11institutetext:  Dublin Institute for Advanced Studies, 31 Fitzwilliam Place,
Dublin 2, Ireland 22institutetext: Max-Planck-Institut für Kernphysik, P.O.
Box 103980, D 69029 Heidelberg, Germany 33institutetext: Landessternwarte,
Universität Heidelberg, Königstuhl, D 69117 Heidelberg, Germany
44institutetext: Kapteyn Astronomical Institute, University of Groningen,
Landleven 12, 9747 AD, Groningen, The Netherlands 55institutetext: Laboratoire
Leprince-Ringuet, École Polytechnique, CNRS, Institut Polytechnique de Paris,
F-91128 Palaiseau, France 66institutetext: University of Namibia, Department
of Physics, Private Bag 13301, Windhoek 10005, Namibia 77institutetext: Centre
for Space Research, North-West University, Potchefstroom 2520, South Africa
88institutetext: DESY, D-15738 Zeuthen, Germany 99institutetext: School of
Physics, University of the Witwatersrand, 1 Jan Smuts Avenue, Braamfontein,
Johannesburg, 2050 South Africa 1010institutetext: Université de Paris, CNRS,
Astroparticule et Cosmologie, F-75013 Paris, France 1111institutetext:
Department of Physics and Electrical Engineering, Linnaeus University, 351 95
Växjö, Sweden 1212institutetext: Institut für Physik, Humboldt-Universität zu
Berlin, Newtonstr. 15, D 12489 Berlin, Germany 1313institutetext: Institut für
Astronomie und Astrophysik, Universität Tübingen, Sand 1, D 72076 Tübingen,
Germany 1414institutetext: Laboratoire Univers et Théories, Observatoire de
Paris, Université PSL, CNRS, Université de Paris, 92190 Meudon, France
1515institutetext: Sorbonne Université, Université Paris Diderot, Sorbonne
Paris Cité, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes
Energies, LPNHE, 4 Place Jussieu, F-75252 Paris, France 1616institutetext:
IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France
1717institutetext: Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen
Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, D 91058 Erlangen,
Germany 1818institutetext: Astronomical Observatory, The University of Warsaw,
Al. Ujazdowskie 4, 00-478 Warsaw, Poland 1919institutetext: Université Savoie
Mont Blanc, CNRS, Laboratoire d’Annecy de Physique des Particules - IN2P3,
74000 Annecy, France 2020institutetext: Instytut Fizyki Ja̧drowej PAN, ul.
Radzikowskiego 152, 31-342 Kraków, Poland 2121institutetext: Université
Bordeaux, CNRS, LP2I Bordeaux, UMR 5797, F-33170 Gradignan, France
2222institutetext: University of Oxford, Department of Physics, Denys
Wilkinson Building, Keble Road, Oxford OX1 3RH, UK 2323institutetext:
Laboratoire Univers et Particules de Montpellier, Université Montpellier,
CNRS/IN2P3, CC 72, Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France
2424institutetext: Institut für Physik und Astronomie, Universität Potsdam,
Karl-Liebknecht-Strasse 24/25, D 14476 Potsdam, Germany 2525institutetext:
School of Physical Sciences, University of Adelaide, Adelaide 5005, Australia
2626institutetext: Aix Marseille Université, CNRS/IN2P3, CPPM, Marseille,
France 2727institutetext: School of Science, Western Sydney University, Locked
Bag 1797, Penrith South DC, NSW 2751, Australia 2828institutetext: Institut
für Astro- und Teilchenphysik, Leopold-Franzens-Universität Innsbruck, A-6020
Innsbruck, Austria 2929institutetext: Universität Hamburg, Institut für
Experimentalphysik, Luruper Chaussee 149, D 22761 Hamburg, Germany
3030institutetext: Obserwatorium Astronomiczne, Uniwersytet Jagielloński, ul.
Orla 171, 30-244 Kraków, Poland 3131institutetext: Institute of Astronomy,
Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University,
Grudziadzka 5, 87-100 Torun, Poland 3232institutetext: Nicolaus Copernicus
Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, 00-716
Warsaw, Poland 3333institutetext: GRAPPA, Anton Pannekoek Institute for
Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The
Netherlands 3434institutetext: Department of Physics, University of the Free
State, PO Box 339, Bloemfontein 9300, South Africa 3535institutetext:
Department of Physics and Astronomy, The University of Leicester, University
Road, Leicester, LE1 7RH, United Kingdom 3636institutetext: Yerevan Physics
Institute, 2 Alikhanian Brothers St., 375036 Yerevan, Armenia
3737institutetext: Department of Physics, Konan University, 8-9-1 Okamoto,
Higashinada, Kobe, Hyogo 658-8501, Japan 3838institutetext: Kavli Institute
for the Physics and Mathematics of the Universe (WPI), The University of Tokyo
Institutes for Advanced Study (UTIAS), The University of Tokyo, 5-1-5 Kashiwa-
no-Ha, Kashiwa, Chiba, 277-8583, Japan 3939institutetext: RIKEN, 2-1 Hirosawa,
Wako, Saitama 351-0198, Japan

# Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.

H.E.S.S. Collaboration  F. Aharonian Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. F. Ait Benkhali Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. J. Aschersleben Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. H. Ashkar Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. M. Backes Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. V. Barbosa Martins Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. R. Batzofin Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. Y. Becherini Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. D. Berge
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S. K. Bernlöhr Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. B. Bi Detection
of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. M. Böttcher Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. C. Boisson Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. J.
Bolmont Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. J. Borowska Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. M. Bouyahiaoui Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. F. Bradascio Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. R. Brose Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. F. Brun
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. B. Bruno Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. T. Bulik Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. C. Burger-
Scheidlin Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S. F. Cangemi Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. S. Caroff ,
Corresponding authors;
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S<EMAIL_ADDRESS>S. Casanova Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. J. Celic Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. M. Cerruti Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. P. Chambery
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. T. Chand Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. S. Chandra Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. A. Chen
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. J. Chibueze Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. O. Chibueze Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. G.
Cotter Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. J. Damascene Mbarubucyeye Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. J.
Devin Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. A. Djannati-Ataï Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. A. Dmytriiev
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. K. Egberts Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. S. Einecke Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. J.-P.
Ernenwein Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S. K. Feijen Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. G. Fichet de
Clairfontaine Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S. M. Filipovic Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. G. Fontaine
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. M. Füßling Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. S. Funk Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. S.
Gabici Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. Y.A. Gallant Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. S. Ghafourizadeh Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. G. Giavitto Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. L. Giunti Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. D. Glawion Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. J.F. Glicenstein Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. P. Goswami Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. G. Grolleron Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. M.-H.
Grondin Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. L. Haerer Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. M. Haupt Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. G. Hermann Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. J.A. Hinton Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. W.
Hofmann Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. T. L. Holch Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. M. Holler Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. D. Horns Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. Zhiqiu Huang Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. M.
Jamrozy Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. F. Jankowsky Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. V. Joshi Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. I. Jung-Richardt Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. E. Kasai Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. K. Katarzyński Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. B. Khélifi Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. W. Kluźniak Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. Nu. Komin Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. K.
Kosack Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. D. Kostunin Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. R.G. Lang Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. S. Le Stum Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. F. Leitl Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. A.
Lemière Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. M. Lemoine-Goumard Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. J.-P. Lenain
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. F. Leuschner Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. T. Lohse Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. A.
Luashvili Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S. I. Lypova Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. J. Mackey
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. D. Malyshev Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. V. Marandon Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. P.
Marchegiani Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S. A. Marcowith Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. P. Marinos
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. G. Martí-Devesa Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. R. Marx Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. G.
Maurin Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. P.J. Meintjes Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. M. Meyer Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. A. Mitchell ,11footnotemark: 1 Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. R.
Moderski Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. L. Mohrmann Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. A. Montanari Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. E. Moulin Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. J. Muller Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. K. Nakashima
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. M. de Naurois Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. J. Niemiec Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. A.
Priyana Noel Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S. P. O’Brien Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. S. Ohm Detection
of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. L. Olivera-Nieto Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. E. de Ona Wilhelmi Detection
of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. M. Ostrowski Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. S. Panny Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. M.
Panter Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. R.D. Parsons Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. G. Peron Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. D.A. Prokhorov Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. G. Pühlhofer Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. A. Quirrenbach Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. A. Reimer Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. O. Reimer Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. M. Renaud Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. B. Reville
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. F. Rieger Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. G. Rowell Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. B. Rudak
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. H. Rueda Ricarte Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. E. Ruiz-Velasco Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. V. Sahakian Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. H. Salzmann Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. A.
Santangelo Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S. M. Sasaki Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. F. Schüssler
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. H.M. Schutte Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. U. Schwanke Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. J.N.S.
Shapopi Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. A. Sinha Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. H. Sol Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. A.
Specovius Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S. S. Spencer Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. Ł. Stawarz
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. S. Steinmassl Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. I. Sushch Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. H.
Suzuki Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. T. Takahashi Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. T. Tanaka Detection of
extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. T. Tavernier Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S. A.M. Taylor Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. R.
Terrier Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. C. Thorpe-Morgan Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. M. Tsirou
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. N. Tsuji Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. M. Vecchi Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. C. Venter
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. J. Vink Detection of extended $\gamma$-ray emission around the
Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S. S.J. Wagner Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. R.
White Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. A. Wierzcholska Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. Yu Wun Wong
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. M. Zacharias Detection of extended $\gamma$-ray emission around
the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S. D. Zargaryan
Detection of extended $\gamma$-ray emission around the Geminga pulsar with
H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S. A.A. Zdziarski Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S. A. Zech Detection of extended
$\gamma$-ray emission around the Geminga pulsar with H.E.S.S.Detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S. S.
Zouari Detection of extended $\gamma$-ray emission around the Geminga pulsar
with H.E.S.S.Detection of extended $\gamma$-ray emission around the Geminga
pulsar with H.E.S.S. N. Żywucka Detection of extended $\gamma$-ray emission
around the Geminga pulsar with H.E.S.S.Detection of extended $\gamma$-ray
emission around the Geminga pulsar with H.E.S.S.

(Received: 23/12/2022 – Accepted: 04/04/2023)

Geminga is an enigmatic radio-quiet $\gamma$-ray pulsar located at a mere 250
pc distance from Earth. Extended very-high-energy $\gamma$-ray emission around
the pulsar was discovered by Milagro and later confirmed by HAWC, which are
both water Cherenkov detector-based experiments. However, evidence for the
Geminga pulsar wind nebula in gamma rays has long evaded detection by imaging
atmospheric Cherenkov telescopes (IACTs) despite targeted observations. The
detection of $\gamma$-ray emission on angular scales $\gtrsim 2^{\circ}$ poses
a considerable challenge for the background estimation in IACT data analysis.
With recent developments in understanding the complementary background
estimation techniques of water Cherenkov and atmospheric Cherenkov
instruments, the H.E.S.S. IACT array can now confirm the detection of highly
extended $\gamma$-ray emission around the Geminga pulsar with a radius of at
least $3^{\circ}$ in the energy range $0.5-40$ TeV. We find no indications for
statistically significant asymmetries or energy-dependent morphology. A flux
normalisation of $(2.8\pm 0.7)\times 10^{-12}$ cm-2s-1TeV-1 at 1 TeV is
obtained within a $1^{\circ}$ radius region around the pulsar. To investigate
the particle transport within the halo of energetic leptons around the pulsar,
we fitted an electron diffusion model to the data. The normalisation of the
diffusion coefficient obtained of $D_{0}=7.6^{+1.5}_{-1.2}\times 10^{27}$
cm2s-1, at an electron energy of 100 TeV, is compatible with values previously
reported for the pulsar halo around Geminga, which is considerably below the
Galactic average.

###### Key Words.:

Geminga; Pulsar wind nebula; Cherenkov Telescopes; halo; diffusion;

##  1 Introduction

The Geminga pulsar (PSR J0633+1746) is a radio-quiet $\gamma$-ray source that
was established as a pulsar in 1992, and at 250 pc is one of the closest
pulsars to Earth (Bignami et al., 1983; Bertsch et al., 1992; Bignami &
Caraveo, 1992; Faherty et al., 2007). Searches for extended $\gamma$-ray
emission around Geminga have been conducted ever since, proving unsuccessful
for many years (Akerlof et al., 1993; Aharonian et al., 1999), until it was
first detected at TeV energies by Milagro in 2007, with a diameter of
$2.8^{\circ}\pm 0.8^{\circ}$ (Abdo et al., 2007). This discovery was
subsequently confirmed by the High Altitude Water Cherenkov (HAWC)
collaboration, presenting evidence for significantly extended emission on
angular scales of up to $\sim 5.5^{\circ}$ radius (Abeysekara et al., 2017b).
However, a detection with imaging atmospheric Cherenkov telescopes (IACTs)
remained elusive (Singh et al., 2009; Finnegan, 2009; Ahnen et al., 2016).
With a spin-down luminosity of $\dot{E}=3.2\times
10^{34}\,\mathrm{ergs}^{-1}$, a spin period of $P=237$ ms and a characteristic
age of $\tau_{c}=342$ kyr, Geminga is one of the oldest pulsars around which
extended very-high-energy $\gamma$-ray emission has been detected, providing
evidence of energetic electron acceleration by middle-aged pulsars
($\tau_{c}\sim 0.1-1$ Myr) (Manchester et al., 2005).111The term ‘electron’ is
used to refer, collectively, to electrons and positrons throughout, unless
explicitly stated otherwise.

The morphology of the emission as seen with HAWC was found to indicate
considerably slower diffusion than values typical for the interstellar medium
(ISM), yielding diffusion coefficients a factor $\sim 100$ lower at 100 TeV
(Abeysekara et al., 2017a). Geminga and the similar nearby pulsar PSR B0656+14
are considerably older pulsars, with longer periods and lower spin-down powers
with respect to other pulsars associated with extended TeV $\gamma$-ray
emission. It has been proposed that the $\gamma$-ray emission regions around
these systems are in a different evolutionary stage to other TeV pulsar wind
nebulae (PWNe) and they constitute a distinct source class of ‘TeV halos’ or
‘pulsar halos’ (Giacinti et al., 2020; Linden et al., 2017).222We adopt the
latter term ‘pulsar halo’ to distinguish from the similar escape phenomenon
occurring in other sources, such as halos around supernova remnants (Brose et
al., 2021).  Within such halos, the $\gamma$-ray emission is due to inverse
Compton (IC) scattering by electrons that have escaped from the PWN and the
energy density of these electrons is lower than the ISM energy density, that
is they do not dominate the surrounding region dynamically or energetically.
As such, in contrast to the many PWNe detected with the High Energy
Stereoscopic System (H.E.S.S.), with 12 firmly identified in the H.E.S.S.
Galactic Plane Survey (H.E.S.S. Collaboration et al., 2018), the detection of
extended $\gamma$-ray emission around the Geminga pulsar with H.E.S.S.
constitutes the first unambiguous detection of a pulsar halo at TeV energies
by IACTs.

Extensive air showers (EAS) caused by cosmic rays are the primary background
source of triggered events for a $\gamma$-ray analysis with IACTs. Despite
state-of-the-art techniques in separating gamma-initiated from hadron-
initiated EAS, there remains an irreducible background of gamma-like hadronic
(mostly proton) showers – those in which a large fraction of the energy is
transferred to neutral pions in the first interaction (Maier & Knapp, 2007).
As the rate of background events can vary based on the atmospheric conditions,
observing direction, and hardware settings, methods to estimate and model the
background level typically rely on counting gamma-like events within a region
of the sky in the same dataset away from the target region (‘Off’), and
subtracting this level from the region of interest (‘On’) (Berge et al.,
2007). In the case of the Geminga halo, the emission fills the field of view,
such that there is no region free from $\gamma$-ray emission available for
such a background estimation. This results in emission belonging to the halo
being counted as background.

Previous IACT observations of the region have resulted in upper limits when
probing angular scales in the range $0.1^{\circ}$ to $0.3^{\circ}$ (Finnegan,
2009; Ahnen et al., 2016). Within the X-ray range a PWN has been identified
with two lateral tails of length $\sim 2-3^{\prime}$ and an axial tail of
length $45^{\prime\prime}$, motivating searches on these angular scales
(Caraveo et al., 2003; Pavlov et al., 2010; Posselt et al., 2017). Analysis
and detection of emission on larger scales is challenging for IACTs, for
reasons outlined above (Berge et al., 2007). Dedicated analysis approaches
developed for this sky region include matching observation conditions between
observations of this region and of empty sky fields, in order to estimate the
background across the full region of interest (Flinders, 2015; Abeysekara,
2019; Hona, 2021).

As shown in a systematic study of differences between the analysis procedures
of the water Cherenkov detector facility HAWC and the H.E.S.S. IACT array
(Abdalla et al., 2021), the two experimental techniques are complementary
approaches, with HAWC providing good sensitivity to high energies ($E\gtrsim
10$ TeV) and extended emission regions. In contrast, H.E.S.S. can provide
detailed morphological and spectral studies, thanks to its few-arcmin angular
and 10-15% energy resolution. Given the extent of the significant emission
detected with HAWC around the Geminga pulsar (out to $\sim 5^{\circ}$ radius),
it is not currently possible for IACTs to measure the full extent of the
emission using standard background estimation techniques. However, as we also
show in the analysis presented here, it is nevertheless possible for IACTs to
significantly detect degree-scale extended emission whilst controlling the
estimated background, and to start probing the morphological and spectral
properties of the inner regions of such sources (Aharonian et al., 2022).

Searches for spectral variation, energy-dependent morphology or asymmetries to
the emission are of particular interest to investigate properties of the large
scale pulsar halo. Similarly, investigating the transport of energetic
particles in the vicinity of the pulsar is important to establish whether the
properties are consistent with suppressed diffusion as seen in previous
analyses.

Figure 1: Surface brightness profile of emission around the Geminga pulsar
measured with HAWC (Abeysekara et al., 2017a). Shaded regions indicate the
parts of the emission profile that are used as ON (radius $\theta\lesssim
1.0^{\circ}$) and OFF normalisation (radius $\theta\gtrsim 3.2^{\circ}$)
regions to estimate the background level and evaluate the significance in the
analysis of the 2019 H.E.S.S. dataset. The region shown for normalisation of
the OFF counts is only accessible with the 2019 dataset, due to the wider
pointing strategy used.

##  2 H.E.S.S. data and analysis

###  2.1 H.E.S.S. observations

H.E.S.S. is an array of five IACTs, located in the Khomas Highland of Namibia
at 1800 m above sea level. Four of the telescopes (CT1-4) have mirror areas of
107 m2 whilst the fifth (CT5) has a mirror area of 612 m2 and correspondingly
lower energy threshold (Holler et al., 2015). Due to its smaller field of
view, of $3.2^{\circ}$ (compared with the $5^{\circ}$ field of view for the
smaller telescopes), CT5 is not used in this analysis. The electronics of the
cameras of the CT1-4 telescopes were upgraded in 2017 (Ashton et al., 2020).
Observation data are collected with H.E.S.S. in ‘runs’ typically 28 minutes in
length, during which telescopes are pointed towards a specific sky position.

H.E.S.S. observed the Geminga region using ‘wobble’ mode offsets of
$0.5^{\circ}$ and $0.7^{\circ}$ around the Geminga pulsar (R.A.
$06$h$33$m$54$s, Dec $+17^{\circ}46^{\prime}13^{\prime\prime}$) over two
seasons in 2006 - 2008, for a total of 14.2 hours of livetime (see Table 1)
(Aharonian et al., 2006). With these default wobble offsets, no $\gamma$-ray
emission has been detected when using background estimation techniques probing
radii of $\lesssim 0.3^{\circ}$.

Table 1: Observation data taken on the Geminga region.

Telescopes | Time period | Livetime | $\theta_{z}$ | Offset
---|---|---|---|---
CT1-4 | Nov 2006 | 7.7 h | 42.2∘ | $\pm 0.5^{\circ}$
CT1-4 | Jan 2008 | 6.5 h | 42.0∘ | $\pm 0.7^{\circ}$
CT1-4 | Jan-Mar 2019 | 27.2 h | 43.5∘ | $\pm 1.6^{\circ}$
333 For each dataset we provide the cumulative livetime after run selection,
average zenith angle $\theta_{z}$ and wobble offset in both R.A. and Dec. of
the observation positions from the location of the Geminga pulsar.

Further observations were taken during the first quarter of 2019, employing a
pointing strategy with much wider pointing offsets of $\pm 1.6^{\circ}$ from
the pulsar in R.A. and Dec., for a total livetime of 27.2 hours. These offsets
are among the widest ‘wobble’ mode pointing offsets yet used with IACTs,
thereby pushing the limits of standard analysis techniques.

###  2.2 Analysis

Considerable advances in gamma-hadron separation were achieved by the use of
template or model-based approaches. In this analysis, a sensitive likelihood-
based template fitting analysis is used (Parsons & Hinton, 2014) and the
results are cross-checked using an independent calibration and analysis chain
(de Naurois & Rolland, 2009).

The Geminga region is observable with H.E.S.S. from November to April only at
large zenith angles $>40^{\circ}$, which raises the intrinsic energy threshold
of the analysis. Additionally, depending on the method of background
estimation and due to systematic uncertainties at the level of the statistical
uncertainties or higher (caused for example by variation in the atmospheric
conditions Hahn et al. 2014) we use a conservative set of selection cuts on
reconstruction quality, known as _hard_ selection cuts, such that only the
best candidate $\gamma$-ray events are retained. This also has the effect of
improving the signal-to-background ratio such that the uncertainty on the
background normalisation is reduced. Correspondingly, the energy threshold is
1 TeV for the spectral analysis; whilst for the morphological analysis, we
used an energy threshold of 0.5 TeV. This is due to the higher event
reconstruction quality and an energy bias of less than 10% required for the
spectral analysis.

Figure 2: Excess counts sky maps of the Geminga region above 0.5 TeV using the
2019 dataset with two different background estimation methods. Left: On-Off
background estimation method. Right: field-of-view background estimation
method. The maps are over-sampled with a 0.08º correlation radius, with
contours at the level of 50 and 100 excess counts from maps over-sampled with
a 0.5º correlation radius. The 68% containment PSF is shown for comparison and
has a value of 0.06º, valid for the innermost $\theta<1^{\circ}$ region around
the pulsar in which the significance is evaluated (indicated by a dashed
circle). The Galactic plane is indicated by a green dashed line and the pulsar
location with a green triangle, whilst an arrow indicates the proper motion
direction of the pulsar (arbitrary length). Background normalisation is
performed using regions at $\theta>3.2^{\circ}$, indicated by a dotted circle.

##  3 Detection of extended $\gamma$-ray emission

A systematic study of analysis differences and background estimation
techniques between H.E.S.S. and HAWC was conducted and applied to the Galactic
plane (Abdalla et al., 2021). By adapting the analysis of H.E.S.S. data to
techniques suitable for angular scales comparable to those probed by HAWC, it
was found that the detectability of large, extended sources could be improved.
This enabled the measurement of $\gamma$-ray emission from a very extended
region and, although the background analysis remains challenging, the H.E.S.S.
data retains the advantage of being able to identify smaller scale structures,
including potential point-like sources.

As the $\gamma$-ray emission is considerably more extended than most other
single TeV $\gamma$-ray sources detected by IACTs to date, a variety of
approaches are used to verify the detection and nature of the emission.
Particular care needed to be taken with the background estimation, which is
described in detail in sections 3.1 and 3.2. It is not yet possible to
evaluate the true extent of the TeV $\gamma$-ray emission, as it extends
beyond the sky region available with the current dataset.

Using an integration region of $1^{\circ}$ radius, extended emission around
the Geminga pulsar is detected at $>6\sigma$ (evaluated using Li & Ma 1983) in
the 2006-2008 dataset and similarly at $\sim 9\sigma$ in the 2019 dataset. Due
to the differences in wobble offset between the 2006-2008 and the 2019
observations, different background estimation techniques are used for the two
datasets.

Table 2: Li & Ma significance for $\gamma$-ray emission within a $1^{\circ}$
radius around the Geminga pulsar obtained with different background methods
and datasets.

Dataset | Background method |  $\sigma$ in $1^{\circ}$ | Cut level
---|---|---|---
2006-08 | Ring | $9\sigma$ | _std_
2006-08 | On-Off | $6.6\sigma$ | _hard_
2019 | Field-of-View | $9.8\sigma$ | _hard_
2019 | On-Off (1) | $9.6\sigma$ | _hard_
2019 | On-Off (2) | $11.6\sigma$ | _hard_
444 For the Field-of-View background method, the Cash statistic was used to
evaluate the significance instead. The level of selection cuts applied (_std_
or _hard_) is also indicated.

###  3.1 Background estimation: 2006-2008 dataset

Revisiting the Geminga observations taken with H.E.S.S. in 2006-2008, the
detection of emission over a much larger angular scale is enabled by
increasing the exclusion region555the region excluded from background
estimation radius from $\sim 0.4^{\circ}$ to $1.5^{\circ}$. The reflected
region background method is well suited to spectral analysis, in which a
region of the same size as the ON region yet reflected across the telescope
pointing direction is used to estimate the background. However, this is not
possible for cases where the ON region size (here of $1^{\circ}$ radius)
exceeds the pointing offset (here of $\pm 0.5^{\circ}-0.7^{\circ}$, see Table
1). Additionally, due to the large exclusion regions, a spectral analysis of
the emission using the reflected background method is not possible on the
Geminga region for any of the current H.E.S.S. datasets (Aharonian et al.,
2001).

Employing the ring background method with a fixed ring thickness of
$0.5^{\circ}$ and a minimum radius in excess of $1.5^{\circ}$, for a 250 pc
distance to Geminga, this corresponds to background estimation from radii
$>$6.5 pc away from the pulsar (Berge et al., 2007). As the background counts
are sampled from the same region of the sky in the ring background method,
standard (_std_) selection cuts are used, whereas more conservative (_hard_)
selection cuts are used for other methods of background estimation. With such
a ring background subtraction, H.E.S.S. is sensitive to the difference in
$\gamma$-ray emission between the ON region at small radii from the pulsar,
and the emission at larger radii that is used to estimate the background. The
limited field of view prevents using a background ring radius large enough
such that the background region does not contain emission from the source. The
measured flux is therefore a relative measurement and will underestimate the
true flux (see also Mitchell et al. 2019).

Two different background methods, Ring and On-Off, are applied to evaluate the
significance of $\gamma$-ray emission within a $1^{\circ}$ radius region
around the pulsar in this dataset, with a consistent detection obtained using
both approaches (see Table 2). When using the On-Off background estimation,
the entire field of view of the observations is considered as the On region,
with Off data taken from observing runs matched for comparable conditions. The
parameters used to match the Off data to On data include the zenith angle of
the observations, the run duration, and the combination of telescopes
participating. The presence of all four telescopes CT1-4 is required for this
analysis in order to provide a smooth acceptance666the probability of
accepting a $\gamma$-ray candidate event reconstructed at a certain position
and energy across the sky region. The Off runs used to estimate the background
are extragalactic observations from 2004-2009 with no significant source
detected in the field of view.

###  3.2 Background estimation: 2019 dataset

Following this detection, observations of the Geminga region were conducted in
2019 at large wobble offsets of $1.6^{\circ}$ around the pulsar, out to which
the camera acceptance remains at $\gtrsim 60\%$ of the on-axis acceptance
(Aharonian et al., 2006). Once again, the emission from the region around
Geminga is found to fill the available field of view, proving a considerable
challenge for background estimation in the analysis. Therefore, two different
methods are used; the so-called On-Off and field-of-view background approaches
(Weekes et al., 1989; Berge et al., 2007).

In order to provide an estimate of the systematic uncertainties of the On-Off
background estimator and cross-check our results, we used two independent
matched lists of Off runs, with different tolerance levels in the matching
criteria (see Table 3). Both lists are comprised of observations from 2017 to
2019. Unless otherwise specified, analysis results from the On-Off background
approach are always presented using the better matched Off List 1, with Off
List 2 contributing to the evaluation of the systematic uncertainty only (see
section 4.2 & Table 5).

Table 3: Matching criteria of Off runs selected for the On-Off background
analysis.

Dataset | Mean $\Delta\theta_{z}$ | Mean $\Delta t$ | Livetime
---|---|---|---
Off List 1 | $0.23^{\circ}\pm 0.04$ |  $5.7\pm 0.7$ s | 17.9 h
Off List 2 | $0.97^{\circ}\pm 0.03$ |  $4.4\pm 0.4$ s | 20.8 h
777 The average discrepancy between On and Off runs in Zenith angle
$\theta_{z}$ and run duration $t$ is indicated for each Off list. The
resulting livetime after run quality selection is also quoted, with the
difference in livetime due to tighter quality cuts and matching criteria used
for Off List 1. We note that the two sets of Off runs are fully independent.

The second background estimation approach used is the field-of-view (FoV)
method, which uses an acceptance model to estimate the expected level of
background counts throughout the field of view (Berge et al., 2007). Regions
of significant $\gamma$-ray emission are excluded and the predicted number of
background counts is normalised to the excess counts outside of these
exclusion regions.

For both background estimation approaches, the background counts are
normalised to the On counts in the region at radii $\theta>3.2^{\circ}$ away
from the pulsar location (that is twice the offset of the observation
positions), as shown in Fig. 1. Excess counts maps constructed using both
background estimation approaches are shown in Fig. 2, using a $0.08^{\circ}$
correlation radius. The observation positions used in 2006-2008 and in 2019
are indicated on the sky map shown in Fig. 3.

Figure 3: Observation positions corresponding to the 2006–2008 data set
(magenta) and the 2019 dataset (green) at offsets of $0.7^{\circ}$ and
$1.6^{\circ}$ from the pulsar respectively. The white dashed circle indicates
the test On region with $1^{\circ}$ radius around the pulsar, whilst the white
dotted circle indicates the $3.2^{\circ}$ radius region beyond which the
background is normalised. Green and magenta circles indicate the radius around
an observation position with the same offset as the pulsar. Counts map and
contours are otherwise as in Fig. 2, left.

Some evidence for run-by-run variation in the Off count rate was investigated
and found to be related to the atmospheric conditions, as indicated by the
Cherenkov Transparency Coefficient, CTC (Hahn et al., 2014). The Pearson
correlation coefficient for the Off count rate and CTC is $\sim 0.7$ for both
Off Lists. A correction for the variation in atmospheric conditions between
the On and Off runs is implemented via two methods; using the ratio of the CTC
between On and Off runs; and using the ratio of the number of hadron-like
events in On and Off runs. Both methods are found to remove the correlation
between the off count rate and the CTC. This correction is applied prior to
the normalisation of On counts to radii $>3.2^{\circ}$. For the remainder of
this paper only this dataset obtained in 2019 is used.

###  3.3 Background estimation: Systematic tests of the On-Off and FoV methods

As the analysis of sources with highly extended $\gamma$-ray emission is
inherently challenging for IACTs, we performed a range of tests to assess the
robustness of the analysis techniques and to quantify systematic
uncertainties. We performed an On-Off background analysis on two extragalactic
regions with no expected $\gamma$-ray emission, using comparable parameters to
the analysis of the Geminga region, that is with the same set of matching
criteria for Off runs. For this purpose we used data obtained in 2017 and 2018
with the telescopes CT1-4 on two Dwarf Spheroidal galaxies, namely Reticulum
II and Tucana III (Abdallah et al., 2020). The average zenith angles of these
datasets are $42.4^{\circ}$ and $39^{\circ}$ respectively, comparable to the
$43.5^{\circ}$ average value of the Geminga dataset. No evidence for
significant emission within the target region is found. The significance
distributions for both dwarf spheroidals are found to be compatible with
background, whereas the significance distribution for the Geminga region is
asymmetric and exhibits a clear tail towards high significance. Also, the mean
of the distribution is greater than zero for the Geminga region, reflecting
the fact that the emission fills the field of view, whereas the distribution
mean is compatible with zero for the test fields. This lack of signal from the
test fields associated with dwarf spheroidal galaxies when using the same
method provides confidence in the clear signal seen from the Geminga region.
These empty fields are also analysed using the field-of-view background
method; again, no significant emission is seen.

Fig. 4 shows the ratio of On counts to background counts with distance from
the FoV centre, with renormalisation to the background at a radial distance of
$>3.2^{\circ}$ from the centre of the field of view. There are indications for
a bias in the Ret II dataset with an On counts to background ratio lower than
1.0 out to beyond 3∘ (that is out to the radius which is used for
normalisation to background). However, for the Tuc III dataset, the background
normalisation results in approximately the expected level, averaging around
1.0. Correcting for a $\sim 10\%$ bias for the region within $1^{\circ}$
radius of the pulsar would lead to a $\sim 30\%$ increase in significance for
the values quoted in Table 2. Given the limited number of tests performed, we
conservatively estimate a systematic error of the order of $30\%$. A similar
bias is found using the FoV background method, averaging $\sim 6\%$ at radii
$<1^{\circ}$.

Figure 4: Ratio of On counts to background counts using the On-Off background
method as a function of radial distance from the centre of the field of view.
Shown are two empty sky regions taken on the dwarf spheroidal galaxies
Reticulum II and Tucana III, as well as Geminga data.

The field of view around the Geminga pulsar contains a bright star of
magnitude 1.9, that leads to increased Night-Sky-Background (NSB) in the
region. We investigated whether any dependence of the excess counts on the NSB
in an On-Off analysis is seen, and found no correlation between the NSB rate
and excess counts per run.

##  4 Analysis results

###  4.1 Morphology of the $\gamma$-ray emission

Fig. 5 shows the On data and background counts as a function of radial
distance from the pulsar. The estimated background from the FoV and On-Off
background methods (using two different Off run lists) are shown, and
normalised to the data at radii $>3.2^{\circ}$, with a statistical uncertainty
on the normalisation at the 1 % level. An excess of On data above background
counts can be seen for all three estimations of the background level,
particularly towards the innermost radii. The levels of background counts from
the different methods are broadly consistent, with the remaining mild
discrepancies providing an indication of the level of background systematics,
estimated as $\leq 5\%$. Fig. 5 provides further confidence that the detected
emission is significant above the level of Galactic diffuse emission, which
would peak towards the Galactic plane (the Geminga pulsar being situated
towards the Galactic anti-centre at a galactic latitude of $4.27^{\circ}$).

Figure 5: Radial profile without acceptance correction applied from
uncorrelated maps, showing counts per unit area with radial distance from the
pulsar at energies $>0.5$ TeV, error bars are statistical. The different
background methods show reasonable agreement.

The extension of the $\gamma$-ray emission is of particular relevance to
studies of energetic particle transport in the region. Therefore, we tested
the extent of the region of significant emission, firstly by varying the
integration region radius (the radius around the pulsar within which the
significance of the $\gamma$-ray emission is computed). Once the significant
$\gamma$-ray emission is fully contained, the significance curve will start to
flatten with increasing radius.

Fig. 6 shows the significance with radius for the two independent background
estimates. Notably, neither of the curves flatten within a radius out to
$3^{\circ}$, indicating that the significant $\gamma$-ray emission is not yet
fully contained by the integration region. Secondly, the shape of the curves
is consistent between the FoV and On-Off background methods; this is
reassuring confirmation in the extent of significant emission provided by
these fundamentally different approaches. Lastly, this analysis shows that the
emission within a radius $<0.25^{\circ}$ around the pulsar is not significant.
This further confirms that although concentrated towards the pulsar, the
majority of the emission is spread out to larger radii.

Figure 6: Significance with integration region radius around the pulsar for
FoV and On-Off background estimation analyses. Emission within 1 degree radius
of the pulsar is detected with a significance of $\gtrsim 8\sigma$ with all
analyses.

Although the analysis is limited by systematic uncertainties, we attempt to
quantify the significance of a possible asymmetry as follows. Firstly, an
acceptance-corrected azimuthal profile of the emission within a $3^{\circ}$
radius around the pulsar is constructed. The peak of the azimuth profile is
found to be at $105^{\circ}\pm 3^{\circ}$ measured anticlockwise from the
north, and is used to divide the region into two halves. From each
semicircular half, radial surface brightness profiles of the emission,
corrected for acceptance, are constructed in the two independent regions.
Indications for a higher level of emission in one region compared to the other
are seen at radii $>1^{\circ}$ (for example Fig. 2), however this is found to
be neither statistically significant nor independently verified in the cross-
check analysis.

To quantify whether the emission is significantly offset from the pulsar,
despite the inability to measure the true extent, we evaluated the barycentre
and 68% containment radius of the excess counts, with acceptance correction
applied. Indications for an offset of the emission centroid from the pulsar
are found for all background methods, yet are nevertheless compatible with the
pulsar location when systematic errors are taken into account. The emission
centroid has an offset of $0.6^{\circ}$, at R.A. $99.1^{\circ}\pm
0.1^{\circ}\pm 0.5^{\circ}$ and Dec $17.7^{\circ}\pm 0.1^{\circ}\pm
0.5^{\circ}$ where errors are statistical and systematic (as estimated from
the difference between the background methods). Evaluating the containment
radius in three energy bands (0.5 - 2 TeV, 2 - 8 TeV, and 8 - 40 TeV) no
evidence for significant energy-dependent morphology is found. These
containment radii in energy bands are summarised in Table 4.

Table 4: Containment radii at 68%, $\theta_{68}$ of the excess emission,
evaluated in different energy bands for both On-Off and FoV background
methods.

Energy Band | On-Off $\theta_{68}$(∘) | FoV $\theta_{68}$(∘)
---|---|---
0.5 - 2 TeV | $2.7\pm 0.2$ | $2.2\pm 1.6$
2 - 8 TeV | $1.9\pm 0.8$ | $1.8\pm 0.7$
8 - 40 TeV | $2.5\pm 1.2$ | $2.5\pm 0.5$
0.5 - 40 TeV | $2.4\pm 0.1$ | $1.8\pm 0.3$
888The energy range of 8 TeV - 40 TeV is chosen to match the energy range of
the analysis by the HAWC collaboration (Abeysekara et al., 2017a).

Although TeV-bright $\gamma$-ray emission may be expected on angular scales
$\lesssim 0.1^{\circ}$ corresponding to the size of the X-ray PWN, with this
analysis we see no indications of a separate component at these scales.
Identifying such a separate emission component is challenging given the
evolved state of the system, the proper motion of the pulsar (which is fast
compared to the cooling time of electrons at the nearby distance of Geminga)
and the predominance of inverse Compton emission from electrons already
escaped from the PWN region.

###  4.2 Spectral results

We perform a spectral analysis for a circular region with 1∘ radius centred on
the pulsar using the On-Off background approach; the resulting spectrum is
shown in Fig. 7 with parameter values for a best fit power law quoted in Table
5. Due to the quality cuts applied for a spectral analysis, the energy
threshold of the analysis is 1 TeV. The spectrum obtained with HAWC for both
the disc morphology of Abeysekara et al. (2017b) and the diffusion model of
Abeysekara et al. (2017a) are also shown in Fig. 7, scaled for comparison to
account for the different sizes of the regions probed. The scaling factor for
the HAWC spectrum from Abeysekara et al. (2017b) corresponds to the ratio of
the disc areas for a $2^{\circ}$ radius and a $1^{\circ}$ radius, the region
probed by this H.E.S.S. analysis. For the HAWC spectrum from Abeysekara et al.
(2017a), we rescale according to the ratio between the diffusion model
integrated out to 1∘ radius and the full integral, assuming that the spectral
index is not radially dependent.

Figure 7: Spectral energy distribution of the $\gamma$-ray emission within
$1^{\circ}$ radius of the Geminga pulsar. Spectra from two HAWC analyses of
Geminga are shown for comparison (Abeysekara et al., 2017a, b), with flux
normalisation scaled to match the sub-region from which the H.E.S.S. spectrum
is extracted. Error bands include systematic uncertainties for H.E.S.S. but
are statistical only for HAWC.  Table 5: Fit parameters to spectra obtained
from a $1^{\circ}$ radius region around the pulsar using the two Off Lists.

Dataset | Index |  $\phi_{0}$ (cm-2s-1TeV-1) | Signif.
---|---|---|---
Off List 1 | $2.76\pm 0.22$ | $(2.81\pm 0.71)\times 10^{-12}$ | 6.7 $\sigma$
Off List 2 | $2.62\pm 0.14$ | $(3.55\pm 0.68)\times 10^{-12}$ | 8.9 $\sigma$
999 A power law spectral model
$\frac{\mathrm{d}N}{\mathrm{d}E}=\phi_{0}\left(\frac{E}{E_{0}}\right)^{-\Gamma}$
is assumed, with index $\Gamma$ and $E_{0}=1$ TeV. The systematic errors are
estimated at $\pm 0.14$ on the spectral index and $\pm 0.7\times
10^{-12}$cm-2s-1TeV-1 on the flux normalisation for both Off Lists.

Fig. 7 shows that the scaled HAWC flux and the H.E.S.S. measurement are
consistent at energies $\gtrsim 5$ TeV and compatible within the systematic
errors of both measurements (Abeysekara et al., 2017a, b). From Abeysekara et
al. (2017a), the best fit spectral measurement has a spectral index of
$2.34\pm 0.07$ and flux normalisation of $13.6^{+2.0}_{-1.7}\times 10^{-15}$
TeV-1cm-2s-1 at 20 TeV. We use this latter spectrum obtained from a dedicated
analysis of HAWC data on the Geminga region for modelling purposes. The
systematic errors of the H.E.S.S. spectral measurement are $\sim 0.2$ on the
index and $\sim 30\%$ on the flux, obtained from the cross-check between the
background models. The $\sim 8$% bias seen at radii $<1.0^{\circ}$ in Fig. 4
corresponds to a $\sim 30\%$ systematic on the flux, assuming these two
contributions are uncorrelated, the total systematic error on the flux is of
the order of $40\%$. The systematic errors of the HAWC spectral measurement
are $0.2$ on the index and $50\%$ on the flux. We note that the measured HAWC
flux is dependent on the morphological model assumed for the emission - here
we show the disc model, the approach most similar to the H.E.S.S. spectral
measurement, which is also approximately a disc with $1^{\circ}$ radius.

###  4.3 Radial profiles

Acceptance-corrected radial profiles of the excess emission are constructed
from the sky maps, using a circular region centred on the pulsar extending out
to $\sim 3.5^{\circ}$ radius. The surface brightness profiles are constructed
from acceptance-corrected excess maps with units of cm-2s-1deg-2. For the
conversion to surface brightness in units of TeV cm-2s-1deg-2, a power law
spectral model is used with index $\Gamma=2.76$ (corresponding to the best-fit
spectral parameters, see Table 5). These profiles are presented in Fig. 8.

Figure 8: Radial profile constructed using three independent methods; from
$\gamma$-ray maps with FoV and On-Off background methods (Off list 1), and
from a spectral analysis in consecutive ring regions around the pulsar. The
latter is limited to radii $<2^{\circ}$ from the pulsar and is shown without
an additional normalisation to background at radii $>3.2^{\circ}$ applied.

Using the On-Off background method, spectral analyses are additionally
performed in concentric rings of $0.5^{\circ}$ width, out to a radius of
$2^{\circ}$. The data is fit in each ring using a power law spectral model,
from which radial profiles of the emission are constructed. The best fit model
is integrated within the energy range $0.5-40$ TeV to obtain the flux as a
function of radius within a specific energy band. This radial profile is also
shown in Fig. 8. The resulting radial surface brightness profiles are
therefore found to be compatible between the normalised profiles from maps
constructed with On-Off background and from maps constructed with the FoV
background. The radial surface brightness profile from a spectral analysis is
however limited to radii $<2^{\circ}$. In order to normalise this radial
profile constructed from spectra, we applied the same normalisation as that
obtained for the radial profile for On-Off background constructed from sky
maps.

These radial profiles also show the normalisation to background at radii
$>3.2^{\circ}$, which implies that the flux is likely underestimated over the
region. The innermost radial bin corresponds to a region $0.2^{\circ}$ around
the pulsar, comparable in size to that of the region probed by X-ray
instruments. A flux measurement is obtained from the surface brightness within
this radial bin, assuming that the spectral index does not vary with radius.
In section 5 these radial profiles, together with the spectral energy
distribution, are fit with a diffusion model.

##  5 Modelling

###  5.1 Diffusion model description

For the particle transport we consider energy-only dependent diffusion, energy
losses, and a continuous point-like source of electrons:

$\displaystyle\partial_{t}N(E,\@vec{r},t)-D(E)\Delta
N(E,\@vec{r},t)+\partial_{E}[b(E)N(E,\@vec{r},t)]$
$\displaystyle=Q(E,t)\,\delta(\@vec{r}-\@vec{r_{s}}),$ (1)

with $N(E,\@vec{r},t)$ the density of electrons per energy, $D(E)$ the spatial
diffusion coefficient, $b(E)$ the energy loss rate, and
$Q(E,t)\delta(\@vec{r}-\@vec{r_{s}})$ the source term with $Q(E,t)$ the rate
of produced electrons per energy and $\@vec{r_{s}}$ the source position. The
assumption of a point-like source of injection is motivated by X-ray
observations of the Geminga pulsar wind nebula (Pavlov et al., 2010) showing
emission up to $0.06$ pc whereas the observed TeV emission extends up to $14$
pc. An advection term is not included in this expression for the particle
transport, as this would generate asymmetries to the $\gamma$-ray emission
that are not seen in this analysis.

####  5.1.1 Diffusion coefficient

The diffusion coefficient $D(E)=D_{0}(E/E_{0})^{\delta}$ is assumed to follow
a power law, with index $\delta$ in the range $\delta\in[0.3,1]$, to cover the
scale between between Kolmogorov turbulence (Kolmogorov, 1991), Kraichnan
turbulence (Kraichnan, 1965), and Bohmian diffusion (Bohm, 1949). For this
model, $E_{0}=100\,$TeV as in Abeysekara et al. (2017a) is chosen to have a
direct comparison of the diffusion coefficient with the HAWC results, whilst
$D_{0}$ is kept as a free parameter of the fit.

####  5.1.2 Energy losses

Synchrotron and Inverse Compton energy losses are considered. We assume the
presence of three different photon fields; the Cosmic Microwave Background,
Infra-Red from dust, and Optical radiation from star light (see also appendix
A.1). Magnetic field values between $1$ $\mu$G and $5$ $\mu$G are tested,
which correspond to inverse-Compton dominant and synchrotron dominant
scenarios respectively.

The typical time of energy losses is defined as:

$t_{E}(E)=-\int_{E}^{+\infty}\frac{1}{b(E^{\prime})}\mathrm{d}E^{\prime}\leavevmode\nobreak\
,$ (2)

where $b(E^{\prime})$ represents the total energy-dependent energy losses. Eq.
(2) can be defined as the maximum age of an electron given its current energy.
This simple calculation enables us to assess the age of the particles that can
be observed, as shown in Fig. 9. Given the H.E.S.S. energy threshold converted
to electron energy, the maximal age of particles is between $140$ kyr for the
$1$ $\mu$G case and $32$ kyr for the $5$ $\mu$G case.

Figure 9: Typical energy loss time versus electron energy. If the particle
loss timescale is longer than the pulsar age (here chosen as $340$ kyr), then
the typical time $t_{E}(E)$ (see equation 2) is taken as the pulsar age. The
difference to the HAWC model is due to the approximation of energy losses made
in the HAWC model.

####  5.1.3 Source term

The total energy released by the pulsar since its birth, $W_{0}$, is defined
as:

$W_{0}=\int^{T_{*}}_{0}L(t)\mathrm{d}t\leavevmode\nobreak\ ,$ (3)

where $T_{*}$ is the current pulsar age and $L$ is the pulsar luminosity
function (see appendix A.2). We assume that the source term $Q(E,t)$ (defined
as the rate of electrons per energy produced by the pulsar) follows a power
law with an exponential cut off, and its normalisation follows the same time-
dependence as the pulsar luminosity:

$Q(E,t)=Q_{0}(1+t/\tau_{0})^{-(n+1)/(n-1)}(E/E_{0})^{-\alpha}\exp{(-E/E_{c})}\leavevmode\nobreak\
,$ (4)

where $\tau_{0}$ is the initial spin-down timescale, $n$ is the braking index,
and $E_{c}$ the energy cut-off of electron spectra (see appendix A.2 for the
definitions of $\tau_{0}$ and $E_{c}$). For simplicity, $E_{c}$ is assumed to
be constant during the lifetime of the pulsar. The normalisation of the source
term can be related to the total energy released by the pulsar by:

$\int^{+\infty}_{E_{min}}\int^{T_{*}}_{0}Q(E,t)E\mathrm{d}E\mathrm{d}t=\eta
W_{0}\leavevmode\nobreak\ ,$ (5)

where the efficiency parameter $\eta$ is the proportion of the pulsar spin-
down energy converted to electron-positron pairs that escaped from the close
source environment to the extended emission region.

###  5.2 Solution

The solution to the diffusion energy loss equation as described in Di Mauro et
al. (2019) is:

$\displaystyle N(E,\@vec{r},t)$ $\displaystyle=$
$\displaystyle\int^{t}_{0}\mathrm{d}t_{0}\frac{b(E_{s}(E,t_{0},t))}{b(E)}\frac{1}{(\pi\lambda^{2}(t_{0},t,E))^{3/2}}$
(6) $\displaystyle\times$
$\displaystyle\exp{\left(-\frac{|\@vec{r}-\@vec{r_{s}}|^{2}}{\lambda^{2}(t_{0},t,E)}\right)}Q(E_{s}(E,t_{0},t),t_{0}),$

where $E_{s}(E,t_{0},t)$ is defined as the energy of an electron that was
produced at time $t_{0}$ and observed at time $t$ with energy $E$. This value
is obtained by solving numerically:

$t-t_{0}=\int_{E}^{E_{s}}\frac{\mathrm{d}E^{\prime}}{b(E^{\prime})}\leavevmode\nobreak\
,$ (7)

while the diffusion radius $\lambda(t_{0},t,E)$ is defined as:

$\lambda^{2}=4\int_{E}^{E_{s}}\mathrm{d}E^{\prime}\frac{D(E^{\prime})}{b(E^{\prime})}\leavevmode\nobreak\
.$ (8)

Since the data we are using to constrain this model are radial profiles, we
compute the mean density over the angle between $\@vec{r}$ and $\@vec{r_{s}}$
of electrons observed today at a given energy and distance from the current
pulsar position $r$ can be written as:

$\displaystyle N(E,r,T_{*})$ $\displaystyle=$
$\displaystyle\int^{T_{*}}_{0}\mathrm{d}t_{0}\frac{b(E_{s}(E,t_{0},T_{*}))}{b(E)}\frac{1}{(\pi\lambda^{2}(t_{0},T_{*},E))^{3/2}}$
(9) $\displaystyle\times$
$\displaystyle\exp{\left(-\frac{r^{2}+r_{s}^{2}(t_{0})}{\lambda^{2}(t_{0},T_{*},E)}\right)}Q(E_{s}(E,t_{0},T_{*}),t_{0})\leavevmode\nobreak\
,$

with $T_{*}$ being the current pulsar age, where $E_{s}(E,t_{0},T_{*})$ is
defined as the energy of an electron that was produced at time $t_{0}$ and
observed at time $T_{*}$ with energy $E$. Since $r_{s}(t)$ is varying as a
function of time, the impact of the asymmetric morphology is taken into
account in our radial profiles. In addition to the source parameters, the
density of electrons depends on the diffusion parameters ($D_{0}$ and
$\delta$) and electron cooling parameters ($B$ field and IC photon field
energy density and temperature). The parameter $r_{s}$ allows us to introduce
the effect of the pulsar proper motion, and is defined as:
$r_{s}(t)=V_{T}(T_{*}-t)$ where $V_{T}$ is the transverse speed of the pulsar.

Figure 10: Comparison of the predicted typical diffusion scale $\lambda$ as a
function of electron energy for different diffusion models and magnetic
fields. The HAWC model is taken from Abeysekara et al. (2017a). It is clear
that the emission is predicted to exceed the H.E.S.S. field of view over most
of the H.E.S.S. energy range (here converted to electron energy).

As shown in Fig. 10, the diffusion radius peaks at the energy where the
cooling time for electrons is equal to the pulsar age. For the Geminga pulsar
with a characteristic age of $T_{c}=342$ kyr, this occurs at $\sim 1$ TeV. At
lower energies, the electrons are uncooled and the diffusion radius $\lambda$
is limited by the diffusion parameters; whereas at higher energies, the
diffusion radius is limited by the cooling time for electrons in the ambient
magnetic field. Fig. 10 demonstrates that the diffusion radius in the energy
range of H.E.S.S. is much larger than the H.E.S.S. field of view, such that
the full extent of the emission cannot be measured.

###  5.3 Model joint fit with H.E.S.S., HAWC and XMM-Newton data

The solution of the diffusion model provides us with the density of electrons
per energy interval. Two more steps are needed to compare the diffusion model
with the multi-wavelength data: Firstly, the electron density needs to be
projected onto a 2D plane and converted to a density versus angle based on the
pulsar distance. Secondly, the obtained electron density profile is then
converted to the $\gamma$-ray flux through Inverse Compton (IC) scattering.
This step makes use of the Naima package (Zabalza, 2015), with a photon field
for IC approximated as three black-body components defined in appendix A.1.

The diffusion model described above is simultaneously fit to radial profiles
and spectral energy distribution (SED) using a chi-square minimisation. The
normalisation of the diffusion coefficient $D_{0}$ is used as a free parameter
while the energy cut-off of the electron spectrum $E_{c}$ is tested as a free
or fixed parameter in order to evaluate its significance in the final result.
We performed this analysis using the combination of H.E.S.S. (radial profile
and SED within $1^{\circ}$) and HAWC 2017 (radial profile) data (see section
4.2). The HAWC SED from Abeysekara et al. (2017a) is not used in the fit but
is superimposed to the model SED obtained with an integration radius of
$10^{\circ}$. The GeV part of the Geminga halo is assumed to be highly offset
compared to the actual pulsar position due to the proper motion of the pulsar.
A detection of such emission is claimed by Di Mauro et al. (2019), where a
template fit, including the effect of pulsar proper motion, enabled the
detection of emission between $10$ GeV and $100$ GeV with an offset position
and extent consistent with that expected from the diffusion model fit to HAWC
data.

We decided not to include these data because our single zone model is not
valid on the full extent of the claimed Fermi-LAT emission. Indeed, a single
zone of diffusion for the Fermi-LAT extent scale cannot take into account the
positron excess of AMS-02 as explained in Martin et al. (2022). A two zone
diffusion model is beyond the scope of this paper, since the emission measured
with H.E.S.S. is not extended enough to constrain such a model. In order to
constrain the synchrotron contribution to the SED, we compared keV upper
limits to the model SED, derived using XMM-Newton data in a $10^{\prime}$
region around the Geminga pulsar as described in Liu et al. (2019).

In order to explore the phase space whilst keeping $D_{0}$ and $E_{c}$ free in
the fit, we conduct a parameter scan over five variables
($n,\eta,\alpha,B,\delta$), because a global minimisation procedure is found
to be degenerate. These scanned parameters are listed in Table 10. All
possible combinations of parameters are tested, leading to 243 different
combinations to explore the full parameter space. Values for the parameters
characterising the Geminga pulsar are taken from the ATNF pulsar catalogue
Manchester et al. (2005). The initial period of the pulsar is assumed to be
$15$ ms since its value is not affecting results above the H.E.S.S. energy
threshold. Two of the free parameters are further constrained by validity
conditions: $P_{0}<P_{*}$ and $\eta<1$.

Table 6: Input parameters for the diffusion model.

Parameter | Description | Value(s)
---|---|---
$L_{*}$ | Current Luminosity |  $3.2\times 10^{34}$ erg/s
$T_{c}$ | Pulsar age |  $342$ kyr
$n$ | Braking index | [$1.5$, $3$, $4.5$]
$P_{0}$ | Initial period |  $15$ ms
$P_{*}$ | Current period |  $0.237$ s
$d$ | Distance |  $250$ pc
$V_{T}$ | Transverse velocity |  $211$ km/s
$\eta$ | Electron efficiency | [$0.01$, $0.1$, $0.5$]
$\alpha$ | index of electron | [$1.8$, $2.0$, $2.2$]
| injection distribution |
$E_{c}$ | Energy cut off of electron | [free,$1$ PeV]
| particle distribution |
$B$ | Ambient magnetic field | [$1$, $3$, $5$] $\mu$G
$D_{0}$ | Normalisation | free
| of diffusion coefficient |
$\delta$ | power law index | [$0.3$, $0.6$, $1$]
| of diffusion coefficient |
101010$D_{0}$ is the normalisation at an electron energy $E_{0}=100$ TeV.

The fits are performed for the results using both On-Off background lists
separately, as well as for the main and cross-check analyses. For each fit
result, the p-value is computed and is used to define the goodness of fit. A
p-value $>0.003$, corresponding to three standard deviations is taken, as a
threshold to define a good fit. This can be considered as a fairly relaxed
threshold since the systematic uncertainties of both H.E.S.S. and HAWC data
are important.

The p-value distribution as a function of the electron efficiency from the
fixed $E_{c}$ scan is presented in Fig. 11, where only those good fit models
with p-value $>0.003$ are retained. The combination of H.E.S.S. and HAWC data
excludes an efficiency of injection and electron-positron pair conversion
lower or equal to $1$ % at the $5\,\sigma$ level. However, no significant
preference is found between the values scanned for the power law index of the
diffusion coefficient $\delta$, the ambient magnetic field $B$, injection
index $\alpha$ and braking index $n$.

The fitted diffusion coefficients are comparable with the one obtained by HAWC
(Abeysekara et al., 2017a) as shown in Fig. 12. The diffusion coefficients
derived from the Boron-to-Carbon ratio (labelled B/C diffusion hereafter) are
obtained under the assumption that the galactic magnetic halo size is $1$ kpc,
$4$ kpc, and $16$ kpc respectively for the three values indicated (Genolini et
al., 2019). The H.E.S.S. result confirms a significantly lower diffusion
coefficient than that obtained with the cosmic ray Boron-to-Carbon ratio.
Moreover, the mean statistical uncertainty of the fits improved from $27\%$
with HAWC only to $17\%$ with the combination of H.E.S.S. and HAWC.

Figure 11: Distribution of p-value for the scanned parameter combinations
presented in the table 10, with different values of $\eta$ parameter, using
the H.E.S.S. and HAWC datasets in combination with $E_{c}$ fixed to 1 PeV.
This distribution is presented as violin plot, illustrating kernel probability
density. The width of the shaded area represents the proportion of the number
of fits located at these position. The upper and lower bar represent the
maximum and minimum values. Models with p-value $<0.003$ are excluded.

A comparison is performed between models both with an energy cut-off fixed to
$1$ PeV and when leaving the energy cut-off free to vary. This value is chosen
as the minimal cut-off value that does not significantly modify the spectral
shape in the energy ranges of H.E.S.S. and HAWC. The cut-off is defined as
significant if the model fit leaving $E_{c}$ free is able to provide a good
fit (p-value $>0.003$) and if the chi-square difference with the $E_{c}$ fixed
model is found to be higher than $25$ (which corresponds to $5$ standard
deviations).

For the H.E.S.S. and HAWC combined data, $53$ models passed this criterion
showing preference for a sub-PeV cut-off. Fig. 12 shows the fitted energy cut-
off compared to the power law index of the injection spectrum and the fitted
diffusion coefficient for good fits obtained with an energy cut-off either
fixed to 1 PeV or left free to vary. A correlation can be observed between the
spectral index and the energy cut-off in the top panel of Fig. 12, with a
Pearson correlation coefficient of $0.64$. A hard injection spectrum, with
index lower than $2$, implies the presence of a sub-PeV energy cut-off to
reproduce the combined H.E.S.S. and HAWC dataset.

Figure 12: Distribution of models with p-value $>0.003$, presented as a violin
plot, illustrating the kernel probability density. The width of the shaded
area represents the proportion of number of fits located at these positions.
The upper and lower bars represent the maximum and minimum values. Top: Fitted
energy cut-off distribution as a function of injection index for the combined
H.E.S.S. and HAWC dataset. The red horizontal line shows the value of energy
cut-off chosen for the fixed $E_{c}$ fit. Bottom: distribution of the fitted
diffusion coefficient in the case of $E_{c}=1$ PeV and $E_{c}$ left free to
vary, for a combined fit to H.E.S.S. and HAWC data. Solid lines indicate the
value ranges for the B/C diffusion coefficient and the statistical uncertainty
of HAWC diffusion coefficient obtained in Abeysekara et al. (2017a).

The expected synchrotron flux in the $10^{\prime}$ region around the pulsar
from the good fit models is compared with the XMM-Newton upper limits with
both a free cut-off and a fixed 1 PeV cut-off. For the models defined as a
good fit of the H.E.S.S. and HAWC data, those with a fixed $E_{c}$ of 1 PeV
result in a synchrotron flux exceeding the upper limit. This leads to their
exclusion, while for the free cut-off, the models with a magnetic field of $1$
$\mu$G and a high-energy cut-off lower than $75$ TeV are able to fulfil the
XMM-Newton constraint. The impact of these two parameters on the SED is
presented in the Fig. 13. A value higher than $1$ $\mu$G overshoots the X-ray
upper limits, as well as a higher injection cut-off energy than $75$ TeV. A
flux point for H.E.S.S. derived from the innermost radial bin of Fig. 8, a
comparable angular scale of $\sim 10^{\prime}$ is also shown for comparison.
We note that due to the background normalisation applied, the absolute flux
value is likely underestimated.

As an example, the best fit from all the scanned parameters with HAWC and
H.E.S.S. data in terms of p-value is shown in the Fig. 14 compared with data.
The list of scanned models that fulfiled the good fit and XMM-Newton
constraints is given in the appendix B.

Figure 13: Comparison of the model SED with the XMM-Newton upper limit for
different parameters. A flux point for H.E.S.S. corresponding to the innermost
radii $\lesssim 10^{\prime}$ is shown for comparison.

Figure 14: Comparison of the fitted SED and profiles with data for the models
with p-value $>0.003$. The best fit is obtained for $n=4.5$, $\eta=0.1$,
$\alpha=1.8$, $\delta=1.0$, $B=1\mu$G and is highlighted compared to other
models, with fitted parameters of $D_{0}=7.6^{+1.5}_{-1.2}\times 10^{27}$
cm2s-1 and $E_{c}=74^{+17}_{-11}$ TeV. The uncertainty bands for H.E.S.S.
indicates the systematic uncertainty with statistical uncertainties indicated
by the error bars, while for HAWC the uncertainty band represents the
systematic and statistical uncertainty.

##  6 Discussion

This H.E.S.S. detection of extended TeV emission around the Geminga pulsar
confirms the results reported by the water Cherenkov detectors Milagro, HAWC,
LHAASO, and Tibet-AS$\gamma$ with IACTs for the first time. Demonstrating this
detection with IACT analysis techniques, potentially opens up the source class
of highly extended ‘halos’ of TeV $\gamma$-ray emission around middle-aged
pulsars to detailed studies by IACTs (Linden et al., 2017; Giacinti et al.,
2020). In contrast to the known population of TeV pulsar wind nebulae, for
which the TeV structures are already known to be typically larger than the
X-ray synchrotron components (Aharonian et al., 1997), in pulsar halos, it is
thought that the emission originates from escaped particles diffusing away
from the canonical PWN.

Geminga is unambiguously in the halo stage of PWN environmental evolution and
as such is the first clear example of extended halo-like TeV emission to be
detected with H.E.S.S. or by IACT-based facilities in general (Giacinti et
al., 2020). Previously, the limited field of view, pointed observations, and
restrictive background estimation approaches of IACTs have prohibited such a
measurement. However, this paper demonstrates that these aspects can be partly
overcome, although a full characterisation of the $\gamma$-ray emission around
Geminga remains challenging. Given the systematic uncertainties, no
significant asymmetry in the morphology of the $\gamma$-ray emission is found.
If an asymmetry were to be confirmed at a significant level, it could provide
valuable information and constraints on the level of magnetic turbulence in
the region (López-Coto & Giacinti, 2018).

Given the observed extent of $\gamma$-ray emission around Geminga by the
survey instruments HAWC and Fermi-LAT of 20-30 pc and $\sim 100$ pc
respectively, we can anticipate that the true extent of TeV $\gamma$-ray
emission within the H.E.S.S. energy range is $\sim 50$ pc, or roughly double
the current H.E.S.S. field of view, following the estimate shown in Fig. 10.
In contrast, the observed extent in other wavebands is just
$\leavevmode\nobreak\ 2^{\prime}$ in X-ray, $10^{\prime\prime}$ in radio or a
mere few arcseconds in optical (de Luca et al., 2006; Pellizzoni et al., 2011;
Shibanov et al., 2006). It is hence apparent that the emission detected in
these wavelength bands are produced by different regions within the PWN-halo
system. Nevertheless, given the $\gamma$-ray evidence for a halo of escaped
electrons far larger than the region indicated for energetic electrons in
X-ray and radio observations to date, one may expect that a larger region of
much weaker low surface brightness synchrotron emission exists, originating
from this electron population. Faint X-ray emission associated with the wider
halo of escaped electrons may be identifiable with instruments sensitive to
low surface brightness emission, such as eROSITA (Predehl et al., 2021).

With the revised energy loss description and analytical solution from Di Mauro
et al. (2019) as described in section 5, a value for $D_{0}$ compatible with
that of the HAWC analysis (Abeysekara et al., 2017a) is obtained. Three
turbulence regimes are probed, Kolmogorov and Kraichnan turbulence, as well as
Bohm diffusion, employing the diffusion index $\delta\in[0.3,0.6,1.]$
respectively (Kolmogorov, 1991; Kraichnan, 1965; Bohm, 1949). No preferred
value is found with the joint fit of H.E.S.S. and HAWC data, as well as no
preference between the ambient magnetic field, injection index, and braking
index over the scanned values. We find that the efficiency with which energy
is converted to electrons has to be higher than $1\%$ giving a direct lower
limit to the proportion of electrons that escape into the pulsar wind nebula
over the history of the pulsar. Even if the spectral index is not directly
constrained, a spectral index lower than $2$ implies the presence of a sub-PeV
energy cut-off in order to explain the combined H.E.S.S. and HAWC data.

Considerable systematic uncertainties are also inherent in this analysis, as
demonstrated by the different background estimation methods used. It was not
possible to measure the true extent of the $\gamma$-ray emission around the
Geminga pulsar, from the currently available H.E.S.S. data. Caution must
therefore be urged in the interpretation of the diffusion model fit in
particular: although the fit is performed simultaneously to both the radial
profile and spectrum, the radial extent of detected emission is limited by the
detector field of view. Given the uncertainty in where the level of background
emission is reached (and necessary assumptions made for background
normalisation), this does not enable us to make an absolute measurement of the
flux or an assumption-free measurement of the diffusion coefficient.

Our study of the X-ray upper limits in a $10^{\prime}$ region around the
pulsar allows us to conclude that the magnetic field, in a one diffusion zone
scenario, and assuming a constant magnetic field over the X-ray to
$\gamma$-ray scale, has to be lower than $1\,\mu G$ in the absence of a sub-
PeV energy cut-off. An energy cut-off lower than $75$ TeV is needed to account
for a magnetic field of $1\,\mu G$; the absence of such a cut-off would imply
a sub 1 $\mu G$ magnetic field leading to a tension with the typical value for
the normalisation of the magnetic field of the interstellar medium
($2.3\,\mu{\rm G}\lesssim B\lesssim 6.1\,\mu{\rm G}$ Delahaye et al. 2010).
This can be seen as another argument favouring the presence of a sub-PeV cut-
off in the pulsar injection. The scenarios where the magnetic field is $\sim
1\,\mu G$ and a strong cut-off is present cannot be disentangled from a
sub-$\mu G$ magnetic field, but an extension towards higher energies of the
current $\gamma$-ray observations would be able to distinguish these two
scenarios.

Several models have considered diffusion as the dominant transport mechanism
for trapped particles within PWNe (Tang & Chevalier, 2012; Porth et al.,
2016). However in older, more evolved systems the halo of energetic particles
probes properties of the ambient medium. Initially, it was surprising that
diffusion at larger distances away from the pulsar continued to be
considerably below the Galactic average value. If this value is representative
of properties of the intervening ISM, it would question the paradigm of
pulsars as the origin of the cosmic-ray positron excess. Several theories have
been proposed to reconcile these aspects, the most popular thus far being that
the region of slow diffusion is constrained to the region around the pulsar in
which accelerated electrons continue to have significant influence on the ISM
(Evoli et al., 2018; Fang et al., 2018, 2019; Profumo et al., 2018). At larger
distances, the lower particle energy densities reduce the influence, such that
the halo emission gradually decreases to join smoothly with the ISM.
Alternatively, it has been noted that although the implied diffusion
coefficient is lower than the Galactic average value, it remains consistent
with predictions of a spiral arm model with non-uniform diffusion coefficient
throughout the Galaxy (Tang & Piran, 2019).

In the future, observing these types of large extended sources will become
more feasible with the Cherenkov Telescope Array (CTA, the next-generation
IACT $\gamma$-ray observatory) due to the anticipated larger telescope field
of view, of up to $\sim 8-9^{\circ}$ towards the highest energies (Acharya et
al., 2013). Additionally, divergent pointing strategies are foreseen to cover
a larger region of the sky at a single observation instant, using the same
number of telescopes but trading an increased field of view for reduced
sensitivity (de Naurois, 2021; Donini et al., 2019). Whilst this is envisaged
in particular for observations of transient phenomena where the location may
be poorly constrained, there are obvious advantages of employing this mode to
observe extended ‘halo’ phenomena. With these approaches and a much larger
number of available telescopes, several of the challenges limiting the
capabilities of current IACT arrays such as H.E.S.S. to observe pulsar halos
and comparable phenomena at TeV energies will be overcome with CTA. Further
survey instruments are also foreseen, that may increase the number of
confirmed pulsar halos dramatically in the near future (López-Coto et al.,
2022).

##  7 Conclusions

With this study, H.E.S.S. confirms the presence of extended very-high-energy
$\gamma$-ray emission around the Geminga pulsar, on angular scales reaching at
least $3.2^{\circ}$ radius. Two different methods of background estimation
were used to evaluate the systematic uncertainties of the measurement; the
field-of-view background method and the On-Off background method, with two
independent lists of Off runs for the latter. No evidence for statistically
significant asymmetries to the emission or energy-dependent morphology is
found. Within a $1^{\circ}$ radius region around the pulsar, a spectral
analysis obtained a flux normalisation at 1 TeV of $(2.8\pm 0.7)\times
10^{-12}$ cm-2s-1TeV-1 with a best-fit power law spectral index of $2.76\pm
0.22$. We fitted an electron diffusion model jointly to the H.E.S.S. data
combined with HAWC and compare to XMM-Newton, taking the different spectral
extraction regions into account. Thanks to the few-arcmin angular resolution
of H.E.S.S., it was possible to extract a flux point for the innermost 10’
radius region around the pulsar, corresponding to the XMM-Newton upper limit.
Due to the large number of free variables in the fit, a parameter scan was
conducted to cycle over specific values for several quantities, whilst leaving
the diffusion coefficient normalisation free. For the best-fit model, a
normalisation of the diffusion coefficient of $D_{0}=7.6^{+1.5}_{-1.2}\times
10^{27}$ cm2s-1 is preferred at an electron energy of 100 TeV, as well as a
cut off energy of the electron spectrum $E_{c}=74^{+17}_{-11}$ TeV. This value
is considerably lower than the Galactic average, yet consistent with results
obtained by the HAWC collaboration (Abeysekara et al., 2017a). We find that
the mean statistical uncertainty on the diffusion coefficient obtained from
our model fit is 17% for the joint fit of H.E.S.S. and HAWC data, whereas it
is 27% for the fit to HAWC data only.

This is a challenging analysis: due to the $\gamma$-ray emission extending
beyond the available sky region, limitations apply such as a lower limit
rather than absolute measurement of the size, and a relative rather than
absolute flux measurement, likely underestimating the true $\gamma$-ray flux.
Despite these caveats, we anticipate that this detection and study of extended
$\gamma$-ray emission around the Geminga pulsar paves the way for further
detailed studies of pulsar halos with current and future generation IACT
facilities.

## Acknowledgements

The support of the Namibian authorities and of the University of Namibia in
facilitating the construction and operation of H.E.S.S. is gratefully
acknowledged, as is the support by the German Ministry for Education and
Research (BMBF), the Max Planck Society, the German Research Foundation (DFG),
the Helmholtz Association, the Alexander von Humboldt Foundation, the French
Ministry of Higher Education, Research and Innovation, the Centre National de
la Recherche Scientifique (CNRS/IN2P3 and CNRS/INSU), the Commissariat à
l’énergie atomique et aux énergies alternatives (CEA), the U.K. Science and
Technology Facilities Council (STFC), the Irish Research Council (IRC) and the
Science Foundation Ireland (SFI), the Knut and Alice Wallenberg Foundation,
the Polish Ministry of Education and Science, agreement no. 2021/WK/06, the
South African Department of Science and Technology and National Research
Foundation, the University of Namibia, the National Commission on Research,
Science & Technology of Namibia (NCRST), the Austrian Federal Ministry of
Education, Science and Research and the Austrian Science Fund (FWF), the
Australian Research Council (ARC), the Japan Society for the Promotion of
Science, the University of Amsterdam and the Science Committee of Armenia
grant 21AG-1C085. We appreciate the excellent work of the technical support
staff in Berlin, Zeuthen, Heidelberg, Palaiseau, Paris, Saclay, Tübingen and
in Namibia in the construction and operation of the equipment. This work
benefited from services provided by the H.E.S.S. Virtual Organisation,
supported by the national resource providers of the EGI Federation.

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##  Appendix A Diffusion model

###  A.1 Energy losses

The energy losses are assumed to be dominated by synchrotron losses and
Inverse Compton losses. For the latter, we use the model developed in Delahaye
et al. (2010) (equation 32), which approximates the intermediate regime
between Thomson and Klein-Nishina regimes. For the Inverse Compton losses, the
three thermal radiation fields considered have the following properties
(Moskalenko & Strong, 1998): Cosmic Microwave Background ($T=2.7$K,
$U_{rad}=0.26$ eV cm-3), Infra-Red ($T=20$K, $U_{rad}=0.3$ eV cm-3) and
Optical ($T=5000$K, $U_{rad}=0.3$ eV cm-3).

The total energy losses are computed as:

$b(E)=b_{\mathrm{sync}}(E)+b_{\mathrm{CMB}}(E)+b_{\mathrm{IR}}(E)+b_{\mathrm{Opt}}(E)\leavevmode\nobreak\
,$ (10)

including losses due to synchrotron $b_{\rm sync}$ and IC scattering on the
three aforementioned radiation fields. Due to the different temperatures and
energy densities, the transition between the Thomson and Klein-Nishina IC
scattering regimes occurs at different energies for the three radiation
fields.

With increasing magnetic field strength, the energy range for which
Synchrotron losses dominate over inverse Compton losses also increases, as
shown in Fig. 15.

###  A.2 Pulsar properties

The pulsar luminosity as a function of time is defined as:

$L(t)=L_{0}(1+t/\tau_{0})^{-(n+1)/(n-1)}\leavevmode\nobreak\ ,$ (11)

where $L_{0}$ is the initial luminosity of the pulsar, $\tau_{0}$ the initial
spin-down timescale, and $n$ is the braking index. The pulsar period as a
function of time is:

$P(t)=P_{0}(1+t/\tau_{0})^{1/(n-1)}\leavevmode\nobreak\ ,$ (12)

and the pulsar age $T_{*}$ is expressed as:

$T_{*}=T_{c}\frac{2}{n-1}(1-P_{0}/P_{*})^{n-1}\leavevmode\nobreak\ ,$ (13)

with $T_{c}$ the characteristic age and $P_{*}$ the actual pulsar period. It
is important to note here that $T_{*}\simeq T_{c}$ for $n=3$ and $P_{*}\gg
P_{0}$ if the pulsar is old enough to have had a significant increase of its
rotation period. Given these previous equations, we can retrieve $\tau_{0}$
solving the equations (12) and (13) with $P(T_{*})=P_{*}$ :

$\tau_{0}=\frac{T_{*}}{(P_{*}/P_{0})^{n-1}-1}\leavevmode\nobreak\ .$ (14)

Figure 15: Energy losses from IC and synchrotron processes for different
magnetic fields used in the diffusion model. IC is the total inverse Compton
losses over all three radiation fields considered.

##  Appendix B Results for the SED and profile fit

Table 7: Fit results on H.E.S.S. and HAWC SED and profiles of the scanned
parameter combinations with a p-value better than 3 standard deviations and
fulfiling the XMM-Newton constraints.

$n$ | $\eta$ | $\alpha$ | $E_{c}$ | $\delta$ | $D_{0}$ | p-value | Chi2/ndof
---|---|---|---|---|---|---|---
|  |  | TeV |  | $10^{27}{\rm cm}^{2}{\rm s}^{-1}$ |  |
$4.5$ | $0.1$ | $1.8$ | $74^{+17}_{-11}$ | $1.0$ | $7.6^{+1.5}_{-1.2}$ | $0.37$ | $1.07$
$4.5$ | $0.5$ | $2.0$ | $47^{+6}_{-5}$ | $0.3$ | $5.7^{+1.1}_{-0.9}$ | $0.35$ | $1.07$
$4.5$ | $0.5$ | $2.0$ | $53^{+9}_{-6}$ | $0.6$ | $10.0^{+1.9}_{-1.6}$ | $0.19$ | $1.21$
$3.0$ | $0.5$ | $2.0$ | $45^{+6}_{-5}$ | $0.3$ | $4.6^{+0.9}_{-0.8}$ | $0.17$ | $1.24$
$3.0$ | $0.5$ | $2.0$ | $49^{+7}_{-5}$ | $0.6$ | $7.3^{+1.4}_{-1.2}$ | $0.15$ | $1.26$
$1.5$ | $0.5$ | $2.0$ | $45^{+6}_{-5}$ | $0.3$ | $4.5^{+0.9}_{-0.8}$ | $0.14$ | $1.28$
$1.5$ | $0.5$ | $2.0$ | $49^{+7}_{-5}$ | $0.6$ | $7.1^{+1.3}_{-1.1}$ | $0.13$ | $1.28$
$3.0$ | $0.1$ | $1.8$ | $71^{+15}_{-10}$ | $1.0$ | $6.5^{+1.2}_{-1.0}$ | $0.13$ | $1.29$
$4.5$ | $0.5$ | $1.8$ | $27^{+3}_{-2}$ | $0.3$ | $12.1^{+2.0}_{-1.7}$ | $0.11$ | $1.32$
$3.0$ | $0.5$ | $1.8$ | $25^{+2}_{-2}$ | $0.3$ | $9.1^{+1.7}_{-1.4}$ | $0.08$ | $1.36$
$1.5$ | $0.5$ | $2.0$ | $57^{+10}_{-7}$ | $1.0$ | $14.0^{+2.5}_{-2.1}$ | $0.08$ | $1.37$
$3.0$ | $0.5$ | $2.0$ | $57^{+10}_{-7}$ | $1.0$ | $14.4^{+2.6}_{-2.2}$ | $0.08$ | $1.37$
$1.5$ | $0.5$ | $1.8$ | $26^{+2}_{-2}$ | $0.3$ | $8.9^{+1.7}_{-1.4}$ | $0.07$ | $1.39$
$3.0$ | $0.5$ | $1.8$ | $27^{+3}_{-2}$ | $0.6$ | $15.6^{+2.8}_{-2.3}$ | $0.05$ | $1.43$
$1.5$ | $0.5$ | $1.8$ | $27^{+3}_{-2}$ | $0.6$ | $15.1^{+2.7}_{-2.3}$ | $0.05$ | $1.45$
$4.5$ | $0.5$ | $2.0$ | $62^{+14}_{-10}$ | $1.0$ | $20.7^{+4.5}_{-3.7}$ | $0.04$ | $1.47$
$4.5$ | $0.5$ | $1.8$ | $30^{+3}_{-3}$ | $0.6$ | $23.9^{+3.9}_{-3.3}$ | $0.03$ | $1.52$
$1.5$ | $0.5$ | $1.8$ | $31^{+4}_{-3}$ | $1.0$ | $32.0^{+5.4}_{-4.6}$ | $0.02$ | $1.59$
$3.0$ | $0.5$ | $1.8$ | $31^{+4}_{-3}$ | $1.0$ | $33.5^{+5.7}_{-4.9}$ | $0.02$ | $1.59$
111111The magnetic field is set to $B=1\,\mu$G for all combinations listed.