The SRG/eROSITA diffuse soft X-ray background

Astronomy
&
Astrophysics
A&A, 690, A399 (2024)
https://doi.org/10.1051/0004-6361/202451045
© The Authors 2024
The SRG/eROSITA diffuse soft X-ray background
I. The local hot bubble in the western Galactic hemisphere
Michael C. H. Yeung1,⋆ , Gabriele Ponti2,1
, Michael J. Freyberg1
, Konrad Dennerl1
, Teng Liu3,4,1
,
Nicola Locatelli2
, Martin G. F. Mayer5
, Jeremy S. Sanders1
, Manami Sasaki5
, Andy Strong1
,
Yi Zhang1
, Xueying Zheng1
, and Efrain Gatuzz1
1
Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, 85748 Garching, Germany
2
INAF-Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italy
3
Department of Astronomy, University of Science and Technology of China, Hefei 230026, China
4
School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China
5
Dr. Karl Remeis Observatory, Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
Sternwartstraße 7, 96049 Bamberg, Germany
Received 9 June 2024 / Accepted 15 August 2024
ABSTRACT
Context. The SRG/eROSITA All-Sky Surveys (eRASSs) combine the advantages of complete sky coverage and the energy resolution
provided by the charge couple device and offer the most holistic and detailed view of the diffuse soft X-ray background (SXRB) to
date. The first eRASS (eRASS1) was completed at solar minimum, when solar wind charge exchange emission was minimal, providing
the clearest view of the SXRB.
Aims. We aim to extract spatial and spectral information from each constituent of the SXRB in the western Galactic hemisphere,
focusing on the local hot bubble (LHB).
Methods. We extracted and analysed eRASS1 spectra from almost all directions in the western Galactic hemisphere by dividing the
sky into equal signal-to-noise bins. We fitted all bins with fixed spectral templates of known background constituents.
Results. We find the temperature of the LHB exhibits a north-south dichotomy at high latitudes (|b| > 30◦
), with the south being
hotter, with a mean temperature at kT = 121.8 ± 0.6 eV and the north at kT = 100.8 ± 0.5 eV. At low latitudes, the LHB temperature
increases towards the Galactic plane, especially towards the inner Galaxy. The LHB emission measure (EMLHB) enhances approx-
imately towards the Galactic poles. The EMLHB map shows clear anti-correlation with the local dust column density. In particular,
we found tunnels of dust cavities filled with hot plasma, potentially forming a wider network of hot interstellar medium. We also
constructed a three-dimensional LHB model from EMLHB, assuming constant density. The average thermal pressure of the LHB is
Pthermal/k = 10 100+1200
−1500 cm−3
K, a lower value than typical supernova remnants and wind-blown bubbles. This could be an indication
of the LHB being open towards high Galactic latitudes.
Key words. ISM: bubbles – ISM: structure – local insterstellar matter – solar neighborhood – X-rays: diffuse background –
X-rays: ISM
1. Introduction
The diffuse soft X-ray background (SXRB) is known to be
a superposition of emission components spanning from the
length scale of AU to cosmic distances (e.g. McCammon &
Sanders 1990; Kuntz & Snowden 2000). All-sky analyses of the
SXRB have largely relied on ROSAT/PSPC broadband count
rates and ratios (e.g. Snowden et al. 1997, 1998; Kuntz &
Snowden 2000), especially for studies on the local hot bubble
(LHB; e.g. Snowden et al. 2000; Liu et al. 2017), as background
photons with energies ≲0.3 keV (R1: 0.11–0.284 keV; R2: 0.14–
0.284 keV) are readily absorbed by the wall of the local bubble
and all R12 counts are effectively of local origin. The bright-
est soft X-ray line feature in the SXRB is the O VII triplet at
0.57 keV. This is an important diagnostic, as it encapsulates
emissions from the LHB; the background Galactic emission,
including the Milky Way circum-galactic medium (MW CGM);
and depending on sight lines, intervening Galactic structures.
⋆
Corresponding author; myeung@mpe.mpg.de
Hence, extracting information for a particular component using
broadband count rates and ratios alone becomes challenging and
heavily dependent on the knowledge and the accurate subtrac-
tion of the rest. Notably, the discovery of the time-variable solar
wind charge exchange (SWCX) component has aggravated the
issue even further (Lisse et al. 1996; Cravens 1997; Dennerl et al.
1997).
X-ray observatories with charge couple device (CCD)
cameras, such as XMM-Newton/EPIC, Chandra/ACIS, and
Suzaku/XIS, have demonstrated the importance of spectral reso-
lution to decompose the SXRB into their respective components
(e.g. Lumb et al. 2002; Markevitch et al. 2003; Henley & Shelton
2008; Yoshino et al. 2009), but they are restricted to limited
‘pencil-beam’ sight lines due to their relatively small fields of
view (FoVs) and observing strategies. Only since the last decade
have a substantial number of pointings been accumulated to
enable a more holistic view of the SXRB (Henley & Shelton
2010, 2012, 2013; Nakashima et al. 2018; Gupta et al. 2023; Pan
et al. 2024). A notable exception is Halosat, where a large por-
tion of the sky is covered in 333 pointings by virtue of its large
A399, page 1 of 31
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Yeung, M. C. H., et al.: A&A, 690, A399 (2024)
FoV (∼10◦
) whilst achieving CCD-type energy resolution using
non-imaging silicon drift detectors (Kaaret et al. 2019, 2020;
Ringuette et al. 2021; Bluem et al. 2022).
Currently, high spectral resolution observations of the SXRB
remain extremely difficult to obtain without telescopes of a large
grasp1
. However, successive sounding rocket launches of the
X-ray Quantum Calorimeter (XQC) (McCammon et al. 2002;
Crowder et al. 2012; Wulf et al. 2019) have shown promise of
being able to separate the SWCX contribution from the LHB
emission beyond CCD energy resolution using C VI Ly-α, γ and
the fine structure lines within the O VII triplet, despite their short
exposure times.
The local interstellar medium (LISM) is known to be a vol-
ume devoid of neutral gas – also known as the local cavity (LC)
or the local bubble (LB) (e.g. see a review by Frisch et al. 2011).
This cavity is instead filled by a ∼0.1 keV (∼106
K) plasma (e.g.
Snowden et al. 1990, 1997; McCammon et al. 2002; Liu et al.
2017; Yeung et al. 2023). The electron density of this plasma
is uncertain without the assumption of its line-of-sight density
profile. However, shadowing studies of molecular clouds on the
wall of the LC at different distances have consistently shown that
a uniform electron density of ∼4 × 10−3
cm−3
is a reasonable
assumption, as probed by various sight lines (Snowden et al.
2014; Yeung et al. 2023). There is a growing number of stud-
ies that support a formation scenario where dozens of supernova
explosions create and sustain the LHB (e.g. Fuchs et al. 2006;
Zucker et al. 2022; Schulreich et al. 2023), employing a com-
bination of star cluster traceback, numerical simulations, and
matching supernova-produced radioisotopes found in Earth’s
crust.
Solar wind charge exchange has long been the biggest source
of uncertainty revolving around the X-ray measurements of the
LHB, as its spectrum resembles that of the LHB despite being
non-thermal in nature (see reviews by Dennerl 2010; Kuntz 2019,
and references therein). SWCX can be broadly separated into
two categories: magnetospheric and heliospheric. The former
refers to the emission from solar wind ions interacting with
neutrals in Earth’s exosphere and the latter with the inflowing
neutral interstellar medium (ISM) into the solar system. Obser-
vations conducted by most X-ray missions are prone to both
kinds of SWCX emissions. An exception is eROSITA (Predehl
et al. 2021), which is on board the Spectrum-Roentgen-Gamma
(SRG) (Sunyaev et al. 2021) observatory. SRG orbits around
the Sun-Earth Lagrangian point L2. SRG/eROSITA always
pointed perpendicular to the Sun-Earth line during all-sky sur-
veys (eRASSs). Hence, it never looked through the Earth’s
exosphere, avoiding the magnetospheric SWCX. No evidence of
SWCX emission from the magnetotail has been found thus far
(Yeung et al. 2023; Dennerl et al., in prep.). Heliospheric SWCX
indeed caused variations in the SXRB in eRASSs, and it corre-
lates with the solar cycle and ecliptic latitudes (Ponti et al. 2023;
Yeung et al. 2023). Studies using other soft X-ray instruments
have found the same correlation between solar cycles and helio-
spheric SWCX (Qu et al. 2022; Ueda et al. 2022; Pan et al. 2024).
The first eRASS (eRASS1) was completed during solar min-
imum and, on average, exhibits a low amount of heliospheric
SWCX (Ponti et al. 2023; Yeung et al. 2023). A parallel work
(Dennerl et al., in prep.) will rigorously present and discuss the
SWCX contributions in eRASSs. In this work, we leverage this
advantage and model the SXRB in eRASS1 without the SWCX
contribution, which became non-negligible in later eRASSs.
1 Grasp is defined as the product of FoV and effective area.
The hot phase of the CGM in a spiral galaxy is believed to
trace gas from both feedback processes and the shock-heated
intergalactic medium out to its virial radius (e.g. see Putman
et al. 2012, for a review). The hot halo gas is also predicted to be
at approximately the virial temperature (for the MW, ∼0.2 keV)
and holds a large fraction of its baryons. Numerous X-ray obser-
vations have confirmed the existence of this phase via both
emission (e.g. Yoshino et al. 2009; Henley & Shelton 2010,
2012, 2013; Miller & Bregman 2015; Nakashima et al. 2018;
Kaaret et al. 2020; Ponti et al. 2023) and absorption studies
(e.g. Bregman & Lloyd-Davies 2007; Yao & Wang 2007; Yao
et al. 2008; Sakai et al. 2012; Miller & Bregman 2013; Fang &
Jiang 2014; Fang et al. 2015). Recently, there have been growing
reports of the presence of an additional thermal component in the
CGM, at ∼0.7 keV (e.g. Das et al. 2019; Bluem et al. 2022; Ponti
et al. 2023), which could alternatively be attributed to coronal
emission from M-dwarfs (Wulf et al. 2019). One of the prime
goals of our spectral analysis is the characterisation of the two
CGM components. The results and discussion concerning the
MW CGM will be presented in Paper II (Ponti et al. in prep.) of
the series.
Our understanding of the cosmic X-ray background (CXB)
at the low-energy end (0.5–2 keV) has taken a giant leap for-
ward with Chandra and XMM-Newton surveys, resolving ∼80%
of the CXB sources (De Luca & Molendi 2004; Luo et al. 2017;
Cappelluti et al. 2017). The resolved sources are mostly identi-
fied as active galactic nuclei (AGNs). Still, contributions from
galaxies, galaxy groups, and clusters are expected to rise below
1 keV, steepening the slope at the softest end (Gilli et al. 2007).
eROSITA is an instrument specialised in the soft X-ray band. As
a result, we expect to be able to measure such a steepening and
subsequently apply the result to refine our measurements on the
Galactic components.
In this work, we report on the spectral analysis of the
eRASS1 data in the western Galactic hemisphere, being as
spatially continuous as possible by extracting a large number
(∼2000 bins) of high S/N spectra. This work is the first of a
series focusing on the various components of the SXRB seen by
eROSITA. In particular, this paper lays out the data extraction
and the general methodology adopted in the spectral modelling
in Sect. 2 and 3, which is the backbone for the series. Then, in
Sect. 4, we report on the results, which are mainly focused on,
but not limited to, the LHB and the hot LISM, with interven-
ing discussions and interpretations. The results on the CGM, the
Galactic corona, and the eROSITA bubbles will mainly be pre-
sented in other works in the series (Ponti et al., in prep.; Yeung
et al., in prep.). Hence, they are only mentioned in this work
when they directly affect results on the LHB.
2. Data selection and spectral extraction
We study the diffuse soft X-ray background from the eRASS1
data of processing version c020 of the western Galactic hemi-
sphere (180◦
< l < 360◦
). We cleaned the data following a few
criteria. First, we discarded the data from TM5 and TM7 due
to optical light-leak (Predehl et al. 2021), and removed good-
time-intervals (GTIs) with count rates >1.435 counts s−1
deg−2
in the 4–9 keV band to minimise contaminations by flares.
The particle background dominates this band, and the thresh-
old corresponds to 1.5 times the level of particle background
in this band (Yeung et al. 2023). The eROSITA-DE consor-
tium has also released a list of temperature-sensitive or bright
pixels that occasionally produce artefacts but are not officially
flagged as bad pixels. They are mostly from TM4, which suffered
A399, page 2 of 31

from a major micrometeoroid hit (Freyberg et al. 2022). We
rejected these sensitive pixels in addition. Last but not least, we
masked regions with overdense source detection (Merloni et al.
2024) and positions of known galaxy clusters with R500 ≳ 3′
as
described in Bulbul et al. (2024, and references therein). The
overdense source detection regions could be regions within or
near extended sources, such as supernova remnants or artefacts
caused by bright point sources, which triggered a high density of
spurious source detections.
Subsequently, we defined our spatial binning of spectral
extraction using the software contbin (Sanders 2006), with the
primary aim of dividing the western Galactic hemisphere into
bins of approximately constant S/N in the diffuse soft X-ray
emission, instead of imposing a regular grid system such as the
skytile system adopted by the standard products of eROSITA. For
our analysis, contbin also has the advantage of defining bins
with edges more closely following distinct features (for example,
from superbubbles, supernova remnants etc.) and being com-
putationally efficient compared to traditional Voronoi binning
codes. The binning was done on the eRASS1 0.2–0.6 keV dif-
fuse emission count map (all detected sources masked2
) after
subtracting the expected counts from the non-X-ray background
measured from the filter-wheel-closed data (Yeung et al. 2023),
as this band contains the bulk of the emissions from the LHB
that eROSITA observes. This can be written explicitly as
S (r) = C(r) − Bnonvig(r) (1)
= C(r) − Enonvig(r) × RFWC(r), (2)
where r, S , C and Bnonvig denote the sky position, signal, total
counts from diffuse emission, non-X-ray or non-vignetted back-
ground counts respectively. Bnonvig can be further written as a
product of non-vignetted exposure time (Enonvig) and the count
rate of the filter-wheel-closed background (RFWC). We estimate
the corresponding noise map N(r) for the S/N calculation using
equation (4) of Sanders (2006) as adopted from Gehrels (1986),
that is,
N(r) =
q
g[C(r)] + g[Bnonvig(r)] , (3)
where
g(c) = 1 +
r
c +
3
4
!2
(4)
is an estimation of the upper limit of the squared uncertainty
on c counts in Poissonian statistics. Before binning, the maps
were projected into the zenithal equal area (ZEA) projection.
Contour-binning yielded 2010 bins larger than 1 deg2
, which we
consider valid bins for spectral analysis. 1 deg2
is approximately
the eROSITA field-of-view. This selection primarily removed
areas near the south ecliptic pole and the Large Magellanic
Cloud where the exposure time is maximal due to the overlap-
ping of the scanning loci in eRASSs but are not representative of
the general SXRB.
2 More precisely, the masking of ‘all’ detected sources is done by
merging the CheeseMask images from the standard eSASS pipeline
from all the skytiles and project them into a HEALPix map of Nside =
4096 (pixel size ⋍51′′
) using nearest-neighbour interpolation. Con-
tribution from the masked pixels is then removed from the final
diffuse emission count map after downsampling the HEALPix map from
Nside = 4096 to Nside = 256.
0°
30°
60°
60°
30°
0°
-30°
-60° -60°
-30°
0 2 4 6 8 10
countss 1
deg 2
Fig. 1. Contour-binned eRASS1 0.2–0.6 keV band surface brightness
map in zenithal equal-area projection. Locations of big galaxy clusters,
overdense regions in source detection, and bins with sky area less than
1 deg2
were masked. Counts from the non-X-ray background and all
eRASS1-catalogued sources (Merloni et al. 2024) were also removed
from this image (but not in the spectra; see Sect. 2 for more details).
Fig. 2. Sky area distribution of the contour bins. It can be approximated
by a log-normal distribution, as shown with the red line.
Fig. 1 shows the contour-binned eRASS1 0.2–0.6 keV band
surface brightness map of the valid bins. Large soft X-ray emit-
ting structures such as the eROSITA bubbles (a pair of bubbles at
l ≳ 290◦
in the north and l ≳ 320◦
in the south), Antlia supernova
remnant (l, b)∼(275◦
, 15◦
), Monogem Ring (l, b)∼(200◦
, 8◦
) and
Orion-Eridanus Superbubble (l, b)∼(205◦
, −30◦
), and the Galac-
tic disc stand out in stark relief. The sky area distribution of the
valid contour bins is shown in Fig. 2. The median bin size is
∼7 deg2
. All bins with an area less than 1 deg2
were removed.
The distribution can be well approximated by a log-normal
function, as shown by the red line.
A399, page 3 of 31

Fig. 3. Example spectra outside (left) and inside (right) of the eROSITA bubbles, overlaid with the best-fit spectral models labelled in the legends.
We modelled the eROSITA bubbles with an additional thermal component in brown. Both spectra have been divided by the effective area, aiming
to bring the ROSAT R1 and R2 count rates into a reasonable range. This has the disadvantage of creating the fictitious jump of the instrumental
background component above the gold absorption edge at ∼2 keV. The horizontal error bars at the top left corners of the figures reflect the width
of the ROSAT R1 and R2 bands.
Possible fluctuations contributed by bright X-ray sources
should be removed from the analysis of diffuse emission. How-
ever, as is discussed in more detail in Sect. 3.2, removing sources
alters the shape and normalisation of the cosmic X-ray back-
ground (CXB). Masking sources above a flux limit threshold
substantially higher than the eRASS1 flux limit (Flim,0.5−2 keV ∼
10−14
erg s−1
cm−2
; Merloni et al. 2024) could avoid the prob-
lem of having an exposure-dependent CXB component in the
spectra. In order words, if one were to mask all detected
sources, the CXB resolve fraction would depend on the expo-
sure depth, causing spatially correlated CXB normalisation and
photon index variations. As such, we chose to mask only sources
with fluxes F0.5−2 keV > 10−12
erg s−1
cm−2
and detection like-
lihood of DET_LIKE_0 > 10 in the eRASS1 source catalogue
(Merloni et al. 2024), using circular masks of radius four times
their aperture photometry extraction radii (4 × APE_RADIUS_1 ≃
2′
)3
during spectral extraction. This choice minimises the spec-
tral fluctuations introduced by bright sources while maintaining
a largely uniform CXB component in all spectra. This masking
was applied during spectral extraction and differs from the one
used to create the diffuse emission map for contour-binning.
We extracted the spectrum and its auxiliary response file for
events of all valid patterns (pat=15) of each contour bin by pro-
viding a point source cheesemask and a contour bin mask that
demarcates the bin profile to the eSASS task srctool. They
serve as the basis of all analyses in this work.
Last but not least, we also utilised publicly available ROSAT
R1 and R2 diffuse background maps in our spectral fitting
(Snowden et al. 1997). Despite ROSAT/PSPC’s poor spectral
resolution, it provides a larger grasp than eROSITA at ener-
gies ≲0.3 keV (see Fig. 10 of Predehl et al. 2021). We found the
addition of ROSAT data helps break the degeneracies between
the LHB and the CGM components that arose in some low
absorption regions (see Sect. 4.3).
The R1 and R2 band count rate and sigma maps were used
to provide two more data points in each spectrum, with some
manipulations. In detail, we began by binning the maps using
3 The masking radius of four times APE_RADIUS_1 is ⋍2′
for most
sources since APE_RADIUS_1 relates directly to the encircled energy
fraction (0.75) set during the source detection pipeline, which corre-
sponds to ⋍0.
′
5 for the point-spread-function of eROSITA.
the same set of contour bins, then we converted the map unit of
count Ms−1
arcmin−2
to a flux unit of erg s−1
cm−2
by assuming
a 0.1 keV apec spectral model and multiplication with the bin
area. The use of a apec model (Smith et al. 2001) is motivated by
the approximation that the contribution from the LHB dominates
R1 and R2 counts. As the last step, we used the ftflx2xsp task
in FTOOLS to create Xspec/PyXspec-ready spectra and diago-
nal response matrices to enable simultaneously fitting with the
eROSITA spectra. The bin edges of the R1 and R2 maps were
taken from (Snowden et al. 1994) where the band response drops
to 10% of the peak values, namely, 0.11–0.284 keV for R1 and
0.14–0.284 keV for R2.
3. Spectral analysis
Attempting to decompose the diffuse emission from half the
sky both spatially and spectrally is an ambitious task. As the
first study to do so, we decided to employ a more conventional
approach in the analysis, whereby treating the contour bins as
independent during spectral fitting. Additionally, we adopted a
fixed number of spectral components to fit the spectra from all
contour bins. These components include four conventional X-
ray background components (e.g. Gupta et al. 2021; Bluem et al.
2022; Ponti et al. 2023; Yeung et al. 2023): (1) the local hot bub-
ble (LHB), (2) the Milky Way’s halo or (warm-hot) CGM, (3) the
Galactic corona or the hot component of the CGM (COR) and (4)
the cosmic X-ray background (CXB), and the non-X-ray back-
ground modelled by (5) the eROSITA filter-wheel-closed (FWC)
background models (Yeung et al. 2023). The SWCX emission
during eRASS1 was weak, as it was the time of solar minimum,
as shown by Dennerl et al. (in prep.) in a detailed eRASS1–
eRASS4 half-sky analysis, Ponti et al. (2023) in the eFEDS field,
and Yeung et al. (2023) in three giant molecular cloud sight lines.
Therefore, we did not include an SWCX component in our spec-
tral fits. In addition, we included an extra thermal component for
contour bins overlapping with the eROSITA bubbles (eROBub)
(Sect. 3.4).
3.1. Description of model components
Fig. 3 shows two example spectra to illustrate our spectral
templates, one outside the eROSITA bubbles and one inside.
A399, page 4 of 31

The discussion in the remainder of this Section deals with the
detailed description of these model components.
3.1.1. Local hot bubble
The LHB is a foreground component and was thus modelled
as an unabsorbed optically thin plasma in collisional ionisation
equilibrium (CIE) using the apec model (Smith et al. 2001)
using AtomDB version 3.0.9 (Foster et al. 2012). Its temperature
is allowed to vary freely only below 0.15 keV, for contour bins
that have log10 (NH/cm−2
) < 20.5, estimated from a combination
of HI4PI and Planck dust radiance map (see Sect. 3.1.2). The
introduction of this bound was to prevent the LHB component
from ‘switching’ with the CGM component in low column den-
sity (NH) regions. We refer the reader to Sect. 4.3 for a detailed
discussion. While this effect was mitigated by introducing the
ROSAT R1 and R2 bands into the spectral fitting, this degener-
acy remained in some low NH regions and necessitated the use
of a hardbound. Its emission measure (EM) was left to vary in
all locations. The abundance of the LHB is assumed to be solar.
In Appendix A, we loosen our CIE prescription of LHB to
see if there is evidence of non-equilibrium ionisation (NEI) in
our data. In short, we did not find a clear indication of NEI
despite it being the general expectation from simulations of the
ISM (e.g. de Avillez & Breitschwerdt 2012; Breitschwerdt & de
Avillez 2021). We attribute this to the insufficient spectral res-
olution to resolve the emission lines and their ratios, which are
crucial diagnostics of NEI.
3.1.2. Absorption of background components
For the X-ray background components (CGM, COR, CXB and
eRObub), we adopted the simplifying assumption that they are
absorbed by the same NH layer within each bin. In addition, their
absorption was modelled by the disnht model (Locatelli et al.
2022), which behaves identically to tbabs (Wilms et al. 2000)
but with a lognormal distribution with mean log NH and width
σlog NH
. We believe this treatment is more realistic than a single
NH as some of our bins cover a large sky area and a range of NH
within the field. While log NH is left to vary during the spectral
fits, we fixed the value of σlog NH
in each contour bin according
to the following estimation.
We estimated NH independently from our X-ray spec-
tral measurement assuming NH = NHI + 2NH2
. We adopted
the NHI information from 21 cm line measurement by HI4PI
Collaboration (2016), while NH2
was estimated using the con-
version given in Willingale et al. (2013):
NH2
= 7.2 × 1020
"
1 − exp −
NHIE(B − V)
3 × 1020 cm−2
!#1.1
, (5)
where E(B − V) was taken from the conversion from the Planck
dust radiance (R) map using E(B − V)/R = 5.4 × 105
given in
Planck Collaboration XI (2014). We refer to this NH estimation
simply by HI4PI NH hereafter for brevity. We then estimated
σlog NH
as the standard deviation of HI4PI NH within the area of
each contour bin. In other words, our estimated value of σlog NH
accounts only for the spatial variation in the column density and
can be treated as a lower limit to the ‘genuine’ variation, which
should include the line-of-sight component.
We have indeed attempted to let σlog NH
freely vary in the fits.
However, this choice brought about two unforeseen issues:
1) We found the fitted log NH to be higher than total HI4PI
NH by ≳0.3 dex, usually at high σlog NH
areas (≳0.6 dex) near
Fig. 4. Values of σlog NH
used in our spectral fitting, considering only
the spatial spread. They were computed using a combination of NHI
information from HI4PI and NH2
information inferred from Planck. (See
Sect. 3.1.2.)
(l,b)∼(300◦
,−20◦
), and consequently, boosted the CXB above a
level allowed by cosmic variance.
2) A significant number of contour bins resulted in a vanishing
LHB component, which we considered unphysical, especially
since their occurrences appear random.
On the other hand, these issues were not present when σlog NH
was fixed using the aforementioned method. Therefore, we kept
σlog NH
fixed throughout our spectral analysis.
Fig. 4 shows the values of σlog NH
in all contour bins. A clear
decreasing trend can be seen as a function of Galactic longitude.
This is primarily caused by the larger contour bins away from
the Galactic centre, capturing larger spatial spread in NH. The
scatter plot in Fig. 5 demonstrates this correlation. Of course,
the areas of the contour bins are dictated by the S/N in the soft
band, affected by both exposure time and soft band intensity in
the sky, which happens to be lower between 250◦
≲ l ≲ 180◦
.
3.1.3. Milky Way’s circum-galactic medium
The CGM component was modelled using an apec model with
the abundance fixed at 0.1 Z⊙. The low abundance was found to
be required by the eFEDS data (Ponti et al. 2023) and is shown to
be stable against the choice of the emission models of optically
thin plasma in CIE (for example, Raymond-Smith, Mekal) in
Paper II (Ponti et al., in prep.). The main impact of assuming a
different abundance is reflected primarily in the emission mea-
sure. The behaviour is fairly linear; for instance, the conventional
choice of Z = 0.3 Z⊙ would lower the emission measure by about
three times compared to Z = 0.1 Z⊙.
Several recent publications have found a Galactic corona or
a hotter CGM component necessary to reproduce the spectra of
the SXRB (e.g. Gupta et al. 2021; Bluem et al. 2022; Ponti et al.
2023). We refer the reader to Paper II for an in-depth discus-
sion. In this paper, it suffices to note that this component was
necessary for reproducing the spectra. We modelled the Galactic
A399, page 5 of 31

100 101 102
Sky area (deg2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
log(N
H
)
(dex)
200
220
240
260
280
300
320
340
l
(deg)
Fig. 5. Scatter plot of σlog NH
against solid angle of contour bins. A posi-
tive correlation between σlog NH
and the contour bin area can be seen, as
larger bins tend to capture a larger spread in NH. The points are coloured
according to their Galactic longitudes.
corona component using an apec model with solar abundance.
Our choice of S/N = 80 at the 0.2–0.6 keV band did not usually
provide sufficient constraints on the Galactic corona component
in the spectral fitting. Hence, throughout this paper, we kept its
temperature fixed at kTCOR = 0.7 keV.
3.1.4. Instrumental background
For the instrumental background, the spectrum for each TM was
modelled by its own FWC model to account for any TM-specific
features. The normalisation of the FWC model in the sky spec-
trum is related to that of the FWC spectrum by the ratio of their
BACKSCAL header keywords and was fixed accordingly in the
spectral fit. However, we noticed the Al-Kα fluorescence line at
1.49 keV in the CXB regions (see Fig. 6 and Sect. 3.2) is always
weaker than that in the FWC data for all TMs, by ∼20%–60%
depending on the TM. The precise cause of this phenomenon
is unknown. Still, it is suspected to be linked to the extra Alu-
minium put above the CCD when the filter wheel was rotated
to the CLOSED position compared to the FILTER position. How-
ever, this explanation cannot explain the relatively large spread in
the deficit between the TMs. To compensate for the Al-Kα line
deficit in the sky spectra, we singled out this line and refitted its
normalisation of all TMs as the first step of all our spectral fits.
It was then frozen in the subsequent optimisation of the rest of
the free parameters.
3.1.5. Summary of model parameters
We elaborate on the description of the CXB in Sect. 3.2. In
summary, two parametrisations (single (Γ = 1.7) and broken
power-law (Γ1 = 1.9, Γ2 = 1.6, Eb = 1.2 keV)) of the CXB com-
ponent could reproduce the data equally well at high Galactic
latitudes. Therefore, both were used in the spectral fits, and the
differences between the two were considered to be systematic
uncertainties.
There are a total of 7 or 9 free parameters in the spectral fits:
kTLHB, EMLHB, kTCGM, EMCGM, EMCOR, normCXB and log NH
for contour bins outside the eROSITA bubbles; and an addition
of kTeRObub and EMeRObub for bins inside the eROSITA bubbles.
Fig. 6. Similar to Fig. 1 but showing only regions selected to determine
the spectral shape of the CXB.
3.2. Treatment of the cosmic X-ray background
In our spectral analysis, detected sources with F0.5−2 keV >
10−12
erg s−1
cm−2
as well as known clusters from X-ray clus-
ter catalogues with R500 ≳ 3′
were masked (Liu et al. 2022;
Merloni et al. 2024; Bulbul et al. 2024, and references therein).
The choice of the flux threshold is more than an order of magni-
tude higher than the eRASS1 flux limit. With this limit, we can
assume eROSITA is complete in detecting sources above this
threshold in all look directions, and the corresponding resolved
fraction across the western Galactic hemisphere is uniform.
Masking of sources is expected to change the photon index of
the cosmic X-ray background (CXB) from the canonical value
of Γ ∼ 1.4–1.5 (Vecchi et al. 1999; Kushino et al. 2002; Hickox
& Markevitch 2006; Cappelluti et al. 2017); however, the high
flux threshold guarantees this change is not spatially dependent.
Fig. 6 shows the regions we chose to determine the CXB
model. These regions correspond to the spatial bins with centres
located above |b| > 30◦
, and are free of large-scale foreground
structures upon binning the western Galactic hemisphere using
contbin (Sanders 2006) with a target S/N of 400. In addi-
tion to the CXB model we focus on, we fitted the spectrum
of each region independently with freely varying LHB, CGM,
Galactic corona and absorption column densities similar to the
description in Sect. 3.1, with the omission of SWCX since the
contribution of SWCX in eRASS1 is low (Dennerl et al., in
prep.).
From the spectral analysis of the regions shown in Fig. 6,
we found that simple power-law and broken power-law mod-
els perform equally well in reproducing the data, but a double
broken power-law was unnecessary. Therefore, both simple and
broken power-laws were adopted as our CXB models. We delay
the details of this CXB analysis to Sect. 4.8.1.
3.3. Fitting procedures
Spectral fitting was done in PyXspec version 2.1.0 (Arnaud
1996; Gordon & Arnaud 2021). Lodders (2003) was used as
A399, page 6 of 31

Fig. 7. Finding chart for features discussed mainly in Sect. 3.4 and
4.2. The background image shows the emission measure of the LHB,
which is a proxy of the extent of the LHB and is relevant for discus-
sions (Sect. 4.2) on interstellar tunnels and anti-correlation with dust.
The demarcation of the eROSITA bubbles is shown by the turquoise
line, based on the 0.6–1.0 keV intensity map.
the reference for abundance, and absorption cross-sections from
Verner et al. (1996) were assumed.
We repeated the following fitting procedure for all the 2010
contour bins. After determining the normalisation of the Al-Kα
line (Sect. 3.1), we made a simultaneous fit of 5 eROSITA spec-
tra (5 TMs) and 2 ROSAT band fluxes (R1 and R2) for each con-
tour bin. The eROSITA spectra were fitted in the energy range
0.2–5 keV using Poissonian statistics (cstat) (Cash 1979), and
as the R1 and R2 data points were fluxes, χ2
-statistics was used.
We minimised the total statistics using the Levenberg-Marquardt
algorithm. Then, we ran a MCMC with 64 walkers, each with
1.6k steps, giving a total of 102.4k steps, using the Goodman-
Weare method (Goodman & Weare 2010). The walkers were
initially distributed in a Gaussian distribution around the best
fit from the minimisation. We found discarding the initial 80k
steps, albeit aggressive, was a conservative and uniform way
to ensure all parameters had moved past the burn-in phase and
reached convergence in the vast majority of bins. In this paper,
we report the median, 16th
and 84th
percentiles of the posterior
distribution as the most probable value, the lower and the upper
errors, respectively. If the posterior distribution is Gaussian,
these reduce to the mean and ±1 σ errors.
3.4. Treatment of the eROSITA bubbles
The enigmatic eROSITA bubbles (Predehl et al. 2020) emit ther-
mally in the soft X-rays at ∼0.3 keV, and we modelled it using the
Fig. 8. Spatial distribution of kTLHB in the high latitude regions
(|b| > 30◦
).
apec model at solar abundance4
. Combining with the LHB and
CGM, the multi-component spectral analysis in eRASS exposure
depth and spectral resolution inescapably entails model degen-
eracies because of their similar spectral shapes (0.1, 0.2 and
0.3 keV plasma in CIE). As such, we implemented a two-step
approach in fitting spectra within the eROSITA bubbles region,
where some spectral parameters outside the eROSITA bubbles
were passed in the form of Gaussian priors into spectral fits
within the eROSITA bubbles.
We began by defining the demarcation of the eROSITA bub-
bles using the 0.6–1.0 keV intensity map, where the eROSITA
bubbles are the most prominent. The turquoise line in Fig. 7
shows the demarcation. We fitted the spectra of the contour
bins outside the eROSITA bubbles before those inside, fol-
lowing the procedure described in Sect. 3.3. Subsequently, we
fitted the contour bins within the eROSITA bubbles by pass-
ing Galactic latitude-dependent priors based on the fit results.
More precisely, for a given contour bin within the eROSITA
bubbles centred at (l,b), we created a Gaussian prior for each
relevant parameter based on fitting results of all the bins out-
side the eROSITA bubbles centred within the range of b ± 5◦
.
This implementation reflects our recurring observation with the
eROSITA data that most spectral parameters are either constant
(for example, the CXB) or exhibit primarily Galactic latitudinal
dependence (for example, kTLHB, see Fig. 8). These priors were
only applied on kTLHB, kTCGM, EMCGM, EMCOR, normCXB. We
did not impose Galactic latitude-dependent priors on EMLHB,
kTeRObub, EMeRObub because in this paper, we primarily focus
on inferring the 3-dimensional (3D) structure of the LHB. The
eROSITA bubbles will be the focus of a forthcoming paper in
the series (Yeung et al., in prep.).
4 We assumed solar abundance for simplicity, following Lallement
et al. (2016). Indeed, there are reports of sub-solar abundances both from
observations (e.g. Miller et al. 2008; Kataoka et al. 2013; Gu et al. 2016)
and as expectations from simulations (e.g. Mou et al. 2023). We defer
the abundance measurement in eROSITA to an ongoing work (Yeung
et al., in prep.).
A399, page 7 of 31

3.5. Limitations of the current method
At the end of this section, we would like to emphasise the
data obtained by eROSITA is rich, especially for diffuse emis-
sion. A weakness of our method is that the inference of spectral
parameters was independent of locations. However, some param-
eters, such as the LHB temperature, are likely correlated for
bins in proximity based on the intuitive consideration that the
local ISM would preferentially be in similar conditions the
closer they are in space. There exist some potential methods
that could take this into account. For instance, one idea is to
use Bayesian hierarchical modelling and treat the proximity of
bins as a hyperprior. Another potentially interesting method is to
utilise an algorithm called the generalised morphological com-
ponents analysis (GMCA) or its variants (sGMCA, pGMCA)
to decompose the spectral components with the help of each
component’s intrinsically unique spatial distribution, which has
recently been demonstrated to disentangle spectral components
remarkably well in extended X-ray sources (Bobin et al. 2015;
Picquenot et al. 2019, 2021, 2023; Carloni Gertosio et al. 2023).
4. Results and discussion
The sheer number of spectra we analysed prevents us from dis-
cussing them individually. Therefore, we provide a webpage
hosted on the eROSITA Data Release 1 server that provides
all the relevant information, including visualisations and the fit-
ting results organised in downloadable tables, for readers who
are interested in the spectra, their associated model parameters,
uncertainties or fit quality of any our contour bins (Sect. 5).
4.1. Evidence of a variable LHB temperature
Fig. 8 shows the spatial distribution of kTLHB in the high
latitude regions, where the Galactic plane (|b| < 30◦
) and
regions overlapping with the Large Magellanic Cloud, known
supernova remnants or superbubbles (Antlia, Orion-Eridanus,
Monogem Ring, Vela; masked also in other parameter maps)
and the recently discovered structure surrounding the LMC,
dubbed the ‘Goat Horn Complex’ (Locatelli et al. 2024), were
ignored. In addition, regions with 1 σ LHB temperature or
emission measure fitting uncertainty σkTLHB
< 5 × 10−3
keV
or σEMLHB
< 5 × 10−5
cm−6
pc were removed. These criteria are
possible indications of sub-optimal spectral fits; the former is to
avoid regions with kTLHB pegged at edges of the uniform prior,
and the latter is useful for removing regions with EMLHB pegged
close to zero. A total of 788 valid bins remained following the
screening (36 bins were screened out).
The histogram of kTLHB of all the valid bins is shown in
the black line in Fig. 9. We report a median (and the 0.16 and
0.84 quantiles) kTLHB of 0.111+0.018
−0.015 keV. The observation of the
mean 1 σ fitting uncertainty (0.010 keV), as shown in the black
error bar, being significantly smaller than the σ of the distri-
bution (0.018 keV) indicates a variable kTLHB. An inspection of
Fig. 8 shows the primary origin of this variation is a large-scale
temperature gradient, with the northern Galactic hemisphere
being cooler than the south by ∼0.02 keV. This is further demon-
strated by dividing the histogram in Fig. 9 into the north (orange)
and south (green). The north-south temperature dichotomy is
evident. By overlaying the typical spectral fitting uncertainties
within each hemisphere on the figure, one can notice the width
of the temperature distributions within both hemispheres could
be attributed mainly to their respective spectral fitting uncer-
tainties, especially in the north. Hence, the dominating factor
0.08 0.10 0.12 0.14 0.16 0.18 0.20
kTLHB (keV)
0
10
20
30
40
50
60
N
valid
bins
Total
Gauss-fit
North
South
N Gauss-fit
S Gauss-fit
tot, typical
N, typical
S, typical
Fig. 9. Distribution of kTLHB in the high latitude regions (|b| > 30◦
).
The unfilled histogram outlined in black displays the distribution of
all the high latitude bins displayed in Fig. 8. The black dashed line
shows the Gaussian best fit for the distribution. The black data point
with error bars shows the typical (median) 1 σ spectral fitting uncer-
tainty, centred at the mean of the fitted Gaussian. It is plotted at a
height of exp (−1/2)Ngauss,peak = 0.606 Ngauss,peak for proper comparison
between the spectral fitting uncertainty and the width of the distribution.
The former (0.010 keV) is approximately half of the latter (0.018 keV),
demonstrating a genuine variation in the LHB temperature. The sam-
ple is divided into the northern (orange) and the southern (green)
Galactic hemispheres to demonstrate the primary source of variation
is a large-scale gradient instead of small-scale fluctuations. The north-
ern hemisphere is cooler (0.100 keV) than the south (0.122 keV), and
comparing their distributions with the respective typical spectral fitting
uncertainties shows that each hemisphere exhibits approximately con-
stant LHB temperature.
of the spread in the LHB temperature is a large-scale tempera-
ture gradient, but not bin-to-bin fluctuations. To complement the
systematic change in kTLHB observed from the projected map
and histogram, Appendix D highlights the spectral signature that
determines the LHB temperature using high-S/N spectra from
each hemisphere.
We tested if the northern and southern temperature distribu-
tions could be drawn from the same underlying distribution using
the two-sample Kolmogorov-Smirnov test. The test returned
a statistic of 0.693 and a p-value in the order of 10−102
≪
0.01. Thus, we can safely reject the null hypothesis that the
temperature dichotomy occurs by chance.
To quantify the intrinsic temperature distributions in each
hemisphere, we used a maximum likelihood approach to iden-
tify the mean and width of the Gaussian distributions, which
best reproduce the observed temperature measurements, includ-
ing their statistical errors. We constrained the mean temperature
in the northern and southern Galactic hemispheres to be kTN =
100.8 ± 0.5 eV and kTS = 121.8 ± 0.6 eV, respectively. We also
extracted the widths (1 σ) of the intrinsic temperature distribu-
tions, which are σN = 2.9+1.0
−1.3 eV in the north and σS = 8.5+0.7
−0.6 eV
in the south. This result reiterates the temperature dichotomy is
highly significant, as is evident from the precision with which
we can determine the mean temperature of each hemisphere. It
is also clear that the southern hemisphere exhibits a larger intrin-
sic temperature scatter than the northern counterpart. We would
like to emphasise that the statistical uncertainty of the means
and the widths of the intrinsic temperature distributions should
A399, page 8 of 31

0°
30°
60°
60°
30°
0°
-30°
-60° -60°
-30°
8 10 12 14 16
Pthermal (103 cm 3 K)
0.08 0.10 0.12 0.14 0.16 0.18 0.20
kTLHB (keV)
Fig. 10. Map of kTLHB including low latitude regions. We note that the
colour bar is scaled differently from Fig. 8. The thermal pressure is also
shown under the assumption of constant ne = 4 × 10−3
cm−3
.
not be confused with the fitting uncertainty of each bin. The
temperature dichotomy is highly significant because of the large
number of bins that sampled each hemispheric distribution well
despite the individual bin having a median fitting uncertainty
(⋍0.01 keV; see Fig. 9) larger than the intrinsic widths of the
distributions.
Having established a temperature dichotomy at high Galac-
tic latitudes, it is natural to ask if a smooth transition across the
Galactic plane connects them. We separated the discussion of
high latitude regions since we believe our measurement there is
relatively secure. Still, it is unclear how close to the Galactic
plane one can venture before one is heavily biased by com-
plexities such as multiple line-of-sight emitting and absorbing
components. We relax the |b| > 30◦
limit and plot all the valid
bins, including those on the Galactic plane in Fig. 10, using the
same screening applied to Fig. 8. Fig. 10 shows a remarkable
temperature enhancement towards the Galactic plane, especially
at l ≳ 270◦
. Inspection of the spectra and their posterior distri-
butions suggests the spectra have significant constraining power
on kTLHB down to at least 10◦
of the Galactic plane, albeit
a subjective choice. The column densities in these regions
(≳ 1021
cm−2
; see Fig. 11 for NH information) are adequate and
peak strongly at the first wall of LB absorption (∼100–200 pc)
(Lallement et al. 2022; Edenhofer et al. 2024), enabling us to
differentiate between the unabsorbed LHB and the absorbed
components, primarily the CGM. Therefore, we argue the tem-
perature enhancement towards the inner Galaxy is a real feature
of the LHB.
First, we must address why earlier studies did not observe
the north-south LHB temperature gradient using ROSAT data
(see Sect. 4.1.2 for a deeper literature discussion). The R2/R1
band ratio is the main tracer of LHB temperature using ROSAT
data, commonly assumed to be only contributed by the LHB
emission. Fig. 12 shows the R2/R1 band ratio as a function of
plasma temperature at solar abundance, using three common
plasma models, Raymond-Smith (Raymond & Smith 1977),
Mekal (Mewe et al. 1985, 1986; Liedahl et al. 1995) and apec
(Smith et al. 2001). The Raymond-Smith model was commonly
used by studies before and around the millennium (e.g. Snowden
et al. 1990, 1997, 1998) to infer the temperature of the LHB.
The calibration curve of the Raymond-Smith model in Fig. 12
shows the R2/R1 ratio is insensitive to temperatures above
0.13 keV. This could be one of the reasons why the LHB temper-
ature was not found to be ∼0.2 keV in earlier LHB temperature
measurements using ROSAT.
Another possibly more important reason is the flawed
assumption that only the LHB emission contributes to the R1
and R2 bands. To demonstrate this, we forward-modelled the
same models used to extract Fig. 10, using the R1 and R2 band
responses. The resulting R2/R1 band ratio map is shown in the
left panel of Fig. 13. The eROSITA bubbles appear distinctly
on the map, indicating that absorbed background structures can
shine through and alter the R2/R1 ratio. The north-south LHB
temperature dichotomy also disappears. It is fair to suggest the
forward-modelled R2/R1 map does not capture the primary mor-
phological features of the kTLHB map, and conversely, kTLHB
directly inferred from R2/R1 without contributions from the
background components could be biased. Furthermore, regions
within the eROSITA bubbles in the forward-modelled map com-
monly show R2/R1 > 1.4, a limit according to the calibration
curves in Fig. 12 that should not be crossed for an unabsorbed
plasma. This is another evidence of the absorbed components
contributing to the R1 and R2 bands, boosting the R2/R1 ratio.
The right panel of Fig. 13 displays the binned R2/R1 ratio of
the original ROSAT maps presented in Snowden et al. (1997),
which can be directly compared to our forward-modelled map.
Their resemblance reiterates that our kTLHB measurement is not
at odds with the ROSAT data, and including background compo-
nents in the modelling is essential to extract information on the
LHB.
Merely showing our measurement is consistent with ROSAT
does not necessarily mean it is robust, especially towards the
inner Galaxy at low latitudes. We want to show that the LHB
emission we measured at low latitudes must be local and not
heavily contaminated by background ISM. Fig. 14 shows the
corresponding EMLHB map. The details of it are discussed in
Sect. 4.2, but for the current purpose, it suffices to note the EM
at low latitudes within the Loop I contour in Fig. 7 is one of
the lowest in this hemisphere. Our LHB component is unlikely
to capture extra emission from the background ISM and returns
a lower EM. Fig. 15 demonstrates the column density reaches
1020
cm−2
at a very close distance (∼100 pc) and inspection of
the local dust radial profiles suggests most of the total NH is
located within the first 250 pc, corroborating that very few back-
ground ISM photons can contaminate our LHB measurement.
Indeed, because of the lower EMLHB, the uncertainties of the
kTLHB is larger (σkTLHB
∼ 0.03 keV), which is a well-known anti-
correlation that can also be readily observed from their posterior
distributions. These regions, excluding the ones that are exactly
lying on the Galactic plane, do not show poorer fit statistics
than the rest of the sky (see Sect. 4.9 and Fig. 16). Even if they
do, they are mostly due to larger residuals above ≳0.6 keV, and
not at energies where the LHB dominates. Therefore, we con-
servatively consider our inferred LHB temperature credible at
|b| > 10◦
and characterise it using spherical harmonics in the
next Section.
4.1.1. Spherical harmonic analysis
In Figs. 8 and 10, an apparent gradient or dichotomy in approx-
imately the north-south direction and an enhancement towards
A399, page 9 of 31

0°
30°
60°
60°
30°
0°
-30°
-60° -60°
-30°
19.5 20.0 20.5 21.0 21.5 22.0
log10(NH/cm 2)
0°
30°
60°
60°
30°
0°
-30°
-60° -60°
-30°
19.5 20.0 20.5 21.0 21.5 22.0
log10(NH/cm-2)
Fig. 11. Comparison of fitted NH and independent estimate of NH from H I and dust measurements. Left: fitted NH. Right: estimate of NH combin-
ing neutral hydrogen information from HI4PI (HI4PI Collaboration 2016) and E(B − V) information derived from Planck radiance map (Planck
Collaboration XI 2014).
Fig. 12. Calibration curves of R2/R1 band ratio of the Raymond-Smith
(Raymond & Smith 1977), Mekal (Mewe et al. 1985, 1986; Liedahl et al.
1995), and apec (Smith et al. 2001) models as a function of the temper-
ature of an unabsorbed plasma at solar abundance.
the Galactic plane can be observed, respectively. One way to
quantify a gradient in the sky is by fitting a combination of
spherical harmonics to the data given the uncertainties, and the
simplest case of a gradient is a dipole. We decided to separate
the analysis of the high latitude regions (|b| > 30◦
) from the ‘full
sample’ down to |b| > 10◦
, because we want to tackle two dif-
ferent issues: 1) quantify the significance of the (high latitude)
north-south gradient as modelled by a dipole, and 2) produce
an empirical multipole model that captures the main kTLHB
profile.
The significance of the high latitude north-south gradient can
be evaluated by independently fitting a monopole and dipole
model to the data. Then, the F-test could determine if the
improvement of using a dipole is statistically significant over the
null hypothesis of a uniform kTLHB model (monopole).
The expansion of any well-behaved functions (in our case,
the LHB temperature) on a sphere can be expanded into spherical
harmonics Ylm up to degree lmax by
kTLHB(θ, ϕ) =
lmax
X
l=0
m=+l
X
m=−l
almYlm, (6)
where alm is the complex coefficient associated with each
Ylm. The question is finding the set of alm that minimises the
χ2
between the observed data and spherical harmonic model
kTLHB(θ, ϕ), which can be written explicitly as
χ2
=
X
i







d(θi, ϕi) −
Plmax
l=0
Pm=+l
m=−l almYlm(θi, ϕi)
σ(θi, ϕi)







2
, (7)
where d(θi, ϕi) and σ(θi, ϕi) represent the ith
data point and uncer-
tainty associated with it. In our convention, ϕi = li and θi =
90 − bi, where li and bi are the Galactic longitude and latitude.
For real-valued functions, the conjugate property of spherical
harmonics means that al(−m) = (−1)m
a∗
lm, where the ∗ denotes
the complex conjugate. It follows that al0 has no imaginary part,
and the number of independent parameters required to describe
each spherical harmonics of degree l is 2l + 1 (each complex alm
requires two).
The most probable alm coefficients and their associated
uncertainties are found by running MCMC using the emcee
package (Foreman-Mackey et al. 2013), with the walkers ini-
tialised (with a small spread) at the minimum χ2
position found
by the Levenberg-Marquardt algorithm implemented in lmfit
(Newville et al. 2016).
We begin with evaluating the significance of the north-south
gradient, using the same regions (|b| > 30◦
) shown in Fig. 8.
A399, page 10 of 31

Fig. 13. Comparison of our forward-modelled R2/R1 band ratio map with the observation. Left: R2/R1 band ratio map calculated by folding our
best-fit (median) spectral models with the ROSAT R1 and R2 band responses. Right: R2/R1 data binned to the same contour-binning scheme
(Snowden et al. 1997).
0°
30°
60°
60°
30°
0°
-30°
-60° -60°
-30°
100 200 300 400
LLHB [(
ne
4 × 10 3 cm 3 )
2 pc]
1 2 3 4 5
EMLHB(10 3 cm 6 pc)
Fig. 14. Spatial distribution of EMLHB. Regions with EMLHB uncertainty
< 5 × 10−5
cm−6
pc were also masked. The solid black line indicates
the position of the ecliptic, and the two dashed lines represent a range
of ±25◦
around it in ecliptic latitude where the solar wind density is
expected to be high. The extent of the LHB under the assumption of
ne = 4 × 10−3
cm−3
is also shown.
We began by modelling kTLHB as a constant (monopole) in
the unmasked regions. We found a median kTLHB of 0.1154 ±
0.0003 keV with χ2
/d.o.f = 6.36 (915 d.o.f), a unacceptable fit.
Subsequently, we fitted the data with a dipole (lmax ≤ 1). The
most probable model and the corresponding residual normalised
by the bins’ fitting uncertainty are shown in Fig. 17. The χ2
/d.o.f
decreased to 4.01 (912 d.o.f) compared to the monopole. Using
the F-test and a significance level of 0.001, we deduced an F-
statistic of 179.7 > Fcrit = 5.5 with a p-value in the order of
10−16
. This suggests the dipole model is strongly preferred over
the constant model, and the presence of a north-south gradient is
statistically significant. However, one could still recognise sys-
tematic residuals in the northern hemisphere by inspecting the
residual image, reflecting the north–south gradient is not simply
a dipole.
Naturally, raising lmax can improve the fidelity of our spher-
ical harmonics model to reproduce the data more closely. We
included data as close as 10◦
from the Galactic plane for this
empirical model. With multiple trials of different lmax values, we
arrived at a lmax = 6 model that captures the main large-scale
features in the data reasonably well, presented in Fig. 18. It has a
χ2
/d.o.f of 3.11 (1560 d.o.f.). We emphasise that we do not asso-
ciate any physical interpretations with the multipoles. It merely
serves as an empirical model for the LHB temperature profile.
However, we speculate on the origin of the gradient in Sect. 4.1.4.
The model parameters can be found in Appendix B. Appendix C
presents the latitudinal profiles of kTLHB, where both the uncer-
tainties and scatters of kTLHB are shown to compare with the
spherical harmonics models.
4.1.2. Comparison with past observations
The most relevant references on the large-scale temperature vari-
ations of the LHB are Snowden et al. (1990), Snowden et al.
(2000) and Liu et al. (2017). Snowden et al. (1990) inferred a
temperature gradient of the LHB for the first time using the
Wisconsin B/C band intensity ratio. They reported a mean tem-
perature of 0.097 keV (106.05
K), and a dipole gradient pointing
towards (l,b) = (348.
◦
7,−11.
◦
2) going from 0.064 keV (105.87
K;
near Galactic anti-centre) to 0.127 keV (106.17
K; near Galac-
tic centre). With the advent of ROSAT All-Sky Survey data
(RASS), Snowden et al. (2000) compiled a catalogue of X-
ray shadows at high Galactic latitudes (|b| > 20◦
). With these
A399, page 11 of 31

Fig. 15. Distance at which the integration of NH reaches 1020
cm−2
in the Lallement et al. (2022) (left) and Edenhofer et al. (2024) (right) dust
cubes, as proxies of the extent of the local bubble. The empty regions in the map indicate that the integration does not reach NH ≥ 1020
cm−2
before
400 pc.
Fig. 16. Map of reduced-χ2
(χ2
/d.o.f) after rebinning each spectrum,
imposing that each spectral bin receives at least ten counts.
X-ray shadows and taking the simplifying assumption similar
to us, that all components except the LHB are absorbed by the
total Galactic NH, they arrived at an LHB temperature dipole in
a similar direction, but with the dipole spanning the range of
0.094(106.04
K)–0.116 (106.13
K), only ∼1/3 of that of Snowden
et al. (1990). We note that our dipole model has the dipole ampli-
tude in between the two studies (Adi = 0.0133 ± 0.0004 keV5
;
5 The full temperature range spans by the dipole model is 2Adi.
Table B.1). Liu et al. (2017) make use of both the RASS R2/R1
band ratio as well as the estimation of the SWCX contribution
from the DXL sounding rocket (Galeazzi et al. 2012) mission to
conclude that kTLHB is fairly uniform over the sky at 0.097 ±
0.019 keV (Bluem et al. (2022) recently lower this estimate
to 0.084 ± 0.019 keV using AtomDB version 3.0.9). Inspection
of the kTLHB map (left panel of their Fig. 6) from Liu et al.
(2017) shows enhanced temperature in the Galactic south pole,
but the enhancement appeared more localised than we observed
using eROSITA, possibly caused by the unsubtracted eROSITA
bubbles component.
Snowden et al. (1990, 2000) and our work show markedly
different dipole direction of the LHB temperature. While a quan-
titative comparison of Snowden et al. (1990, 2000)’s dipole
models with our dipole model is problematic because we found
the LHB temperature profile not fully following a dipole, a qual-
itative comparison suggests our observed dipole axis is almost
∼60◦
away from Snowden et al. (1990, 2000)’s. In contrast,
despite being similar to Snowden et al. (1990, 2000) in using
band ratio as a proxy of LHB temperature, Liu et al. (2017)
reports a temperature map that resembles ours more closely.
For Snowden et al. (1990), Wisconsin B (0.13–0.188 keV) and
C (0.16–0.284 keV) bands match the peak emitting energies of a
∼0.1 keV plasma and are largely (but not completely) unaffected
by the hotter emission from the CGM when taking the band ratio
as a proxy of LHB temperature. However, it was completed by
10 sounding rocket flights in ∼8 years, sampling various parts
of a solar cycle. We note that the analysis was done before the
realisation that the SWCX process could contaminate the SXRB.
Thus, how much of the B/C band ratio genuinely traced the
LHB emission is unclear. ROSAT R1 (0.11–0.284 keV) and R2
(0.14–0.284 keV) bands have similar energy coverage to the B
and C bands and are similarly sensitive to the LHB temperature.
Subtraction of the SWCX (comparing Fig. 11a of Snowden et al.
1997 and Fig. 3 of Liu et al. 2017) and background absorbed
emission (see Sect. 4.1) can significantly change the profile of
A399, page 12 of 31

Fig. 17. Dipole model and residual. Left: dipole model of kTLHB. Right: residual normalised by the 1 σ fitting uncertainty of each valid bin. The
black cross in both panels shows the direction of the dipole towards (l, b) = (291◦
, −26◦
).
Fig. 18. Similar to Fig. 17 but for the spherical harmonics model of lmax = 6. The data were fitted down to 10◦
from the Galactic plane. We note
that the colour bar of the left panel was scaled up to 0.2 keV, identical to Fig. 10 for comparison.
the R2/R1 band ratio. The dipole gradient Snowden et al. (2000)
found could be plagued by SWCX as RASS was conducted near
solar maximum.
Despite only covering the western Galactic hemisphere, it is
a positive sign that our LHB temperature profile shows compat-
ible morphology to the Liu et al. (2017)’s map, but with ade-
quate significance to suggest a gradient. However, the spherical
harmonic models we presented are a partial view without infor-
mation from the eastern Galactic hemisphere. The low-order
spherical harmonics would almost certainly change significantly
should this information become available, as they are most sen-
sitive to features spanning large angular scales by definition.
Nevertheless, we suspect the change is unlikely to reconcile our
model with Snowden et al. (1990, 2000)’s dipole as there is only
a very weak sign of longitudinal temperature dependence from
the eROSITA data.
4.1.3. Temperature anti-correlation with absorption column
density
The left panel of Fig. 11 shows the distribution of
log10(NH/cm−2
) inferred from our spectral fitting.
Neglecting the Galactic plane where both our and HI4PI
NH measurements are not particularly accurate due to modelling
simplification and self-absorption, the two NH maps possess
almost identical morphology, with the X-ray absorption inferring
A399, page 13 of 31

Fig. 19. Map of kTLHB upon fixing the NH to the HI4PI NH. The colour
scaling conforms to Fig. 8 to highlight the high latitude temperature
dichotomy.
a lower NH at high Galactic latitudes. By comparing the kTLHB
maps in Figs. 8 and 10 with the NH maps, one can immediately
notice the anti-correlation between the kTLHB and NH, especially
in the southern Galactic hemisphere. The hottest regions of the
LHB above b < −30◦
are where the NH are the lowest. Inter-
estingly, as is described in more detail in Sect. 4.2, the hottest
regions correspond to the largest EMLHB.
Given the morphological similarity of the southern hot patch
and the low NH region and the observation that the X-ray absorp-
tion column density there is consistently lower than the HI4PI
measurement, we suspected the cause of the hot patch was the
result of our inaccuracies in fitting the NH, biasing kTLHB in the
process. Therefore, we reran the spectral modelling of all con-
tour bins and imposed the condition that the NH must be fixed at
the HI4PI values.
We show the resulting map on kTLHB in Fig. 19. The kTLHB
gradient remains, with similar amplitude and direction. The
anti-correlation with NH remains clear. Not surprisingly, the
quality of the spectral fits became increasingly worse towards
the Galactic plane, as the assumption of using total NH alone the
line-of-sight becomes questionable. The result of this test indi-
cates the LHB temperature gradient is unlikely to be caused by
our NH determination, and in addition, leaving NH free during
spectral fitting was appropriate and necessary.
4.1.4. Speculations on the source of temperature gradient
The mechanism that set up this temperature gradient is unclear.
Still, it is not completely unexpected. Schulreich et al. (2023)
demonstrate in a sophisticated numerical study that sequential
supernova explosions could have created the LHB, with some
possibly exploding in the last ∼1–2 Myr. In their simulation, the
LHB temperature is not uniform in the present day; instead, it
shows a large-scale gradient that could span approximately an
order of magnitude. Therefore, the scenario of recent off-centre
supernova explosions can, in principle, explain our measured
temperature contrast. However, their simulated temperature gra-
dient direction differs from our measurement. They back-traced
the trajectories of the stellar populations, which most likely
hosted massive stars that expanded the LB using Gaia EDR3
data. Further, they predicted the massive stars’ explosion times
and positions by considering the initial mass function and stellar
isochrones. They used these informed explosion sites and times
as input parameters of their simulation; thus, the simulation’s
gradient direction is not arbitrary. Judging from their Fig. 4,
the simulated temperature gradient is primarily along the Galac-
tic centre-anticentre line, in contrast to our measurement in the
north-south direction. Whilst the uncertainty of the stellar trace-
back does not allow for the explosion sites in the south, shock
reflections from the thick LB shell and gas sloshing following
any explosions could easily change the direction of the tem-
perature gradient (Pacicco, M. & Schulreich, M., priv. comm.).
Pressure gradients in the LHB are washed out following the
sound crossing timescale (a few 0.1–1 Myr). Transient shocks
within the LHB exist in the simulation of Schulreich et al.
(2023). These shocks create hotter and denser plasma layers
and can be preferentially picked up in their X-ray emissions as
emission measure scales as n2
. The temperature dichotomy we
observe could be caused by these shocks in the south. It is likely
a matter of fine-tuning the simulations and choosing a correct
time stamp to reproduce the current LHB observables so that
these shocks appear in the right place and time. Last but not
least, the density of the plasma also displays a similar gradient as
the temperature in their simulation, which we do not see or have
not seen with the current instrumental sensitivity in shadowing
experiments (e.g. Yeung et al. 2023) that probed only very few
sight lines. Future observations in this direction will be essential
to confirm or reject this scenario.
The second possible scenario hinges on more assumptions,
including the density of the LHB and magnetic pressure within
it do not vary significantly in different directions. Consider a
simple static scenario where the LHB is not expanding and is
in pressure equilibrium with the surrounding ISM. The temper-
ature gradient, in this case, reflects a pressure gradient set up
by the surrounding ISM, where its profile can be traced by the
thermal pressure of the LHB, given by
Pthermal = nkTLHB (8)
=






ne +
X
A
nA






 kTLHB (9)
≃ 1.92nekTLHB, (10)
where nA is number density of the Ath
element and n = 1.92ne is
a common estimation of the total particle density6
(e.g. Galeazzi
et al. 2007; Snowden et al. 2014). The colour bar in the kTLHB
map in Fig. 10 shows also the resulting thermal pressure, assum-
ing ne = 4 × 10−3
cm−3
(Yeung et al. 2023). An enhancement of
pressure near the Galactic disc can be observed, and the pressure
decreases rapidly away from it. It makes intuitive sense that the
Galactic disc exerts a larger pressure on the LHB. Indeed, the
vertical pressure profile in the Solar neighbourhood is approx-
imately an exponentially decaying function from the midplane,
with a scale height of ∼500 pc (Cox 2005). The smaller high
6 The factor of 1.92 assumes the abundance in Anders & Grevesse
(1989). Opting for the abundance of Lodders (2003) (the abundance
reference we use for spectral fitting) results in a factor closer to 1.9.
The difference is dominated by the difference in Helium abundance:
(nHe/nH)AnGr = 0.0977 versus (nHe/nH)Lodd = 0.0792.
A399, page 14 of 31

latitude temperature or pressure gradient could also be explained
if the initial surrounding medium was not uniform before the
formation of the LHB. This scenario does not necessarily con-
flict with the off-centre supernova explosions scenario in the
simulation of Schulreich et al. (2023) and could both be at work.
4.2. Emission measure and extent of the local hot bubble
Fig. 14 shows the spatial distribution of the emission measure
of the LHB, EMLHB. In addition to masking known large super-
bubbles and supernova remnants, we also excluded regions with
σEMLHB
< 5 × 10−5
cm−6
pc. Regions that make up the latter are
usually biased, possibly spectral fits with vanishing LHB com-
ponents or MCMC chains that did not converge well. Before
delving into the correlation or anti-correlation of EMLHB with
other parameters, it is best to discuss EMLHB in the context of
the extent of the LHB.
EMLHB is directly related to the extent of the LHB in a given
look direction if one knows the line-of-sight density profile,
which is given by the following equation:
EMLHB =
Z
ne(l)nH(l)dl. (11)
By adopting the assumption of a fully ionised solar abundance
plasma (see Leahy et al. 2023, for caveats of this assumption),
ne/nH ≃ 1.2 and further assuming a constant density profile, we
obtained a simple relation between EMLHB and the extent of the
LHB L:
EMLHB =
n2
e L
1.2
. (12)
One could calibrate ne under the constant density assumption
using sight lines through various molecular clouds on the surface
of the LHB. Recent work suggests ne ≃ (4 ± 0.5) × 10−3
cm−3
,
moderately independent of the look direction (Yeung et al.
2023), which is the number we adopted in converting EMLHB
to the extent of the LHB. This is smaller than ne = (4.68 ±
0.47) × 10−3
cm−3
inferred by Snowden et al. (2014), but within
the uncertainties. The assumed electron density gives rise to the
largest systematic uncertainty in our 3D LHB model. The differ-
ence between the two numbers alone entails a 37% difference in
estimating the LHB distance.
Figs. 14 and 20 show the structure of the LHB assum-
ing ne = 4 × 10−3
cm−3
, in the ZEA projection and a 3D-
rendered surface respectively. For the latter, the LHB surface is
smoothed and interpolated into the masked regions using radial
basis function interpolation (scipy.interpolate.rbf(*,
smooth=0.3)) implemented in the Python package scipy
(Virtanen et al. 2020).
Despite its irregular shape, the LHB is systematically more
extended away from the disk, presumably because the denser
medium permeated along the Galactic plane prohibits its expan-
sion. The conventional picture of 1 MK plasma displacing colder
ISM phases in the solar neighbourhood first put forward by
Sanders et al. (1977) appears to explain our result excellently,
especially in the southern Galactic hemisphere. An almost per-
fect anti-correlation with Galactic NH can be observed in the
southern Galactic hemisphere, slightly less so in the northern
hemisphere.
For a more comprehensive view of the multiphase ISM in
the solar neighbourhood, we also compared our spatial model of
the hot phase of the ISM to the local dust maps, inferred pri-
marily from Gaia extinction data by Lallement et al. (2022) and
Edenhofer et al. (2024) independently. A sophisticated method
was used by Pelgrims et al. (2020) to trace the inner surface of
the local bubble (LB) from 3D dust maps, involving identifying
inflexion points in the differential extinction radial profiles and
an iterative refining process. In our analysis, we took a simpler
approach by identifying the LB extent at which the integration of
dust maps reaches a column density of NH = 1020
cm−2
, where
the optical depth is approximately unity at 0.2 keV. Both the
extinction cubes of Lallement et al. (2022) and Edenhofer et al.
(2024) were converted from AV to NH cubes using the conversion
NH = 2.21 × 1021
AV (Güver & Özel 2009). The extent of the LB
inferred from these two cubes is shown in Fig. 15.
We could see excellent agreement in a few areas (see the find-
ing chart in Fig. 7):
1) in the general direction of the Loop I superbubble or the
eROSITA bubbles, both the LHB plasma and dust cubes show
distances in the order of 100 pc. The LHB emission appears to
be absorption-bounded.
2) The LHB is much more extended in the low NH regions in the
southern Galactic hemisphere.
3) Around 240◦
≲ l ≲ 270◦
on the Galactic plane (∼ β Canis
Majoris interstellar tunnel; Gry et al. 1985; Welsh 1991), one
could match the larger extent of the LHB well with the dust
maps, both in position and morphology, especially when com-
pared with Edenhofer et al. (2024).
4) In the northern Galactic polar cap (l ≳ 60◦
), the LHB is more
extended when there is low NH.
5) Towards the constellation Centaurus at (l, b) ⋍ (315◦
, 25◦
),
one can see a hint of an extended tunnel, possibly connecting to
the Loop I superbubble. This feature can also be seen when inte-
grating the Edenhofer et al. (2024)’s cube in the right panel of
Fig. 15.
6) On the other hand, we could see in the region of (180◦
≲
l ≲ 240◦
, 15◦
≲ b ≲ 45◦
), little absorbing material is present
within 400 pc, yet the LHB does not extend as freely as in
regions towards the LoopI/eROSITA bubbles and the Galactic
poles. This could be a hint that the hot plasma is not completely
volume-filling, or its density is lower in this general direction, or
the dust wall of the LB in this direction is very low in density.
Indeed, a peak of low-density dust at ∼150 pc could be present
there as found by the recent work by O’Neill et al. (2024). We
elaborate on the interstellar tunnels in Sect. 4.6.
The anti-correlation of EMLHB and dust can also be appre-
ciated by looking at slices of 3D dust cubes. Fig. 21 shows
the shape of the LHB shell overlaid on the y = 0 slice in the
Edenhofer et al. (2024) dust cube. The extent of the LHB
matches the onset of extinction extremely well, filling gaps of
low dust density.
Liu et al. (2017) produced the latest X-ray LHB model before
our work. Consulting their EM map and great-circle cuts (their
Figs. 6 and 7), one can see despite the difference in method-
ology (spectral-fitting versus band-ratio) and instrument, the
inferred EM and shape of the LHB are remarkably similar. Nev-
ertheless, we have identified some areas that are interestingly
different:
(i) at l ∼ 240◦
we found a more distinct transition in LHB extent
between the Galactic plane (b ∼ 0, region of the β Canis-Majoris
tunnel) and a large-scale protrusion in the southern Galactic
hemisphere (b ≲ −30◦
) than Liu et al. (2017).
(ii) The aforementioned channel around (l,b)=(315◦
, 25◦
) did
not appear in the Liu et al. (2017)’s EM map. It was because
that region was categorised as contaminated by the background
Loop I superbubble and was excluded from the analysis. In fact,
the appearance of the tunnel was seen in the ROSAT R1+R2
A399, page 15 of 31

LHB lower error + kT
LHB upper error
100pc sphere
Sun
0.08
0.1
0.12
0.14
0.16
0.18
0.2
LHB
Temperature
(+-0.01
keV)
Fig. 20. Three-dimensional structure of the LHB in the western Galactic hemisphere assuming a constant density of 4 × 10−3
cm−3
. The inner
(opaque, coloured) and the outer (grey, translucent) surfaces show the ±1 σ uncertainty bounds of the distance under the constant ne assumption.
We note that the two surfaces only reflect the uncertainty in the spectral fitting but not in ne. We note that kTLHB is also colour-coded on the inner
surface. A sphere of 100 pc radius is shown around the Sun (yellow) as a ruler. An interactive version of this figure, including the comparison with
dust maps (not shown) can be accessed from online or the accompanying website (Sect. 5).
Fig. 21. A slice of the Edenhofer et al. (2024) cube at the x-z plane
overplotted with the silhouette of the LHB.
band map even after the SWCX subtraction (Liu et al. 2017, their
Fig. 2).
4.3. Degeneracy between the local hot bubble and the Milky
Way’s circum-galactic medium components
This Section discusses the degeneracy between the LHB and
CGM components and our mitigation methods. We begin by
laying out the observations that demonstrate this degeneracy. In
summary, this led us to incorporate ROSAT R1 and R2 bands
into our analysis, as well as imposing a uniform prior to limit
kTLHB < 0.15 keV for regions of low NH.
Inspecting the spectral fitting results with only eROSITA
data revealed a degeneracy between the LHB and the CGM com-
ponents in some contour bins. An example of this degeneracy
can be seen in Fig. 22, where we show the parameter correla-
tions in the spectral fits of Bin476 and Bin1875, located at (l,
b)=(257.
◦
9, 59.
◦
6) and (296.
◦
7, 47.
◦
6) respectively. Both regions
have log (NH/cm−2
) ≃ 20.5 as traced by the HI4PI survey (HI4PI
Collaboration 2016) using Eq. (5). One can see the posterior
A399, page 16 of 31

0.15+0.01
0.01
0
.
0
8
0
.
1
6
EM
CGM
0.06+0.02
0.01
0
.
1
5
0
.
3
0
kT
LHB
0.09+0.01
0.01
0
.
0
0
4
0
.
0
0
8
EM
LHB
0.002+0.001
0.001
0
.
1
2
0
.
1
6
kTCGM
2
6
8
0
2
7
0
0
Fit-stat
0
.
0
8
0
.
1
6
EMCGM
0
.
1
5
0
.
3
0
kTLHB
0
.
0
0
4
0
.
0
0
8
EMLHB
2
6
8
0
2
7
0
0
Fit-stat
2685+5
4
Bin476
0.18+0.01
0.02
0
.
1
0
0
.
1
5
EM
CGM
0.08+0.02
0.02
0
.
1
5
0
.
3
0
kT
LHB
0.14+0.08
0.03
0
.
0
0
3
0
.
0
0
6
EM
LHB
0.003+0.001
0.002
0
.
1
2
0
.
1
6
kTCGM
2
9
1
0
2
9
4
0
Fit-stat
0
.
1
0
0
.
1
5
EMCGM
0
.
1
5
0
.
3
0
kTLHB
0
.
0
0
3
0
.
0
0
6
EMLHB
2
9
1
0
2
9
4
0
Fit-stat
2935+9
10
Bin1875
Fig. 22. Corner plots showing LHB and CGM parameters are degener-
ate in regions of low NH. Top: Bin476 at (l, b)=(257.
◦
9, 59.
◦
6). Bottom:
Bin1875 at (l, b)=(296.
◦
7, 47.
◦
6).
distributions of kTCGM are bimodal in both bins. The lower
kTCGM peak is contributed by either partially in Bin476, or
completely in Bin1875, by samples that have kTLHB > kTCGM.
Similarly, the posterior distributions of kTLHB show either an
extended tail in Bin476 or a second peak in Bin1875, at tem-
peratures at least doubled of the main peak. We believe these
solutions are unlikely to represent the real picture, but are
caused by the similarity of the LHB’s and CGM’s apec models
observed at eROSITA’s energy resolution. The main differen-
tiating factor of LHB and CGM is the absorption that causes
CGM component to drop off at the low energy end. Therefore,
one would expect the two to exhibit some levels of degeneracy
at regions of low NH. To help break the degeneracy, one would
ideally go to energies lower than 0.2 keV, but eROSITA’s effec-
tive area there is small. However, ROSAT maintains significant
effective area until ∼0.1 keV, and is therefore more sensitive to
detect the drop off caused by absorption in the CGM component.
The details of how the ROSAT data were used were described
Fig. 23. The same plots and panel configuration as Fig. 22 but with
ROSAT R1 and R2 data.
in Sect. 2. We note that ideally, one should subtract the helio-
spheric SWCX contributions from the ROSAT R1 and R2 band
maps as in Uprety et al. (2016), who reported an all-sky aver-
age SWCX contribution of 30 ± 8% in R1 and 8 ± 10% in R2.
However, given that the average relative uncertainties in R1 and
R2 are 37% and 31% in our contour bins and that they only
contributed to two spectral bins in the fitting, we believe the
bias introduced by neglecting SWCX in ROSAT data is vastly
subdominant to the resulting fit parameters of our full spectral
analysis. Perhaps a more pertinent question would be whether a
pair of data points from ROSAT possesses sufficient capability to
resolve the degeneracy between the LHB and CGM components.
The answer is largely affirmative in many contour bins
plagued by the degeneracy. For a specific example and a direct
comparison, the top panel of Fig. 23 shows the corner plot
of Bin476 after including the ROSAT data. The high tempera-
ture tail in the posterior of kTLHB vanished and samples with
kTLHB > kTCGM disappeared. Motivated by the positive results,
we incorporated the ROSAT R1 and R2 bands into the spectral
analysis.
A399, page 17 of 31

Fig. 24. Corner plot of Bin1875 after using ROSAT data and imposing
a uniform prior on kTLHB below 0.15 keV.
Nonetheless, cases remained where this was inadequate,
especially at low NH regions. For instance, in Bin1875 (lower
panel of Fig. 23), the degeneracy remained. The posterior distri-
butions remained highly bimodal, albeit a longer MCMC chain
was needed to sample the second peak (kTLHB > kTCGM). Unfor-
tunately, we could not break this degeneracy using the currently
available data. Therefore, we decided to forbid kTLHB to go above
0.15 keV by imposing a uniform prior between 0.07–0.15 keV on
kTLHB. Not surprisingly, the resulting posterior distributions no
longer show a second peak, as shown in Fig. 24. This prior was
used in the fitting of all the contour bins where log (NH/cm−2
) <
20.5. To our knowledge, the LHB temperature has seldomly
been measured to be higher than 0.15 keV (e.g. Snowden et al.
1990, 2000; McCammon et al. 2002; Liu et al. 2017), and con-
versely, the CGM seldomly below 0.15 keV (e.g. McCammon
et al. 2002; Yoshino et al. 2009; Gupta et al. 2021; Bluem et al.
2022; Ponti et al. 2023), justifying our choice of prior in the low
NH regions.
4.4. Solar wind charge exchange
The halo orbit of SRG/eROSITA around the Sun-Earth
Lagrangian point L2 (Freyberg et al. 2020) avoids much of
the magnetospheric SWCX that originates from the interaction
of the solar wind ions with neutrals in the Earth’s exosphere
(Kuntz 2019). An image of the Earth’s exosphere in Ly-α taken
from 2348 R⊕ away shows that it is contained within 94 R⊕
(Kameda et al. 2017), while SRG never ventured within 175 R⊕
from the Earth (Freyberg et al. 2020), effectively immune to the
magnetospheric SWCX emission since it is always pointing per-
pendicular to the Sun-Earth-L2 line. Even avoiding the bulk of
the exosphere, in principle, SWCX can also occur on the sur-
face of the magnetotail, but we expect its effect to be small since
such a thin interaction and emission layer would incur bright-
ness variation in short timescales (minutes to hours), and we do
not observe obvious stripes along the ecliptic longitude lines in
the soft band in eRASS1 (eROSITA completes scanning a great
circle primarily along this direction every four hours).
We further ignored the effect of heliospheric SWCX in our
spectral analysis, which can indeed contaminate eROSITA data
in general. We verify that this is at least a satisfactory assumption
by looking at the EMLHB map a posteriori. It is well-known that
the solar wind density is higher at low heliographic latitudes than
the poles, especially around solar minima (e.g. Porowski et al.
2022). Therefore, unlike pointed observations, eRASS provides
a way of estimating the heliospheric SWCX contribution from
detecting extra emission preferentially along the ecliptic plane.
In Fig. 14, we overlaid the ecliptic in black on the EMLHB map,
and the dashed black lines mark the range of ±25◦
within the
ecliptic, where the denser, slow solar wind dominates during
solar minima (e.g. McComas et al. 1998, 2003). The fact that
there is no obvious enhancement within this ecliptic latitude
range suggests the heliospheric SWCX contamination in our
results is minor. Indeed, as we gradually approached solar max-
imum, eRASS3 and 4 data clearly show enhancement in this
band, but the presentation and the detailed analysis of this effect
will be presented in another work (Dennerl et al., in prep.).
4.5. Energetics and pressure balance within the local hot
bubble
We estimated the thermal energy of the LHB as follows:
Ethermal =
X
Pthermal∆V (13)
= 1.92ne
X
kTLHB(l, b)∆V, (14)
where ∆V is the volume occupied by the LHB plasma within a
contour bin. The EMLHB information is implicitly passed into ∆V
in the form of the extent of the LHB. We have incomplete cover-
age of the sky. Hence, we scaled up our estimation of Ethermal
to a total of 4π solid angle. Finally, we estimated Ethermal =
1.3+0.9
−0.5 × 1051
erg, where we have assumed ne = 4 × 10−3
cm−3
.
This is in the order of the energy released by a supernova explo-
sion. However, one must consider that the LHB’s size (radius)
is >100 pc, which is too large for a single explosion. There-
fore, the LHB is more likely to have been episodically reheated
and expanded by successive supernova explosions and simulta-
neously radiated its energy away to produce the current energy
content.
We presented the thermal pressure map in Fig. 10, using
Pthermal/k = nT. We found the mean pressure of the LHB is
Pthermal/k = 10 100+1200
−1500 cm−3
K. These numbers are consistent
with pressure measured in the sight line of (l,b)=(144◦
,0◦
) by
Snowden et al. (2014), hence supporting their finding of the LHB
being in pressure equilibrium with the local interstellar clouds
(LICs), after accounting for magnetic pressure as measured by
Voyager I outside of the heliosphere Burlaga & Ness (2014). But
we also point out that Snowden et al. (2014)’s sight line lies
on the Galactic plane, which we found to have systematically
higher thermal pressure. This pressure is also fully consistent
with measurements based on X-ray shadowing using eRASS:4
data (Yeung et al. 2023).
Compared with the latest numerical simulation of the LHB
by Schulreich et al. (2023), our measured pressure is consis-
tent with their simulated present-day pressure of 10 100 cm−3
K.
However, this results from combining their n and T, which are
lower and higher by roughly an order of magnitude compared
to our results, respectively. Despite the apparent coincidence,
their simulation shows how successive supernova explosions in
the Sco-Cen complex can lead to the large-scale temperature
gradient within the LHB, as we found in this work (Sect. 4.1).
The estimated Galactic midplane total ISM pressure is P/k ∼
(2.8 ± 0.7) × 104
cm−3
K, about two to three times the thermal
pressure we estimated from the LHB on the plane (Boulares &
Cox 1990; Cox 2005). As already put forth at the time, one can
A399, page 18 of 31

330° 300°
30°
15°
0°
l
b
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
EM
LHB
(10
3
cm
6
pc)
330° 300°
30°
15°
0°
l
b
0
50
100
150
200
250
300
350
400
Edenhofer+23
L
N
H
1e20
(pc)
Fig. 25. Zoom-in of the Centaurus tunnel in the smoothed EMLHB map (left) and dust distance map by integrating the dAV map up to 1020
cm−2
(right) (Edenhofer et al. 2024). The contour line outlining the region where the NH does not reach 1020
cm−2
in 400 pc is overlaid on the
EMLHB map.
approximate the cosmic rays, magnetic and dynamical (possibly
thermal) pressure as equipartition. This yields a thermal pressure
from the ISM about Pthermal/k ∼ 104
cm−3
K, consistent with
our measurement. Finally, we would like to highlight the LHB
pressure of Pthermal/k ∼ 104
cm−3
K is lower than the typical val-
ues seen in superbubbles, supernova remnants or wind-blown
bubbles (e.g. Oey & García-Segura 2004; Sasaki et al. 2011;
Kavanagh et al. 2012; Sasaki et al. 2022). This may indicate the
LHB being open towards higher Galactic latitudes.
4.6. Interstellar tunnel network
Cox & Smith (1974) famously postulated supernovae can gen-
erate and maintain an interstellar tunnel network in the Galaxy,
filled with ∼0.1 keV gas. Mapping the hot phase structure of the
LB/LHB in emission has more limitations than mapping the cold
phase of the local ISM with stellar extinction or line absorption
because one has to assume a model of the emitting components
and the absorption they are subject to. This makes inference of
the LHB shape on the Galactic plane difficult as one expects
many line-of-sight emitting and absorbing components. Also,
extensions and possible connecting tunnels of the LHB that are
not aligned with the radial direction are difficult to infer. Never-
theless, as mentioned in Sect. 4.2, we found evidence of hot gas
filling nearby channels that lack neutral material.
The most convincing evidence of such a channel is the β CMa
tunnel (l ∼ 250◦
, b ∼ 0◦
), where even the projected morphol-
ogy of the tunnel in EMLHB (see Fig. 14) is similar to the dust
map (especially, the right panel of Fig. 15). Historically, ROSAT
did not provide enough evidence of the tunnel being filled by
hot gas (Welsh & Shelton 2009, and references therein)7
. How-
ever, Dupin & Gry (1998) and Gry & Jenkins (2001) found
the intervening absorbing clouds to the stars β CMa and ϵ CMa
to show absorptions from high ionisation species such as C IV
and Si III, but comparison with the cloud turbulent temperature
(∼7000 K) demonstrated that they are likely partially ionised due
to photoionisation of the hot stars and/or the interaction with
the hot LHB gas in the surrounding. Our observation confirms
the presence of the surrounding hot gas in the tunnel. Indeed,
7 Closer inspection of the R12 band count rate map suggests there is
indeed a weak enhancement, independent of the subtraction of SWCX
(Snowden et al. 1997; Uprety et al. 2016).
both the UV and our soft X-ray observations corroborate the
scenario where the warm gas in LICs forms a conductive layer
between themselves and the surrounding hot gas (see e.g. Cowie
& McKee 1977; Slavin 1989). Welsh (1991) suggested this tun-
nel is connected to the Gum nebula (Gum 1952), which lies in
the same direction at 400 pc (Brandt et al. 1971). The Gum neb-
ula is likely part of a large superbubble GSH 238+00+09 (Heiles
1998), which the LHB could also connect to.
In addition, we report on a possible channel filled by hot
plasma towards the direction (l ∼ 315◦
, b ∼ 25◦
), in the con-
stellation Centaurus. A zoom-in view of the EMLHB and dust
map tracing the neutral matter in the region is shown in Fig. 25.
There is an enhancement in EMLHB in this direction, which anti-
correlates with the amount of dust extinction. This implies the
presence of extra path length of the LHB plasma, possibly filling
the tunnel. It might be another channel connecting to the Loop I
superbubble (Egger & Aschenbach 1995), in addition to the
Lupus tunnels first discovered from Na I absorption lines (Welsh
et al. 1994; Lallement et al. 2003). The latter are, unfortunately,
located on the Galactic plane and, thus, are difficult to identify in
our spectral analysis. The Centaurus tunnel region is located on
the edge of the eROSITA bubbles, further complicating the spec-
tral fitting. A dedicated spectral analysis, with a tailored spectral
extraction region, of this region will likely help disentangle the
emission from the Loop I superbubble (its nature or existence
has become unclear after the discovery of the eROSITA bub-
bles) from the eROSITA bubbles, in terms of spectral properties
and distance. This analysis is currently ongoing.
The Antlia supernova remnant (l, b)∼(275◦
, 15◦
) (e.g. Fesen
et al. 2021), Monogem Ring (l, b)∼(200◦
, 8◦
) (e.g. Knies et al.
2024) and Orion-Eridanus superbubble (l, b)∼(205◦
, −30◦
) (e.g.
Pon et al. 2016) are other nearby bubbles that our data suggest
the LHB could be touching or connected to. Monogem Ring is
arguably even more interesting because there is a lack of dust
(Lallement et al. 2022; Edenhofer et al. 2024) in the line-of-
sight up to its distance of ⋍300 pc (Knies et al. 2024, we refer to
the closest components in the whole Gemini-Monoceros X-ray
enhancement), consistent with the eROSITA spectral analysis of
it (Knies et al. 2024). The LHB might be currently merging or
about to interact with it.
On the other hand, there is the presence of channels of low
NH that do not seem to be filled with soft X-ray-emitting plasma.
The more obvious one is located at (l, b)∼(260◦
, 29◦
), which can
A399, page 19 of 31

Fig. 26. Comparison of the local (hot) bubble extent from X-ray and dust. Left: extent of the local hot bubble under the constant ne = 4 × 10−3
cm−3
assumption. Right: local bubble shell tracing the closest extinction peak inferred by O’Neill et al. (2024). An interactive comparison in 3D can be
accessed from the accompanying website (Sect. 5).
be seen in the right panel of Fig. 15. The EMLHB map in Fig. 14
does not show an anti-correlated enhancement.
In summary, our data demonstrates that the displacement
model in which the LHB plasma fills the LC works well overall.
Only a few regions seemingly do not abide by the displacement
model.
4.7. Beyond constant density of the local hot bubble
We have discussed the close anti-correlation between the local
dust map and the emission measure of the LHB in Sect. 4.2. The
dust maps (Fig. 15) we compared our LHB shape to were inferred
from integrating the NH up to 1020
cm−2
(an optical depth for a
0.2 keV photon) from our position. An alternative way of locat-
ing the shell of the LB is by finding the first peak of the extinction
as adopted by Pelgrims et al. (2020) and O’Neill et al. (2024).
This might be more physically motivated as the expansion on the
LB is expected to create a shell of denser material at its bound-
ary that is not necessarily dense enough to be opaque to soft
X-ray.
A comparison of the inferred LB shell from O’Neill et al.
(2024) using Edenhofer et al. (2024) data to our constant den-
sity LHB model is shown in Fig. 26. For this comparison, we
adopted their A′
0.5 peak density distance as the extent of the LB.
Their shapes are significantly different, most notably at b > 60◦
,
where the apparent enhancement in EMLHB (hence extent) lacks
a counterpart in the LB. Below b ≲ 30◦
, the LHB and the LB
are both more extended but with a fairly different morphol-
ogy. The clear tunnel in the LB at (l, b) ⋍ (250◦
, −20◦
) is not
shared by the LHB. There is a reasonable agreement for regions
between −30◦
≲ b ≲ 60◦
, ignoring the various tunnels discussed
in Sect. 4.6. The simplest reason for the discrepancy between the
two maps is the constant density assumption is inaccurate, given
LLHB ∝ EMLHBn−2
e , which means the extent of the LHB is more
sensitive to ne than EMLHB.
One can work conversely, assuming the LB shell is the true
extent of the LHB and estimate the electron density of the LHB
plasma using Eq. (12), as well as its thermal pressure by Eq. (10)
subsequently. We show the resulting electron density and ther-
mal pressure maps in Fig. 27. Clearly, much of the modulation
in EMLHB is transferred to the variation in density, especially in
regions where the LHB extent derived from the constant den-
sity assumption differs much from the extent of the LB. This
causes the average LHB density at b > 60◦
to be higher than
the rest of the areas by ∼40%. The extensions towards β CMa
(at least the part on the Galactic plane) and Centaurus tunnels
mentioned in Sect. 4.6 do not have direct counterparts in the LB
map, and thus they are taken as high-density regions of the LHB.
The thermal pressure map slightly differs from the constant den-
sity case shown in Fig. 10, where the former displays a weaker
north-south gradient because of the higher electron density in
the northern polar cap balancing the lower LHB temperature,
and it produces a more uniform pressure profile on the largest
scale. Indeed, being in pressure equilibrium with its surround-
ings would be the expectation for an old bubble such as the
LHB. From the two maps, the half-sky median electron den-
sity and thermal pressure are ne = 3.75+0.87
−0.69 × 10−3
cm−3
and
Pthermal/k = 10.5+3.4
−2.5 × 103
cm−3
K.
Assuming the LHB extends up to the first extinction peak is
a sound assumption, but it is not without challenges. The hardest
part to reconcile is the presence of tunnels that seem indepen-
dent of the LB shell. We suggest it is unlikely a coincidence
that EMLHB shows a better anti-correlation to the local dust col-
umn density (Fig. 15) than the extent of the LB shell, especially
when the LB shell peak density is the lowest in these regions (see
Fig. 4 in O’Neill et al. 2024). Not accepting it as a coincidence
entails hot plasma located beyond the first wall of absorption.
They could have the same origin as the LHB or had other heat-
ing mechanisms. One possibility is shown by the LB simulation
by Schulreich et al. (2023) (third and fourth columns of their
Fig. 4), in which pockets of hot plasma could be formed beyond
the main LB cavity because of the LHB anisotropic expansion
into the inhomogeneous surrounding materials that were already
stirred up during the wind-driven phase.
A399, page 20 of 31

Fig. 27. Electron density (left) and thermal pressure (right) of the local hot bubble assuming it extends up to the local bubble shell (O’Neill et al.
2024).
4.8. Cosmic X-ray background
4.8.1. Determination of the cosmic X-ray background photon
indices
Ponti et al. (2023) adopts a double-broken power-law model
to describe the CXB in the eFEDS field, following the CXB
synthesis model by Gilli et al. (2007), which produces a steep-
ening CXB slope ≲ 1 keV from galaxy groups and clusters. The
double-broken power-law8
(bkn2pow in Xspec/PyXspec) can
be written as
f(E) =











K(E/keV)−1.9
if E ≤ 0.4 keV
K0.4−0.3
(E/keV)−1.6
if 0.4 keV ≤ E ≤ 1.2 keV
K 0.4−0.3
1.20.15 (E/keV)−1.45
if E ≥ 1.2 keV,
(15)
where K is the normalisation of the power-law at 1 keV in the
unit of photons s−1
cm−2
keV−1
and the first and second break
energies are located at 0.4 and 1.2 keV.
From the spectral fitting of the regions shown in Fig. 6,
we found that the first break energy at 0.4 keV is redundant
as the photon indexes below (Γ1 = 1.93+0.22
−0.21) and above (Γ2 =
1.91+0.09
−0.16) this energy are consistent within 1 σ when we left
all three photon indexes free to vary in the spectral fitting.
Therefore, we subsequently replace the double-broken power-
law model with a single-broken power-law model, which can be
treated as a special case of Eq. (15). It has the form
f(E) =
(
K(E/keV)Γ1
if E ≤ Eb
K(Eb/keV)Γ2−Γ1
(E/keV)−Γ2
if E ≥ Eb,
(16)
where Eb is the break energy and Γ1 and Γ2 are the photon
indexes below and above Eb. Another iteration of the spec-
tral fitting with the single-broken power-law model indicates
8 Three models of CXB were used in Ponti et al. (2023). We refer to
their ‘CXB’ model here.
Eb = 1.19−0.26
−0.17, Γ1 = 1.81+0.24
−0.21 and Γ2 = 1.61+0.13
−0.07. This result
suggests a change of slope in the CXB at 1.2 keV, below which
the slope is possibly steeper from the contributions from active
galactic nuclei, galaxy groups and galaxy clusters (Hasinger et al.
1993; Smith et al. 2007; Yoshino et al. 2009). Nonetheless, the
transition of the CXB slope is only significant on ∼1 σ level9
and is not strictly required by the data. Additionally, the uncer-
tainties of the parameters are dominated by the spread from
multiple regions instead of the statistical uncertainties within
each region. A simple power-law fit yields Γ = 1.68+0.08
−0.10, in good
agreement with Γ2 in the single-broken power-law model. This is
steeper than the canonical photon index of 1.4–1.5. Calibration
issues could partly cause this, as eROSITA indeed tends to mea-
sure cooler cluster temperature than Chandra and XMM-Newton
for massive clusters (Liu et al. 2023; Migkas et al. 2024)10
.
Both treatments of the CXB reproduce the data almost equiva-
lently, with the simple power-law and single-broken power-law
resulting in an average C-statistic/d.o.f in the Markov chain
Monte Carlo (MCMC) runs of all regions of 3123+249
−99 /2595 and
3092+228
−99 /2588, respectively. Therefore, we adopted both in our
full western Galactic hemisphere analysis and discuss the sys-
tematic uncertainties associated with either choice in Sect. 4.8.3.
Based on this analysis of the CXB, we froze Γ1, Γ2 and Eb to be
1.9, 1.6 and 1.2 keV of the single-broken power-law model, Γ to
1.7 for the simple power-law model as the CXB model in the full
spectral analysis.
4.8.2. Normalisation
Fig. 28 shows the normalisation of the CXB, taken from the
simple power-law model of Γ = 1.7. Ignoring the Galactic
9 Fixing Eb at 1.2 keV would decrease the statistical uncertainties of Γ1
and Γ2 to 1.87+0.16
−0.11 and 1.63+0.08
−0.06, respectively.
10 The currently developing eSASS pipeline processing version c030
will improve the calibration of the effective area and redistribution
matrix above 1 keV.
A399, page 21 of 31

0°
30°
60°
60°
30°
0°
-30°
-60° -60°
-30°
2.5 3.0 3.5 4.0 4.5 5.0
normCXB (10 3 ph cm 2 s 1 keV 1 deg 2 at 1 keV)
Fig. 28. Normalisation of the cosmic X-ray background assuming a sim-
ple power-law of Γ = 1.7.
plane, the normalisation of the CXB is extremely uniform.
At |b| > 30◦
, it has a median of normCXB = 3.54+0.24
−0.17 ×
10−3
photons cm−2
s−1
keV−1
deg−2
at 1 keV. This means the
fluctuation of the CXB is ≲10%, on the angular scales of
∼1.
◦
5–4◦
that we probe set by the contour binning scheme. The
enhancement on the Galactic plane is likely due to the Galactic
ridge emission.
Our main finding on the CXB is that eROSITA observes
a steeper photon index than the conventional value of 1.45
(Cappelluti et al. 2017) (Sect. 4.8.1). It might be due to cali-
bration issues (Migkas et al. 2024) or caused by the steepening
of the CXB slope at soft energies. A newer processing ver-
sion of eROSITA data will potentially mitigate or resolve the
calibration issue. Given the steeper slope of Γ ∼1.6–1.7, the nor-
malisation we found is also slightly higher compared to Chandra
((3.32 ± 0.05) × 10−3
photons cm−2
s−1
keV−1
deg−2
, Cappelluti
et al. 2017). In addition, the uniformity of the CXB demonstrates
the robustness of our spectral fitting to differentiate multiple sky
components.
4.8.3. Systematic uncertainties associated with the choice
of cosmic X-ray background model
As described in Sect. 4.8.1, the power-law and the broken power-
law model could reproduce the diffuse X-ray spectra equally
well. So far, we have been discussing the results using a CXB
described by a simple power-law model. To test the systematic
effect caused by the CXB model, we fitted all spectra again in
the same way but using a broken-power-law CXB component.
We then plotted the 2D maps and inspected the histograms of the
other components. We observed that the distributions of most of
the components were unaffected except for the LHB.
Fig. 29 shows the difference in distributions of kTLHB and
EMLHB by using the two CXB models for regions above |b| >
10◦
. The broken power-law model appears to raise the temper-
ature of the LHB systematically by 0.02 keV in terms of the
median. On the other hand, the impact of such temperature
change on the EM is small (∼0.3 × 10−3
cm−6
pc) compared
to the width of the EMLHB distribution (∼1.5 × 10−3
cm−6
pc).
Inspection of the spectra suggests this is the result of the intri-
cate balancing at the softest end of the spectra (∼0.2 keV). By
increasing the temperature slightly, the LHB could lower its flux
density at this energy to accommodate the steepening slope of
the CXB.
Fig. 30 demonstrates this is indeed a systematic effect by
showing the ubiquitous under-subtraction in the residual map
made from subtracting the LHB temperature map determined
by the simple power-law model from that of the broken power-
law model. The broken power-law model consistently predicts a
higher LHB temperature than the simple power-law model, and
the difference is more enhanced towards the Galactic plane. We
computed the median differences below and above |b| = 30◦
. For
|b| > 30◦
, the median difference between the two CXB models is
0.009 keV, about the same as the 1 σ statistical fitting uncertainty
in this region. Between 10◦
< |b| < 30◦
, the median difference
rises to 0.029 keV, about 1.5 times the statistical uncertainty in
this region. We consider these to be the systematic uncertainties
of the absolute values of the temperature of the LHB. However,
we emphasise that the temperature gradient reported in Sect. 4.1
is still present, as this systematic effect only steers the absolute
value of kTLHB in a single direction and is only very weakly
dependent on location.
4.9. Robustness of the spectral fits
The use of cstat in evaluating the goodness of fit is possible
but not straightforward, as the expected value and the variance
of cstat depend on the number of counts in each spectral bin
(Kaastra 2017). Instead of the rule-of-thumb χ2
/d.o.f. ∼ 1 indi-
cating a good fit, one needs to compare the fitted cstat with the
expected value and variance of cstat in each spectrum, which
is a variable number dependent on counts. Therefore, while the
spectral fitting was done using cstat (the eROSITA part), we
report the χ2
/d.o.f after rebinning each spectrum to have at least
10 counts in each spectral bin using the same model. The rebin-
ning was done following the scheme presented by Kaastra &
Bleeker (2016). We present the map of the resulting χ2
/d.o.f.
values in Fig. 16.
We would like to remind the reader that the relatively small
and narrow range of χ2
/d.o.f. results partly from fitting the
spectra up to 5 keV, where the instrumental background dom-
inates with a small CXB contribution. The wide energy range
between ∼2–5 keV can almost always be well-modelled by the
dominating, but fixed, FWC model, thus inclined to result in
low χ2
/d.o.f. However, this energy range has strong constraining
power on the subdominant CXB component, which is the only
contributing component and was thus included in our spectral
fitting.
To better understand the robustness of the spectral fits, we
show four spectra with increasing values of χ2
/d.o.f. in Fig. 31
and 32, from ∼1.0–1.7. Fig. 31 shows spectra in the range of
1.0 ≲ χ2
/d.o.f. ≲ 1.1, where no major discrepancies between
data and model could be seen. Fig. 16 demonstrates these val-
ues are typical of spectra away from the Galactic plane and the
central part of the Galaxy. On the Galactic plane, poorer spec-
tral fits are expected because of the definite presence of multiple
line-of-sight structures, both in emission and absorption, which
are not part of our model template. Towards the inner part of
the Galaxy (310◦
≲ l ≲ 360◦
), this scenario exacerbates, and
poorer fits begin to be prevalent to even high Galactic latitudes
around −20◦
≲ b ≲ 30◦
. There are many ways to obtain poor
A399, page 22 of 31

Fig. 29. Distributions of kTLHB (left) and EMLHB(right) using either a simple power-law (pow) or a broken power-law (bknpow) parametrisation of
the CXB.
Fig. 30. Difference in the LHB temperature in assuming a simple power-
law CXB model and a broken power-law model, namely, kTLHB,bknpow −
kTLHB,pow.
fits, but one of the most common ways is the poor reproduction
between ∼0.7–1 keV, where the Fe XVII, Ne IX and Ne X lines
dominate. The reproduction is usually imperfect when these lines
are bright, especially at the base of the eROSITA bubbles. How-
ever, it is currently difficult to distinguish between calibration
imperfections and real variations in the astrophysical source.
The upcoming pipeline processing version c030 will address
the former slightly in terms of including a more accurate energy
resolution in the redistribution matrix.
The robustness of each region’s fit and MCMC chain could
be inspected by its associated corner plot and steps plot in the
choropleth maps on our website (Sect. 5).
5. Conclusion
In this work, we binned the western Galactic hemisphere
into ∼2000 regions. We extracted their spectra at solar mini-
mum (eRASS1), where the heliospheric SWCX emission was
negligible (Dennerl et al., in prep.). These spectra were fitted
with a model template consisting of the LHB, the Milky Way’s
CGM, the Galactic corona, the CXB, and, depending on the
location, the eROSITA bubbles. This resulted in maps of the
parameters, primarily the temperature and emission measure
maps for the thermal plasma. We focused on the results regarding
the LHB. We summarise our main findings below:
– We found the median temperature of the LHB to be
0.111+0.018
−0.015 keV. Much of the spread comes from the approx-
imately north-south gradient above Galactic latitudes of 30◦
,
with the south being hotter (121.8 ± 0.6 eV) than the north
(100.8 ± 0.5 eV). Venturing closer to the Galactic plane,
there seems to be an increase in LHB temperature, espe-
cially at l ≳ 270◦
. The origin of the temperature gradient
is unclear, but it could be set up by more recent supernova
explosions within the LHB. The enhancement in tempera-
ture towards the Galactic plane could be due to the enhanced
thermal pressure needed in the Galactic plane for the LHB
to maintain pressure equilibrium with the surroundings;
– The emission measure of the LHB is higher towards high
latitudes in both hemispheres. It entails that the LHB is
more extended towards high latitudes, assuming a constant
electron density. The emission measure is also spatially anti-
correlated with the local dust column density, consistent with
the displacement model put forth by Sanders et al. (1977).
We produced a 3D model of the LHB in the western Galactic
hemisphere;
– We found two tunnels with a low local column density that
appear to be filled by hot plasma. One is the well-known
β CMa tunnel, and the other is towards (l,b)∼(315◦
, 25◦
), in
the constellation Centaurus. This hints at the possibility of a
widespread tunnel network connecting regions filled by the
hot phase of the ISM;
– Both a simple and broken power law could fit the cosmic
X-ray background well, but with a steeper slope (Γ ≃ 1.6–
1.7 above 1.2 keV) than the conventional value (Γ = 1.45).
Whether this represents a genuine steepening of the CXB
slope below ∼1 keV or is caused by calibration inaccuracies
is uncertain. It will likely become clear with a forthcoming
new calibration version.
From this study, it is clear that eROSITA and its all-sky
surveys provide a valuable dataset for studying the SXRB.
Future papers will follow to discuss other aspects of the
SXRB.
A399, page 23 of 31

Fig. 31. Example spectra having χ2
/d.o.f = 0.99 and 1.10.
Fig. 32. Example spectra having χ2
/d.o.f of 1.50 and 1.70, respectively. They are commonly found near the Galactic plane, towards the inner part
of the Galaxy and the base of the eROSITA bubbles. The models usually cannot predict the spectrum between ∼0.7–1 keV well.
Data availability
The results of the spectral fits can be accessed and visualised
as choropleth maps via the website https://erosita.mpe.
mpg.de/dr1/AllSkySurveyData_dr1/DiffuseBkg/ hosted
on the eROSITA DR1 server. An interactive version of the 3D
model of the LHB (Fig. 20) and structures in the solar neigh-
bourhood can also be accessed there, as well as the spherical
harmonics models of LHB temperature presented in Sect. 4.1.1
and Appendix B. Xspec model files are also available through
the website for readers interested in particular regions in the
western Galactic hemisphere. The 3D interactive map in Fig. 20
is also available at https://www.aanda.org.
Acknowledgements. MY thanks Mattia Pacicco and Michael Schulreich for their
in-depth explanations of their LB simulations during the EAS meeting 2024
in Padova, which has greatly improved the interpretation of the results. MY
extends his gratitude to Susanne Friedrich and Harald Baumgartner for their
guidance and help in setting up the accompanying website on the eROSITA
DR1 server. MY appreciates the helpful discussion on cluster masks and
measurement of cluster temperature provided by Ang Liu. MY and MF acknowl-
edge support from the Deutsche Forschungsgemeinschaft through the grant FR
1691/2-1. GP acknowledges funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 865637). MS acknowledges support from the Deutsche
Forschungsgemeinschaft through the grants SA 2131/13-1, SA 2131/14-1, and SA
2131/15-1. This work is based on data from eROSITA, the soft X-ray instrument
aboard SRG, a joint Russian-German science mission supported by the Russian
Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences
represented by its Space Research Institute (IKI), and the Deutsches Zentrum für
Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Asso-
ciation (NPOL) and its subcontractors, and is operated by NPOL with support
from the Max Planck Institute for Extraterrestrial Physics (MPE). The devel-
opment and construction of the eROSITA X-ray instrument was led by MPE,
with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP
(FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leib-
niz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy
and Astrophysics of the University of Tübingen, with the support of DLR and
the Max Planck Society. The Argelander Institute for Astronomy of the Univer-
sity of Bonn and the Ludwig Maximilians Universität Munich also participated
in the science preparation for eROSITA. The eROSITA data shown here were
processed using the eSASS/NRTA software system developed by the German
eROSITA consortium. MY thanks the developers of the following software and
packages that made this work possible: numpy (Harris et al. 2020), matplotlib
(Hunter 2007), astropy (Astropy Collaboration 2013, 2018, 2022), PyXspec
(Arnaud 1996), HEAsoft (Nasa High Energy Astrophysics Science Archive
Research Center (Heasarc) 2014), FTOOLS (Blackburn 1995), contbin (Sanders
2006), lmfit (Newville et al. 2016), emcee (Foreman-Mackey et al. 2013),
corner.py (Foreman-Mackey 2016), plotly (Plotly Technologies Inc. 2015),
folium (Filipe et al. 2021), K3D-jupyter (Trzesiok et al. 2022), OpenCV
(Bradski 2000) and scipy (Virtanen et al. 2020).
A399, page 24 of 31

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A399, page 26 of 31

Appendix A: Test for non-equilibrium ionisation
Non-equilibrium ionisation (NEI) can happen in the LHB if there
were recent rapid heating or cooling events. Electrons respond
quickly to temperature change via collisions, but ionisation and
recombination of the ions lag behind (e.g. see a review by Bre-
itschwerdt & de Avillez 2021). As a result, the line intensities
under the assumption of collisional ionisation equilibrium (CIE)
do not reflect the real temperature of the gas. If a recent super-
nova exploded within the LHB, parts or all of LHB could be
under-ionised due to rapid shock-heating. On the other hand, if
the LHB spawned from a dense cloud, it could undergo rapid
adiabatic cooling when it bursts out of the cloud, resulting
in an over-ionised plasma (Breitschwerdt & Schmutzler 1994).
Henley et al. (2007) investigated both scenarios using a com-
bination of XMM-Newton/EPIC-MOS and ROSAT/PSPC data
and concluded the data could not distinguish between CIE and
under-ionised plasma but disfavoured the over-ionised scenario.
We used a simple test to evaluate whether our data
(eRASS1+ROSAT R12) are sensitive to the signature of NEI.
We chose the same regions that were used to determine the spec-
tral shape of the CXB (Fig. 6) for this analysis. They have high
S/N (400) in the soft band. This minimises the statistical uncer-
tainty and boosts any potential signatures of NEI. We refitted
the spectra by replacing the CIE LHB model (apec) with a NEI
LHB model (nei).
We summarise the fitting results in Table A.1. Six of
the seven regions have the fitted density-weighted ionisation
timescale τ > 1012
cm−3
s. At τ > 1012
cm−3
s, Smith & Hughes
(2010) showed that 90% of carbon and oxygen ions (which
dominate the observed LHB emission) would reach CIE for
plasma at ∼0.1 keV. The one exception is the region centred at
(l, b)=(241◦
, 45◦
), which returns τ = (5.6+0.8
−0.7) × 1011
cm−3
s, sug-
gesting a small under-ionisation. Fig. A.1 compares the CIE and
NEI models in this region. From the residual spectra, one can
see the improvement below ∼0.5 keV. Nevertheless, because the
NEI models of most regions reduce to the CIE case even at a high
S/N of 400, we conclude our data can be sufficiently explained
by CIE and keep this assumption throughout our analysis in the
main text.
Appendix B: Posterior distributions of the dipole
and lmax = 6 models of local hot bubble
temperature
The best-fit parameters of the dipole model in Fig. 17 are pre-
sented in Table B.1. The z-axis points towards the Galactic north
pole in our convention, and the x-axis points towards the Galac-
tic centre. Two equivalent representations are presented: the
complex alm coefficients and the dipole vector. In the complex
representation, each multipole gives 2l + 1 free parameters; Each
al|m| contributes to two, one for the real part and one for the
imaginary part, except from al0 coefficients, which do not have a
complex part for real-valued functions.
For the dipole, it can be shown that a10 and a11 are related to
the dipole vector by the following equations:





























A1 = a10
r
3
4π
cos θmax − 2
r
3
8π
sin θmax f,
tan θmax = −
√
2
a10
f
tan ϕmax = −
Im{a11}
Re{a11}
,
(B.1)
(B.2)
(B.3)
where f = Re{a11} cos ϕmax −Im{a11} sin ϕmax, A is the amplitude
of the dipole and (θmax, ϕmax; colatitude and azimuth in spherical
coordinates) is the direction where this amplitude is obtained.
The posterior distributions of the dipole are shown in Fig. B.1
both in complex and dipole vector representations.
We list the best-fit parameters of the lmax = 6 spherical
harmonics model (Fig. 18) in Table B.2.
Appendix C: Latitudinal profiles of local hot bubble
temperature
Fig. C.1 shows the kTLHB data in another light by dividing the
western Galactic hemisphere in 15◦
-wide longitudinal stripes
and shows kTLHB as a function of latitude. This presentation has
the advantage of enabling visual inspection of the uncertainty
associated with individual contour bins and their comparison
with the large-scale gradient. Lines of the monopole (orange),
dipole (red) and lmax = 6 (blue) models are overlaid in addition.
With the help of the best-fit monopole model line, it is apparent
that kTLHB is higher in the southern hemisphere than in the north.
The dipole model can capture the primary latitudinal gradient,
especially in the south, but fails to follow the data closely above
b > 30◦
. While the model complexity can increase to accommo-
date the temperature variation on a smaller scale, we note the
presence of a temperature gradient in LHB is clear. The scatter
is large in the region (270◦
≲ l ≲ 300◦
,−40◦
≲ b ≲ −10◦
), which
partially covers the Large Magellanic Cloud but cannot be fully
attributed to it. The relatively large scatter below 10◦
< b < 30◦
and l > 300◦
is caused by the lower EMLHB in this direction,
resulting in higher uncertainties in the determination of kTLHB.
Appendix D: Local hot bubble temperature
dichotomy seen from high S/N spectra
Sect. 4.1 demonstrated the north-south dichotomy of the
LHB temperature, mainly by the systematic temperature off-
set between the hemispheres. However, this offset is difficult to
appreciate purely by looking at individual spectra of S/N ≃ 80
because the temperature difference of 0.02 keV is only approxi-
mately twice the typical fitting uncertainty, as shown in Fig. 9.
To highlight the spectral difference, we took two examples
of high-S/N spectra (S/N = 400), one from each hemisphere.
Fig. D.1 shows the spectrum in the south. The difference in the
left and right panels lies only with the model: the left panel has
the kTLHB kept at the northern value, while all model param-
eters were allowed to vary in the right panel. Fig. D.2 shows
the reverse: showing the northern spectrum and keeping the
southern temperature of the LHB in the left panel. The com-
parisons demonstrate the incompatibility of the spectrum from
either hemisphere with the LHB temperature measured from the
opposite hemisphere. This lends further support for a real LHB
temperature dichotomy in the hemispheres.
A399, page 27 of 31

NEI CIE
l (◦
) b (◦
) kTLHB (keV) τLHB (1012
cm−3
s) EMLHB (10−3
cm−6
pc) kTLHB (keV) EMLHB (10−3
cm−6
pc)
228 −72 0.120 ± 0.002 25+16
−17 3.41+0.14
−0.13 0.115 ± 0.002 3.09+0.15
−0.14
202 −61 0.114+0.003
−0.002 30+13
−19 3.11+0.12
−0.16 0.112+0.002
−0.003 2.84+0.13
−0.12
229 −48 0.119 ± 0.003 26 ± 15 4.34+0.27
−0.37 0.121 ± 0.002 4.05+0.27
−0.18
205 39 0.103 ± 0.002 23+19
−17 3.48+0.14
−0.13 0.098 ± 0.002 3.19+0.14
−0.17
241 45 0.116 ± 0.003 0.55+0.08
−0.07 3.12+0.12
−0.11 0.097+0.003
−0.002 2.96+0.18
−0.17
208 66 0.104 ± 0.002 25 ± 16 5.63+0.19
−0.21 0.098 ± 0.002 5.41+0.26
−0.25
258 67 0.104+0.003
−0.002 17+22
−16 4.72+0.18
−0.17 0.097 ± 0.002 4.43+0.21
−0.20
Table A.1. Fitted LHB parameters of the seven high-SN spectra under the NEI and CIE assumptions. All but one region show an ionisation
timescale of τ > 1012
cm−3
s, essentially reducing to the CIE case.
10 3
0.01
0.1
1
Normalised
counts
s
1
keV
1
cm
2
deg
2
Bin62, (l, b) = (241.1, 45.3), area=639.9 deg2
R1
R2
0.1 0.2 0.5 1 2 5
Energy (keV)
5
0
5
data
model
error
10 3
0.01
0.1
1
Normalised
counts
s
1
keV
1
cm
2
deg
2
Bin62, (l, b) = (241.1, 45.3), area=639.9 deg2
R1
R2
0.1 0.2 0.5 1 2 5
Energy (keV)
5
0
5
data
model
error
Fig. A.1. Comparison of spectral fits assuming the LHB in CIE (left) and NEI (right) of the region showing a potential NEI signature. The residual
highlights the improvement below 0.5 keV.
Table B.1. Best-fit parameters of the dipole (lmax = 1) model of kTLHB.
The resulting temperature model is in units of keV.
Complex
a00 0.368 ± 0.002
a10 −0.027 ± 0.001
Re{a11}(i)
0.014 ± 0.001
Im{a11}(i)
−0.036 ± 0.002
Dipole vector
A0
(ii)
0.104 ± 0.001
A1
(ii)
0.030 ± 0.001
ϕmax 291.
◦
1 ± 1.
◦
8
θmax 116.
◦
5+1.
◦
5
−1.
◦
6
Notes. (i)
Re{a} and Im{a} refer to the real and imaginary part of the
complex coefficient a.
(ii)
Al has the unit of keV and can be interpreted as the amplitude of the
corresponding multipole.
A399, page 28 of 31

0.368+0.002
0.002
0
.
0
3
0
0
.
0
2
7
0
.
0
2
4
a
10
0.027+0.001
0.001
0
.
0
1
6
0
.
0
1
2
Re{a
11
}
0.014+0.001
0.001
0
.
3
6
0
0
.
3
6
8
0
.
3
7
6
a00
0
.
0
4
0
0
.
0
3
2
Im{a
11
}
0
.
0
3
0
0
.
0
2
7
0
.
0
2
4
a10
0
.
0
1
6
0
.
0
1
2
Re{a11}
0
.
0
4
0
0
.
0
3
2
Im{a11}
0.036+0.002
0.002
0.104+0.001
0.001
0
.
0
2
8
0
.
0
3
2
A
1
0.030+0.001
0.001
2
8
8
2
9
4
max
(deg)
291.1+1.8
1.8
0
.
1
0
2
0
.
1
0
4
0
.
1
0
6
A0
1
1
5
1
2
0
max
(deg)
0
.
0
2
8
0
.
0
3
2
A1
2
8
8
2
9
4
max (deg)
1
1
5
1
2
0
max (deg)
116.5+1.6
1.5
Fig. B.1. Posterior distributions of the dipole (lmax = 1) model in com-
plex (top) and multipole vector (bottom) representations. The contours
show the 1, 2, and 3 σ uncertainties.
Table B.2. Best-fit parameters of the lmax = 6 model of kTLHB. The
resulting temperature model is in units of keV.
a00 6.165 ± 8.591
a10 −2.583 ± 0.730
Re{a11} −1.425 ± 0.983
Im{a11} 5.622 ± 8.654
a20 −3.707 ± 5.337
Re{a21} −0.452 ± 0.112
Im{a21} −2.798 ± 0.785
Re{a22} −3.858 ± 6.466
Im{a22} −2.161 ± 1.484
a30 2.327 ± 0.624
Re{a31} 0.556 ± 0.346
Im{a31} −1.990 ± 2.931
Re{a32} 2.067 ± 0.566
Im{a32} −0.690 ± 0.165
Re{a33} 1.952 ± 1.294
Im{a33} −1.798 ± 3.709
a40 0.884 ± 1.186
Re{a41} 0.213 ± 0.051
Im{a41} 1.300 ± 0.332
Re{a42} 0.775 ± 1.232
Im{a42} 0.510 ± 0.299
Re{a43} 0.564 ± 0.130
Im{a43} 1.099 ± 0.292
Re{a44} 0.476 ± 1.594
Im{a44} 1.177 ± 0.752
a50 −0.520 ± 0.122
Re{a51} −0.091 ± 0.044
Im{a51} 0.242 ± 0.341
Re{a52} −0.480 ± 0.116
Im{a52} 0.164 ± 0.038
Re{a53} −0.254 ± 0.135
Im{a53} 0.178 ± 0.357
Re{a54} −0.389 ± 0.103
Im{a54} 0.290 ± 0.062
Re{a55} −0.475 ± 0.283
Im{a55} −0.008 ± 0.473
a60 −0.049 ± 0.054
Re{a61} −0.019 ± 0.005
Im{a61} −0.119 ± 0.024
Re{a62} −0.032 ± 0.054
Im{a62} −0.035 ± 0.015
Re{a63} −0.053 ± 0.012
Im{a63} −0.104 ± 0.022
Re{a64} −0.010 ± 0.056
Im{a64} −0.069 ± 0.030
Re{a65} −0.079 ± 0.015
Im{a65} −0.082 ± 0.021
Re{a66} 0.040 ± 0.078
Im{a66} −0.098 ± 0.057
Notes. Re{alm} and Im{alm} refer to the real and imaginary part of the
complex coefficient alm.
A399, page 29 of 31

Fig. C.1. Latitudinal profiles of kTLHB with their 1 σ error bars in all longitudinal stripes of 15◦
width in the western Galactic hemisphere. The
vertical orange, red and blue lines show the corresponding monopole, dipole and lmax = 6 models, respectively. The black region indicates the
Galactic plane region, which we masked for the temperature gradient analysis. Regions plotted on the white background indicate where the dipole
model was fitted (|b| > 30◦
). The lmax model was fitted with the inclusion of data from the shaded region (|b| > 10◦
).
Fig. D.1. North-south temperature dichotomy shown by a high S/N spectra in the south. Left: Spectrum in the southern hemisphere fitted while
keeping kTLHB fixed at the northern value (kTLHB = 0.10 keV). Right: Same as on the left but all the parameters were allowed to vary while fitting.
A399, page 30 of 31

10 3
0.01
0.1
Normalised
counts
s
1
keV
1
cm
2
deg
2
kTLHB = 0.121 keV
kTCGM = 0.160 keV
stat/dof =1.27(2691 dof)
Bin62, (l, b) = (241.1, 45.3), area=639.9 deg2
R1
R2
0.1 0.2 0.5 1 2
Energy (keV)
5
0
5
data
model
error
10 3
0.01
0.1
Normalised
counts
s
1
keV
1
cm
2
deg
2
kTLHB = 0.097 keV
kTCGM = 0.157 keV
stat/dof =1.23(2690 dof)
Bin62, (l, b) = (241.1, 45.3), area=639.9 deg2
R1
R2
0.1 0.2 0.5 1 2
Energy (keV)
5
0
5
data
model
error
Fig. D.2. Similar to Fig. D.1, but showing a high S/N spectra in the north. Left: Spectrum in the northern hemisphere fitted while keeping kTLHB
fixed at the southern value (kTLHB = 0.12 keV). Right: Same as on the left but all the parameters were allowed to vary while fitting.
A399, page 31 of 31

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