ISSN 0010-9525, Cosmic Research, 2006, Vol. 44, No. 6, pp. 500–505. © Pleiades Publishing, Inc., 2006.Original Russian Text © E.I. Daibog, Yu.I. Logachev, K. Kecskeméty, 2006, published in Kosmicheskie Issledovaniya, 2006, Vol. 44, No. 6, pp. 521–526.

500

INTRODUCTION

The rising phases and maxima of the fluxes weremainly considered in many papers dedicated to propa-gation in the interplanetary space of charged particlesaccelerated both in flares on the Sun and by shockwaves related to flares or coronal mass ejections(CME). Considerably less attention was paid to fluxdecays after a maximum though this phase of the eventcarries considerable information on the velocity of thesolar wind, disturbance of the magnetic field, and otherparameters of the interplanetary medium.

Various models of particles propagation in the envi-ronment lead to various laws of the decay of their fluxesat the late stage of an event. The time profile of particlefluxes in solar events has a typical form. For example,at the Earth’s orbit for a solar event related to a singleflare, the particle fluxes first increase quickly enough,reach a maximum, and then gradually decrease to thepre-flare level. Cases are frequent when this picture canbe represented in a diffusion approximation. Then,under the assumption of an impulsive source of parti-cles (the generation time is much less than the time ofparticle propagation from the Sun to the observationpoint), the time profile of particle fluxes

J

(

t

)

at the latestage of the event has a power-law character and is pro-portional to

t

–3/2

. In the case of a long particle injection,one has to take into account the source function of par-ticles, which should lead to an extension of the eventbut with only a small influence on the late phase of itsdecay.

However, the solar wind permanently present in theinterplanetary space provides for convective transfer ofparticles and their adiabatic cooling, and at predomi-nance of these processes at the late stage of events thefluxes decrease exponentially (

J

(

t

) ~

e

–

t

/

τ

[1–4]). The

power law is fairly valid for particles with high energy(> 100 MeV). The convective transfer begins to play aconsiderable role and the particle fluxes decay becomesexponential for particles with lower energies (< 10 MeV).In previous papers we obtained that in 90% of cases, theproton fluxes with low energy (< 10 MeV) have anexponential decay, while for particles with an energy of> 30–60 MeV the exponential decay is observed muchmore seldom [5, 6]. However, particle propagation isoften accompanied by various processes of their addi-tional acceleration, these processes inevitably leadingto a distortion of the “smooth” temporal profile andmaking difficult its description by any single law.

In addition to flares, a shock wave is a source (some-times the only one) of accelerated particles. Due to theshock wave, the regions of quasi-trapped particles canappear. These regions can both be related to the shockwave front and exist between the front and the strongmagnetic field on the Sun [7, 8]. The decay phase of thetrapped particles (in the same way as in the diffusionapproximation) is described by a power-law function[7]. In this case, the further fate at least of those parti-cles which have been earlier accelerated by the shockwave and propagate in the interplanetary space ahead ofits front is determined by the diffusion and processesaccompanying it.

Under predominance of the convective transfer ofparticles and their adiabatic cooling over the diffusionat the late phase of the event, the following expressionfor the characteristic decay time was obtained [1–4]:

τ

= 3

r

/2

V

(2 +

αγ

). (1)

Here

V

,

γ

, and

r

are the solar wind velocity, thepower-law index of the particle energy spectrum, andthe distance from the observation point to the Sun,respectively, and

α ≈

2

for particles of nonrelativistic

Characteristics of the Decay Phase of Proton Fluxes in Solar Events as a Function of Observer’s Heliolongitude

E. I. Daibog

a

, Yu. I. Logachev

a

, and K. Kecskeméty

b

a

Skobeltsyn Institute of Nuclear Physics, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia

b

KFKI Research Institute for Particle and Nuclear Physics, H-1525 Budapest, POB 49, Hungary

Received October 13, 2005

Abstract

—Events in energetic solar protons with the energy > 4 MeV at the stage of their decay are consideredfor the period from 1974 to 2001. It is shown that in the events with the exponential shape of decay for westflares (relative to the observation point), the characteristic decay time

τ

and the power index

γ

of the energyspectrum decrease with an increase in the angular distance between the observer and the source of the particleson the Sun, while this effect is absent for east flares.

PACS numbers: 96.60.Vg

DOI:

10.1134/S0010952506060062

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CHARACTERISTICS OF THE DECAY PHASE OF PROTON FLUXES 501

energies. The analysis performed in [9] showed that inthe considerable part of cases (up to 50%) when

V

wasconstant during the entire decay,

τ

is satisfactorilydescribed by this formula. For other events this depen-dence is not valid. The latter fact testifies that about ahalf of the solar events does not satisfy simultaneouslyall the accepted criteria: the events are not isolated, theparticle injection is not impulsive, etc.

We assume that for the time intervals when thedecay of particle fluxes during a long time (about oneday or more) is described by an exponential law, thenearest interplanetary space is homogeneous and quasi-stationary, this facts providing for a constancy of

τ

. Theconcept of the quasi-stationarity of the environmentassumes a constancy of some totality of its propertiesdetermining the rate of particle flux decay at this spatialpoint.

The analysis of the time properties of events is diffi-cult because of the spatial variations related to the Sun’srotation. Indeed, if a spacecraft were permanentlylocated in some tube with a stationary magnetic field,then all the variations would have only a time character.The real measurements always are conducted in variousfield tubes where the magnetic conditions, as a rule, aredifferent. In some cases the constancy of particle fluxesover longitude is observed only at a very short distance,and then two spacecraft located at a small angular dis-tance (sometimes

≤

10°

) detect absolutely differentfluxes. At the same time, conditions for particles prop-agation are fairly often the same for spatial regionsextensive over heliolongitude, the fact being confirmedby simultaneous observations at several spacecraft. Inthese cases, the particle fluxes are constant at consider-able angular distances (up to angle values of

>100°

) [7,10]. This means that in the first and second cases thespacecraft were located in various field tubes of themagnetic field and in a single wide tube with homoge-neous characteristics of the interplanetary medium,respectively.

It was shown in [5, 6] that for the considerable partof solar cosmic ray (SCR) events (more than a half oftheir total number), the value of

τ

for particles with anenergy of > 4 MeV is 16–20 h, which corresponds to (1)at typical values of the parameters involved. This meansthat no less than a half of the time falling on the periodswith energetic solar particles, the interplanetary spaceis in the state corresponding to this

τ

. About 20% of theevents have larger values of

τ

reaching sometimes 50 hand more.

Neither the distribution over

τ

depends on the powerof a solar event. For protons with energies

E

> 4 MeV,the distribution is almost similar for the events withmaximal fluxes

J

max

> 2–3 particles cm

−

2

s

-1

sr

–1

and

J

max

> 100 particles cm

−

2

s

-1

sr

–1

. The distribution ofintegral values of the spectral indices

γ

for all eventsvaries from 0.7 to 3.5 with the average value

⟨γ⟩

= 1.85.It is also shown in [5, 6] that no considerable variations

in the average values of

τ

and

γ

during a solar activitycycle are observed.

Mutual angular position of the observer and thesource of accelerated particles is one of the importantfactors determining the shape of the time profile of anevent and, in particular, its decay rate. We consider inthis paper events with exponential decays. On the footpoint of these events, the dependence of the typicaldecay time

τ

and the power index of the energy spec-trum

γ

on the location of a flare (i.e., the source of par-ticles on the Sun) is studied.

SELECTION OF THE EVENTS

The decay phase after the maximum in particle fluxincreases initiated by CME and shock waves of variousorigin was studied in this paper. We considered allincreases in proton fluxes with the energy > 4 MeV atthe Earth’s orbit according to the data of the CPMEinstrument onboard the

IMP

-8

satellite during theperiod 1974–2001.

The periods at the decay phase of events when thedecrease in the particle flux could be described by anexponential dependence were chosen for the analysis.In this study, predominantly such time intervals of fluxdecay were taken into account which lasted for no lessthan half a day. Such a choice of the decay intervals isdue to the fact that decays lasting less than 12 h are dif-ficult to identify with an exponential, power, or anyother law. Integral fluxes of the protons with energies

Ep

> 4,

> 10, > 30, and > 60 MeV were considered. Theevents with the flux of protons with

Ep

> 4 MeV in themaximum exceeding 2–3 particles cm

−

2

s

−

1

sr

–1

wereselected for the analysis. Altogether more than 600decays were considered. The power-law index of theenergy spectrum

γ

was determined from the relationbetween fluxes using the formula

J

(>

E

)

~

E

–

γ

and, as arule, was averaged over the results of the first threechannels (> 4, > 10, and > 30 MeV). The 60 MeV chan-nel had an increased background and was not used inthe

γ

determination at small fluxes.

In order to determine the dependence of the charac-teristic decay time on the flare position, the events inenergetic solar particles were studied for which thesource of particles (the flare on the Sun) was detectedreliably. Mainly, the attribution of particle events wasthe same as in the “Catalogs of Solar Proton Events”[11–13]. It should be noted, however, that sometimes incases of long decays the input of the next flares is sig-nificant during this phase. In this case, attribution tolater flares is preferable, though a few decays were notincluded into the statistics just because of this very rea-son, due to an uncertainty in event attribution to flares.In total, 238 events satisfying the criteria describedwere reliably attributed to solar flares.

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DEPENDENCE OF

τ

AND

γ

IN SOLAR EVENTS ON THE FLARE HELIOLONGITUDE

The

τ

value is determined by a number of factors:mainly by the state of the interplanetary plasma andmagnetic field, and also by the disturbance of the latterin the close vicinity (< 0.5 a.u.) of the observationpoint. Besides, there exist additional circumstancesinfluencing the rate of the decay of the particle flux inan event. It was established fairly long ago that the timeof the rising of the flux up to its maximal value in thevast majority of the events related to eastern flares isprolonged as compared to the western flares [14], how-ever, no detailed analysis of the dependence of the rateof the particle flux decay on the flare heliolongitude hasbeen yet performed. Such a dependence should be,apparently, observed even if one assumes an absolutestationarity and regularity of the interplanetary space.

The observed value of

τ

at impulsive injection ofparticles from the source and isotropic interplanetaryspace is a result of at least two effects: 1) a naturaldecrease in the particle flux after the event maximum(the rate of the decrease is governed by the above men-tioned factors) and 2) a decrease (or rising) of the par-ticle flux due to permanent shift of the observationalpoint to other field lines. In other words,

τ

shoulddepend on the longitudinal profile of particle fluxes atthe moment of their injection. In this case, one shouldtake into account that at different field lines of the mag-netic field,

τ

may be different because of different struc-ture of the magnetic field and different degree of its dis-turbance. Moreover, in the case of a long source,

τ

should depend on the time profile of the particle source.Thus, one can see that the observed values of

τ

are acomplicated function of spatial and time parameters ofthe interplanetary medium.

When solar cosmic rays are detected in the vicinityof the Earth, an observer, at the value of the solar windvelocity

V

= 400 km/s, is connected with the Sun by themagnetic field line with a heliolongitude of W60 (itchanges depending on the solar wind velocity). If aflare occurs at this heliolongitude, the particles gener-ated in the flare come to the observer by the shortestway. If a flare occurs at a different longitude, one canexpect a different time of the intensity increase and therate of its decay.

The properties of the decay phase of a few eventswere studied in [4] on the basis of the data of measure-ments onboard the

Pioneer

-6 –

Pioneer-9

spacecraft.According to [4], one may write the rate of changes inthe particle density (

U

) at a given point of space in theform:

dU

/

dt

=

∂

U

/

∂ϕ

d

ϕ

/

dt

–

U

/

τ

d

, (2)

where

ϕ

is the heliolongitude of the foot point (on theSun) of the Archimedean spiral of the magnetic fieldline passing through the observation point,

τ

d

is thecharacteristic time of the exponential decay of particlefluxes caused by the convection and adiabatic decelera-

tion (

τ

d

= 3

r

/2

V

(2 +

αγ)

)

. If the entire change in

U

occurs exponentially (in that case we may take that

dU

/

dt

= −U/τ), we obtain for the characteristic decaytime τ the following expression:

(3)

and if U–1∂U/∂ϕ = 1/ϕ0, then

(4)

where dϕ/dt ~ 0.5 degree per hour is due to the solarrotation. For each out of five events measured onboardPioneer-8 and Pioneer-9, it was shown in [4] (where ϕ0had a scatter of ±60°) that the decay time is a functionof the position of the observer relative to the field lineof the flare.

It follows from (4) that at constant values of τd andϕ0, the characteristic time of the decay τ changes in ajump at a change of the observer position relative to theoptimal heliolongitude from the western to easternposition due to the sign reversal of the term 1/ϕ0dϕ/dtin (4). That is why one should expect an appearance ofa “step” in the distribution of the decay time over theheliolongitude. This very situation was observed duringthe event on September 23, 1978 discussed earlier inrelation to the properties of the temporal and spatialinvariance of events with particles of a solar origin [7,15]. This event was initiated by a flare with coordinatesN35W50 and by a shock wave associated with CME,and it was detected simultaneously by the spacecraftHelios-1, Helios-2, and IMP-8. The flare heliolongituderelative to the foot points of the magnetic field lines ofHelios-1 and Helios-2 in this case was E110 and E145,respectively, while in the beginning of the event, IMP-8 was located at the field line passing very close to theflare. Figure 1 shows the time profiles of the protonswith an energy of 4–13 MeV (Helios-1 and Helios-2)and > 4 MeV (IMP-8). One can see that τ is equal to30 h at the space vehicles Helios-1 and Helios-2 sepa-rated by almost 40° but all the time located at field lineswestward from the flare, while at IMP-8 located duringthe decay phase eastward from the flare, τ was muchshorter and equal to 12 h. It is worth noting that in [7]and some other publications of these authors, the timeprofiles for the September 23, 1978 event at the phaseof rising and maximum of the intensities and the verybeginning of the decays (approximately coinciding dur-ing one and a half–two days) are presented. However,at the later stage (not shown in [7]), the decay rates atHelios-1–Helios-2 and IMP-8 differ radically. Never-theless, the authors of [7] came to the conclusion on thecomplete coincidence of all three profiles.

One can present other examples of the cases, wheneven at a rather large distance between spacecraft but attheir similar western or eastern position relative to theflare, the characteristic decay times coincide [15].However, opposite examples exist not confirming theabove described spatial picture. These examples, in

1/τ– U 1– U/ ϕdϕ/dt 1/τd,–∂∂=

1/τ 1/ϕ0dϕ/dt– 1/τd,+=

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CHARACTERISTICS OF THE DECAY PHASE OF PROTON FLUXES 503

principle, may be caused by other factors influencingthe rate of particle flux decay.

An indirect evidence of the existence of the τ(ϕ)dependence is presented in [16] where the dependenceon ϕ of the total duration of the event ∆T from thebeginning of the intensity increase to its recovery to theundisturbed level is considered. The data of Helios-1and Helios-2 show that within the entire interval of theobserved differences in the heliolongitudes of the flareand the foot points of the magnetic field line passingthrough the observation point ∆ϕ from E180 to W90,there exists a tendency of the ∆T decrease at the changeof ∆ϕ in the westward direction. One should note, how-ever, that in the considerations of the total duration ofthe event, together with the decay, some role is playedalso by the time of the flux increase up to its maximalvalue which also decreases when ∆ϕ changes from eastto west.

Indications were also obtained in [4] that in the fiveevents considered there, at the late (> 2 days) stage, thespectral index γ in the energy range about 10 MeVdepends on the observed heliolongitude relative to thefoot point of the field line connected to the flare. Thepower index of the energy spectrum within the interval�4.5 < γ < �2 changes at the changes of |∆ϕ| from 0° to180°, the effect being explained by the authors by amore effective propagation of high-energy particles tofield lines far distant from the flare in the vicinity of theSun as compared to particles of low energies. The latterstatement agrees qualitatively to all available conceptsof the near-Sun diffusion.

In this connection, a question arises about the gen-eral character of such dependencies for a large numberof events. One cannot exclude a suggestion that theevents discussed in [4] are only a fortunately foundillustration of the claimed physical idea, but on thewhole this dependence is not confirmed. To answer thequestion we analyzed the dependencies of the charac-teristic time of the exponential decay τ of the protonfluxes with an energy > 4 MeV and their spectral indexfor 1974–2001 on the heliolongitudes of the flares, i.e.,sources of the events on the Sun. As we have alreadymentioned, there were 238 such events. The depen-dence of τ on the flare heliolongitude is presented inFig. 2 (top panel). One can see that in the cases whenthe source on the Sun is located eastward from the opti-mal heliolongitude (on the average, W60), τ is statisti-cally independent of the flare heliolongitude. At thesame time, one should note that the arrival of particlesis considerably delayed with an increase in the angulardistance of the flare from the optimal longitude atwhich the field line of the interplanetary magnetic fieldpasses in the vicinity of the observation point. The pic-ture changes in the region of the heliolongitude of theorder of W60. One can see a tendency to a decrease inτ with an increase of the flare heliolongitude at thewestern location of the flare relative to the foot point ofthe magnetic field line connected to the Earth.

The power-law index of the energy spectrum γ in thebeginning of the exponential decay is also shown inFig. 2 as a function of the heliolongitude (bottompanel). The statistical analysis does not make it possibleto confirm unambiguously the result obtained in [4] onthe basis of a few particular events within the entireangular range E90–W90. More probably, γ does notdepend on heliolongitude for the eastern flares. Never-theless, the tendency is seen of γ decrease for the flareswestward relative to the optimal longitude. The charac-teristic time of decay τ = τ(γ) decreases with an increasein γ. Indeed, we obtained in [17] an indication to theexistence of such statistical dependence. However, if auniversal declining dependence γ(ϕ) existed, it wouldbe very difficult to explain the observed cases of con-stancy of τ within a wide range of angles.

Currently available statistics does not make it possi-ble to reduce the considerable scatter in the τ and γ val-ues. It could have been done only under condition thatit would be possible to create a homogeneous series ofdata on all other parameters influencing the values of τand γ (events with similar values of the magnetic fieldstrength, solar wind velocity, τ (for γ) or γ (for τ), etc.).

CONCLUSIONS

The phase of decay of particle fluxes carries someinformation on the state of the interplanetary medium(the solar wind velocity, magnetic field disturbance,and other phenomena). The analysis performed in thispaper showed that in the events with an exponentialdecay of proton fluxes at the eastward position of theflare relative to the foot point of the magnetic field lineof the observer, neither the characteristic time τ nor the

Jp, arb. units

264 268 272 276Day of 1978

103

280

102

101

100

10–1

10–2

10–3

10–4

Fig. 1. Event on September 23, 1978. Protons with theenergy of 4–13 MeV according to the data of Helios-1 andHelios-2 (τ = 30 h) and protons with the energy > 4 MeVaccording to IMP-8 data (τ = 12 h) are shown by circles,crosses, and triangles, respectively.

504

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DAIBOG et al.

power index of the energy spectrum γ depend statisti-cally on the flare heliolongitude, while for the westwardflares there exists a tendency of a decrease of both τ andγ with an increase in the angular distance between theflare and the observation point. An increase in the sta-tistics would make it possible to specify the indepen-dence of γ(ϕ) on ϕ for eastern flares. It has been alreadymentioned above that symmetry of this dependence rel-ative to the foot point of the magnetic field line of theobserver follows from the interpretation of the eventsconsidered in [4]. The asymmetry obtained in this paperneeds an explanation. The scatter in τ values in a certaindegree characterizes the scatter in values of the inter-planetary environment parameters.

Since the effects considered here, apparently, areexclusively manifestations of the solar rotation influ-ence, one may formulate the general conclusion fromthe study performed in the following way: statisticallythe conditions of particle propagation up to 1 a.u. do notdepend on heliolongitude of the flare relative to theobservation point. This fact may be considered as anextra argument in favor of the existence of periodswhen the set of values of the parameters of the inter-planetary space within the inner heliosphere makes ithomogeneous and quasi-stationary at considerableangular distances. This is manifested, first of all, in thevery existence of long exponential decays. A valuablerole in confirmation of this statement could have beenplayed by simultaneous observations in observationpoints widely separated in heliolongitude.

ACKNOWLEDGMENTS

The paper was supported by the International SpaceScience Institute (Bern, Switzerland), Team 77 and bythe Russian Foundation for Basic Research (projectNo. 5-02-17096). The experimental data on the protonfluxes were taken from the website of the CPME instru-ment of the IMP-8 satellite. The authors thank A. V.Belov and V. G. Kurt for the data on attribution of theevents in energetic solar particles of the 23rd solaractivity cycle to solar flares in the hard X-ray radiation.

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COSMIC RESEARCH Vol. 44 No. 6 2006

CHARACTERISTICS OF THE DECAY PHASE OF PROTON FLUXES 505

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