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With Equation 12.11, one can calculate the predicted number of photons arriving at the PMT if they were produced by an isotropic point source. However, real particle tracks differ from this in two ways: the emission of photons occurs along an extended trajectory, and it is not isotropic as emission is most likely close to the Cherenkov angle.

Ideally we could model the extended track as a sum of several point sources: the track is divided into steps, Equation 12.11is calculated for each step, and the contribution from each step is summed to produce the final predicted number of photons: µ = X step i ΦiT (Ri)(ψi) Ω(Ri) 4π (12.12)

To evaluate this contribution we must calculate Φi, the total number of photons

emitted at a given step. The extended and anisotropic nature of the source must be considered when calculating Φi, and this is achieved using functions termed ‘emission

Figure 12.4: Illustration demonstrating the calculation of the emission profile ρ to describe the number of photons a particle emits as a function of the distance along the track. Here, the (fictional) track has been divided into three steps, and ρ is the fraction of photons released in each step.

12.4.2.1 Definition of the emission profiles

The normalisation Φi is expressed in terms of the total number of photons Φ via two

multiplicative factors:

Φi = Φ(E)× ρ(E, si)× g(E, si, cos θi) (12.13)

Here,ρ and g are the linear and angular emission profiles respectively; ρ accounts for dividing the track up into steps, and g accounts for the photons not being emitted isotropically. To simplify the notation, the energy dependence of the emission profiles will be treated as implicit in this Section.

For the sake of illustration, consider a fictional particle that survives for 3 m and emits a thousand photons at angles close to (but spread around) the Cherenkov angle. We wish to divide the track into 1 m steps, and we observe that the particle emits 600 photons in the first metre, 300 photons in the second, and 100 photons in the third. Thus, labelling by the middle of each step, the fraction of photons emitted is given by ρ(0.5 m) = 0.6, ρ(1.5 m) = 0.3, and ρ(2.5 m) = 0.1. This defines ρ(s) and is illustrated in Figure 12.4.

Using ρ alone we can only describe a sum over isotropic sources, whereas the actual emission is peaked around the Cherenkov angle. Figure 12.5 ilustrates the first step of the track: most of the 600 photons are emitted at close to the Cherenkov angle, with very few of them emitted backwards, but for an isotropic source we would expect the same number in each bin. So using ρ alone, for PMTs located close to the Cherenkov angle we would underpredict the number of photons, and for PMTs far from the Cherenkov angle we would overpredict. This is accommodated usingg, which can be thought of as defining a different point source for each bin of emission angle, as illustrated in Figure 12.6.

Reconstruction for CHIPS 149

Figure 12.5: Illustration of a section of a particle track where 600 photons are released, divided into six bins of_{|θ|. The real emission would be peaked around the} Cherenkov angle, with very few photons emitted backwards (left). Scaling by ρ alone would produce the situation shown on the right, with the same density of photons emitted into each bin.

In practice the fitter uses bins of cosθ rather than |θ|. It may also be desirable to use bins of variable width, so instead of multiplying by the number of bins we multiply by the range of possible values (cosθ varies from -1 to +1, so this is 2) and divide by the width of the angular bin in question. The distribution of emission angles changes as the particle propagates (e.g. as the electromagnetic shower develops, or the particle slows down and the Cherenkov angle increases), and so the procedure shown in Figure 12.6 is repeated for each step along the track.

Both ρ and g depend on the type and energy of the particle. They are evaluated from large Monte Carlo samples by producing histograms of the distance and angle at which particles of a given energy emit optical photons. For a discretely-binned histogram, ρ and g are formally defined using:

ρ(si) =

# photons emitted in ith distance bin

total number of photons emitted by particle (12.14)

g(si, cos θj) =

 # photons emitted in ith distance bin and jth cos θ bin # photons emitted in ith distance bin



÷ width ofR+1jth cos θ bin

−1 d(cos θ)

!

(12.15)

Under this formalism, the sum over all bins of theρ histogram is 1, as is the sum of g(s, cos θ)_{× (width of cos θ bin) when taken over all cos θ bins at a given value of}

Figure 12.6: Demonstration of the angular emission profile g for three of the six illustrated bins of_{|θ|. The left circle shows the true angular distribution of photons,} and the three circles on the right show the calculation of g for three different angle bins. As a function of the the angle at which a photon would need to be released in order to hit the PMT, g determines the factor by which the right-hand image from Figure 12.5should be scaled in order to emit the correct number of photons in the relevant direction. In the fitter, the binning scheme is in terms of cos θ, not_|θ|.

Reconstruction for CHIPS 151

Distance along track (cm)

0 50 100 150 200 250 300 350 400 450 500 (s) ρ 0 1 2 3 4 5 6 -3 10 × 1GeV electrons 1GeV muons θ cos 0 0.2 0.4 0.6 0.8 1

Distance along track (cm)

0 50 100 150 200 250 300 350 400 450 500 0 2 4 6 8 10 12 14 ) for 1GeV electron

θ g(s, cos

θ cos

0 0.2 0.4 0.6 0.8 1

Distance along track (cm)

0 50 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25 ) for 1GeV muon

θ g(s, cos

Figure 12.7: Sample emission profiles for electrons and muons. The linear profile ρ(s) is shown in the top plot for 1 GeV electrons and muons, and the angular profile g(s, cos θ) is shown for electrons in the bottom left plot, and muons in the bottom right. The long, sharp emission near to the Cherenkov angle is evident for the muon, compared to the shorter and more diffuse shower from the electron. The narrowing of the muon profile and its peak being slightly displaced from the start of the track are due to the normalisation scheme’s requirement that each row integrates to one.

s. Note also that for a given track vertex, track direction, and PMT position, the value of cosθ required to hit the PMT is a function of si only. Emission profiles for

With these emission profiles in hand, the predicted number of photons at a given PMT due to a given track can be written as:

µ = Φ(E)X

step i

T (Ri)(ψi)

Ω(Ri)

4π ρ(si)g(si, cos θ(si)) (12.16)

Because the vertex and direction of the track, and the position of the PMT are fixed, Ri andψi depend only on the distance si that the particle has travelled, so Equation

12.16 can be recast as:

µ = Φ(E) X

step i

T (si)(si)

Ω(si)

4π ρ(si)g(si, cos θ(si)) (12.17)

12.5 Calculating the components of the predicted