Vol.0 (20xx) No.0, 000–000

11institutetext: University of Chinese Academy of Sciences, Beijing 100049,
China
22institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China; Lei Qian<EMAIL_ADDRESS>
33institutetext: South-Western Institute for Astronomy Research, Yunnan
University Kunming, 650500 Yunnan, P.R. China<EMAIL_ADDRESS>
Received 20xx month day; accepted 20xx month day

#  Quantifying the anisotropic density structure of the Central Molecular Zone
– a 2D correlation function approach

Xiu-Yu Cai 1122 Guang-Xing Li 33 Lei Qian 1122

###### Abstract

The Central Molecular Zone (CMZ) is a ring-like structure sitting at the
center of the Milky Way. Using the 870 $\mu$m continuum map from the APEX
Telescope Large Area Survey of the Galaxy (ATLASGAL), we study the anisotropy
of the density structure of the gas in the Central Molecular Zone (CMZ) using
the 2D correlation function. To quantify the spatial anisotropy, we define the
critical angle $\theta_{\rm half}$, as well as the anisotropy parameter
$A\equiv\frac{\pi}{4\theta_{\rm half}}-1$. We find that the density structure
is strongly anisotropic at the large scale ($\sim$ 100 pc), and the degree of
spatial anisotropy decreases with the decreasing scale. At the scale of $\sim$
10 pc, the structure is still mildly anisotropic. In our analyses, we provide
a quantitative description of the anisotropic density structure of gas in the
CMZ, and the formalism can be applied to different regions to study their
differences.

###### keywords:

anisotropy — galactic centre — shear – correlation function

##  1 Introduction

Situated at the center of the Milky Way, the Central Molecular Zone (CMZ) is
an unusually dense molecular cloud complex with a size of a few hundred
parsecs (Launhardt et al. 2002). A total mass of $3\times 10^{7}M_{\odot}$ is
found inside this region (Dahmen et al. 1998; Molinari et al. 2014).
Observations indicate that the gas in the CMZ has high volume densities (with
a mean value of $\sim 10^{4}{\ \rm cm^{-3}}$, Li & Zhang 2020) and high column
densities (about $\sim 10^{23}{\ \rm cm^{-2}}$, Lis & Carlstrom 1994).

In spite of the widespread presence of dense gas, the star formation
efficiency (SFE) is about $10-100$ times lower than the standard values
(Kauffmann et al. 2013; Longmore et al. 2013a; Emsellem et al. 2015; Jeffreson
et al. 2018). The dynamics in this region can be affected by a variety of
processes, such as gravitational instability, turbulence, tidal force, cloud-
cloud collision and shear etc. (Longmore et al. 2013a; Jeffreson et al. 2018;
Kruijssen et al. 2019).

The shear on a cloud will cause a velocity difference between its near side
and its far side to the Galactic center, which stretches the gas into long
streams. The strength of shear can be quantified using the shear timescale,
which is $t_{\rm shear}=(\partial\Omega/\partial r\times r)^{-1}$. In the
Milky way, shear is believed to be responsible for creating large-scale
filamentary structures (Dobbs & Bonnell 2006). Some first hints on the
importance of shear came from the discovery of kpc-sized filamentary
structures (Li et al. 2013; Goodman et al. 2014; Wang et al. 2015). Further
observations have found that the filamentary structures of sizes of a few to a
few tens of pc tend to stay parallel to the Galactic disk (Li et al. 2016;
Wang et al. 2015), indicating that shear is dynamically important on these
scales. In some cases, shear can play a dominant role in determining the star
formation activity: recent results from Li & Zhang (2020) indicate that shear
alone is responsible for the observed low level of star formation seen in the
CMZ region.

One way to reveal the role of shear is to study the alignment of filamentary
structures (Li et al. 2016; Wang et al. 2015). However, this approach,
although effective, is cumbersome to implements. Besides, the evolution of the
interstellar medium is a multi-scale process, and ideally, we would like to
know the role of shear over a range of scales. As studying the role of shear
using the alignment of filamentary structures only allows us to probe the
scales comparable to the lengths of the filaments, better methods are needed.
In this paper, we develop a formalism to quantify the anisotropy of the
density structure of the CMZ quantitatively using the 2D correlation function,
with which the role of shear can be studied over a range of scales. In section
2, we present the data. In section 3, we describe the method and present the
results. In section 4, we give a conclusion.

(a) column density

(b) anisotropy

(c) column density_3

(d) anisotropy_3

(e) column density_5

(f) anisotropy_5

Figure 1: Left panels: Column density distribution in the Central Molecular
Zone of the Galactic center observed by the survey:the APEX Telescope Large
Area Survey of the Galaxy (ATLASGAL) at $870{\ \mu\rm m}$, with a beam size of
$19.2\arcsec$ and a typical noise level of 50-70 $\rm mJy/beam$. The
horizontal size bar in upper right corner indicates a length of 100 pc. The
yellow contours in Figure 1(c) show the region where the flux intensity is
more than 3 $\rm Jy/beam$ ($8.31\times 10^{21}{\ \rm{cm^{-2}}}$). The yellow
contours in Figure 1(e) shows the region where the flux intensity is more than
5 $\rm Jy/beam$ ($1.385\times 10^{22}{\ \rm{cm^{-2}}}$). Right panels: 2D
correlation functions $C_{\rm auto,\rm n}$. In Figure 1(b), the correlation
function is computed using the emission map. In Figures 1(d) and Figure 1(f),
we plot the correlation function computed from clipped emission maps where we
set the value of regions where I $>$ 3, 5 $\rm Jy/beam$ to $I_{\rm max}$ = 3,
5 $\rm Jy/beam$, which correspond to $N_{\rm H_{2}}=8.31\times 10^{21}{\
\rm{cm^{-2}}}$ and $1.385\times 10^{22}{\ \rm{cm^{-2}}}$.

##  2 DATA

We use the $870{\ \mu\rm m}$ map from the APEX Telescope Large Area Survey of
the Galaxy (ATLASGAL) (Schuller et al. 2009). The observations were carried
out with the APEX 12 m submillimeter telescope in dust emission continuum with
an angular resolution of $19.2\arcsec$ and a sensitivity of 50 $\rm Jy/beam$.
We assume that the CMZ region has a mean distance of 8.2 kpc, estimated from
the updated distance to the Sgr A from Gravity Collaboration et al. (2019).
The corresponding spatial resolution is $\sim$ 0.76 pc. The size of the
selected region is about 477 pc $\times$ 159 pc. The maps contain
contaminations from the fore/background, and this amounts to around 10 % of
the total flux (Li & Zhang 2020). Thus, the contributions from fore/background
emission to the overall correlation should be insignificant.

##  3 METHODS AND RESULTS

###  3.1 The correlation function

Our data is taken from Schuller et al. (2009), from which the mass surface
density can be calculated by using

$N_{\rm H_{2}}=\frac{F_{\nu}R_{0}}{B_{\nu}(T_{\rm D})\Omega\kappa\mu m_{\rm
H}},$ (1)

where ${F_{\nu}}$ is the flux density, $R_{0}\sim 100$ is the gas-to-dust
ratio, $\Omega$ is the solid angle of the telescope beam, $\mu\sim 2.8$ is the
mean molecular weight of the interstellar medium with respect to hydrogen
molecules, (Kauffmann et al. 2008), and ${m_{H}}$ is the mass of an hydrogen
atom. We adopt a uniform temperature of $T_{\rm D}=20{\ \rm K}$ (Ginsburg et
al. 2016; Molinari et al. 2010; Traficante et al. 2011). At 870 $\mu$m,
$\kappa=1.85{\ \rm cm^{2}\ g^{-1}}$ (Ossenkopf & Henning 1994) and

$B_{\nu}(T_{D})={\frac{2h{\nu}^{3}}{c^{2}}{\frac{1}{{e^{\frac{h{\nu}}{kT}}}-1}}}\;.$
(2)

Here, we have assumed a uniform temperature for the whole region. In reality,
the temperature varies by around 5 K, leading to an uncertainty of about 20 %
for individual regions. As the errors of surface density caused by temperature
variations are small compared to the intensity variations, the contribution
from temperature to the overall correlation should be minimal.

To quantify the anisotropy of the density structure, we evaluate $C_{\rm
auto}$, which is the 2D correlation function. Assuming that $f(x,y)$
represents the intensity distribution in the $x$–$y$ plane, its Fourier
transform is

$f(k_{1},k_{2})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)e^{-i(k_{1}x+k_{2}y)}\,dxdy.$
(3)

We first calculated the power spectrum in the $k$-space. $f^{\dagger}$ is the
conjugation of $f$.

$P(k_{1},k_{2})=f(k_{1},k_{2}){f^{\dagger}(k_{1},k_{2})}.$ (4)

Then the correlation function in the real space, $C_{\rm auto}$ is obtained by
the inverse transform

$C_{\rm
auto}(x,y)=\frac{1}{4\pi^{2}}\int_{a}^{b}\int_{a}^{b}P(k_{1},k_{2})e^{i(k_{1}x+k_{2}y)}\,dk_{1}dk_{2}.$
(5)

The column density map of the CMZ is shown in Figure 1(a). The normalized 2-D
correlation function ( $C_{\rm auto,\rm n}\equiv C_{\rm auto}/C_{\rm auto,\rm
max}$) based on the flux data is shown in Figure 1(b). Note that calculations
done in the Fourier space assumes a periodic boundary condition. To minimize
boundary effects, we add zero paddings around our maps before performing the
calculations. We note that in some cases, a very significant amount of
emission is contained in a very small region (e.g. the Sgr B2 region). To
evaluate their contributions to the overall correlation, we experiment with
performing clipping operations to the data at regions where the flux is larger
than a certain threshold of 3 $\rm Jy/beam$ ($8.31\times 10^{21}{\
\rm{cm^{-2}}}$) and 5 $\rm Jy/beam$ ($1.385\times 10^{22}{\ \rm{cm^{-2}}}$),
separately. This is achieved by setting the values of regions where the flux
is above a threshold to that threshold. We set flux data more than 3, 5 $\rm
Jy/beam$ ($8.31\times 10^{21}{\ \rm{cm^{-2}}}$, $1.385\times 10^{22}{\
\rm{cm^{-2}}}$) to $I_{\rm max}$ = 3, 5 $\rm Jy/beam$ ($8.31\times 10^{21}{\
\rm{cm^{-2}}}$, $1.385\times 10^{22}{\ \rm{cm^{-2}}}$). The regions where the
flux data more than 3, 5 $\rm Jy/beam$ ($8.31\times 10^{21}{\ \rm{cm^{-2}}}$,
$1.385\times 10^{22}{\ \rm{cm^{-2}}}$) are marked yellow in Figure 1(c) and
Figure 1(e), separately. The clipping operation effectively reduces the
dynamic range of the maps. The correlation functions computed from these
clipped maps are presented in Figure 1(d) and Figure 1(f). By comparing the
clipped results ( Figure 1(d) and Figure 1(f) ) to the unclipped ones ( Figure
1(b)), we are able to evaluate the contribution to the correlation function
from regions of different surface densities.

From all these correlation functions, we observe that the contours in the
centre (at small scales) are nearly roundish, but at larger scales, the
contours are elliptical where the long axes of the ellipses are aligned with
the mid-plane of the Milky Way. This is an indication that shear is
dynamically important in the region.

###  3.2 Quantifying the spatial anisotropy

To further quantify the spatial anisotropy measured as a function of the
scale, we divide the region into rings of different radii. Each ring is
characterized by the $l$ parameter, which is the distance from the origin (
Figure 2). For each ring, we plot the value of the correlation function
against $\theta$, which is the angle measured with respect to the Galactic
mid-plane. We have considered all the points. And $\theta$ has been
transformed to the range (0,$\frac{\pi}{2}$) (Figure 2). Due to symmetry, the
$\theta$ is between 0 and $\pi$. Additional, we assume that the structure of
the CMZ is symmetric had we turned it upside down (horizontal mirror
symmetry). Thus, we only need to plot $\theta$ between 0 and $\frac{\pi}{2}$.

As an example, we plot the results from a ring at $l\sim 33.4-35.8$ pc. The
ring width $\Delta l$ is 10 pixels, about 2.4 pc. In Figure 3, we plot the
$C_{\rm auto,\rm n}$ against $\theta$. Here, the spatial anisotropy can be
seen from the fact that the the correlation is stronger along the $l$
direction where $\theta=0$, but becomes weaker as $\theta$ increases.

(g)

(h)

Figure 2: (a) The region is divided with different radii and $l$ is the
distance from the origin. (b) The region is divided with different angles, we
symmetrize it into the range of 0 to $\frac{\pi}{2}$, the colorbar is
$\theta/\pi$.

(a)

(b)

Figure 3: (a) The distribution of $C_{\rm auto,\rm n}$ against $\theta$. (b)
$C_{\rm cumulative,\rm n}$\- $\theta/\pi$ relation. $C_{\rm cumulative,\rm n}$
is the normalized cumulative correlation function. The vertical line shows
where the $C_{\rm cumulative,\rm n}$ is 0.5.

(a)

(b)

Figure 4: (a) $\theta_{\rm half}/\pi$ \- $l$ relation. $\theta_{\rm half}$
decreases with $l$ in the CMZ. (b) Anisotropy-$l$ relation. The left vertical
line indicates where the Jeans lengths $l_{\rm J}$ (6 pc) and the right
vertical line represents the Toomre lengths $l_{\rm T}$ (17 pc) estimated by
Henshaw et al. (2016). The solid line is the anisotropy-$l$ distribution
calculated with raw data. The dashed line is the anisotropy-$l$ distribution
calculated by setting raw data more than 3 $\rm Jy/beam$ ($8.31\times
10^{21}{\ \rm{cm^{-2}}}$) to $I_{\rm max}$ = 3 $\rm Jy/beam$ ($8.31\times
10^{21}{\ \rm{cm^{-2}}}$). The dashed line with stars is the anisotropy-$l$
distribution calculated by setting raw data more than 5 $\rm Jy/beam$
($1.385\times 10^{22}{\ \rm{cm^{-2}}}$) to $I_{\rm max}$ = 5 $\rm Jy/beam$
($1.385\times 10^{22}{\ \rm{cm^{-2}}}$). The straight line at scales larger
than 10 pc is the linear fit of data, ${\rm log}_{10}\;A\approx 1.4\;{\rm
log}_{10}\;l-2.2$.

To quantify the spatial anisotropy at scale $l$, we define the so-called half
correlation angle $\theta_{\rm half}$, which is the critical angle within
which half of the correlation function is contained. To derive $\theta_{\rm
half}$, we define the so-called cumulative correlation function $C_{\rm
cumulative}(\theta)=\int C_{\rm auto,\rm n}(\theta){\rm d}\theta$, then we
normalize it, obtaining the normalized cumulative correlation function $C_{\rm
cumulative,\rm n}(\theta)=C_{\rm cumulative}(\theta)/C_{\rm cumulative,\rm
max}(\theta)$. The $\theta_{\rm half}$ is obtained by solving c The procedure
is illustrated in Figure 3. For isotropic structures, $\theta_{\rm
half}=\frac{\pi}{4}$, whereas for structures that are preferentially aligned
with the disk mid-plane, $\theta_{\rm half}<\frac{\pi}{4}$. An example of how
we derive $\theta_{\rm half}$ is presented in Figure 3.

The above-mentioned exercise allow us to study how the spatial anisotropy
evolve as the function of the scale. In Figure 4, we plot $\theta_{\rm half}$
against the scale, and where $\theta_{\rm half}$ decreases with the increasing
scale. This indicates that the spatial anisotropy is stronger at larger
scales.

We further define the the anisotropy parameter

$A\equiv\frac{\pi}{4\;\theta_{\rm half}}-1\;.$ (6)

$A>0$ means that at the scale of interest, the density structure is
anisotropic, and the value of $A$ measures the degree of anisotropy. For an
homogeneous region, the correlation strength is evenly distributed between 0
and $\frac{\pi}{2}$. $\theta_{\rm half}=\frac{\pi}{4}$, $A$ will be 0. As
$\theta=0$ corresponds to the direction along which the correlation
concentrates, in most cases, $\theta_{\rm half}<\frac{\pi}{4}$ and $A>0$. In
the Figure 4, we plot $A$ against the scale. We also plot regions where we
have chosen different $I_{\rm max}$. We note that at scales below 10 pc, the
anisotropy parameter depends on the value of $I_{\rm max}$. Thus, we should
only interpret results from scales larger than 10 pc, as only in this range do
results from different $I_{\rm max}$ converge. At scales larger than 10 pc, we
perform a fit to our data and find ${\rm log}_{10}\;A\approx 1.4\;{\rm
log}_{10}\;l-2.2$. The anisotropy is strong on the large scale, and it
decreases with decreasing scale. At around $l$ = 10 pc, the density structure
is still moderately anisotropic.

We further add the two vertical lines indicating the Toomre length $l_{\rm
T}\approx 17$ pc and the Jeans length $l_{\rm J}\approx 6$ pc (Henshaw et al.
2016), respectively. The Jeans length is the length scale above which the
gravity can induce collapse, and the Toomre length is the length below which
self-gravity is stronger than shear. The density structure is expected to be
anisotropic at $l>l_{\rm T}$, and this is confirmed by our results. Apart from
this, we can still observe a significant amount of anisotropy at $l_{\rm
T}>l>l_{\rm J}$.

##  4 Conclusions

We study the density structure of gas in the Central Molecular Zone by
applying the 2D correlation function. We find that the density structure is
strongly anisotropic where the correlation is strong along the $l$ direction,
suggesting that shear is dynamically important.

To quantify anisotropic density structure, we define the half-correlation
angle $\theta_{\rm half}$ and the anisotropy parameter
$A\equiv\frac{\pi}{4\theta_{\rm half}}-1\;.$ The density structure is strongly
anisotropic at $l=100\;\rm pc$, and is slightly anisotropic at $l=10\;\rm pc$.
The between 10 pc and 100 pc, we find that

${\rm log}_{10}\;A\approx 1.4\;{\rm log}_{10}\,l-2.2\;.$

We propose a picture where at the large scale ($l$ $>$ 10 pc), shear is
dynamically important such that it can change the density structure of the gas
significantly, and its strength diminishes as one moves to smaller scales. At
($l$ $\lesssim$ 10 pc), as shear should have the roughly same strength as the
self-gravity, thus the gas can enter a state called “shear-enabled pressure
equilibrium” (Li & Zhang 2020). The formalism developed here can be used to
study the role of shear in different regions in a quantitative fashion and
reveal the differences.

###### Acknowledgements.

We thank the anonymous referee for a constructive review report that improved
this paper. Lei Qian is supported by NSFC No. U1631237 and the Youth
Innovation Promotion Association of CAS (id. 2018075). Guang-Xing Li is
supported by a starting grant from Yunnan University, and NSFC grant
W820301904. The paper makes use of data from the ATLASGAL survey carried out
by the APEX telescope.

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