Interpretations of the cosmic ray secondary-to-primary ratios measured by DAMPE Peng-Xiong Maa Zhi-Hui Xuab Qiang Yuanab Xiao-Jun Bicd Yi-Zhong Fanab Igor V . Moskalenkoe and Chuan Yuea aKey Laboratory of Dark Matter and Space Astronomy

2025-05-05 0 0 650.99KB 14 页 10玖币

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Interpretations of the cosmic ray secondary-to-primary ratios measured by DAMPE

Peng-Xiong Maa, Zhi-Hui Xua,b, Qiang Yuana,b∗, Xiao-Jun Bic,d, Yi-Zhong Fana,b, Igor V. Moskalenkoe, and Chuan Yuea

aKey Laboratory of Dark Matter and Space Astronomy,

Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China

bSchool of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China

cKey Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

dUniversity of Chinese Academy of Sciences, Beijing 100049, China

eW. W. Hansen Experimental Physics Laboratory and Kavli Institute for Particle

Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA

(Dated: March 7, 2023)

Precise measurements of the boron-to-carbon and boron-to-oxygen ratios by DAMPE show clear hardenings

around 100 GeV/n, which provide important implications on the production, propagation, and interaction of

Galactic cosmic rays. In this work we investigate a number of models proposed in literature in light of the

DAMPE ﬁndings. These models can roughly be classiﬁed into two classes, driven by propagation eﬀects or

by source ones. Among these models discussed, we ﬁnd that the re-acceleration of cosmic rays, during their

propagation, by random magnetohydrodynamic waves may not reproduce suﬃcient hardenings of B/C and B/O,

and an additional spectral break of the diﬀusion coeﬃcient is required. The other models can properly explain

the hardenings of the ratios. However, depending on simpliﬁcations assumed, the models diﬀer in their quality

in reproducing the data in a wide energy range. The models with signiﬁcant re-acceleration eﬀect will under-

predict low-energy antiprotons but over-predict low-energy positrons, and the models with secondary production

at sources over-predict high-energy antiprotons. For all models high-energy positron excess exists.

PACS numbers: 96.50.S-

I. INTRODUCTION

Galactic cosmic rays (GCRs) are energetic particles pro-

duced by powerful astrophysical objects such as the remnants

of supernova explosions. After being accelerated up to very

high energies, they propagate and interact in the Milky Way

before entering the solar system and being recorded by our

detectors. There are typically two types of GCRs, the primary

family (such as protons, helium, carbon, oxygen, neon, mag-

nesium, silicon, and iron) which is produced directly by ac-

celeration at their sources and the secondary family (such as

lithium, berylium, boron, and sub-iron nuclei) which is pro-

duced via fragmentations of primary particles mainly during

the propagation process. Precise measurements of the ratios

between secondary particles and their parent primary particles

are important probe of the propagation of GCRs as well as the

turbulent properties of the interstellar medium (ISM) [1–3].

Among various secondary-to-primary ratios of nuclei, the

boron-to-carbon ratio (B/C) is the best measured and most

widely studied. Measurements of B/C up to kinetic energies1

of hundreds of GeV/n have been achieved with good preci-

sion by many experiments [4–16], which were extensively

used to constrain the propagation of GCR models (e.g., [17–

30]). The B/C ratio above O(10) GV can be well ﬁtted by a

power-law function of rigidity, ∝ R−δ, with δ≈1/3 [12], in

agreement with the prediction of GCR diﬀusion in the ISM

∗yuanq@pmo.ac.cn

1In this paper, we are necessarily using the mixed energy units: discussions

of the injection spectra and cosmic ray transport is done in terms of rigidity,

while a comparison with experiments requires a conversion to the kinetic

energy per nucleon.

with a Kolmogorov type turbulence spectrum [18,31]. Fur-

ther measurements of ratios of secondary lithium, beryllium,

and boron to primary carbon and oxygen by AMS-02 jointly

showed a hardening [4,32]. Non-trivial spectral shapes of the

secondary-to-primary ratios thus challenge the simple produc-

tion and propagation models of GCRs.

Very recently, high-precision measurements up to 5 TeV/n

of the boron-to-carbon (B/C) and boron-to-oxygen (B/O) ra-

tios have been obtained by the Dark Matter Particle Explorer

(DAMPE; [33,34]). The DAMPE results revealed clear hard-

ening of both ratios with high signiﬁcance at nearly the same

kinetic energy of ∼100 GeV/n [35]. A broken power-law ﬁt to

the B/C (B/O) ratio gives a low-energy slope of 0.356 (0.394)

and a high-energy slope of 0.201 (0.187), and the change of

slope is ∆γ=0.155 (0.207). Previous measurements showed

also remarkable hardenings of primary nuclei at similar ener-

gies [36–42]. The slope changes of primary nuclei are about

0.1∼0.2, which are slightly diverse among diﬀerent mea-

surements. These spectral features of GCRs may suggest a

common origin.

A straightforward interpretation of the hardenings of B/C

and B/O is the existence of a break of the diﬀusion coeﬃcient

at a few hundred GV [43–45]. Such a break of the diﬀusion

coeﬃcient may be a consequence of the change of the scale-

dependence of the ISM turbulence, or be due to the nonlin-

ear particle-wave interactions [46]. Other interpretations with

diﬀerent physical models were also proposed (e.g., [47–53]).

These models either employ more complicated propagation

eﬀects or introduce additional sources of (secondary and/or

primary) GCRs beyond the standard paradigm. Some of the

above possibilities have been brieﬂy discussed in Ref. [35].

In this work we further explore these models to test whether

they can explain the DAMPE data satisfactorily. Antiprotons

and positrons from these models will also be discussed as in-

arXiv:2210.09205v3 [astro-ph.HE] 6 Mar 2023

dependent tests of the models.

II. PRODUCTION AND PROPAGATION MODEL OF

GALACTIC COSMIC RAYS

The propagation of GCRs in the Milky Way can be gener-

ally described by the diﬀusion equation

∂ψ

∂t=∇ · (Dxx∇ψ−Vcψ)+∂

∂pp2Dpp

∂

∂p

p2ψ

−∂

∂p˙pψ−p

3(∇ · Vcψ)−ψ

τf

−ψ

τr

+q(r,p),(1)

which includes also the possible convective transportation ef-

fect with velocity Vc, the re-accerlation eﬀect described by a

diﬀusion in the momemtum space with diﬀusion coeﬃcient

Dpp, the energy losses with rate ˙pand adiabatic losses, frag-

mentations with time scale τf, and radioactive decays with

lifetime τr[54]. The source function q(r,p) includes both the

primary contribution from acceleration sources and the sec-

ondary contribution from GCR interactions with the ISM.

The geometry of the propagation halo is assumed to be cyn-

lindrially symmetric, with radial extension Rh=20 kpc and

height ±zhto be determined by the data. The spatial diﬀusion

coeﬃcient is usually assumed to be spatially homogeneous,

and depends on particle rigidity with a power-law form

Dxx(R)=D0βη R

R0!δ

,(2)

where βis the velocity of the particle in unit of light speed,

R0≡4 GV is a reference rigidity, δis the power-law index

describing the properties of the interstellar turbulence. A phe-

nomenological parameter ηis introduced to modify the ve-

locity dependence at low energies, in order to better match the

measurements. We will discuss alternative cases about the dif-

fusion coeﬃcient in this work (see below Sec. III for details).

The convection eﬀect is neglected in this work according to

the ﬁtting to the up-to-date data on GCR primary and sec-

ondary nuclei [17,55]. The momentum diﬀusion coeﬃcient

can be expressed as [56]

Dpp =4p2v2

3δ(4 −δ2)(4 −δ)wDxx

,(3)

where vAis the Alfven speed of magnetized disturbances, wis

the ratio of magnetohydrodynamic (MHD) wave energy den-

sity to the magnetic ﬁeld energy density and can be eﬀectively

absorbed into vA.

The injection spectrum is assumed to be a smoothly broken

power-law function of rigidity

q(R)=q0R−γ0

i=1"1+ R

Rbr,i!s#(γi−1−γi)/s

,(4)

where γ0is the spectral index at the lowest energies, γi−1and

γiare spectral indices below and above break rigidity Rbr,i,

and sdescribes the smoothness of the break which was ﬁxed

to be s=2 throughout this work. Depending on the assump-

tions and purposes of diﬀerent models, diﬀerent numbers of

breaks will be assumed. Speciﬁcally, n=2 will be assumed

in general, except that the high-energy hardening is ascribed

to other physical eﬀects (n=1 in these cases). The spatial

distribution of sources of GCRs is parameterized as

f(r,z)= r

r!α

exp "−β(r−r)

r#exp −|z|

zs!,(5)

where r=8.5 kpc is the distance from the solar system to the

Galactic center, zs=0.2 kpc is the scale width of the vertical

extension of sources, α=1.25, and β=3.56 [24]. Unless

explicitly stated, we will use the GALPROP2code (version

563to calculate the propagation of GCRs [18].

To compare with the low-energy measurements in the solar

system, we use the force-ﬁeld approximation to account for

the solar modulation of GCRs [58]. More sophisticated mod-

els of heliospheric propagation exist, e.g., HELMOD [59], but

using them is beyond the scope of this paper.

III. INTERPRETATIONS OF SPECTRAL BREAKS OF B/C

AND B/O

A. Nested leaky box model

The leaky-box model, which was popular for the most part

of the 20th century, is a simpliﬁed model with uniform dis-

tribution of gas, sources, and cosmic rays where the cosmic

ray transport in the whole Galaxy is described with a sin-

gle parameter, the escape time τesc(R). Neglecting other pro-

cesses such as the convection, re-acceleration, and fragmen-

tation, the solution of the propagation equation is as sim-

ple as ψ(R)=q(R)τesc(R). An extension of the leaky box

model to take into account the residence and secondary pro-

duction in dense regions surrounding the sources, known as

the nested leaky box (NLB) model (denoted as model A), was

proposed to explain more complicated observational proper-

ties of GCRs [47,60,61]. In the NLB model, GCRs were

accelerated to a power-law spectrum R−γ, which diﬀuse in

an energy-dependent way in the immediate vicinity of the

sources (so-called cocoons), and then enter the Galaxy and ﬁ-

nally leak to the extragalactic space in an energy-independent

way. The escape time is assumed to be [47]

(τc

esc =τ1R−ζln R,for cocoons,

τg

esc =τ2≡const,for Galaxy.(6)

For primary GCRs, the propagated spectrum in cocoons is

ψc

pri(R)=q(R)τc

esc ∝ R−γ−ζln R. The propagated spectrum

in the Galaxy is ψg

pri(R)=[ψc

pri/τc

esc]τg

esc =q(R)τg

esc ∝ R−γ,

whose spectral shape is the same as the source spectrum. For

2https://galprop.stanford.edu

3A newer version 57 was recently released [57].

secondary particles, there are two components. The one in co-

coons has a spectrum ψc

sec =ψc

pri ·ncσv·τc

esc. This component

then injects into the Galaxy and experiences a further leak-

age, resulting in a ﬁnal spectrum ψc,g

sec =[ψc

sec/τc

esc]τg

esc. The

other component is directly produced by GCRs in the Galaxy,

whose spectrum is ψg,g

sec =ψg

pri ·ngσv·τg

esc. The total secondary

spectrum is thus ψg

sec =ψc,g

sec +ψg,g

sec =q(R)τg

escσv(ncτc

esc +

ngτg

esc). In the above formulae, ncand ngare the gas densities

in cocoons and the Galaxy, σis the production cross section,

and vis the velocity of the GCR particle.

Here we set nc=1 H cm−3,ng=0.1 H cm−3, and derive

the other parameters through ﬁtting to the data. Since some

complicated physical eﬀects at low energies (e.g., the ioniza-

tion and Coulomb energy losses) are not included in the NLB

model, we focus on the data-model comparison above a few

tens of GeV/n. Several experiments found that the spectra of

carbon and oxygen nuclei are not single power-law, but expe-

rience hardening features around a few hundred GV [36–39].

We therefore assume that the source spectrum is a smoothly

broken power-law form of rigidity with n=1 in Eq. (4).

Fig. 1shows the best-ﬁt B/C, B/O, and C, O ﬂuxes in the

NLB model, compared with the AMS-02 [4,38] and DAMPE

[35] data, where the statistical and systematic errors of the

measurements are added in quadrature. The ﬁtting parameters

are γ0=2.69 ±0.01, Rbr,1=(533 ±292) GV, γ1=2.54 ±0.07

for C, and γ0=2.67 ±0.01, Rbr,1=(864 ±637) GV, γ1=

2.51±0.11 for O. For the secondary-to-primary ratios, we have

τ1σ=(9.42 ±0.43) ×10−12 cm2s, τ2σ=(1.88 ±0.06) ×10−11

cm2s, ζ=−0.07 for the B/C ratio, and τ1σ=(1.04 ±0.06) ×

10−11 cm2s, τ2σ=(1.84 ±0.06) ×10−11 cm2s, ζ=−0.08

for the B/O ratio. There are many channels to produce boron

from fragmentations of carbon and oxygen [62,63]. As an

order of magnitude estimate, we take the total fragmentation

cross section of carbon, ∼250 mb, as a reference and obtain

τ1∼1.3 Myr and τ2∼2.3 Myr.

In the NLB model, the high-energy behaviors of B/C and

B/O asymptotically approach constants due to the energy-

independent leakage in the Milky Way. This energy-

independent leakage predicts a constant dipole anisotropy4of

GCRs above ∼TeV [61], which is at odds with observations.

The NLB model is over-simpliﬁed, neglecting many impor-

tant processes of propagation of GCRs, and fails to reproduce

data in a wide energy range. However, the idea that GCRs may

propagate diﬀerently in diﬀerent regions is important and will

be extended to a spatially-dependent propagation model or a

scenario with conﬁnements and interactions surrounding the

acceleration sources detailed below.

4The density gradient is ignored in the leaky box model. Via an analogy with

the diﬀusion model with diﬀusion coeﬃcient being scaled to the escape

time, the anisotropy in this model was estimated.

B. Re-acceleration during propagation

GCR particles may get re-accelerated via interactions with

randomly moving interstellar MHD waves during their prop-

agation process [56]. This stochastic acceleration process

is usually described by a diﬀusion in momentum space,

with diﬀusion coeﬃcient Dpp. The re-acceleration results in

bump-like spectral features of low-energy (less than tens of

GeV/n) GCRs, and was shown can better explain the peaks of

secondary-to-primary ratios [17,55,64]. The softer spectra of

secondary nuclei experience larger eﬀect of re-acceleration,

leading to a decrease in the secondary-to-primary ratio at low

energies. Fitting to the new measurements of the Li, Be,

B, C, and O ﬂuxes by AMS-02 [4,38] indicates that the

re-acceleration can indeed reproduce well the reported more

signiﬁcant hardenings of the secondary family than the pri-

mary family [48]. We re-visit the question whether the re-

acceleration can explain the even stronger hardenings of the

B/C and B/O ratios measured by DAMPE.

TABLE I: Data used in the ﬁtting.

Experiment Time Ref.

B/C Voyager 2012/12-2015/06 [14]

ACE 2011/05-2016/05 [55]

AMS-02 2011/05-2016/05 [4]

DAMPE 2016/01-2021/12 [35]

B/O ACE 2011/05-2016/05

AMS-02 2011/05-2016/05 [4]

DAMPE 2016/01-2021/12 [35]

C & O Voyager 2012/12-2015/06 [14]

ACE 2011/05-2016/05 [55]

AMS-02 2011/05-2016/05 [38]

10Be/9Be Voyager 1977/01-1998/12 [65]

ACE 1997/08-1999/04 [66]

IMP 1974/01-1980/05 [67]

Ulysses 1990/10-1997/12 [68]

ISOMAX 1998/08-1998/08 [69]

The ﬁtting procedure is similar with Ref. [48]. We include

the Voyager measurements outside the solar system [14], the

5-year AMS-02 secondary-to-primary data and 5-year carbon

and oxygen data [4,38], the ACE-CRIS5measurements with

the same time period of AMS-02, and the DAMPE data. To

reduce the degeneracy between the diﬀusion coeﬃcient and

the halo height, the data of 10Be/9Be from several experiments

are also included [65–69]. The data used in the ﬁtting are

summarized in Table I.

We employ the Markov Chain Monte Carlo (MCMC)

method to do the ﬁtting, using the emcee code [70]. The injec-

tion spectrum takes the form of Eq. (4) with n=2. The best-

ﬁt model parameters are given in Table II (labelled as model

5http://www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA CRIS.html

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Interpretationsofthecosmicraysecondary-to-primaryratiosmeasuredbyDAMPEPeng-XiongMaa,Zhi-HuiXua;b,QiangYuana;b,Xiao-JunBic;d,Yi-ZhongFana;b,IgorV.Moskalenkoe,andChuanYueaaKeyLaboratoryofDarkMatterandSpaceAstronomy,PurpleMountainObservatory,ChineseAcademyofSciences,Nanjing210023,ChinabSchoolofAstrono...

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Interpretations of the cosmic ray secondary-to-primary ratios measured by DAMPE Peng-Xiong Maa Zhi-Hui Xuab Qiang Yuanab Xiao-Jun Bicd Yi-Zhong Fanab Igor V . Moskalenkoe and Chuan Yuea aKey Laboratory of Dark Matter and Space Astronomy

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