# A Magnetic Field Connecting the Galactic Center Circumnuclear Disk with
Streamers and Mini-spiral - Implications from 850 $\micron$ Polarization Data

Pei-Ying Hsieh11affiliationmark:  , Patrick M. Koch11affiliationmark:  ,
Woong-Tae Kim 22affiliationmark:  33affiliationmark:  , Paul T. P.
Ho11affiliationmark:  44affiliationmark:  , Ya-Wen Tang11affiliationmark:  ,
Hsiang-Hsu Wang 55affiliationmark:
<EMAIL_ADDRESS>1 Academia Sinica Institute of Astronomy and
Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan, R.O.C.  2 Department of
Physics & Astronomy, Seoul National University, Seoul 151-742, Korea  3
Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544, USA  4 East Asian Observatory, 660 N. Aohoku Place, University Park,
Hilo, Hawaii 96720, U.S.A.  5 Department of Physics and Institute of
Theoretical Physics, The Chinese University of Hong Kong, Shatin, New
Territories, Hong Kong, China

###### Abstract

Utilizing James Clark Maxwell Telescope (JCMT) 850 $\micron$ SCUPOL dust
polarization data, we investigate the configuration of the magnetic ($B$)
field in the circumnuclear disk (CND) of the Galactic Center (GC). The SCUPOL
data show a highly improved polarization coverage and resolution compared to
earlier 100 $\micron$ observations. The 850 $\micron$ data have a resolution
and coverage similar to previous 350 $\micron$ polarimetry data. However, with
a proper sampling on a 10$"$ grid, we find the 850 $\micron$ data trace the
morphological structures of the CND substantially better. Furthermore, because
the 850 $\micron$ trace the field deeper into the material near Sgr A*, they
represent the highest resolution submillimeter probe to date of the CND
magnetic field. The observed $B$-field morphology is well described by a self-
similar axisymmetric disk model where the radial infall velocity is one
quarter of the rotational velocity. A detailed comparison with higher-
resolution interferometric maps from the Submillimeter Array further reveals
that the $B$-field aligns with the neutral gas streamers connecting to the
CND. Moreover, the innermost observed $B$-field structure also appears to
trace and align with the mini-spiral located inside the CND. This suggests
that there is one underlying $B$-field structure that is connecting the CND
with its streamers and the inner mini-spiral. An estimate of $\beta_{\rm
Plasma}\lesssim 1$ – based on the global B-field morphology that constrains
the azimuthal-to-vertical field strength ratio of around 40 combined with a
measurement of the azimuthal velocity – indicates that the B-field appears
dynamically significant towards the CND and also onwards to the inner mini-
spiral.

###### Subject headings:

Galaxy: center – ISM: magnetic fields – polarization – radio lines: ISM

##  1\. Introduction

###  1.1. Gas Inflow in the Galactic Center

The origin of the 2-pc circumnuclear disk (CND) in the Galactic Center (GC)
has remained unclear in spite of intensive studies over the past decades
(e.g., Güsten et al., 1987; Jackson et al., 1993; Amo-Baladrón et al., 2011;
Harris et al., 1985; Mezger et al., 1989; Etxaluze et al., 2011; Lau et al.,
2013; Dent et al., 1993; Morris & Serabyn, 1996; Wright et al., 2001; Montero-
Castaño et al., 2009; Martín et al., 2012; Herrnstein & Ho, 2002, 2005;
Christopher et al., 2005; Requena-Torres et al., 2012; Mills et al., 2013;
Hsieh et al., 2017). The CND is a ring-like molecular structure rotating with
respect to the supermassive black hole (SMBH) SgrA*, within which are the
ionized gas streamers called SgrA West (mini-spiral) (Roberts & Goss, 1993;
Irons et al., 2012; Lacy et al., 1991; Wang et al., 2010; Paumard et al.,
2004; Scoville et al., 2003; Zhao et al., 2009; Lo & Claussen, 1983; Tsuboi et
al., 2017, 2016). The CND, being the closest molecular reservoir in the GC, is
critical on the understanding of the feeding of the nucleus. The replenishment
of the CND itself, therefore, is an important problem.

Previous studies have shown that multiple dense gas streamers surrounding the
CND may carry gas directly toward the nuclear region. Parts of these streamers
might be captured by the central potential (Okumura et al., 1989; Ho et al.,
1991; Coil & Ho, 2000; McGary et al., 2001; Hsieh et al., 2017). In
particular, our analysis of the gas kinematics in the CND and streamers
presented in Hsieh et al. (2017) indicates that these streamers show a
signature of rotation with an additional inward radial motion with
progressively higher velocities as the gas approaches the CND and finally ends
up co-rotating with the CND. Our results suggest a possible mechanism of gas
feeding towards the CND from 20 pc to around 2 pc. We also find that the
western and eastern streamers can be described by a simple model of Keplerian
rotation and infall. The western streamer also shows kinematic and
morphological evidence of an intersection with the CND, possibly supplying gas
to the CND. Within the CND, the parts with densities lower than the tidal
threshold will be drawn inwards to the GC.

The magnetic field ($B$-field) is expected to be important for material
orbiting within the CND (Aitken et al., 1986; Werner et al., 1988; Hildebrand
et al., 1990, 1993) as the $B$-field can not only make the gas resist more
against gravity but also helps accretion by removing angular momentum.
Moreover, investigating the $B$-field in the GC is necessary because the
energy stored in the $B$-field is expected to be comparable to the gas
kinetic, the radiation, and the cosmic ray energies, if the equipartition
theorem is valid (Chandrasekhar & Fermi, 1953). The observed mG $B$-field
(Plante et al., 1994) at a $100$-pc scale is seen to interact with the
molecular gas, as indicated by the molecular loop (Fukui et al., 2006), the
nonthermal radio filaments (NTFs) (e.g. Yusef-Zadeh et al., 1984, 2004; LaRosa
et al., 2004), and the Galactic Center arc (Serabyn & Guesten, 1987; Serabyn &
Morris, 1994; Tsuboi et al., 1997).

Figure 1.—  Color map shows the SCUBA 850 $\micron$ continuum from Di
Francesco et al. (2008). Color map shows the SCUBA 850 $\micron$ continuum
from Di Francesco et al. (2008), with solid contours spaced at 0.35, 1, 2, 3,
$\cdots$, 14 Jy beam-1 with an interval of 1 Jy beam-1. The beam size is
20$\arcsec$. The SCUPOL polarization segments (yellow) show the orientation of
the magnetic field in $p/dp~{}(=\sigma_{\rm P})\geq 2,3,4$ in the top left,
top right, and bottom panel, respectively. The 450 $\micron$ $B$-field
detected with SPARO (Dotson et al., 1998; Novak et al., 2003) with a
resolution of 6$\arcmin$ is overlaid with red segments in the top leftpanel.
The regions encompassing the 50 MC, the 20 MC, and the CND – magenta boxes in
the top right panel – are compared with the 350 $\micron$ Hertz data in
Appendix A (Figure 10). Figure 2.— (a): SCUPOL $B$-field (red segments)
overlaid on the 850 $\micron$ continuum map. The beam size of 20$\arcsec$ is
shown with a green circle. The segments are sampled on a 10$\arcsec$ grid.
Contour levels are 20$\%$, 30$\%$, …, 100$\%$ of the intensity peak (6.5 Jy
beam-1). We show the $B$-field data associated with the CND (the green box:
$1.86\arcmin\times 2.9\arcmin$). The $B$-field angle is measured north of
west. (b): $B$-field segments overlaid on the JCMT HCN($J=4-3$) map (beam:
14$\arcsec$). Contour levels are 10$\%$, 20$\%$, 40$\%$,.., 100$\%$ of the
peak (835 K m s-1). (c): $B$-field segments overlaid on the SMA HCN($J=4-3$)
map (Montero-Castaño et al., 2009) (beam:$5.9\arcsec\times 4.4\arcsec$).
Contour levels are 57, 115, 229, 459, 918, 1376, 1835, 2294 Jy beam -1 km s-1.
Earlier identified features in the CND – northeast lobe (green dots),
southwest lobe (cyan dots), and southern extension (yellow dots), and western
streamer are labeled. (d): $B$-field segments overlaid on the JCMT
HCN($J=4-3$) map (color) and the VLA 6 cm continuum (contours; beam:
$3.8\arcsec\times 3.2\arcsec$). Contour levels are 0.07, 0.1, 0.157, 0.246,
0.335, 0.424, 0.51, 0.6, 0.69, 0.78, 0.87, 0.95, 1 Jy beam-1 (beam:
$3.8\arcsec\times 3.2\arcsec$ ). SgrA* is labeled with the blue star. The
eastern arm, northern arm, bar, and western arc of the mini-spiral are
labeled.

The large-scale (100 pc) configuration of the $B$-field in the GC region
consists of two structures identified from NIR (Nishiyama et al., 2010) and
submillimeter polarization data (Novak et al., 2000). On a large scale, the
$B$-field shows a vertical structure (poloidal) in the NIR polarimetry above
0.4$\arcdeg$ off the Galactic plane. Near the Galactic plane, the field
orientation gradually becomes toroidal (Dotson et al., 1998; Novak et al.,
2003). In the finer structures revealed by the 350 $\micron$ Hertz data at a
$20\arcsec$ resolution, the B-field lines tend to follow the long axis of the
molecular clouds (Chuss et al., 2003). The 350 $\micron$ flux shows a linear
and positive correlation with the $B$-field position angles (Chuss et al.,
2003), which suggests that the field is more poloidal in less dense regions
while the field is more toroidal in the denser regions. This correlation was
attributed to the tidal shear in the central molecular zone (CMZ) (Aitken et
al., 1991, 1998; Chuss et al., 2003). Moreover, near-infrared polarimetric
data have shown that the orientation of the $B$-field is more parallel to the
Galactic plane (toroidal), but is nearly perpendicular to the plane above
$b\geq 0.4^{\circ}$ (Nishiyama et al., 2010). This provides evidence for a
smooth transition of the large-scale $B$-field. Uchida et al. (1985)
constructed a model to connect poloidal and toroidal field components based on
the assumption of magnetic flux-freezing. They found that differential
rotation can shear an initially poloidal field into a toroidal field in
sufficiently dense regions.

Compared to the large-scale Galactic disk, the configuration and the role of
the $B$-field in the CND are not well understood despite the B-field’s
potential dynamical importance in e.g., the gas inflow that can feed the SgrA*
and its surrounding, the CND (Hildebrand et al., 1990, 1993; Greaves et al.,
2002), the mini-spiral (Aitken et al., 1991; Zylka et al., 1995; Aitken et
al., 1998, 2000), and the collimation of outflows (e.g., Zhao et al., 2014;
Hsieh et al., 2016). Furthermore, recent simulations (Blank et al., 2016) have
also shown that a mG-$B$-field can support the stability of the ionized cavity
for at least 106 yr.

To date, there are only a few attempts to construct models of the $B$-field
configuration in the CND. In particular, Wardle & Königl (1990); Desch &
Roberge (1997) investigated a smooth disk model for the CND including the
$B$-field. They adopted an axially symmetric magnetic model where an initially
poloidal field threading the differentially rotating disk is pulled into a
toroidal configuration. Their models can describe the 100 $\micron$
polarization data reasonably well (Hildebrand et al., 1990, 1993). In their
model, the azimuthal ($B_{\phi}$) and the radial ($B_{\rm r}$) components
exceed the vertical ($B_{\rm z}$) component which implies that the field can
remove the excess angular momentum of the accreted material (Wardle & Königl,
1990).

The CND is undoubtedly a key object for investigating the interplay between
gas dynamics and the magnetic field near SgrA*. Yet, the effects of the
$B$-field on the gas inflow associated with the CND, its streamers, and the
inner mini-spiral are not clearly understood at parsec-scale. Previous
polarization measurements toward the CND have mostly been reported at shorter
wavelengths (350 $\micron$ and 100 $\micron$), while the polarization from
cooler components at 850 $\micron$ has not yet been probed in detail (see
Matthews et al., 2009; Aitken et al., 2000). In this paper, we investigate the
$B$-field configurations and their relationship with the molecular gas in the
GC using the 850 $\micron$ SCUPOL archival data from the James Clark Maxwell
Telescope (JCMT) (Matthews et al., 2009). We will inspect the polarized
emission at 850 $\micron$ at a sub-parcsec resolution (20$\arcsec=0.76$ pc)
toward the CND.

Figure 3.—  (a): Azimuthal variations of the 850 $\micron$ $B$-field lines
(green, blue, and red segments). Sampled elliptical annuli are indicated with
green, blue, and red triangles labelling the outer, middle, and inner annulus,
respectively. The center of the ellipses is SgrA* (blue star). The background
color image is the archival JCMT HCN($J=4-3$) map. (b): Same as (a), but the
gray/contour map shows SMA HCN($J=4-3$). The contour levels are the same as
Figure 2c. (c): $x$\- and $y$-axis are the azimuthal angle ($az$) and the
position angle of the $B$-field segments ($\Phi_{\rm B}$), respectively. The
reference ($0\arcdeg$) of $az$ and $\Phi_{\rm B}$ is west. $\Phi_{\rm B}$ is
shown from $-90\arcdeg$ to $90\arcdeg$ (see the definition for the range of
coordinate in the lower left corner in (a) and (b)). The green, blue, and red
triangles mark the $\Phi_{\rm B}$ sampled along the green, blue, and red
annulus shown in (a) and (b). The locations of the NE-lobe (green dots) , SW-
lobe (cyan dots), and southern extension (yellow dots) are labeled in (b) and
(c). The region “E” and its corresponding $B$-field segments are marked in
(a), revealing a discontinuity in orientations from E to the southern
extension.

###  1.2.  850 $\micron$ SCUPOL Polarization Archival Data

In Figure 1, we show the archival 850 $\micron$ polarization data from the
legacy program of SCUPOL with the JCMT (Matthews et al., 2009) (effective
20$\arcsec$ beam$\sim 0.76$ pc). Matthews et al. (2009) collected the JCMT
data toward the GC, where the $B$-field is sampled on a 10$\arcsec$ grid. The
linearly polarized light from dust grains is frequently used to probe the
integrated plane-of-sky $B$-field morphology. Interstellar dust grains are
elongated, with their minor axes parallel to the $B$-field. The thermal
emission from the aligned dust grains is then polarized with polarization
segments perpendicular to the field lines (Cudlip et al., 1982; Hildebrand et
al., 1984; Hildebrand, 1988; Lazarian, 2000; Andersson et al., 2015). The dust
polarization can, therefore, reveal the plane-of-sky projected $B$-field
orientations. In Figure 1, the SCUPOL $B$-field segments are overlaid on the
JCMT 850 $\micron$ map (Di Francesco et al., 2008). The segments are plotted
with $p/dp~{}(=\sigma_{\rm P})\geq 2,3,4$. The 450 $\micron$ Submillimeter
Polarimeter for Antarctic Remote Observing (SPARO) $B$-field (Novak et al.,
2003) is also overlaid with a resolution of 6$\arcmin$ (corresponding to a
linear scale of 13.7 pc). The low-resolution SPARO map traces the large-scale
$B$-field which is parallel to the plane of our Galaxy. This alignment of
field orientations with the Galactic plane is attributed to a large-scale
toroidal $B$-field configuration (i.e., azimuthal field). The SCUPOL 850
$\micron$ $B$-field as well as the continuum are clearly detected and resolved
along and across the sub-features in the GC at a 20$\arcsec$ resolution
($=0.76$ pc). The detected $B$-field orientations ($\Phi_{\rm B}$) vary
enormously over the entire map, ranging from $-90\arcdeg$ to $90\arcdeg$
(0$\arcdeg$ is west, positive is counterclockwise). Nevertheless, the
$\Phi_{\rm B}$ varies smoothly and systematically along certain sub-
structures, revealing organized patches in, e.g., the CND, the giant molecular
cloud 20/50 MC (e.g. Güsten et al., 1981), and the HVCC CO 0.02-0.02 (Oka et
al., 1999). The azimuthal correlation seen in the CND is providing the link
between the model proposed by Wardle & Königl (1990) and the SCUPOL 850
$\micron$ polarization data. We are exploring this link in the following
sections. A comparison of polarization data is presented in Appendix A. We
find that the morphology of the $B$-field is consistent with different
threshold cuts in $\sigma$. Hence, in order to work with the maximum of
independent data points, we present data with $\sigma_{\rm P}\geq 2$, and we
will focus on the $B$-field structure of the CND.

##  2\. $B$-field of the CND traced with 850 $\micron$ Polarization

###  2.1.  Structures of CND and $B$-field

Figure 2 displays the JCMT 850 $\micron$ continuum map of the CND (green box:
central $1.86\arcmin\times 2.9\arcmin$) with the $B$-field lines overlaid. The
CND consisting of the northeast lobe, SW-lobe, and southern extension is
surrounded by the 20/50 MC. Compared to the 850 $\micron$ continuum, the CND
shows a more prominent ring-like feature in the high-excitation HCN($J=4-3$)
line surrounded by several streamers (Hsieh et al., 2017). The JCMT
HCN($J=4-3$) data are from the JCMT archive (project code: M06BC06). We
reduced the data using the ORAC-DR pipeline software. The $B$-field lines show
complicated but organized structures associated with the CND. In order to
reveal more smaller-scale features that might help to explain the $B$-field
morphology, we additionally present a higher-resolution SMA HCN($J=4-3$) map
(Figure 2c). Known features of the CND and its surrounding (Montero-Castaño et
al., 2009; Martín et al., 2012) are also labeled in Figure 2: (1) northeast
lobe (NE-lobe), (2) southern extension, and (3) southwest lobe (SW-lobe).
These three main features are fitted as three Keplerian components with the
CN($J=2-1$) line (Martín et al., 2012). We also label these 3 kinematic
components in Figure 2 and 3 with green, yellow, and cyan dots. The positions
of these dots are determined by Martín et al. (2012).

A detailed look at the position angles of the $B$-field, $\Phi_{\rm B}$,
reveals systematic changes in $\Phi_{\rm B}$ along individual streamers or
filaments at pc-scale. Overall, the $\Phi_{\rm B}$ associated with the CND can
be described by at least three azimuthal components, i.e., the NE-lobe, the
southern extension, and the SW-lobe (Figure 2b). The $B$-field shows smooth
radial variations along the long axes of individual streamers. The locations
of these three $B$-field components spatially coincide with the three
Keplerian components mentioned above. Besides, several streamers surrounding
the CND were reported to feed the CND (Okumura et al., 1989; Ho et al., 1991;
Coil & Ho, 2000; McGary et al., 2001; Hsieh et al., 2017). These streamers are
the SW-Lobe and western streamer, which show signatures of rotation and inward
radial motion with progressively higher velocities as the gas approaches the
CND and finally ends up co-rotating with the CND (Hsieh et al., 2017).

In Figure 2d, we also show the archival Very Large Array (VLA) 6 cm continuum
tracing the mini-spiral (SgrA West) (e.g., Roberts & Goss, 1993). The mini-
spiral consists of the arc-shaped ionized gas streamers converging onto SgrA*.
These streamers have been proposed to originate from the inner edge of the
CND. The western arc of the mini-spiral might be the boundary of the CND,
ionized by the central cluster (Martín et al., 2008; Amo-Baladrón et al.,
2009). This association is kinematically consistent with the HCN($J=1-0$) and
H92$\alpha$ lines (Christopher et al., 2005). Very interestingly, Figure 2d
reveals that around the mini-spiral, the $B$-field lines curve along the arcs
of the mini-spiral. This may hint an underlying coherent magnetic structure
that is connecting the CND and the mini-spiral. We will discuss their link in
the later sections.

###  2.2. Azimuthal Correlation of $\Phi_{\rm B}$

To further elucidate the $B$-field structures in the CND, we are looking at
the azimuthal correlation of $\Phi_{\rm B}$ in the CND in Figure 3. The
azimuthal angles ($az$) and the $B$-field position angles ($\Phi_{\rm B}$) are
both measured counterclockwise, starting from west. The $az$ angles are
measured for the $B$-field segments selected along the three concentric
ellipses, marked by the green, blue, and red triangles. The center of the
ellipses is SgrA*, where the ellipses are oriented with their major axes
aligned with the major axis of the CND. We find that $\Phi_{\rm B}$ varies
systematically as a function of $az$ along the elliptical annuli (Figure 3c).
The $\Phi_{\rm B}$ of the southern extension of the CND (yellow dots;
$az\sim-100\arcdeg$) is $\sim 70\arcdeg$. The $\Phi_{\rm B}$ then shows a
smooth variation from the SW-lobe (cyan dots; $az\sim-60\arcdeg$) to the NE-
lobe (green dots; $az\sim 100\arcdeg$). This figure demonstrates a clear
azimuthal structure in $\Phi_{\rm B}$ which connects the various features
along the CND. We note that there is an apparent sudden jump in $\Phi_{\rm B}$
from the region E to the southern extension. We have to acknowledge that the
representation with $\Phi_{\rm B}$ can be ambiguous for values close to $\pm
90\arcdeg$ which can mimic a discontinuity in $\Phi_{\rm B}$. Nevertheless,
the sudden jump in $\Phi_{\rm B}$ between the region E and the southern
extension appears real as can be seen in particular for the inner red and blue
segments where a change in orientations of about $90\arcdeg$ is detected. The
close-to-axisymmetric pattern in $\Phi_{\rm B}$ in Figure 3 resembles the
self-similar $B$-field model presented by Wardle & Königl (1990) (hereafter
WK-model), as we will analyze further in Section 3.

Finally, we note that the lower-resolution 100 $\micron$ data
(beam$=45\arcsec$) in Hildebrand et al. (1990, 1993) already revealed a trend.
The first observations in Hildebrand et al. (1990) contained only six data
points with polarization position angles mostly along an east-west direction
in a ”dust ring” around Sgr A*. The later observations in Hildebrand et al.
(1993) tripled the number of data points with new detections in the western
region that started to hint a rotation in position angles, providing a clear
indication for an axisymmetrical model. The high-resolution 850 $\micron$
(beam$=20\arcsec$) SCUPOL data presented and analyzed here reveal a
dramatically improved polarization coverage with significantly more structural
details as compared to the earlier works in Hildebrand et al. (1990, 1993).
This is the reason why we are focusing on the 850 $\micron$ data in our
comparison with the WK-model in the following section. Appendix A gives
further details on the comparison among different polarization data sets.
Similar to the 100 $\micron$ map, the 350 $\micron$ data show a $B$-field with
a north-south orientation. In any case, substantially more detailed sub-
structures are resolved with the 850 $\micron$ polarization data analyzed
here.

Figure 4.—  (a): the $B$-field model (gc1) in the CND is shown in blue
segments overlaid on the JCMT HCN($J=4-3$) map. The $B$-field is sampled along
the same annuli (red, blue, and green triangles) as shown in Figure 3. A scale
bar of 2.5 pc is shown with the blue line. (b) The SCUPOL $B$-field data are
shown with red segments. Note that 74% of the segments are $\sigma_{\rm P}\geq
3$ in the CND and the mean uncertainty is $\sim 7.5^{\circ}$. (c) The
$B$-field model (blue segments) and the data (red segments) are overlaid on
the HCN($J=4-3$) map (color) and the VLA 6 cm continuum contours (the contour
levels are the same as Figure 2d.). The yellow ellipse marks the boundary of
the mini-spiral. (d) $az$-$\Phi_{\rm B}$ correlation of the three annuli (red,
blue, and green triangles). Error bars of $\pm 1\sigma$ are shown. The
$B$-field model (gc1) is overlaid with the blue line.

##  3\. Comparison with the WK-Model

###  3.1. Possible $B$-Field Configurations

In the WK-model, the open $B$-field lines originally perpendicular to the
Galactic plane are wrapped by the rotating radial inflow, and thus the
$B$-field lines within the Galactic plane are parallel to the plane. The
$B$-field structure is axially symmetric and the predicted orientation of
polarization is the same at a constant radius. This self-similar magnetized
disk model with an axially symmetric configuration gives a satisfactory fit to
the far-infrared 100 $\micron$ polarization of the CND (Hildebrand et al.,
1990, 1993). Here, we adopt the 5 self-similar disk models used in Wardle &
Königl (1990), called gc1, gc2, gc3, gc4, and gc5. These 5 models are
representative cases of varying $B_{\rm r}$ (radial $B$-field component),
$B_{\rm z}$ (vertical $B$-field component), and $B_{\rm\phi}$ (azimuthal
$B$-field component). Four dimensionless parameters specify each of these 5
models. They are $\alpha$, $\beta$, $\delta$, and $\epsilon$ (also see Figure
2 and Table 1 in Wardle & Königl, 1990). $\alpha$ and $\beta$ fix the ratios
of $\left|B_{\phi}/B_{\rm z}\right|$ and $\left|B_{\rm r}/B_{\rm z}\right|$,
respectively. $\delta$ describes the thickness of the disk in a thin-disk
approximation ($\delta\ll 1$). $\epsilon$ is the ratio between inflow and
azimuthal velocity ($v_{\rm r}$ and $v_{\phi}$). The main features of these
models are as following. A bar indicates averages along $z$ through the disk.
We note that the ratio $\epsilon$, the inflow-to-rotational velocity, is 0.1
for gc2, and 0.25 for all the other models (Wardle & Königl, 1990).

  * •

gc1: represents the case where $\overline{\left|B_{\rm
r}\right|}\approx\overline{\left|B_{\phi}\right|}>\overline{B_{\rm z}}$
($\alpha=40$, $\beta=27.0$).

  * •

gc2: $\epsilon$ is reduced from 0.25 to 0.1. $\overline{\left|B_{\rm
r}\right|}<\overline{\left|B_{\phi}\right|}$, but $\overline{\left|B_{\rm
r}\right|}>\overline{B_{\rm z}}$ is held ($\alpha=40$, $\beta=16.9$).

  * •

gc3: $\overline{\left|B_{\rm
z}\right|}\gtrsim\overline{\left|B_{\phi}\right|}$ and $\overline{\left|B_{\rm
r}\right|}$ is larger than $B_{\rm z}$ ($\alpha=15$, $\beta=16.5$).

  * •

gc4: $\left|B_{\phi}\right|$ remains unchanged and $\left|B_{\rm r}\right|$ is
smaller than $B_{\rm z}$ everywhere ($\alpha=40$, $\beta=7.5$).

  * •

gc5: $\overline{B_{\rm z}}$ remains larger than
$\overline{\left|B_{\phi}\right|}$ and $\overline{\left|B_{\rm r}\right|}$
everywhere ($\alpha=15$, $\beta=7.5$).

The predicted polarization angle $\Phi_{\rm p}$ ($\chi$ in the original paper)
of the integrated $B$-field is given by the following equations (Wardle &
Königl, 1990):

$\cos{2\Phi_{\rm p}}=\frac{q}{(q^{2}+u^{2})^{1/2}},~{}\sin{2\Phi_{\rm
p}}=\frac{u}{(q^{2}+u^{2})^{1/2}}.$ (1)

The Stokes parameters $q$ and $u$ are

$q/N_{\rm
d}=\sin^{2}{\omega}[\cos^{2}{\xi}(\cos^{2}{\xi}(\cos^{2}{i}+1)-1)]+\cos^{2}{\omega}\sin^{2}{i},$
(2) $u/N_{\rm d}=\cos{i}\sin^{2}{\omega}\sin{2\xi},$ (3) $\xi=\theta+az,$ (4)

where $\theta$ is the angle between the outward cylindrical radius vector and
the projection of $B$ in the upper layer onto the mid-plane (see Figure 4b in
Wardle & Königl, 1990). $N_{\rm d}$ is the column density of dust, $\omega$ is
the angle between $B$ and the $z$-axis, and $az$ is the azimuthal angle
measured from the west in the plane of the sky. $i=70^{\circ}$ is the
inclination of the disk, adopted from Güsten et al. (1987). The resulting
projected polarization angle $\Phi_{\rm p}$ can then be shown as a function of
$az$. The projected $B$-field lines for these 5 self-similar models – rotated
by $90\arcdeg$ with respect to the derived polarization orientations – are
shown in Appendix B (Figure 11). We remark that in Wardle & Königl (1990),
Hildebrand et al. (1990), and Hildebrand et al. (1993), $\Phi_{\rm p}$ is
measured east of north, while in our paper, the $\Phi_{\rm p}$ is measured
north of west. Therefore, the $\Phi_{\rm p}$ determined in the above equations
directly corresponds to the angle of the $B$-field, assuming that the
polarization angle ($E$-field) is perpendicular to the $B$-field for dust
polarization observed in the submillimeter at our densities and temperatures.
Hence, in the following we will compare the observed position angles of the
$B$-field, $\Phi_{\rm B}$, with the ones from the model, $\Phi_{\rm
model}\equiv\Phi_{\rm p}$.

In Appendix B, we present the projected $B$-field configurations predicted by
the WK-models in comparison with the SCUPOL 850 $\micron$ observations. We
further analyze the differences, $\Phi_{\rm B}$-$\Phi_{\rm model}$, between
the predicted and observed angles as a function of $az$. The differences are
minimized for model gc1, suggesting that the magnetic field in the CND is
equally dominated by $B_{\rm r}$ and $B_{\phi}$ with an inflow-to-rotational
velocity ratio of 0.25. In what follows, we adopt model gc1 as the best-fit
model and compare it further with the observations. The reader is referred to
Appendix B for a detailed comparison of the other models.

The result of the predicted $\Phi_{\rm model}$ of gc1 as a function of $az$ is
shown in Figure 4 (other models are shown from Figure 14 to 17 in Appendix B).
The model (panel a) and data points (panel b) are sampled within the same
elliptical annuli as shown in Figure 3. Panels c and d indicate that the
$B$-field data can in general be described by the model gc1 (blue line in
panel d) within $\pm 1\sigma$ uncertainties. However, in the eastern region
(marked as E in panel d), deviations between data and model ranging from
$20\arcdeg-60\arcdeg$ are found. The largest deviations are seen in the outer
ring (green triangles). The SW-Lobe has deviations from 10$\arcdeg$ to
20$\arcdeg$, and the NE-lobe (NE-lobe in panel d) shows deviations from
20$\arcdeg$ to 40$\arcdeg$. In fact, none of the models can give a single fit
to all the data points. The model gc3 is able to fit the southwest-northeast
transition along the CND, but some data points (around $az=0\arcdeg-60\arcdeg$
and in region E) show large deviations around $\geq 30\arcdeg$ Lastly, the
models gc4 (dominating $B_{\phi}$) and gc5 (dominating $B_{\rm z}$) are less
accurate descriptions for their larger and systematic offsets from the data.

###  3.2. Spatial Correlation of CND, Mini-Spiral, and $B$-field

In Figure 2d we have found that the SCUPOL 850 $\micron$ $B$-field lines seem
to connect to and morphologically match the mini-spiral with its northern arm,
its eastern arm, and its western arc. To further investigate the possible
association between the CND, the mini-spiral and the $B$-field, we overlay the
best-fit model gc1 – identical to the model used in Section 3.1 where the most
inner part was not displayed – on the SMA HCN($J=4-3$) and the 6 cm maps in
Figure 5. As argued in the previous section, the model $B$-field orientations
are aligned with the streamers and the CND. Figure 5 suggests that the same
field lines, continuing inward, also align with the three arms of the mini-
spiral. In Figure 6 we display the $az-\Phi_{\rm B}$ plot of the data and
model (gc1) in the inner region of the CND. Similar to the CND, gc1 also gives
the best Chi-square fit ($\chi^{2}$) in the inner region
($\chi^{2}/dof=17.48$, degree of freedom $(dof)=$21, $P=68\%$). This result
provides some evidence that the CND and the mini-spiral are connected through
the same coherent $B$-field, such that the morphology of the mini-spiral
follows the pattern of the $B$-field. Nonetheless, in the western arc, there
are two points (SW1, SW2) that have large deviations ($\geq 80\arcdeg$) from
the model (panel d in Figure 6) . We will discuss these deviations further in
Section 4.

Figure 5.—  Left: $B$-field model of gc1 (blue segments) overlaid on the SMA
HCN($J=4-3$) map. Right: $B$-field model of gc1 (blue segments) overlaid on
the SMA HCN($J=4-3$) map (gray) and the VLA 6 cm contours (orange and
magenta). The known features of the mini-spiral (western arc, bar, northern
arm, and eastern arm) are labeled. The contour levels are 0.03, 0.05, 0.07,
0.1, 0.157, 0.246, 0.335, 0.424, 0.51, 0.6, 0.69, 0.78, 0.87, 0.95, and 1 Jy
beam-1. The model $B$-field orientations are aligned with the streamers and
the CND, which suggests that the same field lines (gc1), continuing inward,
also align with the three arms of the mini-spiral.

##  4\. Discussion

In the previous section we have found that the $B$-field in the CND has
comparatively important $B_{\phi}$ and $B_{\rm r}$ components (gc1), while
$B_{\rm z}$ is less important. We have further suggested that the best-fit
model gc1 can also describe the $\Phi_{\rm B}$ of the mini-spiral. A detailed
comparison between the other models and filaments of the CND is presented in
the following.

###  4.1. The CND – A Magnetized Disk

As already hinted by the earlier lower-resolution 100 $\micron$ polarization
data (Hildebrand et al., 1990, 1993), the $B$-field of the CND traced by the
higher-resolution 850 $\micron$ SCUPOL data at a $20\arcsec$ resolution can be
described as an axially symmetric field, which is generated by the
differentially-rotating radial inflow in the WK-model. The initial Galactic
poloidal field is stretched and wrapped by the inflows with rotational shear
to produce the observed azimuthal field. This scenario is also proposed based
on the measurements done by Chuss et al. (2003) who find that the initial
poloidal field is collapsed to the denser regions near the Galactic plane. The
$B$-field of the CND seems to be predominately azimuthal ($B_{\rm\phi}$).
However, the radial component ($B_{\rm r}$) cannot be ignored when compared to
$B_{\rm\phi}$. The extreme cases of gc4 ($B_{\phi}$ largely dominates) and gc5
($B_{\rm z}$ largely dominates) are less likely the dominant case in the CND.
In the previous sections, we have found that gc1 gives the best fit to the 850
$\micron$ polarization data. A 25$\%$ deviation from circular rotation is
required in gc1 (inflow-to rotational velocity ratio $\epsilon=0.25$). This is
similar to the value we derived in the previous studies where we constrained
the radial inward motion to be about 20-40$\%$ of the circular rotation in the
CND and surrounding streamers, based on a phase plot analysis (Hsieh et al.,
2017). Besides, the neutral streamers are likely converging to the CND. These
streamers are likely to follow the field lines as predicted in the WK-model
shown in Figure 5. However, the detailed comparison in Appendix B also
indicates that model gc2 cannot be rejected. The difference between gc1 and
gc2 is that the inflow velocity in gc2 (10$\%$ deviation from circular
rotation) is smaller than in gc1. This leads to a reduced inward magnetic flux
advection, resulting in $B_{\rm r}$ being smaller than $B_{\phi}$.

We note that Hildebrand et al. (1990) suggested gc1 to be the best-fit
configuration for only six mostly north-south oriented $B$-field segments from
their lower-resolution 100 $\micron$ polarization data. The later data set
with about three times more data points in Hildebrand et al. (1993) still
revealed a prevailing north-south orientation of the $B$-field but with some
additional inward bending. This later work favored the axisymmetrical model
gc2. The $B$-field morphology traced at 850 $\micron$ shows dramatically more
sub-structures and a much larger coverage than at 100 $\micron$. The
observations at shorter wavelengths (100 $\micron$, 350 $\micron$, and 450
$\micron$) possibly trace warmer dust components, while the 850 $\micron$
emission is dominated by a cooler and likely denser dust component, i.e., the
CND and inflow material. Hence, polarization emission coming from different
regions may lead to different interpretations when compared to the models.
Higher-resolution observations, such as e.g., with SOFIA at near-infrared
wavelengths, will provide further important clues on the separation and origin
of different dust components.

We are returning to the SW-lobe in Figure 4d which displays $\sim 20\arcdeg$
deviations from gc1. The complication in the SW-lobe is likely due to the
presence of incoming material that is accreted onto the CND (Hsieh et al.,
2017). We find that gc3 can fit the SW-lobe better than gc1. The model gc1
represents the case where the $B$-field is dominated equally by the azimuthal
and radial components. Since the model gc3 represents a limiting case in which
$B_{\rm z}$ almost dominates near the disk mid-plane at $z/r\leq 0.06$ (but
$B_{\rm\phi}$ cannot be ignored), it is possible that the initial $B_{\rm z}$
from the replenishing inflow is still preserved in the SW-lobe. We also find
that the NE-lobe and the region E can be better fitted by the model gc4. The
model gc4 implies that the azimuthal field ($B_{\phi}$) becomes more important
relative to the radial field ($B_{\rm r}$) as the vertical field ($B_{\rm z}$)
decreases in magnitude in the NE-lobe and region E. The main support against
gravity can then come from the hydrostatic $B$-field pressure in the NE-lobe.
As we have noted in Section 2.2, there is an apparent sudden jump in
$\Phi_{\rm B}$ from the NE-lobe to the southern extension. The different
field-line properties demonstrate that the sudden jump in $\Phi_{\rm B}$
between the NE-lobe (region E) and the southern extension is physical and
real. Moreover, the NE-lobe and region E may form a continuous streamer.

In summary, although the model gc1 can describe the overall large-scale field,
sub-structures associated with filamentary structures show systematic
deviations from the overall axisymmetrical field. Our analysis shows that the
$B$-field in the CND obtained by the 850 $\micron$ polarization reveals more
detailed sub-structures than the far-infrared polarization data. These sub-
structures are associated with the pc-scale streamers, as indicated by our
high-resolution SMA maps, and they are likely at the origin of the systematic
deviations form the overall gc1 model. The different smaller-scale features in
the $B$-field might also be related to the accretion history of the past
incoming flows, i.e., the age of each of the CND structures. This possibility
echoes with the previous report by Martín et al. (2012), who found that the
northeast extension (close to NE-lobe) has been mostly ionized in the northern
arm of the mini-spiral and less molecular gas is left than in the southwest
lobe. The SW-lobe likely will begin to be ionized when it approaches the
central region. Since a single WK-model cannot explain all detailed features
of the observed $B$-field morphologies, those regions that deviate from the
model gc1 may also indicate different stages of accretion.

###  4.2. $B$-field – Linking Mini-Spiral and CND?

In Figure 6, we have found that the $B$-field lines trace not only the outer
neutral streamers towards the CND but also the inner ionized mini-spiral. In
the $az$-plot, the smooth connection of the $B$-field from the CND to the
inner mini-spiral can be described by the same model gc1. This suggests that
the CND and the mini-spiral are connected structures with the same underlying
$B$-field. In Figure 7 we display the color-composite map of the CND and the
mini-spiral with the $B$-field segments overlaid. In earlier work, the
northern arm of the mini-spiral and the NE-lobe of the CND were already shown
to be kinematically connected to the CND with closely agreeing velocities
(Christopher et al., 2005). In agreement with our result, the earlier 10
$\micron$ polarization measurements in Aitken et al. (1986, 1991) and Roche et
al. (2018) also found that the $B$-field within the northern arm of Sgr A West
is parallel to it.

Our comparison of model and data shows that the CND and the mini-spiral might
be linked by field lines. However, two data points (SW1, SW2 in panels c and d
in Figure 6) show large deviations from the model gc1. We note that the
$\Phi_{\rm B}$ of SW1 and SW2 are significantly different from the earlier
published 850 $\micron$ polarization data in Aitken et al. (2000), which show
a clearer north-south orientation in agreement with model gc1. There are
several comments that can be made here: (1) The small $\Phi_{\rm B}$ of SW1
and SW2 as presented in our figures might indeed indicate that the vertical
field component $B_{\rm z}$ can be important here as suggested by gc5; (2) the
SW-lobe (SW3) in the CND also has a large deviation around 40$\arcdeg$ from
the gc1 model. Since the SW-lobe may be undergoing gas replenishment (Hsieh et
al., 2017), the SW1, SW2, and SW3 all together might be contaminated by
superimposed components of the incoming streamer; (3) calibration
uncertainties might explain the differences between Aitken et al. (2000) or
Matthews et al. (2009) (note that Matthews et al. (2009) adopted larger grid
than Aitken et al. (2000)). Although we cannot further elucidate on these
comments with only these few data points and the current limited resolution
and sensitivity, we can still conclude that the $B$-field morphology of the
mini-spiral is overall in agreement with a continuation of the $B$-field
morphology from the CND. Hence, the $B$-field might have the potential to
guide the motions of the ionized blobs/filaments originated from the CND and
to form the mini-spiral. Moreover, the CND and the mini-spiral could possibly
even be understood together as a “disk” instead of a “ring”, with an exterior
molecular phase and an interior ionized phase.

Figure 6.— (a): $B$-field model (gc1) of the interior CND and the mini-spiral
shown in blue segments overlaid on the JCMT HCN($J=4-3$) map. The $B$-field is
sampled with the same annuli (red, blue, and green triangles). A scale bar of
2.5 pc is shown with the blue line. (b) The SCUPOL $B$-field data are shown
with red segments. Note that 19 out of 20 segments have the mean uncertainty
of $B$-field angle $\sim 7.5^{\circ}$ and are $\sigma_{\rm P}\geq 3$. The
segments labeled with SW1, SW2, and SW3 are above $\sigma_{\rm P}\geq 3$. (c)
The $B$-field model (blue segments) and the data (red segments) are overlaid
on the HCN($J=4-3$) map. The yellow-dotted ellipse marks the boundary of the
mini-spiral in green contours. (d) Observed $az$-$\Phi_{\rm B}$ correlation of
the three annuli (red, blue, and green triangles). Error bars of $\pm 1\sigma$
are shown. The $B$-field model (gc1) is overlaid with the blue line.  Figure
7.—  Color-composite image of SMA HCN($J=4-3$) tracing the CND (blue) and VLA
6 cm map tracing the mini-spiral (red). The $B$-field of the JCMT data and
model gc1 are overlaid with the white segments in the left and right panel,
respectively. The location of SgrA* is labeled with the black cross.

###  4.3. How Important is the $B$-Field in CND and Mini-Spiral?

One way to quantify the dynamical importance of the $B$-field is the plasma
parameter, $\beta_{\rm Plasma}$, that measures the ratio between thermal and
magnetic pressure and can be expressed as

$\beta_{\rm Plasma}\equiv\frac{c_{\rm s}^{2}}{v_{\rm A}^{2}}=\frac{4\pi P_{\rm
th}}{B^{2}},$ (5)

where $c_{\rm s}$ and $v_{\rm A}^{2}=B^{2}/(4\pi\rho_{\rm i})$ are the sound
speed and the Alfv$\acute{\rm e}$n velocity, respectively. $\rho_{\rm i}$ is
the total mass density of the charged particles. $P_{\rm th}$ is the mean
thermal pressure of the ionized gas. Taking a $B$-field strength of $B=$1 mG
measured from the Zeeman observation in Plante et al. (1994) and a value for
the thermal pressure $P_{\rm th}/k\sim 4\times 10^{8}$ cm-3 K determined in
Hsieh et al. (2016) (based on an electron density $n_{\rm e}=7.8\times 10^{4}$
cm-3 and an electron temperature $T_{\rm e}=5000$ K which are in the ranges
derived in Zhao et al., 2010), we find $\beta_{\rm Plasma}\sim 0.7$. This
indicates that a milli-Gauss magnetic field can be important for the dynamical
evolution of the gas inflow in the GC.

It has to be noted that the above estimate, strictly speaking, is based on a
single localized assessment of the role of the $B$-field because the
measurement of the $B$-field strength is from a single-pointing Zeeman
observation (Plante et al., 1994) that is covering a limited small region in
the GC. Hence, this estimate does not take into account in any way the
morphological features in the $B$-field that we have presented and analyzed in
the earlier sections. Our goal here is to additionally utilize the overall
B-field morphology to assess the dynamical role of the $B$-field. Making use
of the azimuthal-to-vertical field strength ratio $\alpha$ ($=4\pi\rho_{\rm
i}v_{\phi}v_{\rm r}$/$B^{2}$) and the inflow-to-rotational velocity ratio
$\epsilon$ ($=v_{\rm r}$/$v_{\phi}$) from the WK-model, the plasma parameter
$\beta_{\rm Plasma}$ can be rewritten as

$\beta_{\rm Plasma}\equiv\frac{c_{\rm s}^{2}}{v_{\rm
A}^{2}}=\frac{\alpha}{\epsilon}\frac{c_{\rm s}^{2}}{v_{\phi}^{2}}.$ (6)

For our overall best-fit model gc1, $\alpha=40$ and $\epsilon=0.25$. In Hsieh
et al. (2017) we observationally determined an azimuthal velocity
$v_{\phi}=110$ km s-1. Requena-Torres et al. (2012) estimated the gas
temperature in the CND to be from 175 K to 325 K. The corresponding $c_{\rm
s}$ then ranges from 1.3 to 1.8 km s-1. The $\beta_{\rm Plasma}$ obtained with
this thermal sound speed is $\sim 0.02-0.05$ for model gc1. This number seems
to be too small because if $\beta_{\rm Plasma}\leq 0.1$, the $B$-field would
behave like an almost rigid wire. In reality, the CND is additionally subject
to turbulent motions. We estimate the linewidth of the cloud in the CND with
the velocity linewidth-size relation determined by Miyazaki & Tsuboi (2000).
In the marginally resolved JCMT HCN($J=4-3$) map, the largest cloud size is
$\sim 1$ pc (radius). The corresponding linewidth is $\sim 10$ km s-1 based on
their empirical relation. A $c_{\rm s}$ $\sim$10 km s-1 yields $\beta_{\rm
Plasma}\sim 1.3$. If the linewidth-size relation determined by Miyazaki &
Tsuboi (2000) holds in the CND, adopting a 1 pc cloud size with a 10 km s-1
line widths might be an upper limit because most clumps seen in the SMA map
are smaller. Therefore, smaller line widths in e.g., compact clouds would lead
to an even smaller $\beta_{\rm Plasma}$, and hence might hint at an even more
dominating role of the $B$-field in compact clouds. Besides the local estimate
based on a Zeeman field strength measurement, we can thus conclude that also
globally towards and across the CND and onwards to the mini-spiral the B-field
appears to be dynamically significant when compared to thermal and turbulent
pressure.

Lastly, a comparison between gravity and $B$-field in the WK-model can be
elucidated by the ratio of $v_{\phi}$ and $v_{\rm A}$. The best-fit model gc1
has $\alpha=40$ and $\epsilon=0.25$, leading to $v_{\phi}$/$v_{\rm
A}=(\alpha/\epsilon)^{0.5}\sim 13$. This indicates that the rotation driven by
the enclosed mass is about 10 times faster than the Alfv$\acute{\rm e}$n
velocity, suggesting that the CND is a rotation-dominated disk and the
$B$-field is less important in affecting the disk rotation.

##  5\. Summary

We utilize the JCMT SCUPOL 850 $\micron$ polarization data to investigate the
$B$-field structure of the CND in the GC. The SCUPOL 850 $\micron$ data show a
highly improved and broader polarization coverage at a higher resolution than
earlier 100 $\micron$. The 850 $\micron$ data also provide the highest
resolution (20$\arcsec$) submillimeter polarimetry data that are probing the
B-field morphology towards the CND, its streamers, and the inner mini-spiral.
On a large scale, the $B$-field shows a clear predominantly azimuthal
morphology. In a detailed look, the 850 $\micron$ polarization reveals
filamentary-like structures which are not revealed in the 100 $\micron$ and
350 $\micron$ polarization data. These sub-structures in the $B$-field
morphology are associated with the earlier identified streamers in the CND
(the SW-lobe, the southern extension, and the NE-lobe). These streamers were
previously shown to consist of distinct Keplerian components in the SMA CN
map. Moreover, the innermost observed $B$-field structure also appears to
trace and align with the ionized mini-spiral located inside the CND. This
suggests that there might be one underlying $B$-field structure that is
connecting the CND with its streamers and the inner mini-spiral as shown in
Figure 7.

Overall on a large scale, the observed $B$-field morphology is well described
by a self-similar axisymmetric disk model where the radial infall velocity is
one quarter of the rotational velocity (WK model gc1). However, a detailed
comparison of the deviation from the model gc1 and individual streamers shows
that the model gc3 and gc4 can describe the SW-lobe and the NE-lobe of the CND
better than model gc1. Our interpretation is that in the SW-lobe, the vertical
$B_{\rm z}$ component is still preserved along the incoming streamer. In the
NE-lobe, the $B$-field is dominated by the azimuthal field component
$B_{\phi}$. The variation of the field configurations among these streamers
might suggest an accretion history or different stages of accretion in the
CND. The NE-lobe has been mostly ionized in the northern arm of the mini-
spiral and less molecular gas is left than in the SW-lobe. On the other hand,
the SW-lobe likely will begin to be ionized when it approaches the central
region.

We also estimate the dynamical importance of the $B$-field with the plasma
parameter ($\beta_{\rm Plasma}$). Based on a single-pointing Zeeman field
strength measurement, we derive $\beta_{\rm Plasma}\sim 0.7$. Based on the
global $B$-field morphology that constrains the azimuthal-to-vertical field
strength ratio ($\left|B_{\phi}/B_{\rm z}\right|$) to be around 40 combined
with a measurement of the azimuthal velocity ($v_{\phi}$), we estimate
$\beta_{\rm Plasma}\sim 1.3$ as an upper limit. $\beta_{\rm Plasma}$ being of
order unity indicates that the $B$-field appears to be dynamically significant
towards the CND and also onwards to the inner mini-spiral as compared to
thermal/turbulent pressure. Although the CND is a rotation-dominated disk, the
coherent underlying field geometry and $\beta_{\rm Plasma}\lesssim 1$ suggest
that the $B$-field can help to guide the motions of the ionized
blobs/filaments originated from the CND, possibly form the mini-spiral, and
provide substantial support to structures as compared to thermal and turbulent
pressure.

We emphasize that the $B$-field morphology traced at 850 $\micron$ shows
dramatically more sub-structures than at 100 $\micron$ and 350 $\micron$. The
observations at shorter wavelengths possibly trace warmer dust components,
while the 850 $\micron$ emission is dominated by a cooler and likely denser
dust component, i.e., the CND and the inflow material. Our results reveal the
uniqueness of using 850 $\micron$ polarization to trace cooler disk components
/ streamers, and also show that the $B$-field can play an important role for
the gas flows and accretion in the GC.

We thank the anonymous referee for comments which have helped to improve the
manuscript. P.-Y. H. is supported by the Ministry of Science and Technology
(MoST) of Taiwan through the grants MoST 105-2811-M-001-141, MoST
106-2811-M-001-136. P. M. K. acknowledges support from MoST
104-2119-M-001-019-MY3 and from an Academia Sinica Career Development Award.
W.-T.K. was supported by the National Research Foundation of Korea (NRF) grant
funded by the Korea government (MEST; No. 3348-20160021).

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##  Appendix A Comparison with Previous Data

In Figure 8 we show the 850 $\micron$ SCUPOL histograms of the polarization
percentages ($P$) and position angles ($PA$) of the $B$-field ($\Phi_{\rm B}$)
with $p/dp~{}(=\sigma_{\rm P})\geq 2,3,4$ as presented in the three panels in
Figure 1. The overall distributions do not vary significantly from
$\sigma_{\rm P}$ of 2 to 4 cuts. In particular, this indicates that overall
structural features in the $B$-field, measured by position angles PA, are
preserved from 4$\sigma_{\rm P}$ to 2 $\sigma_{\rm P}$, and the different cuts
in $\sigma_{\rm P}$ are unlikely mimicking or introducing wrong global large-
scale trends in the B-field. We note this finding in order to make best use of
the available data. Therefore, in order to work with the maximum number of
data points, we will present the $B$-field segments using $\sigma_{\rm P}\geq
2\sigma$ in this paper. The majority of the $P$ is around $1.5\%$ with a tail
extending to $10-25\%$. In the right panel in Figure 8, the largest $P$ are
found in the faintest Stokes I regions. This indicates that the generally
observed $P-I$ anti-correlation in star-forming regions (see, e.g., discussion
in Tang et al., 2013) also holds in the GC environment. Hence, similar dust
grain properties and alignment with the magnetic field can probably be
expected in the GC. Histograms of the 350 $\micron$ polarization data (Novak
et al., 2000) are shown in Figure 9. Comparing to the 850 $\micron$ data, we
find that the 350 $\micron$ data indicate mostly a north-south orientation of
the $B$-field with a $PA$ distribution peaking around $\pm 90^{\circ}$. While
this prevailing north-south orientation is still also apparent in the 850
$\micron$ $PA$ distribution, the 850 $\micron$ data additionally show finer
sub-structures which are central for our discussion in this work.

Figure 10 compares the $B$-field from the 850 $\micron$ SCUPOL and the 350
$\micron$ HERTZ data (Novak et al., 2000; Chuss et al., 2003,
beam=20$\arcsec$) toward the 20/50 MC and the CND. The orientation of the
$B$-field in the 20 MC is mostly north-south, both at 850 $\micron$ and at 350
$\micron$. For the 50 MC and the CND, Chuss et al. (2003) mention that the
$B$-field in these two regions shows a coherent $X$-shaped polarization
feature at 350 $\micron$ which traces the arched filaments in the GC. The
$B$-field morphology in the 50 MC at 850 $\micron$ is similar to that at 350
$\micron$. However, the difference of the $B$-field morphology in the CND
between the 350 $\micron$ and the 850 $\micron$ data may suggest different
origins for the polarized signals. In the CND region, Novak et al. (2000)
estimated that one third of the 350 $\micron$ emission is not from the CND.
The $B$-field at 850 $\micron$ seems to trace the inner sub-structures shown
in the continuum map, while the $B$-field at 350 $\micron$ likely traces more
large-scale structures associated with the arched filaments as suggested by
Chuss et al. (2003). Hence, the 350 $\micron$ data likely trace warmer and
possibly upper-layer components of the CND, while the 850 $\micron$ data
likely trace the inner cooler sub-structures (e.g., streamers) of the CND.

Figure 8.— Left: distributions of $B$-field position angles (PA) of SCUPOL 850
$\micron$ data. Middle: distributions of polarization percentages (P). Right:
Stokes $I$, normalized to $I_{\rm max}$, versus polarization percentage $P$.
In all panels, blue, olive, and orange present the data above $\sigma_{\rm
P}\geq$ 2, 3, and 4, respectively. Figure 9.— Left: distribution of the
position angles (PA) of the $B$-field of the HERTZ 350 $\micron$ polarization
data in the GC (Novak et al., 2000). Only data above $\sigma_{\rm P}\geq$ 3
are presented in their paper. Right: distribution of polarization percentages.
Figure 10.— Comparison of the $B$-field probed by 850 $\micron$ SCUPOL data
(yellow segments) and 350 $\micron$ HERTZ data (red segments; sampling with
20$\arcsec$ grid) towards the 20/50 MC and the CND. The contour levels are the
same as Figure 1. Left and right columns show the JCMT data with $\sigma_{\rm
P}\geq$ 2 and 3, respectively. The thin and thick red segments present the
data from Novak et al. (2000) and Chuss et al. (2003), respectively. The
20$\arcsec$ HERTZ and SCUPOL beam is displayed with a blue circle.

##  Appendix B Predicted $B$-field Morphologies with the WK-Model

The 5 WK-models (gc1 to gc5) (see descriptions in Table 1 in Wardle & Königl,
1990) are shown in Figure 11. The model $B$-field lines are sampled on a
10$\arcsec$ grid. Comparisons of the models and the data are shown in Figure
12 and Figure 13. As in Figure 4, Figure 14 to Figure 17 show the $az$-plots
of the comparisons between the models (gc2 to gc5) and the data. The
difference of the orientations between the data and the models, $\Phi_{\rm B}$
\- $\Phi_{\rm model}$, is shown in the middle and right panels of Figure 12.
In the CND, the difference is within $\sim 20\arcdeg$ for the models gc1, gc2,
and gc3. The difference is larger for model gc4 ($\sim 40\arcdeg$) and gc5
($\sim 60\arcdeg$). In all the models, the 20/50 MC show differences larger
than $60\arcdeg$, which likely is due to contaminating polarized emission from
the 20/50 MC away from the CND, and a likely intrinsically different B-field
structure in the 20/50 MC that substantially differs from the axisymmetrical
B-field in the CND. In Figure 13, histograms of the differences of real and
absolute values of these 5 models are displayed. The median and mean values
are calculated. The difference is toward positive or negative offsets from gc3
to gc5 as compared to the gc1 and gc2. We also run a Chi-square test
($\chi^{2}$/$dof$) for each model, where $dof$ is the degree of freedom. To
avoid the regions that might be contaminated by the 20/50 MC, we only compute
$\chi^{2}$/$dof$ within $61\arcsec\times 85\arcsec$ (magenta ellipse in Figure
12). The computed $\chi^{2}$/$dof$ for models gc1 to gc5 are 45.39 ($dof=91$),
115.14 ($dof=91$), 72.60 ($dof=91$), 105.72 ($dof=91$), and 263.71 ($dof=89$),
respectively. The model gc1 appears to be the best fit with the smallest
$\chi^{2}$/$dof$ ($P\geq 99.5\%$).

We note that Hildebrand et al. (1990) suggested gc1 to be the best-fit
configuration for their six polarization segments detected at 100 $\micron$.
Later, with about three times more data points, Hildebrand et al. (1993)
favored model gc2. However, in terms of the mean values (Figure 13), gc1 and
gc2 are indistinguishable in our analysis based on 850 $\micron$ data. In a
more detailed analysis, we find that model gc3 can fit the SW-lobe better than
gc1. Moreover, we also find that the region NE-lobe (middle and inner annuli)
is dominated by $B_{\phi}$ and can best be described by model gc4 (see also
discussions in Section 4).

Figure 11.— The WK-models (blue segments; gc1, gc2, gc3, gc4, gc5) overlaid on
the JCMT HCN($J=4-3$) map. The beam size of 14$\arcsec$ is displayed with the
red circle. Previously identified features associated with the CND – northeast
lobe (NE-lobe), southwest lobe (SW-lobe), and southern extension are indicated
with the green, cyan, and yellow dots, respectively. SgrA* is marked with the
blue star.  Figure 12.— Left: Observed $B$-field (red segments) and WK-models
(blue segments) overlaid on the JCMT HCN($J=4-3$) map. The models gc1 to gc5
are shown from top to bottom. Middle: Differences between observed $B$-field
and model $B$-field ($\phi_{\rm B}$ \- $\phi_{\rm model}$) are shown in color.
Right: Same as in the middle panel, but for absolute differences.  Figure 13.—
Histograms of the differences between models and data shown in Figure 12. A
region within 80$\arcsec$ centered on the CND is used for the calculation of
mean and median values to avoid the possible confusion from the 20 MC. The
mean and median values for gc1 to gc5 are shown from top to bottom. Left and
right columns show the real and absolute values, respectively.  Figure 14.—
(a): $B$-field model (gc2) in blue segments overlaid on the JCMT HCN($J=4-3$)
map. The $B$-field is sampled with the same annuli (red, blue, and green
triangles) as shown in Figure 3. A scale bar of 2.5 pc is shown with the blue
line. (b) The SCUPOL $B$-field data are shown with red segments. (c) $B$-field
model (blue segments) and data (red segments) overlaid on the HCN($J=4-3$)
map. The yellow ellipse marks the boundary of the mini-spiral in green
contours. (d) $az$-$\Phi_{\rm B}$ correlation of the three annuli (red, blue,
and green triangles). Error bars of $\pm 1\sigma$ are shown. The $B$-field
model (gc2) is overlaid with the blue line.  Figure 15.— Identical to Figure
14 but for $B$-field model gc3.  Figure 16.— Identical to Figure 14 but for
$B$-field model gc4.  Figure 17.— Identical to Figure 14 but for $B$-field
model gc5.