# Searching for Synchrotron Emission from the Geminga TeV Halo using the
Planck Satellite

Dan Hooper  Elena Pinetti  and Anastasia Sokolenko

###### Abstract

Pulsars convert a significant fraction of their total spin-down power into
very high-energy electrons, leading to the formation of TeV halos. It is not
yet known, however, whether these sources also efficiently accelerate
electrons at lower energies and, if so, how those particles propagate through
the surrounding environment. If pulsars produce $\sim 50-300\,{\rm GeV}$
electrons, these particles would produce a spatially extended halo of
synchrotron emission in the frequency range measured by Planck. Such emission
could be used to constrain the low-energy diffusion coefficient in the regions
surrounding these pulsars, as well as the spectrum and intensity of the
electrons that are accelerated in this energy range. In this study, we attempt
to use Planck data to constrain the nature of the Geminga pulsar’s TeV halo.
We find no conclusive evidence of this emission in Planck’s frequency range,
however, and calculate that the synchrotron flux from Geminga should be well
below the total flux measured by Planck, even for models with favorable
diffusion parameters or soft injection spectra. At this time, these
measurements are not capable of significantly constraining the values of these
parameters.

##  1 Introduction

Observations by the HAWC, LHAASO, and HESS Collaborations have shown that
young and middle-aged pulsars produce large fluxes of spatially extended
emission at multi-TeV energies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The
measured intensity of this emission indicates that these pulsars convert a
significant fraction of their total spin-down power into very high-energy
electron-positron pairs, which produce the observed gamma rays through inverse
Compton scattering [13, 14, 15, 16, 17, 18, 19]. The resulting TeV halos
extend out to $\sim 25\,{\rm pc}$ in radius, indicating that cosmic rays
propagate much less efficiently in the vicinity of these sources than they do
elsewhere in the interstellar medium (ISM) [13, 20, 21, 22, 23, 24, 25, 26, 1,
12, 27, 28, 14, 29, 30].

There are many open questions regarding the physics of TeV halos. Perhaps most
notably, it is not yet clear why diffusion is so slow within the regions that
surround these sources [31, 32, 33, 34, 35, 36, 24, 25]. In a previous study,
we assessed the ability of the Cherenkov Telescope Array (CTA) to discriminate
between various models of TeV halos, focusing on the specific case of the
Geminga pulsar [37]. While observations of TeV halos at and above TeV-scale
energies can be quite informative, such measurements do not reveal much about
the electrons that these sources generate at lower energies, or how those
particles propagate through the surrounding volume of space.

In this paper, we consider the ability of the Planck satellite to detect and
measure the synchrotron emission from the GeV-TeV electrons that are expected
to be accelerated by TeV halos. For electrons with energies in the range of
$\sim 50-300\,{\rm GeV}$, the resulting synchrotron emission is predicted to
peak in the frequency range measured by Planck, providing us with a way of
constraining the production and diffusion of electrons in this energy range.
We focus in this study on the TeV halo associated with the Geminga pulsar,
finding no conclusive evidence of synchrotron emission from this source. This
information can be used to constrain the physics behind Geminga’s TeV halo.

##  2 Radio Synchrotron From TeV Halos

The spectrum of synchrotron emission from a relativistic electron features a
broad peak at approximately one third of the critical frequency [38]:

$\displaystyle\nu_{\rm peak}\sim\frac{\nu_{c}}{3}$
$\displaystyle=\frac{1}{2}\gamma^{2}_{e}\nu_{g}\sin\alpha_{p}$ (2.1)
$\displaystyle\approx 40\,{\rm GHz}\times\bigg{(}\frac{E_{e}}{60\,{\rm
GeV}}\bigg{)}^{2}\bigg{(}\frac{B}{3\,\mu{\rm
G}}\bigg{)}\bigg{(}\frac{\sin\alpha_{p}}{2/3}\bigg{)},$

where $\nu_{g}=eB/2\pi m_{e}$ is the non-relativistic gyrofrequency, and
$\alpha_{p}$ is the pitch angle. From this expression, we expect electrons
with energies in the range of $\sim 50-300\,{\rm GeV}$ to produce synchrotron
emission that peaks in the frequency range measured by Planck ($30-857\,{\rm
GHz}$).

In the Thomson regime ($E_{e}\ll m^{2}_{e}/\epsilon_{\gamma}$), the processes
of synchrotron and inverse Compton scattering cause electrons to lose energy
at the following rate [38]:

$\displaystyle\frac{dE_{e}}{dt}$
$\displaystyle\approx-\frac{4}{3}\sigma_{t}\rho_{\rm
mag}v_{e}^{2}\gamma^{2}_{e}-\frac{4}{3}\sigma_{t}\rho_{\rm
rad}v_{e}^{2}\gamma^{2}_{e}$ (2.2) $\displaystyle\approx-4.5\times
10^{-13}\,{\rm GeV/s}\times\bigg{(}\frac{E_{e}}{60\,{\rm
GeV}}\bigg{)}^{2}\,\Bigg{[}\bigg{(}\frac{B}{3\,\mu{\rm
G}}\bigg{)}^{2}+\bigg{(}\frac{\rho_{\rm rad}}{1\,{\rm
eV/cm}^{3}}\bigg{)}\Bigg{]},$

where $\sigma_{\rm t}$ is the Thomson cross-section, $\rho_{\rm
mag}=B^{2}/2\mu_{0}$ is the energy density of the magnetic field, and
$\rho_{\rm rad}$ is the energy density of the radiation field. From this
expression, we see that electrons with
$E_{e}\raisebox{-3.69899pt}{~{}\shortstack{$<$ \\\ [-0.07cm]
$\sim$}}~{}100\,{\rm GeV}$ lose less than 10% of their energy over the
lifetime of the Geminga pulsar ($t_{\rm Geminga}\approx 340\,{\rm kyr}$).

Over a time, $t$, an electron is typically displaced via diffusion by a
distance on the order of

$\displaystyle L$ $\displaystyle\sim 2\sqrt{Dt}\approx 40\,{\rm
pc}\times\bigg{(}\frac{D}{4\times 10^{26}\,{\rm cm}^{2}/{\rm
s}}\bigg{)}^{1/2}\bigg{(}\frac{t}{340\,{\rm kyr}}\bigg{)}^{1/2},$ (2.3)

where $D$ is the diffusion coefficient. In this expression, we have neglected
energy losses, as justified for the case of electrons with energies below
$\sim 100\,{\rm GeV}$.

The observed angular extent of TeV halos indicates that diffusion is highly
suppressed in the regions that surround these sources, in particular when
compared to that of the overall ISM [13, 15, 14, 29, 30]. In Ref. [37], we
found that data from the direction of Geminga, as collected by HAWC [1] and
HESS [39], could be well fit by a model with $D(E_{e})=10^{26}\,{\rm
cm}^{2}/{\rm s}\times(E_{e}/{\rm GeV})^{1/3}$, which is a factor of several
hundred times smaller than that of the ISM [40, 41].

From the above considerations, we expect any $\sim 10\,{\rm GeV}-{\rm TeV}$
electrons that are accelerated by young and middle aged pulsars to produce a
spatially extended flux of synchrotron emission at radio frequencies.
Measurements in this frequency range could thus be used to constrain the low-
energy diffusion coefficient in the regions surrounding these pulsars, as well
as the spectrum and intensity of the electrons that are accelerated in this
energy range.

##  3 Planck Data and TeV Halos

The Planck satellite has measured the sky in 9 frequency bands, ranging from
30 to 857 GHz [42, 43]. In this study, we use the raw data from the Planck
Public Data Release 3 in each of these frequency bands, and compare the flux
observed from the direction of the Geminga pulsar (PSR J0633+1746) to that
predicted by different models of its TeV halo.

Figure 1: The intensity and morphology of the synchrotron emission predicted
from the Geminga pulsar at four selected frequencies. Here we have adopted our
default model for Geminga, as described in Ref. [37]. The green arrow in each
frame indicates the proper motion of Geminga over its lifetime.

The Geminga pulsar is measured to have a proper motion of 178.2 mas/yr, which
over 340 kyr corresponds to a total displacement of $16.8^{\circ}$ [44]. We
assume that the rate of Geminga’s proper motion has been constant over its
lifetime and that the halo and pulsar have moved together over this period of
time. Given the fact that electrons in the energy range of interest do not
experience significant energy losses over the lifetime of Geminga, the signal
in the Planck frequency range is expected to extend from the location of
Geminga’s birth to the present location of this pulsar.

In Fig. 1, we plot the intensity and morphology of the synchrotron emission
predicted from the Geminga pulsar at four selected frequencies. In this
calculation, we have adopted our default model for Geminga, as described in
our previous study [37]. This model has been shown to provide a good fit to
the high-energy emission observed by HAWC [1] and HESS [39]. The green arrow
in each frame indicates the proper motion of Geminga over its lifetime. Note
that the brightest flux of synchrotron is not expected from Geminga’s present
location, but from the angular region surrounding its birth. This is due to
the fact that Geminga’s spindown power was much greater when it was young than
it is in the present era.

In Fig. 2, we show the raw flux observed by Planck in eight frequency bands in
the direction of Geminga. The center of these frames is chosen to be ${\rm
RA}=91.6^{\circ}$, ${\rm DEC}=12.8^{\circ}$, which is the midway point between
the birth and current position of Geminga [1, 12, 44]. These frames show no
obvious signs of emission from this source (although these maps do include
some emission that could plausibly originate from the region surrounding
Geminga’s birth, especially at the lowest measured frequencies). In light of
this information, we will proceed in the following section to use the lack of
synchrotron emission observed from Geminga to constrain our model of this
pulsar’s TeV halo.

Note that we have not considered the Monogem pulsar (PSR J0659+1414) in this
study, despite the fact that it is approximately half as bright as Geminga at
multi-TeV energies. The reason for this decision is that Monogem is much
younger than Geminga ($t_{\rm Monogem}\approx 110\,{\rm kyr}$), and thus the
total integrated energy of the emission from this source is likely to be much
lower than that of Geminga. Under the assumption of magnetic dipole breaking,
$\dot{E}(t)\propto[1+(t/\tau)]^{-2}$, and otherwise adopting the same
parameters for Geminga and Monogem, we estimate the following ratio for their
total time-integrated emission:

$\displaystyle\frac{E_{\rm Geminga}}{E_{\rm Monogem}}\approx\frac{F^{\rm
Geminga}_{\rm TeV}}{F^{\rm Monogem}_{\rm TeV}}\times\bigg{(}\frac{t_{\rm
Geminga}}{t_{\rm Monogem}}\bigg{)}^{2}\sim 20,$ (3.1)

where $F_{\rm TeV}^{\rm Geminga}$ and $F_{\rm TeV}^{\rm Monogem}$ denote the
current fluxes of very high-energy emission from these sources. In light of
this comparison, we expect Monogem to be $\sim 20$ times fainter than Geminga
in the frequency range measured by Planck.

##  4 Results

In this section, we use a two-zone model for Geminga’s TeV halo, with
parameters as described in Ref. [37], and use the lack of synchrotron emission
observed from the direction of this object to constrain the parameters of this
model. Throughout this study, we adopt a pulsar braking index of $n=3$
(corresponding to magnetic dipole braking), a spindown timescale of
$\tau=12\,{\rm kyr}$, and normalize the total emission to match the flux
observed by HAWC and HESS at multi-TeV energies. For the diffusion coefficient
within 30 pc of Geminga, we adopt the following broken power-law
parameterization:

$D(E_{e})=\Biggl{\\{}\begin{array}[]{lll}D_{0}\left(E_{e}/{1\,{\rm
TeV}}\right)^{\delta_{1}}&,\;\;E_{e}>1\,{\rm TeV}\\\\[3.0pt]
D_{0}\left(E_{e}/{1\,\rm\text{TeV}}\right)^{\delta_{2}}&,\;\;E_{e}<1\,\text{TeV},\end{array}$
(4.1)

where $D_{0}=10^{27}\,{\rm cm}^{2}/{\rm s}$ and $\delta_{1}=1/3$ are chosen to
accommodate the spectrum observed at multi-TeV energies, as described in Ref.
[37] (see also, Refs. [13, 15, 14, 29, 30]). We treat $\delta_{2}$ as a free
parameter which characterizes the evolution of the diffusion coefficient at
sub-TeV energies. Beyond 30 pc from the pulsar, we adopt $D_{0}=4\times
10^{29}\,{\rm cm}^{2}/{\rm s}$ and $\delta_{1}=1/3$, as inferred from
measurements of the cosmic-ray boron-to-carbon ratio (and other secondary-to-
primary ratios) [40, 41]. Note that 30 pc is comparable to the diffusion
length of $\sim 10-100\,{\rm GeV}$ electrons in the TeV halo, as integrated
over the lifetime of Geminga. For this reason, we do not expect our results to
change dramatically if the radius of the halo were larger than we have
assumed.

In addition to variations in the value of $\delta_{2}$, we also consider
different values for the spectral index of the injected electrons. Much like
the diffusion coefficient, HAWC and HESS data constrain the value of this
parameter at multi-TeV energies, but leave it largely unconstrained at lower
energies, motivating a parameterization in the form of a broken power-law:

$\frac{dN_{e}}{dE_{e}}=\Biggl{\\{}\begin{array}[]{lll}A\left(E_{e}/{1\,{\rm
TeV}}\right)^{-\alpha_{1}}&,\;\;E_{e}>1\,{\rm TeV}\\\\[3.0pt]
A\left(E_{e}/{1\,\rm\text{TeV}}\right)^{-\alpha_{2}}&,\;\;E_{e}<1\,\text{TeV},\end{array}$
(4.2)

where we adopt $\alpha_{1}=1.8$, set $A$ to match the intensity observed by
HAWC [37], and take $\alpha_{2}$ to be a free parameter.

Our main results are shown in Fig. 3, where we compare the synchrotron
spectrum predicted by our model to the measurements of Planck. These results
are shown as integrated over two different regions: a $20^{\circ}$ radius
circle centered on the point ${\rm RA}=91.6^{\circ}$, ${\rm DEC}=12.8^{\circ}$
(left frame) and an $8^{\circ}$ radius circle centered on ${\rm
RA}=85.1^{\circ}$, ${\rm DEC}=8.2^{\circ}$, corresponding to Geminga
birthplace where we expect the brightest synchrotron flux (right frame). For
each of the model parameters we have considered (corresponding to different
values of $\alpha_{2}$ and $\delta_{2}$), the predicted synchrotron flux is
well below the total flux measured by Planck in each of their 9 frequency
bands. These measurements are thus not capable of significantly constraining
the values of these parameters at this time.

Note that we have not attempted to model or subtract any backgrounds from the
Planck data. Such an analysis could potentially provide a stronger constraint
on the model of Geminga’s TeV halo.

Figure 2: The emission measured by the Planck satellite from the direction of
the Geminga pulsar. The arrows indicate the proper motion of Geminga over its
lifetime. These frames show no obvious signs of emission from this source. The
green arrow shows the proper motion of Geminga.

Figure 3: A comparison of the synchrotron spectrum predicted by our model of
Geminga’s TeV halo (lines) to the measurements of Planck (black points). For
each of the model parameters we considered in this study, the predicted
synchrotron flux is well below the total flux measured by Planck. The fluxes
shown are integrated over a $20^{\circ}$ radius circle centered on the point
${\rm RA}=91.6^{\circ}$, ${\rm DEC}=12.8^{\circ}$ (left frame) and an
$8^{\circ}$ radius circle centered on ${\rm RA}=85.1^{\circ}$, ${\rm
DEC}=8.2^{\circ}$ (right frame).

##  5 Summary and Conclusions

Measurements by HAWC, LHAASO, and HESS have shown that young and middle-aged
pulsars convert a significant fraction of their total spin-down power into
very high-energy electrons, leading to the formation of TeV halos. Although
observations of pulsars at very high-energies can reveal a great deal about
the nature of these objects, such measurements do not reveal much about the
electrons that these sources might produce at lower energies, or how those
particles propagate through the surrounding volume of space.

If pulsars efficiently accelerate electrons in $\sim 50-300\,{\rm GeV}$ energy
range, these particles would produce a spectrum of spatially extended
synchrotron emission that peaks in the frequency range measured by Planck.
Such emission could be used to constrain the low-energy diffusion coefficient
in the regions surrounding these pulsars, as well as the spectrum and
intensity of the electrons that are accelerated in this energy range.

In this study, we have attempted to use Planck data to constrain the nature of
Geminga’s TeV halo. We find no conclusive evidence of emission in Planck’s
frequency range from this source, however, and calculate that the synchrotron
flux from Geminga should be well below the total flux measured by Planck, even
for models with favorable diffusion parameters or soft injection spectra.
These measurements are thus not capable of significantly constraining the
values of these parameters at this time.

In a spirit similar to that of the work presented here, the authors of Ref.
[45] have used X-ray measurements of Geminga to place an upper limit on the
synchrotron emission from ultra-high-energy electrons around this source,
reaching the conclusion that the magnetic field within Geminga’s halo must be
weaker than $\sim 2\mu{\rm G}$. Data from the Fermi gamma-ray space telescope
has also been used to search for the inverse Compton emission from GeV-TeV
scale electrons within Geminga’s halo [46].

## Acknowledgements

We would like to thank Tom Crawford, Damiano Caprioli, and Benedikt Schroer
for helpful discussions. The authors acknowledge support from the Fermi
Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S.
Department of Energy, Office of High Energy Physics.

## References

  * [1] HAWC Collaboration, A. U. Abeysekara et al., Extended gamma-ray sources around pulsars constrain the origin of the positron flux at Earth, Science 358 (2017), no. 6365 911–914, [arXiv:1711.06223].
  * [2] T. Linden, K. Auchettl, J. Bramante, I. Cholis, K. Fang, D. Hooper, T. Karwal, and S. W. Li, Using HAWC to discover invisible pulsars, Phys. Rev. D 96 (2017), no. 10 103016, [arXiv:1703.09704].
  * [3] T. Sudoh, T. Linden, and J. F. Beacom, TeV Halos are Everywhere: Prospects for New Discoveries, Phys. Rev. D 100 (2019), no. 4 043016, [arXiv:1902.08203].
  * [4] T. Sudoh, T. Linden, and D. Hooper, The Highest Energy HAWC Sources are Likely Leptonic and Powered by Pulsars, JCAP 08 (2021) 010, [arXiv:2101.11026].
  * [5] HAWC Collaboration, A. Albert et al., 3HWC: The Third HAWC Catalog of Very-High-Energy Gamma-ray Sources, Astrophys. J. 905 (2020), no. 1 76, [arXiv:2007.08582].
  * [6] A. Smith, A Systematic Search for TeV Halos associated with known pulsars, in 36th International Cosmic Ray Conference (ICRC2019), vol. 36 of International Cosmic Ray Conference, p. 797, July, 2019.
  * [7] HESS Collaboration, H. Abdalla et al., The H.E.S.S. Galactic plane survey, Astron. Astrophys. 612 (2018) A1, [arXiv:1804.02432].
  * [8] HESS Collaboration, H. Abdalla et al., The population of TeV pulsar wind nebulae in the H.E.S.S. Galactic Plane Survey, Astron. Astrophys. 612 (2018) A2, [arXiv:1702.08280].
  * [9] LHAASO Collaboration, Z. Cao et al., The First LHAASO Catalog of Gamma-Ray Sources, [arXiv:2305.17030].
  * [10] LHAASO Collaboration, F. Aharonian et al., Extended Very-High-Energy Gamma-Ray Emission Surrounding PSR J0622+3749 Observed by LHAASO-KM2A, Phys. Rev. Lett. 126 (2021), no. 24 241103, [arXiv:2106.09396].
  * [11] K. Fang, S.-Q. Xi, and X.-J. Bi, Self-consistent interpretations of the multiwavelength gamma-ray spectrum of LHAASO J0621+3755, Phys. Rev. D 104 (2021), no. 10 103024, [arXiv:2107.02140].
  * [12] A. U. Abeysekara et al., The 2HWC HAWC Observatory Gamma Ray Catalog, Astrophys. J. 843 (2017), no. 1 40, [arXiv:1702.02992].
  * [13] D. Hooper, I. Cholis, T. Linden, and K. Fang, HAWC observations strongly favor pulsar interpretations of the cosmic-ray positron excess, Physical Review D 96 (nov, 2017).
  * [14] S. Profumo, J. Reynoso-Cordova, N. Kaaz, and M. Silverman, Lessons from HAWC pulsar wind nebulae observations: The diffusion constant is not a constant; pulsars remain the likeliest sources of the anomalous positron fraction; cosmic rays are trapped for long periods of time in pockets of inefficient diffusion, Physical Review D 97 (jun, 2018).
  * [15] K. Fang, X.-J. Bi, P.-F. Yin, and Q. Yuan, Two-zone diffusion of electrons and positrons from geminga explains the positron anomaly, The Astrophysical Journal 863 (aug, 2018) 30.
  * [16] M. Di Mauro, F. Donato, and S. Manconi, Novel interpretation of the latest AMS-02 cosmic-ray electron spectrum, Phys. Rev. D 104 (2021), no. 8 083012, [arXiv:2010.13825].
  * [17] C. Evoli, E. Amato, P. Blasi, and R. Aloisio, Galactic factories of cosmic-ray electrons and positrons, Phys. Rev. D 103 (2021), no. 8 083010, [arXiv:2010.11955].
  * [18] S. Manconi, M. Di Mauro, and F. Donato, Contribution of pulsars to cosmic-ray positrons in light of recent observation of inverse-Compton halos, Phys. Rev. D 102 (2020), no. 2 023015, [arXiv:2001.09985].
  * [19] L. Orusa, S. Manconi, F. Donato, and M. Di Mauro, Constraining positron emission from pulsar populations with AMS-02 data, JCAP 12 (2021), no. 12 014, [arXiv:2107.06300].
  * [20] R. N. Manchester, G. B. Hobbs, A. Teoh, and M. Hobbs, VizieR Online Data Catalog: ATNF Pulsar Catalog (Manchester+, 2005), VizieR Online Data Catalog (Aug., 2005) VII/245.
  * [21] R. López-Coto and G. Giacinti, Constraining the properties of the magnetic turbulence in the Geminga region using HAWC $\gamma$-ray data, Mon. Not. Roy. Astron. Soc. 479 (2018), no. 4 4526–4534, [arXiv:1712.04373].
  * [22] G. Johannesson, T. A. Porter, and I. V. Moskalenko, Cosmic-Ray Propagation in Light of the Recent Observation of Geminga, Astrophys. J. 879 (2019), no. 2 91, [arXiv:1903.05509].
  * [23] M. Di Mauro, S. Manconi, and F. Donato, Evidences of low-diffusion bubbles around Galactic pulsars, Phys. Rev. D 101 (2020), no. 10 103035, [arXiv:1908.03216].
  * [24] C. Evoli, T. Linden, and G. Morlino, Self-generated cosmic-ray confinement in TeV halos: Implications for TeV $\gamma$-ray emission and the positron excess, Phys. Rev. D 98 (2018), no. 6 063017, [arXiv:1807.09263].
  * [25] K. Fang, X.-J. Bi, and P.-F. Yin, Possible origin of the slow-diffusion region around Geminga, Mon. Not. Roy. Astron. Soc. 488 (2019), no. 3 4074–4080, [arXiv:1903.06421].
  * [26] D. Hooper and T. Linden, Evidence of TeV halos around millisecond pulsars, Phys. Rev. D 105 (2022), no. 10 103013, [arXiv:2104.00014].
  * [27] R. Kappl, A. Reinert, and M. W. Winkler, AMS-02 Antiprotons Reloaded, JCAP 10 (2015) 034, [arXiv:1506.04145].
  * [28] Y. Génolini et al., Cosmic-ray transport from AMS-02 boron to carbon ratio data: Benchmark models and interpretation, Phys. Rev. D 99 (2019), no. 12 123028, [arXiv:1904.08917].
  * [29] X. Tang and T. Piran, Positron flux and $\gamma$-ray emission from geminga pulsar and pulsar wind nebula, Monthly Notices of the Royal Astronomical Society 484 (jan, 2019) 3491–3501.
  * [30] G. Jó hannesson, T. A. Porter, and I. V. Moskalenko, Cosmic-ray propagation in light of the recent observation of geminga, The Astrophysical Journal 879 (jul, 2019) 91.
  * [31] B. Schroer, C. Evoli, and P. Blasi, TeV halos and the role of pulsar wind nebulae as sources of cosmic-ray positrons, Phys. Rev. D 107 (2023), no. 12 123020, [arXiv:2305.08019].
  * [32] P. Mukhopadhyay and T. Linden, Self-generated cosmic-ray turbulence can explain the morphology of TeV halos, Phys. Rev. D 105 (2022), no. 12 123008, [arXiv:2111.01143].
  * [33] P. De La Torre Luque, O. Fornieri, and T. Linden, Anisotropic diffusion cannot explain TeV halo observations, Phys. Rev. D 106 (2022), no. 12 123033, [arXiv:2205.08544].
  * [34] R.-Y. Liu, H. Yan, and H. Zhang, Understanding the Multiwavelength Observation of Geminga’s Tev Halo: The Role of Anisotropic Diffusion of Particles, Phys. Rev. Lett. 123 (2019), no. 22 221103, [arXiv:1904.11536].
  * [35] S. Recchia, M. Di Mauro, F. A. Aharonian, L. Orusa, F. Donato, S. Gabici, and S. Manconi, Do the Geminga, Monogem and PSR J0622+3749 $\gamma$-ray halos imply slow diffusion around pulsars?, Phys. Rev. D 104 (2021), no. 12 123017, [arXiv:2106.02275].
  * [36] L.-Z. Bao, K. Fang, X.-J. Bi, and S.-H. Wang, Slow Diffusion is Necessary to Explain the $\gamma$-Ray Pulsar Halos, Astrophys. J. 936 (2022), no. 2 183, [arXiv:2107.07395].
  * [37] D. Hooper, E. Pinetti, and A. Sokolenko, Unraveling TeV Halos with the Cherenkov Telescope Array, [arXiv:2312.10232].
  * [38] G. R. Blumenthal and R. J. Gould, Bremsstrahlung, synchrotron radiation, and compton scattering of high-energy electrons traversing dilute gases, Rev. Mod. Phys. 42 (1970) 237–270.
  * [39] H.E.S.S. Collaboration, F. Aharonian et al., Detection of extended gamma-ray emission around the Geminga pulsar with H.E.S.S, Astron. Astrophys. 673 (2023) A148, [arXiv:2304.02631].
  * [40] A. W. Strong, I. V. Moskalenko, and V. S. Ptuskin, Cosmic-ray propagation and interactions in the Galaxy, Ann. Rev. Nucl. Part. Sci. 57 (2007) 285–327, [astro-ph/0701517].
  * [41] R. Trotta, G. Jóhannesson, I. V. Moskalenko, T. A. Porter, R. R. d. Austri, and A. W. Strong, Constraints on cosmic-ray propagation models from a global Bayesian analysis, Astrophys. J. 729 (2011) 106, [arXiv:1011.0037].
  * [42] Planck Collaboration, Y. Akrami et al., Planck 2018 results. II. Low Frequency Instrument data processing, Astron. Astrophys. 641 (2020) A2, [arXiv:1807.06206].
  * [43] Planck Collaboration, N. Aghanim et al., Planck 2018 results. III. High Frequency Instrument data processing and frequency maps, Astron. Astrophys. 641 (2020) A3, [arXiv:1807.06207].
  * [44] G. Hobbs, D. R. Lorimer, A. G. Lyne, and M. Kramer, A Statistical study of 233 pulsar proper motions, Mon. Not. Roy. Astron. Soc. 360 (2005) 974–992, [astro-ph/0504584].
  * [45] S. Manconi, J. Woo, R.-Y. Shang, R. Krivonos, C. Tang, M. Di Mauro, F. Donato, K. Mori, and C. J. Hailey, Geminga’s pulsar halo: an X-ray view, [arXiv:2403.10902].
  * [46] M. Di Mauro, S. Manconi, and F. Donato, Detection of a $\gamma$-ray halo around Geminga with the Fermi -LAT data and implications for the positron flux, Phys. Rev. D 100 (2019), no. 12 123015, [arXiv:1903.05647]. [Erratum: Phys.Rev.D 104, 089903 (2021)].