# The Effect of Light Deflection by Solar System Objects on High-Precision SKA
Astrometry

Yingjie Li  Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing
210023, China  Ye Xu  Purple Mountain Observatory, Chinese Academy of
Sciences, Nanjing 210023, China  University of Science and Technology of
China, Hefei, Anhui 230026, China  Shaibo Bian  Purple Mountain Observatory,
Chinese Academy of Sciences, Nanjing 210023, China  University of Science and
Technology of China, Hefei, Anhui 230026, China  ZeHao Lin  Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210023, China  University of
Science and Technology of China, Hefei, Anhui 230026, China  JingJing Li
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023,
China  DeJian Liu  Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210023, China  University of Science and Technology of China, Hefei,
Anhui 230026, China  Chaojie Hao  Purple Mountain Observatory, Chinese Academy
of Sciences, Nanjing 210023, China  University of Science and Technology of
China, Hefei, Anhui 230026, China

###### Abstract

We have computed the deflection angles caused by 195 objects in the solar
system, including 177 satellites and eight asteroids. Twenty-one satellites
and six asteroids can bend light from distant compact extragalactic sources by
more than 0.1 $\mu$as, and fourteen satellites and the asteroid Ceres can
deflect light by more than 1.0 $\mu$as. We calculated the zones and durations
of perturbations posed by the gravitational fields of five planets (excluding
Earth, Jupiter, and Saturn), Pluto, and Ceres, where the perturbations would
affect astrometry measured with the Squared Kilometre Array (SKA). Perturbed
zones with deflection angles larger than 0.1 and 1.0 $\mu$as appear as
ribbons. Their widths range from dozens of degrees for Uranus, Neptune, and
Venus to several degrees or less for other objects at 0.1 $\mu$as, and from
$\sim$ 16∘ for Venus to several degrees or less for other objects at 1.0
$\mu$as. From the calculated perturbation durations, the influence of the
gravitational fields of selected objects can be divided into four levels:
hardly affect SKA astrometry (I), may have little effect (II), may have a
great effect (III) on single-epoch astrometry, and may greatly affect both
single- and multi-epoch astrometry (IV). The objects corresponding to these
levels are Ceres (I), Pluto (II), Mercury and Mars (III), and other objects
(IV).

gravitation — quasars — relativity — techniques: interferometric

##  1 Introduction

Testing the theory of general relativity (GR) by observing the light
deflection by the Sun (Dyson et al., 1920) made GR famous and initiated the
“modern era” of astronomy (Johnson, 1983). GR has been tested over 100 years,
especially in terms of gravitational light deflection, relativistic perihelion
advance, and gravitational redshift. Complementary to light deflection is the
Shapiro time delay (see Shapiro, 1964). Celestial bodies in the solar system
(which define Barycentric Celestial Reference System, BCRS) act as
gravitational sources and can play an important role in GR tests (Ivanitskaia
et al., 1986; Soffel, 1989; Damour et al., 1991, 1992, 1993, 1994; Hees et
al., 2014a; Will, 2014; Crosta et al., 2017; Ni, 2017; Bernus et al., 2019;
Crosta, 2019). One important parameter to be tested is the parameterized post-
Newtonian (PPN) parameter, $\gamma$, which is measuring the amount of the
curvature produced by a unit mass-energy and is equal to unity in GR (Damour,
1989; Poisson & Will, 2014; Will, 2014). The accuracy of $\gamma$ obtained by
measurements has rapidly improved in recent years (for more details see the
review by Will, 2015). The highest accuracy of $\gamma$ is $\sim 2\times
10^{-5}$, determined via data obtained with the Cassini spacecraft (Shapiro,
1964; Bertotti et al., 2003). The latest accuracy of $\gamma$ obtained using
the very long baseline interferometry (VLBI) technique has reached $\sim
9\times 10^{-5}$ (Titov et al., 2018). Planets, such as Jupiter, have also
been used to test GR (e.g., Fomalont & Kopeikin, 2003, 2008; Abbas et al.,
2022; Li et al., 2022). Light deflection and the underlying theory (e.g.,
relativistic gravity, gravitational lensing theory, etc.) have become
important tools for both astronomy and cosmology (Will, 2015).

Measurements of $\gamma$ with high accuracy are intertwined with high-
precision astrometry. Obtaining a high accuracy of $\gamma$ benefits from the
rapid improvement of the relevant technologies and instruments. Ultra-precise
($\sim$ 1 $\mu$as or higher) and ultra-sensitive (e.g., several hundred
nJy/beam @ 9.2 GHz for an 8 hr exposure with the Square Kilometre Array, SKA,
Bonaldi et al., 2021) astrometry have been the immediate goals for SKA (Braun
et al., 2015), the next-generation Very Large Array (ngVLA, Murphy et al.,
2018), and their pathfinders (Rioja & Dodson, 2020), and also Gaia Mission
from space (Vecchiato et al., 2003; Butkevich et al., 2022). Such improvements
together with other laboratories and space experiments (e.g., Solar System
Odyssey, Christophe et al. 2009; the series of Astrodynamical Space Test of
Relativity using Optical Devices, ASTROD, Ni 1998, 2009; Astrometric Science
and Technology Roadmap for Astrophysics, ASTRA, Gai et al. 2020; Transiting
Exoplanet Survey Satellite mission, TESS, Gai et al. 2022, etc.; see more
experiments in Ni 2017) have the potential to yield another 3–4 orders of
magnitude of precision in testing GR (Ni, 2017). Development of theoretical
study of gravity is in turn vital to future ultra-precise and ultra-sensitive
astrometry (Sekido & Fukushima, 2006; Petit & Luzum, 2010; Li et al., 2022).

High-precision astrometry has played a leading role in revealing the spiral
structure of the Milky Way at both radio (e.g., Xu et al., 2006; Reid et al.,
2009; Xu et al., 2016; Reid et al., 2019b) and optical (e.g., Xu et al.,
2018a, b, 2021; Poggio et al., 2021) wavelengths, and in producing the 3rd
realization of the International Celestial Reference Frame (ICRF3, Charlot et
al., 2020) and the Gaia DR2/(E)DR3 Celestial Reference Frame (Gaia-CRF2/CRF3,
Gaia Collaboration et al., 2018, 2022a). As the astrometric precision is
stepping into 10 $\mu$as or better (e.g., the recorded parallax precision has
reached $\pm$ 3 $\mu$as by Zhang et al. 2013 using the Very Long Baseline
Array and the predicted parallax precision is 10 $\mu$as for GAIA data release
4, Gaia Collaboration et al. 2022b), the revealed structures of the Milky Way
would be extended to distant regions beyond $\sim$ 10 kpc. High-precision
astrometry is also helpful for studying notable VLBI-Gaia positional offsets
(i.e., the offsets between measured optical and radio positions; Mignard et
al., 2016; Kovalev et al., 2017; Petrov & Kovalev, 2017; Charlot et al.,
2020). High-precision astrometry can also shed light on other broad sources
and problems in astrophysics, e.g., providing electromagnetic counterparts of
gravitational wave sources, monitoring the orbit of binaries and analyzing
their physical parameters, investigating the evolution of galaxies, etc. (see
Reid et al., 2019a).

High-precision astrometry suffers from the effects posed by the gravitational
fields of objects in the solar system. For instance, the predicted deflection
angle (according to the predictions of GR) caused by the planets, Pluto, and
even large satellites exceeds several $\mu$as and can even reach a dozen of
mas (see Fienga et al., 2011; Hees et al., 2014b; Bertone et al., 2014; Crosta
et al., 2015, 2017). The critical angular distances between compact
extragalactic sources (CESs, hereafter) and solar system objects (e.g.,
planets and the Moon), at which the deflection angles are still $\geq$ 1
$\mu$as, are larger than several arcminutes or even tens of degrees or more
(e.g., Crosta & Mignard, 2006; Li et al., 2022).

As a pilot work, we have evaluated the effect of light deflection caused by
objects in the solar system on high-precision SKA astrometry. This evaluation
may be helpful in saving computing resources for the vast data volumes
expected to be collected by SKA (e.g., Quinn et al., 2015). In addition, this
work may contribute to the selection of interesting region and time to test GR
or PPN parameters via light deflections or the Shapiro time delay, especially
when there are unknown weak sources that can only be detected by the ultra-
sensitive SKA (e.g., Bonaldi et al., 2021). Those interesting region and time
may also be helpful in broader researches (Li et al., 2022), e.g., measuring
additional Shapiro delay relative to post-Newtonian corrections (see Will,
2014), testing and developing multi-plane lensing effects (Subramanian &
Chitre, 1984; Erdl & Schneider, 1993; Ramesh et al., 2022), higher-order
(e.g., second-order) PPN formalisms (Crosta & Mignard, 2006; Kopeikin &
Makarov, 2007), etc.

The remainder of this paper is organized as follows. Section 3 overviews the
deflection angle caused by 195 objects in the solar system, using the sample
given in Section 2. In Section 4, we discuss the zones and durations of
perturbations posed by the gravitational fields of objects that revolve around
the Sun and evaluate the impact of those objects on SKA astrometry. Finally,
we summarize our conclusions in Section 5.

##  2 Sample

The sample includes 195 objects in the solar system; i.e., the Sun, all eight
planets, Pluto, 177 satellites (including the Moon), and eight asteroids with
$GM>0.1$ km3 s-2, where $G$ and $M$ are the gravitational constant and the
mass of the solar system object, respectively. The parameters of the Sun were
obtained from IAU 2015 Resolution B3 (see Miley, 2015). 111See
https://www.iau.org/administration/resolutions/general_assemblies/The physical
and orbital parameters of the planets, Pluto, satellites, and asteroids were
obtained from the Jet Propulsion Laboratory (JPL, hereafter). 222Planets and
Pluto: physical parameters see https://ssd.jpl.nasa.gov/?planet_phys_par;
orbital parameters see https://ssd.jpl.nasa.gov/planets/approx_pos.html.
Satellites: physical parameters see https://ssd.jpl.nasa.gov/?sat_phys_par;
and orbital parameters see https://ssd.jpl.nasa.gov/?sat_elem. The parameters
of the asteroids were obtained from
https://ssd.jpl.nasa.gov/tools/sbdb_query.html. The references therein are not
listed here because they are very numerous, and we kindly advise the reader to
the consult the various references found on the aforementioned websites.We
replaced the orbital parameters of the satellites (other than the Moon) with
that of their host planets or Pluto because the ratios of the semi-major axis
of the orbital revolution for these satellites and that of the corresponding
planets or Pluto, $R_{\mathrm{sp}}$, are small; i.e., all of them are
$\lesssim 0.04$ (see Table A1). The other parameters required in this work are
also listed in Table A1.

##  3 Deflection Angle Caused by Solar System Objects

###  3.1 Orbits of the Planets

To obtain the positions of the planets, Pluto, and asteroids, it is convenient
to use the heliocentric plane orbital coordinate system. Assuming that a
planet moves along a fixed elliptical orbit, the position of the planet with
eccentric anomaly, $E$, is

$x=a(\cos E-e);\;\;y=a\sqrt{1-e^{2}}\sin E;\;\;z=0,$ (1)

where $a$ is the semi-major axis, $e$ is the eccentricity (see Table A1), and
$z=0$ represents that the planet is located in its orbital plane. The
calculated coordinates can be transformed into the coordinates in the ecliptic
coordinate system ($x_{\mathrm{ecl}}$, $y_{\mathrm{ecl}}$, $z_{\mathrm{ecl}}$)
by

$\left\\{\begin{aligned}
x_{\mathrm{ecl}}&=(\cos\omega\cos\Omega-\sin\omega\sin\Omega\cos
I)\;\;x&+&\;\;(-\sin\omega\cos\Omega-\cos\omega\sin\Omega\cos I)\;\;y\\\
y_{\mathrm{ecl}}&=(\cos\omega\sin\Omega+\sin\omega\cos\Omega\cos
I)\;\;x&+&\;\;(-\sin\omega\sin\Omega+\cos\omega\cos\Omega\cos I)\;\;y,\\\
z_{\mathrm{ecl}}&=(\sin\omega\sin I)\;\;x&+&\;\;(\cos\omega\sin
I)\;\;y\end{aligned}\right.$ (2)

where $I$, $\omega$, and $\Omega$ are the inclination, perihelion, and
longitude of the ascending node (see Table A1), respectively. These three
parameters are valid for the time interval 1800 AD–2050 AD. The computations
below are based on the rough orbital determinations derived using the above
formulas.

###  3.2 Light Deflection and Its Impact

####  3.2.1 Calculation of Light Deflection

Light will bend when it passes a massive body. When the impact parameter, $b$,
is far less than the distance from the body to both the Earth (observer) and
target source, the deflection angle, $\alpha$, can be calculated as follows
(Hosokawa et al., 1993)

$\alpha=(1+\gamma)\frac{2GM}{c^{2}b},$ (3)

where $\gamma$ is the PPN parameter, which is $=1$ for GR. When $b$ does not
satisfy the described condition above, the expression of $\alpha$ (Cowling,
1984; Will, 1993; Ni, 2017) is

$\alpha=(1+\gamma)\frac{GM}{c^{2}b}(\cos\theta_{1}-\cos\theta_{0}),$ (4)

where $\theta_{0}$ (and $\theta_{1}$) is the angle between the light
propagation vector and the CES (and the Earth; see schematic diagrams of light
deflection under the gravitational field of a lens in Figure 3 of Li et al.,
2022). We denote Equation (3) as “Simplified” and Equation (4) as “Normal”
because Equation (4) shows a wider range of applications. $\theta_{0}\approx
180^{\circ}$ if the observed target is a distant CES, $b=r\sin\beta$ (see
Cowling, 1984) and $\theta_{1}\approx\alpha+\beta$, where $r$ is the distance
from the observer to the celestial body, and $\beta$ is the angle between the
CES (in the absence of gravitational or aberrational bending) and the
celestial body as seen by the observer. When $\theta_{1}$ approaches 0∘,
Equation (4) approaches Equation (3). When the celestial body is a solar
system object, it is valid to replace $\theta_{1}$ with $\beta$ under a
precision of $0.1$ $\mu$as (see Li et al., 2022).

The maximum and minimum distances (i.e., $r_{\mathrm{max}}$ and
$r_{\mathrm{min}}$, respectively) from the Earth to the celestial body are
calculated based on the orbit determination in Section 3.1. The corresponding
angular radii of a celestial body seen from the Earth are
$\Theta_{\mathrm{min}}$ and $\Theta_{\mathrm{max}}$ (see Table 1),
respectively, assuming that the physical size of the celestial body is its
mean radius. For satellites other than the Moon, their values of
$r_{\mathrm{max}}$, $r_{\mathrm{min}}$, $\Theta_{\mathrm{min}}$, and
$\Theta_{\mathrm{max}}$ are replaced with the corresponding values of their
primary component (including Pluto).

In this work, every solar system object is regarded as an isolated individual
lens. The reasons are as follows. Firstly, in the process of the data
correlator, a necessary step that all radio interference data should be
conducted, the total corrected general relativistic time delay is the linear
superposition of the time delay of each individual lens in the solar system
(see Petit & Luzum, 2010). Secondly, it is showed that the effect of
uncertainty of $\beta$ (assumed to be 10 mas, which can be regarded as the
combined deflection angle caused by other objects) would be rapidly decreased
to less than 0.1 $\mu$as with the increase of $\beta$ for Saturn and Jupiter
(see figure 7 of Li et al., 2022). These facts indicate that it is reasonable
to regard every solar system object as an isolated individual lens. The
combined deflection angle may not affect the results of individual, and could
be a linear superposition of the deflection angle (as a vector, e.g., Li et
al., 2022) of each individual lens in this work. Further study considering
multi-plane lensing effects would be conducted to accurately determine the
range of application of “linear superposition of the deflection angle”.

####  3.2.2 Maximum Deflection Angle

The maximum values of $\alpha$ calculated by Equation (3), where $b$ is the
mean radius of the celestial body (see Table A1), are consistent with those
calculated by Crosta & Mignard (2006). All planets can deflect light by up to
$\sim$ 82.0 $\mu$as or more. Pluto can also bend light by $\sim$ 7.0 $\mu$as.
The value of $\alpha_{\max}$ for asteroid Ceres is 1.2 $\mu$as. This asteroid
is the only one with $\alpha_{\max}>1.0$ $\mu$as. The number of asteroids with
$\alpha_{\max}>0.1$ $\mu$as is six.

Large satellites can also bend light by several or dozens of $\mu$as. The
number of satellites with $\alpha_{\max}>1.0$ $\mu$as is 14, and they are the
Moon, Ganymede, Callisto, Io, Europa (four satellites of Jupiter), Titan,
Rhea, Iapetus, Dione (four satellites of Saturn), Titania, Oberon, Ariel,
Umbriel (four satellites of Uranus), Triton (a satellite of Neptune), and
Charon (a satellite of Pluto). The number of satellites with
$\alpha_{\max}>0.1$ $\mu$as is 21, i.e., the Moon, four, seven, five, three,
and one satellite(s) of Jupiter, Saturn, Uranus, Neptune, and Pluto,
respectively. $R_{\mathrm{sp}}<3\%$ and $<0.2\%$ for satellites other than the
Moon with $\alpha_{\max}$ $\geq$ 0.1 and $\geq$ 1.0 $\mu$as, respectively.
Therefore, replacing $r_{\mathrm{max}}$ and $r_{\mathrm{min}}$ of these
satellites with their primaries does not affect the conclusions of this work.

####  3.2.3 Impact Ranges

(a) Jupiter with Maximum Distance from Earth (b) Jupiter with Minimum Distance
from Earth (c) Mars with Maximum Distance from Earth (d) Mars with Minimum
Distance from Earth (e) Sun with Minimum Distance from Earth (f) Sun with
Maximum Distance from Earth

Figure 1: $\alpha$ as a function of $\beta$ under $r_{\mathrm{max}}$ (left)
and $r_{\mathrm{min}}$ (right) for Jupiter (top), Mars (middle), and the Sun
(bottom), respectively. The red and green lines present the results computed
from Equations (3) and (4), respectively, and the blue line shows the
difference between the two results (see Section 3.2.4). The solid and dashed
black lines denote $\alpha$ values of 1.0 and 0.1 $\mu$as, respectively, for
Jupiter and Mars, and 1.0 and 0.1 mas for the Sun, respectively.

Figure 1 presents $\alpha$ as a function of $\beta$ for three objects, i.e.,
Jupiter, Mars, and the Sun under both $r_{\mathrm{max}}$ (left) and
$r_{\mathrm{min}}$ (right). The critical value of $\beta$ where $\alpha$ is
equal to a given value (e.g., 1.0 $\mu$as) under a given condition (e.g.,
$r_{\mathrm{min}}$), can be regarded as the impact range of an event of light
deflection, and thus is denoted as $\beta$-impact for short hereafter. For
instance, $\beta$-impact for $\alpha$ = 1.0 $\mu$as under $r_{\mathrm{min}}$,
denoted as $\beta_{1.0,\mathrm{min}}$, indicates that a celestial body can
deflect light by amount of 1.0 $\mu$as up to $\leq\beta_{1.0,\mathrm{min}}$.
This naming convention also applies to other given values of $\alpha$ or
conditions; e.g., $\beta_{1.0,\mathrm{max}}$, $\beta_{0.1,\mathrm{min}}$, and
$\beta_{0.1,\mathrm{max}}$ represent the critical values of $\beta$ when the
deflection angle equals 1.0 $\mu$as under $r_{\mathrm{max}}$, 0.1 $\mu$as
under $r_{\mathrm{max}}$, and $r_{\mathrm{min}}$, respectively; and
$\beta_{1.0}$ and $\beta_{0.1}$ denote the critical values of $\beta$ when the
deflection angle equals 1.0 and 0.1 $\mu$as, respectively.

Table 1 lists $\beta$-impact for more planets, satellites, and asteroids,
where only objects with $\alpha_{\mathrm{max}}>0.005$ $\mu$as and
$\beta_{0.1,\mathrm{max}}>\Theta_{\mathrm{max}}$ have been presented. The
results of $\beta$-impact for the planets, Pluto, and the Moon are consistent
with those reported by Crosta & Mignard (2006).

Table 1: $\beta$-Impact and $\beta$-Validity

Objects | Index | $\Theta_{\mathrm{min}}$ | $\Theta_{\mathrm{max}}$ | $\beta_{\mathrm{1.0,min}}$ | $\beta_{\mathrm{1.0,max}}$ | $\beta_{\mathrm{0.1,min}}$ | $\beta_{\mathrm{0.1,max}}$ | $\beta_{\mathrm{d,1.0,min}}$ | $\beta_{\mathrm{d,1.0,max}}$ | $\beta_{\mathrm{d,0.1,min}}$ | $\beta_{\mathrm{d,0.1,max}}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | (′′) | (′′) | (′′) | (′′) | (′′) | (′′) | (∘) | (∘) | (∘) | (∘)
Sun | 1 | 943.45 | 975.52 | 647896 | 647900 | 647989 | 647990 | 0.26 | 0.27 | 0.26 | 0.27
Mercury | 2 | 2.32 | 6.13 | 191 | 507 | 1920 | 5078 | 179.95 | 179.86 | 179.47 | 178.59
Venus | 3 | 4.81 | 31.58 | 2368 | 15555 | 23661 | 148812 | 179.34 | 175.68 | 173.43 | 138.66
Moon | 5 | 883.38 | 987.00 | 22859 | 25534 | 208923 | 228881 | 173.65 | 172.91 | 121.97 | 116.42
Mars | 6 | 1.75 | 12.16 | 203 | 1410 | 2035 | 14102 | 179.94 | 179.61 | 179.43 | 176.08
Jupiter | 9 | 14.93 | 24.39 | 223618 | 320605 | 580120 | 606206 | 117.88 | 90.94 | 18.86 | 11.61
Ganymede | 10 | 0.56 | 0.92 | 19 | 31 | 194 | 316 | 179.99 | 179.99 | 179.95 | 179.91
Callisto | 11 | 0.51 | 0.84 | 14 | 23 | 141 | 230 | 180.00 | 179.99 | 179.96 | 179.94
Io | 12 | 0.39 | 0.64 | 11 | 19 | 116 | 191 | 180.00 | 179.99 | 179.97 | 179.95
Europa | 13 | 0.33 | 0.54 | 5 | 10 | 62 | 103 | 180.00 | 180.00 | 179.98 | 179.97
Saturn | 77 | 7.28 | 9.99 | 43350 | 59304 | 334988 | 398610 | 167.96 | 163.53 | 86.95 | 69.27
Titan | 78 | 0.32 | 0.44 | 9 | 13 | 102 | 140 | 180.00 | 180.00 | 179.97 | 179.96
Rhea | 79 | 0.10 | 0.13 | NaN | NaN | 1 | 2 | 180.00 | 180.00 | 180.00 | 180.00
Iapetus | 80 | 0.09 | 0.13 | NaN | NaN | 1 | 1 | 180.00 | 180.00 | 180.00 | 180.00
Dione | 81 | 0.07 | 0.10 | NaN | NaN | NaN | 1 | 180.00 | 180.00 | 180.00 | 180.00
Uranus | 139 | 1.66 | 2.02 | 3477 | 4241 | 34691 | 42269 | 179.03 | 178.82 | 170.36 | 168.26
Titania | 140 | 0.05 | 0.06 | NaN | NaN | 1 | 1 | 180.00 | 180.00 | 180.00 | 180.00
Oberon | 141 | 0.05 | 0.06 | NaN | NaN | 1 | 1 | 180.00 | 180.00 | 180.00 | 180.00
Neptune | 167 | 1.08 | 1.18 | 2762 | 3001 | 27582 | 29965 | 179.23 | 179.17 | 172.34 | 171.68
Triton | 168 | 0.06 | 0.06 | NaN | NaN | 5 | 6 | 180.00 | 180.00 | 180.00 | 180.00
Pluto | 182 | 0.03 | 0.06 | NaN | NaN | 2 | 3 | 180.00 | 180.00 | 180.00 | 180.00
Ceres | 188 | 0.16 | 0.41 | NaN | NaN | 1 | 4 | 180.00 | 180.00 | 180.00 | 180.00
Pallas | 189 | 0.09 | 0.31 | NaN | NaN | NaN | 1 | 180.00 | 180.00 | 180.00 | 180.00

Note. — $\beta$-Impact is the impact range of an event of light deflection
(see Section 3.2.3); $\beta$-Validity is the range of application of Equation
(3) (see Section 3.2.4); NaN represents that $\beta_{\mathrm{1.0,min}}$ or
$\beta_{\mathrm{0.1,min}}$ being less than $\Theta_{\mathrm{min}}$, or
$\beta_{\mathrm{1.0,max}}$ or $\beta_{\mathrm{0.1,max}}$ being less than
$\Theta_{\mathrm{max}}$.

The Sun’s gravitational field could bend light in the direction of
$\beta\in[\Theta,180-\Theta]$ (where $\Theta$ is the apparent radius of the
Sun) with a deflection angle $>$ 10 $\mu$as. Above all, Jupiter can deflect
light by an amount of 0.1 $\mu$as up to $\leq 168.4^{\circ}$ (i.e.,
$\beta_{0.1,\mathrm{max}}\approx 168.4^{\circ}$), and by an amount of 1.0
$\mu$as up to $\leq 89.1^{\circ}$ (i.e., $\beta_{1.0,\mathrm{max}}\approx
89.1^{\circ}$; see Figure 1). Such large values of $\beta_{0.1,\mathrm{max}}$
and $\beta_{1.0,\mathrm{max}}$ make Jupiter an unavoidable issue in
astrometry, like the Sun. For Saturn, the values of $\beta_{0.1,\mathrm{max}}$
and $\beta_{1.0,\mathrm{max}}$ are $\approx 110.7^{\circ}$ and $\approx
16.5^{\circ}$, respectively. For Mercury, Venus, Mars, Uranus, and Neptune,
the values of ($\beta_{0.1,\mathrm{max}}$, $\beta_{1.0,\mathrm{max}}$) are
$\approx$ ($1.4^{\circ}$, $0.1^{\circ}$), ($41.3^{\circ}$, $4.3^{\circ}$),
($3.9^{\circ}$, $0.4^{\circ}$), ($11.7^{\circ}$, $1.2^{\circ}$), and
($8.3^{\circ}$, $0.8^{\circ}$), respectively.

Moon’s effect is comparable to that of the planets, i.e.,
($\beta_{0.1,\mathrm{max}}$, $\beta_{1.0,\mathrm{max}}$) $\approx$
($63.6^{\circ}$, $7.1^{\circ}$) (see Table 1). For Ganymede,
($\beta_{0.1,\mathrm{max}}$, $\beta_{1.0,\mathrm{max}}$) $\approx$ (316′′,
31′′). Eleven satellites (including the Moon) can bend light by 0.1 $\mu$as,
and six can bend light by 1.0 $\mu$as, as long as the corresponding $\beta$ is
$\geq 1^{\prime\prime}$. The $\beta$-impacts of Pluto and several asteroids
(i.e., Ceres and Pallas) are comparable to those of large satellites.

The aforementioned values of ($\beta_{0.1,\mathrm{max}}$,
$\beta_{1.0,\mathrm{max}}$) highlight the challenge of conducting high-precise
astrometry, especially by using next-generation observatories.

####  3.2.4 Validity of the Simplified Equation

Equation (3) is much simpler relative to Equation (4), therefore it may be
helpful in saving computing resources by using Equation (3). The difference
between Equations (4) and (3) is presented to judge the range of application
of Equation (3) (see Figure 1). The critical value of $\beta$ where the
difference of $\alpha$ calculated from the two equations is equal to a given
value (e.g., 1.0 $\mu$as) under a given condition (e.g., $r_{\mathrm{min}}$)
can be regarded as the validity of Equation (3) (which is much simpler
relative to Equation (4)), and is denoted as $\beta$-validity for short
hereafter. The naming convention is similar to that for $\beta$-impact, and
denoted as, e.g., $\beta_{\mathrm{d,1.0,min}}$, $\beta_{\mathrm{d,1.0,max}}$,
$\beta_{\mathrm{d,0.1,min}}$, and $\beta_{\mathrm{d,0.1,max}}$.

The Sun and Mars can be seen as two extremes (see Figure 1). For the Sun,
Equation (3) is invalid even when light grazes its surface under a precision
of 1.0 $\mu$as. For Jupiter and Saturn (see Table 1),
$\beta_{\mathrm{d,0.1,max}}$ is less than 90∘, but
$\beta_{\mathrm{d,1.0,max}}$ exceeds 90∘. For other objects, including the
other five planets, Pluto, and all satellites and asteroids,
$\beta_{\mathrm{d,0.1,max}}$, $\beta_{\mathrm{d,0.1,min}}$,
$\beta_{\mathrm{d,1.0,max}}$ and $\beta_{\mathrm{d,1.0,min}}$ all exceed 90∘,
indicating that Equation (3) is valid even under a precision of 0.1 $\mu$as
when $\beta\leq 90^{\circ}$. If $\beta_{\mathrm{d,0.1,max}}$ or
$\beta_{\mathrm{d,1.0,max}}$ exceeds $90^{\circ}$ when $\beta>90^{\circ}$, no
correction for a positional offset caused by the gravitational fields of the
corresponding objects is required even under a precision level of 0.1 $\mu$as.

##  4 Discussion

###  4.1 Maximum Deflection Angle and Earth-Lens Distance on Impact Range of
Light Deflection

The value of $\beta$-impact is related to $\alpha_{\mathrm{max}}$ and the
distance from the celestial body to the Earth (see Section 3.2). Figure 2
presents the relationship among $\alpha_{\mathrm{max}}$, distance, and the
values of $\beta$-impact (i.e., $\beta_{1.0,\mathrm{min}}$,
$\beta_{1.0,\mathrm{max}}$, $\beta_{0.1,\mathrm{min}}$, and
$\beta_{0.1,\mathrm{max}}$). The results indicate that the values of
$\beta$-impact increase with an increase of $\alpha_{\mathrm{max}}$ and a
decrease of distance.

(a) $\beta_{1.0,\mathrm{min}}$ with Maximum Distance (b)
$\beta_{1.0,\mathrm{max}}$ with Minimum Distance (c)
$\beta_{0.1,\mathrm{min}}$ with Maximum Distance (d)
$\beta_{0.1,\mathrm{max}}$ with Maximum Distance

Figure 2: The value of $\beta$-impact (i.e., $\beta_{1.0,\mathrm{min}}$,
$\beta_{1.0,\mathrm{max}}$, $\beta_{0.1,\mathrm{min}}$, and
$\beta_{0.1,\mathrm{max}}$) as a function of $\alpha_{\mathrm{max}}$ and the
distances from the celestial bodies to the Earth. Only sources with
$0.1\leq\alpha_{\mathrm{max}}<17\,000.0$ $\mu$as and the corresponding values
of $\beta$-impact $\geq 1^{\prime\prime}$ are included.

###  4.2 Zones of Perturbation

This section roughly estimates the specific zones in which light from CESs
would be deflected by a given angle (i.e., 1.0 and 0.1 $\mu$as) by Mercury,
Venus, Mars, Uranus, Neptune, Pluto, and Ceres. The $\beta_{\mathrm{0.1,max}}$
values of these sources are all $\geq 1^{\prime\prime}$. Jupiter and Saturn
are not included because their values of $\beta_{\mathrm{0.1,min}}$ are larger
than $90^{\circ}$. So heavy is their impact on light from CESs, like the Sun,
that they should be taken into consideration in most cases. For satellites,
two orbital centers (both the Sun and primary components including Pluto)
should be considered. In addition, the perturbed zones for satellites other
than the Moon are buried within those for their primary components. For the
Moon, $\beta_{\mathrm{0.1,min}}$ is about $58^{\circ}$. The impact of the Moon
is comparable to that of Saturn. Therefore, the perturbed zones for the many
satellites are not included in this work. We use the position of the Earth-
Moon system as the position of the Earth, and assume that the planets, Pluto,
and asteroids move along a fixed elliptical orbit around the Sun.

We assume that the CES, the Earth and the celestial body (e.g., planets, Pluto
or asteroids) all exist along a single line. The position of a CES can be
calculated by assuming two arbitrary values of $E$ for the Earth and the
celestial body, respectively; see the definition of $E$ in Equation (1). The
distance from the Earth to a CES is fixed to be $10^{14}$ times larger than
the distance from the Earth to the celestial body. This corresponds to a
distance from the Earth to a fiducial CES of $\sim$ 500 Mpc if the distance
from the Earth to the celestial body is $\sim$ 1 au. If the distance from the
Earth to the CES fluctuates between 1 Mpc and 500 Mpc, the corresponding
fluctuation of the computed position of the CES is about dozens of $\mu$as.
This computed position is used to calculate the deflection angle caused by the
gravitational field of solar system objects. The positional uncertainty of
CESs of dozens of $\mu$as does not affect the computed deflection angle under
precision of 0.1 $\mu$as. Therefore, this fluctuation does not affect the
conclusions of this work.

The coverage area of the perturbed zone corresponding to $\beta_{1.0}$ (i.e.,
the zone where light from a CES would be deflected by a celestial body by 1.0
$\mu$as) is a circle, at the given positions of the Earth and the celestial
body; i.e., two given values of $E$. The center of this circle is the computed
CES position projected on the sky as seen from Earth (coinciding with the
projected point on the sky of the trajectory of the corresponding celestial
body), and the radius is $\beta_{1.0}$. Similarly, the perturbed zone for
$\beta_{0.1}$ is also a circle with the same center, but its radius is
$\beta_{0.1}$.

The ecliptic coordinate system was used to compute the positions of the Earth,
celestial bodies, and CESs. These positions can be transformed into the J2000
equatorial coordinate system as follows:

$\left\\{\begin{aligned} x_{\mathrm{eq}}&=x_{\mathrm{ecl}}\\\
y_{\mathrm{eq}}&=\cos\varepsilon\;y_{\mathrm{ecl}}-\sin\varepsilon\;z_{\mathrm{ecl}},\\\
z_{\mathrm{eq}}&=\sin\varepsilon\;y_{\mathrm{ecl}}+\cos\varepsilon\;z_{\mathrm{ecl}}\end{aligned}\right.$
(5)

where $\varepsilon=23.43928^{\circ}$ is the obliquity of the ecliptic.

We have computed the whole zone of perturbation for both $\beta_{1.0}$ and
$\beta_{0.1}$ (see Figure 3). The perturbed zones for $\beta_{1.0}$ and
$\beta_{0.1}$ are ribbons and their widths are denoted as $W_{1.0}$ and
$W_{0.1}$, respectively. For Pluto, $W_{1.0}$ and $W_{0.1}$ are less than
$\sim$ 1∘ (see Figure 3 and Table 2). For Mercury, Mars, and Ceres, $W_{1.0}$
and $W_{0.1}$ are similar and their values are $\sim$ 4∘–9∘, $\sim$ 2∘–6∘, and
$\sim$ 5∘–10∘, respectively. The values of $W_{1.0}$ for Uranus and Neptune
are $\sim$ 3∘ or less, and close to that of Pluto. However, considering
$\beta_{\mathrm{0.1}}$ of $\sim$ 10∘ for Uranus and Neptune, their values of
$W_{0.1}$ range from $\sim$ 10∘ to $\sim$ 23∘, which are larger than those for
Mercury, Mars, and Ceres. In particular, $W_{0.1}$ for Venus is close to
$\sim$ 80∘ and $W_{1.0}$ also reaches $\sim$ 13∘–16∘. These results imply that
the size of the perturbed zone is negatively correlated with the distance from
the Earth to the celestial body and positively correlated with the value of
$\beta_{\mathrm{0.1}}$ or $\beta_{\mathrm{1.0}}$.

(a) Mercury

(b) Venus

(c) Mars

(d) Uranus

(e) Neptune

(f) Pluto

(g) Ceres

Figure 3: Whole perturbed zone for $\beta_{1.0}$ (red) and $\beta_{0.1}$
(blue). The corresponding celestial bodies are denoted in the subhead. The
black points show the projected points on the sky of the trajectory of the
corresponding celestial body. Table 2: Widths of the Ribbon-like Perturbed
Zones

Quantity | Mercury | Venus | Mars | Uranus | Neptune | Pluto | Ceres
---|---|---|---|---|---|---|---
$W_{1.0}$ |  $\sim$ 4∘–7∘ |  $\sim$ 13∘–16∘ |  $\sim$ 2∘–5∘ |  $\sim$ 2∘–3∘ |  $\sim$ 1∘–2∘ |  $\lesssim$ 1∘ |  $\sim$ 5∘–10∘
$W_{0.1}$ |  $\sim$ 5∘–9∘ |  $\sim$ 66∘–80∘ |  $\sim$ 4∘–6∘ |  $\sim$ 14∘–23∘ |  $\sim$ 10∘–17∘ |  $\lesssim$ 1∘ |  $\sim$ 5∘–10∘

Note. — See the detailed distribution of whole perturbed zone in Figure 3.

###  4.3 Duration of Perturbation

It is interesting to investigate the durations when light from CESs can be
deflected by celestial bodies by 1.0 and 0.1 $\mu$as, referred to as
$\tau_{1.0}$ and $\tau_{0.1}$, respectively. Similar to the perturbed zones,
we only focused on calculating the values for Mercury, Venus, Mars, Uranus,
Neptune, Pluto, and Ceres. These bodies have been classified into three
categories: planets inside and outside the Earth’s orbit and others (i.e.,
Pluto and Ceres).

####  4.3.1 Trajectory of the Earth and Celestial Bodies

It is necessary to determine the positions of the Earth and the celestial body
in the solar system at any given time. For celestial bodies inside and outside
the Earth’s orbit, two different methods are used to obtain the positions of
the Earth and the celestial body. For celestial bodies inside the Earth’s
orbit, we divide the orbits of a given celestial body into segments, and
compute the movement time in each segment. The relative position of the Earth
is determined by using this movement time. The reason for choosing the given
celestial body as a reference body is that the angle variation of the Earth
should not exceed 180∘ if that of the given gravitational body is $\leq
180^{\circ}$, because the angular speed of the Earth is slower. Therefore, in
the progress of summing the movement time for all eligible segments (i.e., the
segments where the deflection angle is larger than 1.0 or 0.1 $\mu$as), the
movement time for any segment should not be double-counted when the angular
change of the given celestial body is within $\pm$ 180∘. For celestial bodies
outside the Earth’s orbit, the selected reference body is the Earth for
similar reasons.

Specifically, for the computation of planets inside and outside the Earth’s
orbit, $E$ is divided into 129 601 parts for the reference bodies; i.e., the
resolution of $E$ is about 10′′. This resolution corresponds to a mean time
resolution of $\sim$ 1.0 and $\sim$ 2.5 min for Mercury and Venus (inside the
Earth’s orbit), and $\sim$ 4.1 min for Mars, Uranus, and Neptune (outside the
Earth’s orbit), respectively. When computing the values for the category of
others (i.e., Pluto and Ceres, also outside the Earth’s orbit), $E$ is divided
into 5 184 001 parts, i.e., the resolution of $E$ is about 0.25′′. That is
because $\beta_{\mathrm{0.1,min}}$ is a few arcseconds for Pluto and Ceres.
The corresponding mean time resolution for Pluto and Ceres is about 6.0 s.

As a pilot work, we divided the orbit of a given celestial body into 1000
points (this is enough to show details; see Figures 4–10) as the initial
positions, $\theta_{\mathrm{CES}}$, i.e., $\theta_{\mathrm{CES}}$ =
$\frac{360^{\circ}}{1000}\times 0,1,2,...,1000$. For each point, we calculate
the value of $E$ for the Earth corresponding to the minimum distance from the
Earth to the given gravitational body. We select one of these values of $E$,
and set $E+0^{\circ},10^{\circ},...,180^{\circ}$ as the initial position of
the Earth, $\phi_{\mathrm{CES}}$. The number of total initial combinations of
a given celestial body and the Earth,
$(\theta_{\mathrm{CES}},\phi_{\mathrm{CES}})$, is 19 000. The same method as
presented in Section 4.2 is used to determine the position of a CES under
$(\theta_{\mathrm{CES}},\phi_{\mathrm{CES}})$, where the position of a CES
remains unchanged during the calculation for a given initial combination of
$(\theta_{\mathrm{CES}},\phi_{\mathrm{CES}})$. We then calculate the duration
of perturbation where the angle between the CES and the given celestial body
as seen from the Earth is less than $\beta_{1.0}$ and $\beta_{0.1}$, i.e.,
$\tau_{1.0}$ and $\tau_{0.1}$, respectively.

####  4.3.2 Timescale Criteria Potentially Affecting SKA Astrometry

Here, we assume that the flux density of a CES is 0.15 Jy/beam @ 9200 MHz (the
median flux density in the X band for a short baseline is $\sim$ 0.15 Jy from
ICRF3; see Charlot et al., 2020), and 0.30 Jy/beam @ 1400 and 560 MHz (the
median flux density in the S band is $\sim$ 0.23 Jy from ICRF3). The
sensitivity of SKA is assumed to be 0.43 $\mu$Jy/beam @ 9200 MHz, 0.71
$\mu$Jy/beam @ 1400 MHz, and 2.88 $\mu$Jy/beam @ 560 MHz for an exposure time
of 8 hr, and the full width at half-maximum size in the three wavebands is
assumed to be 0.09′′, 0.6′′, and 1.5′′, respectively (see Bonaldi et al.,
2021).

The integration time needed to achieve a positional accuracy of 1.0 $\mu$as
should be $\sim$ 8.0 min @ 9200 MHz, $\sim$ 4.0 hr @ 1400 MHz, and $\sim$
414.7 hr @ 560 MHz (this one may be meaningless), respectively. These three
timescales can be regarded as the timescale criteria that the predicted event
potentially affects SKA astrometry under a precision level of 0.1 $\mu$as.
These three criteria are denoted as $\tau_{c,0.1,9200}$, $\tau_{c,0.1,1400}$,
and $\tau_{c,0.1,560}$ (see Table 3), respectively. The three timescale
criteria corresponding to a precision level of 1.0 $\mu$as are smaller than
those of 0.1 $\mu$as by a factor of 100, and are denoted as
$\tau_{c,1.0,9200}$, $\tau_{c,1.0,1400}$, and $\tau_{c,1.0,560}$,
respectively. When the duration exceeds 48.0 hr (except for the case at 560
MHz), the events can affect SKA astrometry over multiple epochs. We denote
this criterion as $\tau_{c,m}$.

Table 3: Timescale Criteria Potentially Affecting SKA Astrometry

$\tau_{c,0.1,9200}$ | $\tau_{c,0.1,1400}$ |  $\tau_{c,0.1,560}$$\dagger$$\dagger$footnotemark:  |  $\tau_{c,1.0,9200}$ | $\tau_{c,1.0,1400}$ | $\tau_{c,1.0,560}$ | $\tau_{c,m}$
---|---|---|---|---|---|---
(min) | (hr) | (hr) | (sec) | (min) | (hr) | (hr)
8.0 | 4.0 | 414.7 | 4.8 | 2.4 | 4.1 | 48.0

††footnotetext: This value may be meaningless.

####  4.3.3 Category of Planets inside the Earth’s Orbit

This category includes Mercury and Venus. The computed positions in this work
in this category manifest as a group of closed patterns (see Figures 4 and 5).

(a) $\tau_{1.0}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(b) Distribution of $\tau_{1.0}$

(c) $\tau_{0.1}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(d) Distribution of $\tau_{0.1}$

Figure 4: $\tau_{1.0}$ and $\tau_{0.1}$ for Mercury. The value of
$\phi_{\mathrm{CES}}$ is labeled in the legend; see the main text for
$\theta_{\mathrm{CES}}$ and $\phi_{\mathrm{CES}}$.

(a) $\tau_{1.0}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(b) Distribution of $\tau_{1.0}$

(c) $\tau_{0.1}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(d) Distribution of $\tau_{0.1}$

Figure 5: $\tau_{1.0}$ and $\tau_{0.1}$ for Venus. The description of each map
refers to Figure 4.

Figure 4 shows the distributions of $\tau_{1.0}$ and $\tau_{0.1}$ (right
panel) and they as functions of $(\theta_{\mathrm{CES}},\phi_{\mathrm{CES}})$
(left panel) for Mercury. The minimum value of $\tau_{1.0}$ is 1.3 hr, which
is larger than $\tau_{c,1.0,9200}$ and $\tau_{c,1.0,1400}$. This indicates
that the gravitational field of Mercury affects SKA astrometry over single-
epoch observations at 9200 and 1400 MHz under a precision level of 1.0
$\mu$as. This conclusion also applies to the case for a precision level of 0.1
$\mu$as. $\tau_{1.0}$ is larger than 48.0 hr in a few cases, indicating that
the gravitational field of Mercury hardly affects SKA astrometry over multi-
epoch observations under a precision level of 1.0 $\mu$as. However,
$\tau_{0.1}$ is larger than 48.0 hr in most cases, indicating that the
gravitational field of Mercury is likely going to affect SKA astrometry over
multi-epoch observations under a precision level of 0.1 $\mu$as. Table 4
summarizes our conclusions. It is worth noting that large values of
$\tau_{1.0}$ and $\tau_{0.1}$ do not appear at $\phi_{\mathrm{CES}}=0$ (i.e.,
the minimum distance from the Earth to Mercury), but at $\phi_{\mathrm{CES}}$
being about 20∘–40∘ and about 20∘–50∘, respectively.

Table 4: Effect of Celestial Bodies on SKA Astrometry

Objects | Index | Precision Level | Single-Epoch |  | Multi-epoch
---|---|---|---|---|---
|  | ($\mu$as) | 9200 MHz | 1400 MHz | 560 MHz |  |
Mercury | 2 | 1.0 | Y | Y | N |  | N?
|  | 0.1 | Y | Y | N$\dagger$$\dagger$footnotemark:  |  | Y?
Venus | 3 | 1.0 | Y | Y | Y |  | Y?
|  | 0.1 | Y | Y | Y?$\dagger$$\dagger$footnotemark:  |  | Y
Mars | 6 | 1.0 | Y | Y | Y? |  | N?
|  | 0.1 | Y | Y | N?$\dagger$$\dagger$footnotemark:  |  | Y?
Uranus | 139 | 1.0 | Y | Y | Y |  | Y
|  | 0.1 | Y | Y | Y$\dagger$$\dagger$footnotemark:  |  | Y
Neptune | 167 | 1.0 | Y | Y | Y |  | Y
|  | 0.1 | Y | Y | Y$\dagger$$\dagger$footnotemark:  |  | Y
Pluto | 182 | 1.0 | N | N | N |  | N
|  | 0.1 | Y | N? | N$\dagger$$\dagger$footnotemark:  |  | N?
Ceres | 188 | 1.0 | N | N | N |  | N
|  | 0.1 | N? | N? | N$\dagger$$\dagger$footnotemark:  |  | N

††footnotetext: This result may be meaningless.

Note. — Y = affects SKA astrometry, Y? = likely affects SKA astrometry, N? =
hardly affects SKA astrometry, N = does not affect SKA astrometry.

From Figure 5 it can be seen that the effect of Venus is larger than that of
Mercury. The minimum values of $\tau_{1.0}$ and $\tau_{0.1}$ are 25.4 and
254.3 hr, respectively. The maximum values of $\tau_{1.0}$ and $\tau_{0.1}$
are both larger than 912.5 hr (i.e., exceeding one month). Large values of
$\tau_{1.0}$ and $\tau_{0.1}$ appear at $\phi_{\mathrm{CES}}\sim
0^{\circ}$–30∘ and $\sim 0^{\circ}$–40∘, respectively. Therefore, for Venus,
its gravitational field can largely affect SKA astrometry over both single-
epoch and multi-epoch observations (see the summary in Table 4).

####  4.3.4 Category of Planets outside the Earth’s Orbit

This category includes Mars, Uranus, and Neptune. Unlike the category inside
the Earth’s orbit, the computed points of this category are a group of ribbons
with R.A.(J2000) ranging from 0∘ to 360∘ (see Figures 6–8).

(a) $\tau_{1.0}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(b) Distribution of $\tau_{1.0}$

(c) $\tau_{0.1}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(d) Distribution of $\tau_{0.1}$

Figure 6: $\tau_{1.0}$ and $\tau_{0.1}$ for Mars. The description of each map
refers to Figure 4.

(a) $\tau_{1.0}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(b) Distribution of $\tau_{1.0}$

(c) $\tau_{0.1}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(d) Distribution of $\tau_{0.1}$

Figure 7: $\tau_{1.0}$ and $\tau_{0.1}$ for Uranus. The description of each
map refers to Figure 4.

(a) $\tau_{1.0}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(b) Distribution of $\tau_{1.0}$

(c) $\tau_{0.1}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(d) Distribution of $\tau_{0.1}$

Figure 8: $\tau_{1.0}$ and $\tau_{0.1}$ for Neptune. The description of each
map refers to Figure 4.

The effect of Mars is a little greater than that of Mercury (see Figure 6).
The minimum values of $\tau_{1.0}$ and $\tau_{0.1}$ are 3.9 and 38.8 hr,
respectively. $\tau_{1.0}$ is less than 10 hr and $\tau_{0.1}$ less than 100
hr in most cases. The maximum values of $\tau_{1.0}$ and $\tau_{0.1}$ are
379.4 and 1585.3 hr, respectively. Large values of $\tau_{1.0}$ and
$\tau_{0.1}$ appear at about 10–20∘ and about 0–40∘, respectively. Mars’s
gravitational field can affect astrometry over single-epoch observations at
precision levels of both 1.0 and 0.1 $\mu$as, and will likely affect multi-
epoch observations at a precision level of 0.1 $\mu$as, but will hardly affect
multi-epoch observations at a precision level of 1.0 $\mu$as (see Table 4).

From Figures 7 and 8, it can be seen that the effects of Uranus and Neptune
are larger than that of Venus. The minimum value of $\tau_{1.0}$ is larger
than 994.9 hr (i.e., exceeding one month). The value of $\tau_{0.1}$ is $\sim$
1 yr, which is the period of revolution of Earth around the Sun, indicating
that the obtained value reaches the calculation limit of this work. Therefore,
for Uranus and Neptune, their gravitational fields can largely affect
astrometry for both single-epoch and multi-epoch observations (see Table 4).

####  4.3.5 Category of Others

This category includes Pluto and Ceres. Objects in this category are also
outside the Earth’s orbit, and thus show similar trajectories as the category
of planets outside the Earth’s orbit, i.e., a group of ribbons with
R.A.(J2000) ranging from 0∘ to 360∘ (see Figures 9–10). For these two objects,
the values of $\beta_{\mathrm{1.0,max}}$ are less than their angular radii
seen from the Earth. Therefore, the gravitational fields of Pluto and Ceres do
not affect SKA astrometry under a precision level of 1.0 $\mu$as, and we did
not calculate $\tau_{1.0}$.

(a) $\tau_{1.0}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(b) Distribution of $\tau_{1.0}$

Figure 9: $\tau_{0.1}$ for Pluto. The description of each map refers to Figure
4.

(a) $\tau_{1.0}$ as a function of $\theta_{\mathrm{CES}}$ and
$\phi_{\mathrm{CES}}$

(b) Distribution of $\tau_{1.0}$

Figure 10: $\tau_{0.1}$ for Ceres. The description of each map refers to
Figure 4.

For Pluto, the maximum value of $\tau_{0.1}$ is 108.0 hr, but it only appears
at $\phi_{\mathrm{CES}}\sim 80^{\circ}$ (see Figure 9). The minimum value of
$\tau_{0.1}$ exceeds 8.0 min, and the value of $\tau_{0.1}$ exceeding 4.0 hr
only appears at $\phi_{\mathrm{CES}}$ $\sim$ 60∘–90∘. Therefore, Pluto’s
gravitational field can largely affect astrometry over single-epoch
observations at 9200 MHz, but will hardly affect single-epoch observations at
1400 MHz, and will hardly affect multi-epoch observations (see Table 4).

For Ceres, the maximum value of $\tau_{0.1}$ is 10.2 hr, but it only appears
at $\phi_{\mathrm{CES}}\sim 40^{\circ}$ (see Figures 10). The groups of values
of $\tau_{1.0}$ exceeding 8 min only appear at $\phi_{\mathrm{CES}}$ $\sim$
0∘–60∘. Therefore, its gravitational field will hardly affect SKA astrometry
(see Table 4).

##  5 Summary and Conclusions

We have calculated the maximum deflection angle caused by 195 objects in the
solar system, including the Sun, all planets, 177 satellites (contains the
Moon), and eight asteroids with $GM>0.1$ km3 s-2 (see Table A1). An overview
of the deflection angle caused by these objects is as follows (see Table 1).

  1. 1.

Twenty-one satellites and six asteroids can deflect light from CESs by more
than 0.1 $\mu$as, and 14 satellites and one asteroid (i.e., Ceres) can bend
light by more than 1.0 $\mu$as.

  2. 2.

Jupiter and Saturn can bend light by amount of 0.1 $\mu$as up to 100∘ and by
amount of 1.0 $\mu$as up to dozens of degrees. The ranges of influence under
0.1 $\mu$as and 1.0 $\mu$as for the other planets (other than the Earth) and
the Moon are $1.4^{\circ}$–63.6∘ and 0.1∘–7.1∘, respectively. But for the
satellites and Ceres, the corresponding ranges of influence are all from a few
arcseconds to a few hundred arcseconds.

Further computations, regarding the zones and durations of perturbations
caused by the celestial bodies, were made towards Mercury, Venus, Mars,
Uranus, Neptune, Pluto, and Ceres, and the main results are as follows.

  1. 1.

The computed perturbation zones with deflection angles larger than 1.0 $\mu$as
are ribbons with a width of several degrees or less, except for Venus whose
corresponding ribbon width is $\sim$ 13∘–16∘. For deflection angles larger
than 0.1 $\mu$as, the widths increase to more than a dozen degrees for Uranus
and Neptune, increase to close to 80∘ for Venus, and remain almost unchanged
for Mercury, Mars, Pluto, and Ceres (see Table 2 and Figure 3).

  2. 2.

The computation of the durations of perturbations (see Figures 4–10) posed by
the gravitational field of solar system objects indicates that the
gravitational field of Ceres will hardly affect SKA astrometry, but that of
Pluto may have a small effect on the astrometry of single-epoch observations
under a precision level of 0.1 $\mu$as. The gravitational fields of Mercury
and Mars may have a great influence on astrometry for single-epoch
observations at precision levels of both 1.0 and 0.1 $\mu$as, and those of
other objects can largely affect astrometry for both single and multi-epoch
observations (see Table 4).

We would like to thank the anonymous referee for the helpful comments and
suggestions that helped to improve the paper. This work was sponsored by the
Natural Science Foundation of Jiangsu Province (Grants No. BK20210999), the
Entrepreneurship and Innovation Program of Jiangsu Province, the NSFC Grants
Nos. 12203104, 11933011 and 11873019, the Key Laboratory for Radio Astronomy,
CAS.

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Table A1 presents the maximum deflection angle caused by 195 objects in the
solar system.

Table A1: Maximum Deflction Angle, $\alpha_{\mathrm{max}}$, under the
Gravitational Field of Celestial Bodies in Solar System

Objects | Index | $GM$ | $R_{\mathrm{L}}$ | $a$ | $e$ | $I$ | $\omega$ | $\Omega$ | $R_{\mathrm{sp}}$ | $\alpha_{\mathrm{max}}$
---|---|---|---|---|---|---|---|---|---|---
|  | (km3 s-2) | (km) | (au) |  | (∘) | (∘) | (∘) |  | ($\mu$as)
Sun | 1 | 1.33E+11 | 695700.0 | … | … | … | … | … | 0.00 | 1751190.33
Mercury | 2 | 2.20E+04 | 2439.4 | 0.387 | 0.206 | 7.00 | 29.13 | 48.33 | 0.00 | 82.91
Venus | 3 | 3.25E+05 | 6051.8 | 0.723 | 0.007 | 3.39 | 54.92 | 76.68 | 0.00 | 492.78
Earth$\dagger$$\dagger$footnotemark:  | 4 | 3.99E+05 | 6378.1 | 1.000 | 0.017 | 0.00 | 102.94 | 0.00 | 0.00 | 543.05
Moon | 5 | 4.90E+03 | 1737.5 | 0.003 | 0.055 | … | … | … | 0.00 | 25.90
Mars | 6 | 4.28E+04 | 3389.5 | 1.524 | 0.093 | 1.85 | -25.62 | 49.56 | 0.00 | 116.00
Phobos | 7 | 7.11E-04 | 11.1 | … | … | … | … | … | 0.00 | 0.00
Deimos | 8 | 9.85E-05 | 6.2 | … | … | … | … | … | 0.00 | 0.00
Jupiter | 9 | 1.27E+08 | 69911.0 | 5.203 | 0.048 | 1.30 | -85.75 | 100.47 | 0.00 | 16635.22
Ganymede | 10 | 9.89E+03 | 2631.2 | … | … | … | … | … | 0.00 | 34.50
Callisto | 11 | 7.18E+03 | 2410.3 | … | … | … | … | … | 0.00 | 27.34
Io | 12 | 5.96E+03 | 1821.6 | … | … | … | … | … | 0.00 | 30.04
Europa | 13 | 3.20E+03 | 1560.8 | … | … | … | … | … | 0.00 | 18.84
Himalia | 14 | 4.50E-01 | 85.0 | … | … | … | … | … | 0.02 | 0.05
Amalthea | 15 | 1.38E-01 | 83.5 | … | … | … | … | … | 0.00 | 0.02
Thebe | 16 | 1.00E-01 | 49.3 | … | … | … | … | … | 0.00 | 0.02
Elara | 17 | 5.80E-02 | 43.0 | … | … | … | … | … | 0.02 | 0.01
Pasiphae | 18 | 2.00E-02 | 30.0 | … | … | … | … | … | 0.04 | 0.01
Carme | 19 | 8.80E-03 | 23.0 | … | … | … | … | … | 0.04 | 0.00
Metis | 20 | 8.00E-03 | 21.5 | … | … | … | … | … | 0.00 | 0.00
Sinope | 21 | 5.00E-03 | 19.0 | … | … | … | … | … | 0.04 | 0.00
Lysithea | 22 | 4.20E-03 | 18.0 | … | … | … | … | … | 0.02 | 0.00
Ananke | 23 | 2.00E-03 | 14.0 | … | … | … | … | … | 0.04 | 0.00
Leda | 24 | 7.30E-04 | 10.0 | … | … | … | … | … | 0.02 | 0.00
Adrastea | 25 | 5.00E-04 | 8.2 | … | … | … | … | … | 0.00 | 0.00
Callirrhoe | 26 | 5.80E-05 | 4.3 | … | … | … | … | … | 0.04 | 0.00
Themisto | 27 | 4.60E-05 | 4.0 | … | … | … | … | … | 0.01 | 0.00
Praxidike | 28 | 2.90E-05 | 3.4 | … | … | … | … | … | 0.04 | 0.00
Megaclite | 29 | 1.40E-05 | 2.7 | … | … | … | … | … | 0.04 | 0.00
Iocaste | 30 | 1.30E-05 | 2.6 | … | … | … | … | … | 0.04 | 0.00
Kalyke | 31 | 1.30E-05 | 2.6 | … | … | … | … | … | 0.04 | 0.00
Taygete | 32 | 1.10E-05 | 2.5 | … | … | … | … | … | 0.04 | 0.00
Harpalyke | 33 | 8.00E-06 | 2.2 | … | … | … | … | … | 0.04 | 0.00
Aoede | 34 | 6.00E-06 | 2.0 | … | … | … | … | … | 0.04 | 0.00
Autonoe | 35 | 6.00E-06 | 2.0 | … | … | … | … | … | 0.04 | 0.00
Eukelade | 36 | 6.00E-06 | 2.0 | … | … | … | … | … | 0.04 | 0.00
Helike | 37 | 6.00E-06 | 2.0 | … | … | … | … | … | 0.04 | 0.00
Hermippe | 38 | 6.00E-06 | 2.0 | … | … | … | … | … | 0.04 | 0.00
S/2003 J5 | 39 | 6.00E-06 | 2.0 | … | … | … | … | … | 0.04 | 0.00
Thyone | 40 | 6.00E-06 | 2.0 | … | … | … | … | … | 0.04 | 0.00
Chaldene | 41 | 5.00E-06 | 1.9 | … | … | … | … | … | 0.04 | 0.00
Isonoe | 42 | 5.00E-06 | 1.9 | … | … | … | … | … | 0.04 | 0.00
Aitne | 43 | 3.00E-06 | 1.5 | … | … | … | … | … | 0.04 | 0.00
Arche | 44 | 3.00E-06 | 1.5 | … | … | … | … | … | 0.04 | 0.00
Carpo | 45 | 3.00E-06 | 1.5 | … | … | … | … | … | 0.03 | 0.00
Erinome | 46 | 3.00E-06 | 1.6 | … | … | … | … | … | 0.04 | 0.00
Euanthe | 47 | 3.00E-06 | 1.5 | … | … | … | … | … | 0.04 | 0.00
Eurydome | 48 | 3.00E-06 | 1.5 | … | … | … | … | … | 0.04 | 0.00
Hegemone | 49 | 3.00E-06 | 1.5 | … | … | … | … | … | 0.04 | 0.00
Cyllene | 50 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
Euporie | 51 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.03 | 0.00
Herse | 52 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
Kale | 53 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
Kallichore | 54 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
Kore | 55 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
Mneme | 56 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
Orthosie | 57 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
Pasithee | 58 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2000 J11 | 59 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.02 | 0.00
S/2003 J10 | 60 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2003 J15 | 61 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2003 J16 | 62 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2003 J18 | 63 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.03 | 0.00
S/2003 J19 | 64 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2003 J2 | 65 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.05 | 0.00
S/2003 J23 | 66 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2003 J3 | 67 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.03 | 0.00
S/2003 J4 | 68 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2010 J1 | 69 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2010 J2 | 70 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2011 J1 | 71 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2011 J2 | 72 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
Sponde | 73 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
Thelxinoe | 74 | 1.00E-06 | 1.0 | … | … | … | … | … | 0.04 | 0.00
S/2003 J12 | 75 | 1.00E-07 | 0.5 | … | … | … | … | … | 0.03 | 0.00
S/2003 J9 | 76 | 1.00E-07 | 0.5 | … | … | … | … | … | 0.04 | 0.00
Saturn | 77 | 3.79E+07 | 58232.0 | 9.537 | 0.054 | 2.49 | -21.06 | 113.66 | 0.00 | 5979.49
Titan | 78 | 8.98E+03 | 2574.7 | … | … | … | … | … | 0.00 | 32.01
Rhea | 79 | 1.54E+02 | 764.3 | … | … | … | … | … | 0.00 | 1.85
Iapetus | 80 | 1.21E+02 | 735.6 | … | … | … | … | … | 0.00 | 1.50
Dione | 81 | 7.31E+01 | 561.7 | … | … | … | … | … | 0.00 | 1.19
Tethys | 82 | 4.12E+01 | 533.0 | … | … | … | … | … | 0.00 | 0.71
Enceladus | 83 | 7.20E+00 | 252.1 | … | … | … | … | … | 0.00 | 0.26
Mimas | 84 | 2.50E+00 | 198.2 | … | … | … | … | … | 0.00 | 0.12
Phoebe | 85 | 5.53E-01 | 106.5 | … | … | … | … | … | 0.01 | 0.05
Hyperion | 86 | 3.73E-01 | 135.0 | … | … | … | … | … | 0.00 | 0.03
Janus | 87 | 1.26E-01 | 89.5 | … | … | … | … | … | 0.00 | 0.01
Epimetheus | 88 | 3.51E-02 | 58.1 | … | … | … | … | … | 0.00 | 0.01
Prometheus | 89 | 1.07E-02 | 43.1 | … | … | … | … | … | 0.00 | 0.00
Pandora | 90 | 9.24E-03 | 40.7 | … | … | … | … | … | 0.00 | 0.00
Siarnaq | 91 | 2.60E-03 | 20.0 | … | … | … | … | … | 0.02 | 0.00
Albiorix | 92 | 1.40E-03 | 16.0 | … | … | … | … | … | 0.01 | 0.00
Helene | 93 | 7.60E-04 | 17.6 | … | … | … | … | … | 0.00 | 0.00
Paaliaq | 94 | 5.50E-04 | 11.0 | … | … | … | … | … | 0.01 | 0.00
Atlas | 95 | 4.40E-04 | 15.1 | … | … | … | … | … | 0.00 | 0.00
Pan | 96 | 3.30E-04 | 14.1 | … | … | … | … | … | 0.00 | 0.00
Ymir | 97 | 3.30E-04 | 9.0 | … | … | … | … | … | 0.02 | 0.00
Telesto | 98 | 2.70E-04 | 12.4 | … | … | … | … | … | 0.00 | 0.00
Kiviuq | 99 | 2.20E-04 | 8.0 | … | … | … | … | … | 0.01 | 0.00
Tarvos | 100 | 1.80E-04 | 7.5 | … | … | … | … | … | 0.02 | 0.00
Calypso | 101 | 1.70E-04 | 10.7 | … | … | … | … | … | 0.00 | 0.00
Ijiraq | 102 | 8.00E-05 | 6.0 | … | … | … | … | … | 0.01 | 0.00
Erriapus | 103 | 5.10E-05 | 5.0 | … | … | … | … | … | 0.01 | 0.00
Narvi | 104 | 2.30E-05 | 3.5 | … | … | … | … | … | 0.02 | 0.00
Skathi | 105 | 2.10E-05 | 4.0 | … | … | … | … | … | 0.01 | 0.00
Mundilfari | 106 | 1.40E-05 | 3.5 | … | … | … | … | … | 0.02 | 0.00
Suttungr | 107 | 1.40E-05 | 3.5 | … | … | … | … | … | 0.02 | 0.00
Thrymr | 108 | 1.40E-05 | 3.5 | … | … | … | … | … | 0.02 | 0.00
Daphnis | 109 | 5.20E-06 | 3.8 | … | … | … | … | … | 0.00 | 0.00
Pallene | 110 | 2.20E-06 | 2.5 | … | … | … | … | … | 0.00 | 0.00
Methone | 111 | 6.00E-07 | 1.6 | … | … | … | … | … | 0.00 | 0.00
Polydeuces | 112 | 3.00E-07 | 1.3 | … | … | … | … | … | 0.00 | 0.00
Anthe | 113 | 1.00E-07 | 0.9 | … | … | … | … | … | 0.00 | 0.00
Aegaeon | 114 | 4.00E-09 | 0.3 | … | … | … | … | … | 0.00 | 0.00
Aegir | 115 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
Bebhionn | 116 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.01 | 0.00
Bergelmir | 117 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
Bestla | 118 | 0.00E+00 | 3.5 | … | … | … | … | … | 0.02 | 0.00
Farbauti | 119 | 0.00E+00 | 2.5 | … | … | … | … | … | 0.02 | 0.00
Fenrir | 120 | 0.00E+00 | 2.0 | … | … | … | … | … | 0.02 | 0.00
Fornjot | 121 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
Greip | 122 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
Hati | 123 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
Hyrrokkin | 124 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
Jarnsaxa | 125 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
Kari | 126 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
Loge | 127 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
S/2004 S12 | 128 | 0.00E+00 | 2.5 | … | … | … | … | … | 0.02 | 0.00
S/2004 S13 | 129 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
S/2004 S17 | 130 | 0.00E+00 | 2.0 | … | … | … | … | … | 0.02 | 0.00
S/2004 S7 | 131 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
S/2006 S1 | 132 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
S/2006 S3 | 133 | 0.00E+00 | 2.5 | … | … | … | … | … | 0.02 | 0.00
S/2007 S2 | 134 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.01 | 0.00
S/2007 S3 | 135 | 0.00E+00 | 2.0 | … | … | … | … | … | 0.02 | 0.00
Skoll | 136 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.01 | 0.00
Surtur | 137 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.02 | 0.00
Tarqeq | 138 | 0.00E+00 | 3.0 | … | … | … | … | … | 0.01 | 0.00
Uranus | 139 | 5.79E+06 | 25362.0 | 19.189 | 0.047 | 0.77 | 96.94 | 74.02 | 0.00 | 2097.18
Titania | 140 | 2.28E+02 | 788.9 | … | … | … | … | … | 0.00 | 2.66
Oberon | 141 | 1.92E+02 | 761.4 | … | … | … | … | … | 0.00 | 2.32
Ariel | 142 | 8.64E+01 | 578.9 | … | … | … | … | … | 0.00 | 1.37
Umbriel | 143 | 8.15E+01 | 584.7 | … | … | … | … | … | 0.00 | 1.28
Miranda | 144 | 4.40E+00 | 235.8 | … | … | … | … | … | 0.00 | 0.17
Puck | 145 | 1.93E-01 | 81.0 | … | … | … | … | … | 0.00 | 0.02
Sycorax | 146 | 1.80E-01 | 75.0 | … | … | … | … | … | 0.00 | 0.02
Portia | 147 | 1.12E-01 | 70.0 | … | … | … | … | … | 0.00 | 0.01
Juliet | 148 | 3.72E-02 | 53.0 | … | … | … | … | … | 0.00 | 0.01
Belinda | 149 | 2.38E-02 | 45.0 | … | … | … | … | … | 0.00 | 0.00
Cressida | 150 | 2.29E-02 | 41.0 | … | … | … | … | … | 0.00 | 0.01
Caliban | 151 | 2.00E-02 | 36.0 | … | … | … | … | … | 0.00 | 0.01
Rosalind | 152 | 1.70E-02 | 36.0 | … | … | … | … | … | 0.00 | 0.00
Desdemona | 153 | 1.19E-02 | 35.0 | … | … | … | … | … | 0.00 | 0.00
Prospero | 154 | 6.60E-03 | 25.0 | … | … | … | … | … | 0.01 | 0.00
Bianca | 155 | 6.20E-03 | 27.0 | … | … | … | … | … | 0.00 | 0.00
Setebos | 156 | 5.80E-03 | 24.0 | … | … | … | … | … | 0.01 | 0.00
Ophelia | 157 | 3.60E-03 | 21.4 | … | … | … | … | … | 0.00 | 0.00
Cordelia | 158 | 3.00E-03 | 20.1 | … | … | … | … | … | 0.00 | 0.00
Stephano | 159 | 1.70E-03 | 16.0 | … | … | … | … | … | 0.00 | 0.00
Perdita | 160 | 1.20E-03 | 13.0 | … | … | … | … | … | 0.00 | 0.00
Mab | 161 | 6.00E-04 | 12.0 | … | … | … | … | … | 0.00 | 0.00
Francisco | 162 | 5.60E-04 | 11.0 | … | … | … | … | … | 0.00 | 0.00
Ferdinand | 163 | 4.20E-04 | 10.0 | … | … | … | … | … | 0.01 | 0.00
Margaret | 164 | 4.20E-04 | 10.0 | … | … | … | … | … | 0.01 | 0.00
Trinculo | 165 | 3.10E-04 | 9.0 | … | … | … | … | … | 0.00 | 0.00
Cupid | 166 | 2.00E-04 | 9.0 | … | … | … | … | … | 0.00 | 0.00
Neptune | 167 | 6.84E+06 | 24622.0 | 30.070 | 0.009 | 1.77 | -86.82 | 131.78 | 0.00 | 2548.39
Triton | 168 | 1.43E+03 | 1353.4 | … | … | … | … | … | 0.00 | 9.68
Proteus | 169 | 3.36E+00 | 210.0 | … | … | … | … | … | 0.00 | 0.15
Nereid | 170 | 2.06E+00 | 170.0 | … | … | … | … | … | 0.00 | 0.11
Larissa | 171 | 3.30E-01 | 97.0 | … | … | … | … | … | 0.00 | 0.03
Galatea | 172 | 2.50E-01 | 88.0 | … | … | … | … | … | 0.00 | 0.03
Despina | 173 | 1.40E-01 | 75.0 | … | … | … | … | … | 0.00 | 0.02
Thalassa | 174 | 2.50E-02 | 41.0 | … | … | … | … | … | 0.00 | 0.01
Naiad | 175 | 1.30E-02 | 33.0 | … | … | … | … | … | 0.00 | 0.00
Halimede | 176 | 1.20E-02 | 31.0 | … | … | … | … | … | 0.00 | 0.00
Neso | 177 | 1.10E-02 | 30.0 | … | … | … | … | … | 0.01 | 0.00
Sao | 178 | 4.50E-03 | 22.0 | … | … | … | … | … | 0.01 | 0.00
Laomedeia | 179 | 3.90E-03 | 21.0 | … | … | … | … | … | 0.01 | 0.00
Psamathe | 180 | 3.30E-03 | 20.0 | … | … | … | … | … | 0.01 | 0.00
S/2004 N1 | 181 | 3.00E-04 | 9.0 | … | … | … | … | … | 0.00 | 0.00
Pluto | 182 | 8.70E+02 | 1188.3 | 39.482 | 0.249 | 17.14 | 113.76 | 110.30 | 0.00 | 6.72
Charon | 183 | 1.02E+02 | 603.6 | … | … | … | … | … | 0.00 | 1.56
Hydra | 184 | 6.50E-03 | 30.5 | … | … | … | … | … | 0.00 | 0.00
Nix | 185 | 1.30E-03 | 23.0 | … | … | … | … | … | 0.00 | 0.00
Kerberos | 186 | 1.10E-03 | 14.0 | … | … | … | … | … | 0.00 | 0.00
Styx | 187 | 0.00E+00 | 10.0 | … | … | … | … | … | 0.00 | 0.00
Ceres | 188 | 6.26E+01 | 469.7 | 2.766 | 0.078 | 10.59 | -6.53 | 80.27 | 0.00 | 1.22
Pallas | 189 | 1.43E+01 | 272.5 | 2.774 | 0.230 | 34.90 | 137.52 | 172.92 | 0.00 | 0.48
Vesta | 190 | 1.73E+01 | 262.7 | 2.362 | 0.088 | 7.14 | 47.21 | 103.81 | 0.00 | 0.60
Hygiea | 191 | 7.00E+00 | 203.6 | 3.142 | 0.112 | 3.83 | 29.26 | 283.19 | 0.00 | 0.32
Interamnia | 192 | 5.00E+00 | 153.2 | 3.056 | 0.155 | 17.31 | -185.44 | 280.28 | 0.00 | 0.30
Psyche | 193 | 1.53E+00 | 113.0 | 2.924 | 0.134 | 3.10 | 79.12 | 150.03 | 0.00 | 0.12
Kalliope | 194 | 4.91E-01 | 83.8 | 2.909 | 0.098 | 13.70 | 291.55 | 66.00 | 0.00 | 0.05
Camilla | 195 | 7.48E-01 | 105.2 | 3.489 | 0.065 | 10.00 | 133.37 | 172.58 | 0.00 | 0.07

††footnotetext: The orbital elements are from the Earth-Moon barycenter.

Note. — (1) $GM$, no satellites provides error except Styx. (2)
$R_{\mathrm{L}}$ is the radius of the Objects. (3) $a$ and $e$ are the semi-
major axis and eccentricity, respectively. (4) $I$, $\omega$ and $\Omega$ are
inclination, preihelion, and longitude of ascending node, respectively. (5)
$R_{\mathrm{sp}}$: 0 denotes $<0.005$ except the sun, planets, pluto and the
Moon. (6) $\alpha_{\mathrm{max}}$: 0.00 denotes $<0.005$.