where E is the muon energy, x is the densimetric thickness (path length times average density along the muon path) along the path traversed by the muon, k(E) is the ionization energy loss, and b(E)E is the sum of the energy losses via three stochastic processes. The values for k(E) and b(E)E can be found at the reference for various materials (Groom et al. 2001). By integrating equation 1.1 over the energy range between 0 and the muon’s incident energy, the continuous slowing down approximation (CSDA) range of the muon can be calculated.
Since the range is a function of the incident muon energy, it can be incorporated in the open‐sky muon spectrum. Once both the muon path length and the average density along the path are known, the densimetric thickness (x) can be calculated by multiplying them, and thus the minimum energy (E c) of muons that can penetrate through a material with this thickness can be determined using equation 1.1. By integrating the open‐sky spectrum I (E, θ) from E c to infinity, we obtain the integrated muon intensity N(E c,), which represents the number of muons that have enough energy to escape from the target of interest:
where I(E,θ) is the zenith‐angle‐dependent open‐sky muon energy spectrum. The integrated muon intensities calculated with equation 1.2 are shown for various near‐horizontal angles in Fig. 1.1.
1.2.6 Muon Scattering
When muons travel through matter, the Coulomb force between the muons and the nuclei in the medium leads to numerous small deflections in the muon trajectories. As a result, their angular distribution becomes broader as they propagate through matter. However, owing to the steep nature of the muon spectrum (Taira & Tanaka, 2010) the angular spread is suppressed to ~12 mrad, half width at half maximum after penetrating matter thicker than 500 hg/cm2, which is almost independent of the total thickness of the material that is traversed by muons. This angular spread has the effect of limiting the positioning resolution at the target, for example, 12 m at a distance of 1 km from the detector. The highly penetrating nature of high‐energy muons, coupled with their low divergence, enables the efficient muographic imaging of a distant target, making them suitable for long‐range applications.
Figure 1.1 Integrated muon flux after passing through rock with a given thickness in units of meter water equivalent (mwe). The angles are measured from the zenith.
1.2.7 Background Events in Muography
The background events that degrade the quality of muographic images can be originated from both outside (extrinsic background) and inside (intrinsic background) the observation system (a combination of detectors, radiation shields, and other related equipments). Primary GCRs produce not only muons but also cascades of protons, neutrons, mesons, and other electromagnetic particles. The extrinsic background is generated by these secondary particles. The number of these particles rapidly decreases as they travel in the atmosphere but some of them arrive at sea level. The number of protons, electrons, and pions above 1 GeV are, respectively, two orders, three orders, and four orders of magnitude lower than the number of muons in the same energy range at sea level (Particle Data Group, 2020). However, when kilometric objects are observed in which low muon rates are expected, these backgrounds substantially degrade the image contrast. In particular, the protons are heavier than muons; hence longer MFP is present within the material (~10 cm in Pb), and thus could be a serious source of the background. On the contrary, the electromagnetic components such as electrons and gammas can be rejected relatively simply by requesting a linear trajectory through multiple layers of detectors and radiation shields. The radiation length of electrons is less than 6 mm in Pb.
The intrinsic background is caused by the muons scattered inside the observation system. Increasing the thickness of the radiation shields to reduce more extrinsic background rates would result in more muons being scattered inside the detectors, and thus the resultant images would be blurred. The intrinsic background level can be reduced by improving the tracking quality of the observation system (Kusagaya, 2017).
In an extraterrestrial environment, the condition of the extrinsic background will be completely different from the terrestrial one. Unlike Earth, Mars and small solar system bodies (SSSB) either do not have their own atmosphere or, they have an extremely thin atmosphere. Therefore, GCRs and GCR‐originated gamma rays (continuum gamma rays) will tend to directly hit the ground surface. As a result, the level of the extrinsic background is much higher than that on the Earth’s surface, for example, the proton flux at the ground level of these SSSBs tends to be 100 times higher than the terrestrial muon flux of Earth at sea level (Prettyman et al., 2014). However, in outer space, it is difficult to reject these high‐rate extrinsic backgrounds by using radiation shields, since these shields would add too much weight to the overall payload of the spacecraft. Geometrical configuration of the detectors must be well‐designed when muographers plan extraterrestrial muography.
1.2.8 Required Measurement Times
A muographic image is represented as a function of elevation and azimuth angles (θ, ϕ). These angles of an incident particle can be computed as follows by connecting two points in a space S(x, y, z) with a straight line:
(1.3)
(1.4)
Figure 1.2 Principle of the muon tracking. The top view (a) and the side view (b) of the detectors are shown.
where L is the distance between the upstream (pointed towards the target object) and downstream detectors (pointed away from the target object). The subscripts of x and y indicate the first and second points (Fig. 1.2). The positioning resolution of the points (Δx, Δy) therefore gives the detector’s angular resolution (Δθ,
