[a,b]Hiroko Miyamoto

#  Tests and characterisation of the KI trigger for fast events on the EUSO-
SPB2 Fluorescence Telescope

Matteo Battisti  Alexander Belov  Mario Bertaina  Sylvie Blin  Alxandre
Creusot  Johannes Eser  George Filippatos  Pavel Klimov  Massimiliano Manfrin
Marco Mignone  Etienne Parizot  Lech Wiktor Piotrowski  Guillaume Prévôt
Daniil Trofimov

###### Abstract

The second generation Extreme Universe Space Observatory on a Super-Pressure
Balloon (EUSO-SPB2) mission is a stratospheric balloon mission developed
within the Joint Exploratory Missions for Extreme Universe Space Observatory
(JEM-EUSO) program. The Fluorescence Telescope (FT) is one of the two separate
Schmidt telescopes of EUSO-SPB2, which aims at measuring the fluorescence
emission of extensive air showers from cosmic rays above the energy of 1
$EeV$, looking downwards onto the atmosphere from the float altitude of 33 km.
The FT measures photons with a time resolution of 1.05 ${\mu}$s in two
different modes: single photon counting (PC) and charge integration (KI). In
this paper, we describe the latter and report on the measurements of its
characteristics. We also present a new trigger based on this channel, the so-
called KI trigger, which allows to measure additional types of events, namely
very short and intense light pulses. We report on the tests of this trigger
mode in the laboratory and at the TurLab facility, and its implementation in
the EUSO-SPB2 mission.

##  1 Introduction

The second generation Extreme Universe Space Observatory on a Super-Pressure
Balloon (EUSO-SPB2) mission [1] is a stratospheric balloon mission developed
within the Joint Exploratory Missions for Extreme Universe Space Observatory
(JEM-EUSO) program [2]. It hosts two main telescopes: a fluorescence telescope
(FT), looking towards the nadir and aiming primarily at the detection of
extensive air showers (EAS) from high-energy cosmic rays (above $\sim 10^{18}$
eV) [1], and a Cherenkov telescope (CT), looking almost horizontally within a
few degrees above and below the limb of the Earth, aiming at the detection of
Cherenkov light from air showers induced by cosmic rays (above the limb) or
upward going neutrinos (below the limb) [3].

The FT is composed of 3 photo-detection modules (PDMs), each of which features
9 elementary cells (ECs) composed of 4 Hamamatsu 64-ch Multi-Anode-
Photomultiplier Tubes (MAPMTs), for a total of 2304 pixels. An EC is a compact
assembly of HVPS generator board and front-end electronics based on a
SPACIROC-3 ASIC [5] for each PMTs, potted in a gelatinous compound to prevent
discharge between the various components.

##  2 Photon counting and charge integration

The so-called Gate Time Unit (GTU) is the basic time window of the FT, which
is set to 1.05$\mu$s in EUSO-SPB2. Over each GTU, the ASIC performs two types
of measurements: photon counting (PC) and charge integration (referred to as
“KI”). The PC functionality allows the independent counting of charge signals
on each of the 64 pixels of the MAPMT, with a time resolution of $\delta
t\simeq 6$ ns. The anode signals go through a low input impedance preamplifier
followed by a fast discriminator, optimized for single photon detection. The
threshold is set by a common 10-bit DAC register for each PMT, as well as a
7-bit DAC for each individual channel, used to compensate the MAPMT pixel gain
dispersion. When two photoelectrons are generated at the photocathode within
an interval of time lower than the resolution $\delta t$, only one photon is
counted. Therefore, for light intensities corresponding to more than $\sim
20$–25 photoelectrons, pile-up [7] becomes important and the photon counting
is not anymore proportional to the actual photon flux. An example of the
resulting saturation curve is shown in the right panel of Fig. 1, where the
photon counts are seen to first increase linearly with the photon intensity,
then reach a maximum around 45 per GTU, and finally decrease to zero as more
and more photons are “piled-up” together.

To help solving the ambiguity on the true number of photon counts in case of
large photon counts, the KI channels of the SPACIROC-3 ASIC can be used. There
are 8 KI channels per ASIC (thus, per MAPMT), each of which gathers 8 pixels
of an MAPMT, according to the patterns as shown in the left panel of Fig. 1
(whose shapes are dictated by tight routing constraints on the ASIC board
inserting inside the EC). These channels perform an integration of the charge
collected at the 8 anodes of the corresponding pixels, which is then read
through a time of discharge and coded on 6 bits (i.e. with values from 0 to
63). Although the response of the KI channel is not linear with the photon
flux (due to internal ASIC characteristics as well as limitations of the power
supply inherent to the very low consumption system), it allows a clear
discrimination between low values of the integrated charge (for which the KI
returns its default value, namely 4), intermediate signals for which the KI
value increases from 4 to 63 (with a plateau spreading over a large part of
this range), and very large signals, namely too large to be discharged with
one GTU, for which the KI then returns the value 0. An example of the KI
response as a function of light intensity can be seen in the right panel of
Fig. 1, in the case of a constant intensity. Each individual channel can be
configured with 2 parameters: a gain (amplifying or reducing the input charge
signals) and a subtraction current (to compensate the internal DC current
associated with the channel). Increasing the gain results in a shift of the KI
response curve to the left, while increasing the subtraction current (adjusted
through the so-called “qdcsub” parameter) results in a shift to the right.

An important application of the KI measurements in the framework of EUSO-SPB2
is to allow the detection of very short light pulses, of the order of or lower
than the time resolution of the PC channel. Such a signal would be essentially
counted just 1 in PC mode independent from the charge, while it could be
measured by its intensity through their corresponding KI signal. Such short,
intense signals are for instance expected from the Cherenkov emission of EAS
induced by cosmic rays when the FT camera is pointed towards the sky from the
ground (as during the field tests of EUSO-SPB2). In flight, with nadir
pointing, speculative upward-going events on nanosecond scale, similar to
those reported by ANITA (with steeper emergence angles), could also give rise
to clear KI signals [3]. Therefore, it is interesting to extend the triggering
scheme originally developed in the framework of the previous balloon flight
EUSO-SPB1 [4], to use the KI signals in addition to the photon counting.

##  3 KI trigger

For EUSO-SPB2, two types of triggers have been developed: a Level 1 (L1)
trigger and a KI trigger. The FT operation system allows to work either with
only any one of them, or with both in parallel. The logics of the L1 trigger
has been described elsewhere [6]. Here, we concentrate on the KI trigger
logic, of which a schematic view is shown in Fig. 2.

Figure 1:  Left: distribution of the KI channels on a PMT. Pixels with the
same colour belong to the same KI channel. Right: Photon counts per GTU (in
black) and KI values as a function of light intensity, in number of photons
per GTU. The curves correspond to a KI gain of 8 and values of qdcsub of 38
(light blue), 42 (blue), 46 (magenta) and KI gain of 4 and qdcsub of 42
(yellow).

The KI trigger has two key parameters: a threshold, Pixel_THR, and a time
duration, in numbers of GTUs, Ncounter. A trigger is issued when a KI channel
records a value larger than the threshold, for a number of GTUs _lower_ (not
larger!) than the predefined duration. The idea behind it is to concentrate on
fast signals (pulses or events lasting less than Ncounter GTUs), and not allow
mere bright areas (or occasionally defective pixels) to trigger the system.

More precisely, when one KI channel value exceeds the threshold value
Pixel_THR, or is equal to 0 (even larger signal), the counter [Ncounter] is
incremented. If the signal does not last more than Ncounter GTUs (in the same
EC), a trigger is issued. If the signal lasts more than Ncounter GTUs above
the threshold (or with KI $=0$), the counter is reset and does not trigger.

During the flight of EUSO-SPB2, the Ncounter parameter was set to 2, while
Pixel_THR was given different values at different periods, between 10 and 20.

Figure 2:  SPB2-KI trigger logic

### KI response to light pulses

The above-mentioned KI response to light (see the right panel of Fig. 1)
corresponds to the case of continuous light. In the case of a very short
pulse, unfortunately, the situation is more complicated, for two independent
reasons: one related mostly to the ASIC, the other to the HVPS.

Regarding the ASIC, if the subtraction current parameter (“qdcsub”) is not
perfectly adjusted to exactly compensate the DC current associated with a
particular KI channel, then the actual charge measured is not exactly the
charge that has been received at the 8 anodes of the MAPMT connected to that
KI channel. If the subtraction is too low, then some charge is building up,
even in the absence of actual light, and the KI values are overestimated. On
the other hand, if the subtraction is too large, even actual signal may be
subtracted away, and the KI values are underestimated, compared to what would
be obtained with perfect compensation. Moreover, in the latter case, the KI
value obtained from a given light pulse will be different, depending on when
the pulse arrived within the GTU. It will be lower (larger) if the pulse
arrive earlier (later), because the overcompensation of its signal will have
started earlier (later). The other difficulty is related to the fact that a
very bright pulse of light will result in the production of many
photoelectrons at once, which will then multiply through the MAPMT up to a
very large number of electrons in a very short time (a few ns). This will draw
current at a level which cannot be instantly matched by the powering system.
As a result, the gain will drop, which will lead to saturation. As a
consequence, very bright and short pulses of different intensities (even by
one order of magnitude) can lead to the same amount of charges entering the
ASIC, and thus be integrated to produce the same value of KI. In addition, an
oscillation of the electronics baseline results at the level the PMTs or even
the EC, which can lead to fake, nearly uniform photon counts (typically
between 1 and 4) if the PC threshold is set too close to the pedestal.

Because of the above features, with the current hardware the KI channels
cannot be used to determine precisely the intensity of the incoming light.
Nevertheless, they can indeed be used as clear indicators of intense light
pulse, which would remain unnoticeable through the PC channel. This is the
property that is exploited by the KI trigger. In addition, despite the above-
mentioned saturation effect (which will be discussed and quantitatively
studied in more detail elsewhere), the monotonous (though not proportional)
relation between the light intensity and the KI values allows to derive
comparative estimates of the detected signals.

### KI measurements inside blackboxes

KI measurements were made using pulsed LED inside blackboxes with two
different setups, one at the Physics department of the University of Turin
(Italy) and the other at APC, University of Paris (France). In both cases, the
measurements were made using the same hardware as EUSO-SPB2, including an EC,
the Zynq board and the HVPS. In Paris, we measured the KI response to an LED
producing light pulses of 8 ns duration with an adjustable intensity, from a
few hundreds to a few thousands of hundreds of photons, initially measured at
high frequency with a calibrated NIST. The pulses were shot with a low
frequency of 1 Hz and sent directly to a single pixel using an optical fibre.
In Turin, the light from a sub-nanosecond LED pulse was focused on one pixel
from a distance of $\sim 80$ cm, using an optical system with an Iris, a
collimator and two plano-convex lenses. The same light was also measured by a
SiPM whose output signal was recorded by a digital oscilloscope to estimate
the approximate number of incident photons as a function of the intensity
level of the sub-ns pulses. This setup allows additional light to illuminate
the entire EC continuously, to simulate different levels of background light
(similar to those encountered in flight) and verify their influence.

In Fig. 3, we show the PC and KI response of the EC to a bright sub-ns pulse
for different light intensities: 8.6 (left), 9.0 (center) and 10.0 (right),
which correspond respectively to $\sim 4000$, $\sim 7000$ and $\sim 15000$
photons per pulse. The top line shows the PC and KI maps for each case, in a
single GTU corresponding to when the pulse is produced. The middle line shows
the photon counts as a function of time (in GTUs), for a periodic sequence of
light pulses. The bottom line shows the corresponding KI response.

As expected, the KI values are larger for larger light intensities, which
validates the above-mentioned semi-quantitative use of the KI channels. For
the 3 intensities, the typical KI values are round 7, 17 and 40, respectively.
Individual values can however vary to some extent, which is reminiscent of the
above remark regarding an imperfect DC current compensation and its effect on
the KI values depending on the occurrence of the pulse within a GTU. It can
also be seen that the photon counts remain low, around 2 or 3 at the lowest
intensity, and up to between 5 and 10 at the largest one, as a result of the
discussed baseline oscillation, which is indeed longer and thus allows for
more (fake) photon counts in the case of a very strong current draw.

Figure 3:  KI measurements in the blackbox for a different intensity of sub-ns
pulsed LED light. Top: images of PC (left) and KI (right) of the LED focused
light in a GTU. Bottom: PC (top) and KI (bottom) counts as a function of GTU
in the LED nominal intensities of 8.6, 9.0 and 10.0.

### KI trigger tests at TurLab

Figure 4:  Left: a telescope with EUSO-SPB2 detector and electronics hanging
on the ceiling above the TurLab rotation tank. Bottom-right: examples of
materials arranged in the tank. Right (top): a photo of the setup in the
blackbox, where pulsed light is illuminating single pixel with optical fibre.

The TurLab is a laboratory for geo-fluid dynamics studies, where rotation is a
key parameter such control the Coriolis force and Rossby Number. The tank has
a 5 m diameter and can rotate with periods from 1 to 20 min per rotation. It
is located underground at the fourth basement of the Physics department
building of the University of Turin, where the level of light can be
controlled from almost complete darkness to any desired ambient brightness, to
reproduce different conditions that can be encountered in flight. In practice,
we place the detector under study at the focal surface of a small telescope
attached to the ceiling above the tank, looking down to the the bottom of the
tank where several materials and light sources are arranged. The rotation of
the tank then allows to have these artefacts cross the field-of-view of the
detector and reproduce variations of the background light and dynamical
occurrences of various types of events, mimicking what the instrument can be
expected to see during a flight. The dynamical response of several JEM-EUSO
detectors have already been studied in this way in the framework of
EUSO@TurLab project [8], notably to test their performances and triggers (see
e.g. [6]).

In the specific case of the KI trigger tests at the TurLab facility, we use an
EC with the EUSO-SPB2 Zynq board and HVPS, assembled with a 1” plano-convex
lens tube in a telescope, as shown in Fig 4. At the bottom of the tank,
crossing the field of view during tank rotation, we place various materials
with different albedos (e.g. bricks, sand, ground glass and moss, mimicking
rocks, desert, glacier, forest), as well as light sources such as the above-
mentioned sub-ns pulsed LED (simulating very short and bright events), a strip
of 10 white LEDs driven by an Arduino circuit (where each LED is pulsed
sequentially for 5 ${\mu}s$, overlapping for 1${\mu}s$, mimicking EAS like
events), and Lissajous images from an analogue oscilloscope (mimicking meteor-
like events).

The top panel of Fig. 5 shows the lightcurve measured by the EC during one
entire rotation, with photon counts averaged over 50000 consecutive GTUs, i.e.
50 ms. The middle and bottom panels show respectively the photon counts per
GTU and the corresponding KI values, as a function of GTU, for events
triggered by the dedicated KI trigger during the periods indicated by the
arrows. For this run, the Pixel_THR and Ncounter parameters were set to 15 and
2 respectively.

Figure 5:  Top: Light curve showing the evolution of the photon counts
averaged over 50ms periods, during a tank revolution ($\sim 9$ minutes in
total). Bottom: PC (middle) and KI counts (bottom) as a function of time (in
GTUs), for events triggered by the KI trigger only (see text).

The main result of these measurements is the confirmation of the KI trigger is
working as expected, including in the presence of background light of
different intensity, as well as during transitions from areas with different
average photon counts. In particular, only short pulsed signals produced
triggered events. It is also instructive to compare the triggered events
obtained when the sub-ns LED pulses were in the field of view (periods in
purple on the plot) and those when the Arduino-controlled LED strip were in
the field of view (period in blue). In the case of the LED strip, the signals
are long enough for large values of the photon counts to be measured by the
ASIC, and the KI values are also large, because of the large integrated charge
on the anodes. By contrast, in the case of the sub-ns light pulse, no
significant signal is found in the photon counting mode, because of extreme
pile-up, but the KI properly detects high values on very short time scales.
This is indeed the intended behaviour, which confirms the relevance of this
trigger mode for the operation of the instrument in flight.

##  4 Summary and Outlook

We have studied a new type of trigger for the JEM-EUSO FT instruments,
intended to detect very short and intense flashes of light. This so-called “KI
trigger” is based on the charge integration functionality of the SPACIROC-3
ASIC, whose characteristics were described both in the presence of intense
continuous light or in response to short pulses of light. The tests performed
with pulsed LED in Paris and in Turin, using the full detection chain of the
EUSO-SPB2 ECs, demonstrated consistent behaviour of the KI channels in both
modes, and confirmed the proper implementation of the KI trigger logic
described in these proceedings. Finally, the tests performed at the TurLab
facility allowed to demonstrate the performance and stability of the KI
trigger in the presence of various levels of background light, including
changing ones. Based on these measurements, the KI trigger was implemented and
successfully operated in flight for the first time in the EUSO-SPB2 mission
[10].

##  5 Acknowledgments

The authors would like to acknowledge the support by NASA award 11-APRA-0058,
16-APROBES16-0023, 17-APRA17-0066, NNX17AJ82G, NNX13AH54G, 80NSSC18K0246,
80NSSC18K0473, 80NSSC19K0626, 80NSSC18K0464, 80NSSC22K1488, 80NSSC19K0627 and
80NSSC22K0426 and by National Science Centre in Poland grant n.
2017/27/B/ST9/02162. This research used resources of the National Energy
Research Scientific Computing Center (NERSC), a U.S. Department of Energy
Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.
This work was partially supported by the French space agency CNES. We
acknowledge the NASA Balloon Program Office and the Columbia Scientific
Balloon Facility and staff for extensive support. We acknowledge the ASI-INFN
agreement n. 2021-8-HH.0 and its amendments. We also acknowledge the
invaluable contributions of the administrative and technical staffs at our
home institutions.

## References

  * [1] J. Eser et al., “Overview and First results of EUSO-SPB2” in this conference (2023)
  * [2] E. Parizot et al., “Overview and latest results of the JEM-EUSO program for the study of ultra-high-energy cosmic-ray from space” in this conference (2023)
  * [3] The ANITA collaboration, $\bf{DOI:10.1103/PhysRevLett.117.071101}$ (2016)
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P. Barrillon et al., $\bf{DOI:10.1007/s10686\\!-\\!022\\!-\\!09871\\!-\\!8}$
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D. Trofimov et al., $\bf{DOI:10.1051/epjconf/202328306016}$ (2022)

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G. Filippatos et al., “EUSO-SPB2 Fluorescence Telescope in flight performance
and preliminary results” in this conference (2023)

Full Authors list: The JEM-EUSO Collaboration

S. Abeff, J.H. Adams Jr.ld, D. Allardcb, P. Alldredgeld, R. Aloisioep, L.
Anchordoquile, A. Anzaloneed,eh, E. Arnoneek,el, M. Bagherilh, B. Baretcb, D.
Barghiniek,el,em, M. Battisticb,ek,el, R. Bellottiea,eb, A.A. Belovib, M.
Bertainaek,el, P.F. Bertonelf, M. Bianciottoek,el, F. Biscontiei, C.
Blaksleyfg, S. Blin-Bondilcb, K. Bolmgrenja, S. Brizlb, J. Burtonld, F.
Cafagnaea.eb, G. Cambiéei,ej, D. Campanaef, F. Capeldb, R. Carusoec,ed, M.
Casolinoei,ej,fg, C. Cassardoek,el, A. Castellinaek,em, K. Černýba, M.J.
Christllf, R. Colalilloef,eg, L. Contiei,en, G. Cottoek,el, H.J. Crawfordla,
R. Cremoniniel, A. Creusotcb, A. Cummingslm, A. de Castro Gónzalezlb, C. de la
Tailleca, R. Diesinglb, P. Dinaucourtca, A. Di Nolaeg, T. Ebisuzakifg, J.
Eserlb, F. Fenueo, S. Ferrareseek,el, G. Filippatoslc, W.W. Finchlc, F.
Flaminioeg, C. Fornaroei,en, D. Fuehnelc, C. Fuglesangja, M. Fukushimafa, S.
Gadamsettylh, D. Gardiolek,em, G.K. Garipovib, E. Gazdalh, A. Golzioel, F.
Guarinoef,eg, C. Guépinlb, A. Haungsda, T. Heibgeslc, F. Isgròef,eg, E.G.
Juddla, F. Kajinofb, I. Kanekofg, S.-W. Kimga, P.A. Klimovib, J.F.
Krizmaniclj, V. Kungellc, E. Kuznetsovld, F. López Martínezlb, D. Mandátbb, M.
Manfrinek,el, A. Marcelliej, L. Marcelliei, W. Marszałha, J.N. Matthewslg, M.
Meseef,eg, S.S. Meyerlb, J. Mimouniab, H. Miyamotoek,el,ep, Y. Mizumotofd, A.
Monacoea,eb, S. Nagatakifg, J.M. Nachtmanli, D. Naumovia, A. Neronovcb, T.
Nonakafa, T. Ogawafg, S. Ogiofa, H. Ohmorifg, A.V. Olintolb, Y. Onelli, G.
Osteriaef, A.N. Ottelh, A. Pagliaroed,eh, B. Panicoef,eg, E. Parizotcb,cc,
I.H. Parkgb, T. Paulle, M. Pechbb, F. Perfettoef, P. Picozzaei,ej, L.W.
Piotrowskihb, Z. Plebaniakei,ej, J. Posliguali, M. Pottslh, R. Preveteef,eg,
G. Prévôtcb, M. Przybylakha, E. Realiei,ej, P. Reardonld, M.H. Renoli, M.
Ricciee, O.F. Romero Matamalalh, G. Romoliei,ej, H. Sagawafa, N. Sakakifg,
O.A. Saprykinic, F. Sarazinlc, M. Satofe, P. Schovánekbb, V. Scottief,eg, S.
Selmanecb, S.A. Sharakinib, K. Shinozakiha, S. Stepanofflh, J.F. Sorianole, J.
Szabelskiha, N. Tajimafg, T. Tajimafg, Y. Takahashife, M. Takedafa, Y.
Takizawafg, S.B. Thomaslg, L.G. Tkachevia, T. Tomidafc, S. Toscanoka, M.
Traïcheaa, D. Trofimovcb,ib, K. Tsunofg, P. Vallaniaek,em, L. Valoreef,eg,
T.M. Venterslj, C. Vigoritoek,el, M. Vrabelha, S. Wadafg, J. Watts Jr.ld, L.
Wienckelc, D. Winnlk, H. Wistrandlc, I.V. Yashinib, R. Younglf, M.Yu. Zotovib.

aa Centre for Development of Advanced Technologies (CDTA), Algiers, Algeria
ab Lab. of Math. and Sub-Atomic Phys. (LPMPS), Univ. Constantine I,
Constantine, Algeria
ba Joint Laboratory of Optics, Faculty of Science, Palacký University,
Olomouc, Czech Republic
bb Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic
ca Omega, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France
cb Université de Paris, CNRS, AstroParticule et Cosmologie, F-75013 Paris,
France
cc Institut Universitaire de France (IUF), France
da Karlsruhe Institute of Technology (KIT), Germany
db Max Planck Institute for Physics, Munich, Germany
ea Istituto Nazionale di Fisica Nucleare - Sezione di Bari, Italy
eb Università degli Studi di Bari Aldo Moro, Italy
ec Dipartimento di Fisica e Astronomia "Ettore Majorana", Università di
Catania, Italy
ed Istituto Nazionale di Fisica Nucleare - Sezione di Catania, Italy
ee Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali di Frascati,
Italy
ef Istituto Nazionale di Fisica Nucleare - Sezione di Napoli, Italy
eg Università di Napoli Federico II - Dipartimento di Fisica "Ettore Pancini",
Italy
eh INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo, Italy
ei Istituto Nazionale di Fisica Nucleare - Sezione di Roma Tor Vergata, Italy
ej Università di Roma Tor Vergata - Dipartimento di Fisica, Roma, Italy
ek Istituto Nazionale di Fisica Nucleare - Sezione di Torino, Italy
el Dipartimento di Fisica, Università di Torino, Italy
em Osservatorio Astrofisico di Torino, Istituto Nazionale di Astrofisica,
Italy
en Uninettuno University, Rome, Italy
eo Agenzia Spaziale Italiana, Via del Politecnico, 00133, Roma, Italy
ep Gran Sasso Science Institute, L’Aquila, Italy
fa Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Japan
fb Konan University, Kobe, Japan
fc Shinshu University, Nagano, Japan
fd National Astronomical Observatory, Mitaka, Japan
fe Hokkaido University, Sapporo, Japan
ff Nihon University Chiyoda, Tokyo, Japan
fg RIKEN, Wako, Japan
ga Korea Astronomy and Space Science Institute
gb Sungkyunkwan University, Seoul, Republic of Korea
ha National Centre for Nuclear Research, Otwock, Poland
hb Faculty of Physics, University of Warsaw, Poland
ia Joint Institute for Nuclear Research, Dubna, Russia
ib Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
Russia
ic Space Regatta Consortium, Korolev, Russia
ja KTH Royal Institute of Technology, Stockholm, Sweden
ka ISDC Data Centre for Astrophysics, Versoix, Switzerland
la Space Science Laboratory, University of California, Berkeley, CA, USA
lb University of Chicago, IL, USA
lc Colorado School of Mines, Golden, CO, USA
ld University of Alabama in Huntsville, Huntsville, AL, USA
le Lehman College, City University of New York (CUNY), NY, USA
lf NASA Marshall Space Flight Center, Huntsville, AL, USA
lg University of Utah, Salt Lake City, UT, USA
lh Georgia Institute of Technology, USA
li University of Iowa, Iowa City, IA, USA
lj NASA Goddard Space Flight Center, Greenbelt, MD, USA
lk Fairfield University, Fairfield, CT, USA
ll Department of Physics and Astronomy, University of California, Irvine, USA
lm Pennsylvania State University, PA, USA


Clearly, the quantity $\int_S \rho_S$ is independent of any orthonormal basis used to span $\bR^4$. Suppose $S$ lies in a plane. Then it is straight forward to show that $\int_S \rho_S$ is equal to the area of the surface $S$. Now given any surface $S \subset \bR^4$. We can approximate $S$ by small rectangles or squares $\Delta_i S$. Then, $\int_S \rho_S = \lim \sum_i \int_{\Delta_iS} \rho_{\Delta_i S}$ and since $\int_{\Delta_i S}\rho_{\Delta_i S}$ is the area of $\Delta_i S$, we obtain that $\int_S \rho_S$ is the area of the surface $S$.

§ SOME INEQUALITIES

For $0< x < \epsilon$, $\dfrac{1}{1-x} \leq e^{cx}$ if we choose $c = 1/(1- \epsilon)$.

Let $f(x) =  e^{cx}(1-x)$ such that for $0< x < \epsilon$, $f(x) \geq 1$. Therefore, f'(x) = ce^cx(1-x) - e^cx = (c-1)e^cx - cxe^cx>0 ⇒(c-1) - cϵ> 0 ⇒c> 1/(1-ϵ).

Let $\mathcal{A}^\ast \mathcal{A}$ be a $n \times n$ matrix such that $2\parallel \mathcal{A}^\ast \mathcal{A} \parallel < \epsilon < 1$. Then we have 1/(1 - 2𝒜^∗𝒜) ≤e^2c[𝒜^∗𝒜] whereby $c = (1-\epsilon)^{-1}$.

Let $\epsilon > \lambda_1 \geq \ldots \geq \lambda_n \geq 0$ be the eigenvalues of $\mathcal{A}^\ast \mathcal{A}$. Then, we have from Lemma <ref>,

\begin{align}
\frac{1}{\det(1 - 2\mathcal{A}^\ast \mathcal{A})} =& \prod_{i=1}^n \frac{1}{1-2\lambda_i}
\leq \prod_{i=1}^n e^{2c\lambda_i} \nonumber \\
=& \exp\left[ 2c\sum_{i=1}^n \lambda_i \right] = \exp\left[ 2c\Tr[\mathcal{A}^\ast\mathcal{A}]\right]. \nonumber
\end{align}

Let $A$, $B$ be $N \times N$ matrices.
We have | A^⊗^n - B^⊗^n | ≤n|A- B|[|A| + |B|]^n-1.

Now A^⊗^n - B^⊗^n = (A - B) ⊗A^⊗^n-1 + B ⊗(A-B) ⊗A^⊗^n-2 + ⋯B^⊗^n-2⊗(A - B) ⊗A + B^⊗^n-1⊗(A-B) . Hence,

\begin{align}
\left| A^{\otimes^n} - B^{\otimes^n}\right| \leq |A- B| \left[ |A|^{n-1} + |A|^{n-2}|B| \cdots + |B|^{n-1} \right] \leq n|A- B|[|A| + |B|]^{n-1}. \nonumber
\end{align}

Let $F, G: I^2 \rightarrow M_{N \times N}(\bC)$ and write e^∫_I^2F(s,t) ds dt = ∑_n=0^∞1/n! (∫_I^2F(s,t) ds dt )^⊗^n,  e^∫_I^2G(s,t) ds dt = ∑_n=0^∞1/n! (∫_I^2G(s,t) ds dt )^⊗^n. Then,

\begin{align}
\left| e^{\int_{I^2} F(s,t) dsdt} - e^{\int_{I^2}G(s,t)ds dt}\right| \leq \left| \int_{I^2}F(s,t)ds dt - \int_{I^2}G(s,t) ds dt \right|\ \exp\left[\left|\int_{I^2}F(s,t)ds dt\right| + \left|\int_{I^2}G(s,t) ds dt\right| \right]. \nonumber
\end{align}

Apply Lemma <ref>,

\begin{align}
&\left| e^{\int_{I^2} F(s,t) dsdt} - e^{\int_{I^2}G(s,t)ds dt}\right| \nonumber \\
&\leq \sum_{n=0}^\infty \frac{n}{n!} \left| \int_{I^2}F(s,t)ds dt - \int_{I^2}G(s,t) ds dt \right|\left( \left|\int_{I^2}F(s,t) ds dt\right| + \left|\int_{I^2}G(s,t) ds dt\right| \right)^{n-1} \nonumber \\
&\leq \sum_{n=0}^\infty \frac{1}{(n-1)!} \left| \int_{I^2}F(s,t)ds dt - \int_{I^2}G(s,t) ds dt \right|\left( \left|\int_{I^2}F(s,t) ds dt\right| + \left|\int_{I^2}G(s,t) ds dt\right| \right)^{n-1} \nonumber \\
&\leq \left| \int_{I^2}F(s,t)ds dt - \int_{I^2}G(s,t) ds dt \right|\ \exp\left[\left|\int_{I^2}F(s,t)ds dt\right| + \left|\int_{I^2}G(s,t) ds dt\right| \right]. \nonumber
\end{align}

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The axisymmetric character of Kerr spacetime enforces the tetrad $e^{\mu}_{a}$
to be independent of $t$ and $\phi$. This yields
$\poissonbracket{p_{t,\phi}}{e^{\mu}_{a}}=-\partial_{t,\phi}e^{\mu}_{a}=0$.
Consequently,

$\displaystyle\poissonbracket{p_{t,\phi}}{S^{ab}}=\poissonbracket{p_{t,\phi}}{S^{\mu\nu}}e^{a}_{\mu}e^{b}_{\nu}.$
(296)

Using this last equation and the fact that $\mathcal{Q}_{Y}$ commute with
$x^{\mu}$, we get

$\displaystyle\poissonbracket{\mathcal{Q}_{Y}}{E}$
$\displaystyle=\poissonbracket{p_{t}}{\mathcal{Q}_{Y}}+\frac{1}{2}\nabla_{a}k_{b}\poissonbracket{S^{ab}}{\mathcal{Q}_{Y}}$
(297)
$\displaystyle\overset{\eqref{ss_tetrad}}{=}-4\bigg{[}r\quantity(\Gamma^{[1}_{kt}-\nabla_{k}k^{[1})S^{0]k}$
$\displaystyle\quad+a\cos\theta\quantity(\Gamma^{[3}_{kt}-\nabla_{k}k^{[3})S^{2]k}\bigg{]}$
(298) $\displaystyle=0.$ (299)

#####  $\mathcal{I}_{\hat{A}}=\mathcal{Q}_{R}$.

Using the identity

$\displaystyle\poissonbracket{E^{G}}{E}$
$\displaystyle=\frac{1}{2}\nabla_{\alpha}k_{\beta}\Gamma^{\alpha}_{\lambda
t}S^{\lambda\beta}$ (300)
$\displaystyle=-\frac{1}{2}\nabla_{\beta}k_{\alpha}\nabla_{\lambda}k^{\alpha}S^{\lambda\beta}=0$
(301)

and the chain rule (284), one has

$\displaystyle\poissonbracket{\mathcal{Q}_{R}}{E}$
$\displaystyle=2p_{\mu}K^{\mu\nu}\poissonbracket{p_{\nu}}{E}-2\mu\poissonbracket{S^{\alpha}\partial_{\alpha}\mathcal{Z}}{E}$
(302)
$\displaystyle=-p_{\mu}K^{\mu\nu}\quantity(2\poissonbracket{p_{\nu}}{p_{t}}+\poissonbracket{p_{\nu}}{\nabla_{\alpha}k_{\beta}S^{\alpha\beta}})$
$\displaystyle\quad+\mu\quantity(2\poissonbracket{S^{\alpha}\partial_{\alpha}\mathcal{Z}}{p_{t}}+\poissonbracket{S^{\alpha}\partial_{\alpha}\mathcal{Z}}{\nabla_{\mu}k_{\nu}S^{\mu\nu}}).$
(303)

Making use of the identity

$\displaystyle R_{t\beta\mu\nu}S^{\mu\nu}$
$\displaystyle=\partial_{\beta}\quantity(\nabla_{\mu}k_{\nu})S^{\mu\nu}+2\nabla_{\alpha}k_{\nu}\Gamma^{\alpha}_{\beta\lambda}S^{\nu\lambda},$
(304)

the two first Poisson brackets of this expression can be shown to cancel
mutually, and we are left with

$\displaystyle\poissonbracket{\mathcal{Q}_{R}}{E}$
$\displaystyle=\frac{1}{2}\varepsilon^{\alpha\beta\gamma\delta}S_{\gamma\delta}\,\partial_{\alpha}\mathcal{Z}\quantity[R_{t\beta\mu\nu}-\partial_{\beta}\quantity(\nabla_{\mu}k_{\nu})]S^{\mu\nu}$
(305)
$\displaystyle=\varepsilon^{\alpha\beta\gamma\delta}\partial_{\alpha}\mathcal{Z}S_{\gamma\delta}\nabla_{\rho}k_{\nu}\Gamma^{\rho}_{\lambda\beta}S^{\nu\lambda}$
(306) $\displaystyle=\mathcal{O}(\mathcal{S}^{2}).$ (307)

###  A.4 $\poissonbracket{\mathcal{I}_{A}}{\ell}$-type brackets

The computations are identical to the
$\poissonbracket{\mathcal{I}_{A}}{E}$-type case, but with $k^{\alpha}\to
l^{\alpha}$. We consequently find

$\displaystyle\poissonbracket{\mathcal{Q}_{Y}}{\ell}$ $\displaystyle=0,$ (308)
$\displaystyle\poissonbracket{\mathcal{Q}_{R}}{\ell}$
$\displaystyle=\mathcal{O}(\mathcal{S}^{2}).$ (309)

###  A.5 $\poissonbracket{\mathcal{I}_{A}}{\mathcal{Q}_{Y}}$-type brackets

The final bracket to be computed takes the form

$\displaystyle\poissonbracket{\mathcal{Q}_{R}}{\mathcal{Q}_{Y}}$
$\displaystyle=2p_{\mu}K^{\mu\nu}\poissonbracket{p_{\nu}}{\mathcal{Q}_{Y}}-2\mu\,\partial_{\alpha}\mathcal{Z}\poissonbracket{S^{\alpha}}{\mathcal{Q}_{Y}}$
(310)
$\displaystyle=-2p_{\mu}K^{\mu\nu}\quantity(\partial_{\nu}Y^{*}_{\alpha\beta}S^{\alpha\beta}+2\Gamma^{\alpha}_{\rho\nu}S^{\beta\rho}Y^{*}_{\alpha\beta})-4\varepsilon^{\alpha\beta}_{\phantom{\alpha\beta}\gamma\delta}\partial_{\alpha}\mathcal{Z}p_{\beta}Y^{*\gamma}_{\phantom{*\gamma}\nu}S^{\delta\nu}+\mathcal{O}(\mathcal{S}^{2})$
(311)
$\displaystyle=-2p_{\mu}K^{\mu\nu}\nabla_{\nu}Y^{*}_{\alpha\beta}S^{\alpha\beta}-2\varepsilon^{\alpha\beta\gamma\delta}\partial_{\alpha}\mathcal{Z}p_{\beta}\varepsilon_{\gamma\nu\rho\sigma}Y^{\rho\sigma}S_{\delta}^{\phantom{\delta}\nu}+\mathcal{O}(\mathcal{S}^{2})$
(312)
$\displaystyle=4p_{\mu}S^{\alpha\beta}\quantity(K^{\mu}_{\phantom{\mu}\alpha}\xi_{\beta}+Y^{\mu}_{\phantom{\mu}\alpha}Y^{\lambda}_{\phantom{\lambda}\beta}\xi_{\lambda})+\mathcal{O}(\mathcal{S}^{2}).$
(313)

This expression is generally non-vanishing at order
$\mathcal{O}(\mathcal{S})$.

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