Mean transverse momentum as a mass composition estimator of cosmic rays

TL;DR

This paper introduces a new statistical parameter based on mean transverse momentum to estimate the mass composition of cosmic rays using only a simple surface array, simplifying previous methods that required multiple measurements.

Contribution

A novel parameter utilizing mean transverse momentum is proposed for cosmic ray mass composition analysis without the need for muon detectors.

Findings

The new parameter effectively estimates cosmic ray mass composition.

It simplifies experimental setup by eliminating the need for multiple measurements.

Potential for broader application in surface array experiments.

Abstract

Determination of mass composition of high energy cosmic rays is one of the greatest challenge in modern astrophysics. All of previous methods for finding the mass composition of primary cosmic rays in a surface array at least require two independent measurements (e.g. muon and electron components) of extensive air showers (EAS). Here a new statistical parameter is introduced which can be used to determine the mass composition of vertical downward cosmic rays in a simple surface array. The most interesting thing about the new parameter is that it does not need two independent measurements and can be used in a simple surface array which does not have muon detectors.

Tables

Table 1. Table 1 : EASs’ specifications. Other specifications are CORSIKA default values.

Specifications	Values
energy of primaries	$E=100,\mathrm{\hspace{0.17em}200},\mathrm{\hspace{0.17em}300}\text{ and }400\text{TeV}$
type of primaries	proton and iron
zenith angle of primaries	$\theta =0^{\circ }$
geographical longitude	51 E
geographical latitude	35 N
altitude	1200 m
earth magnetic field ($B_x$)	$28.1\mu$T
earth magnetic field ($B_z$)	$38.4\mu$T
low energy hadronic model	Fluka 2011.2b [15]
high energy hadronic model	QGSJETII-04 [23]

Table 2. Table 2 : Resolution coefficients R 𝑅 R of MCEP parameters.

$R$	100 TeV	200 TeV	300 TeV	400 TeV
$⟨P_T⟩$	2.17	2.29	2.35	2.37
$N_{\mu }/N$	2.21	2.32	2.37	2.38
$N_{\mu }/N_e$	2.06	2.18	2.26	2.30
$⟨d_T⟩$	2.02	2.05	2.079	2.066

Equations

$$⟨P_T^e⟩$$

$$⟨P_T^{\mu }⟩$$

$$⟨P_T⟩$$

$$R=\frac{|m_{\mathrm{P}}-m_{\mathrm{Fe}}|}{{\sigma }_{\mathrm{P}}+{\sigma }_{\mathrm{Fe}}}$$

$$⟨d_T⟩=\frac{1}{N}\sum _{i=1}^N{d}_i,$$

$${\rho }_{Pd}$$

$$cov(⟨P_T⟩,⟨d_T⟩)$$

$$=\frac{1}{N_{EASs}}\sum _{i=1}^{N_{EASs}}(⟨P_T{⟩}_i-{\mu }_P)(⟨d_T{⟩}_i-{\mu }_d),$$

$${\mu }_P$$

$${\mu }_d$$

$${\sigma }_P^2$$

$${\sigma }_d^2$$

Full text

Keywords— Cosmic ray; Extensive air showers; Mass compositions

1 Introduction

Since the beginning of cosmic ray researches, one of the most important questions has been about their nature. After it became clear that most of them are ionized nucleus of common atoms, this question was changed to a question how frequent are individual nucleus in the primary cosmic rays composition. Now, the mass composition of primary cosmic rays is very important for the interpretation of different astrophysical processes and for acceleration mechanisms of ultra high energy cosmic rays.

Up to energies of about $10^{14}$ eV, balloon-borne or satellite-borne experiments directly measured the mass composition of primaries with negligible uncertainties [11, 7]. At higher energies, direct measurement of mass composition is not possible and mass composition estimation is only possible through extensive air showers which is generated by primary cosmic rays. Propagation of an extensive air shower through atmosphere includes lots of intrinsic fluctuations and so ground based experiments can not access the mass composition of every single primary.

Strictly speaking, no EAS experiment measures the mass content of primaries. Instead, they measure one or more of primaries’ mass sensitive parameters of EASs generated by primaries. The events can then be interpreted in terms of primary mass by a comparison to air shower simulations using different hadronic interaction models. Mass composition estimation of primaries (MCEP) can be achieved either in observation of longitudinal development of EAS through atmosphere or simultaneous measurement of the electromagnetic and muonic component of EASs in a surface array. Where, in the former case, people usually determine the EAS maximum development through atmosphere which is a relatively good mass sensitive parameter.

From the EAS maximum development measurements, two observables are derived: $<X_{max}>$, the mean depth of EAS maximum development in $\textrm{g/cm}^{2}$ and its standard deviation, $\sigma(X_{max})$, which are related to $\ln A$ and $\sigma(\ln A)$, where A is the atomic mass of the primary. EAS simulations shows that proton and iron induced EASs’ maximum development are expected to differ by around 100 $\textrm{g/cm}^{2}$ . However, unfortunately even extreme mass content like proton and iron have considerable overlap [21]. Currently, direct observation of the EAS maximum development is only possible by fluorescence light telescopes. At the moment, there are two active experiments that use fluorescence telescopes for MCEP: The Pierre Auger observatory [1] and the Telescope Array (TA) [22]. For the Auger Observatory, EAS maximum depth is determined with a resolution of about 25 $\textrm{g/cm}^{2}$ at low energies decreasing down to about 15 $\textrm{g/cm}^{2}$ above $10^{18}$ eV. For the latter, the resolution is better than 40 $\textrm{g/cm}^{2}$ at energy of $10^{16}$ eV and at higher energies, the accuracy performs better and reaches to 20 $\textrm{g/cm}^{2}$ [25].

Also, non-imaging Cherenkov Telescope can be used for determination of the EAS maximum. It can be deduced either from the measurement of the pulse width at 400 m from the core or from the ratio of the photon densities at two different distances from the shower core [24].

Recently, the Low Frequency Array, LOFAR [26], a radio telescope consisting of thousands of crossed dipoles, reported a radio measurements of EAS maximum with a high resolution, a mean uncertainty of 16 $\textrm{g/cm}^{2}$ [9]. The situation will be even better with the new proposed installations. For the example of the Square Kilometre Array (SKA) which will be operating in Australia in 2023, it is claimed to reach the resolution of 6 $\textrm{g/cm}^{2}$ in determination of EAS maximum [20]. If that is the case, this results in a better mass spectroscopy than ever done before. Although with such a high resolution in determination of $X_{max}$, a far better mass spectroscopy is possible, single event mass determination is nevertheless impossible.

Another method for the MCEP is the measurement of particle densities at a surface array. Unlike the observation of EAS maximum development which takes places in a wide range of Earth’s atmosphere, surface arrays measure EAS profile only in a single plane and so is more susceptible to the EAS fluctuations. Nevertheless, surface arrays are far more common than telescopes, air Cherenkov detectors and recently developed radio detection techniques. Furthermore, unlike telescopes or air Cherenkov detectors, they have a full duty cycle. So MCEP in surface arrays are still widely used.

The most common method for the MCEP in a surface array is the estimation of electron and muon numbers. While the sum of electron and muon numbers at the ground relate to the energy content of an EAS, the ratio of muons number to electrons number is an indication of the primary mass content. The most widely used technique for electron-muon discrimination in a surface array is simultaneous use of unshielded and shielded scintillation detectors (e.g. AGASA [12], CASA-MIA [8], EAS-TOP [4], GRAPES [18], KASCADE [5], KASCADE-Grande [6], Maket-ANI [14], GAMMA [16], and Yakutsk [3]). The Auger observatory uses Cherenkov tanks which also enables a limited muon identification [2].

More recently, Canal et al. in their simulations found that the ratio $r_{\mu e}=n_{\mu}/(E_{\textrm{em}}/0.5\,\textrm{MeV})$ of the muons number to the energy of electromagnetic component of an EAS at the ground level is a good measure for the MCEP. They reported that those vertical EASs with only a value of $r_{\mu e}$ between 0.5 and 3 enable us to reach a $98\%$ efficiency for discrimination of proton from iron primaries [10].

Another method for the MCEP is the analysis of lateral distribution of charged particles of EASs. Most of lateral distribution functions (LDF) which are in use for energy estimation of EASs have an age parameter which is an estimation of EASs’ maximum development depth. EASs of heavy primaries reach their maximum development depth at higher altitudes than EASs of lighter primaries. So the LDF of heavier primaries are flatter and age parameter has higher values than lighter primaries. However, the precision of this method for the MCEP is lower than other methods (for a review of MCEP methods, see [21]).

In this paper, a new parameter is introduced which enable a good MCEP for vertical EASs. The new parameter can be used in a simple surface array that lacks muon detectors and only measures charged particles in general.

2 Mean transverse momentom as a mass composition estimator

From the new results of LHC, it is evident that the more multiplicity of secondary particles of a reaction, the higher transverse momentum per particle of the reaction’s products (e.g. see [13]).

On the other hand, we conclude from extensive simulations of EASs which conducted by different research groups that the higher the mass composition of primaries, the higher the multiplicity of secondary particles (see chapter 3 of [17]).

The overall result of the above discussion is that when the primary particle is heavier, the generated secondary particles have higher transverse momentum. So if we could determine the transverse momentum per particles ($\langle{P_{T}}\rangle$) of an EAS, we may estimate the primary’s rough mass composition.

In order to test this hypothesis, some CORSIKA [19] simulated EASs have been generated whose general properties are summarized in table 1. The primary particles of these EASs are protons or irons. For each type and energy, 10000 EASs have been separately generated.

According to table 1, there are 8 different combinations of types and energies altogether. So the number of all generated EASs are 80000.

For the vertical EASs, $\langle{P_{T}}\rangle$ is calculated from the following equation:

[TABLE]

where $p_{xi}$ and $p_{yi}$ are the components of horizontal momentum of secondary particles.

Figure 1 shows the distribution of the $\langle{P_{T}}\rangle$ for EASs of table 1 on the ground level. As can be seen in this figure, iron primaries are distinguished from protons based on their charged secondary particles’ $\langle{P_{T}}\rangle$s. Actually $\langle{P_{T}}\rangle$ of charged secondary particles are higher for iron EASs than proton EASs which is consistent with our expectation. Based on part b and c of figure 1, it is evident that the good separation of $\langle{P_{T}}\rangle$ distributions for all charged particles, can not be seen for muonic and electronic components separately. Though at first glance it may seem strange, examining equation 3 shows that the contribution of coefficeints of $N_{e}/N$ and $N_{\mu}/N$ should also be taken into account. It should also be apparent that $\langle{P_{T}}\rangle$ for electronic component is one order of magnitude smaller than $\langle{P_{T}}\rangle$ for muonic component. This difference is due to more interaction of electrons with air nucleus along their path compared to the mouns.

In order to evaluate the effect of each term of equation 3 on $\langle{P_{T}}\rangle$ seperately, their distributions have been depicted in figure 2. As is evident, $N_{\mu}\langle{P_{T}^{\mu}}\rangle/N$ distribution has a far better seperation than $N_{e}\langle{P_{T}^{e}}\rangle/N$ distribution. So, the seperation of $\langle{P_{T}}\rangle$ is mainly due to the contribution of muonic term. The electron contribution has negligible effect on the seperation.

Next, we sould evaluate the effect of $N_{\mu}/N$ on the seperation. Figure 3(a) shows the results of the $N_{\mu}/N$ distribution. A comparison of this figure and figure 1(c), convince us that the $N_{\mu}/N$ coefficient has the most important role in the MCEP of the $\langle{P_{T}}\rangle$. Figure 3(b) also shows the results of $N_{\mu}/N_{e}$ for reference.

Up to now, we only have done a qualitative investigation of the MCEP parameters. However, a quantitative comarison will be more informative. Also, the precisions have not been investigated with respect to the energy of EASs.

In order to study the resolution of $\langle{P_{T}}\rangle$ and $N_{\mu}/N$ for the MCEP, we introduce the resolution coefficient with the following equation:

[TABLE]

where $m$ is the mean value of a distribution and $\sigma$ is the standard deviation. Actually, resolution coefficietnt $R$ is a measure of distance between two distributions’ averages as well as the narrowness of the distributions. The merit of such a dimensionless parameter is that it checks not only the sepration of the two distributions’ averages but also the mixing amount of them.

As can be seen in table 2, both of $\langle{P_{T}}\rangle$ and $N_{\mu}/N$ have almost the same precision. Altough, $\langle{P_{T}}\rangle$ has slightly less precision, it has an obvious advantage: It only uses the information of all charged particles and do not need muon specilized detectors. Also according to table 2, $\langle{P_{T}}\rangle$ and $N_{\mu}/N$ have better results than the $N_{\mu}/Ne$.

3 Mean transverse distance from axis

Although, $\langle{P_{T}}\rangle$ is a useful parameter for the MCEP, it can not be directly obtained by the current surface arrays in use (A surface array of hodoscopes is probably appropriate for this purpose). Therefore, we should find a new parameter which has some correlation with the above-mensioned parameters and can be obtained from the information provided by a common surface array.

Although the propagation of an EAS in the atmosphere has lots of stochastic fluctuations, obviously when the $\langle{P_{T}}\rangle$ of an EAS is large, its secondary particles have a higher chance to be found at larger distances from axis. So, a new variable of an EAS which is correlated to $\langle{P_{T}}\rangle$ is the mean transverse distance of particles from the axis ($\langle{d_{T}}\rangle$):

[TABLE]

where $d_{i}$ is the distance of $i$th secondary particle from the axis.

[TABLE]

Table 3 shows the correlation coefficient between $\langle{P_{T}}\rangle$ and $\langle{d_{T}}\rangle$ for the EASs of table 1. This correlation coefficient is given by the following equation:

[TABLE]

where $\langle{P_{T}}\rangle_{i}$ and $\langle{d_{T}}\rangle_{i}$ are the mean transverse momentum and mean transverse distance of $i$th EAS, the $E$ means the ensemble average over all the EASs, $\mu_{P}$ and $\mu_{d}$ are the ensemble average of $\langle{P_{T}}\rangle$ and $\langle{d_{T}}\rangle$ of all EASs, respectively and $N_{EASs}$ parameter is the total number of EASs that is here 10000.

As is evident from table 3, the correlation between $\langle{P_{T}}\rangle$ and $\langle{d_{T}}\rangle$ is nearly perfect. So $\langle{d_{T}}\rangle$ can be safely used instead of $\langle{P_{T}}\rangle$ for MCEP. Figure 4 shows the distribution of $\langle{d_{T}}\rangle$. It is apparent from this figure that $\langle{d_{T}}\rangle$ is a suitable parameter for the MCEP, as it was anticipated.

As we can see from figure 4, $\langle{d_{T}}\rangle$ can also discriminate between P and Fe with a relatively high resolution. Another interesting feature in figure 4 is that the distribution of $\langle{d_{T}}\rangle$ becomes sharper with increasing energy of the primaries (especially for the irons). Also the $\langle{d_{T}}\rangle$ approaches to lower (but distinct) values for irons than protons.

Table 2 also shows the $R$ coefficient for the distributions of figure 4. As can be seen from this table, this parameter have better results in the higher energies.

4 Conclusions

In this paper a new parameter has been introduced which can be applied for the mass composition estimation of vertical primary cosmic rays. This parameter is the transverse momentum per secondary particles of an EAS, $\langle{P_{T}}\rangle$. In an obvious manner, we concluded that $\langle{P_{T}}\rangle$ can be reduced to $N_{\mu}/N$ parameter for the MCEP with a negligible loss of information. It is also shown that a related parameter to $\langle{P_{T}}\rangle$ which can be directly measured in a simple surface array without muonic specialized detectors, is the mean transverse distance from axis, $\langle{d_{T}}\rangle$. It is also evident that the distributions of these parameters are sharper in higher energies.