Gaia22dkvLb: A Microlensing Planet Potentially Accessible to Radial-Velocity Characterization

Subo Dong

0009-0007-5754-6206]Zexuan Wu
Department of Astronomy, School of Physics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Kavli Institute of Astronomy and Astrophysics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

0000-0002-1027-0990]Subo Dong
Department of Astronomy, School of Physics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Kavli Institute of Astronomy and Astrophysics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Tuan Yi
Department of Astronomy, School of Physics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Kavli Institute of Astronomy and Astrophysics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Zhuokai Liu
Department of Astronomy, School of Physics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Kavli Institute of Astronomy and Astrophysics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Kareem El-Badry
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

Andrew Gould
Max-Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany

Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA

Grant W. Christie
Auckland Observatory, Auckland, New Zealand
0000-0001-8179-1147]L. de Almeida
Laboratório Nacional de Astrofísica, Rua Estados Unidos 154, 37504-364 Itajubá – MG, Brazil
L. A. G. Monard
Klein Karoo Observatory, Centre for Backyard Astrophysics, Calitzdorp, South Africa
J. McCormick
Farm Cove Observatory, Centre for Backyard Astrophysics, Pakuranga, Auckland, New Zealand
Huiling Chen
Department of Astronomy, School of Physics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Kavli Institute of Astronomy and Astrophysics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Yang Huang
University of Chinese Academy of Sciences, Beijing 100049,  China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012,  China

0000-0002-7866-4531]Chang Liu
Department of Physics and Astronomy, Northwestern University, 2145 Sheridan Rd, Evanston, IL 60208, USA
Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA
0000-0003-2125-0183]A. Mérand
European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany
0000-0001-7016-1692]Przemek Mróz
Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4,00-478 Warszawa, Poland
Jinyi Shangguan
Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748, Garching, Germany
0000-0001-5207-5619]Andrzej Udalski
Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4,00-478 Warszawa, Poland
J. Woillez
European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany
Huawei Zhang
Department of Astronomy, School of Physics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

Kavli Institute of Astronomy and Astrophysics, Peking University,
Yiheyuan Rd. 5, Haidian District, Beijing, China, 100871

We report discovering an exoplanet from following up a microlensing event alerted by Gaia. The event Gaia22dkv is toward a nearby disk source at $\sim 2.5$ kpc rather than the traditional bulge microlensing fields. Our primary analysis yields a Jovian planet with $M_{\rm p} = 0.50 \pm 0.05\,M_{\rm J}$ at a projected orbital separation $r_{\perp} = 1.63\pm0.17$ AU. The host is a turnoff star with mass $1.24\pm{0.06}\,M_\odot$ and distance of $1.35\pm0.09$ kpc, and at $r'\approx14$, it is far brighter than any previously discovered microlensing planet host, opening up the opportunity of testing the microlensing model with radial velocity (RV) observations.
RV data can be used to measure the planet's orbital period and eccentricity, and they also enable searching for inner planets of the microlensing cold Jupiter, as expected from the “inner-outer correlation” inferred from Kepler and RV discoveries.
Furthermore, we show that Gaia astrometric microlensing will not only allow precise measurements of its angular Einstein radius $\theta_{\rm E}$, but also directly measure the microlens parallax vector and unambiguously break a geometric light-curve degeneracy, leading to definitive characterization of the lens system.

§ INTRODUCTION

Almost 200 exoplanets have been detected with gravitational microlensing. In a microlensing event, the light from a background star (the “source”) is bent in the gravitational field of the foreground mass (the “lens”), and a planet in the lens system can be detected from a short-lasting signal in the microlensing light curve. Most of the exoplanets discovered with microlensing are near or beyond the snow line, and the majority of their host stars are significantly more distant than $\sim$ 1 kpc. In contrast, other detection methods overwhelmingly probe nearby hosts, and the two most prolific methods at present, transit and radial velocity (RV), predominantly find planets orbiting within the snow line. On the one hand, these distinguishing properties of the microlensing planetary systems provide a uniquely powerful probe of the Galactic exoplanet demography (see Zhu & Dong, 2021 for a recent review). On the other hand, microlensing planetary systems are generally inaccessible to follow-up observations by other methods (except for a handful of potentially possible cases near the sensitivity limit of present RV capabilities, such as OGLE-2018-BLG-0740Lb at $V\sim18$ by Han et al., 2019), posing a limitation to further characterizing the detected planets and probing the existence of other planets in those systems.

The recent surge of time-domain surveys covering a considerable portion of the sky are opening up fresh possibilities of searching for microlensing planets in regions other than the traditional Galactic bulge fields.
The first such discovery was a Neptune-mass planet inside the snow line [Nucita et al., 2018, Fukui et al., 2019] found in a near-field (source at $\approx 660$ pc) microlensing event Kojima-1 (a.k.a., TCP J05074264+2447555), whose microlensing nature was recognized with the survey data of All-Sky Automatic Survey for Supernovae (ASAS-SN; Shappee et al., 2014, Kochanek et al., 2017). The interferometric observations [Dong et al., 2019] carried out by the GRAVITY instrument [GRAVITY Collaboration et al., 2017] mounted on the Very Large Telescope Interferometer (VLTI) enabled measuring the physical parameters of Kojima-1, revealing that the planet has mass $M_{\rm planet} = 19.0\pm3.0\,M_\oplus$ at projected planet-host separation $r_\perp = 0.78\pm0.05\,{\rm AU}$ orbiting the host star with mass $M=0.495\pm0.063\,M_\odot$ at $429\pm21\,{\rm pc}$ [Zang et al., 2020].

In this paper, we report the discovery of an extrasolar planet from our follow-up observations of a near-field microlensing event Gaia22dkv detected by the Gaia mission, which has found hundreds of microlensing events over the entire sky (see, e.g., Wyrzykowski et al., 2023). Our primary analysis suggests that Gaia22dkvLb is a Jovian planet inside the snow line, and it is potentially accessible to high-precision radial velocity (RV) facilities.

§ OBSERVATIONS AND DATA REDUCTIONS

Gaia22dkv's light curve and the best-fit planetary microlensing model (black solid line) considering microlens parallax and orbital motion effects. Data points observed by various sites are shown as open circles in different colors respectively. The magnitudes are transformed into $i'$ band using the magnifications from the best-fit model. The inset panel presents the zoomed-in view of the region in the neighborhood of the planetary anomaly. The grey line represents the best-fit binary-source (1L2S) model, which fail to match the anomaly.

Gaia22dkv was discovered by Gaia Science Alerts
[Gaia Collaboration et al., 2016] on 2022 August 16 UT and announced[<http://gsaweb.ast.cam.ac.uk/alerts/Gaia22dkv>] as a microlensing candidate  on 2022 August 19 UT.
The event's equatorial coordinates are
$\left(\alpha,\,\delta\right)_\mathrm{J2000} = \left(10^{\rm h}07^{\rm m}04.56^{\rm s},-66\arcdeg10\arcmin51.20\arcsec\right)$
, corresponding to Galactic coordinates
$\left(l,\,b\right)_\mathrm{J2000} = \left(287.36783 ,\, -8.41019\right)$.

We followed up Gaia22dkv as part of an ongoing program to identify  and characterize bright and long timescale microlensing events to be observed by VLTI-GRAVITY, and the main scientific goal of the program is to search for isolated dark stellar remnants (i.e., neutron stars and black holes).  Our program started in 2018, and we have systematically processed microlensing alerts and performed photometric follow-up observations using the 1-m telescopes of Las Cumbres Observatory global network (LCOGT, Brown et al., 2013) and several small telescopes.

At the time of the Gaia alert, Gaia22dkv was not accessible to LCOGT due to its high airmass. We began to take images in the $Blue$ and $Red$ bands (denoted as $B'$ and $R'$, respectively, throughout the paper) of the Astrodon LRGB filters on 2022 August 20 UT ($\mathrm{HJD}' \coloneqq \mathrm{HJD} - 2450000\approx 9811.89$) using the 0.18-m Newtonian telescope (CHI-18) located at the El Sauce Observatory in Chile on every clear night. We also performed imaging observations with the 40-cm telescope at the Auckland Observatory (AO) in SDSS $i'$ band since 2022 August 22 UT. Our LCOGT observations started on October 06 2022 with a daily cadence in SDSS $i'$ and $r'$ bands.

VLTI-GRAVITY observations with the Auxiliary Telescopes (ATs) under the Target-of-Opportunity (ToO) program 108.220D.007 were carried out on 2022 November 3, 2022 November 30, and 2022 December 15. No fringe was found during the first epoch, and for the latter two epochs, fringes were present only for two baselines, inadequate for inferring the closure phase. Therefore,  we obtain no useful VLTI data.

As the brightness of Gaia22dkv continued to decline and approached the baseline, the prospect of obtaining useful VLTI data with ATs diminished. On 2023 February 01, we thus decided to change the LCOGT observations to a weekly cadence, while the cadence of CHI-18 observations stayed unchanged. We perform real-time, automatic photometry on the LCOGT images. From an inspection of the LCOGT photometry on UT 2023 March 01 ($\mathrm{HJD}' \approx 10004.5$), we noticed a deviation from the single-lens microlensing model. We subsequently confirmed the anomaly using the CHI-18 data after processing the images. On the same day, we alerted to Microlensing Follow-Up Network ($\mu$FUN) requesting high-cadence follow-up observations. Three sites associated with $\mu$FUN observed this event, including the 0.36-m telescope at the Farm Cove Observatory (FCO) with clear ($C$) filter, the 0.36-m telescope at the Klein Karoo Observatory (KKO) with clear ($C$) filter and the 0.6-m and 1.6-m telescopes in Pico dos Dias Observatory (OPD) with Johnson-Cousin $I$-band filter.

We process the images using the  photometric pipeline [Chen et al., 2022], which has a Python interface wrapping several external software packages. We conduct aperture photometry on LCOGT, CHI-18, FCO, and OPD data and point-spread-function (PSF) photometry on the AO data using
[Schechter et al., 1993]. The Gaia $G$ photometric data downloaded from the Gaia Science Alerts website do not have photometric uncertainties, and we adopt $\sigma_G = 0.02$ as suggested by Hodgkin et al., 2021 for a bright source ($G \lesssim 13$). When modeling the light curves, we employ the commonly-adopted procedure of rescaling the photometric errors <cit.> to reach $\chi^2 / \mathrm{dof} \approx 1$ for each data set.

On 2023 April 23 UT, we took spectroscopic follow-up observation using the Magellan Inamori Kyocera Echelle <cit.> spectrograph
mounted on the Magellan 6.5-m telescope. We used a 0.5  slit and took a 900 s exposure. We employ the CarPy [Kelson et al., 2000, Kelson, 2003] package to reduce the raw spectroscopic data, following the standard procedures of bias subtraction, flat-field correction, automatic cosmic-ray removal, 1-D spectral orders extraction, wavelength calibration, order merge, and applying heliocentric RV correction. Then we manually identify and remove un-cleaned cosmic-rays and detector defects. The resulting spectrum covers 3400 Å  - 9000 Å, with a spectral resolution $R\sim40000$ and median signal-to-noise ratio ${\rm SNR}\sim100$ on the red side and $R\sim55000$ with median ${\rm SNR}\sim38$ on the blue side.
Because the SNR is relatively low at blue wavelengths $\lambda\lesssim3800$ Å and has serious telluric contaminations on the red side with $\lambda\gtrsim6800$ Å, we use the spectrum between 3800 Å  – 6800 Å  for subsequent analysis.

Two degenerate microlens parallax solutions from PSPL analysis of the light curve with the region in the neighborhood of the planet anomaly removed. The upper and the lower panels display the $u_0+$ and $u_0-$ grid-search results, respectively. The solutions with $\Delta \chi^2 < 1,\, 4,\, 9,\, 16$ and $25$ of the best fit are color-coded with red, yellow, green, blue, and grey. The insets display zoomed-in views.

The $\Delta \chi^2$ map of the binary-lens point-source grid-search results on the $\left(\log s,\, \log q\right)$ and $\left(\alpha,\, \log q\right)$ planes.
The upper and lower panels show $u_0+$ and $u_0-$ grid searches, respectively.
The solutions with $\Delta \chi^2 < 50,\, 200,\, 450,\, 800,\, 1250$ and $>1250$ compared with the best fits ($\chi^2_{\rm min}\approx868$ for $u_0+$ and  $\chi^2_{\rm min}\approx690$ for $u_0-$) are color-coded with red, yellow, green, blue, grey, and light grey, respectively.

§ MICROLENS MODELING

The overall light curve of Gaia22dkv (see Figure <ref>) has a striking asymmetry, with the post-peak declining at a significantly steeper rate than the pre-peak rising. As discussed later in this section, this long-term deviation from the standard, symmetric point-source point-lens (PSPL) model [Paczynski, 1986] is well explained by the annual microlens parallax effect due to the Earth's orbital motion [Gould, 1992].  There is a short-duration anomaly on the light curve, and in the following, we show that it is the type of planetary microlensing signature as discussed in Gould & Loeb, 1992.

Before performing the more complicated modeling necessary to interpret the anomaly, we fit the light curve by excluding the region in the neighborhood of the anomaly with $\mathrm{HJD}'=[9980, 10020]$
to PSPL models, including microlens parallax. The PSPL model includes 5 parameters $\Theta_{\rm PSPL} = \left(t_0,\, u_0,\, t_\E,\, \pi_{\mathrm{E,N}}, \pi_{\mathrm{E,E}}\right)$ to compute the magnification $A$ of the background source star as a function of time $t$. $\left(t_0,\,u_0,\,t_\E\right)$ denote the time of the closest source-lens approach, the impact parameter normalized by the Einstein radius, and the timescale to cross the Einstein radius, respectively. The microlens parallax vector $\bpi_\E=\left(\pi_\mathrm{E,N}, \pi_{\mathrm{E,E}}\right)$ is expressed in terms of its north and east equatorial coordinates, as defined in Gould, 2004. The observed flux $f_i\left(t\right)$ for each dataset $i$ is modeled as $f_i \left(t\right) = f_{\Sc,i} A \left(t\right)  + f_{\B,i}$, where the source flux $ f_{\Sc,i} $ and the blended (i.e., un-lensed) flux $ f_{\B,i}$ within the PSF are the two flux parameters. The blended flux is referred to as the “blend” throughout the text. The microlens parallax solutions are subject to a 2-fold discrete “constant acceleration” degeneracy [Smith et al., 2003], and the two corresponding sets of solutions ($u_0+$ and $u_0-$) from grid searches are shown in the upper and lower panels of Figure <ref>, respectively.

In  <ref>, we analyze the light curve using the binary-lens[As a common practice in the microlensing literature, a binary lens is a generic term referring to a lens system with two point masses, including a planetary lens for which the mass ratio is low.] (2L1S) model, and we consider the effects of microlens parallax and planetary orbital motion. We also explore alternative models, including the so-called xallarap effect that can mimic microlens parallax [Griest & Hu, 1992, Smith et al., 2002] in  <ref>, and the single-lens binary-source (1L2S) model that can sometimes produce a planet-like signal [Gaudi, 1998, Han & Gould, 1997, Hwang et al., 2013] in  <ref>, respectively.

2L1S model parameters and $1\sigma$ uncertainties for $u_0+$.

Model     t_0     u_0     t_   logρ    s     q     α    π_E, N     π_E, E     w_1     w_2     w_3     P_⊥

$\chi^2$     (HJD')          (day)                ×10^3     (deg)             yr^-1    yr^-1   yr^-1     yr

LPS$-2.2$     9894.66     0.171     213.1     -1.27     0.648     5.832     8.4     0.138     -0.135     6.00     12.64     6.57     0.35

656.0     ^+0.08_-0.06     ^+0.009_-0.004     ^+4.7_-8.8     <-1.14     ^+0.005_-0.010     ^+0.523_-0.511     ^+0.3_-0.8     ^+0.003_-0.002     ^+0.002_-0.004     ^+0.42_-0.60     ^+0.24_-0.46     ^+0.21_-0.14     ^+0.02_-0.01
LPS$-2.9$     9895.48     0.162     222.6     -1.60     0.679     1.361     -0.3     0.117     -0.124     2.33     6.90     3.77     0.71

660.6     ^+0.05_-0.11     ^+0.006_-0.003     ^+3.8_-7.3     <-1.48     ^+0.004_-0.007     ^+0.160_-0.160     ^+0.3_-0.5     ^+0.002_-0.001     ^+0.002_-0.003     ^+0.29_-0.31     ^+0.25_-0.42     ^+0.16_-0.15     ^+0.05_-0.04
UPS$-3.4$     9894.95     0.173     207.6     -1.81     0.673     0.381     -11.7     0.122     -0.129     0.34     -2.84     0.32     1.81

648.9     ^+0.04_-0.04     ^+0.005_-0.005     ^+5.6_-5.2     <-1.68     ^+0.006_-0.006     ^+0.028_-0.022     ^+0.2_-0.2     ^+0.002_-0.002     ^+0.003_-0.002     ^+0.14_-0.08     ^+0.09_-0.06     ^+0.08_-0.05     ^+0.11_-0.09
UFS$-3.4$     9894.96     0.171     209.9     -1.40     0.685     0.383     -11.4     0.121     -0.128     0.52     -2.75     0.41     1.75

658.8     ^+0.04_-0.03     ^+0.004_-0.005     ^+5.6_-4.4     ^+0.01_-0.01     ^+0.006_-0.005     ^+0.028_-0.024     ^+0.2_-0.2     ^+0.002_-0.002     ^+0.002_-0.002     ^+0.19_-0.08     ^+0.09_-0.08     ^+0.09_-0.05     ^+0.08_-0.14
UFS$-3.5$     9894.92     0.179     201.7     -1.42     0.669     0.350     -12.0     0.124     -0.131     -0.20     -3.13     0.28     1.80

656.4     ^+0.03_-0.04     ^+0.005_-0.004     ^+4.4_-4.6     ^+0.01_-0.02     ^+0.005_-0.005     ^+0.020_-0.020     ^+0.2_-0.2     ^+0.001_-0.001     ^+0.002_-0.002     ^+0.05_-0.06     ^+0.09_-0.07     ^+0.14_-0.07     ^+0.09_-0.05
UPS$-4.3$     9894.92     0.168     212.9     -3.26     0.689     0.050     -8.5     0.126     -0.127     -2.46     -0.66     1.45     1.53

660.6     ^+0.03_-0.05     ^+0.006_-0.005     ^+5.7_-6.1     <-2.32     ^+0.006_-0.007     ^+0.007_-0.005     ^+0.2_-0.3     ^+0.002_-0.002     ^+0.003_-0.003     ^+0.06_-0.06     ^+0.06_-0.06     ^+0.27_-0.06     ^+0.05_-0.05

WPS$-3.9$     9894.84     0.174     207.3     -3.64     1.437     0.129     173.3     0.127     -0.129     -0.12     -0.02     10.27     0.61

664.5     ^+0.04_-0.04     ^+0.006_-0.006     ^+6.6_-6.3     <-2.26     ^+0.016_-0.015     ^+0.010_-0.009     ^+0.2_-0.3     ^+0.002_-0.002     ^+0.003_-0.003     ^+0.07_-0.07     ^+0.05_-0.07     ^+0.05_-0.06     ^+0.004_-0.003
WPS$-3.2$     9894.75     0.177     203.3     -4.55     1.398     0.705     170.9     0.123     -0.128     -2.85     2.38     16.50     0.37

668.3     ^+0.03_-0.02     ^+0.005_-0.003     ^+3.0_-4.5     <-2.03     ^+0.012_-0.006     ^+0.016_-0.032     ^+0.1_-0.2     ^+0.002_-0.001     ^+0.001_-0.002     ^+0.11_-0.01     ^+0.08_-0.04     ^+0.04_-0.17     ^+0.004_-0.001

The “FS” or “PS” contained in labels refer to solutions with significant finite source effect or solutions consistent with points source. For PS solutions, we report the 3 $\sigma$ upper limits in $\log \rho$ column. The wide solutions ($s>1$) are labeled with the prefix “W”. The close solutions ($s<1$) are labeled with the prefix “U”(upper planetary caustic) and “L”(lower planetary caustic) based on which planetary caustic the source trajectory crosses/approaches.

2L1S model parameters and $1\sigma$ uncertainties for $u_0-$.

Model     t_0     u_0     t_   logρ    s     q     α    π_E, N     π_E, E     w_1     w_2     w_3     P_⊥

$\chi^2$     (HJD')          (day)                ×10^3     (deg)             yr^-1    yr^-1   yr^-1     yr

LPS-2.8     9894.46     -0.199     -246.5     -1.47     0.640     1.816     12.0     0.055     -0.286     2.10     4.57     6.21     0.76

671.1     ^+0.11_-0.07     ^+0.006_-0.006     ^+7.2_-8.1     <-1.19     ^+0.007_-0.006     ^+0.130_-0.187     ^+0.3_-0.6     ^+0.001_-0.002     ^+0.005_-0.004     ^+0.24_-0.26     ^+0.20_-0.26     ^+0.30_-0.18     ^+0.02_-0.02

LPS$-3.0$     9895.21     -0.186     253.7     -1.61     0.657     0.987     9.3     0.046     -0.270     1.50     3.06     0.73     1.19

657.1     ^+0.07_-0.09     ^+0.004_-0.011     ^+14.7_-7.1     <-1.39     ^+0.004_-0.011     ^+0.077_-0.042     ^+0.4_-0.3     ^+0.002_-0.002     ^+0.003_-0.006     ^+0.19_-0.21     ^+0.16_-0.10     ^+0.10_-0.91     ^+0.11_-0.60

LFS$-3.0$     9895.26     -0.195     245.1     -1.19     0.655     0.935     9.6     0.045     -0.279     0.92     3.58     0.74     1.32

665.1     ^+0.09_-0.11     ^+0.008_-0.007     ^+8.8_-12.7     ^+0.01_-0.02     ^+0.008_-0.008     ^+0.057_-0.047     ^+0.4_-0.4     ^+0.002_-0.002     ^+0.005_-0.005     ^+0.12_-0.18     ^+0.09_-0.11     ^+0.10_-0.07     ^+0.13_-0.05

UPS$-2.5$     9894.34     -0.179     288.2     -1.57     0.653     2.933     -9.0     0.046     -0.294     4.26     -11.04     4.16     0.42

659.0     ^+0.11_-0.06     ^+0.004_-0.003     ^+4.8_-6.7     <-1.45     ^+0.005_-0.004     ^+0.132_-0.121     ^+0.3_-0.1     ^+0.001_-0.001     ^+0.005_-0.004     ^+0.18_-0.15     ^+0.22_-0.13     ^+0.16_-0.08     ^+0.01_-0.01

UPS$-2.6$     9894.62     -0.166     306.7     -1.54     0.669     2.193     -7.7     0.047     -0.279     3.44     -10.20     4.34     0.48

659.2     ^+0.07_-0.09     ^+0.004_-0.007     ^+11.5_-8.6     <-1.32     ^+0.005_-0.008     ^+0.159_-0.122     ^+0.4_-0.4     ^+0.001_-0.002     ^+0.005_-0.006     ^+0.38_-0.22     ^+0.18_-0.25     ^+0.15_-0.13     ^+0.01_-0.02

UFS$-2.7$     9895.11     -0.179     273.3     -1.33     0.660     1.867     -6.1     0.047     -0.276     1.76     -9.73     9.08     0.46

663.5     ^+0.08_-0.08     ^+0.005_-0.005     ^+7.8_-8.5     ^+0.04_-0.11     ^+0.006_-0.007     ^+0.161_-0.153     ^+0.3_-0.3     ^+0.001_-0.001     ^+0.004_-0.005     ^+0.45_-0.14     ^+0.29_-0.22     ^+0.29_-0.21     ^+0.002_-0.005

WPS$-3.1$     9894.66     -0.172     282.2     -3.32     1.381     0.687     -178.4     0.049     -0.265     -2.72     -0.73     15.89     0.39

671.2     ^+0.05_-0.06     ^+0.004_-0.005     ^+7.7_-7.0     <-2.21     ^+0.013_-0.012     ^+0.097_-0.040     ^+0.2_-0.2     ^+0.001_-0.001     ^+0.005_-0.005     ^+0.14_-0.33     ^+0.11_-0.11     ^+0.55_-0.15     ^+0.004_-0.014

WPS$-3.2$     9894.64     -0.177     276.4     -4.89     1.399     0.570     -178.2     0.050     -0.270     -2.24     -0.92     14.98     0.41

671.2     ^+0.06_-0.05     ^+0.005_-0.006     ^+8.1_-7.1     <-2.21     ^+0.017_-0.013     ^+0.040_-0.064     ^+0.2_-0.2     ^+0.001_-0.001     ^+0.005_-0.006     ^+0.25_-0.15     ^+0.10_-0.11     ^+0.17_-0.39     ^+0.01_-0.01

(For $u_0+$ cases) The source trajectories and light curves in the neighborhood of the planet anomaly for several $u_0+$ solutions. Left:
The upper and middle panels show the caustics and source trajectories for solutions UPS$-4.3$ and UPS$-3.4$ (the global best-fit solution), respectively. The bottom panel displays the caustics and trajectories for the UFS$-3.4$ (solid lines) and UFS$-3.5$ (dashed lines) solutions, which have similar binary-lens configurations but different parameters for orbital motion. The trajectories with arrows indicating the direction of motion are plotted with respect to the binary-lens axis. The insets of each sub-panel on the left display the caustics (blue) and source position (red) at $t_{0,\mathrm{kep}} - 1\,{\rm d}$, $t_{0,\mathrm{kep}}$ and $t_{0,\mathrm{kep}} + 1\,{\rm d}$ with increasing opacities. The radii of the red circles indicate the best-fit source sizes (upper limits) for the FS (PS) solutions.
Right: The light curves and best-fit models of the four solutions (UPS$-4.3$: dotted line, UPS$-3.4$: solid lines, UFS$-3.4$: dashed lines, UFS$-3.5$: dotted-dashed lines) are shown in the upper panel. The residuals are displayed in the lower panel. The globally best-fit solution has the only physically reasonable circular orbital parameters (see discussion in  <ref>).

(For $u_0-$ cases) The source trajectories and light curves for three $u_0-$ solutions. Left: The source trajectories and caustics for the LPS$-3.0$ (upper), LFS$-3.0$ (middle) and LPS$-2.8$ (lower) solutions. Right: The light curves in the neighborhood of the planet anomaly and the best-fit models (upper) and corresponding residuals (lower) for the LPS$-3.0$ (dotted lines), LFS$-3.0$ (dotted-dashed lines) and LPS$-2.8$ (dashed lines) solutions.

§.§ 2L1S Modeling

We start the 2L1S modeling analysis by employing the static binary lens model, which includes three parameters $(s,\, q,\, \alpha)$ to describe the binary systems: $s$ is the projected separation between the binary components normalized to the angular Einstein radius $\theta_\E$, $q$ is the binary mass ratio,
and $\alpha$ is the angle between the source trajectory and the binary-lens axis. An additional parameter relative to the PSPL model is $\rho = \theta_{*} / \theta_\E$, which is the angular source radius $\theta_{*}$ normalized by $\theta_\E$, and it is used to model the finite-source effects when the light curve deviates from the point-source approximation ($\rho=0$).

We use the  package
[Bozza et al., 2018]
to compute the binary-lens magnification $ A \left( t;\, \Theta_{\rm 2L1S}\right)$ at a given set of model parameters $\Theta_{\rm 2L1S}$ and time $t$.  uses the optimized root solver by Skowron & Gould, 2012 to perform the point source (PS) calculations, and the advanced contour integration algorithm [Bozza, 2010] for calculating the finite source (FS) effects.

We probe the parameter space of 2L1S models on a fixed
$\left(\log s,\,\log q ,\, \alpha,\, \log \rho\right)$ grid of $-0.8 \le \log s \le 0.8 ,\, -5 \le \log q \le 0 $ and $ 0 \le \alpha \le 2\pi$ with $81$, $51$ and $181$ equally spaced values, respectively. We set $\left(t_0 ,\,  u_0 ,\,  t_\mathrm{eff} \equiv u_0 t_\E\right)$ free, which are fitted using Markov Chain Monte Carlo (MCMC) with the  ensemble sampler
[Foreman-Mackey et al., 2013], and their initial values are seeded using their PSPL model estimates. The flux parameters $(f_{\Sc,i}, f_{\B,i})$ for site $i$ are determined by linear fits <cit.>. The parallax parameters $\left(\pi_\mathrm{EN} ,\, \pi _\mathrm{EE}\right)$ are held fixed using the PSPL values.  Due to the two-fold degeneracy for microlens parallax, we perform two sets of grid searches for $u_0+$ and $u_0-$ solutions separately.

We conduct the 2L1S grid search for PS models and FS models fixed at $\log\rho=\{-2.5,\, -1.5\}$, respectively. Figure <ref> shows the PS grid search results with multiple minima in the planetary regime ($\log q\lesssim -2$). The FS grid searches yield local minima in similar $(\log s,\,\log q ,\, \alpha)$ regions.
The $u_0+$ and $u_0-$ grid-search search results, as separately shown in the upper and lower panels of Figure <ref>, exhibit different patterns. The light-curve profile of the planetary anomaly critically depends on the geometry of the source-lens trajectory when crossing/passing by the caustics, which differs for the $u_0+$ and $u_0-$ cases due to the presence of strong microlens parallax, giving rise to different perturbations.

We next analyze the binary orbital motion, which can also introduce significant distortions in the source-lens trajectory. We assume a circular orbit and use the parameterization adopted in , which  includes three parameters $\left(w_1, \, w_2,\, w_3\right)$ describing the first derivatives of $s$, $\alpha$ and the relative radial velocity at a reference time  $t_{0, \mathrm{kep}}$, respectively:

\begin{equation*}
w_1  = \frac{1}{s}\frac{\mathrm{d} s}{\mathrm{d}t }, \quad
w_2  = \frac{\mathrm{d} \alpha}{\mathrm{d}t }, \quad
w_3  = \frac{1}{s}\frac{\mathrm{d} s_z}{\mathrm{d}t }.
\end{equation*}

See Skowron et al., 2011 and Bozza et al., 2021 for relevant discussions on the parameterization. We choose the reference time ($t_{0, \mathrm{kep}} = \mathrm{HJD}' = 10001.8$) to be around the peak of the planetary bump.
We seed all local minima from the static-binary grid searches to investigate the lens-orbital motion effect using MCMC. They eventually converge to 16 solutions, including 8 for $u_0+$ and 8 for $u_0-$, which are listed in  Table <ref> and <ref>, respectively.

We classify these solutions according to two types of degeneracies and label them accordingly in the first columns of Table <ref> and <ref>.
The first type is a discrete degeneracy regarding the source size. For most solutions, the planetary bump is consistent with a cusp approach by a point source (i.e., $\rho=0$ at $<3\,\sigma$ level), and they have “PS” in the labels. In the table, we report the best-fit $\log\rho$ values and their $3\,\sigma$ upper limits. For other solutions, the source is resolved via a caustic crossing, corresponding to those with “FS” in the labels. The second type of degeneracy is related to the caustic topology. There is a well-known degeneracy between close ($s<1$) and wide ($s>1$) binaries [Griest & Safizadeh, 1998, Dominik, 1999, An, 2005]. The wide-binary solutions are labeled with the prefix “W”. For a close binary, there are a pair of planetary caustics, and depending on whether the source trajectory crosses/approaches the upper or lower planetary caustics, we label the solution with the prefix “U” and “L”, respectively.
We also distinguish the solutions by putting their $\log q$ values with one significant digit as the suffix.

The $\chi^2$ value for each solution is given below its label in the first columns of Table <ref> and <ref>. The globally best-fit solution UPS$-$3.4($u_0+$) is a point-source ($\rho<0.02$ at the 3 $\sigma$ level), close-binary model with $q\approx4\times10^{-4}$. There are three other solutions within $\Delta\chi^2<9$, including a finite-source, close-binary model ($u_0+$, UFS$-$3.5) with $\rho\approx0.038$. All 16 solutions have $\Delta\chi^2<23.3$, with mass ratios covering factor of $\sim120$ $(5\times10^{-5}\lesssim q \lesssim6\times10^{-3})$. We display several models and their corresponding source trajectories with their respective caustics for the $u_0+$ and $u_0-$ cases in Figure <ref> and <ref>, respectively. The circular orbital motion is subject to physical constraints, and as discussed in  <ref>, only the globally best-fit solution UPS$-$3.4($u_0+$) is physically reasonable.

The planetary anomaly occurred follows a long interval $(\sim 100\, \mathrm{d})$ after the event peak, which is a substantial fraction of the orbital period (e.g., circular orbital period $P_{\rm circ}=3.15^{+0.42}_{-0.22}$ yr for the solution UPS-3.4($u_0+$)). This confirms the necessity to use the full orbital parametrization with three parameters rather than the linear approximation with two parameters. The orbital motion parameters may sometimes be strongly correlated with binary-lens parameters; for example, $q$ differs significantly between static models and those considering binary orbital motions in the analysis by Han et al., 2022.
This raises a possible concern that our preceding grid search based on static binaries may miss local minima by not including orbital motion effects. The long-term deviations from rectilinear motion for Gaia22dkv are dominated by the strong microlens parallax, which is taken into account in the static binary search, so the orbital motion effects are unlikely to bias the solution search. Nevertheless, we address this possible concern by performing new grid searches on a unidimensional grid of $\log q$ with the lens-orbital motion effects included. The $\left(\log s,\, \alpha\right)$ are seeded based on the caustic geometry and the U/L degeneracies. We also attempt various initial $\log \rho$ values corresponding to the PS/FS degeneracy. However, no new local minima are found from our extensive searches.

The black crosses connected by solid lines show the $\chi^2$ distributions for best-fit xallarap solutions at fixed binary-source orbital periods $P$. The red open circle at $P=1$ yr shows the $\chi^2$ for the best-fit microlens parallax solution.

Results of xallarap fits by fixing $\beta$ and $\lambda$ at period $P=1\,\mathrm{yr}$ and setting $u_0 > 0$. The colors marked as red, yellow, green, blue and grey represent solutions with $\Delta \chi^2 < 1,\, 4,\, 9,\, 16$ and $25$ of the best fit.
The ecliptic coordinates of Gaia22dkv, which represents the Earth parameter, are indicated by the black circles, with the lower and upper circles corresponding to the $u_0>0$ and $u_0<0$ cases due to perfect symmetry, respectively.

§.§ Xallarap effects

If the source is in a binary system, its orbital motion can induce distortion of the microlensing light curve [Griest & Hu, 1992, Han & Gould, 1997]. This so-called xallarap effect can perfectly mimic parallax signals when mirroring Earth's orbital parameters <cit.>. However, such finely-tuned parameters are a priori unlikely when randomly drawn from broadly distributed binary star orbital distributions. We follow the procedures of the test in Dong et al., 2009 to assess whether the xallarap interpretation is preferable: the xallarap interpretation would be favored if the xallarap solution differed significantly from the Earth and having substantial $\chi^2$ improvements over microlens parallax. Otherwise parallax would be preferred.

We assume that the source is on a circular orbit, which requires 5 additional parameters to describe, namely
the period $P$ of the binary motion, the ecliptic longitude and latitude of the binary source orbit $\left(\lambda,\, \beta\right)$ and the xallarap vector's components $\left(\xi_\mathrm{E,N},\, \xi_\mathrm{E,E}\right)$.

The effects of binary-lens orbital motion can be entangled with parallax/xallarap effects, and to avoid such complications, we fit the light curve to PSPL models.
Like the PSPL+parallax modeling, we mask the light-curve region in the neighborhood of the anomaly region and perform PSPL+xallarap fitting.
We take advantage of the exact degeneracy
$\alpha' = \alpha + 180^\circ,\, {\bdv\xi}_\E = -{\bdv\xi}_\E$
[Poindexter et al., 2005]
and restrict our search to solutions with $180^\circ \le \alpha < 360^\circ$, $-90^\circ<\delta\le 90^\circ$.
We fix $\left(\lambda,\,\beta,\,\log P\right)$ on a dense grid and allow $\left(\xi_\mathrm{E,N},\, \xi_\mathrm{E,E}\right)$ together with PSPL parameters to vary freely in MCMC.
Compared to parallax, the best-fit xallarap solution improves the fit by $\Delta \chi^2 \sim 12$. We find that most of the improvement comes from dataset CHI-18 $R'$ ($\Delta \chi^2 \sim 9$). In contrast, there is only a minor improvement ($\Delta \chi^2 \sim 2$) in CHI-18 $B'$ despite their essentially identical coverages and similar photometric precision. This indicates low-level systematics commonly seen in parallax/xallarap modelings [Poindexter et al., 2005]. To make a further check, we redo the modeling but with only Gaia and LCOGT datasets. The best-fit xallarap solution  improves the fit by a mere $\Delta \chi^2 = 3.5$ with extra 3 degrees of freedom (dof), which has a false alarm probability of $>15\%$, so it is not statistically significant. Figure <ref> shows the $\chi^2$ values of best-fit xallarap solutions as a function of period $P$, and the $\chi^2$ minima are in the range of $P \in \left[1,\, 1.4\right]\,\mathrm{yr}$, which is consistent with Earth's orbital period of 1 yr.
Figure <ref> shows the $\left(\lambda,\,\beta \right)$ $\chi^2$ map for the xallarap solutions at  1 yr period, and the Earth's parameters (black open circle) have $\Delta \chi^2 < 4$.
Given the minor $\chi^2$ improvement and the similarity between xallarap parameters with those of the Earth,
we favor the microlens parallax interpretation over xallarap. Nevertheless, our test does not rule out xallarap or a combination of xallarap and parallax.

§.§ 1L2S Modeling

In this section, we analyze the data using a 1L2S model, involving a stationary, luminous binary companion of the source and a single lens, to assess whether it can explain the short-duration anomaly. The magnification $A$ in the 1L2S model is the flux-weighted mean of the respective PSPL magnifications of the two sources,

\begin{equation}
A_j=\frac{A_1+q_{F,\,j} A_2}{1+q_{F,\,j}},
\end{equation}

where $q_{F,\, j}$ is the flux ratio in the passband $j$ between the two sources. Note that the 1L2S magnification is wavelength-dependent unless the two sources have strictly the same color.

We first perform a linear regression between $R'$ and $B'$ band fluxes and find no evidence for deviation from a constant color, indicating that the two sources, if they existed, would have insignificant color differences.
Therefore, we use the same flux ratio $q_F$ for all bands. Since
the light curve is well-characterized by 2L1S models with no obvious residual; if the light curve in one band, say $B'$-band, is well-fitted by a 1L2S model, the other bands should achieve a similar goodness-of-fit with a flux ratio identical to $q_{F,\, B'}$.
For a 1L2S model to re-produces a short-duration planet-like bump, a high peak magnification of the fainter source is often needed, so we include the finite-source effect, which could be necessary for the high magnification.
The best-fit 1L2S model is worse than the best-fit planetary model UPS$-$3.4($u_0+$) by $\Delta \chi^2=565$ and is shown (grey curve) in Figure <ref>. In particular, the 1L2S model fails to explain the region preceding the rising wing of the anomaly, which is the characteristic negative excess of a planetary anomaly as discussed in  [Gaudi, 1998]. We conclude that the 1L2S interpretation is ruled out.

Physical parameters derived with finite-source effects for FS solutions

Solution    M_Ł    M_p     r_⊥    μ_N, hel     μ_E, hel     D_Ł    θ_    P_⊥

M_⊙     M_J     AU     mas yr^-1     mas yr^-1      kpc     mas     yr

2*$u_0+$   UFS$-$3.4    0.407_+0.015^-0.010     0.166_+0.012^-0.008     0.637_+0.022^-0.015     1.333_+0.055^-0.035     -0.845_+0.026^-0.037     1.595_+0.005^-0.008     0.584_+0.022^-0.012     0.797_+0.027^-0.018

UFS$-$3.5    0.434_+0.016^-0.011     0.159_+0.005^-0.005     0.672_+0.022^-0.015     1.511_+0.041^-0.035     -0.967_+0.024^-0.024     1.572_+0.006^-0.005     0.640_+0.015^-0.012     0.836_+0.026^-0.019

2*$u_0-$   LFS$-$3.0    0.174_+0.005^-0.006     0.171_+0.010^-0.007     0.411_+0.013^-0.013     0.783_+0.024^-0.022     -0.689_+0.026^-0.037     1.568_+0.008^-0.011     0.398_+0.010^-0.009     0.630_+0.020^-0.021

UFS$-2.7$     0.267_+0.056^-0.038     0.533_+0.130^-0.088     0.606_+0.108^-0.078     1.199_+0.231^-0.186     -0.989_+0.160^-0.198     1.508_+0.035^-0.039     0.612_+0.120^-0.089     0.911_+0.149^-0.110

The $r'-i'$ vs. $i'$ color-magnitude diagram (CMD) using stars within 30 of Gaia22dkv. The red asterisk represents the centroid of the red clump (RC). The blue and green filled circles represent the source and the blend (i.e., the un-lensed light within the PSF), respectively. Our primary interpretation assumes that the blend is the lens.

§ PHYSICAL PARAMETERS

§.§ Source Radius and the Finite-Source Effects

The angular source size can be used as a “ruler” to measure or constrain the angular Einstein radius via the finite-source effects, $\theta_{\E} = {\theta_*}/{\rho}$. We follow the standard procedure described in [Yoo et al., 2004] to estimate $\theta_*$. We construct a color-magnitude diagram (CMD) using stars within 30 of the microlens (see Figure <ref>), and the photometric measurements are taken from the ATLAS All-Sky Stellar Reference Catalog (Refcat2; Tonry et al., 2018) calibrated to the Pan-STARRS1 photometric system [Tonry et al., 2012].  We measure the centroid of the Red Clumps (RCs) at $(r' - i')_\mathrm{RC} = 0.360 \pm 0.016$ on the CMD. We adopt the intrinsic color of the red clump as $\left(r' - i'\right)_\mathrm{RC,\,0}=0.290 \pm 0.041$ based on Ruiz-Dern et al., 2018, and thus the reddening coefficient $E\left(r'-i'\right) = 0.070 \pm 0.044$.  We adopt the empirical selective-to-total extinction coefficients $\left(R_{i'},\,R_{r'}\right) = \left(1.791\,, 2.363\right)$ according to Zhang & Yuan, 2023  for our field, and we find $A_{i'} = 0.219 \pm 0.138$. Assuming that the source (the blue filled circle in Figure <ref>) suffers from the same extinction as the RCs, for the best-fit solution, we derive $\left(r'-i',\, i'\right)_\mathrm{s,\,0} = \left( 0.338 \pm 0.044,\,  13.340 \pm 0.138 \right)$. By using the surface brightness-color relation calibrated in the $i'$ and $r'$ bands [Boyajian et al., 2013, Bachelet et al., 2022],
$\log _{10}\left(2 \theta_*\right)=(-0.298 \pm 0.044)(r'-i')_0^2 +(0.919 \pm 0.058)(r'-i')_0-0.2 i'_0+(0.767 \pm 0.010)$, we find that the source angular radius $\theta_* = 11.9 \pm 0.9 \,\mathrm{\mu as}$.

For the FS solutions (i.e., in which $\rho$ is detected at high significance via caustic crossing), the physical parameters can be unambiguously determined with

\begin{equation}
\quad M_{\L}=\frac{\theta_{\E}}{\kappa \pi_{\E}}=\frac{\theta_*}{\kappa \pi_{\E}\rho}, \quad D_{\L}=\frac{\au}{\pi_{\L}}=\frac{\au}{\pi_{\E} \theta_{\E}+\pi_{\Sc}}.
\end{equation}

where $M_{\L}$ is the lens mass, $\pi_{\L}$ and $\pi_{\Sc}$ are the trigonometric parallaxes of the lens at distance $D_{\L}$ and source at distance $D_{\Sc}$, respectively, and $\kappa = 4\pi G/(c^2 \au ) = 8.14 \,{\rm mas}/M_\odot$ is a constant.
We list the derived physical parameters for these solutions in Table <ref>. They all correspond to low-mass lenses with $\sim 0.2 - 0.4\,M_\odot$ at $\sim 1.5-1.6~\mathrm{kpc}$.

§.§ The Blended Light

From the light-curve analysis, we detect significant blending (the green filled circle in Figure <ref>), which is nearly equally bright in Gaia $G$ ($\sim 14$) as the source. The blended light can be due to the lens, a binary companion to the lens or the source, a random interloper within the PSF, or some combination of these possibilities.

From the Gaia Data Release 3 (DR3) catalog [Gaia Collaboration et al., 2016, Gaia Collaboration et al., 2023], the number density is $9.8\times 10^{-5}\,\mathrm{arcsec}^{-2}$ for nearby field stars with $G < 14$. Thus the probability of a random field star falling in within $1$ of the source is $\approx 3 \times 10^{-4}$, suggesting that the blend is unlikely due to a random interloper.

In the following section, we assume that the blend is the lens and subsequently estimate the lens mass. We briefly discuss the possibilities of the blend as a companion of the source or the lens in  <ref>.

§.§ Blend as the Lens

As discussed in  <ref>, the FS solutions yield M-dwarf lenses at $\sim 1.5-1.6~\mathrm{kpc}$, which are far too faint compared to the blended flux. In this section, we consider the scenario in which blend is the lens for the PS solutions.

Physical parameters derived assuming blend as lens for PS solutions

Solution    M_h     M_p     r_⊥    μ_N, hel     μ_E, hel     D_Ł    θ_    P_⊥

M_⊙     M_J     AU     mas yr^-1     mas yr^-1      kpc     mas     yr

6*$u_0+$       LPS$-2.2$     1.219^+0.056_-0.053     7.469^+0.719_-0.773     1.637^+0.174_-0.142     4.774^+0.273_-0.291     -2.797^+0.200_-0.175     1.307^+0.078_-0.066     1.945^+0.087_-0.075     1.897^+0.262_-0.207

LPS$-2.9$     1.295^+0.038_-0.054     1.763^+0.242_-0.262     1.652^+0.134_-0.096     3.972^+0.207_-0.155     -2.509^+0.103_-0.145     1.344^+0.071_-0.038     1.811^+0.068_-0.063     1.866^+0.204_-0.126

UPS$-3.4$     1.244^+0.057_-0.059     0.496^+0.045_-0.038     1.631^+0.174_-0.141     4.128^+0.230_-0.186     -2.622^+0.134_-0.158     1.351^+0.092_-0.064     1.793^+0.085_-0.069     1.869^+0.268_-0.198

UPS$-4.3$     1.259^+0.049_-0.039     0.066^+0.006_-0.006     1.707^+0.181_-0.117     4.285^+0.223_-0.231     -2.613^+0.144_-0.146     1.348^+0.095_-0.061     1.841^+0.078_-0.063     1.989^+0.288_-0.177

WPS$-3.2$     1.230^+0.058_-0.046     0.990^+0.047_-0.054     3.327^+0.354_-0.249     4.115^+0.159_-0.234     -2.611^+0.150_-0.102     1.351^+0.087_-0.055     1.769^+0.073_-0.061     5.473^+0.751_-0.553

WPS$-3.9$     1.201^+0.040_-0.050     0.109^+0.026_-0.035     3.493^+0.346_-0.247     4.370^+0.215_-0.139     -2.724^+0.095_-0.144     1.322^+0.076_-0.066     1.808^+0.065_-0.054     5.960^+0.839_-0.514

6*$u_0-$       LPS$-2.8$     1.063^+0.058_-0.039     1.975^+0.333_-0.238     1.628^+0.154_-0.098     3.829^+0.320_-0.209     2.893^+0.348_-0.226     1.007^+0.083_-0.073     2.523^+0.181_-0.107     2.016^+0.273_-0.164

LPS$-3.0$     1.063^+0.032_-0.029     1.137^+0.084_-0.078     1.675^+0.145_-0.122     3.509^+0.179_-0.165     2.807^+0.248_-0.214     1.071^+0.070_-0.066     2.396^+0.096_-0.082     2.104^+0.250_-0.206

UPS$-2.5$     1.056^+0.033_-0.026     3.216^+0.197_-0.209     1.772^+0.137_-0.119     4.210^+0.157_-0.164     2.407^+0.119_-0.115     1.057^+0.054_-0.050     2.563^+0.064_-0.065     2.296^+0.233_-0.203

UPS$-2.6$     1.099^+0.049_-0.045     2.568^+0.705_-0.231     1.878^+0.170_-0.161     3.990^+0.251_-0.170     2.263^+0.163_-0.113     1.101^+0.057_-0.058     2.555^+0.088_-0.075     2.455^+0.279_-0.264

WPS$-3.1$     1.082^+0.040_-0.036     0.710^+0.159_-0.169     3.728^+0.260_-0.224     3.491^+0.203_-0.160     2.425^+0.152_-0.136     1.111^+0.057_-0.059     2.409^+0.066_-0.061     6.921^+0.654_-0.587

WPS$-3.2$     1.086^+0.037_-0.030     0.702^+0.196_-0.155     3.776^+0.246_-0.271     3.512^+0.166_-0.186     2.439^+0.164_-0.160     1.119^+0.060_-0.060     2.421^+0.058_-0.074     7.043^+0.649_-0.682

For the PS solutions (i.e., in which $\rho$ is consistent with zero at $<3\,\sigma$), we can constrain the lens mass using the measured blended fluxes and $\pi_\E$.
The $\pi_\E$ measurement constrains $M_\L$ and $\pi_{\rm rel}$:

\begin{equation}
\pi_\E= \sqrt{\frac{\pi_\mathrm{rel}}{\kappa M_\L}} = \sqrt{\left(\frac{1}{D_\L} - \frac{1}{D_\Sc}\right) \frac{\au}{\kappa M_\L}},
\end{equation}

where $\pi_{\rm rel}$ is the relative lens-source parallax. The trigonometric parallax of the baseline object measured by Gaia DR3 is $\pi_\text{base} = 0.58 \pm 0.05 \,\mathrm{mas}$,
which is the flux-weighted mean parallax by the lens and source [Dong et al., 2019],

\begin{equation}
\pi_\text{base} = \eta_G \pi_\text{L} + (1-\eta_G)\pi_\text{S},
\end{equation}

where $\eta_G = f_{\B,G}/(f_{\Sc,G} + f_{\B,G})$ is the fraction of blended flux in Gaia $G$. Gaia DR3 reports that it has a high renormalized unit weight error ${\rm RUWE} = 4.38 \gg 1$,  indicating significant departures from the standard 5-parameter astrometric fitting [Lindegren et al., 2021]. The amplitude of astrometric microlensing as discussed in  <ref> is too small to be responsible for the large RUWE, and in particular, the Gaia DR3 measurements are based on data collected between 25 July 2014 and 28 May 2017, during which, the microlensing astrometric signals were negligible.

Without the single-epoch observations, we cannot assess how the Gaia DR3 parallax is biased. In the following analysis, we inflate the error by a factor of $\sqrt{\rm RUWE}=2.09$ and adopt $\pi_{\rm base} = 0.58\pm0.10$ mas.

The measured blend fluxes place another constraint on $D_\L$ and $M_\L$.
We employ the theoretical stellar isochrones in  package <cit.> to compute the expected lens luminosity for a given lens mass $M_\L$ at a certain stellar age and metallicity [Fe/H]. We sample the MCMC posteriors of blended fluxes in $i'$ and $r'$ bands from the light-curve fitting and add the $\chi^2$ compensations using the measured blended fluxes in both $r'$ and $i'$ bands and expected lens fluxes on a grid of stellar ages and metallicities $\left(\mathrm{Age},\,\mathrm{[Fe/H]}\right)$.
We adopt two error terms $\left(e_\mathrm{cal},\,e_\mathrm{iso}\right)= \left(0.01,\, 0.01 \right)\,\mathrm{mag}$ to take into account of the photometric calibration errors and the theoretical uncertainties of the isochrones. We use the three-dimensional extinction map by Guo et al., 2021 to estimate the extinction corrections at a given lens distance. We note that the result from the extinction map is consistent with that from our CMD analysis discussed in  <ref>. We impose flat priors in metallicities in the range of $\mathrm{[Fe/H]}\in \left\{ -0.2,\, 0,\, 0.2\right\}$ and in stellar age between 1 and 10 $\mathrm{Gyr}$. The $u_0+$ solutions generally yield lens masses $M_\L \sim 1.3\,M_\odot$ and distances $D_\L\sim 1.4\,{\rm kpc}$, whereas the $u_0-$ solutions favor $M_\L\sim 1.1\,M_\odot$  at $D_\L\sim 1\,{\rm kpc}$. Figure <ref> illustrates the constraints on $M_\L - D_\L$ from the flux and the $\pi_{\rm E}$ constraints for the globally best-fit solution, and the best-fit $M_\L$ and $D_\L$ are compatible with the lower limit on $\theta_{\rm E}$  from the $\rho$ upper limit. The metallicities and stellar ages are loosely constrained by the photometric data alone. The best-fit physical parameters derived for the PS solutions are given in Table <ref>.

Physical constraints on $M_\L - D_\L$ for the best-fit solution. The contours are $1\,\sigma$ constraints by jointly fitting lens fluxes using isochrones and microlens parallax $\pi_{\rm E}$. The dashed-dotted, solid and dotted contours are constrained by isochrones at [Fe/H] = $-0.2$, $0$ and $0.2$, respectively. The red and blue lines show the isochrones for constraining fluxes in $i'$ and $r'$ bands, respectively. We display isochrones with stellar ages at 4.0 Gyr and 5.2 Gyr, which are the $1\,\sigma$ lower and upper limit of the posterior, respectively. The results are consistent with those from circular orbital motion using $P_\perp$ (blue).  The green shaded region shows the $1\,\sigma$ exclusion region using the upper limit on $\theta_\E$ by fitting the finite-source effects, and it is also in consistency with the lens' physical constraints.

§.§ Physical Constraints from Circular Orbital Motion

The binary-lens orbital motion can place constraints on physical parameters of the lens system <cit.>. We carry out our analysis assuming circular orbits and evaluate the physical constraints under this assumption. We define the “projected” orbital period parameter $P_\perp$, which is directly related to observables and physical parameters,

\begin{equation}{\label{eqn:pperp_raw}}
P_\perp = (M_\L / M_\odot) ^{-1/2} (r_\perp / \mathrm{AU}) ^{3/2}\,\mathrm{yr},
\end{equation}

where $r_\perp/\au = s \theta_\E/\pi _\L = s\kappa M_\L \pi_\E / \pi_\L $.

$P_\perp$ values derived using the above equation are given in the last columns of Table <ref> and Table <ref> for the FS and PS solutions, respectively.

The orbital period of a circular orbit and projected separation  ${r_\perp}$ can be expressed in terms of $\left(w_1, \, w_2,\, w_3\right)$ as follows,

\begin{equation}{\label{eqn:orbital_formula}}
P_{\rm circ}=\frac{2\pi \sqrt{w_1^2+w_3^2}}{w_3 \sqrt{w_1^2+w_2^2+w_3^2}}, \quad \frac{r_\perp}{r}   = \frac{w_3}{\sqrt{w_1^2+w_3^2}}.
\end{equation}

Given that

\begin{equation*}
P_\perp = P_{\rm circ} \left(\frac{r_\perp}{r}\right)^{3/2},
\end{equation*}

$P_\perp$ is thus related to the orbital parameters $\left(w_1, \, w_2,\, w_3\right)$:

\begin{equation}{\label{eqn:pperp2}}
P_{\perp}=\frac{2\pi }{ \sqrt{w_1^2+w_2^2+w_3^2}}{\left(\frac{w_3}{\sqrt{w_1^2+w_3^2}}\right)}^{1/2}.
\end{equation}

The best-fit $P_\perp$ with uncertainties from the light-curve analysis are given in Table <ref> and Table <ref>. We find that only one solution, UPS$-3.4 (u_0+)$, which is also the globally best-fit solution from the light-curve analysis, has $P_\perp$ value derived from Equation <ref> consistent with that from Equation <ref>. We show that the $P_\perp$ constraint (blue band) from Equation <ref> on $\left(M_\L,\, D_\L\right)$ in Figure <ref>.
All other solutions have discrepancies at $>3\,\sigma$, suggesting that they are unphysical under the assumption of circular orbits.

§ SUMMARY AND FUTURE PROSPECTS

Our main interpretation of Gaia22dkv is under the assumption that the blend is the lens star itself and the microlens parallax effects rather than xallarap induce the light-curve distortions. Assuming circular planetary orbital motion, there are multiple degenerate planetary solutions with $\Delta{\chi^2}\lesssim 20$  from the light-curve analysis. Among them, the globally best-fit solution from the light-curve modeling is the only one consistent with the physical constraints imposed by the flux, parallax and orbital motion. From the best fit, the lens is an $M_\L = 1.244^{+0.057}_{-0.059}\,M_\odot$ star at $D_\L = 1.351^{+0.092}_{-0.064}\,{\rm kpc}$ orbited by a planet with $M_{\rm p} = 0.496^{+0.045}_{-0.038}\,M_{\rm J}$ with a projected orbital separation of $r_{\perp} = 1.631^{+0.174}_{-0.141}$ AU and a circular orbital period $P_{\rm circ}=3.15^{+0.42}_{-0.22}$ yr.

However, one or more of these assumptions may break down. The blend can be the binary companion of either the lens or the source, and the lens could be a low-mass star that is fainter than the blend. The lens could be an M dwarf lens as suggested by the FS solutions, and a low-mass lens could also be compatible with the lower limits on $\theta_{\rm E}$ for the PS solutions. Additionally, the large RUWE from Gaia DR3 might hint at the existence of the binary orbital motion of a companion. Besides the common follow-up observations by ground-based Adaptive Optics (AOs) observations (see Gould, 2022 for comprehensive discussions), there are extraordinary opportunities to robustly test our primary interpretation in the near future.

As discussed in  <ref>, the exceptionally bright blend allows unprecedented opportunity for high-precision (on the order of $\mathrm{m\,s^{-1}}$) RV follow-ups of the planetary signal. And more epochs of spectra with RV precision on the order of $\mathrm{km\,s^{-1}}$ can make stringent tests on the lens/source companion scenarios. The astrometric issues such as the high RUWE will be clarified with the release of single-epoch Gaia astrometric observations in the future. As shown in  <ref>, Gaia astrometric data will not only be able to measure the $\theta_{\rm E}$ precisely, but also break the $u_0+/-$ degeneracy and constrain $\bpi_\E$ to  high precision, so it has the potential of definitively measure the physical properties of the lens.

The upper panel displays the predicted radial velocity curves based on the orbital parameters of the best fit. The black lines represent the MCMC sample with $\Delta \chi^2 < 1$. The lower panel displays the RV semi amplitude $K$ and circular orbital period $P_{\rm circ}$ for all solutions. The best-fit solution is shown as a red filled circle with error bar, which is the only physically reasonable solution under the assumption that the blend is the lens and circular planetary orbital motion. Other solutions are shown as the color-coded open circles in red, yellow, green, blue, grey represents other solutions with $\Delta\chi^2 < 1,\,4,\,9,\,16,\,25$ compared with best fit, respectively. One year period is indicated as dashed line.

The spectral broadening function (BF) for the blue (left), green (middle), and red (right) regions of Gaia22dkv's MIKE spectrum.
We identify two distinct peaks on the BFs, which correspond to the source (larger RV) and the lens (smaller RV) respectively.
In each panel, the dots are the computed BF values to which we fit a function (solid color curve) using the sum of two Gaussians (grey curves). The RVs values (i.e., the centers of the Gaussians) are marked with dashed lines. Note that the BFs include the contributions from the instrumental spectral resolution $\sigma_{\rm instr}\sim 5 \, \mathrm{km\,s^{-1}}$, and thus the stellar line widths are smaller than the Gaussians.

Galactic kinematics of the lens (green dot with error bar) and the source (blue dot with error bar) based on the best-fit solution.
Upper: the Galactocentric distance $R$ and height $z$ with respect to the Galactic plane positions (the red dot represents the Sun at $z_{\odot}$ $=$ 25 pc; Jurić et al., 2008).
Lower:  the Toomre diagram. Dashed curves indicate constant peculiar velocities = 50, 100, 150, and 200 $\mathrm{km~s^{-1}}$.

§.§ RV Expectations and Galactic Kinematics

There have been no microlensing planets characterized by RV observations. The primary difficulty is the faintness of the host (lens) star. One of the brightest microlensing planet hosts was OGLE-2018-BLG-0740 reached $V\sim18$ [Han et al., 2019], which is near the limit of achieving 10 ${\rm m\,s^{-1}}$ by the Echelle SPectrograph for Rocky Exoplanet and Stable Spectroscopic Observations (ESPRESSO) of VLT [Pepe et al., 2021], the most sensitive high-precision RV instrument. By contrast, the blend of Gaia22dkv has $r \sim 14$ and $V \sim 14$, and if it is the lens, it is the brightest host for a microlensing planet to date. Furthermore, the best-fit planetary solution of Gaia22dkvLb yields a massive planet $\approx0.5\,M_{\rm J}$ on a relatively close-in orbit $(\approx 1.7\,{\rm AU})$, making it favorable for RV detections. For the globally best-fit solution (UPS$-3.4(u_0+)$), we derive that the expected RV signals will have a semi-amplitude of $\approx 5\, \mathrm{m\,s^{-1}}$ with orbital period of $P_{\rm circ}=3.15^{+0.42}_{-0.22}$ yr (see the upper panel of Figure <ref> for the RV forecast of this solution). As discussed in  <ref>, UPS$-3.4(u_0+)$ is the only physically reasonable solution for a circular orbit; we nevertheless estimate the RV amplitudes and periods for all solutions, which are shown in the lower panel of Figure <ref>. Note that a significant fraction of cold Jovian planets are on nearly circular orbits [Shen & Turner, 2008, Zakamska et al., 2011], but they have a broad eccentricity distribution [Winn & Fabrycky, 2015] with $\bar{e}\sim0.3$. RV follow-up observations will be able to test the microlensing predictions, and in addition, RV data will enable measuring eccentricity and orbital period.

One possible complication for making precision RV observations of the lens is spectroscopically disentangling it from the source within the seeing disk. For Gaia22dkv, the source and blend are nearly equally bright, and thus, the feasibility of carrying out RV observations relies upon their separation in the velocity space.

We plan to derive the stellar parameters, including the chemical abundances, by analyzing the MIKE spectrum in detail in a follow-up work. In this work, we present a preliminary analysis of the MIKE spectrum to primarily estimate the approximate RVs of the blend and the source.

We first make rough velocity estimates by using the spectral broadening function (BF) technique through linear inversion [Rucinski, 1999], which convolves a sharp-line spectral template with BFs to fit the observed spectrum.  We follow similar procedures in Yi et al., 2022 and use a grid of BT-Settl atmospheric models [Allard et al., 2012] ($T_{\mathrm{eff}}:$ 4000K – 7000 K, $\log{g}$: 1.0 dex – 5.0 dex, $\mathrm{[Fe/H]}$:  $-$1.0 dex – 0.5 dex) as our sharp-line templates. We analyze the MIKE spectrum in three wavelength regions separately, 3800 Å–4800 Å (“blue”), 4800 Å–5800 Å (“green”), and 5800 Å–6800 Å (“red”), where the flux ratios between the source and blend vary due to their different temperatures.
We calculate the BFs for all  templates and visually inspect each one, and we consistently find two distinct peaks in most of the BFs for the blue, green, and red regions.
The best-match templates have ($T_{\mathrm{eff}} = 5800$ K, $\log{g} = 3.0$, $\mathrm{[Fe/H]} = -0.5$), ($T_{\mathrm{eff}} = 5200$ K, $\log{g} = 2.5$, $\mathrm{[Fe/H]} = -0.5$), ($T_{\mathrm{eff}} = 4900$ K, $\log{g} = 2.5$, $\mathrm{[Fe/H]} = -0.5$) for blue, green and red regions, respectively. Figure <ref> shows the resulting BFs from the best-match templates. The two peaks correspond to the blend and the source: the peak with smaller radial velocity has smaller relative amplitudes in redder regions compared to the other peak, so the former peak is from the blend, which is bluer than the source. We fit the BFs by the sum of two Gaussian functions and obtain the parameter uncertainties with the bootstrap method. The results from three regions are consistent with each other, and by taking their weighted means, we get ${\rm RV}_\B = 2.4_{-0.1}^{+0.1}\,\mathrm{km~s^{-1}}$ and ${\rm RV}_\Sc = 14.1_{-0.1}^{+0.1}\,\mathrm{km~s^{-1}}$ for the blend and the source, respectively. The standard deviations of the best-fit Gaussian functions for the blend and the source are $\sigma_\B = 6.2_{-0.1}^{+0.1}\,\mathrm{km~s^{-1}}$ and $\sigma_\Sc = 6.3_{-0.1}^{+0.1}\,\mathrm{km~s^{-1}}$, respectively. Note that the instrument spectral resolution ($\sigma_{\rm instr}\sim 5\, \mathrm{km\,s^{-1}}$) significantly contributes to the measured line widths, and thus the stellar line widths should be notably smaller.

Next, we perform MCMC fitting of the MIKE spectrum with a combination of two stellar templates. We adopt the RVs from the BFs results as priors, and for simplicity, we set $\mathrm{[Fe/H]}$ close to zero  (allowing for small variations). The procedures for generating a model template are as follows: (1) For a given set of stellar parameters ($T_{\mathrm{eff}}$, $\log{g}$, $\mathrm{[Fe/H]}$), a  stellar template is derived from linear interpolation on the grid of synthetic spectra.  (2) The source and blend stellar templates are generated independently, shifted by respective RVs, and broadened by Gaussian kernels with $\sigma_{\mathrm{S/B}}$. (3) The two stellar templates are co-added after multiplying the source template by a flux ratio term. (4) The co-added template is re-binned onto the same wavelength grid as the MIKE spectrum and re-normalized by a fitted fourth-order polynomial as the pseudo continuum. The best-fit stellar parameters ($T_{\mathrm{eff, S}} = 4648~\mathrm{K}$, $\log{g}_{\mathrm{S}} = 3.0~\mathrm{dex}$; $T_{\mathrm{eff, B}} = 6032~\mathrm{K}$, $\log{g}_{\mathrm{B}} = 4.2~\mathrm{dex}$) are broadly consistent with the photometry-based analysis in  <ref>.  We obtain the best-fit RVs for the blend and source as ${\rm RV_{B}} = 2.9 \pm 0.1~\mathrm{km~s^{-1}}$ and ${\rm RV_{S}} = 14.3 \pm 0.1~\mathrm{km~s^{-1}}$, respectively. The best-fit standard deviations of the Gaussian profiles are  $\sigma_{\mathrm{B}} = 8.2 \pm 0.2~\mathrm{km~s^{-1}}$ and $\sigma_{\mathrm{S}} = 6.2 \pm 0.1~\mathrm{km~s^{-1}}$, respectively. Again, these line widths are considerably larger than the stellar line widths due to significant contributions by the instrumental spectral resolution ($\sigma_{\rm instr}\sim 5\, \mathrm{km\,s^{-1}}$). Therefore, the blend appears to be distinctly separated from the source by more than $\sim 11\, \mathrm{km\,s^{-1}}$ in the velocity space, and with a high-resolution spectrograph, we may obtain high-precision RV data using the source as the velocity reference. We expect to improve the RV analysis and further assess the feasibility of high-precision RV characterizations in the follow-up work with detailed spectral modeling.

With the RVs available, we derive the Galactic kinematics of the source and the lens for the best-fit solution (blend=lens). The best-fit lens and source parallaxes are $\pi_{\mathrm L} = 0.727_{-0.026}^{+0.039}$ mas, $\pi_{\mathrm S} = 0.404_{-0.032}^{+0.052}$ mas, respectively. Thus, they are at heights $z_{\mathrm L} = -176_{-11}^{+7}$ pc and  $z_{\mathrm S}=-337_{-50}^{+29}$ pc with respective to the Galactic plane (see the upper panel of Figure <ref>). Their velocities relative to the local standard rest (LSR)
are estimated using the  package in  adopting the peculiar velocity of the Sun as
$(U_{\mathrm{LSR}, \odot}, V_{\mathrm{LSR}, \odot}, W_{\mathrm{LSR}, \odot}) = (11.1, 12.24, 7.25)~\mathrm{km~s^{-1}}$ [Schönrich et al., 2010].
The lens and source have best-fit proper motions
$(\mu_{\mathrm{L, N}}, \, \mu_{\mathrm{L, E}}) = (5.210_{-0.175}^{+0.105}, \, -10.130_{-0.074}^{+0.118})\,\mathrm{mas~yr^{-1}}$ and $(\mu_{\mathrm{S, N}}, \, \mu_{\mathrm{S, E}}) = (1.001_{-0.062}^{+0.110}, \, -7.457_{-0.063}^{+0.036})\,\mathrm{mas~yr^{-1}}$, respectively.
Adopting $\mathrm{RV}_{\mathrm{L}}=2.9\pm1.0\,\mathrm{km~s^{-1}}$ and $\mathrm{RV}_{\mathrm{S}}=14.3\pm1.0\,\mathrm{km~s^{-1}}$ derived from the MIKE spectrum[The RV uncertainty ($1.0\,\mathrm{km~s^{-1}}$) is dominated by the drift in the wavelength solution over the course of the night.],
we obtain
$(U_{\mathrm{LSR, L}}, V_{\mathrm{LSR, L}}, W_{\mathrm{LSR, L}}) = (-59.0\pm4.0, -10.8\pm1.5, -4.7\pm1.2)~\mathrm{km~s^{-1}}$ for the lens and
$(U_{\mathrm{LSR, S}}, V_{\mathrm{LSR, S}}, W_{\mathrm{LSR, S}}) = (-60.0\pm10.0, -18.4\pm2.5, -37.0\pm6.0)~\mathrm{km~s^{-1}}$ for the source, respectively.
The lower panel of Figure <ref> shows the locations of the lens and the source in the so-called Toomre diagram. They are near the transition between the thin and thick disk populations with some preference for the former.

The simulated Gaia astrometric data and fitted models of the microlens relative to that of the baseline object (i.e., subtracting the baseline coordinates and the effects of its trigonometric parallax and proper motion).
The black dots and red lines represents the simulated Gaia observations and along-scan error bars. The black solid line represents the best-fit $u_0+$ model.
The blue line shows the best-fit $u_0-$, which has $\Delta{\chi^2} \approx 452$ higher than $u_0+$.

The posteriors of astrometric lensing model parameters from MCMC sampling.
The red, yellow, green, blue area represent $\Delta{\chi^2}$ within $1, 4, 9, 16$ of the best fit. The model parameters with light-curve priors, i.e. $(t_0,\,u_0,\,t_\E,\,f_\B/f_\Sc)$, are not shown.

Host properties of microlensing planets. We use the parameters of the default models (modeldef=1) in the microlensing planet catalog of the NASA Exoplanet Archive for all discoveries in the bulge field (black dots). The host properties of Kojima-1Lb (blue star) are from Zang et al., 2020, and those of Gaia22dkvLb are based on the globally best fit UPS$-$3.4$(u_0+)$ in this work. Panel A): Host  mass ($M_\L$) vs. distance ($D_\L$). Panel B):  Host brightness in $V$ band ($m_V$) vs. distance ($D_\L$). For the systems in the bulge field, we make crude $V$-band brightness estimates by assuming the lens star is a main-sequence star with stellar mass $M_\L$ at $D_\L$. We use the isochrone at a stellar age of $5$ Gyr from the Dartmouth Stellar Evolution Database  [Dotter et al., 2008] to estimate the stellar luminosities. Then we apply the extinction corrections using the near-infrared extinction maps by Gonzalez et al., 2012 and convert it to $A_V$ with the relationship in Cardelli et al., 1989. Panel C): The Galactic longitude ($l$) vs. latitude ($b$) distribution.

Semi-major axis (normalized to snow line) vs. planet mass distribution of microlensing planets (black dots: bulge field, red star: Gaia22dkvLb, blue star: Kojima1-Lb). We scale the projected separation by a factor of $\sqrt{3/2}$ (the averaged value assuming random orbital orientation) to estimate the  semi-major axis. The parameters are from the same sources as in Figure <ref>.

§.§ Gaia astrometric lensing

Our best-fit planetary solution lacks the $\theta_\E$ measurement o ue to the absence of significant finite-source effect. Astrometric microlensing is an alternative channel to measure $\theta_\E$
[Hog et al., 1995, Miyamoto & Yoshii, 1995, Walker, 1995, Dominik & Sahu, 2000].
We perform a simplified PSPL analysis to evaluate the prospect of astrometric microlensing with the Gaia time-series data that will ultimately be released.

We simulate Gaia astrometric observations based on the PSPL parameters from our
best-fit planetary solution (UPS$-$3.4$(u_0+)$) along with the physical parameters estimated by assuming the blend to be the lens as evaluated in Table <ref>. The epochs of observation from the start of Gaia mission to future predictions are based on output from the Gaia Observation Forecast Tool (GOST[<https://gaia.esac.esa.int/gost/>]).
We adopt the along-scan (AL) astrometric precision of $0.08\,\mathrm{mas}$ for each epoch based on the estimate from Rybicki et al., 2018 for a target with $G\sim 13$. See the black dots with red error bars in Figure <ref> for the simulated data. Then we fit the simulated Gaia data with an astrometric microlensing model with 12 parameters, including 4 parameters $(t_0,\,u_0,\,t_\E,\,f_\B/f_\Sc)$ with priors derived from the fit to the light curve. We do not impose light-curve priors on the other 8 free parameters $(\mathrm{{RA_{base}}},\, \mathrm{{Dec_{base}}},\,\theta_\E,\, \pi_\mathrm{E,N},\, \pi_\mathrm{E,E},\, \mu_\mathrm{N,base},\,\mu_\mathrm{E,base},\, \pi_\mathrm{base})$, where $(\mathrm{{RA_{base}}}, \mathrm{{Dec_{base}}})$, $\bmu_\mathrm{base} = (\mu_\mathrm{N,base},\,\mu_\mathrm{E,base})$ are the coordinates and proper motion of the baseline object when un-lensed at the same reference epoch of the Gaia DR3 catalog. In practice, we fit $(\mathrm{\Delta{RA_\mathrm{base}}}, \mathrm{\Delta{Dec_\mathrm{base}}})$, which are differences from the input Gaia DR3 coordinates.

We sample the posteriors using
[Speagle, 2020, Koposov et al., 2023], which implements Nested Sampling on the high-dimension parameter space. The posteriors of the 8 free parameters of the astrometric model are displayed in Figure <ref>, and the centroid's trajectory from the best-fit model is shown as the solid black line in Figure <ref>. The expected $1\sigma$ error of $\theta_\E$ from our MCMC posterior is $\sim 0.1\,\mathrm{mas}$, and the corresponding uncertainty of lens mass is $\sim 15\%$. Interestingly, the astrometric data alone can also constrain $\bpi_\E$ to a high precision ($\sigma_{\pi_{\rm E,N}}\approx0.02$ and $\sigma_{\pi_{\rm E,E}}\approx0.03$). We fit the simulated data to $u_0-$ models and we find that the best-fit $u_0-$ model has a $\Delta \chi^2 = {452}$ compared to $u_0+$. Therefore, the Gaia astrometric microlensing data can completely break the $u_0+/-$ degeneracy and definitively determine the microlens parallax and thus the lens mass and distance.

§ CONCLUSION

Gaia22dkv is an extrasolar planet discovered from a microlensing event alerted by Gaia, and its location is far from the traditional bulge microlensing fields (see the bottom panel of Figure <ref>). The only other non-bulge microlensing planet Kojima-1Lb was found serendipitously, and it is difficult to quantify the detection efficiency and make statistical inferences. Systematic all-sky surveys such as Gaia can open up the largely untapped discovery potential of near-field microlensing [Han, 2008]. Our primary interpretation reveals the lens as to be nearby ($\sim 1.4$ kpc) turn-off star, making it the most massive and brightest microlensing planet host (see the top and middle panels of Figure <ref>). Thanks to the host's close distance and its high effective temperature, the Einstein radius of Gaia22dkv is well within the snow line, unlike most microlensing planets. As a half-Jupiter-mass planet orbiting a turn-off star, Gaia22dkvLb is one of the few microlensing planets detected inside the snow line (see Figure <ref>).  Such near-field events can provide far richer opportunities for follow-up studies than planetary systems found in the bulge field. Gaia astrometric and host spectroscopic observations can yield detailed characterizations on the host's physical and kinematic properties and chemical abundance, which can offer significant clues regarding planet formation and dynamical evolutions [Zhu & Dong, 2021]. The combination of high planet mass, relatively small orbital separation and the exceptionally bright host makes Gaia22dkvLb the most promising microlensing planet for RV follow ups. Besides determining Gaia22dkvLb's orbital period and eccentricity, precision RV data can also be used to search for much shorter-period planets and probe the planet architecture, which is generally impossible for microlensing by itself. While microlensing does occasionally detect two planets, both of which are projected near the Einstein ring typically at 1–4 AU <cit.>, thus it can hardly probe multiple planets with high semi-major-axis ratios. Kepler and RV observations have revealed tantalizing evidence for a strong “inner-outer” correlation: $\sim 90\%$ of cold Jovian planets are found to be accompanied with close-in super Earths in the same systems [Zhu & Wu, 2018, Bryan et al., 2019]. Future RV follow ups can examine whether the cold Jovian planet of Gaia22dkvLb has an associated inner planet as would be likely according to the inner-out correlation.

§ ACKNOWLEDGMENT

We thank Johanna Teske for stimulating discussion and Wei Zhu's team for attempting to observe the event. This work is supported by the National Key R&D Program of China No. 2019YFA0405100, the National Natural Science Foundation of China (Grant No. 12133005) and the science research grants from the China Manned Space Project with No. CMS-CSST-2021-B12. SD acknowledges the New Cornerstone Science Foundation through the XPLORER PRIZE. This research uses data obtained through the Telescope Access Program (TAP), which has been funded by the TAP member institutes. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. These observations are associated with program 16871. This work has made use of data from the European Space Agency (ESA) mission Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC,
<https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. This work uses observations made at the Observatorio do Pico dos Dias/LNA (Brazil).

Gaia, LCOGT, Magellan:Clay (MIKE), Exoplanet Archive
PmPyeasy, DoPHOT, VBBinaryLensing, EMCEE, MIST, PyAstronomy, dynesty

[Allard et al., 2012]
Allard, F., Homeier, D., & Freytag, B. 2012, Philosophical Transactions
of the Royal Society of London Series A, 370, 2765,

[An, 2005]
An, J. H. 2005, , 356, 1409, 10.1111/j.1365-2966.2004.08581.x

[An et al., 2002]
An, J. H., Albrow, M. D., Beaulieu, J. P., et al. 2002, , 572, 521,

[Bachelet et al., 2022]
Bachelet, E., Tsapras, Y., Gould, A., et al. 2022, , 164, 75,

[Bernstein et al., 2003]
Bernstein, R., Shectman, S. A., Gunnels, S. M., Mochnacki, S., &
Athey, A. E. 2003, in Society of Photo-Optical Instrumentation Engineers
(SPIE) Conference Series, Vol. 4841, Instrument Design and Performance for
Optical/Infrared Ground-based Telescopes, ed. M. Iye & A. F. M.
Moorwood, 1694–1704, 10.1117/12.461502

[Boyajian et al., 2013]
Boyajian, T. S., von Braun, K., van Belle, G., et al. 2013, , 771,
40, 10.1088/0004-637X/771/1/40

[Bozza, 2010]
Bozza, V. 2010, , 408, 2188, 10.1111/j.1365-2966.2010.17265.x

[Bozza et al., 2018]
Bozza, V., Bachelet, E., Bartolić, F., et al. 2018, , 479,
5157, 10.1093/mnras/sty1791

[Bozza et al., 2021]
Bozza, V., Khalouei, E., & Bachelet, E. 2021, , 505, 126,

[Brown et al., 2013]
Brown, T. M., Baliber, N., Bianco, F. B., et al. 2013, , 125,
1031, 10.1086/673168

[Bryan et al., 2019]
Bryan, M. L., Knutson, H. A., Lee, E. J., et al. 2019, , 157, 52,

[Cardelli et al., 1989]
Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, , 345, 245,

[Chen et al., 2022]
Chen, P., Dong, S., Kochanek, C. S., et al. 2022, , 259, 53,

[Choi et al., 2016]
Choi, J., Dotter, A., Conroy, C., et al. 2016, , 823, 102,

[Dominik, 1999]
Dominik, M. 1999, , 349, 108, 10.48550/arXiv.astro-ph/9903014

[Dominik & Sahu, 2000]
Dominik, M., & Sahu, K. C. 2000, , 534, 213, 10.1086/308716

[Dong et al., 2009]
Dong, S., Gould, A., Udalski, A., et al. 2009, , 695, 970,

[Dong et al., 2019]
Dong, S., Mérand, A., Delplancke-Ströbele, F., et al. 2019,
, 871, 70, 10.3847/1538-4357/aaeffb

[Dotter et al., 2008]
Dotter, A., Chaboyer, B., Jevremović, D., et al. 2008, , 178,
89, 10.1086/589654

[Foreman-Mackey et al., 2013]
Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP,
125, 306, 10.1086/670067

[Fukui et al., 2019]
Fukui, A., Suzuki, D., Koshimoto, N., et al. 2019, , 158, 206,

[Gaia Collaboration et al., 2016]
Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al.
2016a, , 595, A1, 10.1051/0004-6361/201629272

[Gaia Collaboration et al., 2016]
—. 2016b, , 595, A1, 10.1051/0004-6361/201629272

[Gaia Collaboration et al., 2023]
Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, ,
674, A1, 10.1051/0004-6361/202243940

[Gaudi, 1998]
Gaudi, B. S. 1998, , 506, 533, 10.1086/306256

[Gonzalez et al., 2012]
Gonzalez, O. A., Rejkuba, M., Zoccali, M., et al. 2012, , 543, A13,

[Gould, 1992]
Gould, A. 1992, , 392, 442, 10.1086/171443

[Gould, 2003]
—. 2003, arXiv e-prints, astro.

[Gould, 2004]
—. 2004, , 606, 319, 10.1086/382782

[Gould, 2022]
—. 2022, arXiv e-prints, arXiv:2209.12501, 10.48550/arXiv.2209.12501

[Gould & Loeb, 1992]
Gould, A., & Loeb, A. 1992, , 396, 104, 10.1086/171700

[GRAVITY Collaboration et al., 2017]
GRAVITY Collaboration, Abuter, R., Accardo, M., et al. 2017, , 602,
A94, 10.1051/0004-6361/201730838

[Griest & Hu, 1992]
Griest, K., & Hu, W. 1992, , 397, 362, 10.1086/171793

[Griest & Safizadeh, 1998]
Griest, K., & Safizadeh, N. 1998, , 500, 37, 10.1086/305729

[Guo et al., 2021]
Guo, H. L., Chen, B. Q., Yuan, H. B., et al. 2021, , 906, 47,

[Han, 2008]
Han, C. 2008, , 681, 806, 10.1086/588083

[Han & Gould, 1997]
Han, C., & Gould, A. 1997, , 480, 196, 10.1086/303944

[Han et al., 2019]
Han, C., Yee, J. C., Udalski, A., et al. 2019, , 158, 102,

[Han et al., 2022]
Han, C., Bond, I. A., Yee, J. C., et al. 2022, , 658, A94,

[Hodgkin et al., 2021]
Hodgkin, S. T., Harrison, D. L., Breedt, E., et al. 2021, , 652,
A76, 10.1051/0004-6361/202140735

[Hog et al., 1995]
Hog, E., Novikov, I. D., & Polnarev, A. G. 1995, , 294, 287

[Hwang et al., 2013]
Hwang, K. H., Choi, J. Y., Bond, I. A., et al. 2013, , 778, 55,

[Jurić et al., 2008]
Jurić, M., Ivezić, Ž., Brooks, A., et al. 2008, ,
673, 864, 10.1086/523619

[Kelson, 2003]
Kelson, D. D. 2003, , 115, 688, 10.1086/375502

[Kelson et al., 2000]
Kelson, D. D., Illingworth, G. D., van Dokkum, P. G., & Franx, M.
2000, , 531, 159, 10.1086/308445

[Kochanek et al., 2017]
Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, ,
129, 104502, 10.1088/1538-3873/aa80d9

[Koposov et al., 2023]
Koposov, S., Speagle, J., Barbary, K., et al. 2023, joshspeagle/dynesty:
v2.1.2, v2.1.2,  Zenodo, 10.5281/zenodo.7995596

[Kuang et al., 2023]
Kuang, R., Zang, W., Mao, S., Zhang, J., & Jiang, H. 2023, ,
520, 4540, 10.1093/mnras/stad461

[Lindegren et al., 2021]
Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021, ,
649, A2, 10.1051/0004-6361/202039709

[Miyamoto & Yoshii, 1995]
Miyamoto, M., & Yoshii, Y. 1995, , 110, 1427, 10.1086/117616

[Nucita et al., 2018]
Nucita, A. A., Licchelli, D., De Paolis, F., et al. 2018, , 476,
2962, 10.1093/mnras/sty448

[Paczynski, 1986]
Paczynski, B. 1986, , 304, 1, 10.1086/164140

[Pepe et al., 2021]
Pepe, F., Cristiani, S., Rebolo, R., et al. 2021, , 645, A96,

[Poindexter et al., 2005]
Poindexter, S., Afonso, C., Bennett, D. P., et al. 2005, , 633,
914, 10.1086/468182

[Rucinski, 1999]
Rucinski, S. 1999, in Astronomical Society of the Pacific Conference Series,
Vol. 185, IAU Colloq. 170: Precise Stellar Radial Velocities, ed. J. B.
Hearnshaw & C. D. Scarfe, 82, 10.48550/arXiv.astro-ph/9807327

[Ruiz-Dern et al., 2018]
Ruiz-Dern, L., Babusiaux, C., Arenou, F., Turon, C., & Lallement, R.
2018, , 609, A116, 10.1051/0004-6361/201731572

[Rybicki et al., 2018]
Rybicki, K. A., Wyrzykowski, Ł., Klencki, J., et al. 2018, ,
476, 2013, 10.1093/mnras/sty356

[Schechter et al., 1993]
Schechter, P. L., Mateo, M., & Saha, A. 1993, , 105, 1342,

[Schönrich et al., 2010]
Schönrich, R., Binney, J., & Dehnen, W. 2010, , 403, 1829,

[Shappee et al., 2014]
Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, , 788, 48,

[Shen & Turner, 2008]
Shen, Y., & Turner, E. L. 2008, , 685, 553, 10.1086/590548

[Skowron & Gould, 2012]
Skowron, J., & Gould, A. 2012, arXiv e-prints, arXiv:1203.1034,

[Skowron et al., 2011]
Skowron, J., Udalski, A., Gould, A., et al. 2011, , 738, 87,

[Smith et al., 2003]
Smith, M. C., Mao, S., & Paczyński, B. 2003, , 339, 925,

[Smith et al., 2002]
Smith, M. C., Mao, S., Woźniak, P., et al. 2002, , 336, 670,

[Speagle, 2020]
Speagle, J. S. 2020, , 493, 3132, 10.1093/mnras/staa278

[Tonry et al., 2012]
Tonry, J. L., Stubbs, C. W., Lykke, K. R., et al. 2012, , 750, 99,

[Tonry et al., 2018]
Tonry, J. L., Denneau, L., Flewelling, H., et al. 2018, , 867, 105,

[Walker, 1995]
Walker, M. A. 1995, , 453, 37, 10.1086/176367

[Winn & Fabrycky, 2015]
Winn, J. N., & Fabrycky, D. C. 2015, Annual Review of Astronomy and
Astrophysics, 53, 409, 10.1146/annurev-astro-082214-122246

[Wyrzykowski et al., 2023]
Wyrzykowski, Ł., Kruszyńska, K., Rybicki, K. A., et al. 2023,
, 674, A23, 10.1051/0004-6361/202243756

[Yee et al., 2012]
Yee, J. C., Shvartzvald, Y., Gal-Yam, A., et al. 2012, , 755, 102,

[Yi et al., 2022]
Yi, T., Gu, W.-M., Zhang, Z.-X., et al. 2022, Nature Astronomy, 6,
1203, 10.1038/s41550-022-01766-0

[Yoo et al., 2004]
Yoo, J., DePoy, D. L., Gal-Yam, A., et al. 2004, , 603, 139,

[Zakamska et al., 2011]
Zakamska, N. L., Pan, M., & Ford, E. B. 2011, , 410, 1895,

[Zang et al., 2020]
Zang, W., Dong, S., Gould, A., et al. 2020, , 897, 180,

[Zhang & Yuan, 2023]
Zhang, R., & Yuan, H. 2023, , 264, 14,

[Zhu & Dong, 2021]
Zhu, W., & Dong, S. 2021, , 59, 291,

[Zhu & Wu, 2018]
Zhu, W., & Wu, Y. 2018, , 156, 92, 10.3847/1538-3881/aad22a


# A hybrid deep learning framework for Covid-19 detection via 3D Chest CT
Images

Shuang Liang
University of Science and Technology Beijing
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present a hybrid deep learning framework named CTNet which
combines convolutional neural network and transformer together for the
detection of COVID-19 via 3D chest CT images. It consists of a CNN feature
extractor module with SE attention to extract sufficient features from CT
scans, together with a transformer model to model the discriminative features
of the 3D CT scans. Compared to previous works, CTNet provides an effective
and efficient method to perform COVID-19 diagnosis via 3D CT scans with data
resampling strategy. Advanced results on a large and public benchmarks,
COV19-CT-DB database was achieved by the proposed CTNet, over the state-of-
the-art baseline approach proposed together with the dataset.

##  1 Introduction

The Coronavirus Disease 2019 (COVID-19), a highly infectious disease, has
become a global pademic and posed serious threats to human worldwide. In order
to prevent further spreading of COVID-19 and treat the infected patients
instantly, various examination methods have been proposed for the diagnosis of
COVID-19 [2, 5], of which rRT-PCR is usually considered as the golden standard
for the diagnosis of COVID-19 [11]. However, due to the limitation of sample
collection and transportation, the senstivity of rRT-PCR might encounter some
problems. As reported by the World Health Organization (WHO), the lung
infection was detected in the autosies [13]. Thus, medical imaging of chest
radiography is proved to be useful for rapid COVID-19 detection [16]. Computed
Tomography (CT) was considered as the precise examination tools since it
provided a 3D view of organs especially the lung and could be used to locate
the lesion areas [3]. In the CT examination, large volume of CT scans are
obtained from persons suspected of COVID-19, which poses heavy workload on
physicians and radiologists to diagnose COVID-19. There is a great need to
develop some auxilary reliable methods in the medical imaging analysis of
COVID-19. As demonstrated ub some studies, machine and deep learning methods
might be a potential approach for the detection of COVID-19 [14].

However, there are two main problems in their works. First, the size of
dataset used in these works is limited which makes it hard to avoid the
overfitting problem and reduces the usability of their works. Second, the
preprocess methods of 3D CT scans are not ideal. Some works only take the
slices of CT scans that are suspected to have related sympotoms into account
[12, 6, 1, 15]. But in actual use case, this method require physicians and
radiologists to first screen these slices which might not be applicable. The
other works take full volumn of CT scans into consideration. This method
ensures that the discriminative features were included in the input data but
make it hard to train models since the batch size could only be set to the
single digits which might cause the unstable training process.

In this paper, we proposed a hybrid deep learning framework named CTNet that
combines CNN and transformer together for the detection of COVID-19 via 3D
chest CT images. Based on the hypothese that the relevant lesion area is
continuous in lung and the features of lesion area is discriminative for the
detection of COVID-19, we proposed a data resampling method to reduce the size
of input data while preserving useful information. The contribution of our
paper could be summarized as following:

  1. 1.

A effective data resampling method was proposed for 3D CT scans to reduce the
size of input while preserving sufficent information.

  2. 2.

A hybrid deep learning framework is proposed for the detetion of COVID-19
which combines a CNN to mining effective features and a Transformer to conduct
feature aggregation.

  3. 3.

Good performances were achieved by the proposed framework with a macro F1
score of 88.24% on the validation set of COV19-CT-DB dataset, which is 16
percent lead of that of the baseline method proposed along with the dataset.

The reset of this paper mainly focuses on the introduction of the proposed
method and the corresponding results and discussion.

##  2 Implemetation details

Figure 1: The framework of the proposed CTNet.

The proposed DL framework, called CTNet, is a hybrid DL method consisting of
three componets: the input module with data resampling strategy, the CNN
feature extractor module with SE attention module [4], and the information
aggregation module with the transformer and fully connected (FC) layer. As
shown in Figure 1, the 3D CT scans were resampled to a fixed number of CT
slices as the input data, following is the CNN feature extractor that
extracted sufficient and effective features. The extracted features were then
flatten to vectors and sent to two branches, transformer brach and FC branch,
and the vectors generated by the two branches were fused using element-wise-
add (EWA) operation. The vector is then sent to the softmax function to
predict the probability vector for the detection of COVID-19.

###  2.1 Data resampling strategy

As we introduced in Introduction, the preprocess method of 3D CT scans matters
a lot considering the actual application in clinical. Unlike the previous
works that only take the annotated slices or take all the volumns of CT scans,
we proposed a data resampling strategy that following the uniform
distribution. Since the total number of CT scans varies from different cases,
we firstly set a variable named totalnum to deal with this situation. The
resampled number of CT scans is controled by the variable renumCT which could
be adjusted according to the actual use case (especially the computation power
and memory size). The algorithm contains two branches, one for resampling and
the other for oversampling which is switchable according to the total number
of CT scans and the resampled number of CT scans. The output is the indexes
array of the 3D CT images. The process is summarized as following:

Algorithm 1 Data resampled strategy for 3D CT scans

0: $\rm{totalnum},\rm{n}$

0: $\rm{indexes}$(The indexes array of the resampled CT scans)

$\rm{renumCT}\leftarrow\rm{n}$

$\rm{i}\leftarrow 0$

$\rm{indexes}\leftarrow[]$

if $\rm{totalnum}>=\rm{renumCT}$ then

$\rm{diff}=\rm{totalnum}/(\rm{renumCT}+1)$

while $i<renumCT$ do

$\rm{indexes}.append(i*\rm{diff})$

$i+=1$

end while

else

while $i<renumCT$ do

$\rm{indexes}.append(random.choice(\rm{totalnum}))$

end while

end if

###  2.2 CNN feature extractor with SE attention module

The CNN feature extractor is a multi-stage residual neural network that
incorporate the SE attention module to increase the discriminative ability. As
shown in Figure. 1, the CNN can be seperated to two parts, the input
convolution layer and the residual block. The number of the input channels for
the input convolution layer is the same as the number of the resampled CT
images. The following part is the residual block that adopted the SE module.
The structure of the SE module is also demonstrated in Fig.1, which consists
of a global pooling, two FC layers, the ReLU function, and the sigmoid
function to generate attention vectors with the value range from 0 to 1. Given
the resampled CT images as input, the extractor generates feature maps of N
channels. Global pooling and flatten operations are performed to transform the
feature maps to a feature vector.

###  2.3 Information aggregation module

As introduced previously, the 3D CT scans were resampled and sent to the CNN
feature extractor. The output feature vector is then sent to two braches, the
transformer block and the FC block, to fuse information and generate the final
prediction. The structure of the transformer is shown in Fig.1, which consists
of two layer normalization, a multi-head attention module and the multilayer
perception module. The probability vectors generated by the two branches are
fused using EWA operation to predict the diagnosis result.

###  2.4 Training details

The framework is trained in an end-to-end and from-scratch manner on a local
machine with two 1080ti GPUs. The learning rate is set to 0.01, and is
decreased following the step adjust strategy. The total epoches is set to 120,
and the step epoches are 50 and 100. The number of resampled CT scans is set
to 32 in our experiments and the batch size is set to 32. The size of each CT
scan is resized to 224 $\times$ 224.

##  3 Results

###  3.1 COV19-CT-DB database

The dataset used in this paper for training and validation is the COV19-CT-DB
dataset [9, 8, 10, 7], which consists of 5,000 chest CT scans that are
annotated as COVID-19. Data was collected in the period from September 1, 2020
to March 31, 2021. The data were aggregated from different hospitals,
containing anonymized human lung CT scans with signs of COVID-19 and without
signs of COVID-19. The database was seperated into three set, the training
set, the validation set and the test set, of which the training and validation
set are publicly avaliable currently. The training set consists of 1560 3D CT
scans, including 690 COVID-19 cases and 870 Non-COVID-19 cases. The validation
set contains 374 3D CT scans, including 165 COVID-19 cases and 209 Non-
COVID-19 cases. Each CT scan owns different numbers of CT scans, ranging from
50 to 700. The details were shown in Table. 1.

Table 1: Distribution of the COV19-CT-DB database. KACD | train | val | test | total
---|---|---|---|---
COVID-19 | 690 | 870 | - | 1560+
Non-COVID-19 | 165 | 209 | - | 374+
Total | 855 | 1079 | 3455 | 5389

###  3.2 Evaluation Metrics and experiments

As introduced in the database, the ’macro’ F1 score is used as the evaluation
metric and the score of 0.70 is reported as the baseline score. The ’macro’ F1
score is defined as the unweighted average of the class-wise/label-wise
F1-scores. In our experiments, we conduct two experiments to validate the
proposed framework, one only uses the probability vector generated by the FC
layer and the other uses the fused probability vector to obtain the diagnosis
results. The performance on the validation set can be seen from Table.2. Good
performance was achieved by the CTNet, with a best ’macro’ F1-score of
$88.23\%$, which is $18.23$ percent higher than that of the baseline approach.
Notably, the number of resampled CT scans is set to 32 in our experiments
while that for the baseline is 700. Besides, the inference process of the
CTNet cost only 380ms in diagnosing each case which makes it more applicable
in actual use cases.

Table 2: Performance of the CTNet framework Method | ’macro’ F1 Score
---|---
Baseline | 70.00%
CTNet (FC) | 87.96%
CTNet (Fused) | 88.23%

##  4 Conclusion

We present a hybrid deep learning framework called CTNet for the detection of
COVID-19 via 3D CT scans. The proposed CTNet provides a novel resampling
strategy and network architecture for the effective mining of 3D CT scans.
Experimental results demonstrated that our resampling strategy and framework
achieved a good trade-off between speed and accuracy. Ablation studies showed
CTNet’s superior ability to deal with COVID-19 detection with less requirement
of computation power and data.

## References

  * Afshar et al. [2021] Parnian Afshar, Shahin Heidarian, Nastaran Enshaei, Farnoosh Naderkhani, Moezedin Javad Rafiee, Anastasia Oikonomou, Faranak Babaki Fard, Kaveh Samimi, Konstantinos N Plataniotis, and Arash Mohammadi.  Covid-ct-md, covid-19 computed tomography scan dataset applicable in machine learning and deep learning.  _Scientific Data_ , 8(1):1–8, 2021.
  * Alizadehsani et al. [2021] Roohallah Alizadehsani, Zahra Alizadeh Sani, Mohaddeseh Behjati, Zahra Roshanzamir, Sadiq Hussain, Niloofar Abedini, Fereshteh Hasanzadeh, Abbas Khosravi, Afshin Shoeibi, Mohamad Roshanzamir, et al.  Risk factors prediction, clinical outcomes, and mortality in covid-19 patients.  _Journal of medical virology_ , 93(4):2307–2320, 2021.
  * Dai et al. [2020] Wei-cai Dai, Han-wen Zhang, Juan Yu, Hua-jian Xu, Huan Chen, Si-ping Luo, Hong Zhang, Li-hong Liang, Xiao-liu Wu, Yi Lei, et al.  Ct imaging and differential diagnosis of covid-19.  _Canadian Association of Radiologists Journal_ , 71(2):195–200, 2020.
  * Hu et al. [2018] Jie Hu, Li Shen, and Gang Sun.  Squeeze-and-excitation networks.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 7132–7141, 2018.
  * Khadidos et al. [2020] Adil Khadidos, Alaa O Khadidos, Srihari Kannan, Yuvaraj Natarajan, Sachi Nandan Mohanty, and Georgios Tsaramirsis.  Analysis of covid-19 infections on a ct image using deepsense model.  _Frontiers in Public Health_ , 8, 2020.
  * Ko et al. [2020] Hoon Ko, Heewon Chung, Wu Seong Kang, Kyung Won Kim, Youngbin Shin, Seung Ji Kang, Jae Hoon Lee, Young Jun Kim, Nan Yeol Kim, Hyunseok Jung, et al.  Covid-19 pneumonia diagnosis using a simple 2d deep learning framework with a single chest ct image: model development and validation.  _Journal of medical Internet research_ , 22(6):e19569, 2020.
  * Kollias et al. [2018] Dimitrios Kollias, Athanasios Tagaris, Andreas Stafylopatis, Stefanos Kollias, and Georgios Tagaris.  Deep neural architectures for prediction in healthcare.  _Complex & Intelligent Systems_, 4(2):119–131, 2018.
  * Kollias et al. [2020a] Dimitrios Kollias, N Bouas, Y Vlaxos, V Brillakis, M Seferis, Ilianna Kollia, Levon Sukissian, James Wingate, and S Kollias.  Deep transparent prediction through latent representation analysis.  _arXiv preprint arXiv:2009.07044_ , 2020a.
  * Kollias et al. [2021] Dimitrios Kollias, Anastasios Arsenos, Levon Soukissian, and Stefanos Kollias.  Mia-cov19d: Covid-19 detection through 3-d chest ct image analysis.  _arXiv preprint arXiv:2106.07524_ , 2021.
  * Kollias et al. [2020b] Dimitris Kollias, Y Vlaxos, M Seferis, Ilianna Kollia, Levon Sukissian, James Wingate, and Stefanos D Kollias.  Transparent adaptation in deep medical image diagnosis.  In _TAILOR_ , page 251–267, 2020b.
  * Long et al. [2020] Chunqin Long, Huaxiang Xu, Qinglin Shen, Xianghai Zhang, Bing Fan, Chuanhong Wang, Bingliang Zeng, Zicong Li, Xiaofen Li, and Honglu Li.  Diagnosis of the coronavirus disease (covid-19): rrt-pcr or ct?  _European journal of radiology_ , 126:108961, 2020.
  * Shah et al. [2021] Vruddhi Shah, Rinkal Keniya, Akanksha Shridharani, Manav Punjabi, Jainam Shah, and Ninad Mehendale.  Diagnosis of covid-19 using ct scan images and deep learning techniques.  _Emergency radiology_ , 28(3):497–505, 2021.
  * Sohrabi et al. [2020] Catrin Sohrabi, Zaid Alsafi, Niamh O’neill, Mehdi Khan, Ahmed Kerwan, Ahmed Al-Jabir, Christos Iosifidis, and Riaz Agha.  World health organization declares global emergency: A review of the 2019 novel coronavirus (covid-19).  _International journal of surgery_ , 76:71–76, 2020.
  * Song et al. [2021] Ying Song, Shuangjia Zheng, Liang Li, Xiang Zhang, Xiaodong Zhang, Ziwang Huang, Jianwen Chen, Ruixuan Wang, Huiying Zhao, Yunfei Zha, et al.  Deep learning enables accurate diagnosis of novel coronavirus (covid-19) with ct images.  _IEEE/ACM Transactions on Computational Biology and Bioinformatics_ , 2021.
  * Wu et al. [2021] Xing Wu, Cheng Chen, Mingyu Zhong, Jianjia Wang, and Jun Shi.  Covid-al: The diagnosis of covid-19 with deep active learning.  _Medical Image Analysis_ , 68:101913, 2021.
  * Xiong et al. [2020] Ying Xiong, Dong Sun, Yao Liu, Yanqing Fan, Lingyun Zhao, Xiaoming Li, and Wenzhen Zhu.  Clinical and high-resolution ct features of the covid-19 infection: comparison of the initial and follow-up changes.  _Investigative radiology_ , 2020.