1    T3  .
xT3  T1  x 
r12
 bT3 

(15)
After dividing Eq. 14 on Eq. 15 we obtain
x  T2  T1   T3  T1  1    ,
where
(16)
 bT 

ln  1 x T2  T1  x 3 2 R  T2    T1 
bT 
T T
.
 3 1  2
T2  T1
 bT 

ln  1 x T3  T1  x 3 2 R  T3    T1 
 bT3 

(17)
Eq. 16 can be solved by the iteration method: as the first approximation, we can use x1  T1  Te  500 sec what is
a minimum time of relativistic particles propagation from the Sun to the Earth’s orbit. Then by Eq. 17 we
determine   x1  and by Eq. 16 determine the second approximation x 2 . To put x 2 in Eq. 17 we compute  x2  ,
and then by Eq. 16 determine the third approximation x3 , and so on. After solving Eq. 16 and determining the time
of ejection, we compute very easy diffusion coefficient from Eq. 14 or Eq. 15:
K R   
r12 T2  T1  4 xT2  T1  x 
 bT 

ln  1 x T2  T1  x 3 2 R T2    T1 
 bT2 


r12 T3  T1  4 xT3  T1  x 
 bT 

ln  1 x T3  T1  x 3 2 R T3    T1 
 bT3 

.
(18)
After determining time of ejection and diffusion coefficient it is very easy to determine the FEP spectrum in
source:


N o R   2  1 / 2 bt1  R   t1  Do R  K R t1 3 / 2 exp r12 / 4 K R t1   2  1 / 2 bt 2  R   t 2  Do R 




 K R t 2 3 / 2 exp r12 / 4 K R t 2   2  1 / 2 bt3  R   t3  Do R  K R t3 3 / 2 exp r12 / 4 K R t3 
.
(19)
CHECKING THE MODEL BY CALCULATIONS OF EXPECTED DIFFUSION COEFFICIENT
Fig. 1. The behavior of K R  for R  10 GV with time.
For the checking of used above model of FEP
propagation in the interplanetary space, we
determined in the first values of K R  . These
calculations we made according to the procedure
described above at supposition that K R  does
not depend from the distance to the Sun. Results
are shown in Fig. 1. It can be seen that in the
beginning of event obtained results are not stable
what are caused by a big relative statistical
errors. After few minutes amplitude of CR
intensity increase became many times bigger
than , and we see systematical increase of the
diffusion coefficient with time: really its reflects
the increasing of K R  with the distance to the
Sun.
THE INVERSE PROBLEM FOR THE CASE WHEN THE DIFFUSION COEFFICIENT DEPENDS
FROM THE DISTANCE TO THE SUN
Let us suppose, according to Parker (1963), that the diffusion coefficient K R, r   K1 R   r r1  . In this case
n  R, r , t  
N o R   r13 2    K1 R t 3 2   
2   4    2    3 2   


r1 r 2  

.
 exp 
 2   2 K R  t 
1


(20)
If we know n1 , n2 , n3 at moments of time t1 , t 2 , t3 , the final solutions for  , K1 R  , and N o R  will be
1

 

t t  t 
t t  t 
  2  3ln t 2 t1   3 2 1 ln t3 t1   ln n1 n2   3 2 1 ln n1 n3  ,
t 2 t3  t1 
t 2 t3  t1 

 

K1 R  



(21)

r12 t11  t31
r12 t11  t 21

,
32    ln t 2 t1   2   2 ln n1 n2  32    ln t3 t1   2   2 ln n1 n3 

r12
N o R   n1 2   4    2    3 2   r13 2    K1 R t k 3 2     exp
 2   2 K R  t
1
k

(22)

.


(23)
In the last Eq. 23 index k = 1, 2 or 3.
SIMULATION OF FEP FORECASTING BY USING ONLY NEUTRON MONITOR DATA
By using of the FEP event first few minutes NM data we can determine by Eq. 21 – 23 effective parameters  ,
K1 R  , and N o R  , corresponded to the rigidity about 7 – 10 GV, and then by Eq. 20 we determine the
forecasting curve of expected FEP flux behavior for total neutron intensity. This curve we compare with time
variation of observed total neutron intensity. Really we use data for more than three moments of time by fitting
obtained results in comparison with experimental data to reach the minimal residual (see Fig. 2, which contains 8
figures for time moments t = 110 min up to t = 220 min after 10.00 UT of 29 September, 1989).
Fig. 2. Calculation on line parameters  , K1 R  , N o R  and forecasting of total neutron intensity (time t is in
minutes after 10.00 UT of September 29, 1989; curves – forecasting, circles – observed total neutron intensity) .
From Fig. 2 can be seen that using only the first few minutes of NM data (t = 110 min) does not enough: the
obtained curve forecasts too low intensity. For t = 115 min the forecast shows little bigger intensity, but also not
enough. Only for t = 120 min (15 minutes of increase after beginning) and later (up to t = 140 min) we obtain
about stable forecast with good agreement with observed CR intensity (with accuracy about  10 %).
In Fig. 3 are shown values of parameter K1 R  . From Fig. 3
can be seen that at the very beginning of event (the first
point) the result is unstable: in this period the amplitude of
increase is relatively small, so the relative accuracy is too
low, and we obtain very big diffusion coefficient. Let us
note, that for very beginning step of event the diffusion
model can be very hardly applied (more natural is applying
of kinetic model of FEP propagation). After the first point we
have about stable result with accuracy  20 % (let us
compare with Fig. 1, where diffusion coefficient was found
as effectively increasing with time).
Fig. 3. Diffusion coefficient K1 R  near Earth’s orbit (in
units 10 23 cm 2 sec 1 ) in dependence of time (in minutes
after 10.00 UT of September 29, 1989).
In Fig. 4 are shown values of parameter  . It can be seen
that again the first point is anomaly big, but after the first
point the result became about stable with average value  
0.6 (with accuracy about  20%). Therefore, we came to
conclusion that the model, described by Eq. 20 – 23 reflects
well FEP propagation in the interplanetary space.
Fig. 4. Values of parameter  in dependence of time (in
minutes after 10.00 UT of September 29, 1989).
DISCUSSION AND CONCLUSIONS
Obtained solutions gave possibility on the basis of
determining on-line by one-minute NM data FEP energy spectrum (according to Paper 2, Dorman et al., 2003b)
during 10 – 15 minutes after start of great event (the start is also determined automatically, see Paper 1, Dorman et
al., 2003a) - to determine on-line the diffusion coefficient K1 R  near Earth’s orbit and parameter  (described
the increasing of diffusion coefficient with the distance from the Sun) for FEP propagation in the interplanetary
space, and total FEP flux and energy spectrum N o R  in the source. We check these results by simulation of
forecasting of total neutron intensity. The obtained on-line information can be considered as a basis for the next
working on-line programs “FEP-Forecasting in Space” (for different distances from the Sun), “FEP-Forecasting in
Magnetosphere” (for satellites with different orbits), and “FEP-Forecasting in Atmosphere” for balloons and airplanes on different altitudes at different cut-off rigidities as well as for people and technology on the ground at
different cut-off rigidities (see Paper 4, Dorman et al., 2003c).
REFERENCES
Dorman L.I., Cosmic Ray Variations, Gostekhteorizdat, Moscow, 1957.
Dorman L.I., Cosmic Rays of Solar Origin. VINITI, Moscow ( Ser. "Summary of Science", Space Investigat., Vol.12), 1978.
Dorman L.I. and L.I. Miroshnichenko, Solar Cosmic Rays, Moscow, Fizmatgiz, 1968.
Dorman L.I. and D. Venkatesan “Solar cosmic rays”, Space Sci. Rev., 64, 183-362, 1993.
Dorman L.I. et al., Paper 1, This Issue, 2003a.
Dorman L.I. et al., Paper 2, This Issue, 2003b.
Dorman L.I. et al., Paper 4, This Issue, 2003c.
Duggal, S.P., Rev. Geophys. Space Phys., 17, 1021, 1979.
Parker E.N., Interplanetary Dynamical Processes, New York, Interscience. Publ., 1963.
Stoker P.H. “Relativistic solar cosmic rays”, Space Sci. Rev., 73, 327, 1995.
E-mail address of Lev I. Dorman: lid@physics.technion.ac.il, lid1@ccsg.tau.ac.il
Manuscript received
; revised
, accepted