# Detection of a dense group of hyper-compact radio sources in the central
parsec of the Galaxy

Jun-Hui Zhao11affiliationmark:  , Mark R. Morris22affiliationmark:  & W. M.
Goss33affiliationmark:  1Center for Astrophysics — Harvard-Smithsonian, 60
Garden Street, Cambridge, MA 02138, USA<EMAIL_ADDRESS>2Department of
Physics and Astronomy, University of California Los Angeles, Los Angeles, CA
90095  3NRAO, P.O. Box O, Socorro, NM 87801, USA

###### Abstract

Using the JVLA, we explored the Galactic center (GC) with a resolution of
0.05” at 33.0 and 44.6 GHz. We detected 64 hyper-compact radio sources (HCRs)
in the central parsec. The dense group of HCRs can be divided into three
spectral types: 38 steep-spectrum ($\alpha\leq-0.5$) sources; 10 flat-spectrum
($-0.5<\alpha\leq 0.2$) sources; and 17 inverted-spectrum sources having
$\alpha>0.2$, assuming $S\propto\nu^{\alpha}$. The steep-spectrum HCRs are
likely represent a population of massive stellar remnants associated with
nonthermal compact radio sources powered by neutron stars and stellar black
holes. The surface-density distribution of the HCRs as function of radial
distance ($R$) from Sgr A* can be described as a steep power-law
$\Sigma(R)\propto\/R^{-\Gamma}$, with $\Gamma=1.6\pm 0.2$, along with presence
of a localized order-of-magnitude enhancement in the range 0.1-0.3 pc. The
steeper profile of the HCRs relative to that of the central cluster might
result from the concentration massive stellar remnants by mass segregation at
the GC. The GC magnetar SGR J1745-2900 belongs to the inverted-spectrum sub-
sample. We find that two spectral components present in the averaged radio
spectrum of SGR J1745-2900, separated at $\nu\sim 30$ GHz. The centimeter-
component is fitted to a power-law with $\alpha_{cm}=-1.5\pm 0.6$. The
enhanced millimeter-component shows a rising spectrum $\alpha_{mm}=1.1\pm
0.2$. Based on the ALMA observations at 225 GHz, we find that the GC magnetar
is highly variable on a day-to-day time scale, showing variations up to a
factor of 6. Further JVLA and ALMA observations of the variability, spectrum,
and polarization of the HCRs are critical for determining whether they are
associated with stellar remnants. Unified Astronomy Thesaurus concepts: Center
of the Milky Way; [Galactic center (565)]; Interstellar medium (847); Radio
continuum emission (1340); Black holes (162); Pulsars (1306); Magnetars (992);
Neutron stars (1108); Discrete radio sources (389); Radio transient sources
(1358); Radio interferometry (1346)

††slugcomment: Draft: , started May 12, 2021

##  1 Introduction

One of the outstanding open questions that has challenged astronomers for many
years is the “missing pulsar problem”: there are far fewer pulsars found
toward the Galactic center (GC) than we could expect, given the formation rate
of massive stars in the central molecular zone of the Galaxy implied by the
relative abundance of massive stars produced at the GC over the past ten
million years (e.g., Dong, Wang & Morris., 2012; Lu et al., 2013; Clark et
al., 2021). This can in part be ascribed to the large foreground scatter-
broadening at longer radio wavelengths toward the GC, which can lead to a
large enough pulse broadening that the pulses become indistinguishable (Lazio
& Cordes, 1998). Several other reasons also complicate the discovery of GC
pulsars, as detailed by Eatough et al. (2021). However, the discovery of a
magnetar associated with SGR J1745-2900, located just 3” from Sgr A* (Kennea
et al., 2013; Mori et al., 2013; Rea et al., 2013), indicates that the effect
of the scattering screen could be up to three orders of magnitude smaller than
had previously been expected (Spitler et al., 2014; Bower et al., 2014).
Consequently, the question remains: why haven’t more pulsars been seen toward
the Galactic center? Because massive stars clearly form in abundance at the
GC, and because dynamical friction should cause the more massive stellar
remnants to be concentrated there, neutron stars should be abundant and
continuously produced in the GC region ($R\sim 0.5$ pc) (Bahcall & Wolf, 1976;
Morris, 1993; Miralda-Escudé & Gould, 2000; Pfahl & Loeb, 2004; Alexander &
Hopman, 2009; Merritt, 2010; Antonini & Merritt, 2012; Alexander et al.,
2017).

One obvious answer to this question of where the pulsars are is that the
number of “windows” in the scattering screen is quite small, so that most
pulsars are still too scatter-broadened for their pulses to be detected at the
wavelengths searched. Another, perhaps more interesting answer is that massive
stars that form out of the rather highly magnetized interstellar medium of the
GC (Morris, 2014) tend to themselves be rather strongly magnetized, and
therefore leave strongly magnetized neutron star remnants. That is, pulsars
formed near the GC could frequently be magnetars, which have short lifetimes
as recognizable pulsars ($\sim 10^{3}-10^{5}$ yrs) because of their rapid
spin-down rates (Harding et al., 1999; Espinoza et al., 2011; Kaspi &
Beloborodov, 2017). Such short lifetimes would limit the number of pulsars
that could be detected at any one moment to a small number, although they
could remain detectable as compact radio sources. Radio continuum surveys of
point sources can help distinguish these possibilities. We recently published
a 5.5-GHz survey of GC compact radio sources (GCCRs) within a radius of
$\sim$7.5 arcmin (17 pc) of Sgr A* (Zhao, Morris & Goss, 2020), and concluded
that, of the 110 new compact radio sources observed down to a 10-$\sigma$
sensitivity limit of 70 $\mu$Jy, most of them fall within the high flux
density tail of normal pulsars at the GC (our effort to decrease the 5.5-GHz
flux density limit with existing, additional data is in progress). Of course,
there are several other possible assignations for these sources; 82 of them
are variable or transient and 42 have possible X-ray counterparts.

Limited by the VLA angular resolution and confusion from the HII continuum
emission from Sgr A West, the 5.5-GHz survey focussed primarily on regions
lying beyond a radius of R $\sim$ 1 pc from Sgr A*, that is, on regions
outside the circumnuclear disk. To take the next step in addressing the pulsar
puzzle, we have recently surveyed the central $\pm$0.5 parsecs ($\pm$13”)
around Sgr A* at higher frequencies, using existing JVLA Ka and Q-band data
and X-band observations in the A-array. The high-resolution JVLA observations
at 33 and 44.6 GHz were used to search for hyper-compact (¡ 0.1”) radio
sources (HCRs) in Sgr A West, and to study their radio properties and
distribution near Sgr A*. The motivation for going to higher frequencies in
the context of constraining the magnetar population comes from the discovery
by Torne et al. (2017) that the spectrum of the magnetar near Sgr A*, SGR
J1745-2900, rises at higher frequencies to a millimeter/submillimeter plateau.
Another magnetar, 1E 1547.0$-$5408, has also been seen to display a spectrum
rising at millimeter wavelengths (Chu et al., 2021). If such a rising spectrum
happens to be a general characteristic of magnetars, then this feature can be
used to identify magnetar candidates with higher frequency observations, even
if radio pulses are not detectable.

This paper is organized as follows: Section 2 describes the JVLA observations,
data reduction and imaging procedures used for identifying HCRs within the
central parsec. We also describe there our procedure for data reduction and
imaging using archival ALMA data for measurements of SGR J1745-2900. Section 3
presents a catalog of the HCRs found within Sgr A West. Three selected cases
of HCRs are described in Section 4, including detailed results on the radio
spectrum and variability of the GC magnetar SGR J1745-2900, based on data from
this paper and from prior publications. Section 5 shows a statistical analysis
of HCRs by dividing them into three spectral types. Possible origins of the
HCRs as well as massive stellar remnants as a consequence of the mass
segregation in the central parsec are also discussed in section 5. Finally,
section 6 summarizes our conclusions.

Figure 1: Ka-band image of the central 0.8 parsec of the Galaxy. Sgr A* is
indicated as a white filled circle, and all 64 HCRs that we have identified in
this region are marked with green crosses. This 33-GHz image was made with
baselines from 500 to 4,000 k$\lambda$, giving a FWHM beam of
0.08$\times$0.03” and an rms noise of 8 $\mu$Jy beam-1. The color wedge scales
intensity in units of Jy beam-1. In addition to the marked HCRs, some extended
features corresponding to fine-scale structures in the Northern Arm, Eastern
Arm, and the “mini-cavity” of the Sgr A West HII region, which are present in
this image, are also labelled. The position of Sgr A* is registered at
RA(J2000)=17:45:40.0409, Dec(J2000)= $-$29:00:28.118 (Reid & Brunthaler, 2014)
in the equatorial coordinate system.  Table 1: Log of JVLA datasets and images

|  UV data |  |  Images
---|---|---|---
Project ID | Array | Band | $\nu$ | $\Delta\nu$ | $\Delta t$ | HA range |  Epoch | Weight | ($\theta_{\rm maj}$,$\theta_{\rm min}$, ${\rm PA}$) | rms
|  |  | (GHz) | (GHz) |  | (sec) | (day) | (R) | (arcsec, arcsec, deg) |  ($\mu$Jy beam-1)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
15A-293 | A |  Q† | 44.6 | 8 | 3 |  $-0^{h}.6$ — $+2^{h}.5$ | 2015-09-16 | 0 | 0.078, 0.032, 12 | 17
… | … |  Ka† | 33.0 | 8 | 2 |  $-2^{h}.8$ — $+0^{h}.3$ | 2015-09-11 |  $-$0.3 |  0.079, 0.031, $-11$ | 8
14A-346 | A |  X‡ | 9.0 | 2 | 2 |  $-3^{h}.4$ — $+3^{h}.3$ | 2014-04-17 | 0 |  0.36, 0.15, $-6$ | 4
19B-289 | A |  X‡ | 9.0 | 2 | 2 |  $+0^{h}.6$ — $+2^{h}.6$ | 2019-09-21 | 0 |  0.36, 0.15, $-6$ | 7
20B-203 | A |  X‡ | 9.0 | 2 | 2 |  $-0^{h}.3$ — $+2^{h}.9$ | 2020-11-20 | 0 |  0.36, 0.15, $-6$ | 5
… | … | … | … | … | … |  $-1^{h}.5$ — $+1^{h}.7$ | 2020-12-04 | … | … | …

(1) JVLA program code of PI: Mark Morris for 19B-289 and 20B-203; PI: Farhad
Yusef-Zadeh for 14A-346 and 15A-293. (2) Array configurations. (3) JVLA band
code; ”X”,”Ka”, and ”Q” stand for the VLA bands in the ranges of $8.0-12.0$
GHz, $26.5-40.0$ GHz and $40.0-50.0$ GHz
(https://science.nrao.edu/facilities/vla/docs/manuals/oss2013B/performance/bands).
(4) observing frequencies at the observing band center. (5) bandwidth. (6)
integration time. (7) hour angle (HA) range for the data. (8) date
corresponding to the image epoch. (9) robustness weight parameter. (10) FWHM
of the synthesized beam. (11) rms noise of the image. †Correlator setup: 64
channels in each of 64 subbands with channel width of 2 MHz ‡Correlator setup:
64 channels in each of 16 subbands with channel width of 2 MHz.
---

##  2 Observations, data & imaging

Following the 5.5 GHz JVLA A-array survey of compact radio sources within a
radius of 7.5 arcmin (17 pc) in the Galactic center region, we have carried
out a search for compact radio sources within a radius of 13” (0.5 pc) based
on the existing JVLA high-resolution data at 44.6 GHz (Q-band) and 33.0 GHz
(Ka-band) as well as our recent JVLA A-array observations at 9 GHz. The high-
resolution and sensitive VLA observations at high radio frequencies are
crutial in detections of HCRs at a level of 100 $\mu$Jy to a few mJy in the
vicinity of Sgr A*.

###  2.1 JVLA data, calibration & imaging

New JVLA observations in the A configuration were carried out on 2019-9-21,
2020-11-20 and 2020-12-4 at 9 GHz, in addition to the observation on
2014-04-17 that was used to image the Sgr A West filament (Morris et al.,
2017). The X-band observations were all carried out with the VLA standard
correlator setup for wideband continuum covering 2 GHz bandwidth, produced
from the 8-bit sampler. We also acquired the higher-resolution NRAO archival
data observed with the JVLA in the A-array at 33 and 44.6 GHz on 2015-9-16 and
2015-9-11, respectively. The Q- and Ka-band observations were using the 3-bit
sampler, producing 64 subbands and covering a total of 8 GHz bandwidth. All
the observations were pointed at a sky position111RA(J2000) = 17:45:40.0383,
Dec(J2000) = $-$29:00:28.069, very near Sgr A*. Table 1 summarizes the six
sets of uv data (columns 1 - 8). The data reduction was carried out using the
CASA222http://casa.nrao.edu software package of the NRAO. The standard
calibration procedure for JVLA continuum data was applied. 3C 286 (J1331+3030)
was used for corrections for delay, bandpass and flux-density scale.

J1733-1304 (NRAO 530) and J1744-3116 were used for complex gain calibrations.
In addition, corrections for the time variation of the bandpass across each
baseband due to residual delays were determined using NRAO 530 based on the
model discussed in Zhao, Morris & Goss (2019) . The accuracy of the flux-
density scale at the JVLA is 3%$-$5%, limited by the uncertainty of the flux
density of the primary calibrator, Cygnus A (Perley & Butler, 2017). Following
the procedure for high dynamic range (DR) imaging that we developed recently
(Zhao, Morris & Goss, 2019) and applying it to the Sgr A data with CASA, we
further corrected for the residual errors in phase. The integration time of
the calibrated Q and Ka-band data was averaged into 30-sec time bins, so that
the intensity loss at the 13” outer radius of the search region is less than
10% due to the smearing effect caused by the Earth’s spin. After correcting
for the residual delay, we also binned the spectral channel data to channel
widths of 32 and 16 MHz for the Q- and Ka-band data, respectively, to ensure
that the intensity loss for a point source caused by bandwidth smearing is
less than 10% at the radius of 13”. The short baseline data ($<500{\rm
k}\lambda$) were filtered out in order to avoid contamination by the extended
emission in Sgr A West on scales $>0.4$”. The Ka-band image shown in Fig. 1 is
made with CASA task $CLEAN$ from the baseline data between 500-4,000
k$\lambda$, achieving an rms of 8 $\mu$Jy beam-1 with a FWHM beam of
0.079”$\times$0.031 ($-11^{\rm o}$). This high-resolution image shows numerous
compact radio sources in the central parsec region.

For the X-band data, after corrections for the residual errors, the visibility
data were averaged to a time bin of 16 sec while the original channel width of
2 MHz was kept. We made images at 9 GHz using the multi-frequency synthesis
algorithm (MFS: Rau & Cornwell, 2011) with the 1024 spectral channels covering
the 2 GHz bandwidth. Also we filtered out the short baselines ($<100$
k$\lambda$), and constructed two 9-GHz images for both the 2019 and 2020
epochs. The second epoch image was made with the two observations on
2020-11-20 and 2020-12-04. Hereafter, we use the mean epoch, 2020-11-27, for
this image. The rms noise for the 2020-11-27 X-band image is 5 $\mu$Jy beam-1.
We reconstructed the 2014-04-17 image with the calibrated X-band A-array data
(Morris et al., 2017) by using only the long-baseline data ($>100$
k$\lambda$). The rms noise for the resulting 2014-04-17 X-band image is 4
$\mu$Jy beam-1.

The specified parameters for the final images at Q-, Ka- and X-bands are
summarized in Table 1 from columns 9 to 11.

###  2.2 ALMA data, calibration & imaging

We acquired archival data from the Atacama Large Millimeter Array (ALMA),
observed by Tsuboi et al. (2019) at 225.75 GHz in Cycle 5 (2017.1.00503.S).
Following the ALMA CASA guide for Cycle 5 data reduction, we executed the
pipeline script “scriptForPI.py” to produce calibrated ALMA data in CASA
Measurement Set format. The ALMA datasets are composed of ten 1-hr
observations of IRS 13E in the array configuration of C43-10 within a two-week
interval between 2017-10-6 and 2017-10-20. The observing field covers Sgr A*
and the magnetar SGR J1745-2900 in addition to IRS 13E. The pipelined ALMA
images appear to be marred by severe residual errors. Both IRS 13E and SGRA
J1745-2900 were buried in the sidelobes and artifacts produced by the residual
dirty beam. Then, following the recipe for dynamic range imaging with wideband
data (Zhao, Morris & Goss, 2019), we corrected the residual errors by
utilizing the antenna-based closure relations (Thompson, Moran & Swenson,
2017) and constructed a refined ALMA image from the data observed at the first
epoch on 2017-10-6 as a trial case using CASA task $tCLEAN$ with the MFS
algorithm and robust weighting with Briggs parameter R=0 (Briggs, 1995). An
rms noise of $\sigma=20\leavevmode\nobreak\ \mu$Jy beam-1 was achieved, with a
FWHM beam of 0.024”$\times$0.017” (82o). The magnetar SGR J1745-2900 was
successfully detected with a S/N ratio of 70. We then made 2D-Gaussian fits to
both Sgr A* and the magnetar, finding flux densities of 3.031$\pm$0.013 Jy and
1.41$\pm$0.08 mJy at 225 GHz for Sgr A* and the magnetar, respectively. Fig.
2a shows the ALMA continuum image of the magnetar at 225.75 GHz. Based on the
procedure and input parameters for the CASA tasks that were used for the trial
case, we coded the detailed CASA reduction steps for the ALMA data into a
CASA-Python script. Using this script for corrections of residual errors, we
further processed the datasets from all ten epochs observed on 2017-10-6,
2017-10-7, 2017-10-9, 2017-10-10, 2017-10-11, 2017-10-12, 2017-10-14,
2017-10-17, 2017-10-18, and 2017-10-20. The flux densities for the magnetar
were determined by fitting a 2D-Gaussian and are tabulated in Table 3 (see
Section 4).

Figure 2: (a) Contour image of the magnetar SGR J1745-2900 observed on
2017-10-6 using ALMA at 225.75 GHz with a FWHM beam of 0.024”$\times$0.017”
(PA 82o). Contours are $\sigma_{\rm im}\times$($-$5.7, 5.7, 8, 11.3, 16, 22.6,
32, 45.3 ), where $\sigma_{\rm im}=22\mu$Jy beam-1. (b) ALMA image of the
magnetar SGR J1745-2900 (top-right: a point source) and IRS 21 (bottom-left: a
complex) observed on 2019-10-14 at 318 GHz with a FWHM beam of
0.087”$\times$0.059” (89.5o). Contours are $\sigma_{\rm im}\times$($-$6, 6, 8,
11, 15), where $\sigma_{\rm im}=55\mu$Jy beam-1. The FWHM beams are
illustrated at the bottom-left corner in each image. Numbers on the grey scale
bar are in units of $\mu$Jy beam-1.

We also re-processed ALMA archival data (2015.1.01080.S) for observations at
343.49 GHz by Tsuboi et al. (2017). The observations were carried out at four
epochs 2016-4-23, 2016-8-30, 2016-8-31 and 2016-9-8, for durations of 3h, 1h,
2h and 3h, respectively. The 2016-4-23 observation was in the C36-2/3 array
configuration, and other three observations were in the C40-6 configuration.
The pipeline-calibrated datasets were obtained by executing the pipeline
script “scriptForPI.py”. We then adjusted the input parameters in the CASA-
Python script used for imaging the 225-GHz data, and applied the script to the
343-GHz data. The spatial resolution of the first 343-GHz image from epoch
2016-4-23 is relatively poor, with FWHM beam of 0.35”$\times$0.33” ($-79^{\rm
o}$), and the emission from the magnetar SGR J1745-2900 appears to be
contaminated by surrounding extended emission. The final image of SGR
J1745-2900 was made by applying a high-pass baseline filter, so that only
long-baseline data were included (${\rm>100\leavevmode\nobreak\ k\lambda}$),
thus filtering out extended ($>2$”) emission features. An rms noise of 13 mJy
beam-1 was achieved for the epoch 2016-4-23, and the flux density of
3.29$\pm$0.34 mJy was determined for the magnetar at 343 GHz. The observations
from the later epochs at 343 GHz have a typical angular resolution of 0.1”.
The rms noise values were 0.15, 0.05 and 0.12 mJy beam-1 for the 2017-8-30,
2017-8-31 and 2017-9-8 images, respectively. The magnetar SGR J1745-2900 is
detected in all four epochs at 343-GHz, and the flux densities are determined
at a level of 10$\sigma$ or better using 2D-Gaussian fitting. The measurements
are reported in Table 3, including a 5% error in the flux-density calibration
(Bonato et al., 2018).

A high-resolution (0.087”$\times$0.059”, PA = 89.5o) ALMA observation was
carried out at 320 GHz in the C43-6 array configuration on 2019-10-14
(2018.A.00052.S of P.I. Mark Morris). The data was initially processed with
the pipeline script “scriptForPI.py”. Subsequently, we made further
corrections for the residual errors with the CASA-Python script described
above and then imaged the region containing the GC magnetar, Sgr A*, and IRS
21 with the two lower-frequency sub-bands centered at 318 GHz, which have
relatively stable phase and less contaminations from molecular lines in the
circumnuclear disk (CND). An rms noise of 55 $\mu$Jy beam-1 was achieved. The
magnetar is significantly detected with a flux density of 1.32$\pm$0.15 mJy.
The uncertainties include a 5% error in the determination of the flux-density
scale for ALMA observations. Fig. 2b shows the 318-GHz image of the region
containing SGR J1745-2900 and IRS 21.

Figure 3: Images of individual HCRs, color for 44.6 GHz intensity and contours
for 33 GHz from an image made with a long baseline filter of
$>500\leavevmode\nobreak\ {\rm k}\lambda$ except for HCR39 which was made with
a longer baseline filter $>1000\leavevmode\nobreak\ {\rm k}\lambda$ to
separate the point-like source from the contamination of the extended emission
associated with the radio core IRS 13E. The IR stars (E2, E3, and E4) detected
in the IRS 13E region (Zhu et al., 2020) are marked with white dots (see HCR38
and HCR39 panels). The color wedges scale the intensity of 44.6-GHz emission
in units of Jy beam-1. The contours are $\sigma_{\rm im}\times 2^{\rm n/2}$
where n starts from $\rm n_{1}$ until reaching the source peaks. The values of
$\sigma_{\rm im}$ and $\rm n_{1}$ corresponding to the individual HCR image
panels are listed in Column 11 of Table 2. Bottom-right corners are the FWHM
beams at 44.6 and 33 GHz: 0.08”$\times$0.03” (PA = 12o) in blue and
0.08”$\times$0.03” (PA = -11o) in white, respectively.

##  3 Hypercompact radio sources

###  3.1 Search procedure and selection criteria

The angular resolutions of the A-array observations at Ka- and Q-bands, 0.05”,
are nearly two orders of magnitude smaller in beam area than the 5.5-GHz beam
used in our previous study of the GCCRs (Zhao, Morris & Goss, 2020). At such a
high spatial resolution, most of the emission components in the Sgr A West HII
region have been resolved out. The emission from the HII sources produced
overwhelming confusion at 5.5 GHz, which was the main issue preventing us from
unambiguously detecting GCCRs in Sgr A West, given the relatively low angular
resolution of 0.5”. The Ka-band JVLA, with both its superior angular
resolution and sensitivity ($\sim 8\leavevmode\nobreak\ \mu$Jy beam-1), is
well suited for significant detection of point-like hyper-compact radio
sources (HCRs) at a level of $>$0.1 mJy. Thus, we have an unprecedented
opportunity for study of the compact radio sources associated with massive
stellar remnants, such as pulsars, magnetars and accreting compact stellar
remnants. Unlike the free-free emission of the HII components, the radio
emission from such objects is expected to be nonthermal. Our search has
therefore been focused on using the Ka- and Q-band images to identify
nonthermal HCRs within the Galaxy’s central parsec. We proceeded in three
steps as follows:

(1) We initially found approximately a thousand compact radio sources having
angular size of $\lesssim 0.5$” based on a Ka-band image constructed including
all baselines with a robust weighting parameter, R=0.25 (Briggs, 1995), and a
FWHM beam of 0.12”$\times$0.06” (PA = $-11^{\rm o}$). A sensitivity threshold
of S${}_{\rm 33\/GHz}$/$\sigma_{\rm rms}$ ¿ 6, $\sigma_{\rm rms}\approx
7\leavevmode\nobreak\ \mu$Jy beam-1, was applied in the initial search. Bright
intensity maxima in the HII emission components might have been included in
the compact-source sample, producing false detections for the nonthermal
compact radio sources. To ensure that the ultimate sample contains only
compact nonthermal radio sources, two further steps were carried out:

(2) The Ka-band image was reconstructed with a robust weight R=$-$0.3 to down-
weight the contribution from short baselines and also a high-pass baseline
filter ($>500$ k$\lambda$) was applied. So we can separate point-like sources
from extended emission. The final cleaned image was convolved with a beam
having FWHM = 0.08”$\times$0.03” (PA = $-11^{\rm o}$), similar to the
synthesized beam of the Q-band observations, but keeping the position angle of
the Ka-band synthesized beam determined by the uv-data sampling. We narrowed
the list for those sources that are detected at Ka-band by keeping only those
sources above a S33/$\sigma_{\rm rms}=15$ threshold and limiting the size of
the major axis to be less than 0.1 arcsec ($\theta_{\rm maj}<0.1$”) based on a
2D Gaussian fitting. The number of candidates in the list was thereby reduced
to $\sim$100\. The remaining sub-sample only contains the hyper-compact
members of the initial sample.

(3) Finally, we used the high-resolution Q-band image, with $\sigma_{\rm
rms}=17\leavevmode\nobreak\ \mu$Jy beam-1, FWHM = 0.08”$\times$0.03” (PA =
12o), to narrow down the list of HCRs by imposing a conservative limit of
$S_{44.6}/\sigma_{\rm im}>10$ on the detection significance and an upper limit
on the positional offset between Ka- and Q-band positions less than $3\sigma$,
where $\sigma$ is given in Table 2 with a typical value of a few milli-arcsec
(mas), based on 2D-Gaussian fits at the two frequencies. We note that the
locations of HII peaks depend on the uv-sampling. The uv data in the Q-band
observations were primarily sampled in a positive hour-angle range while the
Ka-band data were sampled in a negative hour-angle range. Consequently, the
sources having a significant offset between the Q- and Ka-band images are
suspected to be HII peaks and are therefore rejected from the HCR sample.

The equal beam sizes of the Q- and Ka-band images allow us to determine
reliable spectral indices that will be used to further distinguish the source
types. We do not use spectral indices as a selection parameter for the HCRs,
given a wide range of values for the spectral indices covered by the radio
sources associated with pulsars, magnetars and stellar-mass black holes. We
note that the primary 15$\sigma$ significance cutoff used for Ka-band sources
and the lower 10$\sigma$ cutoff for Q-band were chosen for the derivation of
significant spectral indices, given the poorer sensitivity of the Q-band data.
Neglecting fitting errors, for a weak point-like source, we expect a maximum
uncertainty of $\sigma_{\alpha}\sim 0.4$ in spectral index.

Those candidates having consistent results from the Gaussian fitting to the
Ka- and Q-band images were included in the final sample of HCRs, consisting of
64 members. We note that the conservative search criteria for HCRs may miss
highly variable and relatively weaker sources having a steep spectrum, given
that the sensitivity of Q-band observations is a factor of 2 poorer than that
of Ka-band data. For example, a hyper-compact radio source in the IRS 7SW
region (see Fig. 1) was rejected from the HCR sample because its Ka-band flux
density below the 15$\sigma$ threshold.

In short, from the JVLA A-array images observed at 33 and 44.6 GHz on
2021-9-11 and 2021-9-11, we have identified 64 hyper-compact radio sources
(HCRs) located inside Sgr A West within a radius of $13"$ from Sgr A* based on
their compactness (a size of $\theta_{\rm maj}<0.1$”) at a conservative
significance level of $S/\sigma>15$ at 33 GHz and $S/\sigma>10$ at 44.6 GHz.
Three exceptions which were not detected at 44.6 GHz on 2015-9-16 are also
included: HCR22, HCR32 and HCR64. HCR22 appears to be a strong candidate for
the object that powers the X-ray PWN G359.95-0.04 (see Sec.4.1). HCR64 is a
compact transient radio source associated with the microquasar discovered
during the 2005 outburst of the X-ray transient XJ174540.0290031 (Muno et al.,
2005; Porquet et al., 2005; Bower et al., 2005; Zhao et al., 2009); further
discussion of this source is given in Sec.4.2. HCR32 appears to be a highly
variable source. It was detected with a flux density of 0.5$\pm$0.1 mJy at 9
GHz at the epoch of 2014-4-17 but no significant detections were made in the
2019 and 2020 epochs at 9 GHz.

One of the sources, HCR49, is associated with the high-velocity head-tail
radio/IR source designated the ”Bullet” (Yusef-Zadeh et al., 1998; Zhao &
Goss, 1998; Zhao et al., 2009); it now appears to consist of at least two
compact components.

###  3.2 Images of hyper-compact radio sources

Fig. 3 shows the images of all 64 HCRs in individual panels labelled with the
HCR identification numbers, the colored images represent the 44.6-GHz HCRs and
the superimposed contours are for the 33-GHz images made with baselines
exceeding $500{\rm k\lambda}$. For those panels containing more than one HCRs,
each of the HCRs is labelled with its HCR ID. The green plus symbols mark the
positions determined from the 44.6-GHz data. For HCR22, HCR32 and HCR64, which
were not detected at 44.6 GHz, the 33-GHz positions are shown.

###  3.3 A catalog for hyper-compact radio sources

Table 2 lists 64 HCRs. Column 1 gives the ID numbers for the members of the
HCR catalog. Columns 2 and 3 are the J2000 coordinates, omitting the common
part of 17h45m in Right Ascension (RA) and $-29^{\rm o}$00’ in Declination
(Dec). With the exception of HCR22, HCR32, and HCR64, the Q-band coordinates
of the HCRs were determined with respect to Sgr A*’s equitorial coordinates at
the J2000 epoch (Reid & Brunthaler, 2014). The positions of non-Q-band sources
are determined using the Ka-band data. The 1$\sigma$ uncertainties in the last
digits of RA and Dec. are given in parentheses. We note that, throughout this
paper, the positions of the HCRs in the figures are labelled in the J2000
equatorial coordinate system with respect to the position of Sgr A*. Columns 4
and 5 list the flux densities at 44.6 and 33 GHz with 1$\sigma$ uncertainties.
Columns 6, 7 and 8 show the source sizes, deconvolved from the telescope beam,
including major ($\theta_{maj}$) and minor ($\theta_{min}$) axes, as well as
the position angle (PA), all with their 1$\sigma$ uncertainties. Column 9
lists the spectral index, $\alpha$ ($S\sim\nu^{\alpha}$), determined from the
flux densities at 44.6 and 33 GHz. The parameters used to plot the contours in
Fig. 3 for individual HCRs are listed in Column 10: $\sigma_{\rm im}$
representing the rms variations of the background regions near the sources and
${\rm n_{1}}$ being the integer corresponding to the multiplicative factor,
$2^{0.5n}$, specifying the lowest contour. Column 11 gives a brief note on
individual HCRs.

Finally, we examined the HCRs that can be spatially associated with the 7mm-
IR(3.8$\mu$m) sources (Yusef-Zadeh et al., 2014) by transfering their
coordinates to the J2000 equatorial coordinate system used in this paper, in
which Sgr A* is located at RA(J2000) = 17:45:40.0409, Dec(J2000) =
$-$29:00:28.118 (Reid & Brunthaler, 2014). After corrections for the system
errors (${\rm\Delta RA=0.0026^{\rm s}}$, ${\rm\Delta Dec=-0.049}$”) that
mainly caused by the difference of the position of Sgr A* between Reid &
Brunthaler (2014) and Yusef-Zadeh et al. (2014), 22 HCRs are identified with
possible 7mm-IR sources within an uncertainty of 25 mas, the typical
positional accuracy given for the 7mm-IR sources. The ID numbers of the
possible 7mm-IR counterparts are given in Column 11 as well.

Figure 4:  (a) The Ka-band image made with the JVLA A-array data using all
available baselines, showing the radio structure of HCR22, a candidate source
of particles and energy powering the PWN. The contours are 5 $\mu$Jy
beam${}^{-1}\times$ ($-4$, 4, 5.7, 8, 11.3, 16) with FWMH =
0.11”$\times$0.04”, $-11^{\rm o}$ (bottom-right). (b) JVLA 9-GHz images at
epochs 2014-4-17 (color) and 2020-11-27 (contours). The contours are 50
$\mu$Jy beam${}^{-1}\times$ (64, 80, 96, 112, 128) with FWMH =
0.36”$\times$0.15”, $-6^{\rm o}$ (bottom-right). The green plus marks the peak
position at 33 GHz. The three black ellipses mark the peak positions of the
counterpart at 9 GHz at the three observing epochs as labelled. The size of
each ellipse scales the 1 $\sigma$ uncertainty of the peak position determined
from a 2D-Gaussian fitting. The color wedge scales the intensity in units of
Jy beam-1 for the JVLA images. (c) The Chandra X-ray image (angular resolution
= 0.5”, from Zhu et al., 2018) of the PWN G359.95-0.04 (Wang et al., 2006).
The green box at the tip of the PWN marks the region of HCR22 as imaged at 9
GHz in panel b.

##  4 Typical cases for hyper-compact radio sources

###  4.1 HCR22 and the X-ray PWN G359.95-0.04

HCR22 is located at the northern termination of the extended X-ray source
G359.95-0.04 (Baganoff et al., 2003) that was suggested to be a pulsar wind
nebula (PWN) (Wang et al., 2006; Muno et al., 2008). The source at 33 GHz can
be characterized by a head (HCR22) with a tail extending $\sim 0.3$” toward
the south (Fig. 4a). The radio morphology is consistent with the X-ray
structure, including the immediate angle of the tail, but with a much smaller
scale. The radio emssion was also detected at 9 GHz in the three epoch images
2014-4-17 (Morris et al., 2017), 2019-9-21 and 2020-11-27 (this paper) with a
larger FWHM of 0.36”$\times$0.15” ($-6^{\rm o}$). At 9 GHz, the source shows
no signficant variation in flux density, with measured flux densities of
1.70$\pm$0.09 mJy, 1.53$\pm$0.06 mJy and 1.69$\pm$0.05 mJy at the three
epochs, respectively. A spectral index of $-1.0\pm 0.1$ is determined by the
flux density of 0.43$\pm$0.03 mJy, integrated over the entire source, at 33
GHz and the mean flux density of 1.64$\pm$0.06 mJy averaged by the three
epochs’ data at 9 GHz. Assuming $\alpha_{33/44.6}\sim-1$, a peak intensity of
$<60\leavevmode\nobreak\ \mu$Jy beam-1 at 44.6 GHz is extrapolated from the 33
GHz image (see Fig. 4). The non-detection of HCR22 at Q-band is consistent
with its steep spectrum. In addition, we noticed that the peak position of the
source at 9 GHz moved towards north as time increases. From a least-squares
fitting of the source position at the three epochs, we find a significant
proper motion of the compact source at 9 GHz in the declination direction:
giving $\mu_{\alpha}=-2.0\pm 1.0$ mas y-1 and $\mu_{\delta}=7.3\pm 1.0$ mas
y-1 (Fig. 4a). For a distance of 8 kpc to the Galactic center, this proper
motion corresponds to a velocity of $270\pm 40$ km s-1 projected onto the
plane of the sky, which is consistent with the source’s projected proximity to
Sgr A*. The location, steep spectrum, head-tail structure and orientation of
HCR22 at 33 GHz along with the significant northward motion of the 9-GHz peak,
suggest that HCR22 is plausibly the candidate source of energetic particles
that are responsible for the X-ray emission of PWN G359.95-0.04.

###  4.2 Microquasar of X-ray transient CXOGC J174540.0-290031

HCR64 is the radio counterpart of the microquasar associated with the X-ray
transient CXOGC J174540.0-290031 that was discovered by the Chandra X-ray
observatory (Muno et al., 2005; Porquet et al., 2005). The radio emission from
this transient was found with the VLA (Bower et al., 2005; Zhao et al., 2009).
Fig. 5 shows the 2015 Ka-band image (contours) overlaid on the color K-band
image of 2005. A double source was detected in the 2005 VLA observations at
22.5 GHz (Zhao et al., 2009) with a SW compact component (K40) associated with
the core at the flux density of 3.4$\pm$0.1 mJy. The position offset between
HCR64 (2015) and the SW component, K40, observed in 2005 appears to be
insignificant, so we follow Zhao et al. (2009) in identifying this as the
core. The bright component 0.5” NE of the core in the 2005 image is not
detected in 2015. The NE component was 3.5$\pm$0.1 mJy during the 2005
outburst, which appeared to be launched from the microquasar (Muno et al.,
2005; Porquet et al., 2005; Bower et al., 2005; Zhao et al., 2009). The core
(K40) was detected with a flux density of 1.1$\pm 0.1$ mJy at 22.5 GHz at the
early epochs in 1991 and 1999 (Zhao et al., 2009), representing quiescent
level of the microquasar.

In our Ka-band image, which was observed 10 years after the 2005 outburst, the
flux density of this compact radio source at 33 GHz was 0.60$\pm$0.07 mJy. The
33-GHz flux density appears to be consistent with that observed at 22.5 GHz
during 1991 and 1995, when the microquasar was apparently in its quiescent
state. From the ratio of the flux densities at 33 and 22.5 GHz, we determine a
spectral index of $\alpha=-1.6\pm 0.4$ assuming that the flux density in the
quiescent state is not variable. The 3-$\sigma$ upper limit of 0.15 mJy at
44.6 GHz is consistent with this derived steep spectrum.

Figure 5: Images of the radio emission from the micro-quasar correspoding to
the X-ray transient CXOGC J174540.0-290031, observed with the JVLA in the
A-array at 33 GHz (contours) and 22.5 GHz (colors) on 2015-9-11 and 2005-9-16,
respectively. At the bottom-right corner, the FWHM beams of 0.11”$\times$0.04”
($-11^{\rm o}$) and 0.2”$\times$0.1” (0o) at 33 and 22.5 GHz are shown with a
white and filled-blue ellipses. Contour values are $\sigma_{\rm im}\times$
(5.7, 8, 11.3, 16, 22.6 and 32), where $\sigma_{\rm im}=7\mu$Jy beam-1. The
two plus symbols mark the positions of K40 at 22.5 GHz in 2005 and of HCR22 at
33 GHz in 2015. Table 2: A catalog of hypercompact radio sources at 44.6 and
33 GHz

ID | RA(J2000) | Dec(J2000) | $\left[S\pm\sigma\right]_{44.6}$ | $\left[S\pm\sigma\right]_{33}$ | $\theta_{maj}\pm\sigma$ | $\theta_{min}\pm\sigma$ |  PA$\pm\sigma$ | $\alpha_{44.6/33}\pm\sigma$ |  $\sigma_{\rm im}$, n1 | Notes
---|---|---|---|---|---|---|---|---|---|---
|  ( s ) | ( ” ) | (mJy) | (mJy) | (mas) | (mas) | (deg) |  |  ($\mu$Jy bm-1) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
HCR01 | 40.6981(3) | 18.323( 6) | $0.95\pm 0.11$ | $1.00\pm 0.16$ | $74.\pm 19.$ | $56.\pm 17.$ | $36.\pm 23.$ | $-0.17\pm 0.65$ | 7.0, 6 |  ID925
HCR02 | 40.5194(4) | 27.445( 7) | $0.20\pm 0.02$ | $0.15\pm 0.02$ | $68.\pm 11.$ | $26.\pm 30.$ | $30.\pm 45.$ | $0.96\pm 0.60$ | 7.0, 5 |
HCR03 | 40.3836(1) | 23.150( 3) | $0.27\pm 0.02$ | $0.20\pm 0.02$ | $57.\pm 14.$ | $40.\pm 8.$ | $150.\pm 36.$ | $1.00\pm 0.48$ | 7.0, 5 |
HCR04 | 40.2641(1) | 33.686( 3) | $0.26\pm 0.02$ | $0.22\pm 0.01$ | $45.\pm 13.$ | $16.\pm 8.$ | $100.\pm 13.$ | $0.55\pm 0.38$ | 7.0, 7 |  IRS 9W21
HCR05 | 40.2679(1) | 29.243( 3) | $0.22\pm 0.02$ | $0.19\pm 0.02$ | unresolved | $0.49\pm 0.52$ | 7.0, 8 |  ID425
HCR06 | 40.2629(1) | 27.217( 1) | $0.80\pm 0.04$ | $0.75\pm 0.04$ | $20.\pm 6.$ | $15.\pm 9.$ | $80.\pm 42.$ | $0.21\pm 0.25$ | 7.0, 5 |  ID225
HCR07 | 40.1708(1) | 29.788( 1) | $2.54\pm 0.13$ | $1.90\pm 0.10$ | $20.\pm 8.$ | $8.\pm 8.$ | $56.\pm 47.$ | $0.96\pm 0.24$ | 7.0, 9 |  Magnetar1
HCR08 | 40.1449(4) | 16.419(13) | $0.23\pm 0.02$ | $0.21\pm 0.02$ | $19.\pm 10.$ | $13.\pm 20.$ | $80.\pm 24.$ | $0.30\pm 0.32$ | 7.0, 6 |
HCR09 | 40.1286(1) | 29.060( 4) | $0.21\pm 0.02$ | $0.27\pm 0.02$ | $49.\pm 23.$ | $18.\pm 11.$ | $84.\pm 30.$ | $-0.83\pm 0.46$ | 7.0, 7 |  ID325
HCR10 | 40.1166(1) | 27.526( 1) | $0.36\pm 0.03$ | $0.33\pm 0.03$ | $15.\pm 16.$ | $4.\pm 17.$ | $22.\pm 44.$ | $0.29\pm 0.36$ | 7.0, 5 |  ID125
HCR11 | 40.1067(6) | 22.277( 9) | $0.11\pm 0.02$ | $0.17\pm 0.01$ | $61.\pm 31.$ | $26.\pm 9.$ | $82.\pm 21.$ | $-1.45\pm 0.68$ | 7.0, 5 |  IRS 7E2
HCR12 | 39.9905(6) | 30.154(12) | $0.19\pm 0.01$ | $0.33\pm 0.03$ | $51.\pm 10.$ | $44.\pm 6.$ | $38.\pm 18.$ | $-1.83\pm 0.42$ | 7.0, 7 |
HCR13 | 39.9683(6) | 29.907(12) | $0.25\pm 0.03$ | $0.40\pm 0.04$ | $95.\pm 18.$ | $34.\pm 6.$ | $94.\pm 28.$ | $-1.56\pm 0.57$ | 8.5, 7 |
HCR14 | 39.9502(1) | 30.636( 3) | $0.31\pm 0.02$ | $0.40\pm 0.04$ | $74.\pm 10.$ | $59.\pm 9.$ | $158.\pm 64.$ | $-0.85\pm 0.36$ | 7.0, 6 |
HCR15 | 39.9380(3) | 31.162( 6) | $0.16\pm 0.02$ | $0.29\pm 0.02$ | $72.\pm 16.$ | $18.\pm 33.$ | $90.\pm 20.$ | $-1.97\pm 0.53$ | 8.0, 6 |  Mini-cavity7
HCR16 | 39.9350(3) | 31.370( 6) | $0.59\pm 0.07$ | $0.82\pm 0.05$ | $81.\pm 17.$ | $62.\pm 18.$ | $134.\pm 32.$ | $-1.09\pm 0.43$ | 8.0, 6 |  Mini-cavity7
HCR17 | 39.9151(3) | 28.563( 7) | $0.52\pm 0.07$ | $0.69\pm 0.11$ | $81.\pm 29.$ | $60.\pm 21.$ | $166.\pm 90$ | $-0.94\pm 0.66$ | 7.0, 6 | $\epsilon^{8}$
HCR18 | 39.9096(1) | 30.895( 4) | $0.17\pm 0.02$ | $0.31\pm 0.03$ | $48.\pm 14.$ | $29.\pm 20.$ | $134.\pm 24.$ | $-1.99\pm 0.50$ | 8.0, 6 |  Mini-cavity7
HCR19 | 39.9013(1) | 30.053( 3) | $0.41\pm 0.03$ | $0.54\pm 0.04$ | $82.\pm 9.$ | $60.\pm 11.$ | $98.\pm 24.$ | $-0.91\pm 0.34$ | 8.0, 6 |  Mini-cavity9, ID1825
HCR20 | 39.8952(6) | 28.230(12) | $0.20\pm 0.03$ | $0.38\pm 0.04$ | $60.\pm 23.$ | $56.\pm 23.$ | $144.\pm 59.$ | $-2.13\pm 0.61$ | 7.0, 6 | $\epsilon^{8}$
HCR21 | 39.8883(3) | 31.088( 7) | $0.32\pm 0.04$ | $0.56\pm 0.03$ | $87.\pm 20.$ | $64.\pm 16.$ | $158.\pm 77.$ | $-1.86\pm 0.49$ | 8.0, 6 |
HCR22 | 39.8408(3) | 19.316( 6) | $<0.06$ | $0.15\pm 0.02$ | $73.\pm 16.$ | $50.\pm 61.$ | $73.\pm 82.$ | N/A | 7.0, 6 |  PWN3
HCR23 | 39.8407(1) | 29.442( 4) | $0.92\pm 0.08$ | $1.00\pm 0.11$ | $84.\pm 12.$ | $44.\pm 30.$ | $14.\pm 8.$ | $-0.28\pm 0.46$ | 8.0, 7 |  Mini-cavity10, ID2025
HCR24 | 39.8345(1) | 29.548( 3) | $0.19\pm 0.01$ | $0.20\pm 0.01$ | unresolved | $-0.17\pm 0.34$ | 8.0, 7 |  Mini-cavity10, ID1925
HCR25 | 39.8272(1) | 29.810( 4) | $0.88\pm 0.08$ | $1.29\pm 0.13$ | $49.\pm 18.$ | $34.\pm 19.$ | $20.\pm 52.$ | $-1.27\pm 0.45$ | 8.0, 7 |  IRS 13E-SE6, ID3225
HCR26 | 39.8247(1) | 29.521( 3) | $0.29\pm 0.02$ | $0.25\pm 0.02$ | unresolved | $0.49\pm 0.42$ | 8.0, 7 |  IRS 13E-NE11, ID2125
HCR27 | 39.8236(3) | 30.039( 7) | $0.21\pm 0.03$ | $0.37\pm 0.04$ | $45.\pm 22.$ | $25.\pm 28.$ | $133.\pm 52.$ | $-1.88\pm 0.64$ | 8.0, 7 |  IRS 13E-SE6
HCR28 | 39.8235(3) | 31.825( 6) | $0.35\pm 0.04$ | $0.45\pm 0.03$ | $73.\pm 16.$ | $36.\pm 8.$ | $20.\pm 16.$ | $-0.83\pm 0.47$ | 7.0, 8 |  Mini-cavity12
HCR29 | 39.8210(1) | 29.311( 4) | $0.30\pm 0.03$ | $0.50\pm 0.03$ | $44.\pm 12.$ | $34.\pm 20.$ | $148.\pm 36.$ | $-1.70\pm 0.41$ | 8.0, 7 |  IRS 13E-NE11, ID2325
HCR30 | 39.8181(1) | 29.475( 4) | $0.20\pm 0.02$ | $0.28\pm 0.02$ | $58.\pm 5.$ | $22.\pm 2.$ | $164.\pm 3.$ | $-1.12\pm 0.47$ | 8.0, 7 |  IRS 13E-NE11, ID2225
HCR31 | 39.8180(1) | 28.902( 6) | $0.63\pm 0.03$ | $0.93\pm 0.06$ | $87.\pm 5.$ | $42.\pm 7.$ | $98.\pm 7.$ | $-1.29\pm 0.26$ | 8.0, 9 |  IRS 13E-NE13
HCR32 | 39.8165(2) | 30.861( 7) | $<0.06$ | $0.42\pm 0.03$ | $40.\pm 5.$ | $11.\pm 22.$ | $72.\pm 4.$ | N/A | 7.0, 7 |  Mini-cavity14
HCR33 | 39.8121(1) | 29.860( 3) | $0.35\pm 0.03$ | $0.58\pm 0.05$ | $48.\pm 14.$ | $16.\pm 10.$ | $14.\pm 22.$ | $-1.68\pm 0.43$ | 8.0, 7 |  IRS 13E-SE6, ID3025
HCR34 | 39.8107(1) | 31.739( 4) | $0.30\pm 0.03$ | $0.45\pm 0.04$ | $93.\pm 14.$ | $45.\pm 6.$ | $8.\pm 12.$ | $-1.35\pm 0.46$ | 7.0, 6 |  Mini-cavity12
HCR35 | 39.8075(1) | 29.300( 3) | $0.42\pm 0.03$ | $0.43\pm 0.05$ | $55.\pm 10.$ | $22.\pm 26.$ | $62.\pm 24.$ | $-0.08\pm 0.42$ | 7.0, 8 |  IRS 13E-NE11, ID2425
HCR36 | 39.8007(3) | 29.020( 6) | $0.31\pm 0.04$ | $0.58\pm 0.04$ | $42.\pm 8.$ | $27.\pm 16.$ | $56.\pm 27.$ | $-2.08\pm 0.50$ | 8.0, 6 |  IRS 13E-N15, ID2825
HCR37 | 39.7967(6) | 30.914( 9) | $0.18\pm 0.04$ | $0.41\pm 0.03$ | $72.\pm 25.$ | $41.\pm 55.$ | $36.\pm 44.$ | $-2.73\pm 0.79$ | 8.0, 7 |  IRS 2-N16
HCR38 | 39.7961(1) | 29.662( 3) | $5.50\pm 0.38$ | $3.35\pm 0.42$ | $78.\pm 18.$ | $42.\pm 18.$ | $52.\pm 21.$ | $1.65\pm 0.47$ | 8.0, 12 |  IRS 13E5, ID3325
HCR39 | 39.7941(3) | 29.844( 9) | $1.27\pm 0.12$ | $1.14\pm 0.09$ | $76.\pm 17.$ | $29.\pm 5.$ | $90.\pm 7.$ | $0.36\pm 0.41$ | 8.0, 6 |  IRS 13E-S17, ID3425
HCR40 | 39.7895(1) | 28.540( 4) | $0.58\pm 0.05$ | $0.69\pm 0.04$ | $94.\pm 9.$ | $54.\pm 3.$ | $91.\pm 5.$ | $-0.58\pm 0.34$ | 8.0, 6 |  Triplet18
HCR41 | 39.7891(3) | 29.508( 7) | $0.90\pm 0.12$ | $1.10\pm 0.07$ | $61.\pm 20.$ | $41.\pm 30.$ | $150.\pm 38.$ | $-0.67\pm 0.49$ | 8.0, 7 |  IRS 13E-W19
HCR42 | 39.7857(3) | 31.239( 6) | $0.67\pm 0.05$ | $1.00\pm 0.06$ | $84.\pm 11.$ | $52.\pm 7.$ | $166.\pm 12.$ | $-1.33\pm 0.32$ | 8.0, 7 | IRS 2
HCR43 | 39.7855(1) | 28.421( 4) | $0.30\pm 0.03$ | $0.48\pm 0.05$ | $68.\pm 16.$ | $46.\pm 11.$ | $161.\pm 30.$ | $-1.56\pm 0.49$ | 8.0, 6 |  Triplet18
HCR44 | 39.7799(1) | 29.587( 4) | $1.23\pm 0.13$ | $1.23\pm 0.10$ | $46.\pm 18.$ | $22.\pm 17.$ | $36.\pm 33.$ | $0.00\pm 0.45$ | 8.0, 7 |  IRS 13E-W19, ID3525
HCR45 | 39.7764(3) | 31.938( 3) | $1.48\pm 0.09$ | $1.15\pm 0.08$ | $96.\pm 10.$ | $40.\pm 4.$ | $1.\pm 5.$ | $0.84\pm 0.31$ | 8.0, 8 |  IRS 2L, ID3625
HCR46 | 39.7721(1) | 29.678( 7) | $0.64\pm 0.06$ | $0.68\pm 0.04$ | $90.\pm 11.$ | $56.\pm 14.$ | $78.\pm 16.$ | $-0.20\pm 0.36$ | 8.0, 7 |  IRS 13E-W19
HCR47 | 39.7721(2) | 31.154( 9) | $0.18\pm 0.03$ | $0.15\pm 0.03$ | $44.\pm 18.$ | $21.\pm 6.$ | $82.\pm 10.$ | $0.61\pm 0.90$ | 7.0, 5 | IRS 2
HCR48 | 39.7699(1) | 29.479( 7) | $0.10\pm 0.02$ | $0.19\pm 0.01$ | $35.\pm 24.$ | $16.\pm 14.$ | $59.\pm 34.$ | $-2.13\pm 0.73$ | 8.0, 7 |  IRS 13E-W19
HCR49 | 39.7697(3) | 25.913( 7) | $0.17\pm 0.02$ | $0.21\pm 0.04$ | $65.\pm 19.$ | $39.\pm 13.$ | $78.\pm 14.$ | $-0.70\pm 0.78$ | 7.0, 7 |  Bullet SE4
… | 39.7671(4) | 25.909( 6) | $0.13\pm 0.02$ | $0.16\pm 0.04$ | $63.\pm 14.$ | $23.\pm 05.$ | $5.\pm 10.$ | $-0.69\pm 1.00$ | … |  Bullet NW4
HCR50 | 39.7677(1) | 29.824( 3) | $0.27\pm 0.02$ | $0.30\pm 0.03$ | $47.\pm 13.$ | $28.\pm 14.$ | $72.\pm 31.$ | $-0.35\pm 0.42$ | 8.0, 7 |  IRS 13E-W19
HCR51 | 39.7672(1) | 31.378( 3) | $0.19\pm 0.01$ | $0.24\pm 0.02$ | unresolved | $-0.78\pm 0.40$ | 7.0, 7 | IRS 2-W
HCR52 | 39.7620(4) | 29.783( 4) | $0.31\pm 0.04$ | $0.47\pm 0.03$ | $24.\pm 1.$ | $16.\pm 1.$ | $60.\pm 3.$ | $-1.38\pm 0.49$ | 8.0, 7 |  IRS 13E-W19
HCR53 | 39.7513(1) | 30.675( 3) | $0.36\pm 0.02$ | $0.30\pm 0.03$ | $80.\pm 11.$ | $14.\pm 41.$ | $20.\pm 3.$ | $0.61\pm 0.34$ | 7.0, 9 |  IRS 13E-SW20
HCR54 | 39.7461(3) | 26.567(10) | $0.11\pm 0.02$ | $0.15\pm 0.02$ | $28.\pm 10.$ | $12.\pm 21.$ | $60.\pm 10.$ | $-1.03\pm 0.78$ | 7.0, 8 |  IRS 34SW22
HCR55 | 39.7439(4) | 31.215( 6) | $0.57\pm 0.08$ | $0.86\pm 0.14$ | $81.\pm 22.$ | $53.\pm 46.$ | $108.\pm 98.$ | $-1.37\pm 0.69$ | 7.0, 7 |
HCR56 | 39.7320(1) | 30.580( 4) | $0.19\pm 0.02$ | $0.36\pm 0.02$ | $58.\pm 7.$ | $38.\pm 6.$ | $55.\pm 10.$ | $-2.12\pm 0.43$ | 7.0, 8 |
HCR57 | 39.7299(1) | 26.595( 3) | $0.33\pm 0.03$ | $0.35\pm 0.03$ | $38.\pm 17.$ | $18.\pm 26.$ | $60.\pm 42.$ | $-0.20\pm 0.48$ | 7.0, 8 |  IRS 34SW22, ID625
HCR58 | 39.6399(1) | 26.651( 4) | $0.19\pm 0.02$ | $0.20\pm 0.02$ | $76.\pm 18.$ | $44.\pm 46.$ | $124.\pm 53.$ | $-0.17\pm 0.54$ | 7.0, 8 |  IRS 6E23
HCR59 | 39.6255(3) | 26.623( 4) | $1.21\pm 0.08$ | $1.40\pm 0.17$ | $94.\pm 22.$ | $68.\pm 7.$ | $132.\pm 35.$ | $-0.48\pm 0.47$ | 8.0, 8 |  IRS 6E23, ID3725
HCR60 | 39.5463(1) | 35.005( 1) | $0.60\pm 0.04$ | $0.50\pm 0.03$ | $31.\pm 12.$ | $16.\pm 50.$ | $90.\pm 23.$ | $0.61\pm 0.30$ | 7.0, 8 |  ID725
HCR61 | 39.4593(1) | 31.736( 4) | $0.34\pm 0.03$ | $0.31\pm 0.03$ | unresolved | $0.31\pm 0.43$ | 7.0, 9 |  ID825
HCR62 | 39.3096(1) | 30.683( 1) | $0.22\pm 0.01$ | $0.18\pm 0.01$ | $30.\pm 5.$ | $9.\pm 2.$ | $10.\pm 2.$ | $0.67\pm 0.33$ | 7.0, 8 |
HCR63 | 39.2771(4) | 27.332(12) | $0.22\pm 0.04$ | $0.27\pm 0.02$ | $51.\pm 33.$ | $36.\pm 34.$ | $42.\pm 48.$ | $-0.68\pm 0.69$ | 7.0, 8 |
HCR64 | 40.0201(9) | 31.178(20) | $<0.15$ | $0.60\pm 0.07$ | $140.\pm 20.$ | $70.\pm 10.$ | $134.\pm 10.$ | N/A | 8.0, 5 |  micro-qso,K4024

1The radio counterpart of SGR J1745-2900 (Eatough et al., 2013), the GC
magnetar, and see Sec.4.3 of this paper. 2Located east of the IRS 7 bow-shock
feature (Yusef-Zadeh & Morris, 1991). 3A candidate cannonball of the PWN
G359.95-0.04 (Wang et al., 2006), and see Sec.4.1 of this paper. 4The head of
the bullet (Zhao et al., 2009), resolved into two components. 5The radio core
associated with IRS 13E. 6Located $\sim$0.5” SE of IRS 13E core (e.g., Tsuboi
et al., 2017). 7Located within the mini-cavity. 8The $\epsilon$ source that
has been resolved into three components (Yusef-Zadeh et al., 1990; Zhao et
al., 1991), S (HCR17), NE and NW (HCR20); the three components correspond to
RS6, RS5 and RS7 detected at 34.5 GHz (Yusef-Zadeh et al., 2016). 9Located in
the northern mini-cavity. 10Located at the northwestern rim of the mini-
cavity. 11Located 0.5” NE of the IRS 13E radio core. 12Located at the
southwestern rim of the mini-cavity. 13Located 1” NE of the IRS 13E radio
core. 14Located at the western rim of the mini-cavity. 15Located 0.7” N of the
IRS 13E radio core. 16Located N of the IRS 2. 17Located 0.2” S of the IRS 13E
radio core. 18A triplet located 1.2” N of the IRS 13E radio core, consisting
of three components S(HCR40), NW(HCR43) and NE with a size larger than the HCR
upper limit. 19Located 0.3” W of the IRS 13E radio core, the HCR41 group
consisting of six members of HCR41, 44, 46, 48, 50 and 52. 20Located 1” SW of
the IRS 13E radio core. 21Located in the IRS 9W region. 22Located in the IRS
34SW region. 23Located in the IRS 6E region. 24Located in the bar, a radio
counterpart of the X-ray transients CXOGC J174540.029003 (Muno et al., 2005;
Bower et al., 2005; Zhao et al., 2009) and see Sec.4.2 of this paper. 25The
7mm-sources are identified with IR stars emitting at 3.8 $\mu$m, among which
HCR05 (ID4), HCR06 (ID2), HCR10 (ID1), HCR57 (ID6), HCR60 (ID7) and HCR61
(ID8) are associated with strong stellar winds (Yusef-Zadeh et al., 2014).
---

###  4.3 The GC magnetar: SGR J1745-2900

HCR07 is one of the most recognizable HCR members, as it is associated with
the GC magnetar and has a high-energy counterpart, SGR J1745-2900. The soft
gamma-ray repeater was discovered by Swift during a large X-ray outburst on
2013 April 24 (MJD 56406), powered by a magnetar close to Sgr A*(Kennea et
al., 2013). The magnetar hypothesis was further supported by NuSTAR detections
of periodic pulsed signal at 3.76 s (Mori et al., 2013). With the observations
by Chandra and Swift, Rea et al. (2013) pinpointed the location of the
magnetar at a projected distance of 2.4$\pm$0.3 arcsec from Sgr A*; and the
authors also determined the source spin period and its derivative with high
precision ($P=$3.7635537(2) s and $\dot{P}=6.61(4)\times 10^{-12}$ s s-1). The
magnetar, SGR J1745-2900, was monitored by the Chandra X-ray observatory for
six years following the X-ray outburst in 2013 April, showing the long-term
properties of the outburst (Rea et al., 2020).

Radio pulses from SGR J1745-2900 were first detected by the Effelsberg radio
telescope and confirmed with other telescopes, including the Nancay telescope,
JVLA, Jodrell Bank Observatory (Eatough et al., 2013), and the Australia
Telescope Compact Array (ATCA) (Shannon & Johnston, 2013) at various
frequencies between 1.5 and 19 GHz. High-frequency pulses were detected at 87,
101, 138, 154, 209 and 225 GHz with the IRAM-30m telescope during the period
between 2014-7-21 (MJD 56859) and 2014-7-24 (MJD 56862) (Torne et al., 2015).
In their follow-up campaign, Torne et al. (2017) detected high-frequency
pulses with the IRAM-30m up to 291 GHz in the interval between 2015-3-4 (MJD
57085) and 2015-3-9 (MJD 57090). The high-frequency pulses were also detected
at 45 GHz with the Green Bank Telescope (GBT) on 2014-4-10 (MJD 56757)
(Gelfand et al., 2017).

####  4.3.1 A collection of flux-densitiy measurements from JVLA and ALMA

In addition to the JVLA and ALMA flux densities determined from the images
reported in this work, we also collected the data from the prior published
literature. Table 3 assembles all the available data from JVLA and ALMA
observations at radio wavelengths. The table is configured into two main
column sections, each of which consists of six sub-columns: (1) observing
date; (2) the corresponding modified Julian day (MJD); (3) band center
frequency in units of GHz; (4) flux densities in units of mJy and (5) the
corresponding 1$\sigma$ uncertainties. For the non-detections, a 3$\sigma$
value is given for the upper limits. Finally, alphabetical codes are
designated for the relevant references that are listed at the bottom notes of
the table. The measurements reported for the first time in this paper are
highlighted with a bold font.

Table 3: JVLA & ALMA flux densities of SGR J1745-2900

Obs. date | MJD |  $\nu_{0}$ (GHz) |  Sν (mJy) |  $\sigma$ (mJy) | Ref. |  | Obs. date | MJD |  $\nu_{0}$ (GHz) |  Sν (mJy) |  $\sigma$ (mJy) | Ref.
---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) |  | (1) | (2) | (3) | (4) | (5) | (6)
2011-08-04 | 55776 | 42 | $<0.24^{\dagger}$ | … | a |  | 2012-10-14 | 56214 | 21.2 | $<0.60^{\dagger}$ | … | a
2012-10-14 | 56214 | 32 | $<0.81^{\dagger}$ | … | a |  | 2012-10-14 | 56214 | 41 | $<0.78^{\dagger}$ | … | a
2012-12-22 | 56283 | 21.2 | $<0.90^{\dagger}$ | … | a |  | 2012-12-22 | 56283 | 32 | $<0.94^{\dagger}$ | … | a
2012-12-22 | 56283 | 41 | $<0.69^{\dagger}$ | … | a |  | 2013-05-10 | 56422 | 9 | 0.56 | 0.011 | g
2013-06-01 | 56444 | 9 | 0.76 | 0.015 | g |  | 2013-06-30 | 56473 | 15 | 0.58 | 0.012 | g
2013-07-13 | 56486 | 9 | 1 .47 | 0.030 | g |  | 2013-10-26 | 56591 | 21.2 | $<1.95^{\dagger}$ | … | a
2013-10-26 | 56591 | 32 | $<1.45^{\dagger}$ | … | a |  | 2013-10-26 | 56591 | 41 | $<0.82^{\dagger}$ | … | a
2013-10-26 | 56591 | 41 | 0.7 | 0.4 | e |  | 2013-11-29 | 56626 | 21.2 | $<2.04^{\dagger}$ | … | a
2013-11-29 | 56626 | 32 | $<0.82^{\dagger}$ | … | a |  | 2013-11-29 | 56626 | 41 | $<0.70^{\dagger}$ | … | a
2013-11-29 | 56626 | 41 | 0.89 | 0.08 | e |  | 2013-12-29 | 56656 | 21.2 | $<1.70^{\dagger}$ | … | a
2013-12-29 | 56656 | 32 | $<0.86^{\dagger}$ | … | a |  | 2013-12-29 | 56656 | 41 | $<1.52^{\dagger}$ | … | a
2013-12-29 | 56656 | 41 | 1.20 | 0.7 | e |  | 2014-01-01 | 56658 | 15 | 2.09 | 0.042 | g
2014-01-01 | 56658 | 9 | 1.18 | 0.024 | g |  | 2014-02-15 | 56703 | 21.2 | 0.84 | 0.33 | a
2014-02-15 | 56703 | 32 | 1.83 | 0.10 | a |  | 2014-02-15 | 56703 | 41 | 1.85 | 0.07 | a
2014-02-15 | 56703 | 41 | 2.1 | 0.4 | e |  | 2014-02-21 | 56709 | 44.6 | 1.62 | 0.04 | a
2014-02-22 | 56710 | 15 | 1.07 | 0.022 | g |  | 2014-02-22 | 56710 | 9 | 0.94 | 0.019 | g
2014-03-09 | 56731 | 34.5 | 1.30 | 0.01 | a |  | 2014-03-22 | 56738 | 21.2 | 2.79 | 0.19 | a
2014-03-22 | 56738 | 32 | 2.64 | 0.05 | a |  | 2014-03-22 | 56738 | 41 | 1.24 | 0.02 | a
2014-03-22 | 56738 | 41 | 2.1 | 0.3 | e |  | 2014-04-03 | 56750 | 23 | 0.92 | 0.019 | g
2014-04-03 | 56750 | 43 | 0.54 | 0.011 | g |  | 2014-04-25 | 56772 | 9 | 1.00 | 0.020 | g
2014-04-25 | 56772 | 15 | 1.22 | 0.025 | g |  | 2014-04-26 | 56743 | 21.2 | 0.90 | 0.14 | a
2014-04-26 | 56743 | 32 | 0.62 | 0.04 | a |  | 2014-04-26 | 56743 | 41 | 1.20 | 0.07 | a
2014-04-26 | 56743 | 41 | 0.91 | 0.30 | e |  | 2014-04-17 | 56764 | 9.0 | 3.50 | 0.08 | b
2014-05-10 | 56787 | 41 | 1.15 | 0.05 | e |  | 2014-05-17 | 56794 | 5.5 | 4.50 | 0.24 | b
2014-05-26 | 56803 | 5.5 | 3.90 | 0.09 | b |  | 2014-05-31 | 56808 | 21.2 | 4.21 | 0.17 | a
2014-05-31 | 56808 | 32 | 2.90 | 0.13 | a |  | 2014-05-31 | 56808 | 41 | 2.94 | 0.12 | a
2014-05-31 | 56808 | 41 | 3.5 | 0.4 | e |  | 2014-08-23 | 56892 | 15 | 0.63 | 0.013 | g
2014-08-30 | 56899 | 23 | 0.26 | 0.006 | g |  | 2014-08-30 | 56899 | 43 | 0.15 | 0.003 | g
2015-02-20 | 57073 | 9.0 | 3.00 | 0.30 | b |  | 2015-09-11 | 57276 | 33.0 | 1.80 | 0.05 | b
2015-09-16 | 57281 | 44.6 | 2.45 | 0.05 | b |  | 2016-04-23 | 57502 | 343 | 2.80 | 0.23 |  b, d
2016-07-12 | 57581 | 44.2 | 5.79 | 0.05 | f |  | 2016-07-15 | 57584 | 226 | 4.70 | 1.29 | f
2016-08-30 | 57630 | 343 | 3.10 | 0.11 |  b, d |  | 2016-08-31 | 57631 | 343 | 3.29 | 0.12 |  b, d
2016-09-08 | 57639 | 343 | 3.90 | 0.13 |  b, d |  | 2017-10-06 | 58032 | 225 | 1.41 | 0.08 |  b, c
2017-10-07 | 58033 | 225 | 1.53 | 0.08 |  b, c |  | 2017-10-09 | 58035 | 225 | 0.79 | 0.07 |  b, c
2017-10-10 | 58036 | 225 | 1.22 | 0.06 |  b, c |  | 2017-10-11 | 58037 | 225 | 1.25 | 0.07 |  b, c
2017-10-12 | 58038 | 225 | 1.44 | 0.07 |  b, c |  | 2017-10-14 | 58040 | 225 | 3.16 | 0.08 |  b, c
2017-10-17 | 58043 | 225 | 5.34 | 0.09 |  b, c |  | 2017-10-18 | 58044 | 225 | 5.42 | 0.11 |  b, c
2017-10-20 | 58046 | 225 | 5.60 | 0.12 |  b, c |  | 2019-09-08 | 58734 | 5.5 | $<0.50^{\dagger}$ | … | b
2019-09-21 | 58747 | 9.0 | 0.40 | 0.10 | b |  | 2019-09-20 | 58748 | 318 | 1.32 | 0.15 | b
2020-11-27 | 59180 | 9.0 | 0.36 | 0.06 | b |  |  |  |  |  |  |

†A 3-$\sigma$ upper limit. Reference: a. Yusef-Zadeh et al. (2015). b. This
paper. c. The ALMA observations at the ten epochs were carried out by the PI
Tsuboi in (Tsuboi et al., 2017). We re-processed the ALMA archival data
(2017.1.00500.S) and determined the flux densities of SGR J1745-2900 at 225.75
GHz. d. Detected in the ALMA image made from observations on 2016-4-23,
2016-8-30/31, and 2016-9-8 (Tsuboi et al., 2017). We re-processed the ALMA
archival data (2015.1.01080.S) and determined the flux densities of SGR
J1745-2900 from each of the 340-GHz images at the four epochs. e. Mean values
determined from two VLA Q-band measurements with a baseline filter ($>500$
k$\lambda$) and all data (Gelfand et al., 2017); for the case of std = 0, the
smaller error in the two measurements is adopted. f. Yusef-Zadeh et al.
(2017). g. Bower et al. (2015/16).
---

Figure 6:  Averaged spectrum of SGR J1745-2900, made with band-averaged flux
densities observed with the JVLA and ALMA (values summarized in Table 3) as
well as with ATCA (Shannon & Johnston, 2013).

####  4.3.2 Radio spectrum of the GC magnetar

In addition to the JVLA and ALMA data, we also collected the flux-density data
of the magnetar from ATCA (Shannon & Johnston, 2013), yielding a total of 91
flux-density measurements of this object at radio wavelengths over the 6.5 yrs
since the 2013 outburst. To investigate the spectrum of the magnetar, we
binned the data into nine bands with frequency ranges corresponding to those
used for the ALMA and JVLA observing bands. We then computed the mean flux
density ($\overline{\rm S}_{\nu}$) and variance ($\sigma_{\nu}^{2}$) in each
of the bands, weighted by $wt(i)=\sigma(i)^{-2}$ where $\sigma(i)$ is the
uncertainty of each individual flux density, $i$. For non-detections, a zero
weight was adopted, assuming that the actual uncertainties of the non-
detections due to the errors in the calibration for the system and atmostheric
issues are much greater than the cited rms errors. For a total of $n$
measurements in each bin, the number of non-zero weighted data points, $m$, is
less than or equal to $n$, (${m\leq n}$), within each bin. The error of the
mean flux density can be determined with

$\displaystyle\sigma_{\overline{\rm
S}_{\nu}}=\sqrt{\sigma_{\nu}^{2}\over{m}}.$

We note that the Ka band shows a lower detection rate ($\sim$55%) than the
Q-band ($\sim$68%), indicating that a spectral minimum is located near the Ka-
band frequency of 33 GHz, given the fact that the rms noise at the Ka-band
center frequency of 33 GHz is about a factor of 2 smaller than that of Q-band
(see Table 1). To reduce the bias due to zero-weighting on the data with
negative detections, we combined the Ka-band and Q-band bins’ data and re-
computed the mean flux density and its uncertainty. Table 4 summarizes the
results. Fig. 6 shows the averaged spectra observed with the JVLA, ALMA and
ATCA over 6.5 yrs since the outburst. The vertical bars on each point reflect
the large intrinsic variation of the flux density from SGR J1745-2900 over the
observed time scale. However, a spectral minimum appears to be present at a
frequency around 30 GHz, hereafter referred to as the transition frequency
$\nu_{\rm t}$, which appears to separate the spectrum into two components
arising from two emission regimes at centimeter and millimeter-submillimeter
wavelengths. The flux density of the centimeter-wave component is typically
$\sim$1 mJy while the millimeter-wave component is about three times more
intense.

Table 4: Mean flux densities of SGR J1745-2900 in the JVLA/ALMA bands

Band code | $n$ | $m$ |  $\overline{\nu}\pm\Delta\nu$ (GHz) |  $\overline{\rm S}_{\nu}\pm\sigma_{\overline{\rm S}_{\nu}}$ (mJy)
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
A7 | 5 | 5 |  338.0$\pm$5.0 |  3.01$\pm$0.41
A6 | 11 | 11 |  225.1$\pm$ 0.1 |  2.14$\pm$0.51
Q-Ka | 34 | 23 |  39.4$\pm$ 0.9 |  0.33$\pm$0.10
K | 15 | 10 |  20.7$\pm$ 0.5 |  0.30$\pm$0.07
Ku | 7 | 7 |  15.3$\pm$ 0.2 |  0.69$\pm$0.13
X | 12 | 12 |  8.9$\pm$ 0.1 |  0.82$\pm$0.10
C | 7 | 6 |  5.5$\pm$ 0.4 |  1.16$\pm$0.37

The band-averaged data are derived from the individual flux-density
measurements with the JVLA and ALMA (Table 3) as well as ATCA (Shannon &
Johnston, 2013).
---

We then carried out a least-squares fitting of the spectrum to both the cm-
and mm-components with two power-law functions. The centimeter data are well
fit with a power-law spectral index of $\alpha_{\rm cm}=-1.5\pm 0.6$
($S_{\nu}\propto\nu^{\alpha_{\rm cm}}$), a steep spectrum similar to that of
radio pulsars. A steep spectrum of $\alpha=-1$ was previously reported between
4.5 and 8.5 GHz for the observations during the 2013-5-30 flare of the cm-
component (Shannon & Johnston, 2013). The spectral data points rising through
millimeter wavelengths can be fit with a spectral index of $\alpha_{\rm
mm}=1.1\pm 0.2$, indicating the presence of an emission bump or a plateau at
higher frequencies. Such a high-frequency plateau appears not to be unique to
the GC magnetar. The radio-active magnetar 1E 1547.0$-$5408 has also been
observed to have a spectrum rising towards millimeter wavelengths (Chu et al.,
2021). Overall, the spectrum of the GC magnetar at the frequencies in the
range between 5 and 310 GHz is described with a combination of the two power-
law functions (see Fig. 6) with a transition frequency $\nu_{\rm t}\approx 30$
GHz corresponding to a minimum flux density of $\sim$0.3 mJy.

Figure 7:  (a) The X-ray luminosity variation after the onset of outburst of
SGR J1745-2900, observations by the Chandra X-ray observatory (Rea et al.,
2020). (b) 100-MJD bin averaged radio light curve of the magnetar J1745-2900
for both the mm- (red dots) and cm- (black dots) components. In addition to
the flux-density measurements from the JVLA and ALMA summarized in Table 3,
other telescope data from literatures (Shannon & Johnston, 2013; Torne et al.,
2015, 2017) are included. (c) Detail of ALMA observations in the interval
between 2017-10-6 and 2017-10-20 (1626 days after the outburst of SGR
1745-2900 on 2013-4-23) show a day-to-day variability of the magnetar SGR
J1745-2900 at 225 GHz displayed as normalized flux densities SMJD-56406/Smin
(red dots), where the minimum flux density S${}_{min}=$0.79$\pm$0.08 mJy for
the magnetar on MJD 58035. The variability of SGR J1745-2900 during the two-
week interval is compared with that of Sgr A* (dark-blue dots); Sgr A*
(Smin=3.015$\pm$0.013 Jy and Save= 3.26$\pm$0.09 Jy) that is three orders in
magnitude more luminous than the magnetar.

####  4.3.3 Radio variability of SGR J1745-2900

We inspected the radio variability of the GC magnetar SGR J1745-2900,
including a total of 161 flux-density measurements since the onset of the 2013
outburst. The measurements by Torne et al. (2015, 2017) are also included in
addition to the JVLA-ALMA (Table 3) and ATCA (Shannon & Johnston, 2013) data.
The magnetar appears to be highly variable on all observed time scales and
wavelengths (e.g. Shannon & Johnston, 2013; Yusef-Zadeh et al., 2015, and this
paper). We utilized a bin-averaging algorithm similar to the analysis of the
magnetar spectrum (Sec.4.3.2). We binned the MJD or the time axis with a
constant interval of 100 days. The two spectral components $--$ low-frequency
and high-frequency $--$ corresponding to the cm-component ($\nu<30$ GHz) and
the mm-component ($\nu>30$ GHz), were examined separately. We then computed
the weighted mean flux densities and the corresponding dispersions in each of
the 56 MJD-bins for both mm- and cm-components. The results for the non-empty
bins are tabulated in Table 5. Columns 1 and 2 show the bin ID (binID) and the
corresponding central MJD. Columns 3 and 4 are the numbers of all the
measurements ($n$) and the non-zero weight data ($m$) in the corresponding
MJD-bins; as above, the measurements with only upper limits are zero-weighted.
Column 5 is the mean MJD of the observing dates. Column 6 gives the mean flux-
density ($\overline{S}_{\rm binID}$), weighted by the inverse variance on each
of the measurements, along with the uncertainty of the mean
$\sigma_{\overline{S}_{\rm binID}}=\sqrt{\sigma^{2}_{\rm binID}/{m}}$. And the
standard deviation ($\sigma_{\rm binID}$), or the dispersion due mainly to the
variation in flux density, is listed in column 7.

Fig. 7b plots the MJD-bin averaged radio light curves for the two spectral
components, red for mm and black for cm. Unlike the X-ray light curve (Fig.
7a) observed by the Chandra X-ray observatory (Rea et al., 2020) that shows a
smooth decrease in X-ray luminosity, both the cm- and mm-components varied
significantly in the first 800 and 1700 days. Our two measurements of flux
density based on the earlier JVLA observations at 5.5 GHz showed a significant
variation from 4.50$\pm$0.24 mJy on 2014-05-17 to 3.90$\pm$0.09 mJy on
2014-05-26, on the time scale of a week, which is consistent with the large
variability at 5 GHz observed during May 2013 (Shannon & Johnston, 2013).

The two most recent JVLA measurements at 9 GHz on 2020-11-27 (MJD 59180) and
2019-09-21 (MJD 58777) indicate that the flux density of the cm-component
decreased to 0.4 mJy from 3.5 mJy in the JVLA observation on 2014-04-17 (MJD
56764). Also, we failed to detect the source at 5.5 GHz based on the JVLA
observations on 2019-09-08 (58734), imposing a 3-$\sigma$ upper limit of 0.5
mJy while the flux density was 4.5 mJy based on our JVLA observation at 5.5
GHz on 2014-5-17 (MJD 56794). The mean flux density of the cm-component of the
magnetar decreased by a factor of 3 over a period of 6.5 yr (Fig. 7b and Table
5).

The Chandra X-ray observations of SGR J1745-2900 following the outburst show a
smooth decrease from 4.9$\times 10^{35}$ erg -1 at the onset of the X-ray
outburst to 0.047$\times 10^{35}$ erg -1 at the most recent epoch (Fig. 7a).
The X-ray luminosity therefore dropped by two orders in magnitude over 6
years. The general trend of decreasing radio flux density is consistent with
the X-ray light curve, although the cm-component has a much slower decline, is
much more variable, and shows a large range of variability on time scales from
days to years.

The mm-component stayed at a relatively low level, with bin-averaged flux
densities of 0.7$\pm$0.40 and 1.53$\pm$0.23 mJy during the second and third
MJD-bins after the outburst (Table 5 and Fig. 7b). During the first 300 days,
the mm-component was difficult to detect (only 5 detections out of 11
observations). We note that the JVLA Q-band data observed during 2013 fall and
2014 spring were reduced by two independent groups (Yusef-Zadeh et al., 2015;
Gelfand et al., 2017). The mm-component reached a maximum of 5.63$\pm$0.19 mJy
in the 7th MJD-bin (601-700 days after the outburst), based on 35 observations
with a detection rate of 100%.

The 17th MJD bin of the mm-component (between 1600-1700 days) contains 10
individual ALMA observations at 225 GHz, all carried out within a two-week
period during October 2017 with a high-resolution (0.02”) configuration
(Tsuboi et al., 2019). The mean flux density for the mm-component in this bin
was 2.25$\pm$0.56 mJy. With this configuration, the extended HII emission, as
well as dust emission from the local medium, are well resolved out and only
hyper-compact sources can be detected. The typical rms noise of the ALMA
images is about 20-30 $\mu$Jy beam-1, and the magnetar has S/N ratios of
50-100. At a distance of $\sim$3” from the magnetar, Sgr A*, with a flux
density near 3 Jy, is an excellent reference source to examine the day-to-day
variability of the magnetar. The ALMA data show the magnetar to be highly
variable at 225 GHz on a time scale of days. The source started at
1.41$\pm$0.08 mJy on 2017-10-6, dropped to a minimum of 0.79$\pm$0.08 mJy
three days later on 2017-10-9, and then increased by a factor 7 within 8 days
reaching 5.34$\pm$0.09 mJy on 2017-10-17, staying at that level for the next
several days. To compare the variability of SGR J1745-2900 with Sgr A*, we
normalized the source flux densities by their own minimum flux densities
during the observing period. Fig. 7c shows the relative variability for both
the magnetar and Sgr A* during the two weeks in bin 17. We define a relative
variability parameter, $\mathscr{RV}$, to quantitatively describe the
magnitude of variability relative to a minimum flux density $S_{\rm min}$,
given a maximum flux density $S_{\rm max}$ of a target source observed in a
period:

$\mathscr{RV}={\left[{\rm S_{\rm max}-S_{\rm min}}\right]\over{\rm S_{\rm
min}}}.$

The magnetar shows the relative variability $\mathscr{RV}\approx$ 6 while Sgr
A* is observed to be only moderately variable in the same ALMA program, with
$\mathscr{RV}\approx$ 0.2..

Finally, the latest available detections of the magnetar in 2019-2020, are
1.32$\pm$0.15 mJy at 318 GHz with ALMA on 2019-09-20 (MJD 58748) and 0.36$\pm
0.06$ mJy with the JVLA on 2020-11-27 (MJD 59180).

Table 5: Mean flux densities of SGR J1745-2900 in 100-day MJD bins

binID | MJD | $n$ | $m$ | $\overline{\rm MJD}$ | $\overline{\rm S}_{\rm binID}\pm\sigma_{\overline{\rm S}_{\rm binID}}$ | $\sigma_{\rm binID}$
---|---|---|---|---|---|---
| (day) |  |  | (day) | (mJy) | (mJy)
(1) | (2) | (3) | (4) | (5) | (6) | (7)
Centimeter-wave component
1 | 56456 | 16 | 16 | 56435.1 |  0.50$\pm$0.06 | 0.24
3 | 56656 | 5 | 3 | 56673.2 |  1.36$\pm$0.26 | 0.45
4 | 56756 | 10 | 10 | 56755.7 |  1.08$\pm$0.14 | 0.44
5 | 56856 | 12 | 12 | 56862.4 |  0.46$\pm$0.09 | 0.31
7 | 57056 | 10 | 10 | 57085.9 |  0.79$\pm$0.13 | 0.41
24 | 58756 | 2 | 1 | 58777.0 |  0.40$\pm$0.10 | 0.10
28 | 59156 | 1 | 1 | 59180.0 |  0.36$\pm$0.06 | 0.06
Millimeter-wave component
2 | 56556 | 3 | 1 | 56591.5 |  0.70$\pm$0.40 | 0.40
3 | 56656 | 9 | 5 | 56678.6 |  1.53$\pm$0.23 | 0.51
4 | 56756 | 10 | 10 | 56742.4 |  1.00$\pm$0.14 | 0.44
5 | 56856 | 26 | 21 | 56853.2 |  2.20$\pm$0.21 | 0.96
7 | 57056 | 37 | 37 | 57087.3 |  5.63$\pm$0.19 | 1.16
9 | 57256 | 2 | 2 | 57278.5 |  2.13$\pm$0.32 | 0.45
11 | 57456 | 1 | 1 | 57502.0 |  2.80$\pm$0.23 | 0.23
12 | 57556 | 2 | 2 | 57582.5 |  5.78$\pm$0.04 | 0.06
13 | 57656 | 3 | 3 | 57633.3 |  3.40$\pm$0.23 | 0.40
17 | 58056 | 10 | 10 | 58038.4 |  2.14$\pm$0.53 | 1.68
24 | 58756 | 1 | 1 | 58770.0 |  1.32$\pm$0.15 | 0.15

The 100-day-averaged data are derived from a total of 161 individual flux-
density measurements with the JVLA, ALMA (Table 3), ATCA (Shannon & Johnston,
2013) and other single-dish telescopes (Torne et al., 2015, 2017) since the
onset of the outburst at the epoch 2013-4-23 (MJD 56406).
---

##  5 Astrophysical implications

A population of hyper-compact radio sources (HCRs) are detected at 33 and 44.6
GHz with the JVLA in the vicinity of Sgr A* within a radius of 13 arcsec. The
new survey was motivated by the previous JVLA detections of 110 Galactic
center compact radio sources (GCCRs) at 5.5 GHz in the radio bright zone
within a radius of 7.5 arcmin from Sgr A* but outside Sgr A West.

###  5.1  Spectral types of HCRs & their distribution in flux-density

Fig. 8 shows the flux-density distribution of the HCRs as compared with that
of the GCCRs. The distribution of the GCCRs is similar to the high-luminosity
tail of the pulsars’ distribution in the Galactic disk (Kramer et al., 1998;
Manchester et al., 2005; Zhao, Morris & Goss, 2020). The HCRs have a
relatively narrow distribution from $-1$ to 0.6 in Log(S [mJy]), or ranging
from 0.1 to 4 mJy, peaked at $-0.4$ in Log(S [mJy]) or 0.4 mJy. We note that
the flux density of the peak in the HCR distribution is close to the minimum
value 0.32 mJy at the transition frequency observed in the band-averaged
spectrum for the GC magnetar J1745-2900 (see Sec.4.3.2).

We divided the HCRs into three sub-types according to their spectral indices
$\alpha_{44.6/33}$ ($S\propto\nu^{\alpha_{44.6/33}}$) derived between 44.6 and
33 GHz: flat ($0.2\geq\alpha_{44.6/33}>-0.5$), steep
($\alpha_{44.6/33}\leq-0.5$) and inverted ($\alpha_{44.6/33}>0.2$).

Of all the HCRs, 58% (38/65) are steep-spectrum sources, 26% (17/65) are
inverted-spectrum sources, and 15% (10/65) have a flat spectrum. We note that
HCR49 is a double; so a total of 65 HCR components are included in the
spectral-index analysis. The inset of Fig. 8 shows the flux-density
distributions for each of the three spectral types with a finer bin,
$\Delta$Log(S [mJy]) = 0.2.

####  5.1.1 Flat-spectrum HCRs

The flux density distribution of the flat-spectrum HCRs appears to be uniform
within the range from 0.16 to 1.6 mJy. Some fraction of the flat-spectrum HCRs
might be unresolved peaks in the HII region components since a flat spectrum
at Ka- and Q-bands is characteristic of free-free emission in optically thin
HII regions.

####  5.1.2 Steep-spectrum HCRs

This sub-sample consists of 38 members, the largest sub-sample among the
three, in which the flux densities are statistically well distributed. The
distribution can be fitted with a Gaussian, with a mean value of $\mu=-0.35$
and a standard deviation of $\sigma=0.22$ in Log(S [mJy]). The mean value of
the steep-spectrum HCRs corresponds to 0.45 mJy in flux density.

The spectral index $\alpha_{44.6/33}$ of the steep-spectrum HCRs is in the
range between $-6.5$ and $-0.6$, giving a mean $\mu_{\alpha}=-1.8\pm 0.2$. The
steep spectrum of this sub-sample differs distinctly from the HII components,
suggesting the presence of a population of hyper-compact nonthermal radio
sources in the central parsec. The nonthermal HCRs are likely associated with
the massive stellar remnants that are expected to be distributed in the close
vicinity of Sgr A* (Morris, 1993; Hailey et al., 2018; Generozov et al.,
2018).

####  5.1.3 Inverted-spectrum HCRs

For this sub-sample, the mean flux density and standard deviation are
$\mu=0.5$ mJy and $\sigma=0.25$ mJy, respectively. The spectral index
$\alpha_{44.6/33}$ of the inverted-spectrum HCRs is in the range between 0.21
and 1.65, with a mean value of $\mu_{\alpha}=0.61\pm 0.08$. Among the three
spectral sub-samples, the distribution of the inverted-spectrum HCRs appears
to most closely follow the GCCR distribution, which matches the high-
luminosity tail of the distribution that normal pulsars would have at the
Galactic center (Zhao, Morris & Goss, 2020). Normal pulsars usually have a
steep spectrum at centimeter wavelengths and are difficult to detect at high
frequencies. However, the GC magnetar, emitting at the Ka- and Q-bands, falls
into this sub-sample. As shown in Sec. 4.3.2, an inverted spectral component
of SGR J1745-2900 is present at millimeter wavelengths in the averaged
spectrum of a large sample of observations. The inverted spectrum cannot be
simply attributed to time variability of the flux density. By analogy with SGR
J1745-2900, the inverted spectral component could be indicative of an
association of the HCRs with magnetars.

On the other hand, hyper-compact HII regions associated with a late-type
massive star may also have a similar characteristics in spectral index, flux
density level and compactness. For example, a group of compact components have
been detected at mm-submm wavelengths in the IRS 21 complex (see Fig. 2b). The
hyper-compact sources associated with IR emission more likely belong to hyper-
compact HII or HC HII regions associated with young stellar objects rather
than old massive stellar remnants. The hyper-compact radio components of IRS
21 are therefore not included in the HCR catalog (Table 2) in this paper. IRS
21 will be discussed in a separate paper.

From cross-correlation between the HCRs of this paper and the nine 7mm-IR
sources that are found to be associated with strong stellar winds (Yusef-Zadeh
et al., 2014), we find that six of the nine candidate IR stars have HCR
counterparts. The spectral indices of the six HCRs – five with inverted
spectra and one with a flat spectrum – are consistent with the radio emission
produced by the ionized winds of hot, massive stars (Panagia & Felli, 1975).
Thus, a large fraction of the 17 inverted-spectrum HCRs might consist of
thermal free-free emission sources. Apparently, a significant portion of the
inverted-spectrum HCRs is associated with late-type massive stars.

We speculate that more magnetars besides SGR J1745-2900 reside in our
inverted-spectrum sample of HCRs. However, we must be able to distinguish such
objects from compact radio sources associated with young massive stars.
Further high-resolution ALMA observations of the variability and polarization
characteristics of this subsample will be crucial for identifying the
nonthermal nature of candidate magnetars and stellar-mass black holes.

Figure 8: Flux-density distribution of the HCRs at 33 GHz within the central
13”$\times$13” region versus the GCCRs (Zhao, Morris & Goss, 2020) detected in
the GC radio bright zone (15’$\times$15’). The grey histogram represents the
110 GCCRs detected at 5.5 GHz outside Sgr A West. The 64 HCRs (65 components,
HCR49 is double), shown by the hatched histograms, are divided into three
types: flat $-0.5<\alpha\leq 0.2$ (red), steep $\alpha\leq-0.5$ (green) and
inverted $\alpha\geq 0.2$ (blue), according to their radio spectral index
determined between 44.6 and 33 GHz, except for three steep-spectrum sources –
HCR22, HCR32 and HCR64 – which were not detected in the Q-band data used in
this paper. The spectral indices for HCR22 and HCR32 are determined using
X-band data discussed in this paper. The spectral index for HCR64, the micro-
quasar, is determined using the K-band data of Zhao et al. (2009) assuming
that the source was in a quiescent state during the 2015 observations. The
inset panels on the right are the more finely binned individual distributions
of the three spectral sub-samples. The distribution of the steep-spectrum HCRs
is fitted with a Gaussian (red curve).  Figure 9: Radial distribution of the
surface density of compact radio compact sources detected at the Galactic
center. The red points are the surface density data $\Sigma^{*}_{\rm crs}$
corrected for a higher equivalent sensitivity cutoff at 5.5 GHz by multiplying
the original 33-GHz surface density data, $\Sigma_{\rm crs}$ (light-grey
points), by a correction factor, $\eta=2.4$, derived from the 33-GHz
observations of the HCRs (see Appendix A of this paper). The black points mark
the data of GCCRs observed at 5.5 GHz. In the annular zones around $R=3"$, a
bump in $\Sigma_{\rm crs}$ is present. The orange line is the best fit using a
power-law with an index $\Gamma=1.6\pm 0.2$ for the inner region outside of
the bump at 1.5 - 7 arcsec; and the blue line indicates the least squares
fitting to the steeper power law with $\beta=2.0\pm 0.2$ for the outer region.
The break radius, $R_{b}$, is $\sim 30"$, or $\sim 1.2$ pc. The top-right
inset shows the surface-density distribution of the HCRs excluding the flat-
spectrum sources.

###  5.2 A dense group of HCRs and the radial distribution of their surface-
density

The surface density of 110 GCCRs, located outside Sgr A West ($>1$ arcmin from
Sgr A*) but within the RBZ ($<7.5$ arcmin from Sgr A*), is $\sim$0.6 counts
arcmin-2. This surface density is an order of magnitude greater than that of
background extragalactic compact radio sources (e.g. Condon et al., 2012; Gim
et al., 2019). The dense group of 64 HCRs, located within a radius of 13” from
SgrA*, has a relatively higher surface density, with an average value of
$\sim$500 counts arcmin-2. We conclude that contamination of our sample by
extragalactic background sources is very likely to be negligible.

In Appendix A, we developed a procedure to construct the surface-density
($\Sigma_{\rm crs}$) distribution of the radio compact sources (RCS) as
function of the projected radial distance ($R$) from Sgr A*, including 64 HCRs
at $R<13$” (this paper) and the 110 GCCRs outside Sgr A West (Zhao, Morris &
Goss, 2020) as well as the Galactic center transient (GCT, see Zhao et al.,
1992). Fig. 9 shows the radial distribution of the surface-density of the RCSs
detected within the RBZ. Excluding the four data points related to a
$\Sigma_{\rm crs}-$bump in the radial distance range between 1.5” and 7”, the
surface-density distribution can be fitted with two power laws (see Fig. 8) as
described by the “Nuker” law (Lauer et al., 1995; Genzel et al., 2003; Fritz
et al., 2016; Schödel, 2018). In the inner region at $R<30$” but excluding the
range between 1.5” and 7”, the surface-density distribution of the HCRs
follows a power law:

$\centering\Sigma_{\rm HCR}(R<30")\approx(5.0\pm 1.8)\times 10^{3}R_{\rm
arcsec}^{-\Gamma}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ({\rm arcmin^{-2}})\@add@centering$
(1)

where the index $\Gamma=1.6\pm 0.2$. At large radii $R\geq 30$”, the surface-
density distribution of the GCCRs shows a steeper power law:

$\centering\Sigma_{\rm GCCR}(R\geq 30")\approx(2.0\pm 0.9)\times 10^{4}R_{\rm
arcsec}^{-\beta}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ({\rm arcmin^{-2}})\@add@centering$ (2)

where $\beta=2.0\pm 0.1$. At $R\sim 30$”, the two power laws intersect. We
note that the power-law distribution derived from HCRs is much steeper than
that $\Gamma\sim 0.6$ of the stellar cusp determined with the $K_{\rm s}$
stars in magnitude range between 12.5 and 18.5 in the radial distance range
between 1” and 50” (Gallego-Cano et al., 2018).

###  5.3 Candidate massive stellar remnants

####  5.3.1 Mass segregation

It has been known for several decades that the central parsec contains a large
number of young massive stars (Krabbe et al., 1991, 1995; Paumard et al.,
2001, 2006; Gillessen et al., 2009; Lu et al., 2009; Bartko et al., 2010; Lu
et al., 2013), with $\sim$100 O and Wolf-Rayet (WR) stars confined within a
radius of $\sim$0.4 pc (Genzel et al., 2003; Ghez et al., 2005). These stars
are relatively young ($\leq 10$ Myr) and are orbiting Sgr A*, the supermassive
black hole (SMBH) at the Galactic center. The early-type stars provide
substantial UV photons to maintain the ionization of the gas within the
central parsec (Zhao et al., 2010).

The stars in the central cusp apparently have a top-heavy present-day mass
function (over-abundance of high-mass stars, see Bartko et al., 2010; Lu et
al., 2013); and the fact that the orbits of many of them collectively define a
coherent disk suggests that the young nuclear cluster may have originated via
in situ star formation (Levin & Beloborodov, 2003; Levin, 2007; Nayakshin et
al., 2007; Collin & Zahn, 2008). The process that formed the young nuclear
cluster near the central black hole is likely to have been a recurring one.
This process follows a limit cycle of activity wherein the star formation is a
violent event coinciding with a heavy accretion episode onto the SMBH. Thus
the inner disk is quickly disrupted and dissipated in the process. Following
the disruption of the inner disk, continued migration of gas from the central
molecular zone toward the center rebuilds the disk, eventually leading to the
next star formation event (Morris, Ghez, & Becklin, 1999). With repeated
instances of this cycle, the remnants of massive stars are produced and left
in place to collect at the bottom of the Galaxy’s gravitational potential
well. Furthermore, the more massive remnants, particularly stellar-mass black
holes, will undergo mass segregation in the dense central star cluster, as a
result of dynamical friction, and become even more concentrated toward the
black hole (Bahcall & Wolf, 1976; Morris, 1993; Miralda-Escudé & Gould, 2000;
Pfahl & Loeb, 2004; Freitag, Amaro-Seoane, & Kalogera, 2006; Alexander &
Hopman, 2009; Merritt, 2010; Antonini & Merritt, 2012; Alexander et al., 2017;
Generozov et al., 2018). The mass segregation may partially explain the
intense flux of gamma rays and X-rays from the Galactic center caused by the
possible presence of a large population of millisecond pulsars (MSPs) and
cataclysmic variables (CVs), as suggested by (Arca-Sedda et al., 2018). The
excess of GeV gamma rays toward the Galactic center may alternatively be
explained by a high supernovae rate, leading to the production of neutron
stars and ultimately to a MSP population (O’Leary et al., 2015, 2016; Calore
et al., 2016).

In the case of stellar-mass BHs that are significantly more massive than the
mean stellar mass expected for an evolved population, such heavier objects
would migrate toward the center and be distributed in a compact cluster around
the SMBH. The nature of the mass segregation depends on the relaxational
coupling parameter $\Delta=4n_{\rm BH}M^{2}_{\rm BH}/\left[n_{\rm
st}M^{2}_{\rm st}(3+M_{\rm BH}/M_{\rm st})\right]$ for black hole (BH) mass
$M_{\rm BH}$ and spatial number density $n_{\rm BH}$ along with stellar (ST)
mass $M_{\rm st}$ and spatial number density $n_{\rm st}$. The value of
$\Delta$ is a measure of the importance of BH-BH scattering relative to BH-ST
scattering for the dynamical friction process (Alexander & Hopman, 2009). Weak
segregation occurs when $\Delta\gg 1$; and when $\Delta\ll 1$, the strong
segregation applies. The latter case leads to a steeper slope in the 3D radial
density distributions of BHs and a larger central concentration of BHs
relative to that of stars: for power-law indices333Throughout this paper, the
lower-case $\gamma$ is used for power-law indices of 3D radial density
distributions while the upper-case $\Gamma$ stands for power-law indices of 2D
radial density, or the surface density, distributions. $\gamma_{\rm BH}$ and
$\gamma_{\rm ST}$ for BHs and stars, respectively, $2\lesssim\gamma_{\rm
BH}\lesssim 11/4$ and $3/2\lesssim\gamma_{\rm ST}\lesssim 7/4$ ($\gamma_{\rm
BH}-\gamma_{\rm ST}\backsimeq 1$) (Alexander et al., 2017). Long-lived stellar
populations usually have $\Delta<0.1$, and the Galactic center is expected to
be strongly segregated (Morris, 1993). Alexander & Hopman (2009) have shown
the effects of strong mass segregation on the density distribution of a model
stellar population around Sgr A* (mass $M_{\rm SMBH}=4\times 10^{6}$ M⊙); the
modelled population includes main sequence stars (MS$\sim 1$ M⊙) and stellar
remnants, including white dwarfs (WD$\sim 0.6$ M⊙), neutron stars (NS$\sim
1.4$ M⊙) and black holes (BH$\sim 10$ M⊙). Their study demonstrates that the
heavier objects produce steeper density distributions via the mass segregation
process.

We note that, in the radial range 0.04-1 pc, the power-law index of the 2D
surface-density profile of the HCRs is $\Gamma_{\rm HCR}=1.6\pm 0.2$ (this
paper) which is much greater than the corresponding value of $\Gamma_{K_{\rm
s}}$ for the $K_{\rm s}$ stars in the same radial range (Gallego-Cano et al.,
2018). The difference between the power-law indices ($\Gamma_{\rm
HCR}-\Gamma_{K_{\rm s}}$) is $\sim$1\. The flat power-law profile for $K_{\rm
s}$ stars provides evidence for the presence of a dynamically relaxed stellar
cusp at the Galactic center (Gallego-Cano et al., 2018; Schödel, 2018). On the
other hand, with the arguments presented in the previous paragraph, the dense
group of HCRs reported in this paper, with its much steeper radial
distribution (see Fig. A2) than that of the faint $K_{\rm s}$ stars, could
represent a population of massive stellar remnants that are mass-segregated in
the nuclear star cluster at the Galactic center. The HCRs are associated with
active massive stellar remnants having relatively higher radio luminosities.
The surface density profile of the HCRs (see Fig. A2) may be subject to change
when a deeper radio survey is carried out with a much lower flux density
cutoff. Nevertheless, the results from our study of HCRs in the central parsec
provides evidence consistent with the presence of a distribution of massive
stellar remnants that is a steep function of the radial distance from Sgr A*.

In addition, if the massive stellar remnants associated with the HCRs had
migrated inward via dynamical friction, the bump in surface density,
$\Sigma_{\rm HCR}$, at 0.1-0.3 pc could be attributed to an accumulation in
that radial range since the dynamical friction force acting on a massive
object ceases at roughly half the radius of the stellar core ($\sim 0.25$ pc)
(Merritt, 2010). A maximum in the density profile of massive remnants is also
predicted to occure at $\sim 0.2$ pc at a time $>$1 Gyr in a dynamical
evolution model, and the bump slowly grows and migrates inward due to the
friction produced by fast-moving stars inside these radii (Antonini & Merritt,
2012). The $\Sigma_{\rm HCR}$-bump may serve as an additional observational
signature of massive stellar remnants as a consequence of stellar dynamical
processes in galactic nuclei.

####  5.3.2 Radiation spectrum

Hailey et al. (2018) recently reported the identifications of a dozen low-mass
back hole X-ray binary candidates within the central parsec, implying the
presence of a large population of X-ray binaries and isolated black holes
residing within that volume (Generozov et al., 2018). Most of them are in a
quiescent state. X-ray and radio flares and outbursts from these stellar
remnants have been discovered over the past three decades, such as the
magnetar SGR J1745-2900 (Kennea et al., 2013; Rea et al., 2020; Eatough et
al., 2013; Shannon & Johnston, 2013; Torne et al., 2015, 2017), the
microquasar of the X-ray transient XJ174540.0290031 (Muno et al., 2005; Bower
et al., 2005; Zhao et al., 2009) and the Galactic center transient (GCT, see
Zhao et al., 1992) as well as the X-ray PWN candidate G359.95-0.04 (Baganoff
et al., 2003; Wang et al., 2006; Muno et al., 2008) which is likely powered by
a neutron star.

With a flux-density range bewteen $\sim$0.1 and a few mJy at 33 and 44.6 GHz,
the HCRs appear to be candidate radio counterparts of the old massive stellar
remnants produced in the end of stellar evolution as expected. Most of them
are in a quiescent state. Although their progenitors and ages are unknown, the
38 steep spectrum HCRs determined from the JVLA observation at Ka and Q bands
provide important clues on the nature of the radio radiation, affirming
nonthermal radiation with a steep power law in the distribution of
relativistic electrons. The nonthermal emission could be produced by
synchrotron jets or outflows that were launched from the compact stellar
remnants powered by accretion from the dense, local medium.

As described above, one of the 17 inverted-spectrum HCRs is the GC magnetar,
SGR J1745-2900, which shows high-frequency pulses up to 291 GHz (Torne et al.,
2017). The continuum emission from this magnetar has been firmly detected at
high radio frequencies up to 340 GHz (Tsuboi et al., 2017, and this paper).
The inverted-spectrum of the mm-component of a magnetar towards the
submillimeter appears to be a remarkable radio wavelength signature, as
predicted by the dynamical model of Beloborodov (2013) using a persistent flow
of electron-position plasma. The configuration of the magnetosphere of
magnetars, created by enforced electric current and radiative drag together,
is subject to two-stream instability. Consequently a relatively hard radio
spectrum that is predicted to emerge, perhaps extending to IR/optical/UV
wavelengths, and is expected to be generated because the instability leads to
a large plasma density, and thus a large plasma frequency (Beloborodov, 2013).
The theory also predicts a large electric current associated with the radio-
submillimeter emission from magnetars (Beloborodov, 2013), producing a bright
radiation beam much broader than the typical pulse width of normal pulsars
with similar periods (Camilo et al., 2006, 2007). A valuable next step will be
to use this theory to formulate predicted shapes of radio spectra and pulse
profiles of magnetars for comparison with observations.

The suggestion has been made that pulsars formed at or near the Galactic
center might mostly to be magnetars, given the rather highly magnetized
interstellar medium of the GC region that could produce highly magnetized
massive stars. Subsequently, strongly magnetized neutron star remnants form
because of the collapse of stellar core and concentration of the flux-frozen
magnetic field (Dexter & O’Leary, 2014; Morris, 2014). However, because the
magnetic flux within the core of a massive star as well as within a neutron
star can undergo considerable evolution owing, for example, to dynamo action
occurring inside the star and the neutron star, (e.g., Duncan & Thompson,
1992; Thompson & Duncan, 1993), this scenario remains rather speculative. In
any case, the discovery of SGR J1745-2900 makes the central parsec a
potentially interesting region to search for new magnetars. Although the
recent formation of a large number of massive stars there could have led to a
population of neutron star remnants, if those remnants are predominantly
highly magnetized, then, as with magnetars in general, they would have short
lifetimes as pulsars ($\sim$ 10${}^{3}-10^{5}$ yrs) because of the powerful
spindown torque associated with the interaction of the neutron star’s magnetic
field with the plasma in its environment (Harding et al., 1999; Espinoza et
al., 2011; Dexter & O’Leary, 2014; Kaspi & Beloborodov, 2017). The time period
during which a magnetar is an observable pulsar is therefore more than two
orders of magnitude shorter than the lifetime of the massive stars observed in
the young nuclear cluster ($\sim$2 - 10 Myrs). So we might therefore expect
only a few (or zero) magnetars to be found as pulsars at any one time in the
central parsec if the above speculation that massive GC stars produce highly
magnetized remnants is correct. Indeed, in addition to the scatter-broadening
that occurs primarily at longer radio wavelengths, the short lifetime of
strongly magnetized pulsars could be the main explanation for the rarity of
pulsars at the Galactic center. The remaining open question is whether
magnetars remain detectable as point radio continuum sources even after they
have spun down to the point at which they can no longer be detectable as
pulsars. If so, then we might consider that some of the HCRs are in that
category.

The inverted spectrum towards short wavelengths found in 1E1547.0-5408 (Chu et
al., 2021) in addition to SGR J1745-2900 is also consistent with the
prediction of spectral hardening at short radio wavelengths from the two-
stream instability model (Beloborodov, 2013). The combination of the inverted
spectrum, high variability and high polarization – nearly 100% for the degree
of linear polarization and 15% for circular polarization (Eatough et al.,
2013) – appears to be unique to magnetars.

Of course, both the inverted and flat spectra of HCRs can also be interpreted
as self-absorbed synchrotron emission from X-ray binaries in the hard state
when the radiation is dominated by the emission from the corona of the compact
object (Coriat et al., 2011).

##  6 Conclusion

Following our 5.5-GHz JVLA survey of the Galactic center compact radio sources
(GCCRs) within the radio bright zone, we have continued to explore Sgr A West
using Ka- and Q-band data obtained by the JVLA in its A-configuration. At an
angular resolution of 0.05 arcsec, we detected a dense group of hyper-compact
(¡0.1”) radio sources (HCRs) within the central parsec of the Galaxy. Based on
a conservative 15$\sigma$ flux-density threshold, corresponding to 150 $\mu$Jy
at 33 GHz, we cataloged 64 HCRs with their J2000 equatorial coordinates, flux
densities at 33 and 44.6 GHz, angular sizes derived from 2D Gaussian fitting,
and spectral index, $\alpha_{44.6/33}$. HCR49 is double.

The surface-density distribution, $\Sigma_{\rm HCR}(R),$ shows a local
enhancement or a density bump in the projected radial distance ($R$) range
1.5”–7” superimposed on a power-law distribution with an index of
$\Gamma=1.6\pm 0.2$. The steeper profile of the HCRs relative to that of the
nuclear stellar cluster might result from the concentration of massive stellar
remnants by mass segregation.

The 65 HCRs divide into three spectral sub-types: 38 steep-spectrum
($\alpha_{44.6/33}\leq-0.5$), 10 flat-spectrum ($-0.5$ ¡
$\alpha_{44.6/33}\leq$ 0.2), and 17 inverted-spectrum sources
($\alpha_{44.6/33}$ ¿ 0.2). Our statistical analysis shows that the
distribution of the steep-spectrum HCRs in Log(S[mJy] can be fitted to a
Gaussian with a mean of $\mu=-0.35$ (corresponding to 0.45 mJy) and a standard
deviation of $\sigma=0.22$. We suggest that the steep-spectrum HCRs be
regarded as candidates for a population of stellar remnants acting as
nonthermal compact radio sources powered by accretion onto neutron stars and
stellar-mass black holes, with the accreted matter supplied either by a binary
companion or by a dense portion of the interstellar medium.

The inverted-spectrum HCRs show a rising spectrum towards high frequencies.
Five of the 17 inverted-spectrum HCRs have compact IR counterparts, suggesting
that they are associated with the ionized stellar wind outflows from hot,
massive stars. A portion of the inverted-spectrum HCRs may consist of X-ray
binaries in the hard state, when the self-absorbed synchrotron emission is
dominated by the corona of the compact object. The GC magnetar, SGR
J1745-2900, belongs to the inverted-spectrum sub-type. Based on our analysis
of 91 flux-density measurements of SGR J1745-2900 observed with the JVLA, ALMA
and ATCA, we find that two distinguishable spectral components contribute to
the averaged spectrum, separated at the transition frequency $\nu_{t}\sim 30$
GHz. The cm-component is fitted to a power-law with a steep spectral index
$\alpha_{\rm cm}=-1.5\pm 0.6$ while the mm-component shows the inverted
spectrum $\alpha_{\rm mm}=1.1\pm 0.2$. Our consolidation of the spectrum from
the interferometer array data is in good agreement with earlier results based
on single-dish observations (Torne et al., 2017).

In addition, we reduced 225-GHz ALMA data observed with a FWHM beam of
0.024”$\times$0.017” on ten individual days within two weeks in October, 2017.
SGR J1745-2900 was detected with a signal-to-noise ratio of $\sim$70 at flux
densities varying widely between 0.79 and 5.60 mJy on day-to-day timescales.
An index of relative variability $\mathscr{RV}\approx$ 6 is found for SGR
J1745-2900, which compares with $\mathscr{RV}\approx$ 0.2 for the more
moderately variable Sgr A*, derived from the same ALMA observations.

Collecting a total of 161 individual flux-density measurements of SGR
J1745-2900 at radio wavelengths from prior published literature and this
paper, we binned data into 100-day bins along the time axis for both cm- and
mm-components separately, starting from the outburst on April 23, 2013 (MJD
56406). The radio light curves with bin-averaged flux density are compared
with the X-ray light curve observed with Chandra (Rea et al., 2020). Except
for the appreciable short-timescale variability at millimeter wavelengths, the
long-term trend at both centimeter and millimeter wavelengths is a slow, but
erratic, decrease in radio power, in contrast to the smoothly decreasing trend
in X-ray luminosity.

Because many HCRs are candidates for being remnants of massive stars, we
considered the origin and fate of such remnants, and speculated on the
possible reasons for the difficulty of finding pulsars there. Neutron stars
formed within the relatively highly magnetized central molecular zone of the
Galaxy could themselves inherit sufficient magnetic flux that they are mostly
born as magnetars. In that case, their ultra-strong magnetic fields (1014 \-
1016 G), acting on plasma trapped in their magnetospheres, apply a spin-down
torque that causes them to have relatively short lifetimes as detectable
pulsars ($10^{3}\leavevmode\nobreak\ -\leavevmode\nobreak\ 10^{5}$ yr)
compared to normal pulsars.

If the inverted spectrum towards high frequencies found in SGR J1745-2900 and
1E1547.0-5408 is a common characteristic of active magnetars, the high-
frequency bands (Ka and Q) of the JVLA and all the ALMA and SMA bands open a
practical window for studying magnetars. Also, dynamical information on the
HCRs can be acquired with high-precision proper motion data that can be
obtained with repeated JVLA and ALMA observations. Such measurements will
supply strong constraints on the candidacy of the HCRs as massive stellar
remnants at the Galactic center.

We are grateful to the anonymous referee and editors for providing their
valuable comments and suggestions. The Jansky Very Large Array (JVLA) is
operated by the National Radio Astronomy Observatory (NRAO). The NRAO is a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. ALMA is a partnership of ESO
(representing its member states), NSF (USA) and NINS (Japan), together with
NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory (JAO) is
operated by ESO, AUI/NRAO and NAOJ. This paper makes use of the data from the
following ALMA programs: ALMA#2018.A.00052.S, ALMA#2017.1.00503.S and
ALMA#2015.1.01080.S. This work has been partially supported by NSF grant
AST1614782 to UCLA. The research has made use of NASA’s Astrophysics Data
System.

##  A Density distribution of HCRs and GCCRs

The stellar density distribution in the Galactic center can be fitted by two
power laws (e.g., Genzel et al., 2003; Fritz et al., 2016), with a flat inner
component and a steeper component at large projected radii (Schödel, 2014,
2018). Usually, the projected surface density can be described by the “Nuker”
profile with two slopes (Lauer et al., 1995):

$\centering\Sigma(R)=\Sigma(R_{b})2^{(\beta-\Gamma)/\alpha}\left(R_{b}\over\/R\right)^{\Gamma}\left[1+\left(R\over\/R_{b}\right)^{\alpha}\right]^{(\Gamma-\beta)/\alpha)}\@add@centering$
(A1)

where $\Sigma(R_{b})$ is the surface density at the break radius $R_{b}$ that
divides the profile into two power laws. At a small radius $R\ll R_{b}$,
$\Sigma(R)\sim\/R^{-\Gamma}$, usually describing the cusp in the distribution
of old stellar population. For $R\gg R_{b}$, $\Sigma(R)\sim\/R^{-\beta}$,
fitting to the outer power law, which is usually steeper than that of the cusp
(Lauer et al., 1995). The break radius $R_{b}$ corresponds both to the point
at which the slope is the mean of $\beta$ and $\Gamma$ and to the radius of
maximum curvature of the distribution in the log10($R$)-log10($\Sigma(R_{b})$)
coordinate system.

In this section, we present the algorithm that we developed to derive the
distribution of the surface density ($\rm\Sigma_{\rm crs}$) for the compact
radio sources (CRS) as function of the projected radial distance from Sgr A*.

###  A.1 Algorithm

First, the radio bright zone (RBZ) is divided into 20 annular zones (AZ), each
of them bounded by two rings at outer and inner radii, $R(i)$ and $R(i+1)$:

$\centering R(i)=2^{-0.5i}R_{\rm RBZ},\@add@centering$ (A2)

where $i=0,1,2...,19\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ 20$ and
$R_{\rm RBZ}$ is the outer radius of the radio bright zone (RBZ). Fig. A1
shows the 20 annular zones along with the distribution of GCCRs and HCRs in
the RBZ (Fig. A1a), Sgr A West (Fig. A1b) and the central parsec (Fig. A1c).
The radius of the outer ring, $R(0)=R_{\rm RBZ}$, corresponds to the largest
radius within which we searched the GCCRs with a 10-$\sigma$ cutoff due to the
JVLA primary beam at 5.5 GHz. The radius of the most inner ring,
$R(20)=0.55"\approx\theta_{\rm FWHM}$, is approximately the size of FWHM beam
at 5.5 GHz. The zone AZ-10 marks the boundary, within which we found 64 HCRs
with the JVLA A-array observations at 33 and 44.6 GHz (this paper). Outside
this boundary, 110 GCCRs are found from the JVLA A-array observations at 5.5
GHz (Zhao, Morris & Goss, 2020).

Then, we counted the CRS in each of the AZs by creating a grid function
$crs(j,k)$ for each of the CRSs, where $k$ is the annular zone ID from 1 to 20
and $j$ is the CRS source ID within an AZ. For given a specific grid cell
($j,k$), a unit value is assigned, i.e. $crs(j,k)=1$. Given a $k$, the AZ-ID,
the number of sources in an annular zone is then derived by summing over $j$:

$\centering N(k)=\sum_{j}{crs(j,k)}.\@add@centering$ (A3)

In addition, we also created grid functions of $s(j,k)$ and $\sigma(j,k)$ for
flux-density and uncertainty, respectively. Corresponding to a specific source
ID: ($j,k$), the values of flux density and 1-$\sigma$ uncertainty of the CRS
are assigned to $s(j,k)$ and $\sigma(j,k)$. The accumulated flux density of
CRS in an annular zone can be determined:

$\centering S(k)=\sum_{i=1}^{N(k)}{s(j,k)},\@add@centering$ (A4)

and

$\centering\sigma(k)=\sqrt{\sum_{j=1}^{N(k)}{\sigma(j,k)^{2}}}.\@add@centering$
(A5)

We note that, for the GCCRs, $s(j,k)$ and $\sigma(j,k)$ correspond to the mean
and 1-$\sigma$ error of the mean, which are determined with the measurements
from the three-epoch observations at 5.5 GHz (Zhao, Morris & Goss, 2020). For
the HCRs, we used Ka-band flux density and 1-$\sigma$ error at 33 GHz (this
paper). The area for the $k$-th AZ can be computed:

$\centering A(k)=\pi\left[R(k)^{2}-R(k+1)^{2}\right].\@add@centering$ (A6)

Furthermore, we also estimated a flux-density-based CRS counts $N_{S}(k)$:

$\centering N_{S}(k)={S(k)\over S_{\mu,5.5\leavevmode\nobreak\ {\rm
GHz}}},\@add@centering$ (A7)

for $k<11$ and

$\centering N_{S}(k)={S(k)\over S_{\mu,33\leavevmode\nobreak\ {\rm
GHz}}}\@add@centering$ (A8)

for $k\geq 11$. Here $S_{\mu,5.5\leavevmode\nobreak\ {\rm GHz}}$ and
$S_{\mu,33\leavevmode\nobreak\ {\rm GHz}}$ are the mean flux-density values
for the GCCRs at 5.5 GHz and HCRs at 33 GHz.

Given the sensitivity cutoff and the variability of the CRSs, the direct
source counts $N(k)$ represents a lower limit of the $k$-th AZ. The flux-
density-based CRS counts $N_{S}(k)$ is usually greater than the direct source
counts, i.e., $N_{S}(k)>N(k)$. Thus, using the averaged values of
$\displaystyle{\left[N_{S}(k)+N(k)\right]/2}$ we derived the surface density
of CRSs as a function of radius from Sgr A*:

$\centering\Sigma_{\rm crs}(k)=\displaystyle{\left[N_{S}(k)+N(k)\right]\over
2A(k)}.\@add@centering$ (A9)

The uncertainty of surface density is estimated as well:

$\centering\Delta_{\Sigma_{\rm
crs}}(k)=\displaystyle{\left[N_{S}(k)-N(k)\right]\over 2A(k)}.\@add@centering$
(A10)

where $k=1,2,3,$ … and 20. Thus, the upper and lower limits in surface density
correspond to the flux-density-based density ($N_{S}(k)/A(k)$) and the actual-
source-counted density ($N(k)/A(k)$), respectively.

We then calculated the mean radius given the $k$-th annular zone:

$\centering R_{\rm AZ}(k)={{R(k)+R(k+1)}\over 2},\@add@centering$ (A11)

and the uncertainty, or more properly half of the annular zone width:

$\centering\Delta_{R_{\rm AZ}}(k)={{R(k)-R(k+1)}\over 2},\@add@centering$
(A12)

where $R(k)$ is defined in Eq.(A1) and $k=1,2,3,$ … and 20.

Fig. A2 shows the distribution of the surface density of CRSs $\Sigma_{\rm
crs}(R)$ as a function of radial distance ($R$) from Sgr A*. The black symbols
represent the data at a large projected radius ($R>13$”), determined from the
JVLA observations at 5.5 GHz (Zhao, Morris & Goss, 2020) while the symbols for
the inner annular zones ($R\leq 13$”) mark the data derived from the Ka-band
observations at 33 GHz. We note that the CRS counts, as well as the surface
density $\Sigma_{\rm crs}$ (light-grey symbols in Fig. A2), may be
underestimated within the inner annular zone (AZ=11 - 20) due to the fact that
the data HCRs observed at 33 GHz corresponds to a higher effective cutoff
limit at 5.5 GHz since most of the HCRs have a steep spectrum. With an
averaged spectral index $\alpha_{\rm HCRs}=-0.42\pm 0.12$ using all 64 HCRs’
data, we converted the 33-GHz flux densities to 5.5-GHz. The equivalent cutoff
limit for the inner AZs is about twice as high as the limit searched for GCCRs
in the RBZ (Zhao, Morris & Goss, 2020). So, we multiplied the surface density
of the inner AZ(k$\geq$11) by a correction factor:

$\centering\eta=(5.5/33)^{\alpha_{\rm
HCR}}\left(\sigma_{33}\over\sigma_{5.5}\right)\approx 2.4,\@add@centering$
(A13)

assuming that the HCR counts are inversely proportional to the cutoff value in
flux density. The corrected surface density is then computed:

$\centering\Sigma^{*}_{\rm crs}=\eta\Sigma_{\rm crs}\@add@centering$ (A14)

as indicated with the red symbols which appear to be aligned better with the
surface density data of the outer radii obtained from the observations at 5.5
GHz.

Figure A1: The radio bright zone (RBZ) is divided into 20 annular zones (AZ)
along the radial direction from Sgr A*. Spatial distribution of GC compact
radio sources or GCCRs (light blue symbols, crosses for variables and dots for
non-variables) in annular zones is displayed over the RBZ (AZ 1 - 6) (top
panel: Fig. A1a) and SgrA West (AZ 7-11) (middle panel: Fig. A1b). The HCRs
(light blue dots) are distributed in the central parsec (AZ 12-20) (bottom
panel: Fig. A1c). The thick red ring marks the 11-th ring (Eq. A2),
$R(11)=13$”, $\sim 0.5$ pc. In Figs. A1a and A1b, the symbol of a black cross-
dot marks the Galactic center transient, the GCT (Zhao et al., 1992). The
magnetar J1745-2900 and IRS 13E are labelled in Fig. 1Ac.

###  A.2 Power-law fit to the $\Sigma_{\rm crs}$ distribution

The surface-density $\Sigma_{\rm crs}$ data at large radii (from AZ01 to AZ09)
can be described by a power law while the data at small radii (from AZ10 to
AZ18) show a bump ($\Sigma_{\rm crs}$-bump), indicating a locally enhanced
source density $\Sigma_{\rm crs}$ at the radial distance range of
$R=1.5$”$-7.0$” (from AZ-17 to AZ-14) superimposed on a slope. We made a
linear regression fit to a function:

$\centering Log_{10}(\Sigma_{\rm crs})=a_{\rm
GCCR}-\beta\/Log_{10}(R),\@add@centering$ (A15)

using the outer nine AZ’s data points (GCCRs) with a weight of
$wt(i)=\Delta^{-2}_{\Sigma_{\rm crs}}(k)$. The best fit derived from the least
squares regression (LSR), as indicated by a straight blue line in Fig. A2,
gives a power law function $\Sigma_{\rm crs}\propto\/R^{-\beta}$ ($R>13$”)
with $\beta=2.0\pm 0.1$ and $a_{\rm GCCR}=4.3\pm 0.2$.

For the inner eight annular-ring data (HCRs), we excluded the four data points
that are involved in the $\Sigma_{\rm crs}$-bump (from AZ14 to AZ17) and fit
the remaining four data points with a linear function:

$\centering Log_{10}(\Sigma_{\rm crs})=a_{\rm
HCR}-\Gamma\/Log_{10}(R).\@add@centering$ (A16)

Based on a least-squares fitting, we derived $\Sigma_{\rm
crs}\propto\/R^{-\Gamma}$ ($R\leq 13$”) with $\Gamma=1.6\pm 0.2$ and $a_{\rm
HCR}=3.7\pm 0.2$, as indicated with the straight orange line in the
logarithmic plot (Fig. A2a). The $\Sigma_{\rm crs}$-bump is characterized by
the high surface-density zones AZ15 and AZ16; the former contains the GC
magnetar J1745-2900 and the latter is associated with IRS 13E. We note that
the surface brightness in the diffuse Br$\gamma$ narrow band filter image
(Fig. 6a of Schödel (2018)) also shows a bump in the range of radius between
$R=1.5$” and $7.0$”, suggesting that the counts of CRSs in the radial range
may be partially caused by the local hyper compact HII sources associated with
younger massive stellar objects. However, excluding the flat-spectrum HCRs,
the $\Sigma_{\rm crs}-$bump is also present (see Fig. A2b). We further checked
the distribution only with the steep-spectrum HCRs and found that the
$\Sigma_{\rm crs}-$bump is still present in the radial range between $R=1.5$”
and $7.0$” (see Fig. A2c).

Due to a small number of data points and at least four free parameters in the
“Nuker” model of Eq(A1), we made no attempt to fit the entire curve for
deriving the break radius $R_{b}$. However, from the intersection between the
two straight lines of Eq(A15) and Eq(A16), we found an approximate break
radius $R_{b}\approx 30$”, or 1.1 pc, which appears to correspond to the
radius of the ionized region. Therefore, in the main text, we use $R_{b}=30$”
as the break radius that separates the two distinctive power-law
distributions.

Figure A2: Radial distribution of surface density of radio compact sources
detected at the Galactic center. The red points are the surface density data
$\Sigma^{*}_{\rm crs}$ corrected for a higher equivalent sensitivity cutoff at
5.5 GHz by multiplying a correction factor $\eta=2.4$ (see Eqs. A13 and A14)
to the original 33-GHz surface density data $\Sigma_{\rm crs}$ (light-grey
points) derived from the 33-GHz observations of the HCRs (this paper). The
black points mark the data of GCCRs observed at 5.5 GHz. In the annular zones
around $R=3"$, a bump in $\Sigma_{\rm crs}$ is present. The orange line shows
the best fitting to the power-law function with $\Gamma=1.6\pm 0.2$ for the
inner region; and the blue line indicates the least squares fitting to the
steeper power law with $\beta=2.0\pm 0.2$ for the outer region. The break
radius $R_{b}$ is $\sim 30"$, or $\sim 1.2$ pc. Three panel show the surface-
density distributions for the three combinations of spectrum classes of HCRs:
(a) inclduing all the HCRs, (b) excluding the flat-spectrum HCRs, and (c) only
including inverted HCRs.

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