- Email: [email protected]
Source composition of cosmic ray nuclei
Adv. Space Res. Vol. 27, No. 4, pp. 749-754.2001 Published by Elsevier Science Ltd on behalf of COSPAR Printed in Great Britain 0273-I 177101$20.00+ 0.00
Pergamon www.elsevier.com/locate/asr PII: SO273-1177(Ol)OOI23-5
SOURCE
COMPOSITION
OF COSMIC
RAY NUCLEI
S.A. Stephens1 and R.E. Streitmatter’
‘Department of Physics, University of Tokyo, 7-3-l Hongo, Bunkyo-ku, Tokyo 113-0033, Japan, 2Laboratory for High Energy Physics, Code 661 NASA/GSFC, Greenbelt, MD 20771, USA
ABSTRACT During propagation
in the Galaxy, cosmic-ray
the time of acceleration
gets modified.
from the observed abundance. variable to examine cross-sections,
nuclei undergo spallation and as a result, their composition
Propagation
at
models have been used to obtain the source composition
We made use of the standard leaky-box model with time as the propagation of all elements. The main ingredients in this method are the spallation
the spectra
which has considerably
show that the spectral
changed over the time.
Using the latest available cross-sections,
we
injection spectrum in rigidity is -2.32. We also derive the The derived abundances differ from those of Webber for and the source abundances.
escape path-length
index of the power-law
some of the elements.
Published by Elsevier Science Ltd on behalf of COSPAR.
INTRODUCTION The observed
cosmic rays contain some universally
heavier cosmic-ray
nuclei while traversing
during the propagation considerably
changed.
is about
using propagation
composition
components.
especially
more accurately, We examined
than before. this problem
model.
cross-sections.
poorly
length,
their composition
has been determined Early attempts
source composition
potential
from the observed
the calculated
with semi-
differed from the solar
is larger than 10 eV. Recently,
However, the observed
spectra are not yet well determined. in the frame work of the simple
the spectral index of the injection
spectrum
instead of the
of the accelerated
spectra with the observed spectra of some of the dominant
nuclei by
nuclei, after correcting
for solar modulation. calculated
B/C
The escape time and its rigidity dependence were obtained from a comparison with the observed data. These are important input parameters and 10Be/gBeratios
are needed for these calculations. them with the observed
We then proceeded
data by adjusting
and present the final source composition PROPAGATION
OF
COSMIC
We follow a simple concept
new
have been made and one is in a position to make these estimates
by making use of the steady state propagation
We determined
gets
to determine the source
flux values at a fixed energy, using slab model
whose first ionization
of
reflects the source composition,
Unlike in the past, we made use of the time as the variable for propagation
matter traversal. comparing
for elements
composition
The derived cosmic-ray
of the spallation cross-sections
by the spallation
Since the amount of matter traversed by them
The source composition
were made from the measured
composition,
leaky-box
cosmic-ray
models and spallation
empirical relations for the cross-section. measurements
rare nuclei, which are produced gas.
the same or larger than the interaction
Thus, the observed
except for the very dominant abundances
interstellar
to determine
the source composition. obtained
of the which
the spectra of all nuclei and compared In this paper, we describe
this method
from this investigation.
RAYS
for the confinement
of cosmic
rays based on the leaky-box
model.
In this
model, cosmic rays diffuse freely in a volume and a small fraction escapes at a constant rate. Over a sufficient period of time, cosmic rays will reach a state of equilibrium. of particles
dE, due to energy 1998) we discussed function
When they are in this steady state, the number
dN that come into any energy interval dE in unit time is the same as the number lost from loss processes,
interaction
and escape.
In our earlier paper (Stephens
in detail the results from the conventional
practice
of describing
of the matter traversed (z) by them. We describe below the propagation 749
and Streitmatter,
the propagation
as a
of cosmic rays in time (t)
750
S. A. Stephens and R. E. Streitmatter
(see Streitmatter and Stephens, 1999). The differential equation describing the propagation of cosmic rays in the Galaxy is written using the general formalism given by Ginzburg & Syrovatskii (1964) as,
- Ni(E,
t)
&(E)7wi
1.0 + T?(E)
1.0 +
+ &
Qi(E,t)
(1)
The left hand side in the above equation is the rate at which the number density of the i-th nucleus varies with time. Here, N(E, t) is the number of particles of energy E in an energy interval dE per unit volume and is related to the flux J(E, t) by the relation Ji(EJ) = Ni(E, t)w;/4 ?r, where vi is the particle velocity at energy E. The first term on the right hand side is the energy loss term, where dE/dt is the ionization loss per unit time. The second term is the loss due to interaction, escape and decay. In this term, oi is the interaction cross-section of the i-th nucleus with an atom of mass < m >, which is the equivalent atomic mass of the interstellar medium (ISM) for one hydrogen atom, and n is the mean number density of hydrogen atoms in the ISM. Observations indicate that mean escape time r?(E) is rigidity dependent and hence depends upon the mass to charge ratio. T: is the decay life time (if radioactive) and y; is the Lorentz factor of the nucleus at energy E. The last term relates to the continuous production of particles in the Galaxy and this production rate term is given by Qi(E, t) = 6
Nj(Ey t)aij(E)vjn
+ C
y
l#i
j>i
(2)
+ qi(E, t) 1
Here, the first term is the production of i-th nucleus by the fragmentation of heavier nuclei, where uij(.E) is the partial fragmentation cross-section to produce the i-th nucleus from interaction of the j-th nucleus of higher mass. The second term is due to the radioactive decay. Contribution from the radioactive decay comes only from nuclei of adjacent charges with the same mass, as the long-lived isotopes decay through beta decay. The last term is the primary source term, which relates to those nuclei that are accelerated in the source. In the above equation Qi(Ey t) is expressed as the particle number density per unit time. Cosmic rays are in a state of equilibrium when 6’NJdt = 0, which would mean that Q(E, t) is independent oft. We have shown earlier (Stephens and Streitmatter, 1998), here after referred to as Paper I, that the set of differential equations given by Eq.(l) can be solved in an elegant manner by integrating these equations simultaneously by the Rungs-Kutta technique until equilibrium is reached, when aNi/& -+ 0 for all nuclei. The equilibrium spectra obtained from such a procedure then represent a full and exact numerical solution of Eq.(l). In the following sub-sections, we discuss some of the input parameters used in this study. Ionization loss In order to evaluate the ionization term in Eq. (l), we adopted a simple technique described in Paper I. Since the ionization loss rate is usually defined per unit gram of material traversed, the partial differential term takes a form in t different from that in z. The ionization loss term in Eq. (1) can be expanded as
&
b(E)g]
= A
[N(Elvpg]
= N(E)pv&
(g)
t N(E)pv
t
(g)
&v
( >p+(E) g
In this investigation, the ionization loss rate was obtained from the range-energy relation. The particle range and its differential energy spectrum can be expressed without loss of accuracy, as power-laws in energy over an energy region larger than the energy spacings used in this numerical integration. Thus, at energy E, if we express the range as R = aEb and the energy spectrum as N(E) cc Ep, then the fist, second and third terms in the above equation can be written as N(E)&1 - b)Evb/ab, N(E)ppvE-‘/ab and N(E)p~E-~/[aby(7 + l.O)]respectively and Eq. (3) can be re-written as
A
k(E):]
= N(Ez-bpu
[P t (1 - b) t 7(7~l~o~]
Source Composition of Cosmic Ray Nuclei
751
Note that in the range-energy relation a is expressed in g/cm 2. The above expression differs from that derived for the propagation in z (Eq. 8 in Paper I) by the addition of the last term l.O/[y(y + l.O)]. We made use of the standard tables for the range of protons in hydrogen and in helium. These range-energy relations for the protons were then scaled by A/Z2 for other isotopes. The mean number density of hydrogen was taken to be 0.2 atom/cm3 (Streitmatter and Stephens, 2000) and the range-energy relation was modified by taking into account the composition of ISM with 80% neutral hydrogen, 10% ionized hydrogen and 10% helium by number (see Paper I). Other parameters We need to make assumptions regarding the form of the escape time as a function of energy and we incorporated a relation, which is equivalent to the mean matter traversed by cosmic rays, namely P(E)
= F”(R/Ro)-” = re*
for
set
set
R > Ro
for
R<
R,,
(5)
where red is the constant escape time, R is the rigidity of the nucleus, R~Jis the rigidity at which rigidity dependent escape commences, and a is the index. Note that there is no velocity dependence in the escape time, because the escape time is related to the commonly used escape length X;(E) in x by the relation Ro and a) in the above equation need to be X;(E) =< m > nvi(E)~:‘(E). The three constants (?, determined.
10
1
10
Kinetic Energy in GeV/n
Fig.
1.
modulation
The
calculated
parameter
B/C
Phi=0.5
ratio
is shown
along
with
the
data
for
a
GV.
Another important parameter needed for this calculation is the fragmentation cross-section as a function of energy. We made use of the cross section tables provided by Webber (private communication) to obtain the required cross-section matrix to be used in solving the set of Eq. (1). The injection spectrum used in this calculation is a power law spectrum in rigidity. RESULTS Propagation parameters The spectral index of the source spectrum and the three constants in Eq. (5) were determined in the following manner. We assumed a power-law spectrum in rigidity as the source spectrum and solved the set of equations described by Eq. (1). The equilibrium spectra of all nuclei were then modulated using spherically symmetric solution (Perko, 1987). From these, we compared the spectra of the dominant nuclei C, 0 and Fe, and the B/C ratio with the observation. The spectral index and the escape parameters in Eq. (5) were adjusted until an excellent fit to all the above data were obtained. The equivalent modulation
752
S. A.
Stephens andR. E. Streitmatter
parameter used in the computation, which gave a consistent fit to the data, was 500 MV. Most of the data used here were taken from experiments carried out during the period of minimum solar activity. In Figure 1 we show the observed B/C ratio as a function of energy from a compilation of data made in Paper I. The solid curve is the best fit that was obtained from this calculation. The derived values of the three parameters in Eq. (5) are: (1) Ro = 2.7GV, (2) th e index a! = 0.5, and (3) the escape life time res corresponds to 14 g/cm2 of ISM traversed by cosmic rays, when v = c. The required spectral index of the injection spectrum, which gave the best fit to the spectra of C, 0 and Fe at high energies (Figure 2), is /? = -2.32. The value of res obtained here is smaller by about 12% than that obtained in Paper I as a result of the new set of cross-sections employed and partly due to the slightly steeper spectrum used in this calculation. Elemental
abundances
0 Simonet al (8) + Muller et al (14) * Carvilleet al (11)
Maehelet al (5)
1
10 -l
lo3
lo2
10
Kinetic Energy in GeVh
Fig.
2.
The
compared
with
are scaled
down
are multiplied
calculated observation. are shown by
spectra The along
of C,
N, 0,
factor
by which
with
the symbols.
Si and the
Fe are spectra
The spectra
E2.
In Figure 2, we show the calculated spectra of C, N, 0, Si and Fe by solid curves. In this figure and in the following figures, the flux values are multiplied by E 2. In order to accommodate many elements in one figure, we scaled down the spectra of N, 0, Si and Fe by factors 2, 100, 100 and 500 respectively, as shown in the figure. Notice that for N, the data by Orth et al (1978) and Simon et al (1980) are well above the rest of the data, while for Si and Fe, Young et al (1981) data branch off from the rest of the data at low energies. As mentioned earlier, all experimental data shown in this and in the following figures were measured during the period of minimum solar activity. Though there is considerable scatter of data points in the figure, especially those for N, the fit to the data is good. The relative abundance of C, N, 0 and Fe
SourceComposition of Cosmic Ray Nuclei
753
with respect to Si (taken to be 100) is 448 : 30 : 524 : 110. These values are given in Table 1 and compared with those given by Webber abundances
(Garrard
(199’7), 1ocal galactic
material
(Grevesse
and Anders,
1989) and solar corona
and Stone, 1993).
The spectra of Mg, Ne, Al, Ar, F and S are shown in Figure 3, by suitably reducing the absolute magnitude by factors 5, 10, 20, 300 and 1500 for the nuclei Ne, Al, Ar, S and F respectively. The solid curves in this figure are the calculated spectra. As in the case of N and Si, the flux values reported by Orth et al. are well above other measurements.
The derived source abundances
of these elements are tabulated
other estimates in Table 1. It can be seen from the figure that the uncertainty of some of the elements are large. knows the systematic
It is difficult to estimate the error in the derived abundance
error in each observation
and in the measured
along with
in deriving the abundances unless one
cross-sections.
1
_r- 10
-I
ii ‘:
-2
k 10 ‘? & p
-3
2 10 0 3
10
-4
.s 2 k
10
-5
2 E _(j “w 10
10
-1
V Orth et al (6) * Caldwell et al (10) 0 Juliusson (2)
1
10
lo3
lo2
1
Kinetic Energy in GeVln
Fig.
3. The observed
are plotted with
the calculated
The observed calculated
spectra
by multiplying
of Mg,
them
10
lo2
lo3
Kinetic Energy in GeVln
Ne, Al, Ar,
F, S
by E2 and compared
spectra.
Fig.
4. The calculated
compared
with
the spectra
spectra
of Na, K, P, Ca, Cl are
the observation.
are scaled,
spectra of elements Na, K, P, Ca and Cl are plotted
spectra
A HEAO (7) 0 Lezniak and Webbe
I(10)
Maehl et al (5)
The factors
are shown
with
by which
symbols.
in Figure 4. For a comparison,
the
are shown by the solid curves.
The absolute magnitude of the spectra for some of the elements K, P, Ca and Cl were reduced by factors 2, 10, 200 and 500 respectively, to accommodate them in a single figure.
It can be seen that the measurements
of Young et al (198l)differ
from the rest by a large
amount at low energies; similar trend existed in all other data, but with smaller magnitude. considerable
paucity of data for some elements.
The input values of the abundances
respect to Si are given in Table 1 along with other estimates. The values of K and Ca given by Webber in Table 1 are solar system abundances.
There is also
of these elements with It can be seen from this
Table that Cl is found to be absent in cosmic ray sources, similar to the solar Corona and local galaxy. The present investigation also shows that the source abundances of P is consistent with zero, while the estimate by Webber
shows a finite value.
could be also consistent
On the other hand, we obtained
with zero. There are also considerable
a small source abundance
for F, which
differences between our estimates
and that
of Webber. Our estimated values are higher for elements N, Na, Al, Ar and Fe than those given by Webber, and for elements Mg and S, our estimates are lower than his values. The implications of these results will
S.A. Stephens andR.E.Streitmatter
154
be examined in a later paper. Table
1. Comparison
Element & charge 6 (C) 7 (N) 8 (0) 9 (F) 10 (Ne) 11 (Na) 12 (Mg) 13 (Al) 14 (Si) 15 (P) 16,(S) 17 (Cl) 18 (Ar) 19 (K) 20 (Ca) 26 (Fe)
CRS Present 448 30 524 0.95 61 5.3 95 9.7 100 0 11.9 0 2.4 0.48 5.7 110
of source
abundances
CRS
Local
Webber 443516 24.1f3 536flO 63.6&4 3.9f0.4 105.5f2 8.3f0.4 100 0.9f0.15 13.6f0.5
Galactic 286f20 78f3 632f20
Solar Corona 1010 313 2380
93f7 7.05.7 123f5 9.550.8 100 0.45fO.l 21.8f0.9
340f48 5.7f.4 107&4 8.5f0.3 100 1.0fO.l 50.5h6.7
1.85f0.2 0.40 6.1 100.5f3
22f0.2 0.33f0.07 7.1f0.3 96.5f6
lO.lf0.6 0.4f0.03 6.150.4 90f2.4
REFERENCES Benegas, J.C., M.H. Israel, J. Klarman, and R.C. Maehl, Proc. 14th Int. Cosmic ray Conf. (Munchen), 1, 251, 1975. Buckley, J., J. Dwyer, D. Muller, and S. Swordy, ApJ, 429, 736, 1994. Caldwell, J.H., and P. Meyer, Proc. 15th Int. Cosmic Ray Conf. (Plovdiv), 1,243, 1977. Caldwell, J.H., ApJ, 218, 269, 1977. Chapell, J.H., and W.R. Webber, Proc. 17th Int. Cosmic Ray Conf. (Paris), 2, 59,1981. Dwyer, R., ApJ, 224, 691, 1978. Garrard, T.L., and E.C. Stone, Proc. 23rd Int. Cosmic Ray Conf. (Calgary), 3, 384, 1993. Garcia-Munoz, M., S.H. Margoiis, J.A. Simpson, and J.P. Wefel, Proc. 16th Int. Cosmic Ray Conf. (Kyoto), 1, 310, 1979. Grevesse, N., and E. Anders, AIP Conf. Proc. No: 183, N.Y. AIP Press, pl. Ichimura, M., M. kogawa, S. Kuromata, H. Mito, T. Murabayashi, et al., Phys. Rev., 48D, 1949,1993. Juliusson, E., ApJ, 191, 331, 1974. Lezniak, J.A., and W.R. Webber, ApJ, 223, 676, 1978. Maehl, R.C., J.F. Ormes, A.J. Fisher, and F.A. Hagen, Ap & Sp. Sci., 47, 163, 1977. Muller, D., S.P. Swordy, P. Meyer, J. L’Heureux, and J.M. Grunsfeld, ApJ, 374, 356,199l. Orth, C., B. Buffington, G.F. Smoot, and T. Mast, ApJ, 226, 1147, 1978. Perko, J.S., A & A, 184, 119, 1987. Simon, M., H. Spiegelhauer, W.K.H. Schmidt, F. Siohan, J.F. Ormes, et al., ApJ, 239, 712, 1980. Young, J.S., P.S. Freier, C.J. waddington, N.R. Brewster, and R.K. Fickle, ApJ, 246, 1014, 1981. Stephens,S.A., and R.E. Streitmatter, ApJ, 505, 266, 1998. Streitmatter,R.E., and S.A. Stephens, Proc. 26th Int. Cosmic Ray Conf. (Salt Lake City), 4, 199,1999. Streitmatter,R.E., and S.A. Stephens, in this issue, 2000. Webber, W.R., Sp. Sci. Rev., 81, 107, 1997.
Pergamon www.elsevier.com/locate/asr PII: SO273-1177(Ol)OOI23-5
SOURCE
COMPOSITION
OF COSMIC
RAY NUCLEI
S.A. Stephens1 and R.E. Streitmatter’
‘Department of Physics, University of Tokyo, 7-3-l Hongo, Bunkyo-ku, Tokyo 113-0033, Japan, 2Laboratory for High Energy Physics, Code 661 NASA/GSFC, Greenbelt, MD 20771, USA
ABSTRACT During propagation
in the Galaxy, cosmic-ray
the time of acceleration
gets modified.
from the observed abundance. variable to examine cross-sections,
nuclei undergo spallation and as a result, their composition
Propagation
at
models have been used to obtain the source composition
We made use of the standard leaky-box model with time as the propagation of all elements. The main ingredients in this method are the spallation
the spectra
which has considerably
show that the spectral
changed over the time.
Using the latest available cross-sections,
we
injection spectrum in rigidity is -2.32. We also derive the The derived abundances differ from those of Webber for and the source abundances.
escape path-length
index of the power-law
some of the elements.
Published by Elsevier Science Ltd on behalf of COSPAR.
INTRODUCTION The observed
cosmic rays contain some universally
heavier cosmic-ray
nuclei while traversing
during the propagation considerably
changed.
is about
using propagation
composition
components.
especially
more accurately, We examined
than before. this problem
model.
cross-sections.
poorly
length,
their composition
has been determined Early attempts
source composition
potential
from the observed
the calculated
with semi-
differed from the solar
is larger than 10 eV. Recently,
However, the observed
spectra are not yet well determined. in the frame work of the simple
the spectral index of the injection
spectrum
instead of the
of the accelerated
spectra with the observed spectra of some of the dominant
nuclei by
nuclei, after correcting
for solar modulation. calculated
B/C
The escape time and its rigidity dependence were obtained from a comparison with the observed data. These are important input parameters and 10Be/gBeratios
are needed for these calculations. them with the observed
We then proceeded
data by adjusting
and present the final source composition PROPAGATION
OF
COSMIC
We follow a simple concept
new
have been made and one is in a position to make these estimates
by making use of the steady state propagation
We determined
gets
to determine the source
flux values at a fixed energy, using slab model
whose first ionization
of
reflects the source composition,
Unlike in the past, we made use of the time as the variable for propagation
matter traversal. comparing
for elements
composition
The derived cosmic-ray
of the spallation cross-sections
by the spallation
Since the amount of matter traversed by them
The source composition
were made from the measured
composition,
leaky-box
cosmic-ray
models and spallation
empirical relations for the cross-section. measurements
rare nuclei, which are produced gas.
the same or larger than the interaction
Thus, the observed
except for the very dominant abundances
interstellar
to determine
the source composition. obtained
of the which
the spectra of all nuclei and compared In this paper, we describe
this method
from this investigation.
RAYS
for the confinement
of cosmic
rays based on the leaky-box
model.
In this
model, cosmic rays diffuse freely in a volume and a small fraction escapes at a constant rate. Over a sufficient period of time, cosmic rays will reach a state of equilibrium. of particles
dE, due to energy 1998) we discussed function
When they are in this steady state, the number
dN that come into any energy interval dE in unit time is the same as the number lost from loss processes,
interaction
and escape.
In our earlier paper (Stephens
in detail the results from the conventional
practice
of describing
of the matter traversed (z) by them. We describe below the propagation 749
and Streitmatter,
the propagation
as a
of cosmic rays in time (t)
750
S. A. Stephens and R. E. Streitmatter
(see Streitmatter and Stephens, 1999). The differential equation describing the propagation of cosmic rays in the Galaxy is written using the general formalism given by Ginzburg & Syrovatskii (1964) as,
- Ni(E,
t)
&(E)7wi
1.0 + T?(E)
1.0 +
+ &
Qi(E,t)
(1)
The left hand side in the above equation is the rate at which the number density of the i-th nucleus varies with time. Here, N(E, t) is the number of particles of energy E in an energy interval dE per unit volume and is related to the flux J(E, t) by the relation Ji(EJ) = Ni(E, t)w;/4 ?r, where vi is the particle velocity at energy E. The first term on the right hand side is the energy loss term, where dE/dt is the ionization loss per unit time. The second term is the loss due to interaction, escape and decay. In this term, oi is the interaction cross-section of the i-th nucleus with an atom of mass < m >, which is the equivalent atomic mass of the interstellar medium (ISM) for one hydrogen atom, and n is the mean number density of hydrogen atoms in the ISM. Observations indicate that mean escape time r?(E) is rigidity dependent and hence depends upon the mass to charge ratio. T: is the decay life time (if radioactive) and y; is the Lorentz factor of the nucleus at energy E. The last term relates to the continuous production of particles in the Galaxy and this production rate term is given by Qi(E, t) = 6
Nj(Ey t)aij(E)vjn
+ C
y
l#i
j>i
(2)
+ qi(E, t) 1
Here, the first term is the production of i-th nucleus by the fragmentation of heavier nuclei, where uij(.E) is the partial fragmentation cross-section to produce the i-th nucleus from interaction of the j-th nucleus of higher mass. The second term is due to the radioactive decay. Contribution from the radioactive decay comes only from nuclei of adjacent charges with the same mass, as the long-lived isotopes decay through beta decay. The last term is the primary source term, which relates to those nuclei that are accelerated in the source. In the above equation Qi(Ey t) is expressed as the particle number density per unit time. Cosmic rays are in a state of equilibrium when 6’NJdt = 0, which would mean that Q(E, t) is independent oft. We have shown earlier (Stephens and Streitmatter, 1998), here after referred to as Paper I, that the set of differential equations given by Eq.(l) can be solved in an elegant manner by integrating these equations simultaneously by the Rungs-Kutta technique until equilibrium is reached, when aNi/& -+ 0 for all nuclei. The equilibrium spectra obtained from such a procedure then represent a full and exact numerical solution of Eq.(l). In the following sub-sections, we discuss some of the input parameters used in this study. Ionization loss In order to evaluate the ionization term in Eq. (l), we adopted a simple technique described in Paper I. Since the ionization loss rate is usually defined per unit gram of material traversed, the partial differential term takes a form in t different from that in z. The ionization loss term in Eq. (1) can be expanded as
&
b(E)g]
= A
[N(Elvpg]
= N(E)pv&
(g)
t N(E)pv
t
(g)
&v
( >p+(E) g
In this investigation, the ionization loss rate was obtained from the range-energy relation. The particle range and its differential energy spectrum can be expressed without loss of accuracy, as power-laws in energy over an energy region larger than the energy spacings used in this numerical integration. Thus, at energy E, if we express the range as R = aEb and the energy spectrum as N(E) cc Ep, then the fist, second and third terms in the above equation can be written as N(E)&1 - b)Evb/ab, N(E)ppvE-‘/ab and N(E)p~E-~/[aby(7 + l.O)]respectively and Eq. (3) can be re-written as
A
k(E):]
= N(Ez-bpu
[P t (1 - b) t 7(7~l~o~]
Source Composition of Cosmic Ray Nuclei
751
Note that in the range-energy relation a is expressed in g/cm 2. The above expression differs from that derived for the propagation in z (Eq. 8 in Paper I) by the addition of the last term l.O/[y(y + l.O)]. We made use of the standard tables for the range of protons in hydrogen and in helium. These range-energy relations for the protons were then scaled by A/Z2 for other isotopes. The mean number density of hydrogen was taken to be 0.2 atom/cm3 (Streitmatter and Stephens, 2000) and the range-energy relation was modified by taking into account the composition of ISM with 80% neutral hydrogen, 10% ionized hydrogen and 10% helium by number (see Paper I). Other parameters We need to make assumptions regarding the form of the escape time as a function of energy and we incorporated a relation, which is equivalent to the mean matter traversed by cosmic rays, namely P(E)
= F”(R/Ro)-” = re*
for
set
set
R > Ro
for
R<
R,,
(5)
where red is the constant escape time, R is the rigidity of the nucleus, R~Jis the rigidity at which rigidity dependent escape commences, and a is the index. Note that there is no velocity dependence in the escape time, because the escape time is related to the commonly used escape length X;(E) in x by the relation Ro and a) in the above equation need to be X;(E) =< m > nvi(E)~:‘(E). The three constants (?, determined.
10
1
10
Kinetic Energy in GeV/n
Fig.
1.
modulation
The
calculated
parameter
B/C
Phi=0.5
ratio
is shown
along
with
the
data
for
a
GV.
Another important parameter needed for this calculation is the fragmentation cross-section as a function of energy. We made use of the cross section tables provided by Webber (private communication) to obtain the required cross-section matrix to be used in solving the set of Eq. (1). The injection spectrum used in this calculation is a power law spectrum in rigidity. RESULTS Propagation parameters The spectral index of the source spectrum and the three constants in Eq. (5) were determined in the following manner. We assumed a power-law spectrum in rigidity as the source spectrum and solved the set of equations described by Eq. (1). The equilibrium spectra of all nuclei were then modulated using spherically symmetric solution (Perko, 1987). From these, we compared the spectra of the dominant nuclei C, 0 and Fe, and the B/C ratio with the observation. The spectral index and the escape parameters in Eq. (5) were adjusted until an excellent fit to all the above data were obtained. The equivalent modulation
752
S. A.
Stephens andR. E. Streitmatter
parameter used in the computation, which gave a consistent fit to the data, was 500 MV. Most of the data used here were taken from experiments carried out during the period of minimum solar activity. In Figure 1 we show the observed B/C ratio as a function of energy from a compilation of data made in Paper I. The solid curve is the best fit that was obtained from this calculation. The derived values of the three parameters in Eq. (5) are: (1) Ro = 2.7GV, (2) th e index a! = 0.5, and (3) the escape life time res corresponds to 14 g/cm2 of ISM traversed by cosmic rays, when v = c. The required spectral index of the injection spectrum, which gave the best fit to the spectra of C, 0 and Fe at high energies (Figure 2), is /? = -2.32. The value of res obtained here is smaller by about 12% than that obtained in Paper I as a result of the new set of cross-sections employed and partly due to the slightly steeper spectrum used in this calculation. Elemental
abundances
0 Simonet al (8) + Muller et al (14) * Carvilleet al (11)
Maehelet al (5)
1
10 -l
lo3
lo2
10
Kinetic Energy in GeVh
Fig.
2.
The
compared
with
are scaled
down
are multiplied
calculated observation. are shown by
spectra The along
of C,
N, 0,
factor
by which
with
the symbols.
Si and the
Fe are spectra
The spectra
E2.
In Figure 2, we show the calculated spectra of C, N, 0, Si and Fe by solid curves. In this figure and in the following figures, the flux values are multiplied by E 2. In order to accommodate many elements in one figure, we scaled down the spectra of N, 0, Si and Fe by factors 2, 100, 100 and 500 respectively, as shown in the figure. Notice that for N, the data by Orth et al (1978) and Simon et al (1980) are well above the rest of the data, while for Si and Fe, Young et al (1981) data branch off from the rest of the data at low energies. As mentioned earlier, all experimental data shown in this and in the following figures were measured during the period of minimum solar activity. Though there is considerable scatter of data points in the figure, especially those for N, the fit to the data is good. The relative abundance of C, N, 0 and Fe
SourceComposition of Cosmic Ray Nuclei
753
with respect to Si (taken to be 100) is 448 : 30 : 524 : 110. These values are given in Table 1 and compared with those given by Webber abundances
(Garrard
(199’7), 1ocal galactic
material
(Grevesse
and Anders,
1989) and solar corona
and Stone, 1993).
The spectra of Mg, Ne, Al, Ar, F and S are shown in Figure 3, by suitably reducing the absolute magnitude by factors 5, 10, 20, 300 and 1500 for the nuclei Ne, Al, Ar, S and F respectively. The solid curves in this figure are the calculated spectra. As in the case of N and Si, the flux values reported by Orth et al. are well above other measurements.
The derived source abundances
of these elements are tabulated
other estimates in Table 1. It can be seen from the figure that the uncertainty of some of the elements are large. knows the systematic
It is difficult to estimate the error in the derived abundance
error in each observation
and in the measured
along with
in deriving the abundances unless one
cross-sections.
1
_r- 10
-I
ii ‘:
-2
k 10 ‘? & p
-3
2 10 0 3
10
-4
.s 2 k
10
-5
2 E _(j “w 10
10
-1
V Orth et al (6) * Caldwell et al (10) 0 Juliusson (2)
1
10
lo3
lo2
1
Kinetic Energy in GeVln
Fig.
3. The observed
are plotted with
the calculated
The observed calculated
spectra
by multiplying
of Mg,
them
10
lo2
lo3
Kinetic Energy in GeVln
Ne, Al, Ar,
F, S
by E2 and compared
spectra.
Fig.
4. The calculated
compared
with
the spectra
spectra
of Na, K, P, Ca, Cl are
the observation.
are scaled,
spectra of elements Na, K, P, Ca and Cl are plotted
spectra
A HEAO (7) 0 Lezniak and Webbe
I(10)
Maehl et al (5)
The factors
are shown
with
by which
symbols.
in Figure 4. For a comparison,
the
are shown by the solid curves.
The absolute magnitude of the spectra for some of the elements K, P, Ca and Cl were reduced by factors 2, 10, 200 and 500 respectively, to accommodate them in a single figure.
It can be seen that the measurements
of Young et al (198l)differ
from the rest by a large
amount at low energies; similar trend existed in all other data, but with smaller magnitude. considerable
paucity of data for some elements.
The input values of the abundances
respect to Si are given in Table 1 along with other estimates. The values of K and Ca given by Webber in Table 1 are solar system abundances.
There is also
of these elements with It can be seen from this
Table that Cl is found to be absent in cosmic ray sources, similar to the solar Corona and local galaxy. The present investigation also shows that the source abundances of P is consistent with zero, while the estimate by Webber
shows a finite value.
could be also consistent
On the other hand, we obtained
with zero. There are also considerable
a small source abundance
for F, which
differences between our estimates
and that
of Webber. Our estimated values are higher for elements N, Na, Al, Ar and Fe than those given by Webber, and for elements Mg and S, our estimates are lower than his values. The implications of these results will
S.A. Stephens andR.E.Streitmatter
154
be examined in a later paper. Table
1. Comparison
Element & charge 6 (C) 7 (N) 8 (0) 9 (F) 10 (Ne) 11 (Na) 12 (Mg) 13 (Al) 14 (Si) 15 (P) 16,(S) 17 (Cl) 18 (Ar) 19 (K) 20 (Ca) 26 (Fe)
CRS Present 448 30 524 0.95 61 5.3 95 9.7 100 0 11.9 0 2.4 0.48 5.7 110
of source
abundances
CRS
Local
Webber 443516 24.1f3 536flO 63.6&4 3.9f0.4 105.5f2 8.3f0.4 100 0.9f0.15 13.6f0.5
Galactic 286f20 78f3 632f20
Solar Corona 1010 313 2380
93f7 7.05.7 123f5 9.550.8 100 0.45fO.l 21.8f0.9
340f48 5.7f.4 107&4 8.5f0.3 100 1.0fO.l 50.5h6.7
1.85f0.2 0.40 6.1 100.5f3
22f0.2 0.33f0.07 7.1f0.3 96.5f6
lO.lf0.6 0.4f0.03 6.150.4 90f2.4
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