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the energy spectrum of cosmic rays , an important key to realize the origin and acceleration mechanism of cosmic rays , can be simply described by a power law over many magnitudes of energy from @xmath1 ev to @xmath2 ev @xcite , except for a few distinctive structures , such as the knee phenomenon around @xmath3 ev .
the spectral power index rapidly steepens from about @xmath4 before the knee to @xmath5 over the knee @xcite .
numerous works on cosmic ray propagation and acceleration mechanisms have been done to investigate the origin of the knee .
these include the diffusive shock acceleration ( dsa ) spectra origin of the knee based on the theory of non - linear diffusive particle acceleration by shock waves from supernova remnants ( snrs ) @xcite , and the contribution of nearby pulsar wind producing very hard spectra in the power - law spectra in the knee region @xcite . however , the origin of the knee structure is still unconfirmed .
currently , the interpretations of this phenomenon are based on phenomenological models from experimental measurements .
many results have been given by direct observations with balloon detectors @xcite and indirect observations with air - shower detectors on high altitude mountains @xcite .
the cream experiment , as a balloon detector , has announced precise measurements of energy spectra for individual nuclei ranging from @xmath6 tev to @xmath7 tev @xcite . as an air - shower ground - based detector located @xmath8 m above sea level
, the tibet-@xmath0 array has presented an all - particle energy spectrum of primary cosmic rays from @xmath9 tev to @xmath9 pev , revealing a sharp knee structure around @xmath10 pev . another air - shower experiment with the same altitude as tibet-@xmath0 , argo - ybj , recently gave a new measurement of the energy spectrum of hydrogen and helium nuclei , exhibiting a clear knee structure @xcite .
the latest measurements released by argo - ybj are investigated by the joint operation of resistive plate chambers ( rpc ) detectors and a cherenkov telescope .
this hybrid detection bridges the gap between balloon detectors and ground - based experiments , improves the shower energy resolution and enhances the capability to discriminate showers induced by light nuclei from events initiated by heavier nuclei .
they yield clear evidence for a knee - like structure in the spectrum of hydrogen and helium nuclei under 1 pev @xcite .    in the following discussions ,
we explain the knee spectra measured by argo - ybj and tibet-@xmath0 by phenomenological models .
the superposition of energy spectrum of individual components described by a simple broken power - law can not reconcile the light components and all - particle energy spectra .
we therefore propose a parametric model to reconcile them . in sec .
@xmath6 we give two different scenarios and parameters to fit the experimental data .
@xmath11 presents the average mass .
finally , sec.@xmath10 gives some discussion and conclusions .
cosmic rays emitted from the sources are most likely accelerated in the strong shock fronts of snrs by the dsa mechanism .
the particles are deflected by chaotic magnetic fields , cross shock fronts frequently , and therefore gain energy up to the pev region .
this acceleration leads to the observed approximate power - law spectrum .
the energy spectrum is modified during diffusive propagation .
the spectra near the sources vary from that observed on the earth , which may due to nuclear spallation or decay , ionization losses , leakage from the galaxy and solar modulation for low energies .
we try to adopt two different phenomenological models to explain the observed energy spectra around knee structure .
one is an expression of a simple broken power - law and the other is the broken power - law with non - linear modification term .
the all - particle primary energy spectra derived from extensive air shower experiments can be described by a broken power - law function .
we therefore use this kind of function to parameterize the observed differential energy spectra for individual elements of cosmic ray , which can be written as @xmath12 where @xmath13 is the differential flux of the individual element with energy @xmath14 , @xmath15 is the normalization constant , @xmath16 represents the break point of the proton spectrum , @xmath17 is the atomic number , @xmath18 is the power index in the energy range @xmath19 , and @xmath20 denotes the power index when @xmath21 .
we try to use this expression to fit both the combined energy spectrum of hydrogen and helium nuclei observed by argo - ybj and the all - particle energy spectrum given by tibet-@xmath0 .
the interaction model sibyll for tibet data is used in this work .
we firstly fix the energy spectra of each component below the knee region , and then extrapolate the spectra beyond the knee .
the observed data of individual elements ( @xmath22 , @xmath23 , @xmath24 , @xmath25 , @xmath26 , @xmath27 , @xmath28 and @xmath29 ) from cream are adopted .
for the spectrum index of each element , see @xcite .
an expression of simple power - law @xmath30 is used to fit the spectrum of each component , with the fitting parameters @xmath31 and spectral indices listed in table 1 .
cccc @xmath17 & element & @xmath32 & @xmath18 + 1 & h & @xmath33 & 2.66 + 2 & he & @xmath34 & 2.58 + 6 & c & @xmath35 & 2.61 + 8 & o & @xmath36 & 2.67 + 10 & ne & @xmath37 & 2.72 + 12 & mg & @xmath38 & 2.66 + 14 & si & @xmath39 & 2.67 + 26 & fe & @xmath40 & 2.63 +    with equation ( [ eq : zy1 ] ) , we fit the combined spectrum of hydrogen and helium nuclei observed by argo - ybj @xcite , and get the fitting result of energy break point @xmath41 and @xmath42 for the proton .
the break point for other elements can be calculated by @xmath43 .
the spectral indices of all elements at the higher energy range beyond the knee region are assumed to be the same value as the result from argo - ybj , i.e. @xmath44 .
we sum up the energy spectra of all elements to get the approximate all - particle spectrum , seen as the red solid line in fig .
[ fig : zy1 ] .
it can be seen from fig .
[ fig : zy1 ] that this all - particle spectrum based on argo - ybj observation with this simple broken power - law model can not match the observed all - particle spectrum given by tibet-@xmath0 .
the simple broken power - law function therefore can not describe the energy spectrum in the knee region , which perhaps is due to the influence of the dsa mechanism .
[ fig : zy1 ]      since the differential energy spectrum in the form of the simple broken power - law can not reconcile the observations of argo - ybj and tibet-@xmath0 , a new correction term is added , so the broken power - law can now be expressed as @xmath45(1+\frac{e}{z\epsilon_{b}})^{-\delta\gamma } , \end{aligned}\ ] ] where @xmath46 is assumed to be a @xmath17-dependent parameter , and @xmath47 is a constant .
the dsa mechanism may result in a spectrum with concave - up curvature , so we multiply the non - linear term @xmath48 into the simple broken power - law to structure a concave - up form as a phenomenological model . without this non - linear term , the superimposed all - particle spectrum from all elements is smooth and can not fit the sharp knee structure as measured by tibet-@xmath0 .
in addition , we have tried some parameters in a non-@xmath17-dependent form , but these can not match the data , so we finally adopt the form in equation ( [ eq : zy2 ] ) .
an equation to express the non - linear effects is proposed in @xcite .
they propose the non - linear process by a modified exponential cut - off expression for the spectra from sources , and then deduce the observed spectra by the superposition of these spectra . from their formula , the model they deduced is a kind of broken power - law multiplied by a complex modified term . while our expression form is simpler , we modify the simple broken power - law directly as the observed spectra and then parameterize this scenario .
combined with the newly published measurements from argo - ybj , our model is easier to quantify the parameters precisely . a model with precisely confirmed parameters
will help us to review its intrinsic physical issues . from equation ( [ eq : zy2 ] ) , we get a good fit result to match the observations from both argo - ybj and tibet-@xmath0 with parameters @xmath49 , @xmath50 , @xmath51 and @xmath52 for the proton .
@xmath53 values of other elements are assumed to obey the formula @xmath54 .
the fitting spectra are plotted in fig .
[ fig : zy2 ] , where the solid red line represents the all - particle spectrum . from fig .
[ fig : zy2 ] we can see that the all - particle spectrum based on argo - ybj observation can match the observed all - particle spectrum given by tibet-@xmath0 . based on this model , the distinctive knee structure is mostly contributed by heavy elements such as @xmath29 .
[ fig : zy2 ]    in order to obtain the best fitting parameters for the non - linear model , we employ the markov chain monte carlo ( mcmc ) @xcite , a very efficient algorithm , to achieve our goal . considering that the calculation of @xmath55 will be affected by the too small errors of all - particle spectrum given by tibet-@xmath0
, we amplify the errors of this spectrum by taking account of energy resolution .
the initial form of data with errors listed in the literature @xcite are expressed by @xmath56 , where @xmath57 denotes the flux and @xmath58 denotes the systematic error . as a new deduced additional error , a new term
is added to this expression , i.e. @xmath59 , where @xmath18 denotes the spectrum index .
we take the energy resolution @xmath60 as a constant .
the following @xmath61 parameters : @xmath62 , @xmath47 , @xmath16 and @xmath53 , are set free for the mcmc algorithm , where @xmath46 consists of free parameters @xmath63 , @xmath64 and redshift @xmath17 .
the parameters @xmath16 and @xmath53 here refer to protons as mentioned above .
because every spectrum index of elements in the higher energy band beyond the knee is supposed to be the same value , we can get the @xmath53 of other components from the @xmath53 of the proton . by this mcmc algorithm
, we obtain the best fitting parameters as follows : @xmath65 , @xmath66 , @xmath67 and @xmath68 for the proton .
the parameters @xmath46 , @xmath47 and @xmath16 decide the shapes of the energy spectra .
different values result in different spectra , as shown in fig .
[ fig : zy3 ] , where the elements @xmath22 and @xmath29 are taken as examples .          [ fig : zy3 ]    the best fit result is shown in fig .
[ fig : zy4 ] .
the measurements of hydrogen and helium nuclei spectra using the sibyll model by the kascade group @xcite are compared with our model in fig .
[ fig : zy4 ] .
the kaskade group claimed that the energy spectra of both proton and helium show a knee - like feature , but in our model , this seems not to be the case .
[ fig : zy4 ]
a commonly - used quantity to characterize the mass composition of cosmic rays is the mean logarithmic mass , which is defined as @xmath69 @xmath70 being the relative fraction of nuclei of mass @xmath71 .
the average mass is reported by many air shower experiments , but the results are divergent because of the poor primary mass resolution .
we calculate the average mass for two cases : one for the former non - linear model and another for the latter best fitting case , as shown in fig .
[ fig : zy5 ] .    from our derived average mass of cosmic rays , we can see @xmath72 goes up between several hundreds of tev and several tens of pev .
it implies that heavier components occupy an increasing proportion with the increase of energy , and predicts an iron - dominant composition above the knee .
but the components above the knee are not all @xmath29 , because @xmath72 for @xmath29 is about 4.0 .
further experimental measurements of the chemical composition of high energy cosmic rays will verify our prediction .
if the light components dominate the knee structure , this model can be excluded , while if the heavy components dominate the knee structure , this model will be confirmed , and the measurements will be helpful to improve the model parameters .     [
fig : zy5 ]
many mechanisms have been discussed to explain the knee structure of cosmic rays , including the diffusive shock acceleration mechanism of cosmic rays in supernova explosion shock fronts , leakage from the galaxy , interactions with background particles in the interstellar medium , and new physics of interactions in the atmosphere .
some models are likely to be excluded , however , such as the leaky box model , interactions with background particles and new types of interactions in the atmosphere .
the leakage from the galaxy does not give a distinct change of the spectral slope at the knee .
the scenario of interactions with background particles yields a very light mean logarithmic mass , while the measurements indicate an increase of mass with energy . for new types of interactions in the atmosphere ,
there are no inconsistencies between the different air shower components by the simultaneous observations from kascade , and the measurements of different air shower components can be interpreted with standard particle physics @xcite .
we propose two scenarios to reconcile the light components and all - particle energy spectra by phenomenological models . in the first scenario
, we use a simple broken power - law model to fit the combined energy spectrum of hydrogen and helium nuclei measured by argo - ybj and the all - particle spectrum measured by tibet-@xmath0 , but the all - particle spectrum can not be fitted by this model . hence , the simple broken power - law fails to explain the knee structure . in the second scenario ,
a non - linear modification term is added to the broken power - law .
this modified broken power - law model with non - linear acceleration effects can describe the sharp knee structure .
the mechanism of non - linear effects at shock fronts of snrs has been explained in earlier works @xcite .
particles with higher energies can scatter farther ahead of the shock , which results in an effectively higher compression ratio , so a more locally hard spectrum is produced , and a concave - up curvature to the accelerated - particle distribution is expected .
in addition , this modified model predicts that heavy elements dominate at the knee . based on this model , some models holding the viewpoints of weak knee structure or protons dominating the knee are likely to be excluded .
precise measurements of energy spectra of individual elements in the knee region will test our model .
the knee phenomenon of the cosmic ray spectrum , which plays an important role in studying the acceleration mechanism of cosmic rays , is still an unsolved mystery . 
 we try to reconcile the knee spectra measured by argo - ybj and tibet-@xmath0 . 
 a simple broken power - law model fails to explain the experimental data . 
 therefore a modified broken power - law model with non - linear acceleration effects is adopted , which can describe the sharp knee structure . 
 this model predicts that heavy elements dominate at the knee .    
 gbksong    energy spectrum , knee , composition    26.40.+r    2