At energies below ~300 MeV/nuc our knowledge of cosmic-ray spectra outside the heliosphere is obscured by the energy loss that cosmic rays experience during transport through the heliosphere into the inner solar system. This paper compares measurements of secondary electron-capture isotope abundances and cosmic-ray spectra from ACE with a simple model of interstellar propagation and solar modulation in order to place limits on the range of interstellar spectra that are compatible with both sets of data

Binns, W. R.

Cummings, A. C.

Davis, A. J.

Israel, M. H.

Leske, R. A.

Mewaldt, R. A.

Scott, L. M.

Stone, E. C.

von Rosenvinge, T. T.

Wiedenbeck, M. E.

English

Caltech Authors - Main

Cosmic-Ray Spectra in Interstellar Space  
   R. A. Mewaldta, M. E. Wiedenbeckb, L. M. Scottc, W. R. Binnsc,          
         A. C. Cummingsa, A. J. Davisa, M. H. Israelc, R. A. Leskea,            
E. C. Stonea, and T. T. von Rosenvinged  
aCalifornia Institute of Technology, Pasadena, CA 91126, USA 
bJet Propulsion Laboratory, California Institute of Technology, Pasadena CA  91190, USA 
cDepartment of Physics, Washington University, St. Louis MO. 63105. USA 
dNASA/Goddard Space Flight Center, Greenbelt MD 20771, USA 
Abstract. At energies below ~300 MeV/nuc our knowledge of cosmic-ray spectra outside the 
heliosphere is obscured by the energy loss that cosmic rays experience during transport through 
the heliosphere into the inner solar system. This paper compares measurements of secondary 
electron-capture isotope abundances and cosmic-ray spectra from ACE with a simple model of 
interstellar propagation and solar modulation in order to place limits on the range of interstellar 
spectra that are compatible with both sets of data.  
INTRODUCTION 
Among the most important clues to the nature of Galactic cosmic-ray sources and 
accelerators are measurements of cosmic-ray energy spectra. There are now high-
resolution measurements of the spectra of almost all of the elements from H to Ni at 1 
AU, but to relate these to the spectrum of cosmic rays accelerated in the cosmic-ray 
source(s) it is necessary to take into account significant changes that take place during 
cosmic-ray transport through the interstellar medium and the heliosphere.   
It is generally believed that most galactic cosmic rays are accelerated by supernova 
shock waves traveling through the interstellar medium (ISM) [1, 2], leading to “source 
spectra” of the form dJ/dT ∝ P–(2+δ), where T is kinetic energy/nuc, P is 
momentum/nuc and δ depends on the Mach number of the strongest shocks 
encountered.  At energies above a few GeV/nuc, the spectra of “primary” species (e.g., 
H, He, C, O, Ne, Mg, Si, and Fe) all have observed slopes of dJ/dT ∝ T–2.75. In 
contrast, “secondary” species that are rare in the source material (e.g., Li, Be, B, and 
products of Fe fragmentation such as Sc, Ti, and V) all have softer spectra, and 
secondary/primary ratios such as B/C and (Sc+Ti+V)/Fe decrease with increasing 
energy due to rigidity-dependent escape from the Galaxy. The cosmic-ray 
transport/solar-modulation model used in this study assumes source spectra 
proportional to P–2.35 for all primary species and a rigidity-dependent mean free path 
for escape from the Galaxy [3, 4] of a form suggested by Soutoul and Ptuskin [5].   
At energies below a few hundred MeV/nuc our knowledge of interstellar cosmic-
ray spectra is obscured by the heliosphere – low-energy cosmic rays are to a large 
extent prevented from entering the inner heliosphere by the interplanetary magnetic 
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field, and a considerable range of interstellar spectral shapes can be shown to be 
consistent with 1-AU observations with appropriate choices for the interplanetary 
diffusion coefficient. In addition, because cosmic rays lose at least several hundred 
MeV/nuc as they traverse the inner heliosphere [6], we have essentially no information 
on the interstellar spectra of cosmic rays below ~200 to ~300 MeV/nuc.  
It is hoped that the Voyagers, or an Interstellar Probe, will eventually escape the 
cosmic-ray modulation region and measure interstellar cosmic-ray spectra directly. 
However, it has recently become clear that a considerable fraction of “cosmic-ray 
modulation” occurs in the heliosheath, well beyond the present location of the 
Voyagers [7, 8, 9, 10]. A comparison of interstellar spectra used in several recent 
studies is shown in Figure 2, along with solar-minimum data from ACE and Voyager.  
For these interstellar spectra to be consistent with the observations, modulation in the 
heliosheath must be between ~1.5 and ~3 times that between 1 AU and 70 AU. 
 
 
 
FIGURE 1.  Comparison of local interstellar carbon spectra used in several recent studies [11, 12, 13, 
14].  Also shown are solar minimum carbon data from ACE at 1 AU and Voyager-1 at ~70 AU [13]. 
 
Recently, a new probe of this energy-loss process was demonstrated that is based 
on studies of the energy spectra of the secondary radioactive isotopes 49V and 51Cr, 
which decay only by electron capture (EC). In cosmic rays the abundances of EC 
isotopes is related to the probability of attaching an electron from the IS gas: at low 
energies attachment and decay is much more likely.  In a recent paper Niebur et al. 
[15] used isotope observations from ACE to provide direct evidence that the amount 
of energy loss increases by 400 to 700 MV (~200 to 300 MeV/nuc) from solar 
minimum to solar maximum (1997-1998 to 2000-2001). In this paper we extend the 
work of Niebur et al. and investigate the range of interstellar spectra that is consistent 
with solar-cycle observations of both the 51V/51Cr ratio and the Fe spectrum at 1 AU 
(note that 51V is the daughter of 51Cr decay, and that Fe fragmentation produces most 
of the Ti, V, and Cr observed in cosmic rays).  
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APPROACH 
Niebur et al. [15] used a cosmic-ray transport and solar-modulation model [4, 3], 
which we also adopt here. The model assumes that primary cosmic rays are 
accelerated to produce source spectra dJ/dT ∝ P–2.35.  In addition, cosmic-ray sources 
are assumed to be uniformly distributed in space and time, with a rigidity-dependent 
mean free path for leakage from the Galaxy [3, 5] and a uniform interstellar density of 
0.3 H atoms/cm3, based on observations of four radioactive clocks [4]. Nuclear 
fragmentation, ionization energy loss, radioactive decay, and electron-attachment and 
stripping are taken into account.  The first stage of the model results in interstellar 
spectra for species from Be to Ni, which then serve as input to a spherically-symmetric 
solar modulation model that includes diffusion, convection, and adiabatic energy loss.   
It is common to characterize the solar modulation level over the solar cycle by the 
“modulation parameter“, φ,  [16], where 
 
φ  ∝  ∫ (Vsw/κ)dr        (in MV). 
 
Here Vsw is the solar wind speed, κ is the diffusion coefficient, and the integral extends 
from 1 AU to the boundary of the modulation region (Rb). In practice, many κ, Vsw and 
Rb combinations have the same effect. There is also a wide range of interstellar spectra 
that can produce the same intensity level at 1 AU with a judicious choice of κ, 
including the spectra in Figure 1. In this paper we will attempt to narrow the range of 
possible interstellar spectra by requiring agreement with EC isotope data.  
Calculations for the P–2.35 source spectra considered by Niebur et al. [15] are 
compared with spectral and isotope data from ACE in Figure 2.  Note that the Fe 
spectra at solar minimum require φ ≈ 400 MV, while solar-maximum Fe requires φ ≈ 
800 to 1000 MV. The right hand panel shows that the solar minimum 51V/51Cr ratio is 
consistent with φ ≈ 400 MV.  Niebur et al. also used the 49Ti/49V ratio in their study; in 
this paper we consider only the 51Cr to 51V decay. 
 
0.2
0.3
0.4
0.5
0.6
0.7
100 200 300 400 500
51
V/
51
Cr
Kinetic Energy (MeV/nucleon)
φ = 0
200
400
600
800
1000
1200
1400
1600
2000
dJ/dE ~ P-2.35
Solar Minimum
10-9
10-8
10-7
10-6
10 100 1000
Fe Spectra at 1 AU
Source Spectra: dJ/dE ∼ P-2.35
Pa
rt
ic
le
s/
(cm
2 s
r-
se
c-
M
eV
/n
u
c)
Kinetic Energy (MeV/nucleon)
Phi = 0
200
400
600
800
1000
1400
2000
 
FIGURE 2.  The results of the model calculations for P–2.35 source spectra (solid lines; labels by solar 
modulation level) are compared with measurements from ACE.  The left panel shows solar minimum 
and maximum Fe spectra (labeled by the modulation parameter φ), while the right panel shows the 
51V/51Cr ratio.  
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In order to study a range of possible interstellar spectra, we consider source spectra 
of the form dJ/dT = Ci(T + To)–2.35 [17, 18] where Ci depends on species, and To is 
independent of species. We consider To values from 200 to 1000 MeV/nuc (Figure 3), 
including spectral shapes both softer and harder than P–2.35. Complete transport and 
solar-modulation calculations were performed for source spectra with To = 200, 400, 
600, 800, and 1000 MeV/nuc. Interstellar spectra for 51V, 51Cr, and Fe were used to 
calculate 1-AU spectra for modulation parameters φ from 0 to 2000 MV in steps of 50 
MV.  
100
101
102
103
200 400 600 800 1000 3000 5000
Cosmic Ray Source Spectra
R
e
la
tiv
e
 
In
te
n
s
ity
 
Kinetic Energy (MeV/nuc)
T
o
 = 0
200
400
600
800
1000
j(T) = K(T+T
o
)-2.35
P-2.35
 
 
FIGURE 3.  Comparison of the source spectra considered in this study. 
 
101
102
103
0 200 400 600 800 1000 1200 1400
Solar Minimum
χ2
Modulation Parameter, φ (MV)
51V/ 51Cr 
Fe Spectrum
at 1 AU
370 ± 20 MV
430 ± 65 MV dJ/dE ~ P
-2.35
10
100
0 250 500 750 1000 1250 1500
Modulation Parameter , φ (MV)
Solar Minimum
420 ± 50 MV
820 ± 20 MV
Fe Spectrum 
at 1 AU
51V/ 51Cr
dJ/dE ~ (T+200)-2.35
 
 
FIGURE 4.  Plots of χ2 vs. the modulation parameter φ for P–2.35 and (T+200)–2.35 source spectra 
at solar minimum, including both the Fe intensity and 51V/51Cr ratio. The bars indicate the range of φ 
over which χ2 varies from a minimum to χ2 +1. There are ten degrees of freedom for the Fe spectra 
evaluations and six degrees of freedom for 51V/51Cr. For P–2.35 source spectra there is consistency 
between the χ2 minima; for (T+200)–2.35 source spectra it is not possible to fit both with the same φ 
value. 
 
The calculated Fe intensity and 51V/51Cr ratios at 1 AU were compared to ACE 
measurements at both solar minimum and maximum and a χ2 goodness of fit was 
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evaluated for each.  For the Fe spectra we used data from both the SIS and CRIS 
instruments on ACE; the isotope data are from CRIS (5). In Figure 4 we show how the 
χ2 values vary with the choice of φ for P–2.35 and (T + 200)–2.35 spectra at solar 
minimum. Comparisons were made for dJ/dT ∝ (T + To)–2.35 spectra with To ranging 
from 200 to 1000 MeV/nuc.  The goodness of fit was evaluated for the Fe-spectra and 
EC-isotope data at both solar minimum and maximum. Figure 4 illustrates an example 
where the Fe and EC-isotope results are consistent and one where they are not. Figure 
5 summarizes the φ and Το values that minimize χ2, and indicates a range allowed by 
χ2+1 that is centered on To = 575 ± 25 MeV/nuc. The χ2 + 4 limits (2σ) give an 
allowable range To = 410 to 630 MeV/nuc. Note that the solar-minimum comparison 
places tighter restrictions on the IS spectra, as would be expected. 
0
200
400
600
800
1000
1200
1400
1600
0 200 400 600 800 1000 1200
Solar Maximum
T
o
 (MeV/nucleon)
φ (Fe)
φ (EC)
T
o
 = 710 ± 190
MeV/nuc
0
200
400
600
800
1000
1200
1400
1600
0 200 400 600 800 1000 1200
Solar Minimum
φ (
MV
)
T
o
 (MeV/nucleon)
φ (Fe)
φ (EC)
T
o
 = 525 ± 75
MeV/nuc
 
FIGURE 5.  Value of φ for which χ2 is a minimum are plotted vs. the spectral parameter To. The left 
panel includes solar-minimum comparisons with the Fe intensity and 51V/51Cr ratios; the right panel is 
for solar maximum. Values of To compatible with both the Fe intensity and 51V/51Cr ratio are indicated.  
DISCUSSION 
The range of Το values compatible with both the Fe spectra and 51V/51Cr 
measurements (To = 575 ± 25 MeV /nuc) is quite narrow. The range of allowed 
spectral shapes (refer to Figure 3)  is consistent with and roughly centered on the P–2.35 
spectra used by Niebur et al. [15], supporting this choice because it provides a self-
consistent interpretation of the spectral and EC isotope data.  We are now in the 
process of evaluating the other IS spectra in Figure 1 to see whether the φ level 
required to fit the solar-minimum C data at 1 AU is consistent with the φ ≈ 430 ± 70 
MV range that we find for the solar-minimum 51V/51Cr data (see Figure 5). It is clear 
that interstellar spectral shapes significantly steeper than that of Moskalenko et al. 
(Figure 1) are not allowed.  
Two caveats should be considered. The steady-state propagation model used here 
assumes cosmic-ray sources uniform in space and time, such that the observed cosmic 
rays have traversed a broad distribution of interstellar pathlengths. However, the solar 
system is embedded within a low-density (~10–3 H atoms/cm3) region (our “Local 
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Bubble”), within which 10-20 supernovae have exploded over the past 10-20 MY 
[19]. If a significant fraction of the observed cosmic rays have passed through very 
little matter without contributing their share of V and Cr secondaries, the assumptions 
of the model are not satisfied.  For example, Davis et al. [3] and Moskalenko et al. 
[14] have considered models in which cosmic rays that originate within our Local 
Bubble contribute ~20% of the intensity of primary species such as Fe at ~300 
MeV/nuc without contributing significantly to the production of Ti, V, and Cr.  
However, this study has shown that changing the spectral shape or intensity of Fe has 
little effect on the IS 51V/51Cr ratio at a given energy. Therefore, it should still be 
possible to test whether these two-component models can satisfy the spectral and EC 
isotope data at 1 AU over the solar cycle with self-consistent solar-modulation levels.  
A second consideration concerns the geometry of the modulation region in the 
heliosheath. The modulation model used here assumes a standard spherically-
symmetric geometry in computing adiabatic energy-loss. It does not include the 
effects of drifts. While we consider it unlikely that the inclusion of drift effects will 
affect these results significantly, it will be important to evaluate if comparable energy 
losses occur in models with improved representations of modulation processes in the 
heliosheath. 
ACKNOWLEDGMENTS 
This work was supported by NASA at Caltech (under grants NAG5-6912 and 
NAG5-12929), JPL, Washington University, and Goddard Space Flight Center.  
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Cosmic-Ray Spectra in Interstellar Space

9 . L a n g n e r , U . W . , P o t g i e t e r

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Cosmic-Ray Spectra in Interstellar Space

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