# Examinations of CO completeness based on three independent CO surveys

Yan Sun Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210008, China  Ji Yang Purple Mountain Observatory, Chinese Academy of
Sciences, Nanjing 210008, China  Qing-Zeng Yan Purple Mountain Observatory,
Chinese Academy of Sciences, Nanjing 210008, China  Zehao Lin  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210008, China  University of
Science and Technology of China, Hefei, Anhui 230026, China  Shaobo Zhang
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008,
China  Yang Su Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210008, China  Ye Xu  Purple Mountain Observatory, Chinese Academy of
Sciences, Nanjing 210008, China  Xuepeng Chen  Purple Mountain Observatory,
Chinese Academy of Sciences, Nanjing 210008, China  Hongchi Wang Purple
Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China  Xin
Zhou Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008,
China

(Accepted to ApJS July 1, 2021)

###### Abstract

We report the global properties recovered by an ongoing CO survey of the Milky
Way Imaging Scroll Painting (MWISP) toward the Galactic outskirts. Our results
are also compared to those extracted by a uniform decomposition method from
the CfA 1.2 m CO survey and the FCRAO 14 m outer Galaxy survey (OGS). We find
that more extended and unseen structures are present in the MWISP data. The
total flux across the disk recovered by the MWISP survey is 1.6 times larger
than those recovered by the CfA and OGS surveys in the case of the same
resolution. The discrepancies are scaling with distance. For example, in the
outermost OSC arm, the flux ratios for MWISP-to-CfA and MWISP-to-OGS increase
up to 43.8 and 7.4, respectively. Nonetheless, the census of molecular gas in
our Galaxy is still far from complete by the MWISP, with flux completeness of
$<$58%. The total mass ratios of the tabulated molecular clouds between
different surveys are similar to the CO flux ratio. The application of these
ratios to the total H2 mass of our Galaxy yields a correction factor of at
least 1.4, meaning that the H2 mass of our Galaxy should be at least 40% more
massive than previously determined. Including the completeness correction, an
even more significant fraction of the matter should be contributed by baryonic
matter. The mass spectrum in the outer Galactic plane is better described by a
non-truncating power-law with $\gamma$=$-$1.83$\pm$0.05, and an upper mass of
$M_{0}$=(1.3$\pm$0.5)$\times$106 $M_{\sun}$.

ISM: clouds – ISM: molecules – Galaxy: general – radio lines: ISM – surveys

††facilities: PMO 13.7 m telescope††software: astropy (Astropy Collaboration
et al., 2013), SciPy (Virtanen et al., 2020).

##  1 Introduction

The $J$ = 1$\rightarrow$0 line of 12CO is the most accessible and widely used
tracer of the molecular interstellar medium due to the combination of high
abundance and low permanent dipole moment (Bolatto et al., 2013; Heyer & Dame,
2015). As a consequence, most scientific time of the single-dish telescopes
(e.g., the 1.2 m, the NANTEN 4 m, the Bell Laboratories 7 m, and the FCRAO 14
m telescopes) was devoted to the CO survey projects (e.g., Heyer et al., 1998;
Dame et al., 2001; Lee et al., 2001; Mizuno & Fukui, 2004; Jackson et al.,
2006) since the discovery of the CO line in 1970. The major scientific
achievements of the large CO surveys were summarized in a recent review by
Heyer & Dame (2015). These CO surveys have been proved essential for improving
our knowledge of the physics of molecular ISM, the physics of star formation,
as well as the structure and dynamics of the Milky Way.

Benefit from the increasing of sensitivity, bandwidth, spectral channels, and
enlarging of field of view by applying multi-beam receivers, more moderate- to
large-size single-dish telescopes became capable of wide-field imaging of CO
and its isotopologues, 13CO and C18O. There are currently several new surveys
being undertaken with new instruments equipped on the Mopra 22 m (e.g., Barnes
et al., 2015, 2018), the Nobeyama 45 m (e.g., Umemoto et al., 2017), and the
Purple Mountain Observatory (PMO) 13.7 m telescopes (e.g., Su et al., 2019).
These new surveys allow at the same time high resolution, high sensitivity,
and wide velocity coverage, enabling the examination of the internal structure
of the clouds and the discovery of the faint and extended cloud envelopes, as
well as the discovery of many new molecular clouds (MCs) with small angular
size, and/or low surface brightness. Indeed, using the data from the ongoing
project Milky Way Imaging Scroll Painting (MWISP) with the PMO 13.7 m
telescope, we have detected many new MCs, and a new segment of the distant
spiral arm (e.g., Sun et al., 2015; Su et al., 2016; Du et al., 2016; Sun et
al., 2017), and have revealed the complex distribution of molecular material
within the parent cloud (e.g., Zhang et al., 2014; Su et al., 2020; Lin et
al., 2020).

While much scientific progress has been achieved from the analysis of CO
survey data, our knowledge of the completeness of these CO surveys is still
very limited. The observational effects on the cloud properties have long been
known. For example, Rosolowsky & Leroy (2006) reported that the bias of
measurements due to the observational effects is at least 40%. Based on the
MWISP data ($l$ = [25$\fdg$8, 49$\fdg$7], and $b$=[$-$5$\fdg$25, +5$\fdg$25]),
Yan et al. (2021b) investigated the dependence of observed properties of MCs
(i.e., brightness temperature, flux) on angular resolution (beam sizes) and
sensitivity by simulating observations with larger beam sizes and lower
sensitivities. They found that the observed brightness temperature is mainly
controlled by the resolution (beam size) of the observation. Rather low
filling factors are reported for small clouds, i.e., only $\sim$50% for cloud
with size of two resolution elements (Yan et al., 2021b). Furthermore, the
flux completeness of a cloud strongly depends on the mean signal-to-noise
ratio (S/N) of the cloud, which is about 0.5 for the case of S/N=3.3 (Yan et
al., 2021b). All these findings strongly indicate that the census of molecular
gas from the CO surveys with finite sensitivities and resolutions is mostly
incomplete.

Currently, the region within Galactic coordinates
$l$=[104$\fdg$75,150$\fdg$25] and $b$=[$-$5$\fdg$25,+5$\fdg$25] has been fully
covered by the MWISP survey (Yan et al., 2021a), as well as partly covered by
the FCRAO outer Galaxy CO survey (OGS; Heyer et al., 1998) and the CfA 1.2 m
CO survey (Dame et al., 2001, hereafter the CfA survey). Such three
independent surveys overlapped over a large area of the sky provide us an
opportunity to investigate the completeness and limits of the CO surveys. The
second quadrant contains four well-separated spiral arms along the line of
sight. Thus it can additionally offer advantages over other Galactic quadrants
when investigating the effects of distance. In this study, we investigate the
recovered flux and its completeness measured by the CO surveys. This paper is
organized as follows. First, the data and signal identification and
decomposition methods are presented in Section 2, and in Section 3 the
integrated cloud properties measured by the MWISP survey are presented along
with a comparison with those measured by other surveys. Next, in Section 4 we
discuss the lessons from the current CO surveys and the correction factor of
total H2 mass of our Galaxy. In Section 5 we summarize the results.

##  2 Data and analysis

This study is on the basis of the overlapped region of the MWISP, CfA, and OGS
surveys, in the region of $l$ = [104$\fdg$75, 141$\fdg$54], $b$ =
[$-$3$\fdg$028, 5$\fdg$007], and $V_{\rm LSR}$ = [$-$137, 40] km s-1.

###  2.1 MWISP data

The MWISP project is an unbiased Northern Galactic Plane CO survey, which is
capable of simultaneously observing J=1-0 12CO, 13CO, and C18O lines (refer to
Zhang et al., 2014; Sun et al., 2015; Su et al., 2019, etc., for project
details and initial results). The MWISP CO data used here were observed during
2011 and 2020 with the PMO 13.7 m telescope. The front end detector was the
Superconducting Spectroscopic Array Receiver (SSAR), a nine-beam, sideband-
separating receiver (Shan et al., 2012). The angular resolution of the 13.7 m
telescope is $\sim$50$\arcsec$ at the frequency of the 12CO(1-0) line. The on-
the-fly (OTF) mode was applied in the MWISP survey. The sample spacing between
adjacent scans is 15$\arcsec$ (see the details in Sun et al., 2018, for
observation setups). Therefore, the MWISP survey fully sampled the mapped
area. The OTF raw data were regridded into 30$\arcsec\times$30$\arcsec$
pixels. The total velocity coverage is $\sim$2600 km s-1 with a channel width
of 0.16 km s-1. A first-order baseline correction was applied to the MWISP
data. Please refer to Su et al. (2019) for details of the observation strategy
and data reduction of the MWISP survey.

The chopper-wheel calibration was adopted to get the antenna temperature,
$T_{\rm A}^{*}$, which has been corrected for atmospheric absorption (Kutner &
Ulich, 1981). Data presented in this paper are further converted to a scale of
main beam brightness temperature, using $T_{\rm MB}$ = $T_{\rm
A}^{*}\,$/$\eta_{\rm MB}$, where the main beam efficiency $\eta_{\rm MB}$ is
typically 0.46 at 115 GHz (see the status report of the 13.7 m telescope
111http://www.radioast.nsdc.cn/mwisp.php). Here, we verify the calibration
accuracy of the MWISP data by comparing it with the FCRAO Galactic ring survey
(GRS; Jackson et al., 2006). Because these two surveys are both fully sampled
and have similar resolution and sensitivity, the difference of calibrated main
beam temperature ($T_{\rm MB}$) would faithfully reflect the calibration
uncertainty. Figure 1 shows the comparison results for the overlapped region
of the two surveys, e.g., in the region of $l$=[29$\fdg$75, 40$\fdg$25],
$b$=[$-$1$\fdg$0, +1$\fdg$0], and $V_{\rm LSR}$=[$-$5, 135] km s-1. The two
data sets are both smoothed to a same channel width of 2 km s-1. The main beam
temperatures of all voxels above 6$\sigma$ are pixel-by-pixel compared, which
can be well fitted by a linear relation of $T_{\rm
MB}$(MWISP)=(0.94$\pm$0.001)$\times$$T_{\rm MB}$(GRS)+(0.08$\pm$0.001), with a
Pearson correlation coefficient of 0.93 (see Fig. 1a). The distribution of the
fractional difference between all valid voxels, [$T_{\rm MB}$ (GRS)$-$$T_{\rm
MB}$ (MWISP)]/$T_{\rm MB}$ (MWISP) is also plotted as a histogram. This
distribution is fitted by a Gaussian function, with a peak of $-$0.014 and a
width of 0.219. These indicate that calibration of the MWISP data is robust,
and is consistent with that of the GRS data within $\sim$20%.

###  2.2 Archival Data

To date, the CO survey conducted with the CfA 1.2 m telescope is the only one
that covers the whole Galactic plane (Dame et al., 2001). The CfA CO data used
here were obtained during 1980 and 2001 (Dame et al., 2001). The angular
resolution of the 1.2 m telescope is $\sim$8.5$\arcmin$. We used the raw data
set where no interpolation was done222Available at
https://www.cfa.harvard.edu/rtdc/CO.. The publicly distributed CfA data are in
units of $T_{\rm MB}$, where $\eta_{\rm MB}$ of 0.82 was used for correction.
The pixel size of the whole data set is 7.5$\arcmin$, slightly less than one
beam size. In the second Galactic quadrant, the velocity coverage of the
survey is $\sim$$-$137 to $\sim$50 km s-1 with a channel width of 1.3 km s-1.

The OGS CO data used in this study are from Heyer et al. (1998), which were
observed during 1994 and 1997 with the FCRAO 14 m telescope. These data,
covering a Galactic coordinates range of $l$ = [102$\fdg$49, 141$\fdg$54] and
$b$= [$-$3$\fdg$03, 5$\fdg$41], were sampled every 50$\arcsec$ with an angular
resolution of $\sim$46$\arcsec$. The publicly distributed OGS data are in
units of $T_{\rm R}^{*}$, corrected for forward spillover and scattering
efficiency of the telescope, using $T_{\rm R}^{*}$ = $T_{\rm
A}^{*}\,$/$\eta_{\rm FSS}$, where $\eta_{\rm FSS}$ is 0.7. We transform the
OGS data to $T_{\rm MB}$ units, using $T_{\rm MB}$ = $T_{\rm
R}^{*}\times$($\eta_{\rm FSS}\,$/$\eta_{\rm MB}$), where $\eta_{\rm MB}$ is
0.45. The velocity coverage is from $-$153 to 40 km s-1 with a channel width
of 0.81 km s-1. Note that unlike the OTF mapping mode of the MWISP survey, the
beam-by-beam mapping mode was applied in the CfA and OGS surveys. Therefore,
these two surveys did not fully sample the mapped area.

Table 1: Summary of the survey data.

Survey | Beam size | Sampling |  $V_{\rm LSR}$range | $\delta_{v}$ | rms noisec | $\eta_{MB}$
---|---|---|---|---|---|---
|  |  | (km s-1) | (km s-1) | (K) |
CfA 1.2 ma | 8.5$\arcmin$ | 7.5$\arcmin$ | [$-$137, +40] | 1.3 | 0.07 | 0.82
OGSb | 46$\arcsec$ | 50$\arcsec$ | [$-$153, +40] | 0.81 | 0.95 | 0.45
MWISP raw | 50$\arcsec$ | 30$\arcsec$ |  $\sim$[-1400, +1200] | 0.16 | 0.46 | 0.46
… smoothed1 | … | 50$\arcsec$ | … | 0.81 | 0.23 | …
… smoothed2 | 8.5$\arcmin$ | 7.5$\arcmin$ | … | 1.3 | 0.04 | …

Note. — ($a$) Dame et al. (2001). ($b$) Heyer et al. (1998). ($c$) The rms
noise are in units of $T_{\rm MB}$

###  2.3 Signal Identification and Decomposition

We estimate the noises of each data set by measuring the rms intensity,
$\sigma_{\rm rms}$, from the signal-free channels of the data cube. The mean
values of $\sigma_{\rm rms}$ are 0.46 K, 0.95 K, and 0.07 K for the data sets
of the MWISP, OGS, and CfA surveys with full resolution, respectively. The
MWISP data are also convolved and regridded to match the resolutions and
sample spacings of the CfA and OGS surveys. The spatial smoothing operation is
performed with the Python package spectral-cube 333https://spectral-
cube.readthedocs.io/en/latest/index.html. The corresponding $\sigma_{\rm rms}$
for the smoothed MWISP data have mean values of 0.04 K, and 0.23 K, which are
larger than predicted by theory. Further investigations are needed to
understand these discrepancies. The cumulative distributions of $\sigma_{\rm
rms}$ for the five data sets of the three CO surveys are shown in Figure 2.

The simplest method for signal identification is “clipping”, in which all
voxels with intensities above some statistical significance level are judge to
be real signals. But we often find ourselves in a dilemma when choosing an
optimum intensity level. A loose criterion could not significantly suppress
the noise, while a strict criterion might filter a substantial amount of real
emission. Therefore, we adopted a more sophisticated approach developed by Yan
et al. (2020) to identify regions of contiguous, significant emission in the
position-position-velocity (PPV) data cubes. This approach is based on the
density-based spatial clustering of applications with noise (DBSCAN) algorithm
444https://scikit-learn.org/stable/auto_examples/cluster/plot_dbscan.html.
This algorithm contains three free parameters: (1) MinPts which defines the
minimum number of voxels of the structure, (2) $\epsilon$ which defines the
neighborhood of voxels, and (3) $T_{\rm cutoff}$ which defines the boundary
isosurface of the structure (Yan et al., 2020). Many combinations of
parameters were made before finalizing on MinPts=4, $\epsilon$=1, and $T_{\rm
cutoff}$=2$\sigma$. Such choice of parameters can identify most of the
significant emission (including both the compact and the extended emission).
The definition of connected regions of emission in PPV space here is generally
consistent with the trunks defined by dendrogram algorithm (Rosolowsky et al.,
2008).

Post selection was also recommended by Yan et al. (2020) to ensure including
as few noise clouds as possible. In this step, a cloud was further rejected
from the raw catalogs if any one of the following criteria is met: (1) the
voxel number below 16, (2) peak brightness temperature below 5$\sigma$, (3)
projection area has no compact 2$\times$2 region, and (4) channel number less
than 3. We then construct a mask according to the signal identification
results, which was applied to the raw data cubes. The emission-free voxels are
blanked. The masked versions of data cubes are used for the following
analysis.

##  3 Results

###  3.1 Large-scale Structure Traced by CO

Figure 3 presents the longitude-velocity ($l$-$v$) maps integrated over $b$ =
[$-$3$\fdg$028, 5$\fdg$007], which are derived from the noise-suppressed data
cubes of the three surveys. The integrated intensity maps of the four spiral
arms derived from the three surveys are also compared in Figure 4. Note that
the maps derived from the MWISP data with the full resolution and the same
resolution as the OGS are almost identical by human eyes, thus the maps
derived from the former are not displayed here. Compared with the moment
masked $l$-$v$ map of the CfA (Fig. 3 of Dame et al., 2001) and the unmasked
map of the OGS (Fig. 4 of Heyer et al., 1998), the structures shown in their
$l$-$v$ maps are generally consistent with those revealed in Fig. 3. Examined
more closely, however, there are some weaker features at the large negative
velocity newly recovered in Fig. 3. We find that these distant features were
actually judged to be real MCs by Digel et al. (1994) and Miville-Deschênes et
al. (2017). The comparison confirms that the DBSCAN performs as well as the
often used moment mask method. Moreover, our choice of parameters assures a
reliable extraction of CO emission, even at a low S/N ratio.

The features revealed by the three independent surveys are also compared. We
find that all data sets with a variety of resolutions and sensitivities have
the ability to resolve major features of the Local arm from $\sim-$25 to
$\sim$15 km s-1and the Perseus arm from $\sim-$65 to $\sim-$25 km s-1, due to
their relatively small distances and/or large angular sizes. And as expected,
the internal sub-structure within each arm can only be well resolved by the
ones with higher resolution (e.g., the OGS and MWISP surveys). The Outer arm
($\sim-$95 to $\sim-$65 km s-1) appears as a more coherent feature in the
MWISP survey than in the other two surveys. Besides, the MWISP data discover a
new component lying beyond the Outer arm at the maximum negative velocity (at
$V_{\rm LSR}$$<\sim-$95 km s-1, namely the Outer Scutum-Centaurus arm,
hereafter OSC), which was firstly reported in our previous work (Sun et al.,
2015). Some new features with small angular size and/or relatively faint
emission are also emerging between each pair of adjacent arms by the MWISP
survey. Considering that these features are extracted by a uniform approach,
these discrepancies would faithfully reflect the observational effects
introduced by the finite sensitivity and resolution of each survey. Our lesson
from such a rough comparison is that the survey with both high sensitivity and
resolution will offer complementary or new insights into the spiral structure
in outer or extreme outer Galaxy regions, the internal structure within each
spiral arm, and the inter-arm structure between each pair of adjacent arms.

###  3.2 Voxel Number and Flux Recovered

The integrated flux of CO emission represents an essential parameter because
it can be well translated into total H2 gas mass. Therefore it is investigated
in detail in the following two sections. The total voxel number can reflect
the volume of the CO emission region in the PPV space. Thus it is also
investigated. Table 2 summarizes the total numbers of voxels (Cols. 2-6),
total fluxes (Cols. 7-11) recovered by the five data sets, the voxel number
ratios and the flux ratios of MWISP-to-OGS (Cols. 12-13), and MWISP-to-CfA
(Cols. 14-15) in the case of full resolution and same resolution,
respectively. Here the total flux is simply the integration of all the
emission: F= $\Sigma_{i}\,T_{i}\delta\theta_{x}\delta\theta_{y}\delta\,v$.

We find that the voxel number and flux recovered by the MWISP survey are
always the largest. In the case of different resolutions, the voxel number
ratio is meaningless and thus is not compared here. The total flux recovered
by the raw MWISP survey is about 1.3 times larger than that of the OGS survey
and 1.1 times larger than that of the CfA survey. Even considering the
uncertainty possibly introduced by calibration, with typically of 10%, the
difference between the MWISP and OGS surveys is still significant. To better
understand which factor is dominant in the discrepancy, we also compared the
results identified by surveys with the fixed resolutions but different
sensitivities. We find that the total fluxes recovered by the smoothed MWISP
data sets (Cols. 8-9) are somewhat larger than that from the raw data (Col.
7). Naturally, the total flux ratios of MWISP-to-OGS and MWISP-to-CfA are both
shifted to an even higher value of 1.6 in the case of the same resolution.

As previously mentioned, the 12CO observation covers a large range of volumes
of the Milky Way (out to Galactocentric radii of $R$=$\sim$25 kpc). The
observational effects would be expected to increase with distance as
decreasing the detection limits of the cloud size and mass. Thus, we also
examine the statistical results across the four spiral arms roughly defined
according to the $V_{\rm LSR}$ distribution (mentioned in § 3.1). Note that
the inter-arm regions are ignored here given their negligible contributions.
It is obvious that both the voxel number and flux drop rapidly beyond the
Perseus arm, which differs by several orders of magnitude in different arms.
This is particularly severe by the CfA and OGS survey. Consequently, the voxel
number ratios and flux ratios rise to as high as 27.9 and 7.4 for MWISP-to-
OGS, and 53.3 and 43.8 for MWISP-to-CfA in the farthest OSC arm. In addition,
we see that the voxel number ratios are always larger than the flux ratios for
all arm components. The voxel number ratios and flux ratios across the four
spiral arms are also shown in Figure 5.

Table 2: Total number of voxel and total flux recovered.

Arm | Total voxel |  | Total flux |  | Voxel ratio&Flux ratio
---|---|---|---|---|---
| MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP/OGS |  | MWISP/CfA
| raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 |  | raw | sm2
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10) | (11) |  | (12) | (13) |  | (14) | (15)
All | 4.2e7 | 5.4e6 | 1.5e5 | 1.4e6 | 9.0e4 |  | 5.2e6 | 6.3e6 | 7.7e6 | 3.9e6 | 4.9e6 |  | –&1.3 | 3.9&1.6 |  | –&1.1 | 1.7&1.6
Loc | 3.2e7 | 4.2e6 | 1.1e5 | 1.0e6 | 6.7e4 |  | 3.9e6 | 4.7e6 | 5.9e6 | 2.7e6 | 3.7e6 |  | –&1.4 | 4.2&1.7 |  | –&1.1 | 1.6&1.6
Per | 8.6e6 | 1.1e6 | 3.7e4 | 3.3e5 | 2.1e4 |  | 1.3e6 | 1.5e6 | 1.7e6 | 1.1e6 | 1.2e6 |  | –&1.2 | 3.3&1.4 |  | –&1.1 | 1.8&1.4
Out | 5.0e5 | 8.8e4 | 4.2e3 | 1.0e4 | 1.6e3 |  | 4.9e4 | 6.7e4 | 1.0e5 | 2.5e4 | 5.0e4 |  | –&2.0 | 8.8&2.7 |  | –&1.0 | 2.6&2.0
OSC | 1.8e4 | 3.9e3 | 1.6e2 | 1.4e2 | 3.0 |  | 1.7e3 | 2.6e3 | 2.8e3 | 3.5e2 | 6.4e1 |  | –&4.9 | 27.9&7.4 |  | –&26.6 | 53.3&43.8

Note. — Columns (2)-(6): total number of voxels recovered by the raw MWISP
survey, the MWISP survey with the same resolutions as the OGS and CfA surveys,
the raw OGS and CfA surveys, respectively. Columns (7)-(11): total flux of the
recovered voxels, in units of K km s-1 arcmin2. Columns (12)-(13): voxel ratio
and flux ratio of MWISP/OGS, with the original resolution and same resolution,
respectively. Columns (14)-(15): voxel ratio and flux ratio of MWISP/CfA, with
the original resolution and same resolution, respectively.

###  3.3 Completeness of the Flux

The fluxes recovered, $F\,$($>$$T_{\rm MB}$), by each survey above a given
main beam temperature, $T_{\rm MB}$, are plotted in Figure 6, both in the case
of the original resolution and sampling (left panel) and the same resolution
and sampling (middle and right panels). The values of $T_{\rm MB}$ range from
$T_{\rm MB}=$3$\sigma_{\rm rms}$ (near the lowest cloud boundary as defined in
§2.3, that is, sensitivity limit) to the peak temperature of the cloud,
$T_{\rm peak}$. To some extent, the high end $T_{\rm MB}$ is mainly affected
by the resolutions and sampling of the observations, and we would expect its
distribution to move to somewhat higher values of $T_{\rm peak}$ with
observations characterized by better resolutions and sampling. Indeed, the
MWISP and OGS surveys with higher resolutions have much higher dynamic ranges
of $T_{\rm MB}$ than the CfA survey. In addition, we find that at a given
$T_{\rm MB}$, the fluxes $F\,$($>$$T_{\rm MB}$) recovered by the MWISP survey
are larger than those by the CfA survey, while are slightly less than those by
the OGS survey even compared under the same angular and velocity resolutions.

Table 3: Flux completeness.

Arm | Parameter $a$ |  | Parameter $b$ |  | Completeness
---|---|---|---|---|---
| MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP
| raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10) | (11) |  | (12) | (13) | (14)
All | 9.0e6 | 8.3e6 | 8.2e6 | 1.0e7 | 5.9e6 |  | 0.38 | 0.38 | 0.58 | 0.34 | 0.87 |  | 0.58 | 0.75 | 0.94
Loc | 7.5e6 | 6.7e6 | 6.4e6 | 9.7e6 | 4.6e6 |  | 0.44 | 0.44 | 0.62 | 0.42 | 0.95 |  | 0.52 | 0.71 | 0.91
Per | 1.8e6 | 1.7e6 | 1.7e6 | 2.0e6 | 1.2e6 |  | 0.25 | 0.25 | 0.52 | 0.22 | 0.63 |  | 0.70 | 0.85 | 1.01
Out | 1.3e5 | 1.2e5 | 1.2e5 | 1.0e5 | 9.9e4 |  | 0.70 | 0.72 | 2.17 | 0.50 | 3.10 |  | 0.38 | 0.61 | 0.80
OSC | 5.3e3 | 5.3e3 | 6.0e3 | 1.2e3 | 1.9e2 |  | 0.84 | 0.95 | 6.51 | 0.43 | 5.00 |  | 0.32 | 0.54 | 0.44

Note. — We fit the $F\,$($>$$T_{\rm MB}$)-$T_{\rm MB}$ relation with an
exponential function, $F\,$($>$$T_{\rm MB})=a\,{\rm exp}(-bT_{\rm MB})$, where
$a$ and $b$ are two parameters. The derived values for $a$ and $b$ are
summarized in Columns (2)-(6) and Columns (7)-(11), respectively. The flux
completeness, $F\,$($>$3$\sigma$)/$F\,$($>$0), for the MWISP survey with the
original resolution (Col. 12) and same resolutions as the OGS (Col. 13) and
CfA (Col. 14) surveys are also tabulated.

Figure 6 shows a nonlinear falling of $F\,$($>$$T_{\rm MB}$) with $T_{\rm MB}$
for all data sets, which can be approximately described by an exponential
function, $F\,$($>$$T_{\rm MB})=a\,{\rm exp}(-b\,T_{\rm MB})$, where $a$ and
$b$ are two parameters. $a$ is equivalent to the flux that we would expect to
measure with perfect sensitivity (a theoretical $T_{\rm MB}$=0 K isosurface).
A larger value of $b$ implies a faster decreasing of $F\,$($>$$T_{\rm MB}$)
with $T_{\rm MB}$. We use curve_fit in the Python package SciPy to fit the
distribution, with flux errors considered. The error of recovered flux is
estimated with $\sqrt{N}\sigma$, where $N$ is the voxel number, and $\sigma$
is the rms noise of each voxel. The derived relations are marked in each
panel, which are apparently different across a variety of data sets. The
values of $a$ and $b$ both seem to strongly depend on the spectral and spatial
resolutions of the data. Indeed, when different data sets are smoothed to the
same resolution, consistent values of both $a$ and $b$ are derived (refer to
middle and right panels of Fig. 6). To avoid chaos, only the exponential
fitting results to the MWISP data are overlapped.

Using these relations, we can also extrapolate the flux to the $T_{\rm MB}$=0
K isosurface. Hence, the completeness of flux can be calculated by the ratio
of the flux at 3$\sigma$ cutoff, $F\,$($>$3$\sigma$), and the expected flux
given infinite sensitivity, $F\,$($>$0 K). We see that the estimated flux
completeness is also suffered observational effects, as higher expected values
of $F\,$($>$0 K), while somewhat lower values of $F\,$($>$3$\sigma$) are found
in the data sets with higher resolutions. Thus the corresponding completeness
estimated here might only represent an upper limit, particularly in the data
set with relatively coarse resolution, at which the majority of clouds are
commonly marginally resolved or detected.

The flux recovered and flux completeness are also investigated in the four
spiral arms with different Galactocentric radii (refer to Fig. 7). In the top
panel, the results are derived from data with the original resolution and
sampling. In comparison, the middle and bottom panels are in the case of the
same resolution and sampling. We see that the conclusions drawn from the whole
region in Fig. 6 are still valid in these four sub-regions. The MWISP and OGS
data have a similar $b$ value for a given arm component, which is smaller than
that of the CfA data. We find that the value of $b$ parameter seems to
increase rapidly beyond the Perseus arm, which implies a rapid falling of
$F\,$($>$$T_{\rm MB}$) with $T_{\rm MB}$. As expected, the flux recovered is
less complete in the more distant spiral arm, which is only 32% in the OSC
arm, even by the MWISP survey. It is important to note that the estimation
only represents a lower limit, as previously discussed. The derived
$F\,$($>$$T_{\rm MB})$-$T_{\rm MB}$ relations and the flux completeness are
all summarized in Table 3.

###  3.4 Recovered Molecular Clouds and Their Area and Mass

In the following two subsections, the molecular gas properties are derived
based on MC samples, i.e., the whole sample and the sub-samples in the four
individual spiral arms. MCs are assigned to each spiral arm based on $V_{\rm
LSR}$ range as mentioned in § 3.1. We use the parallax-based distance
estimator of Reid et al. (2019) to determine the distance to each cloud. Given
the large uncertainties of kinematic distances to the MCs in the Local arm,
these MCs are excluded in the following analysis. The typical distances for
the whole sample and the three sub-samples are summarized in Table 4. The
molecular gas masses are derived by assuming a constant CO-to-H2 conversion
factor of 2.0$\times$1020 cm${}^{\rm-2}{\rm(K\,km\,s^{-1})^{-1}}$ (Bolatto et
al., 2013). Following Yan et al. (2020), the MC’s properties are measured.
Table 5 summarizes the total number of MCs (Cols. 2-6), the total projected
angular area of MCs (Cols. 7-11), the total mass of MCs (Cols. 12-16), and the
area ratio and mass ratio (Cols. 17-20) that recovered by the five data sets
of the three surveys.

Table 4: Summary of the molecular cloud distances.

| Median |  | Mean |  | S.D.
---|---|---|---|---|---
| MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA
| raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10) | (11) |  | (12) | (13) | (14) | (15) | (16)
All | 2.67 | 2.69 | 2.69 | 2.67 | 2.66 |  | 3.35 | 3.53 | 3.90 | 3.05 | 3.33 |  | 1.79 | 1.93 | 2.41 | 1.35 | 1.92
Per | 2.64 | 2.64 | 2.54 | 2.66 | 2.62 |  | 2.77 | 2.80 | 2.71 | 2.76 | 2.67 |  | 0.87 | 0.92 | 0.87 | 0.81 | 0.72
Out | 5.96 | 6.06 | 6.24 | 6.05 | 5.96 |  | 5.96 | 6.24 | 6.54 | 5.78 | 6.20 |  | 1.64 | 1.22 | 1.32 | 1.59 | 1.42
OSC | 10.36 | 10.29 | 10.29 | 10.57 | 17.37 |  | 10.63 | 10.60 | 10.58 | 11.38 | 17.37 |  | 1.74 | 1.82 | 2.58 | 2.34 | –

Note. — The statistical parameters for MC’s distance (in units of kpc) are
summarized, including the median value (Columns 2-6), mean value (Columns
7-11), and standard deviation (Columns 12-16) for the MWISP raw data, same
resolutions as OGS and CfA surveys, raw OGS and CfA surveys, respectively.

Table 5: Summary of the molecular cloud properties.

Arm | Total number |  | Total area |  | Total mass |  | Area ratio&Mass ratio
---|---|---|---|---|---|---|---
| MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP/OGS |  | MWISP/CfA
| raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 |  | raw | sm2
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10) | (11) |  | (12) | (13) | (14) | (15) | (16) |  | (17) | (18) |  | (19) | (20)
All | 7368 | 4599 | 221 | 1946 | 156 |  | 65.0 | 75.0 | 1.9e2 | 27.0 | 1.2e2 |  | 5.2e6 | 6.1e6 | 5.6e6 | 4.2e6 | 4.0e6 |  | 2.4&1.3 | 2.7&1.5 |  | 0.5&1.3 | 1.6&1.4
Per | 6231 | 3752 | 163 | 1773 | 130 |  | 57.8 | 65.9 | 1.7e2 | 26.0 | 1.1e2 |  | 4.3e6 | 4.8e6 | 4.6e6 | 3.7e6 | 3.2e6 |  | 2.2&1.1 | 2.5&1.3 |  | 0.5&1.3 | 1.6&1.4
Out | 1007 | 747 | 48 | 165 | 25 |  | 6.6 | 8.2 | 19.7 | 1.4 | 1.2e1 |  | 8.4e5 | 1.2e6 | 8.1e5 | 4.2e5 | 7.4e5 |  | 4.8&2.0 | 6.0&2.9 |  | 0.5&1.1 | 1.6&1.1
OSC | 130 | 100 | 10 | 8 | 1 |  | 0.4 | 0.6 | 2.5 | 0.02 | 0.08 |  | 1.2e5 | 1.6e5 | 1.6e5 | 2.6e4 | 2.1e4 |  | 19.4&4.6 | 26.1&6.1 |  | 5.5&5.6 | 31.8&7.6

Note. — Columns (2)-(6): total number of objects recovered by the MWISP data
with original resolution, same resolutions as OGS, and CfA surveys, raw OGS
and CfA surveys, respectively. Columns (7)-(11): the total projected area of
the recovered objects, in units of deg2. Columns (12)-(16): the total mass of
the recovered objects, in units of $M_{\sun}$. Columns (17)-(18): area ratio
and mass ratio of MWISP/OGS, with the original resolution and same resolution,
respectively. Columns (19)-(20): area ratio and mass ratio of MWISP/CfA, with
the original resolution and same resolution, respectively.

We find that the number of molecular clouds identified by the MWISP survey
with the full resolution is up to $\sim$2 orders of magnitude larger than that
of the CfA survey, and $\sim$3 times larger than that of the OGS survey. Such
significant discrepancy is perhaps not surprising because the recovered
objects are controlled by the sensitivity and resolution of the data set. As
stated in § 2.3, regions of emission could be successfully recovered as single
entities only if their sizes are larger than the resolutions of the
observations, the separations of the blend sources are larger than the
resolutions, and their brightness temperatures are above a user-defined
threshold. Additionally, the sensitivity of the data set that defines the
boundary of the molecular cloud also has somewhat effects on the decomposition
results, because the higher sensitivity data would tend to merge multiple MCs
into a single MC. When MWISP data are smoothed and resampled to the
resolutions and pixel sizes of the OGS and CfA surveys, we find that
comparable numbers of clouds to the OGS and CfA surveys are identified in
these smoothed MWISP data, with values of 8133 and 286.

The total projected area of MCs recovered by the raw MWISP is 2.0 times larger
than that recovered by the OGS survey, but is only 80 per cent of that
recovered by the CfA survey. Here the deconvolution is performed by
subtracting the beam from the measured projected area. It seems that the
smoothing of the MWISP data to the OGS velocity resolution does not
significantly change the projected area recovered (75 deg2 that is similar to
the raw MWISP). Thus the area ratio of the smoothed MWISP to OGS almost
remains unchanged as compared to that of the raw data. However, the smoothing
of the MWISP to the CfA resolution produces a significant boost of projected
area recovered (up to 190 deg2 that is 2.9 times larger than that of the raw
MWISP), as a consequence, the area ratio of the smoothed MWISP to CfA
increases up to 1.6.

As expected, the total molecular gas mass of the MCs recovered by the MWISP is
the highest, which is 1.5 and 1.4 times those recovered by the OGS and CfA
surveys in the case of the full resolution, respectively. We also examine the
statistical results recovered across the three sub-samples. It shows that the
number, projected area, and mass of the clouds recovered all decrease rapidly
beyond the Perseus arm, particularly significant in results recovered by the
CfA and OGS surveys. As a consequence, the area ratios and mass ratios of
MWISP-to-CfA and MWISP-to-OGS rise with increasing distance (plotted in Fig.
5).

###  3.5 Distributions of physical parameters

Table 6: Statistical parameters of the molecular cloud physical size.

| Min |  | Max |  | Median |  | Mean
---|---|---|---|---|---|---|---
| MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA
| raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10) | (11) |  | (12) | (13) | (14) | (15) | (16) |  | (17) | (18) | (19) | (20) | (21)
All | 0.3 | 0.6 | 6.4 | 0.7 | 5.4 |  | 64.8 | 75.2 | 170.1 | 40.3 | 138.5 |  | 1.3 | 1.9 | 16.4 | 1.8 | 14.7 |  | 2.0 | 2.9 | 23.6 | 2.5 | 19.9
Per | 0.3 | 0.6 | 6.4 | 0.7 | 5.4 |  | 64.8 | 75.2 | 170.1 | 40.3 | 138.5 |  | 1.1 | 1.6 | 12.3 | 1.7 | 11.8 |  | 1.7 | 2.4 | 18.0 | 2.4 | 16.5
Out | 0.5 | 1.0 | 10.4 | 1.1 | 14.6 |  | 31.4 | 57.6 | 101.6 | 14.6 | 78.0 |  | 2.7 | 3.7 | 33.9 | 3.7 | 33.6 |  | 3.6 | 5.0 | 36.6 | 4.5 | 36.3
OSC | 1.9 | 2.1 | 26.2 | 4.0 | 47.8 |  | 24.4 | 27.3 | 76.3 | 10.9 | 47.8 |  | 4.6 | 6.4 | 51.5 | 5.4 | 47.8 |  | 5.3 | 7.2 | 51.1 | 5.9 | 47.8

Note. — The statistical parameters for MC’s size (in units of pc) are
summarized, including the minimum (Columns 2-6), maximum (Columns 7-11),
median (Columns 12-16), and mean (Columns 17-21) values, for the MWISP raw
data, same resolutions as OGS and CfA surveys, raw OGS and CfA surveys,
respectively.

Table 7: Statistical parameters of the molecular cloud mass.

| Min |  | Max |  | Median |  | Mean
---|---|---|---|---|---|---|---
| MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA
| raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10) | (11) |  | (12) | (13) | (14) | (15) | (16) |  | (17) | (18) | (19) | (20) | (21)
All | 1.0 | 6.4 | 1.4e2 | 25.0 | 2.7e2 |  | 1.1e6 | 1.3e6 | 2.6e6 | 5.9e5 | 1.9e6 |  | 34.0 | 95.0 | 2.8e3 | 2.7e2 | 3.2e3 |  | 7.1e2 | 1.3e3 | 2.5e4 | 2.1e3 | 2.6e4
Per | 1.0 | 6.4 | 1.4e2 | 25.0 | 2.7e2 |  | 1.1e6 | 1.3e6 | 2.6e6 | 5.9e5 | 1.9e6 |  | 26.0 | 66.0 | 1.5e3 | 2.4e2 | 2.2e3 |  | 6.8e2 | 1.3e3 | 2.8e4 | 2.1e3 | 2.5e4
Out | 2.7 | 19.0 | 9.3e2 | 55.0 | 2.0e3 |  | 6.6e4 | 1.9e5 | 1.7e5 | 4.1e4 | 1.6e5 |  | 1.2e2 | 2.9e2 | 7.4e3 | 9.4e2 | 1.8e4 |  | 8.3e2 | 1.6e3 | 1.7e4 | 2.5e3 | 3.0e4
OSC | 34.0 | 1.0e2 | 5.6e3 | 9.1e2 | 2.1e4 |  | 2.3e4 | 2.9e4 | 4.8e4 | 1.0e4 | 2.1e4 |  | 3.1e2 | 7.9e2 | 1.3e4 | 1.8e3 | 2.1e4 |  | 9.1e2 | 1.6e3 | 1.6e4 | 3.2e3 | 2.1e4

Note. — The statistical parameters for MC’s mass (in units of $M_{\sun}$) are
summarized, including the minimum (Columns 2-6), maximum (Columns 7-11),
median (Columns 12-16), and mean (Columns 17-21) values, for the MWISP raw
data, same resolutions as OGS and CfA surveys, raw OGS and CfA surveys,
respectively.

Table 8: Statistical parameters of the molecular cloud surface density.

| Min |  | Max |  | Median |  | Mean
---|---|---|---|---|---|---|---
| MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA
| raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10) | (11) |  | (12) | (13) | (14) | (15) | (16) |  | (17) | (18) | (19) | (20) | (21)
All | 1.5 | 2.5 | 0.8 | 10.7 | 2.0 |  | 82.4 | 73.9 | 28.4 | 256.9 | 32.0 |  | 6.2 | 7.6 | 2.8 | 26.2 | 4.7 |  | 7.9 | 9.3 | 3.7 | 29.9 | 5.7
Per | 1.5 | 2.5 | 0.8 | 10.7 | 2.0 |  | 82.4 | 73.9 | 28.4 | 256.9 | 32.0 |  | 6.4 | 7.9 | 3.1 | 26.5 | 5.0 |  | 8.2 | 9.7 | 4.0 | 30.4 | 6.0
Out | 1.8 | 3.0 | 0.8 | 13.2 | 2.4 |  | 27.9 | 27.4 | 10.5 | 66.1 | 9.0 |  | 5.6 | 6.7 | 2.4 | 23.7 | 4.2 |  | 6.5 | 7.5 | 3.0 | 24.9 | 4.7
OSC | 2.3 | 3.1 | 0.8 | 16.3 | 2.9 |  | 21.9 | 20.1 | 2.6 | 34.8 | 2.9 |  | 5.1 | 6.0 | 1.6 | 26.2 | 2.9 |  | 6.1 | 7.0 | 1.8 | 25.0 | 2.9

Note. — The statistical parameters for MC’s surface density (in units of
$M_{\sun}$ pc-2) are summarized, including the minimum (Columns 2-6), maximum
(Columns 7-11), median (Columns 12-16), and mean (Columns 17-21) values, for
the MWISP raw data, same resolutions as OGS and CfA surveys, raw OGS and CfA
surveys, respectively.

The statistical parameters for MC’s size, mass, mass surface density are
summarized in Tables 6-8, including the minimum, maximum, median, and mean
values for each property. We find that all parameters presented here are
different between the five data sets, which seem to be controlled by the data
quality and the distance to MC. As expected, the MWISP MC sample shows the
lowest statistical values of mass surface density, as MWISP data are capable
of detecting faint emission around the periphery of the cloud and a large
number of small and faint clouds that were largely missed by previous surveys.

Figure 8 presents the cumulative distributions of the velocity dispersion
($\sigma_{\rm v}$), effective radius ($R_{\rm eff}$), and mass ($M$) for the
molecular clouds extracted from the five data sets. Both the whole cloud
sample, and the sub-samples in the Perseus, Outer, and OSC arms are compared.
For a better comparison, each property is plotted in figures with a fixed
range of scale. It shows that both the dynamic range of each property and the
shape of each distribution are significantly different between the five data
sets of the three surveys across the Galactic disk. The dynamic ranges of all
these three fundamental properties exhibit the largest range in the MWISP
data, while the smallest range in the CfA data. It seems that those small or
fain MCs could only be detected by those surveys with high resolution and high
sensitivity. We quantify the difference by applying both a truncated power-law
function, $N(x>x{{}^{\prime}})=N_{0}[(x/x_{0})^{\gamma+1}-1]$, and a non-
truncated power-law function, $N(x>x{{}^{\prime}})=[(x/x_{0})^{\gamma+1}]$
(e.g., Heyer & Dame, 2015; Rice et al., 2016; Colombo et al., 2019) to the
cloud size and mass distributions. Here the completeness limits are evaluated
by the mean size and mean mass of objects recovered by each data set (refer to
Tables 6-7), which represent conservative estimations.

Using the maximum-likelihood method of Rosolowsky (2005) and Rosolowsky
(2007), the size and mass spectra are measured. We find that the size and mass
distributions in the outer Galaxy are better fitted by non-truncated power-law
functions in most situations. The only exception is the size spectrum of the
whole cloud sample extracted from the raw MWISP data, which can not be well
fitted at the high-value end. The MCs’ size function fitting results for the
different data sets and different distance components are summarized in Table
9. Similarly, the MCs’ mass function fitting results are summarized in Table
10. Note that the fitting results for clouds number below 10 have large
uncertainties thus are not fitted here.

As an example, Figure 9 shows the fitting results of the size and mass
distributions of the whole cloud sample extracted from the MWISP data with
full resolution. The MCs’ size distribution derives an index of
$\gamma$=$-$2.93$\pm$0.05, and the mass distribution derives an index of
$\gamma$=$-$1.83$\pm$0.05. The index of the mass spectrum is very similar to
the results for catalogs in the first Galactic quadrant extracted by using the
same algorithms and input parameters, as well as the same CO survey data,
e.g., with $\gamma$=$-$1.8 reported by Yan et al. (2020) (their Figure 17) for
the local MC sample, and with $\gamma$=$-$1.74 reported by Su et al. (2021)
for the MC sample located near the tangent points.

We see that the indices fitted by other data sets with different resolutions
and/or sensitivities are quite different from those of the raw MWISP data.
Besides, the exact value of $\gamma$ varies in different distance components.
Generally, the slight steeper $\gamma$ values for both size and mass
distributions are revealed by data with higher resolution and low sensitivity,
and also by cloud samples with larger distances. Nevertheless, all survey data
across all distance components display “bottom heavy” size distributions with
indices $\gamma<-$2, while “top heavy” mass distributions with indices
$\gamma>-$2\. This implies that although the vast majority of clouds are small
ones, the majority of the mass in the Galactic plane is confined to the giant
ones. These properties are similar to what have been found in the inner Galaxy
(refer to the review by Heyer & Dame, 2015).

Compared to other studies toward the same Outer Galaxy, we find that our
indices of mass are much shallower. For examples, the indices reported by
Rosolowsky (2005) and Rice et al. (2016) are both with values of $\gamma<-$2\.
The cloud catalog used by Rosolowsky (2005) was directly from Heyer et al.
(2001) on the basis of the OGS survey, while the catalog used by Rice et al.
(2016) was extracted by dendrogram on the basis of the CfA data. Similar to
this study, both studies identified objects as contiguous regions above a
fixed intensity threshold, but used data sets with different resolutions, and
chose different cutoffs and methods for decomposing structures.

Table 9: Parameters of MC size spectra derived from different data sets.

| Number |  | $\gamma$ |  | $R_{0}$
---|---|---|---|---|---
| MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA
| raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10) | (11) |  | (12) | (13) | (14) | (15) | (16)
All | 2157 | 1392 | 73 | 589 | 44 |  |  $-$2.93$\pm$0.05 |  $-$2.97$\pm$0.05 |  $-$3.01$\pm$0.20 |  $-$3.04$\pm$0.06 |  $-$2.75$\pm$0.20 |  | 115$\pm$12 | 120$\pm$12 | 216$\pm$50 | 60$\pm$ 5 | 184$\pm$39
Per | 1780 | 1055 | 44 | 515 | 43 |  |  $-$2.97$\pm$0.05 |  $-$2.98$\pm$0.08 |  $-$2.57$\pm$0.25 |  $-$3.05$\pm$0.07 |  $-$3.07$\pm$0.54 |  | 76$\pm$6 | 80$\pm$11 | 179$\pm$63 | 50$\pm$5 | 95$\pm$31
Out | 327 | 214 | 18 | 57 | 10 |  |  $-$3.11$\pm$0.11 |  $-$3.05$\pm$0.11 |  $-$3.72$\pm$0.47 |  $-$3.60$\pm$0.32 |  $-$3.52$\pm$0.73 |  | 57$\pm$9 | 69$\pm$9 | 103$\pm$19 | 23$\pm$5 | 92$\pm$22
OSC | 48 | 39 | – | – | – |  |  $-$3.99$\pm$0.54 |  $-$4.18$\pm$0.51 | – | – | – |  | 20$\pm$4 | 23$\pm$4 | – | – | –

Note. — The number of objects above the completeness limit (Columns 2-6),
power-law index of the size spectrum (Columns 7-11), and the upper size of the
distribution (in units of pc, Columns 12-16) recovered by the five data sets
of the three surveys, respectively.

Table 10: Parameters of MC mass spectra derived from different data sets.

| Number |  | $\gamma$ |  | $M_{0}$
---|---|---|---|---|---
| MWISP | OGS | CfA |  | MWISP | OGS | CfA |  | MWISP | OGS | CfA
| raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw |  | raw | sm1 | sm2 | raw | raw
(1) | (2) | (3) | (4) | (5) | (6) |  | (7) | (8) | (9) | (10) | (11) |  | (12) | (13) | (14) | (15) | (16)
All | 505 | 353 | 24 | 203 | 14 |  |  $-$1.83$\pm$0.05 |  $-$1.84$\pm$0.05 |  $-$1.88$\pm$0.22 |  $-$1.99$\pm$0.13 |  $-$1.64$\pm$0.21 |  | (1.3$\pm$0.5)e6 | (1.4$\pm$0.6)e6 | (8.7$\pm$6.3)e5 | (4.7$\pm$2.6)e5 | (1.5$\pm$0.8)e6
Per | 345 | 222 | 10 | 158 | – |  |  $-$1.83$\pm$0.24 |  $-$1.83$\pm$0.29 |  $-$1.71$\pm$0.20 |  $-$1.93$\pm$0.07 | – |  | (7.5$\pm$6.4)e5 | (8.5$\pm$4.7)e5 | (1.2$\pm$1.1)e6 | (4.8$\pm$2.6)e5 | –
Out | 123 | 95 | 14 | 38 | – |  |  $-$1.87$\pm$0.16 |  $-$1.95$\pm$0.20 |  $-$2.91$\pm$0.71 |  $-$2.21$\pm$0.18 | – |  | (2.1$\pm$2.2)e5 | (2.0$\pm$1.5)e5 | (8.1$\pm$5.0)e4 | (5.8$\pm$2.5)e5 | –
OSC | 29 | 25 | – | – | – |  |  $-$2.25$\pm$0.46 |  $-$2.60$\pm$0.64 | – | – | – |  | (1.4$\pm$0.5)e4 | (1.2$\pm$0.8)e4 | – | – | –

Note. — The number of objects above the completeness limit (Columns 2-6),
power-law index of the mass spectrum (Columns 7-11), and the upper mass of the
distribution (in units of $M_{\sun}$, Columns 12-16) recovered by the five
data sets of the three surveys, respectively.

##  4 Discussions

###  4.1 Lessons from the Current CO Surveys

On the basis of three independent large-scale CO survey data with a variety of
resolutions, sensitivities, and observation strategies, we compare the global
or integrated properties of the molecular clouds, particularly the total flux
recovered from these surveys. Since our analyses are employed by using a
uniform decomposition method toward the exact same region, it is
straightforward to estimate the observational effects on the molecular cloud
properties. To isolate the effects of resolutions on the measured properties,
the higher resolution MWISP data are smoothed to the same spatial and/or
velocity resolutions as the OGS and CfA data. To better understand the variety
of observational effects at different distances, we also compare the measured
properties derived across different spiral arms along the line of sight.

In practical observations, $T_{\rm MB}$ is clipped due to the finite
sensitivity. The recovered flux is therefore controlled by the sensitivity of
observation (Yan et al., 2021b). Furthermore, many studies have suggested that
a significant fraction of the observed molecular gas is emitted from regions
with low temperature and low column density (e.g., Heyer et al., 1998;
Goldsmith et al., 2008; Pineda et al., 2008; Sun et al., 2020). All these
naturally predict a significant boost of recovered flux with improving
sensitivity. As expected, the smoothing of MWISP data to the velocity and/or
spatial resolutions of the OGS and CfA surveys decreases the noise level (Tab.
1), which in turn results in increasing of recovered flux (refer to Cols. 7-9
of Tab. 2). For the same reason, the significant sensitivity difference
between the MWISP and OGS surveys (a factor of 4.1) in the case of same
resolution likely leads to the high MWISP collected flux, which exceeds that
of OGS by a factor of 1.6. Similarly, the recovered flux ratio of MWISP-to-CfA
resembles that of the sensitivity ratio (about 1.7). Therefore, our
statistical comparisons may lend support to the view that the recovered flux
is mainly governed by the sensitivity of the observation, as stated at the
beginning.

On the other hand, due to the finite resolution (or beam size) and
inhomogeneous property of CO emission, the observed brightness temperature,
$T_{\rm MB}$ will be inevitably diminished by instruments (Yan et al., 2021b).
We see that larger dynamic ranges of $T_{\rm MB}$ are generally revealed by
surveys having higher resolutions (left panel of Fig. 6). In addition, we find
that the dynamic ranges of $T_{\rm MB}$ for all surveys are decreasing from
the nearby spiral arm to the distant spiral arm (Fig. 7). This can be well
explained by the decreasing angular size of the emission region, and therefore
the decreasing of beam filling factor with increasing of distance for each
survey. All these findings are in agreement with the previous study about
resolution effects, as mentioned at the beginning.

For each individual survey, we find that the level of sensitivity clip effect
changes in different spiral arms. This might be attributed to the changing of
S/N ratios in different spiral arms, as the decreasing of $T_{\rm MB}$ with
increasing distance but not the noise level. All of these effects discussed
here have to be kept in mind when comparing cloud samples suffered from
different levels of observational effects.

Additionally, we find that the produced cloud catalogs are dependent on the
resolution and sensitivity of the observation. The data sets with coarse
resolution often produce catalogs of clouds with much smaller numbers but
larger projected areas by blending adjacent clouds and missing the clouds with
size below the resolution element of the observation. Thus the size and mass
distributions of clouds are shallower than the true distributions. The
observational effects on the produced cloud catalogs will be researched in
near future by Q.-Z. Yan et al. (2021, in preparation).

###  4.2 Correction Factor of H2 Mass of the Galaxy

The total H2 mass of our Galaxy was estimated to be (1.0$\pm$0.3)$\times$109
$M_{\sun}$, which includes a total H2 mass beyond the solar circle of
2.7$\times$108 $M_{\sun}$. These results were based on analyses of the CfA
survey data, and assumed $X_{\rm CO}$=2.0$\times$1020
cm${}^{\rm-2}{\rm(K\,km\,s^{-1})^{-1}}$ and $R_{\sun}$=8.5 kpc (refer to the
review by Heyer & Dame, 2015). Unlike the CfA survey that fairly covers the
entire Galactic plane, the MWISP survey only covers the northern Galactic
plane. Therefore, it is not straightforward to estimate the total H2 mass of
our Galaxy only based on the MWISP data.

In this study, we compare the total flux recovered from the MWISP and CfA
surveys in the second Galactic quadrant, which reveals a flux ratio of 1.6.
Considering the fact that it is the inner Galaxy that makes the largest
contributions to the H2 mass of the Galaxy, we also examine the recovered flux
ratio of the MWISP and CfA surveys in the overlapping region of the first
Galactic quadrant ($l$ = [25$\fdg$8, 49$\fdg$7], and $b$=[$-$5$\fdg$03,
+5$\fdg$03]). The comparison reveals that the flux ratio of the MWISP and CfA
surveys is 1.4 in the first Galactic quadrant. For the third Galactic
quadrant, the flux ratio of these two surveys is not being investigated at the
moment, yet is expected to be much higher than those found in the first and
second Galactic quadrants due to much more severe observational effects. Flux
is, to the zero-order of magnitude, proportional to the H2 mass. Thus the flux
ratio could approximately indicate the total recovered H2 mass ratio by the
two surveys. Actually, the total mass ratio of the tabulated clouds of the
MWISP and CfA surveys is derived to be 1.4 in § 3.4, which is indeed close to
the flux ratio. The discrepancy between the flux and mass ratios might be
mainly contributed by the slight discrepancy of distance derived by different
cloud catalogs, which can be found in Table 4.

The application of these ratios of MWISP and CfA to the total H2 mass of our
Galaxy leads to a correction factor of at least 1.4, which means that the
Galaxy should be at least 40% more massive than previously determined, if the
same $X_{\rm CO}$ and rotation curve were assumed. The comparison of mass
spectra from various surveys reveals that the index (refer to Tab. 10) derived
from the MWISP survey ($\gamma$=$-$1.83) is steeper than the CfA survey
($\gamma$=$-$1.64), also suggesting a higher total H2 mass based on the MWISP
survey.

Furthermore, the flux completeness was estimated to be in the range of 32%-70%
across the disk even based on the improved CO data from the MWISP survey
(refer to § 3.3). It is important to note that the molecular gas at the
distance of $>$20 kpc is outside the current scope of the MWISP survey.
Because in these regions, a typical cloud can only be marginally detected as a
compact source, implying rather low flux completeness. Therefore, we warn that
our knowledge of our Galaxy is still far from complete. If the completeness
correction was considered, an even larger fraction of the matter should be
contributed by baryonic matter. A new instrument with the combination of
higher sensitivity and resolution is demanded and will provide a more accurate
estimation of the H2 mass of the Galaxy and will also benefit a wide range of
science. Such data will also allow us to reveal the integrated properties of
clouds at uniform detection limits of the cloud physical size and mass across
the Galactic plane, in other words, with the same degree of observational
effects.

##  5 Summary

We report the MC integrated properties that the DBSCAN algorithms recovered by
an ongoing CO survey of the MWISP survey within Galactic coordinates $l$ =
[104$\fdg$75, 141$\fdg$54] and $b$ = [$-$3$\fdg$028, 5$\fdg$007]. To better
understand the observational effects, we also compare our results to those
extracted by a uniform decomposition method from the CfA and OGS surveys.
While a survey with higher resolution and sensitivity can reveal more detailed
internal structures in MCs, we focus this work on the integrated properties of
MCs for comparison. The properties recovered from a variety of spiral arms
having different Galactocentric radii are also compared in detail. We draw
several conclusions from the comparison:

(1) The cloud integrated properties are subject to different observational
effects. The level of observational effects changes in both different surveys
and different spiral arms. The effects are particularly severe in surveys with
low-sensitivity, low-resolution, relatively coarse sampling, and in distant
regions, since in these cases, the clouds are either marginally resolved or
have low signal-to-noise ratios.

(2) The total number of voxel and flux integrated over all spiral arms
recovered by the MWISP are 3.9 and 1.6 times larger than those recovered by
the OGS survey, and are 1.7 and 1.6 times larger than those recovered by the
CfA survey, in the case of the same resolution. These discrepancies are
increasing with distance. In the outermost OSC arm, the mass ratio and flux
ratio for MWISP-to-OGS increase up to 27.9 and 7.4, respectively, while
increase up to 53.3 and 43.8 for MWISP-to-CfA.

(3) The census of molecular gas in our Galaxy is incomplete, even in the two
near spiral arms recovered by the MWISP data, which have flux completeness
lower than $\sim$50%-70%.

(4) The total H2 mass of the tabulated clouds based on the MWISP survey is 1.5
and 1.4 times larger than those based on the OGS and CfA surveys,
respectively. These facts may imply that the total H2 mass of our Galaxy
should be at least 40% more massive than that previously estimated based on
the CfA survey.

(5) The mass spectrum in the Galactic outskirts is better described by a non-
truncating power-law with $\gamma$=$-$1.83$\pm$0.05, and an upper mass of
$M_{0}$=(1.3$\pm$0.5)$\times$106 $M_{\sun}$. The index of mass is much
shallower than other studies toward the same Outer Galaxy, which reported
typical values of $\gamma<-$2\. However, this index is consistent with those
found in the inner Galaxy, which are also on the basis of the MWISP data by
using the same criteria and methods for decomposing structures.

We would like to thank other members of the MWISP group, Zhiwei Chen for the
support of the data reduction system, Jixian Sun, and Dengrong Lu for their
support of the MWISP database, Yingjie Li, and Yuehui Ma for their
contribution to the observation. We are extremely grateful to all the staff
members at the PMO 13.7 m telescope, particularly the observer assistants for
their long-term support. Research presented here relied heavily on the use of
the CfA 1.2 m CO survey and the FCRAO 14 m outer Galaxy survey data. We would
like to thank Professor Mark Heyer for providing helpful comments and
suggestions that improved the paper. MWISP was sponsored by the National Key
R&D Program of China with grant no. 2017YFA0402700 and CAS Key Research
Program of Frontier Sciences with grant no. QYZDJ-SSW-SLH047. YS is supported
by the National Natural Science Foundation of China through grant 11773077 and
the Youth Innovation Promotion Association, CAS (2018355). JY is supported by
the National Natural Science Foundation of China through grant 12041305.

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Figure 1: The 13CO(1-0) intensities from the MWISP survey by the PMO 13.7 m
telescope and the GRS by the FCRAO 14 m telescope. The compared area is
$l$=[29$\fdg$75, 40$\fdg$25], $b$=[$-$1$\fdg$0, +1$\fdg$0], and $V_{\rm
LSR}$=[$-$5, 135] km s-1. The two data sets are both smoothed to the same
channel width of 2 km s-1. We apply DBSCAN algotithm to identify signal. All
voxels above 6$\sigma$ noise level are plotted. ($a$) The contours indicate
number density of 2$\times$102, 2$\times$103, 1$\times$104, and 4$\times$104,
respectively. The solid-line shows the best-fit linear function of $T_{\rm
MB}$(MWISP)=(0.94$\pm$0.001)$\times$$T_{\rm MB}$(GRS)+(0.08$\pm$0.001). The
dotted-line shows the $T_{\rm MB}$(MWISP)=$T_{\rm MB}$(GRS). ($b$) Histogram
of the difference between the valid voxel values of the two surveys. The
dashed line shows the best fit of the distribution by a Gaussian function,
with an offset of $-$0.014 and a standard deviation of 0.219.  Figure 2: The
cumulative distributions of $\sigma$ for 12CO emission from the MWISP (blue),
OGS (green), and CfA 1.2 m (red) surveys within the region of
$l$=[104$\fdg$75, 141$\fdg$54] and $b$=[$-$3$\fdg$028, +5$\fdg$007]. The solid
lines are derived from the raw data. The dashed-blue and dotted-blue lines are
derived from the smoothed MWISP data with the same resolution and sampling as
the OGS and CfA surveys, respectively.  Figure 3: The longitude-velocity maps
integrated over Galactic latitude range of $b$ = $-$3$\fdg$028 to 5$\fdg$007\.
The MWISP data are smoothed to match the resolution and sampling of the OGS
and CfA surveys. The images only contain emission identified by DBSCAN. All
emission-free pixels are blanked.  Figure 4: Integrated intensity maps of 12CO
emission. The velocity range is marked in the corner of each map. The MWISP
data are smoothed to match the resolution and sampling of the OGS and CfA
surveys. The images only contain emission identified by DBSCAN. All emission-
free pixels are blanked.  Figure 4: (Continued). Figure 4: (Continued). Figure
4: (Continued). Figure 5: ($a$) The voxel number ratio, flux ratio, projected
area ratio, mass ratio of MWISP-to-OGS in the case of full resolution (plotted
in blue) and same resolution (plotted in black). ($b$) Same as panel ($a$),
but for the ratio of MWISP-to-CfA.  Figure 6: The total flux recovered as a
function of main beam temperature, $T_{\rm cutoff}$ for the MWISP (blue), OGS
(green), and CfA (red) surveys. The MWISP data are also smoothed to match the
resolution and sampling of the OGS data (middle panel), and the CfA data
(right panel), respectively.

Figure 7: Same as Fig. 6 but for different spiral arms that are roughly
defined according to the $V_{\rm LSR}$ distributions: (1) the Local arm
[$-$25, 15 km s-1], (2) the Perseus arm [$-$65, $-$25 km s-1], (3) the Outer
arm [$-$95, $-$65 km s-1], and (4) the OSC arm [$-$125, $-$95 km s-1].

Figure 8: The cumulative distributions of velocity dispersion, effective
radius, and mass of the molecular clouds that extracted from the MWISP (blue),
OGS (green), and CfA (red) surveys. The blue-, green-, and red-solid lines
indicate the results from the MWISP, OGS, and CfA raw data, respectively. The
blue-dashed and blue-dotted lines indicate results from the smoothed MWISP
data with the same resolution as the OGS and CfA data, respectively.

Figure 9: Cumulative effective radius (left) and mass (right) functions for
the whole cloud sample extracted from the raw MWISP data. In each panel, both
a truncated (solid orange) and a non-truncated (dashed cyan) power-law
spectrum are fit. The red-dotted line marks the minimum meaningful value of
each property.


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report. Hence it is certainly worth while to use, as far as possible, standard
LaTeX commands, and to define new ones only to make the source _easier_ to
read and re-use.

  * •

Do not use TeX’s state changing font commands `{\rm ...}`, `{\it ...}`, `{\bf
...}`, , but use the LaTeX equivalents: in text mode `\textrm{...}`,
`\emph{...}`, `\textbf{...}`, and inside math mode, `\mathrm{...}`,
`\mathit{...}`, `\mathbf{...}`,

  * •

Do not reference equations, sectioning commands, figures, or tables by
explicit, hard-coded numbers in your source, but use LaTeX’s cross-reference
mechanism, using a mnemonic key to identify each structural element clearly
(see sec:crossref).

  * •

Do not redefine LaTeX’s basic counters, such as equation, section, Numbering
of such elements is defined by the cernrep class file and should not be
modified by the authors, who should use only LaTeX’s cross-reference mechanism
to refer to equations, figures, tables, (see sec:crossref).

###  .14.2 The amsmath extensions

Recent developments around LaTeX, often sponsored and driven by scientific
publishers, have ensured that structural elements and math constructs in
particular are easily delimited and their sense made clear. The American
Mathematical Society have done a great job with their amsmath extension, which
is loaded by default in the cernrep.cls class. Therefore, we would like our
authors to use the conventions and LaTeX constructs of that package, rather
than the pure TeX primitives. This is all the more important as we want to
take full advantage of the possibilities of hypertext and the Web, where TeX
primitives, because they are not clearly delimited, are much more difficult to
handle (see previous paragraph).

Below we give a few often-occurring mathematics constructs, first in their TeX
form (to be avoided), their LaTeX equivalent (to be preferred), and the
typeset result.

  * •

over to be replaced by frac.

TeX

`${Z^\nu_{i}\over 2 \Lambda_{\rm ext}}$`

LaTeX

`$\frac{Z^\nu_{i}}{2 \Lambda_{\text{ext}}}$`

Result

$\displaystyle\frac{Z^{\nu}_{i}}{2\Lambda_{\text{ext}}}$

  * •

choose to be replaced by binom.

TeX

`${m + k \ choose k}$`

LaTeX

`$\binom{m + k}{k}$`

Result

$\binom{m+k}{k}$

  * •

matrix family to be replaced by corresponding environments.

TeX

`$\matrix{a & b \cr c & d\cr}$`

LaTeX

`$\begin{matrix}a & b\\c & d\end{matrix}$`

Result

$\matrix{a}&b\\\ cd$

TeX

`$\pmatrix{a & b \cr c & d\cr}$`

LaTeX

`$\begin{pmatrix}a & b\\c & d\end{pmatrix}$`

Result

$\pmatrix{a}&b\\\ cd$

TeX

`$\bmatrix{a & b \cr c & d\cr}$`

LaTeX

`$\begin{bmatrix}a & b\\c & d\end{bmatrix}$`

Result

$\bmatrix a&b\\\ cd$

  * •

Cases construct: use the cases environment.

LaTeX without amsmath

        $$
    {\rm curvature~} R = {6K\over a^2(t)} \quad
    \left\{\begin{array}{ll}
    K=-1\quad&{\rm open}\\
    K=0      &{\rm flat}\\
    K=+1     &{\rm closed}
    \end{array}\right.
    $$

LaTeX with amsmath

        \[
    \text{curvature } R = \frac{6K}{a^2(t)} \quad
    \begin{cases}
    K=-1\quad&\text{open}\\
    K=0      &\text{flat}\\
    K=+1     &\text{closed}
    \end{cases}
    \]

Result

$\text{curvature}R=\frac{6K}{a^{2}(t)}\quad\cases{K}=-1\quad&\text{open}\\\
K=0\text{flat}\\\ K=+1\text{closed}$

  * •

amsmath offers symbols for multiple integrals.

TeX

`$\int\!\!\int\!\!\int {\rm div}~\vec{\rm E}\,{\rm d}V`
` = \int\!\!\int\vec{\rm E}\,{\rm d}\vec{\rm S}$`

LaTeX

`$\iiint \text{div}~\vec{\mathrm{E}}\,\mathrm{d}V`
` = \iint\vec{\mathrm{E}}\,\mathrm{d}\vec{\mathrm{S}}$`

Result

$\iiint\text{div}\leavevmode\nobreak\
\vec{\mathrm{E}}\,\mathrm{d}V=\iint\vec{\mathrm{E}}\,\mathrm{d}\vec{\mathrm{S}}$

The cernrep class loads, amongst others, the amsmath and the amssymb packages.
The latter package defines many supplementary commands and symbols that with
TeX would have to be constructed from more basic components. See [1] and [3],
or Chapter 8 of bib:mittelbach2004 for more details.

## References

  * [1] Herbert Voß, _Math mode_ , available at the URL http://www.tex.ac.uk/tex-archive/info/math/voss/mathmode/Mathmode.pdf.
  * [2] Frank Mittelbach and Michel Goossens, _The LaTeX Companion_, 2nd ed. (Addison-Wesley, Boston, 2004).
  * [3] Scatt Pakin, _The Comprehensive LaTeX Symbol List_, available at the URL http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf.


##  Appendix B Striped electrode system

The limits shown in section 3.3 place stringent constraints on the alignment
and shape of the electrode system. Although the simplest way to achieve a
radial $E$-field is to employ a coaxial cylindrical structure, these limits
could be achieved more easily with a more complex setup, e.g. inner and outer
electrode geometries that are composed of individual wires instead of a
uniform cylindrical foil.

Figure 12: Radial electric field for $20\text{\,}\mathrm{m}\mathrm{m}$ inner
charged electrode radius and $40\text{\,}\mathrm{m}\mathrm{m}$ outer grounded
electrode radius when the electrodes are composed of
$1\text{\,}\mathrm{m}\mathrm{m}$ diameter wires. The outer electrode has 120
and the inner 60 wires. The inner electrode high-voltage is
$-6200\text{\,}\mathrm{k}\mathrm{V}$. The black circle shows a
$6\text{\,}\mathrm{m}\mathrm{m}$ displaced orbit.

Such a setup would introduce radial non-uniformity along the circular muon
orbit, which could lead to the accumulation of a geometric phase. However, as
seen in Fig. 3, the limits on the electric field uniformity are quite relaxed
for high-frequency oscillations. For example, separating the electrodes into
100 longitudinal strands would create radial field non-uniformity. If there is
a tilt of the electrode system, that would translate to an $E_{z}$ non-
uniformity as well. However, the frequency of the $E$-field oscillations due
to this non-uniformity will be at about
$40\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. At such high frequency
oscillations of the radial component on the level of even 50% would produce a
negligible geometrical phase accumulation. Note, however, that the geometrical
phase calculations assume adiabatic processes and such extreme conditions
might violate that assumption. FEM simulations of the electric field generated
by an electrode segmented into segmented into 60 parts, each consisting of a
$1\text{\,}\mathrm{m}\mathrm{m}$ diameter wire and a grounded electrode
segmented into 120 wires show that the amplitude of the high-frequency field
oscillations are less than 1% even for a displacement of
$6\text{\,}\mathrm{m}\mathrm{m}$ of the centre of the muon orbit. The
transverse cross section of the electric field and the displaced orbit are
shown in Fig. 12. The geometrical phase effect of such an electrode structure
is shown by the red ellipsoid in Fig. 4.

The main reason to prefer a wired electrode setup is that in practice it would
be easier to construct uniform electrodes compared to using cylindrical foils.
One can devise a setup where the wires are connected to piezoelectric
actuators that allow for very fine control of the position. The actual
position of the wire can then be measured with sub-micrometer precision using
optical or capacitive distance sensors. The produced electric field can then
be precisely simulated using finite-element methods.

Another advantage of a striped system is the reduced material budget, which
would increase the path length of decay positrons and reduce their scattering.
Compared to a cylindrical foil, it would also lead to significantly less eddy
currents from the magnetic kick.