0.26 to 0.58. All of these compositional intervals are at 68% confidence.

  * •

The lack of CH4 in the dayside spectrum is expected from the retrieved
temperature-pressure profiles, even in chemical equilibrium without requiring
chemical quenching because the temperatures are high.

  * •

The inferred heat redistribution efficiency factor for the planet is $f\gtrsim
0.7$ where 0.5 is full redistribution of heat spherically around the planet
and 1.0 is redistribution to the dayside hemisphere only. Equivalently
$\varepsilon<0.76$ in the parameterization of Cowan & Agol (2011). This is
true for the cooler temperature Scattering model given its high implied Bond
Albedo and the higher temperature Two-Region and Cloud Layer models, which
rule out full redistribution of heat at all Bond Albedos.

  * •

The Two Region retrieval and Scattering Model retrieval suggest there is
inhomogeneity in the dayside and that the emission is dominated by a warm
component. This, combined with the inefficient heat redistribution factor and
tentative eclipse mapping signal all point to a planet that has a hot spot at
the substellar point.

  * •

The lack of a temperature inversion rules out absorptive Titan-like hazes that
could warm upper altitudes in the planet’s atmosphere

Together, we find multiple different models and retrievals give evidence that
aerosols are responsible for shaping the spectrum of WASP-69 b. Either these
aerosols are highly reflective, as implied in the Scattering Model, or they
shape the MIRI wavelenghts from 5 µm to 12 µm as compared to NIRCam short
wavelengths. The Scattering Model’s reflective scenario requires an extremely
high Geometric Albedo of 0.64 or higher (assumed to be constant with
wavelength), which is unlikely. The high geometric albedo (equivalent to an 80
ppm eclipse) can be tested with short wavelength eclipse observations such as
CHEOPS, HST WFC3 UVIS or JWST NIRISS SOSS. The TESS observations favor low
geometric Albedos but the eclipse depth uncertainty $\sim$ 50 ppm does not
significantly constrain the inferred Geometric Albedo, whereas the CHEOPS
archive includes 19 eclipses with a larger aperture telescope. Aerosols are
also needed in our favored models: the Cloud Layer model, which includes high
altitude silicates and the Two Region Model, which includes a gray scattering
opacity that mutes some features in the hot region and drastically decreases
the emission from the cold region. While silicates like the MgSiO3 (enstatite)
composition assumed in the Cloud Layer model have the right kind of optical
properties to explain most of the spectrum of WASP-69 b, they condense at far
deeper pressures (10 bar) compared to their inferred high altitudes
$10^{-4.5}$ to $10^{-6}$ bar. Ongoing searches for particles that form
efficiently at 800-1000 K temperatures in a metal-enriched atmosphere and also
change in single scattering albedo near 9.4 µm are encouraged.

Several lines of evidence point to a hot spot near WASP-69 b’s substellar
point. First, the Two-Region model and Scattering models show evidence of an
inhomogeneous dayside that is dominated by warm emission. Second, eclipse map
from the MIRI LRS broadband light tentatively favors a central concentration
in brightness over a uniform dayside. Third, the inferred heat redistribution
efficiency of all of our retrievals rules out a full global redistribution of
heat. Inefficient heat redistribution is also seen in General Circulation
Models of exoplanets that include the feedback effects of clouds (e.g. Roman
et al., 2021; Bell et al., 2024).

WASP-69 b is at an interesting temperature (Teq = 963 K at zero albedo and
full redistribution) where the chemical transition from a CO-dominated to CH4
dominated atmosphere is expected to occur. Probes of the terminator and
nightside of WASP-69 b, with its inferred large day-to-night temperature
contrast will constrain the physics of atmospheres across these temperature
boundaries on the same planet. We note that our retrieved C/O ratio from the
Cloud Layer and Two Region models of 0.65 to 0.94 as well as our inferred
temperature gradients on the planet imply that there is plenty of Carbon and
cool enough temperatures for a large CH4 signature in WASP-69 b’s transmission
spectrum. CH4 has been already detected in high resolution cross correlation
(Guilluy et al., 2022) and will be further constrained in JWST GO programs
3712 and 5924. 800-1000 K is also a temperature range where specific aerosols
may form in planet atmospheres. We find that evidence for aerosols in WASP-69
b and comparisons with planets both warmer and cooler will help illuminate
what kind of chemical and aerosol transitions occur generally in this
temperature regime.

Funding for E Schlawin is provided by NASA Goddard Spaceflight Center. The
authors are grateful for support from NASA through the JWST NIRCam project
though contract number NAS5-02105. T.J.B. and T.P.G. acknowledge funding
support from the NASA Next Generation Space Telescope Flight Investigations
program (now JWST) via WBS 411672.07.05.05.03.02. This research has made use
of the SIMBAD database, operated at CDS, Strasbourg, France; and NASA’s
Astrophysics Data System Bibliographic Services. We respectfully acknowledge
the University of Arizona is on the land and territories of Indigenous
peoples. Today, Arizona is home to 22 federally recognized tribes, with Tucson
being home to the O’odham and the Yaqui. Committed to diversity and inclusion,
the University strives to build sustainable relationships with sovereign
Native Nations and Indigenous communities through education offerings,
partnerships, and community service. This work benefited from the 2023
Exoplanet Summer Program in the Other Worlds Laboratory (OWL) at the
University of California, Santa Cruz, a program funded by the Heising-Simons
Foundation and NASA. Thanks to Jeroen Bouwman for some tips on analyzing MIRI
LRS data. This publication makes use of data products from the Wide-field
Infrared Survey Explorer, which is a joint project of the University of
California, Los Angeles, and the Jet Propulsion Laboratory/California
Institute of Technology, funded by the National Aeronautics and Space
Administration. Numerical computations were in part carried out on PC cluster
at Center for Computational Astrophysics, National Astronomical Observatory of
Japan. The JWST data presented in this article were obtained from the Mikulski
Archive for Space Telescopes (MAST) at the Space Telescope Science Institute.
The specific observations analyzed can be accessed via
https://doi.org/10.17909/v2v9-k243 (catalog DOI: 10.17909/v2v9-k243).

##  Appendix A Illustrative Forward Models

Given the many parameters and model assumptions in this work, described in
Section 5, it is illustrative to show forward model spectra and residuals for
representative models to further understand which parts of the spectrum drive
our atmospheric inferences. We first begin with the fiducial Two-Region model
described in Bullet point 3, Section 5.2.1 and Section 5.3.1. The key
atmospheric abundance parameters from which we report results of the paper are
the atmospheric metal enrichment and the ratio of Carbon-to-Oxygen. These
abundance values allow comparison to Solar System planets as well as planet
formation and evolution models (e.g. Mordasini et al., 2016). As see in Figure
14, this model’s metallicity and C/O are largely constrained by the strength
of the H2O, CO2 and CO absorption, as well as the lack of a strong CH4
feature.

Figure 14: The abundance of WASP-69 b is constrained largely by the H2O
feature at 2.8 µm, the lack of a CH4 feature at 3.3 µm, the CO2 feature at 4.3
µm and the CO feature at 4.5 to 5.0 µm. Upper Left: Three forward models are
shown for 3 different atmospheric enrichments of [M/H]=1.5, [M/H]=1.01 (the
best-fit value) and [M/H]. In these three examples, the relative abundances of
heavy elements, including the C/O ratio is held constant. Lower Left: The
residuals of the models at three different atmospheric enrichments shows the
most significant offsets are due to H2O near 2.8 µm and CO2 at 4.3 µm. Upper
Right: Three forward models are shown for 3 different C/O ratios of 1.0, 0.79
(best-fit value) and 0.3. These three models preserve the overall abundance of
heavy elements. At high C/O ratios, CH4 is expected to be a significant
absorber and causes a pronounced 3.3 µm feature, not observed in WASP-69 b.
Lower Right: High C/O ratios are ruled out because there is no strong 3.3 µm
CH4 feature and low C/O ratios are disfavored because they have large
abundances of H2O and make a deeper 2.8 µm feature as well as shrink the size
of the 4.3 µm CO2 absorption trough.

We also explored whether Sulfur species might be constrained in WASP-69 b and
could improve the fit of the One-Region model described in Section 5.1 without
invoking both clouds and an inhomogeneous dayside. While our retrievals did
not include the Sulfur-to-Hydrogen ratio as a free parameter, we consider the
effects of varying two significant Sulfur-bearing species: H2S and SO2. We
find that increased or reduced H2S abundance cannot significantly improve the
1-region model fit. We find that non-equilibrium SO2, such as produced by
photochemistry (Tsai et al., 2022), can improve the fit around 7.7 µm.
However, it does not significnaly improve the generally high NIRCam flux at
3.5 µm and generally low MIRI flux from 5 to 9 µm.

Figure 15:  Left: We experimented whether enhanced or decreased H2S could help
fit the data better without requiring a temperature gradients on the Dayside.
We show the Single-Region model with equilibrium H2S for reference (black) as
well as highly enhanced and highly diluted abundances of H2S, neither of which
can significantly improve the model as compared to the data. Right: Using the
same single region forward model, we also vary the SO2 abundance to explore
another Sulfur-bearing molecule. While extra SO2 can help fit the data from
7.3 to 7.7 µm, it cannot raise the 2.8 µm flux and lower the 6 to 9 µm flux to
match the data as well as the Two-Region model.

##  Appendix B Posterior Distributions

For completeness, we also include the full stair-step 2D representations of
the Scattering model (Bullet Item 4, Sections 5.2.2 and 5.3.2) and Cloud Layer
model (Bullet item 5, Sections 5.2.3 and 5.3.3). The Stair-Step plot of the
Scattering Model is Shown In Figure 16. The Stair-Step plot of the Cloud Layer
Model is Shown In Figure 17.

Figure 16:  Two-dimensional posterior distributions of the Scattering model
(Bullet Item 4, Sections 5.2.2 and 5.3.2) . The parameters are Temperature-
Pressure Profile parameters: $T_{eq}$ (not necessarily consistent with
planet’s equilibrium temperature), $T_{int}$, $log(g)$, $\kappa_{IR}$ and
$\alpha$, the Carbon to Oxygen ratio C/O, the atmospheric metallicity [M/H],
the free SO2 abundance SO2, the constant geometric albedo $A_{g}$ and the area
fraction of the model emission compared to the full planet disk $s_{dilute}$.
Figure 17:  Two-dimensional posterior distributions of the Cloud Layer model
(Bullet item 5, Sections 5.2.3 and 5.3.3). The parameters are Temperature-
Pressure Profile parameters: $T_{eq}$ (not necessarily consistent with
planet’s equilibrium temperature), $T_{int}$, $log(g)$, $\kappa_{IR}$ and
$\alpha$, the Carbon-to-Oxygen ratio C/O, the atmospheric metallicity [M/H],
the free SO2 abundance SO2, the area fraction of the model emission compared
to the full planet disk $s_{dilute}$, and the cloud parameterization
parameters: the logarithm of the geometric mean radius log(rad), the geometric
standard deviation of radius $\sigma$, the logarithm of the cloud base
pressure in bars log(cld_pres), the log of the sedimentation efficiency
log(${f_{\rm sed}}$) and the normalization factor $ndz$.

##  Appendix C Extra Analysis Details

All NIRCam grism time series in the MANATEE program show a gradual decrease in
flux over the duration of the exposure. While the exact mechanism of this
decline is not fully understood, we have identified an anti-correlation with
detector housing temperature, as measured by the IGDP_NRC_A_T_LWFPAH1,
available in the MAST engineering database. Figure 18 shows the change in
temperature versus the change in flux for out-of-transit and out-of-eclipse
data in the MANATEE program 1185. We note that many different observations of
different brightnesses show this anti-correlation and therefore use the focal
plane housing temperature as a de-trending vector. Another specific data
reduction visual is the aperture used for photometric extraction of the short
wavelength data, which is shown in Figure 19.

Figure 18: The Detector Housing Temperature is anti-correlated with flux
across many different NIRCam observations, so we use it as means to de-trend
the data. Here, the X axis shows the change in the wavelength-integrated flux
from the median in parts per million for all out-of-transit and out-of-eclipse
integrations for 6 different example observations in the MANATEE program
number 1185. The Y axis shows the change in Focal Plane Array Housing (FPAH)
as measured by the telemetry keyword IGDP_NRC_A_T_LWFPAH1. Each observation is
listed with the program number, observation number and which long wavelength
filter was used respectively. The observations describe in this paper are 006
and 007, as listed in Table 1.  Figure 19: Extraction Aperture. The inner and
outer circles are the background annulus radii and the source aperture is the
same as the inner background annulus radius

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11institutetext: Max-Planck-Institut für Kernphysik, Postfach 103980, D 69029
Heidelberg, Germany

# Probing cosmic ray escape from $\eta$ Carinae

S. Steinmassl 11 M. Breuhaus 11 R. White 11 B. Reville 11 J.A. Hinton 11

(Accepted 2023-09-06. Received 2023-xx-xx; in original form 2023-xx-xx)

The binary stellar system $\eta$ Carinae is one of very few established
astrophysical hadron accelerators. It seems likely that at least some fraction
of the accelerated particles escape from the system. Copious target material
for hadronic interactions and associated $\gamma$-ray emission exists on a
wide range of spatial scales outside the binary system. This material creates
a unique opportunity to trace the propagation of particles into the
interstellar medium. Here we analyse $\gamma$-ray data from _Fermi_ -LAT of
$\eta$ Carinae and surrounding molecular clouds and investigate the many
different scales on which escaping particles may interact and produce
$\gamma$-rays. We find that interactions of escaping cosmic rays from $\eta$
Carinae in the wind region and the Homunculus Nebula could produce a
significant contribution to the $\gamma$-ray emission associated with the
system. Furthermore, we detect excess emission from the surrounding molecular
clouds. The derived radial cosmic-ray excess profile is consistent with a
steady injection of cosmic rays by a central source. However, this would
require a higher flux of escaping cosmic rays from $\eta$ Carinae than
provided by our model. Therefore it is likely that additional cosmic ray
sources contribute to the hadronic $\gamma$-ray emission from the clouds.

###### Key Words.:

radiation mechanisms: non-thermal, gamma rays: stars, stars: winds, outflows,
stars: individual: $\eta$ Carinae, (ISM:)cosmic rays

##  1 Introduction

Colliding wind binaries (CWBs) are a well-established class of Galactic
particle accelerator, with non-thermal emission seen from a number of systems
(Moran et al., 1989; Churchwell et al., 1992; Contreras & Rodríguez, 1999;
Dougherty & Williams, 2000; Marcote et al., 2021; del Palacio et al., 2023).
In $\gamma$-rays, only two systems have been detected so far: $\eta$ Carinae
(e.g. Abdo et al., 2010a) and $\gamma^{2}$ Velorum (Martí-Devesa et al.,
2020), located $(2.35\pm 0.05)$ kpc (Smith, 2006) and $(0.34\pm 0.01)$ kpc
(North et al., 2007) from Earth, respectively. $\eta$ Carinae (henceforth
$\eta$ Car) is the more luminous of these two systems and has therefore been
studied in greater detail. Over its 15-year lifetime, the Fermi Large Area
Telescope (LAT) has probed three periastron passages of this highly eccentric
($\epsilon\approx 0.9$, Mehner et al., 2015) CWB with an orbital period of
$\sim$5.5 yr (e.g. Damineli et al., 2008; Teodoro et al., 2016). The high
eccentricity and relatively short orbital period, combined with its dense
matter and photon fields make $\eta$ Carinae a unique source in which to study
high-energy astrophysical processes in a time-dependent environment.

The Astro-Rivelatore Gamma a Immagini Leggero (AGILE) satellite was used to
detect $\eta$ Car for the first time in $\gamma$-rays (Tavani et al., 2009).
The detection was soon confirmed by observations with the _Fermi_ -LAT
satellite (Abdo et al., 2010a; Nolan et al., 2012). Since then, several works
have analysed the $\gamma$-ray emission measured by _Fermi_ -LAT (Abdo et al.,
2010b; Farnier et al., 2011; Reitberger et al., 2012, 2015; Balbo & Walter,
2017; White et al., 2020; Martí-Devesa & Reimer, 2021; Abdollahi et al.,
2022a). Two components are apparent in the emitted spectrum above and below
$\sim 10\text{\,}\mathrm{GeV}$ (see e.g. Abdo et al., 2010b; Farnier et al.,
2011). During the periastron phase when the stars are closest to each other,
the emission increases. The peak emission at $\mathrm{GeV}$ energies around
periastron increases by $\sim 40\text{\,}\mathrm{\char 37\relax}$ relative to
the baseline flux (Martí-Devesa & Reimer, 2021). Emission up to 400 GeV has
now been established using the High Energy Stereoscopic System (H.E.S.S.) (H.
E. S. S. Collaboration et al., 2020).

The $\gamma$-ray emission from $\eta$ Car is generally believed to be emitted
by cosmic rays (CRs) accelerated by the shocks that enclose the wind collision
region (WCR), where the supersonic winds from the two stars collide to form a
pair of standing shocks, separated by a contact discontinuity. The shocks are
approximately located at the surface (cap) where the ram pressures from both
winds are equal. At the shocks, electrons and protons can be accelerated to
relativistic energies and emit $\gamma$-rays via synchrotron and inverse
Compton processes or via proton-proton collisions with subsequent
$\pi^{0}$-decay, respectively. For the latter process a dense target material
is needed to produce a detectable gamma-ray signal. Bednarek & Pabich (2011)
were the first to point out that the two shocks in the system should have
distinct properties due to the different stellar mass loss rates and terminal
wind velocities. These properties result in two different accelerated particle
populations. These ideas were developed further by Ohm et al. (2015) and White
et al. (2020) (henceforth O15 and W20 respectively) where models were
constructed to account for the variable $\gamma$-ray emission observed by
_Fermi_ -LAT and the non-thermal X-ray emission detected with the NuSTAR
satellite (Hamaguchi et al., 2018).

In W20, a non-negligible amount of the accelerated protons escape the WCR.
Such particles would naturally interact with the surrounding material further
out.

The complex environment around $\eta$ Car provides many potential interaction
regions. Though the inner structures, including the binary system and its
immediate vicinity, can not be resolved by _Fermi_ -LAT or H.E.S.S., the
nearby molecular clouds in the Carina Nebula Complex (CNC) and nearby Gum 31
are resolvable. Significant emission associated with these clouds, above the
expectations for the ‘sea’ of diffuse CRs, was reported in W20. Here we
explore this emission in more detail.

Recently, Ge et al. (2022) published an analysis of the excess $\gamma$-ray
emission around $\eta$ Car considering two Gaussian regions, for which
significant emission, likely of hadronic origin was found. They conclude that
the emission could be connected to young massive stellar clusters in the
region, such as Trumpler 14 and 16. Nevertheless, the authors were not able to
rule out, that $\eta$ Car or yet unknown CR sources are the acceleration sites
of the particles producing the $\gamma$-ray emission.

In this article, we follow up on these ideas and perform a detailed analysis
of the emission from $\eta$ Car and the surrounding excess emission
considering not only two but four different regions. These regions correspond
to specific nearby molecular cloud structures connected to the Carina Nebula-
Gum 31 complex. Additionally, we analyse possible $\gamma$-ray emission from
escaping CRs interacting with material on small and large scales around $\eta$
Car.

While this work focuses primarily on escaping particles accelerated in the
WCR, it is also possible that particles have been accelerated in the blast
wave from the great eruption of 1843, when the system released more than
$10\text{\,}\mathrm{M_{\odot}}$ of material (Smith, 2006) with high velocity
(Ohm et al., 2010; Skilton et al., 2012). While the apparent orbital
variability effectively rules out the possibility that all emission from
$\eta$ Car is due to the blast wave, a significant fraction of the steady
emission could still be produced near the blast wave. Though we do not
consider them further in this work, these particles would also contribute to
the $\gamma$-ray emission on larger scales.

The paper is structured as follows: In section 2, we present an overview of
the environment around $\eta$ Car. This is followed by the analysis of the
_Fermi_ -LAT data in section 3. Afterward, we explore possible $\gamma$-ray
emission from escaping particles on scales not resolvable by _Fermi_ -LAT
(section 4) and in the molecular clouds on larger scales (section 5). In the
end, we discuss our findings in section 6.

##  2 Environment around $\eta$ Car

The surroundings in which the $\eta$ Car binary system is embedded are non-
homogeneous with multiple complex structures on a variety of scales. These
provide not only a variety of target conditions for CRs escaping from $\eta$
Car, but also multiple competing sites for particle acceleration. We consider
the following zones, in order of increasing scale, as targets for CR
interactions and gamma-ray emission:

  * •

The shock cap/WCR – the region where the shocks form between the two stars.
Following W20 we refer to the more massive star as $\eta$ Car-A, and the lower
mass companion as $\eta$ Car-B. Depicted as region A in Figure 1.

  * •

The pinwheel/wind region – a high-velocity outflow with gradual mixing of the
high and low-density winds of the two stars, ending on the scale of the little
Homunculus (a possible remnant of an outburst more recent than the great
eruption). Depicted as region B in Figure 1.

  * •

The Homunculus – the expanding shell associated with the great eruption of
1843. Depicted as region D in Figure 1.

  * •

The Carina Nebula / Gum 31 complex – the star formation region containing
several massive stellar clusters and $10^{3}$–$10^{5}$ solar mass molecular
clouds (see e.g. Rebolledo et al., 2016). Depicted as region E in Figure 1.

Figure 1 illustrates these different regions graphically. The approximate
sizes of these regions are indicated by the coloured lines below each graphic.
Additionally, we illustrate the approximate spatial scales associated with the
absorption of the $\gamma$-rays. To compute these scales, the stellar
parameters were taken to be the same as in W20, but additionally the angular
size of the stars was taken into account. While the absorption in W20 was
calculated specifically for the emitting region and integrated for specific
orbital phase ranges, here we adopted a spherical geometry of the $\gamma$-ray
emission region for simplicity and the two stars were taken to be at their
maximum separation, the apastron position. The black line at the uppermost
part of the figure shows the radii for which more than
$50\text{\,}\mathrm{\char 37\relax}$ of $\gamma$-rays with an energy of
$200\text{\,}\mathrm{GeV}$ would be absorbed before reaching Earth. The peak
absorption occurs close to $600\text{\,}\mathrm{GeV}$, and the grey line shows
the radial distances inside which more than $10\text{\,}\mathrm{\char
37\relax}$ of these $600\text{\,}\mathrm{GeV}$ $\gamma$-rays are absorbed.
Therefore, accounting for $\gamma$-ray absorption is essential for the
emission within $\sim 200\text{\,}\mathrm{ua}$ of the stars. A detailed plot
showing the radial dependence for different energies can be found in Appendix
B.

Figure 1: Scales relevant to the possible emission regions around $\eta$ Car.
Because some regions such as the large or the little Homunculus Nebula are
asymmetric, or their size can change over time such as for the shock cap, the
sizes shown cover a large range of values. The different extensions are: $1$
to $20\text{\,}\mathrm{ua}$ (shock cap, A), $10$ to
$6.4\text{\times}{10}^{3}\text{\,}\mathrm{ua}$ (wind region, B),
$2\text{\times}{10}^{3}$ to $6.5\text{\times}{10}^{3}\text{\,}\mathrm{ua}$
(little Homunculus, C), $2.1\text{\times}{10}^{3}$ to
$21.69\text{\times}{10}^{3}\text{\,}\mathrm{ua}$ (Homunculus, D) and
$>50\text{\times}{10}^{3}\text{\,}\mathrm{ua}$ (Carina Nebula, E). The black
and grey bars in the upper left corner indicate the distances beyond which the
absorption at $200\text{\,}\mathrm{GeV}$ is less than
$50\text{\,}\mathrm{\char 37\relax}$ (black) and the absorption at
$600\text{\,}\mathrm{GeV}$ is less than $10\text{\,}\mathrm{\char 37\relax}$
(grey), see text for more details. The individual images are taken from Parkin
et al. (2011), Clementel et al. (2014), Smith (2005), HST, ESO.

The shock cap or WCR is formed by the colliding stellar winds. As discussed
previously, the winds terminate in a pair of shocks separated by a contact
discontinuity. The shock _cap_ refers to its finite lateral extent. The global
size of this region varies with the orbital phase between $\sim
1\text{\,}\mathrm{ua}$ at periastron and $\sim 20\text{\,}\mathrm{ua}$ at
apastron. At the shocks, particles are expected to be accelerated (see e.g.
Eichler & Usov, 1993). A detailed discussion of the particle acceleration and
emission can be found in O15 and W20.

The shock cap and associated pinwheel are embedded in the wind region, which
is created by the outflowing winds from both stars. Due to the orbital motion,
the winds create a pin-wheel like structure, though for $\eta$ Carinae this is
complicated by its eccentric orbit (see Parkin & Pittard, 2008). Simulations
by Madura et al. (2013) (see also Parkin et al. (2011); Clementel et al.
(2014)) find that the fast wind from $\eta$ Car-B carves out a low-density
channel with a large opening angle in the direction of the apastron position.
This is a consequence of the brief periastron passage. As is evident in image
B in Figure 1, the darker region indicates the low-density wind from $\eta$
Car-B. Accelerated particles that enter this region experience negligible
collisions. On the other hand, any particles that encounter denser wind
material from $\eta$ Car-A within $\approx$100 au from the binary will be
collision dominated and have a low probability of escaping the wind region.

The wind region is expected to end as it reaches the little Homunculus Nebula
(see Figure 1 C). This nebula has a total mass of plausibly $\sim
0.1\text{\,}\mathrm{M_{\odot}}$ (Smith, 2005). It is believed to be ejected in
the outburst from 1890 and accelerated by the stellar winds. It has a bipolar
shape, with the equator being $\sim
2\text{\times}{10}^{3}\text{\,}\mathrm{ua}$ and the poles around
$6.5\text{\times}{10}^{3}\text{\,}\mathrm{ua}$ away from the central stars.

The little Homunculus is in turn embedded in the larger Homunculus Nebula, the
large bipolar structure shown in image D of Figure 1. The Homunculus was
ejected during the great eruption of $\eta$ Car in the 1840s. It is mainly
composed of a thin outer shell, with a thickness of up to
$600\text{\,}\mathrm{ua}$, and a thicker inner shell (Smith, 2006). The thin
outer shell contains $\sim 90\text{\,}\mathrm{\char 37\relax}$ of the total
mass of the Homunculus. The total mass is not exactly known, but it is larger
than $10\text{\,}\mathrm{M_{\odot}}$ (Smith et al., 2003) and thought to be
between $15\text{\,}\mathrm{M_{\odot}}$ and $35\text{\,}\mathrm{M_{\odot}}$
(Smith & Ferland, 2007). This makes the Homunculus the most massive structure
in the close proximity of the stars. The equivalent pure atomic hydrogen
densities in the thin outer shell are high: at least
$3\text{\times}{10}^{6}\text{\,}{\mathrm{cm}}^{-3}$ but potentially
${10}^{7}\text{\,}{\mathrm{cm}}^{-3}$ or even higher (Smith, 2006).

The Homunculus lies within the Carina Nebula. As detailed in Rebolledo et al.
(2016) (section 5.3), four nearby clouds with masses of
$\approx$104-$10^{5}\,\mathrm{M_{\odot}}$ are detected in CO maps and also
dust measurements. These are referred to as the Southern Cloud, Northern
Cloud, Southern Pillars, and Gum 31 and are located within 100 pc of $\eta$
Car. From the velocity maps presented in that paper, a clear connection to the
CNC/Gum 31 complex is evident. Whereas the emission of the binary system down
to the scale of the Homunculus is not resolvable with Fermi or H.E.S.S., the
larger clouds allow us to probe the CR density around $\eta$ Car as discussed
in section 3 and section 5.

##  3 Fermi analysis and results

The approach to Fermi data analysis here follows closely that of W20. The data
selection was based on the latest _Fermi_ -LAT Pass 8 data (Bruel et al.,
2018) starting from Aug 4th, 2008 (MET 239557417) to Oct 26th, 2022 (MET
688521600). Events over an energy range of 500 MeV (chosen to avoid poorly
reconstructed events at lower energies) and 500 GeV were included from a
region of interest (ROI) of $10\text{\,}\mathrm{\SIUnitSymbolDegree}$ by
$10\text{\,}\mathrm{\SIUnitSymbolDegree}$, centred at the nominal position of
$\eta$ Car and aligned in galactic coordinates. Data were chosen according to
the SOURCE event class (evclass=128) with FRONT+BACK event types (evtype=3).
Time periods in which the ROI was observed at zenith angle greater than
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ were excluded. Data were analysed
using Fermitools (version 2.2.0), which is the official ScienceTools suite
provided by the Fermi Science Support
Center111https://fermi.gsfc.nasa.gov/ssc/ and FermiPy (version 1.2) (Wood et
al., 2017). The model of sources surrounding $\eta$ Car was taken similar to
W20 and includes all sources in the ROI from the _Fermi_ -LAT 12-year source
catalog (4FGL-DR3) (Abdollahi et al., 2022b), with the exception of the
unidentified point source 4FGL J1046.7-6010, which is located in the centre of
the Southern Pillars and was excluded to avoid a potential miss-assocation of
diffuse flux in this region. The model including the diffuse Southern Pillars
component instead of the point source 4FGL J1046.7-6010 is preferred by
$\Delta TS=27.3$, whereas adding the point source additionally only improves
the model by $\Delta TS=7.0$ with 3 additional degrees of freedom. More
details on the analysis configuration are summarized in Appendix A.

In W20 an additional diffuse source was added to the model based on the CO
survey of Dame et al. (2001). Here, we split the model into four individual
clouds, Southern Cloud, Northern Cloud, Southern Pillars and Gum 31, following
the region definitions as outlined in Rebolledo et al. (2016) and described in
the previous section. The four templates can be seen as an overlay in Figure
2. Each cloud is included in the model as a diffuse component with a power-law
spectral shape. Optimisation of the model was performed in an iterative
fashion making use of the optimize function of FermiPy. First, the
normalisation of a maximum of 5 of the brightest sources with a predicted
number of counts totaling 95% of the total predicted counts of the model are
freed and a simultaneous fit is performed. Next, the normalisation of the
sources not included in the first step was performed individually. Finally,
the shape and normalisation of all sources with a TS exceeding 25 in the
previous fits were freed and a simultaneous fit was performed. The same
process was repeated but allowing up to ten sources for the first optimization
step. After optimising the model, the cloud component spectral shapes were
fixed in a first fitting iteration freeing $\eta$ Car (4FGL J1045.1-5940) and
the normalisation of all sources within a 3∘ radius and with TS values $>$10
and more than 10 predicted counts. As in the 4FGL catalogue, and previous
analysis of the region (W20, Martí-Devesa & Reimer, 2021), $\eta$ Car was
modelled as a point source with a log-parabola spectrum
$\phi=\phi_{0}(E/E_{0})^{-\alpha-\beta\log(E/E_{0})}$. This yielded a best fit
model for $\eta$ Car with $\alpha=2.29\pm 0.02$, $\beta=0.11\pm 0.01$,
$\phi_{0}=(2.44\pm 0.07)\times 10^{-6}\,\,\mathrm{cm^{-2}\,erg^{-1}\,s^{-1}}$
and $E_{0}=2.11\,\,\mathrm{GeV}$ consistent with previous studies (W20, Martí-
Devesa & Reimer, 2021). In a second fitting iteration $\eta$ Car and all other
previously free sources were fixed and the normalisation and spectral shape of
all four clouds were freed. Hence any miss-association of flux between the
close Southern and Northern Clouds and the bright $\eta$ Car could be
minimised. Following the fitting for $\eta$ Car and the molecular clouds, a
spectral energy distribution (SED) was generated for the source of interest.
Whereas a similar event selection approach was taken in Ge et al. (2022),
their main result is based on the addition of two Gaussian disks in the fit
instead of the cloud templates employed in this work.

The four clouds were each detected with significant excesses and TS values of
$>$200\. The exact values are summarised in Table 1. If a photon weighting
procedure as used in the 4FGL catalog (Abdollahi et al., 2020) is applied the
TS values are reduced but still $>$150 for each cloud. The residual emission
in the ROI could be clearly reduced with the addition of the cloud templates,
as shown in Figure 2. The remaining residual emission to the right side of the
region can not be associated with molecular clouds in the CNC/Gum 31 complex
(Rebolledo et al., 2016) but might be connected to diffuse emission around
Westerlund 2 (Yang et al., 2018). In order to account for it in the fit an
additional diffuse component based on the CO map by Dame et al. (2001) was
introduced. The inclusion of this additional component helped to improve the
overall fit but did not affect the properties of the cloud components
significantly. The spectra of the four individual clouds are shown in Figure 3
and the best-fit spectral parameters are also included in Table 1. The
spectral results with and without the photon weighting differ only by up to
0.04 for the spectral index and $3\text{\,}\mathrm{\char 37\relax}$ on the
energy flux, hence being consistent within the statistical uncertainties. The
values without the additional weighting are adopted, whereas the quoted
differences can be interpreted as a contribution to the systematic
uncertainty. For comparison, the spectrum of $\eta$ Car is also shown. The
spectral shapes and normalisations determined for the Southern Cloud, Northern
Cloud and Gum 31 regions show a reasonable level of consistency, with a
spectral index between 2.2 and 2.3. In contrast, the Southern Pillars are
described by a slightly softer spectrum, mostly caused by an apparent cutoff
above photon energies of 20 GeV. Below that energy, the derived flux points
are comparable amongst all four clouds. Different spectral models for each
cloud were tested by fitting the SEDs of each cloud separately. This yielded a
difference of as much as $\leavevmode\nobreak\ 10\%$ for the integrated energy
flux, which can be considered as a contribution to the systematic uncertainty
of the analysis.

Cloud Name | Spectral | Energy Flux | TS
---|---|---|---
| Index | [$\,\mathrm{erg\,cm^{-2}\,s^{-1}}$] |
Southern Cloud | $2.30\pm 0.05$ | $(2.4\pm 0.3)\times 10^{-11}$ | 276
Northern Cloud | $2.25\pm 0.06$ | $(2.0\pm 0.2)\times 10^{-11}$ | 225
Southern Pillars | $2.54\pm 0.06$ | $(1.8\pm 0.2)\times 10^{-11}$ | 348
Gum 31 | $2.24\pm 0.06$ | $(2.4\pm 0.3)\times 10^{-11}$ | 270
Table 1: TS values and power-law spectral properties of the cloud templates as
included in the model and derived by the fit. The energy flux is derived by
integrating from 500 MeV to 100 GeV.

Figure 2: Residual significance maps from the _Fermi_ -LAT analysis in a
zoomed region of the ROI. In the upper panel, the $\sqrt{\rm TS}$ map without
the addition of the cloud templates to the model is shown, whereas they are
included in the lower panel. In each plot the cloud templates are overlaid by
coloured contours, indicating the gas density distribution. Additionally, all
other 4FGL sources are marked together with massive stellar clusters from
Preibisch et al. (2011). Figure 3: Spectra of the four molecular clouds as
derived from the _Fermi_ -LAT analysis. For comparison, the spectrum derived
for $\eta$ Car is also shown in grey. Additionally, a model derived from an
injected CR spectrum is shown. The CR spectrum is modeled as a power law with
an index of 2.0 and an exponential cutoff at 2 TeV. The normalization has been
scaled arbitrarily.

The inner regions of the Carina Nebula are not resolvable with current
$\gamma$-ray instruments. For example, the Homunculus exists on scales
corresponding to a factor 40 smaller than the 68 % containment radius of the
Fermi Point Spread Function (PSF), which is ${\sim}0.1^{\circ}$ above 20 GeV
(Abdollahi et al., 2020). An upper limit on the $\gamma$-ray flux originating
from this inner region (but outside the WCR) can hence only be estimated via
the quiescent baseline component of the $\eta$ Car $\gamma$-ray flux. Long-
term light curves show only mild variability (e.g. W20, Martí-Devesa & Reimer,
2021), with the $\gamma$-ray flux never dropping below $\leavevmode\nobreak\
60\%$ of the mean integrated flux level. This provides a baseline energy flux
of $(5.6\pm 0.3)\times 10^{-11}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$ integrated
from 500 MeV to 100 GeV.

##  4 Propagation of CRs around $\eta$ Car and expected emission

###  4.1 Particle acceleration and emission in the WCR

We summarise in the following the relevant points of the model of O15, W20 for
CR production in the WCR (see also Eichler & Usov, 1993; Bednarek & Pabich,
2011). The model from W20 is then used to obtain the spectrum of escaping CRs.

As mentioned in section 2, the WCR contains two shocks. The wind of the more
massive star $\eta$ Car-A is slower (terminal wind speed
$v_{\infty}\approx$5\text{\times}{10}^{2}\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$)
and more dense
($\dot{M}\approx$4.8\text{\times}{10}^{-4}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{yr}}^{-1}$$)
compared to that of the companion star $\eta$ Car-B
($v_{\infty}\approx$3\text{\times}{10}^{3}\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$,
$\dot{M}\approx$1.4\text{\times}{10}^{-5}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{yr}}^{-1}$$,
see Parkin et al., 2011). At the side of $\eta$ Car-A, the high density leads
to complete cooling of accelerated protons. The maximum proton energy is
limited by inelastic collisions with compressed wind material, with $E_{\rm
max}\sim 200$ GeV, and no energetic particles escape the acceleration region.
On the side of $\eta$ Car-B, the cooling time via collisions greatly exceeds
the flow time, and accelerated particles can leave the acceleration region.
W20 estimated that the maximum proton energy achievable at the wind
termination shock of $\eta$ Car-B is $\approx 30$ TeV.

To match observations, approximately $10\text{\,}\mathrm{\char 37\relax}$ of
the wind power processed by each of the shocks should be converted into
accelerated protons above $1$ GeV. While energetic particles on the side of
$\eta$ Car-A are completely destroyed via hadronic collisions, on the side of
$\eta$ Car-B particles can enter the ballistic region, i.e. the flow
transporting the shocked material laterally away from the WCR. The accelerated
protons from the shock of $\eta$ Car-B produce a non-negligible amount of
$\gamma$-rays in the ballistic region as the shocked material from $\eta$
Car-B mixes with the shocked higher density material from $\eta$ Car-A. The
amount of radiation produced depends on the details of the mixing process,
which was adjusted in W20 to match the _Fermi_ -LAT data. The number of
interacting particles in the W20 model varies with phase and is largest at
periastron.

For $\gamma$-rays produced in the wind region close to the stars, absorption
due to pair production plays an important role. Here, we restrict ourselves to
the assumption of spherically symmetric emission. Due to the change in the
stellar positions, there is an orbital variation in the absorption - in
particular between phases 0.9 and 0.2. However, for emission on radial
distance scales of $20\text{\,}\mathrm{ua}$, the phase dependence drops below
$7\text{\,}\mathrm{\char 37\relax}$ and decreases rapidly for emission on even
larger scales.

In Figure 1 the absorption for spherically symmetric emission at phase $0.5$
for two different energies is shown. The case of $600\text{\,}\mathrm{GeV}$
corresponds roughly to maximum absorption. Whereas at $20\text{\,}\mathrm{ua}$
close to half of the emission is absorbed, at $200\text{\,}\mathrm{ua}$ more
than $90\text{\,}\mathrm{\char 37\relax}$ of the $\gamma$-rays reach Earth.
Thus, the VHE $\gamma$-ray emission reported by H. E. S. S. Collaboration et
al. (2020) can be easier accounted for if it is produced at larger radii.

In the following, we use the average escaping CR spectrum and the stellar wind
parameters quoted above, adapted from the model of W20, to investigate
possible $\gamma$-ray emission produced by the escaping protons in the
surrounding environment. Because the escaping proton spectrum in that work was
not calculated directly, here it was inferred from the $\gamma$-ray data
presented in W20. To obtain the spectral shape, we used the Naima package
(Zabalza, 2015) assuming an exponential cutoff power-law proton spectrum. In
the off-periastron phase range, this resulted in a power-law index of
$\alpha\approx 2$ and cutoff energy of $E_{\rm cut}$\approx
1.8\text{\,}\mathrm{TeV}$$. In W20 the $\gamma$-ray spectrum is calculated
while the particles are advected with the flow out to a distance where the
resulting emission is negligible and the radiative energy losses do not
further change the underlying spectrum. The spectrum of escaping particles is
then assumed to maintain the same shape. The proton spectrum above
$1\text{\,}\mathrm{GeV}$ was then normalised such that the total energy in
escaping protons and transferred into $\gamma$ rays, secondary particles and
directly accelerated electrons is equal to the total power put into particle
acceleration. Approximately $\sim 90\text{\,}\mathrm{\char 37\relax}$ of the
accelerated protons at the side of $\eta$ Car B escape from the WCR.

###  4.2 Propagation in the wind region

The energetic particles that escape in the ballistic flow will eventually
merge with the winds. To avoid substantial adiabatic losses, the transport of
these particles should be diffusion dominated. The ratio of the characteristic
advection and diffusion timescales is a useful measure of which process
dominates the radial transport:

$\displaystyle\frac{t_{\rm adv}}{t_{\rm
diff}}=\frac{R}{u}\times\frac{D_{r}}{R^{2}}=\frac{D_{r}}{uR}$ (1)

where $u$ is the outward flow speed, $D_{r}$ the average radial diffusion
coefficient, and $R$ the radial distance from the binary. Taking $u\sim
1000\,{\rm km\,s}^{-1}$ and $R=100$ au, adiabatic losses can be neglected
provided $D_{r}\gg 10^{23}{\rm cm^{2}\,s^{-1}}$. While the radial diffusion
coefficient is not known, the magnetic fields will be highly disordered in the
ballistic region provided mixing is effective. This can in principle allow
relativistic protons and other nuclei to decouple from the flow and escape the
wind region without undergoing significant adiabatic losses.

If, on the other hand, particles were strongly coupled to the flow, the
maximum energy would be reduced by adiabatic cooling:
$E\propto\rho^{1/3}\propto r^{-2/3}$. Thus in crossing the wind region, the
maximum particle energy would be reduced by a factor of 10 or more. If the VHE
measurements of H. E. S. S. Collaboration et al. (2020) are produced far from
the WCR, this provides a (model-dependent) constraint on the transport.

Protons that move outward, by either diffusion or advection, but remain in the
low-density cavity evacuated by the wind from $\eta$ Car-B, will produce
negligible $\gamma$-ray emission. However, some protons might migrate into the
high-density wind from $\eta$ Car-A, radiate significantly, and if they remain
in the high-density zone, lose all of their energy via hadronic interactions.

The spectrum of the protons migrating into the $\eta$ Car-A wind will depend
on the transport processes, the details of which are currently not well-
constrained. In the most extreme case, when all protons penetrate into the
wind of $\eta$ Car-A, the energy in $\gamma$ rays detected on Earth using a
total energy flux in escaping particles of
$6.5\text{\times}{10}^{35}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$
from the model of W20 would be
${10}^{-9}\text{\,}\mathrm{erg}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{s}}^{-1}$.
This exceeds considerably the flux detected by _Fermi_ -LAT, and the model
parameters of W20 would have to be adapted. In the other extreme case, if a
transport barrier exists at the interface of the two winds, excluding CRs from
the denser target material, no significant emission would be produced in the
wind region.

Variability of the $\gamma$-ray emission in the wind region are likely to be
modest and not necessarily linked to the orbital phase, unless the $\gamma$
rays are produced only very close to the stars. Any potential emission from
escaping protons in the wind region will thus contribute to an approximately
steady baseline flux, modulo turbulent variations. The same should hold true
for larger structures such as the little and large Homunculus nebulae.

###  4.3 The little and large Homunculus nebulae

As shown in the previous section, only the particles diffusing in the low-
density wind from $\eta$ Car-B will leave the pinwheel/wind region and in the
most extreme case, this could be the vast majority of the particles leaving
the WCR. After the wind region, the CRs encounter the little Homunculus
Nebula, which is likely to have a total mass of $\sim
0.1\text{\,}\mathrm{M_{\odot}}$ (Smith, 2005). On larger scales they encounter
the $>$$10\text{\,}\mathrm{M_{\odot}}$ Homunculus Nebula (see section 2). To
obtain a rough estimate of the total $\gamma$-ray emission in each region, we
can use the following approximation:

$\displaystyle F_{\rm Earth}\approx\frac{t_{\rm cross}}{t_{\rm
cool}}\cdot\frac{L_{\rm esc}}{4\pi d^{2}},$ (2)

where $F_{\rm Earth}$ is the total energy flux on Earth, $t_{\rm cross}$ the
time it takes for the particles to cross each region, $t_{\rm cool}$ the
cooling time, $L_{\rm esc}$ the total energy in escaping particles and
$d=$2.3\text{\,}\mathrm{kpc}$$ the distance towards $\eta$ Car. For $L_{\rm
esc}$ we adopt the value of
$6.5\text{\times}{10}^{35}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$
from the model of W20, assuming that energy losses are negligible in the
pinwheel/wind region. We model the little Homunculus and the inner and outer
shells of the large Homunculus as spherical shells around the stars for
simplicity. We first consider the case of no diffusion within these thin
shells, with CRs simply passing through at (close to) the speed of light.

Smith (2005) estimated a thickness of the little Homunculus of $\sim
940\text{\,}\mathrm{ua}$. Using an average inner radius of
$4\text{\times}{10}^{3}\text{\,}\mathrm{ua}$, a thickness of the shell of
$1000\text{\,}\mathrm{ua}$ and a mass of $0.1\text{\,}\mathrm{M_{\odot}}$, the
cooling time at $1\text{\,}\mathrm{GeV}$ is $\sim
4\text{\times}{10}^{5}\text{\,}\mathrm{d}$. If the CRs move through the region
with the speed of light, the resulting total energy flux on Earth is $F_{\rm
Earth}=$1.4\text{\times}{10}^{-14}\text{\,}\mathrm{erg}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{s}}^{-1}$$.
This is orders of magnitudes below the energy flux from _Fermi_ -LAT of
$5.6\text{\times}{10}^{-11}\text{\,}\mathrm{erg}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{s}}^{-1}$.
The inner shell of the large Homunculus contains $10\text{\,}\mathrm{\char
37\relax}$ of the total Homunculus mass (Smith, 2006), which is between
$1\text{\,}\mathrm{M_{\odot}}$ and $3.4\text{\,}\mathrm{M_{\odot}}$. To model
it as a shell, we assume an inner radius of
$1\text{\times}{10}^{4}\text{\,}\mathrm{ua}$ and an outer radius of
$1.5\text{\times}{10}^{4}\text{\,}\mathrm{ua}$. For a mass of $\sim
2\text{\,}\mathrm{M_{\odot}}$, the resulting $F_{\rm Earth}$ is
$3.7\text{\times}{10}^{-14}\text{\,}\mathrm{erg}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{s}}^{-1}$.
Different values of the inner and outer radius of the shell give similar
results. For the outer Homunculus shell with an assumed density of
$1.0\text{\times}{10}^{7}\text{\,}{\mathrm{cm}}^{-3}$ and a thickness of
$600\text{\,}\mathrm{ua}$, one obtains $F_{\rm
Earth}=$6.2\text{\times}{10}^{-13}\text{\,}\mathrm{erg}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{s}}^{-1}$$.
The outer Homunculus shell is therefore the most important region for the
production of $\gamma$ rays by the escaping particles with fluxes at least one
order of magnitude above the fluxes from the little Homunculus and the inner
Homunculus shell. However, if the CRs pass through the outer shell at the
speed of light, the $\gamma$-ray emission produced is still two orders of
magnitude below the _Fermi_ -LAT flux.

The Weigelt knots, located closer to the binary, have not been explicitly
considered in this study. Adopting the typical Weigelt knots parameters from
Remmen et al. (2013), the knots do not appear to have a sufficiently high
over-density relative to the parameters adopted for the Homunculus Nebula to
make a significant difference to the gamma-ray luminosity on those scales.

Cosmic rays are more likely to diffuse slowly through these shells of
material, than pass through at the speed of light. The outer Homunculus shell
is the most massive and the most dense of the three shells, with likely larger
magnetic fields and slower diffusion and hence expected to dominate also in
this case. We therefore restrict the following calculations for different
diffusion coefficients to emission from the outer shell. For simplicity, we
ignore possible spectral changes caused by the propagation through the wind
region, the little Homunculus and the inner Homunculus shell, because the
exact propagation properties are unknown and are impossible to separate
observationally from spectral changes by the propagation through the
Homunculus itself. As before we adopt a value of
$1.0\text{\times}{10}^{7}\text{\,}{\mathrm{cm}}^{-3}$ for the density in the
outer shell. A lower limit on the (parallel) diffusion coefficient can be
estimated by assuming Bohm diffusion. Unfortunately, the magnetic field in the
shell is unknown and difficult to constrain. Values of
$100\text{\,}\mathrm{\SIUnitSymbolMicro G}$ or even higher have been proposed
(Aitken et al., 1995). To provide some numerical estimates, we take the energy
dependence of the radial diffusion coefficient to have the following form:

$\displaystyle
D_{r}(E)=D_{0}\times\left(\frac{E}{$1\text{\,}\mathrm{GeV}$}\right)^{\alpha}.$
(3)

The dotted blue curve in Figure 4 shows the emission produced in the
Homunculus for
$D_{0}=$9\text{\times}{10}^{22}\text{\,}{\mathrm{cm}}^{2}\text{\,}{\mathrm{s}}^{-1}$$
and $\alpha=0.5$. As can be seen, interactions in the Homunculus Nebula of
escaping CRs could produce a significant amount of steady $\gamma$-ray
emission. Thus, models in which particle acceleration at the shock of $\eta$
Car-A is ineffective, for example, due to a low Mach number or a highly
oblique magnetic field, can be replaced with models in which the lower energy
$\gamma$-rays are principally produced in the Homunculus. In Figure 4 the red
curve shows the emission produced by particles accelerated at the shock of
$\eta$ Car-B from the model of W20, the dashed red line is without absorption.
The black dashed-dotted line shows this emission added to the emission from
the Homunculus shown in the blue dashed-dotted curve for
$D_{0}=$9\text{\times}{10}^{21}\text{\,}{\mathrm{cm}}^{2}\text{\,}{\mathrm{s}}^{-1}$$,
$\alpha=1$ and a factor of $1.1$ more escaping CRs compared to the model of
W20. The combined emission reproduces well the _Fermi_ -LAT data over the
whole energy range. In such a scenario, the weak phase-dependent variability
is entirely produced by changes in the emission from $\eta$ Car-B close to the
binary. The assumed diffusion coefficient in the Homunculus is well above the
case of Bohm diffusion for a magnetic field of
$100\text{\,}\mathrm{\SIUnitSymbolMicro G}$.

Figure 4: Model of emission from $\eta$ Car and the Homunculus together with
_Fermi_ -LAT data. The blue curves show emission from the Homunculus for
different diffusion properties of the escaping CRs. The dotted curve (Model A)
is for
$D_{0}=$9\text{\times}{10}^{22}\text{\,}{\mathrm{cm}}^{2}\text{\,}{\mathrm{s}}^{-1}$$
and $\alpha=0.5$. The dashed-dotted curve (Model B) is for
$D_{0}=$9\text{\times}{10}^{21}\text{\,}{\mathrm{cm}}^{2}\text{\,}{\mathrm{s}}^{-1}$$,
$\alpha=1$ and a factor of $1.1$ more escaping CRs compared to the model from
W20. The red solid line shows the emission from $\eta$ Car produced by
particles accelerated at the shock towards $\eta$ Car B from W20 (the red
dashed line is without absorption). The dashed-dotted black curve shows the
combined emission from $\eta$ Car B and Model B.

The above model for particle acceleration occurring solely at the companion
shock together with steady emission from the Homunculus by the escaping CRs
can also account for the hard non-thermal X-ray emission. In the model of W20
the hard X-rays, similar to the $\gamma$-rays, are dominated by particles
accelerated at the shock of $\eta$ Car-B. Since that model reproduced the
variability, we focus here on the phase averaged emission (see also Breuhaus,
2022). While the orbital variability in the $\gamma$-ray emission shows
energy-dependent behaviour that differs between orbits (Balbo & Walter, 2017;
Martí-Devesa & Reimer, 2021), observations are nevertheless consistent with at
least $\sim 60\text{\,}\mathrm{\char 37\relax}$ of the emission being produced
by a steady-state source. Using the existing $\gamma$-ray data, it is
difficult to conclude which of these components dominates the quasi-steady
emission, e.g. emission produced near the wind termination shock of $\eta$
Car-A (W20, Balbo & Walter, 2017), or that produced on larger scales as
presented here (see also Ohm et al., 2010; Skilton et al., 2012, for
additional large scale sources). An observation of a reduction of the steady
part of the emission or a clearly different behavior in different energy bands
could potentially constrain the possible contribution of the Homunculus Nebula
and the wind region to the gamma-ray flux, though the latter could equally be
explained by the turbulent behaviour during the passage as suggested by Martí-
Devesa & Reimer (2021).

##  5 Propagation in the Carina Nebula

CRs that escape both $\eta$ Car and the Homunculus Nebula will eventually
enter the Carina Nebula. There they will encounter the molecular clouds
described in section 2 as potential target material for further $\gamma$-ray
production. The clouds of the CNC and Gum 31 exhibit significant $\gamma$-ray
emission (see section 3) which could be related to $\eta$ Car. Figure 3 also
contains a model $\gamma$-ray spectrum derived from an injected CR
distribution following a power-law with exponential cutoff. In the models of
O15 and W20 a non-linear DSA model was used to account for the required hard
underlying proton spectrum (see subsection 4.1). The assumed CR spectrum has
an index of 2.0 and an exponential cutoff at 2 TeV. To compute a $\gamma$-ray
spectrum the parametrizations from Kappes et al. (2007) were employed and
energy-dependent propagation leading to a softening of $-1/3$ was assumed. The
resulting $\gamma$-ray spectrum has a similar profile to that measured by
_Fermi_ -LAT. The CR density in a certain region can be calculated from the
$\gamma$-ray luminosity and cloud mass following the approximation given in
Aharonian et al. (2019):

$\scriptstyle w_{CR}(\geq 10E_{\gamma})=1.8\times
10^{-2}\left(\frac{\eta}{1.5}\right)^{-1}\left(\frac{L_{\gamma}(\geq
E_{\gamma})}{10^{34}\,\mathrm{erg\,s^{-1}}}\right)\left(\frac{M}{10^{6}M_{\odot}}\right)^{-1}\,\mathrm{eV\,cm^{-3}}$
(4)

Here the parameter $\eta$ accounts for the presence of heavier nuclei and is
assumed to be 1.5. The $\gamma$-ray luminosity, Lγ, above 500 MeV is derived
from integrating the $\gamma$-ray flux and assuming a typical distance of 2.3
kpc, i.e. the same as $\eta$ Car. This translates to a minimum CR energy of 5
GeV. The masses are assumed to follow the dust mass estimate from Rebolledo et
al. (2016), Table 2 based on Herschel infrared Galactic Plane Survey (HiGAL,
Molinari et al. (2010)) data. The mass uncertainty derived from dust maps is
mostly dominated by uncertainties in temperature derivation ($\sim$10$\%$)
(Urquhart et al., 2018) the HiGAL survey flux ($\sim$5$\%$) (Molinari et al.,
2016) and the local gas-to-dust mass ratio assumption. According to Giannetti
et al. (2017) the local variation of this ratio is of order $20\%$. Hence we
adopt an overall uncertainty of $25\%$ on the mass, not reflecting systematic
uncertainties that are common to all of the clouds (which are likely much
larger, but do not impact the shape of the profile).

The resulting CR density profile can be seen in Figure 5. The physical extent
is visualized by the horizontal error bars as the minimum and maximum distance
from $\eta$ Car. The conversion to a physical distance scale assumes that all
clouds are located in a plane at the same distance from the observer. Compared
to the result obtained by Ge et al. (2022) the result from Gum31 is consistent
with their result for region B, whereas their region A centred at $\eta$ Car
comprises both the Southern Cloud and parts of the Northern Cloud and is hence
not directly comparable.

Assuming $\eta$ Car as the origin of the CRs, this suggests a $1/r$ profile,
similar to that observed around some massive stellar clusters (Aharonian et
al., 2019),

$w_{\rm CR}(r)=w_{0}(r/r_{o})^{-1}.$ (5)

Normalising the profile at r${}_{0}=10$ pc, a value of w${}_{0}=0.48\pm
0.09\,\mathrm{eVcm^{-3}}$ in CRs above 5 GeV is obtained from a fit to the
derived CR density profile. Using the maximum energy flux and mass for the
Homunculus, as discussed above, provides an upper limit on the CR energy
density of ${\sim}10^{3}\,\mathrm{eVcm^{-3}}$ at a distance of $\lesssim 0.1$
pc, which does not constrain the postulated $1/r$ behaviour. For an
integration radius of 60 pc, corresponding to the outer edge of the emission
seen in the CNC-Gum 31 complex, the derived CR energy density implies a total
energy of CR protons of $W_{p}=5\times 10^{48}\,\mathrm{erg}$.

The diffusion time can be calculated from the maximum distance, Rmax, that CRs
have propagated for a given diffusion coefficient. Here, Rmax is taken to be
60 pc and the energy-dependent diffusion is assumed to follow Equation 3.
Together with a total power of ${\sim}5\times 10^{35}\,\mathrm{erg\,s^{-1}}$
escaping $\eta$ Car in CRs above an energy of 5 GeV as suggested by the model
in W20, this implies a diffusion coefficient of $D_{0}=5\times
10^{26}\,\mathrm{cm^{2}\,s^{-1}}$. Whilst this is slower than the average
Galactic diffusion coefficient ($3\times 10^{28}\,\mathrm{cm^{2}\,s^{-1}}$),
it is well above the lower limit of $5\times 10^{25}\,\mathrm{cm^{2}\,s^{-1}}$
implied by the age of $\eta$ Car (${\sim}2-3\times 10^{6}$ years (Mehner et
al., 2010)), and so, at first sight, may appear reasonable.

However, this diffusion coefficient implies diffusion times through the clouds
that are on the same scale as the lifetime of the diffusing protons. The
diffusion time through the northern cloud for our estimated diffusion
coefficient and a cloud diameter of 30 pc, estimated from the size of the CO
template, is ${\sim}1.4\times 10^{5}\,\mathrm{yr}$. The life time of the
diffusing protons is approximately $t_{\rm cool}=3\times
10^{7}n^{-1}\,\mathrm{yr}$ (e.g. Hinton & Hofmann, 2009), corresponding to
$\approx 10^{5}\,\mathrm{yr}$ for a spherical cloud of mass
$10^{5}\,\mathrm{M}_{\odot}$ (Rebolledo et al., 2016). Consequently, the thin
target approximation does not hold. Faster diffusion appears to be needed,
leading to a higher total CR power which can not be produced exclusively from
$\eta$ Car in its current state.

Therefore either the CR output from $\eta$ Car was higher in the past by at
least a factor of few, or a contribution from additional sources in the CNC is
needed. $\eta$ Car is known to be a highly variable system on timescales of
$\sim 100\text{\,}\mathrm{yr}$, and little is known about the system prior to
the great eruption in the 19th century. Equally, several candidate sources
exist that could contribute to the observed emission, including massive
stellar clusters (such as the nearby Trumpler 14) and massive binaries
situated in the star-forming regions of the CNC (Ge et al., 2022; Aharonian et
al., 2019).

Figure 5: The CR density for each of the four clouds as a function of distance
to $\eta$ Car. The angular distance has been transformed to a physical using a
distance of 2.3 kpc. Distance errors depict the maximum extent of the cloud
templates. A $1/r$ type profile as described in Equation 5 has been fit to the
data points and is shown by the dashed line. For comparison, the CR densities
as derived in Ge et al. (2022) for their regions A & B are shown in light
grey.

##  6 Summary and conclusion

In this work, we analysed _Fermi_ -LAT data from $\eta$ Car and four
surrounding clouds and investigated CR escape, propagation and $\gamma$-ray
emission from the $\eta$ Car binary system and its neighbourhood. The $\eta$
Car system and its surroundings are extremely complex, with the wind region,
the little and large Homunculus nebulae, and molecular clouds nearby. For
efficient escape of particles from the system, the CRs should diffuse in the
low-density region carved by the wind from $\eta$ Car-B, otherwise, they will
either interact in the wind region or lose their energy via adiabatic losses.
Particles diffusing into the high-density region can lead to an additional
contribution to the total $\gamma$-ray spectrum. Depending on the propagation
properties, the interaction of escaping particles in the Homunculus Nebula can
account for a large fraction of the total $\gamma$-rays measured by _Fermi_
-LAT.

We find that the observed emission of $\eta$ Car on scales not resolvable by
_Fermi_ -LAT can be accounted for by a variety of different models.
Additionally to that put forward by W20, this also includes models solely
accelerating particles at the shock on the side $\eta$ Car-B, with the lower
energy _Fermi_ -LAT emission dominantly produced in the Homunculus (see
section 4). Therefore, it is possible that contributions from several zones
contribute to the overall emission, a fact which should be included in any
future model. However, the determination of the exact amount of emission
produced in each region depends on the details of the CR transport and remains
a challenge.

Escaping CRs from $\eta$ Car may also interact in the molecular clouds of the
Carina nebula. As shown in section 5, the derived radial profile of the CR
densities seems to be indicative of an origin of CRs from $\eta$ Car. However,
to account for the current emission associated with molecular clouds, either
$\eta$ Car must have been more powerful in the past, or additional CR sources
must be present in the region (as proposed by Ge et al., 2022).

Observations of $\eta$ Car at several hundreds of $\mathrm{GeV}$ and
$\mathrm{TeV}$ energies by Imaging Atmospheric Cherenkov Telescopes are also
of great interest. This part of the spectrum is affected by absorption and
potential adiabatic losses. Accurate, potentially time-dependent, observations
around periastron, therefore help to investigate the propagation properties
close to the stars and to constrain the emission regions. Observations from
the H.E.S.S. telescopes of the most recent periastron passage, and
observations from the future CTA Observatory, may provide crucial information
to help unravel the physics of this fascinating binary system $\eta$ Car.

##  7 Acknowledgements

For the numerical calculations in section 4 we made use of the open-source
code GAMERA (Hahn, 2015; Hahn et al., 2022). To estimate the shape of the
escaping proton spectrum, the Naima package was used (Zabalza, 2015).

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##  Appendix A Further _Fermi_ -LAT analysis details

Parameter | Value
---|---
Data release | P8R3
IRFs | P8R3_SOURCE_V3
ROI data width | 10∘
ROI model width | 15∘
Bin size | 0.1∘
zmax | 90∘
Coordinate system | GAL
Minimum energy | 500 MeV
Maximum Energy | 500 GeV
MET start | 239557417
MET stop | 688521600
MET excluded (ASASSN-18fv) | 542144904 – 550885992
evclass | 128
evtype | 3
edisp | True
edisp binning | -1
edisp disabled | Isotropic and diffuse 222Even though disabling energy dispersion for the galactic diffuse component is not the standard procedure with a model based on the 4FGL catalog, this does not affect the result significantly due to the high energy threshold of the analysis.
Galactic diffuse template | gll_iem_v07.fits
Isotropic background component | iso_P8R3_SOURCE_V3_v1.txt
_Fermi_ -LAT catalogue | 4FGL-DR3 (gll_psc_v29.fit)

Table 2: Configuration used for the _Fermi_ -LAT analysis. The time period MET
542144904 – 550885992 was excluded due to the bright nova ASASSN-18fv (or V906
Carinae) (Aydi et al. 2020) within the ROI

.

##  Appendix B $\gamma\gamma$-absorption

For the calculations of $\gamma\gamma$-absorption in the anisotropic stellar
radiation fields, we used the open-source GAMERA code (Hahn 2015; Hahn et al.
2022). It uses the standard cross-section for pair production which can be
found, for example, in Gould & Schréder (1967). In the form found for example
in Vernetto & Lipari (2016, Eq. 1), the cross-section reads:

$\displaystyle\sigma_{\gamma\gamma}=\sigma_{\rm
T}\frac{3}{16}(1-\beta^{2})\left[2\beta(\beta^{2}-2)+(3-\beta^{4})\ln\frac{1+\beta}{1-\beta}\right],$
(6)

where $\beta=\sqrt{1-1/x}$ with $x=s/(4m_{\rm e}^{2}c^{4})$ and
$s=2E_{\gamma}\epsilon(1-\cos\theta)$ the square of the centre-of-mass energy.
Here $\theta$ is the angle between the propagation directions of the two
interacting photons, $E_{\gamma}$ the energy of the $\gamma$-ray photon,
$\epsilon$ the energy of the target photon and $m_{\rm e}$ the electron rest
mass.

The propagation direction of the $\gamma$-ray photon is towards the observer
and the direction of the target photon depends on the position of the stars
inferred from the orbital parameters. The eccentricity of the system was
assumed to be $0.9$ (Damineli et al. 2008), the semi-major axis
$16.64\text{\,}\mathrm{ua}$ (Hillier et al. 2001), the inclination of the
orbit $i=$135\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and the position angle
$\phi=$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (Madura et al. 2012). For
each star, modelled as a black body, Equation 6 is integrated over the stellar
surface, which is assumed to produce photons uniformly. This means that each
point of the stellar surface produces the same amount of photons at the same
temperature. The stellar temperatures are the same as in O15 and W20,
$2.58\text{\times}{10}^{4}\text{\,}\mathrm{K}$ for the primary star and
$3\text{\times}{10}^{4}\text{\,}\mathrm{K}$ for the companion.

Figure 6 shows the transmissivity at $220\text{\,}\mathrm{GeV}$,
$620\text{\,}\mathrm{GeV}$ and $4\text{\,}\mathrm{TeV}$ for different
distances from the centre of mass of the system assuming a spherical uniform
$\gamma$-ray production. As can be seen, the transmissivity increases with
increasing radius as expected due to the decreased radiation energy density at
larger distances from the stars.

While at a distance of $25\text{\,}\mathrm{ua}$, the transmissivity at
$220\text{\,}\mathrm{GeV}$ is larger than at $620\text{\,}\mathrm{GeV}$, the
opposite is the case for distances above $\sim 75\text{\,}\mathrm{ua}$. This
is caused by the relative position of the two stars and their different
temperatures. $\eta$ Car B is hotter than $\eta$ Car A and therefore is more
important at lower energies than at higher energies. The relative absorption
from each star changes with different radii because of the stellar separation,
which is slightly more than $30\text{\,}\mathrm{ua}$, and because the centre
of mass of the system is around five times more distant from $\eta$ Car B than
from $\eta$ Car A. The relative absorption by target photons from $\eta$ Car B
compared to $\eta$ Car A increases with distance for the apastron position
creating the observed effect for the curves at $220\text{\,}\mathrm{GeV}$ and
$620\text{\,}\mathrm{GeV}$.

Figure 6: Transmissivity versus distance from the centre of mass of the binary
system for spherical emission and different energies at an orbital phase of
$0.5$.