11institutetext:  1\. Istituto di Astrofisica e Planetologia Spaziali (IAPS).
INAF. Via Fosso del Cavaliere 100, 00133 Roma, Italy
2\. Laboratoire AIM, Paris-Saclay, CEA/IRFU/SAp - CNRS - Université Paris
Diderot. 91191, Gif-sur-Yvette Cedex, France.
3\. Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische
Astrophysik, Albert-Ueberle-Str. 2, 69120, Heidelberg, Germany.
4\. Universität Heidelberg, Interdiszipliäres Zentrum für Wissenschaftliches
Rechnen, 69120 Heidelberg, Germany.
5\. European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748,
Garching bei München, Germany.
6\. Research School of Astronomy and Astrophysics, The Australian National
University, Canberra, ACT, Australia.
7\. Max-Planck Institute for Astronomy, Königstuhl 17, 69117, Heidelberg,
Germany.
8\. Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK.

# The Galactic dynamics revealed by the filamentary structure in atomic
hydrogen emission

J. D. Soler1 Corresponding author<EMAIL_ADDRESS>M.-A. Miville-
Deschênes2 S. Molinari1 R. S. Klessen3,4 P. Hennebelle2 L. Testi5 N. M.
McClure-Griffiths6 H. Beuther7 D. Elia1 E. Schisano1 A. Traficante1 P.
Girichidis3 S. C. O. Glover3 R. J. Smith8 M. Sormani3 R. Treß3

(Received 16 02 2022 / Accepted 06 05 2022)

We present a study of the filamentary structure in the neutral atomic hydrogen
(Hi) emission at the 21 cm wavelength toward the Galactic plane using the
16.′2-resolution observations in the Hi 4$\pi$ (HI4PI) survey. Using the
Hessian matrix method across radial velocity channels, we identified the
filamentary structures and quantified their orientations using circular
statistics. We found that the regions of the Milky Way’s disk beyond 10 kpc
and up to roughly 18 kpc from the Galactic center display Hi filamentary
structures predominantly parallel to the Galactic plane. For regions at lower
Galactocentric radii, we found that the Hi filaments are mostly perpendicular
or do not have a preferred orientation with respect to the Galactic plane. We
interpret these results as the imprint of supernova feedback in the inner
Galaxy and Galactic rotation and shear in the outer Milky Way. We found that
the Hi filamentary structures follow the Galactic warp and flaring and that
they highlight some of the variations interpreted as the effect of the
gravitational interaction with satellite galaxies. In addition, the mean scale
height of the filamentary structures is lower than that sampled by the bulk of
the Hi emission, thus indicating that the cold and warm atomic hydrogen phases
have different scale heights in the outer galaxy. Finally, we found that the
fraction of the column density in Hi filaments is almost constant up to
approximately 18 kpc from the Galactic center. This is possibly a result of
the roughly constant ratio between the cold and warm atomic hydrogen phases
inferred from the Hi absorption studies. Our results indicate that the Hi
filamentary structures provide insight into the dynamical processes shaping
the Galactic disk. Their orientations record how and where the stellar energy
input, the Galactic fountain process, the cosmic ray diffusion, and the gas
accretion have molded the diffuse interstellar medium in the Galactic plane.

###### Key Words.:

ISM: structure – ISM: kinematics and dynamics – ISM: atoms – ISM: clouds –
Galaxy: structure – radio lines: ISM

##  1 Introduction

The diffuse neutral atomic hydrogen (Hi) is the matrix within which star-
forming clouds form and reside and the medium that takes a large portion of
the energy injected by the massive stars during their lifecycles (see, Dickey
& Lockman, 1990; Kalberla & Kerp, 2009; Ballesteros-Paredes et al., 2020, for
a review). Therefore, the investigation of the Hi distribution and dynamics is
crucial to understanding the cycle of energy and matter in the interstellar
medium (ISM, Ferrière, 2001; Klessen & Glover, 2016). In this paper, we
present a study of the Hi emission toward the Galactic plane, mainly focusing
on the orientation of its filamentary structure and the Galactic dynamics that
it reveals.

The Hi emission at the 21 cm wavelength has been instrumental for the study of
the Milky Way. This radio line is not limited by the dust absorption that
renders optical methods ineffective at distances greater than a few
kiloparsecs (van de Hulst, 1953). Face-on maps of the outer galaxy have been
constructed by using the Doppler shift of the emission line in combination
with the Galactic rotation curve (van de Hulst et al., 1954; Westerhout, 1957;
Oort et al., 1958). The concentration of the Hi gas in spiral arm features has
been evident since the first face-on projections, although the detailed
reconstruction of the density structures is complicated by the departures from
circular motion, the complexity of the velocity field, and the overlapping of
multiple components (Kerr, 1969).

Much of Hi is observed to be either a warm neutral medium (WNM) with $T$
$\approx$ $10^{4}$ K or a cold neutral medium (CNM) with $T$ $\approx$
$10^{2}$ K (Heiles & Troland, 2003). In the classical two-phase picture, both
phases exist in pressure equilibrium with stable phases for temperatures of
$T$ $\approx$ 300 K (CNM) and $T$ $\approx$ 5000 K (WNM) embedded in a hot
ionized phase (McKee & Ostriker, 1977; Wolfire et al., 2003). A portion of the
CNM is sampled by the absorption of background Hi emission by cold foreground
Hi, which is generically known as Hi self-absorption (HISA; Heeschen, 1955;
Gibson et al., 2000; Wang et al., 2020b). However, most of our understanding
of the multiphase structure of Hi comes from the study of absorption toward
continuum sources (see, for example, Strasser et al., 2007; Stanimirović et
al., 2014; Murray et al., 2018). Comparisons between 21-cm Hi emission and
absorption measurements indicate that, in the vicinity of the Sun, the WNM has
roughly the same column density as the CNM (Falgarone & Lequeux, 1973; Liszt,
1983). Observations of the absorption against strong continuum sources
indicate that the CNM to WNM mixture may be constant throughout the Galaxy
(Dickey et al., 2009, 2021).

The first all-sky Hi observations, provided by the 36′-resolution
Leiden/Argentine/Bonn (LAB) survey (Kalberla et al., 2005), displayed the
Milky Way as a nonaxisymmetric spiral system of four arms that are traced out
to at least 25 kpc from the Galactic center (Levine et al., 2006a; Nakanishi &
Sofue, 2016; Koo et al., 2017). These four arms are designated Sagittarius-
Carina (Sgr-Car), Scutum-Centaurus (Sct-Cen), Perseus, and Outer (or Norma-
Cygnus). Their existence, however, is in tension with the star counts toward
the tangent points, which favor the existence of only two spiral arms
(Drimmel, 2000; Benjamin, 2009; Churchwell et al., 2009).

Further studies of the Hi emission have also identified additional spiral arm
structures. Using the LAB survey observations, Dame & Thaddeus (2011)
identified a presumed spiral arm structure lying beyond the Outer Arm in the
first Galactic quadrant (QI) at approximately 15 kpc from the Galactic center.
This arm appears to be the continuation of the Scutum-Centaurus Arm in the
outer Galaxy, as a symmetric counterpart of the nearby Perseus Arm, for which
it is designated as the Outer Scutum-Centaurus (OSC) arm. Using the
2′-resolution Hi observations in the interferometric Southern Galactic Plane
Survey (SGPS, McClure-Griffiths et al., 2005), McClure-Griffiths et al. (2004)
identified a possible distant spiral arm in the fourth quadrant (QIV) of the
Milky Way. This distinct emission feature can be traced for over 70∘ as the
most extreme positive velocity feature in the longitude-velocity diagram.
Assuming circular motions around the Galactic center, this structure is
located at Galactocentric radii ($R_{\rm gal}$) between 18 and 24 kpc.

The Hi emission also reveals the bending of the Galactic plane noticeable at
$R_{\rm gal}$ $>$ 9 kpc and up to 40 kpc (Henderson et al., 1982; Levine et
al., 2006b; Kalberla et al., 2007). This “Galactic warp” is also observed in
the dust emission, in the distribution of Hii regions, in the CO emission, and
the kinematics of stars beyond the solar neighborhood (see, for example,
Drimmel & Spergel, 2001; Sun et al., 2020; Cheng et al., 2020; Paladini et
al., 2004). Furthermore, the Hi emission also reveals that the average disk
thickness shows a pronounced flaring that can be approximated by an
exponential function in the range $5$ $<$ $R_{\rm gal}$ $<$ 35 kpc (Kalberla &
Dedes, 2008).

In addition to the spiral-arm structure, the warp, and the flare, the
distribution of the Hi emission also carries the imprint of the energy input
by main sequence stars, supernovae (SNe), and other energetic process in the
ISM. The first photographic representations of the Hi emission toward the
Galactic plane revealed a series of shells, arcs, and filaments (Weaver &
Williams, 1973; Heiles, 1979). The Hi shells are thought to be formed by the
combined effects of stellar winds and SNe sweeping up gas and dust (Cox &
Smith, 1974; Castor et al., 1975; McKee & Ostriker, 1977). The vertical Hi
filaments, or “worms” seemingly “crawling away from the Galactic plane”, have
been interpreted as the walls of bubbles which have broken through the thin
gaseous disk (Heiles, 1984; Koo et al., 1992).

Dedicated studies of the filamentary structures in the Hi emission, based on
single-dish observations at 4′ and 30′ resolutions, have shown that they have
a preferential orientation parallel to the interstellar magnetic field in the
local ISM (Clark et al., 2014; Kalberla et al., 2016). This alignment may be
the product of magnetically-induced velocity anisotropies, the collapse of
material along field lines, shocks, or anisotropic density distributions (see,
for example, Lazarian & Pogosyan, 2000; Heitsch et al., 2001; Chen & Ostriker,
2015; Inoue & Inutsuka, 2016; Soler & Hennebelle, 2017; Mocz & Burkhart,
2018). More recently, Soler et al. (2020) presented the analysis of the
40″-resolution Hi interferometric observations in The Hi/OH/recombination-line
(THOR) survey (Beuther et al., 2016; Wang et al., 2020a), revealing that the
majority of the filamentary structures in the Hi emission are aligned with the
Galactic plane, but that there are portions of the first Galactic quadrant
dominated by vertical Hi structures that are most likely resulting from the
combined effect of SN feedback and the interstellar magnetic fields.

This paper presents the extension of the Soler et al. (2020) analysis to the
whole Galactic disk, hence extending in Galactic longitude and Galactic
latitude. The filament orientation is a characteristic of the Hi emission that
is less explicitly dependent on the angular resolution than other properties,
such as the filament width and the length. If the filaments are understood as
features in the intensity field and not as physical objects, this analysis
characterizes the anisotropy in the emission distribution. This characteristic
is particularly relevant given the natural axis of symmetry in the Galactic
plane. The gravitational potential of the Galaxy and the stretching due to the
circular motions and the spiral arms could naturally produce horizontal
structures. Departures from this natural anisotropy may indicate energetic
processes that lift the gas in the disk or infall. Thus, rather than defining
the filaments as objects, this study aims at studying the general anisotropy
of the atomic gas distribution across the Galactic disk.

To cover the entire Galactic plane, we used the state-of-the-art Hi emission
observations that we introduce in Sec. 2. We describe the methods for
identifying the filamentary structures and quantifying their preferential
orientation in Sec. 3. We present the main results of our analysis in Sec. 4.
We provide a discussion of our results in Sec. 5 and present our general
conclusions in Sec. 6. We reserve additional analyses for a set of appendices
as follows. Appendix A presents details on the Hessian matrix filament
identification method, detailing the use of circular statistics and the
selection of the analysis parameters. We present the method for de-projecting
the radial velocities into Galactocentric radii and the calculation of the
disk’s warp and flaring in App. B. Appendix C presents the comparison between
the filamentary structures found in the HI4PI and GALFA-HI surveys. Finally,
App. D presents the results of the filament analysis toward two high-velocity
cloud complexes.

##  2 Data

We used the publicly-available observations of the emission by atomic hydrogen
at 21 cm wavelength in the Hi 4$\pi$ (HI4PI) survey (HI4PI Collaboration et
al., 2016). The HI4PI survey is based on data from the Effelsberg-Bonn Hi
Survey (EBHIS, Kerp et al., 2011) and the Galactic All-Sky Survey (GASS,
McClure-Griffiths et al., 2009; Kalberla et al., 2010). This data comprises
21-cm neutral atomic hydrogen data over the whole sky in a radial velocity
range $-600$ $<$ $v_{\rm LSR}$ $<$ $600$ km s-1 for declination $\delta$ $>$
0∘ and $-470$ $<$ $v_{\rm LSR}$ $<$ 470 km s-1 for $\delta$ $<$ 0 ∘, as
observed with the 100-m radio telescope at Effelsberg and the 64-m radio
telescope at Parkes. These observations have full spatial sampling, thus
overcoming the major issue of the LAB survey, and provide a final data product
with an angular resolution of 16.′2 and sensitivity of 43 mK per 1.29-km s-1
velocity channel.

We used the data distributed in FITS-format binary tables containing lists of
spectra sampled on a Hierarchical Equal Area isoLatitude Pixelization
(HEALPix111http://healpix.sf.net) grid with $N_{\rm side}$ = $1024$, which
corresponds to a pixel angular size of 3.′44 (Górski et al., 2005; Zonca et
al., 2019). We arranged these spectra in all-sky HEALPix maps corresponding to
each of the 1.29-km s-1 velocity channels. Using the tools in the Python
healpy package, we produced a Cartesian projection for each one of these
velocity channels to obtain maps covering the range $-180$∘ $<$ $l$ $<$ 180∘
and $-10$∘ $<$ $b$ $<$ 10∘ with a pixel size $\delta l$ $=$ $\delta b$ $=$
3.′44\. We combined these velocity-channel maps into a single position-
position-velocity (PPV) cube, which we segmented for the analysis of
particular $l$ and $b$ ranges using the spectral-cube222http://spectral-
cube.readthedocs.io package.

We also used the publicly-available observations from the Galactic Arecibo
L-Band Feed Array Hi survey (GALFA-Hi) data release 2 (DR2, Peek et al.,
2018). This survey covers the Hi emission from $-650$ to $650$ km s-1, with
0.184 km s-1 channel spacing, 4′ angular resolution, and 150 mK root mean
square (rms) noise per 1 km s-1 velocity channel. The GALFA-Hi DR2
observations cover the entirety of the sky available from the William E.
Gordon 305 m antenna at Arecibo, from $\delta$ $=$ $-1$∘17′ to $+37$∘57′
across the whole right ascension ($\alpha$) range. We chose these data to
extend our analysis to two portions of the Galactic plane at higher angular
and spectral resolution. In the range $|b|$ $\leq$10∘, GALFA-Hi DR2 provides
observations in the ranges 36 .∘5 $<$ $l$ $<$ 70 .∘5 and 175 .∘3 $<$ $l$ $<$
209 .∘4.

We obtained the GALFA-Hi DR2 distributed in FITS-format cubes in the “narrow”
velocity range ($v$ $\leq$188 km s-1). through the GALFA collaboration
website333https://purcell.ssl.berkeley.edu/. We used the Python spectral-cube
and reproject444https://reproject.readthedocs.io packages to arrange these
cubes into two mosaics covering the aforementioned $l$ and $b$ ranges with a
pixel size $\delta l$ $=$ $\delta b$ $=$ 1.′0 and spectral resolution of 0.184
km s-1.

##  3 Methods

###  3.1 Analysis of filamentary structure

We selected the filamentary structures using the Hessian matrix technique as
implemented in Soler et al. (2021) and coded in the hessiananalysis routine of
the magnetar software package555https://github.com/solerjuan/magnetar (Soler,
2020). This method is less computationally demanding than other filament
identification methods, such as the rolling Hough transform (RHT, Clark et
al., 2014) or FILFINDER (Koch & Rosolowsky, 2015), and yields similar results
for the filament orientation, as detailed in Appendix C of Soler et al.
(2020). The main parameters used in the analysis are summarized in Table 1 and
were applied as follows.

The HI4PI data was divided in 3∘ $\times$ 20∘ $\times$ 1.29-km s-1 velocity
channel maps centered on Galactic latitude $b$ $=$ $0$∘ and covering the
Galactic longitude in the range 4∘ $<$ $l$ $<$ 356∘ and the radial velocity
range $-200$ $<$ $v_{\rm LSR}$ $<$ 200 km s-1. Throughout this paper we refer
to these velocity channels as “tiles”. The rectangular shape of the tiles aims
to maximize the $b$ coverage and maintain the fine sampling in $l$, covering
both the Galactic warp and flaring. This selection does not imply any loss of
generality in our results, as we demonstrate for different tile sizes in App.
A.

For each tile centered on the position $l_{0}$ and $b_{0}$ and $v_{\rm LSR}$
$=$ $v_{0}$, $I(l_{0},b_{0},v_{0})$, we estimated the derivatives with respect
to the local coordinates $(x,y)$ and built the Hessian matrix,

$\mathbf{H}(x,y)\equiv\begin{bmatrix}H_{xx}&H_{xy}\\\
H_{yx}&H_{yy}\end{bmatrix},$ (1)

where $H_{xx}$ $\equiv$ $\partial^{2}I/\partial x^{2}$, $H_{xy}$ $\equiv$
$\partial^{2}I/\partial x\partial y$, $H_{yx}$ $\equiv$
$\partial^{2}I/\partial y\partial x$, $H_{yy}$ $\equiv$
$\partial^{2}I/\partial y^{2}$, and $x$ and $y$ are related to the Galactic
coordinates $(l,b)$ as $x$ $\equiv$ $l\cos b$ and $y$ $\equiv$ $b$, so that
the $x$-axis is parallel to the Galactic plane. We obtained the partial
spatial derivatives by convolving $I(l_{0},b_{0},v_{0})$ with the second
derivatives of a two-dimensional Gaussian function. In practice, we used the
gaussian_filter function in the open-source software package SciPy (Virtanen
et al., 2020). We report the results obtained with a 18′ FWHM derivative
kernel, but present the results of different kernel sizes in App. A.

The two eigenvalues ($\lambda_{\pm}$) of the Hessian matrix were found by
solving the characteristic equation,

$\lambda_{\pm}=\frac{(H_{xx}+H_{yy})\pm\sqrt{(H_{xx}-H_{yy})^{2}+4H_{xy}H_{yx}}}{2}.$
(2)

Both eigenvalues define the local curvature of $I(l_{0},b_{0},v_{0})$. In
particular, the minimum eigenvalue ($\lambda_{-}$) highlights filamentary
structures or ridges, as detailed in Planck Collaboration Int. XXXII (2016).
The eigenvector corresponding to $\lambda_{-}$ defines the orientation of
intensity ridges with respect to the Galactic plane, which is characterized by
the angle

$\theta=\frac{1}{2}\arctan\left[\frac{H_{xy}+H_{yx}}{H_{xx}-H_{yy}}\right].$
(3)

We estimated $\theta$ for each of the pixels in $I(l_{0},b_{0},v_{0})$. This
angle, however, is only meaningful in regions of the map that are rated as
filamentary according to selection criteria based on the values of
$\lambda_{-}$ and on the noise properties of the data.

Table 1: Hessian analysis parameters. Parameter |  | Value
---|---|---
Tile size |  | 3∘ $\times$ 20∘ $\times$ 1.29 km s-1
Kernel size |  | 18′ FWHM
Intensity threshold |  | 0.3 K
Curvature threshold ($\lambda^{C}_{-}$) |  |  $-1.8$ K/deg2

###  3.2 Data selection for statistical analysis

In each tile, we selected the filamentary structures based on the criterion
$\lambda_{-}$ $<$ $\lambda^{C}_{-}$ $<$ $0$, where $\lambda^{C}_{-}$ is a
curvature threshold that we determined from the data. Additionally, we chose
regions where $I(l_{0},b_{0},v_{0})$ $>$ 7$\sigma_{I}$. We picked $\sigma_{I}$
$=$ 0.43 mK to match the brightness temperature noise level for a 1.29 km s-1
velocity channel reported in HI4PI Collaboration et al. (2016), which was
estimated using emission line-free spectral channels to compute the standard
deviation and iteratively clip outliers following the procedure described in
Winkel et al. (2016). We empirically tested different signal-to-noise ratio
(S/N) thresholds for a fixed $\lambda^{C}_{-}$ and found no significant
differences in our results for $I(l_{0},b_{0},v_{0})$ $>$ 3$\sigma_{I}$,
5$\sigma_{I}$, and up to 9$\sigma_{I}$, as shown in App. A.2. Most of the
differences introduced by the intensity threshold were found in a few tiles
beyond the terminal velocities, which are dominated by fainter emission and
are dismissed with the highest S/N cutoffs. We found that the distribution of
$V$ across tiles is more sensitive to $\lambda^{C}_{-}$ than to the intensity
S/N, thus, we chose 5$\sigma_{I}$ as a reference value to charaterize the
curvature threshold.

Following the method introduced in Planck Collaboration Int. XXXII (2016), we
estimated $\lambda^{C}_{-}$ in noise-dominated velocity channels. For each
tile group centered on a particular Galactic longitude, we estimated
$\lambda_{-}$ in five velocity channels with low S/N and determined the
minimum value of $\lambda_{-}$ in each of them. We used the median of these
five $\lambda_{-}$ values as the threshold value, $\lambda^{C}_{-}$. We
employed the median to reduce the effect of outliers, but in general the
values of $\lambda_{-}$ in the noise-dominated channels are similar and this
selection does not imply any loss of generality. In the filament orientation
analysis, we exclusively considered regions of each velocity channel map where
$\lambda_{-}$ $<$ $\lambda^{C}_{-}$, which corresponds to the selection of
filamentary structures with curvatures in $I(l_{0},b_{0},v_{0})$ larger than
those present in the noise-dominated channels. We report the results of
applying a common $\lambda^{C}_{-}$ estimated from the values found in all of
the tile groups, but also evaluated the selection of a different
$\lambda^{C}_{-}$ for each tile group, which we present in App. A.2. The
outcome of the two approaches is very similar and the selection of one instead
of the other does not have a significant effect on the results of our
analysis.

###  3.3 Circular statistics

Once the filamentary structures were selected, we used the angles derived from
Eq. 3 to study their orientation with respect to the Galactic plane. For a
systematic evaluation of the preferential orientation, we applied the
projected Rayleigh statistic ($V$) (see, for example, Batschelet, 1981), which
is a test to determine whether the distribution of angles is not uniform and
peaked at a particular angle. This test is equivalent to the modified Rayleigh
test for uniformity proposed by Durand & Greenwood (1958) for the specific
directions of interest $\theta$ $=$ $0^{\circ}$ and $90^{\circ}$ (Jow et al.,
2018), such that $V$ $>$ 0 or $V$ $<$ 0 correspond to preferential
orientations parallel or perpendicular to the Galactic plane, respectively. It
is defined as

$V=\frac{\sum_{ij}w_{ij}\cos(2\theta_{ij})}{\sqrt{\sum_{ij}w_{ij}/2}},$ (4)

where the indices $i$ and $j$ run over the pixel locations in the two spatial
dimensions $(l,b)$ for a given velocity channel and $w_{ij}$ is the
statistical weight of each angle $\theta_{ij}$. In our application, we
accounted for the spatial correlations introduced by the telescope beam by
choosing $w_{ij}$ $=$ $(\delta l/\Delta)^{2}$, where $\delta l$ is the pixel
size and $\Delta$ is the diameter of the derivative kernel that we selected to
calculate the gradients. We present the results of the filament orientation
trends in terms of the mean orientation angle, $\left<\theta\right>$, and the
Rayleigh test for uniformity, $Z$, in App. A.1.

In this paper, we report the results obtained with an 18.′0 FWHM two-
dimensional Gaussian derivative kernel. We chose this kernel size to be close
to the 16.′2 FWHM resolution of the HI4PI survey, which sets the angular scale
of the independent measurements with the Hessian matrix. The selection of a
larger kernel reduces the number of independent gradients and thus, the
significance of the orientation trends quantified by $V$, as illustrated in
App. A.2. We chose a kernel slightly larger than the angular resolution of the
observations motivated by an initial choice that mitigated the effect of noise
in the analysis of the interferometric observations in the THOR survey (Soler
et al., 2020). This early selection was carried forward in this work although
its results are essentially identical to those found using 16.′2 and 17.′0
FWHM derivative kernels.

The values of $V$ indicate a significant preferred orientation if there is
sufficient clustering around the orientations $\theta$ $=$ $0^{\circ}$ and
$90^{\circ}$. The null hypothesis implied in $V$ is that the angle
distribution is uniform. In the particular case of independent and uniformly
distributed angles, and for a large number of samples, values of $V$ $\approx$
$1.64$ and $2.57$ correspond to the rejection of the null hypothesis with a
probability of 5% and 0.5%, respectively (Batschelet, 1972). Thus, a value of
$V$ $\approx$ 2.87 is roughly equivalent to a 3$\sigma$ confidence interval.

The statistical significance represented by $V$ is determined numerically by
taking random samples from a von Mises probability distribution, which is a
circular analog of a Gaussian distribution (Mardia & Jupp, 2009). It is
defined as

$\mathcal{F}(\theta|\mu,\kappa)=\frac{e^{\kappa\cos(\theta)-\mu}}{2\pi
I_{0}(\kappa)},$ (5)

where $I_{0}(\kappa)$ is the modified Bessel function of order 0, $\mu$ is the
mean of the distribution, and $1/\kappa^{2}$ characterizes the dispersion.
Following a procedure like the one described in Jow et al. (2018), the
significance is determined by drawing $N$ samples from a von Mises
distribution and computing $V$ for a given null hypothesis rejection
probability, for example, 5% and 0.5%. In a similar way to the chi-square test
probabilities, $V$ and its corresponding null hypothesis rejection probability
are reported in the classical circular statistics literature as tables of
“critical values”, for example in Batschelet (1972), or computed in the
circular statistics packages, such as circstats in astropy (Astropy
Collaboration et al., 2018).

##  4 Results

Figure 1: Longitude-velocity ($lv$) diagrams of the mean intensity
($\left<I\right>$, left) and the filament orientation quantified by the
projected Rayleigh statistic ($V$, right). Each of the pixel elements in the
diagrams corresponds to a 3∘ $\times$ 20∘ $\times$ 1.29 km s-1 velocity
channel map centered on $b$ $=$ 0∘, which we call tile throughout out this
paper. These results correspond to the Hessian analysis performed using an 18′
FHWM derivative kernel. Values of $V$ $>$ 0 (red) or $V$ $<$ $0$ (blue)
indicate a preferential orientation of the filaments parallel, $\theta$ $=$
$0$∘, or perpendicular, $\theta$ $=$ $90$∘, to the Galactic plane. The
3$\sigma$ statistical significance for these two orientations corresponds to
$V$ $>$ 2.87 or $V$ $<$ $-2.87$, respectively. The overlaid curves correspond
to the main spiral arms features presented in Reid et al. (2016), the outer
Scutum-Centaurus arm (OSC, Dame & Thaddeus, 2011), and the extended outer arm
(M-G2004, McClure-Griffiths et al., 2004).

The general results of the Hi filament orientation analysis are presented in
Fig. 1. The longitude-velocity ($lv$) diagrams show the values of the mean
intensity, $\left<I\right>$, and the filament orientation quantified by the
projected Rayleigh statistic, $V$, for each of the 3∘ $\times$ 20∘ $\times$
1.29 km s-1 tiles around the Galactic plane. For the sake of reference, we
have included lines that indicate the position of the main features that have
been traditionally assigned to the Sgr-Car, Sct-Cen, Perseus, and Outer spiral
arms, as derived from the trigonometric parallaxes and proper motions for
masers associated with young high-mass stars measured in the Bar and Spiral
Structure Legacy (BeSSeL) survey (Reid et al., 2016). We also included the OSC
arm and the distant arm reported in McClure-Griffiths et al. (2004).

The $\left<I\right>$ $lv$-diagram, shown on the left-hand side panel of Fig.
1, corresponds to the average of the intensity over the area of each tile. It
is presented as a reference for the interpretation of the $V$ $lv$ diagram and
not for the particular study of the spiral arm structure. However, the
$\left<I\right>$ $lv$ diagram clearly shows the tracks assigned to the Perseus
and Outer arms in the first and second Galactic quadrants. Other features,
such as the Sct-Cen and the Sgr-Car arms, are less conspicuous.

The right-hand side panel of Fig. 1 suggests that there is no apparent
correlation between the locations of the spiral arms in the $lv$-diagrams and
the preferential filament orientations characterized by $V$. The most
prominent features in the $V$ $lv$-diagram, shown on the right-hand side panel
of Fig. 1, are two groups of filaments parallel to the Galactic plane at the
minimum and maximum radial velocities toward the first two (QI and QII) and
the last two (QIII and QIV) Galactic quadrants, respectively. These bands,
characterized by $V$ $>$ 5, extend over a few tens of kilometers per second
and appear on radial velocities beyond the $\left<I\right>$ peaks and the
locations of the OSC and the McClure-Griffiths et al. (2004) spiral arms. The
high-$V$ band in QIII and QIV appears to extend into QII in the range $0$ $<$
$v_{\rm LSR}$ $<$ 30 km s-1 down to $l$ $\approx$ 110∘. Both high-$V$ bands
are distinguishable around $l$ $\approx$180∘, where they appear to be
separated by a gap of around 20 km s-1. The extension in $l$ of these two
bands depends on the $b$ coverage of the analysis; that is, they appear
truncated or are less prominent when considering ranges $|b|$ $\leq$ $1.5$,
$3.0$, or $5.0$, as shown in App. A.

Around 10% of the Hi emission over the whole sky is coming from intermediate-
and high-velocity clouds (IVCs and HVCs, respectively), which are parcels of
gas at velocities incompatible with a simple model of differential Galactic
rotation (see Wakker & van Woerden, 1997; Putman et al., 2012, for reviews).
Most of these structures are located at $|b|$ $>$ 10∘. The majority of the
HVCs in the range $|b|$ $<$ 10∘ are too small to dominate the trends reported
in this work, for example, the objects in the GALFA-HI Compact Cloud Catalog
(Saul et al., 2012). However, there are two exceptions: (i) the complex H
(Hulsbosch, 1975), located around 110∘ $<$ $l$ $<$ 150∘, $-15$∘ $<$ $b$ $<$
15∘ and $-220$ $<$ $v_{\rm LSR}$ $<$ $-80$ km s-1; (ii) the anticenter (AC)
complexes (Wakker & van Woerden, 1991) around 155∘ $<$ $l$ $<$ 190∘, $-55$∘
$<$ $b$ $<$ $-5$∘ and $-340$ $<$ $v_{\rm LSR}$ $<$ $-135$ km s-1. Figure 1
indicates that both of these regions are beyond the high-$V$ bands and are
composed of mainly horizontal Hi filaments. We present the results of the
filament analysis for these two HVC complexes in App. D.

Figure 2 indicates that there is no evident correlation between the mean
intensity in the tiles, $\left<I\right>$, and the orientation of the
filamentary structure, characterized by $V$. Around 83% of the tiles show
values $V$ $>$ 0 and roughly 23% present $V$ $>$ 2.87. Only 0.34% of the tiles
display $V$ $<$ $-2.87$, which indicates that filamentary structures
perpendicular to the Galactic plane are rare. The tiles with the highest
$\left<I\right>$ generally indicate a preference for filaments parallel to the
Galactic plane, as shown by $V$ $>$ 0 in the range $\left<I\right>$ $\gtrsim$
60 K. However, most of the tiles with a high significance in preferential
orientation, $|V|$ $>$ 2.87, are located in the range $\left<I\right>$
$\lesssim$ 60 K, thus suggesting that the overlapping of multiple components
in the tiles with the highest $\left<I\right>$ randomizes the orientation of
the filamentary structures.

Figure 2 also shows the filament coverage, which we define as the number of
pixels identified as filaments according to the criteria described in Sec. 3.2
over the total number of pixels in each tile. We found that the tiles with the
highest $\left<I\right>$ show a larger coverage of filamentary structures,
which may result from the selection based on the intensity S/N. Still, both
high and low $V$ values appear in tiles with similar filament coverage, which
indicates that the preferential orientations are not the product of the number
of filaments identified in a tile. Tiles with low $\left<I\right>$ and
filament coverage are more likely to be dominated by a single intensity
structure, which explains why they display the most significant values of $V$.

Figure 3 presents an example of the filamentary structure in the emission
toward the high-$V$ bands reported in Fig. 1. The emission maps show that the
filament orientation trends are not dominated by a monolithic object but
rather by a collection of filaments where the orientation parallel to the
Galactic plane seems to be the most frequent. The most prominent horizontal
structures in the emission maps appear over a background of fainter and
seemingly randomly oriented filamentary structures. In general, the width of
the filamentary structures matches the kernel size used to calculate the
Hessian matrix. This fact makes the general definition of the length of the
filaments problematic since the apparent overlap of the most extended
structures may not be a product of spatial coherence but an artifact of the
available angular resolution.

Figure 2: Mean Hi emission intensity ($\left<I\right>$) and projected Rayleigh
statistic ($V$, Eq. 4) for the 3∘ $\times$ 20∘ $\times$ 1.29 km s-1 tiles. The
colors indicate the tile percentage covered by the filamentary structures
selected following the criteria described in Sec. 3.2. The horizontal dotted
lines show $V$ $=$ $\pm 2.87$, which indicate the 3$\sigma$ significance in
the preferential orientation parallel, $\theta$ $=$ $0$∘, or perpendicular,
$\theta$ $=$ $90$∘, to the Galactic plane, respectively. The vertical dashed
line indicates the intensity threshold.

Figure 3: Examples of the Hi emission and filamentary structures toward the
$l$ and $v_{\rm LSR}$ ranges with predominantly horizontal filamentary
structures. The top image in each panel corresponds to the Hi emission
intensity in three velocity-channel maps presented in red, green, and blue
colors. The bottom image corresponds to the second eigenvalue of the Hessian
matrix, $\lambda_{-}$, which highlights the filamentary structures in each
velocity-channel map. The white disk in the lower right corner corresponds to
the size of the derivative kernel used to calculate the Hessian matrix.

###  4.1 Distribution of Hi filamentary structures in the Galactic plane

We analyzed the distribution of Hi filamentary structures in the Galactic
plane by assuming circular motions around the Galactic center and a standard
rotation curve, as detailed in App. B. First, we focused on the behavior of
the filament orientations with heliocentric radius to study the combined
effect of the angular resolution and distance. Second, we compared the
filament orientations at different heliocentric kinematic distances. Finally,
we studied the variations with Galactocentric radius and produced a face-on
view of the Hi filament orientation in the Galactic plane.

Figure 4: Kinematic heliocentric distance ($d$) and projected Rayleigh
statistic (V, Eq. 4) for the 3∘ $\times$ 20∘ $\times$ 1.29 km s-1 tiles. We
report the largest (far) solution to the heliocentric distance equation, as
detailed in App. B. From top to bottom, the panels correspond to each Galactic
quadrant and to the whole Galactic longitude range. The colors indicate the
Galactic longitude of the tiles. The horizontal dashed lines indicate $V$ $=$
$\pm 2.87$, which correspond to the 3$\sigma$ significance in the preferential
orientation around $\theta$ $=$ $0$∘ or $\theta$ $=$ $90$∘, respectively. The
solid line in the bottom panel corresponds to the mean $V$ values in 1-kpc
$R_{\rm gal}$ bins.

####  4.1.1 Filament orientations and heliocentric distance

Figure 4 shows the distribution of $V$ and the far kinematic heliocentric
distance, $d_{\rm far}$, derived for each tile. We chose to report $d_{\rm
far}$ to focus on the effect of distance on $V$, but similar conclusions can
be drawn from the near kinematic distances, $d_{\rm near}$, as shown in App.
B. We found an increase in $V$ with heliocentric distance that is particularly
prominent toward QIII and QIV. However, we did not find that the tiles
corresponding to the largest distances present notably large $V$, thus
suggesting that the increase in $V$ is a product of heliocentric distance
alone.

The mean $V$ values in 1-kpc $R_{\rm gal}$ bins, shown by the black line in
the bottom panel of Fig. 4, reveal a general preference for $V$ $>$ $0$.
However, these mean $V$ values are below the $2.87$ limit corresponding to a
3$\sigma$ significance of a preferential orientation parallel to the Galactic
plane. Most of the scatter in the distribution of $V$ for a particular $R_{\rm
gal}$ seems to come from the azimuthal variations, as can be inferred from the
comparison between the general trend and that toward each Galactic quadrant.
Toward QII and QIII, we found a systematic increase in $V$ up to $d$ $\approx$
15 kpc followed by a systematic decrease at larger distances. This may be the
product of the decrease in the emission at the edge of the Galactic disk.
Toward QI and QIV, we found a decrease in $V$ up to $d_{\rm far}$ $\approx$ 10
kpc, or $d_{\rm near}$ $\approx$ 5 kpc, as illustrated in App. B.

To further establish if the prevalence of $V$ $>$ 0 at large distances is an
effect introduced by the distance, we repeated the analysis using larger
derivative kernels in App. A.2.3. Using derivative kernels with diameters two
and three times larger, which is equivalent to locating the sources two or
three times further away, results in a systematic reduction of $V$ rather than
a bias toward $V$ $>$ 0\. We interpret that result as a strong indication that
the relative orientation trends are not exclusively the product of the
heliocentric distance. However, we also conducted a test using a derivative
kernel fixed to a particular physical scale, which we describe in the
following section.

Figure 5: Mean intensity ($\left<I\right>$, top), projected Rayleigh statistic
($V$, middle), and derivative kernel size ($\Delta$, bottom) for 3∘ $\times$
20∘ $\times$ 1.29 km s-1 toward the indicated positions. The two lines in the
middle panel indicate the results obtained with a derivative kernel with a
fixed angular size and a kernel with a fixed physical size, as indicated on
the legend. The kernel with a fixed physical size is exclusively used in the
radial velocity range corresponding to kinematic distances where the angular
size of the kernel is below 1∘.

####  4.1.2 Filament orientations near and far

By fixing the size of the derivative kernel in the Hessian analysis, we are
effectively selecting a physical size scale that changes with the distance.
For example, at a distance of approximately 4.5 kpc the 18′ FWHM derivative
kernel would select filamentary structures of minimum widths around 25 pc.
Those structures would be blended into 100-pc-width structures sampled at 18
kpc with the same beam.

We evaluated the effect of distance by considering a derivative kernel with a
fixed physical size. We ran this test toward the positions $l$ $\approx$ 90∘
and $l$ $\approx$ 270∘, where the mapping between $v_{\rm LSR}$ and $d$ has
the largest dynamic range. We limited the kernel angular size to $\Delta$ $<$
1∘ to avoid artifacts produced by the limited sampling of the 3∘ $\times$ 20∘
$\times$ 1.29 km s-1 tiles. This limit excludes the radial velocity range
where most of the emission lies but covers the range in which the tiles with
the highest $V$ are found. Figure 5 shows the comparison of the results
obtained with the 18′ kernel and a kernel with varying angular size
corresponding to $\Delta$ $=$ 160 pc at different distances.

We found that the increasing kernel sizes produced lower values of $V$. This
indicates that rather than producing a bias toward higher $V$, as would be the
case if multiple structures with a random orientation were blended into a
single horizontal filament, the smoothing results in a loss of the
preferential orientation parallel to Galactic plane. This trend toward lower
values of $V$ was also found when using derivative kernels with larger angular
sizes, as detailed in App. A. These results strongly suggest that the high-$V$
bands in the $lv$-diagrams in Fig. 1 can not be explained by an effect
exclusively related to the heliocentric distance.

Figure 3 presents the emission toward the range in $l$ and $v_{\rm lsr}$
corresponding to the high-$V$ bands in Fig. 1. Rather than being concentrated,
the emission appears as a series of horizontal filaments distributed in a
broad range of Galactic latitudes. This distribution is the product of the
Galactic flaring and indicates that the smoothing by the derivative kernel is
not blending the emission into a single monolithic structure at large
distances. Therefore, we conclude that the high values of $V$ are not the
result of an observational effect but a property of the Hi structure.

####  4.1.3 Filament orientations and Galactocentric distance

We computed a de-projection of the $\left<I\right>$ and $V$ $lv$-diagrams into
a face-on view of the Milky Way, shown in Fig. 6. This reconstruction does not
aim to produce a complete 3D density distribution but rather identify the
Galactocentric radii and azimuth to which the features in the $V$
$lv$-diagrams belong. Following preceding works, such as Levine et al. (2006b)
and Kalberla et al. (2007), we excluded from the analysis the Galactic
longitude ranges $l$ $<$ 15∘, $l$ $>$ 345∘ and 165∘ $<$ $l$ $<$ 195∘, as these
regions have radial velocities that are too small with respect to their random
velocities to establish reliable distances. We also excluded from this
reconstruction points with $R_{\rm gal}$ $<$ 10 kpc, to avoid the
complications introduced by the near-far distance ambiguity and the artifacts
produced by the emission around $v_{\rm LSR}$ $\approx$ 0 km s-1.

The face-on reconstruction of the mean intensity, shown on the left-hand-side
panel of Fig. 6, displays some of the spiral arm features identified in the
LAB survey around 90∘ $\lesssim$ $\phi$ $\lesssim$150∘ and 200∘ $\lesssim$
$\phi$ $\lesssim$ 250∘ around 10 $<$ $R_{\rm gal}$ $<$ 15 kpc (Levine et al.,
2006a; Nakanishi & Sofue, 2016; Koo et al., 2017). The identification of these
arms is not the main focus of our analysis, but they appear highlighted by the
emission concentration. The arms in QI and QIV are less conspicuous and not
clearly identifiable without additional processing, such as, the median
filtering applied in Levine et al. (2006a) or the spectral peak identification
used in Koo et al. (2017).

The face-on reconstruction of the filament orientation trends, presented on
the right-hand-side panel of Fig. 6, shows a clustering of high $V$ at $R_{\rm
gal}$ $\gtrsim$ 15 kpc on both sides of the Galaxy. This collection of
high-$V$ structures, which corresponds to the bands in the $lv$-diagrams in
Fig. 1, does not appear simply as a ring but seems to have a pitch angle. This
pitch angle impression is produced by the displacement of the peak $V$ values
from $R_{\rm gal}$ $\approx$ 18 kpc at $\phi$ $\approx$ 210∘ to $R_{\rm gal}$
$\approx$ 23 kpc at $\phi$ $\approx$ 270∘. The pitch angle is less noticeable
in the other side of the Galaxy, where there is a prominent exception to the
general $V$ $>$ 0 trend around $R_{\rm gal}$ $\approx$ 17 kpc and $l$
$\approx$ 95∘. This feature, which corresponds to the negative $V$ tiles at
$d$ $\gtrsim$ 25 kpc in Fig. 7, was also identified in the Galactic warp and
flare studies reported in Levine et al. (2006a).

Figure 6 also shows the asymmetry in the Hi between the first two and the last
two quadrants. Toward QI and QII there is a drop in the amount of emission at
$R_{\rm gal}$ $\approx$ 25 kpc, which is noticeable in the lack of filaments
and relative orientation trends beyond that radius in the range 45 $\lesssim$
$\phi$ $\lesssim$ 130∘. However, there is enough emission to suggest a drop in
$V$ at $R_{\rm gal}$ $\approx$ 25 kpc with respect to its peak value, which we
explored further by computing radial profiles.

Figure 6: Face-on view of the mean Hi emission intensity ($\left<I\right>$;
left) and the filament orientation quantified by the projected Rayleigh
statistic ($V$; right). Values of $V$ $>$ 2.87 or $V$ $<$ $-2.87$ indicate a
statistical significance above 3$\sigma$ in the preferential orientation
around $\theta$ $=$ $0$∘ or $\theta$ $=$ $90$∘, respectively. The yellow
circular marker indicates the position of the Sun. The overlaid curves
correspond to the spiral arm models identified in Reid et al. (2019), shown in
black color, and their extrapolation to larger Galactocentric radii, shown in
green.

Figure 7 shows the distribution of $V$ with respect to the $R_{\rm gal}$
derived from the radial velocity of each tile and the average radial profile
computed in 1-kpc-width bins in the range 2 $\leq$ $R_{\rm gal}$ $\leq$ 30
kpc. We included the observations at $R_{\rm gal}$ $<$ 10 kpc since the
$R_{\rm gal}$ estimates are not affected by the heliocentric distance
ambiguity. The general trend is a progressive increase of $V$ with growing
$R_{\rm gal}$ until it reaches its maximum values around $R_{\rm gal}$
$\gtrsim$ 20 kpc. This tendency is equivalent to the continuous transition
from Hi filaments slightly perpendicular or having no preferred orientation to
mostly parallel to the Galactic plane with increasing distance from the
Galactic center. The significance of the relative orientation in the tiles
dominated by mostly perpendicular filaments increases when considering broader
tiles in Galactic longitude, as illustrated in App. A.2.1. However, the
broader tile size reduces the sampling in Galactic longitude and the
resolution in the face-on reconstructions presented in Fig. 6.

Figure 7 also shows the central locations of the Hi shells in the catalogs
produced with the LAB survey (Ehlerová & Palouš, 2013) and the SGPS
observations (McClure-Griffiths et al., 2002) in the range $|b|$ $<$ 10∘. Most
of the Hi shells are found within the range $R_{\rm gal}$ $<$ 10 kpc, and
their number tends to decrease with increasing Galactocentric radius. For
$R_{\rm gal}$ $>$ 15 kpc, where the values of $V$ reveal a very significant
prevalence of Hi filaments parallel to the Galactic plane, there are very few
identified Hi shells. The likely relationship between the values of $V$ and
the density of Hi shells with $R_{\sc i}$ suggest a connection between the Hi
filament orientation and SN feedback, which we discuss further in Sec. 5.

In the vicinity of the solar circle, $R_{\rm gal}$ $=$ 8.15 kpc (Reid et al.,
2019), we found both negative and positive values of $V$ that show a
significant departure from the general trend. Some of these fluctuations are
due to Hi emission in the solar neighborhood, whose proximity makes the local
atomic gas dynamics more prominent than the large scale trend. The peak in
negative $V$ around $R_{\rm gal}$ $\approx$ 10 kpc is most likely associated
with the large Hi shells identified in the range 250∘ $\lesssim$ $l$
$\lesssim$ 360∘ in radial velocities that roughly correspond to that position
in the Galactic plane (McClure-Griffiths et al., 2002). Additional
fluctuations in $V$ are also introduced by noncircular motions, which are
particularly dominant around the anticenter, thus explaining why the largest
$V$ are observed around $l$ $\approx$ 160∘. It is likely that the emission
around that Galactic longitude is produced further away and that the distance
estimates based on the line kinematics are inaccurate (see, for example,
Burton, 1971; Peek et al., 2022).

At $R_{\rm gal}$ $\gtrsim$ 16 kpc the dispersion in $V$ increases and displays
relatively large values of $V$ and an excursion into $V$ $<$ 0 for some of the
tiles. The points corresponding to $V$ $<$ 0 around $R_{\rm gal}$ $\approx$ 20
kpc can be traced into an individual feature around $\phi$ $\approx$ 100∘ that
is also found as an outlier in the Galactic warp (Levine et al., 2006b). Most
of the tiles with high values of $V$ at $R_{\rm gal}$ $\gtrsim$ 20 kpc are
found in QIV, where the high $V$ band in the $lv$-diagram is the broadest, but
for most of the lines of sight $V$ decreases beyond $R_{\rm gal}$ $\gtrsim$ 20
kpc.

Figure 7: Galactocentric radius ($R_{\rm gal}$) and projected Rayleigh
statistic ($V$, Eq. 4) for the 3∘ $\times$ 20∘ $\times$ 1.29 km s-1 tiles.
From top to bottom, the panels correspond to each Galactic quadrant and to the
whole Galactic longitude range. The colors indicate the Galactic longitude of
the tiles. The horizontal dashed lines indicate $V$ $=$ $\pm 2.87$, which
indicate the 3$\sigma$ significance in the preferential orientation around
$\theta$ $=$ $0$∘ or $\theta$ $=$ $90$∘, respectively. The vertical dotted
line indicates the radius of the solar orbit. The solid line in the bottom
panel corresponds to the mean $V$ values in 1-kpc $R_{\rm gal}$ bins. The
vertical solid lines mark the central locations of the Hi shells identified in
the LAB survey (Ehlerová & Palouš, 2013) and the SGPS (McClure-Griffiths et
al., 2002) in the range $|b|$ $<$ 10∘, shown in gray and green colors,
respectively.

###  4.2 Scale height of the Hi filaments

In addition to the analysis of the filament orientation distribution with
$R_{\rm gal}$, we considered two other Hi emission properties that also change
with the distance from the Galactic center: the midplane height and the
dispersion around the midplane. We computed the Galactic height $z$ for each
of the tiles and obtained its first and second intensity-weighted moments,
$\left<z\right>$ and $\sigma_{z}$ following the procedure detailed in App. B.
We estimated $\left<z\right>$ and $\sigma_{z}$ both for all of the Hi emission
in each tile and exclusively for the emission in the structures classified as
filaments; hence we obtained two separate estimates of the warp and the flare.

Figure 8 shows the two face-on reconstructions of $\left<z\right>$. The
midplane height map obtained using all of the Hi emission reproduces some of
the main features obtained in the previous analysis of lower resolution data
(Levine et al., 2006b; Kalberla et al., 2007). First, the warp of the Milky
Way, which corresponds to the variation in midplane height from
$\left<z\right>$ $\approx$ 4 kpc toward QI and QII to $\left<z\right>$
$\approx$ $-1$ kpc at $R_{\rm gal}$ $\lesssim$ 20 kpc toward QIII and QIV.
Second, the finding of $\left<z\right>$ $>$ 0 and $\left<z\right>$ $<$ 0 for
adjacent tiles toward QIII and QIV at $R_{\rm gal}$ $\gtrsim$ 20 kpc, which
has been identified as Galactic disk “scalloping” (Kulkarni et al., 1982;
Levine et al., 2006b). Finally, a sharp dip around $\phi$ $\approx$ 90∘ and
$R_{\rm gal}$ $\gtrsim$ 20 kpc.

The $\left<z\right>$ found with the Hi filaments shows features that are very
similar to the midplane heights obtained using the bulk of the Hi emission.
However, the contrast in the warp, the scalloping, and the sharp dip is much
higher. This result is most likely a consequence of filtering the diffuse
emission implicit in the filament identification.

The $\left<z\right>$ radial profiles, presented in Fig. 9, are very similar
for all the Hi emission and for the Hi filaments in the range $R_{\rm gal}$
$\lesssim$ 20 kpc. For larger $R_{\rm gal}$, the Hi filaments reveal that the
Galactic warp extends up to approximately 4 kpc for tiles in the first
Galactic quadrant. The Hi filaments also show higher values of
$\left<z\right>$ at $R_{\rm gal}$ $\gtrsim$ 20 kpc toward QIII and QIV, which
strongly suggest that specific filamentary structures produce the apparent
disk scalloping.

Figure 8: Face-on view of the Hi midplane height ($\left<z\right>$)
reconstructed from all the emission in the range $|b|$ $<$ 10∘ (left) and the
emission in filamentary structures (right).

Figure 9:  Midplane height ($\left<z\right>$) and Galactocentric radius
($R_{\rm gal}$) for the 3∘ $\times$ 20∘ $\times$ 1.29 km s-1 tiles. The left-
hand side panel corresponds to all the Hi emission in the $|b|$ $<$ 10∘ range.
The right-hand side panel corresponds to the emission in the structures
classified as filaments. The colors indicate the Galactic longitude, $l$. The
vertical dotted line indicates the radius of the solar orbit around the
Galactic center.

Figure 10 shows the two face-on reconstructions of $\sigma_{z}$. The
dispersion of the Hi emission around the midplane shows the flaring of the
Galactic disk at $R_{\rm gal}$ $>$ 15 kpc, both in all the emission and just
in the filaments. However, the radial profiles shown in Fig. 11 indicate that
the average trend is for the filamentary structures to have a lower dispersion
than the bulk of the Hi emission by about a factor of two at $R_{\rm gal}$ $>$
15 kpc. This general trend is followed in most of the $l$ range, particularly
around 250∘ $<$ $l$ $<$ 350∘, but there are clear exceptions where the
$\sigma_{z}$ values traced by the filaments drop abruptly. This can be
attributed to the lower emission in radial velocities corresponding to larger
distances toward QI, which makes difficult the filament identification at
$R_{\rm gal}$ $\gtrsim$ 25 kpc in that azimuth range.

Figure 10: Face-on view of the dispersion around the midplane height
($\sigma_{z}$) reconstructed from all the emission in the range $|b|$ $<$ 10∘
(left) and the emission in filamentary structures (right).

Figure 11: Same as Fig. 9, but for the dispersion around the midplane height
($\sigma_{z}$).

###  4.3 Global properties of the Hi filaments

The results of the Hessian analysis provide a description of the Hi emission
morphology across velocity channels, which does not imply that the highlighted
structures are filaments or sheets in three-dimensional space. Our finding is
that the high-aspect-ratio features in the Hi emission have a preferential
orientation with respect to the Galactic plane. Without any loss of
generality, we can also estimate what fraction of the Hi column density is in
these high-aspect-ratio features.

We calculated the Hi column density in each tile using the assumption of
optically thin emission (see, for example, Kulkarni & Heiles, 1987), such
that,

$N_{\rm H}=1.83\times
10^{18}\left(\frac{T_{b}}{\mbox{K}}\right)\left(\frac{\Delta v}{\mbox{km
s}^{-1}}\right)\mbox{cm}^{-2},$ (6)

where $T_{b}$ is the brightness temperature and $\Delta v$ is the width of the
velocity channel. This assumption reduces our calculation to the ratio between
the sum of $T_{b}$ in the filamentary structures and the total $T_{b}$ in each
tile. We note, however, that the optically-thin assumption underestimates the
amount of Hi in the Galactic plane (Bihr et al., 2015; Wang et al., 2020a).
The optical depth mainly affects the inferred CNM column density, since
typically $\tau$ $\ll$ 1 for the WNM (see, for example, Heiles & Troland,
2003). Insofar as the ratio of CNM to WNM in the filaments may be different
than in the total gas, our results for $N^{\rm fil}_{\rm H}/N_{\rm H}$ are
only indicative of the global trend.

Figure 12 shows the distribution of $N^{\rm fil}_{\rm H}/N_{\rm H}$ as a
function of distance from the Galactic center. The $N^{\rm fil}_{\rm H}/N_{\rm
H}$ radial profile is tightly distributed around 80% for $R_{\rm gal}$
$\lesssim$ 18 kpc. For larger $R_{\rm gal}$, the mean value of $N^{\rm
fil}_{\rm H}/N_{\rm H}$ decreases sharply, although we found a large
dispersion around this general trend. The general decrease in $N^{\rm
fil}_{\rm H}/N_{\rm H}$ for $R_{\rm gal}$ $\lesssim$ 18 kpc is also found when
fixing the $I$ S/N threshold using larger derivative kernels, as illustrated
in App. A.2.3. Thus, it is most likely that this trend represents a transition
toward less structured Hi emission rather than a drop due to the decrease in
the emission.

The azimuthal variations in the $N^{\rm fil}_{\rm H}/N_{\rm H}$ radial profile
are most likely a product of the asymmetry in the Galactic disk. In the range
$l$ $\lesssim$ 90∘ we found a sharp decrease in $N^{\rm fil}_{\rm H}/N_{\rm
H}$ that corresponds to the drop in emission toward that particular direction.
In contrast, we obtained $N^{\rm fil}_{\rm H}/N_{\rm H}$ around 70% up to
$R_{\rm gal}$ $\approx$ 30 kpc for some tiles around $l$ $\approx$ 150∘.

The sharp drop in the mean values of $N^{\rm fil}_{\rm H}/N_{\rm H}$ with
Galactocentric radius roughly coincides with the transition from mostly
horizontal to randomly oriented Hi filaments at $R_{\rm gal}$ $\approx$ 18
kpc, illustrated in Fig. 7. Likely, the large dispersion in $V$ found at
$R_{\rm gal}$ $\gtrsim$ 18 kpc is the product of a reduction in the number of
filamentary structures, as sampled by $N^{\rm fil}_{\rm H}$, and the azimuthal
variations in the emission.

Figure 12: Radial profiles of the ratio between the column density in the
filamentary structures ($N^{\rm fil}_{\rm H}$) and the total column density
($N_{\rm H}$). From top to bottom, the panels correspond to each Galactic
quadrant and to the whole Galactic longitude range. The colors indicate the
Galactic longitude of the tiles. The vertical dotted line indicates the radius
of the solar orbit. The solid line in the bottom panel indicates the mean
values of $N^{\rm fil}_{\rm H}/N_{\rm H}$ in 1-kpc $R_{\rm gal}$ bins.

###  4.4 Comparison with GALFA-HI

We have considered the orientation of the Hi filaments found with the Hessian
method fixing a particular angular scale through the selection of the
derivative kernel size and in velocity channels set by the spectral resolution
of the HI4PI survey. There is no survey covering the same $b$ range across the
whole Galactic plane. Still, the 4′ and 0.184 km s-1 resolution GALFA-HI
provides an insight into the effects of the angular and velocity resolutions,
which we discuss in what follows.

Table 2: Hessian analysis parameters for the GALFA-HI data. Parameter |  | Value
---|---|---
Tile size |  | 3∘ $\times$ 20∘ $\times$ 0.184 km s-1
|  | 3∘ $\times$ 20∘ $\times$ 1.290 km s-1
Kernel size |  | 5′ FWHM
Intensity threshold |  | 0.3 K
Curvature threshold ($\lambda^{C}_{-}$) |  |  $-90.0$ K/deg2

####  4.4.1 Effect of the angular resolution

We performed the Hessian analysis in the two portions of the first and the
third Galactic quadrants covered in the 4′-resolution GALFA-HI data. To
independently study the effects of the difference in angular and spectral
resolution, we projected the GALFA-HI observations into the same spectral axis
of the HI4PI data. We performed the Hessian analysis using an 5′ FWHM
derivative kernel and applied the selection criteria presented in Table 2.
Further details on the implementation of the Hessian matrix method in the
GALFA-HI data and the results obtained with a larger kernel are presented in
App. C.

The results of the GALFA-HI data analysis, presented in Fig. 14, indicate at
first glance that the global trends in relative orientation are roughly the
same in the higher resolution data. The most substantial difference between
the analysis results in the two datasets is the increase in the significance
in the GALFA-HI, which is further illustrated in Fig. 15. The global trend is
for $V$ to increase by approximately a factor of two, which is most likely the
combination of two effects. First, the relative orientation trends observed in
HI4PI are generally the same at the scales sampled by GALFA-HI. If the
primarily horizontal filaments responsible for the highest values of $V$ in
HI4PI resolved into randomly oriented structures, we would find that the
high-$V$ tiles would produce low $V$ in GALFA-HI, which is not what we
observed in most of the tiles. Second, the increase in $V$ is related to the
higher angular resolution, because a smaller beam produces more independent
gradient measurements for the same tile area. This does not imply that there
is a higher number of filaments in the higher resolution but that the
statistical trend is still found when including more independent orientation
samples. We present examples of the filaments identified tiles in the two data
sets in Fig. 13.

Figure 13: Examples of the Hi emission and filamentary structures identified
in the HI4PI and the GALFA-HI data, shown in the top and bottom panels
respectively. The disk in the lower right corner of each panel indicates the
size of the derivative kernels used for the calculation of the Hessian matrix,
18′ and 5′ for the HI4PI and the GALFA-HI observations, respectively.

The higher values of $V$ in the higher resolution data indicate that the
distribution of filament orientations still peaks at 0∘ and the prominence of
that peak is more elevated. These results do not exclude cases in which the
higher resolution observations reveal no preferential orientation, as in some
tiles with $V$ $>$ 2.87 in HI4PI and $V$ $\approx$ 0 in GALFA-HI. Still, the
significance in $V$ is generally higher in the GALFA-HI data by roughly a
factor of two.

The increase in $V$ with the higher resolution observations produces a more
significant portion of the $lv$-diagram, as illustrated in Fig. 14. This
general increase in $V$ does not imply a departure from our observation in the
HI4PI data that the filaments tend to be more parallel to the Galactic plane
with increasing $R_{\rm gal}$. It means that the change in filament
orientation is smoother and more significant at higher angular resolution.
This trend is confirmed by the analysis of the THOR survey (Soler et al.,
2020), where the increase in angular resolution by a factor of 24 with respect
to HI4PI reveals a significant preference for Hi filaments parallel to the
Galactic plane at $v_{\rm lsr}$ $<$ 0 km s-1 in the range 17∘ $<$ $l$ $<$ 67∘
and $|b|$ $<$1 .∘25.

Figure 14: Longitude-velocity ($lv$) diagrams of the filament orientation
quantified by the projected Rayleigh statistic ($V$, right) in the two
portions of the Galactic plane covered in the range $|b|$ $\leq$ 10∘ by both
HI4PI and GALFA-HI. The GALFA-HI observations were analyzed at 1.29 and 0.184
km s-1 resolution using a 5′ FHWM derivative kernel. The curves correspond to
the same spiral arm features introduced in Fig. 1.

Figure 15:  Projected Rayleigh statistic ($V$) calculated for the filamentary
structure in the HI4PI and GALFA-HI observations. Each point corresponds to a
3∘ $\times$ 20∘ $\times$ 1.29 km s-1 tile. The gray dashed lines indicate $V$
$=$ 2.87 and $V$ $=$ $-2.87$, which correspond to a 3$\sigma$ significance in
the preferential orientation around $\theta$ $=$ $0$∘ or $\theta$ $=$ $90$∘,
respectively.

####  4.4.2 Effect of radial velocity resolution

We also performed the Hessian analysis in the GALFA-HI observations at their
native 0.184 km/s resolution using an 5′ FWHM derivative kernel. The results,
also presented in Fig. 15, show no global change in the results of our
analysis when considering the higher radial velocity resolution in the GALFA-
HI data. This is not unexpected given that emission from the narrowest Hi
components, corresponding to the CNM, is at least a few kilometers per second
wide and the emission structures are likely to be coherent across several
narrow velocity channels.

We illustrate the effect of the angular and radial velocity resolution for one
example tile in Fig. 16. At the same 1.29 km s-1 resolution and S/N threshold,
the results of the HI4PI analysis at 18′ FWHM and GALFA-HI at 5′ FWHM indicate
a slight change in the orientation trends. This change is not unexpected, as
the orientation at the large scale does not uniquely determine the orientation
of the structures at the small scale. Most of the differences are found in the
velocity channels where the bulk of the emission is located, $v_{\rm lsr}$
$\approx$ 0 km s-1. At the largest $v_{\rm lsr}$, which also correspond to the
largest $d$ and $R_{\rm gal}$, the increase in angular resolution does not
introduce any significant changes in the relative orientation trends, thus
supporting that the large $V$ values found there are not a product of the
distance to the emission sources. For illustration, we also present the mean
orientation angles $\left<\theta\right>$, which further confirms the dominant
trend of filaments aligned with the Galactic plane at both angular
resolutions.

The results of the GALFA-HI analysis at 0.184 km s-1 resolution are very
similar to those at 1.29 km s-1 resolution. The orientation trends are
coherent across a few kilometers per second in both cases. The most prominent
exception is the loss of signal around at $v_{\rm LSR}$ $\approx$ 80 km s-1
and $v_{\rm LSR}$ $>$ 150 km s-1, which is the result of the S/N decrease in
the narrow velocity channels. In general, we do not find significant
differences in the analysis of the narrow velocity channels that would
critically change the filament orientation trends obtained at 1.29 km s-1
resolution.

Figure 16:  Mean intensity ($\left<I\right>$, top), projected Rayleigh
statistic ($V$, middle), and mean orientation angle of the filaments
($\left<\theta\right>$, bottom) for the tiles centered on $l$ $=$ 202 .∘5 in
the HI4PI and GALFA-HI. The left panel corresponds to the HI4PI observations
analysis with a 18′ FWHM kernel for a 1.29-km s-1 velocity resolution. The
center and right panels correspond to the GALFA-HI observations analysis with
a 5′ FWHM kernel for 1.29-km s-1 and 0.184-km s-1 velocity resolutions,
respectively.

##  5 Discussion

###  5.1 Atomic filament orientations as a tracer of Galactic dynamics

The trends reported in Fig. 1 indicate that most of the filaments in the Hi
emission are parallel to the Galactic plane. This preferential orientation
appears naturally from the combined effect of the gravitational potential and
Galactic rotation, as confirmed by numerical simulations of Galactic disks
(see, for example, Smith et al., 2020; Tress et al., 2020). The gravitational
potential constrains the gas in the disk. The Galactic rotation stretches the
parcels of gas by shearing them apart. However, these effects cannot explain
by themselves the slight prevalence of vertical Hi filaments toward the inner
Galaxy and the clear predominance of horizontal ones toward the outer Galaxy.

One observational effect that may influence the observed Hi filament
orientation change with Galactocentric radius is the pileup of emission from a
broad range of distances into the same channel in velocity space. The portion
of line of sight covered by a fixed-width velocity channel increases with
Galactocentric radius. For example, at $l$ $=$ 90∘ and under the assumption of
circular motions, a 1.29-km s-1 velocity channel corresponds to $\Delta R_{\rm
gal}$ $\approx$ 200 pc and $\Delta R_{\rm gal}$ $\approx$ 500 pc for $v_{\rm
LSR}$ $=$ 100 and 150 km s-1, respectively. The velocity pile up can be a
plausible explanation for the decrease in $V$ at $R_{\rm gal}$ $\gtrsim$ 20
kpc reported in Fig. 7, by mixing portions of the line of sight where Hi
structures are mostly parallel to the Galactic plane and randomly oriented
into the same velocity channel. However, it does not offer a straight forward
explanation for the increase of $V$ with $R_{\rm gal}$ for 10 $\lesssim$
$R_{\rm gal}$ $\lesssim$ 20 kpc or the high-$V$ bands in Fig. 6.

The analysis of the GALFA-HI observations indicates that the regions dominated
by filaments parallel to the Galactic plane, $V$ $>$ 2.87, do not resolve into
structures with random orientations. Instead, the higher angular and velocity
resolution observations revealed that this preferential orientation is even
more significant, as illustrated in Fig. 14. The increase in angular
resolution does not result in a similar increase in the orientation
significance for tiles slightly dominated by filaments perpendicular to the
Galactic plane, $V$ $<$ 0\. We interpret this as indication that vertical Hi
filaments are less frequent than horizontal ones.

The experiments with derivative kernels fixed to a physical length, such as
the one presented in Fig. 5, indicate that the progressive transition from $V$
$<$ 0 to $V$ $>$ 0 with increasing $R_{\rm gal}$ is not due to a mixture of
scales introduced by the fixed angular size of the derivative kernel. Rather
than producing larger $V$, the introduction of derivative kernels with larger
angular sizes results in the reduction of $V$, as further illustrated in App.
A.2.3. Hence, we consider that the progressive transition from $V$ $<$ 0 to
$V$ $>$ 0 with increasing $R_{\rm gal}$ reported in Fig 7 has an origin in the
physical conditions of the Galactic disk.

####  5.1.1 Vertical Hi filaments

Figure 17 shows an example of the emission and the filaments toward one of the
inner Galaxy regions with prevalence of vertical structures, or $V$ $<$ $0$.
The majority of the structures that would be identified as HISA are
constrained to relatively low Galactic longitudes, where the background
emission is brightest, and are unlikely to be the dominant object producing
the global filament orientation. This is also true for other regions in the
inner Galaxy, as expected from the distribution and angular size in HISA
catalogs (see, for example, Gibson et al., 2005). However, it does not dismiss
the importance of a dedicated study of the HISA structures orientation.

Most of the structures contributing to $V$ $<$ $0$ are worms and arcs above
and below the bright filamentary structure that corresponds to the midplane.
These structures extend into high Galactic latitudes, some even beyond the
studied Galactic latitude range, $|b|$ $<$ 10∘. However, the higher-resolution
but lower $b$-coverage Hi observations in the THOR survey indicate that
predominantly vertical filamentary structures are also found in the $|b|$ $<$
1 .∘0 range (Soler et al., 2020).

Figure 17:  Hi emission (top) and filamentary structures (bottom) toward a
region dominated by vertical structures in the indicated radial velocity
range.

One physical explanation for the observed vertical Hi filaments is the
Galactic fountain, in which hot gas is produced by the cumulative effect of
supernovae in the disk; thermal expansion lifts it above the Galactic plane,
where it becomes unstable; cool clouds condense and fall back ballistically
(Shapiro & Field, 1976; Fraternali & Binney, 2006). However, direct SN
feedback events can also cause vertical structures. Strong clustered SN
feedback causes large scale bubbles, which are consecutively elongated along
the vertical direction due to bouyancy and Parker instability (Parker, 1966).
Cosmic rays (CRs) add an additional source of slowly cooling energy, which
builds up a vertical pressure gradient that helps lifting the gas out of the
midplane (Breitschwerdt et al., 1993; Girichidis et al., 2016; Ruszkowski et
al., 2017).

Numerical simulations indicate that CR-lifted gas is primarily warm rather
than hot, which will increase the amount of Hi compared to hot only SN-heated
gas (see, for example, Girichidis et al., 2018). CRs mainly move along the
magnetic field lines, which further amplifies the effects of Parker loops and
the related instabilities (Hanasz & Lesch, 2003). Consequently, the gas along
the buoyantly rising magnetic field lines also shows a preferentially vertical
orientation. Overall, strong SN feedback in the presence of a vertical
stratification of gas and a dynamically relevant magnetic field will favor the
formation of vertical structures. Additionally, spiral shock waves may lift
and stretch the atomic gas in the vertical direction (Suchkov & Shchekinov,
1974; Kim et al., 2010) and Parker instability that can create buoyant loops
of magnetic field at a kiloparsec scale (Rodrigues et al., 2016; Heintz et
al., 2020).

The progressive change from mostly vertical Hi filaments to mostly horizontal
with increasing $R_{\rm gal}$, reported in Fig. 7, indicates that the dominant
process behind the Hi filament orientation also has a radial dependence.
Spiral shocks trace the position of the spiral arm structure, thus they should
carry an azimuthal dependence and produce differences between arm and inter-
arm regions that are not noticeable in our results. The axisymmetric
variations in filament orientation strongly suggest the prevailing role of SN
feedback.

Supernovae are very effective in lifting the gas off from the Galactic plane,
as seen in numerous numerical experiments (see, for example, Joung & Mac Low,
2006; Hennebelle & Iffrig, 2014; Gatto et al., 2015; Kim & Ostriker, 2015a).
The radial variations in the clustering and frequency of SNe with
Galactocentric radius are a plausible explanation for the lack of vertical
filaments in the outer Galaxy. The number of SN remnants in the Milky Way
decreases significantly for $R_{\rm gal}$ $>$ 5 kpc (Case & Bhattacharya,
1998; Green, 2015). The distribution of stars in the Galaxy decreases
exponentially with a characteristic radius $R_{0}$ that depends on the age of
the population and is around 3.5 kpc for young stars (Drimmel & Spergel, 2001;
Frankel et al., 2018). Both of these observations indicate that for $R_{\rm
gal}$ $>$ 10 kpc, the number of SNe is significantly lower than in the inner
Galaxy. At large $R_{\rm gal}$ not only the number of SNe is lower, but the
area of the disk also is more extensive, so the SN rate per unit area falls
off even faster with radius than the number of SNe. Consequently, due to the
lack of events in the outer Galaxy, SNe’s energy and momentum input may not be
the dominant process shaping the atomic gas.

The SNe do not only contribute to the formation of vertical filaments by
lifting the Hi gas, but also through the infall of the previously ejected gas
back on the plane. Marasco & Fraternali (2011) studied the extraplanar Hi gas
of the Milky Way and found that it mainly consists of gas that is falling back
to the MW after being ejected by SNe. The extraplanar Hi is more conspicuous
while falling because it cools while settling down. When it is lifted, it is
hotter and so less prominent in the Hi emission, as can be seen in the
numerical simulations presented in Kim & Ostriker (2015b) and Girichidis et
al. (2016). It is difficult to determine the portion of the vertical filaments
that correspond to outgoing or incoming Hi gas. Still, the filament
orientations provide a roadmap to study these effects in the Galactic plane
and in Galactic-scale simulations that include the impact of stellar feedback.

####  5.1.2 Filaments and holes in other galaxies

Most the face-on nearby galaxies (2 $\lesssim$ $d$ $\lesssim$ 15 Mpc) in The
HI Nearby Galaxy Survey (THINGS, Walter et al., 2008) present holes in their
inner regions, where the gas density and the star formation rate are higher,
and elongated structures at larger radii. A detailed study of one of them, NGC
6946, a well studied spiral galaxy of about one-third of the Milky Way’s size
located at 7.72 Mpc, revealed 121 Hi holes in its inner regions and high-
velocity gas associated with some of them, thus suggesting the presence of a
fountain mechanism (Boomsma et al., 2008). It is not straightforward to
demonstrate that the associations of vertical filaments in the inner Milky Way
correspond to the holes seen in face-on nearby galaxies, particularly due to
the difficulty in disentangling emission components and assigning reliable
distances in our own Galaxy. However, there is significant evidence that
toward some portions of the Galactic plane we are seeing the gas swept by the
expansion of SN remnants, as it is the case of the Hi shells identified in the
range 10 kpc $<$ $R_{\rm gal}$ 15 kpc toward QIV (McClure-Griffiths et al.,
2002).

A recent study of the Hi gas structure in a sample of 33 disk galaxies in
THINGS revealed changes in the delta-variance spectrum in the inner and outer
regions of the disks (Dib et al., 2021). The authors interpret these
variations as the dominant effect of SN feedback in the inner parts of the
galaxies and the prevalence of other large-scale dynamics in the outer parts.
Certainly, SNe are not independent actors in the shaping the ISM. Recent
numerical experiments indicate that feedback exclusively produced by SNe does
not reproduce produce observed star cluster masses and warm gas volume-filling
factors (Kim et al., 2018; Rathjen et al., 2021). Detailed modeling of the
Galactic winds propelled by SNe indicate that cool and hot outflows can lift
significant amounts of mass, energy, and metals (Kim et al., 2020). Our
results indicate that the study of the Hi filaments can shed light in the
study of these Galactic winds and their distribution in the Milky Way.

###  5.2 Atomic filaments as a tracers of Galactic structure

####  5.2.1 Galactic warp

In addition to making the change in orientation with $R_{\rm gal}$ noticeable,
the identification of filaments in the Hi emission highlights the disk warp
and other variations in the midplane height of the Milky Way. Figure 8 shows
that the $\left<z\right>$ features at $R_{\rm gal}$ $>$ 20 kpc known as
scalloping are highlighted by the filaments. This may be an indication that
these features are not the product of global disk perturbations, as considered
in Levine et al. (2006a), but can be the product of events such as gas inflow
or the interaction with Milky Way satellites.

The most prominent variation in $\left<z\right>$, aside from the warp, is the
dip toward $\phi$ $\approx$ 100∘ and 20 $<$ $R_{\rm gal}$ $<$ 25 kpc. This
feature, shown in Fig. 18, corresponds to a horizontal cloud at $b$ $\approx$
$-$8∘ that contrasts with the location of most of the Hi emission, which is
concentrated at 0∘ $<$ $b$ $<$ 10∘. It is located in one of the few regions of
the $lv$-diagram that does not show a preferential filament orientation at 0∘
or 90∘, as it is evidenced by the elongated structures oriented at 45∘, among
which is the giant atomic filament Cattail (Li et al., 2021). The filaments in
this region seem to be pointing in the direction of the Magellanic stream and
their location both in radial velocity and Galactic longitude seem to
reinforce the potential association with the interwoven tail of filaments
trailing the Magellanic Clouds in their orbit around the Milky Way (see
D’Onghia & Fox, 2016, and references therein). Although determining this
association is beyond the scope of this work, it exemplifies the use of the Hi
filament properties as a tracer of the Galactic dynamics.

Figure 18:  Hi emission (top) and filamentary structures (bottom) for the
region around the $\left<z\right>$ dip toward $\phi$ $\approx$ 100∘ and 20 $<$
$R_{\rm gal}$ $<$ 25 kpc in Fig. 8.

####  5.2.2 Filament scale height

Figure 11 indicates a difference of roughly a factor of two in the scale
height traced by the bulk of the Hi emission and just by the Hi filaments at
$R_{\rm gal}$ $>$ 15 kpc. To explain this difference, we need to consider the
physical conditions that may lead to the formation of these structures. Most
likely the filaments are produced by large-scale compressive motions and
thermal instability that produce overdensities in which the Hi can cool
rapidly (Wolfire et al., 2003). Thus, the filaments may correspond to a
mixture of CNM and WNM surrounded by WNM. This scenario is supported by the
relatively narrow emission line widths toward the filamentary structures, as
shown in the example presented in Fig. 19.

Figure 19:  Hi emission (top) and selected spectra toward filamentary
structures (bottom). The spectra correspond to the positions indicated by the
crosses in the upper panel. The colored vertical bars in the bottom panel
correspond to the three velocity channels presented in the corresponding
colors in the top panel.

Kalberla & Kerp (1998) estimated a scale height at the location of the Sun of
about 150 pc for the for the CNM and 400 pc for the WNM, assuming hydrostatic
equilibrium between the components. The authors inferred velocity dispersions
of 4 and 24 km s-1 for the CNM and WNM, which are roughly consistent with the
combination of the turbulent and the thermal sound speeds of the gas in each
phase. That result suggest that each component is separately in vertical
equilibrium, so that the total turbulent and thermal midplane pressures
balances the vertical weight of the ISM (see, for example, Ostriker & Shetty,
2011). However, both the measured scale heights and the measured velocity
dispersions are subject to relatively large uncertainties: for example, Heiles
& Troland (2003) reports that the difference in scale heights of the two
phases is closer to a factor of two, as result of their finding of large
velocity dispersion for the CNM and smaller for the WNM.

Toward the inner Galaxy the scale height of the CNM has also been observed to
be smaller than that of the WNM (Dickey & Lockman, 1990; McClure-Griffiths &
Dickey, 2007). If we assume that the same conditions can be extended to the
outer galaxy, the difference in scale heights reported in Fig. 11 is
consistent with the expected difference for the two Hi components. However,
that conclusion is in tension with the results of the observations of Hi
absorption against strong continuum sources, which indicate that the scale
height of the CNM is similar to that of the WNM, and both increase with
Galactic radius in the outer disk (Dickey et al., 2009).

Liszt (1983) shows that one portion of the WNM must be associated with the
CNM, while the other portion is more widely spread in the neutral medium
between the CNM clouds. Dickey et al. (2009) argues that the scale height
derived from the Hi observed in absorption, which samples mostly the CNM
clouds, and that from the Hi emission, which samples both the CNM and WNM, are
similar in the outer Galaxy, thus suggesting that the scale heights of the two
Hi components are not very different. One reason for the discrepancy with our
results are the azimuthal variations that are evident in the radial profiles
presented in Fig. 12.

Figure 7 of Dickey et al. (2009) shows that the best agreement between the
scale heights sampled by absorption and emission is for the average over the
azimuth range 0∘$<$ $\phi$ $<$180\. In that range our analysis results in
large dispersions, mostly due to the lower emission toward that portion of the
Galaxy at $R_{\rm gal}$ $>$ 20 kpc that results in fewer filaments. In the
range 180∘$<$ $\phi$ $<$360, the agreement between the scale heights at
$R_{\rm gal}$ $>$ 17 kpc in Dickey et al. (2009) is less evident. It is
precisely toward that azimuthal range that we find the clearest difference
between the $\sigma_{z}$ sampled by the filaments and by the bulk of the Hi
emission.

Our results are not conclusive toward 0∘$<$ $\phi$ $<$180, where we do not
find enough filaments to match the results of Dickey et al. (2009). However,
we find a strong indication of different scale heights in the range 180∘$<$
$\phi$ $<$ 360, where only 126 of 650 Hi opacity observations available at the
time of Dickey et al. (2009) are located. Whether or not the difference in the
scale heights derived using the Hi filaments are confirmed by the emission and
opacity studies will depend on the analysis of future high-resolution surveys
that increase the number of observations against continuum sources.

####  5.2.3 Column density in atomic filaments

We found that the ratio $N^{\rm fil}_{\rm H}/N_{\rm H}$ is about $80\%$ up to
$R_{\rm gal}$ $\approx$ 15 kpc, as shown in Fig. 12. The reason for the
decrease in $N^{\rm fil}_{\rm H}/N_{\rm H}$ at $R_{\rm gal}$ $>$ 15 kpc toward
some lines of sight can be attributed to the lack of filamentary structure in
the emission, particularly toward QI and QII. The explanation for the roughly
constant values of $N^{\rm fil}_{\rm H}/N_{\rm H}$ for $R_{\rm gal}$ $<$ 15
kpc is potentially related to the almost constant CNM to WNM mass ratio
throughout the Galaxy. The CNM fraction is almost constant with $R_{\rm gal}$
and corresponds to between 15 and 20% of the atomic gas in the outer disk
(Dickey et al., 2009). The larger values of $N^{\rm fil}_{\rm H}/N_{\rm H}$
can result from the filaments tracing not exclusively CNM but also some WNM
associated with the cool gas. This, however, implies a very coarse average
because on scales up to at least hundreds of parsecs the pressure and density
of the ISM may be highly variable and far from equilibrium

The decrease on the values of $N^{\rm fil}_{\rm H}/N_{\rm H}$ happens at
roughly the same $R_{\rm gal}$ at which we see the end of the regular trend in
filament orientation illustrated in Fig. 7. This two observations can be
interpreted as an indication of the change in the physical conditions of the
atomic gas in the outer galaxy. These changes are not axisymmetric and can be
caused by a variety of conditions that include the passage of the Milky
satellites and variations in the ISM pressure and the stellar potential.

In principle, this observation implies that the formation of the structures
that we observe as Hi filaments is not affected by variations in star
formation and stellar feedback with Galactocentric radius. Potentially
converging flows compensate for the decrease in the SN rate with increasing
distance from the Galactic center in maintaining the density enhancement that
drives the warm gas into the cool structures that we observe as filaments.
However, further interpretation calls for the detailed study of particular
regions and the analysis of numerical simulations of Galactic disks.

###  5.3 Properties of atomic filaments

We have employed the word filaments to designate the elongated structures
found in Hi emission. We used their orientation to describe the structure of
the emission and its anisotropy without assigning their existence to a
particular kind of object in three-dimensional space. The fact that the
preferential orientation is persistent across 1.29-km s-1 velocity channels is
a strong indication that this trend is produced by three-dimensional density
structures in the ISM and not by fluctuations imprinted by the turbulent
velocity field (see Clark et al., 2019, and references therein).

The deep examination of the nature of the filamentary structures in three
dimensions, for example, to distinguish between filaments and sheets (Heiles,
1997), is beyond the scope of this work. However, the characterization of the
emission in terms of filaments highlights a large number of elongated
structures that are reminiscent of the plumes in the Hi emission toward the
outskirts of nearby spiral galaxies (Walter et al., 2008). Recent studies have
identified prominent elongated structures in the Hi emission, for example, the
1.1-kpc-long Cattail filament with roughly 6.5 $\times$ $10^{4}$ M⊙ (Li et
al., 2021) and the 1.2-kpc-long Magdalena filament with approximately 7.2
$\times$ 105 M⊙ (Syed et al., 2022). Our results indicate that these elongated
structures are the norm toward the outer Galaxy.

The Magdalena filament is singular given its orientation, parallel to the
Galactic plane, and its distance from the Hi emission midplane, roughly 500
pc. The kind of perturbations that can produce an object of these
characteristics are currently unknown. However, they may be related to the
horizontal orientation of the VHC complexes AC and H and as such the
orientation of this structures can be used as an indicator of the dynamics of
the Galactic disk and its interaction with the halo.

There are few detections of molecular gas toward the bands dominated by
horizontal Hi filaments in Fig. 1. Carbon monoxide (CO) observations indicate
that there is barely any significant emission at these large radii (Dame et
al., 2001; Miville-Deschênes et al., 2017). Toward the OSC, Dame & Thaddeus
(2011) report ten detections in 220 pointings toward high-Hi density regions.
Sun et al. (2015) reported the detection of 72 molecular clouds in the CO(1
$\rightarrow$ 0) emission toward the extension of the OSC into the second
quadrant. Similar observations for the outer regions toward the outer third
and forth quadrants have not been made. Thus, it is hard at the moment to
estimate if the Hi filaments inherit their elongation and orientation into
molecular clouds that they may help form within them, in contrast with the
general trend for the clouds traced by CO in the inner Galaxy (Soler et al.,
2021). Yet, the outer Galaxy reveals as an attractive laboratory to test our
theories of molecular cloud formation and the coupling between the ISM phases.

##  6 Conclusions

We studied the orientation of the filamentary structure in the Hi emission
toward the Galactic plane. We have found that the orientation of the Hi
filaments with respect to the Galactic plane changes progressively with
Galactocentric distance, from mostly perpendicular or having no preferred
orientation in the inner Galaxy to mostly parallel up to $R_{\rm gal}$
$\approx$ 15 kpc and beyond. The radial dependence in the Hi filament
orientation strongly suggests that SNe’s energy and moment input is
responsible for lifting the atomic gas in the inner Galaxy. Toward the outer
Galaxy, the horizontal Hi filaments are most likely the result of Galactic
rotation and shear. Together these observations indicate that the atomic gas
in the Milky Way follows a similar distribution as in nearby spiral galaxies,
where holes potentially produced by SNe-propelled winds are found toward the
inner portions, and arms and plumes dominate the outer ones.

The characterization of the Hi structure in terms of filaments highlights the
features in midplane height and dispersion around the midplane height that
might be indicative of the local effects perturbing the Galactic disk, such as
variations in the potential or the inflow of gas. We found that the scale
height of the Hi filaments is lower than that traced by the bulk of the Hi
emission by a factor of two. If the filaments are tracing cold gas clouds and
the portion of warm gas associated with them, our results imply that the scale
heights of the cold and warm gas are not close, as previously inferred from Hi
absorption observations.

We estimated that the amount of Hi in filamentary structures is roughly
constant with Galactocentric radius up to $R_{\rm gal}$ $\approx$ 15 kpc. This
observation is potentially related to the mass ratio between the cold and warm
Hi phases. However, conclusively proving that hypothesis calls for further
characterization of the Hi phases, for example, through Gaussian decomposition
and dedicated studies of the filament properties.

In this paper, we have presented a study of the characteristics of the Hi
emission. We did so without assuming that these characteristics define an
idealized physical object in three-dimensional space, such as a spiral arm or
a thin strand of gas. Through this generality, we identified a general
anisotropy related to a physical quantity: the Galactocentric radius. Some of
the emission features that we classified as filaments may correspond to
thread-like gas clouds, but that was not the starting assumption of this
study. Our conclusions were driven by a dominant trend found through the data
analysis rather than adapting the data to a preconceived notion about the gas
structure. Given the diversity of processes shaping the ISM, this data-driven
approach offers a promising path to study the cycles of mass and energy that
lead to the formation and destruction of star-forming molecular clouds. Even
if the observed lack of correlation between star formation and neutral atomic
hydrogen at low surface densities has traditionally excluded the Hi
observations from the global study of star-formation rates in galaxies (see,
for example, Kennicutt, 1989; Leroy et al., 2008; Schruba et al., 2011), the
Hi still carries crucial information on the dynamics that drive the diffuse
gas into the condensations that become new stars.

###### Acknowledgements.

JDS, RKS, and SCOG are funded by the European Research Council via the ERC
Synergy Grant “ECOGAL – Understanding our Galactic ecosystem: From the disk of
the Milky Way to the formation sites of stars and planets” (project ID
855130). RSK, HB and SCOG acknowledge funding from the Deutsche
Forschungsgemeinschaft (DFG) via the Collaborative Research Center (SFB 881,
Project-ID 138713538) “The Milky Way System” (subprojects A1, B1, B2 and B8)
and from the Heidelberg cluster of excellence (EXC 2181 - 390900948)
“STRUCTURES: A unifying approach to emergent phenomena in the physical world,
mathematics, and complex data”, funded by the German Excellence Strategy. RJS
acknowledges funding from an STFC ERF (grant ST/N00485X/1) and HPC from the
DiRAC facility (ST/P002293/1). The team at ZAH acknowledges computing
resources and data storage facilities provided by the State of Baden-
Württemberg through bwHPC and the German Research Foundation (DFG) through
grant INST 35/1134-1 FUGG and INST 35/1503-1 FUGG, and they thank for
computing time from the Leibniz Computing Center (LRZ) in project pr74nu. We
thank the anonymous referee for the thorough review and appreciate the
suggestions, which helped us improve the manuscript. JDS thanks the following
people who helped with their encouragement and conversation: Jonathan Henshaw,
Jonas Syed, Eleonora Zari, Milena Benedettini, Robert Benjamin, and Thomas
Henning. Part of the crucial discussions that lead to this work took part
under the program Milky-Way-Gaia of the PSI2 project funded by the IDEX Paris-
Saclay, ANR-11-IDEX-0003-02. The computations for this work were made at the
Max-Planck Institute for Astronomy (MPIA) astro-node servers. This work has
been written during a moment of strain for the world and its inhabitants. It
would not have been possible without the effort of thousands of workers facing
the COVID-19 emergency around the globe. Our deepest gratitude to all of them.
Software: astropy (Astropy Collaboration et al., 2018), SciPy (Virtanen et
al., 2020), magnetar (Soler, 2020).

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##  Appendix A The Hessian analysis method

###  A.1 Studying angle distributions using circular statistics

Throughout this paper we have characterized the distribution of orientations
using circular statistics, the branch of statistics that deals with
observations that are directions (see, for example, Batschelet 1981; Mardia &
Jupp 2009). The probability distributions against which we are comparing the
data are angles. The most natural characteristic to describe the angle
distributions is the mean, which is defined as

$\left<\theta\right>=\arctan\left(\frac{\sum w_{i}\sin\theta_{i}}{\sum
w_{i}\cos\theta_{i}}\right),$ (7)

where the index $i$ runs over the elements in a group of angle observations
and $w_{i}$ is the statistical weight corresponding to each of them.

The mean angle, $\left<\theta\right>$, is insufficient to fully describe the
distribution of angles. For example, the angle distribution is ill-defined if
the distribution is uniform or not unimodal. To test whether the angle
distribution is uniform and unimodal one can use the Rayleigh statistic, which
is defined as

$Z=\left[\left(\sum w_{i}\cos\theta_{i}\right)^{2}+\left(\sum
w_{i}\sin\theta_{i}\right)^{2}\right]^{1/2}.$ (8)

This value can be understood as the net displacement of a random walk in the
complex plane with step sizes $w_{i}$. For a uniform distribution of angles,
$Z$ $\approx$ 0.

In our application there is one direction that produces an expected anisotropy
in the filament orientation: the Galactic plane. That is why we used the
projected Rayleigh statistic ($V$, Eq. 4) to test the orientation angle
distribution against 0∘ and 90∘, or parallel and perpendicular to the Galactic
plane, respectively. The observations also reveal that these two orientations
are predominant in the Hi filaments.

Figure 20: Same as Fig. 1, but for the filament orientation quantified by the
mean orientation angle ($\left<\theta\right>$, Eq. 7, top) and the Rayleigh
statistic ($Z$, Eq. 8, bottom).

Figure 20 shows the results of our filament orientation analysis in terms of
$\left<\theta\right>$ and $Z$. It is evident that the majority of the tiles
have filamentary structures with mean orientations $\left<\theta\right>$
$\approx$ 0∘. However, this observation alone is not enough to conclude the
significance of this trend. The values of $Z$ indicate that there is a
significant tendency for the angle distributions to be unimodal. The highest
significance is found in two bands located at the minimum and maximum $v_{\rm
LSR}$ toward the first two and the last two Galactic quadrants, respectively.
These bands correspond to the high-$V$ bands shown in Fig. 1.

A summary of the trends obtained using the circular statistic is shown in Fig.
21. We found that the large majority of the tiles with significantly unimodal
distributions, $Z$ $\gtrsim$ 3, correspond to large values of $|V|$. This
confirms that 0∘ and 90∘ correspond to the most represented orientations in
the Hi filaments and justified the use of these two reference angles in the
application of the calculation of $V$.

Figure 21: Two-dimensional histogram of the Rayleigh statistic ($Z$) and
projected Rayleigh statistic ($V$) calculated for the filamentary structures
in the HI4PI observations.

###  A.2 Selection of parameters in the Hessian analysis method

In the main body of this paper we have reported the results of a particular
set of parameters in the Hessian analysis and the study of the filament
orientations. In this section we present the parameter study that led to that
selection. In what follows we detail the considerations related to the
selection of the tile sizes, the derivative kernel size employed for the
calculation of the Hessian matrix, and the thresholds in intensity and
curvature used to select the filamentary structures. We considered the
variations introduced by the changes in one of these parameters and leaving
the other fixed to the values presented in Table. 2.

####  A.2.1 Tile sizes

Figure 22: Longitude-velocity ($lv$) diagrams of the Hi filament orientation
quantified by the projected Rayleigh statistic ($V$, right). Each of the pixel
elements in the diagrams corresponds to a 3∘ $\times$ 3∘ $\times$ 1.29 km s-1
(top), 5∘ $\times$ 5∘ $\times$ 1.29 km s-1 (middle), and 20∘ $\times$ 20∘
$\times$ 1.29 km s-1 (bottom). These results correspond to the Hessian
analysis performed using a 18′ FHWM derivative kernel. The overlaid curves
correspond to the spiral arm features discussed in the main body of the paper.

We selected the $|b|$ $\leq$ 10∘ range for the analysis aiming to resolve the
warp and the flaring of the Galactic disk. However, we examined the dependence
of the Hi filament orientation trends on the tile size. Figure 22 shows the
$V$ $lv$-diagrams corresponding the square tiles centered on $b$ $=$ 0∘ and
with side lengths of 3∘, 5∘, and 20∘.

The 3∘ $\times$ 3∘ tiles cover the range included in the interferometric
Galactic plane surveys THOR, $|b|$ $<$ 1 .∘25, and SGPS, $|b|$ $<$ 1 .∘4\. The
corresponding $V$ $lv$-diagram, shown in the top panel of Fig. 22, indicates
that the high-$V$ band reported in the main body of the paper is not found in
that range toward the first quadrant. This is due to the Galactic warp, which
reaches midplane heights around 3.0 kpc for $R_{\rm gal}$ $\gtrsim$ 15 kpc
toward the first quadrant, as discussed in Sec. 5.2.

The 5∘ $\times$ 5∘ tiles cover the range included in the 1′-resolution
interferometric Canadian Galactic Plane Survey (CGPS Taylor et al. 2003). The
corresponding $V$ $lv$-diagram, shown in the middle panel of Fig. 22,
indicates that the high-$V$ band in the first quadrant is present in this
range, but its significance is relatively low. The high-$V$ band toward the
last two quadrants is clearly distinguishable in both the 3∘ $\times$ 3∘ and
5∘ $\times$ 5∘ tiles analysis, because the Galactic warp in that direction is
lower. Finally, the bottom panel of Fig. 22 shows the results corresponding to
the 20∘ $\times$ 20∘ tiles.

The decrease in the tile size leads to a reduction of the maximum $V$ values,
as it is evident from the comparison of the panels in Figure 22, which is the
result of the lower number of pixels or, equivalently, independent filament
orientations. However, the positive $V$ in radial velocities corresponding to
the outer galaxy and negative $V$ toward the inner galaxy, which are
highlighted as the main results of this paper, are persistent for the
considered tile size selections. The selection of broader tiles washes away
some of the azimuthal variations. Thus, we compromise between $b$-coverage and
resolution in $l$ by using rectangular tiles.

Figure 23 illustrates that the selection of square tiles increases the
significance in the signal produced by mostly vertical structures. However,
this increase comes at the cost of reducing the number of tiles in Galactic
longitude and significantly increasing the signal coming from the local
emission. All things considered, we chose rectangular tiles to highlight the
significance of the filamentary structure in the outer Galaxy and produce a
more detailed face-on reconstruction of the filament orientation trends.

Figure 23: Same as Fig. 7, but for tile sizes 20∘ $\times$ 20∘ $\times$ 1.29
km s-1.

####  A.2.2 Tile aspect ratio

Figure 24: Same as Fig. 22, but for tile sizes 10∘ $\times$ 20∘ $\times$ 1.29
km s-1 and 5∘ $\times$ 20∘ $\times$ 1.29 km s-1, shown in the top and bottom
panels, respectively.

Square tiles are a natural selection to avoid any bias toward one of the
preferential orientations studied in this analysis, that is, 0∘ or 90∘, as
discussed in Soler et al. (2020). Yet the use of rectangular tiles improves
the resolution in $l$ and the study of azimuthal effects. The 3∘ $\times$ 20∘
$\times$ 1.29 km s-1 tiles selected for the analysis in the main body of the
paper result from the compromise between azimuthal resolution and the $V$
significance . To reach that selection, we empirically studied the effect of
the tile aspect ratio on the values of $V$.

Figure 24 presents the results obtained using 10∘ $\times$ 20∘ $\times$ 1.29
km s-1 and 5∘ $\times$ 20∘ $\times$ 1.29 km s-1 tiles. We found that the main
features in the $V$ $lv$-diagram persist with the change in the tile aspect
ratio. The increase in the $l$ resolution resolves the negative-$V$ patches in
the bottom panel of Fig. 22 into regions surrounded by $V$ $=$ $0$ ranges. The
rectangular tiles also constrain the $l$ range to which the dominant features
in the map are assigned. For example, the increase in the $l$ resolution
resolved the separation between the high-$V$ band in the second quadrant and
the HVC complex located around 155∘ $<$ $l$ $<$ 190∘ and $-340$ $<$ $v_{\rm
LSR}$ $<$ $-135$ km s-1.

We found that for tiles with 3∘ coverage in $l$, the values of $V$ are reduced
below the level of significance in most of the $l$ diagram. This effect is
particularly acute for the $V$ $<$ 0, where the narrow tiles reduce the
coverage of close-to-vertical structures and introduce a bias for $V$ $>$ 0\.
The high $V$-bands, which are mostly produced by a series of horizontal
filaments distributed over a relatively broad range in $b$, are less affected
by the tile aspect ratio.

Figure 25 illustrates the effect of a broader tile selection in the trends
reported in Fig. 25. The 10∘ $\times$ 20∘ $\times$ 1.29 km s-1 tiles maintain
a high significance in the regions where the filaments are preferentially
oriented perpendicular to the Galactic plane. This tile aspect ratio
emphasizes the change in the orientation of the filaments, from mostly
perpendicular to mostly parallel to the Galactic plane with increasing
Galactocentric radius. However, the 3∘ $\times$ 20∘ $\times$ 1.29 km s-1 tiles
used to report the results in the mainbody of the paper increase the sampling
in Galactic longitude, thus producing a more robust identification of the
increasing values of $V$ with $R_{\rm gal}$.

Figure 25: Same as Fig. 7, but for tile sizes 10∘ $\times$ 20∘ $\times$ 1.29
km s-1.

####  A.2.3 Kernel sizes

We have chosen a 18′ FWHM derivative kernel to profit from the angular
resolutions of the observations. The use of larger kernels implies necessarily
a lowering of the significance of the filament orientations, given the
reduction in the number of independent filament orientations. However, we
tested the results of the analysis of 3∘ $\times$ 20∘ $\times$ 1.29 km s-1
tiles using larger derivative kernels.

Figure 26 show the $V$ $lv$-diagrams obtained using derivative kernels two and
three times larger than the beam size, 32.′4 and 48.′6 FWHM, respectively.
Besides the expected reduction in the $V$ values, the larger derivative
kernels displace the high-$V$ bands to larger radial velocities. This is due
to the loss of significance produce by the reduction in the number of
independent filament orientation measurements. Rather than producing more
horizontal structures, which would increase the values of $V$, the smoothing
with a larger derivative kernel reduces the preferential orientation in the
filamentary structures.

Figure 26: Same as Fig. 22, but for 3∘ $\times$ 20∘ $\times$ 1.29 km s-1 tiles
analyzed using 32.′4 and 48.′6 FWHM derivative kernels, shown in the top and
the bottom panels, respectively.

Figure 27 illustrates that the larger derivative kernels systematically reduce
the values of $V$ independently of the kinematic distance corresponding to
each tile. Rather than artificially introducing high-$V$ values at closer
distances, the smoothing results in a loss of the significance in the relative
orientation trends. This reinforces the argument that the high-$V$ bands are
not a product of the angular resolution of the HI4PI survey, which we proved
using the 4′ resolution observations in GALFA-HI. For the sake of simplicity,
we present only the results corresponding to the far distance, but the same is
inferred for the nearby solution to the near kinematic distance.

Figure 27: Same as Fig. 4, but for 32.′4 and 48.′6 FWHM derivative kernels,
shown in the top and the bottom panels, respectively.

Figure 28 shows the ratio between the column density in filaments and in the
bulk of the Hi emission ($N_{\rm H}^{\rm fil}/N_{\rm H}$) for the two
derivative kernels. We found that the increase in the derivative kernel size
does not substantially change the fraction of column density in filaments or
the $R_{\rm gal}$ above which we found a drop in $N_{\rm H}^{\rm fil}/N_{\rm
H}$. This result suggests that the drop in $N_{\rm H}^{\rm fil}/N_{\rm H}$ is
not an effect introduced by the distance. If distance was producing the drop,
we would see a systematic shift in the $R_{\rm gal}$ values where $N_{\rm
H}^{\rm fil}/N_{\rm H}$ starts to decrease. The presumed shift in $R_{\rm
gal}$ should be lower for the larger derivative kernel but this is not what is
found in the data, where we found the drop at the same $R_{\rm gal}$ for the
three considered kernel sizes.

Since we fixed the $I$ S/N threshold to test the effect of the derivative
kernel size, we also conclude that the drop in $N_{\rm H}^{\rm fil}/N_{\rm H}$
is not simply caused by a drop in the emission. The derivative kernels
effectively smooth the velocity channel maps, thus bringing a larger portion
of the emission above the $I$ S/N threshold. However, changing this smoothing
by up to a factor of three above the beam size does not substantially change
the trends in $N_{\rm H}^{\rm fil}/N_{\rm H}$. Therefore, we conclude that the
drop in $N_{\rm H}^{\rm fil}/N_{\rm H}$ is related to a transition toward
less-structured Hi emission rather than a change introduced by the decrease in
Hi emission with increasing $R_{\rm gal}$.

Figure 28: Same as Fig. 12, but for 32.′4 and 48.′6 FWHM derivative kernels,
shown in the top and the bottom panels, respectively.

####  A.2.4 Intensity threshold

We empirically evaluated the effect of the intensity threshold in the filament
orientation analysis. Figure 29 shows two examples of this selection above and
below the threshold used for the results presented in the main body of this
paper. The results corresponding to the 3$\sigma_{I}$ intensity threshold show
a vertical stripe around $l$ $\approx$ 140∘. These features are typical for
low-intensity thresholds and are produced by features in low-intensity
velocity channels. Most of these features disappear with a 5$\sigma_{I}$
threshold, as shown in the right-hand side panel of Fig. 1.

We do not find significant differences in the results produced by a
9$\sigma_{I}$ threshold. The increase in the $I$ threshold tends to eliminate
low-intensity velocity channels at the extremes of the velocity range rather
than modifying the general trends in the $V$ $lv$ diagram. This relatively low
dependence on the intensity threshold is partly due to the stringent criterion
in curvature, $\lambda_{-}$, which is dominant in determining the structures
selected for the filament orientations analysis.

Figure 29: Same as Fig. 26, but for 3∘ $\times$ 20∘ $\times$ 1.29 km s-1 tiles
and intensity thresholds of 3$\sigma$ (top), 5$\sigma$ (middle), and 9$\sigma$
(bottom).

####  A.2.5 Curvature threshold

The lowest eigenvalue of the Hessian matrix, $\lambda_{-}$, plays a crucial
role in the identification of the filamentary structure (see, for example,
Planck Collaboration Int. XXXII 2016; Schisano et al. 2014). The filamentary
structures correspond to the pixels where $\lambda_{+}\lambda_{-}$ $<$ 0 and
$\lambda_{-}$ $<$ $0$. In the observations, however, there is a potential
contribution of the noise to the curvature that we account for by choosing a
lower curvature $\lambda^{\rm thres}_{-}$.

We present an example of the results obtained when using no curvature
threshold, just $\lambda_{-}$ $<$ $0$, in Fig. 31. The example tile shows a
collection of low-$I$ and relatively low curvature, $\lambda_{-}$ $\approx$
$0$, filamentary structures. These faint filaments objects appear
systematically over the whole map and do not show an evident preferred
orientation.

The top panel of Fig. 31 shows the net effect of the low-curvature structures.
Globally, the $V$ $lv$-diagram presents the same high-$V$ bands and the low
$V$ regions reported in Fig. 1. However, the $|V|$ values are lower, which
corresponds to a lower significance in the filament orientation trends. We
also found a series of vertical stripes in the $V$ $lv$-diagram which are
particularly prominent in the region $v_{\rm LSR}$ $<$ 0 toward the first two
quadrants and $v_{\rm LSR}$ $>$ 0 toward the last two. These stripes are
mitigated by applying a curvature threshold.

The bottom panel of Fig. 31 shows the results obtained by using a curvature
threshold based on the $\lambda_{-}$ values found in the lowest intensity-S/N
channels toward the position of each tile. This selection increases the
significance of the relative orientation trends, quantified by $V$, We
selected the median of these per-tile $\lambda_{-}$ threshold as the value
used for the results reported in the main body of this paper.

Figure 30: Example of the filamentary structures identified using the Hessian
method with curvature thresholds $\lambda_{-}$ $<$ 0 (top) and $\lambda_{-}$
$<$ $-1.8$ K/deg2 (bottom). The three maps represent: Left. Hi intensity after
the smoothing by a Gaussian kernel with the size of the derivative kernel.
Center. Second eigenvalue of the Hessian matrix, $\lambda_{-}$, calculated
using Eq. 2. Left. Filament orientations, $\theta$, calculated using Eq. 3.
The radial velocity corresponds to a kinematic distance $d$ $=$ $12.7$ kpc and
Galactocentric radius $R_{\rm gal}$ $=$ 16.1 kpc, as derived using the
assumptions in App. B

Figure 31: Same as Fig. 29, but for $\lambda_{-}$ $<$ $0$ (top) and a tile-
dependent $\lambda_{-}$-threshold (bottom).

##  Appendix B De-projection of the spectral observations

###  B.1 Construction of the face-on view

We de-projected the results our analysis results in the $l$, $b$, and $v_{\rm
LSR}$ grid into Galactocentric cylindrical coordinates $R_{\rm gal}$, $\phi$,
and $z$. We calculated a Galactocentric radius, $R_{\rm gal}$ for each tile’s
central position and radial velocity using the Kinematic Distance Utilities:
KDUtils666https://github.com/tvwenger/kd (Wenger et al. 2018). In practice,
these routines solve

$R_{\rm gal}=R_{0}\sin(l)\frac{V(R_{\rm gal})}{v_{p}+V_{0}\sin(l)},$ (9)

where $R_{0}$ and $V_{0}$ are the Galactocentric radius and tangent velocity
of the solar orbit around the Galactic center, $V(r)$ is the rotation curve,
and $v_{p}$ is the radial velocity, which at the center of the tile is equal
to $v_{\rm LSR}$. In this study we used the rotation curve derived in Reid et
al. (2019).

We also evaluated the Galactocentric azimuth angle

$\phi=\arccos\left(\frac{R_{0}-d\cos l}{R_{\rm gal}}\right),$ (10)

where $d$ is the distance given by

$d=R_{0}\cos(l)\pm\sqrt{R_{\rm gal}^{2}-R^{2}_{0}\sin^{2}(l)}.$ (11)

In the inner Galaxy ($R_{\rm gal}$ $<$ $R_{0}$) this expression yields to two
distances along the line of sight, the near and the far kinematic distances,
located on either side of the tangent point. Figure 32 illustrates the near-
and far-distances obtained for the positions of the 3∘ $\times$ 20∘ $\times$
1.29 km s-1 tiles in the $lv$-diagram.

Figure 32: Near and far kinematic distances assigned to the 3∘ $\times$ 20∘
$\times$ 1.29 km s-1 tiles used to construct the $lv$-diagrams in Fig. 1.
These results correspond to the solutions of Eq. 11 using the rotation curve
presented in Reid et al. (2019).

The ambiguity introduced by the two solutions to Eq. 11 makes the
determination of kinematic distances very challenging in the inner Galaxy.
Thus, we restricted the results of our de-projection to the range $R_{\rm
gal}$ $>$ 10 kpc to avoid the artifacts produced by the motions around the
solar orbit. We also exclude the longitude ranges $l$ $<$ 15∘, $l$ $>$ 345∘,
and 165∘ $<$ $l$ $<$ 195∘, where the radial velocities are not dominated by
the circular motions and it is difficult to establish reliable distances
(Levine et al. 2006b).

To test the potential effect of distance in the filament orientation trends,
we presented the $V$ distribution for the “far” heliocentric kinematic
distances, $d_{\rm far}$ in Fig. 4. For the sake of completeness, we present
the same diagram considering the “near” heliocentric distances in Fig. 33,
which differ from $d_{\rm far}$ only toward QI and QIV. We also present the
unmasked face-on reconstruction of Hi filament orientations obtained by using
the far heliocentric distances in Fig. 34.

Figure 33: Same as Fig. 33, but for the near heliocentric kinematic distances.
Figure 34: Same as Fig. 8, but without masking $R_{\rm gal}$ $<$ 10 kpc and
assigning all kinematic heliocentric distances to the largest solution to Eq.
11 .

###  B.2 Galactic warp and flaring