Using Muon Rings for the Optical Throughput Calibration of the Cherenkov Telescope Array

Experimental Astronomy manuscript No. (will be inserted by the editor)

M. Gaug · K. Bernl¨ohr · M.C. Maccarone · P. Majumdar · T. Mineo · A. Mitchell

Received: date / Accepted: date

Abstract The analysis of images produced by muon rings in an Imaging At- mospheric Cherenkov Telescope (IACT) provides a powerful and precise method to calibrate the optical throughput and optical point-spread-function of such a telescope. Being first proposed by Vacanti and collabotors in the early 90’s, this method has been refined in different aspects by the collaborations forming the so- called second generation of IACTs: H.E.S.S., MAGIC and VERITAS. We present here a compilation of the progress that has been made by these instruments and investigate their applicability for the different telescope types forming the future Cherenkov Telescope Array (CTA). We find that several smaller modifications in hardware and analysis need to be made to ensure that such a muon calibration works as precisely as expected from previous telescopes and derive estimates for the statistical and systematic precision of the method.

1 Introduction

The muon ring calibration, ﬁrst advocated by Rowell et al. (1991) and further elabored in detail by Vacanti et al. (1994), has been used as a method to calibrate the optical throughput, ﬁrst of the Whipple telescope (Rose, 1995; Rovero et al., 1996) and later practically all currently operating Imaging Atmospheric

M. Gaug UAB, Barcelona, Spain E-mail: [email protected] K. Bernl¨ohr MPIK, Heidelberg, Germany M.C. Maccarone INAF IASF-Palermo, Italy T. Mineo INAF IASF-Palermo, Italy A. Mitchell MPIK, Heidelberg, Germany 2 M. Gaug et al.

Cherenkov Telescopes (IACTs) (Guy, 2003; Leroy et al., 2003; Humensky, 2005; Shayduk et al., 2003; Meyer et al., 2005; Goebel et al., 2005), plus the optical point-spread-function of the MAGIC telescopes (Meyer et al., 2005; Goebel et al., 2005). In-depth studies of the muon method were furthermore carried out in several PhD theses, dedicated to the optical throughput calibration of the H.E.S.S. telescopes (Bolz, 2004; Leroy, 2004; Mitchell, 2016). The next generation gamma-ray observatory, the Cherenkov Telescope Array (CTA) (Actis et al., 2011; Acharya et al., 2013) is based on the IACT technique, but will largely outperform current installations (?) in terms of sensisivity (improved by at least a factor ten at 1 TeV, but considerably more at higher energies or oﬀ-axis), energy coverage (ranging from several tens of GeV to more than 300 TeV), angular resolution (0.05◦ at 1 TeV), energy resolution (∼7% at 1 TeV) and systematic uncertainties (<10% on the overall energy scale required)1. The CTA is built to boost the number of known Very High Energy (VHE, E > 10 GeV) sources from currently a few more than hundred to over a thousand, and hence move from the discovery mission of its predecessors to perform pop- ulation studies and precision measurements, resolving spectral features and the morphology of gamma-ray sources with unprecedented precision (?). A thorough calibration strategy must hence be included in its design (?). Recently, the CTA site selection has been completed, with as outcome, for its Southern observatory (CTA-S), a plateau of the Cerro Armazones in Northern Chile, close to Paranal, while the “Roque de los Muchachos” Observatory (ORM) on the island of La Palma, Canary Islands, has been selected for its Northern part (CTA-N). Both observatories are located around 2200 m a.s.l. The CTA counts with a large variety of telescope types and sizes (Actis et al., 2011; Acharya et al., 2013), which are listed in Table ??. It is therefore not immediately clear whether the muon calibration method will work for each case, and at which precision and accuracy. To address these issues is the scope of this article. The following points were given special importance:

1. Resume in a comprehensive way the diﬀerent muon analysis algorithms em- ployed by H.E.S.S., MAGIC and VERITAS 2. Determine whether all telescope designs are able to provide suﬃcient muon triggers per night 3. Determine the expected statistical and systematic precision of muon calibration for the CTA telescopes 4. Formulate additional requirements needed to make muon calibration successful for the CTA

We will first introduce the main concepts for muon calibration in Sections 2 and 3. The selection and reconstruction of muon rings is explained in Sections 4, 5 and 6. Systematic effects are investigated in Section 7 and simulations carried out in Section 8. Muon flagging in the camera servers and fast algorithms to achieve this, are treated in Section 9. Expected muon image rates are summarized in Sec- tion 10, and the results are discussed and recommendations given in Section 11. Assumptions and remaining caveats are treated in Section 12. The main conclusions are drawn in Section 13.

1 see also https://www.cta-observatory.org/science/cta-performance/ Title Suppressed Due to Excessive Length 3

primary camera optics mirror FOV reference (m2) (deg ) LST parabolic 387 5 Teshima et al. (2013) MST Davies-Cotton 104 8 Acharya et al. (2013) MST-SC Schwarzschild-Couder 50 8 Vasiliev et al. (2013) SST (ASTRI ) Schwarzschild-Couder 6.0 10 Pareschi et al. (2013) SST (GCT) Schwarzschild-Couder 8.2 10 Zech et al. (2013) SST-1M Davies-Cotton 7.6 10 Moderski et al. (2013)

Table 1 List of the proposed telescopes of the CTA. The following common abbreviations are used: LST: Large-Size-Telescope, MST: Medium-Size-Telescope, SST: Small-Size-Telescope, SC: Schwarzschild-Couder, 1M: single mirror. 4 M. Gaug et al.

2 Muon spectra and expected event rates

The muon spectrum on ground has been measured for momenta from 100 GeV/c to about 1.5 TeV/c with statistical and systematic uncertainties better than 10% per momentum bin for the central 141 to 1122 GeV/c range and better than 15% from 100 to 1500 GeV/c (Schmelling et al., 2013). The data ﬁt very nicely a parameterization for the vertical muon ﬂux from Hebbeker and Timmermans (2002), based on theoretical calculations from (Bugaev et al., 1998):

F (y) = 10H(y)m−2 sr−1 s−1 GeV−1 with: (1)

y = log10(p/GeV) H(y) = 0.133 · (y3/2 − 5y2/2 + 3y) −2.521 · (−2y3/3 + 3y2 − 10y/3 + 1) −5.78 · (y3/6 − y2/2 + y/3) −2.11 · (y3/3 − 2y2 + 11y/3 − 2) .

The spectral index at 100 GeV is then approximately −3.11. The muon ﬂux dependence with altitude has been measured by balloon ex- periments (e.g. Haino et al., 2004) and can be parameterized for muon momenta above 10 GeV and altitudes less than 1000 m as:

F (h) = F (0) · exp(h/L) (2)

with:

L = 4900 + 750 · (p/GeV) m , where h is the altitude in meter. The ﬂux ratio between a location at 2770 m a.s.l. and 1270 m at 10 GeV muon energy results to be ∼13%. The dependency of the 2 muon ﬂux on zenith angle can be described as ∝ cos (θ) at Eµ ≈ 3 GeV, while it steepens with higher energy to reach approximately

3 −2.7 n 1 0.054 o F (cos θ) = 1.4 · 10 · Eµ × + 1.1·Eµ·cos θ 1.1·Eµ·cos θ 1 + 115 GeV 1 + 850 GeV

m−2 sr−1 s−1 GeV−1 (3)

for energies Eµ 115 GeV (Behringer et al., 2012; Sanuki et al., 2002). Monte- Carlo simulations may perfectly simulate a simple power-law for the muon spectra and the events weighted correctly later on in an analysis. The correspondence between Cherenkov angle and muon energy can be established following the Cherenkov equation: 1 cos θc = , (4) p 2 n · 1 − (E0/Eµ)

where E0 ∼ 0.105 GeV is the muon rest mass, and n the refractive index of air, e.g. n ∼ 1.00023 at 2200 m a.s.l. Figure 1 shows the dependency of θc on Eµ, using Eq. 4, for various altitudes. One can see that above ∼30 GeV, the Cherenkov angle approaches its high energy limit, which lies between 1.2◦ and 1.24◦ at our altitudes of interest. Title Suppressed Due to Excessive Length 5 )

° 1.3 ( c θ 1.2 1.1 1 0.9

0.8 1500 m a.s.l. 0.7 2000 m a.s.l. 0.6 2500 m a.s.l. 0.5 0.4 6 7 8 9 10 20 30 40 50 60 70 80 102 Eµ (GeV)

Fig. 1 Dependency of Cherenkov angle on muon energy, for diﬀerent observatory altitudes.

Using the assumptions n = 1 + , 1 and Eµ E0, the Cherenkov angle can be approximated as:

p 2 θc ' θ∞ · 1 − (Et/Eµ) (5)

with : √ θ∞ = 2 , (6) E0 Et(h) = , (7) p1 − 1/n(h)2 where the threshold energy for Cherenkov emission yields Et ≈ 4.9 GeV for a reference altitude of 2200 m a.s.l. 6 M. Gaug et al.

Fig. 2 Sketch of the introduced parameters to describe the geometry of a local muon µ and its image in an IACT camera: The muon emits Cherenkov light under the Cherenkov angle θc along its trajectory, which is inclined by the angle i with respect to the optical axis of the telescope. It ﬁnally hits the telescope mirror, of radius R, at an impact distance ρ from its center (ﬁgure from Bolz, 2004).

3 Review of the muon-calibration method for IACTs

Single muons, which form naturally part of hadronic air showers, emit Cherenkov light in a same way as secondary electrons in a gamma-ray shower do. Since muons in air lose energy almost exclusively through ionization, they have a large pene- tration depth. Hence the telescopes have the possibility to record neat ring images from a “local” muon. Figure 2 recalls the geometry of the system, and introduces the parameters: the impact distance ρ, the inclination angle i, and the Cherenkov angle θc. The telescope mirror has a radius R. As long as (i+θc) is smaller than the ﬁeld-of-view of the camera, the latter will integrate light along the cord D. The cord itself is a function of ρ and the azimuth angle φ, deﬁned with respect to a reference angle φ0, normally chosen as the one at which the cord D is largest. Title Suppressed Due to Excessive Length 7

The number of Cherenkov photons Nc emitted in the wavelength range (λ1, λ2) per unit path length x by a muon in air can then be calculated from the Frank- Tamm formula, provided the muon energy is well above the Cherenkov emission threshold (Behringer et al., 2012):

d2N Z λ2 1 1 c = α · 1 − · dλ m−1 · rad−1 , (8) dx dφ β2(x)n2(λ, x) λ2 λ1

where α is the ﬁne structure constant, β the velocity of the muon, in units of c, and n the refractive index of the surrounding air. The velocity of the muon, in turn, is related to the Cherenkov angle via:

1 cos θ (x, λ) = . (9) c β(x) · n(λ, x)

Since observable muon Cherenkov light is always emitted from distances less than 1.5 km height, even in the case of an LST, the refractive index can be assumed constant along x, but depending on the observatory altitude 2. We can then simplify Eq. 8 to the more convenient form:

dN Z λ2 1 c ' α · sin2 θ · I, with : I = dλ . (10) dx dφ c λ2 λ1

Integrating over the path length L = R/ tan θc · D(ρ, φ − φ0), visible for the camera, we obtain a prediction for the number of observed photo-electrons Npe:

dN pe (ρ, φ ) = α · sin2(θ ) · L · I · T · ε , dφ 0 c µ µ α = · sin(2θ ) · D(ρ, φ − φ ) · I · T · ε , with : 2 c 0 µ µ  q ρ 2 2  2R 1 − R sin (φ − φ0) for : ρ > R D(ρ, φ − φ ) = (11), 0 q  ρ 2 2 ρ R 1 − R sin (φ − φ0) + R cos(φ − φ0) for : ρ ≤ R

Here, we have introduced an eﬀective atmospheric transmission for Cherenkov light from muons Tµ:

Z λ2 T (λ) . Z λ2 1 T = µ dλ dλ , (12) µ λ2 λ2 λ1 λ1

3 and the muon throughput parameter εµ, understood as the average of a combination of mirror reflectivity, camera light guides reflectivities and photon detection efficiencies, ψ(λ), weighted with the spectral form of the arriving Cherenkov light

2 A more detailed discussion of the implications of this assumption is given in Section 7.1.2. 3 This parameter has often been called “muon eﬃciency parameter” in the literature, a mistake which we do not want to repeat here, since it is unrelated with the concept of eﬃciency. 8 M. Gaug et al.

(which takes into account the atmospheric transmission and the emission spectrum scaling with the inverse square of the wavelength):

Z λ2 ψ(λ) · T (λ) . Z λ2 T (λ) ε = µ dλ µ dλ , µ λ2 λ2 λ1 λ1 Z λ2 ψ(λ) . Z λ2 1 ≈ dλ dλ , (13) λ2 λ2 λ1 λ1 where the approximation is valid to better than 1% for typical clear astronomical nights, when the overall transmission is higher than Tµ > 0.96 (see a more detailed treatment of this topic in Section 7.2.1). The muon throughput parameter is related, but not identical, to the gamma- ray throughput parameter εγ :

Z λ2 T (λ) . Z λ2T (λ) ε = ψ(λ) · γ dλ γ dλ γ λ2 λ2 λ1 λ1

= εµ · Cµ−γ

with :

R λ2 2 R λ2 2 (ψ(λ) · Tγ (λ)) /λ dλ Tµ(λ)/λ dλ λ1 λ1 Cµ−γ = · , (14) R λ2 2 R λ2 2 (ψ(λ) · Tµ(λ)) /λ dλ Tγ (λ)/λ dλ λ1 λ1

where Tγ (λ) is the wavelength dependent atmospheric transmission for Cherenkov light from gamma-ray showers, the properties of which depend on the mean emission height of the shower particles, impact parameter and finally the primary gamma-ray energy. Tγ (λ) is always lower and shows a larger wavelength dependency than Tµ(λ), since Cherenkov light from gamma-ray showers is emitted at higher altitudes. Moreover, Tµ(λ) extends to much smaller wavelengths, since both the strong extinction of UV-light by Rayleigh scattering, and the absorption by ozone are greatly reduched during the small path traversed by the muon Cherenkov light. A detector sensitive to UV light can hence yield a considerably biased efficiency correction from a muon analysis, if εµ is not corrected for the spectral differences Cµ−γ . The gamma-ray throughput parameter can then serve to cross-calibrate the telescopes’ response to Cherenkov light from gamma-ray showers of different energy, and observed under different zenith angles. Both muon throughput parameter and the spectral correction Cγ−µ will however change with time. In order to keep to the spectral correction small, we will be obliged to interfere already in the design of the CTA telescopes and cameras (see Section 7.2.7). The integration limits λ1 and λ2 can in principle be chosen freely, as long as λ1 is smaller than the smallest wavelength, at which the light detection efficiency ψ(λlower) is non-zero, and equivalently with λ2. From Eq. 11 follows immediately that for ρ < R, a full ring is visible in the camera (i.e. all values of φ are allowed), while otherwise only a part of the ring is getting imaged: namely up to a maximum azimuth angle:

φmax = arcsin(R/ρ) . (15) Title Suppressed Due to Excessive Length 9

Furthermore, a maximum emission height hmax can be derived from which muon light is visible in the telescope: p hmax = 2R · cot θc ' R · 2/ . (16)

It is important to realize now that the total number of photons depends only on the impact distance, the mirror radius, and the wavelength range, to which the camera is sensitive, apart from shadows by the camera itself, ropes or a secondary mirror, which require a separate treatment. The amount of light received by the camera can be predicted to the same precision as these three parameters are known. While the (effective) mirror radius, and the wavelength range are determined by the chosen hardware, the impact distance has to be retrieved by the modulation of slight intensity along the ring. For full rings, the light intensity reaches a maximum at ◦ φ = φ0, and is proportional to (R+ρ), while the minimum is reached φ = φ0+180 , where the amount of light scales with (R − ρ). If the impact distance is zero, the modulation along the ring becomes flat. Figure 3 (top) shows an example. A central hole in the mirror dish may reduce the amount of light received at the maximum. Shadows by masts and the camera will further reduce the imaged light. In the case of mirrors with deviations from radial symmetry, the shape of D(ρ, φ − φ0) becomes more complicated, but can be found by line integration from a given trial impact point to the mirror edges (Mitchell, 2016). Figure 3 (bottom) shows the true shape of D(ρ, φ − φ0) for the LST, and the one approximated from Eq. 11. All muon parameters, introduced so far, and their geometrical meaning are displayed in Figure 4. One can also derive the amount of light received by one single pixel, characterized by its pixel field-of-view (ω): α ω Ipix(ρ, φ, φ0, ω) ' · · sin(2θc) · Tµ · D(ρ, φ − φ0) · εµ · I, 2 θc ' 213 · εµ · ω/deg · D(ρ, φ − φ0) m , (17)

where the last line has been obtained by setting λ1 to 300 nm, and λ2 to 600 nm, and the Cherenkov angle to 1.22◦. From Eq. 17 follows directly that the amount of light received by each pixel scales linearly with the mirror radius and the pixel ﬁeld-of-view. Using the pixel ﬁeld-of-view, one can also estimate the number of pixels hit by the muon light and in the following a maximum impact parameter to trigger the readout: ( 2φmax · θc/ω if ρ/R > 1 Npix = (18) 2π · θc/ω if ρ/R ≤ 1

If one then asks for an absolute minimum of Nhit hit pixels, required to launch a trigger, the maximum impact parameter results to: R ρ ≤ (19) sin(Nhitω/2θc) Figure 5 shows the maximum impact parameter from Eq. 19 for each of the six telescope types of CTA, as function of Nhit. 10 M. Gaug et al.

Fig. 3 Modulation of the intensity profile along the ring for different impact distances ρ. Top: for a H.E.S.S.-I mirror, the drop in the center is due to the central hole in the H.E.S.S. mirror (figure from Bolz (2004)). Bottom: for an LST mirror, the fine structure (blue) is due the hexagonal shape of the mirror. A spherical mirror with radius similar to the outer edges of the hexagon would yield the black curve. Note the different y-axis scale offsets. THE UPPER FIGURE SHOULD BE REPRODUCED WITH A CTA TELESCOPE GEOM- ETRY!! AXIS TTILES OF THE LOWER FIGURE SHOULD BE REVISED!! Title Suppressed Due to Excessive Length 11

Fig. 4 Introduction of the relevant muon parameters: Top left: geometrical situation, the muon trajectory is deﬁned by the inclination angle i, and α, the impact point by the impact parameter ρ, and the phase angle φ0. Top right: The received photon density at the mirror plane. The red circle marks the mirror dish. Bottom left: the camera image of the muon ring, which is deﬁned by the parameters i, α and θc. Bottom right: intensity distribution along the ring, modulated by the parameters ρ and φ0. Figure from Bolz (2004). THE ENTIRE FIGURE SHOULD BE REPRODUCED WITH A CTA TELESCOPE GEOMETRY!!

The total number of observed photo-electrons can be obtained by integrating Eq. 11:

Z Φ r 2 ρ 2 Ntot(θc, ρ) = 2αR · sin(2θc) · I · Tµ · εµ · 1 − sin φ dφ 0 R arcsin(R/ρ) for : ρ > R with : Φ = (20) π/2 for : ρ ≤ R

≈ U0 · θc · E0(ρ) · Tµ (21)

with :

U0 = 2πα · R · I · εµ (22) Z Φ r 2 2 ρ 2 E0(ρ) = 1 − sin φ dφ , (23) π 0 R 12 M. Gaug et al.

where we have (following the notation of Fegan and Vassiliev, 2007) split the total muon image size into a detector dependent part (U0), the reconstructed Cherenkov angle θc, and a Legendre elliptic integral of second kind which can be evaluated numerically (see e.g. Press et al., 1992, Section 6.11). In the case of full rings, E0(ρ) becomes the complete elliptic integral of second kind E(ρ/R). The impact parameter can, moreover, be related to the distance between the center of gravity of the image and the ring center dc, according to Fegan and Vassiliev (2007): 1 ρ/R dc/θc = . (24) 2 E0(ρ) For the propagation of uncertainties of the reconstructed impact parameter, it is useful to know the derivative of E(ρ/R), namely:

δE(ρ) δρ dE(ρ) = · | | E(ρ) E(ρ) dρ K(ρ/R)/E(ρ/R) − 1 δρ = · 2(ρ/R) R δρ := F (ρ/R) · (25) R where K(x) is the complete elliptic integral of first kind. The function F (x) ranges from F = 1/4 at x = 0 to F ≈ 0.37 at x = 0.5 and F ≈ 0.92 at x = 0.95. At x = 1 it passes through an extremely slim infinity. Figure 6 shows the total number of photons (i.e. using εµ := 1) impinging on the telescope for each of the six telescope types of CTA, as a function of the muon impact distance, according to Eq. 21. The difference between full and dashed lines shows the effect of the central hole in the primary mirrors, subtracted from the full mirror estimate using again Eq. 21. Figure 7 in turn, shows the relative light yield with respect to zero impact as a function of impact parameter. One can directly read off from that figure, until which distance between telescopes the stereo muon trigger efficiency is acceptable with respect to mono triggers, knowing U0 and the typical trigger thresholds. Using our reference observatory altitude of 2200 m a.s.l., approximate muon ◦ rates can be then estimated, requiring a minimum energy of 10 GeV (i.e. θc & 1.1 ), and ρ/R ≤ 1. Such calculations yield rates ranging from ∼0.05 Hz (for an SST observing under a high zenith angle) to ∼20 Hz (for an LST observing at zenith), i.e. sufficiently high for a nightly calibration to yield acceptable statistics. The rates are, moreover, affected by the trigger conditions of each telescope. For the large telescope, we may assume that all fully contained muon rings will always trigger, while in the case of the small telescopes, only upward Poissonian fluctuations may produce as many local photo-electrons as to trigger the readout. If the latter occurs, the muon analysis will become complicated, since the reconstructed optical throughput efficiencies will be biased. More realistic estimates of muon trigger rates – and rates of suitable images for analysis – are addressed in Section 10. Title Suppressed Due to Excessive Length 13

Maximum impact parameter to launch a trigger

LST (m) 2

max 10 MST ρ SCT

SST-ASTRI

SST-GATE

SST-1M

2 2.5 3 3.5 4 4.5 5 5.5 6 Nhit

Fig. 5 Maximum allowed impact parameter, if Npix neighboring pixels are required to launch a trigger. The lower values for the SST-1M stem from the larger pixel size of its camera. 14 M. Gaug et al.

Total number of Cherenkov photons from muons (300 nm-600 nm) γ

N LST 18000 MST

16000 SCT SST-ASTRI 14000 SST-GATE

12000 SST-1M

10000

8000

6000

4000

2000

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ρ(1/R)

Fig. 6 Integrated number of photons from Cherenkov light of muons in the range from 300 nm to 600 nm, impinging on the telescope and focused towards the camerafor the six diﬀerent telescope designs for the CTA. The dashed lines show the case of no central hole, while the full lines the (realistic) situation with the corresponding hole at the center of the primary mirror dish. The results are given as a function of the muon impact distance, in units of the respective telescope radius. Note that no folding with photon detection eﬃciencies has been applied here. Title Suppressed Due to Excessive Length 15

Fig. 7 Relative light yield with respect to zero impact (Eq.23), as a function of impact parameter, expressed in telescope radii. Figure from Fegan and Vassiliev (2007). 16 M. Gaug et al.

4 Muon ring pre-selection

4.1 Tail cuts

Image cleaning levels (also referred to as “tail cuts”) are typically implemented in two levels: A stronger first cut on all pixels, without considering their neighbors. Once a list of pixels is established that fulfill the first condition, their pixel neighbors are searched for signal surpassing a second, less stringent cut level. Ad- ditionally, time constraints may be applied, if available. Typical levels for the first cut lie between 5 and 10 photo-electrons, and between 2 and 5 photo-electrons for the second level. Time constraints depend much on the (charge-dependent) time resolution of the readout and digitization chain. Arrival time constraints for signals between neighboring pixels lie around 1 ns, e.g. in the MAGIC analysis. Note however that the intrinsic time spread of photons from muon images is much smaller, of the order of 200–300 ps. A relatively strong initial tail cut is necessary to make the muon rings visible out of the noise from night-sky background, or residual hadronic shower components, or further muon rings. Only clean rings can then be fit in a robust manner. However, once the ring parameters determined, the tail cut should be released again to a minimum, in order to minimize biases in the distribution of ring sizes. At very low sizes, trigger and selection biases can occur which must be avoided, or otherwise corrected using dedicated Monte-Carlo studies. Also the effect of different levels for the tail cuts on the resulting efficiency parameters must be investigated. In case of a strong enough signal, and a robust analysis, its effect should result negligible.

4.2 Number of pixels after tail cut

A cut on the number of pixels after the tail cut is a standard procedure to reject small, badly reconstructed images. Typical values are Npix > 30 − 50, depending of course on the individual pixel sizes.

4.3 Mean number of next neighbors

The H.E.S.S. muon analysis uses an upper limit on the mean number of next neighbors: < NN > ≤ 3, in order to reject roundish images, which containing many central pixels surrounded by valid neighbors. A well-focused muon image should contain pixels in a small ring, and hence < NN >= 2.

4.4 Number pixels at the camera edges

In order to reject images at the camera edges, typically aﬀected by ineﬃciencies, and aberration, cuts on the number, or fraction of pixels at the camera edges are ◦ used. Typical values are Nedge < 2, or 0.5–1.8 distance between the outer edge of the ring and the camera edge, Title Suppressed Due to Excessive Length 17

4.5 Relative intensity variation

The H.E.S.S. muon analysis uses a cut on the relative intensity variation, deﬁned as the RMS of the signal distribution, after tail cuts, divided by its mean, which has to be smaller than 0.7. 18 M. Gaug et al.

5 Muon ring reconstruction

After pre-selection, only a percentage of all images are left (typically less than 10% for the current IACTs), upon which the muon ring reconstruction can be applied. Several algorithms are in use for this task (Chaudhuri and Kundu, 1993; Chernov and Ososkov, 1984). The MAGIC collaboration uses a simplex minimizer, implemented in the TMinuit class of ROOT. All algorithms calculate the charge- weighted distance of each pre-selected pixel to a supposed ring center, using a candidate ring radius, and minimize its RMS. It is also common to perform two fits: a first fit to find a candidate ring radius and center, followed by a second fit which excludes pixels too far from the candidate ring. Experience has shown that such a two-stage fit is necessary to exclude spurious hadronic shower contamination of the muon image, or further secondary muon rings. VERITAS uses the conjugant gradient search method (Press et al., 1992) ex- 2 ploiting the derivatives of the χ with respect to the Cherenkov angle θc and the two coordinates of the assumed ring center. The value

p 2 ∆θc = χ /Ntot (26)

is directly used as an estimate for the thickness of the ring. To find the impact distance, VERITAS exploits the relation between the distance of the ring center and the center of gravity of the image (Eq. 24). While the MAGIC collaboration only excludes pixels further away from the ring center than a minimum distance, the H.E.S.S. and VERITAS collaborations also exclude pixels closer to the ring center than a certain maximum distance. Both algorithms count the excluded charge and perform cuts on its relative value. The fits may be initialized in a separate step, using the center-of-gravity of the pixel charges as first guess for the inclination, and a typical Cherenkov angle (e.g. 1.2◦) for the radius, but it is not strictly necessary. The MAGIC collaboration has found that neither the precision nor the speed of the ring fit improves by a dedicated initialization.

5.1 Quality of the ring ﬁt

After the fit(s), it is important to reject rings reconstructed with poor quality. Cuts on the χ2/N.d.F are used (e.g. < 0.09 for the ring fit in H.E.S.S.), as well as on the excluded pixels (e.g. H.E.S.S. requires less than 15 inactive pixels on the ring). Conventional analyses also exclude too inclined muons, i.e. establish a maximum distance of the ring center from the camera center. The optimum cut value will depend on the field-of-view of the camera. Ring images with small radius, i.e. small Cherenkov angle, must be rejected since these are more strongly affected by secondary ring-broadening effects (see Section 7.1). VERITAS rejects also Cherenkov angles smaller than 0.3–0.7◦ and larger than 1.4–1.5◦. The H.E.S.S. collaboration rejects also images with a too high number of inactive pixels on the ring, typically Ninactive ≤ 15. The VERITAS collaboration ◦ rejects images with a too broad ring width (obtained via Eq. 26): ∆θc < 0.2 and Title Suppressed Due to Excessive Length 19 the difference between image center of gravity and the reconstructed ring center: dc/θc < 0.75. Cuts may also be applied on the combination of inclination angle and Cherenkov ◦ radius, e.g. i + θc < 2.5 . Finally, a minimum opening angle must have been reconstructed, e.g. ω > 180◦ or ρ/R < 0.9. 20 M. Gaug et al.

6 Determination of impact parameter and muon eﬃciency

6.1 The H.E.S.S. analysis

The H.E.S.S. collaboration (Bolz, 2004) performs a second ﬁt over a smoothed muon ring to determine impact parameter and global optical throughput, the so- called muon eﬃciency: The smoothing procedure is performed over an azimuthal range dΦ of 4 camera pixels width (each of 0.17◦ in the case of H.E.S.S.-I): ω dΦ = 4 ω = 4 (27) θc The obtained light intensity in an azimuthal bin is then re-normalized again to the equivalent of one pixel:

1 Z φi+2ω Ii(θ , ρ, φ ) = I (θ , ρ, φ0) dφ0 b c i 4 b c φi−2ω

φ≤(φi+2ω) 1 X ≡ I (θ , ρ, φ) (28) 4 b c φ≥(φi−2ω) q i i ∆Ib(θc, ρ, φi) = Ib(θc, ρ, φi) (29) (30)

Data points Ib are then constructed in distances of 0.5 ω and ﬁtted to a smoothed version of function 17:

1 Z φi+2ω I = ε · I (θ , ρ, φ , φ0) dφ0 (31) pix,p.e. µ 4 pix c 0 φi−2ω with the free parameters: impact parameter ρ, the phase of the intensity function φ0, and the global muon efficiency parameter εµ = Np.e./Nγ . The muon efficiency εµ comprises the product of mirror reflectivity (including degradations due to missing mirrors) R, efficiency of the light guides Tf and quantum efficiency of the photo-multipliers Q. All three parameters may depend on incidence angles and local impact point, i.e. εµ = εµ(ρ, φ0, α, i) for one single muon image!. The average efficiency εµ over many muon rings, obtained from a random distribution of these parameters then yields an estimate of the overall optical throughput of the telescope, averaged over the muon spectrum:

R λ2 2 R(λ) · Tf (λ) · Q(λ) · 1/λ dλ λ1 εµ = , (32) R λ2 1/λ2 dλ λ1

where the integration limits (λ1, λ2) are typically chosen to cover the full spectral acceptance of the photomultipliers. εµ is then compared to the same number used for the Monte-Carlo simulations. 2 The H.E.S.S. analysis cuts subsequently on the quality of the fit (χ /Npix < 1.1) and on the obtained impact parameter (0.2 < ρ < 0.99), to exclude events which hit the hole in the center of the mirror (and have a lower muon efficiency by definition), and those which wrongly reconstruct an un-closed ring, previously Title Suppressed Due to Excessive Length 21

Fig. 8 Example of a ﬁt of the smoothed intensity function 31 (red) to the smoothed intensity distribution along the ring (Eq. 28, black data points). The error bars seem to be over-estimated here. Figure from Bolz (2004).

Fig. 9 Average number of pixels on the ring vs. reconstructed Cherenkov angle for the H.E.S.S.-I telescopes. Below 1.12◦, the reconstruction algorithm includes pixels with larger distances from the ring. Figure from Bolz (2004).

eliminated.

Finally, a third iteration is performed, after un-doing the tail cut and adopting all pixels within 2ω from the ring (0.15◦ in the case of VERITAS), in order to elim- inate completely any possible biases due to the tail cut. For small Cherenkov angles, the allowed ring width is enlarged, according to the expected broadening of the ring (see Section 7.1). This has an immediate consequence, namely that the number of pixels on the ring falls with decreasing ring radius, until the secondary ring broadening eﬀects become important. At smaller ring radii, the number of pixels increases again, although the ring itself becomes smaller. The distribution of number of pixels on the ring vs. ring radius shows then a minimum at a value which depends on the telescope and camera parameter. In case of the ◦ H.E.S.S.-I telescopes, the minimum is found at θc ∼ 1.12 (see Fig. 9). 22 M. Gaug et al.

Fig. 10 Diﬀerences of reconstructed and simulated impact parameters for the H.E.S.S. telescopes. Figure from Bolz (2004).

This procedure achieves about 13 cm precision in the reconstruction of the impact parameter (see Fig. 10) and 0.1% for the muon efficiency, without taking into account atmospheric effects (see sec. 7.2.1). A lot of care has to be taken when a biased signal extractor is used, like in the case of MAGIC (Albert et al., 2008). In that case, it is desirable to change to an un-biased extractor “on-the-fly”, using the strong arrival-time constraints given by the larger signals on the ring. Apart from O. Bolz’s muon code (Bolz, 2004), a template fitting procedure, based on Le Bohec et al. (1998), is used and described by Guy (2003) and Leroy (2004). 2 th This method constructs a full χ from Eq. 11. Setting Qi the theoretical value of dNobs/dφ, which depends on Cherenkov angle, impact parameter, angle of incidence, Gaussian ring width, azimuth angle and muon efficiency (i.e. 6 free parameters), the following χ2 is constructed:

exp X (Q − Qth)2 χ2 = i i , (33) B + (F − 1) · Qexp + 0.5 · (Qexp + Qth) i i i i i i

where Bi represents the baseline fluctuations of pixel i (mainly due to night sky background), Fi the excess noise factor of the PMT, and the last term in the denominator accounts for the Poissonian fluctuations of the impinging number of photo-electrons onto a pixel.4 That fitting procedure achieves about 50 cm RMS precision (with a bias of 20 cm) in the reconstruction of the impact parameter. The (systematic) uncertainty of the reconstructed muon efficiency is not given in Leroy (2004).

4 This is the χ2 as shown in Leroy (2004) It is however not clear why only the Poissonian ﬂuctuations have been divided symmetrically between the theoretical and the experimental part, and not the excess noise. Title Suppressed Due to Excessive Length 23

Fig. 11 Top: Total number of photo-electrons vs. reconstructed ring radius, for MC simulation (red) and two telescope data sets (black and blue). The slope of each data sets reflects the average muon efficiency. Bottom: Directly reconstructed muon efficiencies (Eq. 32) vs. run number for the 4 H.E.S.S.-I telescopes. The period labeled “1” corresponds to a major hardware intervention on the CT3 telescopes. The line labeled “2” marks the transition from mono to stereo (muon) trigger, whereupon the muon rate decreases by a factor 10, and the error bars become correspondingly larger. The period labeled “3” marks another hardware intervention on the cameras. Figure from Bolz (2004). 24 M. Gaug et al.

6.2 The MAGIC analysis

The analysis of the muon images requires the calculation of the muon image parameters: ring radius, ring width (ArcWidth), its opening angle (ArcPhi) and its total light content, measured in photo-electrons (MuonSize). By comparing the relative ring broadening of muons, taken from observational data, with simulated muon data of different optical point-spread functions (PSF) one can estimate the PSF of the reflector. By comparing the (muon energy dependent) total light content of muons from observational and simulated data the overall light collection efficiency is calculated. The search for the muon ring images, every cleaned (i.e. after tail-cuts applied) image is fitted by a circle. Radius, center of the circle and deviation are calculated, starting at the center of the Hillas ellipse. The distance to every pixel is calculated, and from these the mean value (weighted by the pixel content) with its deviation. The algorithm minimizes the deviation by changing the coordinates of the assumed center. For the minimization, the ROOT class TMinuit is used, employing the simplex algorithm. In a second step, the event before image cleaning is used and two his- tograms are filled: one with the radial and one with the azimuthal intensity distribution. For the azimuthal intensity distribution, all pixels inside a certain margin (default: 0.2◦) around the radius are used. The parameter ArcWidth is defined as the sigma value of a Gaussian fit to the signal region in the radial intensity distribution. The parameter ArcPhi is defined as the sum of connected bins, which lie above a certain threshold, and the MuonSize is defined as the sum of the contents of all pixels along the ring. In order to fit the radial intensity distribution, first a Gaussian plus a baseline is fitted to account for all pixels without Cherenkov photons, but which have a positive signal on average , because the signal extractors may be biased 5. In the next step, the fit is redone by first fixing the baseline to zero and the peak position (mean of the Gaussian) to the previously fit value. The reason for this is that the “baseline” which had been fitted before with a constant term, to accommodate for pixels with no Cherenkov light (but just the bias from the signal extractor), will not be present in a pixel with a true signal. In the second fit, the range of the fit is restricted to ±3σ around the peak position, in order to minimize the influence of pixels having no Cherenkov light. This two-step procedure greatly enhances the quality of the fits to the radial intensity distribution. In order to compare data with Monte Carlo simulations, muons are simulated with the MAGIC Monte Carlo Simulation (MMCS) program, based on the simulation program CORSIKA from the KASKADE group. The muon spectra are simulated from 6 GeV to 80 GeV in 3 subsamples with different slopes, the telescopes pointing towards zenith and impact parameters randomly distributed over 200 meters:

– 6 − 10 GeV: E−2.16 – 10 − 20 GeV: E−2.46 – 20 − 80 GeV: E−2.71

5 The signal extractors used by MAGIC employ peak-ﬁnding algorithms and hence introduce some bias in the signal extraction. Title Suppressed Due to Excessive Length 25

Distribution of ArcWidth 0.2 Quantile vs. overall PSF sigma ) ArcWidth ° 9 0.2 quantile : 0.0714° Entries 278 0.07 Counts 8 Mean 0.0862 For PSF 12.2mm : 0.0485° 7 RMS 0.0205 0.065 0.2 Quantile ( 6 0.06 5

4 0.055 3 2 0.05 1 0.045 0 0 0.05 0.1 0.15 0.2 10 12 14 16 18 20 22 24 W [°] PSF (mm)

Fig. 12 Left: Distribution of the ArcWidth parameter for a real data sample with bad PSF. The obtained 0.2 quantile of 0.0714◦ is much bigger than the 0.0485◦ expected from the rest of the observation period. Right: calibration curve of 0.2 quantile vs. PSF in millimeters. )

° 0.115

0.11

Arc Width ( 0.105

0.1

0.095

0.09

0.085

0.08

0.075 0.6 0.7 0.8 0.9 1 1.1 1.2 Muon Radius (°)

Fig. 13 The muon ArcWidth parameter, as a function of reconstructed muon ring radius. Blue: from real data, green and red: from MC simulations using two diﬀerent light collection eﬃciencies. PSF size in 'mm' 32 30 28 26

PSF size (mm) 24

22 2015-01-01 20 18 16

Time

Mean Muon size 1100 11-2012 12-2012 01-2013 02-2013 03-2013 04-2013 05-2013 06-2013 07-2013 08-2013 09-2013 10-2013 11-2013 12-2013 01-2014 02-2014 03-2014 04-2014 05-2014 06-2014 07-2014 08-2014 09-2014 10-2014 11-2014 12-2014 01-2015 1080 1060 1040 1020 1000 980 960 Fig. 14Mean Muon size [phe] Top: Evolution of the MAGIC-I optical PSF, estimated from the muon analysis. 940 Three periods920 with diﬀerent quality of the telescope optics can be discerned. Points with large error bars900 correspond to very short data sets. Those points which lie signiﬁcantly above the Time average correspond to data where indeed problems with the active mirror control could be found. Bottom: Evolution of the MAGIC-I muon size, estimated from the muon analysis. The red and blue colors only serve to guide the eye.

11-2012 12-2012 01-2013 02-2013 03-2013 04-2013 05-2013 06-2013 07-2013 08-2013 09-2013 10-2013 11-2013 12-2013 01-2014 02-2014 03-2014 04-2014 05-2014 06-2014 07-2014 08-2014 09-2014 10-2014 11-2014 12-2014 01-2015 26 M. Gaug et al.

In order to get an estimate of the optical PSF of the reflector, it is necessary to compare muon simulations with different PSFs to muons extracted from observational data. Typically the positions of the 0.2 quantile of the ArcWidth distribution are compared (see Fig. 12), in order to make the comparison for those rings which are not so strongly affected by other secondary ring broadening effects. Figure 13 shows an example of muon ArcWidth as a function of radius, compared with different MC simulations. To measure the overall light collection efficiency, the MuonSize is plotted as a function of radius for both simulated and observed data. It is seen that the total intensity of the muon ring image behaves nearly linear with the radius. The total number of muons are relatively few when the telescopes are operated in stereoscopic mode. In order to estimate the optical PSF of the telescopes and the light collection efficiency, special muon runs are taken with the telescopes triggered in mono mode where the telescopes typically point to a certain dark patch in the sky. The typical time for such a muon run is 1 hour in order to gather enough statistics. Typical muon rates are about 3 − 5 Hz. Another feature of MAGIC muon analysis is an automatic follow-up procedure of the long-term evolution of the optical PSF. For this purpose, normal data from stereoscopic observations are used. Since the total number of muons when operated in stereoscopic mode, is far less (typical rates are about 0.5 Hz), the data from a full night are required to reliably estimate the PSF of the telescopes. Figure 14 shows the long-term evolution of the PSF for one MAGIC telescope.

6.3 The VERITAS analysis

VERITAS performs only one ring ﬁt, after cutting on standard image selection cuts and using typical tail cuts. In a second attempt, it sums the charge from all pixels which are found at 0.15◦ distance from the ring. It seems that this procedure is still not enough to compensate for biases due to channels suppressed by hardware or the muon reconstruction code, for which they correct by assigning a weight to each channel i: q 2 2 dring−i wi = 1 − (ρ/R) · sin φi + ρ/R · cos φi · exp − 2 (34) 2∆θc and corr X X Np.e. = Np.e. · 1 + wi / wi (35)

i εµ suppr.channels i εµ ring.channels Title Suppressed Due to Excessive Length 27

7 Systematic Eﬀects

7.1 Secondary eﬀects broadening the ring

While Eq. 17 and 31 assume infinitely slim rings, in reality several effects cause a broadening of the ring width, which use to become larger than one pixel. Multiple scattering of the muons on the atoms of the atmosphere causes the strongest effect, however rings with small radius suffer from further effects, like variation of the refractive index with height and wavelength, muon energy loss through ionization and telescope-specific effects: Discrete pixel size and aberrations of the optics. Figure 15 illustrates the increase of ring width, in units of ring radii, as a function of muon energy and Cherenkov angle, for the case of the H.E.S.S.-I tele- ◦ scopes. Here, at energies above 20 GeV, (θc > 1.18 ), the mirror aberrations start to dominate.

7.1.1 Instrumental eﬀects

Mirror aberrations The contribution of the mirror aberrations to the broadening of the ring can be assumed as constant and described as:

∆θ σ c = ab . (36) θc θc

The requirements on the optical aberration θ80 ≈ σab for the LST is much stricter than in the case of the H.E.S.S./MAGIC/VERITAS telescopes, and will reduce this contribution to the ring broadening. All other telescopes however, have less strict requirements on the optical PSF than H.E.S.S. In these cases, the optical aberrations will probably start to dominate the ring broadening already at lower Cherenkov angles. Figure 16 shows an example for its expectation for each telescope type:

Discrete pixel width The discrete pixel with introduces an uncertainty on the reconstruction of the muon ring, which depends on the number of hit pixels:

ω/2 σpix ' √ Nhit ∆θ σ ω c = pix ' 3/2 (37) 1/3 θc θc 2π θc

Effects of non-active pixels Although non-active (de-activated or broken) pixel are typically removed from the image analysis, and the signals approximated by inter- polation from neighboring pixels, a large number of non-active pixels may cause a smearing of the image, especially if the non-active pixels appear in clusters. Bolz (2004) investigated the effects of non-active pixels on the ring for the reconstructed muon efficiency and found stable results up to an average number of 12 pixels on the ring (see Fig. 17). 28 M. Gaug et al.

Fig. 15 Different effects which cause a broadening of the muon ring for the H.E.S.S.-I telescopes: total (red, tot), aberration (black, abb.), multiple scattering (top blue, mult.scat.), variation of refraction index with altitude (top magenta, (h)), variation of refraction index with wavelength ((λ)), finite pixel size (top yellow, pix.), ionization losses (green, ion.). Top: as a function muon energy, figure from Leroy (2004), bottom: as a function of Cherenkov angle, figure from Bolz (2004).

Effects of the stereo trigger Since a single muon cannot or almost never trigger more than one telescope at the same time, stereo muon events contain always an air shower image, plus an accidental local muon. The muon ring images are hence almost always contaminated by the light from the mother air shower. Figure 18 shows the intensity of those pixels which have been excluded by the muon reconstruction algorithm, for mono and stereo events. One can see that all stereo events contain considerably more residual light than the mono events. The H.E.S.S. muon analysis has found out that additional light contaminates not only off-ring pixels, but also the reconstructed muon rings. In this case, a positive bias for the muon efficiency of about 3% has been measured on average, except for one period of the CT2, where a bias of 18% was observed (see Fig. 18 bottom). It is therefore essential to cross-check muon efficiencies obtained with stereo trigger by mono triggered events from time to time. Title Suppressed Due to Excessive Length 29

Effect of optical aberrations on the ring broadening for muons

c 0.8 θ /

c LST θ

Δ 0.7 MST 0.6 SCT 0.5 SST

0.4

0.3

0.2

0.1

0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 θ c (deg.)

Fig. 16 Expected eﬀect of optical aberrations on the broadening of the muon ring for the CTA telescopes.

Fig. 17 Reconstructed muon eﬃciency as a function of the mean number of non-active pixels on the ring, simulated for the H.E.S.S.-I telescopes. In black a random distribution of pixels has been used, in red the observed distribution of pixel clusters, scaled to larger total numbers of non-active pixels. From (Bolz, 2004).

Effects of the pixel baselines Since the individual pixel baselines are subtracted from the extracted signal, (unrecognized) common drifts in the baselines may bias the reconstructed muon efficiency considerably, since the effects may sum up to the overall reconstructed number of photo-electrons in the muon image. Pixel baselines may vary with time and especially temperature. Typically, these offsets are determined regularly, on time scales of a star to transit the pixel. The knowledge of the baseline of each pixel is required to be always better than 0.2 p.e. by requirements B-xST-1370. 30 M. Gaug et al.

Fig. 18 Top: intensity of the light contained in those pixels which have been excluded by the muon ring reconstruction, for mono and stereo events of the H.E.S.S. CT3 telescope. Bottom: Nightly muon efficiencies from stereo triggers, divided by those obtained from the mono trigger. The CT2 values with enhanced stereo efficiencies above 1.1 can be explained by the different ADC readout windows. From (Bolz, 2004).

One method to control such possible biases, is to monitor the oﬀ-ring intensity, i.e. the summed pixel intensity for all pixels which have been excluded from the muon ring reconstruction. The mean oﬀ-ring intensity should then always peak around zero. This works of course only for an AC-couple readout scheme and an unbiased signal extractor. In the case of stereo-triggered muons, the stereo bias needs to be taken into account, however monitoring over time may still re- veal systematic drifts or malfunctioning of the muon analysis code, or the camera hardware, e.g. few slightly defocused mirrors.

Effects of the readout window The Cherenkov light signal of a local muon is much shorter (<1 ns) than typical signals from air showers which spread several nano- seconds over the camera. Also the pixel-wise signals differ significantly. This may Title Suppressed Due to Excessive Length 31

Fig. 19 Muon eﬃciency against the phase angle φ0 for the H.E.S.S. CT3 telescope, ﬁt to ε = ε0 + A · sin(φ − φ0). Figure from Bolz (2004). lead to a shift in the readout window, determined by the arrival time of the trigger signal. If the readout window is short, and adjusted to air shower images, the pure muon image may leak partly out of the digitization range. In the case of the H.E.S.S. CT2 telescope, shifts of up to 3 ns between mean intensity maxima taken with mono and stereo muons have been observed (Bolz, 2004).

Eﬀects of coma aberration Coma aberrations may be relevant for parabolic mirror designs since they distort the muon ring towards the outer part of the camera image. Its eﬀects require further studies involving detailed simulations (see also Section 12).

Non-uniformities of the camera acceptance and mirror reﬂectivity Non-uniformities of the camera acceptance can be controlled by a pixel-wise reconstruction of the muon eﬃciency. To do so, the muon light intensity seen by one pixel can be approximated as:

1 ∆θ2 2∆θ √ −1 Ipix = I0(θ, ρ, φ) · · exp(− 2 ) · ω kc with: kc = 1 + , (38) 2 2π 2σ θc

where ∆θ is the radial distance of the pixel from the ring. Just as for the overall image (Eq. 32), a pixel-wise muon efficiency parameter εpix can be derived which should correlate with the flat-fielding constants from the light flasher calibration. Bolz (2004) finds a good correlation for the H.E.S.S.-I telescopes and 5% RMS deviation. By plotting the mean εpix against the inclination azimuth angle α (see Fig. 4) for a large sample of muons, acceptance gradients in the camera can be found. Doing same against the phase angle φ0 (see again Fig. 4) reveals possible influences in the focused reflectivity of the mirrors. A simple gradient would show up as a sinusoidal curve (see e.g. Fig. 19 where a 3% gradient is found, and Fig. 20). 32 M. Gaug et al.

Fig. 20 Muon eﬃciency vs. ring center positions in the camera (top) and impact points on the mirror (bottom) for two H.E.S.S.-I telescopes. In the top left plot, a gradient is clearly visible, while the bottom plots display the eﬀects of an asymmetrically opening camera lid and the camera support structure. Figure from Bolz (2004).

7.1.2 Atmospheric eﬀects

Multiple scattering Multiple Coulomb scattering of particles in the air along their trajectory cause the emitted Cherenkov photons to appear scattered around their mean positions in the focal plane, according to an approximately Gaussian distribution. The sigma of that distribution can be described as (Highland, 1975, 1979; Lynch and Dahl, 1991):

r 13.6 MeV x · ρair x σmul.scat. ' · · 1 + 0.038 ln( ) rad βcp X0 X0 r ∆θ σ 13.6 MeV Rρ exp(−h/H ) 1 c = mul.scat. ' · 0 0 · 3/2 θc θc pc X0 θc r s R · ρ exp(−h/H ) 1 − (θ /θ )2 1 ' 2.8 · 0 0 · c ∞ · (39), m0 2 2 2 3/2 X0 1 − ( ) · (1 − θ /θ ) Et c ∞ θc

2 where p is the momentum of the particle and X0 its radiation length (≈36.7 g/cm for muons). The density of air can be approximated by ρ0 · exp(−h/H0), where 3 ρ0 ≈ 1.225 g/cm .

It is important to realize here that multiple scattering is the main reason for the almost complete elimination of ring images from energetic electrons. Already at energies of 1 GeV, an electron ring is completely blurred out.

Variation of the refractive index The refraction index of air can varies with altitude, temperature, atmospheric pressure, relative humidity and also with wavelength. Both are discussed in great detail in (Ciddor, 1996, 2002) and later in (Tomasi Title Suppressed Due to Excessive Length 33

et al., 2005). Atmospheric conditions vary with time and altitude, especially what concern its water vapor content, but also, to a smaller extent, on the concentration of CO2. In general, a good description is obtained by assuming a variation of n − 1 with the density of air, which in turn is approximated by the US standard atmosphere (NASA, 1976), as an exponential decrease with a scale height of 6 H0 = 10.3 km. Assuming further ∆H0 ' R/θc, then:

2 ∆θc R · θ∞ ' 3 . (40) θc 2H0θc The wavelength dependency of n − 1 in the range from 230 to 1690 nm for standard air at Sea level can be described well by formula (8) of (Tomasi et al., 2005), assuming CO2 concentration of 300 ppm: 2, 480, 990 17, 477.7 n (p ,T , λ) − 1 · 108 = 8060.51 + + , (41) s s s 132.274 − λ−2 39.32957 − λ−2 where the wavelength λ is used in micrometers. 2 Additionally, since θ∞ ≈ 2 · (n − 1), dependencies of the refractive index and hence the Cherenkov angle, on the atmospheric parameters temperature, pressure and relative humidity, and even concentration of CO2, are expected. The corresponding formulae can be found in Tomasi et al. (2005), and are not reproduced here, due to their complexity. Figure 21 shows some of the calculated drifts of p2 · (n − 1). One can see that dependency on temperature is biggest and effect the Cherenkov angle by up to 6% maximum. On the other hand, no visible dependency of the refractive index on humidity and concentration of CO2 is seen. Since the Cherenkov radius is getting reconstructed directly from the data, the residual effect on the muon efficiency is expected to be smaller. However, Bolz (2004) find effects on the real data of the same size. This means that the muon analysis must control temperature and pressure dependent effects, and understand them during the commissioning phase.

Energy loss of the muon The muons lose energy through ionization, which reduces the Cherenkov angles with path length. It is convenient to write the average rate of muon energy loss as (Barrett et al., 1952):

−dE/dx = a(E) + b(E) · E. (42)

Here, a(E) is the ionization energy loss, and b(E) the sum of e+e−-pair production, bremsstrahlung and photo-nuclear interactions. b(E)·E dominates the energy loss only above 1.1 TeV in air (Groom et al., 2001), where the muon ﬂux is already very small, compared to the energies in the range from 10 to 100 GeV. The ionization energy is given by the Bethe-Bloch formula and amounts to about 1.815 MeV per g/cm−2 for dry air at Sea level (Groom et al., 2001). Deriving Eq. 5 with Eµ, one obtains the variation of Eµ while the particle −2 −1 2 traverses the path length ∆x ' R · d/θc g/cm losing 1.815 MeV g cm , and obtains: ∆θc 1.815 MeV · R · ρair 2 3/2 ' · (Eµ/Et) − 1 , (43) θc mµ

6 Note here that (Vacanti et al., 1994; Leroy, 2004) assumed erroneously a scale height of 8.4 km which is valid only for the atmospheric pressure. 34 M. Gaug et al.

2*(n-1) vs. Temperature 2*(n-1) vs. Pressure 0.0255 0.0255

0.025 0.025

0.0245 0.0245

0.024 0.024

0.0235 0.0235

0.023 0.023 -20 -15 -10 -5 0 5 10 15 20 25 30 1010 1012 1014 1016 1018 1020 1022 1024 1026 1028 1030 Temperature ( °C) pressure (mbar)

2*(n-1) vs. Rel. humidity 2*(n-1) vs. CO2 concentration 0.0255 0.0255

0.025 0.025

0.0245 0.0245

0.024 0.024

0.0235 0.0235

0.023 0.023 0 102030405060708090100 350 360 370 380 390 400 410 420 430 440 450 rel. humidity (%) CO2 concentration (mmpv) p Fig. 21 Expected drifts of 2 · (n − 1) ∝ θ∞ with atmospheric parameters: Top left: temperature, top right: pressure, bottom left: relative humidity, bottom right: CO2 concentration.

where the muon rest mass mµ ' 105 MeV has been used. Figure 22 shows the combined eﬀects of the ring broadening for all 6 telescope types.

7.2 Secondary effects affecting the muon efficiency

7.2.1 Atmospheric Transmission

A photon of wavelength λ emitted at a certain distance r from the telescope suﬀers molecular and aerosol extinction before reaching the mirror. One can write the atmospheric transmission for that photon very generically as:

Z r T (r, λ) = exp − αmol(x, λ) + αaer(x, λ) dx (44) 0 Title Suppressed Due to Excessive Length 35

Different ring broadening effects for the LST Different ring broadening effects for the MST

c 1 c 1 θ θ

/ mirror aberrations / mirror aberrations c c θ 0.9 mult. scattering θ 0.9 mult. scattering Δ Δ 0.8 energy loss 0.8 energy loss 0.7 variation of n(h) 0.7 variation of n(h) discrete pixel width discrete pixel width 0.6 0.6 total natural total natural

0.5 total (incl. aberrations) 0.5 total (incl. aberrations) 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 θ θ c (deg.) c (deg.) Different ring broadening effects for the SCT Different ring broadening effects for the SST-ASTRI

c 1 c 1 θ θ

0.5 total (incl. aberrations) 0.5 total (incl. aberrations) 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 θ θ c (deg.) c (deg.) Different ring broadening effects for the SST-GATE Different ring broadening effects for the SST-1M

c 1 c 1 θ θ

0.5 total (incl. aberrations) 0.5 total (incl. aberrations) 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 θ θ c (deg.) c (deg.) Fig. 22 Expected combined ring broadening eﬀects for the diﬀerent telescope types.

Assuming that the muon emits light uniformly along the track from hmax to the telescope mirror dish, we can write:

1 Z hmax Ttot(λ) = T (L, λ)dL hmax 0 Z hmax Z r 1 = exp − αmol(x, λ) + αaer(x, λ) dx dr (45) hmax 0 0

= Tmol · Taer with:

Z hmax Z r 1 Tmol = exp − αmol(x, λ) dx dr (46) hmax 0 0 Z hmax Z r 1 Taer = exp − αaer(x, λ) dx dr (47) hmax 0 0 36 M. Gaug et al.

The transmission expressed in Eq. 45 must then be folded with the Cherenkov spectrum and the spectral acceptance of the telescope to obtain the weighted transmission: R ∞ 2 Ttot(λ) · εµ(λ)/λ dλ T = 0 (48) weighted R ∞ 2 0 εµ(λ)/λ dλ

Molecular Transmission

The volume Rayleigh scattering cross section for unpolarized light is (see e.g. Bucholtz, 1995; McCartney, 1976; Tomasi et al., 2005):

3 2 2 24π · (ns(λ) − 1) 6 + 3ρ(h) P (h) Ts αmol(λ, h) = 4 2 2 · · · , (49) Ns · λ · (ns(λ) + 2) 6 − 7ρ(h) Ps T (h) where the mean depolarization of the air molecules is ρ ≈ 0.0283, and the correction for the diﬀerent air densities at height h is measured through temperature T and pressure P . Further, ns is the refraction index of air, introduced in Eq. 41, and Ns the number density of molecules per unit volume, both at standard con- 25 −3 ditions (Ns = 2.5469 · 10 m (Bodhaine et al., 1999) at Ts = 288.15 K and Ps = 101.325 kPa). Using these numbers, the volume scattering cross section can be written as:

2 13 (ns(λ) − 1) P (h) Ts −1 αmol(λ, h) ≈ 1.362 · 10 · 4 · · m . (50) λ(nm) Ps T (h)

The relation is precise to at least 0.5%, with the main uncertainty stemming from the unknown water vapor content (Tomasi et al., 2005). Assuming a standard atmosphere (NASA, 1976), the term P (h) · Ts can be Ps T (h) described by a scale height of H0 ≈ 9.7 km, hence:

(n (λ) − 1)2 α (λ, h) ≈ 1.362 · 1013 · s · exp(−h/H ) m−1 . (51) mol λ(nm)4 0

Eq. 51 is not as accurate as Eq. 50 any more, but was used to derive approximate values of Tmol for the wavelength range of interest, for the respective telescope types. The result is displayed in Fig. 23. Inserting Eq. 51 into Eqs. 46 and 48, we obtain the transmission values displayed in Table 27. One can see that only the LST shows non-negligible molecular extinction of muon light. In order to assess the range of variation of the values obtained for LST, we used a full one-year “Global Data Assimilation System” (GDAS) database for the closest grid point to La Palma, whose temperature and pressure predictions show excellent correlation with the measurements of the MAGIC weather station. The full height-resolved temperature and pressure proﬁles were used to derive Tmol,weighted for the LST. The distribution of results is shown in Fig. 24: While

7 The values shown in Table 2 are signiﬁcantly smaller than the claimed 3.45% to 4% of Bolz (2004) (page 62), even if the diﬀerences in observatory altitude and mirror radius are taken into account. They are however compatible if we assume that Bolz (2004) included aerosol transmission in his estimates. Title Suppressed Due to Excessive Length 37

molecular transmission of muon light

mol 1 T

0.98

0.96 LST MST 0.94 SCT 0.92 SST 0.9

0.88 250 300 350 400 450 500 550 600 650 700 wavelength (nm)

Fig. 23 Approximated molecular transmission of muon light for the diﬀerent telescope types, derived from Eqs. 51 and 46 for an observatory altitude of 2200 m a.s.l.

Telescope hmax Tmol(300 nm) Tmol,weighted (m) LST (Hamamatsu PMT) 1120 0.94 0.975 LST (ETE PMT) 0.973 MST (Hamamatsu PMT) 570 0.97 0.987 MST (ETE PMT) 0.986 SCT (SiPM) ∗ 460 0.975 0.993 SST-ASTRI (SiPM)∗ 200 0.99 0.997 SST-GCT (CHEC-M)∗ 190 0.99 0.997 SST-GCT (CHEC-S)∗ 0.995 SST-1M (SiPM) 190 0.99 0.997 Table 2 Approximate molecular transmission for observed Cherenkov light from muons for the diﬀerent telelscope types. The transmission values of those telescopes marked with ∗ have to be considered a lower limit, since the shadow of the secondary mirror has not been taken into account. All values have been derived for an observatory altitude of 2200 m a.s.l.

GDAS the mean of < Tmol >= 0.9752 is compatible with the one obtained from the simple exponential model, the RMS of the distribution is as small as 0.0003, the peak-to-peak diﬀerence between highest and lowest value amounts to 0.0014. We can conclude that the uncertainty on Tmol is absolutely negligible.

Aerosol Transmission

Astronomical sites, like Armazones and La Palma, are characterized by extremely clean environments and small aerosol content close to ground. Patat et al. (2011) 38 M. Gaug et al.

weighted molecular transmission of muon light for LST (using GDAS profiles)

70 Tmol (GDAS)

counts Entries 1460 60 Mean 0.9752 RMS 0.0002925 50

0 0.974 0.9745 0.975 0.9755 0.976 0.9765 Tmol (weighted)

Fig. 24 Distribution of molecular transmission of muon light for the LST (Eqs. 46 and 48) using a one-year database of temperature and pressure proﬁles from the “Global Data Assim- ilation System” (GDAS).

find a median aerosol extinction of 0.045 mag. airmass−1, and the semi-interquartile range is 0.009 mag airmass−1 at 400 nm wavelength for the VLT site at Paranal (about 500 m higher than the CTA site). The height profile of the aerosol extinction is not provided, however. We can nevertheless assume that stratospheric aerosol accounts for about 0.005 mag., and the rest forms part of the nocturnal boundary layer close to ground. Hence about (4±1)% of the Cherenkov light from gamma-ray showers is expected to be scattered out of the field-of-view in the first kilometer on ground, probably slightly more on the CTA site, due to the lower altitude. The wavelength dependency of the aerosol extinction is typically expressed by the Angströmcoefficient˚ A˚, where:

α (λ ) λ ˚ aer 1 = 2 A (52) αaer(λ2) λ1

with A˚ ≈ 1.4 for Paranal (Patat et al., 2011), and ranging from A˚ ≈ 0.8 − 2.0 for La Palma (Whittet et al., 1987). Similar values have been obtained for Tenerife by Maring et al. (2000), ranging from A˚ ≈ 1.5 − 2 during the non-dusty nights, and Andrews et al. (2011), which tend towards A˚ ≈ 1.2 for small scattering coefficients. The MAGIC collaboration, using a 2-years statistics of 532 nm elastic LIDAR data, has found that the altitude profile of the aerosol extinction coefficient for clear nights is exponential with a scale height of Haer ≈ (500 − 700) m, with an AOD on ground of about 0.02 on average, however ranging from practically zero to almost 0.1 for normal clear nights. Only during Saharan dust intrusions, stronger aerosol extinction is possible (called “calima” in the Canary Islands). Such nights show severely enhanced AODs Title Suppressed Due to Excessive Length 39

(up to close to or greater than 1.0), Anströmcoefficients˚ close to zero, and constant aerosol extinction from ground to approximately 5 km altitude. We have calculated the aerosol transmission for muons Taer for an average case (AOD532 nm = 0.02, Haer = 600 m, A˚ = 1.2 for CTA-N, and AOD532 nm = 0.04, Haer = 600 m, A˚ = 1.4 for CTA-S) and two extreme cases of clear nights with the exponential profile, created to yield strong extinction of muon Cherenkov light (AOD532 nm = 0.08, Haer = 500 m, A˚ = 2.0) and tiny extinction (AOD532 nm = 0.01, Haer = 700 m, A˚ = 1.0), just to show the range within which Taer can in principle vary for clean nights. The case of calima is treated as another extreme example: AOD= 0.5 (supposing that this is the absolute limit for observation), A˚ = 0. and constant aerosol extinction up to 5 km. The results are shown in Table 3.

Telescope Taer Taer Taer Taer Taer average average extremely extremely extremely CTA-S CTA-N low high (clear nights) high (calima) LST (Hamamatsu PMT) 0.966 0.984 0.993 0.912 0.947 LST (ETE PMT) 0.965 0.984 0.993 0.907 0.947 MST (Hamamatsu PMT) 0.978 0.990 0.996 0.939 0.973 MST (ETE PMT) 0.977 0.989 0.996 0.937 0.973 SCT (SiPM) ∗ 0.984 0.992 0.997 0.958 0.978 SST-ASTRI (SiPM)∗ 0.992 0.996 0.998 0.979 0.990 SST-GCT (CHEC-M)∗ 0.992 0.996 0.998 0.973 0.991 SST-GCT (CHEC-S)∗ 0.992 0.996 0.999 0.980 0.991 SST-1M (SiPM) 0.992 0.996 0.999 0.980 0.991

Table 3 Approximate aerosol transmission Tweighted,aer for observed Cherenkov light from muons for the different telelscope types. More than two digits behind the comma do not reflect resolution nor accuracy, but are only displayed to highlight the tiny differences between telescope types. Those entries which deviate by more than 3% from the average are marked as bold. The transmission values of those telescopes marked with ∗ have to be considered a lower limit, since the shadow of the secondary mirror has not been taken into account. All values have been derived for an observatory altitude of 2200 m a.s.l.

One can see that only in the case of a strong aerosol densities close to the ground, with unusually high contribution of the accumulation mode particles, a correction needs to be applied for the MSTs and LSTs (possibly the SCTs). This problem can be either circumvented by adequate data selection (either using the LIDAR data, or the CTC), or by introduction of a correction factor, once the aerosol proﬁle is assessed. Such a correction is expected to take place only rarely, when the observatory operates under non-optimal conditions.

7.2.2 Trigger and Selection Biases

If the mirror area is small, the reflected muon light may not be strong enough to ensure a stable trigger above a certain energy threshold. In that case, only Poissonian upward fluctuations will launch a trigger, and the reconstructed muon efficiency distribution εµ will become asymmetric and its mean value gets biased. Mirror degradations of reflectivity will not scale linearly with < εµ > anymore, 40 M. Gaug et al.

aerosol extinction LST aerosol extinction MST

1 1

0.98 0.98

0.96 0.96

0.94 0.94 average CTA-N average CTA-N

0.92 average CTA-S 0.92 average CTA-S

0.9 extreme low 0.9 extreme low extreme high extreme high 0.88 0.88 extreme calima extreme calima 0.86 0.86

300 400 500 600 700 800 300 400 500 600 700 800

aerosol extinction SCT aerosol extinction SST

1 1

0.98 0.98

0.96 0.96

0.94 0.94 average CTA-N average CTA-N

0.92 average CTA-S 0.92 average CTA-S

0.9 extreme low 0.9 extreme low extreme high extreme high 0.88 0.88 extreme calima extreme calima 0.86 0.86

300 400 500 600 700 800 300 400 500 600 700 800

Fig. 25 Approximated aerosol transmission of muon light for the different telescope types, derived from Eq. 47 and the different aerosol models (see text for details), for an observatory altitude of 2200 m a.s.l. and a bias correction needs to be applied. The corresponding correction factor needs to be retrieved from simulations, however since simulations tend to simplify real fluctuations much stronger at the threshold, the correction factors may have a considerable systematic uncertainty. We have tried to simulate the effect using a Gaussian distribution of a certain mean muon image size, and its square root as Gaussian width. Then, the lower parts of the distribution were removed until a given trigger efficiency was reached (see Fig. 26 left). The difference between statistical mean of the new distribution and the original image size has then been defined as a toy-trigger bias (Fig. 26 right). This extremely simplified model may give an order of magnitude of the expected real trigger biases.

However, if the distribution of εµ shows an identiﬁable cut towards lower values, one can try to ﬁt only the peak of that distribution, instead of calculating its mean or median. If such an approach can help, needs further investigation.

7.2.3 Biased signal extractors

Many signal extractors show a small bias towards low signal amplitudes (Albert et al., 2008). Even a small bias, summed over many pixels, will considerably bias the amount of reconstructed muon light, and must be avoided at any time. It is recommended to switch to un-biased extractors at least at the very last analysis step before the muon eﬃciency and impact parameters are extracted from the image. Title Suppressed Due to Excessive Length 41

Size Distribution for mean size=200 p.e. Estimated trigger bias 3500 eff.=0.50, mean=211.3 0.12 mean size=50 p.e. eff.=0.55, mean=210.2 bias

counts mean size=100 p.e. eff.=0.60, mean=209.1 3000 eff.=0.65, mean=208.1 mean size=150 p.e. eff.=0.70, mean=207.1 0.1 mean size=200 p.e. 2500 eff.=0.75, mean=206.0 eff.=0.80, mean=205.0 mean size=250 p.e. eff.=0.85, mean=203.9 0.08 mean size=300 p.e. 2000 eff.=0.90, mean=202.7

0.06 1500

0.04 1000

500 0.02

0 0 160 180 200 220 240 260 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 image size trigger efficiency

Fig. 26 Left: Simulated distributions of image sizes, for a mean size of 200 photo-electrons. The left part of the distribution has been subsequently cut, in order to obtain the efficiency numbers provided in the legend. The corresponding different between statistical mean and the position of the Gaussian peak may give a hint of the expected trigger bias. Right: Distribution of biases as a function of trigger efficiency, for different mean muon image sizes.

7.2.4 Shadows

The H.E.S.S. camera and camera masts cause an average shadowing of 11% of the muon light. In the case of the Small-Size-Telescopes of the CTA, shadows will probably play an even more important role. The exact value needs to be determined by careful MC simulations. Obviously, its contribution is independent of wavelength.

7.2.5 Mirrors

The mirror reflectivity degrades with time, most probably in a wavelength-dependent way. The overall average degradation is a free parameter of the calibration procedure, and is part of the obtained optical throughput. A general worsening of the point spread function of the mirror should be retrieved by a careful analysis of the reconstructed muon arc-width. More difficult is the estimation of the effect of mis-focused mirrors, possibly on very short time scales. These mirrors may reflect part of the muon light into the margins of the ring, and part of the light outside the reconstructed image. In this case, the disentanglement between the worsened point spread function, and the mirror reflectivity gets less precise.

7.2.6 Light funnels

As in the case of the mirrors, the funnels in front of the photo-detector plate may degrade with time, in such a way that the wavelength-dependency of their eﬀective reﬂection changes, and/or the dependency on the incidence angle. 42 M. Gaug et al.

7.2.7 Light detectors

Photomultipliers may have considerable quantum efficiency below 290 nm, if UV- transparent glass (fused silica) is used for the photocathode. This is the range where Cherenkov light from air showers is completely absorbed, but the one from muons still gets through to the camera. Figure 27 shows normalized Cherenkov spectra from muons and gamma-ray showers, after transmission through the atmosphere from 270 m, as would be the case for muons observed in an SST, from 540 m, representative for a muon observed by an MST, and from 8 m height, typical for gamma-ray showers. One can see that light from gamma-ray showers below 290 nm is completely absorbed by the atmosphere, while the muon light reaches its maximum there. Figure 27 shows also the quantum efficiency curves of the Hamamatsu R11920 PMT, and a Silicon PM. While the second does not have practically any quantum efficiency below 290 nm, the PMT does show efficiency down to 200 nm. We conclude that the cameras equipped with Silicon PMs will observe very similar muon and gamma-ray spectra, while the ones equipped with photomultipliers will have efficiency for muon light at a wavelength range where gamma-ray showers do not emit light. To quantify the difference, Figure 28 shows the same normalized Cherenkov spectra, after convolution with the quantum efficiency curve of the Hamamatsu R11920 PMT. The right figure shows the normalized cumulative spectrum, from which one can read off directly a 20–23% acceptance below 300 nm for muon light, compared with only about 5% for light from gamma-ray showers, and 10–12% for muon light below 280 nm, where practically no light is received from gamma-ray showers. This picture may slightly change for gamma-ray showers of TeV energies, which can reach further down the atmosphere such that a part of the light travels less distance until reaching the telescopes. It becomes clear at that point that a 10–15% systematic effect of the muon calibration may be expected unless the degradation of the spectral acceptance in the middle UV is monitored, or this part is cut out in its way through the optical chain. One possibility is the use of Shinkolite #000 from Mitsubishi Rayon,8 for the protection windows of the telescope cameras, as is foreseen in the case of the LST. This PMMA material cuts transmission of wavelengths precisely below 290 nm. Figure 29 shows the obtained Cherenkov spectra for muons and gamma- ray showers, after transmission through such a protecting window, and application of the PMT quantum efficiency. One can see that only a residual 3% of additional muon light is found below 290 nm, if compared to the gamma-ray case. On the contrary the previous default window for NectarCam, the Sunactive® GS 9 from Evonik is too transparent below 290 nm (see Figure 30) and would cause a systematic uncertainty of ±4.8% for the optical throughput retrieved by the muon analysis, which is moreover expected to show a long-term drift.

8 see https://www.mrc.co.jp/shinkolite/. 9 see http://www.per-plex.de/plexiglas-sunactive.html. Title Suppressed Due to Excessive Length 43

Normalized Cherenkov Spectra

Muons SST 1 0.02 Muons MST Gammas (8 km) 0.9

0.018 QE PMT (Hamamtsu) QE PMT (ETE) 0.8 0.016 QE SiPM Sanko Mirror 0.7 0.014 Acrylite Window Sunactive GS (Evonik) Window ASTRI Window 0.6 0.012 0.5

photons per pixel (a.u.) 0.01 0.4 0.008

0.006 0.3

0.004 0.2 Transmission / Reflectivity

0.002 0.1

0 0 200 250 300 350 400 450 500 550 600 650 wavelength (nm)

Fig. 27 Normalized Cherenkov spectra of muons arriving from 270 m distance (green line), representative for the case of a SST, from 540 m distance (red line), representative for the case of an MST, and the Cherenkov spectrum from a gamma-ray shower emitted at 8 km above ground (blue line). In different shades of brown are shown the quantum efficiency of the Hamamatsu R11920 PMT, the new 7-dyn. ETE prototype11, a typical Silicon PM, and the reflectivity of the SANKO mirrors, and three possible camera protection windows.

11 The QE of this PMT is expected to improve in the near future. 44 M. Gaug et al.

Normalized Cherenkov Spectra, convoluted with QE Hamamatsu R11920 Cumulative Normalized Spectrum convoluted with QE Hamamatsu R11920 Muons SST 1 0.025 Muons MST

Gammas (8 km) 0.8 0.02 13.0%@341 nm 0.6 0.015 14.8%@336 nm

photons per pixel (a.u.) Muons SST 0.4 9.4%@290 nm 0.01 11.3%@290 nm Muons MST

0.005 Part of photons below wavelength 0.2 Gammas (8 km)

0.025 Muons SST 1 Muons MST Gammas (8 km) 0.02 0.8 23.0%@311 nm

0.015 0.6 28.8%@306 nm

photons per pixel (a.u.) Muons SST 0.01 0.4 21.8%@290 nm 28.0%@290 nm Muons MST

0.005 Part of photons below wavelength 0.2 Gammas (8 km)

0 0 200 250 300 350 400 450 500 550 600 650 250 300 350 400 450 500 550 600 650 wavelength (nm) wavelength (nm)

Fig. 28 Normalized Cherenkov spectra of muons arriving from 270 m distance (green line), representative for the case of a SST, from 540 m distance (red line), representative for the case of an MST, and the Cherenkov spectrum from a gamma-ray shower emitted at 8 km above ground (blue line), after convolution with the quantum efficiency of the Hamamatsu R11920 PMT (top) and the new 7-dyn. ETE prototype13(bottom). Left: Differential spectrum, right: cumulative. The numbers indicate the difference between two cumulative distributions, at 290 nm and at the place of maximum distance.

7.3 Chromatic degradation of optical elements

Given the different spectra of Cherenkov light from muons and gamma-showers, un-recognized chromatic changes in the telescope optics (including cameras) may lead to un-recognized changes in the conversion from muon to gamma efficiency (Cµ−γ , see eq. 14). In this case, the muon efficiency εµ degrades, but differently to the corresponding gamma efficiency εγ , and corrections only based on εµ may under- or over- estimate the telescopes’ optical throughput for Cherenkov photons from gamma- ray showers. In order to estimate the magnitude of this effect, we have integrated both εµ and εγ from the sensitivity limit at long wavelengths (λ2) to shorter wavelengths, till the sensitivity limit at the short wavelength side (λ1), and plotted their ratio, the cumulative of dCµ−γ /dλ for the different telescope types, using typical assumptions about mirror reflectivity, quantum efficiencies and the protecting plexiglases in front of the camera. The results are shown in figure 32. Since Cµ−γ has shown up to be rather sensitivity to small variation of the integration limit at the small wavelength side (λ1), the limit has been slightly adapted to each case, such that Cµ−γ comes out to be exactly 1 at the left side.

13 The QE of this PMT is expected to improve in the near future. Title Suppressed Due to Excessive Length 45

Normalized Cherenkov Spectra, convoluted with QE Hamamatsu R11920 Cumulative Normalized Spectrum, convoluted with QE Hamamatsu R11920 SANKO Mirror reflectivity and Acrylite transmission SANKO Mirror reflectivity and Acrylite transmission 0.03 Muons SST 1 Muons MST 0.025 Gammas (8 km) 0.8

0.02 7.4%@366 nm 0.6 7.9%@366 nm 0.015 Muons SST 0.4 1.3%@290 nm 0.01 1.4%@290 nm Muons MST

Part of photons below wavelength 0.2 0.005 Gammas (8 km)

0 0 200 250 300 350 400 450 500 550 600 650 250 300 350 400 450 500 550 600 650 wavelength (nm) wavelength (nm) Normalized Cherenkov Spectra, convoluted with QE ETE D573KFLSA 7dyn. Cumulative Normalized Spectrum, convoluted with QE ETE D573KFLSA 7dyn. SANKO Mirror reflectivity and Acrylite transmission SANKO Mirror reflectivity and Acrylite transmission 1 0.03 Muons SST Muons MST

0.025 Gammas (8 km) 0.8

0.02 8.1%@356 nm 0.6 8.6%@356 nm

0.015 Muons SST 0.4 2.0%@290 nm 2.1%@290 nm 0.01 Muons MST

Part of photons below wavelength 0.2 0.005 Gammas (8 km)

0 0 200 250 300 350 400 450 500 550 600 650 250 300 350 400 450 500 550 600 650 wavelength (nm) wavelength (nm)

Fig. 29 Normalized Cherenkov spectra of muons arriving from 270 m distance (green line), representative for the case of a SST, from 540 m distance (red line), representative for the case of an MST, and the Cherenkov spectrum from a gamma-ray shower emitted at 8 km above ground (blue line), after convolution with the quantum efficiency of the Hamamatsu R11920 PMT (top) and the new 7-dyn. ETE prototype15(bottom), the reflectivity of SANKO mirrors, and the transmission of Shinkolite #000 protecting glass. Left: Differential spectrum, right: cumulative. This glass is now the default option for both the LST camera and NectarCam, and effectively cuts out all but 2% of the additional muon light received below 290 nm, if compared to the gamma-ray shower light. A systematic uncertainty of ±1% is acceptable. The absolute maximum possible error due to spectral dependencies is 8–9% for the MSTs and 7–8% for the SSTs. Since the shape of the spectral efficiencies is not expected to change much above 290 nm, this difference can be corrected for, and is only affected by possible very long-term drifts.

This choice reflects the liberty of convention between the MC simulation and the muon analysis (recall that the normalization of µ, eq. 13 and I, eq. 11, cancel out) and does not change any conclusion, it facilitates however direct reading off the effects of possible chromatic degradation of optical elements. How can the results in figure 32 be interpreted? Imagine the hypothetical case that an optical element of the LST suddenly becomes blind to wavelengths below 350 nm, while maintaining its sensitivity in the rest of the wavelength regime. In this case, the muon analysis will detect a sudden drop of the muon efficiency from εµ = 0.21 to 0.135, i.e. by 36%. It will correspondingly instruct the MC simulation to adjust the optical throughput of that LST. However, since Cherenkov light from gamma-ray is redder, a corresponding efficiency adjustment to only εγ = 0.16 would have been adequate, i.e. only by 24%. The “correction” makes a relative error 12%, found in a similar way by the value of the black line at 350 nm: R 650 nm 350 nm dCµ−γ /dλ = 0.88. We will discuss in the following the impact of several types of possible chromatic changes in optical elements of the telescopes: 46 M. Gaug et al.

Normalized Cherenkov Spectra, convoluted with QE Hamamatsu R11920 Cumulative Normalized Spectrum, convoluted with QE Hamamatsu R11920 SANKO Mirror reflectivity and Sunactive GS (Evonik) transmission SANKO Mirror reflectivity and Sunactive GS (Evonik) transmission Muons SST 1

0.025 Muons MST

Gammas (8km) 0.8 0.02 9.6%@351 nm 0.6 10.5%@351 nm 0.015 Muons SST 0.4 4.9%@290 nm 0.01 5.5%@290 nm Muons MST

Part of photons below wavelength 0.2 0.005 Gammas (8 km)

0 0 200 250 300 350 400 450 500 550 600 650 250 300 350 400 450 500 550 600 650 wavelength (nm) wavelength (nm) Normalized Cherenkov Spectra, convoluted with QE ETE D573KFLSA 7dyn. Cumulative Normalized Spectrum, convoluted with QE ETE D573KFLSA 7dyn. SANKO Mirror reflectivity and Sunactive GS (Evonik) transmission SANKO Mirror reflectivity and Sunactive GS (Evonik) transmission Muons SST 1

0.025 Muons MST

Gammas (8km) 0.8 0.02 12.7%@341 nm 0.6 0.015 14.0%@336 nm Muons SST 0.4 9.1%@290 nm 0.01 10.3%@290 nm Muons MST

Part of photons below wavelength 0.2 0.005 Gammas (8 km)

0 0 200 250 300 350 400 450 500 550 600 650 250 300 350 400 450 500 550 600 650 wavelength (nm) wavelength (nm)

Fig. 30 Normalized Cherenkov spectra of muons arriving from 270 m distance (green line), representative for the case of a SST, from 540 m distance (red line), representative for the case of an MST, and the Cherenkov spectrum from a gamma-ray shower emitted at 8 km above ground (blue line), after convolution with the quantum efficiency of the Hamamatsu R11920 PMT (top) and the new 7-dyn. ETE prototype (bottom), the reflectivity of SANKO mirrors, and the transmission of Sunactive® GS from Evonik protecting glass. Left: Differen- tial spectrum, right: cumulative. This material was the previous default option of NectarCam, but effectively cuts out only but 5–10% of the additional muon light received below 290 nm, if compared to the gamma-ray shower light. A systematic uncertainty of up to ±5% is unacceptable. The absolute maximum possible error due to spectral dependencies is up to 14%.

Chromatic degradation of the plexiglas The Shinkolite #000 plexiglas, currently favorite for the use in both LST and MST cameras, can permanently lose up to 45% reflectivity in the range from 290 nm and 450 nm, after strong exposure to UV light, whereas the rest of the light spectrum is unaffected (after some recovery time). In this case, the muon calibration will over-correct the loss by 45% · (1 − 0.82) = 8% for an affected LST, and 45% · (1 − 0.83) = 7.5% for an MST, both for the worst case scenario of an extreme exposure to Sun light.

Chromatic degradation of the focused mirror reflectivity Experience from the H.E.S.S. mirrors has shown that reflectivity losses affect the wavelength range around 300 nm 5–10% stronger than at wavelengths around 500 nm (Förster,2015). Such a scenario would translate into an error in the muon calibration of the order of 0.5–1.5% for MSTs and LSTs, and <1% for the SSTs.

Chromatic degradation of the photomultiplier quantum eﬃciency Experience with the Whipple photomultipliers has shown that the photon detection eﬃciency degraded from 5%–20% (depending on PMT tube), in the wavelength range from 290 nm to 450 nm (Daniel, 2015). If such a behaviour occurs with the CTA PMTs, the muon calibration will over-correct the loss by 1%–3.5% for the photomultiplier-equipped cameras. Title Suppressed Due to Excessive Length 47

Normalized Cherenkov Spectra, convoluted with QE Hamamatsu R11920 Cumulative Normalized Spectrum, convoluted with QE Hamamatsu R11920 SANKO Mirror reflectivity and ASTRI transmission SANKO Mirror reflectivity and ASTRI transmission Muons SST 1

0.025 Muons MST

Gammas (8km) 0.8 0.02 9.9%@351 nm 0.6 10.7%@351 nm 0.015 Muons SST 0.4 5.0%@290 nm 0.01 5.6%@290 nm Muons MST

Part of photons below wavelength 0.2 0.005 Gammas (8 km)

0 0 200 250 300 350 400 450 500 550 600 650 250 300 350 400 450 500 550 600 650 wavelength (nm) wavelength (nm) Normalized Cherenkov Spectra, convoluted with QE ETE D573KFLSA 7dyn. Cumulative Normalized Spectrum, convoluted with QE ETE D573KFLSA 7dyn. SANKO Mirror reflectivity and ASTRI transmission SANKO Mirror reflectivity and ASTRI transmission 0.03 Muons SST 1 Muons MST 0.025 Gammas (8km) 0.8

0.02 12.7%@336 nm 0.6 13.9%@336 nm 0.015 Muons SST 0.4 8.9%@290 nm 0.01 9.9%@290 nm Muons MST

Part of photons below wavelength 0.2 0.005 Gammas (8 km)

0 0 200 250 300 350 400 450 500 550 600 650 250 300 350 400 450 500 550 600 650 wavelength (nm) wavelength (nm)

Fig. 31 Normalized Cherenkov spectra of muons arriving from 270 m distance (green line), representative for the case of a SST, from 540 m distance (red line), representative for the case of an MST, and the Cherenkov spectrum from a gamma-ray shower emitted at 8 km above ground (blue line), after convolution with the quantum efficiency of the Hamamatsu R11920 PMT (top) and the new 7-dyn. ETE prototype (bottom), the reflectivity of SANKO mirrors, and the transmission of plexiglas, as used to protect the ASTRI camera (hypothetical case, since the ASTRI camera does not house PMTs, but SiPMs). Left: Differential spectrum, right: cumulative. This plexiglas cuts out all but 5–10% additional muon light below 290 nm, if compared to the gamma-ray shower light. A systematic uncertainty of up to ±5% is unacceptable. The absolute maximum possible error due to spectral dependencies is of the order to 10–14%.

Required precision for control of chromaticity of degradation If we require unrecognized chromaticity eﬀects to add maximally 3% to the systematic uncertainty of this method, we can establish a formal requirement for the precision with which the chromaticity of the degradataion of optical elements needs to be (externally) controlled. The easiest way is with respect to a reference wavelength, e.g. 400 nm above/below which about half of the Cherenkov light falls. The requirement would then read: Cumulative efficiency and spectral correction for the LST (using Hamamatsu R11290, SANKO Mirror reflectivity and Acrylite window)

λ 0.25 1 / d ε d λ ∫ 48 700 nm 0.2 0.9 M. Gaug et al. γ ε /

0.15 0.8 µ ε Gammas (10 km) 0.1 0.7 Ratio Cumulative efficiency and spectral correction forMuons the MST LST (using Hamamatsu R11920, SANKO Mirror reflectivity and Acrylite window)

λ Ratio 0.250.05 10.6 / d ε d λ ∫

700 nm 0 0.5 0.2 300 350 400 450 500 550 600 0.9 wavelength (nm) γ ε /

0.15 0.8 µ ε Gammas (8 km) 0.1 0.7 Ratio Cumulative efficiency and spectral correction forMuons the SST MST (using SiPM, SANKO Mirror reflectivity and ASTRI window)

λ Ratio 0.250.05 10.6 / d ε d λ ∫

900 nm 0 0.5 0.2 300 350 400 450 500 550 600 0.9 wavelength (nm) γ ε /

0.15 0.8 µ ε Gammas (6.5 km) 0.1 0.7 Ratio Muons SST

Ratio 0.05 0.6

0 0.5 400 500 600 700 800 wavelength (nm)

Fig. 32 Cumulatives of εµ (green) and εγ (red), integrated from the sensitivity limit at long R λ2 wavelengths down to shower wavelengths. In black, the ratio of both λ dCµ−γ /dλ is shown. All curves reach the full efficiency, or chromatic correction factor, at the left side. Mirror reflectivities and transmissions / quantum efficiencies, typical for the shown telescopes, have been assumed. Top: LST, center: MST, bottom: SST. Title Suppressed Due to Excessive Length 49

8 Simulations for the CTA telescopes

The CTA observatory will consist of different telescope types, employing different mirror sizes and field-of-view. Currently prototyped technologies employ tradi- tional single-mirror designs, either with parabolic (LST) or Davis-Cotton reflector (MST, SST-1M), and innovative dual mirror designs (SC-MST, ASTRI, GATE). The cameras will be equipped either with photomultipliers (LST, MST, CHEC-M) or silicon photomultipliers (SC-MST, ASTRI, CHEC-S, SST-1M). This variety of technologies is reflected in the present section, where each telescope team reports its preliminary study of the muon feasibility as ”calibrators”. It has to be noted that a detailed comparison among the outcomes from the various telescopes cannot be complete at this stage, mainly because the telescope simulation baseline does not yet present yet uniformity. A careful process to reach a uniform baseline for muon simulations is currently in progress within the various CTA teams involved in (CCF, MonteCarlo, Cameras, Telescopes). For the moment, the following input parameters have been agreed upon by the involved muon simulation groups:

Site: The reference site for all simulations is Paranal, located at htel = 2150 m a.s.l., using the CORSIKA atmosphere 26. Injection height: Chosen individually for each telescope, according to: hinj = htel + 2R/ tan θc ≡ 1640 + 2R · 55 m, corresponding to the Aar site and a min- ◦ imum Cherenkov angle θc & 0.5 . The primary mirror radius R is then chosen for each telescope individually. The FIXCHI parameter is set to 753 g/cm2 for an SST. FoV: The simulated view cone spans from zero to the nominal field-of-view of each telescope. It is parameterized by the VIEWCONE input parameter for CORSIKA, which needs to be given a value corresponding to half the telescope FOV. Impact parameter: Impact parameters from zero to the radius of the first mirror were simulated. Spectral index: The simulated spectral index is −2.0, in order to save simulation time. Later on, events might be re-weighted to obtain the correct muon spectrum. Zenith angle: All simulations were carried out with the telescopes pointing to zenith. Trigger: All telescopes assumed mono-trigger (in sim telarray this corresponds to the option –C TRIGGER TELESCOPES=1 ). Mirror degradation: Simulations are made with various scales of mirror reflectivity degradation (in sim telarray this corresponds to the option –C MIR- ROR DEGRADED REFLECTION=x.y, where x.y=1.0 is used for default reflectivity).

Moreover, common selection criteria and cuts were adopted for those parameters that are telescope-independent, namely:

Distance to camera edge: Maximum ring distance to camera edge: 0.3◦. Reconstructed impact parameter: Smaller than primary mirror radius. ◦ ◦ Reconstructed ring radius:0 .5 < Rrec < 1.5 . 50 M. Gaug et al.

Furthermore, each team optimized the other parameters according to the telescope conﬁguration. The level of night sky background considered in the simulations was relative to a dark sky, with extra-Galactic source observation. No stars in the FOV were simulated.

8.1 Simulations for the MST and the LST

Due to the proven performance of muons as a calibration method for both the current generation H.E.S.S. (Bolz, 2004) and VERITAS arrays (Fegan and Vas- siliev, 2007) dedicated simulations of muons in the similarly designed MSTs are not required. When employing a stereo event trigger, the rate of muons detected can be quite dependent on the position of an individual telescope in the array. Experience from the H.E.S.S. 1 array, analogous to an MST surrounded by other MSTs, leads us to expect a post-cuts muon detection rate of about 0.06 per second. Addition of a 28 m telescope at the centre of the array significantly increases the muon rate in the 12 m telescopes in the H.E.S.S. II array, giving a rate of around 0.17 muons per second. In order to avoid such large variations in trigger rate across the array, it may be beneficial to identify candidate muon-like events at camera level, keeping these for further analysis (i.e. without the array level stereo trigger requirement). Experience from the H.E.S.S. array has also shown that on a run wise basis an accuracy of 1% can be achieved on the optical efficiency of a given telescope. The CTA MSTs, with their increased field of view and quantum efficiency over the H.E.S.S. instrument, can be expected to detect significantly more muons and hence further improve the expected accuracy. The expected run to run variation of this efficiency parameter however is likely to vary on larger scales than this 1% due to uncertainties in the hardware behaviour and instrument calibration. The approximate cut values currently used for the H.E.S.S. II muon analysis are given in table 4.

H.E.S.S.-I H.E.S.S.-CT5 Cut min max min max pixels in image 30 1.e6 100 1.e6 number broken pixels 0 70 0 50 number pixels on edge of camera 0 10 0 30 Average number neighbouring pixels in image 0 3.5 0 3.5 Muon ring radius (deg.) 1.0 1.5 1.1 1.4 Muon ring outer radius (radius + impact parameter, deg.) 0 2 0 1.7 Muon ring width (deg.) 0.04 0.08 0.04 0.08 Impact parameter (m) 0.9 6.4 2.0 14

Table 4 Muon analysis cuts used for the two diﬀerent H.E.S.S. telescope types.

For the case of the LST, we assume that the H.E.S.S. II analysis can be applied in a very similar way. Dedicated simulations, especially to understand better the eﬀects of coma aberration on the muon images, are underway. Title Suppressed Due to Excessive Length 51

Trigger Efficiency Reconstructed Efficiency heffi_nCp heffi_reco Entries 1000 0.4 Entries 1000 1 Mean -1.595 Mean -1.528 RMS 0.3483 0.35 RMS 0.319 Efficiency Efficiency

0.8 0.3

0.25 0.6 0.2

0.4 0.15

0.1 0.2 0.05

0 0 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 log (E/TeV) log (E/TeV)

Fig. 33 Simulation for SC-MST. First results of the muon trigger eﬃciency vs energy (left) and the image reconstruction eﬃciency (right).

8.2 Simulations for the SC-MST

To understand the muon trigger efficiency on the SC-MST we have simulated one million muons using CORSIKA v6990. Muon energies were distributed between 1 and 100 GeV following a power law of index -2. The telescope optics were simulated using the root-based ray-tracing package GrOptics and the definition of optical surfaces in the current SC-MST design. A software package developed at Georgia Tech (CARE) was used to simulate the camera and readout. Trigger pixels were composed of 2×2 arrays of imaging pixels, and the baseline L1 topological pattern trigger was used, be based on triggering on triplets of adjacent discriminator threshold hits set at 4 photoelectrons. This simulation used optimal performance of all the light collection elements: mirror reflectivity, and photon detection efficiency of the SiPMs. The reconstruction efficiency was evaluated by selecting well-reconstructed rings with 0.5◦ < R < 2◦ and at least 0.3◦ offset from the camera edge. With these selection criteria we find an overall muon reconstruction efficiency of 30-35% for muon energies above 15 GeV, falling rapidly for lower energies. Further work is needed to evaluate the effect of degradation of the total optical throughput in the muon collection efficiency. Different pattern trigger topologies will also be explored to increase the trigger efficiency for muons. These will become especially important for the prototype SC-MST that will start commissioning in Arizona in 2015. The prototype will have a partially-populated secondary mirror and camera which will naturally result in lower muon rates. An additional trigger mode with improved acceptance for muons might be required to increase the number of muon rings available for calibration of the prototype SC-MST.

8.3 Simulations for the SST-2M ASTRI

The end-to-end telescope prototype ASTRI SST-2M has been installed in Italy, at the INAF ”M.C. Fracastoro” observing station (1735 m a.s.l.) located in Serra 52 M. Gaug et al.

La Nave, in September 2014. The completion of the telescope with the camera deployment is foreseen in early Automn 2015; the calibration phase will start soon after, followed by scientific observations of the Crab Nebula, Mrk 421 and Mrk 501. The adopted configuration, based on components not yet definitively tailored for ASTRI/CTA, will be updated in the final design to improve the telescope perfor- mances. Due to the small mirror area, it is expected that ASTRI SST-2M will not cap- ture much muon light. Nevertheless, a detailed study, through dedicated simulations of both positive and negative muons, is currently in progress (as described in the following) to evaluate both the trigger rate and, after “good” reconstruction of the muon ring image parameters, the precision on the efficiency and Point Spread Function calibration. In any case, muons will be acquired in normal Cherenkov events without any specific trigger and they will be surely seen as a cross-check with the results coming from the ASTRI end-to-end calibration strategy (Mac- carone et al., 2014) performed by means of its auxiliary systems. The muon simulations presented in this report are relative to the prototype configuration. The results will be validated from the analysis of real data obtained from the observations in Serra La Nave that will also contribute to drive the design and requirements for the ASTRI/CTA updated configuration.

8.3.1 ASTRI SST-2M conﬁguration

The code developed at INAF/IASF-Palermo for simulating the ASTRI SST-2M end-to-end prototype is mainly used for performance study and calibration pur- poses. It is a stand-alone ray-tracing that checks for the interactions of the input photons with all telescope components and follows them up to the eventual detection (Cusumano et al., 2013). The mirrors and the focal plane are simulated taking into account their modular structure and all their particular features (Pareschi et al., 2013). The primary mirror (4.3 m diameter) is segmented in 18 hexagonal facets; the secondary mirror (1.8 m diameter) is a monolithic one with a radius of curvature of 2.2 m. The focal length of 2.15 m and the F-number f0.5 correspond to a full field-of-view of 9.6◦ with angular resolution of 0.17◦. The mirror reflectivity takes into account that the 3000 nm Al coating is covered by a SiO2 substrate with 100 nm and 150 nm thickness for the primary mirror and the secondary one, respectively. The camera surface is protected by a UV transmitting plexiglas acrylic sheet 3.175 mm thick that reduces the telescope transmission above 300 nm by ∼10%. The camera at the focal plane (Catalano et al., 2014) is composed of a matrix of monolithic multi-pixel Silicon Photo Multipliers (SiPM) managed by a non- conventional front-end electronics. The model of the sensors forming the camera of the prototype is Hamamatsu S11828-3344M. A total of 496 SiPMs, organized in 37 Photon Detection Modules (PDM) covers the full field-of-view of 9.6◦. The Photon Detection Efficiency (PDE) of the SiPMs is derived from measurements performed at the INAF/OACT laboratory in Catania, Italy, and includes the effects of both cross-talk and after-pulse (∼18%). A simplified scheme of the electronics foresees two main outputs: the signal shaped with a function characterized by a shaping time of 50 ns and the trigger with 15 ns shaping time. Triggers are considered valid if they occur within a single PDM module, choice that strongly simplifies the focal plane electronics without Title Suppressed Due to Excessive Length 53

affecting the trigger rate of gamma events above 5 TeV. The trigger topological configuration assumed for the simulations is of 4 contiguous pixels, each of them with a signal above a threshold of 4 photo-electrons. The level of night sky background corresponds to one photo-electron per pixel over the 50 ns integration time of the signal. In order to better compare the ASTRI SST-2M performance in calibrating with muons with respect to the various SST configurations proposed for CTA, we have adopted the common baseline defined within CCF for such a purpose. In particular the CORSIKA simulations are relative to the Aar site (1640 m a.s.l.) and use the CORSIKA atmosphere 24. Muon energies are randomized from a power-law with spectral index 2 in the energy range 6 GeV – 1 TeV with a zenith angle equal to 0◦. The impact parameter ranges within the radius of the primary mirror and the view-cone adopted corresponds to the full camera opening angle. The image cleaning (tail cuts) and the minimum number of pixels used in the analysis are optimized for the ASTRI SST-2M camera, such as the best trigger configuration. The analysis is performed selecting well reconstructed events from complete rings with maximum distance from the camera edge of 0.3◦ and radius in the range 0.5-1.5◦. Adopting such a set of input parameters, common to all the SSTs whenever telescope independent, two millions of µ+ and µ− events have been simulated with CORSIKA v6.99. Figure 34 shows a ring example.

Fig. 34 Muon ring example as triggered by the ASTRI SST-2M telescope. Positive muon at energy 9 GeV embedded in a night sky background level of 20 MHz per pixel.

A ﬁrst strong cut on all pixels is used to make visible the ring respect to the background: all pixels with number of photoelectrons higher than half of the maximum value or 5 times the root mean square, RMS, of the background are considered and the event is analyzed if a number pixels ≥4 pixel are left. To evaluate the ring geometrical parameters, the coordinate of the center (Xc, Yc) and the radius (R) are computed with the Taubin method (Taubin, 1991) minimizing the function ξ given by the following formula:

P 2 2 2 [(X − Xc) + (Y − Yc) − R ] ξ = P 2 2 , (53) [(X − Xc) + (Y − Yc) ] 54 M. Gaug et al.

where X and Y are the image coordinates of the pixels survived to the tail cuts. Events are assumed well fitted if ξ ≤ 0.05. Selecting on radius in the range 0.9◦ ≤ R ≤ 1.5◦, only '6% of events are left, but the selection efficiency is constant at all optical efficiency up to 70% of the nominal value and the direction reconstruction error is 0.14◦ ± 0.07◦. Figure 35 shows the selection efficiency as function of muon energy and impact point, and Figure 36 the trigger efficiency as a function of the optical efficiency of the telescope. The reconstruction of the muon ring radius (not shown here) seems to be possible with a stable bias of smaller than 5%, and if correctly subtracted, will degrade the systematic precision of the average muon efficiency parameter by less than 1%.

Fig. 35 ASTRI SST-2M telescope. Left: muon selection eﬃciency as function of energy for ξ ≤ 0.05 and 0.9◦ ≤ R ≤ 1.5◦, right: as a function of impact point distance.

The same analysis has been performed on an equal number of events using Serra La Nave as site: results are statistically equivalent to those relative to Aar site.

8.3.2 ASTRI/CTA mini-array conﬁguration

As a second step, the ASTRI project is addressed to the implementation of an AS- TRI/CTA mini-array composed of seven SST-2M telescopes (Pareschi et al., 2014); placed at the final CTA Southern Site, the mini-array could represent a precursor of the whole CTA Observatory. The configuration of the ASTRI/CTA telescopes will be updated taking into account the results of the observations and calibra- tions performed with the prototype at Serra La Nave. The main improvements will derive from new SiPM sensors (currently under test) which present better performance, as higher detection efficiency and strong reduction of the cross-talk and after-pulse. Moreover the read-out camera electronics could be modified to better fit the focal plane detectors working specifics. Statistically significant simulations for the muon feasibility study will be then performed after the establishment of such improvements. Title Suppressed Due to Excessive Length 55 ASTRI Muon Relative Trigger Efficiency 0 0.5 1 0.2 0.4 0.6 0.8 1 Optical Efficiency / Nominal

Fig. 36 ASTRI SST-2M telescope. Muon eﬃciency as a function of the optical eﬃciency for muons in the range 0.9◦ ≤ R ≤ 1.5◦.

8.4 Simulations for the SST-1M

The results reported in the following have been obtained from simulations by using the latest telescope and camera parameters (e.g. mirror reflectivity, light collectors, photon detection efficiency of the GAPD sensors forming the SST-1M camera, hexagonal geometry of the camera,...) as detailed in the document (Moderski et al., 2014). The baseline trigger is a digital sum trigger with sampling rate 250 MHz. Muons are simulated with the standard CTA software, namely CORSIKA (version 6990) and sim telarray (version 2013-12-15), and then processed with the Chimp executable and with Mars, the standard software for the MAGIC telescopes data analysis. In this analysis only vertical negative muons resulting in complete ring images (with maximum impact parameter not greater than 2.2 m) have been simulated. Muon energies were distributed between 6 and 1000 GeV following a power law of index -2. The atmosphere model relative to the Aar site, 1640 m a.s.l., has been adopted with the atmospheric refraction for Cherenkov photons included (ATMOSPHERE 24 Y); the parameters of the magnetic field at the Aar site are assumed (MAGNET 10.9 -24.9). A full muon ring reconstruction analysis has also been carried out. In particular, different muon parameters like the impact parameter, the ring center, its radius and profile, have been estimated with this analysis. Furthermore, the ξ parameter (Eq. 53), as defined in ASTRI analysis has been evaluated. We apply the following quality cuts to recognize muons:

– The ring ﬁtting the muon image shoule be fully contained in the camera; – The reconstructed radius should be between 0.5◦ and 1.5◦; – The projected ring width along the ring radius should have a gaussian shape (reduced chi square of the arcwidth ﬁt < 2); 56 M. Gaug et al.

– the ξ parameter should be smaller than 2. There has been no study yet on the eﬃciency of these cuts in the recognition of muons with respect to cosmic rays. An example of a full ring image as it would appear on the SST-1M is shown in Figure 37.

Fig. 37 Muon ring example as triggered by the SST-1M telescope.

Like in the case of other realizations of the SST, a possible trigger bias seem to be the point of biggest concern because of the relatively small muon image sizes. With the purpose of obtaining a muon efficiency curve independent of the optical degradation, the effect of the trigger threshold and that of selection cuts have been investigated. Our simulations have shown that the best scenario is obtained when adopting a trigger threshold of 100 ADC counts (the standard trigger treshold for SST-1M is 127 ADC counts). This solution would provide a higher acquisition rate, however still sustainable by the SST-1M readout system. The effect of cuts on the muon parameters did not provide any improvement to the flatness of the muon efficiency curve. The muon efficiency curves obtained with a trigger threshold of 100 ADC counts as a function of the muon energy above 10 GeV is reported in Figure 38 (left). The effect of the muon energy on the muon efficiency is rather neglibile for all the considered optical efficiencies. The muon efficiency is estimated as the ratio between the observed and simulated muons. The right side of Fig. 38 shows the muon efficiency as a function of the optical efficiency. The muon efficiency has a constant value of around 60% in the region of interest (between 60% to 100% of optical efficiency). Figure 39 shows the resolution of the reconstructed muon ring radius as a function of the muon energy. The bias on the reconstructed ring radius is always smaller than 1%, over the entire investigated energy range. In conclusion, our analysis shows that muons trigger the SST-1M telescope, and the trigger rate seems to be rather insensitive to the optical efficiency of the system if a trigger threshold of 100 ADC counts is adopted. Title Suppressed Due to Excessive Length 57 true 0.3 / R true -R

rec 0.2 R

0.1 Fig. 38 Left plot: Muon efficiency for SST-1M obtained with a trigger threshold of 100 ADC counts as a function0 of energy for different optical efficiencies (from 100%, red dots, to 50%, cyan dots). Right plot: Muon efficiency for SST-1M obtained with a trigger threshold of 100 ADC counts and as-0.1 a function of the optical efficiency for muons with an energy above 10 GeV.

-0.2

-0.3

1 1.5 2 2.5 3 Log10(Energy) [GeV]

Fig. 39 Resolution of the reconstructed muon ring radius (Rrec − Rtrue)/Rtrue as a function of the muon energy. 58 M. Gaug et al.

8.5 Simulations for the SST-2M GATE with CHEC cameras (GCT)

Here we investigate the potential of using muons as a method for optical calibration of the GCT (Gamma Cherenkov Telescope) candidate for a dual mirror SST; for a description of the structure see Dournaux et al. (2015) and a description of the CHEC-M camera using MAPMT with 2048 pixels can be found in De Franco et al. (2015). The muon events were simulated using CORSIKA v6990 and the telescope optics with sim telarray using the same configuration files as the ones used for Monte Carlo production Prod-3 (Armstrong et al. (for an overview see 2015)). We use a single telescope with the default Majority sum trigger of 8.6 p.e. and 11.4 p.e. threshold at pixel level for GCT-M and GCT-S, respectively, and 2 pixels at sector level (where a sector consists of 4 pixels). The per pixel trigger rate from night sky background is set at 14.2 MHz and 41 MHz, respectively. Muons are injected in the energy range between 4 GeV and 1 TeV with an initial power law spectrum with index −2. They were injected from a height of 2.057 km a.s.l. (407 m above the telescope) over the Aar reference site with the telescope pointed at zenith. To estimate the expected rate of muon triggers, 106µ− were simulated in a view cone of 0 to 4.7◦ (field of view of the camera) with an impact parameter between 0 and 2.2 m (radius of primary). The files were then analyzed by a modified version of read hess, in which two levels of tail cuts where applied in order to optimise the selection of the muon rings. First a hard cut of 5, 10 p.e. and then a loose cut of 3, 6 p.e. Each image is then modeled with a simple ring using the Taubin method (Taubin, 1991). This fit is then used to estimate some of the muon parameters. To ensure only complete rings were stored in order to represent the expected muon rate for calibration, we implemented the following cuts:

– Edge Cut: Any ring that was within 0.3◦ of the edge of the camera was rejected – Amplitude Cut: Total amplitude of triggered pixels >20 p.e. – Pixel Cut: Total number of triggered pixels >10 – Ring Radius: Reconstructed ring radius had to be between 0.5◦ and 1.5◦. – χ2/NDF of ring ﬁt <0.05

The amplitude and pixel cut values are currently just initial estimates and need optimizing. Figure 41 shows the resolution of the reconstructed ring radius for GCT-S. The RMS of the distribution is at the level of ±3%, and a bias considerably smaller than 1% (except for the lowest energies well below 10 GeV) is visible. Figure 43 shows the absolute muon trigger efficiency as a function of degradation of the optical efficiency for both GCT-M and GCT-S. The degradation describes the aging of the total optical throughput of the telescope structure, including mirrors. The solid lines denote the trigger efficiency for all events triggering the camera, while the dashed lines indicate the trigger efficiency after cuts have been applied to the triggered events. The dashed lines indicate that the drop in trigger efficiency for both GCT-M and GCT-S is small down to 70% of the default optical efficiency of the telescope. Applying the toy model of Section 7.2.2, we can estimate that the resulting bias on εµ is smaller than 2%. It should be emphasized Title Suppressed Due to Excessive Length 59 0.3 0.25 Efficiency Efficiency 0.25 0.2

0.2

0.15 0.15

0.1 0.1 opt. eff 1.0 opt. eff 0.9 opt. eff 1.0 opt. eff 0.8 opt. eff 0.9 opt. eff 0.8 0.05 opt. eff 0.7 0.05 opt. eff 0.7 opt. eff 0.6 opt. eff 0.6 opt. eff 0.5 opt. eff 0.5 0 0 -2 -1.5 -1 -0.5 0 -2 -1.5 -1 -0.5 0 log10(Energy [TeV]) log10(Energy [TeV])

Fig. 40 Post-cut muon efficiency as a function of energy for different optical efficiencies (from 100%, red dots, to 50%, cyan dots). Left: for GCT-M, right: GCT-s. 1 1 200 Mirror 100 mirror )/R 0.8 180 )/R 0.8 true true -R -R rec 160 rec 80 (R 0.6 (R 0.6 140

0.4 120 0.4 60

100 0.2 0.2 80 40

0 60 0

40 20 -0.2 -0.2 20

-0.4 0 -0.4 0 -2.5 -2 -1.5 -1 -0.5 0 -2.5 -2 -1.5 -1 -0.5 0 Energy (TeV) Energy (TeV)

Fig. 41 The resolution of the muon ring radius as a function of muon energy. Left: for GCT-M, right: for GCT-S.

Fig. 42 An image of a muon ring event, as seen with GCT-S. The colour scale denotes the number of pe detected in a given pixel. The x-y position of the pixels, along with the gaps between the 32 modules, are not representative of the GCT focal plane. however that there is a considerable potential to improve this analysis, and reduce the remaining reconstruction bias. 60 M. Gaug et al.

Fig. 43 The absolute muon trigger efficiency as a function of degradation of the optical efficiency for both GCT-M and GCT-S. The degradation describes the aging of the total optical throughput of the telescope structure, including mirrors. The solid lines denote the trigger efficiency for all events triggering the camera, while the dashed lines indicate the trigger efficiency after cuts have been applied to the triggered events. Title Suppressed Due to Excessive Length 61

9 Muon enrichment in the camera server

During a dedicated interface meeting between ACTL and the telescope cameras in May 2015 in Heidelberg, the cameras committed to accept a common clock distribution and trigger timestamping board (UCTS) board, and to provide an input trigger signal to that board for time-stamping of stereo triggers. The time-stamp is then provided to the software array trigger (SWAT) which takes the ultimate trigger decision. The camera server has however the option to force readout and storage of an individual event. This is the case for local muon images which need to be recognized as such in the camera server (from un-calibrated events) at high efficiency while keeping the number of mis-identified events as low as not to saturate the readout. This task may be challenging for the SSTs, given the relatively small size of their muon images. That’s why the ASTRI telescope has started to develop an efficient and fast algorithm to recognize muons in the camera server.

9.1 Muon enrichment algorithms in the camera server

The Taubin method Taubin (1991) allows to determine the geometrical parameters of the ring in a fast and efficient way. According to preliminary simulations for ASTRI, selection cuts on the reconstructed ring radius, the distance of the ring center from the camera center, and the ring width achieve > 99.5% efficiency for muons while rejecting > 98.8% of the proton-induced triggers. The latter efficiency may be further improved with cuts on the reconstructed Hillas parameters, which are approximately as fast as the Taubin method. Cuts based on the number of pixels in the cleaned image, or the average count per pixel, or the full distribution of counts per pixels, have been shown to be by far inferior to the aforementioned method, at least for the ASTRI camera. 62 M. Gaug et al.

10 Trigger strategies and expected rates

In order to estimate the expected rate of muon images which can be used for the measurement of optical eﬃciencies (ERgood), we combined Eq. 1 and 2, and weight the muon rates with the energy-depending eﬃciencies, including trigger and analysis cuts selection, which we have derived from MC simulations shown in the previous sections:

Z ∞ 2 ERgood = F (E, h) · Rimp · π · 2π · (1 − cos θmax) · η(E) dE, (54) 0 where Rimp is the maximum simulated impact parameter, and θmax the maximum simulated zenith angle. The overall efficiency, as a function of energy, is denoted by η(E), and contains all trigger and analysis cut efficiencies. Eq. 54 is constructed such that the integral ERgood remains unchanged, independently of the values of Rimp and θmax, if these are only chosen big enough, since the efficiency η(E) takes care of removing all simulated, but unused events. Requiring a statistical precision of better than 1% for the derived muon efficiency εµ, we estimate a minimum number of about 50 usable muon images for the larger telescopes, using 5% RMS for the distribution (εµ(data) − εµ(MC))/εµ(MC) (see Bolz, 2004, fig. 3.16 right), and 400 images for the SSTs, using 20% RMS of the distribution of εµ.

Time needed Time needed ERgood to obtain ERgood to obtain at θ = 0◦ suﬃcient at θ = 60◦ suﬃcient comments (Hz) usable events (Hz) usable events at θ = 0◦ at θ = 60◦ LST 19 3 s 6.7 7 s for 50 usable events MST 5 10 s 1.8 28 s for 50 usable events SCT 1.6 1 min 0.6 3 min for 100 usable events SST (ASTRI) 0.2 40 min 0.06 1.8 hr with SiPMs used for the prototype, using strong analysis cuts, for 400 usable events SST (GCT with 0.9 8 min 0.3 22 min preliminary, for 400 usable CHEC-M) events SST-1M 2 3 min 0.7 9 min with very loose analysis cuts, for 400 usable events

Table 5 List of expected good muon rates above 10 GeV for the proposed telescopes of the CTA (assuming an observatory altitude of 2150 m) and the preliminary analyses developed by the individual telescope teams. Title Suppressed Due to Excessive Length 63

Item LSTs MSTs SSTs comments (%) (%) (%)

Instrumental part U0 Determination of average R <0.5 <0.5 <0.5 Account for hexagonal mirrors (see Fig. 3). Shadows <0.5 <1 ?? May require cuts on impact parameter and inclination angle. Still to be assessed by simulations for SSTs.

Reconstr. Cherenkov angle θc Reconstr. bias from analysis <0.5 <0.5 <1 Small correction might be necessary, but independent of ring radius (see Fig. 39 and 41). Ring broadening effects <0.5 <0.5 <1 Modifies ring radius along muon path (see Fig. 22). Coma aberration effects ?? 0 0 Still to be assessed for LST. Light modulation due to impact dis- <1 <1 <3 See Eq. 25 and Fig. 10. Still to be verified for SSTs. tance E0 Atmospheric transmission T

Molecular part Tmol <0.2 <0.1 <0.04 See Sect. 7.2.1. Aerosol part Taer 1–3 <2 <1 Exclusion of very bad nights needed, or correction from atmospheric monitoring data, see Sect. 7.2.1.

Reconstructed image size Ntot Trigger biases 1–3 0 <2 See Sect. 7.2.2, 7.1.1, 8.3, 8.4 and 8.5. Stereo trigger is assumed for the LST, but can be corrected using mono runs. Checks with different levels of NSB still missing. Signal extraction biases 0 0 0 Requires un-biased signal extractors (fixed window). Image selection biases 0 0 <1 See Sect. 7.2.2 and required analysis cuts in Sect. 8.3, 8.4 and 8.5. Pixel baselines <0.1 <0.2 <1 See Fig. 6, assumed requirement B-xST-1370. Non-active pixels <0.5 <0.5 <0.5 Requiring less than 10 broken pixels on the ring, see Fig. 17. Translation muon to gamma efficiency εµ → εγ Chromaticity of degradation <2 <2 <1 Need requirements B-xST-1500 and B-xST-1600. Mis-focused mirrors ?? ?? ?? Exact magnitude of effects needs MC simulations.

Total 3–5 <3.5 <4.5 Missing items not yet accounted Table 6 Summary of systematic uncertainties.

11 Discussion and available precision

The muon image reconstruction algorithm used by H.E.S.S. and as implemented by O. Bolz (Section 6.1) and later adapted for use with multiple telescopes, shall serve as the main muon reconstruction algorithm, cross-checked by an additional algorithm, possibly based on the MAGIC or the VERITAS analysis. From the experience with muon calibration in H.E.S.S., MAGIC and VERITAS and a detailed analysis of the method (Eq. 21), we derive the expected systematic uncertainties listed in Table 6 and the following overall precision of the optical throughput calibration using muon rings: 64 M. Gaug et al.

Calibration of optical throughput: If two new hardware requirements B-xST- 1300 and B-xST-1500 (see below) are met, the optical throughput of each telescope can be calculated with better than 2% systematic uncertainty for any achromatic degradation of the optical throughput and at least on a night- per-night basis. Systematic biases (due to the reconstruction of the Cherenkov angle, trigger biases, etc.) can then be controlled to that precision. For the larger telescopes, faster updates can be easily achieved. Only nights with strong extinction of light due to local aerosols need to be discarded for that analysis. Corrections to the muon calibration: In the case of chromatic degradations, the correction applied by the muon eﬃciency (Eq. 13) might result in an over-correction if used for the Cherenkov light from gamma-ray showers of maximally 13% (7% for the cameras equipped with SiPMs). Since one can always make an educated guess on the chromacity of the degradation, and the experience from past IACTs, the over-correction can be corrected itself with a precision of about 5% (3% for the SiPM cameras). This means that muon calibration without wavelength-dependent direct measurements will result in about 6% systematic precision for the L/MSTs (4% for the SSTs), after a long period of degradation of the optical elements. If the chromaticity of the degradation can be controlled to better than 10% (i.e. the new requirement B-xST-1600 is met), the precision can be improved to 2–3% again.

The obtained muon trigger rates and the precision of the muon analysis strongly depend on two main points:

Trigger schemes: A separate mono trigger for muon images is always recommended. The estimated muon image rates (Table 5) have been obtained assuming a dedicated mono trigger for muons. A stereo trigger would reduce the rates according to the distance between telescopes (cf. Fig. 7). For SSTs, the stereo muon rate would clearly tend to zero, given their small mirror area and large distances to other telescopes. In the case of the MSTs and the LSTs, a pure stereo trigger should yield sufficient images, however the stereo bias (Section 7.1.1) needs to be carefully controlled and monitored with time. This may require dedicated mono runs from time to time. Fully contained muon rings can however be recognized efficiently already from un-calibrated images in the camera server and flagged to be written to disc, even if no stereo coincidence is available. Plexiglas solutions: PMT-based cameras (and possibly cameras using new SiPMs with enhanced photon detection efficiency towards the UV) need to ensure that the camera becomes sufficiently opaque to light below 290 nm wavelength. It seems that the only technically viable way to achieve this consists of the adequate choice of the protecting plexiglas in front of the camera. We have seen that the Shinkolite #000 glass does fulfill this requirement, while the Sunactive® GS does not (see Section 7.2.7). Other materials shall be carefully evaluated in cooperation with the muon calibration group.

To ensure the required precision in the muon analysis, we suggest hence the following new level-B requirements: Title Suppressed Due to Excessive Length 65

B-M/SST-1300 The camera must be able to trigger on, and ﬂag from pre- calibration data, fully contained muon rings impacting the mirror with an energy >20 GeV with an eﬃciency greater than 90%, even if visible in only one telescope camera. Requirement B-M/SST-1300 applies to the SSTs and the MSTs, while it is not strictly necessary (although recommended) for the LSTs. Our study suggests that the LSTs which do not provide a separate mono-trigger for muons during regular data taking shall carry out regular checks between mono and stereo triggered muons, in order to ensure that the bias introduced by the stereo trigger does not degrade the performance of the derived optical throughput beyond 3%. B-xST-1500 The optical elements of the telescope (mirrors and camera) must ensure at all times that the part of the Cherenkov light spectrum from local muons, which stems from wavelengths below 290 nm, must contribute by less than 5% to the observed muon image size, where “size” is understood as the sum of all photo-electrons contained in the ring image. Appropriate monitoring must be implemented.

B-xST-1600 The degradation of the optical throughput of the telescope must always be controlled such that the degradation at a wavelength measured 10 nm around 350 nm is known to better than 10% with respect to the degradation measured 10nm around 450 nm.

Muon calibration allows to monitor additionally, and without much effort the following parameters: Monitoring of flat-fielding: Using the muon images to monitor the flat-fielding of the camera, according to Eq. 38, should be feasible at least for the large telescopes, but could also be tried for the SSTs. An automatic monitoring algorithm shall be implemented in the off-site analysis. Monitoring of mirrors: Similarly, differences in reflectivity among the mirrors can be detected by plotting the muon efficiency as a function of reconstructed inclination angle. This procedure should work at least for the SSTs and the MSTs. Because of the larger mirror area covered by LST muon light, further investigation is necessary to find out whether such an approach is also useful for these telescopes. Monitoring of the optical PSF: Following the approach of the MAGIC analysis (Section 6.2), the 0.2 quantile of the muon ArcWidth distribution shall be used to monitor changes in the telescopes’ optical point-spread-function. Because of the very strong requirements on the optical quality of the mirrors, the muon ArcWidth might be dominated by atmospheric effects in the case of the LSTs and SCTs. Whether such an approach is useful for theses two telescope types, needs to be found out still. An automatic monitoring algorithm shall be implemented in the on-site, as well as the off-site analysis. Monitoring of the time resolution: For those telescopes with record the event arrival times, the muon images will provide the time resolution of each pixel, and can serve to determine the relative time offsets. Once muon calibration implemented, we have found that the following parameters and dependency provided by the muon analysis, shall be monitored regularly, in order to ensure its precision: 66 M. Gaug et al.

The off-ring intensity: needed to control possible biases on the pixel baselines (see Sect. 7.1.1) The average number of pixels on the ring: needed to control the correct working of trigger and selection cuts (see Fig. 9). The muon efficiency: vs. time, temperature, atmospheric pressure, humidity, reconstructed impact parameters and inclination angles, as well as the number of de-activated pixels and the number of pixels (with charge above tail cut) which are excluded from the muon image analysis, to control systematic effects. The latter is especially important for the case of stereo-triggered muons. The optical telescope PSF: derived from the muon ArcWidth vs. time, temperature, humidity and zenith angle, in order to control possible systematic effects on the reconstruction of the PSF.

12 Assumptions and Caveats

Generally, all simulations have been carried out using those parameters and char- acteristics of the telescopes which were state-of-the-art during the redaction of this report. Some of the assumptions may change soon, as the camera designs evolve towards more eﬃcient ﬁnalized products (most of which will make the present assumptions conservative). This is especially true for:

1. the trigger designs some of which are currently being improved in order to catch muon images more eﬃciently. 2. the photon detection eﬃciency of the SiPM, which may considerably increase towards the blue and UV, as the most recent developments in industry suggest.

The level of night sky background assumed in the simulations correspond to extra-galactic night sky background without bright stars in the field-of-view of the telescopes. How much different background light levels, especially those found during observation of Galactic sources, affect the trigger rates and the reconstruction precision, needs to be found out in the future with more detailed simulations. Similarly, pure muon samples have been simulated so far, without making any effort to mix the samples with air shower data and investigate their influence on the derived selection efficiencies and precision of the muon analysis. It may be that analysis cuts need to be further tuned to accomodate with the left-overs of cosmic ray images. Several items, as listed below, need to be investigated by means of more specific simulations:

– The effect of coma aberration needs to be studied in detail for the parabolic mirrors, like the LST. Especially, further studies shall be carried out to find out whether some information on coma aberration can be obtained by comparing the muon arc width for different muon inclination angles. – Some of the camera triggers are still not properly simulated, efforts are still ongoing to incorporate all features in CARE and GrOptics. – All simulations have been carried out for one generic site, namely Aar, located at 1640 m a.s.l. Going to higher altitudes will increase the muon flux at 10 GeV. A rough estimate using Eq. 2 shows that the dependency is very small. Title Suppressed Due to Excessive Length 67

– All simulations have been carried out for vertical incidence. Observations at higher zenith angles will suppress the ﬂux muons at 10 GeV and increase the ﬂux of muons with energies greater than 100 GeV.

The effect of zenith angles other than vertical has been very roughly estimated using Eq. ??, but a better model is certainly required. For the LSTs and the MSTs, the HESS selection of cuts (Table 4) gives a good starting point, and shall be further refined using real data. Finally, the simulations have not yet been confronted with real data, which will be available as soon as the first telescope prototypes are delivering data. 68 M. Gaug et al.

13 Conclusions

We have assessed the viability to use muon images for the calibration of the optical throughput, monitoring of the optical PSF, the flat-fielding of the camera, and uniformity of the mirror reflectivity for the different telescopes currently designed for the CTA. If certain design requirements are met, like separate mono-triggers, possibly adjusted to the expected shapes of muon images, and design of the telescope and camera components which ensure that the transmitted part of the muon spectrum below 290 nm becomes negligible, such a calibration scheme seems viable for all telescopes, using regular data taken in less than one night. Apart from the well-known performance of the larger telescopes (LST and MST), all implementations of an SST are able to provide sufficient high-quality muon images per night to and seem to be free of, or show only negligible, trigger biases, if the mirrors degradation is maintained within current requirements (B- xST-0120 and B-xST-0125), muon images are adequately selected (ASTRI and GCT (MaPMT option)) and a low trigger threshold is adopted (SST-1M), at least for observations of extra-galactic sources. A more detailed investigation of the effects of coma aberration for the parabolic mirror designs (like the LST) is still necessary in the future, as well as more detailed simulations incorporating the most recent telescope designs. This should be done however, once the sites are selected, using the correct site-related input parameters. The precision of muon calibration can be safely estimated to about 2% systematic uncertainties for any achromatic degradation of the optical throughput from Cherenkov photons in the wavelength range between 290 nm and 700 nm. Instead, in the case of chromatic degradation, the correction applied by the degradation of the muon efficiency might result in an over-correction for the Cherenkov light from gamma-ray showers of maximally 13% (7% for the cameras equipped with SiPMs). Since one can always make an educated guess on the chromacity of the degradation, and the experience from past IACTs, the over- correction can be corrected itself with a precision of about 5% (3% for the SiPM cameras). This means that muon calibration without wavelength-dependent direct measurements will result in about 6% systematic precision for the L/MSTs (4% for the SSTs), after a long period of degradation of the optical elements, with such an additional regular wavelength-dependent assessment, the precision can be improved to 2–3%. These numbers can however only be achieved if the new proposed requirements B-xST-1300, B-xST-1500 and B-xST-1600 (see Section 11) are met. Title Suppressed Due to Excessive Length 69

A Derivation of 2◦ mirror shadowing for muon rings

In all the following, a circular mirror shape is assumed. All of the light hitting the mirror from one part of the Cherenkov light cone around the muon will hit the mirror at the same angle, and hence is reflected to the same point on the camera. The variation in light intensity with azimuthal angle φ then depends only on the distance across the mirror dish from which that light is reflected; D(ρ, φ) (see Fig. 2). In the case of a single mirror, you have the two cases according to whether the impact parameter ρ is larger than or less than the mirror radius R (i.e. whether or not the muon hits the mirror). Taking Eq. 11 as a starting point:  q ρ 2 2 2R 1 − R sin φ (ρ > R) D(ρ, φ) = q (55) R 1 − ρ 2 sin2 φ + ρ cos φ (ρ ≤ R)  R R In the case of a dual mirror telescope, you need to consider how much of the chord D is affected by shadowing from the second mirror.

A.1 Assuming i = 0.; i.e. on axis

There are several cases to be considered, as outlined below. The distance is integrated over all φ angles, and whether or not the intensity in a given pixel is aﬀected by shadowing depends on the azimuthal angle φ. An important term to determine which case is relevant is d = ρ sin φ, which is the distance from the center of the mirror to the line D, along the perpendicular intersection with D. The part of D traversing the secondary mirror and therefore aﬀected by shadowing is D0. Also, it is assumed here that the shadowing is complete; if not, then the total distance C is C = D − αD0, where α is the strength of shadowing (here α = 1).

A.1.1 ρ > R and ρ sin φ > R0

The muon falls outside of both mirrors. In this case, there is no shadowing by the secondary mirror, the chord D traverses the primary mirror at a radial distance from the center larger than the radius R0 of the secondary mirror. The expression to use is the ﬁrst one in equation (55) and C = D. r ρ 2 D(ρ, φ) = 2R 1 − sin2 φ (56) R

A.1.2 ρ > R and ρ sin φ < R0

The muon falls outside of both mirrors In this case, the cord traverses both the primary mirror and the secondary mirror. A term similar to that in the previous case expresses the length D0 traversing the secondary mirror and hence aﬀected by shadowing (replacing R by R0).

r ρ 2 D0(ρ, φ) = 2R0 1 − sin2 φ (57) R0 Length D0 should always be less than D, the total distance C = D − D0

A.1.3 ρ < R and ρ sin φ > R0

The muon hits the primary mirror. The chord does not traverse the secondary mirror, hence the expression to use is the second part of equation (55): " # r ρ 2 ρ D(ρ, φ) = R 1 − sin2 φ + cos φ (58) R R

and total distance C = D. 70 M. Gaug et al.

A.1.4 ρ < R and ρ sin φ < R0

In this case, the chord is aﬀected by shadowing again. The basic chord across the primary mirror is given by (58). Note that with ρ sin φ < R0, it is still possible to have ρ > R0 and ρ < R0, so there are two sub-cases for the shadowing term D0:

ρ > R0 In this case, the path traverses across the secondary mirror fully, as in case 2, so the expression for D0 is (57). The full term is then (58) with shadowing term (57) (C = D − D0).

ρ < R0 The last case, in which the term for D0 → D0/2 + ρ cos φ (with D0 retaining it’s previous deﬁnition). Shadowing term in this case is then: " # r ρ 2 ρ D0(ρ, φ) = R0 1 − sin2 φ + cos φ (59) R0 R0

The full term is then (58) with shadowing term (59) (C = D − D0).

A.2 Oﬀ Axis: i 6= 0.

All of the above is derived in the case of on-axis muons. The effect of a muon arriving with an inclination angle i is to shift the muon ring away from the centre of the camera, and this will also shift the position of the shadow from the secondary mirror on the primary mirror dish. As long as the shadow is wholly contained within the primary mirror dish, then the term D0 can be subtracted as per usual; total integration distance C = D − D0, and it doesn’t matter where the shadow is located. The inclination angle i is given by the offset of the centre of the muon ring in the camera from the camera centre. For the inclination angle to be large enough to shift the shadow from the secondary mirror such that it no longer falls entirely on the primary mirror dish, the following must be true: ∆m sin i > R − R0 (60) where ∆m is the distance between the two mirror dishes, and R and R0 are the radii of the primary and secondary mirror dishes respectively. That is - if replacing i by the (radius of the) field of view, you find that:

∆m sin i < R − R0 (61) always, then you never need to worry about inclination in shifting the shadow of the secondary mirror. Title Suppressed Due to Excessive Length 71

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