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where
is in £
© and ¢
¢ £
¦ ¡ £§
is given by the Bethe-Bloch formula [8]:
£
¨ ©
¢
¤¢
in which ¦ ¢ is the sum of the discrete processes’ contributions
and the correction term (from
¡ ¡
knock-on electrons) causes the
energy loss to become constant for , i.e. upon entering
the Fermi plateau [30], which is reached at around 10 GeV for muons. Both ¢ and ¦ ¨¢ can
be
£ © © ¦ ¢¡ © £
£ © ©
approximated with constants. Typically and .
By solving Eq. 32, we get the mean range of a muon starting with energy ¢ and decaying with
energy ¢ :
m
10 4
10 3
10 2
10
1 10 10 2
10 3
10 4
10 5
18
(33)
10
GeV
6
Figure 9: Mean range of a muon in ice as a function of its energy.
¡
¢ ¢ ¢
¦
¢ ¡ ¢ §
¡ ¢ § (34)
¢
¤¢ ¦
©
§¤£ ¢
where is the critical energy above which stochastic processes dominate ¤ over ionization.
Below ¢ § , the muon range increases linearly with the muon energy, and above,
it
¢ ¢
is
proportional
¢ §
to . When , the fluctuations are large and a Monte Carlo simulation £¢
is necessary
to take into account the stochastic nature of the energy loss.
Since the muons considered here have high energies, so do the particles they produce during
their propagation through ice. As a result, they are boosted in the same direction and the
Cherenkov light they emit is sent out at an angle relative to the muon close to as well. The
smearing induced is of the order of a few degrees (see [26, 31]).