So far, we have discussed the (tau) neutrino propagation by means of solving the respective cascade equation(s). Alas, it has become evident that this method requires various approximations, and even so, obtaining the neutrino flux proved to be be somewhat tedious.
However, we may get round the need of discussing integro-differential equations by using a discretized energy. This can formally be implemented in the cascade equations by assuming that the various cross sections involved can be expressed as a sum of delta functions, i.e. that they may be written in the form
dσ(E, E′)
dE =
XM
i=1
Bi(E)δ(E′−Ei(E)), (3.38) where the Bi and Ei(E) are chosen appropriately. As before, we’ll per-form the calculation for an instantaneous tauon decay. Hence, if we rewrite Eq. 3.32 in terms of the (final) energy rather than inelasticity parameter,
∂φ(E, t)
∂t =−σtot(E)φ(E, t) + Z ∞
E
dE′dσ(E′, E)
dE φ(E′, t), (3.39) we see by inserting Eq. 3.38 that the cascade equation simplifies to
∂φ(E, t)
∂t =−σtot(E)φ(E, t) +
M˜
X
i=1
B˜i(E)φ( ˜Ei(E), t). (3.40)
3.3. DISCRETIZING THE CASCADE EQUATION 47 with functions ˜Bi, ˜Ei and an integer ˜M. Now let us consider a setR ofN+ 1 energies (labeled from 0 to N), which are ordered by decreasing value, and let us assume that for any En∈ R, the energies E(En) with a non-vanishing cross section are given by the subset {Ei|i < n}ofR. Then with the notation An≡σtot(En) and Bin ≡Bi(En), Eq. 3.40 can be put in the form
∂φ(En, t)
∂t =−Anφ(En, t) + Xn−1
i=0
Binφ(Ei, t). (3.41) Accordingly the task of solving an integro-diferential equation has been boiled down to that of finding the solution of a set ofN coupled ordinary differential equations. The latter, however, can be performed analytically. Indeed, the neutrino flux φ described by Eq. 3.41 is given by
φ(En, t) = Xn
i=0
φ(i)n e−Ait, (3.42) where the coefficients φ(i)n are defined as
φ(i)n ≡
Xn−1
r=0
Brn
An−Ai
φ(i)r (i < n) φ(En,0)−
Xn−1
k=0
Xn−1
r=0
Brn
An−Akφ(k)r (i=n)
0 (i > n)
. (3.43)
The form of the expression for φ given in Eq. 3.42 shouldn’t be too much of a surprise. After all, for some given energy, we have to consider the fluxes for the energies greater or equal to that energy, and these should change more or less like an exponential function.
In order to prove Eqs. 3.42 and 3.43, we use complete induction. To start with, consider the first (and thus largest) energy E0. Here, the differential equation is simply
∂φ(E0, t)
∂t =−A0φ(E0, t) and hence has the solution
φ(E0, t) =φ(E0,0)e−A0t,
in agreement with our proposition, so that the basis is indeed true. Turning now to an arbitrary n > 0, we see that we have to solve the more general equation Eq. 3.41, which can be done by means of the variation of parameters method: Inserting the ansatzφ(En, t)≡K(t)e−Antinto Eq. 3.41, we see that K must fulfill the condition
dK(t) dt =
Xn−1
i=0
Binφi(t)eAnt.
But the induction hypothesis asserts that the φi on the right hand side are given by Eq. 3.42, so that we can write down the formula for dK/dt more explicitly, integration of Eq. 3.44 immediately yields
K(t) =K0+
from which we get the value of K0. Employing Eq. 3.43, we arrive at φn(t) = φn(0)−
3.3. DISCRETIZING THE CASCADE EQUATION 49 which is just Eq. 3.42. Thus we have successfully completed the induction step and thus proved our assertion.
So far, we have confined ourselves to a single flavor. There is no need for this, though. Looking at the found solution, we note that the energy appears in arguments only, so that its value doesn’t matter as long as we get the Ai
and Bik right. This means, however, that we may introduce formal energy values Eiformal via
Enformal(νe) = En Enformal(¯νe) =En+ 1 2∆E Enformal(νµ) = En+ 1
6∆E Enformal(¯νµ) =En+ 2 3∆E Enformal(ντ) = En+ 1
3∆E Enformal(¯ντ) =En+ 5 6∆E where ∆E is the smallest energy difference between neighboring energy val-ues, ∆E = min({Ei−Ei+1|i= 0, . . . , N−1}) (cf. Fig. 3.2). If we describe the neutrino propagation in terms of these variables, the set of coupled equations describing the propagation with transitions between the various flavors can be put together into a (single) set of equations of the form of Eq. 3.41, which can be solved as in the case of one flavor discussed above. As soon as we have introduced cross sections of the form of Eq. 3.38, the method outlined in this section obviously can be considered to be analytical and thus leads to no further inaccuracy in the computed neutrino fluxes. In other words, the error in the results is solely due to that of the cross section approximation. In order to estimate the latter, one may assume that the energy in a cross sec-tion argument is known up to the difference between adjacent energy values only.
Hence, if the number of discretized energies is increased, the solution should get more accurate. While this is true in principle, however, there is an important caveat: As the energy differences get smaller, the corresponding differences of the An will get smaller as well. This means that the φ(r)i get very large, and hence we finally end up computing small differences of large values. For sufficiently small energy differences, the numerical precision of the variables in a computer program implementing the method thus won’t suffice any longer, and one obtains random results, as illustrated by Fig. 3.3.
E0
νe νµ ντ νe νµ ντ
} E
E2
E1
real energy
E0 E E2
1
formal energy
∆
Figure 3.2: Schematic representation of the formal discretized energy values introduced in the main text. For simplicity, equidistant energies are assumed.
1.5 2 2.5 3 3.5 4 4.5
5 5.5 6 6.5 7 7.5 8
lg(Eν2φν/m-2s-1GeV-1)
lg(Eν/ GeV)
300 Energieintervalle
300 energy intervals
2 3 4 5 6 7 8 9
5 5.5 6 6.5 7 7.5 8
lg(Eν2φν/m-2s-1GeV-1)
lg(Eν/ GeV)
500 Energieintervalle
500 energy intervals
Figure 3.3: Left: Calculation of a neutrino flux with sufficiently large energy differences such that the numerical precision is sufficient. Right: If the energy differences are too small, the numerical precision is not sufficient for computing the small differences of the larges values encountered during the flux calculation.
In this case the result becomes purely random.