# A spatially dependent correction of Gaia EDR3 parallax zero-point offset
based on 0.3 million LAMOST DR8 giant stars

Chun Wang  Corresponding author<EMAIL_ADDRESS> Tianjin Astrophysics
Center, Tianjin Normal University, Tianjin 300387, People’s Republic of China.
Haibo Yuan  Department of Astronomy, Beijing Normal University, Beijing
100875, People’s Republic of China.  Yang Huang  South-Western Institute for
Astronomy Research, Yunnan University, Kunming, Yunnan 650091, People’s
Republic of China.

###### Abstract

We have studied the zero-point offset of Gaia early Data Release 3 (EDR3)
parallaxes based on a sample of 0.3 million giant stars built from the LAMOST
data with distance accuracy better than 8.5%. The official parallax zero-point
corrections largely reduce the global offset in the Gaia EDR3 parallaxes: the
global parallax offsets are $-$27.9 $\mu$as and $-$26.5 $\mu$as (before
correction) and $+$2.6 $\mu$as and $+$2.9 $\mu$as (after correction) for the
five- and six-parameter solutions, respectively. The bias of the raw parallax
measurements is significantly dependent on the $G$ magnitudes, spectral
colors, and positions of stars. The official parallax zero-point corrections
could reduce parallax bias patterns with $G$ magnitudes, while could not fully
account the patterns in the spaces of the spectral colors and positions. In
the current paper, a spatially dependent parallax zero-point correction model
for Gaia EDR3 five-parameter solution in the LAMOST footprint is firstly
provided with the advantage of huge number of stars in our sample.

##  1 Introduction

Gaia Early Data Release 3 (Gaia Collaboration et al., 2021) have released
astrometric and photometric data for over 1.8 billion sources based on
observations from the first 34 months (from 2014 July to 2017 May) by the
European Space Agency’s Gaia mission (Gaia Collaboration et al., 2016). Among
of them, 1.468 billion sources have full astrometric data, including position,
parallax, and proper motions. The data set of Gaia EDR3 is widely used in the
fields of the stellar and Galactic astrophysics.

In Gaia EDR3, the typical uncertainties of parallaxes are 0.03–1.4 mas for
stars with $15<G<21$ mag (Gaia Collaboration et al., 2021). The systematic
parallax errors are inevitable because of the imperfections in the instruments
and data processing (Lindegren et al., 2021a), which will produce large
systematic distance errors especially for more distant stars. Thus,
investigating the systematic bias of the Gaia EDR3 parallax is important for
its further applications. Lindegren et al. (2021b) have constructed a parallax
zero-point correction model, which is a function of the $G$-band magnitude,
spectral shape (colors), and ecliptic latitude of the sources. This correction
model is based on quasars distributed across the entire sky, binary stars, and
stars in the Large Magellanic Cloud (LMC). The correction models are different
for sources with five- and six-parameter solutions. However, the correction
models are mostly appropriate for only sources having similar magnitudes and
colors to quasars. Thus the independent test of parallax zero-point offsets of
Gaia EDR3 using Galactic stars is needed. Several papers have also contributed
to the parallax zero-point offset of Gaia EDR3 estimates using different
tracers, including quasars, RR Lyrae stars, red clump stars, red giant stars
and binaries (Bhardwaj et al., 2021; Liao et al., 2021; Groenewegen, 2021; El-
Badry et al., 2021; Stassun & Torres, 2021; Zinn, 2021; Huang et al., 2021;
Ren et al., 2021). Amongst of them, Huang et al. (2021) and Ren et al. (2021)
estimate the independent parallax zero-point offsets of Gaia EDR3 using $\sim$
0.07 million LAMOST primary red clump (PRC) stars and $\sim$ 0.11 million W
Ursae Majoris (EW)-type eclipsing binary systems in a wide magnitude and color
ranges with the advantage of large stellar sample and their accurate
measurements of distance. They find $\sim-27\,\mu\rm as$ and $\sim+4\,\mu\rm
as$ global zero-point offset of Gaia EDR3 parallaxs before and after
correction, respectively. The official parallax zero-point model of Lindegren
et al. (2021b) could reduce the global bias in the Gaia EDR3 parallax. After
correction, the parallax bias variations with spectral colors and positions
still exist. Thus, it is useful to investigate the zero-point offset in Gaia
EDR3 parallaxes, especially the parallax zero-point offset variations with the
positions.

Here, we use the sources targeted by LAMOST to investigate the zero-point
offset in Gaia EDR3 parallaxes. We will introduce our data set in Section 2.
The main results are presented and discussed in Section 3. Finally, Section 4
presents a summary of our results.

##  2 Data

We use the data of the value-added catalog for LAMOST DR8 low resolution
spectra (Wang et al. 2022, accepted) to test the zero-point offset of Gaia
EDR3. The catalog contains accurate photometric distances for about 7.1
million spectra of 5.17 million unique stars with spectral signal-to-noise
ratios (SNRs) higher than 10 obtained by the Large Sky Area Multi-Object Fibre
Spectroscopic Telescope (LAMOST) Galactic spectroscopic surveys (Deng et al.,
2012; Zhao et al., 2012; Liu et al., 2014). The photometric distance of
individual stars observed by LAMOST are estimated with distance modulus method
using the 2MASS (Skrutskie et al., 2006) $K_{s}$ band apparent magnitudes,
interstellar extinctions and 2MASS $K_{s}$ band absolute magnitudes ($M_{Ks}$)
derived directly from the LAMOST spectra using neural network models. The
photometric distance is accurate to 8.5% for stars with spectral SNRs larger
than 50. Especially, the photometric distance show small dependence on the
apparent magnitudes, positions of stars, absolute magnitudes and spectral
colors. It is a suitable dataset to test the zero-point offset in Gaia EDR3
parallaxes.

###  2.1 Sample selection

We cross-match the value-added catalog with Gaia EDR3 and 2MASS, and use the
following criteria to select stellar sample to test the zero-point offset of
Gaia EDR3 parallax:

  * •

LAMOST spectral signal to noise ratio (SNR) $\geq$50, parallaxes smaller than
1.5 mas, surface gravity $\log g\leq 3.8$, and the uncertainty of 2MASS
$K_{s}$ band apparent magnitude smaller than 0.03 mag.

  * •

The renormalized unit weight error (RUWE) $\leq$ 1.4.

  * •

An effective wavenumber 1.1$\leq\nu_{\rm eff}\leq$1.9 for the five-parameter
solution and 1.24 $\leq$ pseudocolor $\leq 1.72$ for six-parameter solution.

  * •

The spectral color $0.3<B_{p}-R_{p}<1.65$ mag.

Only distant giant stars are selected for this study as shown in the first
selection criterion. The systematic difference between $M_{Ks}$ coming from
Wang et al. (2022, accepted) and that estimated using the distance of Bailer-
Jones et al. (2021) of our test sample are 0.015 mag and 0.0049 mag for dwarf
and giant stars as shown in Fig.1, respectively (Wang et al. 2022, accepted).
Because the typical parallaxes of the dwarf and giant stars in our sample are
1.5 and 0.45 mas, respectively, these systematic differences in $M_{Ks}$ will
typically produce a systematic parallax difference of about $\sim 10.3\,\mu\rm
as$ for the dwarf stars, and $\sim 1.0\,\mu\rm as$ for the giants.
Furthermore, the effect of binary stars on the estimated absolute magnitudes
of dwarf stars is larger than that of giant stars. Thus, we only select
distant giant stars as tracers to check the zero-point offset of Gaia EDR3
parallax in the current work. The RUWE is a quality indicator of Gaia EDR3
data, given by the renormalized square root of the reduced chisquare of the
Gaia astrometric solution. Larger values indicate that the astrometric
solution does not completely describe the source motion (Lindegren et al.,
2021a; Fabricius et al., 2021). The stars with large value of RUWE are also
removed. The effective wavenumber and pseudocolor represent the color
information of the observed targets. Besides, the stars with $B_{p}-R_{p}\leq
0.3$ mag or $B_{p}-R_{p}\geq 1.65$ mag are also removed in order to make sure
that our training sample could cover all kinds of stars in our giant sample
(one can see Section 2.2 for more details). After the above cuts, we select
about 268,003 and 5,070 LAMOST giant stars with five-parameter and six-
parameter solutions in Gaia EDR3, respectively.

Figure 1: The distributions of $M_{Ks}$ difference between $M_{Ks}$ coming
from Wang et al. (2022, accepted) ($M_{Ks}^{W}$) and that estimated using the
distance of Bailer-Jones et al. (2021) ($M_{Ks}^{Gaia}$) of the test sample in
Wang et al. (2022, accepted). The left and right panels are the results of
dwarfs and giants, respectively. The black dashed line represent Gaussian fits
for the difference distributions. The mean values and standard deviations of
the two distributions are labeled in the top left corner of each panel. The
$\Delta M_{Ks}$ is the $M_{Ks}^{Gaia}-M_{Ks}^{W}$.

###  2.2 The parallax quality of our giant sample

As shown in Wang et al. (2022, accepted), the photometric distance are
estimated with the distance modulus method using the 2MASS $K_{s}$ band
apparent magnitudes, interstellar extinctions and $M_{Ks}$ derived directly
from the LAMOST spectra. Amongst of them, the $M_{Ks}$ are derived using
neural network models, which are built up using the LGMWAS (LAMOST–Gaia
EDR3–2MASS–WISE–APASS–SDSS ) common stars as training set. In the LGMWAS
training sample, the absolute magnitudes are estimated using the geometrical
Gaia EDR3 distance (Bailer-Jones et al., 2021), based on the Gaia EDR3
parallaxes corrected according to Lindegren et al. (2021b). We can
nevertheless use the distances in Wang et al. (2022, accepted) to test the
parallax bias in Gaia EDR3 because the mean parallax of the LGMWAS training
set ($\sim$ 2 mas) is much larger than the mean parallax in our giant sample
($\sim$ 0.45 mas). Our sample is therefore 4–5 times more sensitive to a
parallax bias than the training set, which breaks the degeneracy, considering
also that the two samples cover the same ranges of absolute magnitude and
spectral color as shown in Fig. 2.

In Wang et al. (2022, accepted), we only select stars with small extinctions
($E(B-V)<0.02$ mag) as training stars. Almost all of our giant stars have
larger extinction (with a median value of $E(B-V)\sim 0.09$ mag, estimated
using “star-pair” method of Yuan et al. (2013)) and extinction errors (with a
typical value of 0.02 mag) than that of training stars, which will produce
larger errors of the estimated photometric distance. The uncertainties of
0.006 mag for the 2MASS $K_{s}$ band apparent magnitude and 1.2 $\mu\rm as$
for the the estimated parallax will be caused by the typical extinction error.
This parallax error is very small even if it is a systematic error. Thus, the
variations of estimated parallax zero-point offset with spectral color,
apparent magnitude and spatial positions will be less affected considering
also the huge number of stars in our giant sample. In conclusion, although the
estimates of parallax in our giant sample are based on the geometrical Gaia
EDR3 distance of Bailer-Jones et al. (2021), we could use the sample to test
the parallax bias of Gaia EDR3 and its dependence on the spectral color,
apparent magnitude and spatial positions.

PRC stars are standard candles, their absolute magnitudes and distances could
be accurately estimated (Cannon, 1970; Paczyński & Stanek, 1998; Bovy et al.,
2014; Huang et al., 2015; Wan et al., 2015; Chen et al., 2017; Huang et al.,
2020). Huang et al. (2020) have selected $\sim 140,000$ PRCs from LAMOST, the
distances of which could be accurate to 5% for stars with spectral SNRs larger
than 50. We cross-match our giant sample and PRC sample of Huang et al.
(2020), and select 49,459 common stars. Fig. 3 shows the comparison of our
parallax ($\omega_{G}$) and PRC parallax ($\omega_{RC}$) and the parallax
difference variations with the PRC parallax. The global systematic difference
of the two kinds of parallax is 3.6 $\mu\rm as$, which is slightly larger than
the claimed typical parallax systematic difference ($\sim 1.0\mu\rm as$)
between our parallax and official zero-point offset corrected Gaia EDR3
parallax as discussed in the Section 2.1. It is reasonable considering the
systematic difference between the corrected Gaia EDR3 parallax and the
parallax of PRC sample. The standard deviation of the parallax difference is
$\sim 41\,\mu as$, which is also good enough. From Fig. 3, we can find a trend
of parallax difference with the parallax. The parallax difference are $-1$%,
1% and 3% at $\omega_{RC}\sim 200\,\mu\rm as$ (corresponding parallax error is
$\sim-2\,\mu\rm as$), $\omega_{RC}\sim 450\,\mu\rm as$ (corresponding parallax
error is $\sim+4.5\,\mu\rm as$) and $\omega_{RC}\geq 1000\,\mu\rm as$
(corresponding parallax error is $\geq+30\,\mu\rm as$), respectively.
Considering the typical parallax of our giant stars is $\sim 450\,\mu\rm as$,
the several $\mu\rm as$ systematic error of our giant sample and its
variations with parallax will produce small systematic error of our final
estimated parallax zero-point offset, which are supported by the similar
global parallax zero-point offsets of us and other previous works discussed in
Section 3. Besides, the stars with different parallax have similar spatial
distributions, which suggest that the relative spatial variations of the
estimated parallax zero-point offset will not be affected by the trend. In
conclusion, our giant sample is a good sample to test the Gaia EDR3 parallax
zero-point offset, especially its variations with apparent magnitude, spectral
color and spatial positions.

In the current paper, $\omega_{\rm EDR3}$, $\omega_{\rm EDR3}^{\rm corr}$ and
$\omega_{\rm G}$ (here ’G’ mean giant stellar sample) are the Gaia EDR3
parallax, Gaia EDR3 parallax with zero-point correction using the model of
Lindegren et al. (2021b) and the parallax given by photometric distance of our
LAMOST giant stars, respectively.

Figure 2: The absolute magnitude (left panel) and spectral color (right panel)
distributions of the LGMWAS training sample (black lines) and our final giant
sample (blue lines).  Figure 3: Top panel: the comparison between our parallax
and PRC parallax. Bottom panel: the variations of relative parallax difference
between our parallax and PRC parallax with the PRC parallax. The black dashed
lines in the bottom panel are zero minus/plus 1% or 3%.

##  3 Results and Discussions

Using the selected giant stars, we compare the parallaxes derived from our
photometric distances ($\omega_{\rm G}$) with those from Gaia EDR3
($\omega_{\rm EDR3}$) for the five- and six-parameter solutions. As shown in
Fig. 4, the median offsets of Gaia EDR3 parallaxes are respectively $-27.9$
$\mu\rm as$ and $-26.5$ $\mu\rm as$ for the five- and six-parameter solutions,
which are slightly larger than the value of $-17\mu\rm as$ derived from
distant quasars (Lindegren et al., 2021b). Our estimated median offsets are
consistent with that of Huang et al. (2021, $\sim-26\,\mu\rm as$ for both
five- and six-parameter solutions) and Ren et al. (2021, $\sim-28$ and
$-25\,\mu\rm as$ for respectively five- and six-parameter solutions). The
median values of the parallax difference between $\omega_{\rm EDR3}^{\rm
corr}$ and $\omega_{\rm G}$ are only $+2.6$ and $+2.9$ $\mu\rm as$ for the
five- and six-parameter solutions, respectively. The negligible positive
values of median offsets suggest that the official parallax zero-point
correction model (Lindegren et al., 2021b) significantly reduces the global
bias of Gaia EDR3 parallaxes.

Figure 4: Differences distributions of parallaxes between the LAMOST giant
stellar sample and Gaia EDR3 five-parameter solution (left panel) or six-
parameter solution (right panel). The blue and red histograms represent the
$\omega_{\rm EDR3}-\omega_{\rm G}$ and $\omega_{\rm EDR3}^{\rm
corr}-\omega_{\rm G}$ distributions, respectively. The blue and red dashed
lines represent Gaussian fits for the two parallax difference distributions,
respectively. The mean values and standard deviations of the two distributions
are labeled in the top corner using their corresponding colors.

As discussed in Lindegren et al. (2021b), Huang et al. (2021) and Ren et al.
(2021), the bias of Gaia EDR3 parallaxes is dependent on apparent magnitude
($G$ band), effective wavenumber $\nu_{\rm eff}$/pseudocolor and ecliptic
latitude (sin$\beta$). With the advantage of large number of stars, large and
continue coverage in the Galactic plane and wide range of colors and
magnitudes of our giant sample, we test the main dependencies of the Gaia EDR3
parallaxes bias. Fig. 5 shows the parallax bias of $\omega_{\rm
EDR3}-\omega_{\rm G}$ and $\omega_{\rm EDR3}^{\rm corr}-\omega_{\rm G}$, as a
function of G magnitude, effective wavenumber $\nu_{\rm eff}$/pseudocolor,
ecliptic latitude sin$\beta$ for the five- and six-parameter solutions. From
Fig. 5, we find that the $\omega_{\rm EDR3}-\omega_{\rm G}$ are almost
$-30\,\mu\rm as$, the $\omega_{\rm EDR3}^{\rm corr}-\omega_{\rm G}$ are almost
$0\,\mu\rm as$, which is similar with the result shown in Fig. 4.

From Fig. 5, we can find significant patterns of $\omega_{\rm
EDR3}-\omega_{\rm G}$ as a function of $G$ magnitude for both five- and six-
parameter solutions. The “hump-like” feature in $10<G<11$ and $12<G<13$ are
found for both five- and six-parameter solutions, which is similar with the
results of Lindegren et al. (2021b) and Huang et al. (2021). The $\omega_{\rm
EDR3}^{\rm corr}-\omega_{\rm G}$ are almost equal to zero and show smaller
variations with $G$ magnitude. The results suggest that the official parallax
zero-point correction could reduce parallax bias patterns with G magnitudes.
The $\omega_{\rm EDR3}-\omega_{\rm G}$ and $\omega_{\rm EDR3}^{\rm
corr}-\omega_{\rm G}$ as a function of $\nu_{\rm eff}$/ $\rm pseudocolour$
show several patterns for five- or six-parameter solutions, respectively. The
results suggest that the official parallax zero-point correction could not
reduce parallax bias patterns with spectral colors. For the $\omega_{\rm
EDR3}-\omega_{\rm G}$ and $\omega_{\rm EDR3}^{\rm corr}-\omega_{\rm G}$
variations with sin$\beta$, we do not find significant trend for both five-
and six-parameter solutions.

Figure 5: Parallax differences between our LAMOST giant sample and Gaia EDR3
five-parameter solution (left panel) or Gaia EDR3 six-parameter solution
(right panel), as a function of $G$ band magnitudes (top panels), color
information (middle panels: $\nu_{\rm eff}$ for five-paramter solution,
psceudocolour for six-paramter solution) and ecliptic latitude sin$\beta$
(bottom panels). The blue and red points represent the ($\omega_{\rm
EDR3}-\omega_{\rm G}$) and ($\omega_{\rm EDR3}^{\rm corr}-\omega_{\rm G}$),
respectively. In each bin, the number of stars is no less than 100. The black
dashed lines in each panel mark the differences of $-$30 $\mu\rm as$ and 0
$\mu\rm as$.

The bias of Gaia EDR3 parallax is positional dependent (Lindegren et al.,
2021b; Huang et al., 2021; Ren et al., 2021). Our LAMOST giant sample covers a
large and continue volume of the Galaxy and has huge number of stars, thus it
is a better sample to investigate the distribution of the parallax biases as a
function of spatial positions. Fig. 6 shows the maps of the mean parallax
differences for the five-parameter solutions of $\omega_{\rm EDR3}-\omega_{\rm
G}$ and $\omega_{\rm EDR3}^{\rm corr}-\omega_{\rm G}$ in equatorial
coordinates and Galactic coordinates. The $\omega_{\rm EDR3}-\omega_{\rm G}$
are smaller than 0 at almost all position bins, with a median value of
$\sim-30\,\mu\rm as$. The $\omega_{\rm EDR3}^{\rm corr}-\omega_{\rm G}$ are
smaller than $+30\,\mu\rm as$ and larger than $-30\,\mu\rm as$ at almost all
position bins, with a median value of $\sim 0\,\mu\rm as$. The results suggest
that the official corrections for Gaia EDR3 parallaxes are effective in
reducing global offset in the Gaia parallaxes. From this plot, we find that
the Gaia EDR3 parallax bias exhibit significant and clear trend with the
positions. The parallax biases in the Galactic plane ($b\sim 0^{\circ}$) are
largest, which is similar with the result of Lindegren et al. (2021b) and Ren
et al. (2021). Fig. 6 suggests that the parallax biases before and after
correction show similar trends with the positions, which suggest that official
correction for Gaia EDR3 parallaxes could not reduce the parallax bias
variations with positions. A spatially dependent Gaia EDR3 parallax zero-point
correction model is needed.

We will provide a dataset, which contains the position-dependent Gaia EDR3
parallax zero-point corrections for five-parameter solution within the LAMOST
footprint in the Galactic coordinate and equatorial coordinate. The resolution
of each pixel is about 3.36 $\rm deg^{2}$ in HEALPix (Hierarchical Equal Area
isoLatitude Pixelisation; Górski et al., 2005) grid after dividing the whole
sky into 12288 equal area pixels, which is same with that in Fig. 6. For the
stars located in a pixel we obtain the mean ($\mu$) and standard deviation
($\sigma$) of $\omega_{\rm EDR3}-\omega_{\rm G}$ and $\omega_{\rm EDR3}^{\rm
corr}-\omega_{\rm G}$ by fitting a gaussian function. The dataset gives for
each pixel $\mu$ as the Gaia EDR3 parallax zero-point correction in the pixel,
$\frac{\sigma}{\sqrt{N}}$ as the corresponding error, and N as the number of
stars in the pixel. One can use this dataset to correct the spatially
dependent Gaia EDR3 parallax zero-point offset. The dataset will be released
in the website of http://www.lamost.org/dr8/v1.0/doc/vac. Table. 1 shows the
detailed description of the dataset. It is noted that we give both the
position-dependent Gaia EDR3 parallax zero-point corrections in the Galactic
and equatorial coordinates, one can choose one of them according to their
coordinates.

Figure 6: Maps of $\omega_{\rm EDR3}-\omega_{\rm G}$ (top panels) and $\omega_{\rm EDR3}^{\rm corr}-\omega_{\rm G}$ (bottom panels) in equatorial coordinate (left panels) and Galactic coordinate (right panels) for the five-parameter solutions.Each pixel covers equal sky area of about $3.36$ square degrees. Number of stars in each pixel is larger than 50.  Table 1: Descriptions for the dataset to correct spatially dependent Gaia EDR3 parallax zero-point offset for five-parameter solutions.  Field | Description | Unit
---|---|---
hpindex | the HEALPix index number (level 5) | -
ra | Right ascension at the center of each pixel | degree
dec | Declination at the center of each pixel | degree
gl | Galactic longitude at the center of each pixel | degree
gb | Galactic latitude at the center of each pixel | degree
$\Delta\omega^{\rm corr}$ |  $\omega_{\rm EDR3}^{\rm corr}-\omega_{\rm G}$ values in each pixel |  $\mu$as
E$\\_\Delta\omega^{\rm corr}$ | the error of estimated $\Delta\omega^{\rm corr}$ in each pixel |  $\mu$as
$\Delta\omega^{\rm noncorr}$ |  $\omega_{\rm EDR3}-\omega_{\rm G}$ values in each pixel |  $\mu$as
E$\\_\Delta\omega^{\rm noncorr}$ | the error of estimated $\Delta\omega^{\rm noncorr}$ in each pixel |  $\mu$as
$N$ | Number of stars in each pixel | -

##  4 Summary

In the current paper, we use $\sim$ 0.3 million giant stars selected from the
LAMOST to investigate the zero-point offset of the Gaia EDR3 parallax
$\omega_{\rm EDR3}$ and the Gaia EDR3 parallax after correction $\omega_{\rm
EDR3}^{\rm corr}$ using the model of Lindegren et al. (2021b). The distance
accuracy of these giant stars are about 8.5%. The global median zero-point
offset of $\omega_{\rm EDR3}$ are $-27.9\,\mu\rm as$ and $-26.5\,\mu\rm as$
for five-parameter solution and six-parameter solution, respectively. The bias
of $\omega_{\rm EDR3}^{\rm corr}$ are $+$2.6 $\mu\rm as$ and $+$2.9 $\mu\rm
as$ for five-parameter solution and six-parameter solution, respectively.

We also studied the main dependences of the Gaia EDR3 parallax bias before and
after correction. We find that the parallax bias show significant systematic
trends with $G$ magnitude, spectral colors. The official parallax zero-point
correction could reduce the global offfset in the Gaia EDR3 parallaxes and the
significant systematic trends with $G$ magnitude. While the official parallax
zero-point correction could not reduce the parallax bias variations with
spectral colors. With the advantage of the huge number of stars in our giant
stellar sample, we also investigate the parallax bias variations with
positions. Both the parallax bias before and after correction exhibit
significant and clear trend with the positions, which suggest that the
official parallax zero-point correction could not reduce the parallax bias
variations with positions. In the current paper, a spatially dependent Gaia
EDR3 parallax zero-point correction model for five-parameter solutions in the
LAMOST footprint is firstly provided.

This work was funded by the National Key R&D Program of China (No.
2019YFA0405500) and the National Natural Science Foundation of China (NSFC
Grant No.11833006, U1531244 and 11973001, 12173007). We used data from the
European Space Agency mission Gaia (http:// www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC; see
http://www.cosmos.esa. int/web/gaia/dpac/consortium). Guoshoujing Telescope
(the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a
National Major Scientific Project built by the Chinese Academy of Sciences.
Funding for the project has been provided by the National Development and
Reform Commission. LAMOST is operated and managed by the National Astronomical
Observatories, Chinese Academy of Sciences.

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